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1,314,259,995,763 | arxiv | \section{Introduction}
Field emission of electrons from a metallic surface~\cite{Fowler01051928, Forbes08112007, Jensen1528} is an important process in vacuum electronics~\cite{eichmeier2008vacuum, zhu2004vacuum}.
Its primary use is for cold cathodes in applications such as microwave tubes, electron microscopes and flat panel displays.
In addition to this beneficial application of field emission there are also negative implications,
such as vacuum breakdown at high field strengths and dielectric surface breakdown initiated by field emitted electrons~\cite{neuber2001windows}.
The physical basis for field emission is that a strong applied electric field can deform the surface barrier of a
metal-vacuum interface so as to increase the tunnelling probability to such a degree that a considerable current of electrons can be drawn from the surface.
This process was first described by Fowler and Nordheim~\cite{Fowler01051928} and has since been recast and extended for different applications.
A particular area of interest is the effect of space-charge on field emission. Barbour et~al. addressed this issue in 1953~\cite{PhysRev.92.45},
while Lau et~al. conducted a study of transition from field emission to space-charge dominated emission in 1994~\cite{Lau1870603}.
In both cases the approach was based on finding the equilibrium injection current that corresponds to the space-charge modified surface field.
Feng and Verboncoeur~\cite{Feng2006} used a particle-in-cell code to simulate this problem,
in which they used a detailed model that takes into account the effect of image-charge on the potential barrier for field emission.
Rokhlenko et~al.~\cite{Rokhlenko3272690} developed an elegant approach to calculate the effect of space-charge on field emission in a one dimensional system
that included lowering of the potential barrier by an effective work function, that matched the results of Feng and Verboncoeur quite well.
Other recent work includes a three dimensional theory for space-charge effects on field emission from a Spindt type emitter~\cite{Jensen14905}
and on space-charge and quantum effects~\cite{Jensen3692577}.
The work presented in this paper was motivated by the desire to use molecular dynamics methods to simulate electron beams in vacuum micro- and nanoelectronics devices.
An advantage of this approach is that it can account more accurately for collisional effects than simulations based on fluid models or particle-in-cell codes.
A disadvantage is the high computational cost, but in the small systems that are of interest, the number of free electrons is small enough that this is not an issue.
Previous work in this area~\cite{PhysRevLett.104.175002, Jonsson4793451, 6979259} has been based on something akin to photoemission as the source of electrons,
whereby the surface electric field does not directly affect the rate of electron emission from the cathode although space-charge effects do limit the current via virtual cathode formation.
Due to the importance of field emission it is desirable to implement that type of emission process in the molecular dynamics code being used for
simulation of vacuum nanoelectronics devices.
In this paper we present a field emission model based on image-charge considerations that is suitable for the molecular dynamics approach.
We show how this model can replicate already established results, particularly with regard to space-charge effects, and we use our model to simulate field emission in a planar diode of limited emitter area.
The simulation results are then compared to a simple fluid model for field emission from a planar emitter of finite area.
\autoref{sec:method} of this paper gives a description of the model and simulation methodology used.
Simulation results are presented in~\autoref{sec:results} followed by a fluid model description of two-dimensional effects in field emission
in~\autoref{sec:fluid} and finally by a short summary and discussion in~\autoref{sec:summary}.
\section{Methodology\label{sec:method}}
The model used is a planar vacuum diode with gap width
\(d\) as is depicted in~\autoref{fig:system}. The potential at the
cathode is zero and the anode potential is \(V_0\). Field emission
is only allowed to take place from a finite square-shaped area on the
cathode surface. Both the cathode and anode surfaces are infinite.
The side length, \(L\), of this square emitter
region is smaller than the gap spacing \(d\). A molecular dynamics (MD)
approach is used to calculate electron motion and is the basis for
the field emission algorithm. The simulation is of high resolution
in the sense that every single electron present in the vacuum gap
is treated as an individual particle. The Coulomb field due to
every electron in the system along with the image-charge partners on
either side of the cathode/anode are taken into account. Thus the
total field at any point is \(E = E_{sc} + E_0\), where \(E_{sc}\)
is the detailed space-charge field and \(E_0\) is the vacuum field.
Particle advancement is calculated using Verlet integration with a
time step of \(0.1\,\mathrm{fs}\).
\begin{figure}[bt]
\centering
\includegraphics{fig01.pdf}
\caption{The model of the planar nanodiode.}
\label{fig:system}
\end{figure}
Field emission is a quantum mechanical tunnelling process which can be described with the Fowler-Nordheim equation~\cite{Fowler01051928}
\begin{equation}\label{eq:FN-eq}
J = \frac{A}{t^2(\ell)\phi}F^2 \mathrm{e}^{-\nu(\ell)B\phi^{\frac{3}{2}}/F}\, ,
\end{equation}
where \(\phi\) is the work-function and \(F\) is the field
at the surface of the cathode, taken to be positive. \(A = e^2/(16\pi^2\hbar)\;
[\mathrm{A}\,\mathrm{eV}\,\mathrm{V}^{-2}]\)
and \(B = 4/(3\hbar) \sqrt{2m_e e}\;
[\mathrm{eV}^{-\frac{3}{2}}\,\mathrm{V}\,\mathrm{m}^{-1}]\) are the
first and second Fowler-Nordheim constants, while \(\nu(\ell)\)
is called the Nordheim function and arises due to the
image-charge effect. It contains complete elliptic integrals of
the first and second kind and is related to \(t(\ell)\) by the
relation \(t(\ell) = \nu(\ell) - (4/3)\ell\, \mathrm{d} \nu(\ell) / \mathrm{d}
\ell\). Approximations found by Forbes and Deane~\cite{Forbes08112007},
\begin{subequations}\label{eq:nordheim-fun}
\begin{equation}
\nu(\ell) = 1 - \ell + \frac{1}{6} \ell \ln(\ell)
\end{equation}
and
\begin{equation}
t(\ell) = 1 + \ell\left(\frac{1}{9} - \frac{1}{18}\ln(\ell) \right)
\end{equation}
are used, where
\begin{equation}\label{eq:ell}
\ell = \frac{e}{4\pi\varepsilon_0}\frac{F}{\phi^2}\,.
\end{equation}
\end{subequations}
Image-charge is taken into consideration with two objectives in mind. First,
to maintain the proper boundary conditions at the cathode we include
image-charge partners for the space-charge to be found in the gap. Second,
the image-charge of the electron being emitted is taken into account in
calculating the barrier potential which the electron must tunnel through.
This can be seen in~\autoref{fig:barrier} where the bare triangular (BT)
barrier is given by \(U^{BT}(z) = \phi - eFz\)
and the screened Schottky-Nordheim (SN) barrier by
\(U^{SN}(z) = \phi - eFz - e^2/(16\pi\epsilon_0 z)\).
\begin{figure}[tb]
\centering
\includegraphics{fig02.pdf}
\caption{The bare triangular barrier vs. the Schottky-Nordheim barrier for field emission.}
\label{fig:barrier}
\end{figure}
The SN barrier contains an additional term which arises due to the image-charge effect.
Tunnelling is more pronounced due to this term because the barrier height and width are reduced compared to the triangular barrier.
Numerical methods are used to obtain the dynamics in the system. The total number of electrons in the system is not constant since electrons are continuously
entering the system at the cathode and leaving at the anode.
The Metropolis-Hastings algorithm~\cite{Hastings} is used to sample the surface electric field to find a favourable locations for emission.
The probability of emission \(D_F = \exp(-\nu(\ell)B\phi^{\frac{3}{2}}/F)\), is then evaluated for small areas at those locations in order to determine
whether emission occurs.
The current density emitted from the cathode is normalized such that Fowler-Nordheim current density is obtained.
Relativistic effects and radiation caused by acceleration of electrons are safely neglected as all occurring velocities are much smaller than the speed of light.
\section{Results of the MD simulations\label{sec:results}}
\subsection{Current-voltage characteristic}
We start be examining a system with a work function of \(\phi = 4.7\,\mathrm{eV}\). The value is chosen because many metals have a work function in the range
from \(4.5\,\mathrm{eV}\) to \(5.0\,\mathrm{eV}\). This work function might represent Copper~(Cu), Tungsten~(W) or other metals.
The gap spacing \(d = 2500\,\mathrm{nm}\) and voltage \(V = 20\text{--}35\,\mathrm{kV}\) were then selected such that the vacuum field was sufficient to obtain field emission.
Care was taken in selecting the vacuum field such that the parameter \(\ell\) in~\autoref{eq:ell} was less than \(1\). If \(\ell\) is larger than \(1\) then the barrier
in~\autoref{fig:barrier} will be below the Fermi energy.
A higher work function means that a higher surface field is required to obtain field emission.
In~\autoref{fig:JvsV} the current density is plotted as a function of the voltage on a log scale. It shows the exponential increase as is expected from the Fowler-Nordheim
equation. The \textcolor{red}{red} dashed line shows the Fowler-Nordheim theory (\autoref{eq:FN-eq}) without space-charge effects.
The \textcolor{blue}{blue} solid line represents the results from the simulations in the steady state. We see that it is lower than the values given by the Fowler-Nordheim theory.
This is due to space-charge effects lowering the surface field at the cathode and reducing the emission.
The \textcolor{green}{green} dot dashed curve shows the 1D Child-Langmuir (CL)~\cite{PhysRevSeriesI.32.492,PhysRev.2.450} limit and the double dashed \textcolor{violet}{violet} is the 2D CL limit derived
by Lau~\cite{PhysRevLett.87.278301}. We see that under high voltage it is possible to go over the 1D CL limit but the 2D limit is still far off.
\begin{figure}
\centering
\includegraphics{fig03.pdf}
\caption{The current density plotted on a log scale vs. the voltage in the system.
The \textbf{\textcolor{red}{red}} dashed curve shows the results using the value of the vacuum field in the Fowler-Nordheim equation, and
the \textbf{\textcolor{blue}{blue}} solid curve shows the results from the simulations.
While the \textbf{\textcolor{green}{green}} dot dashed and \textbf{\textcolor{violet}{violet}} double dashed curves represent the 1D and 2D~\cite{PhysRevLett.87.278301}
CL limits respectively. The parameters used in the simulation were
\(\Phi = 4.7\,\mathrm{eV}\), \(d = 2500\,\mathrm{nm}\) and \(L = 100\,\mathrm{nm}\).}
\label{fig:JvsV}
\end{figure}
\subsection{Surface field at the cathode}
In order to verify the code we use for our simulations, we compared our results to Particle in Cell (PIC) simulations.
In reference~\onlinecite{Feng2006} Feng and Verboncoeur do 1D PIC simulations to study space-charge limited current with Fowler-Nordheim field emission.
Using the same parameters as they do, \(\phi = 2.0\,\mathrm{eV}\), \(V = 2\,\mathrm{kV}\) and \(d = 1000\,\mathrm{nm}\),
we obtain~\autoref{fig:plot-field}, which shows the average surface field on the cathode similar to Fig.~4 in the paper by Feng and Verboncoeur.
The \textcolor{blue}{blue} lines represent the average surface field for different side lengths \(L\) of the active region on the cathode.
\begin{figure}[t]
\centering
\includegraphics{fig04.pdf}
\caption{Average surface field plotted for different side lengths, \(L = 50,\, 100,\, 200,\, \ldots,\, 1000,\,2500\,\mathrm{nm}\).
The magnitude of the steady-state field decreases with increasing \(L\).
Other parameters were \(\phi = 2.0\,\mathrm{eV}\), \(V = 2\,\mathrm{kV}\) and \(d = 1000\,\mathrm{nm}\). }
\label{fig:plot-field}
\end{figure}
The values used are \(L = 50,\, 100,\, 200,\, \ldots,\,
1000,\,2500\,\mathrm{nm}\). The magnitude of the steady state field
decreases as the side length increases, because the space-charge effects
increase. For a large side length, comparable to the gap space, the field
resulting from our MD simulations approaches the 1D limit obtained in the steady state
by the PIC method, which we plotted as the \textcolor{black}{black}
line.
The surface field shows time dependent oscillations, both in the
MD and PIC simulations, before it settles down into the steady state.
The oscillations resulting from the PIC method are not shown here (see
Figure 4 of Ref. \cite{Feng2006}), but they are very similar to ours
for \(L=2500\,\mathrm{nm}\) and \(t>0.1\,\mathrm{ps}\).
However, an interesting feature
is present in the MD simulations, but not visible in the PIC results:
it is the sudden increase in the surface field seen at around \(t \approx
0.075\,\mathrm{ps}\), like a kink growing with increasing cathode
size. This time corresponds to the transit time of the electrons in the
system and is the moment when the first electrons are
being absorbed at the anode. The large number of electrons that were
emitted in beginning are now being absorbed which then cause the surface
field to increase abruptly as the absorbed electrons leave the
system. The effect is more pronounced for larger side lengths because
of the higher number of electrons in the system being emitted and absorbed.
The Coulomb oscillations become faster and denser for a small cathode size
because of the larger fluctuations of the emitted charge in combination with
the self consistent space-charge. Such oscillations may eventually split the
electron beam into bunches if the emission process can be sufficiently fast
as in the photoemission case \cite{PhysRevLett.104.175002}.
In our present approach the diode is initially empty
and a large number of electrons can be emitted in the first time steps.
In this regime, i. e. for \(t<0.01-0.02\,\mathrm{ps}\), the space-charge effects are small. After the
field reaches its lowest magnitude it starts to increase slightly
again because the electron are moving away from the cathode and towards
the anode. The oscillations in the field slowly die away with time.
In our ~\autoref{fig:plot-field} the vacuum field is represented
by the \textcolor{red}{red} line and the \textcolor{violet}{violet}
line shows the equilibrium surface field obtained by the fluid model described in~\autoref{sec:fluid}.
\subsection{Current vs. cathode size}
In~\autoref{fig:JvsLall}\hyperref[fig:JvsLall]{(a)} we see the current density scaled with the 1D Child-Langmuir limit
and in~\autoref{fig:JvsLall}\hyperref[fig:JvsLall]{(b)} scaled with the 2D Child-Langmuir limit from Lau~\cite{PhysRevLett.87.278301}
as a function of the side length \(L = 10\text{--}100\,\mathrm{nm}\).
\begin{figure}[tb]
\centering
\includegraphics{fig05.pdf}
\caption{(\textbf{a}) The current density scaled with the 1D Child-Langmuir limit.
(\textbf{b}) The current density scaled with the 2D Child-Langmuir from Lau~\cite{PhysRevLett.87.278301} as a function of the side length \(L\) for different voltages.
Other parameters used were \(\Phi = 4.7\,\mathrm{eV}\) and \(d = 2500\,\mathrm{nm}\).}
\label{fig:JvsLall}
\end{figure}
The current density decreases as the size of the active emission
area on the cathode increases, and approaches an asymptotic value for
an infinite area. This can be easily understood since, as is the case with Child-Langmuir emission from a 2D emitter,
the surface field has its lowest magnitude in the center of the emitter and increases towards
the edges~\cite{Luginsland2002}. The contribution of the edge
electrons to the surface field will be less and less as the emitter
area increases due to the inverse square nature of Coulomb's law.
The current density will therefore, asymptotically approach some final
value as the active emission area increases.
\subsection{Beam distribution}
\autoref{fig:transit} shows the distribution of the transit time of the electrons through the gap.
The distribution is approximately a Gaussian with
a \(\mathrm{FWHM} \approx 0.07\,\mathrm{fs}\), and a peak at \(t =
53.5\,\mathrm{fs}\).
From energy conservation the estimated transit time for a single
electron over the
gap is \(\Delta t = \sqrt{2 m_e d^2/(V_0
e)} \approx 53.32\,\mathrm{fs}\), which is not far from the peak
value of the distribution. It is the Coulomb interaction in the system that slightly
shifts the peak from the single electron value and
gives the width of the distribution. The width of the peak is small,
which indicates that the electrons travel quite fast over the gap and
do not have a lot of time to interact and spread out.
\begin{figure}[tb]
\centering
\includegraphics{fig06.pdf}
\caption{Distribution of the transit time of the electrons through the gap for \(V = 25\,\mathrm{kV}\), \(d = 2500\,\mathrm{nm}\) and \(\phi = 4.7\,\mathrm{eV}\).}
\label{fig:transit}
\end{figure}
\autoref{fig:abs-30kv} shows the absorption profile on the anode surface. The inner white square represents the active emission area on the cathode, while the outer white square
shows the boundaries of the absorbed electrons on the anode. The Coulomb interaction slightly rounds the corners of the beam from the square shape of the emitter.
\begin{figure}[tb]
\centering
\includegraphics{fig07.pdf}
\caption{Absorption profile at the anode. Inner box shows the emitter size while the outer box shows the edge of the beam on the absorption plane.
Parameters used were \(\Phi = 4.7\,\mathrm{eV}\), \(d = 2500\,\mathrm{nm}\) and \(V = 30\,\mathrm{kV}\).}
\label{fig:abs-30kv}
\end{figure}
The side length of the outer square is about \(2 \times 32.25\,\mathrm{nm}\) larger than the inner square which has \(L = 100\,\mathrm{nm}\). This means that beam spreading
is small compared with the gap spacing, \({d = 2500\,\mathrm{nm}}\).
\section{Fluid Model\label{sec:fluid}}
A simpler method that can be used to described the field emission is by
calculating the electric field in the diode from the charge density.
The charge density can be estimated by combining the
continuity equation, \(\rho(z)p/m_e = J\), where \(p\) is the momentum,
and the conservation of energy \(p^2/(2m_e) = e V_0 z/d\),
which gives \(\rho(z) = J \sqrt{m_ed/(2eV_0 z)}\).
Note that we have made use of the vacuum potential to calculate the charge density. This proves to be a
sufficient approximation for the situation that is being studies where the current density is still
considerably lower than the 2D Child-Langmuir current density (as can be seen from~\autoref{fig:JvsLall}). When the
current density approaches the Child-Langmuir current density an iterative calculation of the potential as
a function of z can be used instead, at added computational cost.
This charge density is distributed over the whole diode and therefore it behaves more like a fluid than like
a collection of single particles. It also assumes the beam does not spread too much laterally, which is a fair
approximation in our system as can be seen in~\autoref{fig:abs-30kv}.
Once the charge density is known it is easy to write down the equation
for the \(z\)-component of the electric field through the center of
the diode,
\begin{equation}\label{eq:ez-fluid}
\!\hat{E}_{\mathrm{sc}}^{\pm}(\hat{z})\! =\! \frac{\hat{J}}{9\pi}\!\!\int\limits_0^1\!\!\! \int\!\!\!\!\!
\int\limits_{\scalebox{0.5}[1.0]{\( - \)}\frac{L}{2d}}^{\frac{L}{2d}}\!\!\!
\frac{\hat{z}^\prime \pm \hat{z}}{\sqrt{\hat{z}^\prime} (\hat{x}^{\prime 2} + \hat{y}^{\prime 2} + (\hat{z}^\prime\pm \hat{z})^2)^{\frac{3}{2}}}\,
\mathrm{d} \hat{x}^\prime \mathrm{d} \hat{y}^\prime \mathrm{d} \hat{z}^\prime ,
\end{equation}
where \(\hat{E} = E/(-V_0/d)\), \(\hat{J} = J/J_{\mathrm{CL}}^{1\mathrm{D}}\) and \(\hat{x}\), \(\hat{y}\) and \(\hat{z}\) are scaled using the gap spacing \(d\).
The plus sign in the integral is used when calculating the image-charge effect and the minus sign for the field in the diode. The total field is then
\( \hat{E}_z(\hat{z}) = 1 - \hat{E}_{\mathrm{sc}}^+(\hat{z}) - \hat{E}_{\mathrm{sc}}^-(\hat{z})\).
In~\autoref{fig:Ez-an} we see the \(z\)-component of the
electric field through the center of the diode plotted as a
function of the \(z\)-coordinate (distance from the cathode). The
\textcolor{red}{red} dashed curve shows a typical result of an
MD simulation snapshot at a fixed time step, while
the \textcolor{blue}{blue} solid curve shows our fluid like model
calculated using~\autoref{eq:ez-fluid} with numerical integration. The
value of the current density used, \(J\), is taken from the simulation.
The fluctuations in the field from the simulations are due to electrons that are close to the center line where the field is being calculated.
\begin{figure}[tb]
\centering
\includegraphics{fig08.pdf}
\caption{The electric field in the \(z\)-direction through the center of the diode.
\(V = 30\,\mathrm{kV}\), \(d = 2500\,\mathrm{nm}\) and \(\phi = 4.7\,\mathrm{eV}\).
Inset shows the electric field near near the anode.}
\label{fig:Ez-an}
\end{figure}
The fluid model fits quite well with the simulation results.
The derivative of the electric field is proportional to the charge density, \(\rho(z)\),
which is highest near the cathode, where most of the electrons are
located. The electrons are injected with zero-velocity and consequently
spend more time near the cathode before being accelerated over the
gap. The electrons spread out as their velocity increases, and therefore the charge
density decreases rapidly and the electric field levels off after the electrons
have travelled roughly the distance of the side length \(L\) away from the cathode.
The electric field decreases slightly at the end of the gap, near the anode, due
to the absorption of the electrons into the anode. There is no charge
accumulated at the anode, which causes the field to drop slightly. Image-charge
partners are included in the simulations at the anode, but not in the fluid
model. It appears that the image-charge contributes little to the field
near the anode.
\begin{figure}[tb]
\centering
\includegraphics{fig09.pdf}
\caption{Comparison of the current density between the MD simulation (solid curves) and
the fluid model (dashed curves) for \(V = 20\,\mathrm{kV}\) (bottom \textcolor{blue}{blue} curves)
and \(V = 30\,\mathrm{kV}\) (top \textcolor{red}{red} curves).
Other parameters used were \(d = 2500\,\mathrm{nm}\) and \(\phi = 4.7\,\mathrm{eV}\)
}
\label{fig:simvsfluid}
\end{figure}
It is also possible to use the fluid model to calculate the current density from the cathode. This is done by iterations, using~\autoref{eq:FN-eq} and~\ref{eq:ez-fluid}.
The vacuum field is used as an initial value for the field in~\autoref{eq:FN-eq}. The value obtained is then scaled using the Child-Langmuir limit and put
into~\autoref{eq:ez-fluid} with \(z = 0\), which gives a new value for the surface field in center of the cathode. This procedure is repeated until the
current density has converged.
In~\autoref{fig:simvsfluid} we see a comparison between the fluid model and the simulation results for calculating
the current density emitted. The \textcolor{blue}{blue} solid curve shows the simulations results, while the \textcolor{red}{red} dashed the fluid model calculate
using method just described.
It is expected that the fluid model would
give slightly lower results. The model calculates the surface field in the center of the diode and as was explained earlier the field is lower there than at the edges.
The fluid model therefore underestimates the current density emitted from the edges.
In~\autoref{fig:fluid-model} we see the results of the fluid model for different values of the voltage as a function of
\({L = 10\text{--}10.000\,\mathrm{nm}}\) with \(d = 2500\,\mathrm{nm}\) and \(\phi = 4.7\mathrm{eV}\).
The results are qualitatively the same as in~\autoref{fig:JvsLall}\hyperref[fig:JvsLall]{(b)}.
\begin{figure}[tb]
\centering
\includegraphics{fig10.pdf}
\caption{Results of the fluid model for different values of the voltage as a function of the side length.
\(L = 10\text{--}10.000\,\mathrm{nm}\) with \(d = 2500\,\mathrm{nm}\) and \(\phi = 4.7\mathrm{eV}\).}
\label{fig:fluid-model}
\end{figure}
The fluid model allows us to estimate the current density for diodes with a larger emitting area and see the asymptotic behaviour better with calculations that
can be done in a few hours, whereas the MD simulations would require weeks of computational time for the same parameters. The shape of the curves seen is very similar
to the current profile seen in space-charge limited flow from circular cathodes~\cite{Kelley13243474}.
\section{Summary and conclusions\label{sec:summary}}
We performed molecular dynamics (MD) simulations of electrons in a diode
where the electrons are extracted from the cathode by the field emission
mechanism. The emission is essentially governed by the quantum mechanical
tunnelling through the potential barrier associated with the surface of the
metallic cathode, which leads to the well known Fowler-Nordheim law for
an infinite planar diode.
Our results include space-charge effects and are in good agreement
with the results obtained by other authors~\cite{Feng2006, Rokhlenko3272690}.
In particular they are in good agreement with the PIC simulations.
MD simulations are more computationally expensive than PIC simulations,
but are more accurate.
In the MD simulation every particle is tracked precisely and the force on it is
calculated exactly, using the Coulomb interaction from all other particles
in the system. Whereas in PIC simulation the system is divided up into
a grid and the field in each section of the grid is the mean field of
the particles in that section. The only methodological approximation in
the MD simulations is the finite time step, which must be chosen much
smaller than the characteristic time of the beam dynamics.
Time dependent oscillations of the electric field are obtained during
the transient period after the diode is switched on. A kink of cathode
field is obtained as a response to the charge fluctuation produced when
the first electrons are absorbed at the anode.
The MD method is best suited when the number of the electrons in the diode is
not too large, typically below few thousands. For larger numbers it
becomes computationally prohibited, but it those cases the mean field results
like those of PIC calculations, are usually accurate.
We derive a relatively simple fluid model of the electron beam,
which incorporates the essential electrostatics, where the
electric field is derived self consistently with an estimated continuous
charge distribution, and with the image-charge induced at the cathode and at the anode.
The results of the fluid model are good agreement with those of the MD simulations.
We can use this fluid model to estimate the effects of finite emitter area over a wide range of parameters.
We observe that the current density for field emission from a finite emitter with space-charge effects included,
is qualitatively similar to Child-Langmuir emission from a finite emitter area,
in that the current density increases with diminishing emitter area.
\begin{acknowledgments}
This work was financially supported by the Icelandic Research Fund grant number 120009021.
The simulations were performed on resources provided by the Nordic High Performance Computing (NHPC).
\end{acknowledgments}
|
1,314,259,995,764 | arxiv | \section{Introduction}
The random phase approximation (RPA)~\cite{ref:Bo53} is ubiquituous in
many fields of physics including nuclear physics, and it is described
in textbooks such as the much cited monograph by Ring and
Schuck~\cite{ref:Ri80}. Formally it leads to the analysis of a matrix
$$
\ma M = \tbt A B {{-B}^\ast} {{-A}^\ast} = \tbt 1 0 0 {-1} \ma S
$$
with
$$
\ma S = \tbt A B {B^\ast} {A^\ast} ,
$$
where $\ma A$ and $\ma B$ are $n \times n$ matrices and $\ma A$ is
Hermitian and $\ma B$ symmetric so that $\ma S$ is Hermitian. The
matrix $\ma M$ is the \textit{RPA matrix} and $\ma S$ is the
\textit{stability matrix}. When $\ma M$ is constructed from
excitations of a Hartree, Hartree-Fock, Hartree-Bogolyubov or
Hartree-Fock-Bogolyubov self-consistent mean field solution, $\ma S$
is the Hessian matrix of the mean field energy with respect to
variations about self-consistency~\cite{ref:Th60-61}. This
mathematical problem is well analysed when $\ma S$ is \textit{positive
definite}~\cite{ref:Th60-61,ref:Ri80}. Then $\ma M$ has $2 n$
linearly independent eigenvectors $\ma x_j$. The corresponding
eigenvalues $\omega_j$ form pairs of opposite nonvanishing reals and
the eigenvectors can be so normalised that %
$\ma x_j^\dagger \tbt 1 0 0 {-1} \ma x_k =%
(\text{sign\,} \omega_j) \delta_{jk}$.
It often occurs, however, that $\ma S$ is only positive
\emph{semidefinite}. Specifically, this is the case when the mean
field solution violates some continuous symmetry of the many-body
Hamiltonian, such as translational or rotational invariance, because
the mean field solution is then invariant to transformations within
the symmetry group. This leads to vanishing eigenvalues of $\ma M$,
and the number of linearly independent eigenvectors is generally less
than $2 n$. It is usually assumed that the eigenvectors corresponding
to such vanishing eigenvalues can be interpreted as associated, in the
language of classical analytic mechanics, with generalised momenta
whose conjugate coordinates describe local variations within the
symmetry group. These pairs of a cyclic coordinate and its conjugate
momentum form the so-called \textit{Nambu-Goldstone
modes}~\cite{ref:Th60-61,ref:Nm60Go61}. However, it was not, to my
knowledge, until very recently proved that this interpretation is
always consistent with the structure of $\ma M$.
This situation changed due to work by Nakada, who presented an
extensive analysis of $\ma M$ in the most general case when no
definiteness of $\ma S$ at all is assumed~\cite{ref:Nk16}. In an
addendum to this work Nakada derives from his general formalism that
when $\ma S$ is positive semidefinite then the space acted on by %
$\ma
M$ is decomposed into three subspaces: one where the vectors of a
certain basis correspond to pairs of a coordinate and its conjugate
momentum that do not enter the Hamiltonian at all, one where they form
Nambu-Goldstone mode pairs and one where $\ma M$ acts as in the case
of a positive definite $\ma S$~\cite{ref:Nk16a}. I here give a proof
of this result which does not rest on Nakada's general formalism.
\section{Change of basis}
A unitary transformation gives
\begin{gather*}
\ma S' = \tfrac12 \tbt 1 1 {{-i}} i \ma S \tbt 1 i 1 {{-i}}
= \tbt C {E^T} E D , \\
\tfrac12 \tbt 1 1 {{-i}} i \tbt 1 0 0 {{-1}} \tbt 1 i 1 {{-i}}
= \tbt 0 i {{-i}} 0
\end{gather*}
with \emph{real} matrices
$$
\ma C = \Re \ma A + \Re \ma B , \quad
\ma D = \Re \ma A - \Re \ma B , \quad
\ma E = \Im \ma A + \Im \ma B .
$$
Because $\ma S'$ is a real, symmetric matrix, a further real,
orthogonal transformation maps it to a real, positive semidefinite,
diagonal matrix $\ma \Delta$. Applying both transformations
successively results in
\begin{equation}\label{eq:M''}
\ma M'' = \ma N'' \ma \Delta ,
\end{equation}
where $\ma N''$ is imaginary and antisymmetric and obeys %
$\ma N''^2 = \ma 1$. Conversely any such matrix $\ma N''$ is mapped by
inverses of transformations of the above forms to $\tbt 1 0 0 {{-1}}$
and these transformations give, when applied to a real, positive
semidefinite, diagonal $\ma \Delta$, an $\ma S$ of the original
structure. As matrices of the form of $\ma M$ are thus unitary
equivalent to ones of the form of $\ma M''$, I drop the double primes
from now on.
\section{Onedimensional kernel of the stability matrix}
First assume for simplicity that $\ma \Delta$'s kernel is
onedimensional and its first diagonal element is zero while the rest
are positive. Let
$$
\ma x^{(1)} = \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} .
$$
Due to the antisymmetry of $\ma N$ the real vector %
$-i \ma N \ma x^{(1)}$ belongs to the space orthogonal to %
$\ma x^{(1)}$, where orthogonality $\ma x \perp \ma y$ is defined by
$\ma x^T \ma y = 0$. It does not vanish because $\ma N^2 = \ma 1$.
With $\ma \Delta^{-1}$ denoting the diagonal matrix with first
diagonal element zero and the reciprocals of $\ma \Delta$'s diagonal
elements in the following positions (so it is not in a strict sense
$\ma \Delta$'s inverse, which does not exist), let
$$
\ma x^{(2)} = a \ma \Delta^{-1} \ma N \ma x^{(1)}
$$
with an imaginary normalisation factor $a$. Then
$$
\ma M \ma x^{(1)} = 0 , \quad \ma M
\ma x^{(2)} = a \ma x^{(1)} .
$$
When $a$ is chosen such that
\begin{equation}\label{eq:conj}
\ma x^{(1)T} \ma N \ma x^{(2)} = -i ,
\end{equation}
we have $x^{(1)}_i = \{r_i,p\}$ and $x^{(2)}_i = \{r_i,q\}$, where $q$
and $p$ form a pair of a coordinate and its conjugate momentum and
$r_i, i = 1 \dots 2n$, are the canonical coordinates obeying
$\{r_i,r_j\} = -i N_{ij}$ in terms of which %
$\frac12 \sum_{i} \Delta_{ii} r_i^2$ is the Hamiltonian. Here
$\{.,.\}$ denotes the Poisson bracket; the quantal commutators
$[r_i,r_j]$ thus form the matrix $\ma N$. Because $\ma x^{(1)}$
belongs to the kernel of $\ma \Delta$, the Poisson bracket of the
Hamiltonian with $p$ vanishes, so $q$ is a cyclic coordinate. The
condition~\eqref{eq:conj} renders $a$ negative imaginary corresponding
to a positive mass. More precisely $i a$ is the reciprocal mass.
A canonical coordinate $r$ has vanishing Poisson brackets with these
two if and only if the vector $\ma x$ with coordinates %
$x_i = \{r_i,r\}$ satisfies
\begin{equation}\label{eq:res}
\ma x^{(j)T} \ma N \ma x = 0, \quad j = 1, 2 .
\end{equation}
I call the space of such vectors the \emph{residual} space. Being the
orthogonal complement of $\text{span\,} (-i \ma N x^{(j)}, j = 1,2)$,
which is twodimentional because $-i \ma N$ is orthogonal and $\ma
x^{(1)}$ and $\ma x^{(2)}$ are linearly independent by mutual
orthogonality, the residual space has dimension $2n - 2$. The
relations~\eqref{eq:res} are easily seen to imply
$$
\ma x^{(j)T} \ma N \ma M\ma x = \ma x^{(j)T} \ma\Delta \ma x = 0,
\quad j = 1, 2 ,
$$
so the residual space is invariant to $i \ma M$. For $j = 1$ the
relation~\eqref{eq:res} requires that $\ma x$ is orthogonal to %
$-i \ma N \ma \ma x^{(1)}$. For $j = 2$ it is equivalent to
\begin{equation}\label{eq:x_1}
\sum_k \ma N_{1 \cdot} \ma \Delta^{-1} \ma N_{\cdot k} x_k = 0 ,
\end{equation}
where $\ma N_{k \cdot}$ and $\ma N_{\cdot k}$ denote the $k$th row and
column vectors. Because the coefficient of $x_1$ in this equation is
positive the equation can be satified for any $x_k, k = 2, \dots, 2n,$
by adjustment of $x_1$. A basis for the residual space is thus
obtained by selecting a basis for the space orthogonal to %
$\ma x^{(1)}$ and $-i \ma N \ma x^{(1)}$ and supplementing each basic
vector by the first coordinate required by equation~\eqref{eq:x_1}.
Now let $\ma e^{(j)}, j = 1, \dots, 2n,$ be a real, orthonormal basis
for the total space such that
$$
\ma e^{(1)} =\ma x^{(1)} , \quad
\ma e^{(2)} = -i \ma N \ma x^{(1)} ,
$$
and let, for $j = 3, \dots, 2n$, the vector $\ma x^{(j)}$ be obtained
by supplementing $\ma e^{(j)}$ by a first coordinate such as to
satisfy equation~\eqref{eq:x_1}. As these vectors are linearly
independent, they span the residual space. Moreover, because
$$
\ma e^{(j)T} \ma x^{(k)} = \ma e^{(j)T} \ma e^{(k)}
= \delta_{jk}, \quad j,k \ge 3 ,
$$
the matrix
$$
\sum_{j \ge 3} \ma x^{(j)} \ma e^{(j)T}
$$
performs the identity transformation within this space. In the residual
space the transformation $\ma M$ therefore has matrix elements
$$
\ma e^{(j)T} \ma M \ma x^{(k)}
= \sum_l \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma \Delta \ma x^{(k)} \\
= \sum_{l \ge 3} \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma \Delta \ma e^{(k)},
\quad j,k \ge 3 ,
$$
where the reduction of the sum follows from
\begin{equation}\label{eq:x->e}\begin{gathered}
\ma e^{(j)T} \ma N \ma e^{(1)}
\propto \ma e^{(j)T} \ma e^{(2)} = 0 \quad
\text{(or $\ma e^{(1)T} \ma \Delta \ma e^{(j)} = 0$)} , \\
\ma e^{(j)T} \ma N \ma e^{(2)}
\propto \ma e^{(j)T} \ma N^2 \ma e^{(1)}
= \ma e^{(j)T} \ma e^{(1)} = 0, \quad j = 3 , \dots , 2n .
\end{gathered}\end{equation}
These relations also give
\begin{multline*}
\sum_{l \ge 3} \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma N \ma e^{(k)}
= \sum_l \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma N \ma e^{(k)} \\
= \ma e^{(j)T} \ma N^2 \ma e^{(k)}
= \ma e^{(j)T} \ma e^{(k)} = \delta_{jk} , \quad j,k \ge 3 .
\end{multline*}
As the matrix $(\ma e^{(j)T} \ma \Delta \ma e^{(k)}, j,k \ge 3)$ is
positive definite and the matrix %
$(\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \ge 3)$ is imaginary and
antisymmetric, the restriction of $\ma M$ to the residual space is
thus similar to a matrix of the form~\eqref{eq:M''} with a positive
definite $\ma \Delta$. Moreover, because
$$
\ma x^{(j)T} \ma N \ma x^{(k)}
= \left( \ma e^{(j)} + x^{(j)}_1\ma e^{(1)} \right)^T \\
\ma N \left( \ma e^{(k)} + x^{(k)}_1\ma e^{(1)} \right)
= \ma e^{(j)T} \ma N \ma e^{(k)} ,
\quad j,k \ge 3 ,
$$
by the equations~\eqref{eq:x->e} and the antisymmetry of $\ma N$, the
matrix $-i (\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \ge 3)$ is the matrix
of Poisson brackets of the canonical coordinates associated with the
basic vectors $\ma x^{(j)}, j = 3, \dots 2n$.
\section{Multidimensional kernel}
For a generalisation to the case when $\ma \Delta$'s kernel %
$\mathcal K$ has dimension $m$ greater than one, let %
$\ma e^{(j)},j = 1 \dots m,$ be orthonormal basic vectors for
$\mathcal K$ such that $\ma e^{(j)},j = m' + 1 , \dots m,$ span
$\mathcal K \cap i \ma N \mathcal K^\perp$. Assume %
$m' + 1 \le j \le m$. Then $\ma e^{(j)} \in i \ma N \mathcal K^\perp$
implies $\ma e^{(j+m-m')} \mathrel{\mathop:}=%
-i \ma N e^{(j)} \in \mathcal K^\perp$, and because $-i \ma N$ is
orthogonal, the latter vectors are orthonormal. The matrix $(\ma
e^{(j)T} \ma N \ma e^{(k)}, j,k \le m')$ is nonsingular. In fact, if
for some linear combination $\ma x$ of %
$ \ma e^{(j)}, j = 1 , \dots , m',$ the vector $-i \ma N \ma x$ would
be perpendicular to all $\ma e^{(k)}, k= 1 , \dots , m'$, then
because, by $\ma x \perp \ma -e^{(k+m-m')} = i \ma N \ma e^{(k)}$ and
the orthogonality of $-i \ma N$, the vector $-i \ma N \ma x$ is also
perpendicular to all $\ma e^{(k)}, k = m' + 1, \dots , m$, it would be
perpendicular to $\mathcal K$. But then $\ma x$ would belong to %
$i \ma N \mathcal K^\perp$, a contradiction. As %
$(\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \le m')$ is also imaginary and
antisymmetric, it follows that $m'$ is even. The square of %
$(\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \le m')$ is not necessarily the
unit matrix, but $(\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \le m')$ can be
given this property by right and left multiplications by a nonsingular
square matrix and its transposed. This allows defining a basis for
$\text{span\,} (\ma e^{(j)}, j = 1 , \dots , m')$ whose vectors are
associated, in the manner detailed above, with pairs of a coordinate
and its conjugate momentum obeying the canonical Poisson bracket
relations (including vanishing of the Poisson brackets between
coordinates and momenta belonging to different pairs). These canonical
coordinates have vanishing Poisson brackets with the Hamiltonian, so
as parts of a complete set of pairs of a coordinate and its conjugate
momentum obeying the canonical Poisson bracket relations they will be
entirely absent from the Hamiltonian.
Let
\begin{gather*}
\ma x^{(j)} = \ma e^{(j)} , \quad j = 1 , \dots , m , \\
\ma x^{(j)} = \ma \Delta^{-1} \ma e^{(j)} , \quad
j = m + 1, \dots , 2m - m' ,
\end{gather*}
with $\ma \Delta^{-1}$ defined in the way analogous to that above.
These vectors are linearly independent because $\ma \Delta^{-1}$ is
nonsingular in $\mathcal K^\perp$. Like before the vectors $\ma
x^{(k)}, k = m' + 1, \dots , 2m - m'$ span a subspace invariant to %
$i \ma M$. Notice
$$
\ma x^{(j)T} \ma N \ma x^{(k)} \propto \ma e^{(j)T} \ma e^{(k+m-m')}
= 0, \quad m' + 1 \le j,k \le m .
$$
Because the matrix $\ma m$ of elements
\begin{equation}\label{eq:posdef}
m_{jk} = i \ma x^{(j)T} \ma N \ma x^{(k+m-m')} \\
= \ma e^{(j)T} \ma N \ma \Delta^{-1} \ma N \ma e^{(k)} ,
\quad m' + 1 \le j,k \le m ,
\end{equation}
is positive definite, one can make a transformation among the vectors
$\ma x^{(j)}, j = m' + 1, \dots , m,$ to get
\begin{equation}\label{eq:canon}
\ma x^{(j)T} \ma N \ma x^{(k+m-m')} = -i \delta_{jk} ,
\quad m' + 1 \le j,k \le m .
\end{equation}
The transformation
$$
\ma x^{(j)} \mapsto \ma x^{(j)} + \frac i 2 \sum_{k=m'+1}^m
\ma x^{(k)} \ma x^{(k+m-m')T} \ma N \ma x^{(j)} , \\
\quad j = m + 1 , \dots 2m - m' ,
$$
then gives
$$
\ma x^{(j)T} \ma N \ma x^{(k)} = 0,
\quad m + 1 \le j,k \le 2m - m' ,
$$
without destroying the relations~\eqref{eq:canon}. After these
transformations one has
\begin{equation}\label{eq:dyn}
\ma M \ma x^{(j)} = 0 , \quad
\ma M \ma x^{(j+m-m')} = \sum_{k=m'+1}^m a_{jk} \ma x^{(k)} ,
\quad j = m' + 1, \dots , m,
\end{equation}
where the matrix $i \ma a$ with elements $i a_{jk}$ is the inverse of
the matrix $\ma m$ defined by equation~\eqref{eq:posdef} \emph{before}
the transformations. As $i \ma a$ is symmetric and positive definite,
its appearance in equation~\eqref{eq:dyn} is rendered positive
diagonal by application \emph{after} the transformations of one more
orthogonal transformation simultaneously to both sets of vectors %
$\ma x^{(j)}$ and $\ma x^{(j+m-m')}, j = m' + 1 , \dots , m$. This
does not change %
$\ma x^{(j)T} \ma N \ma x^{(k)}, m' + 1 \le j,k \le 2m - m'$, and one
arrives at an interpretation of $\ma x^{(j+m-m')}$ and %
$\ma x^{(j)}, j = m' + 1 , \dots , m,$ as vectors corresponding to
pairs of a cyclic coordinate and its conjugate momentum. The negative
imaginary signs of %
$a_{jj}, j = m' + 1 , \dots , m,$ correspond to positive masses; in
fact the diagonal matrix elements of the transform of $\ma m$ by the
final orthogonal transformation are the masses themselves.
Now let the orthonormal set $(\ma e^{(j)}, j = 1 , \dots , 2m - m')$
be extended to an orthonormal basis for the entire space. Like before
the relations
$$
\ma x^{(j)T} \ma N \ma x = 0, \quad j = 1,\dots , 2m - m',
$$
define another subspace invariant to $i \ma M$, the residual space.
These relations ensure that the canonical coordinates associated with
the two previous spaces have vanishing Poisson brackets with those
associated with the residual space. They are satisfied automatically
for $j = m' + 1 , \dots , m$ when $\ma x$ is a linear combination of %
$\ma e^{(k)}, k = 1 , \dots , m, 2m - m' +1 , \dots , 2n$.
Attempting to satisfy the remaining $m$ relations by supplementing one
$\ma e^{(k)}, k = 2m - m' +1 , \dots , 2n,$ by a linear combination of
$\ma e^{(l)}, l = 1 , \dots , m,$ gives $m$ linear equations, which
can be solved because %
$(\ma e^{(j)T} \ma N \ma e^{(k)}, 1 \le j,k \le m')$ and $\ma m$ are
nonsingular. This results in a basis %
$(\ma x^{(k)}, k = 2m - m' +1 , \dots , 2n)$ for the residual space.
Due to
$$
\ma e^{(j)T} \ma x^{(k)} = \ma e^{(j)T} \ma e^{(k)}
= \delta_{jk}, \quad j,k \ge 2m - m' + 1 ,
$$
the matrix
$$
\sum_{j \ge 2m-m'+1} \ma x^{(j)} \ma e^{(j)T}
$$
performs the identity transformation within the residual space, so the
restriction of $\ma M$ to this space has matrix elements
\begin{multline}\label{eq:product}
\ma e^{(j)T} \ma M \ma x^{(k)}
= \sum_l \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma \Delta \ma x^{(k)} \\
= \sum_{l \ge 2m-m'+1} \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma \Delta \ma e^{(k)},
\quad j,k \ge 2m - m' + 1 ,
\end{multline}
where the reduction of the sum follows from
\begin{gather*}
\ma e^{(l)T} \ma \Delta \ma e^{(k)} = 0 ,
\quad l = 1 , \dots , m , \\
\ma e^{(j)T} \ma N \ma e^{(l)}
\propto \ma e^{(j)T} \ma N^2 \ma e^{(l-m+m')}
= \ma e^{(j)T} \ma e^{(l-m+m')} = 0, \quad
l = m + 1, \dots , 2m - m' .
\end{gather*}
Because the vectors %
$\ma \Delta \ma x^{(j)}, j = 2m - m' + 1 , \dots , 2n,$ are linearly
independent and map to members of the residual space by the orthogonal
matrix $i \ma N$, the restriction of $\ma M$ to the residual space is
nonsingular. So is then the matrix %
$(\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \ge 2m - m' + 1)$ by
equation~\eqref{eq:product}. This matrix is also imaginary and
antisymmetric. Its square is not necessarily the unit matrix, but %
$(\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \ge 2m - m' + 1)$ can be given
this property by right and left multiplications by a nonsingular
matrix $\ma T$ and its transposed $\ma T^T$. Right and left
multiplications of the matrix %
$(\ma e^{(j)T} \ma \Delta \ma e^{(k)}, j,k \ge 2m - m' + 1)$ by %
$(\ma T^{-1})^T$ and $\ma T^{-1}$ then result in a similarity
transformation of the matrix %
$(\ma e^{(j)T} \ma M \ma x^{(k)}, j,k \ge 2m - m' + 1)$. As %
$(\ma e^{(j)T} \ma \Delta \ma e^{(k)}, j,k \ge 2m - m' + 1)$ is
symmetric and positive definite and this property is conserved by the
right and left multiplications by $(\ma T^{-1})^T$ and $\ma T^{-1}$,
the restriction of $\ma M$ to the residual space is thus similar to a
matrix of the form~\eqref{eq:M''} with a positive definite %
$\ma \Delta$.
Only when $m' = 0$ one has
$$
\ma e^{(j)T} \ma N \ma e^{(k)}
\propto \ma e^{(j)T} \ma e^{(k+m)} = 0 , \\
j = 2m + 1 \dots 2n , \ k = 1 , \dots , m ,
$$
so that
\begin{multline*}
\sum_{l \ge 2m+1} \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma N \ma e^{(k)}
= \sum_l \ma e^{(j)T} \ma N \ma e^{(l)}
\ma e^{(l)T} \ma N \ma e^{(k)} \\
= \ma e^{(j)T} \ma N^2 \ma e^{(k)}
= \ma e^{(j)T} \ma e^{(k)} = \delta_{jk} , \quad j,k \ge 2m + 1 .
\end{multline*}
The transformations by $\ma T$ are then not required. Also
$$
\ma x^{(j)T} \ma N \ma x^{(k)}
= \ma e^{(j)T} \ma N \ma e^{(k)} ,
\quad j,k \ge 2m + 1 ,
$$
because $\ma e^{(j)T} \ma N \ma e^{(k)}$ vanishes for %
$j = 1 , \dots , m , 2m + 1 , \dots , 2n$ and $k = 1 , \dots , m$. The
matrix $-i (\ma e^{(j)T} \ma N \ma e^{(k)}, j,k \ge 2m + 1)$ is then
the matrix of Poisson brackets of the canonical coordinates associated
with the basic vectors %
$\ma x^{(j)}, j = 2m + 1, \dots 2n$. For $m' > 0$ the matrix %
$-i \ma T^T (\ma e^{(j)T} \ma N \ma e^{(k)}, %
j,k \ge 2m - m' + 1) \ma T$ is not in general the matrix of Poisson
brackets of the canonical coordinates associated with the basic
vectors %
$\sum_{k \ge 2m - m' + 1} \ma x^{(k)} T_{kj}, j = 2m - m' + 1, \dots 2n$.
\section{Conclusion}
It was shown that when the random phase approximation (RPA) stability
matrix is positive semidefinite, the vector space on which it acts can
be decomposed into three parts: one where the vectors of a certain
basis correspond, in the equivalent formalism of a classical
Hamiltonian homogeneous of second degree in canonical coordinates, to
pairs of a coordinate and its conjugate momentum that do not enter the
Hamiltonian at all, one where they correspond to pairs of a cyclic
coordinate and its conjugate momentum (Nambu-Goldstone modes) and a
residual space where the RPA matrix acts as in the case of a positive
definite stability matrix. This was also proved very recently by
Nakada as a corollary to a general analysis of the most general RPA
matrix without limitations on the definiteness of the stability
matrix. The present proof does not rest on Nakada's general results.
\section*{Acknowledgment}
Discussions with Hitoshi Nakada are warmly appreciated.
\bibliographystyle{ptephy} |
1,314,259,995,765 | arxiv | \section{Introduction}
\label{sec:intro}
An Abelian differential defines a flat structure such that the underlying
Riemann surface can be realized as a plane polygon whose edges are pairwise identified via translation. Varying the shape of the polygon by $\operatorname{GL}_2^+(\mathbb R)$ induces an action on the moduli space of Abelian differentials, called Teichm\"uller dynamics, see Figure~\ref{fig:teich}.
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{c}{$c$}
\psfrag{d}{$d$}
\includegraphics[scale=0.5]{teich.eps}
\caption{\label{fig:teich} $\operatorname{GL}_2^+(\mathbb R)$-action on a flat surface}
\end{figure}
The corresponding $\operatorname{GL}_2^+(\mathbb R)$-orbit closures in the moduli space of Abelian differentials are now known as affine invariant submanifolds. A number of questions about surface geometry boil down to understanding the structures of affine invariant submanifolds. From the viewpoint of algebraic geometry affine invariant submanifolds are of an independent interest, which can provide special subvarieties in the moduli space of curves.
We aim to introduce Teichm\"uller dynamics from the viewpoint of algebraic geometry. In Section~\ref{sec:prelim} we review background material, including translation surfaces, strata of Abelian differentials, the $\operatorname{GL}_2^+(\mathbb R)$-action, and affine invariant submanifolds. Section~\ref{sec:teich} focuses on Teichm\"uller curves formed by closed $\operatorname{GL}_2^+(\mathbb R)$-orbits, where we describe their properties, examples, classification, and invariants. In Section~\ref{sec:affine} we study general affine invariant submanifolds and survey recent breakthroughs about their structures, classification, and boundary behavior. Finally in Section~\ref{sec:higher} we discuss similar questions for meromorphic and higher order differentials.
This article is written in an expository style. We will often highlight motivations and minimize technical details. For further reading, we refer to \cite{Zorich, MoellerSurvey, Wright1, Wright2} for a number of excellent surveys on related topics.
\subsection*{Acknowledgement}
This article is partially based on the lectures given by the author during the Algebraic Geometry Summer Research Institute Bootcamp, July 2015. The author is grateful to the Bootcamp organizers Izzet Coskun, Tommaso de Fernex, Angela Gibney, and Max Lieblich for their invitation and hospitality. The author thanks Alex Wright for carefully reading the article and many helpful comments.
\section{Preliminaries}
\label{sec:prelim}
In this section we introduce basic background material that will be used later.
\subsection{Abelian differentials and translation surfaces}
\label{subsec:abelian}
A \emph{translation surface} (also called a \emph{flat surface}) is a closed, topological surface $X$ together with a finite set
$\Sigma\subset X$ such that:
\begin{itemize}
\item
There exists an atlas of charts $X\backslash \Sigma \to \mathbb C$, where the transition functions are translation.
\item
For each $p\in \Sigma$, under the Euclidean metric of $\mathbb C$
the total angle at $p$ is $(k+1)\cdot (2\pi) $ for some $k\in \mathbb Z^+$.
\end{itemize}
We say that $p$ is a \emph{saddle point of cone angle} $(k+1)\cdot (2\pi)$. Locally one can glue $2k+2$ half-disks consecutively to form a cone of angle $2\pi \cdot (k+1)$, see Figure~\ref{fig:zero-k}.
\begin{figure}[h]
\centering
\psfrag{B1}{$B_1$}
\psfrag{B2}{$B_2$}
\psfrag{Bk+1}{$B_{k+1}$}
\psfrag{A1}{$A_1$}
\psfrag{A2}{$A_2$}
\psfrag{A3}{$A_3$}
\psfrag{Ak+1}{$A_{k+1}$}
\includegraphics[scale=1.2]{zero-k.eps}
\caption{\label{fig:zero-k} A saddle point of cone angle $(k+1)\cdot (2\pi)$}
\end{figure}
Equivalently, a translation surface is a closed Riemann surface $X$ with an \emph{Abelian differential} $\omega$, not identically zero:
\begin{itemize}
\item
The set of zeros of $\omega$ corresponds to $\Sigma$.
\item
If $p$ is a zero of $\omega$ of order $k$, then the cone angle at $p$ is $(k+1)\cdot (2\pi)$.
\end{itemize}
For example, take an octagon $X$ with four pairs of parallel edges, see Figure~\ref{fig:flat}.
\begin{figure}[h]
\centering
\psfrag{a}{$v_1$}
\psfrag{b}{$v_2$}
\psfrag{c}{$v_3$}
\psfrag{d}{$v_4$}
\includegraphics[scale=0.5]{flat.eps}
\caption{\label{fig:flat} An octagon $X$ with four pairs of parallel edges}
\end{figure}
Identifying the edges with the same labels by translation, $X$ becomes a
closed surface. All vertices are glued as one point $p$. By the topological Euler characteristic formula,
the genus of $X$ is two. It is moreover a Riemann surface whose complex structure is induced from $\mathbb C$.
Away from $p$ it admits an atlas of charts with transition functions given by translation: $z' = z + \text{constant}$,
see Figure~\ref{fig:flat-p}.
\begin{figure}[h]
\centering
\psfrag{p}{$p$}
\psfrag{z}{$z$}
\psfrag{w}{$z'$}
\includegraphics[scale=0.5]{flat-p.eps}
\caption{\label{fig:flat-p} Translation structure on $X\backslash p$}
\end{figure}
The differential $\omega = dz$ is well-defined and nowhere vanishing on $X\backslash p$, which further extends to the entire $X$.
The angle at $p$ is $6\pi = 3 \cdot (2\pi)$, hence $\omega$ has a local expression $d (z^3) \sim z^2 dz$ at $p$.
In summary, $\omega$ is an Abelian differential with a unique zero of order two on a Riemann surface of genus two.
The above example illustrates the equivalence between translation surfaces and Abelian differentials in general. Given a translation surface, away from its saddle points, differentiating local coordinates provides a globally defined Abelian differential. Conversely, integrating an Abelian differential away from its zeros provides an atlas of charts whose transition functions are translation, because antiderivatives differ by constants. In addition, a saddle point $p$ has cone angle $(k+1)\cdot (2\pi)$ if and only if $\omega = d(z^{k+1})\sim z^k dz$ under a local coordinate $z$ at $p$, namely, if and only if $\omega$ has a zero of order $k$ at $p$.
Below we provide two more examples. Figure~\ref{fig:torus} represents a nowhere vanishing differential on a torus. Conversely every Abelian differential on a torus give rises to such a parallelogram presentation. Figure~\ref{fig:two-simple} represents an Abelian differential with two simple zeros on a Riemann surface of genus two.
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\includegraphics[scale=0.25]{torus.eps}
\caption{\label{fig:torus} A flat torus}
\end{figure}
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{c}{$c$}
\psfrag{d}{$d$}
\psfrag{e}{$e$}
\includegraphics[scale=0.3]{two-simple.eps}
\caption{\label{fig:two-simple} A flat surface with two simple zeros}
\end{figure}
Note that Abelian differentials are sections of the \emph{canonical line bundle}, hence the study of translation surfaces is naturally connected to algebraic geometry.
\subsection{Strata of Abelian differentials}
\label{subsec:strata}
We identify Riemann surfaces with smooth complex algebraic curves. Let $\mathcal M_g$ be the \emph{moduli space of genus $g$ curves}. Let $\mathcal H$ be the \emph{Hodge bundle} over $\mathcal M_g$ whose fibers parameterize Abelian differentials on a fixed genus $g$ curve.
Let $\mu = (m_1, \ldots, m_n)$ be a tuple of positive integers such that $\sum_{i=1}^n m_i = 2g-2$. We say that
$\mu$ is a \emph{partition} of $2g-2$.\footnote{In Section~\ref{sec:higher} we will consider meromorphic differentials, where the entires $m_i$ are allowed to be negative.}
Define a subset $\mathcal H(\mu)$ of $\mathcal H$ that parameterizes
pairs $(X, \omega)$, where $X$ is a Riemann surface of genus $g$ and $\omega$ is an Abelian differential on $X$ such that the zero divisor of $\omega$ is of type
$\mu$:
$$ (\omega)_0 = m_1 p_1 + \cdots + m_n p_n. $$
We say that $\mathcal H(\mu)$ is the \emph{stratum of Abelian differentials of type $\mu$}.
Equivalently, $\mathcal H(\mu)$ parameterizes translation surfaces with $n$ saddle points, each having cone angle
$(m_i+1) \cdot (2\pi)$. The union of $\mathcal H(\mu)$ over all partitions of $2g-2$ is the Hodge bundle $\mathcal H$ (with the zero section removed).
Take a basis $\gamma_1, \ldots, \gamma_{2g + n -1}$ of the relative homology $H_1(X, p_1, \ldots, p_n; \mathbb Z)$. Integrating
$\omega$ over each $\gamma_i$ provides a local coordinate system for $\mathcal H(\mu)$, called the \emph{period coordinates}. For instance, the complex vectors $v_1$, $v_2$, $v_3$, and $v_4$ in Figure~\ref{fig:flat} above are periods of a translation surface in $\mathcal H(2)$. Under the polygon presentation, locally deforming the periods preserves the number of saddle points, their cone angles, and the way of edge identification. Consequently $\mathcal H(\mu)$ is a $(2g+n-1)$-dimensional manifold.\footnote{More precisely it is an orbifold, because special translation surfaces can have extra automorphisms.}
For special partitions $\mu$, $\mathcal H(\mu)$ can be \emph{disconnected}. Kontsevich-Zorich
(\cite{KontsevichZorich}) classified connected components of $\mathcal H(\mu)$ for all $\mu$, where extra components arise due to hyperelliptic and spin structures. If a translation surface $(X, \omega)$ satisfies that $X$ is hyperelliptic, $(\omega)_0 = (2g-2)z$ or $(\omega)_0 = (g-1)(z_1+z_2)$, where
$z$ is a Weierstrass point of $X$ in the former, or $z_1$ and $z_2$ are hyperelliptic conjugate in the latter, we say that $(X, \omega)$ is a \emph{hyperelliptic translation surface}.\footnote{Being a hyperelliptic translation surface not only requires $X$ to be hyperelliptic, but also imposes a condition on $\omega$.} For a nonhyperelliptic translation surface $(X, \omega)$, if
$(\omega)_0 = 2 k_1 z_1 + \cdots + 2k_n z_n$, then the line bundle
$\mathcal O_X(\sum_{i=1}^n k_i z_i)$
is a square root of the canonical line bundle, namely, it is a \emph{theta characteristic}. Define its \emph{parity} by
$h^0(X, \sum_{i=1}^n k_i z_i) \pmod{2}$, which is deformation invariant (\cite{Atiyah, Mumford}).
A theta characteristic with its parity is called a \emph{spin structure}. In general, $\mathcal H(\mu)$ can have \emph{up to three} connected components, distinguished by these hyperelliptic and spin structures.
\subsection{$\operatorname{GL}_2^+(\mathbb R)$-action and affine invariant submanifolds}
\label{subsec:GL2}
Given $(X, \omega) \in \mathcal H$ and $A\in \operatorname{GL}^{+}_2(\mathbb R)$, varying the polygon presentation of $(X, \omega)$ by $A$
induces a $\operatorname{GL}^{+}_2(\mathbb R)$-action on $\mathcal H$, which is called \emph{Teichm\"uller dynamics}. For example, the $\operatorname{GL}^{+}_2(\mathbb R)$-orbit of a flat torus consists of all parallelogram presentations, see Figure~\ref{fig:torus-GL}.
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{A}{$A$}
\includegraphics[scale=0.4]{torus-GL.eps}
\caption{\label{fig:torus-GL} $\operatorname{GL}^{+}_2(\mathbb R)$-action on a flat torus}
\end{figure}
The number and cone angles of saddle points are preserved under this action, hence the $\operatorname{GL}^{+}_2(\mathbb R)$-action descends to each stratum $\mathcal H(\mu)$. Equip $\mathcal H(\mu)$ the standard topology using its period coordinates. For almost all $(X, \omega) \in \mathcal H(\mu)$, Masur and Veech (\cite{MasurInterval, VeechInterval}) showed that its $\operatorname{GL}^{+}_2(\mathbb R)$-orbit is \emph{equidistributed} in $\mathcal H(\mu)$, hence the orbit closure is the whole stratum (or a connected component if the stratum is disconnected). For special $(X, \omega)$, however, its $\operatorname{GL}^{+}_2(\mathbb R)$-orbit closure can be a \emph{proper} subset of $\mathcal H(\mu)$. Classifying
$\operatorname{GL}^{+}_2(\mathbb R)$-orbit closures in $\mathcal H(\mu)$ is a central question in Teichm\"uller dynamics.
The recent breakthrough of Eskin-Mirzakhani-Mohammadi~(\cite{EskinMirzakhani, EskinMirzakhaniMohammadi}) showed that any $\operatorname{GL}^{+}_2(\mathbb R)$-orbit closure is an \emph{affine invariant submanifold} in $\mathcal H(\mu)$, that is, locally it is a subspace of $\mathcal H(\mu)$ cut out by real linear homogeneous equations of period coordinates.\footnote{Here the term ``affine'' is different from what it usually means in algebraic geometry. It refers to the linear structure on $\mathbb C^n$. Moreover, the closure of an orbit is taken under the standard topology in the Hodge bundle over the interior of the moduli space parameterizing smooth curves.}
Filip (\cite{Filip}) further showed that all affine invariant submanifolds are algebraic varieties defined over $\overline{\mathbb Q}$. In particular, it means that affine invariant submanifolds can be defined and characterized purely in terms of algebraic conditions on the Jacobian. We will elaborate on these results in Section~\ref{sec:affine}.
\subsection{Veech group}
\label{subsec:veech}
Let $(X,\omega) \in \mathcal H(\mu)$ be a translation surface. Suppose a matrix $A\in \operatorname{SL}_2(\mathbb R)$ acts on $(X, \omega)$.
If the resulting translation surface $A\cdot (X, \omega)$ is isomorphic to $(X,\omega)$, that is, if the polygon presentation
of $A\cdot (X, \omega)$ can be cut into pieces and reassembled via translation to represent $(X,\omega)$, we say that
$A$ is a \emph{stabilizer} of $(X, \omega)$. The subgroup of all stabilizers of $(X, \omega)$ in $\operatorname{SL}_2(\mathbb R)$ is called
the \emph{Veech group}, and denoted by $\operatorname{SL}(X, \omega)$. For example,
$A = \big(\begin{smallmatrix}
1 & 1 \\
0 & 1
\end{smallmatrix}\big)$
is in the Veech group of the square torus, see Figure~\ref{fig:veech}.
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{c}{$c$}
\psfrag{e}{$=$}
\psfrag{A}{$\big(\begin{smallmatrix}
1 & 1 \\
0 & 1
\end{smallmatrix}\big)$}
\includegraphics[scale=1.0]{veech.eps}
\caption{\label{fig:veech} An element of the Veech group}
\end{figure}
A line segment under the flat metric that connects two zeros of $\omega$ (not necessarily distinct) is called a \emph{saddle connection}. Since the set of saddle connections of $(X, \omega)$ is preserved by a stabilizer, it follows that $\operatorname{SL}(X, \omega)$ is \emph{discrete}. Without loss of generality, suppose $(X, \omega)$ has a horizontal saddle connection.
Let $g_t = \bigl(\begin{smallmatrix}
e^t & 0\\ 0 & e^{-t}
\end{smallmatrix} \bigr)
\in \operatorname{SL}_2(\mathbb R)$ act on $(X, \omega)$. As $t\to \infty$, the horizontal saddle connection becomes arbitrarily long, hence the
$\operatorname{SL}_2(\mathbb R)$-orbit of $(X, \omega)$ is \emph{unbounded} in $\mathcal H(\mu)$. Consequently $\operatorname{SL}(X, \omega)$
is \emph{not cocompact} in $\operatorname{SL}_2(\mathbb R)$.
Adding the traces of all $A\in \operatorname{SL}(X, \omega)$ to $\mathbb Q$, we obtain a field extension of $\mathbb Q$, called the \emph{trace field} of
$(X, \omega)$. The degree of the trace field over $\mathbb Q$ is bounded by the genus of $X$
(see \cite[Proposition 2.5]{MoellerSurvey}).
\section{Teichm\"uller curves}
\label{sec:teich}
The Hodge bundle $\mathcal H$ maps to the moduli space $\mathcal M_g$ of smooth genus $g$ curves by forgetting the differentials:
$(X, \omega) \mapsto X$. Note that the subgroup $\operatorname{SO}(2)$ acting on an Abelian differential amounts to rotating the corresponding flat surface, hence it does not change the underlying complex structure. Similarly scaling the size of a flat surface preserves the underlying complex structure. It follows that the projection of a $\operatorname{GL}^{+}_2(\mathbb R)$-orbit to $\mathcal M_g$ factors through the upper half plane $\mathbb H$, and the induced map $\mathbb H \to \mathcal M_g$ (or simply its image) is called a \emph{Teichm\"uller disk}. On rare occasions a Teichm\"uller disk forms an algebraic curve in $\mathcal M_g$. In that case we call it a \emph{Teichm\"uller curve}.
\subsection{Properties of Teichm\"uller curves}
\label{subsec:teich}
Teichm\"uller curves are dimensionally minimal affine invariant submanifolds, which possess a number of fascinating properties. To name a few, a Teichm\"uller curve is a local \emph{isometry} from a curve to $\mathcal M_g$ under the
Kobayashi/Teichm\"uller metric (\cite{SmillieWeiss, VeechTeich}). The union of all Teichm\"uller curves is \emph{dense} in moduli spaces (\cite{EskinOkounkov, ChenRigid}). McMullen (\cite{McMullenRigid}) proved that Teichm\"uller curves are \emph{rigid}, hence they are defined over number fields.\footnote{Here the rigidity means as a map it does not deform.} Conversely, Ellenberg-McReynolds (\cite{EllenbergMcReynolds}) showed that every curve over a number field is birational to a Teichm\"uller curve over $\mathbb C$. If $(X, \omega)$ generates an algebraically primitive Teichm\"uller curve (see Section~\ref{subsec:teich-class}), M\"oller (\cite{MoellerTorsion})
showed that the difference of any two zeros of $\omega$ is a torsion in the Jacobian of $X$. M\"oller (\cite{MoellerHodge})
also analyzed the variation of Hodge structures associated to a Teichm\"uller curve and deduced that it parameterizes curves whose Jacobians have \emph{real multiplication}. Teichm\"uller curves are never complete in $\mathcal M_g$, but the closure of a Teichm\"uller curve only intersects certain boundary divisor of the Deligne-Mumford compactification $\overline{\mathcal M}_g$ (see Section~\ref{subsec:slope}).\footnote{Here we restrict to Teichm\"uller curves generated by Abelian differentials. Special quadratic differentials can also generate Teichm\"uller curves, see Section~\ref{subsec:quad}, but they may intersect other boundary divisors of $\overline{\mathcal M}_g$.}
\subsection{Square-tiled surfaces}
\label{subsec:square-tiled}
We first show that Teichm\"uller curves exist. One type of Teichm\"uller curves arises from certain \emph{branched covering construction}. Let $\mu = (m_1, \ldots, m_n)$ be a partition of $2g-2$. Consider a branched cover $\pi: X \to E$ such that
\begin{itemize}
\item $\deg \pi = d$,
\item $g(X) = g$,
\item $E$ is the square torus,
\item $\pi$ has a \emph{unique} branch point $q\in E$,
\item $\pi$ has $n$ ramification points $p_1, \ldots, p_n$ over $q$, each with ramification order $m_i$.
\end{itemize}
Let $\omega = \pi^{*} (dz)$, where $z$ is the standard coordinate on $E$. Then by the Riemann-Hurwitz formula
$$(\omega)_0 = m_1 p_1 + \cdots + m_n p_n,$$
hence $(X, \omega) \in \mathcal H(\mu)$. Such $(X, \omega)$ are called \emph{square-tiled surfaces} (or \emph{origami}). For example, Figure~\ref{fig:cover} exhibits a degree $5$ and genus $2$ branched cover of $E$ with a unique ramification point of order $2$. The resulting $(X, \omega)$ belongs to the stratum $\mathcal H(2)$.
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{C}{$X$}
\psfrag{c}{$c$}
\psfrag{d}{$d$}
\psfrag{E}{$E$}
\psfrag{u}{$u$}
\psfrag{v}{$v$}
\psfrag{e}{$5:1$}
\includegraphics[scale=0.5]{cover.eps}
\caption{\label{fig:cover} A square-tiled surface in $\mathcal H(2)$}
\end{figure}
The $\operatorname{GL}^{+}_2(\mathbb R)$-action on a square-tiled surface
amounts to varying the shape of the square, which is exchangeable with varying the shape of the flat torus first, see Figure~\ref{fig:cover-GL} for an example. Therefore, the Teichm\"uller disk generated by a square-tiled surface corresponds to the one-dimensional \emph{Hurwitz space} parameterizing degree $d$, genus $g$ connected covers of all elliptic curves with a unique branch point of ramification type $\mu$. Since Hurwitz spaces are algebraic varieties, it follows that the $\operatorname{GL}^{+}_2(\mathbb R)$-orbit of a square-tiled surface gives rise to a Teichm\"uller curve.\footnote{When $d$, $g$, and $\mu$ are fixed, the Hurwitz space can still be disconnected. In that case each of its connected components gives a Teichm\"uller curve.} Since $d$ can be arbitrarily large, this way we indeed obtain \emph{infinitely many} Teichm\"uller curves in $\mathcal H(\mu)$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{cover-GL.eps}
\caption{\label{fig:cover-GL} $\operatorname{GL}^{+}_2(\mathbb R)$-action on a square-tiled surface}
\end{figure}
Square-tiled surfaces correspond to \emph{lattice points} under period coordinates of a stratum (\cite[Lemma 3.1]{EskinOkounkov}), see Figure~\ref{fig:lattice-square}.
If $\pi: X\to E$ is a square-tiled surface of type $\mu = (m_1, \ldots, m_n)$, for any $\gamma\in H_1(X, p_1, \ldots, p_n; \mathbb Z)$, $\pi(\gamma)$ represents a closed loop in $E$, because
$p_1, \ldots, p_n$ all map to the unique branch point. Therefore,
$$\int_{\gamma}\pi^{*}(dz) = \int_{\pi_{*}\gamma} dz \in \mathbb Z \oplus \mathbb Z[i]. $$
Conversely if all relative periods of $(X,\omega)$ are lattice points, the map $\pi: X\to \mathbb C/\mathbb Z \oplus \mathbb Z[i]$
induced by
$$x\mapsto \int_{b}^x \omega$$
is well-defined, where $b$ is a fixed base point, hence realizing $(X,\omega)$ as a square-tiled surface. This explains density of
the union of Teichm\"uller curves in moduli spaces. It also provides an approach for analyzing \emph{volume growths} of the strata of Abelian differentials by counting the number of such square-tiled surfaces (\cite{EskinOkounkov}).
\begin{figure}[h]
\centering
\includegraphics[scale=1.0]{lattice-square.eps}
\caption{\label{fig:lattice-square} Square-tiled surfaces as lattice points in a stratum}
\end{figure}
Gutkin-Judge (\cite{GutkinJudge}) showed that the Veech group of a translation surface is
{\em commensurable with $\operatorname{SL}(2,\mathbb Z)$} if and only if the surface is tiled by parallelograms. In this sense Teichm\"uller curves generated by square-tiled surfaces are called \emph{arithmetic} Teichm\"uller curves.
However, there exist Teichm\"uller curves of other type that are \emph{not} generated by square-tiled surfaces. In the next section we will survey known results on the classification of Teichm\"uller curves.
\subsection{Classification of Teichm\"uller curves}
\label{subsec:teich-class}
By definition, the moduli space $\mathcal M_{1,1}$ of elliptic curves is a very first example of Teichm\"uller curves. Arithmetic Teichm\"uller curves generated by square-tiled surfaces are coverings of $\mathcal M_{1,1}$. On the contrary, a Teichm\"uller curve is called (geometrically) \emph{primitive} if it does not arise from a curve in a moduli space of lower genus via such a covering construction.
It is a challenging task to find examples of primitive Teichm\"uller curves. In genus two, Calta and McMullen (\cite{Calta, McMullenWeier})
independently classified primitive Teichm\"uller curves in $\mathcal H(2)$. They found infinitely many primitive Teichm\"uller curves whose constructions have several incarnations. In terms of flat geometry, a translation surface $(X, \omega)$ generating a primitive Teichm\"uller curve in $\mathcal H(2)$ possesses an $L$-shaped polygon presentation whose edges satisfy relations in a real quadratic field. In terms of algebraic geometry, the Jacobian of $X$ admits real multiplication, and the corresponding Teichm\"uller curve lies on a Hilbert modular surface. However for the other stratum $\mathcal H(1,1)$ in genus two, McMullen (\cite{McMullenTorsion}) proved that it contains a unique primitive Teichm\"uller curve generated by the regular decagon with parallel edges identified. McMullen (\cite{McMullenPrym}) further generalized the method of real multiplication by using Prym varieties and discovered infinitely many primitive Teichm\"uller curves in genus three and four. Using quotients of abelian covers of $\mathbb P^1$ by a finite group, in each genus Bouw-M\"oller (\cite{BouwMoeller})
constructed (finitely many) primitive Teichm\"uller curves, generalizing earlier constructions of Veech and Ward (\cite{VeechTriangle, Ward}).
Besides the above results, to date only few sporadic examples of primitive Teichm\"uller curves are known. It is natural to ask if there are only finitely many primitive Teichm\"uller curves in a given stratum $\mathcal H(\mu)$. There is a stronger notion of primitivity called \emph{algebraically primitive}, if the trace field of a translation surface $X$ in the Teichm\"uller curve has degree equal to the genus of $X$ (see Section~\ref{subsec:veech}). For example, the trace field of a square-tiled surface is $\mathbb Q$, hence it is far from being algebraically primitive. Indeed, algebraically primitive Teichm\"uller curves are geometrically primitive, but the converse is not always true (see \cite[Section 5.1]{MoellerSurvey}).
Finiteness results of algebrically primitive Teichm\"uller curves have been established in various cases. M\"oller (\cite{MoellerFinite})
proved that the hyperelliptic component of $\mathcal H(g-1, g-1)$ contains finitely many algebrically primitive Teichm\"uller curves. The strategy is to track the degeneration of flat surfaces along an algebrically primitive Teichm\"uller curve in the horizontal and vertical directions and use it to bound the torsion order of the difference of the two zeros. By studying the boundary of the locus of curves with real multiplication, Bainbridge-M\"oller (\cite{BainbridgeMoellerReal})
showed finiteness of algebrically primitive Teichm\"uller curves in $\mathcal H(3,1)$. Bainbridge-Habegger-M\"oller (\cite{BainbridgeHabeggerMoeller}) further established finiteness of algebrically primitive Teichm\"uller curves for all strata in genus three by a mix of techniques, including the Harder-Narasimhan filtration of the Hodge bundle over Teichm\"uller curves and height bounds for the boundary points of Teichm\"uller curves. Matheus-Wright (\cite{MatheusWright})
proved finiteness of algebrically primitive Teichm\"uller curves in the minimal stratum $\mathcal H(2g-2)$ for each prime genus $g \geq 3$. Their approach is to study orthogonality of Hodge-Teichm\"uller (real) planes in the Hodge bundle that respect the Hodge decomposition along a Teichm\"uller curve. Matheus-Nguyen-Wright (\cite{NguyenWright})
further showed that there are at most finitely many non-arithmetic Teichm\"uller curves in the hyperelliptic component of $\mathcal H(4)$.
One can also study Teichm\"uller curves contained in an affine invariant submanifold of a stratum.
Lanneau-Nguyen (\cite{LanneauNguyen}) showed that there are at most finitely many closed $\operatorname{GL}^{+}_2(\mathbb R)$-orbits (including primitive Teichm\"uller curves) in certain Prym loci in genus three. Lanneau-Nguyen-Wright (\cite{LanneauNguyenWright})
further proved finiteness of closed $\operatorname{GL}^{+}_2(\mathbb R)$-orbits in each non-arithmetic rank $1$ affine invariant submanifold (see Section~\ref{subsec:affine-class}). Apisa (\cite{Apisa})
showed finiteness of algebrically primitive Teichm\"uller curves in the hyperelliptic components of each minimal stratum in $g > 2$, as a byproduct of his classification of affine invariant submanifolds in the hyperelliptic components (see Section~\ref{subsec:affine-class}).
Although a complete classification of Teichm\"uller curves is still missing, the seminal work of Eskin-Mirzakhani-Mohammadi~(\cite{EskinMirzakhani, EskinMirzakhaniMohammadi}) on the structure of $\operatorname{GL}^{+}_2(\mathbb R)$-orbit closures provides us a powerful tool. Indeed some of the above results are built on their work. We hope the classification problem of Teichm\"uller curves (and in general affine invariant submanifolds) can be resolved in the next few decades.
\subsection{Slope, Siegel-Veech constant, and Lyapunov exponent}
\label{subsec:slope}
In this section we discuss several invariants of Teichm\"uller curves and describe a relation between them.
First, the behavior of geodesics on translation surfaces is related to \emph{billiards in polygons}. One can study various counting problems from this viewpoint. Recall that a saddle connection is a line segment connecting two saddle points. Consider counting saddle connections with bounded lengths. For instance, consider the standard torus formed by identifying parallel edges of the unit square and marked at the origin. The number of saddle connections of length $< L$ (counting with direction) equals the number of lattice points in the disk of radius $L$, see Figure~\ref{fig:lattice-saddle}.
\begin{figure}[h]
\centering
\psfrag{L}{$L$}
\includegraphics[scale=0.8]{lattice-saddle.eps}
\caption{\label{fig:lattice-saddle} Saddle connections on the standard torus}
\end{figure}
It has asymptotically \emph{quadratic growth} $\sim \pi L^2$. The leading term $\pi$ is an example of so-called \emph{Siegel-Veech constant}. In general, Siegel-Veech constants relate quadratic growth rates of finite-length trajectories on a translation surface to the volume of the corresponding $\operatorname{SL}_2(\mathbb R)$-orbit. In what follows we will concentrate on one type of Siegel-Veech constants, called the \emph{area} Siegel-Veech constant, which has a connection to dynamics as well as intersection theory on moduli space.
A real geodesic on a translation surface $(X, \omega)$ is called \emph{regular}, if it does not pass through any saddle point, namely, it does not contain any zero of $\omega$. Vary a closed regular geodesic in parallel until it hits a saddle point on both ends. The union of those geodesics fill a \emph{cylinder} $\operatorname{cyl}$, whose boundary circles contain saddle points. For example, the square-tiled surface in Figure~\ref{fig:cylinder} has two cylinders in the horizontal direction.
\begin{figure}[h]
\centering
\psfrag{h1=1}{$h_1 = 1$}
\psfrag{l1=1}{$w_1 = 1$}
\psfrag{h2=1}{$h_2 = 1$}
\psfrag{l2=4}{$w_2 = 4$}
\includegraphics[scale=0.5]{cylinder.eps}
\caption{\label{fig:cylinder} Two horizontal cylinders on a square-tiled surface}
\end{figure}
Let $w$ and $h$ be the \emph{width} and \emph{height} of a cylinder $\operatorname{cyl}$, respectively. The \emph{area} of the cylinder is
$\operatorname{Area}(\operatorname{cyl}) = w \cdot h$.
For $L > 0$, define
$$N(X, L) = \frac{1}{\operatorname{Area}(X)} \cdot \sum_{\operatorname{cyl}\subset X\atop w(\operatorname{cyl}) < L} \operatorname{Area}(\operatorname{cyl}).$$
Veech and Eskin-Masur (\cite{VeechSiegel, EskinMasur}) showed that for any $\operatorname{GL}^{+}_2(\mathbb R)$-orbit closure $\mathcal N$, there exists a constant $c$ such that
$$ \frac{\pi}{3}\cdot \lim_{L \to \infty} \frac{N(X,L)}{L^2} = c $$
for \emph{generic} $(X, \omega)\in \mathcal N$. The constant
$c$ is called the \emph{area Siegel-Veech constant} of $\mathcal N$.\footnote{Here our definition of the Siegel-Veech constant differs from its usual definition by a scalar multiple of $\pi^2/3$. We choose to normalize it this way for the convenience of describing its relation with slope and Lyapunov exponent.} Its value depends on $\mathcal N$.
If $\mathcal N$ is the orbit of a square-tiled surface, namely, if its projection to $\mathcal M_g$ is an arithmetic Teichm\"uller curve, there is a \emph{combinatorial} way to calculate its area Siegel-Veech constant. Let $N$ be the number of square-tiled surfaces in $\mathcal N$. In other words, $N$ is the \emph{Hurwitz number} that counts branched covers of a fixed torus with a unique branch point and prescribed ramification type. Take all square-tiled surfaces in $\mathcal N$. For each one, consider all of its horizontal cylinders. Sum up $h/w$ (the \emph{modulus} of a cylinder) over all of them. Denote the total sum by $M$. For example, the square-tiled surface in Figure~\ref{fig:cylinder} above contributes $ \frac{h_1}{w_1} + \frac{h_2}{w_2} = \frac{1}{1} + \frac{1}{4}$
to the sum $M$.
For an arithmetic Teichm\"uller curve generated by a square-tiled surface,
$$ c = \frac{M}{N}, $$
see~\cite[Appendix B]{EKZ}.
The idea behind the formula is that $c$ measures the average number of cylinders weighted by their moduli $h/w$ in
$\mathcal N$, where
$$\frac{h}{w} = \frac{hw}{w^2} = \frac{\operatorname{Area}(\operatorname{cyl})}{w^2}, $$
hence it is related to the quadratic growth rate of $N(X, L)$ in this case.
For low degree $d$ and low genus $g$, using monodromy of branched covers one can calculate $N$ and $M$ explicitly. Nevertheless, the enumeration of $N$
as $d$ and $g$ increase is a highly non-trivial problem in symmetric group representations. Eskin-Okounkov (\cite{EskinOkounkov})
analyzed the asymptotic behavior of $N$ and calculated the \emph{volume} growth of strata of Abelian differentials.
The enumeration of $M$ is more complicated for large $d$ and $g$. Joint with M\"oller and Zagier (\cite{ChenMoellerZagier}) we are able to understand the asymptotic growth of $M$ and hence $c$ for arithmetic Teichm\"uller curves by using techniques of shifted symmetric functions and quasimodular forms.
Next we define an important index associated to a one-dimensional family of stable curves. The \emph{boundary} $\Delta$ of the
Deligne-Mumford moduli space $\overline{\mathcal M}_g$ parameterizes \emph{stable nodal curves}, where
$\Delta = \bigcup\limits_{i=0}^{[g/2]} \Delta_i$ is a union of irreducible \emph{boundary divisors} $\Delta_i$.
General points of $\Delta_i$ parameterize nodal curves of a given topological type, and they can further degenerate,
see Figure~\ref{fig:boundary}.
\begin{figure}[h]
\centering
\psfrag{i}{$i$}
\psfrag{a}{$\Delta_i$:}
\psfrag{j}{$g-i$}
\psfrag{k}{$g-1$}
\psfrag{1}{$i-1$}
\psfrag{b}{$\Delta_0$:}
\psfrag{c}{$\Delta_0\cap \Delta_i$:}
\includegraphics[scale=0.3]{boundary.eps}
\caption{\label{fig:boundary} Curves in the boundary of $\overline{\mathcal M}_g$}
\end{figure}
The Hodge bundle $\mathcal H$ over $\mathcal M_g$ extends to a rank $g$ bundle $\overline{\mathcal H}$ over $\overline{\mathcal M}_g$, whose fiber over a stable curve $X$ parameterizes sections of the dualizing line bundle $K_X$. Geometrically speaking, the fiber of $\overline{\mathcal H}$ over $X$ can be identified with the space of \emph{stable differentials} that have at worst simple pole at each node of $X$ with opposite residues on the two branches of a node (\cite[Chapter 3.A]{HarrisMorrison}). Denote by $\lambda$ the first Chern class of $\overline{\mathcal H}$ over $\overline{\mathcal M}_g$. Given a one-dimensional family $B$ of stable genus $g$ curves, define its \emph{slope} by
$$ s = \frac{\deg \Delta|_B}{\deg \lambda|_B}. $$
Morally speaking, $\deg \Delta|_B$ counts the number of nodes (with multiplicity) and $\deg \lambda|_B$ measures the variation of complex structures, see Figure~\ref{fig:family}.
\begin{figure}[h]
\centering
\psfrag{B}{$B$}
\includegraphics[scale=0.2]{family.eps}
\caption{\label{fig:family} A one-dimensional family of stable curves}
\end{figure}
Let $\mu = (m_1, \ldots, m_n)$ be a partition of $2g-2$. Define
$$\kappa_{\mu} = \frac{1}{12}\sum_{i=1}^{n} \frac{m_i (m_i+2)}{m_i+1},$$
which depends on $\mu$ only. For a Teichm\"uller curve in $\mathcal H(\mu)$, despite that its area Siegel-Veech constant and slope are defined in different contexts, they
determine each other (\cite{ChenThesis, ChenRigid, ChenMoellerAbelian}):
$$ s = \frac{12c}{c + \kappa_{\mu}}. $$
The upshot to prove the formula consists of the following. First, the constant $\kappa_\mu$ corresponds to the Miller-Morita-Mumford \emph{$\kappa$-class}. Next, a cylinder with modulus $h/w$ contributes $h/w$ to the intersection of the Teichm\"uller curve with the boundary $\Delta$, see Figure~\ref{fig:delta}. Moreover, the numerical relation $\displaystyle 12\lambda \equiv \kappa+ \Delta$ holds on $\overline{\mathcal M}_g$.
\begin{figure}[h]
\centering
\psfrag{h}{$h$}
\psfrag{w}{$w$}
\psfrag{n}{a node}
\includegraphics[scale=0.25]{delta.eps}
\caption{\label{fig:delta} Shrinking the core curve of a cylinder}
\end{figure}
Finally we introduce an important dynamical invariant for an affine invariant submanifold. The diagonal subgroup $\bigl(\begin{smallmatrix}
e^t & 0\\ 0 & e^{-t}
\end{smallmatrix} \bigr)$
defines the \emph{Teichm\"uller geodesic flow} on the Hodge bundle $\mathcal H$ over an affine invariant submanifold $\mathcal N$.
Since $\mathcal H$ is of rank $g$, there are $g$ nonnegative \emph{Lyapunov exponents}
$\lambda_1 \geq \cdots \geq \lambda_g \geq 0$
as logarithms of mean eigenvalues of monodromy of $\mathcal H$ along the flow on $\mathcal N$. Morally speaking, Lyapunov exponents
measure the separation rates of infinitesimally closed trajectories (see \cite{Zorich} for more details). Denote the \emph{sum} of the Lyapunov exponents by
$$L = \lambda_1 + \cdots + \lambda_g.$$
In a seminal work, Eskin-Kontsevich-Zorich (\cite{EKZ})
proved that the sum of Lyapunov exponents and the area Siegel-Veech constant determine each other for \emph{any} affine invariant submanifold:
$$ L = c + \kappa_{\mu}. $$
As a corollary, we thus obtain that for a Teichm\"uller curve,
any one of the three numbers $s$, $c$, and $L$ determine the other two.
As an application, one can deduce a non-varying phenomenon of Teichm\"uller curves in low genus. By computer experiments, Kontsevich and Zorich observed that for many strata $\mathcal H(\mu)$ in low genus, \emph{all} Teichm\"uller curves in the same stratum (component) have \emph{non-varying} sums of Lyapunov exponents. They came up with a conjectural list of such strata:
\begin{itemize}
\item $g =2$: $\mathcal H(1,1)$, $\mathcal H(2)$.
\item $g=3$: $\mathcal H(4)$, $\mathcal H(3,1)$, $\mathcal H(2,2)$, $\mathcal H(2,1,1)$.
\item $g=4$: $\mathcal H(6)$, $\mathcal H(5,1)$, $\mathcal H(4,2)$, $\mathcal H(3,3)$, $\mathcal H(3,2,1)$, $\mathcal H^{\operatorname{odd}}(2,2,2)$.
\item $g=5$: $\mathcal H(8)$, $\mathcal H^{\operatorname{odd}}(6,2)$, $\mathcal H(5,3)$, $\mathcal H^{\operatorname{hyp}}(4,4)$.
\end{itemize}
This phenomenon disappears when $g\geq 6$ except for the \emph{hyperelliptic} strata.
Joint with M\"oller we proved Kontsevich-Zorich's conjecture (\cite{ChenMoellerAbelian}). Let us use $\mathcal H(3,1)$ as an example to explain our method.
For $(X, \omega)\in \mathcal H(3,1)$, it is easy to observe that $X$ is \emph{non-hyperelliptic}. The same holds for any nodal curve $X$ contained in the \emph{boundary} of a Teichm\"uller curve $\mathcal{T}$ in $\mathcal H(3,1)$. Let
$\operatorname{Hyp}$ be the closure of the locus of hyperelliptic curves in $\overline{\mathcal M}_3$, which is a divisor with class
$$\operatorname{Hyp} \equiv 9 \lambda - \Delta_0 - 3\Delta_1, $$
see \cite[Chapter 3.H]{HarrisMorrison}. Since $\mathcal{T}$ is disjoint with $\operatorname{Hyp}$, we conclude that $\mathcal{T} \cdot \operatorname{Hyp} = 0$. Moreover,
$\mathcal{T}\cdot \Delta_i = 0$ for $i>0$, because shrinking the core curve of a cylinder yields a node of type $\Delta_0$ only, see
Figure~\ref{fig:delta} above. By $\mathcal{T} \cdot (9 \lambda - \Delta) = 0$, we see that the slope
$$ s = \displaystyle\frac{\deg \Delta|_\mathcal{T}}{\deg \lambda|_\mathcal{T}} = 9.$$
The above calculation is independent of the choice of Teichm\"uller curves in $\mathcal H(3,1)$. In summary, all Teichm\"uller curves in $\mathcal H(3,1)$ have slope $9$. By the slope, Siegel-Veech, and Lyapunov exponent formula, sums of Lyapunov exponents are the same for all Teichm\"uller curves in $\mathcal H(3,1)$.
In general, the strategy is to find a geometrically meaningful \emph{divisor} $D$ in $\overline{\mathcal M}_g$ (or in $\overline{\mathcal M}_{g,n}$ by marking the zeros of $\omega$) such that $D$ is \emph{disjoint} with all Teichm\"uller curves in $\mathcal H(\mu)$. The divisor class of $D$ determines the invariants of those Teichm\"uller curves.
We remark that Yu-Zuo (\cite{YuZuo}) gave another novel proof of Kontsevich-Zorich conjecture using the \emph{Harder-Narasimhan filtration} of the Hodge bundle over Teichm\"uller curves. Their idea is the following. Suppose $f: \mathcal X\to \mathcal{T}$ is the universal curve over a Teichm\"uller curve. Let $S_1, \ldots, S_n$ be the disjoint sections of zeros of $(X, \omega)\in \mathcal{T}$. For a tuple of integers $a_1, \ldots, a_n$, if $h^0(X, \sum_{i=1}^n a_i z_i) = k$ for all $X$, then the direct image sheaf
$f_{*}\mathcal O_{\mathcal X}(\sum_{i=1}^n a_i S_i)$ is a vector bundle of rank $k$ on $\mathcal{T}$.
Use $\mathcal H(3,1)$ as an example again. Since
$3z_1 + z_2 \sim K_X$, one checks that $h^0(X, z_1 + z_2 ) = 1$ and $h^0(X, 2z_1 + z_2) = 2$ for all $X \in \mathcal{T}$ (including the boundary points). Hence we obtain a filtration
$$ f_{*}\mathcal O_{\mathcal X}(S_1 + S_2)\subset f_{*}\mathcal O_{\mathcal X}(2S_1 + S_2)\subset f_{*}\mathcal O_{\mathcal X}(3S_1 + S_2), $$
where the last term can be identified with the Hodge bundle twisted by the generating differential of $\mathcal{T}$. Then the sum $L$ of
Lyapunov exponents of $\mathcal{T}$ follows from Chern class calculations of the subbundles in the filtration.
We also remark that the slopes of Teichm\"uller curves can provide information to understand the \emph{cone of effective divisors}
of $\overline{\mathcal M}_g$. The idea is that any effective divisor on $\overline{\mathcal M}_g$ cannot contain all arithmetic Teichm\"uller curves, because their union is \emph{dense} in $\overline{\mathcal M}_g$. Hence the limit of slopes of arithmetic Teichm\"uller curves as the degree of the coverings approach infinity bounds the numerical class of the divisor. We refer to the survey (\cite{ChenFaraksMorrison}) for more details on the effective cone of moduli spaces.
\subsection{Miscellaneous}
\label{subsec:misc}
In this section we collect various results with a flavor in algebraic geometry that are related to the preceding discussions.
McMullen (\cite{McMullenSpin}) classified connected components of arithmetic Teichm\"uller curves in $\mathcal H(2)$. The result is that an arithmetic Teichm\"uller curve in $\mathcal H(2)$ is either connected or has two components. The special case when the degree of the coverings is prime was previously established by
Hubert-Leli\`{e}ver (\cite{HubertLelievre}). In general, classifying connected components of arithmetic Teichm\"uller curves is a wide open question. From the viewpoint of algebraic geometry, it amounts to classifying connected components of the Hurwitz space of torus coverings with a unique branch point and prescribed ramification type. Bainbridge (\cite{BainbridgeEuler})
calculated Euler characteristics of Teichm\"uller curves in $\mathcal H(2)$. The idea is to calculate the fundamental classes of these Teichm\"uller curves in certain compactifications of Hilbert modular surfaces. Mukamel (\cite{Mukamel})
further determined the number and type of orbifold points of these Teichm\"uller curves. Kumar-Mukamel (\cite{KumarMukamel})
described algebraic models of Teichm\"uller curves in genus two. Weiss (\cite{Weiss})
calculated the volume of certain twisted Teichm\"uller curves on Hilbert modular surfaces and partially classified their connected components.
Siegel-Veech constants and Lyapunov exponents can be defined similarly for any affine invariant submanifold, say the strata themselves. Eskin-Masur-Zorich (\cite{EskinMasurZorich}) analyzed the principal boundary of the moduli space of Abelian differentials and gave a recursive method to compute Siegel-Veech constants of the strata for any Siegel-Veech configuration. For the cylinder configuration, the limit of area Siegel-Veech constants of arithmetic Teichm\"uller curves in a stratum as the degree of the coverings approach infinity equals the area Siegel-Veech constant of the stratum (\cite[Appendix A]{ChenRigid}).
Previously we discussed the sum of Lyapunov exponents. For arithmetic Teichm\"uller curves generated by cyclic covers of $\mathbb P^1$ branched at four points, Eskin-Kontsevich-Zorich (\cite{EKZ2}) calculated all individual Lyapunov exponents. Yu (\cite{Yu})
conjectured that the polygon of Lyapunov spectrum bounds the Harder-Narasimhan polygon on Teichm\"uller curves, and a proof of this conjecture is recently announced by Eskin-Kontsevich-M\"oller-Zorich. The fundamental work of Forni (\cite{Forni})
showed that for a stratum of Abelian differentials no Lyapunov exponent vanishes. The fundamental work of Avila and Viana (\cite{AvilaViana}) further showed that for a stratum the Lyapunov spectrum is simple, that is, the strict inequality
$\lambda_i > \lambda_{i+1}$ holds for all $i$. For arithmetic Teichm\"uller curves generated by square-tiled surfaces,
Matheus-M\"oller-Yoccoz (\cite{MatheusMoellerYoccoz}) gave a Galois-theoretical criterion for the simplicity of their Lyapunov spectra.
The Lyapunov spectrum can degenerate for a special affine invariant submanifold. If $\lambda_2 = \cdots = \lambda_g = 0$, we say that the Lyapunov spectrum is completely degenerate.
Forni and Forni-Matheus-Zorich (\cite{ForniDegenerate, ForniMatheusZorich})
found examples of Teichm\"uller curves with completely degenerate Lyapunov spectrum in genus three and four, respectively. By studying Teichm\"uller curves that
are also Shimura curves, M\"oller (\cite{MoellerST}) showed that those examples are the only Teichm\"uller curves with completely degenerate Lyapunov spectrum with possible exceptions in genus five. Aulicino (\cite{Aulicino})
showed that an affine invariant submanifold with completely degenerate Lyapunov spectrum can only be an arithmetic Teichm\"uller curve in genus at most five, and he further established finiteness of Teichm\"uller curves with completely degenerate Lyapunov spectrum. Filip (\cite{FilipDegenerate})
described all situations when an affine invariant submanifold can possess a zero Lyapunov exponent by analyzing monodromy of the corresponding Kontsevich-Zorich cocycle. As a higher dimensional analogue of families of curves, Filip (\cite{FilipK3})
considered families of K3 surfaces whose second cohomology groups form a local system, and showed that their top Lyapunov exponents are always rational.
To conclude this section we mention two problems that have broad connections to other fields. Kontsevich-Zorich (\cite{KontsevichZorich2}) conjectured that every connected component of the strata is $K(\pi, 1)$, that is, it has a contractible universal cover and its fundamental group is commensurable with certain mapping class group. For all strata in genus three except $\mathcal H(1^4)$, Looijenga-Mondello (\cite{LooijengaMondello}) determined their (orbifold) fundamental groups by analyzing geometry of canonical curves. Kontsevich-Soibelman (\cite{KontsevichSoibelman}) speculated that moduli spaces of differentials can be identified with
moduli spaces of certain stability conditions, where the $\operatorname{GL}_2^{+}(\mathbb R)$-action and saddle connections are analogues
of the central charge in the theory of stability conditions. The seminal work of Bridgeland-Smith (\cite{BridgelandSmith}) established this identification for moduli spaces of quadratic differentials with simple zeros by relating the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.
\section{Affine invariant submanifolds}
\label{sec:affine}
In the preceding section we discussed Teichm\"uller curves as examples of affine invariant submanifolds.
In this section we consider affine invariant submanifolds in general.
\subsection{Structure of affine invariant submanifolds}
\label{subsec:structure}
The celebrated Ratner's orbit closure theorem in ergodic theory says that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are homogeneous submanifolds. In the context of Teichm\"uller dynamics, the strata of Abelian differentials do not behave like homogeneous spaces. Hence it is unclear whether $\operatorname{GL}_2^{+}(\mathbb R)$-orbit closures can have nice geometric structures. Nevertheless, the recent breakthrough of
Eskin-Mirzakhani-Mohammadi (\cite{EskinMirzakhani, EskinMirzakhaniMohammadi}) showed that
$\operatorname{GL}_2^{+}(\mathbb R)$-orbit closures are locally cut out by homogeneous linear equations of period coordinates with real coefficients, thus justifying that $\operatorname{GL}_2^{+}(\mathbb R)$-orbit closures are affine invariant submanifolds.\footnote{Zorich called it the ``magic wand theorem'', which is part of Mirzakhani's Fields Medal work.} Previously this result was only proved in genus two by the fundamental work of McMullen (\cite{McMullenOrbit}).
Recall that the period coordinates at $(X, \omega) \in \mathcal H(\mu)$ are given by integrating $\omega$ over a basis of the relative homology $H_1(X, \Sigma; \mathbb Z)$, where $\Sigma$ is the set of zeros of $\omega$. Although period coordinates are not canonical, any two choices of period coordinates differ by a matrix in $\operatorname{GL}_n(\mathbb Z)$, where $n = \dim \mathcal H(\mu)$, hence the above theorem does not depend on the choice of period coordinates. The proof of the theorem is remarkably long and technical, which involves many ideas from dynamics on homogeneous space, ergodic theory, and measure theory.\footnote{The paper of Eskin-Mirzakhani (\cite{EskinMirzakhani}) is more than $170$ pages. The author heard from Eskin that even a referee report for the paper is more than $40$ pages.}
Since period coordinates are transcendental, affine invariant submanifolds are apriori only complex-analytic submanifolds.
The hidden algebraic nature has been discovered by Filip (\cite{Filip}), who proved that affine invariant submanifolds are algebraic subvarieties of $\mathcal H(\mu)$, defined over $\overline{\mathbb Q}$. In particular, Filip used tools from variations of Hodge structures and showed that affine invariant submanifolds (except those of full rank,
see Section~\ref{subsec:affine-class}) parameterize curves with non-trivial endormophisms, such as real multiplication on a factor of the Jacobians, which generalizes M\"oller's earlier work (\cite{MoellerHodge, MoellerTorsion}) on torsion and real multiplication for Teichm\"uller curves. As a corollary, the closure of any Teichm\"uller disk (in the standard topology) in $\mathcal M_g$ is a subvariety of $\mathcal M_g$.
\subsection{Classification of affine invariant submanifolds}
\label{subsec:affine-class}
After the structure theorem of Eskin-Mirzakhani-Mohammadi, the classification of affine invariant submanifolds remains to be a central open question in the study of Teichm\"uller curves.
The tangent space of an affine invariant submanifold $\mathcal N$ at $(X, \omega)$ can be identified with $H^1(X, \Sigma; \mathbb C)$ under the period coordinates.\footnote{In Teichm\"uller dynamics $\mathcal M$ is commonly used to denote an affine invariant submanifold, but here we reserve $\mathcal M$ for the moduli space.} Denote by
$p: H^1(X, \Sigma; \mathbb C) \to H^1(X; \mathbb C)$ the projection from the relative cohomology to the absolute cohomology. Avila-Eskin-M\"oller (\cite{AvilaEskinMoeller})
proved that $p(T_{(X, \omega)}\mathcal N)$ is a complex symplectic vector space, hence it has even dimension.
Define the \emph{rank} of $\mathcal N$ to be $\frac{1}{2} \dim_{\mathbb C}p(T_ {(X, \omega)}\mathcal N)$, which is at most $g$ by definition.
If the rank of $\mathcal N$ is bigger than one, we say that $\mathcal N$ is of \emph{higher rank}. If it is equal to $g$, we say that $\mathcal N$
has \emph{full rank}. For example, arithmetic Teichm\"uller curves generated by square-tiled surfaces have rank one.
By analyzing the boundary of affine invariant submanifolds, Mirzakhani-Wright (\cite{MirzakhaniWright})
proved that if an affine invariant submanifold is of full rank, then it is either a connected component of a stratum or the hyperelliptic locus in a connected component of a stratum. Apisa (\cite{Apisa}) further showed that all affine invariant submanifolds of higher rank in the hyperelliptic strata arise from covering constructions, which gives a coarse classification of affine invariant submanifolds in the hyperelliptic strata modulo finitely many non-arithmetic Teichm\"uller curves and their connected components. Based on previously known examples, Mirzakhani conjectured that higher rank affine invariant submanifolds are either connected components of the strata or arise from covering constructions. Recently counterexamples of Mirzakhani's conjecture are announced by McMullen-Mukamel-Wright and Eskin-McMullen-Mukamel-Wright, whose discoveries rely on special Hurwitz spaces of branched covers, moduli spaces of pointed genus one curves, and a computer search.
\subsection{Degeneration of Abelian differentials}
\label{subsec:degeneration}
Despite the analytic definition of Teichm\"uller dynamics, a profound algebro-geometric foundation behind the story has already been revealed by many of the preceding results. In order to apply ideas from algebraic geometry, one upshot is to understanding \emph{degeneration} of Abelian differentials, or equivalently, describing a \emph{compactification} of strata of Abelian differentials, analogous to the Deligne-Mumford compactification $\overline{\mathcal M}_g$ of the moduli space of curves.
Recall that the Hodge bundle $\mathcal H$ extends to a rank $g$ bundle $\overline{\mathcal H}$ over $\overline{\mathcal M}_g$, parameterizing stable differentials that are sections of the dualizing line bundle. Hence it would be natural to compactify the stratum $\mathcal H(\mu)$ by taking its closure in $\overline{\mathcal H}$. Nevertheless, a disadvantage of this Hodge bundle compactification is that it can lose information of the limit positions of the zeros of $\omega$, especially if $\omega$ vanishes entirely on a component of the underlying reducible curve. Alternatively, up to scaling an Abelian differential is determined by the associated canonical divisor. Hence one can consider the
\emph{stratum $\mathcal P(\mu)$ of canonical divisors of type $\mu$} as the projectivization of $\mathcal H(\mu)$. Marking the $n$ zeros of the divisors, $\mathcal P(\mu)$ can be regarded as a subvariety of the Deligne-Mumford moduli space $\overline{\mathcal M}_{g,n}$ of stable genus $g$ curves with $n$ marked points, hence one can study degeneration of canonical divisors of type $\mu$ by taking the closure of $\mathcal P(\mu)$ in $\overline{\mathcal M}_{g,n}$.
Eisenbud-Harris (\cite{EisenbudHarrisLimit}) developed a theory of limit linear series that studies degeneration of line bundles and their sections from smooth curves to nodal curves of compact type. In our context the situation is slightly different, because the zero type $\mu$ of the sections is fixed and the underlying curves may fail to be of compact type. Nevertheless, the upshot of \emph{twisting} line bundles by irreducible components of a nodal curve still works. More precisely, define a \emph{twisted canonical divisor of type
$\mu$} on a nodal curve $X$ to be a collection of (possibly meromorphic) canonical divisors $D_j$ on each irreducible component $X_j$ of $X$ such that the following conditions hold:
\begin{enumerate}[(1)]
\item The support of $D_j$ is contained in the set of marked points and the nodes lying in $X_j$. Moreover, if $p_i$ is a marked point contained in $X_j$, then $\operatorname{ord}_{p_i}(D_j) = m_i$.
\item If $q$ is a node of $X$ lying in two irreducible components $X_1$ and $X_2$, then
$\operatorname{ord}_{q}(D_1) + \operatorname{ord}_{q}(D_2) = -2$.
\item If $q$ is a node of $X$ lying in two irreducible components $X_1$ and $X_2$ such that
$\operatorname{ord}_{q}(D_1) = \operatorname{ord}_{q}(D_2) = -1$, then for any node $q' \in X_1\cap X_2$,
$\operatorname{ord}_{q'}(D_1) = \operatorname{ord}_{q'}(D_2) = -1$. In this case we write $X_1 \sim X_2$.
\item If $q$ is a node of $X$ lying in two irreducible components $X_1$ and $X_2$ such that
$\operatorname{ord}_{q}(D_1) > \operatorname{ord}_{q}(D_2)$, then for any node $q' \in X_1\cap X_2$,
$\operatorname{ord}_{q'}(D_1) > \operatorname{ord}_{q'}(D_2)$. In this case we write $X_1 \succ X_2$.
\item There does not exist a directed loop $X_1 \succeq X_2 \succeq \cdots \succeq X_k \succeq X_1$ unless all the relations are $\sim$, where $\succeq$ means $\sim$ or $\succ$.
\end{enumerate}
We briefly explain the motivation behind these conditions. Since the vanishing order along each zero section remains unchanged in a family, it implies condition (1). Since $K_X|_{X_i}$ is locally generated by differentials with a simple pole at a node $q\in X_1\cap X_2$ for $i=1,2$, when twisting by $X_i$, the vanishing order increases by one on one branch of $q$ and decrease by one on the other branch, hence the sum of the vanishing orders does not vary, which implies condition (2). If there is no twist at all, it is the case corresponding to condition (3). Note that twisting by $X_1 + X_2$ does nothing to the nodes lying in their intersection. Hence one can consider twisting, say by a multiple of $X_1$ only. Then the vanishing orders at all nodes between $X_1$ and $X_2$ increase or decrease simultaneously, hence condition (4) follows. By the same token,
$X_1 \succeq X_2$ means the twisting coefficient of $X_1$ is bigger than or equal to that of $X_2$, which implies the last condition.
By an analytic approach, Gendron (\cite{Gendron}) implicitly derived the above conditions and used them to study the Kodaira dimension of strata of twisted canonical divisors. Motivated by the theory of limit linear series, the author (\cite{ChenBoundary})
considered these conditions for curves of compact type and used them to study Weierstrass point behavior for general elements in the strata.
Farkas and Pandharipande (\cite{FarkasPandharipande}) imposed explicitly these conditions and showed that the corresponding closures in $\overline{\mathcal M}_{g,n}$ are reducible in general, containing extra boundary components of dimension one less compared to the main component. It is thus natural to ask what extra conditions can distinguish the main component from the other boundary components in the closure.
Joint with Bainbridge, Gendron, Grushevsky, and M\"oller (\cite{BCGGM1}), we have found the missing condition that arises from a \emph{residue} constraint. Consider the following example. Suppose a family $\mathcal X$ of Abelian differentials $(X_t, \omega_t)$ degenerate to a nodal curve $X$ at $t=0$. Suppose $X$ has a separating node $q$ joining two components $Y$ and $Z$. Without loss of generality, suppose $\lim\limits_{t\to 0} \omega_t |_Y = \eta_Y$ is a holomorphic differential on $Y$, and
$\lim\limits_{t\to 0} (t^{-\ell}\omega_t) |_Z = \eta_Z$ is a meromorphic differential on $Z$, where $\ell \in \mathbb Z^{+}$. In other words, the twisting at $q$ is given by $\mathcal O_{\mathcal X}(-\ell Z)$ from the viewpoint of limit linear series. Let $v_t$ be the vanishing cycle on $X_t = Y_t\cup Z_t$
that shrinks to the node $q$, where $Y_t\to Y$ and $Z_t\to Z$ as $t\to 0$, see Figure~\ref{fig:vanishing}.
\begin{figure}[h]
\centering
\psfrag{Y}{$Y_t$}
\psfrag{Z}{$Z_t$}
\psfrag{v}{$v_t$}
\includegraphics[scale=1.0]{vanishing.eps}
\caption{\label{fig:vanishing} A surface with a separating vanishing cycle}
\end{figure}
Since $q$ is a separating node, $v_t = 0 \in H_1(X_t; \mathbb Z)$ for $t$ nearby $0$. It follows that
$\int_{v_t} \omega_t = 0$ for $t\neq 0$ and hence $\int_{v_t} t^{-\ell} \omega_t = 0$. Taking the limit as $t\to 0$ and restricting to the component $Z$, we conclude that $\operatorname{Res}_{q} (\eta_Z) = 0$. Conversely, once such a residue condition holds, along with conditions (1)--(4) we are able to prove that the limit differential is smoothable by plumbing techniques in complex-analytic geometry as well as by constructions of flat surfaces.
\subsection{Cycle classes of strata of Abelian differentials}
\label{subsec:cycle}
As mentioned before, affine invariant submanifolds can provide special subvarieties in the moduli space of curves. It is natural to ask if one can calculate their \emph{cycle classes} in the Chow ring of the moduli space. The first step is to calculate the cycle classes of the strata of Abelian differentials. Tarasca and the author (\cite{ChenTarasca}) calculated the cycle classes of several minimal strata in low genus. Mullane (\cite{Mullane}) obtained a closed formula for the classes of all strata whose projections are effective divisors in $\overline{\mathcal M}_g$. Both results rely on classical intersection theory on the moduli space of curves. Modulo a conjectural relation to Pixton's formula of the double ramification cycle,
Janda-Pandharipande-Pixton-Zvonkine (\cite[Appendix]{FarkasPandharipande}) obtained a recursive method to compute the cycle classes of all strata in $\overline{\mathcal M}_{g,n}$, which suggests that the strata classes are \emph{tautological}. Their approach relies on analyzing the closure of a stratum of twisted canonical divisors in $\overline{\mathcal M}_{g,n}$ by imposing conditions (1)--(4) in the preceding section. Although the closure may have extra components contained in the boundary, those extra components are products of simpler strata of (possibly meromorphic) differentials, hence one can calculate the class of the main component recursively.
One can also consider the cycle class calculation in the Hodge bundle compactification. Korotkin-Zograf (\cite{KorotkinZograf}) applied Tau functions to compute the divisor class of the closure of the stratum $\mathcal P(2, 1^{2g-4})$ in the projectivized Hodge bundle over $\overline{\mathcal M}_g$. The author (\cite{ChenCycle}) gave another proof of this divisor class using intersection theory. Recently Sauvaget-Zvonkine have announced that they are able to compute the cycle classes of all strata closures in the Hodge bundle compactification.
\section{Meromorphic and higher order differentials}
\label{sec:higher}
In this section we generalize the discussion of Abelian differentials to higher order differentials possibly with poles.
\subsection{Quadratic differentials}
\label{subsec:quad}
First, if $q$ is a quadratic differential, that is, a section of $K^{\otimes 2}$, then it also induces a flat structure, if one allows \emph{reflection} in addition to translation as transition functions. The flat structure can be defined locally by taking a square root $\omega$ of $q$, which is up to $\pm$, and that is the reason why reflection needs to be part of the transition functions. Moreover if $q$ has at worst simple poles, integrating $\omega$ along a path always provides finite length, hence the corresponding flat surface has finite area. In general, quadratic differentials with at worst simple poles are called \emph{half-translation surfaces}. For example, Figure~\ref{fig:pillow} presents a quadratic differential with four simple poles on $\mathbb P^1$ as a pillowcase.
\begin{figure}[h]
\centering
\includegraphics[scale=0.8]{pillow.eps}
\caption{\label{fig:pillow} A quadratic differential with four simple poles on $\mathbb P^1$}
\end{figure}
Suppose $(X, q)$ is a half-translation surface. One can take the unique \emph{canonical double cover} $\pi: \hat{X}\to X$ branched at the odd singularities of $q$ (zeros of odd order and simple poles). Then there exists a global Abelian differential $\hat{\omega}$ on $\hat{X}$ such that $\pi^{*}q = \hat{\omega}^2$. From this viewpoint, all questions for Abelian differentials can be similarly asked for quadratic differentials, and indeed many of them can be similarly answered.
In what follows we mention several results analogous to the case of Abelian differentials. Fix a partition $\nu$ of $4g-4$ such that all entries of $\nu$ are $\geq -1$. Let $\mathcal Q(\nu)$ be the stratum of quadratic differentials of type $\nu$ that are not global squares of Abelian differentials. Lanneau (\cite{LanneauQuad})
classified the connected components of $\mathcal Q(\nu)$ for all $\nu$. In $g\geq 5$, it can have at most two connected components, where extra components are caused by hyperelliptic structures. In particular, the spin parity in the Abelian case does not give rise to additional components in the quadratic case (\cite{LanneauSpin}).
In genus three and four, there are several \emph{exceptional} disconnected strata. Joint with M\"oller (\cite{ChenMoellerQuadratic})
we found an algebraic parity arising from geometry of canonical curves that distinguishes these exceptional components. We further discovered a number of strata $\mathcal Q(\nu)$ in low genus whose Teichm\"uller curves have non-varying sums of Lyapunov exponents by similar techniques as in the Abelian case. The relation between the area Siegel-Veech constant and the sum of Lyapunov exponents for affine invariant submanifolds in $\mathcal Q(\nu)$ also holds by the seminal work of
Eskin-Kontsevich-Zorich (\cite{EKZ}). Eskin-Okounkov (\cite{EskinOkounkovQuad}) analyzed the volume growth of $\mathcal Q(\nu)$ by enumerating covers of $\mathbb P^1$ with certain ramification profile determined by $\nu$. Athreya-Eskin-Zorich (\cite{AthreyaEskinZorich}) proved an explicit formula for the volume of $\mathcal Q(\nu)$ in genus zero. Goujard (\cite{Goujard2}) obtained explicit values for the volume of $\mathcal Q(\nu)$ in all low dimensions. Masur-Zorich (\cite{MasurZorich}) studied the principal boundary of $\mathcal Q(\nu)$, aiming at a recursive way to calculate Siegel-Veech constants of saddle connections. Goujard (\cite{Goujard1}) proved an explicit formula that relates volumes of $\mathcal Q(\nu)$ and Siegel-Veech constants. Grivaux-Hubert (\cite{GrivauxHubert}) constructed explicit affine invariant submanifolds in $\mathcal Q(\nu)$ of arbitrarily large dimension with completely degenerate Lyapunov spectrum.
\subsection{Differentials with poles}
\label{subsec:pole}
Previously when we talked about an Abelian differential $\omega$, we assumed that $\omega$ is holomorphic, so the corresponding translation surface has finite area. In many cases it would be useful to consider \emph{meromorphic} differentials, in particular when we study the boundary structure of strata of Abelian differentials.
First, we describe the local flat geometry around a pole. Suppose $p$ is a \emph{simple} pole of $\omega$. Then the flat neighborhood of $p$ can be viewed as a \emph{half-infinite cylinder}. The \emph{width} of the cylinder corresponds to the \emph{residue} of $\omega$ at $p$. Figure~\ref{fig:pole-simple} exhibits a meromorphic differential that has two simple poles with opposite residues, where the poles locate at the positive infinity and negative infinity.
\begin{figure}[h]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{l}{$l_1$}
\psfrag{m}{$l_2$}
\includegraphics[scale=0.8]{pole-simple.eps}
\caption{\label{fig:pole-simple} A flat surface with two simple poles}
\end{figure}
If $p$ is a pole of order $m \geq 2$, Boissy (\cite{Boissy})
showed that one can glue $2m-2$ \emph{broken half-planes} consecutively to form a flat-geometric presentation for a neighborhood of $p$. The boundary of each broken half-plane consists of a half-line to the left and a half-line to the right, which are connected by finitely many broken line segments (saddle connections). The idea behind Boissy's description is the following. If one glues $2m-2$ half-disks consecutively to form a zero of order $m-2$, the local expression of the differential would be $z^{m-2} dz$. Now change coordinates by $z = 1/w$. Then the differential becomes
$\sim w^{-m} dw$, which has a pole of order $m$, and the half-disks turn into half-planes. In particular, the residue of $\omega$ at $p$ is determined by the complex lengths of the boundary line segments and the gluing pattern of the broken half-planes.
The lower right flat surface of infinite area in Figure~\ref{fig:two-tori} below shows the flat-geometric presentation of a meromorphic differential with a double zero and a double pole on a torus, where the pole locates at infinity. It is constructed by removing the interior of a parallelogram $Z$ from the Euclidean plane and then identifying parallel edges by translation. If we slit the plane along a diagonal of $Z$, we thus recover a pair of broken half-planes as in Boissy's description. In particular, the double pole has no residue.
Building on earlier work of Kontsevich-Zorich and Lanneau (\cite{KontsevichZorich, LanneauQuad}), Boissy (\cite{Boissy}) classified the connected components of strata of meromorphic differentials with prescribed numbers and multiplicities of zeros and poles. Similarly to the holomorphic case, the strata of meromorphic differentials can have at most three connected components, distinguished by hyperelliptic and spin structures.
Let us illustrate an interesting viewpoint using flat geometry of meromorphic differentials to study the boundary of strata of Abelian differentials. The flat surface on the left side of Figure~\ref{fig:two-tori} lies in $\mathcal H(2)$, which is constructed by removing a parallelogram $Z$ from the interior of a parallelogram $Y$ and identifying parallel edges. If we shrink $Z$ to a point, we obtain a holomorphic differential $(Y, \eta_Y) \in \mathcal H(0)$, where the marked point encodes the limit position of the inner square. Alternatively, modulo scaling this procedure amounts to expanding $Y$ to be arbitrarily large, hence the limit object represents a meromorphic differential $(Z, \eta_Z) \in \mathcal H(2, -2)$.
\begin{figure}[h]
\centering
\psfrag{Y}{$Y$}
\psfrag{Z}{$Z$}
\psfrag{S}{Shrink $Z$}
\psfrag{E}{Expand $Y$}
\includegraphics[scale=0.8]{two-tori.eps}
\caption{\label{fig:two-tori} Shrinking $Z$ versus expanding $Y$}
\end{figure}
Note that these two perspectives correspond to exactly the two \emph{aspects} in the theory of limit linear series, which applied to this case says that a family of curves of genus two with a marked double zero of a canonical divisor (Weierstrass point) degenerates to two elliptic curves joined at one node, where the limit marked point is $2$-torsion respect to the node.
From the viewpoint of $\operatorname{GL}^{+}_2(\mathbb R)$-action and Teichm\"uller dynamics, the strata of meromorphic differentials somehow display different properties compared to the case of Abelian differentials (see \cite[Appendix]{Boissy}).
\subsection{Higher order differentials}
\label{subsec:higher}
We conclude the survey by describing some future directions. From the viewpoint of algebraic geometry, an Abelian differential is a section of the canonical line bundle $K$, and a quadratic differential is a section of $K^{\otimes 2}$. Therefore, it is natural to consider \emph{higher order differentials} arising from sections of $K^{\otimes k}$ for a fixed positive integer $k$. Suppose $\mu = (k_1, \ldots, k_n)$ is a partition of $k(2g-2)$, where we allow $k_i$ to be possibly negative. In other words, we want to take meromorphic differentials into account. Let $\mathcal H^k(\mu)$ be the \emph{stratum of $k$-differentials of type $\mu$}, which parameterizes (possibly meromorphic) sections of $K^{\otimes k}$ with zeros and poles of type $\mu$ on genus $g$ Riemann surfaces. All previous questions regarding Abelian and quadratic differentials can be asked similarly for $k$-differentials. In particular, what are the dimension, connected components, compactification, invariants, and cycle class of
$\mathcal H^k(\mu)$? In a forthcoming work (\cite{BCGGM2}), we will treat these questions systematically.
|
1,314,259,995,766 | arxiv | \section{Introduction}
The term ``program synthesis''
refers to automatically generating code to satisfy some specification. That specification describes {\emph{what}} the code should do, without going into details about {\emph{how}} it should be done. The specification could be given as a set of constraints~\cite{MannaW79,KuncakMPS10}, it can be deduced from the program and its environment~\cite{GveroKKP13,Feng:2017:CSC:3009837.3009851}, or it can be inferred from a large corpus~\cite{BalogGBNT17,SantolucitoZDSP17}.
One paradigm of program synthesis is called \textit{programming by example}~\cite{Cypher:1993:WIP:168080} (PBE). In the PBE approach, a user only provides a set of pairs of input-output examples that illustrate the desired behavior of the code. From these examples, the PBE engine should then generate code that generalizes from the examples to create a program which covers the unspecified examples as well.
The idea of automated code synthesis is an area of research with a long history (cf. the Church synthesis problem~\cite{Church1963-CHUAOR}).
However, due to the problem's undecidability and high computational complexity for decidable fragments, for almost 50 years the research in program synthesis was mainly focused on addressing theoretical questions and the size of synthesized programs was relatively small. However, the state of affairs has drastically
changed in the last decade. By leveraging advances in automated reasoning and formal methods, there has been a renewed interest in software synthesis. The research in program synthesis has recently focused on
developing efficient algorithms and tools, and synthesis has even been used in industrial software~\cite{gulwani2011automating}. Today, machine learning plays a vital role in modern software synthesis and there are numerous tools and startups that rely on machine learning and big data to automatically generate code~\cite{codataWeb,BalogGBNT17}.
With numerous synthesis tools and formats being developed, it was difficult to empirically evaluate and compare existing synthesis tools. The Syntax Guided Synthesis (SyGuS) format language~\cite{alur2013syntax,sygusWeb}
was introduced in an effort to standardize the specification format of program synthesis, including PBE synthesis problems. The SyGuS language specifies synthesis problems through two components - a set of constraints (eg input-output examples), and a grammar (a set of functions).
The goal of a SyGuS synthesis problem is to construct a program from functions within the given grammar that satisfies the given constraints. With this standardized synthesis format and an ever expanding set of benchmarks, there is now a yearly competition of synthesis tools~\cite{sygusCompetition2019}, which pushes the frontier of scalable synthesis further.
The SyGuS Competition splits synthesis problems into tracks, for example PBE Strings or PBE BitVectors, assigning a different grammar for each track - and sometimes even varying the grammar within a single track.
As the grammar defines the search space in SyGuS, this allows benchmark designers to ensure problems are relatively in-scope of current tools.
However, when synthesis is deployed in real-world applications, we must allow for larger grammars that account for the wide range of use-cases users require~\cite{chi19}.
While larger grammars allow for more expressive power in the synthesis engine, it also slows down the whole synthesis process.
In our own experimentation, we found that by manually removing some parts of the grammar from the SyGuS Competition benchmarks, we can significantly improve synthesis times.
Accordingly, we sought to automate this process.
Removing parts of a grammar is potentially dangerous though, as we may remove the possibility of finding a solution altogether.
In fact, understanding the grammar's impact on synthesis algorithms is a complex problem, connected to the concept of overfitting~\cite{cav19overfitting}.
In this paper, we utilize machine learning to automate an analysis of a SyGuS grammar and a set of synthesis constraints.
We generate a large number of SyGuS problems, and use this data to train a neural network.
Given a new SyGuS problem, the neural network predicts how likely it is for a given grammar element to be critical to synthesizing a solution to that problem.
Our key insight is that, in addition to criticality, we predict how much time we expect to save by removing this grammar element.
We combine these predictions to efficiently filter grammars to fit a specific synthesis problem, in order to speed up synthesis times.
Even with these reduced grammars, we are still able to find solutions to the problems.
We implemented our approach in a modular tool, GRT\xspace, that can be attached to any existing SyGuS synthesis engine as a blackbox.
We evaluated GRT\xspace by running it on the SyGuS Competition Benchmarks from 2019 in the PBE Strings track.
We found GRT\xspace outperformed CVC4, the winner of the SyGuS Competition from 2019, reducing the overall synthesis time by $47.65\%$.
Additionally, GRT\xspace was able to solve a benchmark for which CVC4 timed out.
In summary, the core contributions of our work are as follows:
\begin{enumerate}
\item A methodology to generate models that can reduce time needed to synthesize PBE SyGuS problems.
In particular, our technique reduced the grammar by identifying which functions to try to eliminate to increase the efficiency of a SyGuS solver.
It also learns a model to predict which functions are critical for a particular PBE problem.
\item A demonstration of the effectiveness of our methodology. We show experiments on existing SyGuS PBE Strings track that demonstrates the speed up resulting from using our filtering as a preprocessor for an existing SyGuS solver. Over the set of benchmarks, our techniques decreases the total time taken by synthesis by $47.65\%$.
\end{enumerate}
\section{Related}
One approach to SyGuS is to directly train a neural network to satisfy the input/output examples~\cite{andrychowicz2016learning,devlin2017robustfill,graves2014neural,joulin2015inferring,kaiser2015neural,ChenLS17a}.
However, such approaches struggle to generalize, especially when the number of examples is small~\cite{devlin2017neural}.
Some existing work~\cite{wang2018execution,bunel2018leveraging} aims to represent aspects of the syntax and semantics of a language
in a neural network.
In contrast to these existing approaches, which aim to outright solve SyGuS problems, our work acts as a preprocessor for a separate SyGuS solver.
However, one could also explore using our work as a preprocessor for one of these existing neural network directed synthesis approaches.
Other works have explored combining logic-directed and machine learning guided synthesis approaches~\cite{nye2019learning}.
This work sought to split synthesis tasks between generating high level sketches with neural networks, and fill in the holes of the sketch with an enumerative solver.
Our work could be complementary to this, by assisting in pruning of the search space needed to fill in the holes.
Like our work, DeepCoder~\cite{BalogGBNT17} and Neural-Guided Deductive Search (NGDS)~\cite{kalyan2018neural}
identify pieces of a grammar that should be removed from the grammar.
However, in our parlance, these works only consider \textit{criticality}, which measures how important a part of the grammar is to completing synthesis.
Unlike our work, they do not consider the time savings from removing or keeping a part of the grammar.
NGDS~\cite{kalyan2018neural} does note that different models could be trained for different pieces of a grammar,
however, it provides no means of automating this process.
Rather, the user would have to manually elect to train individual neural networks for different grammatical elements.
Work by Si et al~\cite{si2018learning} aims to learn an efficient solver for a SyGuS from scratch, rather than, as in our work, acting as a preprocessor for a separate solver.
\section{Background}
A SyGuS synthesis problem is a tuple $(\ensuremath{C}\xspace, \ensuremath{G}\xspace)$ of constraints, \ensuremath{C}\xspace, and a context-free grammar, \ensuremath{G}\xspace.
In our case we restrict the set of constraints to the domain of PBE, so that all constraints are in the form of pairs $(i,o)$ of input-output examples.
We write \drop{\ensuremath{G}\xspace}{\ensuremath{g}\xspace} to denote the grammar \ensuremath{G}\xspace, but without the terminal symbol \ensuremath{g}\xspace.
The set of terminal symbols are the component functions that can be used in constructing a program (e.g. +, -, str.length).
We also use the notation, \setOf{\ensuremath{G}\xspace}, to denote the projection of \ensuremath{G}\xspace into its set representation, which is the set of the terminal symbols in the grammar.
The problem statement of syntax-guided synthesis (SyGuS) is; given a grammar, \ensuremath{G}\xspace, and a set of constraints \ensuremath{C}\xspace, find a program, $P \in \ensuremath{G}\xspace$, such that the program satisfies all the constraints -- $\forall c \in \ensuremath{C}\xspace. P \vdash c$.
For brevity, we equivalently write $P \vdash \ensuremath{C}\xspace$.
If our synthesis engine is able to find such a program in $t$ seconds or less, we write that $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \synthesizesInTime{t} P$.
We use the notation $\synthTime{\ensuremath{C}\xspace}{\ensuremath{G}\xspace}$ to indicate the time to run $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \synthesizesInTime{t} P$.
If the SyGuS solver is not able to find a solution within the timeout ($\synthTime{\ensuremath{C}\xspace}{\ensuremath{G}\xspace} > t$), we denote this as $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \not \synthesizesInTime{t} P$.
We typically set a timeout on all synthesis problems of 3600 seconds, the same value of the timeout used in the SyGuS competition.
We write $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \rightsquigarrow P$ and $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \not \rightsquigarrow P$
as shorthand for $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \synthesizesInTime{3600} P$ and $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \not \synthesizesInTime{3600} P$, respectively.
We define \ensuremath{G}\xspace as the grammar constructed from the maximal set of terminal symbols we consider for synthesis.
We call a terminal, \ensuremath{g}\xspace, within a grammar, \textit{critical} for a set of constraints, \ensuremath{C}\xspace, if $(\drop{\ensuremath{G}\xspace}{g}, \ensuremath{C}\xspace) \not \rightsquigarrow P$.
For any given set of constraints, if a solution exists with \ensuremath{G}\xspace, there is also a grammar, \ensuremath{\grammar_{crit}}\xspace, that contains exactly the critical terminal symbols required to find a solution.
More formally, \ensuremath{\grammar_{crit}}\xspace is constructed such that
\begin{equation*}
(\ensuremath{\grammar_{crit}}\xspace, \ensuremath{C}\xspace) \rightsquigarrow P \land
\forall g \in \ensuremath{\grammar_{crit}}\xspace. \ (\drop{\ensuremath{G}\xspace}{g}, \ensuremath{C}\xspace) \not \rightsquigarrow P
\end{equation*}
Note that \ensuremath{\grammar_{crit}}\xspace is not unique.
The goal of our work is to find a grammar, \ensuremath{\grammar^{\star}}\xspace, where $\setOf{\ensuremath{\grammar_{crit}}\xspace} \subseteq \setOf{\ensuremath{\grammar^{\star}}\xspace} \subseteq \setOf{\ensuremath{G}\xspace}$.
This will yield a grammar that removes some noncritical terminal symbols so that the search space is smaller, but still sufficient to construct a correct program.
\section{Overview}
\begin{figure}
\centering
\tikzstyle{module} = [style={draw,rectangle}, text width=2.2cm,align=center]
\tikzstyle{data} = [style={draw,rectangle}, text width=2.2cm,align=center, fill=blue!5!white]
\usetikzlibrary{calc}
\begin{tikzpicture}[node distance=1.5cm,auto,>=latex']
\node [data] (grammar) {Grammar \ensuremath{G}\xspace};
\node [data, right of=grammar, node distance=3cm] (constraints) {Constraints \ensuremath{C}\xspace};
\node [module, below of=grammar] (time) {Predict Time};
\node [module, below of=constraints] (crit) {Criticality};
\node [module, below of=crit] (combo) at ($(time)!0.5!(crit)$) {Combo};
\node [data, below of=combo] (synthG) {Predicted Grammar \ensuremath{\grammar^{\star}}\xspace};
\path[->] (grammar) edge (time);
\path[->] (constraints) edge (time);
\path[->] (grammar) edge (crit);
\path[->] (constraints) edge (crit);
\path[->] (time) edge (combo);
\path[->] (crit) edge (combo);
\path[->] (combo) edge (synthG);
\end{tikzpicture}
\caption{GRT\xspace uses the grammar \ensuremath{G}\xspace and constraints \ensuremath{C}\xspace to predict how critical each function is,
and the amount of time that would be saved by eliminating it from the grammar.
Then, it outputs a new grammar \ensuremath{\grammar^{\star}}\xspace, which it expects will speed up synthesis over the original grammar (that is, it expects that $T^C_{\ensuremath{\grammar^{\star}}\xspace} < T^C_\ensuremath{G}\xspace$).}
\label{fig:model}
\end{figure}
Our system, GRT\xspace, works as a preprocessing step for a SyGuS solver.
The goal of GRT\xspace is to remove elements from the grammar and thus, by having a smaller search space, save time during synthesis.
To do this we combine two metrics, as shown in Figure~\ref{fig:model}: our predicted confidence that a grammar element is not needed, and our prediction of how much time will be saved by removing that element.
We focus on removing only elements where we are both confident that the grammar element is noncritical, and that removing the grammar element significantly impacts synthesis times.
By giving the constraints and the grammar definition to GRT\xspace, we predict which elements of the grammar can be safely removed.
By analyzing running times we predict which of these elements are benefical to remove.
We describe GRT\xspace in three sections, addressing dataset generation, the training stage, and our evaluation.
\section{Data Generation}
In order to learn a model for GRT\xspace, we need to generate a labelled dataset that maps constraints to grammar components in \ensuremath{\grammar_{crit}}\xspace.
This will allow us to predict, given a new set of constraints $\ensuremath{C}\xspace '$, which grammar elements are noncritical for synthesis, and accordingly prune our grammar.
The generation of data for application to machine learning for program synthesis is a nontrivial problem, requiring careful construction of the dataset~\cite{shin2018synthetic}.
We break the generation of this dataset into two stages: first, we generate a set of programs, $\mathcal{P}$ from \ensuremath{G}\xspace.
Then, for each program in $\mathcal{P}$, we generate constraints for that program.
We additionally need a dataset of synthesis times, in order to predict how long synthesis takes for a given set of constraints.
\subsection{Criticality Data}
To generate a set of programs $\mathcal{P}$, that can be generated from a grammar \ensuremath{G}\xspace, we construct a synthesis query with no constraints.
We then run CVC4 with the command \texttt{--sygus-stream}, which instructs CVC4 to output as many solutions as it can find.
With no constraints, all functions satisfy the specification, and CVC4 will generate all permutations of (well-formed and well-typed) functions in the grammar, until the process is terminated (we terminate after generating $n$ programs).
Because CVC4 generates solutions of increasing size, we collect all generated programs, then shuffle the order to prevent data bias with respect to the order (size) in which CVC4 generated programs.
After generating programs, we generate corresponding constraints (in the form of input-output examples for PBE) for these functions.
To do this, for each program, $P$, we randomly generate a set of inputs $I$, and compute the input-output pairs $\ensuremath{C}\xspace = \{(i, P(i))\ |\ i \in I\}$.
We then form a SyGuS problem $(\ensuremath{G}\xspace, \ensuremath{C}\xspace)$, where we know that the program $P$ satisfies the constraints, and is part of the grammar: $P \vdash \ensuremath{C}\xspace$ and $P \in \ensuremath{G}\xspace$.
This amounts to programs that \textit{could} be synthesized from the constraints (i.e. $(\ensuremath{G}\xspace,\ensuremath{C}\xspace) \rightsquigarrow_{\infty} P$).
It is important that our dataset represent programs that \textit{could} be synthesized, as opposed to what \textit{can} be synthesized (i.e. $(\ensuremath{G}\xspace,\ensuremath{C}\xspace) \rightsquigarrow_{3600} P$).
This is important because we will use this data set to try to learn the ``semantics'' of constraints, and we do not want to use this data set to additionally, inadvertently learn the limitations of the synthesis engine.
At this point, we have now constructed a dataset of triples of grammars (fixed for all benchmarks), constraints, and programs, $D = \{(\ensuremath{G}\xspace, \ensuremath{C}\xspace_1, P_1) \ldots (\ensuremath{G}\xspace, \ensuremath{C}\xspace_n, P_n)\}$.
In order to use $D$ to helps us predict \ensuremath{\grammar_{crit}}\xspace, we break up each triple by splitting each constraint set \ensuremath{C}\xspace into its individual constraints.
For a triple $(\ensuremath{G}\xspace,\ensuremath{C}\xspace, P)$, where $\ensuremath{C}\xspace = \{c_1 \ldots c_m\}$, we generate a new set of triples $\{(\ensuremath{G}\xspace, c_1, P) \ldots (\ensuremath{G}\xspace, c_m, P)\}$.
The union of all these triples of individual constraints form our training set, \ensuremath{\mathcal{TR}_{crit}}, that will be used to predict critical functions in the grammar for a given set of constraints.
\subsection{Timing Data}
In addition to a training set for predicting \ensuremath{\grammar_{crit}}\xspace, we also need a separate training set for predicting the time that can be saved by removing a terminal from the grammar.
This dataset maps grammar elements $\ensuremath{g}\xspace \in \ensuremath{G}\xspace$ to the effect on synthesis times, $\ensuremath{\mathbb{R}}$, when $\ensuremath{g}\xspace$ is dropped from the grammar.
To do this we require synthesis problems that more closely model the types of constraints that humans typically write.
We collect these set of benchmarks from users of the live coding interface for SyGuS~\cite{chi19}.
Because we had limited number of human-generated constraint examples, we augmented this with constraints generated from \ensuremath{\mathcal{TR}_{crit}}.
We run synthesis for each problem with the full grammar, as well as with all grammars constructed by removing one element, $\ensuremath{g}\xspace$.
For every synthesis problem benchmark, $1 \leq i \leq m$, we record the difference in synthesis times between running with the full grammar, and removing $\ensuremath{g}\xspace$:
\begin{equation}
T^{\ensuremath{C}\xspace_i}_\ensuremath{G}\xspace - T^{\ensuremath{C}\xspace_i}_{\drop{\ensuremath{G}\xspace}{\ensuremath{g}\xspace}}
\end{equation}
Thus, we create a training set, \ensuremath{\mathcal{TR}_{time}}, relating each terminal $\ensuremath{g}\xspace \in \setOf{\ensuremath{G}\xspace}$ and a set of constraints, to the time it takes to synthesize a solution without that terminal.
\section{Training}
\subsection{Predicting criticality}
Our goal is to predict, given a set of constraints \ensuremath{C}\xspace, if a terminal \ensuremath{g}\xspace belongs to the set of terminals \setOf{\ensuremath{\grammar_{crit}}\xspace} for \ensuremath{C}\xspace.
To do this, we use a Feedforward Neural Network (Multi-Layer Perceptron), with an extra embedding layer to encode the string valued input-output examples into feature vectors.
We train the neural network to predict the membership of each terminal $\ensuremath{g}\xspace \in \setOf{\ensuremath{G}\xspace}$ to the critical set $\setOf{\ensuremath{\grammar_{crit}}\xspace}$, based on a single constraint $c \in \ensuremath{C}\xspace$.
This prediction produces a 1D binary vector of length $|\setOf{\ensuremath{G}\xspace}|$, where 1 at position $i$ in the binary vector indicates the terminal in position $i$ is predicted to belong to the critical set.
When a SyGuS problem has multiple ($|\ensuremath{C}\xspace| \geq 2$) constraints, we run our prediction on each constraint individually.
We then use a voting mechanism to come to consensus on the construction of \ensuremath{\grammar^{\star}}\xspace.
After computing $|\ensuremath{C}\xspace|$ binary vectors across all constraints, the vectors are summed to produce a final voting vector.
The magnitude of each element in this final voting vector represents the number of votes ``from each constraint'' that the terminal represented by that element is in the critical set.
We then use this final voting vector in combination with our time predictions.
\subsection{Predicting time savings}
It is only worthwhile to remove a terminal symbol \ensuremath{g}\xspace from a grammar \ensuremath{G}\xspace if
$T^{\ensuremath{C}\xspace}_{\drop{\ensuremath{G}\xspace}{\ensuremath{g}\xspace}}$ is less than $T^{\ensuremath{C}\xspace}_\ensuremath{G}\xspace$.
If a \ensuremath{g}\xspace stands to only give us a small gain in synthesis times, it may not be worth the risk that we incorrectly predicted its criticality.
To predict the amount of time saved by removing a terminal \ensuremath{g}\xspace we examine the distribution of times in our training set \ensuremath{\mathcal{TR}_{time}}.
For each terminal \ensuremath{g}\xspace, we calculate $A_\ensuremath{g}\xspace$, the average time increase that results from removing \ensuremath{g}\xspace from the grammar.
Denoting the time to run $(\ensuremath{G}\xspace, \ensuremath{C}\xspace) \rightsquigarrow P$ as $T^\ensuremath{C}\xspace_\ensuremath{G}\xspace$, we can write $A_\ensuremath{g}\xspace$ as:
\begin{equation*}
A_\ensuremath{g}\xspace = \dfrac{\sum_{i = 1}^n T^{\ensuremath{C}\xspace_i}_\ensuremath{G}\xspace - T^{\ensuremath{C}\xspace_i}_{\drop{\ensuremath{G}\xspace}{\ensuremath{g}\xspace}}}{n}
\end{equation*}
If a terminal \ensuremath{g}\xspace has a negative $A_\ensuremath{g}\xspace$,
then removing it from the grammar actually slows down synthesis, on average.
As such, dropping the terminal from the grammar is not generally helpful.
Thus, we only consider those terminals with a positive $A_\ensuremath{g}\xspace$ in our second step.
\subsection{Combining predictions}
With our predictions of the criticality a terminal \ensuremath{g}\xspace and of time saved by removing \ensuremath{g}\xspace, we must make a final decision on whether or not we should remove \ensuremath{g}\xspace.
To do this, we take the top three terminals with the greatest average positive impact on synthesis time over the training set, as computed with $A_\ensuremath{g}\xspace$. These tended to be terminals that mapped between types which saved more time due to the internal mechanisms and heuristics of the CVC4 solver.
We then use the final voting vector from our criticality prediction to choose only two out of the three to remove from \ensuremath{G}\xspace to form \ensuremath{\grammar^{\star}}\xspace.
We chose to remove only two terminals from \ensuremath{G}\xspace in order to minimize the likelihood of generating a \ensuremath{\grammar^{\star}}\xspace, such that $\setOf{\ensuremath{\grammar^{\star}}\xspace} \subseteq \setOf{\ensuremath{\grammar_{crit}}\xspace}$. We conjecture that the number of terminals removed is a grammar-dependent parameter that must be selected on a per grammar basis, just as the number of terminals with $A_\ensuremath{g}\xspace > 0$ is grammar specific.
\subsection{Falling back to the full grammar}
\label{sec:fallingback}
There is some danger that \ensuremath{\grammar^{\star}}\xspace will, in fact, not be sufficient to synthesize a program.
Thus, we propose a strategy that
\begin{itemize}
\item first, tries to synthesize a program with the grammar \ensuremath{\grammar^{\star}}\xspace
\item second, if synthesis with \ensuremath{\grammar^{\star}}\xspace is unsuccessful, falls back to attempting synthesis with the full grammar \ensuremath{G}\xspace.
\end{itemize}
We determine how long to wait before switching from \ensuremath{\grammar^{\star}}\xspace to \ensuremath{G}\xspace by finding an $x$ that minimizes:
\begin{equation}
\sum_{i = 1}^n \begin{Bmatrix}
T^{C_i}_{\ensuremath{\grammar^{\star}}\xspace} & T^{C_i}_{\ensuremath{\grammar^{\star}}\xspace} < x\\
\min(x+T^{C_i}_\ensuremath{G}\xspace, t)& T^{C_i}_{\ensuremath{\grammar^{\star}}\xspace} > x
\end{Bmatrix}
\end{equation}
where $C_1 \ldots C_n$ are the constraints from the training set, and $t$ is the timeout for synthesis.
Ideally, as captured in the first line of the sum, $(\ensuremath{C}\xspace_i, \ensuremath{\grammar^{\star}}\xspace) \synthesizesInTime{x} P$ will finish before $T^{C_i}_{\ensuremath{\grammar^{\star}}\xspace}$ = x.
However, if a benchmark does not finish in that time, it will fall back on the full grammar.
Then, either $(\ensuremath{C}\xspace_i, \ensuremath{\grammar^{\star}}\xspace) \synthesizesInTime{t - x} P$ will succeed, and synthesize the expression in total time $x+T^{C_i}_{\ensuremath{G}\xspace}$, or synthesis will timeout, in total time $(t - x) + x = t$.
\section{Experiments}
\begin{figure*}[t!]
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ylabel=Seconds to complete,
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\legend{CVC4, GRT+CVC4\xspace}
\end{axis}
\end{tikzpicture}
\caption{The top 20 problems with longest synthesis time for CVC4 (excepting timeouts), and the corresponding synthesis times for GRT+CVC4\xspace.}
\label{fig:top20}
\end{figure*}
The SyGuS competition~\cite{sygusCompetition} provides public competition benchmarks and results from previous years.
In particular, the PBE Strings dataset provides a collection of PBE problems over a grammar that includes string, integer, and Boolean manipulating functions.
First, we describe our approach to generating a training set of PBE problems over strings.
Then, we present our results running GRT\xspace against the 2019 competition's winner in the PBE Strings track, CVC4~\cite{notzli2019syntax,barrett2011cvc4,sygusCompetition2019}.
We are able to reduce synthesis time by $47.65\%$ and synthesize a new solution to a benchmark that was left unsolved by CVC4.
\subsection{Technical details}
The data triples generated during our initial data generation process of \ensuremath{\mathcal{TR}_{crit}} \, are triples of strings.
However, the neural network cannot process input-output pairs of type string as input.
Thus, this data must be encoded numerically before it can be utilized to train the neural network.
Each character in the input-output pairs is converted to its ASCII equivalent integer value.
The size of each pair is then standardized by adding a padding of zeros to the end of each newly encoded input and output vector respectively.
This creates two vectors: the encoded input and the encoded output, both of which have a length of 20.
These two vectors are then concatenated to give us a single vector for training.
By the end of this process the triples created in our first data generation step are now one vector of type $\mathbb{N}^{40}$ representing the input-output pair and a correct label $P$ that will be predicted.
To generate the training set for predicting synthesis times, \ensuremath{\mathcal{TR}_{time}}, we combine human generated and automatically generated SyGuS problems.
Specifically, we use 10 human generated SyGuS problems, and 20 randomly selected problems from \ensuremath{\mathcal{TR}_{crit}}.
The overall architecture of our model can be categorized as a multi-layer perceptron (MLP) neural network.
More specifically, our model is made up of five fully connected layers: the input layer, three hidden layers, and the output layer.
By using the Keras Framework, we include an embedding layer along with our input layer which enables us to create unique vector embeddings of length 100 for any given input-output pair in the dataset.
This embedding layer learns the optimal weights used to create these unique vectors through the training process.
Thus, we create an encoding of the input-output pairs for training, while simultaneously standardizing the scale of the vector before it reaches the first hidden layer.
The hidden layers of the model are all fully connected, and all use the sigmoid activation function.
In addition, we implement dropout during training to ensure that overfitting does not occur.
The size of the hidden layers was calculated using a geometric series to ensure that there was a consistent decrease in layer size as the layers get closer to the output layer.
Specifically, the size of each hidden layer was calculated by:
\begin{equation}
\mathit{HL}_{\mathit{size}}(n) = {\mathit{input}}_{\mathit{size}} \big( \frac{\mathit{output}_{\mathit{size}}}{\mathit{input}_{\mathit{size}}} \big) ^ {\frac{n}{L_{\mathit{num}}+1}}
\end{equation}
where $L_{num}$ represents the total number of layers in the network.
Our model used the Adam optimization method and the binary-cross entropy loss function as it is well suited for multi-label classification. Overall, our model was trained on 124928 data points for 15 epochs with a batch size of 200 producing a training time of 228 seconds.
\subsection{Results}
After generating our data sets and training our model, we wrote a wrapper script to run GRT\xspace as a preprocessor for CVC4's SyGuS engine.
We compared the synthesis results of GRT+CVC4\xspace with the synthesis results of running CVC4 alone.
All experiments were run on MacBook Pro with a 2.9 GhZ Intel i5 processor with 8GB of RAM.
CVC4 uses a default random seed, and is deterministic over the choice of that seed, so the results of synthesis from CVC4 on a given grammar and set of constraints are deterministic.
We note that our training data in no way used any of the SyGuS benchmarks.
GRT+CVC4\xspace outperformed directly calling CVC4 on 32 out of 64 benchmarks (50\%), with a reduction in total synthesis time over all benchmarks from 1304.87 seconds with CVC4 to 683.09 seconds with GRT+CVC4\xspace.
On one benchmark, CVC4 timed out and was not able to find a solution (even when the timeout was increased to 5000 seconds), while GRT+CVC4\xspace found a solution within the timeout specified by the SyGuS Competition rules (3600 seconds).
On one benchmark, both CVC4 and GRT+CVC4\xspace timeout (TO) and are not able to find a solution.
On the other 31 benchmarks, CVC4 performed the same (within $\pm0.1\text{s}$) with and without the preprocessor.
All the benchmarks for which CVC4 performed the same as GRT+CVC4\xspace finish in under 2 seconds, and 28 of the 31 finish in under a second.
In these cases there was little room for improvement even with GRT+CVC4\xspace.
Figure~\ref{fig:top30} shows the exact running times with both the full and reduced grammars from the benchmarks with the 30 largest running times with the full grammar.
These are the benchmarks for which the synthesis times and size of the solution diverge most meaningfully, however all other data is available in the supplementary material for this paper.
Figure~\ref{fig:top30} also shows $|P|$ and $|P^*|$, the sizes of the programs found by the CVC4 and GRT+CVC4\xspace, respectively.
We define size of a program as the number of nodes in the abstract syntax tree of the program.
In terms of the grammar \ensuremath{G}\xspace, this is the number of terminals (including duplicates) that were composed to create the program.
\begin{figure}
\begin{tikzpicture}
\begin{axis}[
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height = 6cm,
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xticklabels={ 49, 61, 62, 58, 48
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scaled y ticks = false,
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ymin=0,
ylabel=Length of sythesized solutions,
xlabel=Benchmark Id,
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coordinates {
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\caption{When the GRT+CVC4\xspace found a different solution than CVC4, it was on average shorter than the solution found with the full grammar.}
\label{fig:sizes}
\end{figure}
\begin{filecontents}{localResults.csv}
34, lastname-small.sl, 1.80, 1.84, 4, 4
35, bikes-long.sl, 1.97, 1.76, 3, 3
36, bikes-long-repeat.sl, 2.08, 1.71, 3, 3
37, lastname.sl, 2.31, 1.83, 4, 4
38, phone-6-short.sl, 3.23, 1.22, 11, 11
39, phone-7-short.sl, 3.26, 1.26, 11, 11
40, initials-long-repeat.sl, 3.33, 2.54, 7, 7
41, phone-5-short.sl, 3.72, 1.51, 9, 9
42, phone-7.sl, 4.57, 2.03, 11, 11
43, phone-8.sl, 4.72, 2.17, 11, 11
44, phone-6.sl, 4.85, 1.97, 11, 11
45, phone-5.sl, 4.88, 2.20, 11, 11
46, phone-9-short.sl, 4.88, 4.73, 52, 52
47, phone-10-short.sl, 8.81, 8.28, 49, 49
48, phone-9.sl, 12.08, 4.86, 56, 52
49, phone-10.sl, 31.23, 8.49, 97, 49
50, lastname-long.sl, 32.40, 25.49, 4, 4
51, lastname-long-repeat.sl, 32.49, 24.92, 4, 4
52, phone-6-long-repeat.sl, 83.59, 25.31, 11, 11
53, phone-5-long-repeat.sl, 84.77, 33.68, 11, 11
54, phone-7-long.sl, 87.83, 26.15, 11, 11
55, phone-7-long-repeat.sl, 89.13, 26.23, 11, 11
56, phone-5-long.sl, 90.81, 30.01, 11, 11
57, phone-8-long-repeat.sl, 91.04, 35.64, 11, 11
58, phone-9-long-repeat.sl, 91.19, 77.02, 47, 50
59, phone-6-long.sl, 98.15, 24.75, 11, 11
60, phone-8-long.sl, 108.06, 29.94, 11, 11
61, phone-10-long-repeat.sl, 149.53, 129.43, 49, 65
62, phone-10-long.sl, 153.32, 133.22, 49, 65
63, initials-long.sl, TO, TO, -, -
64, phone-9-long.sl, TO, 3516.21, -, 49
\end{filecontents}
\begin{figure}[t]
\footnotesize
\def1.1{1.1}
\setlength{\tabcolsep}{0.55em}
\begin{tabular}{c|c|c|c|c|c}%
\bfseries id & \bfseries file & $T^{\ensuremath{C}\xspace}_\ensuremath{G}\xspace$ & $T^{\ensuremath{C}\xspace}_{\ensuremath{\grammar^{\star}}\xspace}$ & $|P|$ & $|P^\star|$
\csvreader[no head]{localResults.csv}{
{\\\hline \csvcoli&\csvcolii&\csvcoliii&\csvcoliv&\csvcolv&\csvcolvi}
\\\hline
\end{tabular}
\caption{Synthesis results over the 30 longest running benchmarks from SyGuS Competition's PBE Strings track.}
\label{fig:top30}
\end{figure}
In Figure~\ref{fig:top20}, we present a visual comparison of the results for the 20 functions that took CVC4 the longest, while still finishing in the 3,600 second time limit.
We note that we have the largest gains on the problems for which CVC4 is the slowest.
Problems that CVC4 already handles quickly stand to benefit less from our approach.
In order to get a better baseline to understand the impact of GRT on running times, we ran a version of GRT with only the criticality prediction, which we call GRTC.
In this case, GRTC+CVC4 actually performed worse than CVC4 by itself, increasing the running time on 53 out of the 62 benchmarks that did not timeout on CVC4.
On all but 5 benchmarks, CVC4 synthesized the same program when running with \ensuremath{G}\xspace and \ensuremath{\grammar^{\star}}\xspace.
The sizes of the programs (in terms of the number of terminal symbols used) for the benchmarks on which CVC4 synthesized different programs are shown in Figure~\ref{fig:sizes}.
While on some benchmarks GRT+CVC4\xspace produced a larger solution than CVC4, as a whole the sum of the size of all solutions for CVC4 was 806, while for GRT+CVC4\xspace it was 789.
Thus, overall, we were able to outperform CVC4 on size of synthesis as well.
The SyGuS competition scores each tool using the formula: $5N+3F+S$, where $N$ is the number of benchmarks solved (non-timeouts), $F$ is based on a ``pseudo-logarithmic scale''~\cite{sygusCompetition} indicating speed of synthesis, and $S$ is based on a ``pseudo-logarithmic scale'' indicating size of the synthesized solution.
On all three of these measurements, GRT+CVC4\xspace performed better than CVC4.
There are number of other synthesis tracks available in the SyGuS competition, which do not involve PBE constraints.
We note that our approach can selectively be applied as a preprocessing step for input in the PBE track without incurring an overhead on other synthesis tasks.
Although we implemented a strategy to manage a switch from the reduced grammar back to the full grammar, we found in practice that the optimal strategy for our system was to exclusively use the reduced grammar.
Because we had conservatively pruned the grammar, we had no need to switch back to the full grammar.
\section{Conclusions}
In a way, by training on a dataset we generate from the output of the interpreter of the language, we are encoding an approximation of the semantics into our neural network.
While the semantic approximation is too coarse to drive synthesis itself, we can use it to prune the search space of potential programs.
By predicting terminals impact on synthesis time, we more conservatively remove only terminals likely to have a positive impact.
In conjunction with analytically driven tools, we can then significantly improve synthesis times with very little overhead.
While we have presented GRT\xspace, which demonstrates a significant gain in performance over all existing SyGuS solvers, we still have many opportunities for further improvement.
In our prediction of the potential time saved by removing a terminal from the grammar, we have simply used the average expected value over all samples in the dataset.
By using a neural network here, we may be able to leverage some property of the SyGuS problem constraints to have more accurate potential time savings predictions.
This would allow us, possibly in combination with a more advance prediction combination strategy, to more aggressively prune the grammar.
The drawback to this approach is that we may then potentially remove too much from the grammar.
One of the key features of GRT\xspace is that it introduces no new timeouts, that is, it does not remove any critical parts of the grammar.
Additionally, our prediction of criticality of a terminal uses a voting mechanism to combine the prediction based on each constraint.
While this worked well in practice, this strategy ignores the potential for interaction between constraints.
In our preliminary exploration, we were not able to construct a model that captures this inter-constraint interaction in a useful way.
This may be a path for future work.
In a similar vein, there exist a number of other works that define a criticality measure for each terminal in the SyGuS grammar~\cite{BalogGBNT17,kalyan2018neural}.
It may be possible to leverage these in place of our criticality measure, and in combination with our time savings prediction, to achieve better results.
So far we have only explored the PBE Strings track of the SyGuS Competition.
The competition also features a PBE BitVectors track where our technique may have significant gains as well.
This would require a new encoding scheme, but the overall approach would remain similar.
In general, extending this work to allow for other PBE types, as well as more general constraints, would broaden the potential real-world application of SyGuS.
\paragraph{Acknowledgments}
This work was supported in part by NSF grants CCF-1302327, CCF-1715387, and CCF-1553168.
\bibliographystyle{aaai}
|
1,314,259,995,767 | arxiv | \section{Introduction}
Large and complex Cyber-Physical Systems (CPSs), such as intelligent buildings, transportation, and energy systems, cannot be designed in a monolithic manner. Instead, designers use hierarchical and compositional methods, which allow assembling a large and complex system from smaller and simpler components, such as pre-defined library blocks. Contract-based design is emerging as a unifying formal compositional paradigm for CPS design and has been demonstrated on several applications~\cite{Benveniste2013,Nuzzo15b}. It supports requirement engineering by providing formalisms and mechanisms for early detection of integration errors, for example, by checking compatibility between components locally, before performing expensive, global system verification tasks.
However, while a number of contract and interface theories have appeared to support deterministic system models~\cite{Benveniste08,Alfaro01_2}, the development of contract frameworks for stochastic systems under probabilistic requirements is still in its infancy.
Deterministic approaches fall short of accurately capturing those aspects of practical systems that are subject to variability (e.g., due to manufacturing tolerances, usage, and faults), noise, or model uncertainties. While trying to meet the specifications over the entire space of uncertain behaviors, they tend to produce worst-case designs that are overly conservative.
Moreover, several design requirements in practical applications cannot be rigidly defined, and would be better expressed as probabilistic constraints, e.g., to formally capture that
``the room temperature in a building shall be in a comfort region with a confidence level larger than 80\% at any time during a day.''
Providing support for reasoning about probabilistic behaviors and for the development of robust design techniques that can avoid over-design is, therefore, crucial. This need becomes increasingly more compelling as a broad number of safety-critical systems, such as autonomous vehicles, uses machine learning and statistical sensor fusion algorithms to infer information from the external world.
An obstacle to the development of stochastic contract frameworks and their adoption in system design stems from the computational complexity of the main verification and synthesis tasks for stochastic systems (see, for example, \cite{KNP07a,KNP11}), which are needed to perform concrete computations with contracts.
A few proposals toward a specification and contract theory for stochastic systems
have recently appeared, e.g., based on
Interactive Markov Chains~\cite{gossler12}, Constraint Markov Chains~\cite{Caillaud10}, and Abstract Probabilistic Automata~\cite{delahaye2011abstract,delahaye2011apac}. However, these frameworks mostly use contract representations based on automata, which are more suitable to reason about discrete-state discrete-time system abstractions. They tend to favor an imperative specification style, and may show poor scalability when applied to hybrid systems.
A declarative specification style is often deemed as more practical for system-level requirement specification and validation, since it retains a better correspondence between informal requirements and formal statements.
In this paper, we develop an A/G contract framework for automated design of CPSs modeled as closed-loop control systems under probabilistic requirements. We aim to identify formalisms for contract representation and manipulation that effectively trade expressiveness with tractability: (i) they are rich enough to represent \emph{hybrid system behaviors} using a \emph{declarative style}; (ii) they are amenable to algorithms for \emph{efficient computation} of contract operations and relations.
We address these challenges by leveraging an extension of Signal Temporal Logic (STL)~\cite{MalerN04}, namely, Stochastic Signal Temporal Logic (StSTL), to support the specification of probabilistic constraints in the contract assumptions and guarantees. We show that the main verification tasks for bounded StSTL contracts on stochastic linear systems, i.e., compatibility, consistency, and refinement checking, as well as the synthesis of stochastic Model Predictive Control (MPC) strategies can all be translated into mixed integer programs (MIPs) which can be efficiently solved by state-of-the-art tools. Since probabilistic constraints on stochastic systems cannot be expressed in closed analytic form except for a small set of stochastic models~\cite{nemirovski2006convex}, we propose conservative approximations to provide optimization problem formulations that are both sound and tractable. We illustrate the effectiveness of our approach with a few examples, including the synthesis of controllers for an aircraft electric power distribution system.
\emph{\textbf{Related Work.}} A generic assume-guarantee (A/G) contract framework for probabilistic systems that can also capture reliability and availability properties using a declarative style has been recently proposed~\cite{delahaye2010probabilistic}. Our work differs from this effort, since it is not based on a probabilistic notion of contract satisfiability. In our approach, probabilistic constraints appear, instead, as predicates in the contract assumptions and guarantees.
We express assumptions and guarantees using StSTL, which is an extension of STL~\cite{MalerN04}. STL was proposed for the specification of properties of continuous-time real-valued signals and has been previously used in CPS design~\cite{Nuzzo15b}.
A few probabilistic extensions of temporal logics have been proposed over the years to express properties of stochastic systems.
Among these, Probabilistic Computation Tree Logic (PCTL) was introduced
to expresses properties over the realizations (paths) of finite-state Markov
chains and Markov decision processes~\cite{hansson1994logic} by extending
the Computation Tree Logic (CTL)~\cite{clarke1986automatic}. While PCTL can reason about global system executions and uncertainties about the times of occurrence of certain events,
certain applications are rather concerned with capturing the uncertainty on the value of a signal at a certain time. This is the case, for instance, in the deployment of stochastic MPC schemes in different domains.
By using StSTL, we can express requirements where uncertainty is restricted to probabilistic predicates and does not involve temporal operators.
While being expressive enough to cover the applications of interest, this restriction is also convenient, since it allows directly translating design and verification problems into optimization and feasibility problems with chance (probabilistic) constraints that can be efficiently solved using off-the-shelf tools.
Closely related to StSTL, Probabilistic Signal Temporal Logic (PrSTL)~\cite{sadigh2016} has been recently proposed
to specify properties and design controllers for deterministic systems in uncertain
environments, captured by Gaussian stochastic processes.
Our work is different since it focuses on developing a comprehensive contract framework that supports both verification and control synthesis tasks. Our framework can reason about a broader class of systems, including linear systems with additive and control-dependent noise and Markovian jump linear systems.
Moreover, it supports non-Gaussian probabilistic constraints
that cannot be captured in closed analytic form, by formulating encodings of synthesis and verification tasks that can produce sound and efficient approximations.
\section{Preliminaries}
\label{sec:background}
As we aim to extend the \emph{Assume-Guarantee (A/G) contract} framework~\cite{Benveniste2013} to stochastic systems, we start by providing some background on A/G contracts and Stochastic Signal Temporal Logic (StSTL).
\subsection{Assume-Guarantee Contracts: An Overview}
\seclabel{agc}
The notion of contracts originates from
\emph{assume-guarantee reasoning}~\cite{Clark99}, which has been known for a
long time as a hardware and software verification technique. However, its adoption in the context of reactive systems, i.e., systems that maintain an ongoing interaction with their environment, such as CPSs, has been advocated only recently~\cite{Benveniste2013,Sangiovanni-Vincentelli2012a}.
We provide an overview of A/G contracts starting with a generic
representation of a component. We associate to it a set of properties that
the component satisfies, expressed with contracts. The contracts will be used to
verify the correctness of the composition and of the refinements.
A component is an element of a design,
characterized by a set of \emph{variables} (input or output), a set of \emph{ports} (input or output), and a set of \emph{behaviors} over its variables and
ports. Components can be connected together by sharing certain ports under
constraints on the values of certain variables.
Behaviors are generic and could be continuous functions that result from solving
differential equations, or sequences of values or events recognized by an
automaton. To simplify, we use the same term ``variables'' to
denote both component variables and ports.
We use $\sem{M}$ to denote the
set of behaviors of component $M$.
A \emph{contract} $C$ for a component $M$ is a triple $(V, A, G)$, where
$V$ is the set of component variables, and $A$ and $G$ are sets of behaviors
over $V$~\cite{Benveniste08}. $A$ represents the \emph{assumptions} that $M$
makes on its environment, and $G$ represents the \emph{guarantees} provided by
$M$ under the environment assumptions.
A component $M$ satisfies a contract $C$ whenever $M$ and $C$ are
defined over the same set of variables, and all the behaviors of $M$ are
\emph{contained} in the guarantees of $C$ once they are composed (i.e.,
intersected) with the assumptions, that is, when $\sem{M} \cap A \subseteq G$. We
denote this \emph{satisfaction} relation by writing $M \models C$, and we
say that $M$ is an \emph{implementation} of $C$.
However, a component $E$ can also be associated to a contract $C$ as an
\emph{environment}. We say that $E$ is a legal environment of $C$, and
write $E \models_E C$, whenever $E$ and $C$ have the same variables
and $\sem{E} \subseteq A$.
A contract $C = (V, A, G)$ is in \emph{canonical form} if the \emph{union}
of its guarantees $G$ and the complement of its assumptions $A$ is coincident
with $G$, i.e., $G = G \cup \overline{A}$, where $\overline{A}$ is the
complement of $A$. Any contract $C$ can be turned into a contract in
canonical form $C'$ by taking $A'=A$ and $G' = G \cup \overline{A}$.
We observe that
$C$ and $C'$ possess identical sets of environments and implementations.
Such two contracts $C$ and
$C'$ are then \emph{equivalent}. Because of this equivalence, in what follows, we assume that all
contracts are in canonical form.
A contract is \emph{consistent} when the set of implementations satisfying it is
not empty, i.e., it is feasible to develop implementations for it. This amounts to verifying that $G \neq \emptyset$, where
$\emptyset$ denotes the empty set. Let $M$ be any implementation; then $C$ is \emph{compatible} if there exists a legal
environment $E$ for $M$, i.e., if and only if $A \neq \emptyset$. The intent is
that a component satisfying contract $C$ can only be used in the context
of a compatible environment.
Contracts can be combined according to
different rules. \emph{Composition} ($\otimes$) of contracts can be used to construct complex global
contracts out of simpler local ones.
Let $C_1$ and $C_2$ be contracts over the same set of variables $V$.
Reasoning on the compatibility and consistency of the composite contract $C_1 \otimes C_2$ can then be used to assess whether there exist components $M_1$ and $M_2$ such that their composition is valid, even if the full implementation of $M_1$ and $M_2$ is not available.
To reason about consistency between different abstraction layers in a design,
contracts can be ordered by establishing a \emph{refinement} relation. We say that $C$ refines $C'$, written
$C \preceq C'$, if and only if $A \supseteq A'$ and $G \subseteq G'$. Refinement amounts to
relaxing assumptions and reinforcing guarantees.
Clearly, if $M \models C$ and $C \preceq C'$, then $M \models
C'$. On the other hand, if $E \models_E C'$, then $E \models_E
C$. In other words, contract $C$ refines $C'$, if $C$
admits less implementations than $C'$, but more legal environments than
$C'$. We can then replace $C'$ with $C$.
Finally, to combine multiple requirements on the same component that need to be
satisfied simultaneously, the \emph{conjunction} ($\wedge$) of contracts can
also be defined so that, if a component $M$ satisfies the conjunction of $\mathcal{C}_1$
and $\mathcal{C}_2$, i.e., $M \models C_1 \wedge C_2$, then it also satisfies
each of them independently, i.e., $M \models C_1$ and $M \models
C_2$.
We refer the reader to the literature~\cite{Benveniste2013} for the formal definitions and mathematical expressions of contract composition and conjunction.
In the following, we provide concrete representations of some of these operations and relations using operations on StSTL formulas.
\subsection{Stochastic Signal Temporal Logic (StSTL)}\label{sec:StSTL}
We use StSTL to formalize requirements for discrete-time stochastic system and express both contract assumptions and guarantees. However, similarly to STL, StSTL also extends to continuous-time systems.
\begin{figure}[t]
\centering
\includegraphics[width=0.3\textwidth]{sys_structure1}
\caption{Components in the control loop and their interactions.}
\label{fig:sys_structure}
\end{figure}
\textbf{Stochastic System.} We consider a discrete-time stochastic system in a classic closed-loop control configuration as shown in Fig.~\ref{fig:sys_structure}. The system dynamics are given by
\begin{equation}\label{eq:sys}
x_0 = \bar{x}_0, \quad x_{k+1} = f(x_k,u_k,w_k), \quad k=0,1,\ldots
\end{equation}
where $f$ is an arbitrary measurable function~\cite{Durrett10}, $x_k\in\mathbb{R}^{n_x}$ is the system state, $\bar{x}_0$ is the initial state, $u_k\in\mathbb{R}^{n_u}$ is
the (control) input, and $\{w_{k}\}_{k=0}^\infty$ is a random process on a complete
probability space, which we denote as $(\Omega, \mathcal{F}, \mathcal{P})$, using the standard notation, respectively, for the sample space, the set of events, and the probability measure on them~\cite{Durrett10}.
Each element $\mathcal{F}_k$ of the filtration $\mathcal{F}$ denotes the $\sigma$-algebra generated by the sequence $\{w_{t}\}_{t=0}^k$, while we set $\mathcal{F}_{-1} = \{\emptyset,\Omega\}$ as being the trivial $\sigma$-algebra.
We assume that the input $u_k$ is
a function of the system states $\{x_{t}\}_{t=0}^k$ and both $x_k$ and $u_k$
are $\mathcal{F}_{k-1}$-measurable random variables~\cite{Durrett10}. We also denote as $z_k = (x_k,u_k,w_{k})$ the vector of all the system variables at time $k$.
Finally, we abbreviate as $\boldsymbol{z} = z_0, z_1, \ldots$ a system \emph{behavior} and as $\boldsymbol{z}^H = z_0, \ldots, z_{H-1}$ its truncation over the horizon $H$.
\textbf{StSTL Syntax and Semantics.} StSTL formulas are defined over atomic predicates represented by \emph{chance constraints} of the form
\begin{equation}\label{eq:atomic_prop}
\mu ^{[p]} := \mathcal{P}\{\mu(v) \le 0\} \ge p,
\end{equation}
where $\mu(\cdot)$ is a real-valued measurable function, $v$ is a random
variable on the probability space $(\Omega, \mathcal{F}, \mathcal{P})$, and
$p \in [0,1]$. The truth value of $\mu ^{[p]}$ is interpreted based on the satisfaction of the chance constraint, i.e.,
$\mu ^{[p]}$ is true (denoted with $\top$) if and only if $\mu(v) \le 0$ holds
with probability larger than or equal to $p$. StSTL also supports deterministic predicates as a particular case. If $\mu(v)$ is deterministic, then $\mu^{[p]}$ holds for any value of $p$ if and only if $\mu(v) \le 0$ holds. In this case, we can omit the superscript $[p]$.
We define the syntax of an StSTL formula as follows:
\begin{equation} \label{eq:formula_form}
\psi := \mu ^{[p]} \;|\; \neg \psi \;|\; \psi \vee \phi \;|\; \psi\ \mathcal{U}_{[t_1,t_2]} \phi
\;|\; {\bf G}_{[t_1,t_2]} \psi,
\end{equation}
where $\mu ^{[p]}$ is an atomic predicate, $\psi$ and $\phi$ are StSTL formulas, $t_1, t_2 \in \mathbb{R}_+ \cup \{+\infty\}$, and $\mathcal{U}$ and ${\bf G}$
are, respectively, the \emph{until} and \emph{globally} temporal operators. Other operators, such as \emph{conjunction} ($\land$), \emph{weak until} (${\bf W}$), or \emph{eventually} (${\bf F}$) are also supported and can be expressed using the operators in~\eqref{eq:formula_form}.
The semantics of an StSTL formula can be defined recursively as follows:
\begin{equation*}\label{eq:semantics}
{\small
\begin{aligned}
(\boldsymbol{z},k) &\models \mu ^{[p]} &\leftrightarrow\;&
\mathcal{P}\{\mu(z_k) \le 0\} \ge p, \\
(\boldsymbol{z},k) &\models \neg \psi &\leftrightarrow\;&
\neg( (\boldsymbol{z},k) \models \psi) \\
(\boldsymbol{z},k) &\models \psi \vee \phi &\leftrightarrow\;&
(\boldsymbol{z},k) \models \psi \vee (\boldsymbol{z},k) \models \phi, \\
(\boldsymbol{z},k) &\models \psi \mathcal{U}_{[t_1,t_2]} \phi &\leftrightarrow\;&
\exists i \in [k+t_1,k+t_2]: (\boldsymbol{z},i) \models \phi \land \\
& & & (\forall j \in {[k+t_1,i-1]}: (\boldsymbol{z},j) \models \psi),\\
(\boldsymbol{z},k) &\models {\bf G}_{[t_1,t_2]} \psi &\leftrightarrow\;&
\forall i \in [k+t_1,k+t_2]: (\boldsymbol{z},i) \models \psi.
\end{aligned}
}
\end{equation*}
As an example, $(\boldsymbol{z},k) \models {\bf G}_{[t_1,t_2]} \phi$ means that
$\phi$ holds for all times $t$ between $t_1$ and $t_2$.
Intervals may also be open or unbounded, e.g., of the form $[t_1,+\infty)$.
In this paper, we focus on \emph{bounded} StSTL formulas, that is, formulas that contain no unbounded operators.
StSTL reduces to STL for deterministic systems, with the exception that the atomic predicate has the form $\mu(v)\le 0$ rather than $\mu(v) > 0$, as in STL.
A difference between StSTL and PrSTL is in the interpretation of the negation of an atomic predicate. In PrSTL the semantics of negation is \emph{probabilistic}, i.e., if $(\boldsymbol{z},t) \models \lambda^{\epsilon_t}_{\alpha_t}$ holds for an atomic PrSTL predicate $\lambda^{\epsilon_t}_{\alpha_t}$, which is equivalent to stating that $\mathcal{P}\{\lambda_{\alpha_t} (z_t)<0\} > 1 - \epsilon_t$, then $(\boldsymbol{z},t) \models \tilde{\neg} \lambda^{\epsilon_t}_{\alpha_t}$ is interpreted as
$\mathcal{P}\{\lambda_{\alpha_t} (z_t)>0\} > 1 - \epsilon_t$, so that $\tilde{\neg} \lambda^{\epsilon_t}_{\alpha_t}$
and $\lambda^{\epsilon_t}_{\alpha_t}$ can be true at the same time.
StSTL keeps, instead, the standard semantics of \emph{logic negation}.
\section{Problem Formulation}
We can concretely express the sets of behaviors $A$
and $G$ in a contract using temporal logic formulas~\cite{Nuzzo15b} and, in particular, StSTL formulas. We then define an StSTL A/G contract as a triple $(V,\phi_A,\phi_G)$, where
$\phi_A$ and $\phi_G$ are StSTL formulas over the set of variables $V$. The canonical form of $(V,\phi_A,\phi_G)$ can be achieved by setting $\phi_G := \phi_A \to \phi_G$.
The main contract operators can then be mapped into entailment of StSTL formulas.
We define below the verification and synthesis problems addressed in this paper.
\begin{problem}[Contract Consistency and Compatibility Checking] \label{prob:1}
Given a stochastic system representation $\pfr{S}$ as in~\eqref{eq:sys} and a bounded StSTL contract $C = (V, \phi_A, \phi_G)$ on the system variables $V$, determine whether $C$ is consistent (compatible), that is, whether $\phi_G$ ($\phi_A$) is satisfiable.
\end{problem}
\begin{problem}[Contract Refinement Checking] \label{prob:2}
Given a stochastic system representation $\pfr{S}$ as in~\eqref{eq:sys} and bounded StSTL contracts $C_1 = (V, \phi_{A1}, \phi_{G1})$ and $C_2 =
(V, \phi_{A2}, \phi_{A2})$ on the system variables $V$, determine whether $C_1 \preceq C_2$, that is,
$\phi_{A2} \rightarrow \phi_{A1}$ and $\phi_{A1} \rightarrow
\phi_{G2}$ are both valid.
\end{problem}
\begin{problem}[Synthesis from Contract] \label{prob:3}
Given a stochastic system representation $\pfr{S}$ as in~\eqref{eq:sys}, a bounded StSTL contract $C = (V, \phi_A, \phi_G)$ on the system variables $V$, and time horizon $H$, determine a control trajectory $\boldsymbol{u}^H $ such that $(\boldsymbol{z}^H,0) \models \phi_A \to \phi_G$.
\end{problem}
\begin{example}\label{sec:motiv_exmp}
We consider the following system description:
\begin{equation} \label{eq:motivdyn}
\begin{split}
x_{k+1} = \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} x_{k} + \begin{bmatrix} 1 + 0.3w_{k,1} & -0.2w_{k,2} \\ -0.2w_{k,2} & 1 + 0.3w_{k,1}\end{bmatrix} u_k,
\end{split}
\end{equation}
where $w_k = [w_{k,1},w_{k,2}]^T$ follows a standard Gaussian distribution, i.e., $w_k \sim \mathcal{N}(0,I)$ for all $k$, $I$ being the identity matrix. We assume that the first state variable at time $0$, $[1,0] x_{0}$, is in the interval $[1,2]$ and require that with probability smaller than $0.7$ the first state variable at time $2$ does not exceed $1$. We can formalize this requirement with the following StSTL contract $C_1 = (\phi_{A1}, \phi_{G1})$ in canonical form:
\begin{equation} \label{eq:motivcon}
\begin{split}
\phi_{A1} & := (1 \leq [1,0] x_0) \land ([1,0] x_0 \leq 2), \\
\phi_{G1} & := \phi_{A1} \rightarrow \neg (\mathcal{P}\{[1,0]x_{2} \le 1\} \ge 0.7),
\end{split}
\end{equation}
where,
for brevity, we drop the set of variables in the contract tuple.
Assumptions and guarantees are expressed by logical combinations of arithmetic constraints over real numbers and chance constraints, all supported by StSTL.
We intend to verify the \emph{consistency} of $C_1$.
Given the assumption on the distribution of $w_k$, it is possible to show that there exists a constant matrix $ \Lambda_1^{1/2} \in \mathbb{R}^{3\times 3}$ such that the constraint $\mathcal{P}\left\{[1,0] x_{2} \le 1\right\} \ge 0.7$ translates into a deterministic constraint\footnote{Details on how to compute such a matrix $\Lambda_1^{1/2}$ are provided in Sec.~\ref{sec:encoding}.}
$f(x_0,u_0,u_1) \leq 0$, where
\begin{align}\label{eq:exmp_chance_cons_equi}
f(.) = & [1,2] x_0 + [1,1,1,0]\begin{bmatrix} u_0 \\ u_1\end{bmatrix} -1 + \\
& + F^{-1}(0.7) \left\| \Lambda_1^{1/2}\begin{bmatrix} u_0 \\ u_{1} \notag \\
\end{bmatrix}\right\|_2,
\end{align}
$F^{-1}$ is the inverse cumulative distribution of a standard normal random variable, and $\left\| . \right\|_2$ is the $\ell_2$ norm. Hence, the contract is consistent if and only if there exists $(x_0,u_0,u_1)$ that satisfies
\begin{equation}\label{eq:exmp_consis2}
([1,0]x_0 < 1) \vee ([1,0]x_0 > 2) \vee f(x_0,u_0,u_1) > 0.
\end{equation}
To solve this problem, we can translate~\eqref{eq:exmp_consis2} into a mixed integer program by applying encoding techniques proposed in the literature~\cite{raman2014model}. However, since one of the constraints in~\eqref{eq:exmp_consis2} is non-convex, using a nonlinear solver may be inefficient and usually requires the knowledge of bounding boxes for all the decision variables. Moreover, analytical expressions of chance constraints may not be even available in general~\cite{nemirovski2006convex}.
Similar considerations hold for the problems of checking compatibility, refinement, and for the generation of MPC schemes.
\end{example}
Sec.~\ref{sec:encoding} addresses the issue highlighted in Example~\ref{sec:motiv_exmp} by providing techniques for systematically computing mixed integer linear approximations of chance constraints and bounded StSTL formulas for three common classes of stochastic linear systems. To effectively perform the verification and synthesis tasks in Problem~\ref{prob:1}-\ref{prob:3}, we look for both under- and over-approximations of StSTL formulas. For example, if the under-approximation of~\eqref{eq:exmp_consis2} is feasible, then we can conclude that $C_1$ is consistent. However, infeasibility of the under-approximation is not sufficient to conclude about contract inconsistency; for this purpose, we need to prove that the over-approximation of~\eqref{eq:exmp_consis2} is infeasible.
\ignore{
\secref{encoding} For instance, sufficient and necessary conditions for the satisfiability of $\mathcal{P}\left((1,0)x_{2} \le 1\right) \ge 0.7$ in~\eqref{eq:exmp_chance_cons_equi} can be, respectively, expressed by the following linear constraints:
{\small
\begin{equation}\label{eq:suffi_neces_exmp}
\begin{split}
& (1,2)x_0 + [1,1,1,0]\begin{bmatrix} u_0 \\ u_1\end{bmatrix} - 1 + F^{-1}(0.7) \sum_{j=1}^5 \left|e_j^T T \begin{bmatrix} u_0 \\ u_1 \\ 1\end{bmatrix}\right| \le 0, \\
& (1,2)x_0 + [1,1,1,0]\begin{bmatrix} u_0 \\ u_1\end{bmatrix} - 1 + \frac{F^{-1}(0.7)}{\sqrt{5}} \sum_{j=1}^5 \left|e_j^T T \begin{bmatrix} u_0 \\ u_1 \\ 1\end{bmatrix}\right| \le 0,
\end{split}
\end{equation}
}
\noindent where $e_j^T$ is the $j$th row of the identity matrix $I$. The constraints in~\eqref{eq:suffi_neces_exmp} can be easily linearized and are, therefore, more tractable than the one in~\eqref{eq:exmp_chance_cons_equi}.
}
\section{MIP Encoding of Bounded StSTL}\label{sec:encoding}
We present algorithms for the translation of bounded StSTL formulas into mixed integer constraints on the variables of a stochastic system.
A MIP \emph{under-approximation} of an StSTL formula $\psi$ is a set of mixed integer constraints $\mathcal{C}^S(\psi)$ whose feasibility is sufficient to ensure the satisfiability of $\psi$.
A MIP \emph{over-approximation} of $\psi$ is a set of mixed integer constraints $\mathcal{C}^N(\psi)$ which must be feasible if $\psi$ is satisfiable.
When tractable closed-form translations of chance constraints are available, the formula under- and over-approximations coincide and provide an \emph{equivalent} encoding of the satisfiability problem. Otherwise, our framework provides under- and over-approximations in the form of mixed integer linear constraints.
We start by discussing the translation of atomic predicates.
\subsection{MIP Translation of Chance Constraints}
\label{sec:handlingCons}
Our goal is to translate chance constraints into sets of
deterministic constraints that can be efficiently solved and provide a sound
formulation for our verification and synthesis tasks.
Since approximation techniques depend on the structure of the function
$\mu(\cdot)$ and the distribution of $z_k$ at each time $k$,
we detail solutions for three classes of dynamical systems and chance
constraints that arise in various application domains.
We denote by $S(\mu^{[p]}) \le 0$ the \emph{under-approximation} of the chance constraint, i.e., the set of mixed integer constraints
whose feasibility is sufficient to guarantee the predicate satisfaction.
Similarly, we denote by $N(\mu^{[p]})\le 0$ the chance constraint \emph{over-approximation}, i.e., the set of constraints whose feasibility is necessary for the predicate satisfiability.
For simplicity, we present approximations of nonlinear constraints consisting of single linear constraints. Piecewise-affine approximations can also be used to arbitrarily improve the approximation accuracy~\cite{bradley1977applied} at higher computation costs.
\subsubsection{Linear Systems with Additive and Control-Dependent Noise}\label{sec:class1}
We consider the class of stochastic linear systems governed by the following dynamics
\begin{equation} \label{eq:sys1}
\begin{split}
x_{k+1} &= A x_k + B_k u_k + \zeta_k, \\
[B_k,\zeta_k] &= [\bar B_k,\bar\zeta_k] + \sum_{l=1}^N [\tilde B_l,\tilde \zeta_l] w_{k,l},
\end{split}
\end{equation}
where $w_k = [w_{k,1},\ldots,w_{k,N}]^T \in \mathbb{R}^N$ follows the normal distribution $\mathcal{N}(\bar w_k, \Theta_{k})$, and $\bar B_k$ and $\bar\zeta_k$, for each $k$, and $\tilde B_l$ and $\tilde \zeta_l$, for each $l \in \{1,\ldots,N\}$, are constant matrices and vectors, respectively. The resulting matrix $B_k$ and vector $\zeta_k$ are stochastic and model, respectively, a multiplicative and and additive noise term.
This model has been used, for instance, to represent motion dynamics under corrupted control signals~\cite{harris1998signal} or
networked control systems affected by channel fading~\cite{elia2005remote}.
Requirements such as policy gains or bounds on the states
for these systems are often expressed by the following chance constraint:
\begin{equation}\label{eq:type1_chance_cons}
\mathcal{P}\{\mu(z_k) \le 0\} \ge p, \;
\mu(z_k) = a^T x_k + b^T u_k + c.
\end{equation}
The next result provides an exact encoding for~\eqref{eq:type1_chance_cons}. Let $\boldsymbol{u}_{[0,k]}=\left[ u_0^T,\ldots,u_{k}^T \right]^T$ be the vector of the control inputs from $u_0$ to $u_{k}$.
We denote by $\Theta_{k}^{(l_1 l_2)}$ the $l_1$-th row and $l_2$-th column element of the covariance matrix $\Theta_{k}$, and by $F^{-1}$ the inverse cumulative distribution function of a standard normal random variable.
\begin{thm}
The chance constraint \eqref{eq:type1_chance_cons} on the behaviors of the system in~\eqref{eq:sys1} is equivalent to
\begin{equation}\label{eq:linear_chance_cons_deter}
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2 (x_0, \boldsymbol{u}_{[0,k]}) \le 0,
\end{equation}
where $\lambda_1$ is given by
\begin{equation}\label{eq:Lambda1}
\begin{split}
\lambda_1(x_0, \boldsymbol{u}_{[0,k]}) & = a^T A^k x_0 + b^T u_k + c \\
&+ \sum_{t=1}^{k} a^T A^{k-t} (\bar\zeta_{t-1} + \bar B_{t-1} u_{t-1}) \\
&+ \sum_{t=1}^{k} \sum_{l=1}^N a^T A^{k-t}
(\tilde\zeta_{l} + \tilde B_{l} u_{t-1}) \bar w_{t-1,l},
\end{split}
\end{equation}
and $\lambda_2$ is an $\ell_2$-norm of the system inputs
\begin{equation}\label{eq:Lambda2}
\lambda_2(x_0, \boldsymbol{u}_{[0,k]}) = \left\|\Lambda_{k-1}^{1/2} \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right]^T \right\|_2.
\end{equation}
The scaling matrix $\Lambda_{k-1}^{1/2}$ is deterministic for the given dynamics~\eqref{eq:sys1} and chance constraint~\eqref{eq:type1_chance_cons} and can be computed as a square root matrix of $\Lambda_{k-1}$, obtained as follows:
\begin{equation}\label{eq:Lambda_k_minus_1}
\begin{split}
\Lambda_{k-1} & = \begin{bmatrix} \Lambda_{1,1} & \Lambda_{1,2} \\ \Lambda_{1,2}^T & \Lambda_{2,2} \end{bmatrix}, \\
\Lambda_{1,1} & = \mathrm{diag}(\alpha_{k-1}, \ldots, \alpha_0), \quad \Lambda_{1,2} = [\beta_{k-1}, \ldots, \beta_0]^T,\\
\Lambda_{2,2} & = \sum_{t = 1}^k \sum_{l_1 = 1}^N \sum_{l_2 = 1}^N a^T A^{k-t} \tilde{\zeta}_{l_1} a^T A^{k-t} \tilde{\zeta}_{l_2} \Theta_{t-1}^{(l_1 l_2)}, \\
& \forall t\in \{0,\ldots,k-1\}: \\
\alpha_t & = \sum_{l_1 = 1}^N \sum_{l_2 = 1}^N \tilde{B}_{l_1}^T (A^t)^T a a^T A^t \tilde{B}_{l_2} \Theta_{k-1-t}^{(l_1 l_2)}, \\
\beta_t &= \sum_{l_1 = 1}^N \sum_{l_2 = 1}^N a^T A^t \tilde{\zeta}_{l_1} a^T A^t \tilde{B}_{l_2} \Theta_{k-1-t}^{(l_1 l_2)}. \\
\end{split}
\end{equation}
\end{thm}
\begin{proof}
The state $x_k$ of the stochastic system \eqref{eq:sys1} is known to be a linear function of the Gaussian sequence $\{w_t\}_{t=0}^{k-1}$, hence it follows a Gaussian distribution. This also applies to $\mu(z_k)$. In fact, by substituting \eqref{eq:sys1} into the expression for $\mu(z_k)$, we obtain
\begin{equation}\label{eq:sys_random_linear_cons_linear}
\begin{split}
\mu(z_k) = {} & a^T A^k x_0 + b^T u_k + c \\
&+ \sum_{t=1}^{k} a^T A^{k-t} (\bar\zeta_{t-1} + \bar B_{t-1} u_{t-1}) \\
&+ \sum_{t=1}^{k} \sum_{l=1}^N a^T A^{k-t} (\tilde\zeta_{l} + \tilde B_{l} u_{t-1}) w_{t-1,l}.
\end{split}
\end{equation}
Therefore, $\mu(z_k)$ is linear in the random variables $w_{t-1,l}$, $l\in\{1,\ldots,N\}$ and also follows a Gaussian distribution. Next, we derive the mean and the standard deviation of $\mu(z_k)$.
Since the random vector $w_{t-1}$ follows the Gaussian distribution $\mathcal{N}(\bar w_{t-1}, \Theta_{k})$, the expectation of its $l$-th element $w_{t-1,l}$ is $\bar w_{t-1,l}$. Let $\lambda_1 = \mathbb{E} \{\mu(z_k)\}$ be the expectation of $\mu(z_k)$. Then, we obtain
\begin{align*}
\lambda_1 & = a^T A^k x_0 + b^T u_k + c + \sum_{t=1}^{k} a^T A^{k-t} (\bar\zeta_{t-1} + \bar B_{t-1} u_{t-1}) \\
&+ \sum_{t=1}^{k} \sum_{l=1}^N a^T A^{k-t}
(\tilde\zeta_{l} + \tilde B_{l} u_{t-1}) \bar w_{t-1,l},
\end{align*}
which is \eqref{eq:Lambda1}. To derive the standard deviation of $\mu(z_k)$, we first write $\tilde{\mu} = \mu(z_k) - \mathbb{E} \{\mu(z_k)\}$ into a more compact form,
\begin{equation*}
\tilde{\mu} = \mathcal{B}_{k-1} \boldsymbol{u}_{[0,k-1]} + \mathcal{Z}_{k-1} = \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix},
\end{equation*}
where $\mathcal{B}_{k-1}$ and $\mathcal{Z}_{k-1}$ are random matrices defined as follows
\begin{equation*}
\begin{split}
\mathcal{B}_{k-1} & = \sum_{l=1}^N \left[ a^T A^{k-1} \tilde B_{l} \tilde w_{0,l},\; \ldots, \; a^T \tilde B_{l} \tilde w_{k-1,l} \right], \\
\mathcal{Z}_{k-1} & = \sum_{t=1}^k \sum_{l=1}^N a^T A^{k-t} \tilde\zeta_{l} \tilde w_{t-1,l}, \\
\tilde w_{t-1,l} & = w_{t-1,l} - \bar w_{t-1,l}.
\end{split}
\end{equation*}
Then, we obtain
\begin{equation*}
\begin{split}
\mathbb{E}\{\tilde{\mu}^2\} & = \mathbb{E} \left\{ \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right] \begin{bmatrix}\mathcal{B}_{k-1}^T \\ \mathcal{Z}_{k-1}^T\end{bmatrix} \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix} \right\} \\
& = \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right] \mathbb{E}\left\{ \begin{bmatrix}\mathcal{B}_{k-1}^T \\ \mathcal{Z}_{k-1}^T\end{bmatrix} \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \right\} \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix}
\end{split}
\end{equation*}
and, by renaming the positive semidefinite matrix
\begin{equation}\label{eq:Lambda}
\Lambda_{k-1} = \mathbb{E}\left\{ \begin{bmatrix}\mathcal{B}_{k-1}^T \\ \mathcal{Z}_{k-1}^T\end{bmatrix} \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \right\},
\end{equation}
we can finally write
\[
\mathbb{E}\{\tilde{\mu}^2\} = \left\|\Lambda_{k-1}^{1/2} \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right]^T \right\|_2^2 = \lambda^2_2,
\]
saying that $\lambda_2$ in \eqref{eq:Lambda2} corresponds to the standard deviation of $\mu(z_k)$.
The full expression for $\Lambda_{k-1}$ in \eqref{eq:Lambda} can be obtained by computing the expectation $\mathbb{E}\{\cdot\}$ and observing that $\mathbb{E}\{\tilde w_{t,l}\} = 0$ and $\mathbb{E}\{\tilde w_{t,l_1} \tilde w_{t,l_2}\} = \Theta_t^{(l_1 l_2)}$, which leads to \eqref{eq:Lambda_k_minus_1}.
Finally, the chance constraint \eqref{eq:type1_chance_cons} on the random variable $\mu(z_k)$ following the distribution $\mathcal{N}(\lambda_1, \lambda_2)$ is equivalent to
\[
\lambda_1 + F^{-1}(p) \lambda_2 \le 0,
\]
which corresponds to \eqref{eq:linear_chance_cons_deter}, as we wanted to prove.
\end{proof}
In~\eqref{eq:linear_chance_cons_deter}, $\lambda_1$ is a linear function of its variables, and $\lambda_2$ is an $\ell_2$-norm of the system inputs. While~\eqref{eq:linear_chance_cons_deter} is convex when $p \ge 0.5$, this is no longer the case for $p < 0.5$.
In both cases, we provide an efficient linear approximation by applying a classical norm inequality
to derive lower and upper bound functions $\lambda_2^u$ and $\lambda_2^l$ for $\lambda_2(.)$ as follows:
\begin{equation*}
\begin{split}
\lambda_2^u (x_0, \boldsymbol{u}_{[0,k]}) &=
\sum_{j=1}^{k n_u + 1} \left|e_j^T \Lambda_{k-1}^{1/2} \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix}\right|, \\
\lambda_2^l (x_0, \boldsymbol{u}_{[0,k]}) &= \frac{1}{\sqrt{k n_u + 1}}\lambda_2^u
(x_0, \boldsymbol{u}_{[0,k]}),
\end{split}
\end{equation*}
where $e_j^T$ is the $j$-th row of the identity matrix $I$ and $n_u$ is the dimension of $u_k$.
Then, an under-approximation $S(\mu^{[p]}) \le 0$ for~\eqref{eq:linear_chance_cons_deter} is given by
{\small
\begin{equation}\label{eq:linear_chance_cons_deter_suffi}
\begin{cases}
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^u (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p \ge 0.5 \\
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^l (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p < 0.5.
\end{cases}
\end{equation}
}
Similarly, an over-approximation $N(\mu^{[p]}) \le 0$ can be obtained as follows:
{\small
\begin{equation}\label{eq:linear_chance_cons_deter_neces}
\begin{cases}
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^l (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p \ge 0.5 \\
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^u (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p < 0.5.
\end{cases}
\end{equation}
}
\begin{table*}[t]
\centering
\caption{Deterministic encodings of the chance constraint $\mathcal{P}\{\mu(z_k)\le 0\}\ge p$}
\label{tab:chance_cons_formu}
\begin{tabular}{c|c|c|c|c|c}
\hline
System dynamics
& Constraint function $\mu(z_k)$
& Distribution of $w_k$
& Exact
& \begin{tabular}{@{}c@{}} Under-approx \\ $S(\mu^{[p]})(z_k)\le 0$ \end{tabular}
& \begin{tabular}{@{}c@{}} Over-approx \\ $N(\mu^{[p]})(z_k)\le 0$ \end{tabular} \\
\hline
$\begin{aligned}
x_{k+1} &= A x_k + B_k u_k + \zeta_k, \\
[B_k,\zeta_k] &= [\bar B_k,\bar\zeta_k] + \textstyle\sum_{l=1}^H [\tilde B_l,\tilde \zeta_l] w_{k,l}
\end{aligned}$
& $a^T x_k + b^T u_k + c$
& Normal $\mathcal{N}(\bar w_k, \Theta_{k})$
& \eqref{eq:linear_chance_cons_deter}
& \eqref{eq:linear_chance_cons_deter_suffi}
& \eqref{eq:linear_chance_cons_deter_neces} \\
\hline
$\begin{aligned}
x_{k+1} &= A_k x_k + B_k u_k + \zeta_k, \\
[A_k,B_k,\zeta_k] &= [A(w_k),B(w_k), \zeta(w_k)]
\end{aligned}$
& $a^T x_k + b^T u_k + c$
& \begin{tabular}{@{}c@{}}
Discrete-time finite-state \\
Markov chain
\end{tabular}
& \eqref{eq:chance_cons_Markov_jump_equiv}
& \eqref{eq:chance_cons_Markov_jump_equiv}
& \eqref{eq:chance_cons_Markov_jump_equiv} \\
\hline
$\begin{aligned}
x_{k+1} &= A x_k + B u_k, \\
\xi_k &= \left[ x_k^T ,\; u_k^T \right]^T
\end{aligned}$
& $w_{k}^T \xi_k + c$
& Normal $\mathcal{N}(\bar w_k, \Theta_{k})$
& \eqref{eq:deter_sys_Gaus_cons_equi}
& \eqref{eq:deter_sys_Gaus_cons_suffi}
& \eqref{eq:deter_sys_Gaus_cons_neces} \\
\hline
\end{tabular}
\end{table*}
\subsubsection{Markovian Jump Linear Systems}
\label{sec:class2}
Markovian jump linear systems are frequently used to model discrete transitions, for instance, due to component failures, abrupt disturbances, or changes in the operating
points of linearized models of nonlinear systems~\cite{de2006mode}. They are characterized by the following dynamics
\begin{equation}\label{eq:sys2}
\begin{split}
x_{k+1} &= A_k x_k + B_k u_k + \zeta_k, \\
[A_k,B_k,\zeta_k] &= [A(w_k),B(w_k),\zeta(w_k)],
\end{split}
\end{equation}
where $A_k, B_k, \zeta_k$ are all functions of $w_k$, and the sequence
$\{w_k\}_{k=0}^\infty$ is a discrete-time finite-state Markov chain. We assume that,
for all $k$, $w_k$ takes a value
$w^{l_k}\in\{w^{0}, \ldots, w^{N}\}$.
We use $\boldsymbol{w}_{[0,k-1]}$ and $\boldsymbol{w}^{[l_0,l_{k-1}]}$ to denote, respectively, the random trajectory $w_0,\ldots,w_{k-1}$ and a particular scenario $w^{l_0},\ldots,w^{l_{k-1}}$. $\mathcal{P}\{\boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\}$ is the probability of occurrence of the scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$. Moreover, for each scenario, we introduce a binary variable $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$ which evaluates to $1$ if and only if $\mu(z_k) \le 0$ holds for the scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$. Then, an exact encoding for the chance constraint~\eqref{eq:type1_chance_cons} on a Markovian jump linear system is given by the following result.
\begin{thm}
The chance constraint~\eqref{eq:type1_chance_cons} on the behaviors of the system in~\eqref{eq:sys2} is equivalent to the following MIL constraints
\begin{equation}\label{eq:chance_cons_Markov_jump_equiv}
{\small
\begin{cases}
\sum\limits_{t = 0}^{k-1} \sum\limits_{l_{t} = 0}^N b(\boldsymbol{w}^{[l_0,l_{k-1}]})
\mathcal{P}\{\boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\} \ge p, \\
\lambda(x_0, \boldsymbol{u}_{[0,k]}, \boldsymbol{w}^{[l_0,l_{k-1}]}) \le 0 \leftrightarrow b(\boldsymbol{w}^{[l_0,l_{k-1}]}) = 1, \\
\end{cases}
}
\end{equation}
where $\lambda(x_0, \boldsymbol{u}_{[0,k]}, \boldsymbol{w}^{[l_0,l_{k-1}]}) \le 0$ enforces that the particular scenario satisfies the chance constraint. $\lambda(\cdot)$
can be computed as follows:
\begin{equation}\label{eq:markov_jump_lambda}
{\small
\begin{split}
\lambda(\cdot) ={} & a^T \mathcal{A}_{k-1} x_0 + \mathcal{B}_{k-1} \boldsymbol{u}_{[0,k]} + \mathcal{Z}_{k-1} + c \\
\mathcal{A}_{k-1} ={} & \left[ A(w^{l_{k-1}}), \cdots, A(w^{l_0})\right], \\
\mathcal{B}_{k-1} ={} & \left[ a^T \mathcal{A}_{k-1} B(w^{l_0}),\; \ldots,\; a^T B(w^{l_{k-1}}), b^T \right] \\
\mathcal{Z}_{k-1} ={} & a^T \mathcal{A}_{k-1} \zeta(w^{l_0}) + \ldots + a^T \zeta(w^{l_{k-1}}),
\end{split}
}
\end{equation}
with $\boldsymbol{u}_{[0,k]} = [u_{0}^T,\ldots,u_{k}^T]^T$.
\end{thm}
\begin{proof}
For a given scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$ for the Markovian jump linear system in~\eqref{eq:sys2}, the system state $x_k$ is a deterministic function of $\boldsymbol{u}_{[0,k-1]} = [u_{0}^T,\ldots,u_{k-1}^T]^T$. We can then express the constraint $\mu(z) = a^T x_k + b_i^T u_k + c \le 0$ as in~\eqref{eq:markov_jump_lambda}. The probability $\mathcal{P}\{a^T x_k + b^T u_k + c \le 0\}$ can be computed by considering all the possible scenarios for $\boldsymbol{w}_{[0,k-1]}$ as follows:
\begin{equation}\label{eq:chanc_cons_Markov_jump}
\begin{split}
& \mathcal{P}\{a^T x_k + b^T u_k + c \le 0\} \\
& \;\; = \sum_{t = 0}^{k-1} \sum_{l_{t} = 0}^N
\mathcal{P}\{a^T x_k + b^T u_k + c \le 0, \boldsymbol{w}^{[l_0,l_{k-1}]}\} \\
& \;\; = \sum_{t = 0}^{k-1} \sum_{l_{t} = 0}^N
\mathcal{P}\{a^T x_k + b^T u_k + c \le 0 | \boldsymbol{w}^{[l_0,l_{k-1}]}\}\cdot \\
& \phantom{\;\; = \sum_{t = 0}^{k-1} \sum_{l_{t} = 1}^H}
\; \mathcal{P}\{\boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\}.
\end{split}
\end{equation}
Whether the constraint $a^T x_k + b^T u_k + c \le 0$ is satisfied or not under a given scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$ is a deterministic event, hence the probability $\mathcal{P}\{a^T x_k + b^T u_k + c \le 0 | \boldsymbol{w}^{[l_0,l_{k-1}]}\}$ is either $1$ or $0$, and corresponds to the value of the binary indicator variable $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$. By introducing $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$ into~\eqref{eq:chanc_cons_Markov_jump}, the chance constraint
$\mathcal{P}\{a^T x_k + b^T u_k + c \le 0\} \ge p$ reduces to the first constraint in~\eqref{eq:chance_cons_Markov_jump_equiv}, where the probability
$\mathcal{P}\{ \boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\}$
is given by the transition probability matrix of the Markov chain. The second constraint in~\eqref{eq:chance_cons_Markov_jump_equiv} directly descends from the definition of $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$.
Therefore, constraints~\eqref{eq:chance_cons_Markov_jump_equiv} and~\eqref{eq:markov_jump_lambda}
provide an exact encoding of the chance constraint~\eqref{eq:type1_chance_cons} for a Markovian jump linear system, which is what we wanted to prove. The implication in~\eqref{eq:chance_cons_Markov_jump_equiv} can be translated into MIL constraints using standard techniques~\cite{l2008operations}.
\end{proof}
\subsubsection{Deterministic Systems with Measurement Noise}\label{sec:class3}
We consider a system
\begin{equation*}
x_{k+1} = A x_k + B u_k, \quad
\xi_k = \begin{bmatrix}
x_k \\
u_k
\end{bmatrix},
\end{equation*}
subject to constraints of the form
\begin{equation}\label{eq:chance_cons_deter}
\begin{split}
\mathcal{P}\{\mu(z_k)\le 0\} \ge p, \quad \mu(z_k) = w_{k}^T \xi_k + c,\;
\end{split}
\end{equation}
where $w_k$ follows the normal distribution $\mathcal{N}(\bar w_k, \Theta_{k})$. This setting can be used to represent uncertainties in perception, e.g., in the detection of environment obstacles to the trajectory of autonomous systems~\cite{sadigh2016}.
As for the system in Sec.~\ref{sec:class1}, an exact translation of~\eqref{eq:chance_cons_deter}~\cite{sadigh2016} leads to
\begin{equation}\label{eq:deter_sys_Gaus_cons_equi}
\begin{split}
\bar w_{k}^T \xi_k + c + F^{-1}(p) \left\|\Theta_k^{1/2} \xi_k\right\|_2 \le 0,
\end{split}
\end{equation}
which may result in non-convex constraint. Again, by using
a norm inequality to bound the $\ell_2$-norm in~\eqref{eq:deter_sys_Gaus_cons_equi}, we provide an under-approximation of~\eqref{eq:chance_cons_deter} in the form
\begin{equation}\label{eq:deter_sys_Gaus_cons_suffi}
{\small
\begin{split}
\begin{cases}
\bar w_{k}^T \xi_k + c + F^{-1}(p) \sum\limits_{j=1}^{n_z} \left|e_j^T \Theta_k^{1/2} \xi_k\right|
\le 0, \;\; p \ge 0.5, \\
\bar w_{k}^T \xi_k + c + \frac{F^{-1}(p)}{\sqrt{n_\xi}} \sum\limits_{j=1}^{n_\xi}
\left|e_j^T \Theta_k^{1/2} \xi_k\right| \le 0, \;\; p < 0.5,
\end{cases} \\
\end{split}
}
\end{equation}
where $e_j$ is the $j$-th column of the identity matrix, and an over-approximation in the form
\begin{equation}\label{eq:deter_sys_Gaus_cons_neces}
{\small
\begin{cases}
\bar w_{k}^T \xi_k + c + \frac{F^{-1}(p)}{\sqrt{n_\xi}} \sum\limits_{j=1}^{n_\xi} \left|e_j^T \Theta_k^{1/2} \xi_k\right| \le 0, \;\; p \ge 0.5, \\
\bar w_{k}^T \xi_k + c + F^{-1}(p) \sum\limits_{j=1}^{n_\xi} \left|e_j^T \Theta_k^{1/2} \xi_k\right| \le 0, \;\; p < 0.5.
\end{cases}
}
\end{equation}
Table~\ref{tab:chance_cons_formu} provides a summary of the encodings in this section.
\subsection{MIP Under-Approximation}\label{sec:suffi_encode}
We construct a MIP under-approximation $\mathcal{C}_k^S(\psi)$ of a formula $\psi$ by assigning a binary variable $b^{S}_{k}(\psi)$ to the formula such that
$b^{S}_{k} (\psi) = 1 \to (\boldsymbol{z},k) \models\psi$.
We then traverse the parse tree of $\psi$ and associate binary
variables with all the sub-formulas in $\psi$.
Following the semantics in Sec.~\ref{eq:semantics}, the logical
relation between $\psi$ and its sub-formulas is then recursively captured using mixed integer constraints. The translation terminates when all the atomic
predicates are translated.
Our encoding is different from the ones previously proposed for deterministic STL formulas~\cite{raman2014model}, in that the truth value of the Boolean variable $b$ associated to each atomic predicate $(\mu \leq 0)$ is not equivalent to the predicate satisfaction. Instead, $b = 1$ is only a sufficient condition for predicate satisfaction, as we are only able to associate $b$ with an under-approximation $S(\mu^{[p]})(z_{k}) \le 0$. Because $b=0$ cannot encode the logical negation of the predicate, we deal with atomic predicates and their negations separately. Specifically, we convert any formula into its negation normal form and associate distinct Boolean variables, e.g., $b$ and $\bar{b}$, to each atomic predicate and its negation, respectively.
We use both $b$ and $\bar{b}$ to translate any Boolean and temporal operator involving the predicate or its negation in the formula. We illustrate this approach on some special cases below.
$\bm{\psi = \mu^{[p]}}$: We requires that $b_{k}^S(\mu^{[p]}) = 1$ implies the feasibility of a sufficient condition for $(\boldsymbol{z},k)\models \mu^{[p]}$ by the following constraint
\begin{equation}\label{eq:atomic_milp}
S(\mu^{[p]})(z_{k}) \le (1 - b_{k}^S(\mu^{[p]}))M,
\end{equation}
where $M$ is a sufficiently large positive constant (``big-$M$'' encoding technique)~\cite{l2008operations}, and $S(\mu^{[p]})(z_{k}) \le 0$ is the chance constraint under-approximation.
$\bm{\psi = \neg\mu^{[p]}}$:
If an under-approximation $S(\neg\mu^{[p]})(z_{k}) \le 0$ is available, then we require
\begin{equation}\label{eq:neg_atomic_milp}
\begin{split}
S(\neg\mu^{[p]})(z_{k}) \le (1 - b_{k}^S(\neg\mu^{[p]}))M.
\end{split}
\end{equation}
Otherwise, we recall that $\mathcal{P}(\mu(z_k) \leq 0) < p$
is equivalent to $\mathcal{P}(\mu(z_k) > 0) > 1-p$.
To bring this predicate into a standard form, we require that $\mathcal{P}(-\mu(z_k) + \epsilon\leq 0) \ge 1-p + \epsilon$, where $\epsilon > 0$ is a sufficiently small real constant. We can then use the encoding in~\eqref{eq:atomic_milp} to obtain
\begin{equation}\label{eq:neg_atomic_milp3}
S((-\mu + \epsilon)^{[1-p + \epsilon]})(z_k) \le (1 - b_{k}^S(\neg\mu^{[p]}))M.
\end{equation}
$\bm{\psi = {\bf G}_{[t_1,t_2]} \phi}$: To encode the bounded \emph{globally} predicate we add to $\mathcal{C}_k^S(\psi)$ the mixed integer linear constraint
\begin{equation}\label{eq:global_milp}
b_k^S({\bf G}_{[t_1,t_2]} \phi) \leftrightarrow \wedge_{i=t_1}^{t_2} b_{k+i}^S(\phi),
\end{equation}
requiring that $b_k^S({\bf G}_{[t_1,t_2]} \phi) = 1$ if and only if $b_{k+i}^S(\phi) = 1$
for all $i \in [t_1, t_2]$.
The conjunction of the $b_{k+i}^S(\phi)$ is then translated into
mixed integer linear constraints using standard techniques~\cite{raman2014model}.
$\bm{\psi = \neg {\bf G}_{[t_1,t_2]} \phi}$: When \emph{globally} is negated, we augment $\mathcal{C}_k^S(\psi)$ with the mixed integer linear
constraint
\begin{equation}\label{eq:neg_global_milp}
b_k^S(\neg ({\bf G}_{[t_1,t_2]} \phi)) \leftrightarrow \vee_{i=t_1}^{t_2} b_{k+i}^S(\neg\phi),
\end{equation}
showing how we push the negation of a formula to its sub-formulas in a recursive fashion until we reach the atomic predicates.
For brevity, we omit the encoding for the other temporal operators, which directly follows from the semantics in Sec.~\ref{eq:semantics} and the approach in~\eqref{eq:global_milp} and~\eqref{eq:neg_global_milp}.
If~\eqref{eq:atomic_milp} and~\eqref{eq:neg_atomic_milp3} are linear, then $\mathcal{C}_k^S(\psi)$ is a mixed integer linear constraint set.
Based on the above procedure, the following theorem summarizes the property of the MIP under-approximation.
\begin{thm}\label{thm:suffi_encoding}
$\mathcal{C}_k^S(\psi)$ is a MIP under-approximation of $\psi$, i.e., if $\mathcal{C}_k^S(\psi)$ is feasible and $\boldsymbol{z}^*$ is a solution, then $\psi$ is satisfiable and $(\boldsymbol{z}^*, k)\models\psi$.
\end{thm}
\begin{proof}
We first prove the theorem for the atomic predicates $\mu^{[p]}$ and $\neg\mu^{[p]}$. We observe that $\mathcal{C}_k^S(\mu^{[p]})$ is equivalent to the conjunction of the constraints $(b_k^S(\mu^{[p]}) = 1)$ and~\eqref{eq:atomic_milp}.
If $\mathcal{C}_k^S(\mu^{[p]})$ is feasible, then $S(\mu^{[p]})(z_k)\le 0$ must hold.
Since $S(\mu^{[p]})(z_k)\le 0$ is a sufficient condition for the satisfaction of the predicate, we conclude $(\boldsymbol{z}^*,k)\models \mu ^{[p]}$. Similarly, the feasibility of $\mathcal{C}_k^S(\neg\mu^{[p]})$ implies $(\boldsymbol{z}^*,k)\models \neg \mu ^{[p]}$ using constraint~\eqref{eq:neg_atomic_milp}.
We now consider a formula $\psi$ such that Theorem \ref{thm:suffi_encoding} holds for all its sub-formulas. Without loss of generality, we discuss $\psi = \phi_1 \mathcal{U}_{[t_1,t_2]}\phi_2$; the same proof structure can be applied to other temporal or logical operators.
$\mathcal{C}_k^S(\psi)$ contains the following constraints
\begin{equation*}
\begin{split}
& b_k^S(\psi) = 1,\; b_k^S(\psi) = \vee_{i=t_1}^{t_2} (b_{k+i}^S(\phi_2)
\wedge_{j=t_1}^{i-1}b_{k+j}^S(\phi_1)), \\
& \mathcal{C}_{k+i}^S(\phi_1)\setminus \{b_{k+i}^S(\phi_1) = 1\},\;
\mathcal{C}_{k+j}^S(\phi_2)\setminus \{b_{k+j}^S(\phi_2) = 1\},
\end{split}
\end{equation*}
for all $i\in [t_1,t_2]$ and $j\in [t_1,t_2-1]$. We use $\mathcal{C}_{k+i}^S(\phi_1)\setminus \{b_{k+i}^S(\phi_1) = 1\}$ to denote the set of constraints in $\mathcal{C}_{k+i}^S(\phi_1)$ except for the constraint $(b_{k+i}^S(\phi_1) = 1)$. If $\mathcal{C}_k^S(\psi)$ is feasible, then $b_k^S(\psi) = 1$ must hold, hence there exists $i\in [t_1,t_2]$ such that
$b_{k+i}^S(\phi_2) \wedge_{j=t_1}^{i-1}b_{k+j}^S(\phi_1) = 1$.
We then obtain that $b_{k+i}^S(\phi_2) = 1$ holds as well as $b_{k+j}^S(\phi_1) = 1$, $\forall \ j \in [t_1,i-1]$. This ensures that $\mathcal{C}_{k+i}^S(\phi_1)$ and $\mathcal{C}_{k+j}^S(\phi_2)$, $\forall \ j \in [t_1,i-1]$, are feasible.
Since Theorem \ref{thm:suffi_encoding} holds for $\phi_1$ and $\phi_2$, we also have
$(\boldsymbol{z}^*,k+i)\models\phi_2$ and
$(\boldsymbol{z}^*,k+j)\models\phi_1$ $\forall \ j \in [t_1,i-1]$, hence
$(\boldsymbol{z}^*,k)\models\phi_1 \mathcal{U}_{[t_1,t_2]}\phi_2$, which is what we wanted to prove.
\end{proof}
It is possible that both the $\mathcal{C}_k^S(\psi)$ and $\mathcal{C}_k^S(\neg\psi)$
under-approximations are infeasible, in which case
we cannot make any conclusion on whether $\psi$ or $\neg\psi$ are satisfiable.
To conclude on the unsatisfiability of a formula, we resort to a MIP over-approximation.
\subsection{MIP Over-Approximation}\label{sec:neces_encoding}
To generate an over-approximation of $\psi$, we associate a binary variable $b^{N}_{k} (\psi)$ to $\psi$ and seek for a set of mixed integer constraints $\mathcal{C}_k^N(\psi)$ so that
$(\boldsymbol{z},k) \models\psi \rightarrow b^{N}_{k} (\psi) = 1$.
Creating an over-approximation only differs in
the interpretation of the atomic propositions, since we now use deterministic
mixed integer constraints that are necessary for the satisfaction of the chance constraints in the formula. As in Sec.~\ref{sec:suffi_encode}, we deal with an atomic predicate and its negation separately, and provide necessary condition for their satisfaction as follows.
$\bm{\psi = \mu^{[p]}}$: We assign a binary variable $b_{k}^N(\mu^{[p]})$ so that, if the over-approximation $N(\mu^{[p]})(z_k) \le
0$ is not satisfied, then $b_{k}^N(\mu^{[p]})$ is false. We, therefore, add the following mixed integer constraint:
\begin{equation}\label{eq:atomic_milp_neces}
\begin{split}
N(\mu^{[p]})(z_k) \le (1 - b_k^N(\mu^{[p]}))M,
\end{split}
\end{equation}
where $M$ is a large enough positive constant~\cite{l2008operations}.
$\bm{\psi = \neg\mu^{[p]}}$: If an over-approximation $N(\neg\mu^{[p]})(z_k) \le 0$ is available, then we add a binary variable $b_{k}^N(\neg\mu^{[p]})$ and the mixed integer constraint
\begin{equation}\label{eq:neg_atomic_milp_neces}
\begin{split}
N(\neg\mu^{[p]})(z_k) \le (1 - b_k^N(\neg\mu^{[p]}))M.
\end{split}
\end{equation}
Otherwise, since $\mathcal{P}(\mu(z_k) \leq 0) < p$
implies $\mathcal{P}(-\mu(z_k) \leq 0) \ge 1- p$ we require
\begin{equation}\label{eq:neg_atomic_milp_neces2}
\begin{split}
N((-\mu)^{[1-p]})(z_k) \le (1 - b_k^N(\neg\mu^{[p]}))M.
\end{split}
\end{equation}
Other logic and temporal operators are encoded as in Sec.~\ref{sec:suffi_encode}.
By similar arguments, we obtain the result below.
\begin{thm}\label{thm:neces_encoding}
$\mathcal{C}_k^N(\psi)$ is a MIP over-approximation for the formula $\psi$, i.e., if $\mathcal{C}_k^N(\psi)$ is infeasible, then $\psi$ is unsatisfiable.
\end{thm}
\begin{proof
We need to prove that $(\boldsymbol{z},k)\models\psi$ is sufficient for the feasibility of $\mathcal{C}_k^N(\psi)$. Let first $\psi$ be the atomic proposition $\mu^{[p]}$.
Since $N(\mu^{[p]})(z_k) \le 0$ is a necessary condition for the satisfaction of $\mu^{[p]}$, we obtain
$(\boldsymbol{z},k)\models \mu^{[p]} \to N(\mu^{[p]})(z_k) \le 0$.
Then, if $\mu^{[p]}$ is satisfiable, the conjunction of \eqref{eq:atomic_milp_neces} and $b_k^N(\mu^{[p]}) = 1$ holds, which is equivalent to the feasibility of $\mathcal{C}_k^N(\psi)$. A similar argument can be used for $\neg\mu^{[p]}$.
When $\psi$ is a generic formula, let Theorem \ref{thm:neces_encoding} hold for the sub-formulas of $\psi$. Then, if a sub-formula is satisfiable, its over-approximation is feasible. Without loss of generality, we consider $\psi = \neg(\phi_1 \mathcal{U}_{[t_1,t_2]} \phi_2)$. $(\boldsymbol{z},k)\models\psi$ is equivalent to
\[
\wedge_{i=t_1}^{t_2} ((\boldsymbol{z},k+i)\models\neg\phi_2 \vee_{j=t_1}^{i-1}
(\boldsymbol{z},k+j)\models\neg\phi_1)
\]
being true, meaning that for all $i \in [t_1,t_2]$ either $(\boldsymbol{z},k+i)\models\neg\phi_2$ holds or there exists $j \in [t_1,i-1]$
such that $(\boldsymbol{z},k+j)\models\neg\phi_1$. Since both $\neg\phi_1$ and $\neg\phi_2$ are sub-formulas of $\psi$, $(\boldsymbol{z},k+i)\models\neg\phi_2$ and $(\boldsymbol{z},k+j)\models\neg\phi_1$ imply, respectively, that $\mathcal{C}_{k+j}^N(\neg\phi_1)$ and $\mathcal{C}_{k+i}^N(\neg\phi_2)$ are feasible.
We deduce that for all $i \in [t_1,t_2]$ either $b_{k+i}^N(\neg\phi_2)= 1$ or there exists $j \in [t_1,i-1]$ such that $b_{k+j}^N(\neg\phi_1) = 1$.
Since the relation between $b_{k}^N(\psi)$, $b_{k+j}^N(\neg\phi_1)$, and $b_{k+i}^N(\neg\phi_2)$, as encoded in $\mathcal{C}_k^N(\psi)$, is
\begin{equation}\label{eq:logic_relation2}
b_k^N(\psi) = \wedge_{i=t_1}^{t_2} (b_{k+i}^N(\neg\phi_2)
\vee_{j=t_1}^{i-1}b_{k+j}^N(\neg\phi_1)),
\end{equation}
we infer that $b_{k}^N(\psi) = 1$ is feasible. The feasibility of $\mathcal{C}_k^N(\psi)$ is then proved since a feasible solution for $\mathcal{C}_k^N(\psi)$ can be obtained by solving the conjunction of the constraints $\mathcal{C}_{k+j}^N(\neg\phi_1) \setminus \{b_{k+j}^N(\neg\phi_1) = 1\}$ for all $j\in[t_1, t_2-1]$, $\mathcal{C}_{k+i}^N(\neg\phi_2) \setminus \{b_{k+i}^N(\neg\phi_2) = 1\}$ for all $j\in[t_1, t_2]$, constraint~\eqref{eq:logic_relation2}, and $b_{k}^N(\psi) = 1$.
\end{proof}
\section{Contract-Based Verification and Synthesis}\label{sec:contract_check}
We formulate verification and synthesis procedures that leverage under- and over-approximations of bounded StSTL contracts to solve Problem~\ref{prob:1}-\ref{prob:3} for the classes of stochastic systems introduced in Sec.~\ref{sec:handlingCons}.
A first result provides sound procedures to check contract consistency and compatibility (Problem~\ref{prob:1}).
\begin{thm}\label{thm:compati_consis}
Let $\pfr{S}$ be a stochastic system belonging to one of the classes introduced in Sec.~\ref{sec:handlingCons} (Table~\ref{tab:chance_cons_formu}); let $C = (\phi_A,\phi_G)$ be an A/G contract where $\phi_A$ and
$\phi_G$ are bounded StSTL formulas over the system variables. If over- and under-approximations are available for both $\phi_A$ and $\neg \phi_A \vee \phi_G$, then the following hold:
\begin{enumerate}
\item If $\mathcal{C}_0^S(\phi_A)$ is feasible, then $C$ is compatible.
\item If $\mathcal{C}_0^N(\phi_A)$ is infeasible, then $C$ is not
compatible.
\item If $\mathcal{C}_0^S(\neg \phi_A \vee \phi_G)$ is feasible, then $C$
is consistent.
\item If $\mathcal{C}_0^N(\neg \phi_A \vee \phi_G)$ is infeasible, then $C$
is not consistent.
\end{enumerate}
\end{thm}
\begin{proof}
By Theorem~\ref{thm:suffi_encoding}, if $\mathcal{C}_0^S(\phi_A)$ is feasible, then $\phi_A$ is satisfiable, which indicates that $C$ is
compatible. On the other hand, by Theorem \ref{thm:neces_encoding}, if $\mathcal{C}_0^N(\phi_A)$ is
infeasible, then $\phi_A$ is unsatisfiable, hence $C$ is incompatible.
The results on consistency can be obtained in the same way.
\end{proof}
The following result addresses refinement checking
(Problem~\ref{prob:2}).
\begin{thm}\label{thm:refine}
Let $\pfr{S}$ be a stochastic system belonging to one of the classes introduced in Sec.~\ref{sec:handlingCons} (Table~\ref{tab:chance_cons_formu}); let $C_1 = (\phi_{A1},\phi_{G1})$ and $C_2 = (\phi_{A2}, \phi_{G2})$ be A/G contracts whose assumptions and guarantees are
bounded StSTL formulas over the system variables. If over- and under-approximations are available for $\psi_1 = \neg \phi_{A2} \vee \phi_{A1}$ and
$\psi_2 = (\phi_{A1} \wedge \neg \phi_{G1}) \vee (\neg \phi_{A2} \vee \phi_{G2})$, then the following hold:
\begin{enumerate}
\item If $\mathcal{C}_0^N(\neg\psi_1)$ and $\mathcal{C}_0^N(\neg\psi_2)$ are
infeasible, then $C_1 \preceq C_2$.
\item If $\mathcal{C}_0^S(\neg\psi_1)$ or $\mathcal{C}_0^S(\neg\psi_2)$ are
feasible, then $C_1 \not\preceq C_2$.
\end{enumerate}
\end{thm}
\begin{proof
The proof proceeds as in Theorem~\ref{thm:compati_consis}, by directly applying the definition of contract refinement. By Theorem~\ref{thm:neces_encoding}, if $\mathcal{C}_0^N(\neg\psi_1)$ and $\mathcal{C}_0^N(\neg\psi_2)$ are infeasible, then $\neg\psi_1$ and $\neg\psi_2$ are unsatisfiable, hence $\psi_1$ and $\psi_2$ are valid. We therefore obtain than $\phi_{A2} \rightarrow \phi_{A1}$ and $(\neg \phi_{A1} \vee \phi_{G1}) \rightarrow (\neg \phi_{A2} \vee \phi_{G2})$ are valid, hence $C_1 \preceq C_2$ by definition.
Similarly, $\mathcal{C}_0^S(\neg\psi_1)$ or $\mathcal{C}_0^S(\neg\psi_2)$ being
feasible implies that either $\psi_1$ or $\psi_2$ are not valid formulas by Theorem~\ref{thm:suffi_encoding}. We therefore conclude that $C_1 \not\preceq C_2$ holds.
\end{proof}
The above decision procedures are not complete.
For instance, it is possible that $\mathcal{C}_0^S(\phi_A)$ is infeasible and $\mathcal{C}_0^N(\phi_A)$
is feasible, in which case we are not able to conclude on the satisfiability of $\phi_A$.
In this case, we increasingly refine piecewise-affine under- and over-approximations of chance constraints until we obtain an answer.
Finally, as an application of Theorem~\ref{thm:compati_consis}, we provide a framework for the design of stochastic MPC schemes using StSTL contracts. We show how a stochastic optimization problem can be generated by enforcing contract consistency on the system in Fig.~\ref{fig:sys_structure} to obtain a control trajectory which solves Problem~\ref{prob:3}.
\begin{example}[Generation of Stochastic MPC Schemes] \label{ex:mpc}
In stochastic MPC, the controller measures the plant state $x_k$ at time $k$ and derives a control input $u_k$ by solving a stochastic optimization problem. The plant state $x_{k+1}$ is a function of $u_k$ and the random external signal $w_k$ according to the system dynamics.
Given a stochastic system described as in~\eqref{eq:sys}, where the environment input (disturbance) $w_k$ at each time $k$ follows a distribution $\mathcal{D}$, let the bounded StSTL contract $C = (Q x_0 \leq r,\phi)$ capture the system requirement that $\phi$ be satisfied if the initial state $x_0$ is in the polyhedron represented by set of linear inequalities $Q x_0 \leq r$ for a fixed matrix $Q$ and vector $r$.
Control synthesis can then be formulated as the problem of finding a control trajectory $\boldsymbol{u}$ that makes $C$ consistent and optimizes a predefined cost. For a finite horizon $H$, this translates into requiring that the guarantees of $C$ are satisfiable in the context of its assumptions, hence the conjunction of the following constraints
\begin{align*}
& (\boldsymbol{z}^H,0) \models (Q\bar{x}_0 \leq r) \to \phi, \; x_{k+1} = f(x_k,u_k,w_k), \\ & w_k \sim \mathcal{D}, x_0 = \bar{x}_0,
\end{align*}
for $k=0,1,\ldots, H-1$, must be feasible, while optimizing a cost function $J_H(x_0, \boldsymbol{u}^H)$.
By calling $\psi := (Qx_0 \leq r) \to \phi$ and using Theorem~\ref{thm:compati_consis}, we can finally solve this problem using the under-approximation $\mathcal{C}_0^S(\psi)$ obtained as described in Sec. \ref{sec:encoding} over the horizon $H$, which provides the following stochastic optimization problem:
\begin{equation}\label{eq:MPC_opti1}
\begin{split}
\min_{\boldsymbol{u}^H} \quad J_H(x_0, \boldsymbol{u}^H), \quad
\mathrm{s.t.} \quad \mathcal{C}_0^S(\psi)
\end{split}
\end{equation}
to be executed in a receding horizon fashion. It is then possible to extend previous results on MPC from STL specifications~\cite{raman2014model} to stochastic linear systems.
\end{example}
\section{Case Studies}\label{sec:sim_exam}
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{tool}
\caption{The \textsc{ScanS} Flow.}
\label{fig:tool}
\end{figure}
We implemented the verification and synthesis procedures in Sec.~\ref{sec:contract_check} in the \textsc{Matlab} toolbox \textsc{SCAnS} (Stochastic Contract-based Analysis and Synthesis).
As shown in Fig.~\ref{fig:tool}, \textsc{SCAnS} receives as inputs a system description in one of the classes of Sec.~\ref{sec:handlingCons}, a set of bounded StSTL contracts, a time horizon $H$, and a set of verification or synthesis tasks. In the verification flow, \textsc{SCAnS} computes under- and over-approximations of contract assumptions and guarantees and perform consistency, compatibility, and refinement checking of user-defined contracts using the results in Theorem~\ref{thm:compati_consis} and Theorem~\ref{thm:refine}. In the synthesis flow, \textsc{SCAnS} follows the procedure in Example~\ref{ex:mpc} to generate a stochastic optimization problem from a user-defined contract, which can be executed in a receding horizon scheme.
We illustrate the effectiveness of our approach on two examples.
The first example utilizes both under- and over-approximations of StSTL formulas to perform contract compatibility, consistency, and refinement checking.
The second example uses a formula under-approximation to synthesize an MPC controller for an aircraft power distribution network. \textsc{SCAnS} uses \textsc{Yalmip}~\cite{lofberg2004yalmip} to formulate mixed integer programs, \textsc{Gurobi}~\cite{gurobi} to solve mixed integer linear programs, and \textsc{bmibnb} (in \textsc{Yalmip}) to solve mixed integer nonlinear programs. All experiments ran on a $3.2$-GHz Intel Core i5 processor with $4$-GB memory.
\subsection{Contract-Based Verification}
We check compatibility and consistency for the contract and system in Example~\ref{sec:motiv_exmp}. By applying Theorem~\ref{thm:compati_consis} and the under-approximation in Sec.~\ref{sec:suffi_encode}, we find that $\mathcal{C}_0^S(\phi_{A1})$ is feasible, and so is $\mathcal{C}_0^S(\neg \phi_{A1} \vee \phi_{G1})$. Therefore, contract $(\phi_{A1},\phi_{G1})$ is both compatible and consistent.
Since the system is in the class of Sec.~\ref{sec:class1}, our encoding uses~\eqref{eq:linear_chance_cons_deter_suffi} and~\eqref{eq:linear_chance_cons_deter_neces}.
Given a contract $C_2$ defined as follows:
\begin{equation*}
\begin{split}
\phi_{A2} &:= [1,0]x_0 \leq 3, \\
\phi_{G2} &:= \phi_{A2} \rightarrow {\bf G}_{[1,3]}\neg (\mathcal{P}\{[1,0]x_{2} \le 2\} \ge 0.6),
\end{split}
\end{equation*}
we can also check that $C_2 \preceq C_1$ by using the results in Theorem~\ref{thm:refine}. Moreover, to show the effectiveness of the proposed approximation, we increase the system dimension by redefining the dynamics as follows:
\begin{equation*}
\begin{split}
x_{k+1} &= A x_k + B_k u_k, \\
B_k &= I + 0.3\begin{bmatrix} w_{k,1} & & \\ & \ddots & \\ & & w_{k,1}\end{bmatrix}
-0.2\begin{bmatrix} & & w_{k,2} \\ & \iddots & \\ w_{k,2} & & \end{bmatrix}
\end{split}
\end{equation*}
where $A$ is a Jordan matrix constructed using blocks of dimension $2$ as in \eqref{eq:motivdyn}. Contract refinement checking on a system with $100$ state variables took about $20$ ms using the proposed approximate encoding, which is a $20\times$ reduction in execution time with respect to the exact encoding.
\subsection{Requirement Analysis and Control Synthesis for Aircraft Electric Power Distribution}
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{EPS_bus}
\caption{Simplified diagram of an aircraft power distribution system.}
\label{fig:epsbus}
\end{figure}
An aircraft power system distributes power from
generators (engines) to loads by configuring a set of electronic control switches
denoted as contactors~\cite{nuzzo2014contract}. As shown in the simplified diagram of Fig.~\ref{fig:epsbus}, physical components of a power system include
generators, AC and DC buses, Transformer and Rectifier Units (TRUs), contactors (C1-C11), loads, and batteries. The controller, which is also denoted as Load Management System (LMS) and is not shown in the figure, determines the configuration of the contactors at each time instant, in order to provide the required power to the loads, while being subject to a set of constraints, e.g.,
on the battery charge level.
A hierarchical LMS structure was proposed for aircraft power systems, which adopts two controller levels and is based on a deterministic model of the system~\cite{maasoumy2013optimal}. A high-level LMS (HL-LMS) operates at a lower frequency (e.g., 0.1~Hz) and provides
advice on the contactor configuration as obtained by
solving an optimization problem. The control objective is to provide power to the highest number of loads at each time (minimize load shedding) and reduce the switching frequency of contactors, hence the wear-and-tear associated with switching. A low-level LMS (LL-LMS), working at a
faster frequency (e.g., 1~Hz) takes critical decisions to place the system in safety mode by
shedding non-essential loads every time a generator fails. The LL-LMS accepts the suggestion of the HL-LMS only if it is safe.
We adopt the same model for the system architecture and the dynamics as in this reference design~\cite{maasoumy2013optimal}. The system state
is represented by the state of charge of the batteries, which are allowed to,
respectively, discharge or charge when the generator power
is insufficient or redundant with respect to the load power.
The system contains a number of generators $N_s = 3$ and a number of AC (DC) buses
$N_b = 2$, where each bus must be connected to a functional generator or TRU to receive
power. Each DC bus has $N_{sl} = 10$ sheddable loads and $N_{nsl} = 10$
non-sheddable loads, which are shown as lumped components in Fig.~\ref{fig:epsbus}. The maximum power supplied by the three generators is
$100$~kW (GEN1), $100$~kW (GEN2), and $85$~kW (GEN3). However, differently from the reference design~\cite{maasoumy2013optimal}, the power demand of each load is now a Gaussian random
variable. The average power demand assumes the values in Table II of our reference~\cite{maasoumy2013optimal}, while the variance is $0.1$ times larger than the average value. A controller based on stochastic MPC has been recently proposed for a similar power system model~\cite{shahsavari2015stochastic}. In this section, we show that \textsc{SCAnS} is able to \emph{automatically} design a controller that follows the same approach
but can handle a \emph{richer} set of specifications.
We use StSTL to express the control specification $\psi$ for the HL-LMS, involving both
deterministic constraints on the network connectivity~\cite{maasoumy2013optimal} and stochastic constraints
on the battery levels. Sample requirements in $\psi$, over a time horizon of $20$ steps, are formalized as follows:
\begin{itemize}
\item
The battery charge level $B_j$ shall not be less than $0.3$ with
probability larger than or equal to $0.95$, i.e.,
\begin{equation}\label{batt_spec1}
\square_{[1,20]} (0.3 - B_j)^{[0.95]}, \; j = 1,\ldots,N_b,
\end{equation}
\item
If the battery level $B_j$ at time $0$ is less than or equal to $0.25$, then there exists a time in at most 5 steps at which $B_j$ equals or exceeds $0.4$ with probability larger than or equal to $0.95$, i.e., for all $j = 1,\ldots, N_b$:
\begin{equation}\label{batt_spec2}
(B_j - 0.25 \leq 0) \rightarrow \top \ \mathcal{U}_{[0,5]} (0.4 - B_j)^{[0.95]},
\end{equation}
\item
If a generator is unhealthy, then it is disconnected from the buses.
By denoting with
$\boldsymbol{h} = (h_1,\ldots,h_{N_s})$
the binary vector indicating the health status of the generators, where $1$ stands for ``healthy," and with
$\boldsymbol{\delta}_j = [\delta_{1,j},\ldots,\delta_{N_s,j}]^T$ the vector whose component $\delta_{i,j}$ is $1$ if and only if generator $i \in \{1,\ldots,N_s\}$ is connected to bus $j$, this requirement can be translated as
\begin{equation}\label{batt_spec3}
\square_{[0,20]} (\delta_{i,j} -{h}_i \leq 0), \qquad \forall \ i \in \{1, \ldots, N_s \}.
\end{equation}
\end{itemize}
By calling $\psi$ the conjunction of all system requirement assertions, such as the ones above, the system-level contract is
\[
C_S = ( (\forall j \in \{1,\ldots,N_b\}\!\!: B_{j0} \in [0.2,1]) \wedge \textstyle\sum_{j=1}^{N_s} h_j \ge 2, \psi),
\]
stating that the specification $\psi$ must be satisfied if the initial battery
level is between $0.2$ and $1$ ($20\%$ and $100\%$ of the full level of charge) and if there are at least two healthy generators.
\textsc{SCAnS} was able to verify the consistency of $C_S$ using the result in Theorem~\ref{thm:compati_consis} and generate a
stochastic MPC scheme for the HL-LMS. We relied on the mixed integer linear under-approximation of $\psi$ into the constraint set
$\mathcal{C}_0^S(\psi)$ because of the large number of variables (more than $400$) in the optimization problems. When parsing $\psi$, deterministic constraints encoding the atomic propositions $(0.3 - B_j)^{[0.95]}$
were formulated using~\eqref{eq:linear_chance_cons_deter_suffi}.
$\mathcal{C}_0^S(\psi)$ and the control objective
formed the optimization problem solved by the HL-LMS every $10$~s to provide suggestions to the LL-LMS. We observe that constraint~\eqref{batt_spec2}, capturing more complex transient behaviors, was not present in previous formulations~\cite{shahsavari2015stochastic}, while it could be easily expressed in StSTL and automatically accounted for in our MPC scheme.
In every simulation run, GEN2 is shut down at time $34$ to test the response of the LMS.
The contactor signals indicating the connection of the 3 generators to the 2 AC buses are in Fig.~\ref{fig:HL_LL_engine}. First, we observe that the LL-LMS connects GEN3 to
bus 2 at time $34$ to immediately replace the
faulty generator GEN2, before the HL-LMS can respond to this event at time 40. Meanwhile,
because the average total power consumption of
either bus 1 or bus 2 exceeds 85~kW (the maximum power supplied by GEN3), the LL-LMS sheds the loads at time $34$ in Fig.~\ref{fig:LL_bus2}. Conversely, the HL-LMS
does not detect this shutdown until time $40$. Once a new optimal configuration is
computed, as shown in Fig.~\ref{fig:HL_LL_engine}, the HL-LMS realizes that GEN2 must indeed be disconnected from bus 2 (requirement~\eqref{batt_spec3}) and proposes a configuration that connects GEN1 and GEN3 alternatively to the two buses. This prevents load shedding (all loads are now powered again) and better resource utilization, since the battery can now be effectively charged
when GEN1 is connected and then used to provide extra power when
GEN3 is connected. While the switching activity increases in this new configuration, the switching frequency is always compatible with the requirements and minimized by the MPC scheme.
\begin{figure}[t]
\centering
\includegraphics[width=0.3\textwidth]{HL_LL_eng}
\caption{Contactor signals for the connection between generators (engines) and buses. The connection is present when the signal evaluates to $1$.}
\label{fig:HL_LL_engine}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.3\textwidth]{LL_bus2}
\caption{Contactor signals for the connection between sheddable loads and DC Bus 2. The connection is present when the signal evaluates to $1$.}
\label{fig:LL_bus2}
\end{figure}
The trajectories of the battery charge level from 50 simulation runs
are shown in Fig.~\ref{fig:battery}. We see that the constraint~\eqref{batt_spec1} is
effective since the battery level mostly remains above $0.3$ after time $0$. Moreover,
most of the battery profiles starting from the initial condition
$B_{1,0} = B_{2,0} = 0.225$ climbs above $0.4$ before time $5$, which is consistent
with requirement \eqref{batt_spec2}. Finally, the rate of satisfaction of the constraint
$B_j \ge 0.3$, as estimated using 500 simulation runs, is larger than 0.95 at all times, which is consistent with requirement~\eqref{batt_spec1}.
One optimization run takes 0.05~s on average
and 0.24~s in the worst case.
\begin{figure}[t]
\centering
\includegraphics[width=0.3\textwidth]{bat}
\caption{Battery charge level over time for 50 simulation runs.}
\label{fig:battery}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
We developed an assume-guarantee contract framework and a supporting tool for the automated verification of certain classes of stochastic linear systems and the generation of stochastic Model Predictive Control (MPC) schemes. Our approach leverages Stochastic
Signal Temporal Logic to specify system behaviors and contracts, and algorithms that can efficiently encode and solve contract compatibility, consistency, and refinement checking problems using conservative approximations of probabilistic constraints. We illustrated the effectiveness of our approach on a few examples, including the control of aircraft electrical power distribution
systems. Our tool can automatically design stochastic MPC schemes for a richer set of specifications than in previous work. Future work includes the investigation of mechanisms to improve the accuracy and scalability of our framework.
\bibliographystyle{IEEEtran}
{\small
|
1,314,259,995,768 | arxiv | \section{Introduction}
\begin{figure}[t!]
\footnotesize
\begin{center}
\begin{tabular}{@{}cc@{}}
\includegraphics[width = 0.4\linewidth]{{figures/single-v2.png}} &
\includegraphics[width = 0.4\linewidth]{{figures/finetune-v2.png}} \\
{\footnotesize (a) Single-Dataset Training} &
{\footnotesize (b) Finetuning}
\end{tabular}
\begin{tabular}{@{}cc@{}}
\includegraphics[width = 0.4\linewidth]{{figures/remap-v2.png}} &
\includegraphics[width = 0.4\linewidth]{{figures/cdcl-v2.png}}\\
{\footnotesize (c) Label Remapping} &
{\footnotesize (d) CDCL (Ours)}
\end{tabular}
\end{center}
\caption{Illustration of prior baselines and the proposed algorithm in the cross-dataset setting. (a) Single-dataset training: each network is trained on each dataset separately. (b) Finetuning: each network is pretrained on source and finetuned on target with shared structure. (c) Label remapping: each network is trained on the combination of the target dataset and a subset of the source dataset with common classes. (d) CDCL (Ours): a unified network is trained on all the datasets with data-aware block (DAB) and dataset alternation training (DAT). Notation: D: Dataset, O: Output, C: Classifier layer, DS: Dataset-Specific layer.}
\label{figure: idea}
\end{figure}
Semantic segmentation is a fundamental task in scene understanding, which aims to partition the image into semantically meaningful parts. It has been widely applied to practical vision-based systems, such as self-driving cars and robotics, medical aided diagnosis and augmented reality.
Owing to the development of deep learning, semantic segmentation has achieved remarkable improvements in recent years. The success is mainly attributed to discriminative features learned from network backbone, tailored encoder-decoder framework and dense annotations for supervised learning. Various deep networks and their variants (e.g., AlexNet ((Krizhevsky et al. 2017), ResNet \cite{he2016deep}) provide discriminative features to recognize semantics in images. Different encoder-decoder frameworks are developed for semantic segmentation. Especially, the mIoU performance on the Cityscapes benchmark \cite{cordts2016cityscapes} is improved from 65.3\% by FCN (Long et al. 2015) to 82.7\% by DeepLabv3+ \cite{chen2018encoder}. Moreover, multiple semantic segmentation datasets blossom (e.g., Cityscapes \cite{cordts2016cityscapes}, BDD100K \cite{yu2018bdd100k}), which provide rich labeled data for network training. However, how to build a unified system by simultaneously training from several datasets has not been well investigated by the community. In this work, we consider cross-dataset semantic segmentation and expect to learn a general model from multiple datasets. Such cross-dataset setting is useful for practical applications. For example, in the autonomous driving scenarios, each dataset can be collected from different scenes, weather and illumination. Learning a unified model may exploit information from several datasets and help improve each other's performance, particularly for those with limited data.
Figure \ref{figure: idea} illustrates prior single-dataset and cross-dataset baseline methods. Most of existing semantic segmentation methods follow the single-dataset setting, i.e., the model is trained and tested on a single dataset only (Figure \ref{figure: idea} (a)). When directly apply the model to another dataset, it tends to yield inferior performance because of the distribution gap between the two datasets. A straightforward approach to improve the performance on the target dataset is finetuning the pretrained model with shared structure except the final prediction layer (Figure \ref{figure: idea} (b)). However, finetuning requires extra training epochs on target and may lead to degraded performance on source. It is not a unified model for maintaining high performance for both source and target datasets. Another cross-dataset solution is to exploit image samples with common classes from other datasets so as to enrich the training set (Figure \ref{figure: idea} (c)). Although training with more data sometimes can boost the performance, it counts on the assumption of class overlap among different datasets and requires carefully remapping the labels. Moreover, it often encounters inconsistent taxonomies and annotation criteria, leading to limited application.
To alleviate these problems, we propose a Cross-Dataset Collaborative Learning (CDCL) algorithm that is capable of learning from multiple datasets (Figure \ref{figure: idea} (d)). First, we present a sightful investigation on how a model differs by training on different datasets separately. Our key observations lie in: (1) The convolution (Conv) filters can be shared for all datasets to maintain network efficiency without accuracy loss. (2) The batch normalization (BN) layers are not appropriate to share across different datasets due to the bias of statistics. Second, motivated by the observations, we present a unified network to preserve the commonality and particularity of different datasets. Specifically, we introduce a Dataset-Aware Block (DAB) as the fundamental computing unit of the network, which helps capture homogeneous convolutional representations and heterogeneous statistics across different datasets. The proposed block is composed of a dataset-invariant convolution layer, multiple dataset-specific batch normalization layers, and an activation layer. Moreover, we propose a Dataset Alternation Training (DAT) mechanism to facilitate the collaborative optimization procedure. Our method is simple, easy-to-implement, and compatible with mainstream semantic segmentation frameworks. As the Conv layers are shared across different datasets, our network does not introduce extra computation cost compared to the single-dataset baseline\footnote{The number of BN parameters is far less than that of Conv weights in our networks.}. We conduct extensive experiments on diverse semantic segmentation datasets for autonomous driving. Experimental results demonstrate our method consistently outperforms prior single-dataset and cross-dataset training methods. Particularly, with the same architecture of PSPNet (ResNet-18), our method surpasses the single-dataset baseline by 5.65\%, 6.57\%, and 5.79\% mIoU on the validation sets of Cityscapes, BDD100K, CamVid, respectively.
The main contributions of this paper are summarized as follows. (1) We analyze the limitations of existing single-dataset and cross-dataset semantic segmentation methods, and provide insights on how a model differs by training on different datasets separately. (2) We propose a simple, flexible, and general cross-dataset collaborative learning algorithm, which can alleviate the distribution shift problem in the training phase and introduce no extra computation cost in the inference phase for each dataset. (3) We demonstrate the effectiveness of the proposed approach on diverse semantic segmentation datasets for autonomous driving. Our method consistently outperforms prior single-dataset and cross-dataset training baselines with the same computation budget. We also achieve improved performance for point cloud 3D semantic segmentation, which further validates the superiority and generality of our method.
\section{Related Work}
\subsection{Semantic Segmentation}
Semantic segmentation is a dense image prediction task, which plays a key role in high-level scene understanding. FCN \cite{long2015fully} and its follow-ups \cite{zhao2017pyramid,chen2014semantic,chen2017deeplab,chen2018encoder} have achieved impressive performance for semantic segmentation. Recent transformer-based methods \cite{zheng2021rethinking,xie2021segformer,liu2021swin,strudel2021segmenter} have also shown promising results on semantic segmentation benchmarks with self-attention architectures. In addition, crucial strategies have been developed to further improve the performance, including atrous convolution \cite{chen2017rethinking}, pyramid pooling module \cite{zhao2017pyramid}, attention mechanism (Hu et al. 2018; Fu et al. 2019b) and context encoding \cite{zhang2018context}. In parallel, light-weight networks \cite{mehta2018espnet,mehta2019espnetv2,paszke2016enet} arouse great research interest owing to their high speed and wide applications on resource-constrained devices. However, these semantic segmentation methods typically follow the setting of single-dataset training and can not maintain high accuracy on other datasets without finetuning.
\subsection{Cross-Dataset Training}
Recurrent Assistance (Perrett et al. 2017) first proposed the cross-dataset training mechanism, which is used for frame-based action recognition during the pretraining stage. Recent work explores cross-dataset training for object detection \cite{yao2020cross,wang2019towards}. The work of \cite{yao2020cross} proposed to generate a hybrid dataset by simple label concatenation or label mapping since the number and identity of classes are different for each dataset. Inspired by Squeeze-and-Excitation (Hu et al. 2018), the work of \cite{wang2019towards} introduced a domain attention module to activate a single network for universal object detection tasks. Our work is a vanguard in cross-dataset semantic segmentation, which aims to learn a unified model by simultaneously training from multiple datasets.
\subsection{Transfer Learning}
Both of transfer learning and our cross-dataset training methods address the distribution shift problem between different datasets. The main differences are summarized below. (1) Domain adaptation (DA) methods aim to adapt a model from the source domain with adequate labeled data to the known target domain (i.e., training images are available) with little or no labeled data (Argyriou et al. 2016; Li et al. 2017; Gkioxari et al. 2014). (2) Domain generalization (DG) emphasizes generalization on the unknown domain (i.e., training images are not available). Different from DA and DG tasks, we focus on learning a unified model from multiple datasets simultaneously
and improving the performance of all known datasets.
\begin{figure}[t]
\footnotesize
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/pipeline-new-v3.png}
\end{center}
\caption{An overview of the proposed cross-dataset collaborative learning (CDCL) method. See Section \ref{sec:cdcl} for details.}
\label{fig:pipeline}
\end{figure}
\subsection{Multi-Task Learning}
Another related research topic is multi-task learning (MTL) \cite{argyriou2006multi,li2017integrated,gkioxari2014r}. The work of \cite{argyriou2006multi} jointly trained classification, detection, and segmentation tasks on a single dataset. It requires annotations for all tasks on a single dataset. The work of \cite{li2017integrated} trained an integrated face analysis model (facial landmark, facial emotion, and face parsing) by using multiple datasets, where each dataset is only labeled for one task. MTL methods can boost the performance by explicitly modeling the interaction of different tasks. In this paper, we aim to jointly train multiple datasets for a similar task. Furthermore, we analyze the commonality and particularity of training on different datasets and build an alternation training method for effective collaborative optimization.
\begin{comment}
\begin{figure*}[t!]
\begin{center}
\begin{tabular}{@{}cc@{}}
\includegraphics[width = 0.35\linewidth]{{figures/demo_imgs_v2.png}} & \hspace{1mm}
\includegraphics[width = 0.50\linewidth]{{figures/conv_bn_ana_v6.png}} \\
{\footnotesize (a) Visualization for samples of different datasets} &
{\footnotesize (b) The statistic distributions of BN and Conv for different datasets.}
\end{tabular}
\end{center}
\caption{Comparisons of different datasets. (a) Visualization for samples of different datasets. (b) The distributions of
BN and Conv of single-dataset semantic segmentation networks on four datasets. First row: running mean of BN. Second row: running variance of BN. Third row: L2 norm of Conv weights. Here, we take two low-level layers (\emph{bn2} and \emph{layer2}) and one high-level layer (\emph{layer4}) of PSPNet (ResNet-18) network as examples to show the results.}
\label{fig:conv_bn_ana}
\end{figure*}
\end{comment}
\begin{figure}[t!]
\footnotesize
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/distribution-v2.png}
\end{center}
\caption{Parameter distributions of Conv and BN layers on three datasets (red: Cityscapes, green: BDD100K, blue: CamVid). The networks are trained with PSPNet (ResNet-18) on each dataset separately. We sample three layers (left: \texttt{conv2} \& \texttt{bn2}, middle: \texttt{layer2}, right: \texttt{layer4}) and provide more curves of layers in the supplementary material.}
\label{fig:conv_bn_ana}
\end{figure}
\section{Approach}
In this section, we first analyze the most popular single-dataset and cross-dataset solutions for semantic segmentation. We then explain the motivation of the proposed cross-dataset collaborative learning algorithm and introduce each algorithmic component in detail.
\subsection{Analysis of Semantic Segmentation Methods}
\subsubsection{Single-Dataset Baseline.}
The encoder-decoder is used as the most popular architecture for semantic segmentation. It employs deep convolutional neural networks as the encoder to extract hierarchical features, and exploits a simple sub-network as the decoder to refine the segmentation results. For segmentation on a single dataset, recent work (e.g., PSPNet \cite{zhao2017pyramid} and DeepLab \cite{chen2018encoder}) has extended this basic architecture to improve the performance. For segmentation on multiple datasets, suppose we have $N$ datasets in total: $\{D_i\}_{i=1}^N$, the simplest solution is to train and evaluate a network for each dataset separately. This single-dataset baseline is inefficient since it needs to save $N$ sets of network parameters that are not shared. Furthermore, each network is only responsible for its corresponding dataset and the distribution shift between different datasets is not considered.
\subsubsection{Cross-Dataset Baseline.} Finetuning is a straightforward solution to handle the distribution shift problem and improve the accuracy on the target dataset. Specifically, it first pretrains the network on $D_i$ and then finetunes it on $D_j$. The finetuning baseline often needs to adjust hyper-parameters and requires extra training epochs to reach a satisfactory accuracy on the target dataset. In addition, it may lead to degraded performance on the source dataset as it does not incorporate source samples for joint optimization. Another solution is to exploit image samples with common classes from other datasets to enrich the training set of target dataset. For example, we denote $D_i^*$ as a subset of $D_i$ by remapping the common labels from $D_i$ to $D_j$. Training on combination of $D_i^*$ and $D_j$ may lift the performance owing to the usage of more data. However, such label remapping baseline has three limitations. (1) It counts on the assumption of class overlap among different datasets and not applicable to the case where datasets have disjoint classes. (2) It requires carefully remapping the labels when encountering inconsistent taxonomies and annotation criteria. For example, ``drivable'' and ``non-drivable'' areas are separately labeled in $D_j$ but only ``road'' is labeled in $D_i$, which incurs class conflict. (3) Naive combination of training samples from different datasets is likely to yield poor performance due to the discrepancy of image compositions, illuminations, scenes, etc.
\subsection{Cross-Dataset Collaborative Learning}
\label{sec:cdcl}
\subsubsection{Rethinking Convolution and Batch Normalization.}
To explore how to effectively alleviate the above dilemmas, we analyze the distribution shift problem across different semantic segmentation datasets. Intuitively, images from different datasets may vary greatly. Even though images come from a similar autonomous driving scenario, there exists inevitable appearance discrepancies as they are usually collected under different conditions (e.g., scenes and illuminations) in practice. That makes challenging to directly combine these images for joint training. Furthermore, we analyze the parameter distributions of Conv and BN layers when training on different datasets separately. Figure \ref{fig:conv_bn_ana} presents these parameter distributions of sampled low-/middle-/high-level layer on three different autonomous driving datasets (i.e., Cityscapes, BDD100K and CamVid). We summarize two key observations. (1) Conv weights hold similar distributions and most values are concentrated near zero. From the perspective of perception, each neuron in Conv layers only attends a local region. Such local information is less different than global appearance, which implies that Conv weights can be shared during the optimization of different datasets. (2) Both running mean and running variance of BN hold different distributions for different datasets, even for those sharing the same label space (e.g., Cityscapes and BDD100K). This is because the statistical moments (mean and variance) used in BN are relevant to each specific dataset. In detail, during training, the BN layer first calculates the mean and variance of the activations through the exponential moving average method (Ioffe et al. 2015), and then normalizes activations and applies a linear transformation to derive the layer's output. During inference, BN uses the estimated mean and variance for normalization and uses the learned linear transformation parameters to recover the representation ability of activations. We assume that the mean and variance used in BN compute the global statistics of a specific dataset, and thus both the normalization and linear transformation operations are sensitive to the dataset itself. These observations motivate us to use shared Conv and independent BN parameters for joint training on different datasets.
\subsubsection{Dataset-Aware Block.}
Inspired by the above analysis, we propose a simple and flexible framework for training on different semantic segmentation datasets simultaneously. As shown in Figure \ref{fig:pipeline}, we design a dataset-aware block (DAB) as the fundamental computing unit of the encoder-decoder architecture. Specifically, DAB consists of a dataset-invariant Conv layer, multiple dataset-specific BN layers, and an activation layer.
The weights of dataset-invariant Conv layers are shared, while the parameters of each dataset-specific BN layer are not shared across different datasets.
In each block, we learn $N$ BN layers for $N$ different datasets and each BN layer is responsible for a specific dataset. A switch is automatically to determine which BN should be activated based on the data source.
Therefore, our dataset-specific BN can be formulated by:
\begin{equation}
\mbox{BN}\{D_i\}(X_i;\gamma_i,\beta_i) = \gamma_i {\bar{X_i}} + \beta_i \\
\label{eq:dsbn}
\end{equation}
where,
\begin{equation}
\bar{X_i} = \frac{X_i - \mu_i}{\sqrt{\sigma_i^{2} + \epsilon}} \\
\mu_i = \frac{1}{B}\sum_{j=1}^{B} X_i^{j},\ \sigma_i^{2} = \frac{1}{B}\sum_{j=1}^{B} (X_i^{j} - \mu_i)^2
\end{equation}
Here, $\mu_i$ and $\sigma_i^{2}$ denote the running mean and running variance, respectively. $\gamma_i$ and $\beta_i$ denote the linear transformation parameters. $B$ denotes the batch size and $\epsilon$ is a small constant added for numerical stability. The proposed DAB brings two main advantages: (1) Dataset-invariant Conv layers preserve the commonality and derive homogeneous representations for different datasets, which help maintain the efficiency of network without introducing extra FLOPs and parameters for segmentation on different datasets. (2) Dataset-specific BN layers preserve the particularity and capture heterogeneous statistics across different datasets, which help alleviate the distribution shift problem during the joint optimization process.
For final predictions on different datasets, we append several dataset-specific classifiers to the output of the encoder-decoder network. Each classifier is only responsible for its corresponding dataset. Unlike the label remapping method, we do not rely on the assumption of class overlap and can exploit the out-of-the-box labels of each dataset for training.
\subsubsection{Dataset Alternation Training.}
To facilitate the collaborative optimization procedure, we introduce a dataset alternation training (DAT) mechanism to train the network. As shown in Figure \ref{fig:pipeline}, a complete iteration of DAT includes $N$ forward passes and 1 backward pass. In each forward pass, we construct the batch with samples from a single specific dataset and compute the loss separately. After all the datasets are counted, we accumulate the loss of each dataset and backpropagate the entire gradient through each dataset flow. We repeat such training procedures till network converges. To better understand the effect of DAT, we provide analysis in the following aspects. (1) Compared to backpropagating the gradients immediately after computing the loss of each dataset, DAT can reduce the training instability caused by the discrepancy of different feature distributions between two consecutive iterations. (2) Alternating between different datasets may help sufficiently leverage information from other datasets to lift the performance.
We follow common segmentation methods to use the conventional multi-class cross-entropy loss for training each dataset. Our final objective function for cross-dataset training can be formulated as below and can be minimized end-to-end.
\begin{equation}
L = -\sum_{i=1}^{N}\sum_{j=1}^{M}w^{i}y^{i}_j{\emph{l}og}(p^{i}_j) \\
\label{eq:loss}
\end{equation}
where, $N$ denotes the number of datasets, $M$ denotes the number of image pixels, $p^{i}_j$ and $y^{i}_j$ refer the predicted probability and corresponding label for the $j$-th pixel on the $i$-th dataset, respectively. $w^{i}$ denotes the loss weight and we set $w^{i}=1$ to make these loss values comparable.
\section{Experiments}
\subsection{Experimental Setting}
\subsubsection{Datasets.} We apply our CDCL method on three semantic segmentation datasets for autonomous driving: Cityscapes, BDD100K, CamVid. Dataset details are provided in the supplementary material.
\subsubsection{Implementation Details.} We use the architecture of PSPNet with the pretrained ResNet-18 / ResNet-101 on ImageNet as our baseline.
The networks are trained using stochastic gradient descent (SGD) with momentum of 0.9, weight decay of 0.0001, and batch size of 8. The initial learning rate is set to 0.01 and multiplied by $(1 - \frac{iter}{maxiter})^{0.9}$ with a polynomial decaying policy. Unless specified otherwise, we randomly crop the images into $512\times512$ for training, and use random scaling (0.5 $\sim$ 2.1) and random flipping for data augmentation. We use the standard metric of mean IoU (mIoU) to evaluate the segmentation accuracy for each dataset. More implementation details are provided in the supplementary material.
\begin{table}[t!]
\small
\centering
\begin{tabular}{l|c|c|c}
\toprule
Method & \multicolumn{2}{|c|}{Cityscapes (\%)} & BDD100K(\%) \\
\cmidrule{2-3}
& Val. & Test & Val.\\
\midrule
Single-dataset & 67.52 & 67.75 & 53.88 \\
Finetuning & 67.79 & 66.52 & 58.30 \\
Label remapping & 66.23 & 66.39 & 58.74 \\
CDCL (Ours) & \textbf{72.63} & \textbf{71.55} & \textbf{60.47} \\
\bottomrule
\end{tabular}
\centerline{\small{(a) Cityscapes + BDD100K}}
\begin{tabular}{l|c|c|c|c}
\toprule
Method & \multicolumn{2}{|c}{Cityscapes (\%)}& \multicolumn{2}{|c}{CamVid (\%)} \\
\cmidrule{2-5}
& Val. & Test & Val. & Test \\
\midrule
Single-dataset & 67.52 & 67.75 & 73.05 &70.41 \\
Finetuning & 67.35 & 67.87 & 74.83 & 71.16\\
Label remapping & 67.13 & 68.22 & 78.03 & 76.86 \\
CDCL (Ours) & \textbf{69.77} & \textbf{68.56} & \textbf{78.45} & \textbf{77.34}\\
\bottomrule
\end{tabular}
\centerline{\small{(b) Cityscapes + CamVid}}
\caption{Performance comparisons using the same ResNet-18 backbone in the two-dataset setting.}
\label{tab:cs_bdd_cam}
\end{table}
\begin{table}[t!]
\small
\begin{center}
\begin{tabular}{l|c|c|c}
\toprule %
Method & Cityscapes (\%) & BDD100K (\%) & CamVid (\%) \\
\midrule
Single-dataset & 67.52 & 53.88 & 73.05\\
CDCL (Ours) & 73.17 (\textbf{+5.65}) & 60.45 (\textbf{+6.57})& 78.84 (\textbf{+5.79})\\
\bottomrule %
\end{tabular}
\end{center}
\caption{Performance comparisons using the same ResNet-18 backbone in the three-dataset setting.}
\label{tab:cs_bdd_cam}
\end{table}
\subsection{Results in Cross-Dataset Settings}
We compare the proposed CDCL algorithm with three baseline methods on multiple datasets $\{D_i\}_{i=1}^N$:
\begin{itemize}
\item Single-dataset: Each network $net_j$ is trained on the target dataset $D_j$ separately.
\item Finetuning: Each network $net_j$ is pretrained on a source dataset $D_i$ and then finetuned on the target dataset $D_j$.\footnote{Before training on the segmentation datasets, we use the ImageNet pretrained model as initialization in all the experiments.}
\item Label remapping: Each network $net_j$ is trained on the combination of the target dataset $D_j$ and a subset of $D_i$ by remapping the common labels from $D_i$ to $D_j$.
\end{itemize}
For fair comparisons, our CDCL method uses the same network structure, basic loss function and training epoch as the baselines.
\subsubsection{Cityscapes + BDD100K.} In the cross-dataset setting of Cityscapes + BDD100K, the two datasets share the same semantic categories. Table \ref{tab:cs_bdd_cam} (a) shows that our CDCL method significantly outperforms all the baseline methods on both benchmarks, e.g., +5.11\% and +6.59\% mIoU over the single-dataset baseline on the validation sets of Cityscapes and BDD100K, respectively. All the three baselines produce multiple networks, each for a dataset. Differently, our method learns a unified model for all datasets without introducing extra FLOPs, and performs more efficiently without repeating the training process multiple times. We also note that label remapping even performs slightly worse than the single-dataset baseline, despite the usage of more data. We analyze that a simple combination of two datasets may cause disturbation. In contrast, our method effectively eases dataset bias and brings consistent accuracy gains on both datasets. The results demonstrate that our method can sufficiently utilize the complementarity between different datasets for performance improvement on each dataset in the case of the same label space.
\subsubsection{Cityscapes + CamVid.} In this cross-dataset setting, the two datasets have different label spaces: Cityscapes has 19 classes and CamVid has 11 classes. Table \ref{tab:cs_bdd_cam} (b) shows that our method can achieve consistent performance improvement compared to all the baseline methods, e.g., +2.25\% and +5.40\% mIoU over the single-dataset baseline on the validation sets of Cityscapes and CamVid, respectively. We note that label remapping also obtains promising results in this setting. Compared to this baseline, our method still can obtain better performance without the careful label preprocessing step. The results demonstrate the effectiveness of our CDCL method in the case of different label spaces.
\subsubsection{Cityscapes + BDD100K + CamVid.} We also evaluate our method in the three-dataset setting where Cityscapes, BDD100K and CamVid are used. Table \ref{tab:cs_bdd_cam} shows that CDCL enables over 5-point gains on all the validation sets, which validates the effectiveness of our method in the more challenging multi-dataset case.
\subsection{Ablation Study}
\subsubsection{Effect of DAB.} We use the cross-dataset setting of Cityscapes + BDD100K with the ResNet-18 backbone to examine the effect of DAB. Table \ref{tab:cs_bdd_for_dab_dat} compare four different configurations of Conv and BN layers. Using unshared BN and unshared Conv is equivalent to the single-dataset baseline. Using shared BN yields inferior performance on both benchmarks. This is because BN represents the statistics of each specific dataset, which is in line with our observation in Figure \ref{fig:conv_bn_ana}. Our DAB uses shared Conv and unshared BN and achieves the best result in these configurations. The results validate that (1) dataset-invariant Conv can obtain similar performance and reduce parameters compared to dataset-specific Conv, (2) dataset-specific BN is crucial for good performance in joint training of different datasets.
\subsubsection{Effect of DAT.} Table \ref{tab:cs_bdd_for_dab_dat} also compare the results with or without our DAT strategy in last two rows. Without DAT, the gradients are backpropagated immediately after computing the loss of each batch constructed from a single dataset. The results demonstrate that DAT can facilitate collaborative optimization and achieve improved performance on both benchmarks, e.g., +4.08\% on Cityscapes.
\subsubsection{Effect of Backbone.} We also conduct experiments using different network backbones in the cross-dataset settings of Cityscapes + BDD100K and Cityscapes + CamVid. Table \ref{tab:r101_cs_bdd_cam} shows that our CDCL method still can achieve notable improvement compared to the single-dataset baseline using a larger backbone of ResNet-101, e.g., +6.37\% mIoU on BDD100K.
\subsection{Comparisons to the State-of-the-Art Methods}
Table \ref{tab:r101_cs_sota} compares the state-of-the-art methods in terms of computation complexity (FLOPs) and segmentation performance (mIoU) on the Cityscapes test set. Through collaborative training with BDD100K, the proposed CDCL method improves PSPNet with both light-weight backbone (ResNet-18) and large backbone (ResNet-101), without increasing FLOPs in the inference phase. In addition to PSPNet, we also apply CDCL for another segmentation baseline of HANet, and also achieve improved performance with the same FLOPs. The results highlight that our method is effective, flexible and extensible for different semantic segmentation baselines and can improve the performance of state-of-the-art approaches without extra computation budgets.
\begin{table}[t!]
\small
\begin{center}
\begin{tabular}{l|c|c|c|c}
\toprule %
Conv & BN & DAT & Cityscapes & BDD100K \\
\midrule
Not Shared & Not Shared & \xmark & 67.75\% & 53.88\% \\
Not Shared & Shared & \xmark & 62.05\% & 53.50\% \\
Shared & Shared & \xmark & 62.10\% & 53.34\% \\
Shared & Not Shared & \xmark & 68.55\% & 58.93\% \\
Shared & Not Shared & \cmark & \textbf{72.63\%} & \textbf{60.47\%} \\
\bottomrule %
\end{tabular}
\end{center}
\caption{Ablation studies on DAB and DAT with ResNet-18 on the validation sets of Cityscapes and BDD100K.}
\label{tab:cs_bdd_for_dab_dat}
\end{table}
\begin{table}[!t]
\small
\centering
\begin{tabular}{l|c|c|c}
\toprule
Method & \multicolumn{2}{|c|}{Cityscapes (\%)} & BDD100K (\%) \\
\cmidrule{2-3}
& Val. & Test & Val.\\
\midrule
Single-dataset & 73.51 & 74.45 & 57.47 \\
CDCL (Ours) & \textbf{75.83} & \textbf{75.95} & \textbf{63.84} \\
\bottomrule
\end{tabular}
\centerline{\small{(a) Cityscapes + BDD100K}}
\begin{tabular}{l|c|c|c|c}
\toprule
Method & \multicolumn{2}{|c}{Cityscapes (\%)}& \multicolumn{2}{|c}{CamVid (\%)} \\
\cmidrule{2-5}
& Val. & Test & Val. & Test \\
\midrule
Single-dataset & 73.51 &74.45 & 75.86 & 74.72 \\
CDCL (Ours) & \textbf{75.00} & \textbf{74.77} & \textbf{81.15} & \textbf{79.33} \\
\bottomrule
\end{tabular}
\centerline{\small{(b) Cityscapes + CamVid}}
\caption{Performance comparisons using the same ResNet-101 backbone in the two-dataset setting.}
\label{tab:r101_cs_bdd_cam}
\end{table}
\begin{table}[t!]
\small
\begin{center}
\scalebox{0.95}{
\begin{tabular}{l|c|c}
\toprule %
Method & GFLOPs & mIoU (\%) \\
\midrule
\multicolumn{3}{c}{Current state-of-the-art results} \\
\midrule
SegNet (Badrinarayanan et al. 2017) & 286.0 & 56.10 \\
ENet \cite{paszke2016enet} & 7.6 & 58.30 \\
ESPNet \cite{mehta2018espnet} & 8.9 & 60.30 \\
ESPNetv2 \cite{mehta2019espnetv2} & 5.4 & 65.10 \\
FCN-8s (Long et al. 2015) & 1335.6 & 65.30 \\
ERFNet \cite{romera2017erfnet} & 25.6 & 68.00 \\
RefineNet \cite{lin2017refinenet} & 2102.8 & 73.60 \\
Axial-DeepLab-L \cite{wang2020axial} & 1374.8 & 79.50 \\
HRNet \cite{wang2020deep} & 5843.8 & 81.10 \\
BFP \cite{ding2019boundary} & 2157.3 & 81.40 \\
DANet \cite{fu2019dual} & 39647.0 & 81.50 \\
OCRNet (Yuan et al. 2019) & 5843.8 & 81.60 \\
\midrule
\multicolumn{3}{c}{Results w/o and w/ our scheme} \\
\midrule
PSPNet (ResNet-18) \dag & 512.8 & 67.75 \\
PSPNet (ResNet-18) (Ours) & 512.8 & 71.55 \\
PSPNet (ResNet-18) (Ours)\ddag & 1730.7 & 72.52 \\
\midrule
PSPNet (ResNet-101)\dag & 2299.8 & 77.79 \\
PSPNet (ResNet-101) (Ours) & 2299.8 & 78.74 \\
PSPNet (ResNet-101) (Ours)\ddag & 7762.0 & 79.73 \\
\midrule
HANet (ResNet-101) (Choi et al. 2020) & 3160.0 & 80.90 \\
HANet (ResNet-101) (Ours) & 3160.0 & 81.62 \\
\bottomrule
\end{tabular}}
\end{center}
\caption{Comparisons with the state-of-the-art methods on the Cityscapes test set. Both PSPNet and HANet with the ResNet-101 backbone are trained by $769\times769$ input. $\dag$ refers our reproduced results. $\ddag$ refers multi-scale testing.}
\label{tab:r101_cs_sota}
\end{table}
\subsection{Comparisons to Domain Adaption Methods}
The dataset-specific BN in our DAB is similar to existing unsupervised domain adaptation methods~\cite{chang2019domain}. Both of \cite{chang2019domain} and our method tackle the domain shift problem. The main difference lies in the optimization goals and strategies. \cite{chang2019domain} aims to improve the performance on the target domain without annotations by leveraging information from the source domain. It uses a two-stage training process, i.e., first training the network on source and then finetuning the network on target with newly initialized BN layers. Differently, our goal is to improve the performance of all datasets by collaborative training. Table \ref{tab:da_cs_bdd} compares CDCL with this two-stage baseline in our supervised cross-dataset setting of Cityscapes + BDD100K, where both methods use separate BN layers for these two datasets. With freezing Conv when training on the target domain, DA can maintain the performance of the source domain and achieves better results than directly testing (37.02\% vs. 40.28\%), but performs worse than training on target (53.88\% vs. 40.28\%). With updating Conv, the performance on target can be further improved (40.28\% $\to$ 51.16\%), but fails on source (only 1.98\% on Cityscapes). This is not surprising as Conv and BN layers for Cityscapes are not jointly trained in the second stage. Our method can obtain superior performance compared to the two-stage DA baseline on both source and target domains, which verifies the non-trivial design of DAB and DAT for collaborative optimization.
\begin{table}[t!]
\small
\begin{center}
\begin{tabular}{l|c|c}
\toprule
Method & Cityscapes (\%) & BDD100K (\%) \\
\midrule
Train on C only & 67.52 & 37.02 \\
Train on B only & 45.95 & 53.88 \\
C $\rightarrow$ B, freeze Conv & 67.52 & 40.28\\
C $\rightarrow$ B, update Conv & 1.98 & 51.16\\
CDCL (Ours) & \textbf{72.63} & \textbf{60.47}\\
\bottomrule
\end{tabular}
\end{center}
\caption{Performance comparisons with domain adaptation training strategy on the validation sets of Cityscapes and BDD100K. C: Cityscapes. B: BDD100K.}
\label{tab:da_cs_bdd}
\end{table}
\begin{table}[t]
\small
\begin{center}
\begin{tabular}{l|c}
\toprule
Method & KITTI (\%) \\
\midrule\midrule
Single-dataset (Train on C only) & 51.40 \\
Finetuning (B $\to$ C) & 47.52 \\
Label remapping (C + B) & 53.79\\
CDCL (C + B) & \bf54.05\\
\midrule\midrule
Single-dataset (Train on C only) & 54.65 \\
Finetuning (B $\to$ C) & 54.16 \\
Label remapping (C + B) & 55.89\\
CDCL (C + B) & \textbf{59.05}\\
\bottomrule
\end{tabular}
\end{center}
\caption{Zero-shot domain generalization on the KITTI dataset. First group: Directly test using pretrained BN on Cityscapes. Second group: Test with precise BN \cite{wu2021rethinking} using updated mean and variance of BN on KITTI. C: Cityscapes. B: BDD100K.}
\label{tab:dg_cs_bdd}
\end{table}
\subsection{CDCL for Different Scenes} In addition to evaluations on multiple datasets from the similar autonomous driving scenes, we also conduct experiments on different scenes with a larger data distribution gap. We use the setting of Cityscapes + Pascal Context, where Cityscapes is a driving dataset and Pascal Context \cite{mottaghi2014role}
is an everyday object dataset.
Our method achieves gains of 1.87\% and 1.27\% mIoU on the validation set of Cityscapes (67.75\% vs. 69.62\%) and Pascal Context (40.56\% vs. 41.83\%) compared to the single-dataset baseline, respectively.
Although it is more usual that different datasets from similar scenes are used to boost performance in practice, we believe that CDCL can be applied to the general case of multi-dataset joint training.
\subsection{CDCL for Zero-Shot Domain Generalization} Zero-shot domain generalization (DG) aims to generalize the model on the unknown domain where training images are not available. Table \ref{tab:dg_cs_bdd} compares different baselines with our method for this challenging task. With directly testing the trained model on the unseen KITTI dataset \cite{geiger2013vision}, CDCL achieves the best result compared to all the baselines. We also apply precise BN \cite{wu2021rethinking} which updates the mean and variance of BN through forwarding the network on the entire set of KITTI. Results show that our method still outperforms all the baselines for DG, which exhibits the potentiality of CDCL for zero-shot segmentation.
\subsection{CDCL for 3D Segmentation}
Our framework can be readily extended to point cloud 3D segmentation. Point cloud 3D segmentation is another challenging task in autonomous driving, which greatly relies on a mass of annotated data to cover diverse scenes. We use two autonomous driving datasets for evaluations: SemanticKITTI \cite{behley2019semantickitti} and nuScenes \cite{caesar2020nuscenes}.
We adopt SalsaNext as the 3D segmentation baseline with the same hyper-parameter setting in (Cortinhal et al. 2020). Our CDCL method achieves 60.0\% and 67.3\% mIoU on the validation set of SemanticKITTI and nuScenes, improving the SalsaNext baseline by 1.0\% and 0.3\%, respectively. The results validate the effectiveness of CDCL for 3D segmentation and further exhibit the generality of our method.
\section{Conclusion}
In this work, we propose a unified cross-dataset collaborative learning segmentation algorithm that is capable of learning from multiple datasets. The dataset-aware block can capture heterogeneous statistics across different datasets and maintain high performance without introducing extra FLOPs. The dataset alternation training mechanism can facilitate the collaborative optimization procedure. Our method is simple, flexible and compatible with different encoder-decoder segmentation frameworks and applicable for both 2D and 3D segmentation. We consider applying our method for Transformer-based segmentation and further enhance the role in zero-shot domain generalization in future work. We also expect that CDCL can inspire new insights for more computer vision tasks beyond segmentation.
|
1,314,259,995,769 | arxiv | \section*{Nomenclature}
\label{sec:nomenclature}
\printnomenclature
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{T}{he} interest on optimal power system expansion planning has increased worldwide. In developing countries of Latin America, Asia and Africa, with high load growth and limited financial resources, the emphasis is on the most cost-effective expansion plan \cite{ondraczek2014we,deichmann2011economics,kaygusuz2012energy}. In developed countries, load growth is usually flat. In these cases, Renewable Energy Sources (RES) are being built as part of decarbonization policies and to displace more expensive thermal plants \cite{haller2012decarbonization,oberthur2015decarbonization,capros2014european,ekins2004step}. In both cases, selecting the "best" of a group of alternatives is what characterizes the combinatorial nature of the expansion planning problem.
The main objective of the expansion planning process is to guarantee an appropriate balance between electricity supply and demand, i.e. to determine the optimal set of generating plants and transmission routes that should be constructed to meet the demand requirements along a study horizon (mid and long term), while minimizing a cost function considering: (i) investment (capital) and operation (fuel, O\&M, etc.) costs of generation plants and (ii) penalties of energy not supplied, also called deficit costs.
In general terms, this decision process involves meeting economic, reliability and environmental criteria, within the framework of national policies on energy. In addition, one of the greatest challenges is how to deal with the uncertainties inherent in the planning process, such as the load growth, the hydrological inflows and the generation availability, especially in renewable based systems. Taken all the aforementioned facts into account, the expansion planning problem is modeled as a large and complex mixed integer multistage stochastic problem that must be solved by specialized optimization algorithm.
This paper presents a description of the methodology associated with the OptGen model \cite{optgenpsr}, a commercial computational tool for energy systems expansion planning, where two "Solution Strategies" are available:
\begin{itemize}
\item The benders decomposition strategy, proposed in \cite{campodonico2003expansion}: A decomposition of the investment and operation problem, where the master is a MILP investment problem and the slave is a multistage stochastic optimization of the operational problem that is solved using the SDDP algorithm, first proposed in \cite{pereira1991multi};
\item The co-optimization strategy, which is the methodology described in this paper
\end{itemize}
The main characteristics of the model are:
\begin{itemize}
\item Study horizons from 1 year up to several decades;
\item Many different candidate projects may be contemplated in the study, such as:
\begin{itemize}
\item Production components: hydro, thermal and renewable plants (wind, solar, biomass, etc.);
\item Interconnection links and transmission circuits (lines, transformers, DC links etc.);
\item Storage devices such as Batteries, Hydro pump stations, etc.
\item Gas pipelines, production nodes, regasification stations.
\end{itemize}
\item Detailed project's financial data, such as, investment costs, payment schedules, life-time, construction time;
\item Detailed project specific data, such as, decision type (obligatory or optional), decision variable type (binary, integer or continuous), maximum and minimum en-trance dates, generating unit entrance schedule, etc.;
\item Additional constraints, such as, firm energy/capacity constraints, exclusivity, association and precedence between projects, minimum and maximum additional capacity, generation capacity targets and so on;
\item Unit commitment constraints
\item Ramping constraints
\item Co-optimization of energy and reserves
\end{itemize}
In summary, the objective of OptGen is to determine a least-cost investment schedule for the construction of new plant capacity (hydro, thermal and renewable projects), regional inter-connections (or detailed transmission circuits), gas production sources and gas pipelines. This is obtained by optimizing the trade-off between investment costs to build new projects and the expected value of operating and deficit costs.
This paper is organized as follows. In the Section \ref{sec:litreview}, a review of the current state-of-the-art policies, methodologies and models regarding systems with a high level of renewable penetration is presented. In Section \ref{sec:solmet}, we discuss the assumptions used by the methodology in order to make it computational tractable. In Section \ref{sec:uncertain}, we analyze how the uncertainties are taken into account in the proposed model. In Section \ref{sec:problemform}, we provide a detailed formulation of the proposed methodology. Finally, in Section \ref{sec:conclusion}, the final conclusions are presented.
\section{Literature review}\label{sec:litreview}
The increasing economic competitiveness of wind and solar generation sources, also called variable renewable energy sources (VRE), has widely studies in the literature. These energy resources reduces green-house gas emissions, as studied in \cite{renewableenergyeurope}. Besides that, \cite{irenageopolitical} showed that in a renewable energy economy, since renewable energy potential is available everywhere, the countries that heavily depends on fossil fuel imports will be able to use renewable energy as a manner to achieve energy independence, i.e, they will have greater energy security and more freedom to take the energy decisions that suit them, reducing its vulnerability to import fossil fuels (particularly, oil and natural gas).
However, the fast penetration of these new sources has also raised some concerns for both planners and operators that are highly studies in the literature: (i) most of these sources are non-dispatchable, i.e., their generation cannot be controlled by the system operator \cite{denholm2011grid,lund2015review,perera2017electrical}; and (ii) their energy production presents strong variability, i.e., the production can change significantly from one hour to the next \cite{halamay2011reserve,bird2013integrating,golestaneh2016very,hoeltgebaum2018generating}.
As can be seen, the VRE penetration ends up causing representative impacts on the net demand profile. In addition to the change in the profile, it is worth noting the raise of net demand ramps and their respective inclinations with the greater renewable penetration. These impacts lead to new operational challenges, which stand out:
\begin{itemize}
\item \textbf{Over-supply situations}: periods when the renewable generation is higher than the demand to be met (for example, in the middle of the night in regions with strong night winds or during the day in regions with a significant solar power capacity). \cite{su2017impact} in hydropower-dominated regions;
\item \textbf{Fast upward and downward ramps}: dispatchable plants must have the ability to fast respond to the increase and decrease of intermittent renewable generation to maintain supply reliability and system stability;
\item \textbf{Increasing thermal cycling}: possible increase in the number of startups and shutdowns of thermal plants in the system due to renewable generation intermittency;
\end{itemize}
There are several studies in the literature that adresses these challenges. For example, \cite{schaber2013managing} analyzes forms of efficiently curtail renewable generation in over-supply situations in Germany and \cite{bird2014wind} analyzes the historical operation and current practices of curtailment in the United States. Besides that, several works analyze needs for thermal flexibility due to renewable generation \cite{eser2016effect,alizadeh2016flexibility,kondziella2016flexibility}
Because of its importance, expansion planning problems are vastly discussed in the literature. There are several decomposition approaches model in the literature. In \cite{campodonico2003expansion} a Benders decomposition between investment problem and SDDP algorithm is proposed. Since this methodology is very scalable, stochastic and produces optimal solution, it has been used in several real-case studies, such as \cite{oliveira2007value,barroso2005integrated,drayton2004transmission}. The work in \cite{pina2013high} also proposes a decomposition where the master is a MILP investment problem and the slave a short-term deterministic operating model, not taking into account uncertainties.
Since those decomposition algorithm requires convexity in the operation problem, there are some constraints such as unit commitment, that requires binary variables. In order to deal with that, there are several studies proposing the co-optimization between investment and operation, so that it could be solved with a MILP. For example, \cite{pineda2018chronological,koltsaklis2015multi,rashidaee2018linear,zhang2018mixed} proposes the co-optimization with some assumptions in order to make the MILP computational tractable. Since it may be hard to solve a huge MILP, most of the co-optimization methodology proposes to aggregate the days of the year into representative days, reducing computational time. \cite{liu2018hierarchical} showed that the clustered model (using representative days) leads to expansion planning results very similar to the unclustered model (considering all of the days of the year).
\section{Solution methodology}\label{sec:solmet}
Similar to the works in \cite{pineda2018chronological,koltsaklis2015multi,rashidaee2018linear,zhang2018mixed}, the proposed model uses the co-optimization between the investment and operation. Also, some assumptions to cluster the days of the years into representative days are used in order to reduce computational effort. Besides that, a rolling horizon scheme is implemented, in order to split the horizon into windows of a year, also in order to reduce computational effort. The next section introduces the concept of the rolling horizon, seasons and typical days.
\subsection{Horizon Decomposition Heuristic}
Since the planning horizons are long, in order to solve the expansion problem when applying co-optimization of investment and operation, the horizon is decomposed into annual sub-horizons through the forward strategy in time, that is, a problem of co-optimization of the investment and operation is solved for each year in a rolling horizon scheme. An optimum expansion plan is calculated per year, this decision is fixed, and a new optimization problem is set for the following year, considering the investment decisions taken in the previous year as fixed and completing the expansion plan, when necessary, as shown in Figure \ref{fig:hr}
\begin{figure}
\centering
\includegraphics[scale=0.7]{horizonte_rodante.png}
\caption{Horizon decomposition heuristic}
\label{fig:hr}
\end{figure}
\subsection{Typical days and seasons}
Since the operation is solved with hourly representation, it may result in a large and computationally intractable problem, given the size of studies that envision long-term horizons in the planning process, and since the proposed model solves a MILP that aims to minimize investment costs and the expected value of operating costs, subject to uncertainties in hydrology and generation of intermittent renewable sources.
As a way of exemplifying this issue, taking a real energy system into account, the Table \ref{tab:comp} summarizes the size of the optimization problems for 1 month and 5 blocks versus 744 hours.
\begin{table}[]
\caption{Comparing blocks and hours resolution size of the problems}
\label{tab:comp}
\begin{tabular}{@{}|l|c|c|@{}}
\toprule
\multicolumn{1}{|c|}{\textbf{Constraints}} & \textbf{Blocks} & \textbf{Hours} \\ \midrule
Water balance constraints & 114 & +80,000 \\ \midrule
Load balance constraints & 30 & +4,000 \\ \midrule
Maximum generation \& turbining constraints & 1499 & +290,000 \\ \midrule
Maximum \& minimum volume constraints & 228 & +165,000 \\ \midrule
Total & 1461 & +520,000 \\ \bottomrule
\end{tabular}
\end{table}
As can be seen, the size of the optimization problems increases significantly. In addition to that, while evaluating real systems' expansion, it is also necessary to use multiple scenarios to incorporate the uncertainties to which the system will be exposed (hydrology, renewable generation, etc.) and, consequently, the addition of all constraints per scenario in the optimization problem. For this reason, it is necessary to create a strategy that reduces the size of the problem, but without compromising the quality of the results.
In order to reduce the computational effort required by these optimization problems, it is necessary to introduce the concepts of seasons and representative (typical) days, which in addition to enabling the solution of these problems in acceptable computational times, captures the effects of intermittent generation in the system.
\begin{figure}
\centering
\includegraphics[scale=0.7]{seasons.png}
\caption{Mapping months into seasons}
\label{fig:seasons}
\end{figure}
The first step of this strategy is to group the months of the year into seasons, as shown in Figure \ref{fig:seasons}. Once the seasons are defined, the representative days within each of them, here referred to as typical days, should be defined. This type of representation aims to reduce the number of days analyzed within each season, since the daily demand profiles are not usually so different, especially within the pre-defined seasons. The Figure \ref{fig:typicaldays} illustrates this grouping of real days on typical days for a set of seasons in a given year. The allocation presented in the figure was made in a generic way, with illustrative purposes.
\begin{figure}
\centering
\includegraphics[scale=0.7]{typicaldays.png}
\caption{Mapping typical days inside each season}
\label{fig:typicaldays}
\end{figure}
\section{Handling uncertainties}\label{sec:uncertain}
In SDDP model, the long-term production costing decision making process (generation of each plant, interconnections between regions, circuit flows, etc.) consists in a stochastic optimization problem that seeks to balance the immediate cost and the expected value of the future cost (the expected value comes from the uncertainty about future hydrology, wind, consumption, availability of equipment, etc.). This problem is intrinsically related to storage devices that create a time-coupling between stages. Therefore, today's operating decisions, such as storage levels, may impact the mid and long-term operation, affecting thus the future operating costs. For further details, please refer to the SDDP Methodology Manual.
Taking the aforementioned explanation into account and given that this expansion approach performs the investment and operation co-optimization within the same problem, the operational policy is not calculated through SDDP algorithm, the proposed model does not consider the calculation of a Future Cost Function (FCF) for the system in each stage of the operation, since its calculation would require iterations of the system operation which reflects in the simulation of the operation in each stage several times until the FCF is sufficiently well approximated. It is intuitive to see that the SDDP application to calculate the FCF is the most realistic way to simulate the operation of the system, but, as it is intended to apply co-optimization, the operation of the hydro reservoirs throughout the year should be simplified.
The formulation proposed ensures that the initial storage of the reservoir of each hydroelectric plant at the beginning of each year of the study horizon will be equal to the final storage of that year. This operating strategy prevents the model from completely depleting the reservoirs present in the system during the year, optimizing its use throughout the year. The concept behind this modeling is a multi-deterministic operation, where the operation of the reservoirs is optimized for each separate scenario, without the incorporation of hydrological uncertainty into the decision-making process of the system operation in each scenario. It is plausible to explain that this simplification of the operation of large hydropower plants with large reservoirs has an optimistic bias, however, its application indicates that it is an approximation that presents satisfactory results for investment decision making and calculation of the expansion plan.
\section{Problem Formulation}\label{sec:problemform}
The expansion planning problem of an energy system is primarily formulated as a mathematical
programming problem, expressed by the formulation below. We suppose, for the sake of
simplicity, that all plants are candidate projects to the expansion problem.
\subsection{Investment Constraints}\label{sec:investmentconstraint}
\subsubsection{Precedence between projects}
\nomenclature[V]{$x_\omega$}{ Decision variable of generic project $\omega$}
\nomenclature[V]{$x_\zeta$}{ Decision variable of generic project $\zeta$}
\nomenclature[S]{$R^{pre}$}{ Set of precedence constraints.}
\nomenclature[S]{$P_k^{pre}$}{ Set of projects that belong to precedence constraint $k$}
\nomenclature[S]{$R^{ctr}$}{ Set of minimum / maximum constraints.}
\nomenclature[S]{$P_k^{ctr}$}{ Set of projects that belong to the constraint $k$.}
\nomenclature[C]{$w_\zeta^k$}{ Installed capacity, firm energy or firm capacity of generic project $\zeta$ relative to constraint $k$.}
\nomenclature[C]{$\underline{w_k}$}{ Lower-bound of min/max constraint $k$.}
\nomenclature[C]{$\overline{w_k}$}{ Upper-bound of min/max constraint $k$.}
\begin{align}
\label{con:precedence}
&x_{\omega} - x_{\zeta} \geq 0 &\forall \omega, \zeta\ \in P_k^{pre},\ \forall k \in R^{PRE}
\end{align}
\subsubsection{Mutually exclusive projects}
\nomenclature[S]{$R^{ex}$}{ Set of exclusivity constraints.}
\nomenclature[S]{$P_k^{ex}$}{ Set of projects that belong to exclusivity constraint $k$.}
\begin{align}
\label{con:exclusivity}
&\sum\limits_{\omega \in P_{k}^{ex}}x_{\omega} \leq 1 & \forall k \in R^{ex}
\end{align}
\subsubsection{Associated projects}
\nomenclature[S]{$R^{as}$}{ Set of association constraints.}
\nomenclature[S]{$P_k^{as}$}{ Set of projects that belong to association constraint $k$.}
\begin{align}
\label{con:associated}
&x_{\omega} - x_{\zeta} \geq 0 &\forall \omega, \zeta\ \in P_k^{as},\ \forall k \in R^{as}
\end{align}
\subsubsection{Minimum and maximum installed capacity / firm energy / firm capacity}
\nomenclature[S]{$R^{ctr}$}{ Set of min/max constraints.}
\nomenclature[S]{$P_k^{ctr}$}{ Set of projects that belong to the constraint $k$.}
\nomenclature[C]{$w_\zeta^k$}{ Installed capacity, firm energy or firm capacity of generic project $\zeta$ relative to min/max constraint $k$.}
\nomenclature[C]{$\underline{w_k}$}{ Minimum value (RHS) of min/max constraint $k$.}
\nomenclature[C]{$\overline{w_k}$}{ Maximum value (RHS) of min/max constraint $k$.}
\begin{align}
\label{con:capacity_min}
&\sum\limits_{\zeta \in P_k^{cap}}w_\zeta^k x_{\zeta} \geq \underline{w_k} &\forall k \in R^{ctr}
\end{align}
\begin{align}
\label{con:capacity_max}
&\sum\limits_{\zeta \in P_k^{cap}}w_\zeta^k x_{\zeta} \leq \overline{w_k} &\forall k \in R^{ctr}
\end{align}
\subsection{Thermal plants constraints}
\label{sec:thermal_powerplants}
\nomenclature[C]{$\overline{g_j}$}{ Maximum generation of thermal plant $j$}
\nomenclature[C]{$\underline{g_j}$}{ Minimum generation of thermal plant $j$}
\nomenclature[V]{$\gamma_{j,t,d,h,s}$}{ Commitment decision of thermal plant $j$, season $s$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$g_{j,t,d,h,s}$}{ Generation decision of thermal plant $j$, season $s$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$x_j$}{Investment decision of thermal plant $j$}
\nomenclature[V]{$st_{j,t,d,h,s}$}{Startup decision of thermal plant $j$, season $s$, typical day $d$, hour of the day $h$ and scenario $s$}
\subsubsection{Minimum and maximum energy generation}
\begin{align}
& \underline{g_j}\gamma_{j,t,d,h,s} \leq g_{j,t,d,h,s} \leq \overline{g_j}\gamma_{j,t,d,h,s} & \forall j,t,d,h,s
\end{align}
\subsubsection{Ramp up and ramp down generation}
\begin{align}
& g_{j,t,d,h,s} - g_{j,t,d,h-1,s} \leq \Delta_j^{UP} & \forall j,t,d,h,s\\
& g_{j,t,d,h-1,s} - g_{j,t,d,h,s} \leq \Delta_j^{DN} & \forall j,t,d,h,s
\end{align}
\subsubsection{Unit commitment}
\begin{align}
& st_{j,t,d,h,s} \geq \gamma_{j,t,d,h,s} - \gamma_{j,t,d,h-1,s} & \forall j,t,d,h,s\\
& \gamma_{j,t,d,h,s} \leq x_j & \forall j,t,d,h,s \label{eq:invest}\\
& \gamma_{j,t,d,h,s} \in \{0,1\} & \forall j,t,d,h,s
\end{align}
The constraint \eqref{eq:invest} model the relation between commitment and investment decisions, preventing a thermal plant to be committed without being invested before. This constraint make continuous investment decisions to be incompatible with thermal commitment constraints (because it requires binary variables).
\subsection{Hydro plants constraints}
\nomenclature[V]{$v_{i,t,s}$}{ Storage of hydro plant $i$, season $t$ and scenario $s$}
\nomenclature[V]{$u_{i,t,s}$}{ Turbining of hydro plant $i$, season $t$ and scenario $s$}
\nomenclature[V]{$s_{i,t,s}$}{ Spilling of hydro plant $i$, season $t$ and scenario $s$}
\nomenclature[V]{$a_{i,t,s}$}{ Lateral streamflow arriving at hydro plant $i$, season $t$ and scenario $s$}
\nomenclature[S]{$M_{i}$}{ The set of plants upstream of hydro plant $i$}
\nomenclature[V]{$g_{i,t,d,h,s}$}{ Generation decision of hydro plant $i$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[C]{$\rho_i$}{ Mean production factor of hydro plant $i$}
\nomenclature[C]{$\overline{V_i}$}{ Maximum storage of hydro plant $i$}
\nomenclature[C]{$\underline{V_i}$}{ Minimum storage of hydro plant $i$}
\nomenclature[C]{$\overline{U_i}$}{ Maximum turbining of hydro plant $i$}
\nomenclature[C]{$\overline{g_i}$}{ Maximum generation of hydro plant $i$}
\nomenclature[C]{$\underline{u_i}$}{ Minimum turbining of hydro plant $i$}
\nomenclature[C]{$\underline{q_i}$}{ Minimum total outflow of hydro plant $i$}
\nomenclature[C]{$D_{t,d}$}{ Duration of typical day $d$ at season $t$}
\nomenclature[V]{$\delta^v_{i,t,s}$}{ Minimum storage violation decision of hydro plant $i$, season $t$ and scenario $s$}
\nomenclature[V]{$\delta^u_{i,t,s}$}{ Minimum turbining violation decision of hydro plant $i$, season $t$ and scenario $s$}
\nomenclature[V]{$\delta^1_{i,t,s}$}{ Minimum total outflow violation decision of hydro plant $i$, season $t$ and scenario $s$}
\subsubsection{Water storage balance}
Since the model does not consider the Future Cost to go Function (FCF), it forces water reservoir levels of all hydro plants to finish at the same level they started (initial storage = final storage), preventing the system to deplete all water in the reservoir at the end of the horizon, in order to avoid thermal operative costs. This strategy forces the model to optimize reservoir operation in order to utilize all the water inflows that arrived in the analyzed period.
\begin{dmath}
v_{i,t+1,s} = v_{i,t,s} + a_{i,t,s} - \left( u_{i,t,s} + s_{i,t,s} \right) + \sum\limits_{m \in M_i}\left( u_{m,t,s} + s_{m,t,s} \right)\ \ \ \ \forall i,t,s
\end{dmath}
\begin{align}
&v_{i,T,s} = v_{i,0,s} & \forall i,s \label{con:vi=vf}
\end{align}
\subsubsection{Energy production}
The equation \eqref{eq:regulation} guarantees that the hourly energy production of the hydro plants is equal to the total energy turbined in the season. This equation assumes that the hydro plants have total regulation within season, i.e, they may freely transfer water, from an hour to another.
\begin{align}
& \sum\limits_{d,h}D_{t,d}g_{i,t,d,h,s} = \rho_i u_{i,t,s} & \forall i,t,s\label{eq:regulation}\\
&g_{i,t,d,h,s} \leq \overline{g_i}x_i & \forall i,t,d,h,s \label{con:prodmax}
\end{align}
\subsubsection{Minimum and maximum storage}
\begin{align}
&v_{i,t,s} \leq \overline{v_i}x_i & \forall i,t,s\\
&v_{i,t,s} + \delta^v_{i,t,s} = \underline{v_i}x_i & \forall i,t,s \label{con:vmax}
\end{align}
\subsubsection{Minimum and maximum turbining}
\begin{align}
&u_{i,t,s} \leq \overline{u_i}x_i & \forall i,t,s\\
&u_{i,t,s} + \delta^u_{i,t,s} = \underline{u_i}x_i & \forall i,t,s \label{con:vmax}
\end{align}
\subsubsection{Minimum total outflow}
\begin{align}
&u_{i,t,s} + s_{i,t,s} \leq \overline{q_i}x_i & \forall i,t,s\\
&u_{i,t,s} + s_{i,t,s} + \delta^q_{i,t,s} = \underline{q_i}x_i & \forall i,t,s \label{con:vmax}
\end{align}
\subsection{Renewables contraints}
\nomenclature[V]{$g_{l,t,d,h,s}$}{ Generation decision of renewable plant $l$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[C]{$\psi_{l,t,d,h,s}$}{ Renewable generation scenario for plant $l$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$x_l$}{ Investment decision of renewable plant $l$}
Renewable plants generation decision must be lower than renewable generation scenarios.
\begin{align}
& g_{l,t,d,h,s} \leq \psi_{l,t,d,h,s}x_l & \forall l,t,d,h,s
\end{align}
\subsection{Batteries}
\nomenclature[V]{$v_{b,t,d,h,s}$}{ Storage of battery $b$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$q^{+}_{b,t,d,h,s}$}{ Charge of battery $b$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$q^{-}_{b,t,d,h,s}$}{ Discharge of battery $b$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$x_b$}{ Investment decision of battery $b$}
\nomenclature[C]{$\eta^{+}_b$}{ Charge efficiency of battery $b$}
\nomenclature[C]{$\eta^{-}_b$}{ Discharge efficiency of battery $b$}
\nomenclature[C]{$\overline{V_b}$}{ Maximum storage of battery $b$}
\nomenclature[C]{$\overline{q^{+}_b}$}{ Maximum charge capacity of battery $b$}
\nomenclature[C]{$\overline{q^{-}_b}$}{ Maximum discharge capacity of battery $b$}
\subsubsection{Energy storage balance}
Battery storage balance has hourly time steps, as in equation \eqref{con:battery_balance}. Like the hydro plants, batteries also have regulation constraints \eqref{con:battery_vivf}, where the initial energy storage is equal the final energy storage.
\begin{align}
&v_{b,t,d,h+1,s} = v_{b,t,d,h,s} + \eta^{+}_b q^{+}_{b,t,d,h,s} - q^{-}_{b,t,d,h,s} & \forall b,t,d,h,s \label{con:battery_balance}\\
&v_{b,t,d,24,s} = v_{b,t,d,0,s} & \forall b,t,d,s \label{con:battery_vivf}
\end{align}
\subsubsection{Maximum storage, charge and discharge}
\begin{align}
&v_{b,t,d,h,s} \leq \overline{V_b}x_b & \forall b,t,d,h,s \label{con:vbmax}\\
&q^{+}_{b,t,d,h,s} \leq \overline{q^{+}_b}x_b & \forall b,t,d,h,s \label{con:chmax} \\
&q^{-}_{b,t,d,h,s}s \leq \overline{q^{-}_b}x_b & \forall b,t,d,h,s \label{con:dischmax}
\end{align}
\subsection{Transmission lines constraints}
\nomenclature[V]{$f_{k,t,d,h,s}^{+}$}{From$\rightarrow$To flow in transmission line $k$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$f_{k,t,d,h,s}^{-}$}{To$\rightarrow$From flow in transmission line $k$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[C]{$\overline{f_k^{+}}$}{ From$\rightarrow$To maximum flow capacity in transmission line $k$.}
\nomenclature[C]{$\overline{f_k^{-}}$}{ To$\rightarrow$From maximum flow capacity in transmission line $k$.}
\nomenclature[V]{$x_k$}{Investment decision of transmission line $k$}
\nomenclature[C]{$\gamma_k$}{ Susceptance of transmission line $k$.}
\nomenclature[V]{$\theta_{b_k^{+},t,d,h,s}$}{ Nodal angle of bus $b_k^{+}$ (From bus of transmission line $k$.}
\nomenclature[V]{$\theta_{b_k^{-},t,d,h,s}$}{ Nodal angle of bus $b_k^{-}$ (To bus of transmission line $k$.}
\nomenclature[C]{$M$}{ Disjunctive constant.}
\nomenclature[S]{$K^p$}{ Set of circuits projects.}
\nomenclature[S]{$K_a^{+}$}{ Set of transmission lines that arrive at area $a$ (To bus is in the area $a$ and the From bus is in a different area)}
\nomenclature[S]{$K_a^{-}$}{ Set of transmission lines that leave at area $a$ (From bus is in the area $a$ and the To bus is in a different area)}
\nomenclature[C]{$\overline{Imp_a}$}{ Maximum import amount of area $a$.}
\nomenclature[C]{$\underline{Imp_a}$}{ Minimum import amount of area $a$.}
\nomenclature[C]{$\overline{Exp_a}$}{ Maximum export amount of area $a$.}
\nomenclature[C]{$\underline{Exp_a}$}{ Maximum import amount of area $a$.}
\subsubsection{Maximum flow}
The flow variables for the network representation are $f_{k,t,d,h}^{+s}$ and $f_{k,t,d,h}^{-s}$, where these two positive variables represent the flow in both direction of each line, where $+$ means positive oriented and $-$ means negative oriented:
\begin{align}
&f_{k,t,d,h,s}^{+} \leq \overline{f_k^+}x_k & \forall k,t,d,h,s \label{con:fpmax}\\
&f_{k,t,d,h,s}^{-} \leq \overline{f_k^-}x_k & \forall k,t,d,h,s \label{con:fnmax}
\end{align}
\subsubsection{Second Kirchhoff law}
\label{sec:second_law}
The model considers two types of transmission lines: DC-Links and Circuits. Second Kirchhoff law will only be represented for circuits.
\begin{dmath}
\label{con:second_kirchoff_projects}
f_{k,t,d,h,s}^{+} - f_{k,t,d,h,s}^{-} - \gamma_k \left( \theta_{b_k^+,t,d,h,s} - \theta_{b_k^-,t,d,h,s}\right) \geq -M(1-x_k)\ \ \forall k\in K^p,t,d,h,s
\end{dmath}
\begin{dmath}
\label{con:second_kirchoff_projects}
f_{k,t,d,h,s}^{+} - f_{k,t,d,h,s}^{-} - \gamma_k \left( \theta_{b_k^+,t,d,h,s} - \theta_{b_k^-,t,d,h,s}\right) \leq M(1-x_k)\ \ \forall k\in K^p,t,d,h,s
\end{dmath}
\subsubsection{Area import/export constraints}
Area import/export constraints can limit the maximum amount of energy that enters or leave a specific electrical area.
For import constraints
\begin{align}
&\sum\limits_{k\in K^{+}_a}f_{k,t,d,h,s}^{+} + \sum\limits_{k\in K^{-}_a}f_{k,t,d,h,s}^{-} \leq \overline{Imp_a} & \forall a,t,d,h,s
\end{align}
\begin{align}
&\sum\limits_{k\in K^{+}_a}f_{k,t,d,h,s} + \sum\limits_{k\in K^{-}_a}f_{k,t,d,h,s} \geq \underline{Imp_a} & \forall a,t,d,h,s
\end{align}
For export constraints
\begin{align}
&\sum\limits_{k\in K^{-}_a}f_{k,t,d,h,s}^{+} + \sum\limits_{k\in K^{+}_a}f_{k,t,d,h,s}^{-} \leq \overline{Exp_a} & \forall a,t,d,h,s
\end{align}
\begin{align}
&\sum\limits_{k\in K^{-}_a}f_{k,t,d,h,s}^{+} + \sum\limits_{k\in K^{+}_a}f_{k,t,d,h,s}^{-} \geq \underline{Exp_a} & \forall a,t,d,h,s
\end{align}
\subsection{Generation constraint}
\nomenclature[V]{$\delta^g_{c,t,d,h,s}$}{ Violation decision of generation constraint $c$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[S]{$J_c^G$}{ Set of thermal plants that belongs to generation constraint $c$}
\nomenclature[S]{$I_c^G$}{ Set of hydro plants that belongs to generation constraint $c$}
\nomenclature[C]{$\underline{g_c}$}{Minimum value of generation constraint $c$}
\nomenclature[C]{$\overline{g_c}$}{Maximum value of generation constraint $c$}
Generation constraint is an operative constraint which guarantees that a certain group of generators (thermal and hydro plants) always generate energy above or below a threshold.
\begin{align}
&\sum\limits_{j\in J_c^G}g_{j,t,d,h,s} + \sum\limits_{i\in I_c^G}g_{i,t,d,h,s} + \delta^g_{c,t,d,h,s} \geq \underline{g_c} & \forall c,t,d,h,s\label{con:generation_above}
\end{align}
\begin{align}
&\sum\limits_{j\in J_c^G}g_{j,t,d,h,s} + \sum\limits_{i\in I_c^G}g_{i,t,d,h,s} + \delta^g_{c,t,d,h,s} \leq \overline{g_c} & \forall c,t,d,h,s\label{con:generation_below}
\end{align}
\subsection{Reserve balance constraints}
\nomenclature[V]{$r_{j,t,d,h,s}$}{ Reserve allocated by thermal plant $j$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$r_{i,t,d,h,s}$}{ Reserve allocated by hydro plant $i$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$r_{b,t,d,h,s}$}{ Reserve allocated by battery $b$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[S]{$J_c^R$}{ Set of thermal plants that belongs to reserve constraint $c$}
\nomenclature[S]{$I_c^R$}{ Set of hydro plants that belongs to reserve constraint $c$}
\nomenclature[S]{$B_c^R$}{ Set of batteries that belongs to reserve constraint $c$}
\nomenclature[V]{$\delta^R_{c,t,d,h,s}$}{ Violation decision of reserve requirement constraint $c$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[C]{$R_{c,t,d,h,s}$}{ Reserve requirement $c$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\begin{align}
&g_{j,t,d,h,s} + r_{j,t,d,h,s} \leq \overline{g_j}\gamma_{j,t,d,h,s} & \forall j,t,d,h,s\label{con:thermal_reserve}\\
&r_{j,t,d,h,s} \leq \Delta_j^{UP} & \forall j,t,d,h,s\\
&g_{i,t,d,h,s} + r_{i,t,d,h,s} \leq \overline{g_i}x_i & \forall i,t,d,h,s \label{con:hydro_reserve}\\
&\eta_b^{-} q^{-}_{b,t,d,h,s} + r_{b,t,d,h,s} \leq \eta_b^{-} \overline{q^{-}_b}x_b & \forall b,t,d,h,s \label{con:battery_reserve1}\\
&r_{b,t,d,h,s} \leq \eta_b^{-} v_{b,t,d,h,s} & \forall b,t,d,h,s \label{con:battery_reserve2}
\end{align}
\begin{dmath}
\sum\limits_{j\in J_c^R}r_{j,t,d,h,s} + \sum\limits_{i\in I_c^R}r_{i,t,d,h,s} + \sum\limits_{b\in B_c^R}r_{b,t,d,h,s}+ \delta^R_{c,t,d,h,s} \geq R_{c,t,d,h,s} \forall c,t,d,h,s\label{con:reserve_market}
\end{dmath}
\subsection{Load balance constraints}
\nomenclature[S]{$J_n$}{Set of thermal plants that belong to bus $n$}
\nomenclature[S]{$I_n$}{Set of hydro plants that belongs to bus $n$}
\nomenclature[S]{$B_n$}{Set of batteries that belongs to bus $n$}
\nomenclature[S]{$L_n$}{Set of renewable plants that belongs to bus $n$}
\nomenclature[S]{$K_n^{+}$}{Set of transmission lines that arrive at bus $n$}
\nomenclature[S]{$K_n^{-}$}{Set of transmission lines that leave bus $b$}
\nomenclature[V]{$DE_{n,t,d,h,s}$}{Elastic demand associated to bus $n$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[V]{$d_{n,t,d,h,s}$}{Deficit at bus $n$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\nomenclature[C]{$DI_{n,t,d,h,s}$}{Inelastic demand associated to bus $n$, season $t$, typical day $d$, hour of the day $h$ and scenario $s$}
\begin{dmath}
\label{con:load_balance}
\sum\limits_{j\in J_n} g_{j,t,d,h,s} + \sum\limits_{i\in I_n} g_{i,t,d,h,s} + \sum\limits_{l \in L_n} g_{l,t,d,h,s} + \sum\limits_{b\in B_n}\left(\eta_b^{-} q^{-}_{b,t,d,h,s} - q^{+}_{b,t,d,h,s} \right) + \sum\limits_{k\in K_n^{+}} \left( f_{k,t,d,h,s}^{+} - f_{k,t,d,h,s}^{-} \right) -\sum\limits_{k\in K_n^{-}}\left( f_{k,t,d,h,s}^{+} - f_{k,t,d,h,s}^{-} \right) - DE_{n,t,d,h,s} +d_{n,t,d,h,s} = DI_{n,t,d,h,s}\ \forall n,t,d,h,s
\end{dmath}
\subsection{Objective function}
\nomenclature[C]{$p_s$}{Probability of scenario $s$}
\nomenclature[C]{$rt$}{Discount rate at the season}
\nomenclature[C]{$co_j$}{The operation cost of thermal plant $j$}
\nomenclature[C]{$sc_j$}{The start-up cost of thermal plant $j$}
\nomenclature[C]{$co_i$}{O\&M cost of hydro plant $i$}
\nomenclature[C]{$c_{\delta_i^v}$}{Minimum storage violation penalty of hydro plant $i$}
\nomenclature[C]{$c_{\delta_i^u}$}{Minimum turbining violation penalty of hydro plant $i$}
\nomenclature[C]{$c_{\delta_i^q}$}{Total outflow violation penalty of hydro plant $i$}
\nomenclature[C]{$c_{\delta_c^G}$}{Violation penalty of generation constraint $c$}
\nomenclature[C]{$c_{\delta_c^R}$}{Violation penalty of reserve requirement constraint $c$}
\nomenclature[C]{$c_d$}{Deficit cost}
\nomenclature[C]{$P_n^E$}{Elastic demand price of bus $n$}
\nomenclature[C]{$ci_j$}{Investment cost of thermal project $j$}
\nomenclature[C]{$ci_i$}{Investment cost of hydro project $i$}
\nomenclature[C]{$ci_l$}{Investment cost of renewable project $l$}
\nomenclature[C]{$ci_b$}{Investment cost of battery project $b$}
\nomenclature[C]{$ci_k$}{Investment cost of transmission line project $k$}
\nomenclature[S]{$J_x$}{Set of thermal projects}
\nomenclature[S]{$I_x$}{Set of hydro projects}
\nomenclature[S]{$L_x$}{Set of renewable projects}
\nomenclature[S]{$B_x$}{Set of battery projects}
\nomenclature[S]{$K_x$}{Set of transmission line projects}
Let's define $\beta_{t,d,s}$ as:
\begin{equation}
\beta_{t,d,s} = \dfrac{p_sD_{t,d}}{(1+rt)^{t-1}}
\end{equation}
Then the problem's objective function is the minimization of the following costs:
\subsubsection{Generation Cost}
\begin{dmath}
\sum\limits_{t,d,s}\beta_{t,d,s}\left( \sum\limits_{j,h}(co_jg_{j,t,d,h,s} + cs_jst_{j,t,d,h,s}) + \sum\limits_{i,h}co_ig_{i,t,d,h,s}\right)
\end{dmath}
\subsubsection{Violation Cost}
\begin{dmath}
\sum\limits_{i,t,s}\dfrac{p_s}{(1+rt)^{t-1}}(c_{\delta_i^v}\delta_{i,t,s}^v + c_{\delta_i^u}\delta_{i,t,s}^u + c_{\delta_i^q}\delta_{i,t,s}^q) + \sum\limits_{t,d,s}\beta_{t,d,s}\left( \sum\limits_{c,h}(c_{\delta_c^G}\delta^G_{c,t,d,h,s} + c_{\delta_c^R}\delta^R_{c,t,d,h,s})\right)
\end{dmath}
\subsubsection{Deficit Cost}
\begin{dmath}
\sum\limits_{t,d,s}\beta_{t,d,s}\sum\limits_{n,h}c_dd_{n,t,d,h,s}
\end{dmath}
\subsubsection{Elastic Demand Gain}
\begin{dmath}
\sum\limits_{t,d,s}\beta_{t,d,s}\sum\limits_{n,h}P_n^EDE_{n,t,d,h,s}
\end{dmath}
\subsubsection{Investment Costs}
\begin{dmath}
\sum\limits_{j\in J_x}ci_jx_j + \sum\limits_{i\in I_x}ci_ix_i + \sum\limits_{l\in L_x}ci_lx_l + \sum\limits_{b\in B_x}ci_bx_b + \sum\limits_{k\in K_x}ci_kx_k +
\end{dmath}
\section{Conclusions}\label{sec:conclusion}
The model proposed here considers explicit operative constraints in the investment model. As a result, it can represent non-convexities in the operative constraints (such as commitment decisions). On the other hand, due to the increase of the problem's complexity, some simplifications have to be made. In this approach, we consider yearly time steps opposed to full horizon steps and representative (typical) days instead of real days within a year.
Typical days are days within a season that are considered representative of the input data. Thus, instead of representing all days of a season, the user selects a certain number of typical days to represent the season and associates these typical days with actual days. For instance, it is common to differentiate weekdays from Saturdays and Sundays, but the number of typical days and their definitions are flexible and chosen by the user.
The great advantages of this model are:
\begin{itemize}
\item The co-optimization of investment and operating problems inside the same MILP al-lows the representation of unit commitments and other binary variables;
\item The hourly chronological representation in the operation enables to capture the production variability of intermittent renewable sources and the generation ramps.
\end{itemize}
Besides the great advantages of this solution strategy, it's also important to remember its caveats. As explained in Section \ref{sec:uncertain}, the operative simulation is performed in a multi-deterministic way, where the operation of the reservoirs is optimized for each scenario individually, without the incorporation of hydrological uncertainty into the decision-making process of the system operation (as it is done when the SDDP methodology is applied and the FCF is calculated for each time stage). It is plausible to explain that this simplification of the operation of large hydropower plants with large reservoirs has an optimistic bias, however, its application indicates that it is an approximation that presents satisfactory results for investment decision making and calculation of the expansion plan.
Furthermore, it's also worth noting that since investment and operation problems are co-optimized in this solution strategy, then the more scenarios are contemplated in the problem, the higher computational effort will be demanded to solve the MILP. As a consequence, for large scale systems, the computational time might limit the number of scenarios that can be contemplated.
Finally, the proposed model is suitable for most real-case studies of expansion planning of renewable-dominated regions, representing hourly chronology, short-term constraints such as unit commitment and ramping, co-optimizing energy and reserves and with assumptions and approximations to make it computational tractable.
\bibliographystyle{IEEEtran}
|
1,314,259,995,770 | arxiv | \section{Introduction}
We are interested in studying a class of Schr\"odinger operators
\begin{align*}
\mathcal{L} = - \Delta_{x,y} + V(x,y).
\end{align*}
This operator acts on functions defined on the bounded, convex domain $\Omega \subset \mathbb{R}^2$, and $V(x,y)$ is a convex potential.
The operator $\mathcal{L}$ has an increasing sequence of Dirichlet eigenvalues
\begin{align*}
\lambda_1 < \lambda_2 \leq \cdots \leq \lambda_j \nearrow \infty,
\end{align*}
with corresponding eigenfunctions $u_j(x,y)$ satisfying
\begin{eqnarray*}
\left\{ \begin{array}{rlcc}
(-\Delta_{x,y} + V(x,y))u_j(x,y) & = \lambda_j u_j(x,y) && \text{in } \Omega \\
u_j(x,y) & = 0&& \text{on } \partial \Omega.
\end{array} \right.
\end{eqnarray*}
Our main focus will be to study the first eigenvalue $\lambda = \lambda_1$ and eigenfunction $u(x,y) = u_1(x,y)$. The first eigenfunction $u(x,y)$ does not change sign inside $\Omega$ and so we normalise $u(x,y)$ so that it is positive inside $\Omega$, and attains a maximum of $1$. In Definitions \ref{def:Om} and \ref{def:V} below, we will define the class of convex domains $\Omega$ and potentials $V(x,y)$ that we are interested in. We will see that one consequence of the assumptions on $\Omega$ and $V(x,y)$ is that it ensures that the superlevel sets of $u(x,y)$,
\begin{align*}
W_c \coloneqq \{ (x,y) \in \Omega: u(x,y) \geq c \},
\end{align*}
are convex subsets of $\Omega$ for all $0 \leq c \leq 1$.
A theorem of John, \cite{Jo}, therefore implies that for each $c$ we can find an ellipse $E_c$ contained within this superlevel set $W_c$, such that a dilate of $E_c$, with scaling factor bounded by an absolute constant contains $W_c$. We are interested in determining the shape of the level sets of $u(x,y)$, and to do this we will study the lengths and orientation of the axes of the ellipse $E_c$. One of the main steps in establishing the shape of the level sets of $u(x,y)$ will be to prove sufficiently precise bounds on the first eigenvalue $\lambda$.
We know that the level set $\{(x,y)\in\Omega:u(x,y) = 0\}$ is equal to the boundary, $\partial\Omega$, and so in particular the shape of this level set is determined solely by the geometry of $\Omega$. However, we will see that, in general, for the intermediate level sets, for example $\{(x,y)\in\Omega:u(x,y) = \tfrac{1}{2}\}$, it is not solely the shape of $\partial\Omega$ that governs its shape, but instead the two length scales $L_1$ and $L_2$. These length scales $L_1$ and $L_2$ will be given in Definitions \ref{def:L1} and \ref{def:L2}, but the key feature of their definitions is the following: The length scale $L_1$ will be defined purely in terms of the geometry of $\Omega$ and properties of the potential $V(x,y)$, but the length scale $L_2$ will also depend on a family of associated one dimensional Schr\"odinger operators. Moreover, the definition of $L_2$ will also describe the orientation of these level sets of $u(x,y)$.
Our motivation for studying this problem is as follows: First, $\lambda$ and $\Psi(t,x,y) = e^{\lambda t}u(x,y)$ are the lowest energy and ground state eigenfunction of the quantum system governed by the Schr\"odinger operator
\begin{align*}
\partial_t \Psi(t,x,y) + \mathcal{L}\Psi(t,x,y) = 0 .
\end{align*}
The main motivation comes from the series of papers \cite{J1}, \cite{GJ1}, \cite{GJ2}. There, the authors study the first two Dirichlet eigenfunctions on two dimensional convex domains $\Omega$, normalised so that the inner radius is comparable to $1$, and the diameter is equal to the large parameter $N$. We will describe their results and techniques in more detail below, but for now we will briefly describe one of the techniques used that is most relevant for us: Using their normalisation of the domain $\Omega$, they write it as
\begin{align*}
\Omega = \{ (x,y): f_1(x) < y < f_2(x), a<x<b \},
\end{align*}
for functions $f_1(x)$ and $f_2(x)$, which are convex and concave respectively, and they consider the concave \textit{height function} $h(x)$,
\begin{align*}
h(x) = f_2(x) - f_1(x),
\end{align*}
with $\max_{x\in[a,b]}h(x) = 1$. This allows us to define a large parameter $L$, purely in terms of the function $h(x)$ (and hence just depending on the geometry of the domain). This number $L$ is the largest value such that
\begin{align} \label{eqn:L}
h(x) \geq 1 - L^{-2}
\end{align}
on an interval $I$ of length at least $L$. Rather than the length of the diameter $N$, this parameter $L$ is the relevant length scale to study the low energy eigenfunctions. Since the inner radius of their domain is comparable to $1$, while the projection of the domain onto the $x$-axis is large compared to $1$, it is natural to study the two dimensional problem via an approximate separation of variables. For each fixed $x$, the domain $\Omega$ consists of the interval $[f_1(x),f_2(x)]$ of length $h(x)$, which has first eigenvalue $\pi^2h(x)^{-2}$. Thus, the ordinary differential operator on the interval $[a,b]$, which is naturally associated with this separation of variables is
\begin{align} \label{eqn:1dimJ}
-\frac{d^2}{dx^2} + \frac{\pi^2}{h(x)^2},
\end{align}
with zero boundary conditions. In \cite{J1} the eigenvalues and eigenfunctions of this operator are used to generate appropriate test functions to provide bounds on the first eigenvalue in terms of $L$, and to estimate the location and width of the nodal line of the second eigenfunction. In \cite{GJ1}, they give a sharper estimate on the nodal line, and in \cite{GJ2} they study the location of the maximum of the first eigenfunction of $\Omega$, and its behaviour near this maximum where they use this approximate separation of variables to relate it to the first eigenfunction of the one dimensional operator. As a straightforward consequence of their work, it is this length scale $L$ and orientation of the domain $\Omega$ given above, which determines the shape of the level sets of the eigenfunction $u(x,y)$ in this special case.
The papers \cite{J1}, \cite{GJ1}, \cite{GJ2} also provide more motivation for studying the operators $\mathcal{L}$. In the same way that the one dimensional Schr\"odinger operator in \eqref{eqn:1dimJ} is used in a crucial way to study the eigenfunctions of two dimensional convex domains, it will be important to understand the properties of the eigenfunctions of $\mathcal{L}$ when considering the eigenfunctions of three (and higher) dimensional convex domains.
Before stating our results, let us define precisely the class of domains $\Omega$ and potentials $V(x,y)$ that we will be considering here.
\begin{defn}[The Domain $\Omega$] \label{def:Om}
The domain $\Omega$ is a bounded, convex two dimensional domain with inner radius $N_1$, and diameter $N_2$. We assume that the diameter is large compared to an absolute constant, while the inner radius is bounded below by an absolute constant.
\end{defn}
\begin{rem}
Throughout, the constants that appear will depend on these absolute constants, but the dependence of any bounds on the diameter and inner radius themselves (and the other parameters introduced below) will be explicitly stated.
\end{rem}
We now state the class of potentials of interest.
\begin{defn}[The Potential $V(x,y)$] \label{def:V}
The potential $V(x,y)$ on the domain $\Omega$ satisfies
\begin{align*}
V(x,y) = \frac{1}{h(x,y)^{2}},
\end{align*}
where $h(x,y)$ is a concave function with $0\leq h(x,y) \leq 1$ and $\max_{\Omega}h(x,y) = 1$. In other words, $V(x,y)^{-1/2}$ is concave on $\Omega$ and
\begin{align*}
\min_{\Omega}V(x,y) = 1.
\end{align*}
In particular, this also ensures that $V(x,y)$ is convex.
\end{defn}
We see that this ensures that the first derivatives of $V$ are bounded almost everywhere, and that the second derivatives of $V$ are positive measures. However, we do not impose any further regularity assumptions on the potential. Before continuing, let us briefly discuss the motivation behind Definition \ref{def:V}.
\begin{enumerate}
\item One allowed potential is the constant potential $V(x,y) = 1$. In this case, our operator is analogous to the purely two dimensional operator studied in \cite{J1}. In particular, we can renormalise our domain $\Omega$ to ensure that the inner radius is comparable to $1$. Note in general, our potential $V(x,y)$ is not scale invariant, and so this is not as useful a normalisation for us.
\item The assumption that $V(x,y)^{-1/2}$ is concave is a natural one when we recall the motivation for studying this class of Schr\"odinger operators. In the same way that the operator in \eqref{eqn:1dimJ} has been used to study the eigenfunctions of two dimensional domains, the potential $V(x,y)$ that we are considering is naturally related to the three dimensional domain with height function proportional to $h(x,y)$. This assumption that $V(x,y)^{-1/2}$ is concave also appears in the work of Borell, \cite{B1}, \cite{B2}, when studying the concavity properties of the Green's functions associated to these Schr\"odinger operators.
\item We do not claim that this is the only class of potentials for which the results below will be valid. In fact, many of the results can be restated to hold for a more general class of convex potentials (including those related to the harmonic oscillator). However, at times we will see that it is convenient to restrict to those potentials given in Definition \ref{def:V}, and so we will only state the results for this class of potentials.
\end{enumerate}
We can now introduce the crucial parameters $L_1$ and $L_2$ that will appear as important length scales in our study of the first eigenfunction $u(x,y)$. For each $c\geq0$, let us define the sublevel sets of $V(x,y)$ by
\begin{align*}
\Omega_{c} \coloneqq \{ (x,y) \in \Omega: V(x,y) \leq 1+c\}.
\end{align*}
Since $V(x,y)$ is convex, these sublevel sets $\Omega_{c}$ are convex subsets of $\Omega$.
\begin{defn}[The Parameter $L_1$] \label{def:L1}
Let $L_1$ be the largest value such that the sublevel set $\Omega_{L_1^{-2}}$ has inner radius at least equal to $L_1$.
\end{defn}
\begin{rem}
This definition is analogous to the definition of the parameter $L$ from \cite{J1} described above, and roughly speaking is equal to the largest length scale $L_1$ on which the potential increases by at most $L_1^{-2}$ from its minimum.
\end{rem}
With $L_1$ fixed, we let $\tilde{L}_1$ be the diameter of the set $\Omega_{L_1^{-2}}$. If $L_1$ and $\tilde{L}_1$ are comparable in size, then we define $L_2$ to be equal to $L_1$, but if
\begin{align*}
\tilde{L}_1 \gg L_1,
\end{align*}
then we now describe how to find $L_2$.
\begin{rem}
Throughout, the notation $A \gg B$ denotes $A\geq \tilde{C}B$, for some large fixed absolute constant $\tilde{C}>0$, and if this, and the converse $B \gg A$, do not hold then we say that $A$ and $B$ are comparable. In particular, we are not interested in the exact values of $L_1$ and $L_2$, but instead are interested in knowing whether any length scale is, or is not, comparable to $L_1$ and $L_2$. We will use the notation $C$ to represent an absolute constant, that is small compared to $\tilde{C}$, which may change from line to line.
\end{rem}
To obtain a value for $L_2$, we first rotate our domain $\Omega$, so that the projection of $\Omega_{L_1^{-2}}$ onto the $y$-axis is of the smallest length amongst the projections onto any line. In particular, this means that the projection of $\Omega_{L_1^{-2}}$ onto the $x$-axis is comparable to $\tilde{L}_1$, while the projection of $\Omega_{L_1^{-2}}$ onto the $y$-axis is comparable to $L_1$. This also fixes the orientation of $\Omega$.
For each fixed $x$, let the interval $\Omega(x)$ be the cross-section of $\Omega$ at $x$, and consider the ordinary differential operator
\begin{align} \label{eqn:L(x)}
\mathcal{L}(x)\coloneqq -\frac{d^2}{dy^2} + V(x,y),
\end{align}
with zero boundary conditions on $\Omega(x)$. We let $\mu(x)$ be the first eigenvalue of $\mathcal{L}(x)$, and define the minimum of these eigenvalues,
\begin{align*}
\mu^* \coloneqq \min_{x}\mu(x).
\end{align*}
We can now define the parameter $L_2$.
\begin{defn}[The Parameter $L_2$] \label{def:L2}
We define $L_2$ to be the largest value such that
\begin{align*}
\mu^* \leq \mu(x) \leq \mu^* + L_2^{-2},
\end{align*}
for all $x$ in an interval $I$ of length at least $L_2$.
\end{defn}
\begin{rem} \label{rem:L2def}
Note that in this definition of $L_2$, we have used the orientation of $\Omega_{L_1^{-2}}$ fixed above. Therefore, from now on, whenever we consider any property of the eigenvalue or eigenfunction that depends on the value of $L_2$, we will have to use this orientation of $\Omega_{L_1^{-2}}$. In contrast, the definition of $L_1$ does not depend on the orientation of $\Omega_{L_1^{-2}}$.
\end{rem}
Our main aim in the study of the first eigenfunction is to give precise information about the shape of the level sets $\{(x,y)\in\Omega:u(x,y)=c\}$ which are near to the point where $u(x,y)$ attains its maximum of $1$. Since the potential $V(x,y)$ is a convex function and $\Omega$ is a convex set, Theorem 6.1 in \cite{BL2} tells us that $u(x,y)$ is log concave. Alternative proofs of this result have also been given in \cite{CF}, \cite{K}, \cite{KL}. In particular, this tells us that the superlevel sets are all convex. Since $\{(x,y)\in\Omega :u(x,y) \geq 0 \} = \Omega$, one way of viewing this result is that
\begin{align*}
\{(x,y)\in\Omega :u(x,y)\geq0 \} \text{ convex } \Rightarrow \{(x,y)\in\Omega :u(x,y) \geq c \} \text{ convex}
\end{align*}
for all $0\leq c \leq 1$.
We will use the convexity of the superlevel sets of $u(x,y)$ in a crucial way to describe their shape near its maximum.
\begin{thm} \label{thm:shape}
Let $\Omega$ and $V(x,y)$ be a domain and potential from Definitions \ref{def:Om} and \ref{def:V}. Fix a small absolute constant $c_1>0$, and let $L_1$ and $L_2$ be as in Definitions \ref{def:L1} and \ref{def:L2}. In particular, this means that we have fixed the orientation of the set $\Omega_{L_1^{-2}}$. Then, for any fixed absolute constant $c$, with $c_1<c<1-c_1$, the level set $\{(x,y)\in\Omega:u(x,y) = c\}$ has the following shape: There exists an ellipse $E$ with minor axis in the $y$-direction of length comparable to $L_1$ and major axis in the $x$-direction of length comparable to $L_2$, such that $E$ is contained inside this level set, and a dilate of $E$, with a scaling factor bounded by an absolute constant, contains this level set.
\end{thm}
\begin{rem}
The level set $\{(x,y)\in\Omega:u(x,y) = 0\}$ is equal to $\partial\Omega$, the boundary of $\Omega$. We will see that in general the parameters $L_1$ and $L_2$ are not comparable to the inner radius and diameter of the original domain $\Omega$. Thus, the result of Theorem \ref{thm:shape} does not remain valid when $c$ becomes close to $0$.
\end{rem}
\begin{cor}
For a convex set $W$, we define the eccentricity of $W$, \emph{ecc}$(W)$ in the usual way:
\begin{align*}
\emph{ecc}(W) = \frac{\emph{diam}(W)}{\emph{inradius}(W)}.
\end{align*}
For $c=0$, the eccentricity of the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq c\}$ is equal to the eccentricity of $\Omega$, but as $c$ increases (while bounded above by $1-c_1$), the eccentricity of the superlevel set becomes comparable to $L_2/L_1$.
\end{cor}
The log concavity of the eigenfunction, and resulting convexity of its superlevel sets has been used previously in various situations. For example, in \cite{AC} moduli of convexity and concavity are introduced. Under certain conditions on the potential $V$, it is then possible to strengthen the log concavity of the first eigenfunction by finding an appropriate modulus of concavity. This allows the spectral gap for a class of Schr\"odinger operators to be compared to the case where the potential is identically zero, and allows them to prove the Fundamental Gap Conjecture. In \cite{FJ} the convexity of the superlevel sets of the Green's function are used in a crucial way to prove third derivative estimates on the eigenfunction which are valid up to the boundary of the convex domain.
As well as the convexity of the superlevel sets of $u(x,y)$, a very important part of the proof of Theorem \ref{thm:shape} will be to obtain sufficiently precise eigenvalues bounds for the first eigenvalue $\lambda$. For $\mu(x)$ equal to the first eigenvalue of the operator $\mathcal{L}(x)$, we consider the ordinary differential operator
\begin{align} \label{eqn:A}
\mathcal{A} = - \frac{d^2}{dx^2} + \mu(x),
\end{align}
and let $\mu$ be the first eigenvalue of this operator. Our eigenvalue bounds relate the value of $\lambda$ to this eigenvalue $\mu$.
\begin{thm} \label{thm:eigenvalue}
Let $\Omega$ and $V(x,y)$ be a domain and potential from Definitions \ref{def:Om} and \ref{def:V}. If $L_2$ is defined as in Definition \ref{def:L2} and $\mu$ is the first eigenvalue of the operator $\mathcal{A}$ in \eqref{eqn:A}, then the first eigenvalue $\lambda$ of the operator $\mathcal{L}$ satisfies
\begin{align*}
\mu \leq \lambda \leq \mu + CL_2^{-2},
\end{align*}
for an absolute constant $C$.
\end{thm}
\begin{rem} \label{rem:uniformconstants}
Theorems \ref{thm:shape} and \ref{thm:eigenvalue} are valid for all domains and potentials satisfying the assumptions of Definitions \ref{def:Om} and \ref{def:V}, and the bounds are uniform for domains $\Omega$ and potentials $V$ leading to the same values for $L_1$ and $L_2$.
\end{rem}
While it is much more straightforward to locate the eigenvalue $\lambda$ to an interval of length comparable to $L_1^{-2}$, we will see that the more precise bound obtained in Theorem \ref{thm:eigenvalue} is necessary to obtain sharp information about the length scale on which the eigenfunction $u(x,y)$ decays in the $x$-direction, and hence prove Theorem \ref{thm:shape}.
Theorem \ref{thm:eigenvalue} locates the first eigenvalue $\lambda$ to an interval of length comparable to $L_2^{-2}$, provided we know the value of $\mu$. However, $\mu$ is also an eigenvalue of a differential operator, and so it may seem like we have only been able to locate the unknown $\lambda$ in terms of another unknown $\mu$. Another reason why this theorem still has value is that whereas $\lambda$ is the first eigenvalue of a two dimensional partial differential operator (with a potential), $\mu$ is the first eigenvalue of an ordinary differential operator $\mathcal{A}$. Thus, from a computational standpoint, it is much easier to accurately approximate the value of $\mu$ compared to $\lambda$. Also, we notice that the parameter $L_2$ depends on the geometric properties of the domain $\Omega$ and potential $V(x,y)$, together with the eigenvalues of the differential operator $\mathcal{L}(x)$ given in \eqref{eqn:L(x)}. In other words, $L_2$ also only depends on knowledge of ordinary differential operators. Thus, the bound given in Theorem \ref{thm:eigenvalue} gives information about the eigenvalue of a two dimensional partial differential operator purely in terms of ordinary differential operators.
The idea of relating the eigenfunctions and eigenvalues of a two dimensional problem to an associated ordinary differential operator has also been used extensively by Friedlander and Solomyak in \cite{FS1}, \cite{FS2}, \cite{FS3}. In these papers, they use this approximate separation of variables to obtain asymptotics for the eigenvalues, and the resolvent of the Dirichlet Laplacian. They use a semiclassical method by sending a small parameter $\epsilon$ to $0$ in order to give a one-parameter of \textquoteleft narrow' domains, and then write asymptotics in terms of this small parameter.
\\
Let us now describe how we will proceed in the sections below.
In Section \ref{sec:L1} we study the parameters $L_1$ and $L_2$ from Definitions \ref{def:L1} and \ref{def:L2} in more detail. In particular, we will obtain bounds on $L_1$ and $L_2$ in terms of the diameter and inner radius of the domain and the potential, and construct domains $\Omega$ and potentials $V(x,y)$ to show to what extent these estimates are sharp. We will also give a straightforward bound on $\lambda$ in terms of $L_1$ by using the variational formulation for the first eigenvalue.
In Section \ref{sec:la} we will prove the eigenvalue bounds in Theorem \ref{thm:eigenvalue}. For each fixed $x$, $u(x,y)$ is an admissible test function for the operator $\mathcal{L}(x)$ from \eqref{eqn:L(x)}, and the lower bound on $\lambda$ will follow straightforwardly from this. The proof of the upper bound on $\lambda$ in Theorem \ref{thm:eigenvalue} is more involved. The starting point of the proof is to use the first eigenfunction, $\psi^{(x)}(y)$, of the operator $\mathcal{L}(x)$ to construct a suitable test function in the variational formulation for the first eigenvalue. To obtain the required upper bound on $\lambda$ it will be necessary to study the first variation of $\psi^{(x)}(y)$ in the cross-sectional variable $x$. To do this, we will derive the ordinary differential equation that this first variation satisfies for each fixed $x$. The bounds then follow from using the method of variation of parameters. It will be particularly important to have estimates on the relative size of the first derivative of the potential $V(x,y)$ and the size of $\psi^{(x)}(y)$.
Once we have established the bounds on $\lambda$ in Theorem \ref{thm:eigenvalue}, in Section \ref{sec:L2} we use them to study the first eigenfunction $u(x,y)$ itself. Our first aim is to prove a $L^2(\Omega)$-bound on $u(x,y)$ which is consistent with the shape of the level sets required in Theorem \ref{thm:shape}. We begin by using Theorem \ref{thm:eigenvalue} to prove a Carleman-type estimate to show how the $L^2(\Omega(x))$-norm of the cross-sections of $u(x,y)$,
\begin{align*}
H(x) = \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y,
\end{align*}
decays from its maximum exponentially on a length scale comparable to $L_2$. To find the required bound on the $L^2(\Omega)$-norm of $u(x,y)$, we then need to estimate the size of the maximum of $H(x)$. We will do this by proving $L^2(\Omega)$-bounds on the first derivatives of $u(x,y)$, which are again consistent with Theorem \ref{thm:shape}. We finish Section \ref{sec:L2} by proving an Agmon-type estimate to give an indication of the behaviour of $u(x,y)$ at points at a large distance from its maximum.
In Section \ref{sec:shape} we study the shape of the level sets of $u(x,y)$ and complete the proof of Theorem \ref{thm:shape}. To do this we will use the results of Section \ref{sec:L2} on the $L^2(\Omega)$-norms of $u(x,y)$ itself, and also its first derivatives. We will also use the log-concavity of the eigenfunction $u(x,y)$ in a crucial way, since it is this that ensures that the superlevel sets are convex.
Theorem \ref{thm:shape} gives information about the level sets, $\{(x,y)\in\Omega:u(x,y) = c\}$ whenever $c$ is bounded away from $0$ and $1$. In Section \ref{sec:max}, we want to study the behaviour of the eigenfunction $u(x,y)$ near its maximum. In particular, we will relate the location of the maximum to the region where $V(x,y) - \lambda$ is bounded above by $-c^*L_1^{-2}$, for an absolute constant $c^*>0$. We will do this by first using a maximum principle to restrict attention to the part of $\Omega$ where $V(x,y)-\lambda$ is at most comparable to $L_1^{-2}$. This will then be used to convert the $L^2(\Omega)$-bounds on $\nabla_{x,y}u(x,y)$ from Section \ref{sec:L2} into pointwise bounds near the maximum of $u(x,y)$. These bounds are then in turn used to prove the sharper estimate on the location of the maximum. We finish by giving two consequences of this estimate of the location of the maximum. The first is that we obtain sharper bounds on the derivative $\partial_yu(x,y)$ as we approach the maximum, and we also obtain an improved pointwise bound on $\partial_xu(x,y)$ in a region around the maximum of height comparable to $L_1$ in the $y$-direction, and length comparable to $L_2$ in the $x$-direction.
\subsection{Acknowledgements}
I would like to thank David Jerison for suggesting this problem to me and for many enlightening conversations. I would also like to thank my advisor Charles Fefferman for many useful discussions and for his help in improving the exposition in this paper.
\section{The Parameters $L_1$ and $L_2$} \label{sec:L1}
Before proving Theorems \ref{thm:eigenvalue} and \ref{thm:shape}, we first give some more properties of the parameters $L_1$ and $L_2$ defined in Definitions \ref{def:L1} and \ref{def:L2}.
We first want to give upper and lower bounds for $L_1$, where we recall that $L_1$ is the largest value for which the sublevel set $\{(x,y)\in\Omega: V(x,y) \leq 1+ L_1^{-2} \}$ has inner radius at least $L_1$. We can think of this as being analogous to the parameter $L$ from \cite{J1}, which we described earlier in \eqref{eqn:L}. In \cite{J1}, it was shown that this parameter $L$ satisfies
\begin{align*}
N^{1/3} \leq L \leq N,
\end{align*}
where $N$ is the diameter of the two dimensional domain. The upper bound on $L$ is attained by an exactly rectangular domain, $[0,N]\times[0,1]$, and the lower bound is attained by a right triangle of height $1$ and length $N$. Moreover, any intermediate value for $L$ can be attained by interpolating between these two extreme cases and forming the appropriate trapezoidal shape.
We now give an analogous description for the possible values of $L_1$. Rather than the potential $V(x,y)$, it will be more convenient to work with the \textit{height function}
\begin{align} \label{eqn:h}
h(x,y) = V(x,y)^{-1/2},
\end{align}
which, by the assumptions on the potential, is a concave function, satisfying
\begin{align*}
0 \leq h(x,y) \leq 1,
\end{align*}
and attaining its maximum of $1$ at the minimum of $V(x,y)$.
\begin{prop} \label{prop:L1bounds}
Recalling that $N_1$ is the inner radius of the domain $\Omega$, we have the bounds
\begin{align*}
cN_1^{1/5} \leq L_1 \leq N_1,
\end{align*}
for some absolute constant $c>0$.
\end{prop}
\begin{rem}
We will see in the proof of the proposition, that we are using the stronger assumption that $h(x,y) = V(x,y)^{-1/2}$ is concave, instead of just the convexity of $V(x,y)$.
\end{rem}
\begin{proof}{Proposition \ref{prop:L1bounds}}
The proposition follows easily when the inner radius $N_1$ is comparable to a constant, and so throughout we will assume that $N_1\gg 1$.
The upper bound follows trivially from the definition of $L_1$, and is attained, for example, when $V(x,y)$ (and hence $h(x,y)$) is identically equal to $1$.
Before proving the lower bound, we recall the following theorem of John, \cite{Jo}:
\begin{thm} \label{thm:john}
Let $K \subset \mathbb{R}^m$ be a convex domain. Then, there exists an ellipsoid $E$ such that if $c^* \in \mathbb{R}^m$ is the centre of $E$, then we have
\begin{align*}
E \subset K \subset c^*+m(E-c^*).
\end{align*}
That is, the ellipsoid $E$ is contained within the convex set $K$, but if it is dilated by a constant depending only on the dimension, then it contains $K$.
\end{thm}
We will also need the following simple property of concave functions:
\begin{lem} \label{lem:concave1}
Suppose $g(x)$ is a concave function on an interval of length $M$, with $0 \leq g(x) \leq 1$, and $g(0) = 1$. Let $0<\beta<1$ and suppose that $g(z) = 1 - \beta$ at some point $z \in (0,M)$. Then, we have the bound
\begin{align*}
M \leq \beta^{-1} z .
\end{align*}
\end{lem}
\begin{proof}{Lemma \ref{lem:concave1}}
By the assumptions on the function $g(x)$, it decreases by at most $1$ over an interval of length $M$. Thus, since it is a concave function, it must satisfy
\begin{align*}
g(x) \geq 1-\frac{x}{M}.
\end{align*}
Since $g(z) = 1- \beta$, this gives
\begin{align*}
1- \beta \geq 1 - \frac{z}{M}, \qquad \text{or equivalently} \qquad M \leq \beta^{-1} z,
\end{align*}
as required.
\end{proof}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figure1Eigenfunctions19Sep14.pdf}
\caption{The Domain $\Omega$, and other sets appearing in the proof of Proposition \ref{prop:L1bounds}}
\label{fig:L1bounds}
\end{center}
\end{figure}
We can now prove Proposition \ref{prop:L1bounds}. Let $E$ be the ellipse coming from Theorem \ref{thm:john} for our two dimensional domain $\Omega$, and let $(x^*,y^*)$ be a point where $h(x,y)$ attains its maximum of $1$. Consider the ray $J$ which is the intersection of our domain $\Omega$, and the line containing the point $(x^*,y^*)$ and the centre of the ellipse $E$ (see Figure \ref{fig:L1bounds}).
Since $\Omega$ has inner radius equal to $N_1$, by the properties of the ellipse $E$, we know that the ray $J$ has length $M$ with
\begin{align} \label{eqn:concave1}
M \geq c_1 N_1,
\end{align}
for some small absolute constant $c_1>0$. Now consider the intersection of $J$ with the interior of the sublevel set
\begin{align*}
\Omega_{L_1^{-2}} = \{(x,y)\in\Omega: V(x,y) \leq 1 + L_1^{-2} \}.
\end{align*}
Let $J_1$ be this interval. If $V(x,y) = 1+ L_1^{-2}$, then $1- h(x,y) = 1- V(x,y)^{-2}$ will be comparable to $L_1^{-2}$, and so applying Lemma \ref{lem:concave1} with $\beta = L_1^{-2}$, we see that $J_1$ will be of length $A$, where
\begin{align} \label{eqn:concave2}
M \leq C_1L_1^{2} A,
\end{align}
for a large absolute constant $C_1$.
Combining \eqref{eqn:concave1} and \eqref{eqn:concave2} gives us
\begin{align} \label{eqn:concave2a}
c_1N_1 \leq M \leq C_1L_1^{2} A.
\end{align}
Thus, the lower bound of the proposition is established unless
\begin{align} \label{eqn:concave3}
A \geq C_2L_1^3,
\end{align}
for a large constant $C_2>0$.
Therefore, we will assume that \eqref{eqn:concave3} holds, and so in particular, $A$ is large compared to $L_1$. Let $E_{L_1^{-2}}$ be the ellipse from Theorem \ref{thm:john} for the set $\Omega_{L_1^{-2}}$, and rotate so that the minor axis of $E_{L_1^{-2}}$ lies in the $y$-direction. Then, by the definition of $L_1$, the minor axis of $E_{L_1^{-2}}$ has length comparable to $L_1$.
This means that the ray of length $A$ must approximately lie in the $x$-direction. $\Omega$ is a convex set with inner radius $N_1$, and the original ray, $J$, through $\Omega$ is of length $M$. Therefore, if we pick a point $(x_1,y_1)$ in the interval $J_1$, which is at a distance of at least $A/4$ from the ends of $J_1$, then the height of $\Omega$ in the $y$-direction at $x=x_1$ must be at least
\begin{align} \label{eqn:concave4}
c_2 A N_1/M,
\end{align}
for a constant $c_2>0$. In contrast, the height of $\Omega_{L_1^{-2}}$ at $x=x_1$ must be bounded above by $C_3 L_1$, since the minor axis of $E_{L_1^{-2}}$ lies in the $y$-direction and has length comparable to $L_1$.
Moreover, the concave function $h(x,y)$ varies from $1$ to $1- L_1^{-2}$ in the interval $J_1$ of length $A$. Thus, using Lemma \ref{lem:concave1} again, we have
\begin{align} \label{eqn:concave5}
h(x_1,y_1) \geq 1 - \frac{3}{4L_1^2},
\end{align}
at this point on the ray.
Thus, combining \eqref{eqn:concave4} and \eqref{eqn:concave5}, we see that, for $x=x_1$ fixed, $h(x_1,y)$ is a concave function of $y$, which decreases by at most $1$ on an interval of length comparable to $AN_1/M$, and decreases by $\tfrac{1}{4}L_1^{-2}$ on an interval of length comparable to $L_1$. Thus, using Lemma \ref{lem:concave1} one more time, we see that
\begin{align} \label{eqn:concave6}
\frac{AN_1}{M} \leq C_4L_1^2L_1 = C_4L_1^3,
\end{align}
for a constant $C_4$. Combining \eqref{eqn:concave2a} and \eqref{eqn:concave6} we see that
\begin{align*}
M \leq C_1L_1^{2} A \leq C_1L_1^{2}C_4L_1^3 \frac{M}{N_1} = C_5 L_1^5 \frac{M}{N_1} ,
\end{align*}
for a constant $C_5>0$. Rearranging this inequality gives the desired lower bound on $L_1$.
\end{proof}
We noted in the proof of Proposition \ref{prop:L1bounds} that it is straightforward to give an example showing that the upper bound on $L_1$ is sharp. We now want to construct an example showing that the lower bound on $L_1$ is also optimal.
\begin{lem} \label{lem:sharpL1}
We can find a domain $\Omega$ and potential $V(x,y)$ satisfying the assumptions of Definitions \ref{def:Om} and \ref{def:V} such that
\begin{align*}
L_1 \geq cN_1^{1/5},
\end{align*}
for some absolute constant $c>0$.
\end{lem}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figure2Eigenfunctions19Sep14.pdf}
\caption{The Domain in Lemma \ref{lem:sharpL1}}
\label{fig:sharpL1}
\end{center}
\end{figure}
\begin{proof}{Lemma \ref{lem:sharpL1}}
We first construct the domain $\Omega$. We have remarked earlier, that for the two dimensional domain case in \cite{J1}, a right triangle gives the smallest possible value for $L$. Motivated by this, we let $\Omega$ be a right triangle of side lengths $N_1$ in the $y$-direction, and side length $N_2$ in the $x$-direction (see Figure \ref{fig:sharpL1}). We note that while the inner radius of this domain is not identically to $N_1$, it is comparable to $N_1$ (independently of the size of $N_2$), and this is all we need.
We now define the potential $V(x,y)$, via the function $h(x,y) = V(x,y)^{-2}$. We let $h(x,y) = 1$ at the point where the hypotenuse joins the side of length $N_2$, and set $h(x,y) = 0$ at the midpoint of the side of length $N_1$. We then require $h(x,y)$ to decay linearly on the interval connecting these two points. Finally, $h(x,y)$ decays linearly to $0$ in the $y$-direction as we move away from this interval. This defines $h(x,y)$ everywhere on $\Omega$, and also ensures that $h(x,y)$ is a concave function. Thus the potential $V(x,y)$ satisfies the required properties.
We define $L_1$ as usual from Definition \ref{def:L1} for this domain $\Omega$ and potential $V(x,y)$. Consider the line segment $J$ joining the vertex where $h(x,y) = 1$ to the midpoint of the opposite side, and let $M$ be the length of the line segment $J_1\subset J$ on which $h(x,y) \geq 1 - L_1^{-2}$. Then, since $h(x,y)$ decays linearly, and the whole of $J$ has length comparable to $N_2$, it is easy to see that
\begin{align} \label{eqn:sharp1}
M = c_1L_1^{-2}N_2,
\end{align}
for a constant $c_1>0$.
By the definition of $L_1$, the set $\{ (x,y)\in\Omega: h(x,y) = 1-L_1^{-2}\}$ has inner radius comparable to $L_1$. Thus, at the point $(x_1,y_1)$ on the line segment $J$ with
\begin{align} \label{eqn:sharp2}
h(x_1,y_1) = 1 - \tfrac{1}{2}L_1^{-2},
\end{align}
this set has height comparable to $L_1$ in the $y$-direction for $x=x_1$ fixed. Moreover, the point $(x_1,y_1)$ is at a distance comparable to $M$ from the vertex where $h(x,y) = 1$, and so the height of $\Omega$ at this point is equal to
\begin{align} \label{eqn:sharp3}
c_2M \frac{N_1}{N_2},
\end{align}
for $c_2>0$. Thus, for $x=x_1$ fixed, $h(x_1,y)$ decays linearly to $0$ on an interval of length comparable to $L_1N_2/N_1$, and by \eqref{eqn:sharp2} decreases linearly by $\tfrac{1}{2}L_1^{-2}$ on an interval of length comparable to $L_1$. This tells us that
\begin{align} \label{eqn:sharp4}
L_1^3 = c_3 M \frac{N_1}{N_2}.
\end{align}
Combining \eqref{eqn:sharp1} and \eqref{eqn:sharp4} gives
\begin{align*}
L_1^3 = c_3c_1L_1^{-2}N_2 \frac{N_1}{N_2} = c_3c_1L_1^{-2}N_1,
\end{align*}
and rearranging gives the desired estimate for $L_1$.
\end{proof}
\begin{rem}
By combining the two examples which show that the upper and lower bounds on $L_1$ from Proposition \ref{prop:L1bounds} are sharp, it is easy to construct examples where $L_1$ attains any intermediate length scale.
\end{rem}
We now want to consider the parameter $L_2$ introduced in Definition \ref{def:L2}. Before describing the bounds that $L_2$ must satisfy, we first give a simple bound on the eigenvalue $\lambda$.
\begin{prop} \label{prop:simpleeigenvalue}
The first eigenvalue $\lambda$ satisfies
\begin{align*}
1 \leq \lambda \leq 1 + C_1L_1^{-2},
\end{align*}
for an absolute constant $C_1>0$.
\end{prop}
\begin{proof}{Proposition \ref{prop:simpleeigenvalue}}
We will establish these bounds by using the variational formulation of the first eigenvalue, $\lambda$. That is,
\begin{align} \label{eqn:variation}
\lambda = \inf \left\{ \frac{\int_{\Omega} \left|\nabla\psi(x,y)\right|^2 \,\mathrm{d} x \,\mathrm{d} y + \int_{\Omega} V(x,y)\psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y}{\int_{\Omega} \psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y} \bigg| \psi \in W^{1,2}(\Omega), \psi|_{\partial\Omega} = 0, \psi \not\equiv 0 \right\}
\end{align}
Since $V(x,y) \geq 1$ for all $(x,y) \in \Omega$, the lower bound, $\lambda \geq1$ follows immediately.
To prove the upper bound, we need to construct a suitable test function $\psi(x,y)$ to use in \eqref{eqn:variation}. By the definition of $L_1$, we know that the sublevel set
\begin{align*}
\Omega_{L_1^{-2}} = \{(x,y):V(x,y) \leq 1 + L_1^{-2}\}
\end{align*}
has inner radius equal to $L_1$. Thus, we can choose a point $(x_0,y_0)$ and a constant $c>0$, such that the set
\begin{align*}
R = \{(x,y):|x-x_0| \leq cL_1, |y-y_0| \leq cL_1\}
\end{align*}
is contained in the interior of $\Omega_{L_1^{-2}}$. We then define $\psi(x,y)$ as
\begin{align*}
\psi(x,y) = \cos\left(\frac{\pi (x-x_0)}{2cL_1}\right)\cos\left(\frac{\pi (y-y_0)}{2cL_1}\right)
\end{align*}
inside the square $R$, and set $\psi(x,y) = 0$ for all other $(x,y) \in\Omega$. It is then clear that
\begin{align*}
\frac{\int_{\Omega} \left|\nabla\psi(x,y)\right|^2 \,\mathrm{d} x \,\mathrm{d} y}{\int_{\Omega} \psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y} \leq C_2L_1^{-2},
\end{align*}
and since $V(x,y) \leq 1+ L_1^{-2}$ on the support of the test function $\psi(x,y)$, we also have
\begin{align*}
\frac{ \int_{\Omega} V(x,y)\psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y}{\int_{\Omega} \psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y} \leq 1 + C_3L_1^{-2}.
\end{align*}
Using these inequalities in \eqref{eqn:variation} gives the desired upper bound on $\lambda$.
\end{proof}
We now consider the parameter $L_2$ from Definition \ref{def:L2}. We recall that the sublevel set $\Omega_{L_1^{-2}}$ has inner radius $L_1$ and diameter $\tilde{L}_1$, and that we set $L_2$ to be equal to $L_1$ unless $\tilde{L}_1 \gg L_1$. The upper and lower bound for $L_2$ from Definition \ref{def:L2} that we want to establish is the following:
\begin{prop} \label{prop:L2bound}
The parameter $L_2$ satisfies
\begin{align*}
c_1 \tilde{L}_1^{1/3} L_1^{2/3} \leq L_2 \leq \frac{1}{c_1}\tilde{L}_1,
\end{align*}
for some absolute constant $c_1>0$.
\end{prop}
\begin{rem}
In particular, the lower bound shows us that if we have $\tilde{L}_1 \gg L_1$, then also $L_2 \gg L_1$.
\end{rem}
\begin{proof}{Proposition \ref{prop:L2bound}}
The value of $L_2$ depends on the function $\mu(x)$, where $\mu(x)$ is the first eigenvalue of the operator
\begin{align} \label{eqn:Lx2}
\mathcal{L}(x) = - \frac{d^2}{dy^2} + V(x,y).
\end{align}
$L_2$ is the largest value such that $\mu(x)$ increases by $L_2^{-2}$ from its minimum value, $\mu^*$, on an interval of length at least $L_2$. Therefore, before proving the bounds on $L_2$, we first want to study the properties of the function $\mu(x)$.
We have rotated $\Omega$ so that the projection of the set $\Omega_{L_1^{-2}}$ onto the $y$-axis is of the smallest length amongst the projections onto any line. One immediate consequence of this is that if we set $J$ to be the interval which is the projection of $\Omega_{L_1^{-2}}$ onto the $x$-axis, then the length of $J$ is comparable to $\tilde{L}_1$, the diameter of $\Omega_{L_1^{-2}}$.
We now give a bound on the eigenvalues $\mu(x)$ for $x \in J$.
\begin{lem} \label{lem:mubound}
For $x$ in the middle half of the interval $J$, there exists an absolute constant $C_1>0$ such that
\begin{align*}
1 + \frac{1}{C_1L_1^{2}} \leq \mu(x) \leq 1 + \frac{C_1}{L_1^2}.
\end{align*}
\end{lem}
\begin{proof}{Lemma \ref{lem:mubound}}
Since $\mu(x)$ is the first eigenvalue in the ordinary differential operator in \eqref{eqn:Lx2}, we want to apply Lemma 2.4 (a) in \cite{J1}. This lemma implies that
\begin{align} \label{eqn:Lem2.4}
1 + \frac{1}{C_1L(x)^2} \leq \mu(x) \leq 1 + \frac{C_1}{L(x)^2},
\end{align}
where $L(x)$ in the length scale associated to $V(x,y)$. In other words, for each fixed $x$, $L(x)$ is the largest value such that $V(x,y)$ varies from its minimum by $L(x)^{-2}$ on an interval of length at least $L(x)$. Thus, to prove the lemma it is enough to show that $L(x)$ is comparable to $L_1$ whenever $x$ is in the middle half of the interval $J$.
The projections of $\Omega_{L_1^{-2}}$ onto the $x$ and $y$-axes have lengths comparable to $\tilde{L}_1$ and $L_1$ respectively. It follows from Theorem \ref{thm:john} that, for those $x$ in the middle half of $J$, the height of $\Omega_{L_1^{-2}}$ in the $y$-direction is comparable to $L_1$. Since the potential $V(x,y)$ is convex, attains its minimum of $1$, and is equal to $1+L_1^{-2}$ on the boundary of $\Omega_{L_1^{-2}}$, we know that for all $x$ in the middle half of $J$, we must have $V(x,y) \leq 1 + \tfrac{1}{2}L_1^{-2}$ for some $y$.
As a result, for all $x$ fixed in the middle half of $J$, the potential $V(x,y)$ varies by an amount comparable to $L_1^{-2}$, for $y$ in an interval of length comparable to $L_1$. Therefore, for each $x$ fixed the length scale $L(x)$ is comparable to $L_1$, and hence using \eqref{eqn:Lem2.4} we have the required bound.
\end{proof}
\begin{rem}
Since Lemma 2.4 (a) in \cite{J1} played a key role in the above, let us say a few words about its proof. The upper bound in \eqref{eqn:Lem2.4} follows easily by choosing the appropriate test function, just as in the proof of Proposition \ref{prop:simpleeigenvalue}. The proof of the lower bound is slightly more complicated and makes use of the convexity of the potential to ensure that it grows at a sufficiently fast rate once we move away from its minimum.
\end{rem}
Before completing the proof of Proposition \ref{prop:L2bound}, we need one more property of the function $\mu(x)$.
\begin{lem} \label{lem:mu(x)convex}
The first eigenvalue $\mu(x)$ is a convex function of $x$.
\end{lem}
\begin{proof}{Lemma \ref{lem:mu(x)convex}}
This convexity property follows from Corollary 1.15 in \cite{BL1}. The convexity of the eigenvalue is deduced from the log concavity of the fundamental solution of the associated diffusion operator.
\end{proof}
\begin{rem}
Although in the assumptions of Corollary 1.15 in \cite{BL1}, the potential does not depend on the $x$-variable, the proof of the log concavity of the fundamental solution (and hence the convexity of the first eigenvalue) follows in the same way if $V(x,y)$ is allowed to depend on $x$, provided it remains a convex function.
\end{rem}
We can now combine Lemmas \ref{lem:mubound} and \ref{lem:mu(x)convex} to complete the proof of Proposition \ref{prop:L2bound}: Since the interval $J$ is of length comparable to $\tilde{L}_1$, Lemma \ref{lem:mubound} tells us that $\mu(x)$ varies by an amount at most comparable to $L_1^{-2}$ for $x$ in an interval of length comparable to $\tilde{L}_1$. Thus, since $\mu(x)$ is a convex function, applying the same logic as in Lemma \ref{lem:concave1}, we immediately obtain the lower bound
\begin{align} \label{eqn:L2bound1}
L_2 \geq c_1\tilde{L}_1^{1/3}L_1^{2/3} .
\end{align}
By the convexity of $V(x,y)$, given $C_2>0$, we can find $C_3>0$ to ensure that
\begin{align*}
V(x,y) \geq 1 + C_2L_1^{-2} ,
\end{align*}
whenever the point $(x,y)$ is at least $C_3\tilde{L}_1$ from $\Omega_{L_1^{-2}}$. This means that $\mu(x)$ certainly must increase by an amount comparable to $L_1^{-2}$ when $x$ is a distance comparable to $\tilde{L}_1$ from $J$, and this gives us the upper bound
\begin{align} \label{eqn:L2bound2}
L_2 \leq \frac{1}{c_1} \tilde{L}_1.
\end{align}
Combining the inequalities in \eqref{eqn:L2bound1} and \eqref{eqn:L2bound2} completes the proof of the proposition.
\end{proof}
\section{The Bound On The First Eigenvalue $\lambda$} \label{sec:la}
We recall from Proposition \ref{prop:simpleeigenvalue} that the first eigenvalue $\lambda$ satisfies
\begin{align*}
1 \leq \lambda \leq 1 + C_1L_1^{-2}.
\end{align*}
In this section we will assume that we have $\tilde{L}_1 \gg L_1$ (and hence $L_2\gg L_1$ also), and then prove the improved upper and lower bound on the eigenvalue $\lambda$ from Theorem \ref{thm:eigenvalue}. That is, we will show that $\lambda$ satisfies
\begin{align} \label{eqn:lamu}
\mu \leq \lambda \leq \mu + CL_2^{-2},
\end{align}
where $\mu$ is the first eigenvalue of the ordinary differential operator
\begin{align} \label{eqn:Adefn}
\mathcal{A} = -\frac{d^2}{dx^2} + \mu(x) .
\end{align}
The lower bound in \eqref{eqn:lamu} is more straightforward, and so we establish this bound first.
\begin{prop}[Lower bound on $\lambda$] \label{prop:lalower}
The first eigenvalue $\lambda$ satisfies
\begin{align*}
\lambda \geq \mu.
\end{align*}
\end{prop}
\begin{proof}{Proposition \ref{prop:lalower}}
As before, for each $x$ fixed, let $\Omega(x)$ be the cross-section of $\Omega$ at $x$. Then, the first Dirichlet eigenfunction $u(x,y)$ satisfies $u(x,y) = 0$ whenever $y$ is at the endpoints of the interval $\Omega(x)$. In particular, for each fixed $x$, the function $u(x,\cdot)$ is an admissible test function for the variational formulation of the first eigenvalue of the operator $\mathcal{L}(x)$. Thus,
\begin{align*}
\int_{\Omega(x)} (\partial_yu(x,y))^{2} + V(x,y)u(x,y)^2 \,\mathrm{d} y \geq \mu(x) \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y.
\end{align*}
Integrating this over $x$, and using
\begin{eqnarray*}
\left\{ \begin{array}{rlcc}
(-\Delta_{x,y} + V(x,y))u(x,y) & = \lambda u(x,y) && \text{in } \Omega \\
u(x,y) & = 0 && \text{on } \partial \Omega,
\end{array} \right.
\end{eqnarray*}
we see that
\begin{align*}
\lambda\int_{\Omega} u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y &= \int_{\Omega} (\partial_xu(x,y))^{2} + (\partial_yu(x,y))^{2} + V(x,y)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \\
& \geq \int_{\Omega} (\partial_xu(x,y))^{2} + \mu(x) u(x,y)^2 \,\mathrm{d} x\,\mathrm{d} y \\
& \geq \mu \int_{\Omega} u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y.
\end{align*}
To get the final inequality, we have defined $u(x,y)=0$ outside $\Omega$, used Fubini to calculate
the interval in $x$ first, and then used the variational formulation for the first eigenvalue $\mu$ of the operator $\mathcal{A}$ in \eqref{eqn:Adefn}. This gives us the bound $\lambda \geq \mu$, as required.
\end{proof}
We now turn to the upper bound and prove:
\begin{prop}[Upper bound on $\lambda$] \label{prop:eigbound}
We have an upper bound on the first eigenvalue $\lambda$ of the form,
\[ \lambda \leq \mu + CL_2^{-2}, \]
for an absolute constant $C>0$.
\end{prop}
\begin{rem}
From Lemma 4.2 (e) in \cite{J1}, the operator $\mathcal{A}$ defined in \eqref{eqn:Adefn} has spectral gap bounded from below by a multiple of $L_2^{-2}$. Therefore, obtaining bounds on $\lambda$ up to a precision of $CL_2^{-2}$ is important if we want this separation of variables in the $x$ and $y$ variables to be of use to us.
\end{rem}
\begin{proof}{Proposition \ref{prop:eigbound}}
As in the proof of the simple bound on $\lambda$ in Proposition \ref{prop:simpleeigenvalue}, we will again make use of the variational formulation for $\lambda$ given in \eqref{eqn:variation}. To do this we need to construct an appropriate test function, and our motivation will come from performing an approximate change of variables in the $x$ and $y$-directions. Before stating our test function, we need some definitions.
\begin{defn} \label{def:psi1}
For each fixed $x$, we define $\psi_1^{(x)}(y)$ to be the $L^2$-normalised first eigenfunction of the ordinary differential operator $\mathcal{L}(x)$. That is, $\psi_1^{(x)}(y)$ is $L^2$-normalised on the cross-section $\Omega(x)$, and satisfies
\begin{eqnarray*}
\left\{ \begin{array}{rlcc}
\left(-\frac{d^2}{dy^2} + V(x,y)\right)\psi_1^{(x)}(y) & = \mu(x) \psi_1^{(x)}(y) && \text{in } \Omega(x) \\
\psi_1^{(x)}(y) & = 0 && \text{on } \partial \Omega(x).
\end{array} \right.
\end{eqnarray*}
\end{defn}
\begin{defn} \label{def:chi}
Let $I$ be the interval of length $L_2$ from Definition \ref{def:L2}. We define the cut-off function $\chi(x)$ to be a positive function which is comparable to its maximum in the middle half of the interval $I$, and supported in the middle three quarters of $I$, such that it decays smoothly to zero from its maximum. We also require that $\chi(x)$ is $L^2$-normalised on the interval $I$. In particular, this allows us to ensure that
\begin{align*}
|\chi'(x)| \leq C_1L_2^{-3/2},
\end{align*}
for some absolute constant $C_1$.
\end{defn}
We can now define the test function $f(x,y)$ which we will use in \eqref{eqn:variation}.
\begin{defn} \label{def:f(x,y)}
We define the test function $f(x,y)$ by
\begin{align*}
f(x,y) \coloneqq \chi(x)\psi_1^{(x)}(y).
\end{align*}
\end{defn}
As a first step towards proving Proposition \ref{prop:eigbound}, we prove the following intermediate step.
\begin{prop} \label{prop:eigboundinter}
We have an upper bound for $\lambda$ of the form
\begin{align*}
\lambda \leq \mu + \int_{\Omega} \chi(x)^2(\pa_x\psi^{(x)}_1(y))^2 \,\mathrm{d} x \,\mathrm{d} y + C_1L_2^{-2},
\end{align*}
for a constant $C_1$.
\end{prop}
\begin{proof}{Proposition \ref{prop:eigboundinter}}
To obtain an upper bound on the first eigenvalue $\lambda$, we will calculate the quotient from \eqref{eqn:variation}
\begin{align} \label{eqn:eigquot}
\frac{\int_\Omega |\nabla f(x,y)|^2 \,\mathrm{d} x\,\mathrm{d} y + \int_\Omega V(x,y)|f(x,y)|^2 \,\mathrm{d} x\,\mathrm{d} y}{\int_\Omega |f(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y},
\end{align}
with $f(x,y)$ as in Definition \ref{def:f(x,y)}. Since $\psi^{(x)}(y)$ is $L^2(\Omega(x))$-normalised in $y$ for any fixed $x$, and $\chi(x)$ is $L^2(I)$-normalised in $x$, first computing the integral in $y$, and then the integral in $x$, we see that the denominator in \eqref{eqn:eigquot} is equal to $1$. Thus, we have the bound
\begin{align} \label{eqn:eigquot1}
\lambda \leq \int_\Omega |\nabla_{x,y}\left(\chi(x)\psi_1^{(x)}(y)\right)|^2 \,\mathrm{d} x \,\mathrm{d} y + \int_\Omega V(x,y) \chi(x)^2 \psi^{(x)}_1(y)^2 \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
For each $x$, the function $\psi^{(x)}(y)$ satisfies
\begin{align} \label{eqn:L2norm}
\int_{\Omega(x)} \psi^{(x)}_1(y)^2 \,\mathrm{d} y = 1,
\end{align}
and it is equal to $0$ at the endpoints of the interval $\Omega(x)$. Therefore, differentiating \eqref{eqn:L2norm} with respect to $x$, we obtain the orthogonality relation
\begin{align*}
\int_{\Omega(x)} \pa_x\psi^{(x)}_1(y) \psi^{(x)}_1(y) \,\mathrm{d} y = 0.
\end{align*}
Thus, calculating the derivatives in the first integral in \eqref{eqn:eigquot1}, and using this orthogonality relation, we see that \eqref{eqn:eigquot1} becomes
\begin{align*}
\lambda \leq &\int_\Omega \chi'(x)^2\psi^{(x)}_1(y)^2 \,\mathrm{d} x \,\mathrm{d} y + \int_\Omega \chi(x)^2(\pa_x\psi^{(x)}_1(y))^2 \,\mathrm{d} x \,\mathrm{d} y \\
&+ \int_{\Omega}\chi(x)^2(\partial_y\psi^{(x)}_1(y))^2 \,\mathrm{d} x \,\mathrm{d} y + \int_\Omega V(x,y) \chi(x)^2 \psi^{(x)}_1(y)^2 \,\mathrm{d} x \,\mathrm{d} y.
\end{align*}
The eigenfunction $\psi^{(x)}_1(y)$ of $\mathcal{L}(x)$ has eigenvalue $\mu(x)$, and so we have the inequality
\begin{align*}
\lambda \leq \int_I \chi'(x)^2 \,\mathrm{d} x + \int_\Omega \chi(x)^2(\pa_x\psi^{(x)}_1(y))^2 \,\mathrm{d} x \,\mathrm{d} y +\int_{I} \chi(x)^2\mu(x) \,\mathrm{d} x.
\end{align*}
From Definition \ref{def:L2} we know that
\begin{align*}
|\mu(x) - \mu| \leq L_2^{-2}.
\end{align*}
Therefore, combining this with the bound on $\chi'(x)$ given in Definition \ref{def:chi}, we obtain the desired upper bound on $\lambda$ of
\begin{align*}
\lambda \leq \mu + \int_\Omega \chi(x)^2(\pa_x\psi^{(x)}_1(y))^2 \,\mathrm{d} x \,\mathrm{d} y +C_1L_2^{-2}.
\end{align*}
\end{proof}
As a result of Proposition \ref{prop:eigboundinter}, to obtain an upper bound on $\lambda$, we need to consider the derivative with respect to $x$ of the eigenfunction $\psi^{(x)}_1(y)$. In particular, we want to bound
\begin{align*}
\int_{\Omega(x)} (\partial_x\psi^{(x)}_1(y))^2 \,\mathrm{d} y.
\end{align*}
We will prove the following proposition:
\begin{prop} \label{prop:paxeigupperbound}
Let $x$ be fixed in the support of the cut-off function $\chi(x)$. Then,
\begin{align*}
\int_{\Omega(x)} (\pa_x\psi^{(x)}_1(y))^2 \,\mathrm{d} y \leq C_1L_2^{-2},
\end{align*}
with the constant $C_1$ independent of $x$.
\end{prop}
\begin{rem}
Combining Proposition \ref{prop:eigboundinter} with Proposition \ref{prop:paxeigupperbound} establishes
\begin{align*}
\lambda \leq \mu + C_1L_2^{-2},
\end{align*}
and finishes the proof of Proposition \ref{prop:eigbound}.
\end{rem}
\begin{proof}{Proposition \ref{prop:paxeigupperbound}}
Throughout the proof of this proposition, $x \in I$ will be fixed in the support of the cut-off function $x$, and all bounds that appear will be uniform in $x$. We will also suppress the dependence of certain functions on $x$ where this simplifies the notation.
Since,
\begin{align*}
\left( -\frac{d^2}{dy^2} + V(x,y)\right)\psi^{(x)}_1(y) = \mu(x) \psi^{(x)}_1(y),
\end{align*}
differentiating with respect to $x$ we find that for $y\in \Omega(x)$, we have
\begin{align} \label{eqn:paeigx}
\left( -\frac{d^2}{dy^2} + V(x,y) - \mu(x)\right)\partial_x\psi^{(x)}_1(y) = \mu'(x) \psi^{(x)}_1(y) - \partial_xV(x,y) \psi^{(x)}_1(y),
\end{align}
where the notation $'$ denotes differentiation with respect to $x$. Although, for each fixed $x$, $\psi^{(x)}_1(y)$ is equal to zero at the endpoints on $\Omega(x)$, the function $\pa_x\psi^{(x)}_1(y)$ will not in general be zero here.
Therefore, we will also need to take into account its boundary values. For those $x$ in the support of the cut-off function $\chi(x)$, we can write the two parts of $\partial \Omega$ below and above in the $y$-direction as $\{ y= g_1(x)\}$ and $\{ y= g_2(x)\}$, where $g_1(x)$ and $g_2(x)$ are convex and concave functions respectively. We set $\alpha = \partial_x\psi_1^{(x)}(g_2(x))$, and define
\begin{align} \label{eqn:gdef}
g(y) \coloneqq \pa_x\psi^{(x)}_1(y) - \alpha.
\end{align}
Our aim is to find an expression for the function $g(y)$ using \eqref{eqn:paeigx}. To do this we need to make the following definitions (again suppressing the dependence on $x$ throughout).
\begin{defn} \label{def:F}
We define the function $F(y)$ by,
\begin{align*}
F(y) \coloneqq V(x,y) -\mu(x).
\end{align*}
\end{defn}
We know that $\mu(x) \leq 1+C_1L_1^{-2}$, and that $\min_yV(x,y) \leq \mu(x)$ for all $x$ in the support of $\chi(x)$. This allows us to define the three points $y_1$, $y_2$ and $y_3$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figure3Eigenfunctions19Sep14.pdf}
\caption{The Points $y_1$, $y_2$ and $y_3$ from Definition \ref{def:y}}
\label{fig:y}
\end{center}
\end{figure}
\begin{defn} \label{def:y}
We fix an absolute constant $C$. We define $y_1$ to be the middle point of the \textquoteleft centre', where the centre is the interval on which $V(x,y) \leq \min_yV(x,y) + CL_1^{-2}$. We then choose $y_2\geq y_1$ to be the largest value such that $[y_1,y_2]$ is contained in the middle half of the centre. Finally, we define $y_3\geq y_2$ to be the value of $y$ for which $F(y_3) = V(x,y_3) - \mu(x) = 0.$ (See Figure \ref{fig:y})
\end{defn}
\begin{defn} \label{def:phitilde}
We set $\phi(y)$ to be the first eigenfunction of $\mathcal{L}(x)$, but this time normalised to be positive with a maximum of $1$. Note that this function is equal to a multiple of $\psi^{(x)}_1(y)$ (where the multiple depends on the fixed value of $x$).
For $y \geq y_1$, we define the function $\tilde{\phi}(y)$ by
\begin{align*}
\tilde{\phi}(y) \coloneqq \phi(y) \int_{y_1}^{y} \phi(t)^{-2} \,\mathrm{d} t.
\end{align*}
\end{defn}
We can now write down an expression for the function $g(y)$.
\begin{lem} \label{lem:gexpression}
Let $c_0(x)$ be the value such that
\begin{align*}
g(y) - c_0(x)\psi^{(x)}_1(y) = 0
\end{align*}
at $y=y_1$. Then, for $y \geq y_1$, the function $g(y)$ satisfies
\begin{align} \label{eqn:gexpression}
g(y) - c_0(x)\psi^{(x)}_1(y) = \phi(y)\int_{y_1}^y \tilde{\phi}(t)G(x,t) \,\mathrm{d} t + \tilde{\phi}(y)\int_y^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t,
\end{align}
where $G(x,y)$ is equal to
\begin{align*}
G(x,y) = \mu'(x) \psi^{(x)}_1(y) - \partial_xV(x,y) \psi^{(x)}_1(y) + (V(x,y) - \mu(x))\alpha .
\end{align*}
\end{lem}
\begin{proof}{Lemma \ref{lem:gexpression}}
We see from the definition of $g(y)$ from \eqref{eqn:gdef} and the equation that $\pa_x\psi^{(x)}_1(y)$ satisfies in \eqref{eqn:paeigx}, that we have
\begin{align*}
\left( -\frac{d^2}{dy^2} + V(x,y) -\mu(x)\right)g(y) = \mu'(x) \psi^{(x)}_1(y) - \partial_xV(x,y) \psi^{(x)}_1(y) + (V(x,y) - \mu(x))\alpha.
\end{align*}
The right hand side of the above equation is equal to $G(x,y)$, so that
\begin{align} \label{eqn:geqn}
( \mathcal{L}(x) -\mu(x))(g(y)-c_0(x)\phi(y)) = G(x,y).
\end{align}
Since $\mathcal{L}(x)$ is a second order ordinary differential operator, to find an expression for $g(y)$ we will apply the method of variation of parameters to \eqref{eqn:geqn}. From Definition \ref{def:phitilde}, we know that
\begin{align*}
( \mathcal{L}(x) -\mu(x))\phi(y) = 0,
\end{align*}
with $\phi(g_2(x)) = 0$. It is straightforward to check that the function $\tilde{\phi}(y)$ from Definition \ref{def:phitilde} also satisfies
\begin{align*}
( \mathcal{L}(x) -\mu(x))\tilde{\phi}(y) = 0,
\end{align*}
for $y\geq y_1$, and is equal to $0$ at $y=y_1$. Thus, since the function $g(y) - c_0(x)\phi(y)$ is equal to $0$ at $y=y_1$ and $y=g_2(x)$, using \eqref{eqn:geqn} and variation of parameters, we can write
\begin{align*}
g(y) - c_0(x)\psi^{(x)}_1(y) = \phi(y)\int_{y_1}^y \tilde{\phi}(t)G(x,t) \,\mathrm{d} t + \tilde{\phi}(y)\int_y^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t.
\end{align*}
\end{proof}
Looking at this expression for $g(y)$, we see that we will need to study how the magnitude of the functions $\phi(y)$ and $\tilde{\phi}(y)$ depends on the size of the potential $V(x,y)$, and its derivative with respect to $x$, $\partial_xV(x,y)$. Also, since $g(y) = \pa_x\psi^{(x)}_1(y) - \alpha$, where $\alpha = \partial_x\psi_1^{(x)}(g_2(x))$, we will also need to estimate the size of $\pa_x\psi^{(x)}_1(y)$ at the endpoints of the interval $\Omega(x)$.
\subsection{Properties of $\phi(y)$}
We first study the function $\phi(y)$, where we recall that it satisfies
\begin{align*}
\left(-\frac{d^2}{dy^2} + V(x,y) - \mu(x)\right) \phi(y) = 0.
\end{align*}
For $x$ fixed in the support of $I$, let us set $L(x)$ to be the largest value such that $V(x,y)$ varies from its minimum value by $L(x)^{-2}$ on an interval in $y$ of length at least $L(x)$. Then, as we remarked in the proof of Lemma \ref {lem:mubound}, $L(x)$ is comparable to $L_1$. Thus, from Lemma 2.4 (b), (d) in \cite{J1}, we immediately get the following estimates on $\phi(y)$ (uniformly in $x$).
\begin{lem} \label{lem:phibasic}
There exists an absolute constant $C_1$ such that the eigenfunction $\phi(y)$ (which we recall will depend on $x$) satisfies
\begin{align*}
|\phi'(y)| \leq C_1/L_1 \text{ for all } y \in \Omega(x),
\end{align*}
and
\begin{align*}
\phi(y) \leq C_1e^{-c|y-y_1|/L_1} ,
\end{align*}
where $y_1$ is the point in the \textquoteleft centre' given in Definition \ref{def:y}.
\end{lem}
This second inequality gives an $L^{\infty}$ exponential decay estimate for $\phi(y)$ as we move away from the minimum of $V(x,y)$ on a length scale comparable to $L_1$. In particular, this means that the $L^2(\Omega(x))$ norm of $\phi(y)$ is bounded above by a multiple of $L_1^{1/2}$. (In fact, it follows from Lemma 2.4 in \cite{J1} that the $L^2(\Omega(x))$-norm also has a lower bound that is comparable to $L_1^{1/2}$.)
We now want to sharpen this $L^{\infty}$ exponential decay estimate for $\phi(y)$ as $V(x,y)$ increases from its minimum.
\begin{prop} \label{prop:phi}
Define the interval $J_k$ by,
\begin{align} \label{eqn:Jk}
J_k= [t_k,t_{k+1}] \coloneqq \{ t\geq y_3: \partial_tV(x,t) \in [2^{-k}, 2^{-k+1}] \}.
\end{align}
Then, for all $t_k\leq t \leq g_2(x)$,
\begin{align*}
\phi(t) \leq \phi(t_k) \exp(-(t-t_k)2^{-k/3}/10),
\end{align*}
for all $y_3 \leq t\leq t_{k+1}$,
\begin{align*}
\phi(t_{k+1}) \leq \phi(t) \exp(-(t_{k+1}-t)2^{-k/3}/10)
\end{align*}
and for all $t \in J_k$,
\begin{align*}
\phi(t) \leq |\phi'(t)| 2^{k/3}.
\end{align*}
For the interval $\tilde{J}_k$ defined by,
\begin{align} \label{eqn:Jktilde}
\tilde{J}_k = [\tilde{t}_k,\tilde{t}_{k+1}] \coloneqq \{ t\geq y_3: V(x,t) - \min_{t}V(x,t) \in [2^{-2k/3}, 2^{-2(k-1)/3}]\},
\end{align}
we have the analogous bounds on $\phi(t)$.
\end{prop}
\begin{rem}
We have the analogous decay estimates for $\phi(y)$ as we move away from the region where $V(x,y) \leq \min_yV(x,y) + L_1^{-2}$ in the other direction.
\end{rem}
\begin{rem}
We recall that $y=y_3$ is the point where $V(x,y) - \mu(x) = 0$. Since $\min_{y}V(x,y) - \mu(x) \leq -cL_1^{-2}$, by convexity, $J_k$ and $\tilde{J}_k$ are only non-empty for those $k$ satisfying $2^k \leq CL_1^3$, for some absolute constant $C>0$.
\end{rem}
\begin{proof}{Proposition \ref{prop:phi}}
The proposition follows from the key inequality given in the proof of Theorem A in \cite{J1},
\begin{align*}
\left|\left(\log \phi(t)\right)' \right|= |\phi'(t)|/\phi(t) \geq 2^{-k/3}/10\qquad \text{ for all } t \in J_k.
\end{align*}
Integrating this inequality from both $t=t_k$ and $t=t_{k+1}$ gives all of the desired estimates involving the intervals $J_k$.
By the definition of the intervals $\tilde{J}_k$, we have $V(x,t)-\mu(x) \geq 2^{-2k/3}$ for $t \in \tilde{J}_k$. Therefore, it is straightforward to obtain the same bounds for $\left(\log \phi(t)\right)' $, and hence $\phi(t)$ itself on $\tilde{J}_k$ as for the intervals $J_k$.
\end{proof}
We now show to what extent $\phi'(y)$ inherits this exponential decay as we move away from the centre.
\begin{prop} \label{prop:phiy}
Let the intervals $J_k$ be defined as in Proposition \ref{prop:phi}. Then, for all $t\geq t_k$,
\begin{align*}
|\phi'(t)| \leq C|\phi'(t_k)|\exp(-c|t-t_k|2^{-k/3}),
\end{align*}
for some absolute constants $c$ and $C>0$.
\end{prop}
\begin{proof}{Proposition \ref{prop:phiy}}
The function $\phi(t)$ satisfies the equation
\begin{align*}
\phi''(t) = F(t)\phi(t),
\end{align*}
with the function $F(t) = V(x,t) - \mu(x)$ as before. On the intervals $J_k$, we know that $t \geq y_3$, and so certainly $F(t) \geq 0$. Also, $\phi'(t) \leq 0$, and so this mean that $|\phi'(t)|$ is decreasing. Thus, for $t \geq t_k$, we have
\begin{align*}
|\phi'(t)| \leq |\phi'(t_k)|.
\end{align*}
If $|t-t_k| \leq 2^{k/3}$, then this is enough to establish the required bound.
Now suppose that $|t-t_k| \in [N2^{k/3},(N+1)2^{k/3}]$ for some $N\geq1$. Then, by Proposition \ref{prop:phi}, we know that $\phi(t)$ satisfies
\begin{align*}
\phi(t) \leq C2^{k/3}|\phi'(t_k)| \exp(-cN2^{-k/3}).
\end{align*}
In particular, $\phi(t)$ changes by at most $C2^{k/3}|\phi'(t_k)|\exp(-cN2^{-k/3})$, as $t$ ranges over this interval of length $2^{k/3}$. Since $\phi'(t)$ is negative here, this gives us a bound on the integral of $|\phi'(t)|$ over this interval.
Moreover, as we noted above, by convexity, $|\phi'(t)|$ decreases as $t$ increases. In particular, since the interval $[N2^{k/3},(N+1)2^{k/3}]$ has length $2^{k/3}$, this means that
\begin{align*}
|\phi'(t)| \leq C2^{k/3}|\phi'(t_k)|\exp(-cN2^{-k/3}). 2^{-k/3} = C|\phi'(t_k)|\exp(-cN2^{-k/3}),
\end{align*}
for $t$ at the right endpoint of the interval. This concludes the proof of the proposition.
\end{proof}
It will often be important to measure the distance of a point $(x,y)$ from the level sets $\{(x,y)\in\Omega: V(x,y) = 1+L_1^{-2}\}$.
\begin{defn} \label{def:ystar}
Fix a large absolute constant $C^*$. Then, suppressing the dependence on $x$, let $y^*\geq y_1$ be the first point where $V(x,y) \geq 1+C^*L_1^{-2}$.
\end{defn}
We can now write down an immediate corollary of Proposition \ref{prop:phiy}.
\begin{cor} \label{cor:phiy}
For any $t\geq t_k$, we have the first derivative estimate
\begin{align*}
|\phi'(t)| \leq CL_1^{-1}\exp(-c|t-t_k|2^{-k/3})\exp(-c|t_k-y^*|/L_1).
\end{align*}
\end{cor}
\begin{proof}{Corollary \ref{prop:phiy}}
We can apply Proposition \ref{prop:phiy} with $t$ replaced by $t_k$ and $t_k$ replaced by $y^*$ to obtain a bound on $|\phi'(t_k)|$ of the form
\begin{align*}
|\phi'(t_k)| \leq CL_1^{-1} \exp(-c|t_k-y^*|/L_1).
\end{align*}
We then use this bound in the right hand side of the estimate for $|\phi'(t)|$ in Proposition \ref{prop:phiy} to get the desired result.
\end{proof}
\subsection{Properties of $\tilde{\phi}(y)$}
From Lemma \ref{lem:gexpression}, we see that as well as $\phi(y)$, it will also be important to study the properties of $\tilde{\phi}(y)$, where we recall that for $y \geq y_1$, we have
\begin{align*}
\tilde{\phi}(y) = \phi(y) \int_{y_1}^{y} \phi(t)^{-2} \,\mathrm{d} t.
\end{align*}
We recall from Definition \ref{def:y} that $y_2\geq y_1$ is the largest value of $y_2$ such that $[y_1,y_2]$ is contained in the middle half of the \textquoteleft centre', where $V(x,y) \leq \min_{t}V(x,t) + CL_1^{-2}$ and that $y_3\geq y_2$ is the value of $y$ for which $F(y_3) = V(x,y_3) - \mu(x) = 0.$ We now prove:
\begin{lem} \label{lem:tphi}
The function $\tilde{\phi}(y)$ satisfies
\begin{align*}
\tilde{\phi}(y) \leq C_1L_1,
\end{align*}
for $y_1\leq y \leq y_3$ and
\begin{align*}
\tilde{\phi}(y) \leq C_1L_1 + C_1|\phi'(y)|^{-1},
\end{align*}
for $y_3\leq y \leq g_2(x)$.
\end{lem}
\begin{proof}{Lemma \ref{lem:tphi}}
We first consider the interval $[y_1,y_2]$. By the definition of the point $y_2$, Lemma 2.4 in \cite{J1} implies that we have an absolute lower bound on $\phi(t)$ for $t\in[y_1,y_2]$, and we know that this interval is of length comparable to $L_1$. Thus, for $y\in[y_1,y_2]$, we have
\begin{align*}
\tilde{\phi}(y) \leq C_1L_1 \phi(y).
\end{align*}
Before considering $y\in[y_2,y_3]$, we first assume that $y\geq y_3$. Here $F(y) \geq 0$, and so $|\phi'(y)|$ is decreasing ($\phi'(y)$ is becoming less negative). Therefore, for $t\in[y_3,y]$, we have the lower bound
\begin{align*}
\phi(t) \geq \phi(y) + |\phi'(y)|(y-t).
\end{align*}
This gives us the bound
\begin{align} \label{eqn:tphilem1}
\int_{y_3}^y \phi(t)^{-2} \,\mathrm{d} t \leq C_1\phi(y)^{-1}|\phi'(y)|^{-1}.
\end{align}
We now want to bound
\begin{align*}
\int_{y_2}^{y_3} \phi(t)^{-2} \,\mathrm{d} t.
\end{align*}
Since $\phi''(y) = F(y)\phi(y)$, we have
\begin{align*}
\phi'(y) = \int_{\tilde{y}}^yF(t)\phi(t) \,\mathrm{d} t,
\end{align*}
where $\phi(y)$ attains its maximum of $1$ at $y=\tilde{y}$. For $t\in[y_2,y_3]$, $F(t) \leq 0$, and so $|\phi'(t)|$ is increasing from $0$, $\phi(t)$ is decreasing from $1$, and $|y_3-y_2| \leq C_1L_1$. Therefore, either $\phi(t)$ is bounded below by an absolute constant or else $|\phi'(y)| \geq C_1L_1^{-1}$. This gives us the bound
\begin{align} \label{eqn:tphilem2}
\int_{y_2}^{y_3} \phi(t)^{-2} \,\mathrm{d} t \leq C_1L_1.
\end{align}
Combining the bounds in \eqref{eqn:tphilem1} and \eqref{eqn:tphilem2} shows that
\begin{align*}
\tilde{\phi}(y) \leq C_1L_1,
\end{align*}
for $y \in[y_2,y_3]$, and
\begin{align*}
\tilde{\phi}(y) \leq C_1L_1 + C|\phi'(y)|^{-1}
\end{align*}
for $y\geq y_3$, as required.
\end{proof}
\subsection{An Estimate For $\pa_x\psi^{(x)}_1(y)$ At The Boundary}
We can now bound $\pa_x\psi^{(x)}_1(y)$ at the endpoints of the interval $\Omega(x)$. For each fixed $x$, $\psi^{(x)}_1(y)$ has zero boundary conditions on $\Omega(x)$. However, since the interval $\Omega(x)$ will in general depend on $x$, $\partial_x\psi^{(x)}_1(y)$ will not necessarily be zero when $y$ is at the end-points of $\Omega(x)$.
We recall from Definition \ref{def:ystar}, that $y^*\geq y_1$ is the first point where $V(x,y) \geq 1+ C^*L_1^{-2}$, for a fixed large constant $C^*$. The upper endpoint of the interval $\Omega(x)$ is equal to $g_2(x)$, and we set
\begin{align} \label{eqn:Mdef}
M \coloneqq g_2(x) - y^*,
\end{align}
which is the distance between the endpoint of $\Omega(x)$ and the region where the potential $V(x,y)$ is less than $1+C^*L_1^{-2}$. We can prove a bound on $\partial_x\psi_1^{(x)}(g_2(x))$ in terms of $M$.
\begin{prop} \label{prop:boundary}
For $y = g_2(x)$ equal to the upper endpoint of the interval $\Omega(x)$, we have the bound
\begin{align*}
|\alpha| = \left|\partial_x\psi_1^{(x)}(g_2(x))\right| \leq CL_2^{-1}L_1^{-3/2}(L_1+M)\exp(-cML_1^{-1}).
\end{align*}
We also have an analogous bound for $y$ equal to the lower endpoint of $\Omega(x)$.
\end{prop}
\begin{proof}{Proposition \ref{prop:boundary}}
We can view $\psi^{(x)}_1(y)$ as a function of two variables on the domain $\Omega$, with $\psi^{(x)}_1(y)$ identically equal to $0$ on $\partial \Omega$. In particular, for those $x$ in the support of the cut-off function $\chi(x)$, we have written the upper boundary of $\Omega$ as the graph of the function $y=g_2(x)$, and so $\psi_1^{(x)}(g_2(x))$ is identically zero as a function of $x$. Differentiating this with respect to $x$ gives
\begin{align} \label{eqn:gradbound}
\partial_x\psi_1^{(x)}(g_2(x)) = -g_2'(x)\partial_y\psi_1^{(x)}(g_2(x)).
\end{align}
Thus, to obtain a bound on $\partial_x\psi_1^{(x)}(g_2(x))$, it is enough to consider $\partial_y\psi_1^{(x)}(y)$, and the slope of $\partial \Omega$ at $(x,g_2(x))$.
We remarked in the definition of $\phi(y)$ in Definition \ref{def:phitilde} that the eigenfunction $\psi^{(x)}_1(y)$ is equal to a multiple of $\phi(y)$. Since $\phi(y)$ has $L^2(\Omega(x))$-norm comparable to $L_1^{1/2}$, whereas $\psi^{(x)}_1(y)$ is $L^2(\Omega(x))$-normalised, this multiple is comparable to $L_1^{-1/2}$. Thus, by the bound on $\phi'(y)$ from Proposition \ref{prop:phiy}, with $2^k$ comparable to $L_1^3$, we have the bound
\begin{align*}
\left|\partial_y\psi_1^{(x)}(g_2(x))\right| \leq CL_1^{-3/2}\exp\left(-cML_1^{-1}\right).
\end{align*}
Therefore, by \eqref{eqn:gradbound}, to conclude the proof of the proposition it is enough to show that
\begin{align} \label{eqn:boundary1}
|g_2'(x)| \leq C(L_1+M)L_2^{-1},
\end{align}
for an absolute constant $C>0$. Recall the set $\Omega_{L_1^{-2}} = \{(x,y)\in\Omega: V(x,y) \leq 1 + L_1^{-2}\}$. This is a convex subset of $\Omega$ with height comparable to $L_1$ in the $y$-direction, and length comparable to $\tilde{L}_1$ in the $x$-direction. Moreover, for $x$ fixed in the support of $\chi(x)$, we are at a distance at least comparable to $L_2$ from the left and right ends of $\Omega_{L_1^{-2}}$. Therefore, if we write the upper boundary of $\Omega_{L_1^{-2}}$ of this set as the graph of a function $y=v(x)$, then certainly we have the derivative bound
\begin{align*}
|v'(x)| \leq CL_1L_2^{-1}.
\end{align*}
In particular, if the distance $M$ is bounded above by a multiple of $L_1$, then by convexity, the part of $\partial\Omega$ for $x$ contained in the support of $\chi(x)$ has slope bounded by a multiple of $L_1L_2^{-1}$. This gives the desired bound for $g_2'(x)$ in \eqref{eqn:boundary1},
\begin{align*}
|g_2'(x)| \leq CL_1L_2^{-1}
\end{align*}
If the distance $M$ is large compared to $L_1$, then the domain $\Omega$ is convex, and contains an ellipse of height comparable to $M$ in the $y$-direction, and length comparable to $L_2$ in the $x$-direction. Thus, the part of $\partial\Omega$ with $x$ in the support of $\chi(x)$ has slope bounded by a multiple of $ML_2^{-1}$. Again we get a bound for $g_2'(x)$,
\begin{align*}
|g_2'(x)| \leq CML_2^{-1}
\end{align*}
which implies the bound in \eqref{eqn:boundary1}.
This establishes the estimate in \eqref{eqn:boundary1} in all cases, and completes the proof of the proposition.
\end{proof}
We have now established the properties of the functions $\phi(y)$ and $\tilde{\phi}(y)$ together with the bound required on $\alpha = \partial_x\psi_1^{(x)}(g_2(x))$. Thus, we return to the expression for
\begin{align*}
g(y) = \pa_x\psi^{(x)}_1(y) - \alpha
\end{align*}
that we derived in Lemma \ref{lem:gexpression}:
\begin{align} \label{eqn:gexpression2}
g(y) - c_0(x)\psi^{(x)}_1(y) = \phi(y)\int_{y_1}^y \tilde{\phi}(t)G(x,t) \,\mathrm{d} t + \tilde{\phi}(y)\int_y^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t,
\end{align}
where $G(x,y)$ equals
\begin{align} \label{eqn:Gexpression}
G(x,y) = \mu'(x) \psi^{(x)}_1(y) - \partial_xV(x,y) \psi^{(x)}_1(y) + (V(x,y) - \mu(x))\alpha .
\end{align}
We will use \eqref{eqn:gexpression2} to obtain the desired bound on $\pa_x\psi^{(x)}_1(y)$:
\begin{prop} \label{prop:paxeigpointwise}
As usual, for $2^k \leq CL_1^3$, let the intervals $J_k$ be given by
\begin{align*}
J_k = [t_k,t_{k+1}] = \{t\geq y_3: \partial_tV(x,t) \in [2^{-k},2^{-k+1}] \}.
\end{align*}
We have the pointwise bound
\begin{align*}
\left| \pa_x\psi^{(x)}_1(y) - c_0(x)\psi_1^{(x)}(y) \right| \leq F_1(y) + F_2(y)
\end{align*}
for all $y\in \Omega(x)$ with $y\geq y_1$. Here $F_1(y)$ is a positive function on $\Omega(x)$, with a maximum comparable to $L_2^{-1}L_1^{-1/2}$ and decaying exponentially from this maximum on a length scale comparable to $L_1$ as $y$ moves away from the interval where $V(x,y) \leq 1+ L_1^{-2}$. The function $F_2(y)$ is also a positive function on $\Omega(x)$, with a maximum comparable to $L_2^{-1}L_1^{-1/2}$ but it decays exponentially from this maximum within each interval $J_k$ on a length scale comparable to $2^{k/3}$. We also have the analogous exponential decay estimate on the corresponding intervals as we move away from the \textquoteleft centre' region where $V(x,y) \leq 1 + L_1^{-2}$ in the opposite direction with $y \leq y_1$.
\end{prop}
Before we prove this proposition, let us show how it implies the $L^2(\Omega(x))$-bound on $\pa_x\psi^{(x)}_1(y)$ given in Proposition \ref{prop:paxeigupperbound}: We saw in the proof of Proposition \ref{prop:eigboundinter} that $\pa_x\psi^{(x)}_1(y)$ and $\psi^{(x)}_1(y)$ satisfy the orthogonality relation
\begin{align*}
\int_{\Omega(x)} \pa_x\psi^{(x)}_1(y) \psi^{(x)}_1(y) \,\mathrm{d} y = 0.
\end{align*}
Thus, since $\psi^{(x)}_1(y)$ is $L^2(\Omega(x))$-normalised, we have the expression
\begin{align*}
c_0(x) = \int_{\Omega(x)} \left(c_0(x)\psi^{(x)}_1(y) - \pa_x\psi^{(x)}_1(y)\right)\psi^{(x)}_1(y) \,\mathrm{d} y .
\end{align*}
Using the bound on $c_0(x)\psi^{(x)}_1(y) - \pa_x\psi^{(x)}_1(y)$ in Proposition \ref{prop:paxeigpointwise} we obtain
\begin{align} \label{eqn:c0bound}
|c_0(x)| \leq C_1L_2^{-1}L_1^{-1/2}\int_{\Omega(x)} \psi^{(x)}_1(y) \,\mathrm{d} y \leq C_1L_2^{-1},
\end{align}
where the final inequality holds since $\psi^{(x)}_1(y)$ has $L^2(\Omega(x))$-norm equal to $1$, and decays exponentially away from its maximum on a length scale comparable to $L_1$.
Combining this bound on $c_0(x)$ in \eqref{eqn:c0bound} with Proposition \ref{prop:paxeigpointwise}, we see that $\pa_x\psi^{(x)}_1(y)$ can be bounded by functions $F_1(y) + F_2(y)$ with the same properties as in the statement of Proposition \ref{prop:paxeigpointwise}. This gives us an $L^2(\Omega(x))$-bound on $\pa_x\psi^{(x)}_1(y)$ of the form
\begin{align*}
\int_{\Omega(x)} \left(\pa_x\psi^{(x)}_1(y) \right)^2 \,\mathrm{d} y \leq C_1L_2^{-2} + \sum_{2^k \leq CL_1^3} 2^{k/3}L_2^{-2} L_1^{-1} \leq C_1L_2^{-2}.
\end{align*}
This completes the proof of Proposition \ref{prop:paxeigupperbound}.
\end{proof}
Since Proposition \ref{prop:paxeigupperbound} implies the desired upper bound on the eigenvalue $\lambda$ in Proposition \ref{prop:eigbound}, we just need to prove Proposition \ref{prop:paxeigpointwise}.
\begin{proof}{Proposition \ref{prop:paxeigpointwise}}
From Proposition \ref{prop:boundary} we know that
\begin{align*}
|\pa_x\psi^{(x)}_1(y) - g(y)| = | \partial_x\psi_1^{(x)}(g_2(x))| = |\alpha| \leq CL_2^{-1}L_1^{-3/2}(L_1+ M)\exp\left(-cML_1^{-1}\right),
\end{align*}
and this bound has the same properties as the function $F_1(y)$ in the statement of the proposition. Therefore, to prove Proposition \ref{prop:paxeigpointwise}, it is enough to show that
\begin{align*}
g(y) - c_0(x)\psi_1^{(x)}(y)
\end{align*}
has the desired bounds.
To do this, we want to bound the right hand side of \eqref{eqn:gexpression2}, which contains the functions $\phi(y)$, $\tilde{\phi}(y)$ together with $G(x,y)$. The two remaining functions which we have not discussed above are the functions $\mu'(x)$ and $\partial_xV(x,y)$ appearing in $G(x,y)$. Therefore, let us prove two simple lemmas concerning these functions, and then we will be in a position to bound \eqref{eqn:gexpression2}.
\begin{lem} \label{lem:muprime}
Let $x$ be in the support of the cut-off function $\chi(x)$. Then, we have the bound
\begin{align*}
|\mu'(x)| \leq C_1L_2^{-3},
\end{align*}
for an absolute constant $C_1>0$.
\end{lem}
\begin{proof}{Lemma \ref{lem:muprime}}
We recall from Lemma \ref{lem:mu(x)convex} that the function $\mu(x)$ is a convex function of $x$. Moreover, by the definition of the parameter $L_2$, we know that $\mu(x)$ varies by $L_2^{-2}$ for $x$ in an interval of length at least $L_2$. Since the support of $\chi(x)$ is contained within the middle half of this interval, we immediately obtain the bound
\begin{align*}
|\mu'(x)| \leq C_1L_2^{-3},
\end{align*}
by convexity.
\end{proof}
\begin{lem} \label{lem:paxV}
Let $x$ be in the support of $\chi(x)$, and as in Definition \ref{def:ystar} let $y=y^*$ be the first point where $V(x,y) \geq 1+C^*L_1^{-2}$. Then, for $y\geq y^*$,
\begin{align*}
|\partial_xV(x,y)| \leq C_1(|y-y^*|+L_1)L_2^{-1}|\partial_yV(x,y)|,
\end{align*}
and for $y_1\leq y\leq y^*$,
\begin{align*}
|\partial_xV(x,y)| \leq C_1L_1^{-2}L_2^{-1} + C_1L_1L_2^{-1}|\partial_yV(x,y)|.
\end{align*}
\end{lem}
\begin{proof}{Lemma \ref{lem:paxV}}
Given, $c$, let $y=f(x)$ be a parameterisation of the upper part of the level set $\{(x,y) \in \Omega : V(x,y) = c\}$, so that
\begin{align*}
V(x,f(x)) = c = \text{constant}.
\end{align*}
Differentiating this with respect to $x$, we see that
\begin{align} \label{eqn:paxV1}
\partial_xV(x,f(x)) = -f'(x)\partial_yV(x,f(x)).
\end{align}
Assume first that $y=f(x) \geq y^*$. The sublevel set $\{(x,y)\in\Omega :V(x,y) \leq 1+ C^*L_1^{-2}\}$ is convex with height comparable to $L_1$ in the $y$-direction and length comparable to $\tilde{L}_1$ in the $x$-direction, and $x$ is at distance comparable to $L_2$ from the ends of this set. Thus, by the convexity of the sublevel sets, we certainly have a bound on the slope of
\begin{align*}
|f'(x)| \leq C_1(|y-y^*|+L_1)L_2^{-1}.
\end{align*}
Using this bound in the right hand side of \eqref{eqn:paxV1} gives the desired bound for $y\geq y^*$.
We now suppose that $y_1 \leq y=f(x) \leq y^*$. If $y$ is in the middle half of the interval $\{t: V(x,t) \leq 1+L_1^{-2}\}$, then we certainly have the bound
\begin{align*}
|\partial_xV(x,f(x))| \leq C_1L_1^{-2}L_2^{-1},
\end{align*}
by the convexity of the potential $V(x,y)$. For the remaining points $(x,f(x))$ of interest, we can again use the shape of the level set to obtain the desired bound
\begin{align*}
|\partial_xV(x,f(x))| \leq C_1L_1^{-2}L_2^{-1} + C_1L_1L_2^{-1}|\partial_yV(x,f(x))|.
\end{align*}
This is because for these points we can find a direction $\mathbf{e}$, such that the directional derivative of $V$ at $(x,f(x))$ is bounded by $L_1^{-2}L_2^{-1}$, and this direction makes an angle comparable to $L_1L_2^{-1}$ with the $x$-axis.
\end{proof}
Combining Lemmas \ref{lem:muprime} and \ref{lem:paxV}, we see from \eqref{eqn:Gexpression} that
\begin{align} \label{eqn:Gexpression2} \nonumber
|G(x,t)| & \leq C_1L_1^{-2}L_2^{-1}\psi_1^{(x)}(t) + C_1(|t-y^*|+L_1)L_2^{-1}|\partial_tV(x,t)|\psi_1^{(x)}(t) + |V(x,t) - \mu(x)|\alpha \\
& \leq C_1L_1^{-5/2}L_2^{-1}\phi(t) + C_1(|t-y^*|+L_1)L_1^{-1/2}L_2^{-1}|\partial_tV(x,t)|\phi(t) + |V(x,t) - \mu(x)|\alpha.
\end{align}
The final inequality comes from
\begin{align*}
\psi_1^{(x)}(t) \leq C_1L_1^{-1/2}\phi(t),
\end{align*}
which holds since $\psi_1^{(x)}(t)$ is $L^2(\Omega(x))$-normalised, whereas $\phi(t)$ has $L^{2}(\Omega(x))$-norm comparable to $L_1^{1/2}$.
Everything is now set up to show that the two integrals in \eqref{eqn:gexpression2} have the bounds required in the statement of Proposition \ref{prop:paxeigpointwise}.
\subsection{A Bound on $\phi(y)\int_{y_1}^y \tilde{\phi}(t)G(x,t) \,\mathrm{d} t$}
We start by considering the first integral in \eqref{eqn:gexpression2},
\begin{align} \label{eqn:key1}
\left| \phi(y)\int_{y_1}^y\tilde{\phi}(t)G(x,t) \,\mathrm{d} t \right| \leq \phi(y)\int_{y_1}^y \tilde{\phi}(t)|G(x,t)| \,\mathrm{d} t .
\end{align}
Using \eqref{eqn:Gexpression2}, it is enough to bound
\begin{align} \label{eqn:key2}
\phi(y) \int_{y_1}^y \tilde{\phi}(t) \left(C_1L_1^{-5/2}L_2^{-1}\phi(t) + C_1(|t-y^*|+L_1)L_1^{-1/2}L_2^{-1}|\partial_tV(x,t)|\phi(t) + |V(x,t) - \mu(x)|\alpha \right) \,\mathrm{d} t.
\end{align}
We now bound the three terms in equation \eqref{eqn:key2}.
\begin{lem} \label{lem:G1}
We have a bound on the first term in \eqref{eqn:key2},
\begin{align*}
\phi(y) \int_{y_1}^y \tilde{\phi}(t)L_1^{-5/2}L_2^{-1} \phi(t) \,\mathrm{d} t \leq C_1L_1^{-1/2}L_2^{-1}.
\end{align*}
\end{lem}
\begin{rem}
We will see in the proof of the lemma that the function decays exponentially from its maximum away from the region where $V(x,y) \leq 1+ L_1^{-2}$ on a length scale comparable to $L_1$. Therefore we can include this term in the function $F_1(y)$ in the statement of Proposition \ref{prop:paxeigpointwise}.
\end{rem}
\begin{proof}{Lemma \ref{lem:G1}}
By Lemma \ref{lem:tphi}, we can bound the left hand side by
\begin{align*}
\phi(y)\int_{y_1}^{y_3} C_1L_1L_1^{-5/2}L_2^{-1} \phi(t)\,\mathrm{d} t + \phi(y)\int_{y_3}^{y} (C_1L_1+C_1|\phi'(t)|^{-1})L_1^{-5/2}L_2^{-1} \phi(t)\,\mathrm{d} t.
\end{align*}
Using Proposition \ref{prop:phi} we have the bound, $\phi(t) \leq 2^{k/3}|\phi'(t)| \leq C_1L_1|\phi'(t)|$ for $t\in J_k$, and so these integrals can be bounded by
\begin{align} \label{eqn:G1a}
C_1 \phi(y)\int_{y_1}^y L_2^{-1}L_1^{-3/2} \,\mathrm{d} t.
\end{align}
The eigenfunction $\phi(y)$ has a maximum of $1$, and decays exponentially away from this maximum on a length scale comparable to $L_1$. Thus, we can bound \eqref{eqn:G1a} by $C_1L_1^{-1/2}L_2^{-1}$ as required, and it also has the decay properties of the function $F_1(y)$.
\end{proof}
\begin{lem} \label{lem:G2}
We have a bound on the second term in \eqref{eqn:key2},
\begin{align*}
\phi(y) \int_{y_1}^y \tilde{\phi}(t)(|t-y^*|+L_1)L_1^{-1/2}L_2^{-1}|\partial_tV(x,t)|\phi(t) \,\mathrm{d} t \leq C_1L_1^{-1/2}L_2^{-1}.
\end{align*}
\end{lem}
\begin{rem}
We will again see in the proof of the lemma that the function decays exponentially from its maximum on a length scale comparable to $2^{k/3}$ within each interval $J_k$. Therefore we can include this term in the function $F_2(y)$ in the statement of Proposition \ref{prop:paxeigpointwise}.
\end{rem}
\begin{proof}{Lemma \ref{lem:G2}}
We first consider the part of this integral over $[y_1,y_3]$. Here, $\tilde{\phi}(t) \leq C_1L_1$, and by the convexity of the potential
\begin{align*}
\int_{y_1}^{y_3} |\partial_tV(x,t)| \,\mathrm{d} t \leq 2C_1L_1^{-2}.
\end{align*}
Therefore, we immediately obtain a bound of $C_1L_1^{-1/2}L_2^{-1}\phi(y)$. This is certainly at most $C_1L_1^{-1/2}L_2^{-1}$, and by the properties of $\phi(y)$ it also has the decay properties of the function $F_2(y)$.
We now consider the part of the integral over $[y_3,y]$. Let $y$ be in the interval $J_{k^*}$ for some $k^*$, where as usual the intervals $J_k$ are as in \eqref{eqn:Jk}. We decompose the integral between $y_3$ and $y$ as an integral over the relevant intervals $J_k$ where $k\geq k^*$.
By Proposition \ref{prop:phi},
\begin{align*}
\phi(t) \leq |\phi'(t)|2^{k/3},
\end{align*}
and so using the bound on $\tilde{\phi}(t)$ from Lemma \ref{lem:tphi}, to estimate the contribution to the integral from $J_k$, we have to bound
\begin{multline} \label{eqn:G2}
\phi(y) \int_{J_k} \phi(t)(L_1+|\phi'(t)|^{-1})(|t-y^*|+L_1) L_1^{-1/2}L_2^{-1}|\partial_tV(x,t)| \,\mathrm{d} t \\
\leq C_1\phi(y)\int_{J_k} 2^{k/3}|\phi'(t)| (L_1+|\phi'(t)|^{-1})(|t-y^*|+L_1) L_1^{-1/2} L_2^{-1}2^{-k} \,\mathrm{d} t \\
\leq C_12^{-2k/3}\phi(y)\int_{J_k} (|t-y^*|+L_1)L_1^{-1/2}L_2^{-1} \,\mathrm{d} t .
\end{multline}
Using Proposition \ref{prop:phi} again, we find that for any $k\geq k^*$,
\begin{align*}
\phi(y) \leq \phi(t_{k^*})\exp(-(y-t_{k^*})2^{-k^*/3}/10)\leq 2^{k^*/3}|\phi'(t_{k^*})|\exp(-(y-t_{k^*})2^{-k^*/3}/10).
\end{align*}
By Corollary \ref{cor:phiy}, we can bound the factor of $|\phi'(t_{k^*})|$ as
\begin{align*}
|\phi'(t_{k^*})| \leq CL_1^{-1} \exp(-c|t-t_k|/2^{k/3}) \exp(-c|t_k-y^*|/L_1).
\end{align*}
Inserting these estimates into the integral in \eqref{eqn:G2} and integrating over the interval $J_k$, we have the bound
\begin{align*}
C\exp(-(y-t_{k^*})2^{-k^*/3}/10)2^{k^*/3}2^{-k/3}L_1^{-1/2}L_2^{-1}.
\end{align*}
Summing over $k\geq k^*$ gives a bound for the integral over $y_3\leq t\leq y$ of the form
\begin{align*}
C_1 L_1^{-1/2}L_2^{-1}\exp(-(y-t_{k^*})2^{-k^*/3}/10).
\end{align*}
Note that this quantity is bounded by a multiple of $L_1^{-1/2}L_2^{-1}$, and has the required decay properties that we can include it in the function $F_2(y)$.
\end{proof}
\begin{lem} \label{lem:G3}
We have a bound on the final term in \eqref{eqn:key2},
\begin{align} \label{eqn:G3a}
\phi(y) \int_{y_1}^y \tilde{\phi}(t)|V(x,t) - \mu(x)||\alpha| \,\mathrm{d} t \leq C_1L_1^{-1/2}L_2^{-1}.
\end{align}
\end{lem}
\begin{rem}
We will see that the function decays exponentially from its maximum away from the region where $V(x,y) \leq 1+ L_1^{-2}$ on a length scale comparable to $L_1$. Therefore we can include this term in the function $F_1(y)$ in the statement of Proposition \ref{prop:paxeigpointwise}.
\end{rem}
\begin{proof}{Lemma \ref{lem:G3}}
We recall from Proposition \ref{prop:boundary} that we have an estimate on the boundary value of $\pa_x\psi^{(x)}_1(y)$ of the form
\begin{align} \label{eqn:G3b}
|\alpha| = |\partial_x\psi_1^{(x)}(g_2(x))| \leq CL_2^{-1}L_1^{-3/2}(L_1+M)\exp(-cML_1^{-1}).
\end{align}
Here $M$ is the distance from $g_2(x)$ to the point $y^*$ where $V(x,y^*) = 1+C^*L_1^{-2}$.
For the part of the integral in \eqref{eqn:G3a} over $[y_1,y_3]$, we know that $|V(x,t) - \mu(x)| \leq C_1L_1^{-2}$, $|y_3-y_1| \leq C_1L_1$ and $\tilde{\phi}(t) \leq C_1L_1$. Combining this with the bound on $\alpha$ from \eqref{eqn:G3b}, immediately gives us the desired bound of $L_2^{-1}L_1^{-1/2}\exp(-cML_1^{-1})$ for this part of the integral in \eqref{eqn:G3a}.
For $t\geq y_3$, we decompose $[y_3,y]$ into the intervals $\tilde{J}_k$ as in \eqref{eqn:Jktilde}:
\begin{align*}
\tilde{J}_k = [\tilde{t}_k,\tilde{t}_{k+1}] = \{ t\geq y_3: V(x,t) - \min_{t}V(x,t) \in [2^{-2k/3}, 2^{-2(k-1)/3}]\} .
\end{align*}
Since $\mu(x) \geq \min_{t}V(x,t)$, on $\tilde{J}_k$ we know that
\begin{align*}
|V(x,t) - \mu(x)| \leq 2^{-2k/3}.
\end{align*}
So, for the part of the integral in \eqref{eqn:G3a} over $\tilde{J}_k$, combining this with the bound on $\alpha$ in \eqref{eqn:G3b} and the usual bound on $\tilde{\phi}(t)$ from Lemma \ref{lem:tphi}, we have
\begin{align} \label{eqn:G3c} \nonumber
\phi(y) & \int_{\tilde{J}_k} \tilde{\phi}(t)|V(x,t) - \mu(x)||\alpha| \,\mathrm{d} t \\
& \leq C_1L_2^{-1}L_1^{-3/2} \phi(y)\int_{\tilde{J}_k}(L_1+|\phi'(t)|^{-1})2^{-2k/3}(L_1+M)\exp(-cML_1^{-1}) \,\mathrm{d} t.
\end{align}
Similarly to the proof of Lemma \ref{lem:G2}, let us assume that $y \in \tilde{J}_{k^*}$ for some $k^*$. Then,
using Proposition \ref{prop:phi} and then Proposition \ref{prop:phiy} twice, we obtain
\begin{align*}
\phi(y) \leq C_12^{k^*/3}|\phi'(y)| & \leq C_12^{k^*/3}|\phi'(t_{k^*})| \exp\left(-c|y-t_{k^*}|2^{-k^*/3}\right) \\
& \leq C_12^{k^*/3}|\phi'(t)| \exp\left(-c|y-t_{k^*}|2^{-k^*/3}\right) \exp\left(-c|t_{k^*}-t|2^{-k/3}\right)
\end{align*}
Inserting this bound for $\phi(y)$ into the right hand side of \eqref{eqn:G3c} and integrating over $\tilde{J}_k$ gives us the bound for the part of the integral over $\tilde{J}_{k}$ of
\begin{align*}
C_12^{k^*/3}2^{-k/3}L_2^{-1}L_1^{-1/2}\exp(-cML_1^{-1}/2).
\end{align*}
We finally sum over those $k$ with $k \geq k^*$ to get the desired bound on the part of the integral \eqref{eqn:G3a} with $y_3 \leq t \leq y$.
\end{proof}
Combining Lemmas \ref{lem:G1}, \ref{lem:G2} and \ref{lem:G3}, we see that the part of $g(y) - c_0(x)\psi^{(x)}_1(y)$ coming from
\begin{align*}
\phi(y)\int_{y_1}^{y}\tilde{\phi}(t)G(x,t) \,\mathrm{d} t
\end{align*}
has the bounds required in Proposition \ref{prop:paxeigpointwise}.
Therefore to finish the proof of Proposition \ref{prop:paxeigpointwise} we need to establish the analogous estimates for the other part of $g(y) - c_0(x)\psi^{(x)}_1(y)$ in \eqref{eqn:gexpression2},
\begin{align*}
\tilde{\phi}(y)\int_y^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t .
\end{align*}
\subsection{A Bound on $\tilde{\phi}(y)\int_y^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t$}
The estimates for the various parts of this integral will be similar to the estimates we used above. However, there will be places where we have to use different methods to obtain the desired bounds.
We want to bound
\begin{align} \label{eqn:key1b}
\left| \tilde{\phi}(y)\int_{y}^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t \right| \leq \tilde{\phi}(y)\int_{y}^{g_2(x)} \phi(t)|G(x,t)| \,\mathrm{d} t .
\end{align}
We again use \eqref{eqn:Gexpression2} to bound this by
\begin{align} \label{eqn:key2b}
\tilde{\phi}(y) \int_{y}^{g_2(x)} \phi(t) \left(C_1L_1^{-5/2}L_2^{-1}\phi(t) + C_1(|t-y^*|+L_1)L_1^{-1/2}L_2^{-1}|\partial_tV(x,t)|\phi(t) + |V(x,t) - \mu(x)|\alpha \right) \,\mathrm{d} t,
\end{align}
and we split this into three terms that we need to estimate.
\begin{lem} \label{lem:G1a}
We have a bound on the first term in \eqref{eqn:key2b}
\begin{align*}
\tilde{\phi}(y) \int_{y}^{g_2(x)} \phi(t) L_1^{-5/2}L_2^{-1}\phi(t) \,\mathrm{d} t \leq C_1L_1^{-1/2}L_2^{-1}
\end{align*}
\end{lem}
\begin{rem}
The function also decays exponentially from its maximum away from the region where $V(x,y) \leq 1+ L_1^{-2}$ on a length scale comparable to $L_1$. Therefore we can include this term in the function $F_1(y)$ in the statement of Proposition \ref{prop:paxeigpointwise}.
\end{rem}
\begin{proof}{Lemma \ref{lem:G1a}}
We know that $\tilde{\phi}(y) \leq \tilde{\phi}(t)$, and $\phi(t)$ decays exponentially on a length scale comparable to $L_1$ as we move away from $y^*$. Therefore, this bound follows in a very straightforward manner.
\end{proof}
Before bounding the second term in \eqref{eqn:key2b}, we first want to establish the following lemma.
\begin{lem} \label{lem:energy}
For any $ \tilde{y}\geq y_3$, we have the bound
\begin{align*}
\int_{\tilde{y}}^{g_2(x)}\phi(t)^2 \partial_tV(x,t) \,\mathrm{d} t \leq (\phi'(\tilde{y}))^{1/2}.
\end{align*}
\end{lem}
\begin{proof}{Lemma \ref{lem:energy}}
To prove this lemma, we will consider the \textquoteleft energy'
\begin{align} \label{eqn:energy}
\mathcal{E}(t) \coloneqq (\phi'(t))^2 - F(x,t)\phi(t)^2 .
\end{align}
Differentiating $\mathcal{E}(t)$ we find that
\begin{align*}
\mathcal{E}'(t) = 2\phi'(t)(\phi''(t) - F(x,t)\phi(t)) - \partial_tF(x,t)\phi(t)^2 = - \partial_tF(x,t)\phi(t)^2,
\end{align*}
where the final equality holds because $\phi''(t) = F(x,t)\phi(t)$. Since $F(x,t) = V(x,t)-\mu(x)$, we have
\begin{align*}
\partial_tF(x,t) =\partial_tV(x,t),
\end{align*}
and so
\begin{align*}
\int_{\tilde{y}}^{g_2(x)} \partial_tV(x,t) \phi(t)^2 \,\mathrm{d} t = -\int_{\tilde{y}}^{g_2(x)} \mathcal{E}'(t) \,\mathrm{d} t = \mathcal{E}(\tilde{y}) - \mathcal{E}(g_2(x)).
\end{align*}
Since $F(x,t) \geq 0$ for $t\geq y_3$, we know that
\begin{align*}
\mathcal{E}(\tilde{y}) = (\phi'(\tilde{y}))^2 - F(x,\tilde{y})\phi(\tilde{y})^2 \leq (\phi'(\tilde{y}))^2.
\end{align*}
Thus, to finish the proof of the lemma we need to show that $\mathcal{E}(g_2(x)) \geq 0$. We know that
\begin{align*}
\mathcal{E}(g_2(x)) \geq -F(x,g_2(x))\phi(g_2(x))^2,
\end{align*}
and that $\phi(g_2(x)) = 0$. However, we are not assuming that the potential $V(x,y)$ remains bounded as $y$ approaches $g_2(x)$, and so we cannot immediately deduce that $F(x,g_2(x))\phi(g_2(x))^2 = 0$. Instead we argue as follows. The function $\phi'(t)$ is in $L^{\infty}(\Omega(x))$, and this has two consequences. First, the eigenfunction $\phi(y)$ decays at least linearly to $0$ at $y=g_2(x)$. It also means that $\phi''(y)$ is in $L^1(\Omega(x))$, and hence $F(x,y)\phi(y)$ is in $L^1(\Omega(x))$. This means that $F(x,y)\phi(y)$ cannot grow as fast as $(y-g_2(x))^{-1}$ as we approach the boundary and so
\begin{align*}
\liminf_{y\to g_2(x)} \phi(y)F(x,y)\phi(y) = 0.
\end{align*}
This implies that $\mathcal{E}(g_2(x)) \geq0$ and concludes the proof of the lemma.
\end{proof}
\begin{rem}
This energy $\mathcal{E}(t)$ has also been used in \cite{GJ2} in their proof of Theorem 2.1 (B). There they obtain a pointwise estimate comparing the first eigenfunction of the two dimensional domain with the first eigenfunction of the associated ordinary differential operator.
\end{rem}
We can now bound the contribution from the second term in $G(x,t)$.
\begin{lem} \label{lem:G2b}
We have a bound on the second term in \eqref{eqn:key2b},
\begin{align*}
\tilde{\phi}(y) \int_{y}^{g_2(x)} \phi(t)^2(|t-y^*|+L_1)L_1^{-1/2}L_2^{-1}|\partial_tV(x,t)| \,\mathrm{d} t \leq C_1L_1^{-1/2}L_2^{-1}.
\end{align*}
\end{lem}
\begin{rem}
We will see in the proof that the function also decays exponentially from its maximum away from the region where $V(x,y) \leq 1+ L_1^{-2}$ on a length scale comparable to $L_1$. Therefore we can include this term in the function $F_1(y)$ in the statement of Proposition \ref{prop:paxeigpointwise}.
\end{rem}
\begin{proof}{Lemma \ref{lem:G2b}}
If $y \leq y_3$, we first consider the part of the integral where $t$ lies in the interval $[y,y_3]$ of length at most $C_1L_1$. In this case, we know that $\tilde{\phi}(y) \leq C_1L_1$ and the estimates follow easily.
For $t\geq y_3$, we first consider the integral between $\tilde{y}$ and $\tilde{y}+L_1$, where $\tilde{y}$ is some point with $\tilde{y}\geq y_3$ and $\tilde{y}\geq y$. Since $\partial_tV(x,t)\geq 0$ here, we have
\begin{align} \label{eqn:G2b1} \nonumber
\tilde{\phi}(y) \int_{\tilde{y}}^{\tilde{y}+L_1} & \phi(t)^2(|t-y^*|+L_1)L_2^{-1}|\partial_tV(x,t)| L_1^{-1/2}\,\mathrm{d} t \\
& \leq C_1\tilde{\phi}(y)(|\tilde{y}-y^*|+L_1)L_2^{-1}L_1^{-1/2}\int_{\tilde{y}}^{g_2(x)}\phi(t)^2 \partial_tV(x,t) \,\mathrm{d} t.
\end{align}
Applying Lemma \ref{lem:energy}, we can bound the right hand side of \eqref{eqn:G2b1} by
\begin{align*}
C_1\tilde{\phi}(y) (|\tilde{y}-y^*|+L_1)L_2^{-1}L_1^{-1/2}(\phi'(\tilde{y}))^2 .
\end{align*}
Lemma \ref{lem:tphi} shows that
\begin{align*}
\tilde{\phi}(\tilde{y})|\phi'(\tilde{y})| \leq C_1,
\end{align*}
and using Proposition \ref{prop:phiy} with $2^k$ comparable to $L_1^{3}$, we have the derivative bound
\begin{align*}
|\phi'(\tilde{y})| \leq C_1L_1^{-1}\exp\left(-c|\tilde{y}-y^*|/L_1\right).
\end{align*}
Thus the right hand side of \eqref{eqn:G2b1} has the bound
\begin{align*}
C_1L_2^{-1}L_1^{-1/2} \exp\left(-c|\tilde{y}-y^*|/L_1\right).
\end{align*}
Summing over $\tilde{y}$ between $y$ and $g_2(x)$ at intervals of length comparable to $L_1$ then gives the desired bound.
\end{proof}
We finally have to bound the contribution from the third term in $G(x,t)$.
\begin{lem} \label{lem:G3b}
We have a bound on the third term in \eqref{eqn:key2b}
\begin{align} \label{eqn:G3b1}
\tilde{\phi}(y) \int_{y}^{g_2(x)} \phi(t)|V(x,t) - \mu(x)||\alpha| \,\mathrm{d} t \leq C_1L_1^{-1/2}L_2^{-1}.
\end{align}
\end{lem}
\begin{rem}
As for the previous two lemmas, we will see in the proof that the function also decays exponentially from its maximum away from the region where $V(x,y) \leq 1+ L_1^{-2}$ on a length scale comparable to $L_1$. Therefore we can include this term in the function $F_1(y)$ in the statement of Proposition \ref{prop:paxeigpointwise}.
\end{rem}
\begin{proof}{Lemma \ref{lem:G3b}}
From Proposition \ref{prop:boundary} we have the estimate on $\alpha$ of the form
\begin{align} \label{eqn:G3b1}
|\alpha| = |\partial_x\psi_1^{(x)}(g_2(x))| \leq CL_2^{-1}L_1^{-3/2}(L_1+M)\exp(-cML_1^{-1}).
\end{align}
For the part of the integral in \eqref{eqn:key2} for $t$ between $y$ and $y_3$, we know that $|V(x,t) - \mu(x)|$ is at most $C_1L_1^{-2}$, and $\tilde{\phi}(y) \leq C_1L_1$. Thus, we immediately get a bound of
\begin{align*}
C_1L_1^{-1/2}L_2^{-1}\exp(-cML_1^{-1})
\end{align*}
for this part.
For $t \geq y_3$, we know that $F(x,t) = V(x,t) - \mu(x) \geq 0$. Thus, we can bound this part of the integral in \eqref{eqn:key2} by
\begin{align} \label{eqn:G3b2}
C_1\tilde{\phi}(y)L_2^{-1}L_1^{-3/2}(L_1+M)\exp(-cML_1^{-1}) \int_{\max\{y_3,y\}}^{g_2(x)} \phi(t)(V(x,t) - \mu(x)) \,\mathrm{d} t.
\end{align}
Since
\begin{align*}
\phi''(t) = (V(x,t) - \mu(x))\phi(t),
\end{align*}
and $|\phi'(t)|$ is decreasing for $t \geq y_3$, we find that \eqref{eqn:G3b2} can be bounded by
\begin{align*}
C_1\tilde{\phi}(y)L_2^{-1}L_1^{-3/2}(L_1+M)\exp(-cML_1^{-1}) |\phi'(y)| \leq C_1L_1^{-1/2}L_2^{-1}\exp(-cML_1^{-1}/2),
\end{align*}
where the last inequality comes from Lemma \ref{lem:tphi} as usual. This concludes the proof of the lemma.
\end{proof}
We recall that
\begin{align*}
g(y) - c_0(x)\psi^{(x)}_1(y) = \phi(y)\int_{y_1}^y \tilde{\phi}(t)G(x,t) \,\mathrm{d} t + \tilde{\phi}(y)\int_y^{g_2(x)}\phi(t)G(x,t) \,\mathrm{d} t.
\end{align*}
Then by the bounds on the right hand side in Lemmas \ref{lem:G1}, \ref{lem:G2}, \ref{lem:G3} and Lemmas \ref{lem:G1a}, \ref{lem:G2b}, \ref{lem:G3b}, we have shown that
\begin{align} \label{eqn:eigenvaluefinal}
\left| g(y) - c_0(x)\psi^{(x)}_1(y) \right| \leq F_1(y) + F_2(y).
\end{align}
Here the functions $F_1(y)$ and $F_2(y)$ have the desired properties from the statement of Proposition \ref{prop:paxeigpointwise}. As we remarked at the beginning of the proof, by the bound on $\alpha$ that we obtained in Proposition \ref{prop:boundary}, the estimate in \eqref{eqn:eigenvaluefinal} is sufficient to conclude the proof of Proposition \ref{prop:paxeigpointwise}.
\end{proof}
After the statement of Proposition \ref{prop:paxeigpointwise}, we showed that this implied Proposition \ref{prop:paxeigupperbound} and the bound
\begin{align*}
\int_{\Omega(x)} \left( \pa_x\psi^{(x)}_1(y) \right)^2 \,\mathrm{d} y \leq C_1L_2^{-2}.
\end{align*}
Combining this with the estimate on the first eigenvalue $\lambda$ from Proposition \ref{prop:eigboundinter}
\begin{align*}
\lambda \leq \mu + \int_{\Omega} \chi(x)^2\left( \pa_x\psi^{(x)}_1(y) \right)^2 \,\mathrm{d} x \,\mathrm{d} y + C_1L_2^{-2}
\end{align*}
gives
\begin{align*}
\lambda \leq \mu + CL_2^{-2}.
\end{align*}
This completes the proof of the upper bound on $\lambda$ in Proposition \ref{prop:eigbound}.
\end{proof}
By Propositions \ref{prop:lalower} and \ref{prop:eigbound}, we see that the first eigenvalue $\lambda$ satisfies
\begin{align*}
\mu \leq \lambda \leq \mu + CL_2^{-2},
\end{align*}
and so we have established Theorem \ref{thm:eigenvalue}.
\section{$L^2(\Omega)$ Bounds For The First Eigenfunction $u(x,y)$} \label{sec:L2}
Now that we have established the improved eigenvalue bound on $\lambda$ in Theorem \ref{thm:eigenvalue}, we want to use it to study the corresponding eigenfunction $u(x,y)$. We recall that $u(x,y)$ satisfies
\begin{eqnarray*}
\left\{ \begin{array}{rlcc}
(-\Delta_{x,y} + V(x,y))u(x,y) & = \lambda u(x,y) && \text{in } \Omega \\
u(x,y) & = 0&& \text{on } \partial \Omega,
\end{array} \right.
\end{eqnarray*}
and is normalised to be positive inside $\Omega$ with a maximum of $1$. Our main aim is to prove Theorem \ref{thm:shape} and show that the level sets $\{(x,y)\in\Omega: u(x,y) = c\}$ have lengths comparable to $L_2$ and $L_1$ in the $x$ and $y$-directions respectively, whenever $c$ is bounded away from $0$ and $1$.
Before we prove this theorem, in this section we will first establish an $L^2(\Omega)$-bound for $u(x,y)$. More precisely, we will prove the following proposition.
\begin{prop} \label{prop:uL2bound}
There exists an absolute constant $C>0$ such that
\begin{align*}
\int_{\Omega} u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq CL_1L_2.
\end{align*}
\end{prop}
\begin{rem}
Note that this $L^2(\Omega)$ bound is consistent with the shape of the level sets described in Theorem \ref{thm:shape}. We will use the eigenvalue bound on $\lambda$ from Theorem \ref{thm:eigenvalue} in a critical way in the proof.
\end{rem}
\begin{proof}{Proposition \ref{prop:uL2bound}}
Before beginning the proof of this proposition, we make the following definition.
\begin{defn} \label{def:H}
We define the function $H(x)$ by
\begin{align*}
H(x) \coloneqq \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y.
\end{align*}
That is, $H(x)$ is equal to the square of the $L^2(\Omega(x))$-norm of the cross-section of the eigenfunction $u(x,\cdot)$.
\end{defn}
To prove Proposition \ref{prop:uL2bound} we will first study the rate at which the function $H(x)$ decays from its maximum, and we will then prove an estimate for the maximum of $H(x)$.
To study the decay of $H(x)$, we prove a Carleman-type inequality. For the convex function $\mu(x)$ let $x^*$ be a point where it achieves its minimum of $\mu^*$. We now prove:
\begin{prop} \label{prop:H(x)decay}
For any $x$ we have the differential inequality,
\begin{align*}
H''(x) \geq 2(\mu(x) - \lambda)H(x).
\end{align*}
In particular, for $|x-x^*| \geq CL_2$, with $C$ a sufficiently large absolute constant, we have
\begin{align*}
H''(x) \geq \frac{1}{L_2^2}H(x).
\end{align*}
\end{prop}
\begin{rem}
This type of Carleman inequality has been used frequently in the study of the ground state Dirichlet eigenfunction of Schr\"odinger operators. For example, in Lemma 3.9 \cite{GJ2} it has been used to establish the exponential decay of the first Fourier mode of the ground state eigenfunction of the two dimensional convex domain. This first Fourier mode comes from a Fourier decomposition of the cross-section of the domain at each fixed $x$. A similar argument has also been used in Section 3 of \cite{FS1} to study the decay of the $L^2$-norm of the cross-section at $x$ of the eigenfunction for a two dimensional domain which is periodic in the $x$-direction and with height in the $y$-direction depending on a small parameter $\epsilon>0$.
\end{rem}
\begin{proof}{Proposition \ref{prop:H(x)decay}}
The eigenfunction $u(x,y)$ is equal to $0$ when $y$ is at the endpoints of the interval $\Omega(x)$. This allows us to differentiate $H(x)$ twice and pass the derivative inside the integral to obtain
\begin{align*}
H''(x) &= 2\int_{\Omega(x)} u(x,y)\partial_{x}^2u(x,y) + (\partial_xu(x,y))^2 \,\mathrm{d} y \\
&= 2\int_{\Omega(x)} (V(x,y) - \lambda)u(x,y)^2 - u(x,y)\partial_y^2u(x,y) +(\partial_xu(x,y))^2 \,\mathrm{d} y.
\end{align*}
Integrating by parts one time in $y$ in the term containing a factor of $\partial_y^2u(x,y)$, we can rewrite this as
\begin{align} \label{eqn:H(x)decayA} \nonumber
H''(x) & = 2\int_{\Omega(x)} (V(x,y) - \lambda)u(x,y)^2 + (\partial_yu(x,y))^2 + (\partial_xu(x,y))^2 \,\mathrm{d} y \\
& \geq 2\int_{\Omega(x)} (V(x,y) - \lambda)u(x,y)^2 + (\partial_yu(x,y))^2 \,\mathrm{d} y
\end{align}
Since $\mu(x)$ is the first eigenvalue of the operator
\begin{align*}
\mathcal{L}(x) = - \frac{d^2}{dy^2} + V(x,y),
\end{align*}
and $u(x,\cdot)$ vanishes at the endpoints of $\Omega(x)$, \eqref{eqn:H(x)decayA} gives us the lower bound
\begin{align} \label{eqn:H(x)decay1}
H''(x) \geq 2(\mu(x) - \lambda)\int_{\Omega(x)} u(x,y)^2 \,\mathrm{d} y = 2(\mu(x)-\lambda)H(x) .
\end{align}
Since $\mu(x^*) = \mu^*$ is the minimum value of the function $\mu(x)$, by the definition of the length scale $L_2$, we know that
\begin{align*}
|\mu(x^*) - \mu| \leq C_1L_2^{-2}.
\end{align*}
Thus, applying Theorem \ref{thm:eigenvalue}, we have the bound
\begin{align} \label{eqn:H(x)decay2}
|\lambda - \mu(x^*)| \leq C_1L_2^{-2}.
\end{align}
The function $\mu(x)$ increases from its minimum by $L_2^{-2}$ as $x$ varies in an interval of length comparable to $L_2$ from $x^*$. Moreover, $\mu(x)$ is a convex function. Therefore, provided we choose $C>0$ sufficiently large, we have
\begin{align} \label{eqn:H(x)decay3}
\mu(x) - \mu(x^*) \geq (C_1+1)L_2^{-2}
\end{align}
whenever $x$ satisfies $|x-x^*| \geq CL_2$. Combining the inequalities in \eqref{eqn:H(x)decay2} and \eqref{eqn:H(x)decay3} shows that
\begin{align*}
\mu(x) - \lambda \geq L_2^{-2},
\end{align*}
and using this bound in \eqref{eqn:H(x)decay1} gives
\begin{align*}
H''(x) \geq 2L_2^{-2}H(x)
\end{align*}
as required.
\end{proof}
Before giving a corollary of this proposition, we recall the generalised maximum principle.
\begin{prop} \label{prop:GMP}
Suppose that the functions $v_1$ and $v_2$ satisfy
\begin{align*}
\Delta v_1 + c(x)v_1 = 0, \qquad \Delta v_2 + c(x)v_2 \leq 0,
\end{align*}
in a bounded domain $D$, where $c(x)$ is a continuous function. If in addition $v_1$ and $v_2$ are continuous in $\bar{D}$, $v_1>0$ in $D$ and $v_2>0$ in $\bar{D}$, then
\begin{align*}
\max_{\bar{D}} v_1/v_2 \leq \max_{\partial D} v_1/v_2 .
\end{align*}
\end{prop}
This is proven in \cite{PW}, Theorem 10, page 73, and follows from applying the usual maximum principle to the function $v_1/v_2$. We now prove a corollary of Proposition \ref{prop:H(x)decay}.
\begin{cor} \label{cor:H(x)decay}
Let $A \coloneqq \max_{x}H(x)$. Then, the function $H(x)$ satisfies the upper bound
\begin{align*}
H(x) \leq C_1A\exp{(-c|x-x^*|/L_2)}.
\end{align*}
\end{cor}
\begin{proof}{Corollary \ref{cor:H(x)decay}}
With $C>0$ as in the statement of Proposition \ref{prop:H(x)decay}, let $x_1 = x^* + CL_2$. We also define the function $R(x)$ for $x>x_1$ by
\begin{align*}
R(x) \coloneqq Ae^{-(x-x_1)/L_2}.
\end{align*}
Then, $R(x)$ satisfies $R''(x) = L_2^{-2}R(x)$, and $H(x_1) \leq A = R(x_1)$. By Proposition \ref{prop:H(x)decay} we know that
\begin{align*}
H''(x) \geq L_2^{-2}H(x)
\end{align*}
for all $x \geq x_1$. Therefore, setting $D$ to be the interval $\{x\geq x_1\}$, the conditions of the generalised maximum principle are satisfied and hence
\begin{align*}
H(x) \leq R(x)
\end{align*}
for all $x \geq x_1$. There is also an analogous bound for $x \leq x^* - CL_2$, and this completes the proof.
\end{proof}
\begin{rem}
In fact, we see from the proof that we can replace $A$ by $H(x_1)$ and conclude that for any $x_1\geq x^* + CL_2$ we have the bound
\begin{align} \label{rem:H(x)decay}
H(x) \leq H(x_1)e^{-(x-x_1)/L_2}
\end{align}
for all $x>x_1$.
\end{rem}
In particular, as a result of this corollary, we see that $H(x)$ decays exponentially from its value at $x=x^*$ at least at a length scale comparable to $L_2$.
Our next aim is to obtain an upper bound for
\begin{align} \label{eqn:maxH}
A = \max_x H(x) = \max_x \int_{\Omega(x)} u(x,y)^2 \,\mathrm{d} y.
\end{align}
Suppose that we can show that $A$ satisfies
\begin{align*}
A \leq C_1L_1
\end{align*}
for an absolute constant $C_1$. Then, by Proposition \ref{prop:H(x)decay} we have
\begin{align*}
H(x) \leq C_1L_1 e^{-c|x-x^*|/L_2},
\end{align*}
and so integrating over $x$ gives
\begin{align*}
\int_{\Omega}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y = \int H(x) \,\mathrm{d} x \leq CL_1L_2.
\end{align*}
Therefore to complete the proof of Proposition \ref{prop:uL2bound}, it is sufficient to prove this upper bound on $A$. To do this we first define a cut-off function $\chi_1(x)$ as follows.
\begin{defn} \label{def:chi1}
We define $\chi_1(x)$ to be a smooth cut-off function, which satisfies
\begin{align*}
0 \leq \chi_1(x) \leq 1,
\end{align*}
and is equal to $1$ on the interval $[x^*-2CL_2,x^*+2CL_2]$ of length $4CL_2$, with $C$ as in the statement of Proposition \ref{prop:H(x)decay}. Moreover, the function $\chi_1(x)$ is supported on the interval $[x^*-3CL_2,x^*+3CL_2]$ and has the derivative estimate
\begin{align*}
\left | \partial^k \chi_1(x) \right| \leq (CL_2)^{-k},
\end{align*}
for $k=1,2$.
\end{defn}
We now prove the following.
\begin{prop} \label{prop:ydecayupper}
Let $\chi_1(x)$ be the cut-off function above in Definition \ref{def:chi1}. Then,
\begin{align*}
\int_{\Omega} \chi_1(x)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1L_2,
\end{align*}
for an absolute constant $C_1>0$. Note that this is consistent with $u(x,y)$ decaying on a length scale comparable to $L_1$ in the $y$-direction.
\end{prop}
\begin{proof}{Proposition \ref{prop:ydecayupper}}
The first eigenfunction $u(x,y)$ satisfies
\begin{align*}
-\Delta_{x,y}u(x,y) + (V(x,y) - \lambda)u(x,y) = 0 \text{ in } \Omega
\end{align*}
with zero boundary conditions on $\partial\Omega$. We integrate this against the function $\chi_1(x)u(x,y)$ to obtain
\begin{align*}
\int_{\Omega} -\chi_1(x)u(x,y)\Delta_{x,y}u(x,y) + \chi_1(x)(V(x,y) - \lambda)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y = 0,
\end{align*}
and integrating by parts one time in $x$ and $y$ gives
\begin{align} \label{eqn:ydecayupper1} \nonumber
\int_{\Omega} \chi_1(x)|\nabla_{x,y}u(x,y)|^2 & \,\mathrm{d} x \,\mathrm{d} y + \int_{\Omega}\chi_1'(x) \partial_xu(x,y)u(x,y) \,\mathrm{d} x \,\mathrm{d} y \\
& + \int_{\Omega} \chi_1(x)(V(x,y) - \lambda)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y = 0.
\end{align}
In the second integral in \eqref{eqn:ydecayupper1} we can write
\begin{align*}
\chi_1'(x)\partial_xu(x,y)u(x,y) = \tfrac{1}{2}\chi_1'(x)\partial_x(u(x,y)^2)
\end{align*}
and integrate by parts in $x$ again to rewrite this integral as
\begin{align*}
-\frac{1}{2} \int_{\Omega} \chi_1''(x)u(x,y)^2\,\mathrm{d} x \,\mathrm{d} y.
\end{align*}
Thus, from \eqref{eqn:ydecayupper1} we have
\begin{align} \label{eqn:ydecayupper2} \nonumber
& \int_{\Omega} \chi_1(x)|\nabla_{x,y}u(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y + \int_{\Omega} \chi_1(x)(V(x,y) - \lambda)_{+}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \\ & = \frac{1}{2}\int_{\Omega}\chi_1''(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y
+ \int_{\Omega} \chi_1(x)(V(x,y) - \lambda)_{-}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y,
\end{align}
where we have decomposed $V(x,y) - \lambda$ into its positive and negative parts via
\begin{align*}
V(x,y) - \lambda = (V(x,y) - \lambda)_{+} - (V(x,y)-\lambda)_{-}.
\end{align*}
By the simple eigenvalue bound for $\lambda$ from Proposition \ref{prop:simpleeigenvalue}, we know that
\begin{align*}
(V(x,y) - \lambda)_{-} \leq C_1L_1^{-2}.
\end{align*}
This also means that for any fixed $x$, we can only have $V(x,y) - \lambda \leq 0$ for $y$ in an interval of length at most comparable to $L_1$. Since the eigenfunction is normalised to have a maximum of $1$, and $\chi_1(x)$ is only non-zero in an interval of length comparable to $L_2$, this gives us a bound on the final term in the right hand side of \eqref{eqn:ydecayupper2} of
\begin{align} \label{eqn:ydecayupper3}
\int_{\Omega} \chi_1(x)(V(x,y) - \lambda)_{-}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1^{-2}L_1L_2 = C_1L_1^{-1}L_2.
\end{align}
We now turn to the second integral on the left hand side of \eqref{eqn:ydecayupper2}
\begin{align*}
\int_{\Omega} \chi_1(x)(V(x,y) - \lambda)_{+}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y .
\end{align*}
Fix a large constant $C_2>0$. For each fixed $x$, $V(x,y) - \lambda$ is only bounded above by $C_2L_1^{-2}$ on an interval in $y$ of length comparable to $L_1$. Therefore, again combining this with the bound $u(x,y) \leq 1$, we can write
\begin{align} \label{eqn:ydecayupper4}
C_2L_1^{-2} \int_{\Omega}\chi_1(x)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y - C_1L_1^{-1}L_2 \leq \int_{\Omega} \chi_1(x)(V(x,y) - \lambda)_{+}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y .
\end{align}
Inserting the estimates in \eqref{eqn:ydecayupper3} and \eqref{eqn:ydecayupper4} back into \eqref{eqn:ydecayupper2} we see that
\begin{align} \label{eqn:ydecayupper5} \nonumber
& \int_{\Omega} \chi_1(x)|\nabla_{x,y}u(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y + C_2L_1^{-2} \int_{\Omega} \chi_1(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \\ & \leq \frac{1}{2}\int_{\Omega}\chi_1''(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y +C_1L_1^{-1}L_2.
\end{align}
The first integral in \eqref{eqn:ydecayupper5} is positive, and so we can drop it from the estimate. Therefore, dividing by $C_2L_1^{-2}$ gives us
\begin{align} \label{eqn:ydecayupper6}
\int_{\Omega} \chi_1(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq \frac{1}{2} C_2^{-1}L_1^2\int_{\Omega}\chi_1''(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y + C_1L_1L_2.
\end{align}
To conclude the proof of the proposition, we will use Corollary \ref{cor:H(x)decay} and the remark following it. By \eqref{rem:H(x)decay}, for any $x_1 \geq x^* + CL_2$ and any $x \geq x_1$, we have
\begin{align*}
\int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y \leq e^{-(x-x_1)/L_2} \int_{\Omega(x_1)}u(x_1,y)^2 \,\mathrm{d} y.
\end{align*}
Therefore, we certainly have the estimate
\begin{align} \label{eqn:ydecayupper7}
\int_{x^*+2CL_2}^{x^*+3CL_2} \int_{\Omega(x)}u(x,y)^2\,\mathrm{d} x \,\mathrm{d} y \leq \int_{x^*+CL_2}^{x^*+2CL_2} \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y,
\end{align}
and an analogous estimate for $x_1 \leq x^*-CL_2$ and $x \leq x_1$. By the definition of the cut-off function $\chi_1(x)$, the second derivative $\chi_1''(x)$ is supported on the intervals $[x^*-3CL_2,x^*-2CL_2]$ and $[x^*+2CL_2,x^*+3CL_2]$, and is of order $L_2^{-2}$ here. Also, $\chi_1(x)$ is equal to $1$ on the intervals $[x^*-2CL_2,x^*-CL_2]$ and $[x^*+CL_2,x^*+2CL_2]$. Therefore, using the estimate in \eqref{eqn:ydecayupper7} the integral on the right hand side of \eqref{eqn:ydecayupper6} is certainly at most $\tfrac{1}{2}$ the size of the integral on the left hand side. This means that in \eqref{eqn:ydecayupper6} we can bring over the integral to the left hand side and get the bound
\begin{align*}
\int_{\Omega} \chi_1(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1L_2,
\end{align*}
as required.
\end{proof}
\begin{cor} \label{cor:ydecayupper}
We have the derivative bound
\begin{align*}
\int_{\Omega} \chi_1(x) |\nabla_{x,y}u(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1^{-1}L_2.
\end{align*}
\end{cor}
\begin{proof}{Corollary \ref{cor:ydecayupper}}
In the proof of Proposition \ref{prop:ydecayupper} in \eqref{eqn:ydecayupper5} we established the estimate
\begin{align} \label{cor:ydecayupper1} \nonumber
& \int_{\Omega} \chi_1(x)|\nabla_{x,y}u(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y + C_2L_1^{-2} \int_{\Omega} \chi_1(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \\ & \leq \frac{1}{2}\int_{\Omega}\chi_1''(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y +C_1L_1^{-1}L_2.
\end{align}
We also showed that
\begin{align*}
L_1^2\int_{\Omega}\chi_1''(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y
\end{align*}
is bounded by $\int_{\Omega} \chi_1(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y $, and hence by Proposition \ref{prop:ydecayupper} is bounded by $C_1L_1^{-1}L_2$. Using this estimate in \eqref{cor:ydecayupper1} gives the desired result.
\end{proof}
The derivative bound
\begin{align*}
\int_{\Omega} \chi_1(x) |\nabla_{x,y}u(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1^{-1}L_2
\end{align*}
is of order $L_1^{-2}$ smaller than the bound we obtained for the eigenfunction $u(x,y)$ itself in Proposition \ref{prop:ydecayupper}. For the $y$-derivative $\partial_yu(x,y)$, this bound is consistent with our eventual aim to show that $u(x,y)$ decays away from its maximum on a length scale comparable to $L_1$. However, in the $x$-direction, our aim is to show that $u(x,y)$ decays away from its maximum on a length scale comparable to $L_2$. Therefore, we want to improve the bound on $\partial_xu(x,y)$ given in Corollary \ref{cor:ydecayupper}.
\begin{prop} \label{prop:xdecayupper}
Let $\chi_1(x)$ be as in Definition \ref{def:chi1}. Then, there exists an absolute constant $C_1>0$ such that
\begin{align*}
\int_{\Omega} \chi_1(x) (\partial_xu(x,y))^2 \,\mathrm{d} x \,\mathrm{d} y \leq CL_1L_2^{-1}.
\end{align*}
Note that for $L_2\gg L_1$ this is an improvement on the bound in Corollary \ref{cor:ydecayupper}.
\end{prop}
\begin{proof}{Proposition \ref{prop:xdecayupper}}
We begin by proceeding as in the proof of Proposition \ref{prop:ydecayupper} to obtain the equality in \eqref{eqn:ydecayupper2}:
\begin{align} \label{eqn:xdecayupper1} \nonumber
\int_{\Omega} \chi_1(x)|\nabla_{x,y}u(x,y)|^2 \,\mathrm{d} x \,\mathrm{d} y& + \int_{\Omega} \chi_1(x)(V(x,y) - \lambda)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \\
& - \frac{1}{2}\int_{\Omega}\chi_1''(x) u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y = 0 .
\end{align}
We know that the integral of $\chi_1(x)u(x,y)^2$ is at most $C_1L_1L_2$. Since $|\chi_1''(x)| \leq C_1L_2^{-2}$ this means that
\begin{align*}
\frac{1}{2}\int_{\Omega}\left|\chi_1''(x)\right| u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1L_2^{-1},
\end{align*}
and so from \eqref{eqn:xdecayupper1} we have
\begin{align} \label{eqn:xdecayupper2} \nonumber
& \int_{\Omega} \chi_1(x) (\partial_xu(x,y))^2 \,\mathrm{d} x \,\mathrm{d} y + \int_{\Omega} \chi_1(x)(\partial_{y}u(x,y))^2 \,\mathrm{d} x \,\mathrm{d} y \\
& + \int_{\Omega} \chi_1(x)(V(x,y) - \lambda)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1L_2^{-1} .
\end{align}
For each fixed $x$, the eigenfunction $u(x,y)$ is an admissible test function for our usual ordinary differential operator
\begin{align*}
\mathcal{L}(x) = - \frac{d^2}{dy^2} + V(x,y).
\end{align*}
Since this operator has first eigenvalue equal to $\mu(x)$, we obtain the lower bound
\begin{align} \label{eqn:xdecayupper3}
\int_{\Omega(x)} (\partial_{y}u(x,y))^2 + (V(x,y) - \lambda)u(x,y)^2 \,\mathrm{d} y \geq (\mu(x) - \lambda)\int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y.
\end{align}
Multiplying the inequality in \eqref{eqn:xdecayupper3} by $\chi_1(x)$ and integrating over $x$, \eqref{eqn:xdecayupper2} becomes
\begin{align} \label{eqn:xdecayupper4}
\int_{\Omega} \chi_1(x) (\partial_xu(x,y))^2 \,\mathrm{d} x \,\mathrm{d} y + \int_{\Omega} \chi_1(x)(\mu(x) - \lambda)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1L_2^{-1}.
\end{align}
By the definition of $L_2$, we have
\begin{align*}
\mu(x) - \mu \geq -C_1L_2^{-2},
\end{align*}
and by the eigenvalue bounds in Theorem \ref{thm:eigenvalue}, we know that
\begin{align*}
\mu - \lambda \geq -C_1L_2^{-2}.
\end{align*}
Therefore, \eqref{eqn:xdecayupper4} tells us that
\begin{align*}
\int_{\Omega} \chi_1(x) (\partial_xu(x,y))^2 \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_2^{-2} \int_{\Omega}\chi_1(x)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y + C_1L_1L_2^{-1}.
\end{align*}
Applying Proposition \ref{prop:ydecayupper} then gives the desired bound.
\end{proof}
Now that we have established $L^2$-bounds for the first derivative, $\nabla_{x,y}u(x,y)$, in Propositions \ref{prop:ydecayupper} and \ref{prop:xdecayupper}, we can return to establishing the required upper bound for
\begin{align*}
A = \max_{x}H(x) = \max_{x}\int_{\Omega(x)} u(x,y)^2 \,\mathrm{d} y.
\end{align*}
\begin{prop} \label{prop:Aupper}
$A$ is bounded by $L_1$ multiplied by an absolute constant.
\end{prop}
\begin{proof}{Proposition \ref{prop:Aupper}}
Suppose that we have
\begin{align} \label{eqn:Aupper1}
\max_{x}H(x) = H(x^*) = \int_{\Omega(x^*)} u(x^*,y)^2 \,\mathrm{d} y \geq C^* L_1,
\end{align}
where $C^*>0$ is a large absolute constant that we will specify later. Then, for any $(x,y)$, extending $u(x,y)$ to be $0$ outside of $\Omega$, we can write
\begin{align*}
u(x,y) = u(x^*,y) + \int_{x^*}^{x} \partial_tu(t,y) \,\mathrm{d} t,
\end{align*}
and so
\begin{align} \label{eqn:Aupper2}
u(x,y)^2 \geq \tfrac{1}{2}u(x^*,y)^2 - C_1|x-x^*|\int_{x^*}^x (\partial_tu(t,y))^2 \,\mathrm{d} t,
\end{align}
for a fixed constant $C_1$. Integrating the inequality in \eqref{eqn:Aupper2} over $y$ we find that
\begin{align*}
H(x) \geq \tfrac{1}{2}H(x^*) - C_1|x-x^*|\int_{x^*}^x\int_{\Omega(t)} (\partial_tu(t,y))^2 \,\mathrm{d} y \,\mathrm{d} t,
\end{align*}
and so by the assumption on $H(x^*)$ in \eqref{eqn:Aupper1}, this gives
\begin{align} \label{eqn:Aupper3}
H(x) \geq \tfrac{1}{2}C^*L_1- C_1|x-x^*|\int_{x^*}^x\int_{\Omega(t)} (\partial_tu(t,y))^2 \,\mathrm{d} y \,\mathrm{d} t.
\end{align}
Let us restrict to those values of $x$ with $|x-x^*|\leq c_1L_2$ for a small constant $c_1>0$. Then by the derivative bound on $\partial_tu(t,y)$ in Proposition \ref{prop:xdecayupper}, we can ensure that the second term in \eqref{eqn:Aupper3} is small compared to $\tfrac{1}{4}C^*L_1$. Moreover, this constant $c_1$ can be chosen to be independent of $C^*$. Therefore, this tells us that for all $x$ in an interval of length $2c_1L_2$, we have the lower bound
\begin{align*}
H(x) \geq \tfrac{1}{4}C^*L_1 .
\end{align*}
In particular, this shows that
\begin{align*}
\int_{\Omega}\chi_1(x)u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \geq \int_{x^*-c_1L_2}^{x^*+c_1L_2}H(x) \,\mathrm{d} x \geq \tfrac{1}{2}c_1C^*L_1L_2.
\end{align*}
Since $c_1$ is independent of $C^*$, we can contradict the $L^2(\Omega)$-bound from Proposition \ref{prop:ydecayupper} by choosing $C^*$ sufficiently large.
\end{proof}
By the discussion after the proof of Corollary \ref{cor:H(x)decay}, this upper bound on $A$ from Proposition \ref{prop:Aupper} implies the $L^2(\Omega)$-bound
\begin{align*}
\int_{\Omega} u(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y \leq CL_1 L_2.
\end{align*}
This completes the proof of Proposition \ref{prop:uL2bound}.
\end{proof}
In Proposition \ref{prop:uL2bound} we derived an $L^2(\Omega)$-bound for the first eigenfunction $u(x,y)$. For our purposes of studying the shape of the level sets of $u(x,y)$ near to its maximum this will be sufficient. However, another interesting question is to study the rate at which $u(x,y)$ decays from its maximum. Therefore, before continuing with our study of the level sets, let us give some indication about the decay of $u(x,y)$ as we move away from its maximum.
We will do this by using an Agmon-type estimate, but first we need some definitions.
\begin{defn} \label{def:Om1}
Fix a large absolute constant $C>0$, and let $\Omega_1$ be the subset of $\Omega$ given by
\begin{align*}
\Omega_1 \coloneqq \{ (x,y) \in \Omega: V(x,y) \geq 1 + CL_1^{-2} \} .
\end{align*}
Note that the boundary of $\Omega_1$ consists of parts of the two convex curves coming from $\partial\Omega$ and the level set $\{ (x,y)\in\Omega :V(x,y) = 1+ CL_1^{-2} \}$.
\end{defn}
\begin{defn} \label{def:h(x,y)}
With $ \Omega_1 \subset \Omega$ as above, we also define the distance function
\begin{align*}
h^* : \Omega_1 \to [0,\infty)
\end{align*}
as follows. We first define the function $\nu^*(x,y)$ to be equal to $V(x,y)-\lambda$. For $(x,y)$ in $\Omega_1$ we then define $h^*(x,y)$ by
\begin{align*}
h^*(x,y) = \inf_{\gamma} \frac{1}{2} \int_{0}^{1} \nu^*(\gamma(t))^{1/2}|\gamma'(t)| \,\mathrm{d} t,
\end{align*}
where the infimum is taken over all paths $\gamma:[0,1]\to \Omega_1$ between the inner boundary of $\Omega_1$ and $(x,y)$.
\end{defn}
We are now in a position to state our Agmon-type estimate.
\begin{prop} \label{prop:Agmon}
For $\Omega_1$ and $h^*(x,y)$ defined as above, we have
\begin{align*}
\int_{\Omega_1} u(x,y)^2 e^{2h^*(x,y)} \,\mathrm{d} x \,\mathrm{d} y \leq C_2L_1L_2 ,
\end{align*}
for some absolute constant $C_2>0$.
\end{prop}
\begin{rem}
Since we certainly have the lower bound $V(x,y)-\lambda \geq C_1L_1^{-2}$ on $\Omega_1$, roughly speaking this proposition shows that, in an $L^2(\Omega)$-sense, the function $u(x,y)$ decays at least on a length scale comparable to $L_1$ as we move away from the region where $V(x,y) \leq 1+CL_1^{-2}$. However, as $V(x,y)-\lambda$ grows, this rate of exponential decay also increases.
\end{rem}
\begin{proof}{Proposition \ref{prop:Agmon}}
This proposition will follow from a classical Agmon estimate in \cite{Ag}. Let us restate Theorem 1.5 from \cite{Ag} (using slightly different notation).
\begin{thm}[Theorem 1.5 in \cite{Ag}] \label{thm:Agmon}
Let $D$ be a bounded connected open set in $\mathbb{R}^2$. Let $q(x,y)$ be a real valued function on $D$, and suppose that $\nu(x,y)$ is a positive continuous function on $D$ such that
\begin{align} \label{eqn:Agmonnu}
\int_{D} |\nabla_{x,y} \psi(x,y)|^2 + q(x,y)\psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y\geq \int_{D} \nu(x,y)\psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y
\end{align}
for all $\psi\in C_0^{\infty}(D)$.
Fix a point $(x_0,y_0) \in D$, and define the distance $\rho_\nu(x,y)$ by
\begin{align} \label{eqn:Agmonrho}
\rho_\nu(x,y) \coloneqq \inf_{\gamma} \int_{0}^{1} \nu(\gamma(t))^{1/2} |\gamma'(t)| \,\mathrm{d} t,
\end{align}
where the infimum is taken over all continuous paths $\gamma:[0,1]\to D$ in $D$ between $(x_0,y_0)$ and $(x,y)$. We also define $\rho_{\nu}((x,y),\{\infty\})$ to be the distance from the point $(x,y)$ to $\partial D$ under the distance function $\rho_{\nu}(x,y)$, and define $D_s$ by
\begin{align*}
D_s \coloneqq \{(x,y)\in D: \rho_{\nu}((x,y),\{\infty\}) > s\} .
\end{align*}
Finally, suppose that
\begin{align*}
-\Delta_{x,y} W(x,y) + q(x,y)W(x,y) = 0
\end{align*}
and that the function $g(x,y)$ satisfies
\begin{align} \label{eqn:Agmong}
|\nabla_{x,y} g(x,y)|^2 < \nu(x,y)
\end{align}
in $D$. Then, we have the estimate
\begin{align} \label{eqn:AgmonW} \nonumber
\int_{D_s} & W(x,y)^2 (\nu(x,y) - |\nabla_{x,y}g(x,y)|^2)e^{2g(x,y)} \,\mathrm{d} x \,\mathrm{d} y \\
& \leq \frac{2(1+2s)}{s^2} \int_{D\backslash D_s} W(x,y)^2 \nu(x,y)e^{2g(x,y)} \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
\end{thm}
We will now apply this theorem with $W(x,y) = u(x,y)$ and $q(x,y) = V(x,y) - \lambda$. We will choose the set $D$ as follows: We recall that $\Omega_1$ consists of those points $(x,y)$ with $V(x,y)-\lambda \geq 1 + CL_1^{-2}$. We then define $D$ to be all points in $\mathbb{R}^2$ outside of the inner boundary of $\Omega_1$.
Since $u(x,y) = 0$ on $\partial\Omega$, we can extend $u(x,y)$ to $D$ by setting it to be $0$ for $D \backslash \Omega_1$, and we extend the potential $V(x,y)$ to $D$ arbitrarily.
We clearly have the estimate
\begin{align*}
\int_{D} |\nabla_{x,y} \psi(x,y)|^2 + (V(x,y)-\lambda) \psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y\geq \int_{D} (V(x,y)-\lambda) \psi(x,y)^2 \,\mathrm{d} x \,\mathrm{d} y\end{align*}
for all $\psi\in C_0^{\infty}(D)$. Also, $V(x,y)-\lambda\geq C_1L_1^{-2}$ for $(x,y)\in D$. As a result of this, from \eqref{eqn:Agmonnu} we see that we we can set $\nu^*(x,y)$ to be equal to the function described in the definition of $h^*(x,y)$ in Definition \ref{def:h(x,y)}.
In Theorem \ref{thm:Agmon} we are free to choose the value for $s$, and we will choose $s=1$. Then, we see that
\begin{align*}
D\backslash D_{1} = \{(x,y)\in D: \rho_{\nu}((x,y),\{\infty\}) \leq 1 \}
\end{align*}
consists of the region near the inner boundary of $D$ with width comparable to at most $L_1$. This is because we have ensured that $\nu(x,y) \geq cL_1^{-2}$ when the point $(x,y)$ is within a distance $L_1$ of the boundary of $D$.
We finally need to choose $g(x,y)$ to ensure that \eqref{eqn:Agmong} holds, and so we need
\begin{align*}
|\nabla_{x,y} g(x,y)|^2 < \nu^*(x,y) = V(x,y) -\lambda.
\end{align*}
We can achieve this by setting $g(x,y)$ to be equal to the function $h^*(x,y)$ as in Definition \ref{def:h(x,y)}. This certainly satisfies the required derivative bound.
Thus, we can apply Theorem \ref{thm:Agmon} to get
\begin{align} \label{eqn:Agmonfinal} \nonumber
\int_{D_1} & u(x,y)^2 (\nu(x,y) - |\nabla_{x,y}h^*(x,y)|^2)e^{2h^*(x,y)} \,\mathrm{d} x \,\mathrm{d} y \\
& \leq 6\int_{D\backslash D_1} u(x,y)^2 \nu(x,y)e^{2h^*(x,y)} \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
On $D\backslash D_1$, we know that $\nu(x,y) \leq L_1^{-2}$, and $h^*(x,y) \leq 1$. Therefore, by the $L^2$ bound on $u(x,y)$ from Proposition \ref{prop:uL2bound}, the right hand side of \eqref{eqn:Agmonfinal} is bounded by
\begin{align*}
C_1L_1^{-2}L_1L_2 = C_1L_1^{-1}L_2.
\end{align*}
Since for $(x,y) \in D$, we have $ \nu(x,y) - |\nabla_{x,y}h^*(x,y)|^2 \geq c_1L_1^{-2}$, we can therefore conclude from \eqref{eqn:Agmonfinal} that
\begin{align*}
\int_{\Omega_2} u(x,y)^2 e^{2h^*(x,y)} \,\mathrm{d} x \,\mathrm{d} y \leq C_1L_1L_2
\end{align*}
as required.
\end{proof}
\section{The Shape Of The Level Sets Of $u(x,y)$} \label{sec:shape}
We now return to the problem of studying the shape of the level sets of the first eigenfunction $u(x,y)$. As we have mentioned earlier, since the potential $V(x,y)$ is convex, a theorem of Brascamp and Lieb, \cite{BL2} tells us that $u(x,y)$ is log concave. In particular, this means that the superlevel sets of $u(x,y)$ are convex subsets of $\Omega$.
We will use the results of the previous section to estimate the lengths of the projections of these level sets onto the $x$ and $y$-axis. In particular, in this section we will establish Theorem \ref{thm:shape} about the shape of the level sets. Throughout this section we let $c_1>0$ be a small absolute constant as in the statement of Theorem \ref{thm:shape}. The constant $c>0$ which appears in the propositions below is bounded away from $0$ and $1$ by satisfying
\begin{align*}
c_1 < c < 1-c_1,
\end{align*}
and all other constants will depend on the choice of $c_1$.
We first use the bound on $A$ from Proposition \ref{prop:Aupper} to find an upper bound on the behaviour of the level sets of $u(x,y)$ in the $y$-direction.
\begin{prop} \label{prop:levelyupper}
Let $0<c<1$ be a fixed absolute constant. Then, for any fixed $x$, the cross-section of the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq c\}$ at $x$ consists of an interval of length at most $L_1$ multiplied by an absolute constant.
\end{prop}
\begin{proof}{Proposition \ref{prop:levelyupper}}
By Proposition \ref{prop:Aupper} we know that
\begin{align*}
A = \max_{x}H(x) = \max_{x} \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y \leq C_1L_1.
\end{align*}
If $u(x,y) \geq c$ for $y$ in an interval of length $CL_1$ for $C$ sufficiently large, this immediately gives a contradiction.
\end{proof}
We can also prove an upper bound on the length of the projection of the level sets of $u(x,y)$ in the $y$-direction.
\begin{prop} \label{prop:firstlocation}
For sufficiently small $\delta>0$ fixed, there exists an $\eta>0$ such that if the point $(x,y)$ is within a distance $\eta L_1$ of the level set $\{ (x,y)\in\Omega: V(x,y) = 1+ \eta^{-1}L_1^{-2}\}$, then
\begin{align*}
u(x,y) \leq \delta.
\end{align*}
In particular, the level sets $\{(x,y)\in\Omega: u(x,y) = c\}$ are at a distance comparable to $L_1$ away from the level set $\{ (x,y)\in\Omega: V(x,y) = 1+CL_1^{-2}\}$, for some absolute constant $C>0$.
\end{prop}
\begin{rem}
The proof of this proposition follows closely the proof of Lemma 3.17 in \cite{GJ2}, where an analogous property has been established for the first eigenfunction of a two dimensional convex domain.
\end{rem}
Before proving this proposition, let us show the following corollary:
\begin{cor} \label{cor:firstlocationA}
Let $0<c<1$ be a fixed absolute constant. Then, the projection of the level set $\{(x,y)\in\Omega:u(x,y) = c\}$ onto the $y$-axis has length bounded from above by an absolute constant multiplied by $L_1$.
\end{cor}
\begin{proof}{Corollary \ref{cor:firstlocationA}}
By the definition of the length scale $L_1$ and the orientation of the level set $\Omega_{L_1^{-2}} = \{(x,y)\in\Omega:V(x,y) = 1+L_1^{-2}\}$ we used when defining $L_2$, we know that the projection of the level set $\Omega_{L_1^{-2}}$ onto the $y$-axis has length comparable to $L_1$. Moreover, by the convexity of the potential $V(x,y)$, this is true for any level set $ \{(x,y)\in\Omega:V(x,y) = 1+CL_1^{-2}\}$, for any absolute constant $C>0$. Therefore, the upper bound on the length of the projection of the level sets $\{(x,y)\in\Omega:u(x,y) = c\}$ onto the $y$-axis follows from Proposition \ref{prop:firstlocation}.
\end{proof}
\begin{proof}{Proposition \ref{prop:firstlocation}}
Let $(x',y')$ be a point which is within a distance $\eta L_1$ of the level set $\{(x,y)\in\Omega :V(x,y) = 1+ \eta^{-1}L_1^{-2}\}$. After a rotation, we may assume that the nearest point of $\{(x,y)\in\Omega : V(x,y) = 1+ \eta^{-1}L_1^{-2}\}$ to $(x',y')$ is equal to $(x',y_1)$, with $y_1<y'$ and $y'-y_1 <\eta L_1$.
We will need to use two properties of the potential $V(x,y)$. Firstly, by the simple eigenvalue bounds on $\lambda$ in Proposition \ref{prop:simpleeigenvalue} we have seen before that
\begin{align} \label{eqn:firstlocation1}
\Delta_{x,y}u(x,y) = (V(x,y) - \lambda)u(x,y) \geq -\frac{C_1^2}{L_1^2}u(x,y)
\end{align}
for all values of $(x,y)$, for some absolute constant $C_1$. Moreover, $V(x,y)$ has convex sublevel sets and by the rotation we made above we have $V(x,y_1) = 1+\eta^{-1}L_1^{-2}$. Therefore,
\begin{align} \label{eqn:firstlocation2}
\Delta_{x,y}u(x,y) = (V(x,y) - \lambda)u(x,y) \geq \frac{1}{2\eta L_1^2}u(x,y),
\end{align}
whenever $y \leq y'-\eta L_1 < y_1$.
We define the comparison function $v_1(x,y)$ by
\begin{align} \label{eqn:firstlocation3}
v_1(x,y) = \sin\left(\frac{C_1(y-y')}{2L_1} + \frac{C_1\eta}{2} + C_1\delta\right)
\end{align}
for $y \geq y'-\eta L_1$, and by
\begin{align} \label{eqn:firstlocation4}
v_1(x,y) = (\sin(C_1\delta))\exp\left(\frac{\delta}{2}+\frac{(y-y')}{2\delta L_1}\right)
\end{align}
for $y < y'-\eta L_1$. We make the choice $\eta = \delta^2$, and this ensures that $v_1(x,y)$ is continuous at $y = y'-\eta L_1$ for all values of $x$.
For $\delta>0$ sufficiently small, using $\sin(C_1\delta) >C_1\delta \cos(C_1\delta)$, we find that $\partial_y^2 v_1(x,y)$ has a negative delta function along $y = y'-\eta L_1$. Everywhere else, calculating $\Delta_{x,y} v_1(x,y)$ from its definition in \eqref{eqn:firstlocation3} and \eqref{eqn:firstlocation4}, and using the inequalities for $\Delta_{x,y}u(x,y)$ in \eqref{eqn:firstlocation1} and \eqref{eqn:firstlocation2}, we see that
\begin{align*}
\frac{ \Delta_{x,y} v_1(x,y)}{v_1(x,y)} \leq \frac{ \Delta_{x,y} u(x,y)}{u(x,y)}.
\end{align*}
Moreover, for those $(x,y) \in \partial\Omega$, with $y \leq y' + \left(\frac{\pi}{C_1} - \eta -2\delta\right)L_1$ we have
\begin{align*}
v_1(x,y) > 0 = u(x,y)|_{\partial\Omega}
\end{align*}
and for $(x,y) \in \Omega$ with $y= y' + \left(\frac{\pi}{C_1} - \eta -2\delta\right)L_1$, we have
\begin{align*}
v_1\left(x,y' + (\pi/C_1 - \eta -2\delta)L_1\right) = 1 \geq u\left(x,y' + (\pi/C_1 - \eta -2\delta)L_1\right).
\end{align*}
Thus, applying the generalised maximum principle in Proposition \ref{prop:GMP} to those $(x,y)$ in $\Omega$ with $y \leq y' + \left(\frac{\pi}{C_1} - \eta -2\delta\right)L_1$ , $v_1(x,y)$ is a positive supersolution, and in particular
\begin{align*}
u(x',y') \leq v_1(x',y') = \sin\left(\frac{C_1\eta}{2}+C_1\delta\right) \leq C_2\delta.
\end{align*}
Thus, repeating the argument with a suitable multiple of $\delta$ gives the desired result.
\end{proof}
We now want to obtain a lower bound on the height of the level sets in the $y$-direction.
\begin{prop} \label{prop:levelylower}
Let $0<c<1$ be a fixed absolute constant. Then, the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq c\}$ has inner radius bounded below by an absolute constant multiplied by $L_1$. In particular, the projection of the level set $\{(x,y)\in\Omega:u(x,y) = c\}$ onto the $y$-axis has length bounded from below by an absolute constant multiplied by $L_1$.
\end{prop}
\begin{rem}
The proof of this proposition only considers the parameter $L_1$, and does not use any properties of the eigenvalue or eigenfunction that depend on $L_2$. In particular, this means that we do not need to fix the orientation of the level set $\Omega_{L_1^{-2}} = \{(x,y)\in \Omega:V(x,y) = 1+ L_1^{-2}\}$, and we are free to rotate $\Omega$ in the course of the proof.
\end{rem}
\begin{proof}{Proposition \ref{prop:levelylower}}
Let us consider the case $c=1/4$, and study the level set $\{(x,y)\in\Omega: u(x,y) = \tfrac{1}{4}\}$. Suppose that the shortest projection of the set onto any direction is of length $\alpha$. By the convexity of the superlevel sets of $u(x,y)$, after a rotation and a translation, we may then assume that this level set lies between the two lines $y=0$ and $y=\alpha$.
We will use the comparison function
\begin{align*}
W(x,y) \coloneqq \frac{1}{2}\sin \left(\frac{\pi}{6}+ \frac{2\pi}{3\alpha}y \right).
\end{align*}
This function is equal to $1/4$ when $y=0$ or $y = \alpha$, and satisfies
\begin{align} \label{eqn:levelylower1}
(\Delta_{x,y} -V(x,y)+\lambda)W(x,y) = -\left(\frac{2\pi}{3\alpha}\right)^2W(x,y) + (\lambda-V(x,y))W(x,y).
\end{align}
Since $V(x,y) \geq 1$, by the straightforward eigenvalue bound on $\lambda$ from Proposition \ref{prop:simpleeigenvalue} we have
\begin{align*}
\lambda - V(x,y) \leq C^2L_1^{-2},
\end{align*}
for an absolute constant $C>0$. Therefore, from \eqref{eqn:levelylower1} we obtain
\begin{align} \label{eqn:levelylower2}
(\Delta_{x,y} -V(x,y)+\lambda)W(x,y) \leq \left(-\left(\frac{2\pi}{3\alpha}\right)^2 + C^2L_1^{-2}\right)W(x,y) .
\end{align}
Let us assume that
\begin{align} \label{eqn:levelylower3}
\alpha < \frac{2\pi L_1}{3C}.
\end{align}
Then, from \eqref{eqn:levelylower2} we see that
\begin{align*}
(\Delta_{x,y} -V(x,y)+\lambda)W(x,y) < 0,
\end{align*}
while $ (\Delta_{x,y} -V(x,y)+\lambda)u(x,y) = 0$ in $\Omega$. Also, for all points $(x,y)$ with $y=0$, $\alpha$ we have
\begin{align*}
u(x,y) \leq W(x,y) = \frac{1}{4},
\end{align*}
and $u(x,y) = 0 < W(x,y)$ for $(x,y) \in \partial \Omega$, with $0 \leq y \leq \alpha$. Therefore, by the generalised maximum principle in Proposition \ref{prop:GMP} we find that
\begin{align*}
u(x,y) \leq W(x,y) \qquad \text{ for }(x,y)\in D \text{ with } 0 \leq y \leq \alpha.
\end{align*}
However, $W(x,y) \leq \tfrac{1}{2}$, while $u(x,y)$ attains its maximum of $1$ at some point $(x,y)$ with $0 \leq y \leq \alpha$. This gives a contradiction, and so from \eqref{eqn:levelylower3} we must have
\begin{align*}
\alpha > \frac{2\pi L_1}{3C}.
\end{align*}
Therefore the projection of the superlevel set $\{(x,y)\in \Omega: u(x,y) \geq \tfrac{1}{4}\}$ onto any direction has length at least comparable to $L_1$, and this gives us the required lower bound on the inner radius of this superlevel set. We can also repeat the argument above for the superlevel set $\{(x,y)\in\Omega: u(x,y) \geq c\}$ for any fixed absolute constant $c$ with $c_1<c<1-c_1$ to obtain the same result.
\end{proof}
\begin{cor} \label{cor:levelylower}
As an immediate consequence of Proposition \ref{prop:levelylower}, we see that
\begin{align*}
A = \max_{x} \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y \geq \tilde{c}L_1,
\end{align*}
for an absolute constant $\tilde{c}>0$.
\end{cor}
Combining Propositions \ref{prop:levelyupper} and \ref{prop:levelylower}, the height of the level set $\{(x,y)\in\Omega:u(x,y) = c\}$ in the $y$-direction is comparable to $L_1$. We now turn to the studying the length of the level sets of $u(x,y)$ in the $x$-direction. We first use Corollary \ref{cor:H(x)decay} to obtain an upper bound on the length of the level sets.
\begin{prop} \label{prop:levelxupper}
Let $0<c<1$ be a fixed absolute constant. Then, the projection of the level set $\{(x,y)\in\Omega: u(x,y) = c\}$ onto the $x$-axis has length bounded by an absolute constant multiplied by $L_2$.
\end{prop}
\begin{proof}{Proposition \ref{prop:levelxupper}}
Suppose that the length of the projection of $\{(x,y)\in\Omega: u(x,y) = c\}$ onto the $x$-axis is bounded below by $2CL_2$, where $C>0$ is a large absolute constant that we will specify later in the proof. For each fixed $x$, the cross-section of the superlevel set $\{(x,y) \in\Omega: u(x,y) \geq c\}$ at $x$ consists of an interval. Since the superlevel set is convex, the length of this interval is greater than half of its maximum length for $x$ lying in an interval of length $CL_2$.
By Proposition \ref{prop:levelylower}, this maximum length is bounded below by $2C_1L_1$ for an absolute constant $C_1>0$. In other words, $u(x,y) \geq c$ for all $(x,y)$ in a rectangle of height $C_1L_1$ and width $CL_2$.
As a result of this, we have
\begin{align} \label{eqn:levelxupper1}
H(x) = \int_{\Omega(x)}u(x,y)^2 \,\mathrm{d} y \geq c^2C_1L_1,
\end{align}
for all $x$ in an interval of length $CL_2$. By Proposition \ref{prop:Aupper}, $A$ is bounded by an absolute constant multiplied by $L_1$, and by Corollary \ref{cor:H(x)decay} we have the bound
\begin{align} \label{eqn:levelxupper2}
H(x) \leq A e^{-c|x-x^*|/L_2}.
\end{align}
Therefore, combining \eqref{eqn:levelxupper1} and \eqref{eqn:levelxupper2}, we obtain a contradiction if we choose $C$ to be sufficiently large. This completes the proof of the proposition.
\end{proof}
To complete the proof of Theorem \ref{thm:shape} we finally want to obtain a comparable lower bound on the length of the level set of $u(x,y)$ in the $x$-direction. To do this we will use the $L^2$-bound on the first derivative $\partial_xu(x,y)$ from Proposition \ref{prop:xdecayupper}.
\begin{prop} \label{prop:levelxlower}
Let $0<c<1$ be a fixed absolute constant. Then, the projection of the level set $\{(x,y)\in\Omega: u(x,y) = c\}$ onto the $x$-axis has length bounded from below by an absolute constant multiplied by $L_2$.
\end{prop}
\begin{proof}{Proposition \ref{prop:levelxlower}}
We first prove the proposition for $c=1/4$. By applying Proposition \ref{prop:levelylower} with $c=\tfrac{1}{2}$, there exists a point $x=x_0$, and an interval $J$ of length equal to $2c^*L_1$ for a constant $c^*>0$, such that $u(x_0,y) \geq \tfrac{1}{2}$ for all $y$ in $J$. Therefore,
\begin{align} \label{eqn:levelxlower1}
\int_{J}u(x_0,y)^2 \,\mathrm{d} y \geq \tfrac{1}{4}c^*L_1,
\end{align}
Extending $u(x,y)$ to be zero outside of $\Omega$, for any other $x$, we can write
\begin{align*}
u(x,y) = u(x_0,y) + \int^x_{x_0}\partial_tu(t,y) \,\mathrm{d} t,
\end{align*}
and so,
\begin{align*}
u(x,y)^2 \geq \tfrac{3}{4}u(x_0,y)^2 - C_1|x-x_0|\int_{I(x)}(\partial_tu(t,y))^2 \,\mathrm{d} t,
\end{align*}
where $I(x)$ consists of those points between $x_0$ and $x$. Integrating this over $y \in J$, we find that
\begin{align} \label{eqn:levelxlower2}
\int_{J} u(x,y)^2 \,\mathrm{d} y \geq \tfrac{3}{4} \int_{J} u(x_0,y)^2 \,\mathrm{d} y - C_1|x-x_0|\int_{J} \int_{I(x)}(\partial_tu(t,y))^2 \,\mathrm{d} t \,\mathrm{d} y
\end{align}
By \eqref{eqn:levelxlower1}, the first term on the right hand side of \eqref{eqn:levelxlower2} is bounded from below by $\tfrac{3}{8}c^*L_1$. Provided $|x-x_0| \leq c_2L_2$ for $c_2>0$ sufficiently small, we can use Proposition \ref{prop:xdecayupper} to show that the second term on the right hand side of \eqref{eqn:levelxlower2} is bounded above by
\begin{align*}
C_1|x-x_0|L_1L_2^{-1},
\end{align*}
for an absolute constant $C_1>0$. Thus, if $|x-x_0| \leq c_3L_2$ for $c_3>0$ sufficiently small, we can ensure that
\begin{align*}
\int_{J}u(x,y)^2 \,\mathrm{d} y \geq \tfrac{3}{16}c^*L_1 - \tfrac{1}{16}c^*L_1 = \tfrac{1}{8} c^*L_1.
\end{align*}
Since the interval $J$ has length equal to $c^*L_1$, this means that for each $x$ with $|x-x_0|\leq c_3L_2$, $u(x,y)$ must be at least $\tfrac{1}{4}$ at some point $y\in J$. In particular, the level set $\{(x,y)\in\Omega: u(x,y) = 1/4\}$ must have length in the $x$-direction of at least $c_3L_2$ as required. A lower bound on the length in the $x$-direction of the other level sets of $u(x,y)$ follows in an analogous way.
\end{proof}
Combining Corollary \ref{cor:firstlocationA} and Proposition \ref{prop:levelylower} concerning the height of the level sets in the $y$-direction with Propositions \ref{prop:levelxupper} and \ref{prop:levelxlower} concerning the length of the level sets in the $x$-direction we have established the following: For any $c$ with $c_1<c<1-c_1$, the projections of the level sets $\{(x,y)\in\Omega: u(x,y) = c\}$ onto the $y$ and $x$-axes are of lengths comparable to $L_1$ and $L_2$ respectively, and moreover, the inner radius of the corresponding superlevel set is comparable to $L_1$ while the diameter is comparable to $L_2$. This implies that the level sets have the desired shape and completes the proof of Theorem \ref{thm:shape}.
\section{The Location Of The Maximum Of $u(x,y)$} \label{sec:max}
In Theorem \ref{thm:shape} we have described the shape of the level sets $\{(x,y)\in\Omega:u(x,y)=c\}$ where $c$ is bounded away from $0$ and $1$ by $c_1< c< 1-c_1$. In Proposition \ref{prop:Agmon}, we gave an indication of the behaviour of $u(x,y)$ as $c$ becomes small. In this section, we instead want to focus on the behaviour of the eigenfunction near its maximum.
The main aim in this section is to prove the following:
\begin{prop} \label{prop:location}
Suppose that the eigenfunction $u(x,y)$ attains its maximum at the point $(x^*,y^*)$. Then, there exists an absolute constant $c^*>0$ such that
\begin{align*}
V(x^*,y^*) - \lambda \leq -c^*L_1^{-2}.
\end{align*}
\end{prop}
\begin{proof}{Proposition \ref{prop:location}}
To prove this proposition, we first notice that from Proposition \ref{prop:firstlocation}, we can restrict our attention to the region where $V(x,y) \leq 1+CL_1^{-2}$, for a sufficiently large absolute constant $C>0$. Before we can prove the sharper estimate on the location of the maximum in Proposition \ref{prop:location} we need more information about the first derivatives of $u(x,y)$. We will use Proposition \ref{prop:firstlocation} to obtain a pointwise bound on the first derivatives of $u(x,y)$ near its maximum.
To do this, we need to introduce the following function:
\begin{defn} \label{def:J(r)}
Let $K_0(r)$ be the $0$th modified Bessel function of the second kind. Then, for $r>0$, we define the function $J(r)$ as follows: Let $0<c_1<c_2$ be small absolute constants. We first set
\begin{align*}
J(r) \coloneqq K_0(r/L_1),
\end{align*}
for $0< r \leq c_1L_1$, and then we require that $J(r)$ decays smoothly to $0$ on a length scale comparable to $L_1$ for $c_1L_1 \leq r \leq c_2L_1$, and is identically $0$ for $r>c_2L_1$.
\end{defn}
\begin{lem}[Properties of $J(r)$] \label{lem:J(r)}
For $0<r \leq c_1L_1$, $J(r)$ satisfies the equation,
\begin{align*}
\frac{1}{r^{2}} \left( r^2 \frac{d^2}{dr^2} + r\frac{d}{dr} \right) J(r) = L_1^{-2} J(r),
\end{align*}
and for $c_1L_1\leq r \leq c_2L_1$,
\begin{align*}
\frac{d^m}{dr^m}J(r) \leq CL_1^{-m}
\end{align*}
for $m\leq 3$. Moreover, $J(r)$ has a singularity equal to a multiple of $\log r$ as we approach $r=0$.
\end{lem}
\begin{proof}{Lemma \ref{lem:J(r)}}
These properties follow immediately from the definition of $J(r)$ in Definition \ref{def:J(r)} and the corresponding properties of the modified Bessel function $K_0(r)$.
\end{proof}
We will use the function $J(r)$ with $r$ defined by
\begin{align} \label{eqn:rdefn}
r^2 \coloneqq (x-x')^2 + (y-y')^2
\end{align}
to obtain a pointwise bound on the first derivatives of $u(x,y)$.
\begin{prop} \label{prop:pointwisederivy}
Fix an absolute constant $c$ with $0<c<1$. There exists an absolute constant $C>0$ such that for any point $(x',y')$ with $u(x',y') \geq c$, we have the bound
\begin{align*}
|\nabla_{x',y'}u(x',y')| \leq CL_1^{-1}.
\end{align*}
\end{prop}
\begin{proof}{Proposition \ref{prop:pointwisederivy}}
The strategy of the proof to obtain this pointwise estimate for $\nabla_{x',y'}u(x',y')$ is to use the function $J(r)$ together with the eigenfunction equation
\begin{align*}
-\Delta_{x,y}u(x,y) + (V(x,y) - \lambda)u(x,y) = 0,
\end{align*}
to obtain an expression for the first derivatives of $u(x',y')$.
We fix $(x',y')$ as in the statement of the proposition, and by Proposition \ref{prop:firstlocation}, we see that by choosing $c_1<c_2$ sufficiently small in the definition of $J(r)$, the support of $J(r)$ is contained in the region where $V(x,y) \leq 1+C_1L_1^{-2}$.
With $r$ as in \eqref{eqn:rdefn}, we begin by considering the integral
\begin{align} \label{eqn:J(r)1}
\lim_{\epsilon\to0}\int_{r>\epsilon}\nabla_{x,y}J(r).\nabla_{x,y}\partial_xu(x,y) \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
Since $J(r)$ has a singularity of the form $\log r$ at $r=0$, $\nabla_{x,y}J(r)$ is integrable and the above limit is well-defined.
Integrating by parts one time to move the derivative away from $\partial_{x}u(x,y)$, we obtain a boundary term at $r=0$, and the integral in \eqref{eqn:J(r)1} becomes
\begin{align} \label{eqn:J(r)2}
\partial_xu(x',y') - \lim_{\epsilon\to0}\int_{r>\epsilon} \Delta_{x,y}J(r) \partial_x u(x,y) \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
Note that by the support properties of $J(r)$, there are no other boundary terms appearing from this integration by parts.
By Lemma \ref{lem:J(r)}, $\Delta_{x,y}J(r) = L_1^{-2}J(r)$ for $r\leq c_1L_1$ and is $\leq C_1L_1^{-2}$ elsewhere. Thus, we can integrate by parts again in the integral in \eqref{eqn:J(r)2} to get
\begin{align} \label{eqn:J(r)3}
- \lim_{\epsilon\to0}\int_{r>\epsilon} \partial_x\Delta_{x,y}J(r) u(x,y) \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
The function $ \Delta_{x,y}J(r) = L_1^{-2}J(r)$ only has a logarithmic singularity at $r=0$, and so for this integral we do not get a boundary term at $r=0$ in the integration by parts. Instead, since $J(r)$ is supported in a region of area $L_1^2$ and $u(x,y) \leq 1$ everywhere, this integral is bounded by $CL_1^{-1}$ for an absolute constant $C>0$.
Using this bound we see that the integral in \eqref{eqn:J(r)1} is equal to $\partial_{x}u(x',y')$ plus a contribution which is bounded by $CL_1^{-1}$.
We can also write the integral in \eqref{eqn:J(r)1} as
\begin{align*}
\lim_{\epsilon\to0}\int_{r>\epsilon}\partial_xJ(r)\partial^2_xu(x,y) + \partial_y J(r) \partial_y\partial_xu(x,y) \,\mathrm{d} x \,\mathrm{d} y.
\end{align*}
Using the eigenfunction equation, we can rewrite this as
\begin{align} \label{eqn:J(r)4}
\lim_{\epsilon\to0}\int_{r>\epsilon}-\partial_xJ(r)\partial^2_yu(x,y) +\partial_xJ(r)(V(x,y) - \lambda)u(x,y) + \partial_y J(r) \partial_y\partial_xu(x,y) \,\mathrm{d} x \,\mathrm{d} y .
\end{align}
We consider the contribution to this integral from
\begin{align} \label{eqn:J(r)5}
\lim_{\epsilon\to0}\int_{r>\epsilon}-\partial_xJ(r)\partial^2_yu(x,y) + \partial_y J(r) \partial_y\partial_xu(x,y) \,\mathrm{d} x \,\mathrm{d} y.
\end{align}
If we integrate by parts in $y$ in the first term and in $x$ in the second term, we find that that the boundary terms at $r=0$ vanish, and there are no other boundary terms. Therefore, the integral in \eqref{eqn:J(r)5} is equal to
\begin{align*}
\lim_{\epsilon\to0}\int_{r>\epsilon}\partial_y\partial_xJ(r)\partial_yu(x,y) - \partial_x\partial_y J(r) \partial_yu(x,y) \,\mathrm{d} x \,\mathrm{d} y = 0.
\end{align*}
Thus, from \eqref{eqn:J(r)4}, the original integral in \eqref{eqn:J(r)1} becomes
\begin{align*}
\lim_{\epsilon\to0}\int_{r>\epsilon} \partial_xJ(r)(V(x,y) - \lambda)u(x,y) \,\mathrm{d} x \,\mathrm{d} y = \int_{\Omega} \partial_xJ(r)(V(x,y) - \lambda)u(x,y) \,\mathrm{d} x \,\mathrm{d} y .
\end{align*}
This means that we have shown that
\begin{align} \label{eqn:J(r)6}
\partial_xu(x',y') = \int_{\Omega} \partial_xJ(r)(V(x,y) - \lambda)u(x,y) \,\mathrm{d} x \,\mathrm{d} y,
\end{align}
plus a contribution which is bounded by $CL_1^{-1}$.
Since $u(x',y') \geq c$, we know from Proposition \ref{prop:firstlocation} that $|V(x,y) - \lambda| \leq C_1L_1^{-2}$ on the support of $J(r)$. Therefore, the integral on the right hand side of \eqref{eqn:J(r)6} is bounded by
\begin{align*}
C_1L_1^{-2} \int_{\Omega} \left| \partial_xJ(r) \right| u(x,y) \,\mathrm{d} x \,\mathrm{d} y \leq CL_1^{-1}.
\end{align*}
This gives the required bound for $\partial_xu(x',y')$, and the bound for $\partial_yu(x',y')$ follows in exactly the same way.
\end{proof}
We recall from Theorem \ref{thm:shape} that the level sets of $u(x,y)$ are of height comparable to $L_1$ in the $y$-direction and of length comparable to $L_2$ in the $x$-direction. Therefore, this is consistent with the derivative bound for $\partial_yu(x,y)$ from Proposition \ref{prop:pointwisederivy} above. However, since in general we have $L_2\gg L_1$, we want to improve the bound given for $\partial_xu(x,y)$.
To do this we first prove a corollary of Propositions \ref{prop:firstlocation} and \ref{prop:pointwisederivy} about the location of the level sets of $u(x,y)$ in the $x$-direction.
\begin{cor} \label{cor:firstlocation}
Fix an absolute constant $c$, with $0<c<1$. Then, there exists an absolute constant $C>0$ such that for any point $(x,y)$ in the level set $\{(x,y)\in\Omega:u(x,y) = c\}$, there exists points $(x_1,y_1)$ with $x_1-x$ both positive or negative that $|x_1 - x|$ is comparable to $L_2$ and $V(x_1,y_1) \leq 1+CL_1^{-2}$.
\end{cor}
\begin{proof}{Corollary \ref{cor:firstlocation}}
Let $(x',y')$ be the left most point of the level set $\{(x,y)\in\Omega:u(x,y) = c\}$. That is, $u(x',y') = c$, and $u(x,y) < c$ for any point $(x,y)$ with $x<x'$. Then, by the bound on $\partial_yu(x,y)$ from Proposition \ref{prop:pointwisederivy}, we know that
\begin{align*}
u(x',y) \geq \tfrac{1}{2}c,
\end{align*}
for all $y$ in an interval $J$ of length comparable to $L_1$. In particular,
\begin{align*}
\int_{J}u(x',y)^2 \,\mathrm{d} y \geq \tilde{c}L_1,
\end{align*}
for some absolute constant $\tilde{c}>0$. Therefore, using the $L^2$-bound on $\partial_xu(x,y)$ from Proposition \ref{prop:xdecayupper}, exactly as in the proof of Proposition \ref{prop:levelxlower}, we find that
\begin{align*}
\int_{J}u(x,y)^2 \,\mathrm{d} y \geq \tilde{c}L_1/2,
\end{align*}
for all $x<x'$ with $x'-x>c_2L_2$, for an absolute constant $c_2>0$. Thus, for any such $x=x_1$, there exists a $y_1 \in J$ such that $u(x_1,y_1)$ is bounded below by an absolute constant. In particular, by Proposition \ref{prop:firstlocation}, this means that $V(x_1,y_1) \leq 1+ CL_1^{-2}$, and this completes the proof of the corollary.
\end{proof}
This corollary allows us to partially improve the estimate on $\partial_xu(x,y)$ from Proposition \ref{prop:pointwisederivy}. We recall that the maximum of $u(x,y)$ of $1$ is achieved at $(x,y) = (x^*,y^*)$.
\begin{prop} \label{prop:pointwisederivx}
Consider the level set $\{(x,y) \in\Omega: u(x,y) = c\}$ for a fixed constant $c$, with $c_1<c<1-c_1$. Then, on part of the upper and lower boundaries of this level set, with $x$ in an interval around $x=x^*$ of length comparable to $L_2$, we have the pointwise derivative bound
\begin{align*}
|\partial_xu(x,y)| \leq CL_2^{-1},
\end{align*}
for some absolute constant $C>0$.
\end{prop}
\begin{proof}{Proposition \ref{prop:pointwisederivx}}
For fixed $c$, with $c_1<c<1-c_1$, the level set $\{(x,y)\in\Omega: u(x,y) = c\}$ extends a distance comparable to $L_2$ in the $x$-direction on either side of the point $x^*$. Let $y= g(x)$ be a parametrisation of the upper boundary of the level set. Since this level set is the boundary of a convex set of height comparable to $L_1$ in the $y$-direction, this means that
\begin{align} \label{eqn:pointwisederivx1}
|g'(x)| \leq C_1L_1L_2^{-1}
\end{align}
for all $x$ in an interval around $x^*$ of length comparable to $L_2$. On this part of the level set we have
\begin{align*}
u(x,g(x)) = c,
\end{align*}
and so differentiating this with respect to $x$ gives
\begin{align*}
\partial_x u(x,g(x)) = - g'(x)\partial_yu(x,g(x)).
\end{align*}
Combining the estimate for $g'(x)$ in \eqref{eqn:pointwisederivx1} with the pointwise bound on $\partial_yu(x,y)$ from Proposition \ref{prop:pointwisederivy} gives the required bound for $\partial_xu(x,y)$.
\end{proof}
To prove that the maximum of $u(x,y)$ has the required properties of Proposition \ref{prop:location}, we will also need to improve the derivative estimate of Proposition \ref{prop:pointwisederivy} for those points $(x,y)$ near the maximum. To do this we will prove the following two propositions.
\begin{prop} \label{prop:yloweruniform}
Let $\epsilon>0$ be sufficiently small. Then, the convex superlevel set $\{(x,y)\in\Omega: u(x,y) \geq 1-\epsilon\}$ has inner radius at least $c {\epsilon}^{1/2}L_1$, for a small absolute constant $c>0$, which is independent of $\epsilon$.
\end{prop}
\begin{prop} \label{prop:maxderivy}
Let $u(x',y') = 1-\epsilon$, where $\epsilon>0$ is sufficiently small. Then, there exists an absolute constant $C>0$, which is independent of $\epsilon$, such that
\begin{align*}
|\nabla_{x,y}u(x',y')| \leq C{\epsilon}^{1/2}L_1^{-1}.
\end{align*}
In particular, this is an improvement on Proposition \ref{prop:pointwisederivy} for small $\epsilon$.
\end{prop}
\begin{rem}
Consider a function $f(x)$ defined on the interval $[-L_1,L_1]$ by
\begin{align*}
f(y) \coloneqq 1 - L_1^{-2}y^2.
\end{align*}
Then, $f(y) = 1-\epsilon$ for $y = \pm \epsilon^{1/2}L_1$. In other words, the interval on which $f(y) \geq 1-\epsilon$ has length $2\epsilon^{1/2}L_1$. Thus, the lower bound on the inner radius of the superlevel set of $u(x,y)$ in Proposition \ref{prop:yloweruniform} is consistent with the eigenfunction $u(x,y)$ being bounded from below by such a parabola as we move away from the maximum in the $y$-direction.
Also, at the two points where $f(y)$ is equal to $1-\epsilon$ we have the derivative bound
\begin{align*}
f'(y) = -2L_1^{-2}y = \mp 2\epsilon^{1/2}L_1^{-1}.
\end{align*}
Therefore, the derivative bound in Proposition \ref{prop:maxderivy} is again consistent with the eigenfunction $u(x,y)$ being bounded from below by such a parabola as we move away from the maximum in the $y$-direction. It is also consistent with having a bound comparable to $L_1^{-2}$ on the second derivatives of the eigenfunction.
\end{rem}
\begin{proof}{Proposition \ref{prop:yloweruniform}}
Suppose that the proposition does not hold. Then, after a translation and rotation, we may assume that the level set $\{(x,y)\in\Omega:u(x,y) = 1-\epsilon\}$ lies between the lines $y=\pm\alpha$, where $\alpha < c_1 \epsilon^{1/2}L_1$ for a small absolute constant $c_1$ to be chosen later. Note that we are considering the length scale $L_1$, and we will not use any of the properties of $\lambda$ and $u(x,y)$ that depend on $L_2$. This means that we do not have to fix the orientation of $\Omega_{L_1^{-2}} = \{(x,y)\in\Omega:V(x,y)=1+L_1^{-2}\}$ and so there is no issue in applying the rotation above.
We will use the comparison function
\begin{align*}
v_2(x,y) \coloneqq \left(1-\tfrac{1}{2}\epsilon \right) \sin\left(\frac{\pi}{2} + \frac{{\epsilon}^{1/2}y}{C_1\alpha}\right),
\end{align*}
where $C_1>0$ is chosen so that
\begin{align*}
v_2(x,y) \geq 1-\epsilon
\end{align*}
for all $(x,y)$ with $y = \pm\alpha$. This means that
\begin{align} \label{eqn:yloweruniform1}
u(x,y) \leq v_2(x,y)
\end{align}
for any $(x,y)\in\Omega$ with $y=\pm \alpha$. Also, for all $(x,y)\in\partial\Omega$ with $-\alpha \leq y \leq \alpha$, we see that
\begin{align} \label{eqn:yloweruniform1A}
u(x,y) = 0 \leq v_2(x,y).
\end{align}
Moreover, the function $v_2(x,y)$ satisfies
\begin{align*}
\left(\Delta_{x,y} - V(x,y) + \lambda \right)v_2(x,y) = - \left(\frac{\epsilon}{C_1^2\alpha^2}\right)v_2(x,y) + (\lambda-V(x,y))v_2(x,y).
\end{align*}
We know that $\lambda-V(x,y) \leq C_2^2L_1^{-2}$, for some absolute constant $C_2>0$, and so
\begin{align} \label{eqn:yloweruniform2}
\left(\Delta_{x,y} - V(x,y) + \lambda\right) v_2(x,y) <\left(- \left(\frac{\epsilon}{C_1^2\alpha^2}\right) +C_2^2L_1^{-2}\right)v_2(x,y) .
\end{align}
Provided $\alpha <c_1\epsilon^{1/2}L_1$, for $c_1$ sufficiently small (depending only on $C_1$ and $C_2$), we can ensure from \eqref{eqn:yloweruniform2} that
\begin{align} \label{eqn:yloweruniform3}
\left(\Delta_{x,y} - V(x,y) + \lambda\right)v_2(x,y) < 0.
\end{align}
Combining \eqref{eqn:yloweruniform1}, \eqref{eqn:yloweruniform1A} and \eqref{eqn:yloweruniform3}, we see that by the generalised maximum principle in Proposition \ref{prop:GMP} that
\begin{align*}
u(x,y) \leq v_2(x,y) \qquad \text{ for } (x,y)\in\Omega \text{ with } -\alpha \leq y \leq \alpha.
\end{align*}
However, $v_2(x,y) \leq 1-\frac{1}{2}\epsilon$ everywhere, whereas we know that $u(x,y)$ attains its maximum of $1$ for some $(x,y) \in\Omega$ with $-\alpha \leq y \leq \alpha$. This contradiction completes the proof of the proposition.
\end{proof}
\begin{proof}{Proposition \ref{prop:maxderivy}}
In the proof of Proposition \ref{prop:maxderivy}, as well as Proposition \ref{prop:yloweruniform}, we will also make use of the following proposition:
\begin{prop} \label{prop:A4}
Suppose that the function $v(x,y)$ satisfies
\begin{align*}
\Delta_{x,y} v(x,y) + W_1(x,y)v(x,y) = W_2(x,y),
\end{align*}
in a convex domain $D$, with $|W_1(x,y)|$, $|W_2(x,y)| \leq C_1$. Let $(x_0,y_0) \in \partial D$, and let $B_s$ be the disc of radius $s$ around $(x_0,y_0)$. If we have $v \geq 0$ in $D$, then
\begin{align*}
|\nabla_{x,y}v(x_0,y_0)| \leq C \sup_{B_{1/4}\cap D}v.
\end{align*}
\end{prop}
\begin{proof}{Proposition \ref{prop:A4}}
This proposition is a variation of Proposition A.4 in \cite{GJ2}, and is proved in exactly the same way.
\end{proof}
We can now begin the proof of Proposition \ref{prop:maxderivy}. After a translation, we may assume that $u(x,y) = 1$ at the point $(x,y) = (0,0)$, and we define the function $u_1(x,y)$ by
\begin{align*}
u_1(x,y) \coloneqq u\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right).
\end{align*}
This satisfies the equation
\begin{align*}
-\Delta_{x,y}u_1(x,y) + \epsilon L_1^2\left(V\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right) - \lambda\right)u_1(x,y) = 0.
\end{align*}
We next define $u_2(x,y)$ by
\begin{align*}
u_2(x,y) \coloneqq \epsilon^{-1}(u_1(x,y) - (1-\epsilon)).
\end{align*}
Consider the convex superlevel set $\{(x,y)\in\Omega:u_1(x,y) \geq 1-\epsilon\}$. By Proposition \ref{prop:yloweruniform} and the definition of the function $u_1(x,y)$, this set has inner radius bounded from below by an absolute constant.
Moreover, on the boundary of this set the function $u_2(x,y)$ is equal to $0$ and inside the superlevel set it takes all values between $0$ and $1$. It satisfies the equation
\begin{align} \label{eqn:maxderivy1} \nonumber
-\Delta_{x,y}u_2(x,y) & + \epsilon L_1^2\left(V\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right) - \lambda\right)u_2(x,y) \\
& + (1-\epsilon)L_1^2\left(V\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right) - \lambda\right) = 0.
\end{align}
From Proposition \ref{prop:firstlocation}, we certainly know that inside the convex set $\{(x,y)\in\mathbb{R}^2:u_1(x,y) = 1-\epsilon\}$ we have the bounds
\begin{align*}
\left|L_1^2 \left(V\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right) - \lambda\right)\right| \leq C_1,
\end{align*}
for an absolute constant $C_1>0$. Therefore, in \eqref{eqn:maxderivy1} we have the bounds
\begin{align*}
\left|\epsilon L_1^2\left(V\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right) - \lambda\right)\right| \leq C_1, \qquad \left|(1-\epsilon)L_1^2\left(V\left({\epsilon}^{1/2}L_1x,{\epsilon}^{1/2}L_1y \right) - \lambda\right)\right| \leq C_1.
\end{align*}
We can thus use Proposition \ref{prop:A4} with $D = \{(x,y)\in\mathbb{R}^2:u_1(x,y) \geq 1-\epsilon\}$ to conclude that
\begin{align*}
|\nabla_{x,y}u_2(x,y)| \leq C,
\end{align*}
for $(x,y)$ on the boundary of $D$. Recalling the definitions of $u_1(x,y)$ and $u_2(x,y)$, this shows that
\begin{align*}
|\nabla_{x,y}u(x',y')| \leq C \epsilon(\sqrt{\epsilon}L_1)^{-1} = C\sqrt{\epsilon}L_1^{-1}
\end{align*}
as required.
\end{proof}
We need to write down one more consequence of the log concavity of $u(x,y)$, and then we can start to complete the proof of Proposition \ref{prop:location}.
\begin{lem} \label{lem:logcon}
We have an upper bound on $\partial_x^2u(x,y)$ of the form
\begin{align*}
\partial_x^2u(x,y) \leq \frac{(\partial_xu(x,y))^2}{u(x,y)},
\end{align*}
and an analogous upper bound for $\partial^2_yu(x,y)$.
\end{lem}
\begin{proof}{Lemma \ref{lem:logcon}}
Differentiating the function $\log u(x,y)$ twice with respect to $x$, we find that
\begin{align} \label{eqn:logcon1}
\partial_x^2(\log u(x,y)) =\frac{ \partial_x^2u(x,y)}{u(x,y)} - \frac{(\partial_xu(x,y))^2}{u(x,y)^2}.
\end{align}
However, the eigenfunction $u(x,y)$ is log concave, and so
\begin{align} \label{eqn:logcon2}
\partial_x^2(\log u(x,y)) \leq 0.
\end{align}
Combining \eqref{eqn:logcon1} and \eqref{eqn:logcon2} gives the desired bound.
\end{proof}
To complete the proof of Proposition \ref{prop:location} we split into two cases.
\subsection{Case $1$: $L_2 \gg L_1$}
We will first assume that $L_2 \gg L_1$, or in other words, $L_2 \geq \tilde{C}L_1$ for a large absolute constant $\tilde{C}>0$, which we will specify below.
After a translation, we may assume that $u(x,y)$ attains its maximum at the point $(0,0)$. Let $u(0,-\alpha L_1)$, $u(0,\beta L_1) = 1/2$, where we know from Theorem \ref{thm:shape} that $\alpha$ and $\beta$ are both comparable to $1$. Moreover, without loss of generality, we may assume that $\partial_yV(0,0) \geq 0$. We want to study the integral
\begin{align} \label{eqn:max1}
\int_0^{\beta L_1} (\beta L_1 - y)(V(0,y) -\lambda)u(0,y) \,\mathrm{d} y.
\end{align}
We will prove the following two lemmas:
\begin{lem} \label{lem:max1lower}
The integral in \eqref{eqn:max1} is bounded from below by
\begin{align*}
\tfrac{1}{2}\beta^2L_1^2(V(0,0) - \lambda).
\end{align*}
\end{lem}
\begin{lem} \label{lem:max1upper}
The integral in \eqref{eqn:max1} is bounded from above by $-\tfrac{1}{4}$.
\end{lem}
Combining Lemmas \ref{lem:max1lower} and \ref{lem:max1upper}, we see that
\begin{align*}
V(0,0) - \lambda \leq -\tfrac{1}{2}\beta^{-2}L_1^{-2},
\end{align*}
and so we have established Proposition \ref{prop:location} in the case where $L_2\gg L_1$. Thus, we are left to prove these two lemmas.
\begin{proof}{Lemma \ref{lem:max1lower}}
Since $\partial_yV(0,0) \geq 0$, and $V(x,y)$ is convex, we must have $\partial_yV(0,y) \geq0$ for all $y \geq0$. Thus, for $0 \leq y\leq \beta L_1$ we have
\begin{align*}
V(0,y)-\lambda \geq V(0,0) -\lambda
\end{align*}
and $u(0,y) \geq \tfrac{1}{2}$. The lower bound then follows immediately.
\end{proof}
\begin{proof}{Lemma \ref{lem:max1upper}}
Since $u(x,y)$ satisfies the eigenfunction equation, we can rewrite \eqref{eqn:max1} as
\begin{align} \label{eqn:max1upper1}
\int_0^{\beta L_1} (\beta L_1 - y)\Delta_{x,y}u(0,y) \,\mathrm{d} y.
\end{align}
Let us first consider the term containing a factor of $\partial_y^2u(0,y)$. Since $\partial_yu(0,0) = 0$, integrating by parts, this becomes
\begin{align*}
\int_0^{\beta L_1} \partial_yu(0,y) \,\mathrm{d} y = -u(0,0) + u(0,\beta L_1) = -\tfrac{1}{2}.
\end{align*}
We are left to bound the contribution to \eqref{eqn:max1upper1} from
\begin{align} \label{eqn:max1upper2}
\int_0^{\beta L_1} (\beta L_1 - y)\partial_x^2u(0,y) \,\mathrm{d} y,
\end{align}
and to do this we will use Proposition \ref{prop:maxderivy} and Lemma \ref{lem:logcon}.
We first fix $0<c_1<\beta$ such that $u(0,c_1L_1) = 1- c_2$ for a small constant $c_2>0$. Then, by Proposition \ref{prop:maxderivy}, we have the bound
\begin{align*}
|\partial_xu(0,y)| \leq Cc_2^{1/2}L_1^{-1}
\end{align*}
for all $y$ with $0 \leq y \leq c_1L_1$, with $C$ independent of $c_2$. In particular, by Lemma \ref{lem:logcon}, we have
\begin{align*}
\partial_x^2u(x,y) \leq Cc_2L_1^{-2},
\end{align*}
and so
\begin{align} \label{eqn:max1upper3}
\int_0^{c_1 L_1}(\beta L_1 - y)\partial_x^2 u(0,y) \,\mathrm{d} y \leq \frac{1}{8},
\end{align}
provided $c_1>0$ is sufficiently small. We now need to consider the part of the integral in \eqref{eqn:max1upper2} with $y$ between $c_1L_1$ and $\beta L_1$. For $y$ in this range, we can use the derivative bound on $\partial_xu(x,y)$ from Proposition \ref{prop:pointwisederivx}, which after applying Lemma \ref{lem:logcon} gives
\begin{align*}
\partial_x^2u(0,y) \leq CL_2^{-2}.
\end{align*}
Therefore, provided $L_2/L_1$ is sufficiently large, we also have the bound
\begin{align} \label{eqn:max1upper4}
\int_{c_1L_1}^{\beta L_1} (\beta L_1 - y)\partial_x^2u(0,y) \,\mathrm{d} y \leq \frac{1}{8}.
\end{align}
Combining \eqref{eqn:max1upper3} and \eqref{eqn:max1upper4}, we see that the integral in \eqref{eqn:max1upper2} is bounded above by $\tfrac{1}{4}$, and hence
\begin{align*}
\int_0^{\beta L_1} (\beta L_1 - y) \Delta_{x,y}u(0,y) \,\mathrm{d} y < -\frac{1}{4}.
\end{align*}
This completes the proof of the lemma.
\end{proof}
As we discussed after the statement of Lemmas \ref{lem:max1lower} and \ref{lem:max1upper}, this completes the proof of Proposition \ref{prop:location} in the case where $L_2\gg L_1$.
\subsection{Case $2$: $L_1$ and $L_2$ are comparable}
In this case, we assume that $V(x,y)$ attains its minimum of $1$ at $(0,0)$, and we rescale the eigenfunction $u(x,y)$ by $L_1$ in the $x$ and $y$-directions. Then, $\tilde{u}(x,y) = u(L_1x,L_1y)$ satisfies the equation
\begin{align*}
\Delta_{x,y}\tilde{u}(x,y) = \tilde{F}(x,y)\tilde{u}(x,y),
\end{align*}
where $\tilde{F}(x,y) = L_1^2\left(V(L_1x,L_1y) - \lambda\right)$. We know that $\tilde{u}(x,y)$ must attain its maximum at some point inside the region where $\tilde{F}(x,y) \leq 0$. We now want to improve this estimate on the location of the maximum of $\tilde{u}(x,y)$.
\begin{lem} \label{lem:umaxL1}
In the case where $L_1$ and $L_2$ are comparable there exists a small absolute constant $\epsilon>0$ such that $\tilde{u}(x,y)$ attains its maximum at a distance at least $\epsilon^{2/3}$ from the boundary of the region where $\tilde{F}(x,y) \leq 0$.
\end{lem}
The function $\tilde{F}(x,y)$ is convex, has a minimum of $-c$ for some $c>0$, and the inner radius and diameter of the region where $\tilde{F}(x,y) \leq 0$ are comparable to $1$. Therefore, this lemma implies that
\begin{align*}
\tilde{F}(x,y) \leq -c_1
\end{align*}
for some constant $c_1>0$ at the point where $\tilde{u}(x,y)$ attains its maximum. Returning to $u(x,y)$ and $F(x,y) = V(x,y) - \lambda$, this proves Proposition \ref{prop:location} in the case where $L_1$ and $L_2$ are comparable. Thus, we are left to prove Lemma \ref{lem:umaxL1}.
\begin{proof}{Lemma \ref{lem:umaxL1}}
Suppose that $\tilde{u}(x,y) = 1$ at a point $(x,y)$ within $\epsilon^{2/3}$ of the boundary of the region $\{(x,y):\tilde{F}(x,y) \leq 0\}$, where $\epsilon>0$ is a small constant that we will specify later.
The function $\tilde{F}(x,y)$ attains a negative minimum, is convex, and is negative on a region with diameter comparable to $1$. Therefore, $|\nabla_{x,y}\tilde{F}(x,y) | \geq c_1$ on the set where $\tilde{F}(x,y) = 0$. In particular, we have the lower bound
\begin{align} \label{eqn:umaxL11}
\tilde{F}(x,y) \geq \epsilon^{2/3}
\end{align}
when we are at a distance comparable to $\epsilon^{2/3}$ outside the region where $\tilde{F}(x,y) \leq 0$. Also, by the pointwise derivative bounds on $u(x,y)$ from Proposition \ref{prop:pointwisederivy}, $\tilde{u}(x,y)$ is comparable to $1$ at a distance of $\epsilon^{2/3}$ from its maximum.
By the log concavity of $u(x,y)$, we know that
\begin{align*}
\Delta_{x,y}\log u(x,y) = \frac{\Delta_{x,y}u(x,y)}{u(x,y)} - \frac{|\nabla_{x,y}u(x,y)|^2}{u(x,y)^2} \leq 0.
\end{align*}
Rearranging, and using the eigenfunction equation, this tells us that
\begin{align} \label{eqn:umaxL12}
|\nabla_{x,y}u(x,y)|^2 \geq (V(x,y) - \lambda)u(x,y)^2 = F(x,y)u(x,y)^2.
\end{align}
Thus, from \eqref{eqn:umaxL11} and \eqref{eqn:umaxL12}, we have the lower bound
\begin{align} \label{eqn:umaxL13}
|\nabla_{x,y}\tilde{u}(x,y)| \geq \tilde{c}\epsilon^{1/3}
\end{align}
for some point $(x_1,y_1)$ which is at a distance comparable to $\epsilon^{2/3}$ from the point where $\tilde{u}(x,y)$ attains its maximum.
However, by Proposition \ref{prop:maxderivy}, we know that when $\tilde{u}(x',y') = 1-\epsilon$, we have the derivative bound
\begin{align} \label{eqn:umaxL14}
|\nabla_{x,y}\tilde{u}(x',y')| \leq C\epsilon^{1/2}.
\end{align}
For $\epsilon>0$ sufficiently small, we have $\tilde{c}\epsilon^{1/3} > C\epsilon^{1/2}$, and so from \eqref{eqn:umaxL13} and \eqref{eqn:umaxL14}, we see that $\tilde{u}(x_1,y_1) < 1-\epsilon$.
In other words, the function $\tilde{u}(x,y)$ changes from $1$ to $1-\epsilon$ on a line segment of length comparable to $\epsilon^{2/3}$. However, using Proposition \ref{prop:maxderivy} again, we know that
\begin{align*}
|\nabla_{x,y}\tilde{u}(x,y)| \leq C\epsilon^{1/2}
\end{align*}
whenever $\tilde{u}(x,y) \geq 1- \epsilon$, and so $\tilde{u}(x,y)$ can only change by an amount comparable to
\begin{align*}
\epsilon^{1/2}\epsilon^{2/3} = \epsilon^{7/6},
\end{align*}
on this line segment of length $\epsilon^{2/3}$. For $\epsilon>0$ sufficiently small, we see that $\epsilon^{7/6} \ll \epsilon$, and so this gives us a contradiction.
\end{proof}
As we discussed after the statement of Lemma \ref{lem:umaxL1} this also completes the proof of Proposition \ref{prop:location} in the case where $L_1$ and $L_2$ are comparable.
\end{proof}
Let us finish by giving two consequences of the location of the maximum of $u(x,y)$ derived in Proposition \ref{prop:location}. The first is to show that the lower bound on the inner radius of the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq 1-\epsilon\}$ given in Proposition \ref{prop:yloweruniform} is sharp.
\begin{cor} \label{cor:yupperuniform}
For $\epsilon>0$ sufficiently small, the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq 1-\epsilon\}$ has inner radius at most $C_1\epsilon^{1/2}L_1$, where $C_1>0$ is an absolute constant.
\end{cor}
\begin{proof}{Corollary \ref{cor:yupperuniform}}
By Proposition \ref{prop:location}, we know that $V(x,y) -\lambda \leq -c^*L_1^{-2}$ at the maximum of $u(x,y)$. Moreover, by Proposition \ref{prop:firstlocation}, inside the level set $\{(x,y)\in\Omega:u(x,y) =1/2\}$, we have the bound
\begin{align*}
V(x,y) - \lambda \leq CL_1^{-2}.
\end{align*}
Since this level set has height comparable to $L_1$ in the $y$-direction and length comparable to $L_2$ in the $x$-direction, by the convexity of the potential, we have
\begin{align} \label{eqn:yupperuniform1}
V(x,y) - \lambda \leq -\frac{1}{2}c^*L_1^{-2}
\end{align}
on a region of height $c_1L_1$ and length $c_2L_2$ in the $y$ and $x$-directions around the maximum.
Suppose that the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq 1-\epsilon\}$ has inner radius at least $\alpha$, where $\alpha = C_1\epsilon^{1/2}L_1$ for some large absolute constant $C_1>0$. Then, this superlevel set contains a circle of radius $\alpha$, and after a translation, centre at $(0,0)$.
Let $J_0(r)$ be the $0$th Bessel function of the first kind for $r>0$. This satisfies $J_0(0) = 1$, $J_0'(0) = 0$ and $J_0''(0) = -1/2$, as well as the equation
\begin{align} \label{eqn:yupperuniform2}
r^2J_0''(r) + rJ_0'(r) = -r^2J_0(r) .
\end{align}
Setting $r^2 = x^2+y^2$, we will use the comparison function
\begin{align*}
v(x,y) = (1+\epsilon)J_0(C\epsilon^{1/2}\alpha^{-1}r)
\end{align*}
for $x^2+y^2 \leq \alpha$. Here $C>0$ is chosen so that $(1+\epsilon)J_0\left(C\epsilon^{1/2}\right) \leq 1-\epsilon$. This is possible for $\epsilon>0$ sufficiently small, since for small $r$, $J_0(r)$ satisfies
\begin{align*}
J_0(r) = 1-\tfrac{1}{2}r^2 + O(r^4).
\end{align*}
In particular, this ensures that
\begin{align} \label{eqn:yupperuniform3}
v(x,y) \leq u(x,y),
\end{align}
for $x^2+y^2 = \alpha^2$. By \eqref{eqn:yupperuniform2}, the function $v(x,y)$ also satisfies the equation
\begin{align*}
\Delta_{x,y}v(x,y) = -\frac{C^2\epsilon}{\alpha^2}v(x,y) .
\end{align*}
Thus,
\begin{align} \label{eqn:yupperuniform4}
\Delta_{x,y}v(x,y) - (V(x,y) - \lambda)v(x,y) = -\frac{C^2\epsilon}{\alpha^2}v(x,y) - (V(x,y) - \lambda)v(x,y).
\end{align}
Provided that we take $\epsilon>0$ sufficiently small, we can ensure from \eqref{eqn:yupperuniform1} that
\begin{align*}
V(x,y) - \lambda \leq -\frac{1}{2}c^*L_1^{-2}
\end{align*}
for $x^2 + y^2 \leq \alpha^2$. Therefore, provided $\alpha= C_1\epsilon^{1/2}L_1$ for $C_1$ sufficiently large, and $x^2+y^2 \leq \alpha^2$, we have
\begin{align*}
-\frac{C^2\epsilon}{\alpha^2} - (V(x,y) - \lambda) \geq \frac{1}{4}c^*L_1^{-2} \geq 0,
\end{align*}
and so from \eqref{eqn:yupperuniform4} we see that
\begin{align} \label{eqn:yupperuniform5}
\Delta_{x,y}v(x,y) - (V(x,y) - \lambda)v(x,y) \geq 0
\end{align}
for $x^2+y^2 \leq \alpha^2$. Combining \eqref{eqn:yupperuniform3} and \eqref{eqn:yupperuniform5}, we can apply the generalised maximum principle from Proposition \ref{prop:GMP} to conclude that
\begin{align*}
v(x,y) \leq u(x,y)
\end{align*}
whenever $x^2 + y^2 \leq \alpha^2$. However, $v(0,0) = 1+\epsilon$, while $u(x,y) \leq 1$ everywhere, and so this gives us a contradiction.
\end{proof}
The second consequence of Proposition \ref{prop:location} is to improve the pointwise bound on $\partial_xu(x,y)$ from Proposition \ref{prop:pointwisederivx} in the case where $L_2 \gg L_1$.
\begin{cor} \label{cor:2pointwisederivx}
There exists a constant $c>0$ such that we have the derivative bound
\begin{align*}
|\partial_xu(x,y)| \leq CL_2^{-1},
\end{align*}
for an absolute constant $C$, for all $(x,y)$ in a rectangle of side lengths $cL_2$ and $cL_1$ around the maximum of $u(x,y)$.
\end{cor}
\begin{proof}{Corollary \ref{cor:2pointwisederivx}}
From Corollary \ref{cor:yupperuniform} above, the superlevel sets $\{(x,y)\in\Omega:u(x,y) \geq 1-\epsilon\}$ have inner radius bounded by $C\epsilon^{1/2}L_1$. Let the maximum of $u(x,y)$ be attained at $(0,0)$. Then, we saw in the proof of Corollary \ref{cor:yupperuniform} that the sublevel set
\begin{align*}
\{(x,y) \in \Omega: V(x,y) - \lambda \leq -\tfrac{1}{2}c^*L_1^{-2}\},
\end{align*}
contains a rectangle $R$, with centre at $(0,0)$, and of side lengths comparable to $L_2$ and $L_1$ in the $x$ and $y$-directions. We then construct a set $U \subset \Omega$ as follows: It consists of the part of the superlevel set $\{(x,y)\in\Omega:u(x,y) \geq 1-\tilde{c}\}$ with $x$ restricted to an interval of length $L_2$ around $0$, and $\tilde{c}>0$ sufficiently small so that $U$ is contained within the middle half of the rectangle $R$.
The boundary of this set $U$ then consists of parts of the upper and lower boundaries of the level set $\{(x,y)\in\Omega:u(x,y) = 1-\tilde{c}\}$, and two vertical lines with $x$ fixed. Moreover, by choosing $\tilde{c}$ to be sufficiently small, $U$ is contained between the two lines $y=\pm\tfrac{1}{2} c_1L_1$. We then define a comparison function $W(x,y)$ by
\begin{align*}
W(x,y) = \frac{1}{c_2L_2}\cosh \left(\frac{x\log(L_2/L_1)}{c_3L_2}\right)\cos \left(\frac{\pi y}{2c_1L_1}\right).
\end{align*}
Here $c_2$ and $c_3$ are small absolute constants depending on $c_1$ that we will specify below.
Firstly, we choose $c_2>0$ sufficiently small so that for all $|y| \leq c_1L_1/2$, we have
\begin{align*}
W(x,y) \geq C_1L_2^{-1}.
\end{align*}
This absolute constant $C_1$ is chosen so that
\begin{align} \label{eqn:2pointwisederivx1}
|\partial_xu(x,y)| \leq W(x,y)
\end{align}
for all points $(x,y)$ on the curved portion of $\partial U$ consisting of part of the upper and lower boundaries of $\{(x,y)\in\Omega:u(x,y) = 1-\tilde{c}\}$. This is possible due to Proposition \ref{prop:pointwisederivx}.
We now let $x = cL_2$, where $c>0$ is chosen so that $x=\pm2cL_2$ is contained in the projection of the set $U$ onto the $x$-axis. Then, for all $|y| \leq c_1L_1/2$, we have the lower bound
\begin{align*}
W(cL_2,y) \geq \frac{1}{2c_2L_2}\cosh \left(\frac{c}{c_3}\log(L_2/L_1)\right)\geq \frac{1}{4c_2L_2}\exp\left(\frac{c}{c_3}\log(L_2/L_1)\right) = \frac{1}{4c_2L_2}\left(\frac{L_2}{L_1}\right)^{c/c_3}.
\end{align*}
We can thus choose $c_3>0$ sufficiently small, depending on $c$ only, so that
\begin{align*}
W(cL_2,y) \geq C_2L_1^{-1}.
\end{align*}
Here $C_2$ is chosen so that for $|x|\geq cL_2$, $(x,y)\in U$, we have
\begin{align} \label{eqn:2pointwisederivx2}
|\partial_xu(x,y)| \leq W(x,y).
\end{align}
This is possible due to Proposition \ref{prop:pointwisederivy}.
The function $W(x,y)$ satisfies the equation
\begin{align} \label{eqn:2pointwisederivx3}
\Delta_{x,y}W(x,y) = \left(\left( \frac{\log(L_2/L_1)}{c_3L_2}\right)^2 - \left(\frac{\pi^2}{4c_1^2L_1^2}\right)\right)W(x,y) \leq -\frac{\pi^2}{8c_1^2L_1^2}W(x,y),
\end{align}
provided $L_2/L_1$ is sufficiently large.
The first derivative $\partial_xu(x,y)$ satisfies
\begin{align} \label{eqn:2pointwisederivx4}
(-\Delta_{x,y} + V(x,y) - \lambda)\partial_xu(x,y) = -\partial_xV(x,y)u(x,y).
\end{align}
By the convexity of $V(x,y)$, we have the bound $|\partial_xV(x,y)u(x,y)| \leq C_3L_2^{-1}L_1^{-2}$ for all $(x,y) \in U$. Also, $|V(x,y) - \lambda| \leq C_4L_1^{-2}$.
We will apply the maximum principle to the functions
\begin{align*}
\Psi_{\pm}(x,y) \coloneqq ( (\partial_xu)_\pm(x,y)+L_2^{-1})/W(x,y),
\end{align*}
where $\pm$ signifies taking the positive or negative part of the function.
Let $U_{\pm}$ be any connected component of the support of $(\partial_{x}u)_{\pm}$ in $U$. Then, inside $U_{\pm}$, the functions $\Psi_{\pm}(x,y)$ satisfy
\begin{align*}
\Delta_{x,y}\Psi_{\pm}(x,y) &+ 2\nabla_{x,y}\log W(x,y).\nabla_{x,y}\Psi_{\pm}(x,y) =\\ \nonumber
&W(x,y)^{-1}\left( \Delta_{x,y}\left(\left(\partial_x u\right)_{\pm}(x,y) + L_2^{-1}\right)- \left(\left(\partial_xu\right)_{\pm}(x,y)+L_2^{-1}\right)W(x,y)^{-1} \Delta_{x,y}W(x,y)\right),
\end{align*}
which by \eqref{eqn:2pointwisederivx3} and \eqref{eqn:2pointwisederivx4} implies
\begin{align} \label{eqn:2pointwisederivx5}
& \Delta_{x,y}\Psi_{\pm}(x,y) + 2\nabla_{x,y}\log W(x,y).\nabla_{x,y}\Psi_{\pm}(x,y) \\ \nonumber
& \geq W(x,y)^{-1}\left(\partial_xV(x,y)u(x,y)+(V(x,y) - \lambda)\left((\partial_xu)_{\pm}(x,y)+L_2^{-1}\right) + \tfrac{1}{8}\pi^2 c_1^{-2}L_1^{-2}((\partial_xu)_{\pm}(x,y)+L_2^{-1})\right).
\end{align}
By the bound above on $|\partial_xV(x,y)u(x,y)|$, provided $c_1>0$ is sufficiently small, the right hand side of \eqref{eqn:2pointwisederivx5} is $\geq 0$. Combining this with the bounds from \eqref{eqn:2pointwisederivx1} and \eqref{eqn:2pointwisederivx2} on the boundary of $U$, we can apply the maximum principle to conclude that
\begin{align*}
|\partial_xu(x,y)| \leq W(x,y).
\end{align*}
The function $W(x,y)$ satisfies $W(0,y) \leq CL_2^{-1}$, and we can repeat the argument above with $W(x,y)$ shifted by an amount comparable to $L_2$ in the $x$-direction. This gives us the required bound on $\partial_xu(x,y)$ and concludes the proof of the corollary.
\end{proof}
|
1,314,259,995,771 | arxiv | \section{Introduction}
Recent advances in experiments on cold atomic gases have raised interest in systems consisting of several flavors of interacting fermions, which can be realized for example as different hyperfine states of alkali atoms~\cite{wu2003,honerkamp2004} or nuclear spin states of ytterbium~\cite{cazalilla2009} or alkaline-earth atoms.~\cite{wu2003,gorshkov2010} A model Hamiltonian to describe such systems is the $N$-flavor fermionic Hubbard model given by
\begin{equation}
H = -t \sum_{\langle i,j \rangle} \sum_{\alpha} c_{i \alpha}^\dagger c_{j \alpha} + H.c. + U \sum_i \sum_{\alpha, \beta} n_{i \alpha} n_{i \beta},
\end{equation}
where $\alpha$, $\beta$ run over the different flavors, $\langle i,j \rangle$ runs over pairs of nearest neighbors on the lattice and $i$ runs over all sites of the lattice.
The two-flavor case corresponds to the spin-$\frac{1}{2}$\xspace Hubbard model. It is generally accepted that for sufficiently large $U$ and at half filling, i.e. when each lattice site is occupied by exactly one fermion, the ground state is an antiferromagnetic Mott insulator. In experiments on cold atoms, the transition to a Mott insulator has recently been observed;\cite{jordens2008,schneider2008} the observation of the antiferromagnetic spin order is still open. The spin-$\frac{1}{2}$\xspace Heisenberg model is believed to be a good low-energy model for the spin degrees of freedom.
For the more general case $N > 2$, it is expected that, at certain fillings, Mott insulating states will also emerge.\cite{gorelik2009,miyatake2010} However, the spin order (or flavor order) in this case is not understood yet. This has raised interest in generalizations of the spin-$\frac{1}{2}$\xspace Heisenberg model, namely \su{N} Heisenberg models. Analogous to the $N=2$ case, these are obtained as second-order expansion of the above $N$-flavor fermionic Hubbard model in $t/U$ at a filling such that each site is occupied by exactly one particle.
The Hamiltonian is
\begin{equation} \label{eqn:su3}
H = J \sum_{\langle i,j \rangle} \sum_{\alpha, \beta} |\alpha_i \beta_j \rangle \langle \beta_i \alpha_j |
\end{equation}
where the first sum runs over pairs of nearest neighbors and the second sum over flavors. In this paper, we will focus on the case of the \su{3} Heisenberg model, where $\alpha, \beta \in \lbrace A,B,C \rbrace$.
Note that this model is different from the \su{N} Heisenberg models studied in Refs.~\onlinecite{affleck1988-sun,marston1989,read1989,read1990,harada2003,kawashima2007,beach2009,hermele2009,hermele2011} where other irreducible representations of \su{N} have been considered, which can be labelled by different Young tableaus. The corresponding Young tableau of our model has one single box, i.e. the fundamental representation at each site. In Refs.~\onlinecite{affleck1988-sun, marston1989} the large-N limit with $N/2$ particles per site has been studied (a Young tableau with $N/2$ boxes in one column).
In Refs.~\onlinecite{hermele2009,hermele2011} the large-N limit with representations with $m$ rows and $n_c$ columns with fixed $N/m$ and $n_c$ are considered ($n_c=1$ in Ref.~\onlinecite{hermele2009}). Another possibility is to use conjugate representations on two sublattices,~\cite{marston1989,read1989,read1990,harada2003,kawashima2007,beach2009} which is accessible by Quantum Monte Carlo simulations without a sign problem,\cite{harada2003,kawashima2007,beach2009} in contrast to the \su{3} model considered in this paper.
\begin{figure}
\centering
\includegraphics{su3-fig1.pdf}
\caption{Proposed three-sublattice order for the \su{3} Heisenberg model on the square and triangular lattice. The triangular lattice is obtained from the square lattice by adding couplings along the dashed bonds shown above. Blue boxes indicate the sites that are pinned to a specific flavor in our DMRG simulations in order to explicitly break \su{3} symmetry. \label{fig:su3_order}}
\end{figure}
This \su{3} model is equivalent to the spin-1 bilinear-biquadratic model,
\begin{equation}
\label{eq:bb}
H = \sum_{\langle i,j \rangle} \left[ \cos{\theta} (\vec{S}_i \cdot \vec{S}_j )+ \sin\theta (\vec{S}_i \cdot \vec{S}_j)^2 \right].
\end{equation}
with $\theta=\pi/4$, thus when bilinear and biquadratic terms are equal and positive.
In a pioneering work, Papanicolaou~\cite{papa1988} studied the phase diagram of this model on the square lattice as a function of $\theta$ by a semiclassical analysis. Using a site-factorized variational ansatz (product state) he proposed that the case $\theta = \pi/4$ corresponds to a phase transition from the antiferromagnetically ordered phase (adiabatically connected to the purely bilinear case) to a "semi-ordered phase" with infinitely many degenerate ground states, including states with 2- or 3-sublattice order. For recent progress on the nature of the phases in the "semi-ordered" region and its vicinity, see Ref.~\onlinecite{toth2011}.
The situation changes for the closely related case of the triangular lattice (equivalent to introducing a coupling on one of the diagonals of each plaquette of the square lattice), where a site-factorized ansatz already predicts a three-sublattice order as depicted in Fig.~\ref{fig:su3_order}. \cite{tsunetsugu06, laeuchli2006-triang} This state is stable upon adding quantum fluctuations at the level of linear flavor wave theory (LFWT), and is supported by exact diagonalization results.\cite{laeuchli2006-triang} In contrast, on a zig-zag chain (the one-dimensional analog of the triangular lattice) the system undergoes spontaneous trimerization.~\cite{Corboz07}
A recent study based on LFWT indicates that on the square lattice a similar type of three-sublattice order is selected by quantum fluctuations.~\cite{toth2010} This type of order is further supported by exact diagonalization revealing a tower of states compatible with the continuous symmetry breaking of SU(3).~\cite{toth2010} The ordered moment of the symmetry broken state, however, cannot be computed at the level of LFWT because fluctuations are divergent. Note that at this point this is an
artifact of the linear spin wave theory, and it is open whether higher order flavor wave corrections would lead to a finite or absent ordered moment. The only estimate of the ordered moment so far was obtained with exact diagonalization from the real-space correlation functions of an 18-site cluster, suggesting an ordered moment of $60\%-70\%$ of the saturation value, which is expected to decrease with system size. To further establish the three-sublattice order on the square lattice it is important to have an estimate of the ordered moment in the thermodynamic limit.
While on both lattices three-sublattice order has been suggested, the mechanism how this order is selected is quite different: on the triangular lattice it is already favored at the classical level, and quantum fluctuations only renormalize the ordered moment, which is a situation similar to that of the \su{2} Heisenberg model on bipartite lattices. In the case of the square lattice, on the other hand, the three-sublattice order is one among the many degenerate states in the classical limit, which is selected by quantum fluctuations. Note that thermal fluctuations may select a different order.~\cite{toth2010}
In the previous semiclassical studies quantum fluctuations at the level of LFWT have been taken into account, and higher order terms have been neglected. Thus, it is still an open question if the three-sublattice order is stable upon including higher-order quantum fluctuations, or if in this case another state is selected.
An example of such a scenario has recently been observed in the SU(4) Heisenberg model on the square lattice,\cite{corboz11-su4} where low-order quantum fluctuations select a plaquette state, but additional higher-order quantum fluctuations finally favor a dimerized state. For the SU(3) model, exact results on small systems suggest that the order is stable,\cite{laeuchli2006-triang, toth2010} but an accurate numerical study for larger systems in two dimensions is so far missing.
In this paper we study the stability of the three-sublattice order of the model \eqref{eqn:su3} on the triangular and square lattice with state-of-the-art numerical simulations. We present results for finite 2D systems with open boundaries up to a size $8\times 8$ using the density matrix renormalization group (DMRG) method, and infinite 2D systems with infinite projected entangled-pair states (iPEPS). Both methods belong to the class of tensor network algorithms, enabling to compute ground state properties with an accuracy which can be systematically controlled by a refinement parameter, called the bond dimension. Both methods confirm that the ground state has three-sublattice order for both type of lattices,
and we provide an estimate of the ordered moment in the thermodynamic limit. Finally we discuss an alternative approach based on Schwinger bosons. Unfortunately, this approach turns out to be unable to describe spontaneous SU(3) symmetry breaking, and, as a consequence, its results disagree with those of all other approaches regarding the type of ordering and the value of the ordered moment.
The outline of this paper is as follows. In Sec.~\ref{sec:methods} we give a short summary of linear flavor wave theory and provide details on the DMRG and iPEPS simulations. In Sec.~\ref{sec:triangular} we first present the results for the triangular lattice, where the three-sublattice order is expected to be more robust than on the square lattice, since this order is already favored at the classical level. We compare and discuss results for the energies and the ordered moment obtained with DMRG, iPEPS and LFWT. In Sec.~\ref{sec:square} we provide a similar study for the square lattice case, where we find an ordered moment which is also finite, but stronger suppressed by quantum fluctuations than on the triangular lattice. Finally, Sec.~\ref{sec:summary} summarizes our results. In Appendix~\ref{sec:app} we report on our attempt to extend the Schwinger boson mean-field theory to SU(3).
\section{Methods}
\label{sec:methods}
\subsection{Linear flavor wave theory}
\label{sec:LFWT}
The linear flavor wave theory is the extension of the usual SU(2) spin wave theory to SU(N) models. It has been formulated in Refs.~\onlinecite{N1984281} and \onlinecite{papa1988} for the SU(3) case and in Ref.~\onlinecite{PhysRevB.60.6584} for the SU(4) case. For completeness, here we give some details for the three--sublattice order on the triangular and square lattice in the SU(3) Heisenberg model --- the cases under scrutiny in this paper. We note that for the triangular lattice, an analogous calculation has been presented by Tsunetsugu and Arikawa.\cite{tsunetsugu06}
We begin by extending the Hamitonian (\ref{eqn:su3}) to the case where on each site the states belong to the symmetrical irreducible representation of the SU(3) algebra that can be represented by Young-tableaux drawn with $M$ boxes arranged horizontally. The SU(3) spin operators in such a symmetrical irreducible representation can be expressed as
\begin{equation}
S_{\beta}^{\alpha}(l) = b^{\dagger}_{\beta}(l) b^{\phantom{\dagger}}_{\alpha}(l) ,
\end{equation}
using Schwinger bosons with 3 flavors, where $l$ is the site index and the number of bosons on each site is
\begin{equation}
\sum_{\alpha\in\{A,B,C\}} b^{\dagger}_{\alpha}(l) b^{\phantom{\dagger}}_{\alpha}(l) =M,
\end{equation}
equal to the number of boxes in the Young tableau.
The $S_{\beta}^{\alpha}(l)$ operators satisfy the SU(3) Lie algebra,
\begin{equation}
\left[S_{\beta}^{\alpha},S_{\beta'}^{\alpha'} \right] =
S_{\beta}^{\alpha'}\delta_{\beta'}^{\alpha} - S_{\beta'}^{\alpha}\delta_{\beta}^{\alpha'}
\label{eq:Scommrel}
\end{equation}
where $\delta_{\beta}^{\alpha}$ is the Kronecker $\delta$ function.
For $M=1$ the
$S^\alpha_\beta(l)$ operators act on the 3-dimensional, fundamental
representation $|\alpha\rangle $ (where $\alpha=A, B,$ or $C$) of the
SU(3) algebra as $S^\alpha_\beta |\alpha\rangle =|\beta\rangle $ and $S^\alpha_\beta |\alpha'\rangle = 0$
if $\alpha' \neq \alpha$, with $i$ being the site index.
The Hamiltonian now can be written as
\begin{equation}
{\cal H } = J \sum_{\langle i,j \rangle} S^\alpha_\beta(i) S^\beta_\alpha(j) \;,
\label{eq:H_bosons}
\end{equation}
where the sum is over the nearest neighbor lattice sites, and over the repeated $\alpha$
and $\beta$ flavor indices. To draw a parallel to the SU(2) case, the Hamiltonian (\ref{eqn:su3}) in the fundamental irreducible representation corresponds to the spin--1/2 Heisenberg model, while the Hamiltonian (\ref{eq:H_bosons}) to the Heisenberg model of spins with length $S$ (actually for the SU(2) case $S=M/2$).
In the following, we consider an ordered state where the spins on the sites $l$, that belong to sublattice $\Lambda_\alpha$, point in the direction $\alpha$. Following the analogy with the spin wave theory that is a 1/S expansion, we take the $M \rightarrow \infty$ limit and do a $1/M$ expansion. Starting from the ordered state we can use the following expansion for the $S^\alpha_\beta(l)$ operators in the large--$M$ limit:
\begin{eqnarray}
S^\alpha_\alpha(l) &=& M - \mu_\alpha(l), \\
S^\alpha_\beta(l) &=& b^{\alpha\dagger}_{\beta}(l) \sqrt{M- \mu_\alpha(l)}
\approx \sqrt{M} b^{\alpha\dagger}_{\beta}(l) ,
\\
S^\beta_\alpha(l) &=& \sqrt{M - \mu_\alpha(l)} b^{\alpha}_{\beta}(l)
\approx \sqrt{M} b^{\alpha}_{\beta}(l), \\
S^{\beta'}_\beta(l) &=& b^{\alpha\dagger}_{\beta}(l) b^{\alpha}_{\beta'}(l),
\end{eqnarray}
where we have introduced the shorthand notation
\begin{equation}
\mu_\alpha(l) = \sum_{\beta \neq \alpha} b^{\alpha\dagger}_{\beta}(l) b^{\alpha}_{\beta}(l) .
\end{equation}
The $b^{\alpha\dagger}_{\beta}(l)$ operators with $\beta\neq \alpha$ now correspond to the Holstein--Primakoff bosons on sublattice $\Lambda_\alpha$, and the superscript $\alpha$ keeps track of the sublattice. We replace the expressions above into Hamiltoniam~(\ref{eq:H_bosons}).
Expanding in $1/M$ and keeping the quadratic terms only, for the exchange
term between sites $l \in \Lambda_\alpha$ and $l' \in \Lambda_{\alpha'}$ we get
\begin{eqnarray}
\sum_{\beta,\gamma} S_{\beta}^{\gamma}(l) S_{\gamma}^{\beta}(l') &=& M
\bigl[
b^{\alpha\dagger}_{\alpha'}(l) b^{\alpha}_{\alpha'}(l)
+ b^{\alpha'\dagger}_{\alpha}(l') b^{\alpha'}_{\alpha}(l') \bigr. \nonumber\\
&&
\bigl. + b^{\alpha\dagger}_{\alpha'}(l) b^{\alpha'\dagger}_{\alpha}(l')
+ b^{\alpha}_{\alpha'}(l) b^{\alpha'}_{\alpha}(l')
\bigr] .
\end{eqnarray}
in leading order in $M$ --- note that the bosons with flavor different from the ordered $\alpha$ and $\alpha'$ flavor are missing from the bond expression. Assuming a three-sublattice ordered state, we define the following
Fourier transformation:
\begin{equation}
b^{\alpha}_{\beta,\mathbf{k}} = \sqrt{\frac{3}{N_\Lambda}} \sum_{l\in \Lambda_\alpha} b^{\alpha}_\beta(l)
e^{i \mathbf{k} \mathbf{r}_l}
\end{equation}
where the summation is over the $N_\Lambda/3$ sites of the $\Lambda_\alpha$ sublattice ($N_\Lambda$ is the number of lattice sites).
The Hamiltonian between the sublattices $\Lambda_\alpha$ and $\Lambda_\beta$ in $\mathbf{k}$--space reads
\begin{eqnarray}
{\cal H }_{\alpha \beta} &=& \frac{z J M}{2}\sum_{k}
\left[
b^{\beta\dagger}_{\alpha,\mathbf{k}} b^{\beta}_{\alpha,\mathbf{k}} +
b^{\alpha\dagger}_{\beta,-\mathbf{k}} b^{\alpha}_{\beta,-\mathbf{k}}
\right.
\nonumber\\
&& \left.
+ \gamma_\mathbf{k}
b^{\alpha\dagger}_{\beta,-\mathbf{k}} b^{\beta\dagger}_{\alpha,\mathbf{k}}
+ \gamma^*_\mathbf{k}
b^{\alpha}_{\beta,-\mathbf{k}} b^{\beta}_{\alpha,\mathbf{k}} \right] ,
\end{eqnarray}
where $z$ is the coordination number of the lattice ($z=4$ for the square and $z=6$ for the triangular lattice). The factor $\gamma_\mathbf{k}$ reads
\begin{equation}
\gamma_\mathbf{k} = \frac{1}{3}
\left(
e^{i k_x} + 2 e^{-i k_x/2} \cos \frac{\sqrt{3}k_y}{2}
\right)
\end{equation}
for the triangular lattice and
\begin{equation}
\gamma_\mathbf{k} = \frac{1}{2}
\left(
e^{i k_x} + e^{i k_y}
\right)
\end{equation}
for the square lattice, with $\gamma^*_\mathbf{k} = \gamma_{-\mathbf{k}} $.
The full Hamiltonian is $\mathcal{H} = \sum_{\alpha<\beta} \mathcal{H}_{\alpha\beta}$.
It can be diagonalized via a Bogoljubov transformation:
\begin{equation}
\left(
\begin{array}{c}
\tilde b_{\alpha,\mathbf{k}}^{\beta\dagger} \\
\tilde b_{\beta,-\mathbf{k}}^{\alpha}
\end{array}
\right)
=
\left(
\begin{array}{cc}
\cosh \theta(\mathbf{k}) &
\sinh \theta(\mathbf{k}) \\
\sinh \theta(\mathbf{k}) &
\cosh \theta(\mathbf{k})
\end{array}
\right)
\left(
\begin{array}{c}
b_{\alpha,\mathbf{k}}^{\beta\dagger} \\
b_{\beta,-\mathbf{k}}^{\alpha}
\end{array}
\right)
\end{equation}
with
$ \tanh 2 \theta(\mathbf{k}) = \gamma_\mathbf{k}$ ,
leading to
\begin{equation}
\mathcal{H} = - \frac{z}{2} J M N_{\Lambda}
+ M \sum_{\mathbf{k}\in \textrm{RBZ}} \sum_{\alpha} \sum_{\beta\neq\alpha}
\omega(\mathbf{k})
\left[
\tilde b^{\alpha\dagger}_{\beta,\mathbf{k}} \tilde b^{\alpha}_{\beta,\mathbf{k}} +\frac{1}{2}
\right]
\;.
\end{equation}
The dispersion of the flavor waves is given by
\begin{equation}
\omega(\mathbf{k})
= \frac{z}{2}J \sqrt{ 1 - |\gamma_\mathbf{k}|^2}
\end{equation}
There are 6 degenerate branches in the reduced Brillouin zone, which is equivalent to 2 branches in the normal Brillouin zone. The dispersion agrees with the result of Tsunetsugu and Arikawa\cite{tsunetsugu06} for the triangular lattice. For the square lattice, it is given in Ref.\onlinecite{toth2010}.
The energy per site due to quantum fluctuations is given by the expression
\begin{eqnarray}
\left( 2 \left\langle \frac{\omega(\mathbf{k})}{2} \right\rangle_{BZ} -\frac{z}{2}J\right) M \label{eq:Ezp},
\end{eqnarray}
where we take into account that there are two modes per lattice site. The $\left\langle \dots \right\rangle_{BZ}$ denotes the average over the Brillouin zone.
Quantum fluctuations lower the energy from 0 to $-0.630 J$ per site for the triangular and to $-0.727J$ for the square lattice.
Note that the energy per site of the triangular lattice is higher than the one of the square lattice despite the larger coordination
number of the former lattice.
The reduction of the ordered moment is calculated as
\begin{equation}
\langle S^\alpha_\alpha(l) \rangle =
M - \langle \mu_{\alpha}(l) \rangle
= M - \left\langle \frac{1}{\sqrt{ 1 - |\gamma_\mathbf{k}|^2}}-1 \right\rangle_{\rm BZ}.
\end{equation}
In the triangular lattice $\langle S^\alpha_\alpha(l) \rangle = M - 0.516$, so that the on--site moment is reduced from 1 to $0.484$. In the square lattice, the reduced moment diverges due to the zero line in the spectrum. Thus, LFWT is unable to make a prediction for the ordered moment. We have tried to use a Schwinger boson mean-field theory (SBMFT)
to restore a gap along this line and remove the divergence (see Appendix \ref{sec:app}).
Unfortunately, the SBMFT turned out to be unsatisfactory in several respects, and its results regarding the ordered moment are not reliable.
\subsection{DMRG for finite two-dimensional systems}
\subsubsection{Setup}
For our DMRG simulations, we map the two-dimensional system to a chain following a "TV screen" method (sweeping along the vertical direction). We will generally refer to the extent in the horizontal direction as length, and in the vertical direction as width of the system. We use a single-site optimization scheme augmented by the improvement suggested in Ref.~\onlinecite{white2005}. We perform the simulation starting from different initial states and increase the bond dimension very quickly with the number of sweeps to avoid getting trapped in local minima. This is particularly important for the case of the square lattice, where an insufficient bond dimension may lead to unphysical states. Together with the large number of operators, this limits the bond dimension that we can reach with our computational resources to about $D \sim 5000$ states. Due to the huge growth of entanglement with the width of the system, this allows us to obtain sufficient accuracy for systems up to width 8.
\subsubsection{Calculation of the order parameter}
We expect that, in the thermodynamic limit, the \su{3} symmetry is spontaneously broken. If an appropriate basis is chosen (i.e. after an appropriate \su{3} rotation), one flavor becomes stronger on each site, i.e.
\begin{equation}
n_\alpha > n_\beta = n_\gamma \ \ \ \ \ \alpha,\beta,\gamma \in \lbrace A,B,C \rbrace.
\end{equation}
In this case, we can define the local moment
\begin{equation} \label{eqn:moment}
\langle m \rangle = \frac{3}{2} \left( \max_{\alpha = A,B,C} \langle n_\alpha \rangle - \frac{1}{3} \right),
\end{equation}
which should acquire a finite value in the range $\langle m \rangle \in [0,1]$.
On finite systems, the symmetry is never broken spontaneously and one would conventionally use the relation
\begin{equation}
\langle n_\alpha \rangle^2 =\lim_{d \rightarrow \infty} \left( \langle n_{\alpha,i} n_{\alpha,i+3d} \rangle - \langle n_{\alpha,i} \rangle\langle n_{\alpha,i+3d} \rangle \right)
\end{equation}
to extract information about the moments. This however requires large systems and very accurate estimates for the correlation functions, which are hard to obtain from a DMRG simulation in two dimensions. We therefore follow the prescription of Refs.~\onlinecite{white2007, stoudenmire2011} and break \su{3} symmetry explicitly by introducing fields at the boundaries of the system. The local moments can then be measured locally, preferably on sites far away from the pinning fields. The pinning fields also fix the direction of the symmetry breaking to be along the basis vectors.
We introduce a column of pinned sites at each end of the system, as shown in Fig.~\ref{fig:su3_order}. We choose the system sizes such that the unpinned sites form a square, i.e. the system size including pinned sites is $\sys{L}{L}$. Pinning is done with a flavor-specific chemical potential of magnitude 1. In addition, such pinning fields reduce the entanglement in the system. Simulations were performed for both open and cylindrical boundary conditions.
\subsubsection{Boundary conditions}
An important question when performing finite-size DMRG simulations is the appropriate choice of boundary conditions. From an entanglement point of view, open boundary conditions appear favorable; also, these will have fewer long-range operators in the mapping to a chain. From a physical point of view, on the other hand, periodic boundary conditions are often preferred as they eliminate boundary effects. A compromise suggested e.g. in Ref.~\onlinecite{white2007} is to use cylindrical boundary conditions, which are favorable from an entanglement point of view.
Physically, such boundary conditions are compatible with the approach of pinning two columns, which preserves translational invariance in the vertical direction. In order not to frustrate the three-sublattice order, such boundary conditions should only be chosen for systems whose width is a multiple of three. For other system sizes, shifted cylindrical boundary conditions can be used. For example, for a system of width 5, the bottom site of column $i$ must be connected to the top site of column $i+1$ to obtain a system without additional frustration.
We find numerically that cylindrical boundary conditions favor a state that is a product of periodic length-6 chains wrapped around the cylinder. A calculation for the same model on a periodic chain of length 6 shows that the energy per site is very low in this case, making it favorable for small clusters. Such a state shows significantly reduced local moments. By choosing open boundary conditions in all directions, we suppress this effect.
Another subtlety occurs for the square lattice system sizes $L=3 n-1$, with $n$ a positive integer number, for which the pattern of our pinning fields allows two different ordered states, corresponding to the two different orientation of the diagonal
stripes.\cite{toth2010} For these cases, at a sufficiently large bond dimension, a superposition of both types of order will occur and lead to a significant decrease of the local moment (except on sites where the two types of order coincide) and a significant increase of the entropy. In these cases, we pin two additional sites to uniquely select the order.
\subsubsection{Extrapolation}
The reliable extrapolation of results obtained for a limited bond dimension to the limit of infinite bond dimension, where DMRG becomes exact, remains a challenge. While some reliable results are known for one-dimensional critical systems,\cite{tagliacozzo2008,pollmann2009} one has to resort to heuristic techniques in more general situations, such as two-dimensional systems. Such techniques include the extrapolation in the truncated weight~\cite{white2007} or in the variance. Since we use a single-site optimization method, the truncated weight cannot be obtained reliably, and the calculation of the variance is only possible for smaller system sizes. We therefore resort to an extrapolation of the magnetization in the bond dimension using the values obtained for the three largest values of $D$; for bond energies, we use only the result obtained for the largest value of $D$. While most simulations were performed using up to $D=4800$ states, we have confirmed the accuracy of our results with up to $D=6400$ states for some selected systems.
Similarly, an accurate finite-size extrapolation is difficult given the few system sizes we can access. Also, the dependence of the order parameter and the energy on the system size, boundary conditions and aspect ratio is not known. In fact, previous studies have even observed surprising cases such as non-monotonic behavior for very small systems.~\cite{white2007}
\subsection{Infinite projected entangled-pair states (iPEPS)}
\subsubsection{Setup}
An iPEPS is a tensor network made of a set of rank-5 tensors periodically repeated on a two-dimensional lattice to efficiently represent ground state wave functions in the thermodynamic limit. \cite{sierra1998, nishino1998, verstraete2004, nishio2004, murg2007, jordan2008} Each tensor has four auxiliary bonds which connect to the four nearest-neighbor tensors, and a fifth index carrying the local Hilbert space of a lattice site. The accuracy of the ansatz can be controlled by the dimension of the auxiliary bonds, called the bond dimension $D$. As the optimization scheme for the tensors we perform an imaginary time evolution with the so-called simple update (see Refs.~\onlinecite{jiang2008,corboz2010}) adopted from the time-evolving block decimation method in one dimension.~\cite{vidal2003-1, orus2008} For the square lattice we verified the results up to $D=8$ also with the full update (see e.g.~\onlinecite{corboz2010}), which is optimal but has a higher computational cost. The triangular lattice simulations are done with the same ansatz as for the square lattice, but now with an additional next-nearest neighbor interaction along one of the diagonal directions. The update scheme for this case is explained in Ref.~\onlinecite{corboz2010-nn}.
We performed simulations with different rectangular unit cells of size $L_x \times L_y$ in iPEPS.\cite{corboz2011} To represent the state with three-sublattice order efficiently, a $3 \times 3$ cell is used, with 3 different tensors $T_A, T_B,$ and $T_C$ for the three sublattices respectively. We verified that the same state is obtained by using a similar cell with 9 different tensors. The $2\times 2$ unit cell is used to enforce a state with two-sublattice order.
To contract the tensor network efficiently, e.g. for the computation of observables, the corner transfer matrix scheme\cite{orus2009-1} adapted to large unit cells\cite{corboz2010-nn} is used. The accuracy of the approximate contraction can be controlled by the so-called boundary dimension $\chi$. For large values of $D$ a $\chi$ up to 250 is used, where quantities of interest are extrapolated in $\chi$, with an extrapolation error being small compared to symbol sizes. For a better efficiency we use tensors with $\mathbb{Z}_q$ symmetry, a discrete abelian subgroup of SU(3).\cite{cincio2008,singh2010-1,bauer2011}
\subsubsection{Calculation of the order parameter}
Since iPEPS is an ansatz for the wave function in the thermodynamic limit, the SU(3) symmetry may be spontaneously broken, leading to a finite local moment $m$ defined in Eq.~\ref{eqn:moment}. In order to pin the direction of the moment in SU(3) color space an initial field is applied, which is taken to zero at a later stage of the imaginary time evolution. We verified that we obtain the same results without initial field, and by computing the moment taking all generators of SU(3) into account (see Eq.~\eqref{eq:m} in appendix \ref{sec:app}).
\subsubsection{Extrapolation}
\label{sec:extrapD}
For highly entangled systems quantities of interest such as the energy or the local moment are typically not converged as a function of the bond dimension $D$ at the maximal value of $D$ used, and thus an extrapolation to the infinite $D$ limit is desirable. However, in general the dependence of observables on $D$ is (still) unknown, which limits the accuracy of such extrapolations.
Since the approach is variational the energy decreases with increasing $D$, and therefore the energy at the largest value of $D$ provides an upper bound of the exact energy. Empirically, the exact value lies between the linear extrapolated value and the value at the largest $D$, and thus we take the middle of these two values as an estimate and the difference between the two values as an error bar.
The same holds for the local moment, which is typically suppressed with increasing $D$, since more quantum fluctuations are taken into account with increasing $D$ which renormalize the ordered moment. Typically, the energy converges faster than the local moment.
\section{Results for the triangular lattice}
\label{sec:triangular}
We first present the results for the energy and the ordered moment for the model on the triangular lattice, obtained with LFWT, ED (energies only), DMRG and iPEPS. As mentioned in the introduction, on the triangular lattice a three sub-lattice order is already obtained from a simple product state ansatz. Inclusion of quantum fluctuations via LFWT does not destroy the order,\cite{laeuchli2006-triang} but renormalizes the local moment. In the following we show that this holds even when including further quantum fluctuations.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{res_su3t_papere.pdf}\\ \vspace{0.2cm}
\includegraphics[width=\columnwidth]{res_su3t_paperm.pdf}
\caption{Comparison of the energy per site (upper panel) and the local moment (lower panel) of the SU(3) model \eqref{eqn:su3} on the triangular lattice, obtained from LFWT, ED, DMRG and iPEPS. In each plot, the DMRG results are shown as a function of the inverse system length $1/L$ (lower x-axis), whereas the iPEPS results are shown as a function of inverse bond dimension $1/D$ (upper x-axis). Dotted lines are only guides to the eye.}
\label{fig:t_em}
\end{figure}
\subsection{Energy per site}
Figure~\ref{fig:t_em}a) shows a comparison of the energy per site obtained with the four methods, where linear flavor-wave theory predicts a value of $E_s=-0.6295 J$. Extrapolating
ED energies for symmetric clusters consisting of $N= 9$, $12$, and $21$ sites using a standard $1/N^{3/2}$ form, we obtain $E_s \approx -0.69(1) J$.%
Since we use open boundary conditions in the DMRG simulations, the energy per site is not uniform in the system. Figure~\ref{fig:conf_t} shows that the energy per bond close to the boundary is lower than far away from the boundaries. To obtain an estimate of the energy per site in the "bulk" we average over the six bond energies around the central site for odd systems sizes. For even system sizes, we average over the four sites at the center of the system. This estimate is plotted in Fig.~\ref{fig:t_em}a) for different system sizes. The energy first increases with system size, and decreases slightly from $L=7$ to $L=8$. For the largest system $L=8$ the estimated energy per site in the bulk is $E_s=-0.6775J$.
Comparing the iPEPS energies obtained with different unit cell sizes, we find that the $3\times3$ unit cell yields a considerably lower variational energy than the $2\times2$ unit cell, which indicates that the symmetry breaking in the ground state is compatible with the 3-sublattice order. The energy per site has not converged yet as a function of bond dimension $D$. Since the energy typically converges faster than linearly in $1/D$ we (empirically) expect the energy to lie in between the value for the largest $D$, $E_s^{D=10}=-0.672 J$, and the energy obtained from linear extrapolation of the last three data points in $1/D$, $E_s^{\text{ex}}=-0.708 J$. Taking the mean of these two values yields an estimate of $E_s=-0.69(2)$, which is compatible with the DMRG result for the largest system.
\begin{figure}
\centering
\includegraphics[width=7cm]{res_su3dmrg.pdf}
\caption{Bond energies and local color densities in the triangular $(7+2)\times7$ lattice obtained from DMRG. The thickness of the bonds is proportional to the magnitude of the bond energy. An external potential is applied on the first and the last column to pin the sites to a specific color.
}
\label{fig:conf_t}
\end{figure}
\subsection{Local moment}
In Fig.~\ref{fig:t_em}b) we present the results for the local moment $m$ obtained with the various approaches, where $m=1$ for the fully polarized case.
As mentioned in Sec.~\ref{sec:LFWT}, linear flavor-wave theory predicts a value of $m=0.484$.
The DMRG results correspond to the local moment at the central site of the system for odd system sizes. For even system sizes, the magnitude of the ordered moment is averaged over the four sites that make up the central plaquette of the system. Variations depending on the distance to the boundaries in x- and y- direction can be observed, as shown in Fig.~\ref{fig:dmrg_mi}a). The value is decreasing with increasing distance from the pinning sites (in x-direction), whereas the value is seen to increase away from the boundary in y-direction. As a function of system size the local moment is increasing. As mentioned before, an accurate extrapolation to the thermodynamic limit is challenging, but a value in the range $m \approx 0.43-0.6$ seems compatible with the DMRG data.
The local moment obtained with iPEPS decreases with increasing $D$, an effect which can also be observed e.g. in the SU(2) Heisenberg model. With increasing $D$ more quantum fluctuations are taken into account which reduce the magnetic moment from its value in the classical (product-state) limit, corresponding to $D=1$. For the largest bond dimension used, $D=12$, we find a value $m=0.58$, whereas a linear extrapolation in $1/D$ suggests a value of $m=0.52$. As discussed in Sec.~\ref{sec:extrapD} the exact scaling behavior of $m$ with $1/D$ is not known, but empirically, we expect $m$ to lie in between these two values
\subsection{Discussion}
Even though we cannot determine $m$ up to a high precision all three methods clearly suggest that the ground state has three-sublattice order with a large local magnetic moment in the range $m= 0.43 - 0.6$. As in the case of the SU(2) Heisenberg model on the square lattice, linear flavor wave theory (spin wave theory) already gives a good estimate of the local moment.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{plateaux.pdf}
\caption{DMRG results with $D=4800$ states: Local moments as defined in Eqn.~\eqnref{eqn:moment} for the triangular (top panel) and square (bottom panel) lattice for a system size $\sys{7}{7}$. The plateau is very flat in the case of the square lattice, while corrections from the boundary are more pronounced for the triangular lattice. \label{fig:plateaux} }
\label{fig:dmrg_mi}
\end{figure}
\label{sec:square}
\section{Results for the square lattice}
We next consider the SU(3) Heisenberg model on the square lattice. As explained in the introduction a site factorized ansatz leads to an infinite number of degenerate ground states and quantum fluctuations (with LFWT) selects the three-sublattice state.\cite{toth2010} Thus, quantum fluctuations seem to play a more dominant role on the square lattice, and it is conceivable that another ground state is selected when further quantum fluctuations beyond LFWT are taken into account. However, we show in the following that this is not the case here, i.e. that the three-sublattice order is stable and that additional quantum fluctuations only further renormalize the local moment.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{res_su3_paper3e.pdf}\\ \vspace{0.2cm}
\includegraphics[width=\columnwidth]{res_su3_paper3m.pdf}
\caption{Same plot as in Fig.~\ref{fig:t_em} but for the SU(3) model \eqref{eqn:su3} on the square lattice.}
\label{fig:s_em}
\end{figure}
\subsection{Energy per site}
In Fig.~\ref{fig:s_em}a), the value of the energy per site from linear flavor-wave theory, $E_s=-0.725J$, has previously been calculated in Ref.~\onlinecite{toth2010}, and is low compared to the numerical results. We note the LFWT energy is not variational, so that it can be lower than the exact ground state value. We have also included an ED estimate of the energy per site $E_s = -0.63185 J$, which is based on an extrapolation using square samples with $N=9$ and $18$ sites.
The DMRG energy per site is $E_s=-0.625$ for the largest system, and seems to further increase as a function of system size. As in the triangular lattice case we estimate the bulk energy by taking the mean value over the bonds adjacent to a central site for odd system sizes and four sites for even system sizes. This energy seems higher than the LFWT and iPEPS result, which could indicate that boundary effects are large so that we do not get a good estimate for the "bulk" energy, or it could be that for larger systems the energy as a function of system size decreases again. We further note that an anisotropy in the bond energies can be observed, with stronger bonds in y-direction than in x-direction, shown in Fig.~\ref{fig:conf_sq}.
Comparing the energies from different unit cell sizes in iPEPS, we observe a similar behavior as on the triangular lattice, namely that the $3\times3$ unit cell provides a better variational energy than the $2\times2$ unit cell for all values of $D$. The estimated energy per site in the limit $D\rightarrow \infty$ is $E_s=-0.66(1)$.
\begin{figure}
\centering
\includegraphics[width=7cm]{res_su3dmrgsq.pdf}
\caption{Bond energies and local color densities in the square $(7+2)\times7$ lattice obtained from DMRG. The thickness of the bonds is proportional to the magnitude of the bond energy. An external potential is applied on the first and the last column to pin the sites to a specific color.}
\label{fig:conf_sq}
\end{figure}
\subsection{Local moment}
Figure~\ref{fig:s_em}b) summarizes our results for the local moment, obtained from DMRG and iPEPS. As explained in Ref.~\onlinecite{toth2010}, the ordered moment cannot be computed within LFWT.
The finite size effects observed in DMRG are qualitatively different from the triangular lattice case. The local moment as a function of distance of $x$ in Fig.~\ref{fig:dmrg_mi} reaches a plateau already after 3 sites away from the border, which could suggest that finite size effects on the ordered moment are smaller than on the triangular lattice. In Fig.~\ref{fig:s_em}b) the local moment in the middle of the system first decreases and then increases with system size with, however, a smaller slope than in the triangular lattice case. Thus, the present data is compatible with a non-vanishing local moment in the thermodynamic limit, which is smaller than on the triangular lattice. The ordered moment of the largest system is $m= 0.368$.
The local moment obtained with iPEPS decreases with increasing bond dimension but is not seen to extrapolate to zero in the limit $D \rightarrow \infty$. The value for the largest bond dimension, $D=16$, is $m=0.3422 $ which is close to the DMRG result for the largest system. However, in the limit $D\rightarrow \infty$ the data suggests a lower value of roughly $m=0.25(5)$, which is lower than the prediction from DMRG.
\subsection{Discussion}
Both the DMRG and iPEPS results are compatible with the proposed 3-sublattice N\'eel ordered ground state. From the present data we can only give a rough estimate of the ordered moment in the thermodynamic limit of $m=0.2-0.4$, which is clearly finite, but smaller than on the triangular lattice.
\section{Summary}
\label{sec:summary}
Our study confirms that the ground state of the SU(3) Heisenberg model exhibits a three-sublattice order on both the triangular and the square lattice, in accordance with previous predictions by LFWT and exact diagonalization. \cite{tsunetsugu06, laeuchli2006-triang,toth2010}
The situation on the triangular lattice resembles the one of the SU(2) Heisenberg model on the square lattice. In both cases the ground state can already be understood at the classical level, and quantum fluctuations simply renormalize the ordered moment. These fluctuations are well captured already within linear flavor wave theory (i.e. spin wave theory in the SU(2) case). With iPEPS the ordered moment decreases with increasing bond dimension $D$, which can intuitively be understood because the bond dimension controls the amount of quantum fluctuations taken into account. All three methods used in this study yield a finite ordered moment in the range $m=0.43-0.6$. The uncertainty in this value stems from the error in the extrapolation to the thermodynamic limit in the case of DMRG, and from the extrapolation to the infinite $D$ limit in the case of iPEPS.
In the case of the square lattice, the order cannot be predicted at the classical level. Quantum fluctuations hence play a very different role than in the case of the triangular lattice: instead of renormalizing the mean-field result, they stabilize the three-sublattice order against other competing states. Quantum effects are therefore more important both qualitatively and quantitatively, and an estimate of the ordered moment in the thermodynamic limit has previously been lacking. Both DMRG and iPEPS predict a finite value in the range $m=0.2-0.4$, i.e. the ordered moment is more strongly suppressed than on the triangular lattice, but clearly finite.
\acknowledgments
We acknowledge helpful discussions with S. Manmana and U. Schollw\"ock, and the financial support of the Swiss National Fund and of MaNEP, and of the Hungarian OTKA Grant No. K73455. The DMRG code was developed with support from the Swiss platform for High-Performance and High-Productivity Computing (HP2C)~\cite{hp2c} and based on ALPS libraries.~\cite{ALPS_2,bauer2011-alps} Simulations were performed on the Brutus cluster at ETH Zurich.
|
1,314,259,995,772 | arxiv | \section{Introduction}
Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than
$\tau$.
The problem of computing the topology of the planar algebraic curve
\[
V_{\mathbb{R}}
(P) := \{ (x,y) \in \mathbb{R}^2, P (x,y)
= 0 \}
\]
implicitly defined by $P$, that is, the computation of straight-line planar graph isotopic to
$V_{\mathbb{R}} (P)$ inside $\mathbb{R}^2$, is a classical problem in
algorithmic real algebraic geometry with many
applications in Computer Aided Geometric Design. It is extensively studied in
the context of symbolic
or semi-numerical computation; for instance, see~\cite{AM1,AM2,AMW,BEKS,BCGY,CLPPRT,DMR,EKW,GE1,GI,KS,KoS,MSW2,WM} for recent references.
Almost all certified algorithms are based on some variant of
Cylindrical Algebraic Decomposition (C.A.D.): Decompose the $X$-axis into a finite number of
open intervals and points above which the curve has a cylindrical structure (i.e.~the curve decomposes into disjoint function graphs above each of these intervals).
The \emph{special} values are the projections of the $X$-critical and singular points
onto the $X$-axis, and the special
fibers are the points of the curve above these special values. Taking
additional points between two special values defines additional
\emph{regular} fibers.
Computing a straight-line planar graph isotopic to $V_{\mathbb{R}} (P)$ inside $\mathbb{R}^2$ then
essentially amounts to connect the points of a regular fiber to the
points of its neighboring special fibers. The above approach requires :
\begin{itemize}
\item computing the special and regular fibers, and
\item computing the number of half branches of the curve that go to each of the
points of the special fiber, to the left and to the right.
\end{itemize}
One difficulty is the computation of the special fibers,
which amounts for computing the real roots of univariate polynomials with real
algebraic coefficients, which are not square-free.
The method used for computing the number of half branches of the curve going
to a singular point plays another key role in the algorithm.
A usual strategy
(see \cite{BPRbook2,DMR,EKW,GE1,GI,DBLP:phd/de/Kerber2009,KoS,MSW2}),
consists in putting the curve in a so-called generic position. This is typically achieved by considering a (random) \emph{shearing} that maps $X$ to $X+s\cdot Y$ for some (small) integer $s$. As a consequence, the sheared curve has no asymptotes and each special fiber contains at most one $X$-critical or singular point, which considerably eases the above steps.
Since a shearing does not change the topology of the curve, the graph returned by such algorithms is still isotopic to the curve. However,
these algorithms do not compute a C.A.D. of the curve itself but only of the sheared curve. This might be critical in some applications, for instance, when computing the topology of a surface~\cite{AlbertiMT09,BerberichKS10} $S\subset\mathbb{R}^3$ that is implicitly defined by some polynomial $F(X,Y,Z)\in\mathbb{Z}[X,Y,Z]$. Typically, this is done by computing a C.A.D. of the projection $V_{\mathbb{R}} (P)$ of the so-called silhouette curve of $S$ given as the vanishing set of the polynomial $P(X,Y):=\operatorname{Res}_Z(G,\frac{\partial}{\partial Z} G)\in\mathbb{Z}[X,Y]$, followed by a lifting of the C.A.D. information. In general, the computation of the resultant $\operatorname{Res}_Z(G,\frac{\partial}{\partial_Z} G)$ does not commute with the shearing $X\mapsto X+s\cdot Y$, which means that we cannot directly work with the C.A.D. information of a shearing of $V_{\mathbb{R}} (P)$.
There exists algorithms~\cite{EKW,DBLP:phd/de/Kerber2009} that go one step further by performing a shearing of the curve in the first step and an inverse shearing in the second step in order to eventually compute an isotopic graph whose vertices are located on the curve. Such algorithms compute the C.A.D. of the given curve, however the bit complexity of these methods falls clearly behind the best algorithms~\cite{KoS,MSW2} for computing the topology of a planar algebraic curve, which achieve the complexity bound\footnote{The algorithm from~\cite{KoS} is deterministic, whereas~\cite{MSW2} uses randomization. Both algorithm consider a shearing of the original curve and only return the C.A.D. information of the sheared curve.} $\tilde{O}(d^5\tau+d^6)$. Our algorithm achieves the same complexity bounds, but it never performs any coordinate transformation and yields the C.A.D. information of the original curve.
\begin{theorem}[Topology]\label{finaltopology}
Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$
and integer coefficients of bitsize bounded by $\tau$. There is a deterministic and certified algorithm\footnote{We do not only prove existence of such an algorithm, but also present the algorithm in this paper.} that uses $\tilde{O}(d^5\tau+d^6)$ bit operations to compute the topology of the curve $$V_{\mathbb{R}} (P)= \{(x, y) \in \mathbb{R}^2 \mid P (x, y) = 0\},$$ in terms of a straight-line
planar graph $\mathcal{G}$ that is isotopic to $V_{\mathbb{R}}(P)$ inside $\mathbb{R}^2$. In addition, $\mathcal{G}$ yields the C.A.D. information of the curve $V_{\mathbb{R}}(P)$.\footnote{We remark that our algorithm returns a purely combinatorial representation of the C.A.D., however isolating intervals of all points in all special and regular fibers are computed in sub-steps of the algorithm. By refining these intervals, an isotopic graph whose vertices are arbitrarily close to the curve can be derived.}
\end{theorem}
For our result, we use two main ingredients: The first one is an efficient algorithm for computing the roots of a bivariate system in triangular form.
\begin{theorem}[Bivariate Root Isolation]\label{firstmain}
Let $R\in\mathbb{Z}[X]$ and $F\in \mathbb{Z}[X,Y]$ be polynomials of total degrees $N$ and $n$, and with integer coefficients of bitsize less than $\Lambda$ and $\tau$, respectively. Using a total number of bit operations bounded by
\[
\tilde{O}(N^2\Lambda+N^3+n^5\tau+n^6+n\cdot\max(n^2,N)\cdot (N\tau+n\Lambda+Nn)),
\]
we can compute
\begin{itemize}
\item[(a)] $\deg F(z,-)$ as well as $\deg \gcd (F(z,-),\partial_Y F(z,-))$ for all complex roots $z$ of $R$,
\item[(b)] isolating disks $D_{z,z'}\subset\mathbb{C}$ for all complex roots $z'$ of all polynomials $F(z,-)$ for all complex roots $z$ of $R$, as well as the corresponding multiplicities $\mu(z,z'):=\operatorname{mult}(z', F(z,-))$
\item[(b')] for a given subset $V\subset\{(z,z')\in\mathbb{C}^2:R(z)=0\text{ and }F(z,z')=0\}$ and $L\in\mathbb{N}$, we can further refine all isolating disks ${\mathcal D}_{z,z'}$, with $(z,z')\in V$, to a size less than $2^{-L}$, in an additional number of bit operations bounded by
$$ \tilde{O}(N^2\Lambda+N^3+n\cdot\max(n^2,N)\cdot(N\tau+n\Lambda+Nn)+L\cdot(N\cdot\mu+n^2\cdot\sum_{z \in \pi_X(V)}\mu_{z})),$$
where $\mu_{z}:=\max_{(z,z') \in V}\mu(z',F(z,-))$, and $\mu=\max_{z \in \pi_X(V)}\mu_{z}$.
\end{itemize}
\end{theorem}
Using Theorem~\ref{firstmain} as our main tool, we obtain an efficient Refined Cylindrical Algebraic Decomposition for an algebraic curve; see Theorem~\ref{sing-fibers}.
Compared to the existing literature, the results in Theorem~\ref{firstmain} are much more general as no assumptions on the given polynomials are made. For instance, recent work~\cite{ST2018} makes two strong assumptions, namely that the degree of the polynomials $F(z,-)$ stays invariant for all roots $z$ of $R$ and that each $F(z,-)$ has only simple roots. In~\cite{KS,KoS,MSW2}, the polynomials $F(z,-)$ are allowed to have multiple roots, but the degree of $F(z,-)$ is assumed to be the same for all $z$. In addition, the analysis restricts to the special case, where $R$ is either the resultant polynomial $\operatorname{Res}(F,\partial_YF,Y)$ or its derivative.
Our second ingredient is an algorithm for computing the number of branches reaching a singular point inside an adjacency box (see Section \ref{sec:top} for a precise definition) from information computed on the boundary of the box (see Algorithm \ref{algoconnect}). In order to achieve the complexity bounds we are aiming at, we need to accept that some of the desired information stays ambiguous,\footnote{For instance, we aim to compute the location of the intersections of the curve with the four boundary edges of the adjacency box. However, we were not able to show how to distinguish between the special case, where the curve passes exactly the corner of a box, and the generic case, where the curve intersects one of the neighboring edges close to the corner, using only $\tilde{O}(d^6+d^5\tau)$ bit operations. In such cases, the information at the corner of the box stays ambiguous.} however we will prove
that these ambiguities do not prevent us from computing the correct connectivity; see Proposition \ref{ambigousnoproblem}.\\
For both of the above ingredients, we make essential use of amortized quantitative bounds for polynomials with algebraic coefficients by considering adaptive algorithms that make it possible to exploit this amortization; see Proposition \ref{generalunivariate} and Proposition \ref{generalbivariate}. Finally a precise combinatorial description of the information needed to draw the graph isotopic to the curve is given.
Since we perform our computations without any change of variables, vertical lines and asymptotes need to be dealt with.
\paragraph{Organization of the paper}
The detailed description and complexity analysis of each
step of our algorithm computing the isotopy type is given in Section \ref{sec:top}.
The two preceding sections are devoted to univariate and bivariate results about roots of polynomials. In each of the two sections, a part is devoted to recalling well known results, but there are also several new results, particularly amortized quantitative bounds about algebraic numbers weighted with multiplicities (see Proposition \ref{generalunivariate} and Proposition \ref{generalbivariate}) that play a key role in the proof of Theorem \ref{firstmain}
or in the results from Section \ref{sec:top}.
\section{Univariate results} \label{univariateres}
In our complexity analysis we are going to use some quantitative results related to the geometry of the roots. We first fix some convenient notations and definitions.
\begin{notation} \label{not:fk}
Let $f \in \mathbb{C} [X]$ be a polynomial of degree $n$.
Defining
\begin{equation}
f^{[i]}:=\frac{f^{(i)}}{i!} \label{fk},
\end{equation}
the Taylor formula writes as
$$f(\alpha+X)=\sum_{i=0}^{n} f^{[i]}(\alpha)X^{i}$$
\end{notation}
\begin{definition}
\label{def0}
For a polynomial $f = \sum^n_{i = 0} a_i X^i \in \mathbb{C} [X], \text{with }a_n \neq 0$, the \emph{length} of $f$ is defined as
$$\operatorname{Len} (f) := \sum_{i = 0}^n | a_i | .$$
The \emph{norm} of $f$ is defined as
$$\| f \| := \sqrt{\sum_{i = 0}^n | a_i |^2} .$$
Given $z \in \mathbb{C}$, we denote by
$\operatorname{mult}(z,f)$ the \emph{multiplicity} of $z$ as a root of $f$, i.e. the index $i$ such that
$(X-z)^i$ divides $f$ and $(X-z)^{i+1}$ does not divide $f$. If $z$ is not a root of $f$, then
$\operatorname{mult}(z,f)=0$. If $z$ is a root of $f$,
we have
$$\bigwedge_{i=0}^{\operatorname{mult}(z,f)-1}( f^{[i]}(z)=0) \wedge (f^{[\operatorname{mult}(z,f)]}(z))\not=0.$$
We denote by $\mathrm{V}_{\mathbb{C}}(f)$
the set of the distinct roots of $f$ in $\mathbb{C}$ and by
$\operatorname{mult}(z,f)$ the multiplicity of $z$ as a root of $f$, so that
$$f = \operatorname{lc}(f) \cdot\prod_{z\in \mathrm{V}_{\mathbb{C}}(f)} (X - z)^{\operatorname{mult}(z,f)}.$$
The \emph{square-free part} $f^\star$ of $f$ is defined as
$$f^\star = \operatorname{lc}(f) \cdot\prod_{z\in \mathrm{V}_{\mathbb{C}}(f)} (X - z),$$
where $\operatorname{lc}(f)$ denotes the leading coefficient of $f$.
The \emph{Mahler measure} of $f$ is defined as
$$\operatorname{Mea} (f) := | \operatorname{lc}(f) | \cdot\prod_{z\in \mathrm{V}_{\mathbb{C}}(f)} \max (1, | z|)^{\operatorname{mult}(z,f)}.$$
We further define $$\widehat{\operatorname{Mea}} (f) := \prod_{z\in \mathrm{V}_{\mathbb{C}}(f)} \max (1, | z|)^{\operatorname{mult}(z,f)}.$$
The \emph{separation of $f$ at $z$} is defined as
$$\operatorname{sep}(z,f):= \min_{y \in\mathrm{V}_{\mathbb{C}}(f), \atop y \not{=} z} |y-z |.$$
\hide{
and the \emph{separation of $f$} is defined as
$$\operatorname{sep}(f):=\min_{z\in \mathrm{V}_{\mathbb{C}}(f)}\operatorname{sep}(z,f)$$
}
\end{definition}
The following results are straightforward:
\begin{lemma}
\label{subst1} If $f\in \mathbb{C}[X]$ is of
degree $n$ and $z \in \mathbb{C},$ then
$$|f (z) | \le \operatorname{Len} (f)\cdot \max (1, | z |)^{n}.$$
\end{lemma}
Let $f(X)=a_qX^q+\cdots + a_nX^n$, with $q\le n$, be a univariate polynomial of degree $n$ with coefficients in $\mathbb{C}$, such that $a_q \neq 0$ and $a_n \neq 0$.
\begin{notation}
\label{cauchybounds}
We define
$$C(f):=\sum_{q \leq i\leq n} \left| \frac{a_i}{a_n} \right|.$$
\end{notation}
\begin{proposition} (see for example \cite{BPRbook2})
\label{cauchybound}
The absolute value of any
root of $f$ in $\mathbb{C}$ is
smaller than $C(f)$.
\end{proposition}
We briefly review the classical notion of resultant, subresultant, discriminant, subdiscriminant.
\begin{definition} \label{def:resultant}
Let $\mathbb{D}$ be a domain, and let $f,g$ be two polynomials in $\mathbb{D}[X]$ of respective degrees $n,n'$ with respect to $X$.
The \emph{resultant} $\operatorname{Res}(f,g)$ is an element of $\mathbb{D}$ defined as the determinant of the matrix of the linear mapping $\mathbb{D}_{<n'}[X]\times\mathbb{D}_{<n}[X]\mapsto \mathbb{D}_{<n+n'}[X]$ associating two polynomials $u,v$ of respective degree at most $n'-1,n-1$, the polynomial $uf+vg$,
expressed in the basis $X^{n+n'-1}\ldots,1$. If $\mathbb{D}$ is a field, $\operatorname{Res}(f,g)=0$ if and only if $\deg(\gcd(f,g))>0$. Moreover if $\mathbb{D}=\mathbb{C}$ and $\mu(z)$ is the multiplicity of $z$ as a root of $f$, it holds that
\begin{equation}\label{equ:res}
\operatorname{Res}(f,g)=\operatorname{lc}(f)^{n'} \prod_{z \in V_{\mathbb{C}}(f)} g(z)^{\mu(z)}
\end{equation}
When considering $uf+vg \mod X^k$, for two polynomials $u,v$ of respective degree at most $n'-k-1,n-k-1$, define
the matrix of a new linear mapping
$\mathbb{D}_{<n-k'}\times\mathbb{D}_{<n-k}\mapsto
\mathbb{D}_{<n+n'-2k}$ whose determinant defines the so called $k$-th
(principal) subresultant coefficient, denoted by $|\mathrm{sr}_{k}(f,g)|$. In particular
$|\mathrm{sr}_{0}(f,g)|=|\operatorname{Res}(f,g)|$.
If $\mathbb{D}[X]$ is a GCD domain, for example when
$\mathbb{D}$ is a field, $k$ is the smallest index such that $\mathrm{sr}_{k}(f)\not= 0$ if and only $k=\deg(\gcd(f,g))$.
The \emph{discriminant} $\operatorname{Disc}(f)$ is the element of $\mathbb{D}$ such that $$ \operatorname{lc}(f)\cdot \operatorname{Disc}(f)=\operatorname{Res}_X(f,f').$$ If $\mathbb{D}=\mathbb{C}$, $$| \operatorname{Disc}(f) |= | \operatorname{lc}(f)|^{2n-2}\cdot\prod_{i,j:i\neq j} |z_i-z_j|=|\operatorname{lc}(f)|^{n-2}\cdot\prod_{1\le i \le n} |f'(z_i)|,$$
where $z_1,\ldots,z_n$ denote the (not necessarily distinct) complex roots of $f$. Hence, we have $\operatorname{Disc}(f)=0$ if and only if $f$ has a multiple root.\\
The \emph{$k$-th subdiscriminant coefficient} $\operatorname{sDisc}_{k}(f)$ is the element of $\mathbb{D}$ such that $$\operatorname{lc}(f)\cdot \operatorname{sDisc}_{k}(f)=\mathrm{sr}_{k}(f,f').$$ If $\mathbb{D}=\mathbb{C}$, $k$ is the smallest index such that $\operatorname{sDisc}_{k}(f)\not= 0$ if and only if $f$ has $n-k$ distinct roots in $\mathbb{C}$, and
$$|\operatorname{sDisc}_{k}(f)|=\left(\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)} \mu(z) \right)\cdot |\operatorname{Disc}(f^\star)|.$$
\end{definition}
\begin{proposition}\label{DavenportMahler}
Let $f\in\mathbb{C}[X]$ be a polynomial of degree $n$.
Defining $m$ as the number of distinct complex roots of $f$, it holds
\begin{align}
\prod_{z \in V_{\mathbb{C}}(f)}
\operatorname{sep}(z,f) \geq | \operatorname{sDisc}_{n-m}(f)| \operatorname{Mea}(f)^{2(1-m) }\frac{\sqrt{3}^m}{m^{2m}}
\left(\frac{1}{3}\right)^{\min(n,2n-2m)/3}.
\label{DavenportMahlerb}
\end{align}
\end{proposition}
\begin{proof}
The claim is a special case of the results in \cite{Ein,EPe}.
\end{proof}
Now we introduce the generalized discriminant which is a very natural quantity in our context. We are not aware of the occurrence of this notion earlier in the literature.
\begin{definition} \label{def:gdisc}
Let $\mathbb{D}$ be a domain, and let $f$ in $\mathbb{D}[X]$ of degree $n$ with respect to $X$.
The \emph{generalized discriminant} of $f$, $\operatorname{GDisc}(f)$ is the element of $\mathbb{D}$
such that
$$ \operatorname{lc}(f)\cdot \operatorname{GDisc}(f)=\operatorname{tcoeff}(\operatorname{Res}_X(f,\sum_{k=1}^n U^{k-1}f^{[k]} ))$$ where $\operatorname{tcoeff}(g)$ is the tail coefficient of the polynomial $g\in\mathbb{D}[U]$, i.e. the coefficient of its non zero term of lowest degree in $U$.
\end{definition}
\begin{proposition} \label{prop:gdisc}
If $\mathbb{D}=\mathbb{C}$,
$$\begin{array}{rcl}| \operatorname{GDisc}(f) |&= &\displaystyle{|\operatorname{lc}(f)|^{n-2}\cdot
\prod_{z \in V_{\mathbb{C}}(f)} |f^{[\mu(z)]}(z)|^{\mu(z)}}\\
&=&\displaystyle{ | \operatorname{lc}(f)|^{2n-2}\cdot
\prod_{z,z' \in V_{\mathbb{C}}(f),z\not= z'} |z-z'|^{\mu(z) \mu(z')}},
\end{array}$$
with $\mu(z)$ the multiplicity of $z$ as a root of $f$.
\end{proposition}
\begin{proof}
The equality
$$|\operatorname{lc}(f)|^{n-2}\cdot
\prod_{z \in V_{\mathbb{C}}(f)} |f^{[\mu(z)]}(z)|^{\mu(z)}= | \operatorname{lc}(f)|^{2n-2}\cdot
\prod_{z,z' \in V_{\mathbb{C}}(f),z\not= z'} |z-z'|^{\mu(z) \mu(z')},$$ is clear.
The equality $$| \operatorname{GDisc}(f) |= |\operatorname{lc}(f)|^{n-2}\cdot
\prod_{z \in V_{\mathbb{C}}(f)} |f^{[\mu(z)]}(z)|^{\mu(z)}$$
follows from the fact that
$$|\operatorname{Res}_X(f,\sum_{k=1}^n U^{k-1}f^{[k]} )|=|\operatorname{lc}(f)|^{n-1}\cdot
\prod_{z \in V_{\mathbb{C}}(f)} |\sum_{k=1}^n U^{k-1}f^{[k]}(z)|^{\mu(z)}$$
by (\ref{equ:res}).
\end{proof}
Notice that the generalized discriminant is never $0$ and coincide with the discriminant in the special case where all the roots of $f$ are simple.
\begin{definition}
\label{def}
For a polynomial $f \in \mathbb{C} [X]$, we define
\begin{eqnarray*}
\operatorname{logLen} (f) & := & \max(1,|\log(\operatorname{Len}(f))|),\\
\operatorname{logMea} (f) & := &
\max(1,\log(|\operatorname{lc}(f)|)+ \sum_{z\in \mathrm{V}_{\mathbb{C}}(f)}\operatorname{mult}(z,f)\cdot \log (\max (1, |z|))),\\
\widehat{\operatorname{logMea}} (f) & := &
\max(1, \sum_{z\in \mathrm{V}_{\mathbb{C}}(f)}\operatorname{mult}(z,f)\cdot \log (\max (1, |z|))),\\
\operatorname{logsep} (f) & := & \max(1,
\sum_{z\in \mathrm{V}_{\mathbb{C}}(f)} \operatorname{mult}(z,f)\cdot |\log (\operatorname{sep} (z, f))|),\\
\operatorname{logsep}^\star(f) & := & \operatorname{logsep}(f^\star)=
\max(1, \sum_{z\in \mathrm{V}_{\mathbb{C}}(f)} |\log (\operatorname{sep} (z, f))|).\\
\widehat{\operatorname{logGDisc}}(f) &:=& \max(1,\sum_{z\in \mathrm{V}_{\mathbb{C}}(f)} \operatorname{mult}(z,f)
|\log (|f^{[\operatorname{mult}(z,f)]}(z)|)|).
\end{eqnarray*}
\end{definition}
The various $\max(1,-)$ in the preceding definitions ensure that the corresponding quantities are at least of size $1$, which simplifies some of the complexity statements.
\begin{proposition}(see for example \cite{BPRbook2})\label{compareMahlerCoeff0}
Let $f = \sum^n_{i = 0} a_i X^i \in \mathbb{C} [X], \text{with }a_n \neq 0$.
The norm of $f$, its length and its Mahler measure are related as follows :
$$2^{-n} \operatorname{Len} (f) \leqslant \operatorname{Mea} (f) \leqslant \| f \| .$$
As a consequence,
\begin{equation}
\label{comparelog30}
\operatorname{logLen} (f) \le \operatorname{logMea} (f)+\deg(f).
\end{equation}
\end{proposition}
In the next step, we give a bound for $\operatorname{logsep}(f)$ relating it to $\widehat{\operatorname{logMea}}(f)$, $\operatorname{logLen}(f)$ and $\widehat{\operatorname{logGDisc}}(f),$ and give a bound for $\operatorname{logsep}(f)+\widehat{\operatorname{logGDisc}}(f)$ in the square-free case.
\begin{proposition}\label{GammaDeltaSigma}
Let $f\in\mathbb{C}[X]$ be a polynomial of degree $n$.
\begin{itemize}
\item[(a)] It holds:
\begin{equation}\label{GammaDeltaSigmab}
\operatorname{logsep}(f)\in O(n(n+\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f))+\widehat{\operatorname{logGDisc}}(f))
\end{equation}
\item[(b)] If $f$ is square-free, then
\begin{equation}\label{GammaDeltaSigmac}
\operatorname{logsep}(f)+\widehat{\operatorname{logGDisc}}(f)\in\tilde{O}(n(\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f))+|\log(|\operatorname{Disc}(f)|)|).
\end{equation}
\end{itemize}
\end{proposition}
For the proof of Proposition~\ref{GammaDeltaSigma}, we need the following lemma:
\begin{lemma}\label{6neighbours}
Let $S$ be a finite subset of $\mathbb{C}$. Consider a mapping $\phi: S\mapsto S$ that maps any element $z$ of $S$ to an arbitrary element $z'\in S\setminus\{z\}$ that minimizes the distance to $z$. Then, each element $z'\in S$ has at most $6$ pre-images under the mapping $\phi$.
\end{lemma}
\begin{proof}
As the distance between two pre-images must be greater than the distances between $z'$ and each of its pre-images, the claim follows directly from the fact that if $OAB$ is a triangle and $AB\ge\max(OA,OB)$, then the angle
$\widehat{BOA}$ has a measure at least $2\pi/6$. The value 6 is obtained in the case of a regular hexagon and its center.
\end{proof}
\begin{proof}[Proof of Proposition \ref{GammaDeltaSigma}]
For (a) we first prove that, for any root $z$ of $f$ of multiplicity $\mu(z)=\operatorname{mult}(z,f)$, it holds
\begin{equation}\label{GDS}
\operatorname{sep}(z,f)^{\mu(z)}>|f^{[\mu(z')]}(z')|\cdot 2^{-n}\cdot\max(1,|z'|)^{-n}\cdot \operatorname{Len} (f) ^{-1}
\end{equation}
where $z'\neq z$ is an arbitrary root of $f$ that minimizes the distance to $z$. The inequality
(\ref{GDS}) then follows directly from the following computation:
\begin{align*}
|f^{[\mu(z')]}(z')|&=|\operatorname{lc}(f)|\cdot \prod_{
y \in \mathrm{V}_{\mathbb{C}}(f)\setminus\{z'\} }|z'-y|^{\mu(y)}\\
&=\operatorname{sep}(z,f)^{\mu(z)}\cdot|\operatorname{lc}(f)|\cdot \prod_{
y \in \mathrm{V}_{\mathbb{C}}(f)\setminus\{z,z'\} }|z'-y|^{\mu(y)}\\
&\le \operatorname{sep}(z,f)^{\mu(z)}\cdot\operatorname{Mea}(f(X+z'))
\end{align*}
and
\begin{align*}
\operatorname{Mea}(f(X+z'))&\le 2^n\cdot \operatorname{Len} (f) \cdot\max(1,|z'|)^n.
\end{align*}
We now consider a mapping $\phi: \mathrm{V}_{\mathbb{C}}(f)\mapsto \mathrm{V}_{\mathbb{C}}(f)$ that maps an arbitrary root $z$ of $f$ to an arbitrary root $z'$ that minimizes the distance to $z$. By Lemma \ref{6neighbours} each element $z'\in \mathrm{V}_{\mathbb{C}}(f)$ has at most $6$ pre-images under the mapping $\phi$. We thus conclude that
\begin{align*}
&\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)
} \operatorname{sep}(z,f)^{\mu(z)}\\
&\quad>2^{-n^2}\cdot \operatorname{Len} (f)^{-n}\cdot \prod_{z \in \mathrm{V}_{\mathbb{C}}(f)
}\max(1,|\phi(z)|)^{-n}\cdot \prod_{z \in \mathrm{V}_{\mathbb{C}}(f)
} |f^{[\mu(\phi(z))]}(\phi(z))|\\
&\quad \ge 2^{-n^2}\cdot \operatorname{Len} (f)^{-n}\cdot\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)} \max(1,|z|)^{-6n}\cdot\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)
}
\min(1,|
f^{[\mu(\phi(z))]}(\phi(z))|)
\\
&\quad \ge 2^{-n^2}\cdot \operatorname{Len} (f)^{-n}\cdot \widehat{\operatorname{Mea}}(f)^{-6n}
\cdot\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)} \min(1,|f^{[\mu(z)]}(z)|)^6,
\end{align*}
which shows that
$$-\sum_{z \in \mathrm{V}_{\mathbb{C}}(f)\atop \operatorname{sep}(z,f)<1}\mu(z)\cdot \log (\operatorname{sep}(z,f))\in O(n(n+\operatorname{logLen} (f)+\widehat{\operatorname{logMea}}(f))+\widehat{\operatorname{logGDisc}}(f)).$$
It remains to estimate
$$\sum_{z \in \mathrm{V}_{\mathbb{C}}(f)\atop \operatorname{sep}(z,f)\ge 1}\mu(z)\cdot \log (\operatorname{sep}(z,f)).$$
It holds that
\begin{equation} \label{usefulsep}
\operatorname{sep}(z,f)=|z-\phi(z)|\le 2 \cdot\max(1,|z|)\cdot\max(1,|\phi(z)|).
\end{equation}
Thus, we get
\begin{align*}
\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)\atop \operatorname{sep}(z,f)\ge 1} \operatorname{sep}(z,f)^{\mu(z)}&\le 2^{n}\cdot \widehat{\operatorname{Mea}}(f)^{7}
\end{align*}
and finally
\begin{equation}\label{GDS1}
\sum_{z:\operatorname{sep}(z,f)>1} \mu(z)\log(\operatorname{sep}(z,f))\in O(n + \widehat{\operatorname{logMea}}(f)).
\end{equation}
Part (b) follows almost immediately from (\ref{GDS1}) and Equation (\ref{DavenportMahlerb}).
Namely,
\begin{align*}
\operatorname{logsep}(f)&=\max(1,-\sum_{z\in V_{\mathbb{C}}(f)}\log(\operatorname{sep}(z,f))+2\cdot\sum_{z:\operatorname{sep}(z,f)>1}\log(\operatorname{sep}(z,f)))\\
&\le \log \left(\prod_{z\in V_{\mathbb{C}}(f)}\operatorname{sep}(z,f)^{-1} \right)+ O(n\cdot\widehat{\operatorname{logMea}}(f))\\
&\in \tilde{O}( \widehat{\operatorname{logGDisc}}(f)+n
\operatorname{logMea}(f))\\
&\in \tilde{O}( \widehat{\operatorname{logGDisc}}(f)+n(\widehat{\operatorname{logMea}}(f)+\operatorname{logLen}(f))).
\end{align*}
It remains to estimate $\widehat{\operatorname{logGDisc}}(f)$
as well. For any complex value $z$, it holds that
$|f'(z)|\le n\cdot\operatorname{Len}(f)\cdot\max(1,|z|)^n$ by Lemma \ref{subst1}, and thus
\[
\sum_{z\in V_{\mathbb{C}}(f):|f'(z)|>1}\log (|f'(z)|)\in \tilde{O}(n(
\operatorname{logLen}(f))+\widehat{\operatorname{logMea}}(f)).
\]
Hence, it holds that
\begin{align*}
\widehat{\operatorname{logGDisc}}(f)&=\log (\prod_{z\in V_{\mathbb{C}}(f)} |f'(z)|^{-1})+2\cdot \sum_{z\in V_{\mathbb{C}}(f):|f'(z)|>1}|\log (|f'(z)|)\\
&=\log (\prod_{z\in V_{\mathbb{C}}(f)} |f'(z)|^{-1})+\tilde{O}(n (\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f)))\\
&=\log(\frac{|\operatorname{lc}(f)|^{n-2}}{|\operatorname{Disc}(f)|})+\tilde{O}(n (\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f)))\\
&\in\tilde{O}( n (\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f))+|\log(|\operatorname{Disc}(f)|)|).
\end{align*}
\end{proof}
For polynomials $f$ with integer coefficients, we introduce the notation of the \emph{magnitude} of $f$.
\begin{definition}
A polynomial $f\in \mathbb{Z}[X]$ is of \emph{magnitude} $(n,\tau)$ if its degree is bounded by $n$ and its bitsize is bounded by $\tau$, that is, the absolute value of each of its coefficients is bounded by $2^\tau$.
\end{definition}
The following lemma follows directly from the fact that $\binom{n}{k}<2^n$ for all $k=0,1,\ldots,n$.
\begin{lemma}
\label{littleremark} If $f\in \mathbb{Z}[X]$ is of magnitude $(n,\tau)$,
then
$f^{[k]}$ is of magnitude $(n,\tau+n)$ (using Notation \ref{not:fk}).
\end{lemma}
\begin{proposition} (see for example \cite{BPRbook2}) \label{compareMahlerCoeff}
If $f\in \mathbb{Z}[X]$
is of magnitude $(n,\tau)$, then
\begin{equation}
\label{compare}
2^{\tau- n}
\leqslant \operatorname{Mea} (f)
\leqslant \sqrt{n+1}\cdot 2^\tau.
\end{equation}
Hence,
\begin{equation}
\label{comparelog1} \operatorname{logMea} (f) \in {O} (\tau+\log (n)),
\end{equation}
and
\begin{equation}
\label{comparelog2}
\widehat{\operatorname{logMea}} (f) \in {O} (\tau+\log (n)).
\end{equation}
\end{proposition}
We aim to give bounds on $\operatorname{logsep}(f)$ that depend on the magnitude $(n,\tau)$ of $f$. In Proposition \ref{GammaDeltaSigma}, we have already derived a bound on $\operatorname{logsep}(f)$ that is related to $n$, $\operatorname{logLen}(f)$, $\widehat{\operatorname{logMea}}(f)$ and $\widehat{\operatorname{logGDisc}}(f)$. It is clear that $\operatorname{Len}(f)\le (n+1)\cdot 2^{\tau}$, and we already know that $\widehat \operatorname{Mea}(f)\le \sqrt{n+1} \cdot 2^{\tau}$ by Proposition \ref{compareMahlerCoeff} (\ref{compare}). Hence, we are left to bound $\widehat{\operatorname{logGDisc}}(f)$ in terms of $n$ and $\tau$. For this, we first derive a general bound on the product of the absolute values that a sequence of integer polynomials $g_1,\ldots,g_m$ takes at corresponding roots of $f$. A corresponding bound on $\widehat{\operatorname{logGDisc}}$ will then follow by applying our result to the sequence given by the first non-vanishing derivatives at the roots of $f$.
\begin{proposition}
\label{generalunivariate}
Let $f$ be a polynomial in $\mathbb{Z}[X]$
of magnitude $(n_1,\tau_1)$
and let $(g_i)_{1 \leq i \leq m}$ be a sequence of polynomials in $\mathbb{Z}[X]$
of magnitude $(n_2,\tau_2)$.
As above, we use $\mu(z)=\operatorname{mult}(z,f)$ to denote the multiplicity of $z$ as a root of $f$.
\begin{itemize}
\item[a)]
Let $A \subset \mathrm{V}_{\mathbb{C}}(f)$ be an arbitrary subset of the set of roots of $f$. In addition, for each $z\in A$, let $i(z)\in\{1,\ldots,m\}$ be such that $g_{i(z)}(z)\not=0$. Then, it holds
\begin{equation}\label{boundgeneralunivar}
\sum_{z\in A} \mu(z)
\log (|g_{i(z)} (z)|) \in \tilde O(n_1\tau_2+n_2\tau_1).
\end{equation}
\item[b)] Suppose moreover that, for every root
$z\in \mathrm{V}_{\mathbb{C}}(f)$,
there exists an $i$ such that $g_i(z)\not=0$. Denoting by $i(z)$ the smallest value of $i$ such
that $g_i(z)\not=0$, it holds
\begin{equation}\label{boundgeneraluniv}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(f)}
\mu(z) |\log (|g_{i(z)}(z)|) | \in \tilde O (n_1 \tau_2+n_2 \tau_1).
\end{equation}
\end{itemize}
\end{proposition}
In the proof of the above proposition, we will make use of the following elementary lemma.
\begin{lemma}
\label{elem}
Let $a,b,c,d$ be non negative numbers and suppose $a\le c$, $b-a\le d$
then $a+b\le 2c+d$.
\end{lemma}
\begin{proof}[Proof of Proposition \ref{generalunivariate}]
\noindent a)
For any $z \in A,$ we have
\begin{equation}
\label{basis}
\left|g_{i(z)}(z)\right|\leq2^{\tau'_2}\cdot \max(1,\left|z\right|)^{n_2};
\end{equation}
by applying Proposition \ref{subst1}, with $\tau'_2$ defined by $2^{\tau'_2}=(n_2+1)2^{\tau_2}$.
Then, $$\prod_{z \in A}\left|g_{i(z)}(z)\right|^{\mu(z)} \leq \prod_{z\in A} 2^{\tau'_2 \mu (z)}\cdot \max(1,\left|z\right|)^{n_2 \mu (z)} \leq \prod_{z\in \mathrm{V}_{\mathbb{C}}(f)} 2^{\tau'_2 \mu (z)}\cdot \max(1,\left|z\right|)^{n_2 \mu(z)};$$
using (\ref{basis}).
Finally, we obtain
\begin{equation}\label{less}
\prod_{z\in A}\left|g_{i(z)}(z)\right|^{\mu(z)} \leq 2^{\tau'_2 \sum \mu(z)}
\widehat{\operatorname{Mea}} (f)
^{n_2} \in 2^{O(n_1 \log (n_2)+n_2 \tau_1+n_1 \tau_2+n_2 \log (n_1))};
\end{equation}
as $\sum_{z\in A} \mu(z)\leq n_1$, $\widehat{\operatorname{Mea} (f)} \leq 2^{\tau_1+\log (n_1)}$, and $2^{\tau'_2}=(n_2+1)2^{\tau_2}$.\medskip
\noindent b) We want to prove that
\begin{equation}
\label{greater}
\prod_{z\in \mathrm{V}_{\mathbb{C}}(f)}\left|g_{i(z)}(z)\right|^{\mu(z)}\in 2^{-O(\tau_1 n_2)}.
\end{equation}
We set $$g (X,U)= g_1 (X)+ U g_2(X)+\cdots +U^{m-1} g_{m} (X)$$ and consider
$$\operatorname{Res}_X(f,g)= \operatorname{lc}(f)^{n_2}\cdot\prod_{z\in \mathrm{V}_{\mathbb{C}}(f)}g(z,U)^{\mu(z)}.$$
The polynomial $\operatorname{Res}_X(f, g)$ is then a polynomial in $U$ with integer coefficients. The coefficient of its term of lowest degree, which is an element of $\mathbb{Z}\backslash\{0\}$, has absolute value equal to $$|\operatorname{lc}(f)|^{n_2}\cdot \prod_{z\in \mathrm{V}_{\mathbb{C}}(f)}\left|g_{i(z)}(z)\right|^{\mu(z)}.$$
In particular, this shows that the latter term has absolute value at least $1$.
It is further clear that
$$|\operatorname{lc}(f)|^{n_2}\in 2^{O(\tau_1 n_2)},$$
and thus (\ref{greater}) holds.
We now define $$A:=\lbrace z | f(z)=0, |g_{i(z)}(z)|\geq 1 \rbrace.$$
Since for any $z \in A,~\left|g_{i(z)}(z)\right| \geq 1,$ it is clear by (\ref{less}) that
\begin{equation}\label{+OA}
0\le \sum_{z \in A}\mu(z)\cdot \log (|g_{i(z)}(z)|)\le c \in O(n_1 \tau_2+n_2 \tau_1+n_2 \log (n_1)+n_1 \log (n_2))
\end{equation}
It follows by (\ref{greater})
that
\begin{equation}\label{-OB}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(f)}-\mu(z) \cdot \log (|g_{i(z)}(z)|)\in O(n_2 \tau_1).
\end{equation}
Hence, using Lemma \ref{elem}, we obtain
\begin{equation}\label{-OBB}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(f)} \mu(z) |\log (|g_{i(z)}(z)|)|\in O(n_2 \tau_1+n_1 \tau_2).
\end{equation}
\end{proof}
The following result has already been proven in \cite{KoS}
but we give here a simpler proof based on Proposition \ref{generalunivariate}.
\begin{proposition}
\label{corodisc}
Let $f\in\mathbb{Z}[X]$ be a polynomial of magnitude $(n,\tau)$, then
\begin{itemize}
\item[(a)] $\operatorname{logsep}^\star(f)\in\tilde{O}(n\tau).$
\item[(b)] $\operatorname{logsep}(f)\in\tilde{O}(n\tau+n^2).$
\end{itemize}
\end{proposition}
\begin{proof}
(a) We consider a mapping $\phi: \mathrm{V}_{\mathbb{C}}(f)\mapsto \mathrm{V}_{\mathbb{C}}(f)$ that maps an arbitrary root $z$ of $f$ to (one of) the roots $z'$ that minimizes the distance to $z$.
Using Equation (\ref{usefulsep}) and Lemma \ref{6neighbours} we have
$$\prod_{z \in \mathrm{V}_{\mathbb{C}}(f)} \operatorname{sep}(z,f)\le 2^n \cdot \prod_{z \in \mathrm{V}_{\mathbb{C}}(f)} (\max(1,|z|)\cdot \max(1,|\Phi(z)|))\le 2^n \cdot
\widehat{\operatorname{Mea}}(f)^7.
$$
The claim then follows directly from
Equation
(\ref{DavenportMahlerb}) and Proposition
\ref{compareMahlerCoeff}.
(b) Using Proposition \ref{generalunivariate} b) (for the sequence of higher order derivatives of $f$) we obtain
$$\widehat{\operatorname{logGDisc}}(f)\in\tilde{O}(n\tau+n^2).$$
Since $\widehat{\operatorname{logMea}}(f)\in O(\tau+\log (n))$ by
(\ref{comparelog2}), we conclude by Proposition \ref{GammaDeltaSigma} (b).
\end{proof}
Hereafter, we recall some quantitative and complexity results which will be used in the
complexity analysis of our algorithms.
\begin{proposition}
\label{Mignotte}(see for example \cite{BPRbook2}) Let $f\in \mathbb{Z}[X]$ be of magnitude $(n,\tau)$ and $g$ of degree $n_{1}$ dividing $f$.
Then, $g$ is of magnitude $(n_1,n_{1} + \tau + \log (n +
1))$.
\end{proposition}
\begin{proposition}
\label{gcd-comp}\cite{BLMPRS16,GG} Let $f, g \in \mathbb{Z} [X]$ be two polynomials of respective magnitude $(n_{1},\tau_{1})$ and $(n_{2},\tau_{2})$. Computing
their gcd has a bit complexity of $$\tilde{O} (\max(n_{1},n_{2})\cdot (n_{1}\tau_{2}+n_{2}\tau_{1})).$$
\end{proposition}
\begin{proposition}
\label{exact_division_comp}\cite[Ex.~10.21]{GG} Let $f\in \mathbb{Z} [X]$ be a polynomial of
magnitude $(n,\tau)$. Given a polynomial $g$ that divides $f$, computing the quotient of $f$ divided by $g$
has bit complexity of $\tilde{O} (n\tau+n^2)$.
\end{proposition}
\begin{notation}
\label{bitsize}
We denote by $\lambda(p)$ the bitsize of a rational number, defined by the sum of the bitsizes of its numerator and denominator.
For an interval $I=[a,b]$, $a<b$ with rational endpoints, we denote by $|I|=b-a$ its length and by $\lambda(I)$ the maximum of $\lambda(a)$ and $\lambda(b)$.
\end{notation}
\begin{proposition}
\label{univariate-evaluation}\cite{BaZa,KS15} Let $f \in \mathbb{Z} [X]$ be a
polynomial of magnitude $(n,\tau)$.
Let $r$ a rational number of bitsize $\lambda(r)$. Then, the evaluation of $f$ at $r$
can be performed using $\tilde{O} (n (\tau + \lambda(r)))$ bit operations and the bitsize of
the output $f (r)$ is $\tilde{O} (\tau + n\cdot \lambda(r))$.
\end{proposition}
We now focus on the problem of computing the roots of a given univariate polynomial. Here, we consider the two different but related problems of the computation of disjoint isolating regions and the approximation of the roots to a certain precision. Notice that isolating regions allow us to distinguish between two distinct roots, and thus also to determine the number of distinct roots. However, in general, isolating regions do not allow us to estimate the actual distance between two distinct roots as such regions might be considerably larger than the actual separation of the isolated root.
In order to overcome this issue, we are aiming for the computation of so called well-isolating regions from which we can derive a good estimate for the separation of a root $z$ or, more generally, for the distance from $z$ to any other root $z'$.
\begin{definition}
\label{well-isolating}
Let $f \in \mathbb{C} [X]$ be a polynomial of degree $n$. Then, we define:
\begin{itemize}
\item[(a)] A \emph{well-isolating} interval ${\mathcal I}=(a,b)$ for a real root $z$ of $f$ contains $z$ and no other root of $f$, and it holds that $|b-a|<\frac{{\operatorname{sep}(z,f)}}{32n}$
\item[(b)] A \emph{well-isolating} disk ${\mathcal D}_{r}(m)=\{z\in \mathbb{C} \mid |z-m|\leq r\}$ for a complex root $z$ of $f$ contains $z$ and no other root of $f$, and it holds that $r<\frac{\operatorname{sep}(z,f)}{64n}$.
\end{itemize}
\end{definition}
The following proposition summarizes the best known complexity bounds for computing the roots of a polynomial $f$ with arbitrary complex coefficients. Here, it is assumed that the polynomial is given by means of an oracle that provides arbitrarily good approximations of the coefficients at the cost of reading the approximations. That is, for any given positive integer $L$, we may ask the oracle to provide an $L$-bit approximation of $f$; see Definition~\ref{Lbitapproximation} for details. The cost for reading such an $L$-bit approximation is $\tilde{O}(n(\max(1,\log(\|f\|))+L))$. The root finding algorithm in~\cite{MSW2} recursively asks for $L$-bit approximations with $L=1,2,4,\ldots$ until it succeeds in computing the roots for some $L=L'$. The given bounds on the needed input precision $L'$ are thus worst-case bounds on the precision that is needed in the last call of the oracle; for more details, see~\cite{CIsolate,MSW2,SM16}.
In what follows, we often have to deal with approximations of polynomials and to compute approximations of an exact value that a given polynomial takes at a certain point. The following definitions will turn out to be useful in order to specify these computations.
\begin{definition}\label{Lbitapproximation}
For a complex number $a\in\mathbb{C}$ and an integer $L$, we say that a dyadic Gaussian number of the form $\tilde{a}={c}\cdot {2^{-L-1}}+\mathbf{i}\cdot {d}\cdot {2^{-L-1}}\in\mathbb{Q}+\mathbf{i}\cdot \mathbb{Q}$, with $c,d\in\mathbb{Z}$, is an \emph{(absolute) $L$-bit approximation} of $a$ if $|a-\tilde{a}|<2^{-L}$.
For a polynomial $f=a_0+\cdots+ a_n\cdot X^n\in\mathbb{C}[X]$ with arbitrary complex coefficients and an integer $L$, we say that a polynomial $\tilde{f}=\tilde{a}_0+\cdots +\tilde{a}_n\cdot X^n$
is an \emph{(absolute) $L$-bit approximation} of $f$ if for every $i$, $\tilde{a}_i$ is an (absolute) $L$-bit approximation of $a_i$.
\end{definition}
The following proposition and corollary summarizes the results on root isolation and approximation for a complex polynomials we use in the paper.
\begin{proposition}
\label{sagraloff-isolation}(\cite[Thm.~4]{MSW2}\footnote{See also
\cite{CIsolate,MSW1,Pa,SM16} for comparable results.}) Let $f \in \mathbb{C} [X]$ be a
polynomial of degree $n$ with $1/4\le |\operatorname{lc}(f)| \le 1$.
Suppose that the number $m$ of distinct roots of $f$ is given, then it holds:
\begin{itemize}
\item[(a)] Using a number of bit operations bounded by
\begin{align}\label{complexityisol}
\tilde{O}\left(n\cdot(n^2+n\operatorname{logMea}(f)+\operatorname{logsep}(f)+\widehat{\operatorname{logGDisc}}(f))\right)
\end{align}
we can compute, for all $z\in \mathrm{V}_{\mathbb{C}}(f)$,
the multiplicities $\mu(z)$ as well as well-isolating disks ${\mathcal D}_{r(z)}(m(z))\subset\mathbb{C}$ with dyadic centers $m(z)$ and dyadic radii $r(z)$ such that
the bitsizes of all $m(z)$ and $r(z)$ sum up to $\tilde{O}(n+\operatorname{logMea}(f)+\operatorname{logsep}^*(f))$.
As input, we need an oracle giving an absolute $L'$-bit approximation of $f$, where $L'$ is bounded by
\begin{align}\label{complexityprec}
\tilde{O}\left(n\operatorname{logMea}(f)+\operatorname{logsep}(f)+\widehat{\operatorname{logGDisc}}(f)\right).
\end{align}
\item[(b)] Let $\mathrm{V}^*\subset \mathrm{V}_{\mathbb{C}}(f)
$ be a subset of the roots of $f$, $\mu=\max_{z\in \mathrm{V}^*}\mu(z)$, and let $L$ be a given positive integer. Then, we can further refine the isolating disks ${\mathcal D}_{r(z)}(m(z))$ for all roots $z$ in $\mathrm{V}^*$ to a size less than $2^{-L}$ using
\begin{align}\label{complexityref}
\tilde{O}\left(n\cdot\left(L\cdot\mu+n^2+n\operatorname{logMea}(f)+\operatorname{logsep}(f)+\widehat{\operatorname{logGDisc}}(f)\right)\right)
\end{align}
bit operations.\footnote{\cite[Thm.~4]{MSW2} only provides a bound for the refinement of \emph{all} isolating disks (i.e. for $\mathrm{V}^*=\mathrm{V}_{\mathbb{C}}(f)$), however, from the proof of \cite[Thm.~4]{MSW2}, the claimed bound directly follows. In addition, in~\cite[Theorem 4]{MSW2}, the additive term $nL\cdot \mu$ appears in the bound on the needed input precision. We remark that this is a typo and that the actual bound is better by a factor $n$. The proof of~\cite[Theorem 4]{MSW2} clearly shows this fact.} As input, we need an oracle giving an absolute $L'$-bit approximation of $f$, where $L'$ is bounded by
\begin{align}\label{complexityprec1}
\tilde{O}\left(L\cdot \mu+n\operatorname{logMea}(f)+\operatorname{logsep}(f)+\widehat{\operatorname{logGDisc}}(f)\right).
\end{align}
\end{itemize}
\end{proposition}
We obtain the following corollary, which follows directly from Proposition \ref{sagraloff-isolation} and Proposition~\ref{GammaDeltaSigma} (b).
\begin{corollary}\label{rootisolation:squarefree}
If $f \in \mathbb{C} [X]$ is a square free polynomial of degree $n$ with $1/4 \leq |\operatorname{lc} (f)|\leq 1$, then it holds:
\begin{itemize}
\item[(a)]
Using a number of bit operations bounded by
\begin{align}\label{complexityisolsf}
\tilde{O}(n(n^2+n(\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f))+|\log(|\operatorname{Disc}(f)|)|)
\end{align}
we can compute, for all $z\in \mathrm{V}_{\mathbb{C}}(f)$,
well-isolating disks ${\mathcal D}_{r(z)}(m(z))\subset\mathbb{C}$ with dyadic centers $m(z)$ and dyadic radii $r(z)$ such that
the bitsizes of all $m(z)$ and $r(z)$ sum up to $\tilde{O}(n+\operatorname{logMea}(f)+\operatorname{logsep}^*(f))$.
As input, we need an oracle giving an absolute $L'$-bit approximation of $f$, where $L'$ is bounded by
\begin{align}\label{complexityprecsf}
\tilde{O}(n(\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f))+|\log(|\operatorname{Disc}(f)|)|).
\end{align}
\item[(b)] Let $\mathrm{V}^*\subset \mathrm{V}_{\mathbb{C}}(f)
$ be a subset of the roots of $f$ and $L$ be a given positive integer. Then, we can further refine the isolating disks ${\mathcal D}_{r(z)}(m(z))$ for all roots $z$ in $\mathrm{V}^*$ to a size less than $2^{-L}$ using
\begin{align}\label{complexityref_sqfr}
\text{ }&\tilde{O}\left(n\cdot(L+n^2+n(\operatorname{logLen}(f)+\widehat{\operatorname{logMea}}(f))+|\log(|\operatorname{Disc}(f)|)|\right)
\end{align}
bit operations.
As input, we need an oracle giving an absolute $L'$-bit approximation of $f$, where $L'$ is bounded by
\begin{align}\label{complexityprec2}
\tilde{O}\left(L+n(\operatorname{logLen}(f))+\widehat{\operatorname{logMea}}(f))+|\log(|\operatorname{Disc}(f)|)| \right).
\end{align}
\end{itemize}
\end{corollary}
\medskip
\begin{remark}
Notice that, from the computation of well-isolating disks for all complex roots of a polynomial $f\in\mathbb{R}[X]$, we may immediately deduce well-isolating intervals for all real roots of $f$ with real valued coefficients. Namely, as the roots of a polynomial with real coefficients appear in complex conjugate pairs, a well-isolating disk isolates a real root of $f$ if and only if it intersects the real axis, in which case, the intersection constitutes a well-isolating interval.
However, it is preferable to use a dedicated method for computing the real roots only, if $f$ is known to be square-free. Such a method with comparable running times as above has been presented in~\cite{SM16}. Recent work~\cite{ANewDsc} also reports on a highly efficient implementation of this method.
\end{remark}
The rough idea of the proof of Proposition \ref{sagraloff-isolation} (coming from~\cite{MSW2}) is to consider an approximation $\tilde f$ of the polynomial $f$ and to compute approximations of its complex roots (e.g. using Pan's asymptotically fast approximation algorithm~\cite{Pa}). If $\tilde{f}$ is a sufficiently good approximations of $f$, its roots appear as clusters of size corresponding to the multiplicities of the roots of $f$. So what is done in a first step is to cluster the roots of $\tilde f$ in a meaningful manner (this takes into account that a root of multiplicity $k$ splits into a cluster of roots of size ~$c\cdot\sqrt[k]{\varepsilon}$, where $\varepsilon$ is the approximation quality and $c$ some constant). Finally, once the solutions are clustered, it is shown that $f$ must have $k$ roots (counted with multiplicity) close to the center of each cluster of size $k$, using Rouch\'e's Theorem. If the number of clusters equals the number $m$ of distinct roots of $f$, the algorithm stops. Otherwise, it starts over with a better approximation of $f$ and tries again.
We further remark that the stated complexity bounds do not take into account the cost for the actual computation of the approximations of the coefficients, which might be considerably larger than the cost for just reading the approximation. This might for instance be the case if the coefficients are implicitly given as algebraic numbers
of large degree.
In the literature, the special case of an integer polynomial $f$ with coefficients of bitsize at most $\tau$ has attracted a lot of interest.
The following result, which provides bounds on the isolation of the roots as well on the problem of further refining the isolating disks, is an almost straight forward consequence of Proposition~\ref{sagraloff-isolation} (applied to the polynomial $f\cdot\operatorname{lc}(f)^{-1}$).
\begin{proposition}
\label{sagraloff-isolation-integer}\cite[Thm.~5]{MSW2} \footnote{See also
\cite{CIsolate,MSW1,Pa,PT,SM16}} Let $f \in \mathbb{Z} [X]$ be a
polynomial of magnitude $(n,\tau)$.
Using $\tilde{O}(n^2\tau+n^3)$ bit operations, one can compute
\begin{itemize}
\item[(a)] well-isolating disks ${\mathcal D}_{r(z)}(m(z))\subset\mathbb{C}$ for all complex roots $z$ of $f$ with dyadic centers $m(z)$ and dyadic radii $r(z)
$ such that
the bitsizes of all $m(z)$ and $r(z)$ sum up to $\tilde{O} (n \tau)$, and
\item[(b)] the multiplicities $\mu(z)$ of each of the roots $z$.
\item[(c)] For an arbitrary positive integer $L$, one can further refine all isolating disks to a size less than $2^{-L}$ using $\tilde{O}(n^2\tau+n^3+nL)$ bit operations.\footnote{Notice that, in contrast to the general case, where the coefficients of $f$ are not necessarily integers, the additional factor $\max \mu(z)$ is missing. This is due to the fact that, within the given complexity, we can first compute the square-free part $f^\star$ of $f$ and then work with $f^\star$ to refine the isolating disks.}
\end{itemize}
\end{proposition}
Finally we can also identify common roots of a polynomials $f$ with polynomials of a family $g_1,\ldots,g_m$.
\begin{proposition}\label{comparingroots}
Let $f,g_1,\ldots,g_m\in \mathbb{Z} [X]$ be polynomials of magnitudes $(d,\tau)$, $(d_1,\tau_1),\ldots,(d_m,\tau_m)$, respectively, and $N,\Lambda$ be positive integers such that $d+d_1+\cdots+d_m<N$ and $\tau+\tau_1+\cdots+\tau_m<\Lambda$. Then,
we can isolate all roots of $f$ and all polynomials $g_i$, and identify all common roots of each pair $(f,g_i)$ using no more than
$\tilde{O}(N^2\Lambda+N^3)$ bit operations.
Within the same complexity, we can also determine the signs ($0$, $1$ or $-1$) of the $g_i$ at the real zeroes of $f$.
\end{proposition}
\begin{proof}
We
first compute the square free part $f^\star$
using
$$\tilde{O}(d^2 \tau)\in \tilde{O}(N^2\Lambda)$$
bit operations using Proposition~\ref{gcd-comp} and Proposition \ref{exact_division_comp}.
The magnitude
of $f^\star$
is bounded by $(d,O(d+\tau))$.
Then, we may compute $h_i:=\gcd(f^\star,g_i)$ for all $i$ using
$$\tilde{O}(\sum\nolimits_{i=1}^m \max(d,d_i)\cdot(d\tau_i+d d_i+d_i \tau))\in \tilde{O}(N(d\Lambda+N\tau+dN)$$
bit operations due to Proposition~\ref{gcd-comp}. In the next step, we compute well-isolating disks for all complex roots of the polynomial $f^\star$ as well as for the complex roots of all polynomials $h_i$. We then refine the isolating disks for the roots of all polynomials $h_i$ to a size less than $\min_{z\in V_\mathbb{C}(f)}\operatorname{sep}(z,f)/4$. Since $\operatorname{logsep}(f)$ is bounded by $\tilde{O}(N\Lambda)$, this can be achieved using $\tilde{O}(N^2\Lambda+N^3)$ bit operations
according to Proposition \ref{sagraloff-isolation-integer}.
Notice that, after this refinement, each isolating disk $D'$ for a root of $h_i$ intersects exactly one isolating disk $D$ for a root of $f$, and thus $D$ and $D'$ isolate one common root. Hence, in order to identify common roots of $f$ and $g_i$, we just have to determine all intersections between the isolating disks for the roots of $h_i$ and those for the roots of $f$. For this, we first compute a lexicographic sorting of all centers of the isolating disks $D$ for the roots of $f$, which uses $\tilde{O}(d^2\cdot \tau)$ bit operations as we need $O(d\log d)$ many comparisons, each of precision $\tilde{O}(d\tau)$. Then, for a given isolating disk $D'$ for a root of $h_i$, we can determine the unique disk $D$ that intersects $D'$ using $\tilde{O}(N\Lambda)$ bit operations as the needed precision is bounded by $\tilde{O}(N\Lambda)$ and only $O(\log d)$ many comparisons are needed. Hence, the total complexity for this step is also bounded by $\sum_{i=1}^m d_i\cdot\tilde{O}(N\Lambda)\in \tilde{O}(N^2\Lambda)$.
For the sign determination part, we simply compute sufficiently good $L$-bit approximations $\gamma_{i,z}$ of $g_i(z)$ for all $i$ and all real roots $z$ of $f$ with $g_i(z)\neq 0$. That is, we have to compute $L$-bit approximations $\gamma_{i,z}$ for $L=1,2,4,\ldots$ such that $|\gamma_{i,z}|>2^{-L}$, which then implies that $\operatorname{sign} g_i(z)=\operatorname{sign}\gamma_{i,z}$. Obviously, we succeed in doing so as soon as $L$ is larger than $|\log (|g_i(z)|)|$. Hence, for a specific real root $z$ of $f_i:=f^\star/\gcd(f^\star,g_i)$, the cost for this step is bounded by
$\tilde{O}(d_i(|\log (|g_i(z)|)|+d_i\cdot\log(\max(1, |z|))+\tau_i))$ bit operations; see the following Proposition~\ref{univariate-evaluationbis}. The total cost is thus bounded by
\[
\sum_{i=1}^m d_i^2\cdot\sum_{z\in V_{\mathbb{R}}(f)}\log(\max(1,|z|))\quad+\quad N\cdot\sum_{i=1}^m d_i\tau_i \quad+\quad \sum_{i=1}^m d_i\sum_{z\in V_{\mathbb{R}}(f_i)}|\log( |g_i(z)|)|
\]
The first term is upper bounded by $N^2\cdot\widehat{\operatorname{logMea}}(f)\in \tilde{O}(N^2\Lambda)$, and the second term is upper bounded by $N\cdot\sum_{i=1}^m d_i\cdot\sum_{i=1}^m\tau_i\in O(N^2\Lambda)$. For the last term, we use Proposition~\ref{generalunivariate} b) to show that $\sum_{z\in V_{\mathbb{R}}(f_i)}|\log (|g_i(z)|)|\in \tilde{O}(N\Lambda+N^2)$ for all $i$, and thus also $\sum_{i=1}^m d_i\sum_{z\in V_{\mathbb{R}}(f_i)}|\log (|g_i(z)|)|\in \tilde{O}(N^2\Lambda+N^3)$.
\end{proof}
\begin{proposition}
\label{univariate-evaluationbis}\cite{BZ,KS15} Let $f \in \mathbb{Z} [X]$ be a
polynomial of magnitude $(n,\tau)$, $z \in\mathbb{C}$ be an arbitrary complex value, and $L$ be a positive integer.
We can compute
a dyadic approximation $\tilde{\beta}=
{b}\cdot {2^{-L-1}}$ of $\beta:=f(z)$ , with $b\in\mathbb{Z}$ and $|\beta-\tilde{\beta}|<2^{-L}$, using $\tilde{O}(n(L+n\cdot\log(\max(1, |z|))+\tau))$ bit operations.
\end{proposition}
\section{Bivariate results}\label{bivariateresults}
Similar to our definition of the magnitude of a univariate polynomial, we introduce the following definitions for bivariate polynomials:
\begin{definition}
A polynomial $F\in \mathbb{Z}[X,Y]$ is of \emph{magnitude} $(n,\tau)$ if its degree is bounded by $n$ and the
bitsize of each of its coefficients is bounded by
$\tau$.
\end{definition}
\begin{proposition}
\label{subst} Let $F(X, Y) \in \mathbb{Z} [X, Y]$ be a bivariate polynomial with coefficients of
bitsize at most $\tau$
and degrees $n_x=\deg_X (F), n_y=\deg_Y (F)$. Let $(z, z') \in \mathbb{C}^2,$ then
$$|F (z, z') | \leqslant (n_x + 1) (n_y + 1)
2^{\tau} \max (1, | z |)^{n_x} \max (1, | z' |)^{n_y}.$$
\end{proposition}
The following result on \emph{approximate bivariate evaluation} follows directly from Proposition~\ref{univariate-evaluation}; see also~\cite{KoS}.
\begin{proposition}
\label{bivariate-evaluation}\cite{KoS} Let $F \in \mathbb{Z} [X,Y]$ be a
bivariate polynomial of magnitude $(n,\tau)$. Let $(z,z')\in\mathbb{C}^2$ be a pair of arbitrary complex values and $L\in\mathbb{N}_{\ge 1}$ be a positive integer, then we can compute
a dyadic approximation $\lambda$ of $F(z,z')$, with $2^{L+1}\cdot \lambda\in\mathbb{Z}+\mathbf{i}\cdot\mathbb{Z}$ and $|F(z,z')-\lambda|<2^{-L}$, using $$\tilde{O}(n^2(L+n\cdot(\log(\max(1,|z|))+\log(\max(1,|z'|)))+\tau))$$ bit operations. For the computation, we need an oracle giving as input dyadic approximations $\tilde{z}$, $\tilde{z}'$ of $z$, $z'$, with $2^{L+1}\cdot \tilde{z},2^{L+1}\cdot \tilde{z}'\in\mathbb{Z}+\mathbf{i}\cdot \mathbb{Z}$ and $|z-\tilde{z}|<2^{-L'}$, $|z'-\tilde{z}'|<2^{-L'}$, where $L'$ is bounded by $$\tilde{O}(L+n\cdot(\log(\max(1,|z|))+\log(\max(1,|z'|)))+\tau).$$
\end{proposition}
We further state the following result on the \emph{exact computation} of a bivariate polynomial $F \in \mathbb{Z} [X,Y]$ at a rational point $(r_1,r_2)$.
\begin{proposition}
\label{bivariate-evaluation-rational} Let $F \in \mathbb{Z} [X,Y]$ be a
bivariate polynomial of
magnitude $(n,\tau)$.
Let $(r_1,r_2)\in\mathbb{Q}^2$ be a pair of arbitrary rational values with numerators and denominators of
bitsize at most $\lambda$.
Then, we can exactly compute $F(r_1,r_2)$ using $\tilde{O}(n^2(\tau+\lambda))$ bit operations.
\end{proposition}
\begin{proof}
Let $F=f_0(X)+\cdots+f_{n}(X)Y^n$ with polynomials $f_i(X)\in\mathbb{Z}[X]$. Then, according to Proposition~\ref{univariate-evaluation}, we can exactly compute $f_i(r_1)$ for all $i=0,\ldots,n$ using $\tilde{O}(n^2(\tau+\lambda))$ bit operations. Suppose that $r_1=p/q$ with coprime integers $p$ and $q$, then $q^n\cdot f_i(r_1)$ is an integer of bitsize $O(\tau+n\lambda)$. The computation of each term $q^n\cdot f_i(r_1)$ needs $\tilde{O}(\tau+n\lambda)$ bit operations, hence we can compute $q^n\cdot F(r_1,Y)\in\mathbb{Z}[Y]$ using $\tilde{O}(n^2(\tau+\lambda))$ bit operations. Since $q^n\cdot F(r_1,Y)$ has integer coefficients of bitsize $O(\tau+n\lambda)$, we conclude that we can compute $q^n\cdot F(r_1,r_2)$ using $\tilde{O}(n^2(\tau+\lambda))$ bit operations. Hence, our claim follows.
\end{proof}
\begin{proposition}
\label{subres-comp}(\cite[Prop.~8.46]{BPRbook2},\cite[\S 11.2]{GG} Let $F,G \in \mathbb{Z}[X,Y]$ be polynomials of
magnitude $(n,\tau)$.
The subresultant
coefficients of $F$ and $G$ (considered as polynomials in $Y$ with coefficients in $\mathbb{Z}[X]$), which are polynomials in $X$ of magnitude $(O(n^2),O(n\tau))$, can be computed in $\tilde O (n)$
arithmetic operations between univariate polynomials of degree $O (n^2)$ and
of bitsize $\tilde O (n \tau)$, so with a bit complexity $\tilde{O} (n^4
\tau)$. The bitsize of the output is $\tilde O (n^4 \tau)$.
In particular, the resultant of $F$ and $G$ is of magnitude $(O(n^2),O(n\tau))$ and can be computed with a bit complexity $\tilde{O} (n^4 \tau)$.
\end{proposition}
We consider an arbitrary polynomial $R\in\mathbb{Z}[X]$ of magnitude $(N,\Lambda)$. Let $z$ be a complex root of $R$, and let $\mu(z):=\operatorname{mult}(R,z)$ be the corresponding multiplicity of $z$ as a root of $R$. We further consider a bivariate polynomial
$$F(X,Y)=f_{n_y}(X)\cdot Y^{n_y}+\cdots+f_0(X) \in\mathbb{Z}[X,Y]$$
of magnitude $(n,\tau)$, such that
$$V_{\mathbb{C}}(R,F)=\{(z,z')\in \mathbb{C}\mid R(z)=F(z,z')=0\}$$
is finite.
For $\ell\le n_y$, we define $$F_\ell(X,Y):=f_\ell(X)\cdot Y^\ell+\ldots+f_0(X).$$
For a given root $z$ of $R$, we denote by $n(z)$ the degree of $F(z,-)$, which might be smaller than $n_y$ but at least $0$, since $V_{\mathbb{C}}(R,F)$ is finite. Notice that, for $n(z)\neq n_y$, we have $f_{n(z)}(z)\neq 0$ and $f_{n(z)+1}(z)=\cdots=f_{n_y}(z)=0$.
For a given root $z$ of $R$, we further denote by $\mu(z):=\operatorname{mult}(R,z)$ its multiplicity. In addition, for a root
$z'$ of $F(z,-)$, we denote by $\mu(z,z')=\operatorname{mult}(F(z,-),z')$ the multiplicity of $z'$ as a root of $F(z,-)$.
The following proposition provides amortized complexity bounds on the sum of lengths and Mahler measures of the polynomials $F(z,-)$.
\begin{proposition}\label{pro:bounds}
Let $R\in\mathbb{Z}[X]$ and $F\in\mathbb{Z}[X,Y]$ be polynomials of magnitude $(N,\Lambda)$ and $(n,\tau)$ such that $V_{\mathbb{C}}(R,F)$ is finite. Then, it holds that
\begin{equation}\label{pro:bounds1}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}\mu(z)\cdot \operatorname{logLen}(F(z,-)) \in \tilde{O}(N\tau+n\Lambda+n N)
\end{equation}
\begin{equation}\label{pro:bounds2}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}\mu(z)\cdot\operatorname{logMea}(F(z,-)) \in \tilde{O}(N\tau+n\Lambda+n N)
\end{equation}
\begin{equation}\label{pro:bounds3}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}\mu(z)\cdot \widehat{\operatorname{logMea}} (F(z,-)) \in \tilde{O}(N\tau+n\Lambda
+n N
).
\end{equation}
\end{proposition}
\begin{proof}
For each root $z$ of $R$ in $\mathbb{C}$, we have
$$2^{-n(z)}\cdot\operatorname{Len}(F(z,-))\leq \operatorname{Mea}(F(z,-))\leq \| F(z,-)\|
$$ using Definition \ref{def} and Proposition \ref{compareMahlerCoeff}.\\
Since $n(z)\leq n$ and $\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}\mu(z) = N$, it holds that $$\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}n(z)\mu(z) \leq n N$$ and thus $$2^{-n N}\leq 2^{-\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}n(z)\mu(z)}.$$
Let $\ell(z)$ be such that
$$| f_{\ell(z)}(z)|=\max_{j=0,\ldots,n_y} | f_{j}(z)|.$$
We have $$ | \operatorname{lc}(F(z,-))|=|f_{n(z)}(z)| \leq \operatorname{Len}(F(z,-))\leq (n_y+1)\cdot | f_{\ell(z)}(z)|$$
$$ \| F(z,-)\|\leq \sqrt{n_y+1} \cdot | f_{\ell(z)}(z)|,$$
hence
$$2^{-n}\prod_{z\in \mathrm{V}_{\mathbb{C}}(R)}
|f_{n(z)}(z)
|^{\mu(z)}
\leq \prod_{z\in \mathrm{V}_{\mathbb{C}}(R)} \operatorname{Mea}(F(z,-))^{\mu(z)}\leq \sqrt{n_y+1}^N \cdot \prod_{z\in \mathrm{V}_{\mathbb{C}}(R)} | f_{\ell(z)}(z)|^{\mu(z)}
$$
and
$$\prod_{z\in \mathrm{V}_{\mathbb{C}}(R)}
|f_{n(z)}(z)
|^{\mu(z)}
\leq \prod_{z\in \mathrm{V}_{\mathbb{C}}(R)} \operatorname{Len}(F(z,-))^{\mu(z)}\leq (n_y+1)^N \cdot \prod_{z\in \mathrm{V}_{\mathbb{C}}(R)} | f_{\ell(z)}(z)|^{\mu(z)}
$$
The claims (\ref{pro:bounds1}) and (\ref{pro:bounds2}) follow by Proposition \ref{generalunivariate} applied to $R$ and the family $f_j$.
For the claim (\ref{pro:bounds3}), it remains to notice that
$$\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} \mu(z) |\log (|f_{n(z)}(z)|)|\in\tilde{O}((N\tau+n\Lambda)),$$ which is an immediate consequence of Proposition \ref{generalunivariate} applied to $R$ and the family $f_i$ and to use (\ref{pro:bounds2}).
\end{proof}
We now give results on the multiplicities of the roots of $F(z,-)$ at the $X$-critical points of $F$.
Consider a bivariate polynomial
$$F(X,Y)=f_{n_y}(X)\cdot Y^{n_y}+\cdots+f_0(X) \in\mathbb{C}[X,Y].$$
Denoting by
$$V_{\mathbb{C}}(F)=\{(x,y)\in \mathbb{C}^2\mid F(x,y)=0 \}$$
notice that $V_{\mathbb{C}}(F)$ contains no vertical line if and only if the polynomials $f_i(X)$, $i=0,\ldots,n_y$, do not share a common non-trivial factor.
\begin{proposition}\label{multmult0}
Let $F\in\mathbb{C}[X,Y]$ be such that $V_{\mathbb{C}}(F)$ contains no vertical line.
Let $$\operatorname{Crit} (V_{\mathbb{C}} (F)) = \{(x, y) \in \mathbb{C}^2 \mid F (x, y) =\partial_Y F(x,y)= 0\}$$
be the set of $X$-critical points of $F$
and
$$\operatorname{Crit} (V_{\mathbb{C}} (F))_z = \{z' \in \mathbb{C} \mid F (z, z') =\partial_Y F(z,z')= 0\}$$
its fiber above $z$.
Given $z\in \mathbb{C}$,
\begin{equation}\label{mult5}
\sum_{z'
\in \operatorname{Crit} (V_{\mathbb{C}} (F))_z} (\operatorname{mult}(z', F_{n(z)}(z,Y))-1)
\le \operatorname{mult}(z,\operatorname{Disc}_Y(F_{n(z)})).
\end{equation}
\end{proposition}
The proof of Proposition \ref{multmult0} uses the following lemmas.
\begin{lemma}
\label{mult1bis}Let $F$ and $G$ be two bivariate polynomials.
Given $z\in \mathbb{C}$,
\begin{equation}\label{mult5bis}
\deg(\gcd(F(z,Y),G(z,Y)))
\le \operatorname{mult}(z,\operatorname{Res}_Y(F,G)).
\end{equation}
\end{lemma}
Lemma \ref{mult1bis} follows clearly from the two following lemmas
\begin{lemma}
Let $f,g$ be two univariate polynomials of respective degrees
$p,q$.
Let $\phi$ be the mapping from $K_{< p} [X] \times K_{< q} [X]$ to $K_{<
p+q} [X]$ sending $(U, V)$ to $Uf + Vg$. Then $\operatorname{Im}(\phi)$ is the set of multiples of the greatest common divisor $h=\operatorname{gcd}(f,g)$ of $f$ and $g$, and the rank of $\phi$ is $p+q-\deg(h)$.
\end{lemma}
\begin{proof} It is clear that $(m\cdot g/h,-m\cdot f/h)$ is in the kernel of $\phi$ for every polynomial $m$ of degree $<\deg(h)$. In the other direction, every $(U,V)$ in the kernel of $\Phi$ is such that there exists $m$ of degree $<\deg(h)$ such that $U=m\cdot g/h$ and $V=-m\cdot f/h$. This implies that the dimension of $\operatorname{Ker}(\phi)$ is equal to $\deg(h)$ and the dimension of $\operatorname{Im}(\phi)$ equal to $p+q-\deg(h)$. But every element of $\operatorname{Im}(\phi)$ is a multiple of $h$, and the vector space of multiples of $h$ of degree $<p+q$ is also of dimension $p+q-\deg(h)$. It follows that $\operatorname{Im}(\phi)$ coincides with the set of multiple of $h$, and the rank of $\phi$ is $p+q-\deg(h)$.
\end{proof}
\begin{lemma}
Let $M(X)$ be an $n\times n$ matrix with coefficients in $K[X]$. If the rank of $M(x_0)$ is equal to $n-k$, then $x_0$ is a root of $\det(M(X))$ of multiplicity at least $k$.
\end{lemma}
\begin{proof}
The proof is by induction on $k$.
If $k=0$ the statement is true.
If $k>0$, then $M(x_0)$ is not invertible, and
$\det(M(x_0))=0$.
Denote by $m_{i,j}(X)$ the $(i,j)$-th entry of $M(X)$, and by $M_{i,j}(X)$ the $(n-1)\times (n-1)$-matrix obtained by removing thee $i$-th row and $j$-th column from $M(X)$.
The rank of $M_{i,j}(x_0)$ is
$r(x_0) \le n-k$, hence by induction hypothesis $x_0$ is a root of $\det(M_{i,j}(X))$ of multiplicity at least equal to $(n-1)-r(x_0) \ge k-1$.
According to Jacobi's formula, we have
$$\frac{d}{dX}\det(M(X))=\sum_{i,j} (-1)^{i+j} m'_{i,j}(X) \det(M_{i,j}(X)),$$
and thus the claim follows by induction since $x_0$ is a root of $\det(M(X))$ and a root of
multiplicity at least $k-1$ of its derivative.
\end{proof}
\begin{proof}[Proof of Proposition \ref{multmult0}]
Use Lemma \ref{mult1bis} with $F=F_{n(z)}$, $G=\partial_YF_{n(z)}(X,Y)$ noting that
\begin{equation}\label{mult6}
\sum_{z'
\in \operatorname{Crit} (V_{\mathbb{C}} (F))_z} (\operatorname{mult}(z', F_{n(z)}(z,Y))-1)
=\deg(\gcd(F_{n(z)}(z,Y),\partial_Y F_{n(z)}(z,Y))).
\end{equation}
\end{proof}
\begin{proposition}\label{multmult1}
Let $F\in \mathbb{Z}[X,Y]$ be a square free polynomial of magnitude $(n,\tau)$ such that $V_{\mathbb{C}}(F)$ has no vertical line,
Let $\operatorname{Crit} (V_{\mathbb{C}} (F))$ be the set of $X$-critical points of $F$.
There exists a polynomial $R_Y\in \mathbb{Z}[Y]$ of magnitude $(O(n^2),O(n\tau+n^2))$ such that
given $z'\in \mathbb{C}$
\begin{equation}\label{mult1}
\sum_{z \mid (z,z')
\in \operatorname{Crit} (V_{\mathbb{C}} (F))} (\operatorname{mult}(z'
, F_{n(z)}(z,Y))-1)\le \mathrm{mult}(z',
R_Y).
\end{equation}
In particular, the zeroes of $R_Y$ contain the projection of $\operatorname{Crit} (V_{\mathbb{C}} (F))$ on the $Y$-axis.
\end{proposition}
\begin{proof}
a) We suppose first that there is only one $z$ such that $(z,z')\in \operatorname{Crit} (V_{\mathbb{C}} (F))$, $\operatorname{mult}(z'
, F_{n(z)}(z,Y))> 1$ and denote $\mu=\operatorname{mult}(z'
, F_{n(z)}(z,Y))$.
Defining
\begin{equation}\label{UV}
R_Y(Y):=\operatorname{Res}_X(F,\partial_YF)(Y),
\end{equation}
we have
\begin{equation}\label{UV1}
R_Y(Y)=U(X,Y)F(X,Y)+V(X,Y)\partial_YF(X,Y).
\end{equation}
Derivating (\ref{UV}) $1,\ldots,\mu-2$ times with respect to $Y$ and using that
$$F(z,z')=\partial_YF(z,z')=\ldots=\partial_Y^{(\mu-1)}F(z,z')=0$$
we obtain
\begin{equation}\label{multimult}
R_Y(z')= \partial_Y R_Y(z')=\ldots= \partial_Y^{(\mu-2)}R_Y(z')=0.
\end{equation}
b) We reduce the general case to the preceding situation by the change of variable
$Y\leftarrow Y+\varepsilon X$ where $\varepsilon$ is a new variable. We use for the proof the field $\mathbb{R}\langle \varepsilon \rangle$ of algebraic Puiseux series \cite{BPRbook2}, which is a real closed field containing $\mathbb{R}(\varepsilon)$, totally ordered with the other $0_+$ making $\varepsilon$ positive and smaller that any positive element of $\mathbb{R}$. We denote
$\mathbb{C}\langle \varepsilon \rangle ^2=\mathbb{R}\langle \varepsilon \rangle ^2[i]$.
Notice that each $X$-critical point
$(z,z')$ of $F$ in $\mathbb{C}^2$ yields a $X$-critical point $(z,z'-\varepsilon z)$ of $\hat{F}(X,Y):=F(X,Y+\varepsilon X)$ in $\mathbb{C}\langle \varepsilon \rangle ^2$ , and
that $z'-\varepsilon z$ is a root of $\hat{F}(z,-)$ of multiplicity $\mu(z,z')$ as
$\partial_Y^{(i)}\hat{F}(z,z')=\partial_Y^{(i)}F(z,z'-\varepsilon z)$. Moreover there are no two distinct critical points of $\hat{F}(x,y)$ sharing
the same $y$-coordinate. Hence, it holds by a) that
$z'-\varepsilon z$ is a root of $\operatorname{Res}_X(\tilde F,\partial_Y \tilde F)(Y)$ of multiplicity $\nu(z'-\varepsilon z)$ at least $\mu(z,z')$.
Then
$$\operatorname{Res}_X(\tilde F,\partial_Y \tilde F)(Y)=A(\varepsilon)\tilde R(Y,\varepsilon) B(Y,\varepsilon)$$
with $A(\varepsilon)\in \mathbb{C}(\varepsilon)$,
$\tilde R(Y,\varepsilon)$ monic in $Y$,
$$\tilde R(Y,\varepsilon)=\prod_{(z,z')\in \operatorname{Crit} (V_{\mathbb{C}} (F))}(Y-z'+\varepsilon z)^{\nu(z'-\varepsilon z)}B(Y,\varepsilon),$$
with
\begin{equation}\label{mult1fin}
\operatorname{mult}(z'-\varepsilon z
, \tilde F_{n(z)}(z,Y))-1\le \nu(z'-\varepsilon z),
\end{equation}
and
$B(Y,\varepsilon)\in \mathbb{C}[X,\varepsilon]$ such that $B(Y,0)\in \mathbb{Z}[X]$ is a non zero polynomial.
Hence, denoting by $$\nu(z')=\sum_{z \mid (z,z')\in \operatorname{Crit} (V_{\mathbb{C}} (F))}\nu(z'-\varepsilon z),$$
$$R_Y:=\tilde R(Y,0)=\prod_{
z'\in \pi_Y(\operatorname{Crit} (V_{\mathbb{C}} (F))) }
(Y-z')^{\nu(z')}B(Y,0),$$
we have
\begin{equation}\label{mult1next}
\sum_{z \mid (z,z')\in \operatorname{Crit} (V_{\mathbb{C}} (F))} (\operatorname{mult}(z'
, F_{n(z)}(z,Y))-1)\le \nu(z')\le \mathrm{mult}(z',
R_Y).
\end{equation}
Finally, notice that $f$ is of magnitude $(O(n^2),O(n\tau+n^2))$.
\end{proof}
\begin{remark}
When $\deg_X(F)\ge \deg_Y(F)$, it turns out that $$f(Y)=\operatorname{Res}_X(F,\partial_YF)(Y)$$ because the Sylvester-matrix of $F,\partial_YF$ and $\tilde F,\partial_Y \tilde F$ have the same dimension, since $\deg_X(F)=\deg_X(\tilde F)$.
However, when $\deg_X(F)<\deg_Y(F)$, it can happen that $f(Y)\not=\operatorname{Res}_X(F,\partial_YF)(Y)$.
\end{remark}
We now give some details on the method
that we use for determining $\deg_Y(F(z,Y))$ for every $z\in V_{\mathbb{C}}(R)$ when $V_{\mathbb{C}}(R,F)$ is finite. For this, we first compute a family of polynomials $R^\star_{\ell}$
that we are going to define next. We denote by $R^\star$ the square-free part of $R$ and further define:
\begin{equation}
R^\star_{\le n_y} (X) :=R^\star(X),\label{factorisation-degree1}
\end{equation}
and
\begin{equation}
R^\star_{\le \ell-1} (X) := \gcd ( R^\star_{\le \ell}
(X), \tilde f_{\ell}(X)),\quad R^\star_{\ell-1} (X) := \frac{ R^\star_{\le \ell} (X)}{ R^\star_{\le \ell-1}
(X)}.\label{factorisation-degree2}
\end{equation}
for all $\ell \in \{n_y, \ldots, 1\}.$
The following result is straightforward.
\begin{lemma}\label{lemmafixdegree}
Let $z\in \mathbb{C}$ be a root of $R.$
Then, for any integer $\ell \in [0,n_y]$, it holds
\label{fixdegree}
\[ \deg_Y(F (z, Y)) \le \ell
\Longleftrightarrow R^\star_{\le \ell} (z) = 0. \]
\[ \deg_Y(F (z, Y)) = \ell
\Longleftrightarrow R^\star_{\ell} (z) = 0. \]
\end{lemma}
\begin{proposition}
\label{generalbivariate}
Let $R(X)\in \mathbb{Z}[X]$ be of magnitude $(N,\Lambda)$ and $F(X,Y)\in \mathbb{Z}[X,Y]$
be of magnitude $(n_1,\tau_1)$, and suppose that
$$V_{\mathbb{C}}(R,F)=\{(z,z')\in \mathbb{C}^2\mid R(z)=F(z,z')=0\}$$
is finite.
Let $G_1,\ldots,G_m$ in $\mathbb{C}[X,Y]$ be polynomials of degree bounded by $n_2$ and with coefficients
of absolute value bounded by $2^{\tau_2}$.
Further denote by $\mu(z)$ the multiplicity of $z$ as a root of $R$.
\begin{itemize}
\item[(a)] Suppose that $A\subset V_{\mathbb{C}}(R,F)$ and that, for every $(z,z')\in A$, we have chosen an $i(z,z')\in\{1,\ldots,m\}$ such that $G_{i(z,z')}(z,z')\not=0$. Then, it holds that
\begin{equation}\label{boundgeneralmultivar-a}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} \mu(z)
\sum_{z' \in A_{z}}
\log (|G_{i(z,z')} (z, z')|) \in \tilde O((\tau_2n_1+\tau_1n_2)N+(\Lambda+N) n_1n_2).
\end{equation}
where
$$A_{z}:=\{z' \in \mathbb{C} \mid (z,z')\in A \}.$$
\item[(b)] Suppose that $G_1,\ldots,G_m\in\mathbb{Z}[X,Y]$ have integer coefficients and that, for every $(z,z')\in V_{\mathbb{C}}(R,F)$,
there exists $i$ such that $G_i(z,z')\not=0$.
Denoting by $i(z,z')$ the smallest value of $i$ such
that $G_i(z,z')\not=0$, we have
\begin{equation}\label{boundgeneralmultivar-b}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} \mu(z)
\sum_{z' \in V_{\mathbb{C}}(R,F)_{z}}
| \log (|G_{i(z,z')} (z, z')|) |\in \tilde O((\tau_2n_1+\tau_1n_2)N+(\Lambda+N) n_1n_2),
\end{equation}
where
$$V_{\mathbb{C}}(R,F)_{z}:=\{z'\in \mathbb{C}\mid F(z,z')=0\}.$$
\end{itemize}
\end{proposition}
\begin{proof}
\noindent a)
According to
Proposition \ref{subst}, we have
$$|G_{i(z,z')}(z,z')|\le
2^{\tau'_2}\max(1,\vert z \vert)^{n_2}
\max(1,\vert z' \vert)^{n_2},$$
where $\tau'_2:=\lceil \log( (n_2+1)^2 2^{\tau_2})\rceil\in O(\tau_2+\log (n_2))$.
Further notice that
\begin{equation}
\label{b-coef-Mea}
\prod_{(z,z')\in A}
2^{\tau'_2 \mu(z)}\le \prod_{z\in \mathrm{V}_{\mathbb{C}}(R)} 2^{\tau'_2 n_1 \mu(z)} \in 2^{ O((\tau_2+\log (n_2)) n_1N)}
\end{equation}
and
\begin{eqnarray}
\label{Meal-Sy}
\prod_{(z,z')\in A} \max(1,\vert z\vert)^{\mu(z) n_2}& \le &
\prod_{z\in \mathrm{V}_{\mathbb{C}}(R)}\max(1,\vert z\vert)^{\mu(z) n_1n_2} \\
&\le&
\widehat{\operatorname{Mea}}(R)^{n_1n_2}\in 2^{O((\Lambda+\log( N)) n_1 n_2)}.
\end{eqnarray}
Hence,
since
\begin{equation}
\prod_{(z,z')\in A}
\max(1,\vert z' \vert)^{\mu(z)}\leq
\prod_{z\in \mathrm{V}_{\mathbb{C}}(R)} \widehat{\operatorname{Mea}}(F(z,-))^{\mu(z)}
\end{equation}
it follows that
\begin{equation}
\label{Meal-Sx}
\prod_{(z,z')\in A}
\max(1,\vert z' \vert)^{\mu(z)n_2}\in 2^{\tilde O((\tau_1 N+\Lambda n_1+n_1 N)n_2)}
\end{equation}
by
Proposition \ref{pro:bounds} and Proposition \ref{generalunivariate}.
We thus conclude that
\begin{equation}
\label{b-p}
\prod_{(z,z')\in A} |G_{i(z,z')}(z,z')|^{\mu(z)}\in 2^{\tilde O((\tau_2 n_1+\tau_1 n_2)N+(\Lambda+N) n_1n_2)}
\end{equation}
and
\begin{equation}
\label{valeur-N}
\sum_{(z,z')\in A} \mu (z) \log (|G_{i(z,z')} (z,z')|) \in \tilde O((\tau_2 n_1+\tau_1 n_2)N+(\Lambda+N) n_1n_2).
\end{equation}
\noindent
b) In the first step, we aim to prove that $$\prod_{(z,z')\in V_{\mathbb{C}}(R,F)} |G_{i(z,z')}(z,z')|^{\mu(z)}\ge \frac{1}{E''}$$
where $E''$ is a natural number of
bitsize $O(\Lambda n_1n_2+\tau_1 n_2 N)$.
Let $$F(X,Y):=f_{n_y}(X)Y^{n_y}+\ldots+f_0(X),$$ with $n_y=\deg_Y(F)\le n_1$, and $R^\star$ the square free part of
$R$.
We define the polynomial sequence $(R^\star_{\ell} (X))_{\ell\in[1,n_y]}$ as in Equations (\ref{factorisation-degree1}) and (\ref{factorisation-degree2}), such that
\[ \deg (F (z, Y) =\ell
\Longleftrightarrow R^\star_{\ell} (z) = 0. \]
We further define:
\[ \Psi_{\ell,>1} (X) :={\gcd(R',R^\star_\ell)}, \Psi_{\ell,1} (X):=\frac{R^\star_\ell}{\Psi_{\ell,>1}(X)} \]
and,
for all $i \in \{1, \ldots, N \},$ $$\Psi_{\ell,>i} (X) := \gcd (\Psi_{\ell,i-1}
(X), R^{(i)}(X)), \quad\Psi_{\ell,i} (X) := \frac{\Psi_{\ell,>i-1} (X)}{\Psi_{\ell,>i}
(X)}.$$
It is clear using Lemma \ref{lemmafixdegree} that
$\deg (F (z, Y)) =\ell$ and $z$ is a root of multiplicity $\mu$ of $R$ if and only if
$\Psi_{\ell,\mu} (z) = 0.$
Notice that
\begin{eqnarray}
R^\star&=&\prod \Psi_{\ell,\mu}\\
R&=&\prod \Psi_{\ell,\mu}^\mu
\end{eqnarray}
Let $$F_\ell(X,Y):=f_\ell(X)Y^\ell+\ldots+f_0(X),$$
and
$$G=G_{1}+U G_{2}+\ldots+U^{m-1}G_m.$$
We further define
$$Z_{\ell,\mu}:=\{(z,z')\in V_{\mathbb{C}}(R,F) \mid \Psi_{\ell,\mu}(z)=0\},$$
and
$$A_{\ell,\mu}(U):=\operatorname{Res}_X(\operatorname{Res}_Y(F_\ell,G),\Psi_{\ell,\mu})\in \mathbb{Z}[U].$$
Let $$Q_\ell(U,X):=\operatorname{Res}_Y(F_\ell,G)$$ and notice that
$$Q_\ell(U,z)=f_\ell(z)^{O(n_2)} \prod_{z'\in Z_{\ell,\mu,z}} G(z,z').$$
Denoting by $\delta\le n_1n_2$ the degree of $Q(U,X)$ with respect to $X$ and $n_\ell$ the degree of $f_\ell$ with respect to $X$,
$$A_{\ell,\mu}(U)=\operatorname{lc}(\Psi_{\ell,\mu})^{\delta-n_2n_\ell}\prod_{z\mid \Psi_{\ell,\mu}(z)=0} f_\ell(z)^{O(n_2)} \prod_{(z,z')\in Z_{\ell,\mu}} G(z,z')
$$
The coefficient of the term of smallest degree in $U$ of $A_{\ell,\mu}(U)$ is a non-zero integer and equal to
$$\operatorname{lc}(\Psi_{\ell,\mu})^{\delta-n_2 n_\ell}\operatorname{Res}_X(f_\ell,\Psi_{\ell,\mu})^{n_2}\prod_{(z,z')\in Z_{\ell,\mu}} G_{i(z,z')}(z,z').$$
Since ${\delta-n_2 n_\ell}\in O(n_1n_2)$, $\prod_{\ell,\mu} \operatorname{lc}(\Psi_{\ell,\mu}^\mu)= \operatorname{lc}(R)\in 2^{O(\Lambda)}$, and $$\prod_{\ell,\mu}\operatorname{Res}_X(f_\ell,\Psi_{\ell,\mu}^\mu)=\operatorname{Res}_X(f_\ell,R)\in 2^{O(\tau_1 N+\Lambda n_1)},$$
we conclude that
\begin{equation}
\label{alphagamma}
\prod_{(z,z')\in V_{\mathbb{C}}(R,F)} |G_{i(z,z')}(z,z')|^{\mu(z)}\ge \frac{1}{E''}
\end{equation}
with $E'':=\operatorname{lc}(R)^{n_1n_2} \operatorname{Res}_X(f_\ell,R)^{n_2}\in 2^{O(\Lambda n_1n_2+\tau_1 n_2 N)}$.
Finally,
we have
\begin{equation}
\label{b-n}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} -\mu(z)\sum_{z' \in V_{\mathbb{C}}(R,F)_z} \log( |G_{i(z,z')}(z,z')| ) \le b \in O(\Lambda n_1n_2+\tau_1 n_2 N)
\end{equation}
with $b=\log( E'').$
Defining $Z:=\{(z,z')\in V_{\mathbb{C}}(R,F)\mid |G_i(z,z')|\ge 1\},$
and using (\ref{b-p}), (\ref{b-n}) and Lemma \ref{elem}, we obtain
\begin{equation}
\label{bit-Abar1}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} \mu(z)\sum_{z' \in V_{\mathbb{C}}(R,F)_z}| \log (|G_{i(z,z')}(z,z')|) | \leq 2a+b
\end{equation}
with $a$ as defined in (\ref{valeur-N})
and finally
\begin{equation}
\label{bit-Abar2}
\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} \mu(z)\sum_{z' \in V_{\mathbb{C}}(R,F)_z}| \log (|G_{i(z,z')}(z,z)| )| \in \tilde O((\tau_2n_1+\tau_1n_2)N+(\Lambda+N) n_1n_2)
\end{equation}
\end{proof}
The following proposition provides amortized complexity bounds on the sum of the Mahler measures of the polynomials $F(z_i,-)$, the separators of the roots $z_{i,j}$ as well as the absolute values of the first non-vanishing derivatives of $F(z_i,-)$ at the roots $z_{i,j}$:
In what follows, we denote
\begin{equation}\label{Fk}
F^{[k]}=\frac{\partial_Y^{k}F}{k!}.
\end{equation}
\begin{proposition}\label{thm:bounds}
Let $R(X)\in \mathbb{Z}[X]$ be of magnitude $(N,\Lambda)$ and $F(X,Y)\in \mathbb{Z}[X,Y]$
be of magnitude $(n,\tau)$, and suppose that
$$V_{\mathbb{C}}(R,F)=\{(z,z')\in \mathbb{C}^2\mid R(z)=F(z,z')=0\}$$
is finite.
Then, it holds that
\begin{itemize}
\item[(a)] $\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)} \mu(z)\cdot \widehat{\operatorname{logGDisc}}(F(z,-))\in
\tilde{O}(n(N\tau+n\Lambda+Nn)).$
\item[(b)] $\sum_{z\in \mathrm{V}_{\mathbb{C}}(R)}\mu(z)\cdot\operatorname{logsep}(F(z,-)) \in \tilde{O}(n(N\tau+n\Lambda+Nn)),$
\end{itemize}
\end{proposition}
\begin{proof}
(a) is an immediate consequence of Proposition \ref{generalbivariate}.
(b) follows directly from (a), Proposition~\ref{GammaDeltaSigma} and Proposition \ref{pro:bounds} (\ref{pro:bounds1}) and (\ref{pro:bounds3}).
\end{proof}
We now give details on how to determine the degree of $F(z,Y)$ and its number of distinct complex roots
for a root $z$ of $R$. Our aim is to prove the following proposition:
\begin{proposition}\label{prop:computinggcddeg}
Let $R(X)\in \mathbb{Z}[X]$ be of magnitude $(N,\Lambda)$ and $F(X,Y)\in \mathbb{Z}[X,Y]$
be of magnitude $(n,\tau)$, and suppose that
$$V_{\mathbb{C}}(R,F)=\{(z,z')\in \mathbb{C}^2\mid R(z)=F(z,z')=0\}$$
is finite.
Computing $$n(z):=\deg(F(z,Y)),k(z):=\deg
(\gcd (F (z, Y), \partial_Y (F(z, Y))$$ for every root $z \in \mathbb{C}$ of $R$
has bit complexity
$$\tilde{O} (n\max(N,n^2)(N\tau+n\Lambda+N n)+n^5 \tau + n^6).$$
\end{proposition}
Note that $n(z)-k(z)$ is the number of distinct complex roots of $F(z,Y)$.
We start by estimating the cost of computing the polynomials $R^\star_{\ell}$ defined in Equation (\ref{factorisation-degree1}) and Equation (\ref{factorisation-degree2}).
\begin{lemma}\label{lemmacompdeg}
The computation of all the polynomials $$( R^\star_{\ell} (X))_{\ell\in[1,n_y]}$$ uses
$\tilde O(\max(n,N)( N\tau+n \Lambda+Nn)+n^3\tau+n^4)$ bit operations.
\end{lemma}
\begin{proof}
Since $R^\star$ is of degree at most $N$ and bitsize bounded by $\Lambda+N$, and $f_{n_y}(X)$ is of degree
at most $n$ and bitsize $O(\tau)$, the computation of their gcd needs $\tilde O(\max(n,N)(N\tau+n\Lambda+Nn))$ bit operations according to Proposition \ref{gcd-comp}.
Since, for all $\ell\in [0,n_y-1]$, $\deg( R^\star_{\le \ell})\le n$ and the bitsize of the coefficients of the $ R^\star_{\le \ell}$ are at most $n+\tau$,
the complexity of computing all $( R^\star_{\le \ell})_{\ell\in[0,n_y-1]}$ is in $O(n^3\tau+n^4)$.
It remains to compute the $R^\star_{\ell}$ themselves by performing the exact division of $ R^\star_{\le \ell}$ by $R^\star_{\le \ell-1}$. This takes $O(N\Lambda+N^2)$ binary operations for $\ell=n_y$ and
$O(n\tau+n^2)$ binary operations for each $\ell<n_y$.
\end{proof}
The proof of Proposition \ref{prop:computinggcddeg} uses a family of polynomials $R^\star_{\ell,k}$
that we are going to define next. For any non-negative integer $\ell \le n_y$, define $$F_{\ell}:=\sum_{i=0}^\ell f_i(X)Y^i.$$
We further denote
$\operatorname{sDisc}_{\ell,k} (X)$ the $k$-th subdiscriminant of $F_\ell$ considered as a polynomial in $Y$.
We also define:
\[ R^\star_{\ell,\geq 0} (X) := R^\star_{\ell} ,\]
and for all
$k \in \{ 0, \ldots, \ell - 1 \},$
\begin{equation}
R^\star_{\ell,\geq k+1} (X) := \gcd ( R^\star_{\ell,\geq k}
(X), \operatorname{sDisc}_{\ell,k} (X)),\quad
R^\star_{\ell,k} (X) := \frac{ R^\star_{\ell,\geq k} (X)}{ R^\star_{\ell,\geq k+1}
(X)}.\label{factorisation-degree-gcd1}
\end{equation}
The following result is then straightforward.
\begin{lemma} \label{lemmacompmult}
Given $\ell \in[1,n_y]$ and $z \in \mathbb{C}$ such that $\deg_Y(F (z, Y)) = \ell$ (i.e. $R^\star_{\ell}(z)=0$), then it holds
\label{compmult}
\[ \deg_Y(\gcd (F_{\ell} (z, Y), \partial_Y F_{\ell} (z, Y))) \geq k
\Longleftrightarrow R^\star_{\ell,\geq k} (z) = 0. \]
\[ \deg_Y(\gcd (F_{\ell}(z, Y), \partial_Y F_{\ell} (z, Y))) = k
\Longleftrightarrow R^\star_{\ell,k} (z) = 0. \]
\end{lemma}
\begin{lemma}\label{computesubres}
The computation of all the polynomials $$(\operatorname{sDisc}_{\ell,k} (X))_{\ell\in[1,n_y],k \in [0, \ell - 1
]}$$ needs $\tilde{O} (n^5 \tau)$ bit operations.
\end{lemma}
\begin{proof} The claim follows clearly from Proposition
\ref{subres-comp} since there are $n_y\le n$ lists of sudiscriminants (i.e. subresultant coefficients) to compute.
\end{proof}
\begin{lemma}\label{pro:degrees}
The computation of all the polynomials $(R^\star_{\ell,k} (X))_{\ell\in[1,n_y],k \in [0, \ell - 1]}$ needs a number of bit operations bounded by
$$\tilde{O} (n\max(N,n^2)(N\tau+n\Lambda+N n)+n^5 \tau + n^6).$$
\end{lemma}
\begin{proof}
We compute first all the polynomials $$(\operatorname{sDisc}_{\ell,k} (X))_{\ell\in[1,n_y],k \in [0, \ell - 1
]}$$ using \ref{computesubres}.
Let $\varphi_{\ell,k}$ be the degree of $R^\star_{\ell,\geq k}$, and let $\tau_{\ell,k}$ be the maximal bitsize of the coefficients of $R^\star_{\ell,\geq k}$.
The roots of $R^\star_{n_y,0}$ are exactly the roots $z$ of $R$ with
$$\deg_Y(F(z,Y))=n_y,\quad\text{and } \deg_Y(\gcd (F(z, Y), \partial_Y F (z, Y))) = 0.$$
The computation of $R^\star_{n_y,0}$ needs $\tilde{O} (\max(N,n^2)(Nn\tau+n^2\Lambda+N n^2))$ bit operations according to Proposition \ref{gcd-comp} and Proposition \ref{exact_division_comp}.
From Proposition~\ref{multmult0}, we conclude that each root of $R^\star_{n_y,\geq k}$ for $k>0$ is also a root of $\operatorname{Disc}_Y(F)$ of multiplicity at
least $k$. Hence, $(R^{\star}_{n_y,\geq k})^{k}$ divides $\operatorname{Disc}_Y(F)$, and thus
$$\varphi_{n_y,k} \leqslant \frac{n (2 n - 1)}{k},\quad\text{and } \sum_{k = 1}^{n_y}
\varphi_{n_y,k} \in \tilde{O} (n^2) .$$
Moreover, Proposition
\ref{compareMahlerCoeff0} (\ref{comparelog30}) yields that $\tau_{n_y,k}
\leqslant \operatorname{logMea} (R^\star_{n_y,\geq k}) + \varphi_{n_y,k}$. Since $(R^{\star}_{n_y,\geq k})^k$ divides
$\operatorname{Disc}_Y(F)$ and since the Mahler measure is multiplicative, we have $$\sum_{k = 1}^{n_y} \operatorname{logMea} (R^\star_{n_y,\geq k}) \leqslant \operatorname{logMea} (\operatorname{Disc}_Y(F)) \in \tilde{O} (n
\tau),$$ and thus
$$\sum_{k = 1}^{n_y} \tau_{n_y,\ge k} \in \tilde{O} ( n \tau+n^2).$$
On the other hand, for $\ell \in[1,n_y-1]$, $R^\star_{\ell,\geq k}$ is a divisor of $R^\star_\ell$, so that
$$\sum_{k = 0}^\ell \varphi_{\ell,k} \le (\ell+1) \cdot \deg(R^\star_{\ell}),\sum_{\ell=1}^{n_y-1}\deg(R^\star_{\ell})\le n_y\le n,$$
and
$$\sum_{\ell=1}^{n_y-1} \sum_{k = 0}^\ell
\varphi_{\ell,k} \in \tilde{O} (n^2).$$
As above, we use Proposition
\ref{compareMahlerCoeff0} (\ref{comparelog30}) to show that
$\tau_{\ell,k}
\leqslant \operatorname{logMea} (R^\star_{\ell,\geq k}) + \varphi_{\ell,k}$. Since $R^\star_{\ell,\geq k}$ is a divisor of $R^\star_\ell$, which is a divisor of
$f_{\ell}$, and $$\sum_{k =0}^\ell \operatorname{logMea} (R^\star_{\ell,\geq k}) \leqslant \operatorname{logMea} (f_\ell) \in \tilde{O} (\tau),$$ we conclude that
$$\sum_{\ell=1}^{n_y-1}\sum_{k = 0}^\ell \tau_{\ell,k}\in \tilde{O} (n^2 + n \tau).$$
Finally
\begin{equation}\label{phi}
\sum_{k = 1}^{n_y}
\varphi_{n_y,k}+\sum_{\ell=1}^{n_y-1} \sum_{k = 0}^\ell
\varphi_{\ell,k} \in \tilde{O} (n^2)
\end{equation}
\begin{equation}\label{tau}
\sum_{k = 1}^{n_y-1}
\tau_{n_y,k}+\sum_{\ell=1}^{n_y-1} \sum_{k= 0}^\ell
\tau_{n_y,k} \in \tilde{O} ( n \tau+n^2) .
\end{equation}
The computation of $R^\star_{\ell,\geq k+1} (X)=\gcd (R^\star_{\ell,\geq k} (X), \operatorname{sDisc}_{\ell, k} (X))$ for $(\ell,k)\not=(n_y,0)$ uses
$$ \tilde{O} (\max (\varphi_{\ell,k},n^2)(n^2 \tau_{\ell,k }+\varphi_{\ell,k} n \tau))\in \tilde{O} (n^2(n^2 \tau_{\ell,k }+\varphi_{\ell,k} n \tau))$$ bit operations
according to Proposition \ref{gcd-comp}.
Finally, the computation of all
the $R_{\ell,\geq k}$ needs
$$\tilde{O} (\max(N,n^2)(Nn\tau+n^2\Lambda+N n^2)+n^5 \tau + n^6)$$ bit operations using (\ref{phi}) and (\ref{tau}).
It remains to compute the $R^\star_{\ell,k}$ themselves by performing the exact division of $ R^\star_{\ell,\geq k}$ by $R^\star_{\ell,\geq k+1}$. This takes $O(N\Lambda+N^2)$ binary operations for $(\ell,k)=(n_y,0)$,
$O(n^2\tau+n^3)$ binary operations for each $(n_y,k),$ with $k \not=0$,
and $O(n\tau+n^2)$ binary operations for each $(\ell,k),$ with $\ell <n_y$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:computinggcddeg}]
In order to determine $n(z)$ for every $z\in V_{\mathbb{C}}(f)$, we use Lemma \ref{lemmafixdegree}, Lemma \ref{lemmacompdeg} and use Proposition \ref{comparingroots}.
Similarly in order to determine $k(z)$ for every $z\in V_{\mathbb{C}}(f)$, we use Lemma \ref{lemmacompmult}, Lemma \ref{pro:degrees} and use Proposition \ref{comparingroots}.
\end{proof}
We now use
Proposition \ref{sagraloff-isolation}
to bound the complexity of computing the roots of all polynomials $F(z,-)$, where $z$ runs over all roots of $R$.
\begin{proposition}\label{thm:costisolation}
Let $R\in\mathbb{Z}[X]$ be of magnitude $(N,\Lambda)$, let $F\in\mathbb{Z}[X,Y]$ be of magnitude $(n,\tau)$, and suppose that
$$V_{\mathbb{C}}(R,F)=\{(z,z')\in \mathbb{C}^2\mid R(z)=F(z,z')=0\}$$
is finite. Suppose that
$$n(z):=\deg\gcd(F(z,-),k(z):=\deg\gcd(F(z,-),\partial_Y F(z,-))$$ is part of the input for all roots $z$ of $R$.\footnote{Notice that, according to Proposition~\ref{prop:computinggcddeg}, this computation needs $$\tilde{O} (n^5 \tau + n^6+n\max(N,n^2)(N\tau+n\Lambda+N n))$$ bit operations. Further notice that $F(z,-)$ has $m(z)=n(z)-k(z)$ distinct complex roots.} Then, it holds:
\begin{itemize}
\item[(a)] Using $$ \tilde{O}(N^2\Lambda+N^3+n\cdot\max(n^2,N)\cdot(N\tau+n\Lambda+Nn))$$
bit operations, we can compute
\begin{itemize}
\item[(a.1)] well-isolating disks ${\mathcal D}_{z,z'}\subset \mathbb{C}$ for all complex roots $z'$ of all polynomials $F(z,-)$, where the sum of the bitsizes of the radii and centers of all disks ${\mathcal D}_{z,z'}$ is bounded by $\tilde{O}(n(N\tau+n\Lambda+Nn))$,
\item[(a.2)] the corresponding multiplicities $\mu(z,z')$ for each of the complex roots $z'$ of all polynomials $F(z,-)$, and
\item[(a.3)] dyadic approximation $\tilde{\sigma}_{z,z'}$ of the separations $\operatorname{sep}(z',F(z,-))$ such that $$\frac{1}{2}\cdot\operatorname{sep}(z',F(z,-))<\tilde{\sigma}_{z,z'}<2\cdot \operatorname{sep}(z',F(z,-))$$ for all roots $z$ (resp. $z'$) of $R$ (resp. $F(z,-)$).
\end{itemize}
\item[(b)] Let $V\subset\{(z,z')\in\mathbb{C}^2:R(z)=0\text{ and }F(z,z')=0\}$ , and $L$ be a positive integer. Then, we can further refine all isolating disks ${\mathcal D}_{z,z'}$, with $(z,z')\in V$, to a size less than $2^{-L}$, in a number of bit operations bounded by
$$ \tilde{O}(N^2\Lambda+N^3+n\cdot\max(n^2,N)\cdot(N\tau+n\Lambda+Nn)+L\cdot(N\cdot\mu+n^2\cdot\sum_{z \in \pi_X(V)}\mu_{z})),$$
where $\mu_{z}:=\max_{(z,z') \in V}\mu(z',F(z,-))$, and $\mu=\max_{z \in \pi_X(V)}\mu_{z}$.
\item[(c)]
Let $V\subset\{(z,z')\in\mathbb{C}^2:R(z)=0\text{ and }F(z,z')=0\}$ such that $\mu(z,z')=1$ for every $(z,z')\in V$, and $L$ be a positive integer. Then, we can further refine all isolating disks ${\mathcal D}_{z,z'}$, with $(z,z')\in V$, to a size less than $2^{-L}$, in a number of bit operations bounded by
$$ \tilde{O}(N^2\Lambda+N^3+n\cdot\max(n^2,N)\cdot(N\tau+n\Lambda+Nn)+L\cdot(N+n^2\cdot\operatorname{card}(\pi_X(V)),$$
\end{itemize}
\end{proposition}
\begin{proof}
Let $z$ be a fixed complex root of $R$, $\tau_{z}\in\mathbb{Z}$ such that $2^{-\tau_z-2}<\operatorname{lc}(F(z,-))\le 2^{-\tau_z}$, and $F_z:=2^{\tau_z}\cdot F(z,-)$. Notice that $F_z$ has the same roots as $F(z,-)$, and the leading coefficient of $F_z$ has absolute value in between $1/4$ and $1$. Then according to Proposition~\ref{sagraloff-isolation}, we can compute well-isolating disks ${\mathcal D}_{z,z'}$ for the roots of $F_z$ (and thus also for the roots of $F(z,-)$) as well as the multiplicities $\mu(z,z')$ in a number of bit operations that is bounded by
\begin{align}\label{cost:isolation}
\tilde{O}(n(n^2+n\cdot\operatorname{logMea}(F_z)+\operatorname{logsep}(F_z)+\widehat{\operatorname{logGDisc}}(F_z)))
\end{align}
and
\begin{align}\label{cost:isolation1}
\tilde{O}(n(n^2+n\cdot\operatorname{logMea}(F_z)+\widehat{\operatorname{logGDisc}}(F_z))),
\end{align}
using Proposition~\ref{GammaDeltaSigma} (a).
For this, we need an approximation of $F_z$ to an absolute precision that is bounded by
\begin{align}\label{error:input}
\rho_z
\in\tilde{O}(n\operatorname{logMea}(F_z)+\widehat{\operatorname{logGDisc}}(F_z)).
\end{align}
Using Proposition \ref{pro:bounds}, Proposition~\ref{thm:bounds} now yields
the bound $
\tilde{O}(n^2\cdot(N\tau+n\Lambda+Nn))
$
for the sum of the bound in (\ref{cost:isolation1}) over all roots $z$ of $R$, and
\[
\sum_{z \in V_{\mathbb{C}}(R)} \rho_z\in\tilde{O}(n\cdot (N\tau+n\Lambda+Nn))
\]
for the sum of the bound (\ref{error:input}) for the needed input precision over all $z\in V_{\mathbb{C}}(R)$.
It remains to show that we can compute sufficiently good approximations of the polynomials $F_z$ in a number of bit operations that is bounded by $\tilde{O}(n^2\cdot(N\tau+n\Lambda+Nn))$. In order to compute a $\rho_z$-bit approximation of $F _z$, we first compute a $\tau_z$ with $2^{-\tau_z-2}<\operatorname{lc}(F(z,-))\le 2^{-\tau_z}$ as well as a $(\rho_z+\tau_z)$-bit approximation of $F(z,-)$ and then shift the coefficients of the latter approximation of $F(z,-)$ by $\tau_z$ bits. We first estimate the cost for the computation of the $\tau_z$'s. According to Proposition~\ref{univariate-evaluationbis} a), we can compute an approximation $\tilde{c}_z$ of $c_z:=|\operatorname{lc}(F(z,-))|$ with $|c_z-\tilde{c}_z|<2^{-L}$ using $\tilde{O}(n(L+n\log(\max(1,|z|)))+\tau)$ bit operations, and as input we need an $\tilde{O}(L+n\log\max(1,|z|)+\tau)$-bit approximation of $z$.
Hence, when choosing $L=2,4,8,\ldots$, we succeed in computing $\tau_z$ for an $L_z$ of size $$L_z\in O(|\log (|\operatorname{lc}(F(z,-))|)|+\tau+n+n\log(\max(1,|z|))), $$ and the cost for the evaluation is bounded by
$\tilde{O}(nL_z)$
bit operations. When summing up the latter bound over all $z$ and using
Proposition~\ref{generalunivariate} (b) (with $g_i=f_{n_y-i}$, the sequence of coefficients of $F\in\mathbb{Z}[X][Y]$), we obtain
\begin{align*}
\sum_{z \in V_{\mathbb{C}}(R)}\tilde{O}(nL_z)&\in n\cdot \tilde{O}(\sum_{z \in V_{\mathbb{C}}(R)} (|\log( |\operatorname{lc}(F(z,-))|)|+\tau+n+n\log(\max(1,|z|))))\\
&\in \tilde{O}(n^2(N\tau+n\Lambda+Nn))
\end{align*}
Notice that the above computation further implies that $\sum_{z\in V_{\mathbb{C}(R)}}\tau_z\in\tilde{O}(n(N\tau+n\Lambda+Nn))$.
For estimating the cost of computing sufficiently good approximations of $F(z,-)$, we can again use Proposition~\ref{univariate-evaluationbis} a). We conclude that the cost for computing a $(\rho_z+\tau_z)$-bit approximation of $F(z,-)$ is bounded by $\tilde{O}(n^2(\tau+n+n\log(\max(1,|z|))+\rho_z+\tau_z))$ bit operations and that we need a $\tilde{O}(\tau+n+n\log(\max(1,|z|))+\rho_z+\tau_z)$-bit approximation of $z$. Summing up the cost over all $z$ then yields
\begin{align*}
&\tilde{O}(n^2N(\tau+n)+n^3\sum_{z \in V_{\mathbb{C}}(R)} \log(\max(1,|z|))+n^2\sum_{z \in V_{\mathbb{C}}(R)} (\rho_z+\tau_z))
\end{align*}
which is in
\begin{align*}n^3\cdot\tilde{O}(N\tau+n\Lambda+Nn)).
\end{align*}
Finally, we have to bound the cost for computing sufficiently good approximations of the roots $z$ of $R$. Notice that each term $\tau+n+n\log(\max(1,|z|))+\rho_z+\tau_z$ is bounded by $\tilde{O}(n\cdot(N\tau+n\Lambda+Nn))$ for each root $z$ of $R$; in fact, the latter bound even applies to the sum of all these terms. Hence, it suffices to compute approximations of all roots of $R$ to an absolute precision of size $\tilde{O}(n\cdot (N\tau+n\Lambda+Nn))$. According to Proposition~\ref{sagraloff-isolation-integer}, the cost for the computation of well-isolating disks of corresponding size is bounded by $\tilde{O}(N^2\Lambda+N^3+Nn(N\tau+n\Lambda+Nn))$. The bound on the sum of the bitsizes of the radii and the centers of the disks $D_{z,z'}$ follows directly from Proposition~\ref{pro:bounds} and Proposition~\ref{thm:bounds}.
This concludes the proof of Parts a) and b).
For Part (c), notice that $$\frac{\min_{z''\neq z':F(z,z'')=0}|m_{z,z'}-m_{z,z''}|}{\operatorname{sep}(z',F(z,-))}\in (1-1/32,1+1/32),$$ where $m_{z,z'}$ denotes the center of $\mathcal{D}_{z,z'}$. Hence, approximations $\tilde{\sigma}_{z,z'}$ with the required properties can directly be obtained from the distances between the centers $m_{z,z'}$. Since the sum of the bitsizes of all centers $m_{z,z'}$ is bounded by $\tilde{O}(\sum_{z \in V_{\mathbb{C}}(R)} \operatorname{logMea}(F_z)+\operatorname{logsep}^*(F_z))\in\tilde{O}(n(N\tau+n\operatorname{logMea}+Nn))$, the claim follows.
It remains to prove the last claim on the cost for refining the disks $\mathcal{D}_{z,z'}$, with $(z,z')\in V$, to a size less than $2^{-L}$. For this, we use Proposition~\ref{sagraloff-isolation-integer} (c), which shows that, for a fixed $z$, we can refine all disks $\mathcal{D}_{z,z'}$ using
$$\tilde{O}(n(L\cdot \mu_{z}+n^2\cdot\operatorname{logMea}(F_z)+n\widehat{\operatorname{logGDisc}}(F_z)+n^3))$$
bit operations. Now, summing the latter bound over all $z$ yields $$\tilde{O}(n^2(N\tau+n\Lambda+Nn)+nL\cdot\sum_{z \in V_{\mathbb{C}}(R)}\mu_{z}).$$ For the input precision $\rho_z$ to which we need to approximate the polynomial $F_z$, we obtain the bound
$$\rho_z
\in\tilde{O}(L\cdot \mu+n\cdot\operatorname{logMea}(F_z)+\widehat{\operatorname{logGDisc}}(F_z)+n^2).$$
Again, using the same argument as above, it follows that sufficiently good approximations of the polynomials $F_z$ can be computed using
$$
\tilde{O}(n^3\cdot(N\tau+n\Lambda+Nn)+L(N\mu+ n^2\cdot\sum_{z \in V_{\mathbb{C}}(R)} \mu_{z})
$$
bit operations.
\end{proof}
Theorem \ref{firstmain} follows from Proposition~\ref{prop:computinggcddeg}
and Proposition \ref{thm:costisolation}.
\begin{proposition}\label{thm:comparinginfibers}
Let $R\in\mathbb{Z}[X]$ be of magnitude $(N,\Lambda)$, $F,G\in\mathbb{Z}[X,Y]$ be of magnitude $(n,\tau)$, and $H:=F\cdot G$. Suppose moreover that
$$V_{\mathbb{C}}(R,H)=\{(z,z')\in \mathbb{C}^2\mid R(z)=H(z,z')=0\}$$
is finite.
Using a number of bit operations bounded by
\[
\tilde{O}(N^2\Lambda+N^3+n^5\tau+n^6+n\cdot\max(n^2,N)\cdot (N\tau+n\Lambda+Nn))
\]
we can can carry out the following computations for all complex roots $z$ of $R$:
\begin{itemize}
\item[(a)] computing well-isolating disks $\mathcal{D}_{z,z'}$ for all complex roots $z'$ of the polynomial $H(z,-)$ together with the corresponding multiplicities $\mu(z,z')=\operatorname{mult}(z',H(z,-))$.
\item[(b)] determining for each root $z'$ of $H(z,-)$ whether $z'$ is a root of $F(z,-)$, $G(z,-)$, or both. If $z$ as well as $z'$ are real, we can further determine the sign of $F(z,z')$ and $G(z,z')$.
\end{itemize}
\end{proposition}
\begin{proof}
Part (a) already follows from Lemma~\ref{pro:degrees} and Proposition~\ref{thm:costisolation} as $H$ has magnitude $(O(n),O(\log (n)+\tau))$. We
may further assume that, for all complex roots $z$ of $R$, we have already computed
\begin{itemize}
\item well-isolating disks $\mathcal{D}^F_{z,z''}$
and $\mathcal{D}^G_{z,z'''}$ for all complex roots $z''$ and $z'''$ of the polynomials $F(z,-)$ and $G(z,-)$, respectively,
\item the corresponding multiplicities $\operatorname{mult}(z'',F(z,-))$ and $\operatorname{mult}(z''',G(z,-))$,
\item the degrees of the polynomials $F(z,-)$ and $G(z,-)$, and
\item the signs of the leading coefficients of the polynomials $F(z,-)$ and $G(z,-)$ in case that $z$ is a real root of $R$.
\end{itemize}
For (b), we now refine each disk $\mathcal{D}^F_{z,z''}$ such that it intersects with exactly one of the disks $\mathcal{D}_{z,z'}$. If this is the case, then it holds that $z'=z''$. In addition, $z'$ is also a root of $G(z,-)$ if and only if $\mu(z,z')>\operatorname{mult}(z'',F(z,-))$ as $\operatorname{mult}(z',F(z,-))+\operatorname{mult}(z',G(z,-))=\mu(z,z')$. Hence, for each root $z'$ of $H(z,-)$, we also know its multiplicity as a root of $F(z,-)$ and $G(z,-)$. When restricting to the real roots of $R$, this implies that we can directly deduce the sign of $F(z,z')$ (resp. $G(z,z')$) at each real root $z'$ of $H(z,-)$ that is not a root of $F(z,-)$ (resp. $G(z,-)$). Namely, from the sign of the leading coefficient of $F(z,-)$ (resp. $G(z,-)$) and its degree, we know its sign at $\pm\infty$, and the polynomial changes signs exactly at those roots of $H$ that are roots of $F(z,-)$ (resp. $G(z,-)$) of odd multiplicity.
It remains to bound the cost for the refinement of the disks $\mathcal{D}^F_{z,z''}$. We proceed in rounds enumerated by $\ell=1,2,3,\ldots$: Initially, we set $V^1:=V_{\mathbb{C}}(R,F)$. In the $\ell$-th round, we refine each of the isolating disks $\mathcal{D}^F_{z,z''}$ for all $(z,z'')\in V^\ell$ to a size less than $2^{-2^{\ell}}$ and check whether it intersects exactly one of the isolating disks $\mathcal{D}_{z,z'}$ for the roots of $H(z,-)$. If this is the case, we know that $z'=z''$.
After having treated all points in $V^{\ell}$, we set $V^{\ell+1}$ to be the set of all $(z,z'')$ in $V^\ell_z$ for which the isolating disk $\mathcal{D}^F_{z,z''}$ intersects more than one of isolating disks for $H(z,-)$. That is, $V^{\ell}$ is the set of all $(z,z'')\in V^1$ for which we have not determined a corresponding root $z'$ of $H(z,-)$ with $z'=z''$ after the $\ell$-th round. We then proceed with the $(\ell+1)$-st round. We stop as soon as $V^{\ell}$ becomes empty, in which case, each root of each $F(z,-)=0$ is matched to a corresponding root of $H(z,-)$.
Notice that, for each $(z,z'')$, we must succeed in round $\ell_{z,z''}$, for some $\ell_{z,z''}$ with
$$2^{\ell_{z,z''}}\le O(|\log(\operatorname{sep}(z'',H(z,-)))|).$$ From the amortized bounds on the separation of the roots (Proposition~\ref{thm:bounds}), we have that $|\log(\operatorname{sep}(z'',H(z,-)))|\in \tilde{O}(n(N\tau+n\Lambda+Nn))$, thus
we are done after $\ell_{\max}$ rounds for
\begin{equation}\label{kappa0}
\ell_{\max}=\max_{z,z''} \ell_{z,z"}+1\in O(\log (n(N\tau+n\Lambda+Nn))).
\end{equation}
According to Proposition~\ref{thm:costisolation}, the cost for refining the disks $\mathcal{D}^F_{z,z''}$ for all $(z,z'')\in V^\ell$ to a size less than $2^{-2^{\ell}}$ is bounded by
\begin{align*}
\sum_{\ell=1}^{\ell_{\max}} \tilde{O}(N^2\Lambda+N^3+n\max(n^2,N)(N\tau+n\Lambda+Nn)+2^{\ell}(N\mu^{[\ell]}+n^2\sum_{z\in\mathbb{C}:R(z)=0}\mu_z^{[\ell]}))
\end{align*}
bit operations,
where
\[
\mu_{z}^{[\ell]}:=
\begin{cases}
\max_{(z,z'')\in V^{\ell}} \operatorname{mult}(z'',F(z,-))&\text{if there exists }(z,z'')\in V^{\ell}\\
0 &\text{otherwise.}
\end{cases}
\]
and $\mu^{[\ell]}:=\max_{z:R(z)=0}\mu^{[\ell]}_z.$
Since $\ell_{\max}$ is bounded by $O(\log (n(N\tau+n\Lambda+Nn)))$, we are left to bound the sum
\begin{align}\label{sumalongfiber}
\sum_{\ell=1}^{\ell_{\max}} 2^{\ell}\cdot (N\cdot\mu^{[\ell]}+ n^2\cdot \sum_{z\in\mathbb{C}:R(z)=0}\mu_z^{[\ell]}).
\end{align}
If, for a fixed root $z$ of $R$, there exists a $(z,z'')\in V^{\ell}$, then let $(z,z''_{\ell,z})\in V^\ell$ be a point in $V^\ell$ with $\operatorname{mult}(z''_{\ell},F(z_\ell,-))=\mu^{[\ell]}$. In other words, $z''_{\ell,z}$ maximizes the multiplicity within the fiber for all roots $z''$ of $F(z,-)$ with $(z,z'')\in V^{\ell}$. In addition, let $(z_\ell, z''_\ell)$ be a point in $V^\ell$ that maximizes the multiplicity over all fibers. Thus, the sum in (\ref{sumalongfiber}) can be rewritten as
\[
N\cdot\sum_{\ell=1}^{\ell_{\max}} \mu(z''_\ell,F(z_\ell,-))\cdot 2^{\ell}+n^2\cdot\sum_{\ell=1}^{\ell_{\max}} \sum_{z\in\mathbb{C}:R(z)=0}\mu(z''_{\ell,z},F(z,-))\cdot 2^\ell.
\]
Notice that a pair $(z,z'')\in V^{\ell_z}$ can appear at most $\ell_{\max}$ many times in each of the above sums. In addition, it holds that $(z,z'')\notin V^{\ell}$ if $\ell>\ell_{z,z''}$ for some $2^{\ell_{z,z''}}\in O(|\log(\operatorname{sep}(z'',H(z,-)))|)$. Hence, the above sum is upper bounded by
\begin{align}\label{sumalongfiber2}
O(\ell_{\max}\cdot (n^2+N)\cdot \sum_{(z,z'')\in V^1}\mu(z'',F(z,-))\cdot |\log(\operatorname{sep}(z'',H(z,-)))|)
\end{align}
Since $\mu(z'',F(z,-))\le \mu(z'',H(z,-))$ and $\ell_{\max}\in O(\log (n(N\tau+n\Lambda+Nn)))$, we thus conclude from Proposition~\ref{thm:bounds} that (\ref{sumalongfiber2}) is upper bounded by
\[
\tilde{O}(n\cdot(n^2+N)\cdot (N\tau+n\Lambda+Nn)).
\]
Finally, the cost of checking whether an isolating disk $\mathcal{D}^F_{z,z''}$ intersects exactly one of the isolating disks $\mathcal{D}_{z,z'}$ for the roots of $H(z,-)$ is bounded by $O(n\cdot(\operatorname{logsep}^*(H(z,-))+\operatorname{logMea}(H(z,-)))$ as there are $n$ comparisons between disks with radii and centers of bitsize $O(\operatorname{logsep}^*(H(z,-))+\operatorname{logMea}(H(z,-))$. The total cost for all comparisons is thus bounded by
\[
O(\kappa\cdot n^2\cdot \sum_{z:R(z)=0} [\operatorname{logsep}^*(H(z,-))+\operatorname{logMea}(H(z,-))]=\tilde{O}(n^3(N\tau+n\Gamma+Nn)),
\]
where the latter inequality follows from Proposition~\ref{pro:bounds} and Proposition~\ref{thm:bounds}.
Hence, our claim follows.
\end{proof}
We remark that the above considerations immediately yield an algorithm for solving a bivariate polynomial system that achieves the current record complexity bound for this problem and differs from previous approaches with comparable complexity:
\begin{corollary}\label{cor:bivariatesystems}
Let $F,G\in\mathbb{Z}[X,Y]$ be coprime polynomials of magnitude $(n,\tau)$. Then, we can compute isolating regions for all complex solutions of the system $F=G=0$ using $\tilde{O}(n^5\tau+n^6)$ bit operations.
\end{corollary}
\begin{proof}
Let $R=\operatorname{Res}_X(F,G)$ be the resultant polynomial of $F$ and $G$, which can be computed using $\tilde{O}(n^4\tau+n^5)$ bit operations. Any common solution $(x_0,y_0)\in\mathbb{C}$ of $F=G=0$ yields a root $x_0$ of $R$, and then $y_0$ is a common root of $F(x_0,-)$ and $G(x_0,-)$. Vice versa, any common root $y_0$ of $F(x_0,-)$ and $G(x_0,-)$ yields a solution of $F=G=0$. According to Corollary~\ref{thm:comparinginfibers}, we can compute all common roots of $F(z,-)$ and $G(z,-)$ for all complex roots $z$ of $R$ using $\tilde{O}(n^5\tau+n^6)$ bit operations. Hence, the claim follows.
\end{proof}
\begin{remark}\label{thankstotest}
In the proof of Proposition~\ref{thm:comparinginfibers}, we needed to recursively refine the isolating disks for the roots of $F(z,-)$ until a certain test applies. That is, we needed to check whether a disk intersects exactly one of the isolating disks of $H(z,-)$. While this test itself is rather simple (and cheap), its success is directly related to a hidden parameter, which is in this case the separation of some specific (but unknown) root of the polynomial $H(z,-)$.
In the worst case, the cost for refining an isolating disks for the polynomial $F(z,-)$ until the test applies is rather large (i.e. it is comparable to our bound for the overall computation), and thus it would be very costly to refine all isolating disks to such a small size. Fortunately, this is not necessary as, for most roots of $F(z,-)$, the success of the test is related to a root of $H(z,-)$ with larger separation. In order to exploit this fact, we need to design adaptive algorithms in each of these cases and to use our amortized bounds on the separation of the roots (Proposition \ref{corodisc} and Proposition~\ref{thm:bounds}).
Three instances of such a situation appear in this paper, the first in Proposition \ref{thm:comparinginfibers} and the two others in Proposition
\ref{indexofcritical} and Proposition \ref{intermedfibers_smallenough}.
\end{remark}
\section{Computation of the topology}\label{sec:top}
\subsection{Definitions and notations}\label{sec:basic}
Let $P \in \mathbb{Z} [X, Y]$ be a square-free polynomial and magnitude $(d,\tau)$. In addition, let
$$V_{\mathbb{R}} (P) = \{(x, y) \in \mathbb{R}^2 \mid P (x, y) = 0\}$$ be the real
algebraic curve defined by $P$, and
$$V_{\mathbb{C}}(P) = \{(x, y) \in \mathbb{C}^2 \mid P (x, y) = 0\}$$
be the corresponding complex algebraic curve.
We first decompose
\begin{equation}\label{tildeP}
P(X,Y)=c(X)\cdot\tilde P(X,Y)
\end{equation}
with $c(X)\in \mathbb{Z} [X]$ and $\tilde P(X,Y)\in \mathbb{Z} [X, Y]$ such that $\tilde P(z,Y)$
never identically vanishes for any $z \in \mathbb{C}.$ In more geometric terms, we separate the vertical lines contained in $V_{\mathbb{C}} (P)$ from the remaining part $V_{\mathbb{C}} (\tilde{P})$ of the curve, which does not contain any vertical lines.
\begin{proposition}\label{removeverticallines}
We can compute $c(X)$ and $\tilde{P}(X,Y)$ using $\tilde{O}(n^4+n^3\tau)$ bit operations. The polynomials $\tilde{P}(X,Y)$ and $c(X)$ have magnitude bounded by $(n,\tau+n+\log (n+1))$.
\end{proposition}
\begin{proof}
Let $P(X,Y):=c_{d_y}(X)Y^{d_y}+\ldots+c_0(X),$ with $d_y=\deg_Y(P)\le d$. Let $c(X)$ the gcd of all the coefficients $c_i(X)$. For getting rid of vertical lines of the zero set of $P(X,Y)$, we first compute $c(X)$ then write $P(X,Y)= c(X)\tilde P(X,Y)$.
The claimed bounds on the cost for computing $c(X)$ and $\tilde{P}(X,Y)$ as well as on the magnitude of these polynomials follow immediately from Propositions~\ref{Mignotte} and~\ref{gcd-comp}.
\end{proof}
We now write
\begin{equation}\label{barP}
\tilde P(X,Y)=d(Y)\cdot Q(X,Y)
\end{equation}
with $d(Y)\in \mathbb{Z} [Y]$ and $Q(X,Y)\in \mathbb{Z} [X, Y]$ such that $Q(X,z')$
never identically vanishes for any $z' \in \mathbb{C}.$ In more geometric terms, we separate the horizontal lines contained in $V_{\mathbb{C}} (\tilde P)$ from the remaining part $V_{\mathbb{C}} (\bar{P})$ of the curve, which does not contain any horizontal lines.
\begin{proposition}\label{removehorizontalllines}
We can compute $d(Y)$ and $\bar{P}(X,Y)$ using $\tilde{O}(n^4+n^3\tau)$ bit operations. The polynomials $\bar{P}(X,Y)$ and $d(X)$ have magnitude bounded by $(n,\tau+n+\log (n+1))$.
\end{proposition}
\begin{proof}
Let $\tilde P(X,Y):=d_{d_x}(Y)X^{d_x}+\ldots+d_0(X),$ with $d_x=\deg_X(\tilde P)\le d$. Let $d(X)$ the gcd of all the coefficients $d_i(X)$. We write $\tilde P(X,Y)=
d(Y)\bar{P(X,Y)} $.
The claimed bounds on the cost for computing $d(Y)$ and $\bar{P}(X,Y)$ as well as on the magnitude of these polynomials follow immediately from Propositions~\ref{Mignotte} and~\ref{gcd-comp}.
\end{proof}
From now, we study the zero set of $\tilde P(X,Y)$, which contains no vertical lines.
We suppose moreover that $\deg_X(\tilde P(X,Y))>0$ because otherwise $V_{\mathbb{R}} (\tilde P)$ is a finite number of horizontal lines and its topology, very easy to describe, is treated as a special case in sub-section \ref{topology}.
We also describe in sub-section \ref{topology} how to add back vertical lines contained in $V_{\mathbb{R}} (P)$ to obtain the topology of $V_{\mathbb{R}} (P)$ from the topology of $V_{\mathbb{R}} (\tilde P)$.
We first introduce the following definitions
\begin{eqnarray}
D_X(X)& := &\operatorname{Disc}_Y (\tilde P) (X), \label{eqdisc}\\
D_Y(Y)& := &\operatorname{Disc}_X (\tilde P) (Y),
\label{barD}\\
S_X(X) &=& D_X(X) \cdot \operatorname{Res}_Y (Q, \partial_X Q) (X), \label{eqYcrit}\\
S_Y(Y) &:=& D_Y(Y) \cdot \operatorname{Res}_X (\tilde P, \partial_Y \tilde P) (Y), \label{eqYcritbis},\\
T_X(X)&:=& S_X\cdot (S_X^{\star})', \label{def:RX}\\
T_Y(Y)&:=& S_Y\cdot (S_Y^\star)'.\label{def:RY}
\end{eqnarray}
where $(S_X^\star)'$ and $(S_X^\star)'$ are the derivatives of the square-free parts of $S_X$ and $S_Y$, respectively.
\begin{lemma}\label{magni}
All the polynomials $D_X,D_Y,S_X,S_Y,T_X,T_Y$ are of magnitude \\$(O(d^2), O(d\tau+d^2))$ and can be all computed with a bit complexity $\tilde{O}(d^4 \tau+d^5)$.
\end{lemma}
\begin{proof}.
Use Proposition \ref{gcd-comp} and Proposition \ref{subres-comp}.
\end{proof}
The special case where $D_X(X)$ has no real root is considered separately in sub-section \ref{topology}.
We further denote by
\begin{equation}
\alpha_1<\ldots<\alpha_{N}
\end{equation}
the real roots of $D_X(X)$. A point $(\alpha, \beta) \in V_{\mathbb{R}} (\tilde P)$
is called:
\begin{itemize}
\item an \emph{$X$-critical} point if $\partial_Y \tilde P(\alpha, \beta) = 0$,
\item a \emph{$Y$-critical} point if $\partial_X \tilde P (\alpha, \beta) = 0$,
\item a \emph{singular} point if $\partial_X \tilde P (\alpha, \beta) = \partial_Y \tilde P(\alpha, \beta) = 0$,
\item a \emph{regular} point if $\partial_Y \tilde P (\alpha, \beta)\neq 0$ and
$\partial_X \tilde P (\alpha, \beta) \neq 0$.
\end{itemize}
We denote by $\operatorname{Crit} (V_{\mathbb{R}} (\tilde P))$
the set
of $X$-critical points of $V_{\mathbb{R}} (\tilde P)$.
Notice that a singular point of $V_{\mathbb{R}} (\tilde P)$ is also an $X$-critical and a $Y$-critical point.
Further notice that the $X$-coordinate (resp $Y$-coordinate) of an $X$-critical point is a zero of $D_X$ (resp. $S_Y$),
and the $Y$-coordinate (resp. $X$-coordinate) of a $Y$-critical point is a zero of $D_Y$ (resp. $S_X$).
We also denote by $\xi_1,\ldots,\xi_{N'}$, with
\begin{equation}
\xi_1<\ldots<\xi_{N'},
\end{equation}
the real roots of $(S_X^\star)'(X)$, and
$\xi_0=-(C(T_X))$, $\xi_{N'+1}=C(T_X))$ using Notation \ref{cauchybounds}, such that $\xi_0$ (resp.
$\xi_{N'+1}$ are smaller (resp. bigger) than all real roots of $S_X$ and $(S_X^\star)'$. Remark that $\xi_0$ and $\xi_{N'+1}$ are rational numbers of bitsize $\tilde{O} (d\tau+d^2)$.
For every $i=1,\ldots,N$, we denote
$\alpha^-_i$ and $\alpha^+_i$ the elements of $$\{\xi_0,\xi_1,\ldots,\xi_{N'},\xi_{N'+1}\}$$
such that $\alpha_i\in (\alpha^-_i,\alpha^+_i)$.
Notice that in each interval $(\alpha^-_i,\alpha^+_i)$, $\alpha_i$ is the only root of $S_X$ (hence of $D_X$).
In the $Y$-direction, we denote by $\gamma_1,\ldots,\gamma_M$, with
$$\gamma_1<\ldots<\gamma_M,$$ the real roots of $S_Y(Y)$.
We also denote by
\begin{equation}
\eta_1<\ldots<\eta_{M'},
\end{equation}
the real roots of $(S_Y^\star)'(Y)$, and
$\eta_0=-(C(T_Y))$, $\eta_{M'+1}=C(T_Y))$ using Notation \ref{cauchybounds}, such that $\eta_0$ (resp.
$\eta_{M'+1}$ are smaller (resp. bigger) than all real roots of $S_Y$ and $(S_Y^\star)'$. Remark that $\eta_0$ and $\eta_{M'+1}$ are rational numbers of bitsize ${O} (d\tau+d^2)$.
For every $k=1,\ldots,M$,
we denote by
$\gamma^-_k$ and $\gamma^+_k$ the elements of $$\{\eta_0,\eta_1,\ldots,\eta_{M'},\eta_{M'+1}\}$$
such that $\gamma_k\in (\gamma^-_k,\gamma^+_k)$.
For every $i=1,\ldots,N$, we denote by $\beta_{i,1},\ldots,\beta_{i,m(i)}$, with
\begin{equation}
\beta_{i,1}<\ldots<\beta_{i,m(i)},
\end{equation}
the real roots of $\tilde P(\alpha_i,Y)$.
For every $X$-critical point $(\alpha_i,\beta_{i,j})$, notice that $\beta_{i,j}$ is a root of $S_Y$. We write $k(i,j)$ for the index such that
$\gamma_{k(i,j)}=\beta_{i,j}$.
\subsection{Topology inside an adjacency box}\label{connectingalgo}
The adjacency box associated to a singular point $(\alpha,\beta)=(\alpha_i,\beta_{i,j})$ is defined as
$[\alpha^-,\alpha^+]\times [\gamma^-,\gamma^+]$
where
$\alpha^-=\alpha^-_i,\alpha^+=\alpha^+_i$
and
$\gamma^-=\gamma^-_{k(i,j)},\gamma^+=\gamma^+_{k(i,j)}$, using the Notation introduced in
Subsection \ref{sec:basic}.
The aim of this subsection is to explain how counting the intersection points of $V_{\mathbb{R}} (P)$ with some specific parts of the boundary of the adjacency box makes it possible to compute the number $\textsc{Left}$ of the segments ending at
$(\alpha,\beta)$ to the left of $\alpha$ as well as the number
$\textsc{Right}$ of the segments ending at
$(\alpha,\beta)$ to the right of $\alpha$.
We introduce some definitions.
\begin{notation}
Denote by
\begin{itemize}
\item $L_{\alpha^-}=V_{\mathbb{R}} (\tilde P)\cap \{\alpha^-\} \times (\gamma^-,
\gamma^+)$
(resp. $L_{\alpha^+}=V_{\mathbb{R}} (\tilde P)\cap \{\alpha^+\} \times (\gamma^-, \gamma^+)$), ordered by increasing value of $y$\medskip
\item $L_{\alpha^-}^{<\beta}= L_{\alpha^-}\cap \{\alpha^-\} \times (\gamma^-,
\beta)$ (resp. $L_{\alpha^-}^{>\beta}= L_{\alpha^-}\cap \{\alpha^-\} \times (\beta,\gamma^+)$, ordered by increasing value of $y$\medskip
\item $L_{\alpha^+}^{<\beta}= L_{\alpha^+}\cap \{\alpha^+\} \times (\gamma^-,
\beta)$ (resp. $L_{\alpha^+}^{>\beta}= L_{\alpha^+}\cap \{\alpha^+\} \times (\beta,\gamma^+)$, ordered by increasing value of $y$\medskip
\item $L_{\alpha}^{<\beta}=V_{\mathbb{R}} (\tilde P)\cap \{\alpha\} \times (\gamma^-,
\beta)$ (resp. $L_{\alpha}^{>\beta}=V_{\mathbb{R}} (\tilde P))=\cap \{\alpha\} \times (\gamma^-,
\beta)$), ordered by increasing value of $y$,
\end{itemize}
and
\begin{itemize}
\item $L_{\gamma^-}^{ < \alpha}=V_{\mathbb{R}} (\tilde P)\cap (\alpha^-,\alpha) \times \{\gamma^-\}$, $L_{\gamma^-}^{ >\alpha}=V_{\mathbb{R}} (\tilde P)\cap (\alpha,\alpha^+) \times \{\gamma^-\}$,
(resp. $L_{\gamma^+}^{ < \alpha}=V_{\mathbb{R}} (\tilde P)\cap (\alpha^-,\alpha) \times \{\gamma^+\}$, $L_{\gamma^+}^{ >\alpha}=V_{\mathbb{R}} (\tilde P)\cap (\alpha,\alpha^+) \times \{\gamma^+\}$), ordered by increasing value of $x$.\medskip
\item $L_{\gamma^-}^{= \alpha^-}=V_{\mathbb{R}} (\tilde P)\cap \{(\alpha^-,\gamma^-)\}$ (resp. $L_{\gamma^-}^{= \alpha}=V_{\mathbb{R}} (\tilde P)\cap \{(\alpha,\gamma^-)\}$, $L_{\gamma^-}^{= \alpha^+}=V_{\mathbb{R}} (\tilde P)\cap \{(\alpha,\gamma^-)\}$),\medskip
\item $L_{\gamma^+}^{= \alpha^-}=V_{\mathbb{R}} (\tilde P)\cap \{(\alpha^-,\gamma^+)\}$ (resp. $L_{\gamma^+}^{= \alpha}=V_{\mathbb{R}} (\tilde P)\cap \{(\alpha,\gamma^+)\}$, $L_{\gamma^+}^{= \alpha^+}=V_{\mathbb{R}} (\tilde P)\cap \{(\alpha,\gamma^+)\}$).
\end{itemize}
\end{notation}
\begin{figure}
\centering
\includegraphics[width=9cm,height=7cm]{Dessins/essaidessin0.png}
\caption{\label{description_lists}Illustration of the notations}
\end{figure}
The boundary points of $[\alpha^-,\alpha) \times [\gamma^-,
\gamma^+]$ (resp. $(\alpha,\alpha^+] \times [\gamma^-,
\gamma^+]$)
are the elements of
$[\alpha^-,\alpha)\times \{\gamma^-\}
\cup
\{\alpha^-\} \times [\gamma^-,
\gamma^+]
\cup
[\alpha^-,\alpha)\times \{\gamma^+\}
$
(resp.
$(\alpha,\alpha^+]\times \{\gamma^-\}
\cup
\{\alpha^+\} \times [\gamma^-,
\gamma^+]
\cup
(\alpha,\alpha^+]\times \{\gamma^+\}
$).
Since $[\alpha^-,\alpha)$ (resp. $(\alpha,\alpha^+]$) contains no root of $S_X$, there is no $X$-critical point of of $ V_{\mathbb{R}} (\tilde P)$ or $Y$-critical point of $ V_{\mathbb{R}} (Q)$ inside
$[\alpha^-,\alpha)\times \mathbb{R}$ (resp. $(\alpha^-,\alpha]\times \mathbb{R}$).
Similarly since $[\gamma^-,\gamma)$ (resp. $(\gamma,\gamma^+]$) contains no root of $S_Y$, there is no $X$-critical point or $Y$-critical point of $ V_{\mathbb{R}} (\tilde P)$ inside
$\mathbb{R}\times [\gamma^-,\beta) $ (resp. $(\beta,\gamma^+]\times \mathbb{R}$).
We denote by $\mathrm{Slope}(\tilde P)(X,Y)$ the rational fraction
$$\mathrm{Slope}(\tilde P)(X,Y)=-\frac{\partial_X \tilde P(X,Y)}{\partial_Y \tilde P(X,Y)}.$$
The slope of the
tangent to a level line of $\tilde P$ at a regular point $(x,y)$ is
given by
$$\mathrm{Slope}(\tilde P)(x,y)=-\frac{\partial_X \tilde P(x,y)}{\partial_Y \tilde P(x,y)}.$$
Note that any point $x,y)$ of $[\alpha^-,\alpha^+]\times [\gamma^-,\gamma^+]\setminus \{(\alpha,\beta)\}$ intersected with $V_{\mathbb{R}} (\tilde P)$, $\mathrm{Slope}(\tilde P)(x,y)$ is well defined and not $0$, except if $Y=\beta$ is a line contained in $V_{\mathbb{R}} (\tilde P)$ (i.e. $d(\beta)=0$).
Given a boundary point $(x, y)$ of $[\alpha^-,\alpha) \times [\gamma^-,
\gamma^+]$ (resp. $(\alpha,\alpha^+] \times [\gamma^-,
\gamma^+]$) in $V_{\mathbb{R}} (\tilde P)$,
there is one and
only one analytic arc of $V_{\mathbb{R}} (\tilde P)\cap [\alpha^-,\alpha)\times [\gamma^-,\gamma^+]$
(resp. $V_{\mathbb{R}} (\tilde P)\cap (\alpha,\alpha^+]\times [\gamma^-,\gamma^+]$),
with exactly one of the
following properties
- type 1: the arc ends at another boundary point $\operatorname{Match}((x,y))$ of $[\alpha^-,\alpha) \times [\gamma^-,
\gamma^+]$ (resp. $(\alpha,\alpha^+] \times [\gamma^-,
\gamma^+]$), called the matching point of $(x, y)$;
- type 2: the arc ends at a regular point $\operatorname{Match}((x,y))$ of $\{\alpha\}\times[\gamma^-,\gamma^+]\setminus\{\alpha,\beta\}$, called the matching point of $(x, y)$;
- type 3: the arc ends at $\left(\alpha, \beta \right)$, called the matching point of $(x,y)$.
The matching point $\operatorname{Match}((x,y))$ has the same slope sign as $(x,y)$ except for type 3.
Moreover if $(x,y)\not= (x',y')$, $\operatorname{Match}((x,y))\not=\operatorname{Match}((x',y'))$ except if
$\operatorname{Match}((x,y)=\operatorname{Match}((x',y'))=(\alpha,\beta)$.
Note that an arc of type 1 or type 2 does not meet any other arc, and that an arc of type 3
meets only other arcs of type 3, at
$(\alpha, \beta)$.
Given a list $L = [x_1, \ldots, x_n]$, we denote by
\[ L [i] = x_i, \bar L
:= [x_n, \ldots, x_1] . \]
Given two lists $L = [x_1, \ldots, x_n]$ and $M = [y_1, \ldots, y_m]$ we
denote their concatenation by
$ L + M : = [x_1, \ldots, x_n, y_1, \ldots, y_m] $.
\begin{proposition}
\label{forcorrectness}
It holds
\begin{itemize}
\item [a)] If $L_{\alpha^-}^{>\beta}+L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}\not=[]$, all its points have the same slope sign, which we denote by $\sigma^+$. In addition,\medskip
\begin{itemize}
\item[a.1)] if $\sigma^+>0$,
$\operatorname{Match}(L^+[i])=L'^+[i]$ for every $i=1,\ldots \#L^+$, with
$L^+=L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}$ and $L'^+=\bar L_{\alpha^-}+L_{\gamma^-}^{=\alpha^-}+L_{\gamma^-}^{<\alpha}$, ;
\item[a.2)] if $\sigma^+<0$, $\operatorname{Match}(L^+[i])=L'^+[i]$ for every $i=1,\ldots \#L^+$
with $L^+=\bar L_{\alpha}^{>\beta}$ and
$L'^+=\bar L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha^-}+\bar L_{\alpha^-}^{>\beta}$.\medskip
\end{itemize}
\item [b)] If $\bar L_{=\alpha^-}^{<\beta}+L_{\gamma^-}^{=\alpha^-}+L_{\gamma^-}^{<\alpha}+L_{\gamma^-}^{=\alpha}+L_{\alpha}^{<\beta}\not=[]$, all its points have the same slope sign, which we denote by $\sigma^-$. In addition,\medskip
\begin{itemize}
\item[b.1)] if $\sigma^->0$, $\operatorname{Match}(L^-[i])=L'^-[i]$
for every $i=1,\ldots \#L^-$
with $L^-=L_{\alpha}^{<\beta}$ and
$L'^-=\bar L_{\gamma^-}^{<\alpha}+L_{\gamma^-}^{=\alpha^-}+L_{\alpha^-}^{<\beta}$;
\item[b.2)] if $\sigma^-<0$, $\operatorname{Match}(L^-[i])=L'^-[i]$ for every $i=1,\ldots \#L^-$ with $L^-=L_{\gamma^-}^{<\alpha}+L_{\gamma^-}^{=\alpha}+ L_{\alpha}^{<\beta}$ and
$L'^-= L_{\alpha^-}+L_{\gamma^-}^{=\alpha^-}+L_{\gamma^-}^{<\alpha}$.
\end{itemize}\medskip
\item [c)] If $L_{\alpha^+}^{>\beta}+L_{\gamma^+}^{=\alpha^+}+\bar L_{\gamma^+}^{>\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}\not=[]$, all its points have the same slope sign, which we denote by $\tau^+$. In addition,\medskip
\begin{itemize}
\item[c.1)] if $\tau^+>0$, $\operatorname{Match}(M^+[i])=M'^+[i]$
for every $i=1,\ldots \#M^+$
with $M^+=\bar L_{\alpha}^{>\beta}$ and
$M'^+=L_{\gamma^+}^{>\alpha}+L_{\gamma^+}^{=\alpha^+}+\bar L_{\alpha^+}^{>\beta}$;
\item[c.2)] if $\tau^+<0$, $\operatorname{Match}(M^+[i])=M'^+[i]$ for every $i=1,\ldots \#M^+$
with $M^+=\bar L_{\gamma^+}^{>\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}$ and
$M'^+=\bar L_{\alpha^+}+L_{\gamma^-}^{=\alpha^+}+\bar L_{\gamma^-}^{>\alpha}$.
\end{itemize}\medskip
\item [d)] If $\bar L_{\alpha^+}^{<\beta}+L_{\gamma^-}^{=\alpha^+}+\bar L_{\gamma^-}^{>\alpha}+L_{\gamma^-}^{=\alpha}+ L_{\alpha}^{<\beta}\not=[]$, all its points have the same slope sign, which we denote by $\tau^-$. In addition,\medskip
\begin{itemize}
\item[d.1)] if $\tau^->0$, $\operatorname{Match}(M^-[i])=M'^-[i]$ for every $i=1,\ldots \#M^-$ with $M^-=\bar L_{\gamma^-}^{>\alpha}+L_{\gamma^-}^{=\alpha}+L_{\alpha}^{<\beta}$ and
$M'^-=L_{\alpha^+}+L_{\gamma^-}^{=\alpha^+}+\bar L_{\gamma^-}^{>\alpha}$.
\item[d.2)] if $\tau^-<0$, $\operatorname{Match}(M^-[i])=M'^-[i]$ for every $i=1,\ldots \#M^-$
with
$M^-= L_{\alpha}^{<\beta}$ and
$M'^-=L_{\gamma^-}^{>\alpha}+L_{\gamma^-}^{=\alpha^+}+L_{\alpha^+}^{<\beta}$.
\end{itemize}
\end{itemize}
\end{proposition}
\begin{proof}
We prove only a), the proofs for b), c) and d) being similar.
Suppose that $L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}$ has at least two elements, and that $\left( x_1, \gamma^+ \right)$ and $\left( x_2, \gamma^+ \right)$ $x_1<x_2$ are two
points of $L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}$ with different slope signs and denote by $C_1$ and $C_2$ the connected components of $V_{\mathbb{R}}(\tilde P)$ inside $[\alpha^-,\alpha)\times \mathbb{R}$ such that $(x_1,\gamma^+)\in \bar C_1$ and
$(x_2,\gamma^+)\in \bar C_2$.
Since $[\alpha^-,\alpha)$ contains no root of $S_X$, $C_1$ and $C_2$ are the graphs of two monotonous semi-algebraic continuous functions $\varphi_1$ and $\varphi_2$ defined on $[\alpha^-,\alpha)$. At $\alpha$, $\varphi_1$ and $\varphi_2$ have limits of opposite signs, so that the sign of the limit of $\varphi_1-\varphi_2$ is well defined at $\alpha$.
The signs of $\varphi_1(x_1)-\varphi_2(x_1)$ at $x_1$ and the sign of the limit of $\varphi_1-\varphi_2$ at $x_2$ are opposite, so $\varphi_1(x)=\varphi_2(x)$ for a value $x\in (x_1,\alpha)$, which is impossible because such a point would be a singular point of $V_{\mathbb{R}}(\tilde P)$ inside $[\alpha^-,\alpha)\times \mathbb{R}$.
Suppose that $L_{\alpha}^{>\beta}$ (resp. $L_{\alpha^-}^{>\beta}$) has at least two elements, and that $\left(\alpha, y_1\right)$ and $\left(\alpha,y_2\right)$
(resp. $\left(\alpha^-, y_1\right)$ and $\left(\alpha^-,y_2\right)$ ), $y_1<y_2$, are two
points of $L_{\alpha}^{>\beta}$ (resp. $L_{\alpha}^{>\beta}$) with different slope signs and denote by $C'_1$ and $C'_2$ the connected components of $V_{\mathbb{R}}(\tilde P)$ inside $\mathbb{R}\times (\beta,\gamma^+)$ containing them.
Since $(\beta,\gamma^+]$ contains no root of $S_Y$, $C'_1$ and $C'_2$ are the graphs of two monotonous semi-algebraic continuous functions $\psi_1$ and $\psi_2$ defined on
$(\beta,\gamma^+]$.
The signs of $\psi_1-\psi_2$ at $y_1$ and $y_2$ are opposite, so $\psi_1(y)=\psi_2(y)$ for a value $y\in (y_1,y_2)$, which is impossible because such a point would be a singular point of $V_{\mathbb{R}}(\tilde P)$ inside $\mathbb{R}\times (\beta,\gamma^+]$.
Suppose finally that $L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}\not=[]$ and $L_{\alpha}^{>\beta}\not=[]$ (resp. $L_{\alpha^-}^{>\beta}\not=[]$), and let $(\alpha,y)$ be the first element of $\bar L_{\alpha}^{>\beta}$ (resp. $L_{\alpha^-}^{>\beta}\not=[]$).
(i) Suppose that the slope sign of $(\alpha,y)$ is negative (resp. positive) and the slope sign of the elements of $L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}$ is positive. Let $(x,\gamma^+)$ be the last (resp. first) element of $L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}$. Denote by $C'_1$ and $C'_2$ the connected components of $V_{\mathbb{R}}(\tilde P)$ inside $\mathbb{R}\times (\beta,\gamma^+]$ such that $(\alpha,y)\in \bar C_1$ and $(x,\gamma^+)\in \bar C_2$. Since $(\beta,\gamma^+]$ contains no root of $S_Y$, $C'_1$ (resp.) $C'_2$ is the graph of an increasing (resp. decreasing) semi-algebraic continuous functions $\psi_1$ (resp. $\psi_2$) defined on
$(\beta,\gamma^+]$ . The matching point of $(\alpha,y)$ is not $(x,\gamma^+)$ because these two points have opposite slope signs so that the limit of $\varphi_1$ at $\gamma^+$ is strictly less than $x$.
The signs of the limit $\psi_1-\psi_2$ at $\beta$ (resp. $\gamma^+$) is positive (resp. negative) and at $y_2$ is negative (resp. positive) , so $\psi_1(y)=\psi_2(y)$ for a value $y\in (\beta,\gamma^+]$, which is impossible because such a point would be a singular point of $V_{\mathbb{R}}(\tilde P)$ inside $\mathbb{R}\times (\beta,\gamma^+)$.
(ii) Suppose now that the slope sign of
$(\alpha,y)$ is positive and the slope sign of the elements of $L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}$ is negative.
If $\tilde P(\alpha,\gamma^+)=0$ (resp.$\tilde P(\alpha^-,\gamma^+)=0$) , this is impossible by (i), using the symmetry with respect to the line $X=\alpha$.
Otherwise there is an element $(x,\gamma^+)$ in $L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}$. Denote by $C_1$ (resp. $C_2$) the connected components of $V_{\mathbb{R}}(\tilde P)$ inside $[\alpha^-,\alpha)\times \mathbb{R}$ such that $(x,\gamma^+)\in\bar C_1$ (resp.
$(\alpha,y)\in \bar C_2$).
Since $[\alpha^-,\alpha)$ contains no root of $S_X$, $C_1$ (resp. $C_2$) is the graph of an increasing (resp. decreasing) semi-algebraic continuous functions $\varphi_1$ (resp. $\varphi_2$) defined on $[\alpha^-,\alpha)$. The matching point of $(x,\gamma^+)$ is not $(\alpha,y)$ because these two points have opposite slope signs so that the limit of $\varphi_2$ at $\alpha$ is strictly less than $y$. So the limit of $\varphi_1-\varphi_2$ at $\alpha^-$ (resp. $\alpha$) is negative (resp. positive). It follows that $\varphi_1(x)=\varphi_2(x)$ for a value $x\in (\alpha^-,\alpha)$, which is impossible because such a point would be a singular point of $V_{\mathbb{R}}(\tilde P)$ inside $[\alpha^-,\alpha)\times \mathbb{R}$.
We now prove Parts a.1) and a.2):
a.1)
The matching point of $L^+[i]$ is a point of the boundary to the left of $\alpha$ which
does not belong to $L$ since $S_Y$ has no zero on $(\beta,\gamma^+]$: it is a point of $L'^+=\bar L_{\alpha^-}+L_{\gamma^-}^{=\alpha}+L_{\gamma^-}^{<\alpha}$.
Consider the
first point $L^+ \left[ i \right]$ which is matched to a point
$L'^+\left[ j \right]$ of $L'$ with $j > i$. Then $L'^+ \left[ i \right]$ cannot
be matched with a point of $L^+$ since otherwise the arcs through $L^+ \left[ i \right]$ and $L^+ \left[ j \right]$ would have an
intersection in the adjacency box. This is impossible because such a point would be a singular point of $V_{\mathbb{R}}(\tilde P)$ different from $\alpha,\beta)$ in the adjacency box. So we obtain a contradiction and the matching point of $L^+
\left[ i \right]$ is $L'^+ \left[ i \right]$.
a.2)
The matching point of $L^+[i]$ is a point of the boundary to the left of $\alpha$ which
does not belong to $L$ since $S_X$ has no zero on $[\alpha^+,\alpha)$ and whose $y$-coordinate is bigger than the $y$-coordinate of $L^+ \left[ i \right]$: it is a point of $L'^+$.
Consider the
first point $L^+ \left[ i \right]$ which is matched to a point
$L'^+\left[ j \right]$ of $L'$ with $j > i$. Then $L'^+ \left[ i \right]$, whose $y$-coordinate is at least the $y$-coordinate of $L^+[i]$ cannot
be matched with a point of $L_\alpha$ with a $y$-coordinate smaller than the $y$-coordinate of $L^+[i]$ since otherwise the arcs through $L^+ \left[ i \right]$ and $L ^+\left[ j \right]$ would have an
intersection in the adjacency box. So we obtain a contradiction and the matching point of $L^+
\left[ i \right]$ is $L'^+ \left[ i \right]$.
\end{proof}
\begin{notation}
We denote by $\sigma^+$ (resp. $\sigma^-$) the slope signs of
the elements of $L_{\alpha^-}^{>\beta}+L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}$ (resp.
$\bar L_{\alpha^-}^{<\beta}+L_{\gamma^-}^{=\alpha^-}+L_{\gamma^-}^{<\alpha}+L_{\gamma^-}^{=\alpha}+\bar L_{\alpha}^{<\beta}$)
and by
$\tau^+$ (resp. $\tau^-$)
\hide{the signs
$\rm{Slope}(\tilde P)(L_{\gamma^-}^{> \alpha} [1])$ and $\rm{Slope}(\tilde P)(L_{\gamma^+}^{> \alpha} [1])$.
}
the slope signs of the elements of $L_{\alpha^+}^{>\beta}+L_{\gamma^+}^{=\alpha^+}+L_{\gamma^+}^{>\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}$ (resp.
$\bar L_{\alpha^+}^{>\beta}L_{\gamma^-}^{=\alpha^+}+L_{\gamma^-}^{>\alpha}+L_{\gamma^-}^{=\alpha}+\bar L_{\alpha}^{<\beta}$).
By convention when $L_{\alpha^-}^{>\beta}+L_{\gamma^+}^{=\alpha^-}+L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}$ (resp. $\bar L_{\alpha^-}^{<\beta}+L_{\gamma^-}^{=\alpha^-}+L_{\gamma^-}^{<\alpha}+L_{\gamma^-}^{=\alpha}+\bar L_{\alpha}^{<\beta}$, $L_{\alpha^+}^{>\beta}+L_{\gamma^+}^{=\alpha^+}+L_{\gamma^+}^{>\alpha}+L_{\gamma^+}^{=\alpha}+\bar L_{\alpha}^{>\beta}$,
$\bar L_{\alpha^+}^{<\beta}+L_{\gamma^-}^{=\alpha^+}+L_{\gamma^-}^{>\alpha}+L_{\gamma^-}^{=\alpha}+\bar L_{\alpha}^{<\beta}$) is empty, we define $\sigma^+$ (resp. $\sigma^-$, $\tau^+$, $\tau^-$) to be $>0$.
\end{notation}
\begin{algorithm}[Number of segments arriving at a critical point]
\label{algoconnect}
\noindent {\\\\\bf 1. Number of segments arriving to the left} \\
{\bf Input:} $\#L_{\alpha^-}$,$\#L_{\gamma^-}^{= \alpha^-}$, $\#L_{\gamma^-}^{ < \alpha}$, $\#L_{\gamma^-}^{= \alpha}$, $\#L_{\gamma^+}^{= \alpha^-}$, $\#L_{\gamma^+}^{ < \alpha}$, $\#L_{\gamma^+}^{= \alpha}$, $\#L_{\alpha}^{>\beta}$, $\#L_{\alpha}^{<\beta}$, $\sigma^-$ and $\sigma^+$ \\
{\bf Output:}
the number $\textsc{Left}$ of the segments ending at
$(\alpha,\beta)$ to the left of $\alpha$
\begin{itemize}
\item If $\sigma^+>0$ and $\sigma^->0$, compute
\begin{equation}\label{totheleft1}
\textsc{Left}=\#L_{\alpha^-}-(\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha})+\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha^-}-(\#L_\alpha^{>\beta }+\#L_\alpha^{<\beta})
\end{equation}
\item If $\sigma^+>0$ and $\sigma^-<0$, compute
\begin{equation}\label{totheleft2}
\textsc{Left}=\#L_{\alpha^-}-(\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha})-(\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha})-(\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta})
\end{equation}
\item If $\sigma^+<0$ and $\sigma^->0$, compute
\begin{equation}\label{totheleft3}
\textsc{Left}=\#L_{\alpha^-}+\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha^-}+\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha^-}-(\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta})
\end{equation}
\item If $\sigma^+<0$ and $\sigma^-<0$, compute
\begin{equation}\label{totheleft4}
\textsc{Left}=\#L_{\alpha^-}+\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha^-}-(\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha})-(\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta})
\end{equation}
\end{itemize}
\noindent {\bf 2. Number of segments arriving to the right} \\
{\bf Input:} $\#L_{\alpha^+}$,$\#L_{\gamma^-}^{= \alpha^+}$, $\#L_{\gamma^-}^{ > \alpha}$, $\#L_{\gamma^-}^{= \alpha}$, $\#L_{\gamma^+}^{= \alpha^+}$, $\#L_{\gamma^+}^{ > \alpha}$, $\#L_{\gamma^+}^{= \alpha}$, $\#L_{\alpha}^{>\beta}$, $\#L_{\alpha}^{<\beta}$, $\sigma^-$ and $\sigma^+$ \\
{\bf Output:}
the number $\textsc{Right}$ of the segments ending at
$(\alpha,\beta)$ to the right of $\alpha$.
\begin{itemize}
\item If $\tau^+<0$ and $\tau^-<0$, compute
\begin{equation}\label{totheright1}
\textsc{Right}=\#L_{\alpha^+}-(\#L_{\gamma^+}^{>\alpha}+\#L_{\gamma^+}^{= \alpha})+\#L_{\gamma^-}^{>\alpha}+\#L_{\gamma^-}^{= \alpha^+}-(\#L_\alpha^{>\beta }+\#L_\alpha^{<\beta})
\end{equation}
\item If $\tau^+<0$ and $\tau^->0$, compute
\begin{equation}\label{totheright2}
\textsc{Right}=\#L_{\alpha^+}-(\#L_{\gamma^+}^{ > \alpha}+\#L_{\gamma^+}^{= \alpha})-(\#L_{\gamma^-}^{ > \alpha}+\#L_{\gamma^-}^{= \alpha})-(\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta})
\end{equation}
\item If $\tau^+>0$ and $\tau^-<0$, compute
\begin{equation}\label{totheright3}
\textsc{Right}=\#L_{\alpha^+}+\#L_{\gamma^+}^{ > \alpha}+\#L_{\gamma^+}^{= \alpha^+}+\#L_{\gamma^-}^{ > \alpha}+\#L_{\gamma^-}^{= \alpha^+}-(\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta})
\end{equation}
\item If $\tau^+>0$ and $\tau^->0$, compute
\begin{equation}\label{totheright4}
\textsc{Right}=\#L_{\alpha^+}+\#L_{\gamma^+}^{ > \alpha}+\#L_{\gamma^+}^{= \alpha^+}-(\#L_{\gamma^-}^{ > \alpha}+\#L_{\gamma^-}^{= \alpha})-(\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta})
\end{equation}
\end{itemize}
\end{algorithm}
\begin{proof}[Proof of Corectnesss of Algorithm \ref{algoconnect}]
The correctness of Algorithm \ref{algoconnect} follows from Proposition
\ref{forcorrectness}. Indeed, denoting by $N_1$ (resp. $N_2$, $N_3$) the number of arcs of type 1 (resp. 2, 3) in $[\alpha^-,\alpha)\times [\gamma^-,\gamma^+]$, we notice that $\textsc{Left}=N_3$, $N_2=\#L_{\alpha}^{<\beta}+\#L_{\alpha}^{<\beta}$.
Defining $N=2 N_1+2N_2+N_3$.
\begin{itemize}
\item[(i)] If $\sigma^+>0$ and $\sigma^->0$
$$ N=\#L_{\alpha^-}+\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha}+\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha^-}+\#L_\alpha^{>\beta }+\#L_\alpha^{<\beta}.$$
Moreover, all the points of $L_{\gamma^-}\setminus \operatorname{Match}(L^-)$ are matched to $(\alpha,\beta)$. Finally the points of $L_{\alpha}\setminus \operatorname{Match}(L^+)$ are also matched to $(\alpha,\beta)$.
\item[(ii)] If $\sigma^+>0$ and $\sigma^-<0$
$$N=\#L_{\alpha^-}+\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha}+\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha}+\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta}.$$
The $y$- coordinate of each elements of $\operatorname{Match}(L_{\alpha^-}^{>\beta})$ (resp. $\operatorname{Match}(L_{\alpha^-}^{<\beta})$) is bigger (smaller) than $\beta$ and $\operatorname{Match}(L_{\alpha^-}^{>\beta})=L_{\gamma^+}^{<\alpha}+L_{\gamma^+}^{=\alpha}+L_{\alpha}^{>\beta})$ (resp. $\operatorname{Match}(L_{\alpha^-}^{<\beta})=L_{\gamma^-}^{<\alpha}+L_{\gamma^-}^{=\alpha}+L_{\alpha}^{<\beta}$). So
\begin{itemize}
\item $\textsc{Left}=1$ if $P(\alpha^-,\beta)=0$, which corresponds to an horizontal line $Y=\beta$ contained in $V_\mathbb{R}(\tilde P)$
\item $\textsc{Left}=0$ if $P(\alpha^-,\beta)\not=0$.
\end{itemize}
\item[(iii)] If $\sigma^+<0$ and $\sigma^->0$
$$N=\#L_{\alpha^-}+\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha^-}+\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha^-}+\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta}.$$
Moreover, all the points of $L_{\gamma^+}\setminus \operatorname{Match}(L^+)$ (resp. $L_{\gamma^-}\setminus \operatorname{Match}(L^-)$) are matched to $(\alpha,\beta)$. Finally the points of $L_{\alpha}\setminus (\operatorname{Match}(L^+)\cup \operatorname{Match}(L^-))$ are also matched to $(\alpha,\beta)$.
\item[(iv)] If $\sigma^+<0$ and $\sigma^-<0$
$$N=\#L_{\alpha^-}+\#L_{\gamma^+}^{ < \alpha}+\#L_{\gamma^+}^{= \alpha^-}+(\#L_{\gamma^-}^{ < \alpha}+\#L_{\gamma^-}^{= \alpha})+\#L_{\alpha}^{>\beta }+\#L_{\alpha}^{<\beta}.$$
Moreover, all the points of $L_{\gamma^+}\setminus \operatorname{Match}(L^+)$ are matched to $(\alpha,\beta)$. Finally the points of $L_{\alpha}\setminus \operatorname{Match}(L^-)$ are also matched to $(\alpha,\beta)$.
\end{itemize}
The correctness of the computation of $\textsc{Right}$ by Algorithm \ref{algoconnect} is entirely similar.
\end{proof}
\begin{exemple}
In the case of Figure \ref{description_lists}, for the left side we have
$\#L_{\alpha^-}=7$,$\#L_{\gamma^-}^{= \alpha^-}=0$, $\#L_{\gamma^-}^{ < \alpha}=1$, $\#L_{\gamma^-}^{= \alpha}=0$, $\#L_{\gamma^+}^{= \alpha^-}=0$, $\#L_{\gamma^+}^{ < \alpha}=2$, $\#L_{\gamma^+}^{= \alpha}=0$, $\#L_{\alpha}^{>\beta}=2$, $\#L_{\alpha}^{<\beta}=2$. Hence, from formula (\ref{totheleft3}) $\textsc{Left}= 2$. For the right side we have:
$\#L_{\alpha^+}=5$, $\#L_{\gamma^-}^{= \alpha^+}=0$, $\#L_{\gamma^-}^{ > \alpha}=0$, $\#L_{\gamma^-}^{= \alpha}=0$, $\#L_{\gamma^+}^{= \alpha^+}=0 $, $\#L_{\gamma^+}^{ > \alpha}=1$, $\#L_{\gamma^+}^{= \alpha}=0$, $\#L_{\alpha}^{>\beta}=2$, $\#L_{\alpha}^{<\beta}=0$.
If $\sigma^+>0$, $\sigma^->0$, $\tau^+>0$, $\tau^->0$. we have from formula (\ref{totheleft3}) $\textsc{Left}= 2$ and from formula (\ref{totheright3})
$\textsc{Right}= 2$. This is illustrated by the following picture.
\includegraphics[width=9cm,height=7cm]{Dessins/essaidessin2.png}
Always in the case of Figure \ref{description_lists},
if we have $\sigma^+>0$, $\sigma^-<0$, $\tau^+>0$, $\tau^-<0$
it follows from formula (\ref{totheleft2}) $\textsc{Left}= 0$ and from formula (\ref{totheright3})
$\textsc{Right}= 2$. This is illustrated by the following picture.
\includegraphics[width=9cm,height=7cm]{Dessins/essaidessin3.png}
\end{exemple}
\subsection{Computing the Refined Cylindrical Algebraic Decomposition}
For each $i=1,\ldots,N$, we denote by
\begin{equation}
\eta^-_{i,1}<\ldots<\eta^-_{i,m^-_i} (\mathrm{resp. } \eta^+_{i,1}<\ldots<\eta^+_{i,m^+_i})
\end{equation}
the real roots of $\tilde P(\alpha_i^-,Y)$ (resp. $\tilde P(\alpha_i^+,Y)$).
For each $k=1,\ldots,M$, we denote by
\begin{equation}
\xi^-_{k,1}<\ldots<\xi^-_{k,n^-_k} (\mathrm{resp. } \xi^+_{k,1}<\ldots<\xi^+_{k,n^+_k})
\end{equation}
the real roots of $\tilde P(X,\gamma_k^-)$ (resp. $\tilde P(X,\gamma_k^+)$).
For each fixed $i$, we define the set of \emph{indices of $X$-critical points above $i$} as
\[
\textsc{CritInd}_i:=\{j\in\{1,\ldots,m_i\}:(\alpha_i,\beta_{i,j})\in\operatorname{Crit}(V_{\mathbb{R}})\}.
\]
Theorem \ref{sing-fibers} (Refined Cylindrical Algebraic Decomposition) now gives
\begin{itemize}
\item[(a)] a cylindrical decomposition of $\tilde P$ with some extra information for the projection on the $X$-axis
\item[(b)] a partial cylindrical decomposition of $\tilde P$ for the projection on the $Y$-axis
\item[(c)] refinements of the preceding computations giving compatibilities between (a) and (b).
\end{itemize}
\begin{theorem}[Refined Cylindrical Algebraic Decomposition]\label{sing-fibers}
Using $\tilde{O} (d^5 \tau + d^6)$ bit operations, we can carry out the following computations:
\begin{itemize}
\item[(a.1)]
dyadic intervals $I_i,I_{i,j}$
for $1\le i \le N, 1 \le j \le m_i$
such that
$ I_i$
is well-isolating for the root $\alpha_i$ of $D_X$ as a root of $T_X$,
and $ I_{i,j}$
is well isolating for $\beta_{i,j}$ as a root of $\tilde{P}(\alpha_i,Y)$,
\item[(a.2)] $\deg(\tilde P(\alpha_i,Y))$ as well as $\deg\gcd(\tilde P(\alpha_i,Y),\partial_Y \tilde P(\alpha_i,Y))$ for $1\le i \le N$,
\item[(a.3)] for every $1\le i \le N, 1 \le j \le m_i$, $\operatorname{mult}(\beta_{i,j},\tilde P(\alpha_i,-))$, the multiplicity of $\beta_{i,j}$ as a root of $\tilde P(\alpha_i,Y)$ and
$\textsc{CritInd}_i\subset \{1,\ldots, m_i\}$, the set
of indices of $X$-critical points above $\alpha_i$.
Moreover if $j\notin \textsc{CritInd}_i$,
$I_{i,j}$ is well isolated as a root of
$$\tilde{P}(\alpha_i,Y) \partial_X\tilde{P}(\alpha_i,Y)\cdot \partial_Y\tilde{P}(\alpha_i,Y).$$
\item[(a.4)]
dyadic intervals $I^-_i, I^-_{i,j}$, (resp. $I^-_i, I^-_{i,j}$)
for $1\le i \le N, 1 \le j \le m^-_i$ (resp. $j\le m^+_i$)
such that
$ I^-_i$ (resp $ I^+_i$)
is well-isolating for $\alpha^-_i$ (resp. $\alpha^+_i$) as a root of $T_X$,
and $I^-_{i,j}$ (resp. $I^+_{i,j}$)
is well isolating for $\eta^-_{i,j}$ (resp. $\eta^+_{i,j}$) as a root of $$\tilde{P}(\alpha^-_i,Y) \partial_X\tilde{P}(\alpha^-_i,Y)\cdot \partial_Y\tilde{P}(\alpha^-_i,Y)$$
$$ (\text{resp. } \tilde{P}(\alpha^+_i,Y) \partial_X\tilde{P}(\alpha^+_i,Y)\cdot \partial_Y\tilde{P}(\alpha^+_i,Y)).$$
\item[(b)] dyadic intervals $J^-_{k}, J^-_{k,\ell}$ (resp. $J^+_{k}, J^+_{k,\ell}$)
for $1\le k \le M, 1 \le \ell \le n^-_k$ (resp. $\ell \le n^+_k$) such that
$J^-_{k}$ (resp. $J^+_{k}$) is well-isolating for $\gamma^-_{k}$ (resp. $\gamma^+_{k}$) as a root of
$T_Y$
and $J^-_{k,\ell}$ (resp. $J^+_{k,\ell}$) is well isolating for $\xi^-_{k,\ell}$ (resp. $\xi^+_{k,\ell}$) as a root of $$\tilde{P}(X,\gamma^-_{k})\partial_X\tilde{P}(X,\gamma^-_{k})\partial_Y\tilde{P}(X,\gamma^-_{k})$$
$$(\text{resp. }\tilde{P}(X,\gamma^+_{k})\partial_X\tilde{P}(X,\gamma^+_{k})\partial_Y\tilde{P}(X,\gamma^+_{k})$$
\item[(c.1)] dyadic intervals $J_k$, for $k=1,\ldots,M$, that are well isolating for $\gamma_k$ as roots of $T_Y$ and for each $X$-critical point $(\alpha_i,\beta_{i,j})$ the index $k(i,j)$ such that $\beta_{i,j}=\gamma_{k(i,j)}$
\item[(c.2)] For each $i=1,\ldots,N$, $j\in
\textsc{CritInd}_i$,
the intervals $J^-_{k(i,j),\ell}$ for $\ell=1,\ldots,n^-_{k(i,j),\ell}$ (resp. $J^+_{k(i,j),\ell}$ for $\ell=1,\ldots,n^+_{k(i,j),\ell}$)
contain at most one of the three points $\alpha^-_i$, $\alpha_i$, $\alpha^+_i$.
\item[(c.3)] For each $i=1,\ldots,N$, $j\in
\textsc{CritInd}_i$,
the intervals $I^-_{i,j'}$ for $j'=1,\ldots,m^-_{i}$ (resp. $I^+_{i,j'}$ for $j'=1,\ldots,m^+_{i}$) contain at most one of the two points $\gamma^-_{k(i,j)}$ $\gamma^+_{k(i,j)}$.
\end{itemize}
Moreover, it holds that
\begin{equation}
\sum^{N}_{i = 1} \lambda (I_i) (\text{resp. }\ \sum^{N}_{i = 1} \lambda (I^-_i),\sum^{N}_{i = 1} \lambda (I^+_i))
\in \tilde O (d^3 \tau+d^4), \label{eqboxa1}
\end{equation}
\begin{equation}
\sum^{N}_{i = 1} \sum^{m_i}_{j = 1} \lambda (I_{i, j}) (\mathrm{resp.}\ \sum^{N}_{i = 1} \sum^{m^-_i}_{j = 1} \lambda (I^-_{i, j}),\sum^{N}_{i = 1} \sum^{m^+_i}_{j = 1} \lambda (I^+_{i, j}))
\in \tilde O (d^3
\tau+d^4) \label{eqboxa2} .
\end{equation}
\begin{equation}
\sum^{M}_{k = 1} \lambda (J_k)(\mathrm{resp.}\ \sum^{M}_{k = 1} \lambda (J_k^-) \sum^{M}_{k = 1} \lambda (J_k^+)) \in \tilde O (d^3
\tau+d^4) \label{eqboxe1} .
\end{equation}
\begin{equation}
\sum^{M}_{k = 1} \sum^{n^-_{k}}_{\ell = 1} \lambda (J^-_{k,\ell}) (\mathrm{resp.}\ \sum^{M}_{k = 1} \sum^{n^+_{k}}_{\ell = 1} \lambda (J^+_{k,\ell}))
\in \tilde O (d^3
\tau+d^4) \label{eqboxe2} .
\end{equation}
\end{theorem}
\begin{remark}\label{amb}
Note that Theorem \ref{sing-fibers} (c.2) does not decide whether
$\xi^+_{k,\ell}<\alpha^-_i$,
$\xi^+_{k,\ell}=\alpha^-_i$ or $\xi^+_{k,\ell}=\alpha^-_i$
in the case where
$J^+_{k(i,j),\ell}$ contain $\alpha^-_i$,
Indeed, we do not know the sign of $\tilde P(\alpha_i,\gamma_k^-)$.
It would be of course possible to obtain this information using exact computations, but not within the complexity bounds we are aiming for in this paper:
to the best of our knowledge, the computation for this decision would exceed $\tilde{O} (d^5 \tau + d^6)$ bit operations.
The same remark holds for similar statements covering the other cases considered in (c.2) and (c.3)).
\end{remark}
\begin{proof}[Proof of Theorem \ref{sing-fibers}]
For (a.1,2,3) first note, using Lemma \ref{magni}, that $T_X$ is a polynomial of magnitude $(N,\lambda)\in (O(d^2),\tilde{O}(d\tau+d^2))$. Hence, Proposition \ref{sagraloff-isolation-integer} and Proposition \ref{comparingroots},
we can compute well isolating intervals for the real roots of $T_X$ and identify the real roots of $D_X$ in a number of bit operations bounded by $\tilde{O}(d^5\tau+d^6)$.
\hide{ such that the
bitsizes of the endpoints of the isolating intervals sums up to a value bounded by $\tilde{O}(d^4+d^3\tau)$.} According to
Proposition~\ref{prop:computinggcddeg} (with $R:=D_X$ and $F:=\tilde{P}(X,Y)$), we may further compute $\deg \tilde P(\alpha_i,Y)$ as well
as $\deg\gcd(\tilde P(\alpha_i,Y),\partial_Y \tilde P(\alpha_i,Y))$ for all $i$ from $1$ to $N$ using $\tilde{O}(d^5\tau+d^6)$ bit operations. Now, from Proposition~\ref{thm:costisolation}, we conclude that using also $\tilde{O}(d^5\tau+d^6)$ bit operations we can further compute well isolating intervals for all real roots of the
polynomials $\tilde{P}(\alpha_i,Y)$ as well as the corresponding multiplicities for all $i$ from $1$ to $N$. Each root $\beta_{i,j}$ of multiplicity larger than one then
corresponds to an $X$-critical point $(\alpha_i,\beta_{i,j})$, which defined $\textsc{CritInd}_i$ for $i=1,\ldots,N$.
We also use Proposition \ref{thm:comparinginfibers} with
$$H=\tilde{P}(X,Y)\cdot \partial_X\tilde{P}(X,Y)\cdot \partial_Y\tilde{P}(X,Y),$$
to get well-isolating intervals for the non-multiple roots of
$$\tilde{P}(\alpha_i,Y) \cdot \partial_X\tilde{P}(\alpha_i,Y)\cdot \partial_Y\tilde{P}(\alpha_i,Y),$$
and identify the relevant roots of $\tilde{P}(\alpha_i,Y)$ for each $i=1,\ldots,N$.
The bound on the sum of the bitsizes of the
intervals $I_i,I_{i,j}$ follows from Part (a.1) of Proposition~\ref{thm:costisolation} and Proposition \ref{thm:comparinginfibers}.
For (a.4), using Proposition \ref{sagraloff-isolation-integer} and Proposition \ref{comparingroots},
we can compute well isolating intervals for the real roots of $T_X$ and identify the real roots of $(D_X^\star)'$ in a number of bit operations bounded by $\tilde{O}(d^5\tau+d^6)$.
According to
Proposition~\ref{prop:computinggcddeg} (with $R:=(T_X^\star)'$ and $F:=\tilde{P}(X,Y)$), we may further compute $\deg \tilde P(\xi_i,Y)$.
Now, from Proposition~\ref{thm:costisolation}, we conclude that within a number of bit operations bounded by $\tilde{O}(d^5\tau+d^6)$ we can further compute well isolating intervals for all real roots of the
polynomials
$$\tilde{P}(\xi_i,Y) \cdot \partial_X\tilde{P}(\xi_i,Y)\cdot \partial_Y\tilde{P}(\xi_i,Y),$$
and identify the roots of $\tilde{P}(\xi_i,Y)$ for each $i=1,\ldots,N'$ by Proposition \ref{thm:comparinginfibers}.
It remains to isolate to compute isolating intervals for all the
roots of the
polynomials
$$\tilde{P}(\xi_0,Y) \cdot \partial_X\tilde{P}(\xi_0,Y)\cdot \partial_Y\tilde{P}(\xi_0,Y),$$
and
$$\tilde{P}(\xi_N',Y)\cdot \partial_X\tilde{P}(\xi_N',Y)\cdot \partial_Y\tilde{P}(\xi_N',Y).$$
It is then easy to identify $\alpha_i^-$ and $\alpha_i^+$ as well as $I^-_i,I^-_{i,j'},I^+_i,I^+_{i,j'}$ as part of the results of the preceding computations.
The bound on the sum of the bitsizes of the
intervals $I^-_i,I^-_{i,j'},I^+_i,I^+_{i,j'}$
follows from Part (a.1) of Proposition~\ref{thm:costisolation} and Proposition \ref{thm:comparinginfibers}.
Part (b) is entirely similar to Part (a.4), exchanging the role of $X$ and $Y$.
In Part (c.1), the computation of $T_Y$ takes $\tilde{O}(d^4 \tau+d^5)$ according to Lemma \ref{magni}, and the computation of the $J_k$, $k=1,\ldots,M$ uses a number of bit operations bounded by $\tilde{O}(d^5\tau+d^6)$ from Proposition \ref{sagraloff-isolation-integer} and Proposition \ref{comparingroots} since $T_Y$ is of magnitude $(O(d^2),O(d\tau+d^2))$.
The determination of indices $k(i,j)$ for $i=1,\ldots,N, j\in \textsc{CritInd}_i$ follows from Proposition \ref{indexofcritical}.
Part (c.2,c.3) follows from Proposition \ref{intermedfibers_smallenough}.
\end{proof}
\begin{proposition}
\label{indexofcritical}
Using $\tilde{O}(d^5\tau+d^6)$ bit operations, we can compute integers $k(i,j)\in\{1,\ldots,m\}$ for all $X$-critical points $(\alpha_i,\beta_{i,j})\in\operatorname{Crit}(V_{\mathbb{R}}(\tilde{P}))$ with $\gamma_{k(i,j)}=\beta_{i,j}$.
\hide{
More precisely, we can find for every $i$ and every $j\in \textsc{CritInd}_i$ an interval $J_{k(i,j)}$ which is simultaneously isolating for $\gamma_{k(i,j)}$ a root of $T_Y$ and for $\beta_{i,j}$ as a root of $\tilde P(\alpha_i,Y)=0$.
Moreover
\begin{equation}
\sum^{\delta}_{i = 1} \sum_{j \in \textsc{CritInd}_i} \lambda (J_{k(i, j)})\in \tilde O (d^3
\tau+d^4).
\end{equation}
}
\end{proposition}
\begin{proof}
We define $V^1_i:=\textsc{CritInd}_i$.
In order to compute the numbers $k(i,j)$ for each
$\beta_{i,j}), j \in \textsc{CritInd}_i$, we proceed in rounds enumerated by $\ell=1,2,3,\ldots$. In the $\ell$-th round, we refine the isolating intervals for all $i$ and all roots $\beta_{i,j}$ with $j \in V^\ell_i$ to a size less than $2^{-2^{\ell}}$. If the corresponding isolating interval
$I_{i,j}$
intersects with at most one isolating interval $J_k$ for the roots of
$\operatorname{Res}_X(\tilde P,\partial_Y\tilde P)$, we know that $k=k(i,j)$. After having treated all elements in $V^{\ell}_i$, we set $V^{\ell+1}_i$ to be the set of all critical indices in $V^\ell_i$ for which the isolating interval
$I_{i,j}$ for $\beta_{i,j}$ intersects more than one of the intervals $J_k$. That is, $V^{\ell}_i$ is the set of all critical indices for which $k(i,j)$ is not known after the $\ell$-th round. We then proceed with the $(\ell+1)$-st round. We stop as soon as $V^{\ell}_i$ becomes empty for every $i=1,\ldots,N$, in which case, $k(i,j)$ is determined for all critical points.
We use the polynomial $R_Y$ defined in Proposition \ref{multmult1}, with $F=\tilde P$, remembering that the roots of $R_Y$ contain the projections of the $Y$-critical points of $\tilde P$.
Notice that, for each critical point $(\alpha_i,\beta_{i,j})$, we succeed in round $\ell_{i,j}$, where
$2^{\ell_{i,j}}$ is bounded by $O(|\log(\operatorname{sep}(\beta_{i,j},T_Y R_Y))|)$. That is, $j \notin V^{\ell}_i$ for any $\ell>\ell_{i,j}$.
In addition, the cost of the test for checking whether the interval $I_{i,j}$ intersects with exactly one isolating interval $J_k$ is bounded by $\tilde{O}(d^3\tau+d^4)$ bit operations in each round. Indeed,
we need to consider only $O(\log (d))$ comparisons between corresponding
endpoints of the occurring intervals and each comparison is carried out with a precision bounded by $\tilde{O}(d^3\tau+d^4)$, using that, from the amortized bounds on the separation of the roots (Proposition \ref{corodisc}) $$O(|\log(\operatorname{sep}(\beta_{i,j},T_Y R_Y))|)\in O(d^3\tau+d^4).$$ Since there are $O(d^2)$ many critical points, the
total cost for the comparisons is thus bounded by $\tilde{O}(d^5\tau+d^6)$.
It remains to estimate the cost for refining the intervals $I_{i,j}$ to a width less than $2^{-2^{\ell_{i,j}}}$ for all $i,j$.
Using again $O(|\log(\operatorname{sep}(\beta_{i,j},T_Y R_Y))|)\in O(d^3\tau+d^4)$,
we are done after $\kappa$ rounds for
\begin{equation}\label{kappa1}
\kappa=\max_{i,j} \ell_{i,j}+1 \in O(\log(d^4+d^3 \tau)).
\end{equation}
According to Proposition~\ref{thm:costisolation}, this cost is bounded by
\[
\sum_{\ell=1}^\kappa \tilde{O}(d^5\tau+d^6+ 2^{\ell}d^2\cdot \sum_{i=1}^{N}\mu_{i}^{[\ell]})\in \tilde{O}(d^5\tau+d^6)+\tilde{O}(d^2\cdot\sum_{\ell} 2^{\ell}\cdot \sum_{i=1}^{N}\mu_{i}^{[\ell]}),
\]
where
\[
\mu_{i}^{[\ell]}:=
\begin{cases}
\max_{j \in V^{\ell}_i} \mu(\beta_{i,j},\tilde{P}(\alpha_i,-))&\text{if } V^\ell_z\not= \emptyset\\
0 &\text{otherwise.}
\end{cases}
\]
Hence, it suffices to show that
$$\sum_{\ell=1}^\kappa 2^{\ell}\cdot \sum_{i=1}^{N}\mu_{i}^{[\ell]}\in\tilde{O}(d^3\tau+d^4).$$
If $V^\ell_i\not=\emptyset$, let $j^{[\ell]}_i\in V^\ell_i$ be such that $\mu(\beta_{i,j^{[\ell]}_i},\tilde{P}(\alpha_i,-))=\mu_{i}^{[\ell]}$. In other words, $j^{[\ell]}_i$ is the critical index $V^{\ell}_i$ over $\alpha_i$ which maximizes the multiplicity within the fiber. We may thus write
\[
\sum_{\ell=1}^\kappa 2^{\ell}\cdot \sum_{i=1}^{N}\mu_{i}^{[\ell]}=\sum_{\ell=1}^\kappa 2^{\ell}\sum_{i,V^\ell_i\not=\emptyset}\mu(\beta_{i,j^{[\ell]}_i},\tilde{P}(\alpha_i,-))
\]
Obviously, each critical point $ (\alpha_i,\beta_{i,j})$ appears in the latter sum a number of times that is bounded by $\kappa\in O(\log(d^3\tau+d^4))$. In addition, since $ (\alpha_i,\beta_{i,j})\notin V_{\kappa}$
and $2^{\ell_{i,j}}\in O(|\log(\operatorname{sep}(\beta_{i,j},T_Y R_Y))|)$, it follows that
\[
\sum_{\ell=1}^\kappa 2^{\ell}\sum_{i=1}^N \sum_{j\in V^\ell_i}\mu(\beta_{i,j},\tilde{P}(\alpha_i,-))
\in \tilde{O}(\sum_{i=1}^N \sum_{j\in \textsc{CritInd}_i}\mu(\beta_{i,j},\tilde{P}(\alpha_i,-))\cdot|\log(\operatorname{sep}(\beta_{i,j},T_Y R_Y))|).
\]
Since
\[
\sum_{\ell=1}^\kappa 2^{\ell}\sum_{i,V^\ell_i\not=\emptyset}\mu(\beta_{i,j^{[\ell]}_i},\tilde{P}(\alpha_i,-))
\le \sum_{\ell=1}^\kappa 2^{\ell}\sum_{i=1}^N \sum_{j\in V^\ell_i}\mu(\beta_{i,j},\tilde{P}(\alpha_i,-)),\]
we have
\[
\sum_{\ell=1}^\kappa 2^{\ell}\sum_{i,V^\ell_i\not=\emptyset}\mu(\beta_{i,j^{[\ell]}_i},\tilde{P}(\alpha_i,-))
\in \tilde{O}(\sum_{i=1}^N \sum_{j\in \textsc{CritInd}_i}\mu(\beta_{i,j},\tilde{P}(\alpha_i,-))\cdot|\log(\operatorname{sep}(\beta_{i,j},T_Y R_Y))|).
\]
According to Proposition~\ref{multmult1} , it holds that, for any fixed value $\beta_{i,j}$,
\[
\sum_{(\alpha_i,\beta_{i,j}):\beta_{i,j}=\gamma}(\operatorname{mult}(\beta_{i,j},\tilde{P}(\alpha_{i},-))-1)\le\operatorname{mult}(\gamma,R_Y)).
\]
Hence, since $\operatorname{mult}(\beta_{i,j},\tilde{P}(\alpha_{i},-))\le 2\cdot(\operatorname{mult}(\beta_{i,j},\tilde{P}(\alpha_{i},-))-1)$, we conclude that
$$\sum_{(i,j):\beta_{i,j}=\gamma}\operatorname{mult}(\beta_{i,j},\tilde{P}(\alpha_i,-))\le 2\cdot \operatorname{mult}(\gamma,T_Y R_Y).$$
This shows that
\[
\sum_{\ell=1}^\kappa 2^{\ell} \sum_{i=1}^{N}\mu_{i}^{[\ell]}\in \tilde{O}(
\operatorname{logsep}(T_Y R_Y))
\in\tilde{O}(d^3\tau+d^4).
\]
using
Proposition \ref{corodisc}.
\end{proof}
For each fixed $i$, we define the set of \emph{critical places above $i$} as
\[
\textsc{CritPl}_i:=\{k\in\{1,\ldots,M\}:(\alpha_i,\gamma_k)\in\operatorname{Crit}(V_{\mathbb{R}})\}.
\]
Similarly for each fixed $k$, we define the set of \emph{indices of critical places at level $k$} as
\[
\textsc{CritPl}^k:=\{i\in\{1,\ldots,N\}:(\alpha_i,\gamma_k)\in\operatorname{Crit}(V_{\mathbb{R}})\}.
\]
\begin{proposition}
\label{intermedfibers_smallenough}
Using $\tilde{O}(d^5\tau+d^6)$ bit operations, and without modifying the estimates
(\ref{eqboxa1}), (\ref{eqboxa2}), (\ref{eqboxe1}) and (\ref{eqboxe2}),
a) we can refine all intervals $I^-_{i,j'}$ (resp. $I^+_{i,j'}$) such that if $I^-_{i,j'}$ (resp. $I^+_{i,j'}$) intersects
an interval $J_{k}^+$ (resp. $J_{k}^-$) for some $k\in \textsc{CritPl}_i$, then it contains $J_{k}^+$ (resp. $J_{k}^-$) and does not intersect $J_{k}^-$ (resp. $J_{k}^+$).
b) we can refine all intervals $J^-_{k,\ell}$ (resp. $J^+_{k,\ell}$) such that if $J^-_{k,\ell}$ (resp. $J^+_{k,\ell}$) intersects
an interval $I_{i}^+$ (resp. $I_i$; resp. $I_{i}^-$) for $i\in \textsc{CritPl}^k$, then it contains $I_{i}^+$ (resp. $I_i$; resp. $I_{i}^-$) and does not intersect $I_i$ and $I_{i}^-$ (resp. $I_i^+$ and $I_{i}^-$ ; resp. $I_i$ and $I_{i}^+$).
\end{proposition}
\begin{proof}
We treat in details the case of intervals $I^-_{i,j'}$ which is the first half of a).
Let $I^-_{i,j'}$ be the intervals as computed according to Theorem~\ref{sing-fibers}. In what follows, we restrict to the set $V^1$ of all so-called \emph{bad pairs} $(i,j')$ of indices such that the
interval $I^-_{i,j'}$ intersects two intervals $J_k^-$ and $J_k^+$, and define $V^1_i$ as the set of corresponding indices above $i$.
We fix a mapping $\phi$ that maps
each such bad pair $(i,j')$ to an arbitrary index $k$ (there might exist more than one such index)
such that $I^-_{i,j'}$ intersects the intervals $J_k^-$ and $J_k^+$ and moreover $\gamma_k^+-\gamma_k^-$ is minimal with this property. Then, the size of the pre-image of each $k$ is upper bounded by $\textsc{CritPl}_k$. Namely, for a fixed $i$, there can be at most one interval $I^-_{i,j'}$ intersecting the two intervals $J_k^-$ and $J_k^+$. We thus conclude that
$$|\Phi^{-1}(k)|\le |\textsc{CritPl}_k|\le \mu(\gamma_k,S_Y)\le \mu(\gamma_k,T_Y),$$
which further implies that $V^1$ contains at most $O(d^2)$ many elements. Further notice that a pair $(i,j')$ cannot be bad if the width of the corresponding interval $I^-_{i,j'}$ is smaller than $\frac{1}{2}\cdot \operatorname{sep}(\gamma_k,T_Y)$ as the distance between the intervals $J_k^-$ and $J_k^+$ is at least $\frac{1}{2}\cdot\min(|\gamma^+_k-\gamma^-_k|)\le \frac{1}{2}\cdot \operatorname{sep}(\gamma^-_k,T_Y)$. Hence, in order to guarantee that the $I^-_{i,j'}$ does not intersects two intervals $J_k^-$ and $J_k^+$, it is enough to refine the intervals $I^-_{i,j'}$ with $(i,j')\in V_1$ to a width smaller than $\frac{1}{2}\cdot \operatorname{sep}(\gamma^-_k,T_Y)$.
For this, we proceed in rounds enumerated by $\ell=1,2,3,\ldots$, where, in the $\ell$-th round, we refine all intervals $I^-_{i,j'}$ with $(i,j')\in V^{\ell}_i$ to a width less than $2^{-2^{\ell}}$. Then, we remove all pairs $j'$ from $V^\ell_i$ that are not bad anymore to obtain $V^{\ell+1}_i$. In other words, $V^{\ell+1}_i$ is the set of all pairs for which $I^-_{i,j'}$ intersects two intervals $J_k^-$ and $J_k^+$ after the $\ell$-th round. We then proceed with the $(\ell+1)$-st round. We stop as soon as all $V^{\ell}_i$ becomes empty, in which case, none of the intervals $I^-_{i,j'}$ violates the condition.
Notice that an interval $I^-_{i,j'}$ is removed after $\ell_{i,j'}$ rounds, where $2^{\ell_{i,j'}}$ is bounded by $O(|\log(\operatorname{sep}(\gamma_{\phi((i,j'))},T_Y))|)$. That is, $j'\notin V^{\ell}_i$ for any $\ell>\ell_{i,j'}$. This further implies that each root contained in $\phi(V_\ell)$ has separation smaller than $2^{1+2^{-\ell}}$. Further notice that, from the amortized bounds on the separation of the roots (Proposition \ref{corodisc})
$$O(|\log(\operatorname{sep}(\gamma_{\phi((i,j'))},T_Y))|)\in O(d^3\tau+d^4),$$ so that the test checking whether $j'\in V^\ell_i$ is bounded by $\tilde{O}(d^3\tau+d^4)$ bit operations in each round as we need to consider only $O(\log(d))$ comparisons between corresponding
endpoints of the occurring intervals and each comparison is carried out with a precision bounded by $\tilde{O}(d^3\tau+d^4)$. Since there are $O(d^2)$ many element in each $V$, the
total cost for the comparisons is thus bounded by $\tilde{O}(d^5\tau+d^6)$.
It remains to estimate the cost for refining the intervals $I^-_{i,j'}$ to a width less than $2^{-2^{\ell_{i,j'}}}$ for all $(i,j')\in V$.
Using again $O(|\log(\operatorname{sep}(\gamma_{\phi((i,j'))},T_Y))|)\in O(d^3\tau+d^4),$
$$\kappa=\max_{i,j'} \ell_{i,j'}+1\in O(\log(d^3 \tau+d^4)).$$
According to Proposition~\ref{thm:costisolation}, this cost is bounded by
\[
\sum_{\ell=1}^\kappa \tilde{O}(d^5\tau+d^6+ 2^{\ell}d^2\cdot \lambda_{\ell})\in \tilde{O}(d^5\tau+d^6)+\tilde{O}(d^2\cdot\sum_{\ell} 2^{\ell}\cdot|\{i\mid V^\ell_i\not= \emptyset\}|,
\]
Hence, it suffices to show that
$$\sum_{\ell=1}^\kappa 2^{\ell}\cdot |\{i\mid V^\ell_i\not= \emptyset\}|\in\tilde{O}(d^4+d^3\tau),$$
or alternatively that
\begin{align}\label{sumoverellandgammas}
\sum_{\ell=1}^\kappa \sum_{k:\log(\operatorname{sep}(\gamma_k,T_Y))<2^{1-2^\ell}}\mu(\gamma_k,T_Y)\cdot|\log(\operatorname{sep}(\gamma_k,T_Y))| \in\tilde{O}(d^3\tau+d^4).
\end{align}
as each root $\gamma_k$ has at most $\mu(\gamma_k,T_Y)$ pre-images under the mapping $\phi$ and $\phi(V^\ell_i)$ contains only roots $\gamma_k$ of separation smaller than $2\cdot 2^{1-2^\ell}$.
Since $\kappa\in O(\log(d^3\tau+d^4))$ and since $\sum_{k=1}^M \mu(\gamma_k,T_Y)\cdot|\log(\operatorname{sep}(\gamma_k,T_Y))| \in\tilde{O}(d^3\tau+d^4)
$, we conclude that the inequality in (\ref{sumoverellandgammas}) holds.
The case of intervals $I^+_{i,j'}$ which is the second half of a) is entirely similar.
We also omit the proof of b) which is similar, exchanging the role of $X$ and $Y$ and treating first the case $\alpha_i^-,\alpha_i$ and second the case $\alpha_i,\alpha_i^+$.
\end{proof}
We now explain how to deal with the vertical asymptotes.
The vertical asymptotes occur at values of $\alpha$ where $\deg(\tilde P(\alpha,Y))<d_y=\deg_Y(\tilde P)$, so only at roots of $c_{d_y}(X)$ which are also roots of $D_X$. The indices $i = 1,\ldots,N$ such that $c_{d_y}(\alpha_i)=0$ are part of the output of Theorem \ref{sing-fibers}.
For vertical asymptotes at $-\infty$, we isolate the roots of $\tilde P(X,\gamma^-_1)=0$ and decide the sign slope at the roots of
$\tilde P(X,\gamma^-_1)=0$. This can be done in $\tilde{O}(d^4\tau+d^5)$ bit-operations since
$\tilde P(X,\gamma^-_1)$ is a polynomial of degree at most $d$ and bit size $O(d^2 \tau+d^3)$.
We also compare the roots of $\tilde P(X,\gamma^-_1)$ with the roots of $D_X$. Note first that $\tilde P(X,\gamma^-_1)$ and $D_X$ have no roots in common. Since the separators of $\tilde P(X,\gamma^-_1)$ and $D_X$ are both bounded by $2^{\tilde O(d^3 \tau+d^4)}$ this can be done by Proposition
\ref{sagraloff-isolation-integer} with complexity $\tilde{O}(d^5\tau+d^6)$.
On each open interval $(\alpha_i,\alpha_{i+1})$, $i=1,\ldots,N$ delimited by roots of $D_X$, let $\textsc{Right}_{i,0}$ the number of roots of $\tilde P(X,\gamma^-_1)=0$ such that the slope sign is $>0$ and $\textsc{Left}_{i+1,0}$ the number of roots of $\tilde P(X,\gamma^-_1)=0$ such that the slope sign is $<0$.
We also denote by let $\textsc{Right}_{N,0}$ the number of roots of $\tilde P(X,\gamma^-_1)=0$ such that the slope sign is $>0$ on $(\alpha_N,+\infty)$ and $\textsc{Left}_{1,0}$ the number of roots of $\tilde P(X,\gamma^+_M)=0$ such that the slope sign is $<0$ on $(-\infty, \alpha_1)$.
Note that all the roots $\tilde P(X,\gamma^-_1)=0$ on $(\alpha_i,\alpha_{i+1})$ with positive slope sign are bigger than all the roots $\tilde P(X,\gamma^-_1)=0$ on $I_i$ with negative slope sign. Note also that if $\alpha_i$ is not a root of $c_{d_y}$,
$\textsc{Left}_{1,0}=\textsc{Right}_{1,0}=0$.
The situation at $+\infty$ is entirely similar and we define
$\textsc{Right}_{i,m_i+1}$ the number of roots of $\tilde P(X,\gamma^+_M)=0$ on $(\alpha_i,\alpha_{i+1})$ such that the slope sign is $>0$ and $\textsc{Left}_{i+1,m_{i+1}+1}$ the number of roots of $\tilde P(X,\gamma^+_M)=0$ on $(\alpha_i,\alpha_{i+1})$ such that the slope sign is $<0$.
We also denote by let $\textsc{Right}_{N,m_N+1}$ the number of roots of $\tilde P(X,\gamma^+_M)=0$ such that the slope sign is $<0$ on $(\alpha_N,+\infty)$ and $\textsc{Left}_{1,m_1+1}$ the number of roots of $\tilde P(X,\gamma^+_M)=0$ such that the slope sign is $>0$ on $(-\infty, \alpha_1)$.
Finally we have
\begin{proposition}\label{asympt}
The number of asymptotic branches tending to $-\infty$ (resp $+\infty$) to the left of $\alpha_i$ is $\textsc{Left}_{i,0}$
(resp. $\textsc{Left}_{i,m_i+1}$) and the number of asymptotic branches tending to $-\infty$ (resp $+\infty$) to the right of $\alpha_i$ is $\textsc{Right}_{i,0}$ (resp. $\textsc{Right}_{i,m_i+1}$).Moreover the complexity of computing these numbers is $\tilde{O}(d^5\tau+d^6)$.
\end{proposition}
\subsection{Computing the topology inside adjacency boxes}\label{adjacencyboxes}
Our aim is to define for all adjacency boxes associated to critical points $(\alpha_i,\beta_{i,j})$, the numbers $\textsc{Left}_{i,j}$ and $\textsc{Right}_{i,j}$ of segments arriving at
$(\alpha_i,\beta_{i,j})$ to the left and to the right using the formulas in Algorithm \ref{algoconnect}(\ref{totheleft1}, \ref{totheleft2}, \ref{totheleft3}, \ref{totheleft4}) and (\ref{totheright1}, \ref{totheright2}, \ref{totheright3}, \ref{totheright4}).
Consider the adjacency box $[\alpha^-,\alpha^+]\times [\gamma^-,\gamma^+]$ associated to a singular point $(\alpha,\beta)=(\alpha_i,\beta_{i,j})$, with
$\alpha^-=\alpha^-_i,\alpha^+=\alpha^+_i$
and
$\gamma^-=\gamma^-_{k(i,j)},\gamma^+=\gamma^+_{k(i,j)}$, where
$\alpha^-,\alpha,\alpha^+$ and $\gamma^-,\gamma^+$ are given by dyadic intervals $(a^-,a'^-),(a,a'),(a^+,a'^+)$ and $(c^-,c'^-),(c^+,c'^+)$.
Denote $\textsc{Left}=\textsc{Left}_{i,j}$ and $\textsc{Right}=\textsc{Right}_{i,j}.$
The information computed in Theorem \ref{sing-fibers} is not always sufficient to determine the quantities involved in formulas (\ref{totheleft1}, \ref{totheleft2}, \ref{totheleft3}, \ref{totheleft4}) and (\ref{totheright1}, \ref{totheright2}, \ref{totheright3}, \ref{totheright4}). For example,
if at the corner $(\alpha^-,\gamma^+)$, the interval $(x,x')$
isolating a root $\xi$ of $\tilde P(X,\gamma^+)$ contains
$\alpha^-$ and the interval $(y,y')$
isolating a root $\eta$ of $\tilde P(\alpha^-,Y)$ contains
$\gamma^+$, we do not know the sign $\tilde P(\alpha^-,\gamma^+)$ so we do not know whether $\xi<\alpha^-$, $\xi=\alpha^-$ or $\xi>\alpha^-$ (resp.
$\eta<\gamma^+$, $\eta=\gamma^+$ or $\eta>\gamma^+$)
(see Remark \ref{amb})
and we cannot determine
$\#L_{\alpha^-}$, $\#L_{\gamma^+}^{= \alpha^-}$, $\#L_{\gamma^+}^{ < \alpha}$.
So we introduce the following definition to deal with such situations.
\begin{definition}
The corner $(\alpha^-,\gamma^+)$ is
{\bf ambiguous}
if there are intervals
$(x,x')$ and $(y,y')$ output by Theorem \ref{sing-fibers}
such that
$(x,x')$,
isolating a root $\xi$
of $\tilde P(X,\gamma^+)$,
contains
$\alpha^-$ and
$(y,y')$,
isolating a root $\eta$
of $\tilde P(\alpha^-,Y)$, contains
$\gamma^+$. We omit similar definitions for the
three
corners $(\alpha^-,\gamma^-)$,
$(\alpha^+,\gamma^-)$, $(\alpha^+,\gamma^+) $.
Similarly the midpoint $(\alpha,\gamma^+)$ is
{\bf ambiguous}
if there are intervals
$(x,x')$ and $(y,y')$ output by Theorem \ref{sing-fibers}
sachet that
$(x,x')$,
isolating a root $\xi$,
of $\tilde P(X,\gamma^+)$
contains
$\alpha$ and
$(y,y')$,
isolating a root $\eta$
of $\tilde P(\alpha,Y)$, contains
$\gamma^+$ .
We omit a
similar definition for the
other midpoint $(\alpha,\gamma^-)$.
\end{definition}
At the ambiguous corner $(\alpha^-,\gamma^+)$, it is not possible to know the cardinals of
$L_{\gamma^+}^{<\alpha}\cap [x,x']$, $L_{\gamma^+}^{=\alpha}\cap [x,x']$ and $L_{\alpha^-}\cap [y,y']$ since we do not know whether $\xi \in L_{\gamma^+}^{<\alpha}$ (resp. $\eta \in L_{\alpha^-}$).
However we note that
\begin{itemize}
\item If $\sigma^+>0$
(with $\partial_X\tilde P(\alpha^-,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)<0$) then $\tilde P(x,\gamma^+)<0$ and $\tilde P(x',\gamma^+)>0$ while
$\tilde P(\alpha^-,y)>0$ and $\tilde P(\alpha^-,y')<0$.
\begin{itemize}
\item If $\tilde P(\alpha^-,\gamma^+)>0$, then $\xi<\alpha^-$ and $\eta>\gamma^+$. \begin{center}
\includegraphics[width=4.5cm,height=4cm]{Dessins/corner6.png}
\end{center}
So that $\#L_{\gamma^+}^{<\alpha}\cap [x,x']=\#L_{\alpha^-}\cap [y,y']=0
$
\item If $\tilde P(\alpha^-,\gamma^+)<0$, then $\xi>\alpha^-$ and $\eta<\gamma^+$. \begin{center}
\includegraphics[width=4.5cm,height=4cm]{Dessins/corner5.png}
\end{center}
So that $\#L_{\gamma^+}^{<\alpha}\cap [x,x']=\#L_{\alpha^-}\cap [y,y']=1,
$
\item If $\tilde P(\alpha^-,\gamma^+)=0$, then $\xi=\alpha^-$ and $\eta=\gamma^+$.
\begin{center}
\includegraphics[width=4.5cm,height=4cm]{Dessins/corner4.png}
\end{center}
So that $\#L_{\gamma^+}^{<\alpha}\cap [x,x']=\#L_{\alpha^-}\cap [y,y']=0
$
\end{itemize}
In all cases, when $\sigma^+>0$ (with $\partial_X\tilde P(\alpha^-,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)<0$)
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']-\#L_{\alpha^-}\cap [y,y']=0.$$
\item If $\sigma^+<0$ (with $\partial_X\tilde P(\alpha^-,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)>0$), then $\tilde P(x,\gamma^+)<0$ and $\tilde P(x',\gamma^+)>0$ while
$\tilde P(\alpha^-,y)<0$ and $\tilde P(\alpha^-,y')>0$.
\begin{itemize}
\item If $\tilde P(\alpha^-,\gamma^+)>0$, then $\xi<\gamma^-$ and $\eta<\alpha^+$.
\begin{center} \includegraphics[width=4.5cm,height=4cm]{Dessins/corner1.png}
\end{center}
So that
$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=\#L_{\gamma^+}^{=\alpha^-}\cap [x,x']=0,\#L_{\alpha^-}\cap [y,y']=1
$
\item If $\tilde P(\alpha^-,\gamma^+)<0$,
then $\xi>\gamma^-$ and $\eta>\alpha^+$.
\begin{center} \includegraphics[width=4.5cm,height=4cm]{Dessins/corner2.png}
\end{center}
So that
$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=1,\#L_{\alpha^-}\cap [y,y']=\#L_{\gamma^+}^{=\alpha^-}\cap [x,x']=0
$
\item If $\tilde P(\alpha^-,\gamma^+)=0$ then $\xi=\alpha^-$ and $\eta=\gamma^+$.
\begin{center} \includegraphics[width=4.5cm,height=4cm]{Dessins/corner3.png}
\end{center}
So that
$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=\#L_{\alpha^-}\cap [y,y']=0,\#L_{\gamma^+}^{=\alpha^-}\cap [x,x']=1
$
\end{itemize}
In all cases, when $\sigma^+<0$ (with $\partial_X\tilde P(\alpha^-,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)>0$) $$\#L_{\gamma^+}^{<\alpha}\cap [x,x']+\#L_{\alpha^-}\cap [y,y']+\#L_{\alpha^-}\cap [y,y']=1.$$
\item Details for the remaining cases $\sigma^+>0$ with $\partial_X\tilde P(\alpha^-,\gamma^+)<0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)>0$ (resp. $\sigma^+<0$ with $\partial_X\tilde P(\alpha^-,\gamma^+)<0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)<0$) are entirely similar, and omitted.
\end{itemize}
Summarizing the situation in all the cases, the conclusion is as follows
\begin{itemize}
\item If $\sigma^+>0$ then
$$\#L_{\alpha^-}\cap [y,y']-\#L_{\gamma^+}^{<\alpha}\cap [x,x']=0
$$
\item If $\sigma^+<0$ then
$$\#L_{\alpha^-}\cap [y,y']+\#L_{\gamma^+}^{<\alpha}\cap [x,x']+\#L_{\gamma^+}^{=\alpha}\cap [x,x']=1$$
\end{itemize}
so that the sign of $\tilde P(\alpha^-,\gamma^+)$ has no influence on $\textsc{Left}$
given the formulas of Algorithm \ref{algoconnect}.
\medskip
We now analyze the situation at an ambiguous midpoint $(\alpha,\gamma^+)$.
\begin{itemize}
\item If $\sigma^+>0$ with $\partial_X\tilde P(\alpha,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha,\gamma^+)<0$, then $\tilde P(x,\gamma^+)<0$ and $\tilde P(x',\gamma^+)>0$ while
$\tilde P(\alpha,y)>0$ and $\tilde P(\alpha,y')<0$.
\begin{itemize}
\item If $\tilde P(\alpha,\gamma^+)>0$, then $\xi<\alpha$ and $\eta>\gamma^+$. So that
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=1,\#L_{\gamma^+}^{=\alpha}\cap [x,x']=\#L_\alpha^{>\beta}\cap [y,y']=0
$$
\item If $\tilde P(\alpha,\gamma^+)<0$, then $\xi>\alpha$ and $\eta<\gamma^+$. So that
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=\#L_{\gamma^+}^{=\alpha}\cap [x,x']=0,\#L_\alpha^{>\beta}\cap [y,y']=1
$$
\item If $\tilde P(\alpha,\gamma^+)=0$, then $\xi=\alpha$ and $\eta=\gamma^+$. So that
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=0, \#L_{\gamma^+}^{=\alpha}\cap [x,x']=1,\#L_{\alpha}^{>\beta}\cap [y,y']=0
$$
\end{itemize}
\hide{
\begin{tabular}{lll}
$\tilde P(\alpha,\gamma^+)>0$ & $\tilde P(\alpha,\gamma^+)<0$ &$\tilde P(\alpha,\gamma^+)=0$\\
\includegraphics[width=4.5cm,height=4cm]{Dessins/ambiguous_midpoint-1.pdf} &
\includegraphics[width=4.5cm,height=4cm]{Dessins/ambiguous_midpoint1-1.pdf}&
\includegraphics[width=4.5cm,height=4cm]{Dessins/ambiguous_midpoint3-1.pdf}
\end{tabular}
}
In all cases, if $\sigma^+>0$
(with $\partial_X\tilde P(\alpha^-,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)<0$) then
$$-\#L_{\gamma^+}^{<\alpha}\cap [x,x']-\#L_{\gamma^+}^{=\alpha}\cap [x,x']-\#L_{\alpha}^{>\beta}\cap [y,y']=-1.$$
\item If $\sigma^+<0$ with $\partial_X\tilde P(\alpha,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha,\gamma^+)>0$, then $\tilde P(x,\gamma^+)<0$ and $\tilde P(x',\gamma^+)>0$ while
$\tilde P(\alpha,y)<0$ and $\tilde P(\alpha,y')>0$.
\begin{itemize}
\item If $\tilde P(\alpha,\gamma^+)>0$, then the root $\xi$ is to the left of $\alpha$ and the root $\eta$ is under $\gamma^+$. So that
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=1,\#L_\alpha^{>\beta}\cap [y,y']=1
$$
\item If $\tilde P(\alpha,\gamma^+)<0$,
then the root $\xi$ is to the right of $\alpha$ and the root $\eta$ is above $\gamma^+$. So that
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']=0,\#L_\alpha^{>\beta}\cap [y,y']=0
$$
\item If $\tilde P(\alpha,\gamma^+)=0$ then $\xi=\alpha$ and $\eta=\gamma^+$. So that
$$\#L_{\gamma^+}^{< \alpha}\cap [x,x']=\#L_{\alpha}^{>\beta}\cap [y,y']=0
$$
\end{itemize}
\hide{
\begin{tabular}{lll}
$\tilde P(\alpha,\gamma^+)>0$ & $\tilde P(\alpha,\gamma^+)<0$ &$\tilde P(\alpha,\gamma^+)=0$\\
\includegraphics[width=4.5cm,height=4cm]{Dessins/ambiguous_midpoint4-1.pdf} &
\includegraphics[width=4.5cm,height=4cm]{Dessins/ambiguous_midpoint5-1.pdf}&
\includegraphics[width=4.5cm,height=4cm]{Dessins/ambiguous_midpoint6-1.pdf}
\end{tabular}
}
In all cases, when $\sigma^+<0$
(with $\partial_X\tilde P(\alpha^-,\gamma^+)>0$ and $\partial_Y\tilde P(\alpha^-,\gamma^+)>0$),
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']-\#L_{\alpha}^{>\beta}\cap [y,y']=0.$$
\item Details for the remaining cases $\sigma^+>0$ with $\partial_X\tilde P(\alpha,\gamma^+)<0$ and $\partial_Y\tilde P(\alpha,\gamma^+)>0$ (resp. $\sigma^+<0$ with $\partial_X\tilde P(\alpha,\gamma^+)<0$ and $\partial_Y\tilde P(\alpha,\gamma^+)<0$) are omitted.
\end{itemize}
Summarizing the situation in all the cases, the conclusion is as follows
\begin{itemize}
\item If $\sigma^+>0$ then
$$-\#L_{\gamma^+}^{<\alpha}\cap [x,x']-\#L_{\gamma^+}^{=\alpha}\cap [x,x']-\#L_{\alpha}^{>\beta}\cap [y,y']=-1
$$
\item If $\sigma^+<0$ then
$$\#L_{\gamma^+}^{<\alpha}\cap [x,x']-\#L_{\alpha}^{>\beta}\cap [y,y']=0
$$
\end{itemize}
so that the sign of $\tilde P(\alpha,\gamma^+)$ has no influence on $\textsc{Left}$
given the formulas of Algorithm \ref{algoconnect}).
The analysis for the other ambiguous corners and midpoints is similar.
So we can conclude
\begin{proposition}\label{ambigousnoproblem}
The sign of $P$ at an ambiguous corner or midpoint has no influence on $\textsc{Left}$ and $\textsc{Right}$.
\end{proposition}
So we can decide arbitrarily that $\tilde P$ is zero at all ambiguous corners and midpoints. In order to obtain the quantities
$$
\#L_{\alpha^-},\#L_{\gamma^-}^{= \alpha^-}, \#L_{\gamma^-}^{ < \alpha}, \#L_{\gamma^-}^{= \alpha}, \#L_{\gamma^+}^{= \alpha^-}, \#L_{\gamma^+}^{ < \alpha}, \#L_{\gamma^+}^{= \alpha}, \#L_{\alpha}^{>\beta}, \#L_{\alpha}^{<\beta}
$$
we now use the output of Theorem \ref{sing-fibers} and the extra information that $\tilde P$ is zero at all ambiguous corners and midpoints.
Finally, we use these quantities to compute correctly $\textsc{Left}$
according to the formulas of Algorithm \ref{algoconnect}. The situation is similar for $\textsc{Right}$.
All in all, we proved
\begin{proposition}\label{leftright}
Using $\tilde O(d^5\tau+d^6)$ bit-operations, we can compute $\textsc{Left}_{i,j}$
and $\textsc{Right}_{i,j}$ for every $i,j$, $i=1,\ldots,N$, $j\in \textsc{CritInd}_i$.
\end{proposition}
\subsection{Final topology}\label{topology}
For $(\alpha_i,\beta_{i,j})$ critical, i.e. $j\in \textsc{CritInd}_i$, we denote as before by $\textsc{Left}_{i,j}$ (resp. $\textsc{Right}_{i,j}$) the number of segments arriving at $(\alpha_i,\beta_{i,j})$ inside $[a_i, \alpha_i] \times [c_{i,j}, d_{i,j}]$ (resp. $[\alpha_i, b_i] \times [c_{i,j}, d_{i,j}]$). Note that this information has been computed in Proposition \ref{leftright} using $\tilde{O}(d^5\tau+d^6)$ bit operations.
For $j\notin \textsc{CritInd}_i$, we take $\textsc{Left}_{i,j}=\textsc{Right}_{i,j}=1$.
We denote as before by $\textsc{Left}_{i,0}$ (resp. $\textsc{Right}_{i,0}$) the number $\ell_{i}^{-\infty}$ (resp $\ell_{i}^{-\infty}$) of vertical asymptotes tending to $-\infty$ at the left (resp. the right) of $\alpha_i$ and by $\textsc{Left}_{i,m_i+1}$ (resp. $\textsc{Right}_{i,m_i+1}$) the number of vertical asymptotes tending to $+\infty$ at the left (resp. the right) of $\alpha_i$. Note that this information has been already determined in Proposition \ref{asympt}.
The topology of $V_{\mathbb{R}}(\tilde P)$ is encoded by the finite list $$\tilde{\mathcal{L}}(\tilde P)=[m'_0,L_1,\ldots,L_N ,m'_N]$$ where
\begin{itemize}
\item[-] $L_i=[m_{i},[[\textsc{Left}_{i,j},\textsc{Right}_{i,j}], 0 \le j\le m_{i}+1]]$ for $i=1,\ldots,N$,
\end{itemize}
In the special case where $\deg_X(\tilde P(X,Y))=0$ and $V_{\mathbb{R}} (\tilde P)$ is a finite number of horizontal lines , we compute the number $m$ of real roots of $\tilde P(X,Y)$ which is a polynomial in $Y$.
Similarly in the special case where $S_X(X)$ has no real root, we compute the number $m$ of real roots of $\tilde P(0,Y)$.
In both special cases, the topology of $V_{\mathbb{R}}(\tilde P)$ is encoded by $\tilde{\mathcal{L}}(\tilde P)=[m]$.
\begin{exemple}\label{exemple1}
Now we illustrate the previous result by taking an example, where
$$P(X,Y)= (4X+1)(8X-1)(16X-1)(XY-1)(4Y^2-4X-1)(4Y^2+4X-1),$$
and
$$\tilde P(X,Y)= (XY-1)(4Y^2-4X-1)(4Y^2+4X-1).$$
We have
$$D_X=-2^{28}\,X^4\,\left(4\,X-1\right)\,\left(1+4\,X\right)\,\left(4-
X^2+4\,X^3\right)^2\,\left(-4+X^2+4\,X^3\right)^2.$$
Since
$4-
X^2+4\,X^3$ (resp. $-4+X^2+4\,X^3$) has one real root on $[-1,-1/2]$ (resp. on $[1/2,1]$), we have $N=5$.
We obtain
$$\tilde{\mathcal{L}}(\tilde P)=[3,L_1,3,L_2,5,L_3,5,L_4,3,L_5 ,3]$$ where
\begin{itemize}
\item[-] $L_1=[2,[[0,0],[2,2],[1,1],[0,0]]]$
\item[-] $L_2=[4,[[0,0],[1,1],[1,1],[0,2],[1,1],[0,0]]]$
\item[-] $L_3=[2,[[1,0],[2,2],[2,2],[0,1]]]$
\item[-] $L_4=[4,[[0,0],[1,1],[2,0],[1,1],[1,1],[0,0]]]$
\item[-] $L_5=[2,[[0,0],[1,1],[2,2],[0,0]]]$
\end{itemize}
For example the list $L_3$ means that above $\alpha-3=0$ there is
one asymptote going to $-\infty$ at the left of $\alpha_3$, then two singular points with two branches to the left and two branches to the right and one asymptote going to $+\infty$ at the right of $\alpha_3$.
\begin{figure}[!htp]
\hfill\hbox to 0pt{\hss\includegraphics[width=8cm,height=8cm]{Dessins/sagetopology1.pdf}\hss}\hfill\null
\end{figure}
\end{exemple}
A straight-line planar graph $\widetilde{\mathrm{Gr}}(\tilde P)$
can be obtained as follows,
\begin{itemize}
\item define $d'=\max(\max_{i=0,\ldots, N} m'_i,\max_{i=1,\ldots,N} m_i))$,
\item for each $i=0,\ldots, N$ include the points $$I_{i,j}=\left(2i+1,\frac{j(d'+1)}{m'_i+1}\right)$$ for $j$ from $1$ to $m'_{i}$,
\item for each $i=1,\ldots, N$ include the points $$P_{i,j}=\left(2i,\frac{j(d'+1)}{m_i+1}\right)$$ for $j$ from $0$ to $m_{j}+1$,
\item include the points $$P_{-\infty,j}=\left(0,\frac{j(d'+1)}{m'_0+1}\right)$$ for $j$ from $1$ to $m'_{0}$ as well as the points $$P_{+\infty,j}=\left(2(N+1),\frac{j(d'+1)}{m'_N+1}\right)$$ for $j$ from $1$ to $m'_{N}$,
\item for $i=1,\ldots,N$, add the segment $I_{i-1,\ell}P_{i,j}$ (resp. $P_{i,j}I_{i,r}$) if $$\sum_{k=0}^{j-1} \textsc{Left}_{i,k}<\ell \le \sum_{k=0}^{j}\textsc{Left}_{i,k} (\mbox{resp.} \sum_{k=0}^{j-1} \textsc{Right}_{i,k}<r \le \sum_{k=0}^{j} \textsc{Right}_{i,k})$$
\item add the segments $P_{-\infty,j}I_{0,j}$ for $j=1,\ldots, m'_0$ and the segments
$I_{N,j} P_{\infty,j}$ for $j=1,\ldots, m'_N$
\end{itemize}
It is clear that
\begin{proposition}
$\widetilde{\mathrm{Gr}}(\tilde P)\subset\mathbb (0,2 N)\times (0,d'+1)$
is homemorphic to $V_{\mathbb{R}}(\tilde P)\subset\mathbb{R}^2$.
\end{proposition}
We finally explain how to add vertical lines back.
We remind that $c(X)$ is the gcd of all the coefficients $c_i(X)$ of $P(X,Y)$ written as an element of $\mathbb{Z}[X][Y]$
and we consider the square free part $c^\star(X)$ of $c(X)$.
Define:
\begin{itemize}
\item $c_1(X):=$ gcd $(c^\star(X),D_X(X))$ and $c_2(X):= $quo$(c^\star(X),c_1(X))$,
\item $\mathcal{V}_{1} := \{(x,y)\in \mathbb{R}^2 | c_{1}(x)=0\}$ and $\mathcal{V}_{2} := \{(x,y)\in \mathbb{R}^2 | c_{2}(x)=0\}$.
\end {itemize}
Hence $\mathcal{V}_{1}$ is the subset of vertical lines of $V_{\mathbb{R}} ({P})$ passing through critical values of $V_{\mathbb{R}} (\tilde{P})$ while $\mathcal{V}_{2}$ is the subset of those passing between critical values of $V_{\mathbb{R}} (\tilde{P})$. The computation of $c_{1}(X)$ and $c_{2}(X)$ has respectively bit complexities of $\tilde{O}(d^4 \tau + d^5)$ and $\tilde{O}(d\tau + d^2)$ according to Proposition \ref{gcd-comp} and Proposition \ref{exact_division_comp}.
To add back the lines in $\mathcal{V}_{1}$ to $V_{\mathbb{R}} (\tilde{P})$, it suffices to identify the real roots of $c_{1}(X)$ as roots of $D_X(X)$, i.e. to decide whether the vertical line defined by $X=\alpha_i$ belongs to $V_{\mathbb{R}}(P)$. Such identification has bit complexity $\tilde{O}(d^5 \tau + d^6)$ according to Proposition \ref{comparingroots}. To add back the lines in $\mathcal{V}_{2}$ to $V_{\mathbb{R}} (\tilde{P})$, it suffices to count the number of real root of $c_{2}(X)$ on $\mathcal{I}_i$. This has bit complexity $\tilde{O} (d^5 \tau + d^6)$ according to Proposition \ref{comparingroots}.
\begin{proposition}
Let $P \in \mathbb{Z} [X, Y]$ a square-free polynomial of total degree $d$
and integer coefficients of bitsize bounded by $\tau$. Adding back the vertical lines of $V_{\mathbb{R}} (P)$ to $V_{\mathbb{R}} (\tilde{P})$ has bit complexity $\tilde{O}
(d^5 \tau + d^6)$.
\end{proposition}
For a complete combinatorial description of the topology of $V_{\mathbb{R}}(P)$ we define the finite list $${\mathcal{L}}( P)=[N'_0,L'_1,\ldots,L'_\delta ,N'_\delta]$$ where
\begin{itemize}
\item[-] $L'_i=[[m_{i},w_i],[[\ell_{i,j},r_{i,j}], 0 \le j\le m_{i}+1]]$ for $i=1,\ldots,\delta$,
\item[-] $N'_i=[m'_i,v_i]$ for $i=0,\ldots,\delta$,
\end{itemize}
where $w_i=1$ if the line $X=\alpha_i$ belongs to $V_{\mathbb{R}}(P)$ for $i=1,\ldots, \delta$ and $w_i=0$ otherwise and $v_i$ is the number of distinct vertical lines $X=x$ with $\alpha_i<x<\alpha_{i+1}$ for $i=1,\ldots,\delta-1$ and $v_0$ (resp. $v_\delta$) is the number of distinct vertical lines $X=x$ with $x<\alpha_1$ (resp. $x>\alpha_\delta$).
Finally the straight-line planar graph $\mathrm{Gr}(P)$
is defined as $$\mathrm{Gr}(\tilde P)\cup \bigcup_{i=1,\ldots,\delta\atop w_i=1} V_i \cup \bigcup_{i=0,\ldots,\delta\atop \ell=1,\ldots,v_i} V_{i,\ell}$$
with $V_i$ the vertical segment defined by $X=2i, 0<Y<d'+1$ and $V_{i,\ell}$ for $i=0,\ldots,\delta,\ell=1,\ldots,v_i$ is the vertical segment defined by the equation
$$X=2i+\frac{2\ell}{v_i+1},0<Y<d'+1.$$
It is clear that
\begin{proposition}
$\mathrm{Gr}(P)\subset\mathbb (0,2\delta)\times (0,d'+1)$
is homemorphic to $V_{\mathbb{R}}(P)\subset\mathbb{R}^2$.
\end{proposition}
\begin{exemple}\label{exemple2}
Continuing Example \ref{exemple1},
we have
three vertical lines of equation
$$X=\frac{-1}{4},X=\frac{1}{8},X=\frac{1}{16}$$
We obtain
$${\mathcal{L}}(P)=[N'_0,L'_1,N'_1,L'_2,N'_2,L'_3,N'_3,L'_4,N'_4,L'_5 ,N'_5]$$ where
\begin{itemize}
\item[-] $N'_0=[3,0]$
\item[-] $L_1=[[2,0],[[0,0],[2,2],[1,1],[0,0]]]$
\item[-] $N_1=[3,0]$
\item[-] $L_2=[[4,1],[[0,0],[1,1],[1,1],[0,2],[1,1],[0,0]]]$
\item[-] $N_2=[5,0]$
\item[-] $L_3=[[2,0],[[1,0],[2,2],[2,2],[0,1]]]$
\item[-] $N_3=[5,2]$
\item[-] $L_4=[[4,0],[[0,0],[1,1],[2,0],[1,1],[1,1],[0,0]]]$
\item[-] $N_4=[3,0]$
\item[-] $L_5=[[2,0],[[0,0],[1,1],[2,2],[0,0]]]$
\item[-] $N_5=[3,0]$
\end{itemize}
\begin{figure}[!htp]
\hfill\hbox to 0pt{\hss\includegraphics[width=8cm,height=8cm]{Dessins/sagetopology2.pdf}\hss}\hfill\null
\end{figure}
\end{exemple}
Finally, summarizing our results, we proved Theorem \ref{finaltopology}.
\newpage
\bibliographystyle{plain}
|
1,314,259,995,773 | arxiv | \section{}
Present-day technology pushes device dimensions toward limits
where the traditional semiclassical or Boltzmann theory~\cite{ST} can
no longer be applied, and more rigorous quantum-kinetic
approaches are imperative \cite{QT}. However, in spite of the
quantum-mechanical nature of electron and photon dynamics in the core region
of typical solid-state nanodevices ---e.g., superlattices \cite{SL} and quantum-dot structures \cite{QD}--- the overall behavior of such quantum systems is often governed by a complex interplay between phase coherence and energy
relaxation/dephasing \cite{RMP}, the latter being also due to
the presence of spatial boundaries \cite{Frensley}. Therefore, a
proper treatment of such novel nanoscale devices requires a
theoretical modeling able to properly account for both coherent
and incoherent ---i.e., phase-breaking--- processes on the same
footing.
The wide family of so-called solid-state quantum devices can be schematically divided into
two main classes:
(i) a first one which comprises
low-dimensional nanostructures whose electro-optical response may be
safely treated within the semiclassical picture \cite{SP} (e.g., quantum-cascade lasers \cite{QCL}),
and (ii) a second one grouping solid-state devices
characterized by a genuine quantum-mechanical behavior of their
electronic subsystem (e.g., solid-state quantum logic gates \cite{QLG}) whose quantum evolution is only weakly disturbed by
decoherence processes.
For purely atomic and/or photonic quantum logic gates, decoherence phenomena are successfully described via adiabatic-decoupling procedures \cite{QO}
in terms of extremely simplified models via phenomenological parameters; within such effective treatments, the main goal/requirement is to identify a suitable form of the Liouville superoperator, able to ensure/maintain the positive-definite character of the corresponding density-matrix operator \cite{QOS}.
This is usually accomplished by identifying proper Lindblad-like decoherence superoperators \cite{QOS,Lindblad}, expressed in terms of a few crucial system-environment coupling parameters \cite{constrains}.
In contrast, solid-state devices are often characterized by a complex many-electron quantum evolution, resulting in a non-trivial interplay between coherent dynamics and
energy-relaxation/decoherence processes; it follows that for a
quantitative description of such coherence/dissipation
coupling the latter need to be treated via fully microscopic
models.
To this aim, motivated by the power and flexibility of the semiclassical kinetic theory \cite{ST} in describing a large variety of interaction mechanisms, a quantum generalization of the standard Boltzmann collision operator has been proposed \cite{RMP}; the latter, obtained via the conventional Markov limit, describes the evolution of the reduced density matrix in terms of in- and out-scattering superoperators.
However, contrary to the semiclassical case, such collision superoperator does not preserve the positive-definite character of the density-matrix operator.
To overcome this serious limitation, in this Letter we shall propose an alternative adiabatic procedure which (i) in the semiclassical limit reduces to the standard Fermi's golden rule \cite{FGR}, and (ii) describes a genuine Limblad evolution, thus providing a reliable/robust treatment of energy-dissipation and dephasing processes.
In order to discuss the main features and intrinsic limitations of the conventional adiabatic or Markov limit, let us recall its general derivation following the fully operatorial approach proposed in \cite{PRB}.
Given a generic physical quantity $A$ ---described by the operator
${\hat A}$--- its quantum plus statistical average value is given by
$A = {\rm tr}\left\{{\hat A} {\hat \rho}\right\}$,
where ${\hat \rho}$ is the so-called density-matrix operator. Its time evolution is dictated by the total (system plus environment) Hamiltonian. Within the usual interaction picture, the latter can be regarded as the sum of a noninteracting (system plus environment) contribution plus a system-environment coupling term:
${\hat H} = {\hat H}_\circ + {\hat H}'$;
the corresponding equation of motion for the density-matrix operator ---also known as Liouville-von Neumann equation--- in the interaction picture is given by:
\begin{equation}\label{LvN_i}
{d{\hat \rho}^i \over dt} = -i \left[\hat{\cal H}^i, {\hat \rho}^i\right]\ ,
\end{equation}
where $\hat{\cal H}$ denotes the interaction Hamiltonian $\hat{H}'$ written in units of $\hbar$.
The key idea beyond any perturbation approach is that the effect
of the interaction Hamiltonian ${\hat H}'$ is ``small'' compared
to the free evolution dictated by the noninteracting Hamiltonian
${\hat H}_\circ$.
Following this spirit, by formally integrating Eq.~(\ref{LvN_i}) from $-\infty$ to the current time $t$,
and inserting such formal solution
for ${\hat \rho}^i(t)$
on the right-hand side of Eq.~(\ref{LvN_i}),
we obtain an integro-differential equation of the form:
\begin{widetext}
\begin{equation}\label{IDE}
{d \over dt} {\hat \rho}^i(t) = -i \left[\hat{\cal H}^i(t), {\hat \rho}^i(-\infty)\right]
-
\int_{-\infty}^t dt'
\left[\hat{\cal H}^i(t), \left[\hat{\cal H}^i(t'), {\hat \rho}^i(t')\right]\right]\ .
\end{equation}
\end{widetext}
We stress that so far no approximation has been introduced: Equations (\ref{LvN_i}) and
(\ref{IDE}) are fully equivalent, we have just isolated the first-order contribution from the full
time evolution in Eq.~(\ref{LvN_i}).
Let us now focus on the time
integral in Eq.~(\ref{IDE}). Here, the two quantities to be integrated over $t'$ are the interaction
Hamiltonian $\hat{\cal H}^i$ and the density-matrix operator ${\hat \rho}^i$. In the spirit of the perturbation
approach previously recalled, the time variation of ${\hat \rho}^i$ can be considered adiabatically slow
compared to that of the Hamiltonian $\hat{\cal H}$ written in the interaction picture, i.e.,
$\hat{\cal H}^i(t') = {\hat U}^\dagger_\circ(t') \hat{\cal H} {\hat U}^{ }_\circ(t')$;
indeed, the latter exhibits rapid oscillations due to the noninteracting evolution operator ${\hat U}_\circ(t) = e^{-{i{\hat H}_\circ t \over \hbar}}$.
As a result, the density-matrix operator ${\hat \rho}^i$ can be taken out of the time integral and
evaluated at the current time $t$.
Following such prescription, the second-order contribution to the system dynamics
written in the Schr\"odinger picture
for the case of a time-independent interaction Hamiltonian $\hat{\cal H}$ comes out to be:
\begin{equation}\label{LvN-eff}
{d{\hat \rho} \over dt} =
-{1 \over 2} \left[\hat{\cal H},
\left[\hat{\cal K},{\hat \rho}\right]\right]
\end{equation}
with
\begin{equation}\label{calK}
\hat{\cal K}
=
2 \int_{-\infty}^{0} dt'
\hat{\cal H}^i(t')
=
2 \int_{-\infty}^{0} dt' {\hat U}^{ }_\circ(t') \hat{\cal
H} {\hat U}^\dagger_\circ(t') \ .
\end{equation}
As discussed extensively in \cite{PRB}, the operator $\hat{\cal K}$ describes
energy-conserving scattering events (real processes) as well as energy-renormalization contributions (virtual processes); the latter are known to play a minor role, and in general may be safely neglected; such approximation amounts to impose the following time-reversal symmetry on the system dynamics:
$\hat{\cal H}^i(t) = \hat{\cal H}^i(-t)$ \cite{renormalizations}.
Within such approximation scheme, the operator $\hat{\cal K}$ in (\ref{calK}) may be rewritten extending the time integration from $-\infty$ to $+\infty$:
\begin{equation}\label{calK-bis}
\hat{\cal K} = \int_{-\infty}^{+\infty} dt' {\hat U}^{ }_\circ(t') \hat{\cal
H} {\hat U}^\dagger_\circ(t') \ .
\end{equation}
The effective equation in (\ref{LvN-eff}) has still the double-commutator structure in (\ref{IDE}) but it is now local in time.
The Markov limit recalled so far leads to significant modifications
to the system dynamics: while the exact quantum-mechanical
evolution in (\ref{LvN_i}) corresponds to a fully reversible and
isoentropic unitary transformation, the instantaneous
double-commutator structure in (\ref{LvN-eff}) describes, in
general, a non-reversible (i.e., non unitary) dynamics characterized by energy dissipation and dephasing.
However, since any effective Liouville superoperator should describe correctly the time evolution of
$\hat\rho$
and since the latter, by definition, needs to be trace-invariant and positive-definite at any time,
it is imperative to determine if the Markov superoperator
in (\ref{LvN-eff}) fulfills this two basic requirements.
As far as the first issue is concerned, in view of its commutator structure, it is easy to show that this effective superoperator is indeed trace-preserving.
In contrast, as discussed extensively in \cite{PRB}, the latter does
not ensure that for any initial condition the density-matrix
operator will be positive-definite at any time.
This is by far the most severe limitation of the conventional Markov approximation.
By denoting with $\{\vert \lambda \rangle\}$ the eigenstates of the noninteracting Hamiltonian $\hat{H}_\circ$, the effective equation (\ref{LvN-eff}) written in this basis is of the form:
\begin{equation}\label{LvN-eff-lambda}
{d\rho_{\lambda_1\lambda_2} \over dt} =
{1 \over 2} \sum_{\lambda'_1\lambda'_2}
\left[{\cal P}_{\lambda_1\lambda_2,\lambda'_1\lambda'_2}
\rho_{\lambda'_1\lambda'_2}
-
{\cal P}_{\lambda_1\lambda'_2,\lambda'_1\lambda'_1}
\rho_{\lambda'_2\lambda_2} \right] + {\rm H.c.}
\end{equation}
with generalized scattering rates given by:
\begin{equation}\label{calP}
{\cal P}_{\lambda_1\lambda_2,\lambda'_1\lambda'_2} = {2\pi \over \hbar} H'_{\lambda_1\lambda'_1} H^{\prime *}_{\lambda_2\lambda'_2} \delta(\epsilon_{\lambda_2} - \epsilon_{\lambda'_2}) \ ,
\end{equation}
$\epsilon_\lambda$ denoting the energy corresponding to the noninteracting state $\vert \lambda \rangle$.
The well-known semiclassical or Boltzmann theory~\cite{ST} can be easily derived from the quantum-transport
formulation presented so far, by introducing the so-called
diagonal or semiclassical approximation. The latter corresponds to
neglecting all non-diagonal density-matrix elements (and therefore
any quantum-mechanical phase coherence between the generic states
$\lambda_1$ and $\lambda_2$), i.e.,
$\rho_{\lambda_1\lambda_2} = f_{\lambda_1} \delta_{\lambda_1\lambda_2}$,
where the diagonal elements $f_\lambda$ describe the semiclassical
distribution function over our noninteracting basis states.
Within such approximation scheme, the quantum-transport equation (\ref{LvN-eff-lambda}) reduces to the well-known Boltzmann equation:
\begin{equation}\label{BTE}
{d f_\lambda \over dt} =
\sum_{\lambda'} \left(
P_{\lambda\lambda'} f_{\lambda'} - P_{\lambda'\lambda} f_\lambda
\right)\ ,
\end{equation}
where
\begin{equation}\label{P}
P_{\lambda\lambda'} = {\cal P}_{\lambda\lambda,\lambda'\lambda'} =
{2\pi \over \hbar} |H'_{\lambda\lambda'}|^2 \delta\left(\epsilon_{\lambda}-\epsilon_{\lambda'}\right)
\end{equation}
are the conventional semiclassical scattering rates given by the well-known
Fermi's golden rule \cite{FGR}.
At this point it is crucial to stress that, contrary to the non-diagonal density-matrix description
previously introduced, the Markov limit combined with the semiclassical or diagonal approximation ensures that at any time $t$ our semiclassical distribution function $f_\lambda$
is always positive-definite.
This explains the ``robustness'' of the Boltzmann transport equation (\ref{BTE}), and its extensive application in solid-state-device modeling as well as in many other areas, where quantum effects play a very minor role.
In contrast, in order to investigate genuine quantum-mechanical phenomena, the conventional Markov superoperator in (\ref{LvN-eff}) cannot be employed, since it does not preserve the positive-definite character of the density matrix $\rho_{\lambda_1\lambda_2}$.
As anticipated, aim of the present Letter is to propose an alternative formulation of the standard Markov limit, able to provide a Lindblad-like scattering superoperator, thus preserving the positive-definite character of our density matrix.
To this end, let us go back to the integro-differential equation (\ref{IDE}). As previously discussed, the crucial step in the standard derivation is to replace $\hat\rho^i(t')$ with $\hat\rho^i(t)$. Indeed, since in the adiabatic limit the time variation of the density matrix within the interaction picture is negligible, the latter can be evaluated at any time. Based on this remark, what we propose is the following time symmetrization: given the two times $t'$ and $t$, we shall introduce the ``average'' or ``macroscopic'' time $T = {t+t' \over 2}$ and the ``relative'' time $\tau = t-t'$. The basic idea is that the relevant time characterizing/describing our effective system evolution
is the macroscopic time $T$. Following this spirit, it is easy to rewrite the second-order contribution in Eq.~(\ref{IDE}) in terms of the new time variables $T$ and $\tau$:
\begin{widetext}
\begin{equation}\label{IDE-new}
{d \over dT} {\hat \rho}^i(T) =
-
\int_0^\infty d\tau
\left[\hat{\cal H}^i\left(T+ {1 \over 2} \tau\right), \left[\hat{\cal H}^i\left(T- {1 \over 2}\tau\right), {\hat \rho}^i\left(T-{1 \over 2}\tau\right)\right]\right]\ .
\end{equation}
\end{widetext}
Following again the spirit of the adiabatic decoupling, we shall now replace $\hat\rho^i\left(T-{1 \over 2}\tau\right)$ with $\hat\rho^i(T)$; the resulting effective equation rewritten in the original Schr\"odinger picture comes out to be:
\begin{equation}\label{LvN-eff-new1}
{d \over dT} {\hat \rho}(T) =
-
\int_{0}^{\infty} d\tau
\left[\hat{\cal H}^i\left({1 \over 2} \tau\right), \left[\hat{\cal H}^i\left(-{1 \over 2}\tau\right), {\hat \rho}(T)\right]\right]\ .
\end{equation}
Neglecting again renormalization contributions (see note \cite{renormalizations}),
Eq. (\ref{LvN-eff-new1}) may be rewritten by extending the time integration over $\tau$ from $-\infty$ to $+\infty$:
\begin{equation}\label{LvN-eff-new2}
{d \over dT} {\hat \rho}(T) =
-{1 \over 2}
\int_{-\infty}^{+\infty} d\tau
\left[\hat{\cal H}^i\left({1 \over 2} \tau\right), \left[\hat{\cal H}^i\left(-{1 \over 2}\tau\right), {\hat \rho}(T)\right]\right]\ .
\end{equation}
By Fourier expanding the above symmetric convolution integral we finally get the desired Lindblad-like scattering superoperator:
\begin{equation}\label{Lindblad}
{d {\hat \rho}\over dT} =
-{1 \over 2}
\int d\omega \left[\hat{\cal L}(\omega), \left[\hat{\cal L}(\omega), {\hat \rho}\right]\right]
\end{equation}
with
\begin{equation}\label{calL}
\hat{\cal L}(\omega) = {1 \over \sqrt{2\pi}} \int_{-\infty}^\infty d\tau {\hat U}^{ }_\circ\left(-{1 \over 2}\tau\right) \hat{\cal
H} {\hat U}^\dagger_\circ\left(-{1 \over 2}\tau\right) e^{i\omega\tau} \ .
\end{equation}
We stress how the proposed time symmetrization gives rise to a fully symmetric Lindblad-like superoperator (expressed in terms of the operator $\hat{\cal L}$ only), compared to the strongly asymmetric Markov superoperator in (\ref{LvN-eff}).
If we now rewrite the new Markov superoperator in (\ref{LvN-eff-new2}) in our noninteracting basis $\lambda$, we obtain again the effective equation of motion in
(\ref{LvN-eff-lambda}), but now the generalized scattering rates in (\ref{calP}) are replaced by the following symmetrized quantum scattering rates:
\begin{equation}\label{calPtilde}
\tilde{\cal P}_{\lambda_1\lambda_2,\lambda'_1\lambda'_2} = {2\pi \over \hbar} H'_{\lambda_1\lambda'_1} H^{\prime *}_{\lambda_2\lambda'_2} \delta\left({\epsilon_{\lambda_1}+\epsilon_{\lambda_2}\over 2} - {\epsilon_{\lambda'_1}+\epsilon_{\lambda'_2}\over 2} \right) \ .
\end{equation}
The above scattering superoperator can be regarded as the quantum-mechanical generalization of the conventional Fermi's golden rule; indeed, in the semiclassical limit ($\lambda_1=\lambda_2,\lambda_1'=\lambda_2'$)
the standard formula in (\ref{P}) is readily recovered.
At this point a few comments are in order.
As discussed extensively in \cite{PRB}, also for the simplest case of a standard two-level system ---i.e., a generic quantum bit--- the standard Markov superoperator predicts a non-trivial coupling between level population and polarization described by the so-called $T_3$ contributions.
In contrast, for a two-level system coupled to its environment, the proposed quantum Fermi's golden rule does not predict any $T_3$ coupling term, thus providing a rigorous derivation of the well-known and successfully employed $T_1 T_2$ dephasing model \cite{SL}.
Moreover, it is imperative to stress that in the presence of a strong system-environment interaction the adiabatic decoupling investigated so far needs to be replaced by more realistic treatments, expressed via non-Markovian integro-differential equations of motion (i.e., with ``memory effects'') \cite{RMP}.
Again, while for purely atomic and/or photonic systems it is possible to identify effective non-Markovian evolution operators \cite{nonmarkovian}, for solid-state quantum devices this is still an open problem.
To summarize, we have critically reviewed the standard adiabatic or Markov procedure, showing its intrinsic failure in describing the proper quantum-mechanical evolution of a generic subsystem interacting with its environment. More specifically, we have shown that within the Markov approximation the density-matrix operator is not necessarily positive-definite, thus leading to highly non-physical results. To overcome this serious limitation, we have identified an alternative adiabatic procedure which (i) in the semiclassical limit reduces to the standard Fermi's golden rule, and (ii) describes a genuine Limblad evolution, thus providing a reliable/robust treatment of energy-dissipation and dephasing in state-of-the-art quantum devices.
\medskip\par\noindent
We are grateful to David Taj for stimulating and fruitful discussions.
|
1,314,259,995,774 | arxiv | \section{Introduction}
\label{intro} In quantum information processing (QIP), it is
desirable to achieve a high-fidelity transfer of quantum states
between different parts, such as the core processor, storage, etc.,
of a quantum computer. To this end, a variety of solid-state spin
networks with always-on interactions have been proposed [1-16].
Particularly, Christandl et al. showed that with elaborately
designed modulated exchange couplings between neighboring spins, one
can implement perfect quantum state transfer (QST) over arbitrary
distances between the opposite ends of a XX spin chain or between
the two antipodes of the one-link and the two-link hypercubes with
however the maximum perfect communication distance $\rm2log_{3}\it
N$ [3,4]. In addition, they also showed that these modulated spin
structures can distribute arbitrary entanglement between two distant
parties. Zhang and Long et al. [13] realized this perfect state
transfer algorithm in a three-qubit XX chain using liquid NMR
system. Later, Shi et al. presented a class of more general
pre-engineered perfect spin channels [6] according to the
spectrum-parity-matching condition (SPMC) they deduced. Then, Kostak
et al. [14] established a general formalism for engineering spin
Hamiltonians for perfect state transfer in networks of arbitrary
topology and coupling configuration. Christandl's innovative works
were extended by Jafarizadeh and Sufiani in a recent work [15], in
which they adopted distance-regular graphs as spin networks and
found that any such network (not just the hypercube) can achieve
unit fidelity of state transfer over arbitrarily long distances.
Moreover, D'Amico et al. [10] showed that one can create and
distribute entanglement with an interaction-modulated Y-shaped spin
network, particularly, with a slightly complicated bifurcation
structure, the distributed entanglement can be frozen when a phase
flip is applied to one spin out of each pair.
In addition to the above-mentioned protocols which mainly
concentrated on spin chains with nearest-neighbor (NN) couplings, in
Ref. [17] Paternostro et al. studied QST in imperfect artificial
spin networks with all the qubits are mutually coupled (in which the
usually assumed NN coupling is invalid). They presented a strategy
to avoid the spoiling effects of these redundant connections with a
modification of the couplings of the first and the last qubits in
the chain, which enables nearly optimal state transfer. Then in Ref.
[18] Kay demonstrated that perfect state transfer is also possible
in the presence of next-nearest-neighbor (NNN) couplings. Moreover,
compared to the case where the system contains only two-spin
interactions, the authors in Ref. [19] presented a scheme of QST by
introducing the three-spin interaction, and showed that they can
significantly increase the speed of QST in an XY chain. Besides the
spin-half systems, state and entanglement transfer driven by a
bilinear-biquadratic (BB) spin-1 Heisenberg chain was also discussed
recently [20], in which the authors concentrated on the relations
between the transfer efficiency and the quantum phase transitions.
Most recently, a milestone work appears in Ref. [21] presented a
control-limited scheme [22] for perfect state transfer through a
pre-engineered spin chain with the help of local end-chain
single-qubit operations. While nearly all of the previous schemes
whose achievements of perfect state transfer relies crucially on the
preparation of the spin medium in a fiducial pure state, the authors
in Ref. [21] demonstrated that state initialization of the spin
medium is inessential to the performance of the protocol if proper
encoding at the end of the chain is performed. The key requirements
for their scheme are the arrangement of proper time evolution and
the performance of clean projective measurements on the two end
spins. This innovative work considerably relaxes the prerequisites
for obtaining reliable QST across interacting-spin systems.
Stimulated by this innovative work, in Ref. [23] Markiewicz and
Wie\'{s}niak proposed a special type of two-qubit encoding strategy
for perfect state transfer, where no remote-cooperated global state
initialization and any additional communication are needed.
Apart from these exciting progresses, we noted that although there
are several works [24-36] concerning the decoherence effects on
entanglement dynamics, studies thus far has seldom consider the
influence of different kinds of decoherence scenarios on transfer of
quantum states due to the complex and unclear mechanism of its
interaction with the environments. However, from a practical point
of view, all the real physical systems, especially a solid-state
system, will unavoidably be influenced by its surrounding
environments. This influence can cause the initial state of the
system of interest becomes entangled with the environment in an
uncontrollable way, and it is just this entanglement of the system
with the environment that causes decoherence. The decoherence can
greatly affects the transfer efficiency of quantum states, as well
as generation and distribution of entanglement, and thus becomes one
of the dominating obstacles baffling the physical implementation of
QIP. It is therefore of great importance and fundamentally
interesting to find ways to prevent or minimize the detrimental
effects in the practical realization of QIP.
The standard way to investigate decoherence is to consider the
system of interest as a part of a larger closed system involving the
environment, and the density operator of the system can then be
obtained by tracing out all other degrees except quantum states of
the system. In the present paper, however, we would like to resort
to a different approach, i.e., the scenario of the so-called
intrinsic decoherence proposed by Milburn [37], who modified the
Schr\"{o}dinger equation in such a way that quantum coherence is
automatically destroyed as the system evolves. Such a consideration
is fed by two motivations. First, this model is amenable to exact
analytical treatment as we will see, one can determine the density
operator of the system at arbitrary time $t$ by the sole knowledge
of the eigenvalues and eigenvectors of the system. Second, although
the absence of unitarity for a closed system in this model makes it
unlikely to be a fundamental description of decoherence, its
stochastic behavior in time evolution may still be an effective
approximation for describing the phenomenon of the system. For
example, it has been applied to describe decoherence of a single
trapped ion due to intensity and phase fluctuations in the exciting
laser pulses [38]. Dynamics of the mutual entropy of two-coupled
Josephson charge qubits with intrinsic decoherence has also been
studied recently [39]. Moreover, as pointed by the authors of Ref.
[29,40], this model may be available in approximately describing the
non-dissipative decoherence of several physical systems in the
presence of white noise.
\section{General formalism}
\label{sec:2} In this paper, we consider quantum state transfer
properties in the model of Milburn's intrinsic decoherence [37]. The
kernel of this decoherence scenario is the postulate that on
sufficiently short time steps the system does not evolve
continuously under unitary evolution but rather in a stochastic
sequence of identical unitary transformation, which can account for
the disappearance of quantum coherence as the system evolves. Based
on this assumption, Milburn obtained the master equation (in units
of $\hbar$) governing the time evolution of the system
\begin{equation}
{d\rho\over dt}={1\over\gamma} {[\exp(-i\gamma \hat{H})\rho
\exp(i\gamma \hat{H})-\rho]},
\end{equation}
where $\gamma$ is the intrinsic decoherence parameter (the mean
unitary time step). Expanding Eq. (1) to the first order in
$\gamma$, one finds
\begin{equation}
{d\rho\over
dt}={-i[\hat{H},\rho]}-{\gamma\over2}[\widehat{H},[\hat{H},\rho]].
\end{equation}
The first term on the right-hand side of Eq. (2) generates a
coherent unitary time evolution of the system, while the second
term, which does not commute with the Hamiltonian, represents the
decoherence effect on the system and generates an incoherent
dynamics of the system. In the limit of $\gamma\rightarrow 0$, the
ordinary Schr\"{o}dinger equation is recovered.
To solve Eq. (2), one can define three auxiliary superoperators
$\hat J$, $\hat S$ and $\hat L$, which satisfy
\begin{equation}
\hat J\rho=\gamma\hat H\rho\hat H,\quad\hat S\rho=-i[\hat H,
\rho],\quad \hat L\rho=-\frac{\gamma}{2}\{\hat H^{2},\rho\}.
\end{equation}
From Eq. (3) it is straightforward to show that
\begin{eqnarray}
\exp(\hat J\tau)\rho(t)&=&\sum_{l=0}^{\infty}\frac{(\gamma
\tau)^{l}}{l!}\hat H^l\rho(t)\hat H^l, \nonumber\\
\exp(\hat S\tau)\rho(t)&=&\exp(-i\hat H\tau)\rho(t)\exp(i\hat H\tau), \nonumber\\
\exp(\hat L\tau)\rho(t)&=&\exp\left(-\frac{\gamma\tau}{2}\hat
H^2\right)\rho(t)\exp\left(-\frac{\gamma\tau}{2}\hat H^2\right).
\end{eqnarray}
Thus Eq. (2) simplifies to $d\rho/dt=(\hat J+\hat S+\hat L)\rho$,
and its formal solution can be written in terms of the Kraus
operators $\hat M _{l}(t)$ as
\begin{equation}
\rho(t)=\sum_{l=0}^{\infty}\hat M_{l}(t)\rho(0)\hat M_{l}^{\dag
}(t),
\end{equation}
where $\rho(0)$ denotes the initial state of the system, $\hat
M_{l}(t)=(\gamma t)^{l/2}\hat H^l\exp(-i\hat Ht)\exp(-\gamma t\hat
H^2/2)/\sqrt{l!}$ satisfies the relation
$\sum_{l=0}^{\infty}\hat{M}_l^{\dag}(t)\hat{M}_l(t)=1$ for all time
$t$.
If we rewrite $\rho(0)$ in forms of the energy eigenstate basis as
$\rho(0)=\sum_{kk'}a_{kk'}|\psi_{k}\rangle\langle \psi_{k'}|$, then
we obtain
\begin{equation}
\rho(t)=\sum_{kk'}a_{kk'}\exp[-it(E_k-E_{k'})-\frac{\gamma
t}{2}(E_k-E_{k'})^2]|\psi_{k}\rangle\langle \psi_{k'}|,
\end{equation}
where $a_{kk'}=\langle\psi_k|\rho(0)|\psi_{k'}\rangle$, $E_k$ and
$|\psi_k\rangle$ are eigenvalue and the corresponding eigenvector of
the considered system.
For the special case that $\rho(0)$ is an eigenstate of the system,
$a_{kk'}\neq0$ only when $k=k'$, Thus from Eq. (6) one can obtain
$\rho(t)=\sum_{k}a_{kk}|\psi_k\rangle\langle\psi_k|\equiv\rho(0)$,
the system will be unaffected by the intrinsic decoherence during
the time evolution process.
Furthermore, for a spin chain Hamiltonian commutes with the total
$z$ component of the spin, i.e., $[\hat H,\sigma_{\rm tot}^{z}]=0$,
where $\sigma_{\rm tot}^{z}=\sum_{i}\sigma_{i}^{z}$, the
$2^N\otimes2^N$ Hilbert space can be decomposed into $N+1$ different
invariant subspaces, each of which is a distinct eigenspace of the
operator $\sigma_{\rm tot}^{z}$, and a system prepared in these
subspaces will remains in them. In the single-excitation invariant
subspace $\mathcal{H}_{\rm 1}$ spanned by the site basis
$|n\rangle=\sigma_n^+|0\rangle^{\otimes N}$ $(n=1, 2, ..., N)$, one
can rewrite $\rho(t)$ as $\rho(t)=\sum_{nm}b_{nm}|n\rangle\langle
m|$, then in the standard basis $\{|00\rangle, |01\rangle,
|10\rangle, |11\rangle\}$, the single qubit reduced density matrix
can be obtained as
\begin{equation}
\rho_{i}(t)=\left(\begin{array}{cc}
1-b_{ii}& 0 \\
0 & b_{ii}
\end{array}\right).
\end{equation}
Similarly, one can obtain the two-qubit reduced density matrix
between qubits $i$ and $j$ as
\begin{equation}
\rho_{ij}(t)= \left(\begin{array}{cccc}
1-b_{ii}-b_{jj}& 0& 0& 0 \\
0 & b_{jj}&b_{ji}&0 \\
0&b_{ij}&b_{ii}&0 \\
0&0&0&0
\end{array}\right).
\end{equation}
In this paper, we use the fidelity
$F=\langle\psi(0)|\rho_i(t)|\psi(0)\rangle$ as an estimation of the
quality of the state transfer from the sender to the destination
qubits [1], and adopt the concept of concurrence $C=\rm{max}\{0,
\lambda_1-\lambda_2-\lambda_3-\lambda_4\}$ as a measure of the
pairwise entanglement [31,32]. Here the quantities $\lambda_i$
$(i=1, 2, 3, 4)$ are the square roots of the eigenvalues of the
product matrix
$R=\rho(\sigma^y\otimes\sigma^y)\rho^*(\sigma^y\otimes\sigma^y)$ in
decreasing order.
From Eqs. (7), (8) and the above definitions about transfer fidelity
and concurrence, one can obtain directly that $F(N,t)=|b_{NN}|$ and
$C_{ij}(N,t)=2|b_{ij}|$ for a state initially prepared in the
$N$-dimensional subspace $\mathcal {H}_{\rm1}$.
Another quantity related to the efficiency of the quantum spin
channel of interest is the fidelity averaged over all pure states in
the Bloch sphere. The state of the whole system at the initial time
$t=0$ can be written as
\begin{equation}
|\psi(0)\rangle=\cos{\frac{\theta}{2}}|\textbf{0}\rangle+e^{i\phi}\sin{\frac{\theta}{2}}|s\rangle,
\end{equation}
where $|\textbf{0}\rangle=|00...0\rangle$,
$|s\rangle=\sigma_s^+|0\rangle^{\otimes N}$, $\theta$ and $\phi$ are
arbitrary phase angles.
For this type of initial state, its dynamics is completely
determined by the evolution in the zero and single excitation
subspace $\mathcal {H}_{\rm0\oplus1}$. From Eq. (6) one can obtain
the state at time $t$ as
\begin{eqnarray}
\rho(t)&=&\cos^2\frac{\theta}{2}|\textbf{0}\rangle\langle\textbf{0}|+\sin^2\frac{\theta}{2}\sum_{n,m=1}^{N}a_{nm}|n\rangle\langle
m|\nonumber\\&& +\left(e^{i\phi}\sin\frac{\theta}{2}
\cos\frac{\theta}{2}\sum_{n=1}^{N}b_{n}|n\rangle\langle\textbf{0}|+\rm
H.c.\right),
\end{eqnarray}
with the coefficients $a_{nm}$ and $b_n$ given by
\begin{eqnarray}
a_{nm}&=&\sum_{k,k'=1}^{N}c_{k,s}c_{k',s}c_{k,n}c_{k',m}\times\nonumber\\&&
\exp\left[-it(E_k-E_{k'})-\frac{\gamma t}{2}(E_k-E_{k'})^2\right],\nonumber\\
b_{n}&=&\sum_{k=1}^{N}c_{k,s}c_{k,n}\exp\left(-itE_k-\frac{\gamma
t}{2}E_k^2\right),
\end{eqnarray}
where $c_{k,n}$ is the amplitude of coefficient for the state
$|n\rangle$ in the eigenstate
$|\tilde{k}\rangle=\sum_{n=1}^{N}c_{k,n}|n\rangle$.
Then by tracing off the states of all other spins except $i$ from
$\rho(t)$, one has
\begin{equation}
\rho_{i}(t)=\left(\begin{array}{cc}
1-a_{ii}\sin^2\frac{\theta}{2}& b_i^*e^{-i\phi}\sin\frac{\theta}{2}\cos\frac{\theta}{2} \\
b_ie^{i\phi}\sin\frac{\theta}{2}\cos\frac{\theta}{2} & a_{ii}\sin^2\frac{\theta}{2}
\end{array}\right).
\end{equation}
From Eqs. (9), (12), the fidelity
$F=\langle\psi(0)|\rho_i(t)|\psi(0)\rangle$ can be obtained as
\begin{eqnarray}
F&=&\cos^2\frac{\theta}{2}\left(1-a_{ii}\sin^2\frac{\theta}{2}+2|b_i|\sin^2\frac{\theta}{2}\cos\alpha\right)+\nonumber\\&&
a_{ii}\sin^4\frac{\theta}{2},
\end{eqnarray}
where $\alpha=\rm{arg}\it(b_i)$ denotes the argument of the complex
number $b_i$.
Thus the average fidelity $\bar
F=\frac{1}{4\pi}\int\langle\psi(0)|\rho_i(t)|\psi(0)\rangle d\Omega$
can be calculated as
\begin{equation}
\bar F=\frac{|b_i|\cos(\alpha)}{3}+\frac{a_{ii}}{6}+\frac{1}{2}.
\end{equation}
From Eq. (11) one can see that in the absence of intrinsic
decoherence (i.e., $\gamma=0$), the equality $a_{ii}=|b_i|^2$ holds,
thus Eq. (14) reduces to Eq. (6) in Ref. [1], which describes
average fidelity in the non-disturbed case.
\section{State transfer in decoherence spin channels}
\label{sec:3} We first consider quantum state transfer via spin
chain governed by the XX Hamiltonian
\begin{eqnarray}
\hat{H}=\frac{J}{2}\sum_{n=1}^{N-1}(\sigma_n^x\sigma_{n+1}^x+\sigma_n^y\sigma_{n+1}^y),
\end{eqnarray}
where $\sigma_n^\alpha$ $(\alpha=x, y, z)$ are the usual Pauli
matrices of the $n$th qubit.
For this model, Christandl et al. have shown that perfect state
transfer from one end of the chain to another is only possible for
the case of chain length $N=2$ and $N=3$, respectively [3,4]. Here
we show that this ideal communication channel will be destroyed
under the influence of intrinsic decoherence.
The eigenvalues and eigenvectors of the Hamiltonian (15) can be
obtained as
\begin{eqnarray}
E_{k}&=&2J\cos\frac{\pi k}{N+1},\nonumber\\
|\tilde{k}\rangle&=&\sqrt{\frac{2}{N+1}}\sum_{n=1}^{N}\sin\left(\frac{\pi
kn}{N+1}\right)|n\rangle.
\end{eqnarray}
We first consider transfer of an excitation across the chain. For
this purpose, we assume the system is initially prepared in the
state $|n_0\rangle$. In the energy eigenstate basis, $|n_0\rangle$
can be expressed as
\begin{eqnarray}
|n_0\rangle=\sqrt{\frac{2}{N+1}}\sum_{k=1}^{N}\sin\left(\frac{\pi
kn_0}{N+1}\right)|\tilde{k}\rangle.
\end{eqnarray}
Thus one has
\begin{eqnarray}
\rho(0)=\frac{2}{N+1}\sum_{k,k'=1}^{N}\sin\left(\frac{\pi
kn_0}{N+1}\right)\sin\left(\frac{\pi
k'n_0}{N+1}\right)|\tilde{k}\rangle\langle \tilde{k}'|.\nonumber\\
\end{eqnarray}
Combination of Eqs. (6) and (18) gives rise to
\begin{eqnarray}
\rho(t)&=&\frac{4}{(N+1)^2}\sum_{n,m=1}^{N}\sum_{k,k'=1}^{N}\sin\left(\frac{\pi
kn}{N+1}\right)\sin\left(\frac{\pi k'm}{N+1}\right) \nonumber\\&&
\times\sin\left(\frac{\pi kn_0}{N+1}\right)\sin\left(\frac{\pi
k'n_0}{N+1}\right)\times \nonumber\\&&
\exp\left[-i2Jt\left(\cos{\frac{\pi k}{N+1}}- \cos{\frac{\pi
k'}{N+1}}\right)\right]\times\nonumber\\&& \exp\left[-2J^2\gamma
t\left(\cos{\frac{\pi k}{N+1}}- \cos{\frac{\pi
k'}{N+1}}\right)^2\right]|n\rangle \langle m|.\nonumber\\
\end{eqnarray}
For initial state $|1\rangle$ prepared in the input node A, the
transfer fidelity of the output state in node B can be obtained from
Eq. (19), and typical plots for the cases of $N=2$ and $N=3$ with
different decoherence rates are shown in Fig. 1, where the coupling
constant $J$ is chosen to be 1. In big contrast to the ideal case
(i.e., $\gamma=0$), one can see that the transfer fidelity $F$
behaves as a damped oscillation as the time $t$ evolves. This
phenomenon can be understood from Eq. (19), where the product of the
first five terms on the right-hand side causes the oscillations, and
the last term introduces the amplitude damping. With the increase of
the decoherence rate $\gamma$, or the chain length $N$, the
detrimental effects becomes more severe and therefore more quantum
state information will be lost. Thus for spin networks with
identical neighboring qubit couplings, even if for the one-link and
two-link hypercube geometries, perfect transfer of an excitation is
still impossible in the intrinsic decoherence environments.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure1.eps}}
\caption{(Color online) Dynamics of the state transfer fidelity $F$
for the XX chain with identical interactions. The decoherence rate
is given by $\gamma=0.1$ (black), $\gamma=0.2$ (red) and
$\gamma=0.3$ (blue).} \label{fig:1}
\end{figure}
For infinite time $t$, the system evolves into a steady state with
the transfer fidelity arrives at an asymptotic value $F^{\rm
steady}(N)$, which can be obtained by combination of Eqs. (7), (19)
and taking the infinite-time limit. After a tedious computation, we
obtain
\begin{eqnarray}
F^{\rm steady}(N)=\frac{3}{2(N+1)}.
\end{eqnarray}
Clearly, this steady state transfer fidelity is independent of the
decoherence rate $\gamma$, and it solely decreases with the increase
of the chain length $N$.
Next we consider time-dependence of the average fidelity for the XX
spin chain with identical interactions and subject to intrinsic
decoherence environments, with initial state prepared in the form of
Eq. (9) in node A, i.e., $s=1$. From Fig. 2 one can see clearly that
the average fidelity $\bar F$ also behaves as a damped oscillation
as the time evolves. Here the relative small value for the case of
$N=2$ is due to the fact that the phase of the state at node B is
uncorrected, i.e., $\alpha$ is not a multiple of $2\pi$. When
$t\rightarrow \infty$, the average fidelity also arrives at a steady
state value, which is independent of the decoherence rate $\gamma$,
and can be obtained analytically by taking the infinite-time limit
of $\bar F$ from Eqs. (11), (14), and (16) as
\begin{equation}\label{eq:21}
\bar{F}^{\rm steady}(N)=\left\{
\begin{aligned}
&\frac{6N+17}{12N+12}\quad\rm if\quad\it N\in\rm odd,\\
&\frac{2N+3}{4N+4}\quad\quad\rm if\quad\it N\in\rm even.
\end{aligned} \right.
\end{equation}
Contrary to that of the initial state $|1\rangle$, this steady value
does not decrease monotonously with the increase of the chain length
$N$. However, as can be seen from Eq. (21), they decrease with the
increase of the odd and even $N$, respectively, and approach to the
asymptotic value 0.5 in the limit of $N\rightarrow \infty$.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure2.eps}
} \caption{(Color online) Dynamics of the average fidelity $\bar{F}$
for the XX spin chain with identical interactions, where the
decoherence rate is given by $\gamma=0.1$.} \label{fig:2}
\end{figure}
In the following we discuss quantum state transfer in intrinsic
decoherence spin channels with fixed but different couplings between
qubits. We consider the following modified Hamiltonian
\begin{eqnarray}
\hat{H}=\sum_{n=1}^{N-1}\frac{J_{n,n+1}}{2}(\sigma_n^x\sigma_{n+1}^x+\sigma_n^y\sigma_{n+1}^y),
\end{eqnarray}
where $J_{n,n+1}=\lambda\sqrt{n(N-n)}$ is the modulated exchange
coupling, and $\lambda$ is a scaling constant.
The above Hamiltonian is identical to the representation of the
Hamiltonian $\hat{H}_s$ of a fictitious spin $S=(N-1)/2$ particle:
$\hat{H}_s=\lambda S_x$, where $S_x$ is its angular momentum
operator in $x$-direction and $\lambda$ is a scaling constant. For
this Hamiltonian, its eigenvalues and corresponding eigenvectors can
be obtained as [43]
\begin{eqnarray}
E_{k}=(-N+2k-1)\lambda,\quad
|\tilde{k}\rangle=\sum_{n=1}^Nc_{k,n}|n\rangle.
\end{eqnarray}
where the coefficient $c_{k,n}$ is given by the following recursion
relations
\begin{eqnarray}
c_{1,1}&=&1/2^{(N-1)/2}, c_{k,1}=(-1)^{k+1}c_{1,k}\nonumber\\
c_{k,n}&=&\frac{2E_kc_{k,n-1}-\sqrt{(n-2)(N-n+2)}c_{k,n-2}}{\sqrt{(n-1)(N-n+1)}}
\quad(n\geqslant 2).\nonumber\\
\end{eqnarray}
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure3a.eps}}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure3b.eps}
} \caption{(Color online) Dynamics of the state transfer fidelity
$F$ for the XX chain with modulated interactions. (a) chain length
$N=100$ with different decoherence rate; (b) decoherence rate
$\gamma=0.15$ with different chain length.} \label{fig:3}
\end{figure}
For this modulated chain, it has been shown that one can achieve
perfect state transfer between the input node $n$ and the output
node $N-n+1$ after a time $t_0=\pi/2\lambda$ and at intervals of
$\pi/\lambda$ thereafter in the absence of decoherence environment
[3,4]. When the intrinsic decoherence is present, however, this
ideal spin channel will be destroyed, and it acts as an amplitude
damping quantum channel as the rescaled time $\lambda t$ evolves. As
can be seen from Fig. 3, the transfer fidelity $F$ oscillates around
a steady state value, with the amplitude decreases gradually. This
detrimental effects becomes more and more severe with the increase
of the decoherence rate and the spin chain length, which is in
consistent with the cases of the two- and three-site spin chains
with identical interactions (In fact, they are two special cases of
the interaction-modulated spin chain). This puts new constraints on
these spin chains for long distance quantum state transfer. When
$t\rightarrow \infty$, the transfer fidelity reaches a steady state
value, which can be obtained from Eqs. (6), (7), (23), and (24) as
\begin{eqnarray}
F^{\rm{steady}}(N)=\frac{1}{2^{2N-2}}\prod_{k=2}^N\left(4-\frac{2}{k-1}\right).
\end{eqnarray}
The steady state transfer fidelity of the interaction-modulated spin
chain is still independent of the decoherence rate $\gamma$, and its
magnitude is larger than its unmodulated counterparts [cf. Eqs. (20)
and (25)], thought it still decreases with the increase of the chain
length $N$.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure4.eps}
} \caption{Dependence of $\lambda t_{\rm op}$ on decoherence rate
$\gamma$. Note that the magnitudes of $\lambda t_{\rm op}$ is
independent of the chain length $N$.} \label{fig:4}
\end{figure}
On the other hand, since the detrimental effects become severe as
the rescaled time $\lambda t$ evolves, one may expect there exists
an optimal time $\lambda t_{\rm op}$ at which the state transfer
fidelity $F$ gets its maximum value. In Fig. 4 we show $\lambda
t_{\rm op}$ versus the intrinsic decoherence rate $\gamma$, from
which one can see that $\lambda t_{\rm op}$ is shifted to the
left-hand side of $\lambda t_0=\pi/2\simeq 1.57$, and it decreases
with the increase of $\gamma$. Our numerical results also revealed
that the magnitudes of $\lambda t_{\rm op}$ is independent of the
chain length $N$.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure5.eps}
} \caption{(Color online) Dynamics of the average fidelity $\bar{F}$
for the XX chain with modulated interactions, where the decoherence
rate $\gamma=0.15$. The black lines from top to bottom correspond to
chain length $N=101$, 151, and 201; whereas the red lines from top
to bottom correspond to chain length $N=100$, 150, and 200. The
inset shows the asymptotic value of the average fidelity for
infinite time $t$ versus chain length $N$.} \label{fig:5}
\end{figure}
When considering the average fidelity, the numerical results
calculated from Eqs. (11), (14), (23), and (24) show that it
displays qualitatively the similar behaviors with that displayed in
Fig. 3. The average fidelity decreases with increasing value of both
odd and even $N$, respectively, and the chain with odd-number qubits
seems to be more robust on creating high-fidelity state transfer in
the presence of intrinsic decoherence (see Fig. 5). Moreover, as can
be seen from the inset of Fig. 5, the average fidelity goes to a
steady state value in the limit of $t\rightarrow \infty$, which has
no relation with the decoherence rate $\gamma$. They decrease with
the increase of both odd and even $N$, and approach the asymptotic
value 0.5 in the limit of $N\rightarrow \infty$.
In the absence of intrinsic decoherence (i.e., $\gamma=0$), the
above interaction-modulated spin chain can also be used to perfectly
transfer an entangled state from one end of the chain to another
[4]. When the decoherence is present, however, this ideal spin
channel will be destroyed. For example, If one start with the Bell
state $|\psi^\pm\rangle=(|01\rangle\pm|10\rangle)/\sqrt{2}$ on the
first two qubits of the chain, the temporal evolution of the
concurrence will behaves similarly as the state transfer fidelity,
i.e., it acts as an amplitude damping channel. When the rescaled
evolution time $\lambda t$ approaches infinite, from the formulae
described in Section 2 one can obtain
\begin{eqnarray}
C_{1,2}^{\rm {steady}}(N)=C_{N-1,N}^{\rm
{steady}}(N)=\prod_{n=3}^{N}\frac{2n-5}{2n-4}.
\end{eqnarray}
In fact, one can show that for the initial state
$|\psi\rangle=a|01\rangle\pm b|10\rangle$ $(|a|^2+|b|^2=1)$ prepared
on the first two qubits, the following relation holds
\begin{eqnarray}
C_{1,2}^{\rm {steady}}(N)=C_{N-1,N}^{\rm {steady}}(N)=C_{1,2}^{\rm
{initial}}\prod_{n=3}^{N}\frac{2n-5}{2n-4},
\end{eqnarray}
where $C_{1,2}^{\rm {initial}}=2|ab|$ denotes the concurrence of the
initial state of the first two qubits. This indicates that when the
rescaled evolution time $\lambda t$ approaches infinite, the system
goes to a steady mirror-symmetric state with two-qubit reduced
density matrix $\rho_{nm}(t)=\rho_{N-n+1,N-m+1}(t)$, and the steady
state value $C_{N-1,N}^{\rm {steady}}(N)=C_{1,2}^{\rm {steady}}(N)$
decreases as the chain length $N$ increases.
We now investigate entanglement distribution between two distant
parties through the intrinsic decoherence spin channel. For this
purpose, we assume the entangled state
$|\psi\rangle=(|01\rangle+|10\rangle)/\sqrt{2}$ is initially
prepared between a noninteracting qubit NI and the first qubit A on
the chain, then after some time $t$, the entanglement will be
established between NI and the target spin B. The overall
Hamiltonian of the system can be written as
$\hat{H}'=I\otimes\hat{H}$, and with the same method used above, one
can demonstrate that the concurrence $C_{\rm{NI,B}}(N,t)$ (Note that
here $N$ denotes the length of the interacting-spin chain, and does
not include the noninteracting qubit NI) also behaves as a damped
oscillation, and when $t\rightarrow\infty$, we obtain
\begin{eqnarray} \label{eq:28}
C_{\rm{NI,A}}^{\rm{steady}}(N)&=&C_{\rm{NI,B}}^{\rm{steady}}(N)\nonumber\\
&=&\left\{\begin{aligned}
&0\quad\rm \quad\quad\quad\quad\quad\quad if\quad\it N\in\rm odd, \\
&\prod_{n=3}^{(N+4)/2}\frac{2n-5}{2n-4}\quad\rm if\quad\it N\in \rm
even.
\end{aligned} \right.
\end{eqnarray}
This equation shows clearly that the XX chain with even-number
qubits is more robust than its counterpart with odd-number qubits on
distributing quantum entanglement. This is somewhat different from
that of the average fidelity (see Fig. 5), where the chain with
odd-number qubits is more efficient on creating high-fidelity state
transfer in the presence of intrinsic decoherence.
\section{Creating entanglement in decoherence environments}
\label{sec:4} In this section, we see intrinsic decoherence effects
on the creation of entanglement in various kinds of spin networks.
For this purpose, we consider the multiarm structure
$M(l_1,l_2,N_A)$ of the XX Hamiltonian (22) with the addition of the
exchange couplings between the hub site and its nearest-neighbor
output sites satisfy the branching rule [10]. Here $l_1$ and $l_2$
denote the number of sites in the input and output arms,
respectively, and $N_A$ is the number of output arms (see Fig. 6).
It has been shown that in the absence of decoherence environment,
this structure can be employed to create multi-qubit entangled $W$
state at the ends of the outgoing arms.
\begin{figure}
\centering
\resizebox{0.35\textwidth}{!}{%
\includegraphics{figure6.eps}}
\caption{(Color online) Sketch of the multiarm structure of the spin
network, where the green circle denotes the hub. The number of sites
in the input and output arms are $l_1=3$, $l_2=4$, and the number of
output arms is $N_A=3$.}
\label{fig:6}
\end{figure}
The Hamiltonian in the subspace $\mathcal {H}_1$ spanned by the
basis vectors $|n\rangle$ ($n=1, 2,... , N$ ) is
\begin{eqnarray}
\hat{H}=\sum_{(ij)}J_{ij}|i\rangle\langle j|+\rm H.c.,
\end{eqnarray}
where the summation runs over all pairs of neighboring spins. For
the sake of simplicity, we first consider the Y-shaped structure
$Y(l_1,l_2,2)$. The total number of sites now is $N=l_1+2l_2+1$. To
examine temporal evolution of the concurrence of the prepared
initial state, we make the following basis transformation for spins
just in the same position of each arm
\begin{eqnarray}
|n^\pm\rangle=\frac{1}{\sqrt {2}}(|n\rangle\pm|n'\rangle),
\end{eqnarray}
where $n, n'>l_1+1$. Then in the subspace spanned by $|n\rangle$
$(n\leqslant l_1+1)$ and $|n^\pm\rangle$ $(n>l_1+1)$, the
Hamiltonian can be rewritten as
\begin{eqnarray}
\hat{H}&=&\sum_{n<l_1+1}J_{n,n+1}|n\rangle\langle n+1|+\nonumber\\&&
\sum_{n>l_1+1}\sum_{r=+,-}J_{n,n+1}|n^r\rangle\langle
(n+1)^r|+\nonumber\\&& \sqrt{2}J_{l_1+1,l_1+2}|l_1+1\rangle\langle
(l_1+2)^+|+\rm H.c.
\end{eqnarray}
Clearly, under the transformation (30) the Y-shaped structure is
transformed into a linear chain consisting of the input arm, the hub
and one output arm while the other output arm is decoupled (see Fig.
7), i.e., this structure is identical to the interaction-modulated
one-dimensional XX chain with chain length $l=l_1+l_2+1$.
\begin{figure}
\centering
\resizebox{0.35\textwidth}{!}{%
\includegraphics{figure7a.eps}}
\resizebox{0.35\textwidth}{!}{%
\includegraphics{figure7b.eps}}
\caption{(Color online) (a) Sketch of the Y-shaped spin network. (b)
Under the transformation (30) the Y-shaped structure is transformed
into a linear chain consisting of the input arm and one output arm
while the other output arm is decoupled.}
\label{fig:7}
\end{figure}
For this spin network, if we prepare initial state $|1\rangle$ in
the first node of the input arm, then after some time $t$,
entanglement will be established between the end nodes of the output
arms (for Fig.7, it corresponds to node 8 and $8'$). From the
formalism described in Section 2 one can obtain
$C(l_1,l_2,\lambda,t)=F(l,\lambda,t)$. This implies that under the
influence of intrinsic decoherence, the concurrence of the created
entanglement between the end nodes of the output arms also behaves
as a damped oscillation as the rescaled time $\lambda t$ evolves.
For infinite rescaled evolution time $\lambda t\rightarrow \infty$,
the concurrence goes to a steady state value $C^{\rm
{steady}}(l_1,l_2)=2^{2-2l}\prod_{k=2}^{l}[4-2/(k-1)]$, which can be
obtained directly from Eq. (25).
Similarly, for the multiarm structure $M(l_1,l_2,N_A)$, using the
same method, one can obtain that the concurrence measuring pairwise
entanglement between arbitrary two qubits of the end nodes of the
output arms is given by
$C(l_1,l_2,N_A,\lambda,t)=2F(l,\lambda,t)/N_A$ (when $N_A=2$, this
equality reduces to that describing the Y-shaped structure), which
observes the similar behaviors as the Y-shaped structure, i.e., it
behaves as a damped oscillation as the rescaled time $\lambda t$
evolves, and when $\lambda t\rightarrow\infty$, it goes to a steady
value $C^{\rm
{steady}}(l_1,l_2,N_A)=2^{3-2l}\prod_{k=2}^{l}[4-2/(k-1)]/N_A$.
\section{Modified spin chains for high-fidelity state transfer}
\label{sec:5} From the above arguments one can see that the
interaction-modulated ideal spin channels for perfect state transfer
are destroyed in the presence of intrinsic decoherence environments.
Though there exists an optimal rescaled time at which one can get a
relative high transfer fidelity, however, this transfer fidelity
(including the average fidelity) decreases as the chain length $N$
increases, which puts great constrains for long distance
communication in interacting-spin systems.
Here we demonstrate that a minor modification of the exchange
interactions between the first and the last two nodes of the above
structure can fulfill the requirements of long distance and near
perfect state transfer (see Fig. 8a). To see this, we display our
numerical results for chain length $N=11$ and $N=51$ in Fig. 9(a),
from which one can see that for all decoherence rate $\gamma$, the
maximum transfer fidelity $F_{max}$ approaches unity if $J_0$ is
small enough (note that when $J_0=0$, $F\equiv0$), which indicates
that even in the presence of intrinsic decoherence environments, one
can still achieve near perfect transfer of an excitation between the
opposite ends of a XX chain by varying the strength of the exchange
interactions between the first and the last two nodes of the
modulated spin chain.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure8.eps}}
\caption{ (Color online) Sketches of the two modified seven-site
spin chains which may serve as spin channels for near perfect state
transfer in the presence of intrinsic decoherence environments. Here
we choose $N$ odd for it enables more efficient (i.e., high speed)
state transfer than its counterpart with even $N$.}
\label{fig:8}
\end{figure}
Another structure which may serve as near perfect spin channel for
long distance transfer of an excitation in intrinsic decoherence
environments is the XX quantum wire with the neighboring couplings
except those between the first and the last two nodes are the same
(see Fig. 8b). This chain can serve as spin channel for an almost
perfect state transfer in the absence of decoherence environments
[11]. When the intrinsic decoherence is present, from Fig. 9(b) one
can see that a long distance transfer of an excitation whose
fidelity can be arbitrarily close to unity is also possible for very
small but nonzero $J_0$, even for large decoherence rate $\gamma$.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure9a.eps}}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure9b.eps}}
\caption{(Color online) Dependence of $F_{max}$ on $J_0$ for the two
modified spin channels displayed in Fig. 8. In both figures the
black lines from top to bottom correspond to chain length $N=11$ and
decoherence rate $\gamma$=0.15, 0.30 and 0.45; whereas the red lines
correspond to $N=51$ and $\gamma$=0.15. Note that when $J_0=0$,
$F\equiv0$.}
\label{fig:9}
\end{figure}
To understand the above phenomenon, we sketch dynamics of $c_{\rm
i}(t)$ in Fig. 10, where $c_{\rm i}(t)=\sqrt{\langle
i|\rho_i(t)|i\rangle}$ denotes the amplitude of the coefficient for
the state $|i\rangle$. From these two figures one can see that for
very small but nonzero $J_0$, except the two end spins (here is node
1 and 11), the spins at the odd-number sites remain almost unexcited
during the time evolution process (i.e., $c_{\rm i}(t)|_{\rm{i\in
odd;i\neq1,N}}\simeq 0$), as if the excitation is transferred only
through the even-number nodes. In fact, a more detailed analysis
show that if $J_0\rightarrow 0^+$ and $t\leqslant t_{\rm{op}}$, the
mixedness of the system remain almost unchanged ($1-tr\rho^2\simeq
0$), i.e., the state here is very close to a pure state during the
evolution process and thus can be described approximately by
$|\psi(t)\rangle=c_1(t)|1\rangle+c_N(t)|N\rangle+\sum_{\rm{i\in
even}}c_i(t)|i\rangle$, where $c_1(t)+c_N(t)\simeq 1$ (note that
$c_1^2(t)+c_N^2(t)<1$ since $c_{\rm i}(t)|_{\rm{i\in even}}\neq 0$).
In order to better understand why the spins located at odd-number
sites (the two end spins are exceptions) remain almost unexcited
during the time evolution process for very small values of $J_0$, we
graph the effects of decreasing $J_0$ on the eigenvector population
$c_{k,1}^2=\{|\langle\hat{k}|1\rangle|^2\}_{E_k}$ for the two
modified spin chains in Fig. 11, from which one can see that with
decreasing values of $J_0$, the distribution of the eigenvectors
becomes narrower and narrower. Particularly, in the limitation
$J_0\rightarrow 0^+$, $\{|\langle\hat{k}|1\rangle|^2\}_{E_k}\neq 0$
only for the three central eigenvectors of the system, i.e.,
$c_{k,1}\neq0$ only when $k_1=(N-1)/2$, $k_2=(N+1)/2$ and
$k_3=(N+3)/2$. Moreover, for these three values of $k$, it can be
shown that when $J_0\rightarrow 0^+$, $c_{k,n}\neq0$ only when
$n=1$, $N$ and $n\in \rm{even}$ for $k=k_1$, $k_3$, and $n=1$, $N$
for $k=k_2$. For example, when $N=11$, the eigenvectors of the
Hamiltonian describing the first modified spin chain for the above
three values of $k$ can be obtained as
\begin{equation} \label{eq:32}
\left\{\begin{aligned}
&|\hat{k}_{1,3}\rangle=\frac{1}{2}(|1\rangle+|N\rangle)\mp\sqrt{\frac{5}{42}}(|2\rangle+|10\rangle)\mp \\
& \quad\quad\quad \sqrt{\frac{1}{12}}|6\rangle\pm \sqrt{\frac{15}{168}}(|4\rangle+|8\rangle),\\
&|\hat{k}_{2}\rangle=\sqrt{\frac{1}{2}}(|1\rangle-|N\rangle).
\end{aligned}\right.
\end{equation}
Similarly, for the second modified spin chain, its eigenvectors for
the above three values of $k$ with arbitrary chain length $N$
($N\in$ odd) can be obtained as
\begin{equation} \label{eq:33}
\left\{ \begin{aligned}
&|\hat{k}_{1,3}\rangle=\frac{1}{2}[|1\rangle+(-1)^{(N+1)/2}|N\rangle]\pm \\
& \quad\quad\quad \frac{1}{\sqrt{N-1}}\sum_{n\in\rm{even}}(-1)^{n/2}|n\rangle,\\
&|\hat{k}_{2}\rangle=\frac{1}{\sqrt{2}}[|1\rangle+(-1)^{(N-1)/2}|N\rangle].
\end{aligned} \right.
\end{equation}
On the other hand, for initial state $|1\rangle$ prepared in the
input node A, the density matrix at arbitrary time $t$ can be
obtained by choosing $\theta=\pi/2$, $\phi=0$ of Eqs. (10) and (11)
as
\begin{eqnarray}
\rho(t)&=&\sum_{n,m=1}^N\sum_{k,k'=1}^Nc_{k,1}c_{k',1}c_{k,n}c_{k',m}\nonumber\\&&
\exp\left[-it(E_k-E_{k'})-\frac{\gamma
t}{2}(E_k-E_{k'})^2\right]|n\rangle\langle m|.\nonumber\\
\end{eqnarray}
Combination of Eq. (34) with the above arguments, one can conclude
that the spins located at odd-number sites remain almost unexcited
during the time evolution process, i.e., the excitation is
transferred only through the even-number nodes for very small but
nonzero values of $J_0$.
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure10a.eps}}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure10b.eps}}
\caption{(Color online) Dynamics of $c_{\rm{i}}(t)$ for the two
modified spin channels displayed in Fig. 8 with $J_0=0.02$,
$\gamma=0.15$ and $N=11$. Here the curves for even-number $i$
(denoted by blue lines) are almost overlapped.}
\label{fig:10}
\end{figure}
\begin{figure}
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure11.eps}}
\caption{The eigenvector populations
$\{|\langle\hat{k}|1\rangle|^2\}_{E_k}$ (denoted by the relative
height of the vertical lines) for the system of 11 spins. The graphs
from left to right in every plot correspond to the values of $k$
increases one by one (or equivalently, the eigenvalues increase one
by one). The left seven panels correspond to the first modified spin
chain with $J_0$=3, 2.5, 2, 1.5, 1, 0.5 and $J_0\rightarrow 0^+$
(from top to bottom), while the right six panels correspond to the
second modified spin chain with $J_0$=1, 0.8, 0.6, 0.4, 0.2 and
$J_0\rightarrow 0^+$ (from top to bottom).}
\label{fig:11}
\end{figure}
In fact, from the formalism described in Section 2, one can see that
the transfer fidelity of an excitation from one end of the chain to
another is completely determined by
\begin{eqnarray}
a_{NN}&=&\sum_{k,k'=k_1}^{k_3}c_{k,1}c_{k',1}c_{k,N}c_{k',N}\nonumber\\&&
\exp\left[-it(E_k-E_{k'})-\frac{\gamma t}{2}(E_k-E_{k'})^2\right],
\end{eqnarray}
where the coefficients of the eigenvectors for the three values of
$k$ are given by
\begin{eqnarray}
c_{k_1,1}&=&c_{k_3,1}=\frac{1}{2},\quad
c_{k_1,N}=c_{k_3,N}=\frac{(-1)^{(N+1)/2}}{2},\nonumber\\
c_{k_2,1}&=&\frac{1}{\sqrt{2}},\quad
c_{k_2,N}=\frac{(-1)^{(N-1)/2}}{\sqrt{2}}.
\end{eqnarray}
On the other hand, the eigenvalues of the two modified spin chains
correspond to the three eigenvectors
$\{|\hat{k}_1\rangle,|\hat{k}_2\rangle,$ $|\hat{k}_3\rangle\}$ can
be written as $\{-E_0,0,E_0\}$ ($E_0$ can be obtained numerically),
thus from the above two equations and the formalism described in
Section 2, one can obtain the transfer fidelity of one excitation as
\begin{eqnarray}
F&=&\frac{3}{8}+\frac{1}{8}\exp(-2\gamma
E_0^2t)\cos(2E_0t)-\nonumber\\&& \frac{1}{2}\exp\left(\frac{-\gamma
E_0^2t}{2}\right)\cos(E_0t)\quad (N\in\rm{odd}).
\end{eqnarray}
Since $E_0$ is very small in the limit of $J_0\rightarrow 0^+$, from
Eq. (37) one can see that an almost perfect transfer of an
excitation from one end of the chain to another occurs after time
$t_c\sim\pi/E_0$. Moreover, we noted that $E_0$ decreases with the
decrease of $J_0$, thus the critical time at which the transfer
fidelity $F$ gets its maximum value increases with the decrease of
$J_0$, and is independent of the decoherence rate $\gamma$. These
conclusions can be corroborated by numerical results displayed in
Fig. 10. For the system parameters adopted there (i.e., $N=11$,
$J_0=0.02$), $E_0$ can be obtained numerically as $E_{01}\simeq
0.013801$ and $E_{02}\simeq 0.012648$, thus one has $t_{c1}\sim 227$
and $t_{c1}\sim 248$. Clearly, these results agree well with those
displayed in Fig. 10.
A similar analysis shows that for the two modified spin chains with
even-number qubits, the behavior of the transfer fidelity is
determined only by the two central eigenvectors of the system in the
limit of $J_0\rightarrow 0^+$, i.e., $c_{k,1}\neq0$ only when
$k_1=N/2$ and $k_2=N/2+1$. The eigenvectors of the Hamiltonian
describing both the two modified spin chains for the above two
values of $k$ can be obtained as
\begin{equation} \label{eq:1}
\left\{ \begin{aligned}
&|\hat{k}_{1}\rangle=\frac{1}{\sqrt{2}}[|1\rangle+(-1)^{N/2}|N\rangle],\\
&|\hat{k}_{2}\rangle=\frac{1}{\sqrt{2}}[|1\rangle-(-1)^{N/2}|N\rangle],
\end{aligned} \right.
\end{equation}
with the corresponding eigenvalues $-E_0$ and $E_0$. Combination of
these with the formalism in Section 2, the transfer fidelity of one
excitation can be obtained as
\begin{eqnarray}
F=\frac{1}{2}-\frac{1}{2}\exp(-2\gamma E_0^2t)\cos(2E_0t)\quad
(N\in\rm{even}).
\end{eqnarray}
From Eq. (38) one can see that in the limit of $J_0\rightarrow 0^+$,
except the two spins located at the end nodes, all the other spins
remain almost unexcited during the time evolution process, as if the
excitation is transferred only between the two end nodes. Moreover,
from Eq. (39) one can see that an almost perfect transfer of an
excitation from one end of the chain to another occurs after time
$t_c\sim\pi/2E_0$. However, our numerical results show that the
values of $E_0$ for odd $N$ is much larger than that for even $N$
(e.g., for $J_0=0.001$, the values of $E_0$ for $N=11$ is about 2524
and 2804 times larger than that for $N=10$ and 12), thus the chain
with odd $N$ enables a more efficient (i.e., high speed) state
transfer than its counterpart with even $N$.
As a final discussion, it is worthwhile to investigate the
efficiency of the above two modified spin channels, i.e., whether
they can serve as near perfect spin channels for transfer of an
arbitrary one-qubit state by varying the strength of $J_0$. For this
purpose, we compute the average fidelity. From the above analysis
one can obtain straightforwardly that in the limit of
$J_0\rightarrow 0^+$, the average fidelity can be expressed as
\begin{eqnarray}
\bar{F}=\frac{|b_N|\cos(\alpha)}{3}+\frac{a_{NN}}{6}+\frac{1}{2},
\end{eqnarray}
where the coefficients for odd and even $N$ are given by
\begin{eqnarray}
a_{NN}&=&\frac{3}{8}+\frac{1}{8}\exp(-2\gamma E_0^2t)\cos(2E_0t)-
\nonumber\\&& \frac{1}{2}\exp\left(\frac{-\gamma
E_0^2t}{2}\right)\cos(E_0t),
\nonumber\\
b_N&=&\frac{(-1)^{(N-1)/2}}{2}\left[1-\exp\left(\frac{-\gamma E_0^2t}{2}\right)\cos(E_0t)\right],\nonumber\\
\end{eqnarray}
and
\begin{eqnarray}
a_{NN}&=&\frac{1}{2}-\frac{1}{2}\exp(-2\gamma
E_0^2t)\cos(2E_0t),\nonumber\\
b_N&=&(-1)^{N/2}i\exp\left(\frac{-\gamma
E_0^2t}{2}\right)\sin(E_0t).
\end{eqnarray}
As pointed above, $E_0$ is an infinitesimal in the limit of
$J_0\rightarrow 0^+$, thus from Eqs. (40) and (41) one can see
obviously that for odd $N$, the average fidelity approaches unity
after time $t_c\sim\pi/E_0$. For even $N$, however, due to the fact
that $\alpha=\pi/2$ is not a multiple of $2\pi$, the average
fidelity can only reach its maximum value $2/3$ (equals to the
classical average fidelity) after time $t_c\sim\pi/2E_0$. This
implies that for the above two modified spin channels with even $N$,
one cannot achieve near perfect state transfer of an arbitrary
one-qubit state simply by varying the strength of $J_0$. But if one
can apply an external magnetic field $B$ along the $z$ axis of every
spin (this does not change the eigenvectors of the system since
$[\hat{H},\sigma_{\rm{tot}}^z]=0$), the phases of the received state
at the destination node may be corrected. With this method, we
performed numerous calculations and the numerical results revealed
that the average fidelity can also approaches unity by choosing
appropriate strength of the magnetic field (e.g., for $N=10$,
$J_0=0.01$ and $B=\pm0.0003$, the two modified spin channels give
rise to $\bar{F}_{max}=0.9998$ and $\bar{F}_{max}=0.9996$,
respectively).
\section{Summary}
\label{sec:6} To summarize, we have investigated quantum state
transfer, generation and distribution of entanglement in the model
of Milburn's intrinsic decoherence environment. We focused on
diverse interaction-modulated spin networks which may serve as
perfect spin channels in the absence of decoherence. As one
expected, the state transfer fidelity as well as the amount of the
generated and distributed entanglement will be significantly lowered
by the intrinsic decoherence environment, and this detrimental
effects become severe as the decoherence rate $\gamma$ and the spin
chain length $N$ increase. For infinite evolution time $t$, we show
analytically that both the state transfer fidelity (including the
average fidelity) and the concurrence of the generated and
distributed entanglement approach steady state values, which are
independent of the decoherence rate $\gamma$. This brings great
constraints on these structures as spin channels for long distance
and high-fidelity communication. Finally, as alternative schemes to
diminish the detrimental effects, we presented two modified spin
chains which may serve as spin channels for long distance and near
perfect state transfer in the intrinsic decoherence environments.
Our results revealed that in the limit of $J_0\rightarrow 0^+$,
these two modified spin channels generate maximum fidelity 1 after
certain time $t_c\sim\pi/E_0$ for spin chains with odd-number
qubits. For spin chains with even-number qubits, however, one needs
to apply an external magnetic field in order to achieve near perfect
state transfer. \\
\\
\textbf{Acknowledgements \\} \\ This work was supported by the
National Natural Science Foundation of China under Grant No.
10547008, the Specialized Research Program of Education Bureau of
Shaanxi Province under Grant No. 08JK434, and the Youth Foundation
of Xi'an Institute of Posts and Telecommunications under Grant No.
ZL2008-11.
|
1,314,259,995,775 | arxiv | \section{Introduction}
\label{sec_int}
The development of data acquisition methods allow the analysis of complex systems in several operating conditions as never before. Several examples may be found in the current Industry 4.0 framework, which is reshaping the variety of signals and measurements that can be gathered during manufacturing processes. Experimental data are more and more characterized by complex and novel formats, like images, videos, dense point clouds. These data may be acquired not only off line, during post-process inspections on the product, but also in line, during the production process, by exploiting a variety of sensors installed and embedded into the system. The rich information enclosed in such big data streams allows one to monitor and optimize industrial processes, as well as to improve the productivity and efficiency of production plants and enable several benefits of the ongoing digital transition. As a consequence, the focus of many applications in industrial statistics is moving from product quality characteristics to in-line process measurements, thanks to enhanced sensing and monitoring capabilities. Moreover, novel production paradigms are characterized by several controllable factors and complex process dynamics that impose the need for effective and efficient experimental approaches to determine optimal process conditions and gather deeper comprehension of underlying physical phenomena.
In this framework, a number of novel challenges shall be faced, with respect to how the quality of products is monitored, modelled and continuously improved. In many cases, statistical methods require a transformation of input data that are characterized by complex and/or high dimensional formats (e.g., multi-channel signals, images, videos, point clouds) into a format that is easier to handle and, at the same time, able to capture the information content and in order to draw reliable and robust decisions. A family of statistical methods suitable to tackle this problem is known as functional data analysis (FDA). For a comprehensive overview of FDA methods and applications we refer the reader to \cite{ramsay2005functional,horvath2012inference,kokoszka2017introduction} and, for further theoretical insights, to \cite{hsing2015theoretical,bosq2012linear}. FDA allows the representation of observation units in terms of functions in a 1D, 2D or higher dimensional domain with a general validity this is not limited to manufacturing applications. Such functional representation makes statistical inference methods applicable also in cases where the complexity of the input data goes far beyond traditional univariate or multivariate domains.
A large variety of industrial applications where sensor signals and metrology data can be represented and analyzed as functional data have been presented so far \citep{noorossana2011statistical}. Examples include signals with cyclic patterns, calibration curves and coordinate measurements of profiles that can be treated as 1D functions \citep{Paynabar20131235, Guo2019, Qiu2010265, Colosimo20101575, Grasso20146110}. Other examples include spatial measurements and surface data that can be treated as 2D functions \citep{Zang2018379, Colosimo201495}. FDA resulted to be effective in modelling complex spatial or spatio-temporal patterns of image and video-image data as well, with various applications. Examples from this research line were reviewed by \cite{Megahed201183}. Other examples include process monitoring and quality modelling applications where data are modelled as functional data and lead to effective anomaly detection \citep{Wang2005677, Menafoglio2018497, Wells20131267,capezza2021functional1, centofanti2021functional, capezza2021clus, Colosimo2021}.
One example of the use of FDA to translate video-image data into a functional form is presented here below and motivated the present study. It regards the analysis of process stability in a metal additive manufacturing process known as laser powder bed fusion (L-PBF) by means of high speed videos acquired during the process \citep{colosimo2018opportunities,colosimo2020machine,Grasso2021}. L-PBF is an additive process suitable to produce metal parts by means of a laser beam that selectively melts thin layers of metal powder. The process is repeated layer by layer, with the material solidified in one layer being welded to the material in underneath layers, enabling the fabrication of products with complex geometries and innovative properties \citep{gibson2014additive}. Fig. \ref{fig_introduction_1}, left panel, shows an example of video frame acquired during this process. The small white particles, which are visible in the image, are spatters produced by the laser-material interaction, whereas the bigger white spot is the heat affected region where the laser is melting the material. This is just one frame of a high-speed (1000 frames per second) video, where spatters exhibit a complex time-variant dynamic pattern that is representative of the process stability. It is evident that the application of statistical inference methods to video-image data like these can be applied only if the information content is transformed, modelled or synthesised into a different format. One possible way consists of estimating synthetic quantities (like the number of spatters, their size, etc.) and translating the original video frame into a multivariate vector of descriptors \citep{yang2020monitoring,andani2017spatter,repossini2017use}. This approach entails an intrinsic information loss and an arbitrary and problem dependent choice of descriptors. Another approach consists in transforming the image into a functional format. An example of this transformation is shown in Fig. \ref{fig_introduction_1}, right panel, where a 2D function depicts spatter spread in space over the video frame. This function, which will be referred to as \textit{spatter intensity} function in this study, maps the amount of spatters observed in any region of the bi-dimensional video-frame space, $(s,t)$. The term \textit{intensity} here refers to the occurrence of spatters in a given location. A high spatter intensity at given spatial coordinates $(s,t)$ means that a large amount of spatters was captured in the video image stream in that specific location. Such representation allows one to capture spatial information on spatter spread in space and to make inference in a FDA fashion.
The example shown in Fig. \ref{fig_introduction_1} can be regarded as just one of many real applications where a functional data representation is suitable to deal with complex patterns and data types. A functional representation similar to that of Fig. \ref{fig_introduction_1} can be suitable in all processes where spatters and hot ejections are generated, like welding or laser cutting.
\begin{figure}
\centering
\begin{tabular}{cc}
\vspace{0.32cm} \includegraphics[width=0.3\textwidth]{Figures/frame_video1-eps-converted-to.pdf} &
\vspace{0pt} \includegraphics[width=0.335\textwidth]{Figures/Case-study_introduction_heat-eps-converted-to.pdf}\\
(a) & (b)\\
\end{tabular}
\caption{\label{fig_introduction_1}Example of a video frame acquired during an L-PBF process showing ejected spatters as bright spots (a) and corresponding spatter intensity function (b). }
\end{figure}
A classical statistical problem consists in the identification of significant differences in group functional means belonging to a sample with varying experimental conditions. In the literature, this problem is known as functional analysis of variance (FANOVA) that is the FDA extension of the classical (non-functional) ANOVA problem. Referring to the example in Fig. \ref{fig_introduction_1}, the FANOVA approach may be used to study the effect of different process conditions on the spatter behaviour, which is a problem that attracted great interest in the additive manufacturing community, because the spatter behaviour can be regarded as a proxy of process stability and quality \citep{yang2020monitoring,andani2017spatter,repossini2017use,ly2017metal,bidare2018fluid}.
\cite{ramsay2005functional} proposed a functional ANOVA test, based on a pointwise $ F $-test statistic, that relies on the normality assumption of the error function. If the observed statistics is larger than the critical value, calculated as a percentile of the Fisher distribution, for each domain value, then the hypothesis of no differences among the groups can be safely rejected.
\cite{cuevas2004anova} proposed a FANOVA test based on the integrated squared difference among group functional means, for both the homoscedastic and heteroscedastic cases.
The $ L^2 $-norm-based test proposed by \cite{faraway1997regression,zhang2007statistical} uses a statistic based on the integrated squared differences between the group mean and the global mean, whose distribution is approximately proportional to a chi-squared random variable.
\cite{shen2004f,zhang2011statistical} proposed an $ F $-type test based on the fraction of the sum of the integrated squared differences between the group means and the global mean, and, the sum of the integrated squared differences between the functional observations and the group means. Under certain conditions, this statistic has a Fisher distribution. Bootstrap versions of both $ L^2 $-norm-based and $ F $-type tests were proposed by \cite{zhang2013analysis}. Finally, \cite{zhang2014one} introduced a globalized version of the pointwise $ F $-test. Note that all the aforementioned works deal with the one-way FANOVA design.
The multi-way functional ANOVA design has been much less studied than the one-way counterpart. In particular, \cite{brumback1998smoothing,guo2002inference,gu2013smoothing} proposed tests that are able to deal with more complicated designs that rely on the use of smoothing splines (SS-ANOVA). A simple technique was proposed by \cite{cuesta2010simple} who transform functional data into univariate data by means of random projections. \cite{pini2018domain} proposed a non‐parametric domain‐selective multi-way functional ANOVA able to identify the specific subdomains where group functional means differ.
In this study, we address the functional analysis of variance in the presence of nuisance effects associated to outlying patterns in the experimental dataset. The proposed real-case study in additive manufacturing highlights the need for novel and effective methods in this framework.
In the motivating case study considered by this paper, an outlying spatter ejection behaviour may be observed as a consequence of a variety of possible root causes.
Fig. \ref{fig_introduction} shows an example of an outlying pattern in the spatter intensity function. For the sake of graphical clarity, functions corresponding to different realizations under the same experimental treatment are compared by looking at their cross-sections at a fixed coordinate $t$. The cross-section shown with a solid thick line in Fig. \ref{fig_introduction} represents an outlying spatter behaviour, consisting of a lower amount of spatters spread in space, possibly caused by a transient laser beam attenuation that occurred at a given point in time. Additional details about the real-case study can be found in Section 4.
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{Figures/Case-study_introduction_heat-eps-converted-to.pdf} &
\includegraphics[width=0.3\textwidth]{Figures/Case-study_introduction_slicesingle-eps-converted-to.pdf} &
\includegraphics[width=0.3\textwidth]{Figures/Case-study_introduction-eps-converted-to.pdf} \\
(a) & (b)& (c)\\
\end{tabular}
\caption{\label{fig_introduction} Example of a spatter intensity function (a), one cross-section of the spatter intensity function at t = 0.75 (b) and a superimposition of cross-sections corresponding to different experimental realizations of the spatter intensity function, where an outlying pattern is highlighted with a thick black line (c).}
\end{figure}
From a design-of-experiments perspective, outlying patterns like the one in Fig. \ref{fig_introduction} represent a nuisance, as they may inflate the variability and mask effects of potential interest. From a statistical process monitoring perspective, instead, outliers commonly drive relevant information, being potential indicators of anomalies and flaws. In this study, we refer to the former perspective, aiming at proposing an effective approach for the analysis of variance in the presence of outliers that contaminate the experimental functional data.
Due to the many different dynamics involved in the process, determining whether an experimental point is an outlier and identifying its root cause can be a difficult task, but similar challenges can be faced in many different manufacturing applications, due to the complex nature of the response variables and the complex underlying physical phenomena.
All the one-way and multi-way FANOVA design cited above combine in a different quadratic fashion the functional mean to obtain the test statistic.
However, as in the case of finite dimensional data, the functional mean, as well as quadratic forms, are shown to be highly sensitive to the presence of outliers.
\cite{hubert2015multivariate} set up a taxonomy of functional outliers.
To deal with outliers, the \textit{diagnostic} and the \textit{robust} approaches are the two common alternatives.
The diagnostic approach is based on standard estimates after the removal of sample units identified as outliers. Even though it is criticized as it is subject to the analyst's personal decision, it can often be safely applied, such as in the case depicted in Fig. \ref{fig_introduction}, where the marked curve can be safely deleted. However, as we will see below, it is not always easy to label an observation as outlier, especially when complex process dynamics and lack of measurable covariates make the search for root causes a difficult task.
On the contrary, the robust approach produces parameter estimators as well as associated tests and confidence intervals that limit the influence of outliers on final results and decisions without the need for searching and explicitly removing them before the estimation. For a general perspective on this topic in the classical setting see \cite{huber2004robust,hampel2011robust,maronna2019robust}.
In the very last years, several works have explored robust estimation for functional data.
\cite{fraiman2001trimmed} defined trimmed means for functional data based on a functional depth defined as an integral of the univariate depths for each domain value.
To obtain robust estimates of the center of a functional distribution, \cite{cuesta2006impartial} extended the notion of impartial trimming to a functional data framework.
Other location estimators based on depth functions for functional data were proposed by \cite{cuesta2008random,cuevas2009depth,lopez2009concept,lopez2011half}.
The above methods are all extensions of the classical linear combination type estimators (i.e., $ L $-estimator) \citep{maronna2019robust} to the functional setting.
More recently, \cite{sinova2018m} extended the notion of maximum likelihood type estimators (i.e., $ M $-estimators) to the functional data setting. $ M $-estimators \citep{huber1964robust}
are less influenced by outliers than the standard least-squares or maximum likelihood estimators, because they are based on loss functions that increase less rapidly than the usual square loss.
These estimators have been applied by \cite{kalogridis2019robust} to the functional linear model.
The FANOVA methods are not necessarily robust against outliers, as they rely on both the functional mean and quadratic forms, which are known to be highly sensitive to outlying observations. In the classical setting, robust ANOVA methods have been proposed by \cite{schrader1977robust,schrader1980robust}, who adapted Huber's $ M $-estimates to be used in a modified $ F $-statistic and a likelihood ratio type test.
However, to the best of our knowledge, no robust ANOVA has been introduced so far in the functional setting.
In this paper, we propose a robust functional ANOVA method (RoFANOVA) that is able to test, in a nonparametric fashion, the differences among group functional means. The RoFANOVA method is based on a functional generalization of the test statistic proposed by \cite{schrader1977robust} included in a permutational framework \citep{good2013permutation,pesarin2010permutation}. Applications of nonparametric methods in FDA can be found in \cite{ramsay2005functional,corain2014new,pini2017interval,pini2018domain}.
Moreover, to obtain the test statistic, we introduce a functional extension of the normalized median absolute deviation (NMAD) estimator, referred to as functional normalized median absolute deviation (FuNMAD) estimator, as well as an equivariant version of the functional $ M $-estimator proposed by \cite{sinova2018m}.
An extensive Monte Carlo simulation study is presented to quantify the performance of the RoFANOVA with respect to FANOVA tests already appeared in the literature before, both in one-way and two-way designs. The application of the proposed approach to the real-case study in the additive manufacturing field also highlights its effectiveness over competing methods in identifying interaction effects that are relevant to get deeper insights about the functional response variable of interest.
The paper is organized as follows. In Section \ref{sec_met}, the robust functional analysis of variance is introduced together with the functional normalized median absolute deviation and the scale equivariant functional $ M $-estimator. Section \ref{sec_sim} presents a Monte Carlo simulation study that compares the RoFANOVA with competing methods both in one-way and two-way designs. Then, in Section \ref{sec_real} the RoANOVA is applied to the real-case study devoted to the study of the spatter behaviour in the L-PBF process.
Conclusion is provided in Section \ref{sec_con}.
All computations and plots have been created by using R software \citep{Rcoreteam2020}. The RoFANOVA method is implemented in the \textnormal{\sffamily R} package \textnormal{\sffamily rofanova},
openly available online at \url{https://github.com/unina-sfere/rofanova}.
\section{The robust functional analysis of variance }
\label{sec_met}
\subsection{The scale equivariant functional $ M $-estimator and the functional normalized median absolute deviation estimator}
\label{subsec_mestimator}
This section introduces the equivariant functional $ M $-estimator and the functional normalized median absolute deviation estimators.
Let us consider the random element $ X $ with value in $ L^2\left(\mathcal{T}\right) $, the Hilbert space of square integrable functions defined on the compact set $ \mathcal{T}\subset \mathbb{R}^p $, with the usual norm $ ||f||=\left(\int_{\mathcal{T}}f^2\left(t\right) dt\right)^{1/2} $, for $f\in L^2\left(\mathcal{T}\right)$, having mean function $ \mu\left(t\right)=\Ex\left[X\left(t\right)\right] $ and covariance function $ \gamma\left(s,t\right)=\Cov\left[X\left(s\right),X\left(t\right)\right] $, for $ s,t\in\mathcal{T}$. Moreover, let $ \bm{X}=\left(X_1,\dots,X_n\right)^T $ be a vector whose elements $X_i$ are independent realizations of $ X $.
Recently, \cite{sinova2018m} proposed a functional $ M $-estimator of location defined as
\begin{equation}\label{eq_sin}
\hat{\mu}_{s}=\argmin_{y\in L^2\left(\mathcal{T}\right) }\sum_{i=1}^{n}\rho\left(||X_i-y||\right),
\end{equation}
where $ \rho:\mathbb{R}^+\rightarrow \mathbb{R}$ is the \textit{loss function}, which is continuous, non-decreasing and satisfies $ \rho\left(0\right)=0 $. As shown by \cite{sinova2018m}, each version of $ \hat{\mu}_{s} $ is well-defined and enjoys good theoretical properties, e.g., it has maximal breakdown value and is strong consistent under suitable model assumptions.
Unfortunately, these estimators are not scale equivariant. This means that, if all $ X_i $ are equally scaled, the resulting robust estimator is not necessarily equally scaled, in analogy with the multivariate case \citep{maronna2019robust}.
Following \cite{maronna2019robust}, we propose a scale equivariant $ M $-estimator of location defined as
\begin{equation}\label{eq_scaleq}
\hat{\mu}=\argmin_{y\in L^2\left(\mathcal{T}\right) }\sum_{i=1}^{n}\rho\left(\Big\lVert\frac{X_i-y}{\sigma}\Big\lVert\right),
\end{equation}
where $ \sigma\left(t\right)=\sqrt{\gamma\left(t,t\right) }$, for $ t\in\mathcal{T} $. If $ \sigma $ is known, the problem can be reduced to the case of a $ L^2 $ random element with $ \sigma=1 $. However, $ \sigma $ can be rarely assumed as known, and thus it should be substituted by a robust scale estimator.
In this regard, we define the FuNMAD estimator of $\sigma$ as follows
\begin{equation}\label{eq_funmad}
\FuNMAD\left(\bm{X}\right)=\frac{1}{c}\Med\left(|\bm{X}-\hat{\mu}_{s,med}|\right),
\end{equation}
with $c=0.6745$ and where $ \hat{\mu}_{s,med}$ denotes the functional generalization of the median obtained as the solution of the optimization problem in equation \eqref{eq_sin} with $ \rho^{med}\left(\cdot\right)= |\cdot| $; $|\bm{X}-\hat{\mu}_{s,med}|=\left(|X_1-\hat{\mu}_{s,med}|,\dots,|X_n-\hat{\mu}_{s,med}|\right)^T$ and $ \Med\left(\cdot \right) $ transforming a vector of functions to a function of pointwise medians. The constant $ c $ makes $ \FuNMAD $ an asymptotically pointwise consistent estimator of $\sigma$ as shown in the Supplementary Material.
Because the minimization problem in equation \eqref{eq_sin} has not a closed-form solution, \cite{sinova2018m} proposed a standard iteratively re-weighted least-squares algorithm to approximate $ \hat{\mu}_{s} $. The algorithm is specifically modified to approximate $ \hat{\mu} $ in equation \eqref{eq_scaleq} with $ \sigma $ estimated through $\FuNMAD\left(\bm{X}\right)$, and can be summarized in the following steps.
\begin{enumerate}[Step 1.]
\item Select initial weight vector $\bm{w}^{\left(0\right)}=\left(w_1^{\left(0\right)},\dots,w_n^{\left(0\right)}\right)\in \mathbb{R}^n$ such that $ w_i^{\left(0\right)}\geq 0 $ and $ \sum_{i=1}^{n}w_i^{\left(0\right)}=1$.
\item Generate a sequence $ \lbrace \hat{\mu}^{\left(k\right)}\rbrace_{k\in\mathbb{N}} $ iterating the following procedure:
\begin{equation*}
\hat{\mu}^{\left(k\right)}=\sum_{i=1}^{n}w_i^{\left(k-1\right)}X_i,\quad \quad w_i^{\left(k\right)}=\frac{\psi\left(\Big\lVert\frac{X_i-\hat{\mu}^{\left(k\right)}}{\sigma}\Big\lVert\right)}{\sum_{i=1}^{n}\psi\left(\Big\lVert\frac{X_i-\hat{\mu}^{\left(k\right)}}{\sigma}\Big\lVert\right)},
\end{equation*}
where $ \psi $ is the first derivative of the loss function $ \rho $.
\item Terminate the algorithm when, for a tolerance $ \varepsilon>0 $, the following condition is met
\begin{equation*}
\frac{|J\left(\hat{\mu}^{\left(k\right)}\right)-J\left(\hat{\mu}^{\left(k-1\right)}\right)|}{J\left(\hat{\mu}^{\left(k-1\right)}\right)}<\varepsilon,
\end{equation*}
where $ J\left(h\right) =\sum_{i=1}^{n}\rho\left(\Big\lVert\frac{X_i-h}{\hat{\sigma}}\Big\lVert\right)$.
\end{enumerate}
The initial weight vector can be chosen with $ w_i^{\left(0\right)}=\frac{\psi\left(\Big\lVert\frac{X_i-\hat{\mu}^{\left(0\right)}}{\sigma}\Big\lVert\right)}{\sum_{i=1}^{n}\psi\left(\Big\lVert\frac{X_i-\hat{\mu}^{\left(0\right)}}{\sigma}\Big\lVert\right)} $ where $ \hat{\mu}^{\left(0\right)} $ is a robust initial estimate of $ \mu $.
The loss function $ \rho $ in equation \eqref{eq_scaleq} defines the properties of the resulting estimator $ \hat{\mu} $.
For instance, the \emph{Huber's family} of loss functions \citep{huber1964robust}, which generates monotone functional
$ M $-estimators of location, is given by
\begin{equation*}\label{eq_huber}
\rho_{a}^{HU}\left(x\right)=\begin{cases}
x^2/2 & \text{if } 0\leq x\leq a \\
a\left(x-a/2\right) & \text{if }a< x ,
\end{cases}
\end{equation*}
with tuning parameter $ a>0 $. It gives less importance to large errors compared to the standard least-squares loss function $ \rho^{sqr}\left(x\right)=x^2 $.
Functional $ M $-estimators arise from the \emph{bisquare} or \emph{Tukey's biweight family} of loss functions \citep{beaton1974fitting} defined as
\begin{equation*}\label{eq_bisquare}
\rho_{a}^{BI}\left(x\right)=\begin{cases}
a^2/6\left[1-\left(1-\left(x/a\right)^2\right)^3\right] & \text{if } 0\leq x\leq a \\
a^2/6 & \text{if }a< x ,
\end{cases}
\end{equation*}
with tuning parameter $ a>0 $. $ M $-estimators obtained by using $ \rho_{a}^{BI} $ are redescending, that is values of $ x>a $ give the same contribution to the loss, regardless of their distance from $ a $.
Another very used family of loss functions is the \emph{Hampel's one} \citep{hampel1974influence}, which is defined as
\begin{equation*}\label{eq_hampel}
\rho_{a,b,c}^{HA}\left(x\right)=\begin{cases}
x^2/2 & \text{if } 0\leq x<a \\
a\left(x-a/2\right) & \text{if } a\leq x<b\\
\frac{a\left(x-c\right)^2}{2\left(b-c\right)}+a\left(b+c-a\right)/2 & \text{if } b\leq x<c \\
a\left(b+c-a\right)/2& \text{if }c\leq x,
\end{cases}
\end{equation*}
with tuning parameter $ a,b,c>0 $. $ M $-estimators obtained by using $ \rho_{a,b,c}^{HA} $ are redescending as well.
Finally, the \emph{optimal family} of loss functions \citep{maronna2019robust} is defined as
\begin{equation*}\label{eq_opt}
\rho_{a}^{OP}\left(x\right)=\int_{0}^{x}\left(-\frac{\Phi'\left(|x|\right)+a}{\Phi\left(|x|\right)}\right)_{+}dx,
\end{equation*}
where $ \Phi $ is the standard normal density, $ a>0 $ is a tuning parameter and $ \left(t\right)_{+} $ denotes the positive part of $ t $.
The tuning parameters used in $ \rho_{a}^{HU},\rho_{a}^{BI},\rho_{a,b,c}^{HA} $ and $ \rho_{a}^{OP} $ are chosen in order to ensure given asymptotic efficiency with respect to the normal distribution \citep{maronna2019robust}.
\subsection{The proposed robust method for the functional analysis of variance}
\label{subsec_prop}
The aim of this section is to describe the proposed RoFANOVA for the multiway functional ANOVA design. Without loss of generality, and for ease of notation, we will focus on the two-way functional ANOVA design with interaction, but the extension to more complex designs is straightforward.
To introduce the two-way functional ANOVA design with interaction, let us consider a functional response $ X $, which is a random element with values in $ L^2\left(\mathcal{T}\right) $, $ \mathcal{T}\subset \mathbb{R}^p $, and is possibly affected by two factors, say A and B (with $ I $ and $ J $ levels, respectively). In this model, $ X $ will be expressed as the sum of two main effects and an interaction between them, plus a random error. Our aim is to test the statistical significance of the main effects and interaction term.
For $ k=1,\dots,n_{ij} $, let $ X_{ijk} $, denote the realizations of $ X $ at level $ i $ of the factor A, $ i=1,\dots,I $, and level $ j $ of the factor B, $ j=1,\dots,J $. Then, the two-way functional ANOVA model with interaction to be tested is
\begin{equation}\label{eq_modanova}
X_{ijk}\left(t\right)=m\left(t\right)+f_i\left(t\right)+g_j\left(t\right)+h_{ij}\left(t\right)+\varepsilon_{ijk}\left(t\right) \quad t\in \mathcal{T},
\end{equation}
where $ m $ is the functional grand mean, which describes the overall shape of the process, $ f_i $ and $ g_j $ are the functional main effects and $ h_{ij} $ is the interaction term. All these terms have values in $ L^2\left(\mathcal{T}\right) $.
The functional errors $ \varepsilon_{ijk} $ are assumed to be independent and identically distributed random functions with zero-mean and covariance function $ \gamma$. They are not required to be Gaussian.
In order to make the model identifiable, we will assume that $ \sum_{i=1}^{I}\sum_{j=1}^{J}n_{ij}f_i\left(t\right)=\sum_{j=1}^{J}\sum_{i=1}^{I}n_{ij}g_j\left(t\right)=\sum_{i=1}^{I}\sum_{j=1}^{J}n_{ij}h_{ij}\left(t\right)=0 $.
To test the significance of the coefficients in the model \eqref{eq_modanova}, (that is, to extend the classical ANOVA test to the functional data setting), we consider the following null and alternative hypotheses
\begin{align}
\label{eq_H0A}& H_{0,A}:f_1=\dots=f_I=\bm{0}, \quad H_{1,A}: \left(H_{0,A}\right)^{C},\\
\label{eq_H0B}&H_{0,B}:g_1=\dots=g_J=\bm{0}, \quad H_{1,B}: \left(H_{0,B}\right)^{C},\\
\label{eq_H0AB} &H_{0,AB}:h_{11}=\dots=h_{IJ}=\bm{0}, \quad\quad H_{1,AB}: \left(H_{0,AB}\right)^{C},
\end{align}
where $\bm{0}$ is a function almost everywhere equal to zero.
The hypotheses $ H_{0,A} $ against $ H_{1,A} $ and $ H_{0,B} $ against $ H_{1,B} $ involve the effects of the main factors A and B, respectively, whereas, the hypothesis $ H_{0,AB} $ against $ H_{1,AB} $ involves the interaction term between them.
Each test is carried out through a nonparametric permutational approach.
In this regard, we introduce a test statistic that is a functional extension of the robust F-statistic proposed by \cite{schrader1977robust}. The authors considered a robust version of the classical $ F $-test statistic, defined as the fraction of the drop in residual sum of squares between the full model (i.e., the model when $ H_0 $ is false) and the reduced model (i.e., the model when $ H_0 $ is true), and the standard deviation of the error distribution, where all quantities are estimated by using the least-squares approach. The $ F $-test statistic was modified by a specific residual sum of dispersions corresponding to a loss function as those described in Section \ref{subsec_mestimator} in place of the residual sum of squares, and a robust estimate, instead of the least-squares estimate, of the standard deviation of the error distribution.
Specifically, to test the hypotheses \eqref{eq_H0A}, we propose the following test statistic
\small
\begin{multline*}
F_A=\left(I-1\right)^{-1}\left[\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-\bar{X}_{r}-\bar{X}_{r,ij}+\bar{X}_{r,i\cdot}}{\hat{\sigma}_{r,e}}\Big\lVert\right)\right.\\-\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-\bar{X}_{r,ij}}{\hat{\sigma}_{r,e}}\Big\lVert\right)\Bigg],
\end{multline*}\normalsize
where $ \rho $ is a given loss function, $ \hat{\sigma}_{r,e} $ is a robust estimate of the functional standard deviation of the error distribution, and $ \bar{X}_{r} $, $ \bar{X}_{r,i\cdot} $, and $ \bar{X}_{r,ij} $ are, respectively, scale equivariant functional $ M $-estimators (Section \ref{subsec_mestimator}) of the functional grand mean $ m $, group means of $ \lbrace X_{ijk} \rbrace_{k=1,\dots n_{ij},i=1,\dots I } $ and $ \lbrace X_{ijk} \rbrace_{k=1,\dots n_{ij} } $.
In detail, $ \bar{X}_{r} $, $ \bar{X}_{r,i\cdot} $, $ \bar{X}_{r,ij} $ and $ \hat{\sigma}_{r,e} $ are defined as
\begin{align*}
\bar{X}_{r}&=\argmin_{y\in L^2\left(\mathcal{T}\right) }\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-y}{\hat{\sigma}_{r}}\Big\lVert\right), \quad & \hat{\sigma}_{r}&=\FuNMAD\left(\lbrace X_{ijk} \rbrace_{\begin{aligned}
\scriptscriptstyle i= & \scriptscriptstyle 1,\dots, I\\[-1em]
\scriptscriptstyle j= & \scriptscriptstyle 1,\dots, J\\[-1em]
\scriptscriptstyle k= & \scriptscriptstyle 1,\dots, n_{ij}\\[-1em]
\end{aligned}} \right),\\
\bar{X}_{r,i\cdot}&=\argmin_{y\in L^2\left(\mathcal{T}\right) }\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-y}{\hat{\sigma}_{r,i\cdot}}\Big\lVert\right),\quad & \hat{\sigma}_{r,i\cdot}&=\FuNMAD\left(\lbrace X_{ijk} \rbrace_{\begin{aligned}
\scriptscriptstyle j= & \scriptscriptstyle 1,\dots, J\\[-1em]
\scriptscriptstyle k= & \scriptscriptstyle 1,\dots, n_{ij}\\[-1em]
\end{aligned} } \right),\\
\bar{X}_{r,ij}&=\argmin_{y\in L^2\left(\mathcal{T}\right) }\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-y}{\hat{\sigma}_{r,ij}}\Big\lVert\right),\quad &\hat{\sigma}_{r,ij}&=\FuNMAD\left(\lbrace X_{ijk} \rbrace_{\begin{aligned}
\scriptscriptstyle k= & \scriptscriptstyle 1,\dots, n_{ij}\\[-1em]
\end{aligned}} \right),
\end{align*}
\vspace{-0.2cm}
\begin{equation*}
\hat{\sigma}_{r,e}=\frac{1}{0.6745}\Med\left(|\lbrace X_{ijk}-\bar{X}_{r,ij}\rbrace_{\begin{aligned}
\scriptscriptstyle i= & \scriptscriptstyle 1,\dots, I\\[-1em]
\scriptscriptstyle j= & \scriptscriptstyle 1,\dots, J\\[-1em]
\scriptscriptstyle k= & \scriptscriptstyle 1,\dots, n_{ij}\\[-1em]
\end{aligned}} |\right).
\end{equation*}
The test statistic $F_A$ represents the mean difference between the standardized residual sum of dispersions under the reduced model and the full model, and is analogous to that used by \cite{schrader1977robust} in the classical setting.
Intuitively, it is a measure of the discrepancy between residuals of the model under $H_{0,A} $ and under $ H_{1,A} $, obtained through robust statistics.
Analogously, to test the hypotheses \eqref{eq_H0B} and \eqref{eq_H0AB}, we define
\small{
\begin{multline*}
F_B=\left(J-1\right)^{-1}\left[\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-\bar{X}_{r}-\bar{X}_{r,ij}+\bar{X}_{r,\cdot j}}{\hat{\sigma}_{r,e}}\Big\lVert\right)\right.\\-\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-\bar{X}_{r,ij}}{\hat{\sigma}_{r,e}}\Big\lVert\right)\Bigg],
\end{multline*}
\begin{multline*}
F_{AB}=\left(\left(I-1\right)\left(J-1\right)\right)^{-1}\left[\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-\bar{X}_{r,i\cdot}-\bar{X}_{r,\cdot j}+\bar{X}_{r}}{\hat{\sigma}_{r,e}}\Big\lVert\right)\right.\\-\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-\bar{X}_{r,ij}}{\hat{\sigma}_{r,e}}\Big\lVert\right)\Bigg],
\end{multline*}}
\normalsize
where
\small{
\begin{align*}
\bar{X}_{r,\cdot j}&=\argmin_{y\in L^2\left(\mathcal{T}\right) }\sum_{i=1}^{I}\sum_{k=1}^{n_{ij}}\rho\left(\Big\lVert\frac{X_{ijk}-y}{\hat{\sigma}_{r,\cdot j}}\Big\lVert\right),\quad & \hat{\sigma}_{r,\cdot j}&=\FuNMAD\left(\lbrace X_{ijk} \rbrace_{\begin{aligned}
\scriptscriptstyle i= & \scriptscriptstyle 1,\dots, I\\[-1em]
\scriptscriptstyle k= & \scriptscriptstyle 1,\dots, n_{ij}\\[-1em]
\end{aligned} } \right).
\end{align*}
}
\normalsize
Different versions of the proposed test statistics may emerge by the choice of the loss function $ \rho $ as defined in Section \ref{subsec_mestimator}, and by the use of $ \hat{\sigma}_{r,ij}=\hat{\sigma}_{r,e} $ to estimate $ \bar{X}_{r,ij} $.
Another element to choose in a permutation test is the approximation method for the distribution of the considered statistic under the null hypothesis. In our case, we selected the Manly's scheme \citep{gonzalez1998analysis,manly2006randomization} that consists of simply permuting the raw data without restrictions.
Although other schemes could be used, the Manly's one has demonstrated good performance and simplicity, especially when the sample size, at given factor levels, is small. See \cite{gonzalez1998analysis} and \cite{anderson2001permutation} for further details.
Lt $ F $ generically denotes the statistic (resp., $ F_A $ or $ F_B $ or $ F_{AB} $) to test, at level $ \alpha $, $ H_0 $ against $ H_1 $ (resp., $H_{0,A} $ against $ H_{1,A} $; or $ H_{0,B} $ against $ H_{1,B} $; or $ H_{0,AB} $ against $ H_{1,AB} $). Then, the proposed permutation test can be outlined by the following steps.
\begin{enumerate}[Step 1.]
\item Compute the observed value of the test statistic $ F_{obs} $, by considering the original sample $ \lbrace X_{ijk} \rbrace_{k=1,\dots n_{ij},i=1,\dots I,j=1,\dots J } $.
\item Randomly permute the data, among the Factor A and Factor B combinations, $ B $ times, and for each permuted sample compute the value $ F^*_1,\dots, F^*_B $ of the statistic $ F $.
\item Compute the approximated p-value as
\begin{equation*}
p=\frac{1}{B}\sum_{i=1}^{B}I\left(F^*\geq F_{obs}\right),
\end{equation*}
where $ I\left(E\right) $ takes values 1 or 0 depending on whether E is true or
false.
\item Accept $ H_0 $ if $ p>\alpha $, otherwise reject $ H_0 $.
\end{enumerate}
This is an approximate (asymptotically exact) level-$ \alpha $ test for $ H_0 $ against $ H_1 $ \citep{anderson2001permutation}.
The larger the number of permutations $B$, the lower the approximation error. We suggest to select the number of permutations $B$ equal to or larger than 1000 \citep{good2013permutation}.
\section{Simulation study}
\label{sec_sim}
In this section, by means of an extensive Monte Carlo simulation study, the performance of the proposed method is assessed in terms of empirical size and power of the test. In particular, the following two scenarios are investigated:
\begin{enumerate}[label=Scenario \arabic*]
\item A one-way FANOVA model (i.e., model \eqref{eq_modanova} with $ m=0 $, $ g_1=\dots=g_J=0$ and $ h_{11}=\dots=h_{IJ}=0$) is considered (Section \ref{sec_oneway}).
\item A two-way FANOVA model (i.e., model \eqref{eq_modanova}) is considered (Section \ref{sec_twoway}).
\end{enumerate}
In each scenario, the FANOVA model is contaminated by different type of outlying curves. To do so, we use the same contamination models as in previous works on robust FDA \citep{fraiman2001trimmed,lopez2009concept,sinova2018m}.
All the details about the data generation process are provided in the Supplementary Materials.
\subsection{One-way functional analysis of variance}
\label{sec_oneway}
The proposed simulation study framework for one-way FANOVA has been inspired by \cite{cuevas2004anova,gorecki2015comparison}.
Three different model M1, M2 and M3, with 3 level main effect $ f_i$, $i=1,2,3$, are considered, and without loss of generality, we assume the curve domain $\mathcal{T}=\left[0,1\right]$.
Model M1 corresponds to $ H_0 $: $ f_1=f_2=f_3$ true, whereas, M2 and M3 provide examples, with $ H_0 $ false, of monotone functions with different increasing patterns. In particular, M2 simulates $f_i$ differences that are smaller than M3, where $ f_i $ are quite separated.
In model M1, we use as performance measure the empirical size, whereas in M2 and M3 we use the empirical power.
Moreover, to simulate different types of outlying curves, seven contamination models denoted by C0-6 are considered.
The model C0 is representative of no contamination. C1-4 represents magnitude contaminations, i.e., curves are generated far from the center, with, in particular, C1-2 (resp., C3-4) representing symmetric and partial trajectory contamination models, that are independent (resp.,
dependent) from the level of the main effect.
Models C5-6 are shape contamination models \citep{lopez2009concept,sinova2018m}.
In all the cases considered, the response curves are independent realizations of a Gaussian process with covariance function $ \gamma\left(s,t\right)=\sigma^2 e^{\left(-|s-t|10^{-5}\right)}$ and are observed through $ 25 $ evenly spread discrete points with $ \sigma $ equal to $ \sigma_{1}=1/25 $, $ \sigma_{2}=1.8/25 $, $ \sigma_{3}=2.6/25 $, $ \sigma_{4}=3.4/25 $, $ \sigma_{5}=4.2/25 $, $ \sigma_{6}=5/25 $ \citep{cuevas2004anova}.
We expect that the higher $\sigma$, the worse the performance in terms of both empirical size and power.
Five versions of the RoFANOVA method are considered, which are defined by different choices of the loss function, viz., the RoFANOVA with median loss $ \rho^{med} $, referred to as RoFANOVA-MED, Huber loss $ \rho_{a}^{HU}$, referred to as RoFANOVA-HUB, bisquare loss $ \rho_{a}^{BI} $, referred to as RoFANOVA-BIS, Hampel loss $ \rho_{a,b,c}^{HA} $, referred to as RoFANOVA-HAM, and, optimal loss $ \rho_{a}^{OP} $, referred to as RoFANOVA-OPT. The tuning constants are chosen to achieve $ 95\%$ asymptotic efficiency, the number of permutations $ B $ are set equal to $ 1000 $ and the functional $0.8\%$ deepest curve following the FM criteria \citep{febrero2012fdausc} is chosen as starting value to compute the robust equivariant functional $ M $-estimators (Section \ref{subsec_mestimator}).
The proposed tests are compared with some non-robust methods already appeared in the literature before. In particular, we consider the method proposed by \cite{gorecki2015comparison}, referred to as FP, which is a permutation test that relies on a basis function representation of the response function; the method proposed by \cite{zhang2014one}, referred to as GPF, based on a globalized version of the pointwise $ F $-test; the method proposed by \cite{zhang2007statistical}, referred to as L\textsuperscript{2}B, a $ L^2 $-norm-based test with the bias-reduced method to estimate the unknown parameters; and the method proposed by \cite{zhang2011statistical}, referred to as FB, which is an $ F $-type test based on the bias-reduced estimation method. All these methods are implemented with the default settings of the R package \texttt{fdANOVA} \citep{gorecki2018fdanova}. In addition, the method proposed by \cite{cuesta2010simple}, based on randomly chosen one-dimensional projections, with both the Bonferroni (referred to as TRPbon) and the false discovery rate (referred to as TRPfdr) corrections, is considered. The TRPbon and TRPfdr are run with 30 random projections through the R package \texttt{fda.usc} \citep{febrero2012fdausc}.
For each triplet (M$l$,C$m$,$ \sigma_n $), $ l=1,\dots,3 $, $ m=0,\dots,6 $, $ n=1,\dots,6 $, the five proposed and the seven competing methods are applied $ N=500 $ times to the generated functional sample to test $ H_0 $: $ f_1=f_2=f_3$ against $ H_1 $: $ (H_0)^C $ at level $ \alpha=0.05 $.
Then, for each case, the empirical sizes (for model M1) and powers (for models M2 and M3) of the tests were computed as the proportion of rejections out of the $ N $ replications whose standard deviation is equal at most to 0.0224, which corresponds to the case of probability of rejection equal to 0.5.
Fig. \ref{fig_M1} displays the results for model M1, that is the empirical size of the eleven tests as a function of $ \sigma_n $, $ n=1,\dots,6 $, for contamination models C0-6.
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}}
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M1_C6_p2M3-eps-converted-to.pdf}\\
\end{tabular}}
\caption{\label{fig_M1}Empirical size of all tests for $ H_{0} $ against $ H_{1} $ (at level $ \alpha=0.05 $) as a function of $ \sigma_n $, $ n=1,\dots,6 $, for contamination models C0-6 in model M1 of Scenario 1. The proposed and competing tests are displayed as black and grey lines, respectively. }
\end{figure}
In this case, the tests provide satisfactory results in controlling the level $ \alpha $, i.e., the empirical size is approximately less than or equal to $ 0.05 $, in case of no contamination (C0), symmetric magnitude contamination (C1-2) and shape contamination, both symmetric (C5) and asymmetric (C6).
On the contrary, for asymmetric magnitude contamination (C3-4), only the RoFANOVA tests based on redescending loss functions, i.e., RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT, are able to control the level $ \alpha $ by ensuring an empirical size approximately less or equal than $ 0.05 $. This was somehow expected, as it is known that rededescending estimators give no weight to observations that are far from the center \citep{maronna2019robust}.
The estimators used in RoFANOVA-MED and RoFANOVA-HUB tests do not have this property and, thus, they suffer from the presence of contaminations depending on the level of the main factor. Note that, among the competitors, the TRPbon approximately controls the level for contamination model C4, while it is slightly affected by outliers in model C5. This comes from the Bonferroni correction property of being conservative for high-dimensional multiple comparisons \citep{lehmann2006testing}.
Fig. \ref{fig_M2} shows the results for model M2 in terms of empirical power. These tend to get worse as $ \sigma_n$ increases. In case of no contamination (C0), the FP test achieves the largest empirical power, even though all RoFANOVA tests have comparable results. For contamination model C1-6, it is extremely clear the proposed RoFANOVA tests outperform all competitors. In particular, among RoFANOVA tests, those based on redescending functional $ M $-estimators (viz., RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT) are the best ones. Note that, for contaminations C3-4, only the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT tests and the TRPbon test (only for C4) should be considered, because all the other methods are not able to successfully control the level $ \alpha $ (see Fig. \ref{fig_M1}).
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}}
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M2_C6_p2M3-eps-converted-to.pdf}\\
\end{tabular}}
\caption{\label{fig_M2}Empirical power of all tests for $ H_{0} $ against $ H_{1} $ (at level $ \alpha=0.05 $) as a function of $ \sigma_n $, $ n=1,\dots,6 $, for contamination models C0-6 in model M2 of Scenario 1. The proposed and competing tests are displayed as black and grey lines, respectively. }
\end{figure}
Fig. \ref{fig_M3} shows the empirical power for model M2 generally become smaller when $ \sigma_n $ increases.
The results are similar to those for model M2, even though the empirical power tends to be larger, due to the more apparent separation of the main effects.
Again, the proposed RoFANOVA tests outperform the competitors in case of contamination (C1-6), and have satisfactory power in case of no contamination (C0). The best results are achieved by the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT tests.
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}}
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/One-way_M3_C6_p2M3-eps-converted-to.pdf}\\
\end{tabular}}
\caption{\label{fig_M3}Empirical power of all tests for $ H_{0} $ against $ H_{1} $ (at level $ \alpha=0.05 $) as a function of $ \sigma_n $, $ n=1,\dots,6 $, for contamination models C0-6 in model M3 of Scenario 1. The proposed and competing tests are displayed as black and grey lines, respectively.}
\end{figure}
\subsection{Two-way functional analysis of variance}
\label{sec_twoway}
In this section, we consider the two-way FANOVA model introduced in equation \eqref{eq_modanova}. The simulation design is inspired by that of \cite{cuesta2010simple}.
As for Scenario 1, we assume $ \mathcal{T}=\left[0,1\right] $ and the functional response depending on a grand mean $ m $, 2 level main effects $ f_i$ and $ g_i $, and on an interaction term $ h_{ij} $ through two parameters $ a $ and $ b $ with values in $ \lbrace 0,0.05,0.10,0.25,0.50\rbrace $. Here, the larger the values of $a$ and $b$ the more $ f_i$ and $g_i$ deviate, respectively, from the grand mean $ m $. Thus, the empirical power should be an increasing function of $a$ and $b$. The empirical size is studied for $a=0$ or $b=0$.
As in Scenario 1, seven contamination models C0-6 are considered, and the response curves are assumed as independent realizations of a Gaussian process with covariance function $ \gamma\left(s,t\right)=\sigma^2 e^{\left(-|s-t|10^{-5}\right)}$. Data are observed through $ 25 $ evenly spread discrete points with $ \sigma=0.3 $.
Also in this scenario, we consider the five versions of the proposed method, viz., RoFANOVA-MED, RoFANOVA-HUB, RoFANOVA-BIS, RoFANOVA-HAM, and RoFANOVA-OPT, with tuning parameters chosen as in Scenario 1.
As competitors we consider: (i) the permutation version of the method proposed by \cite{zhang2011statistical}, referred to as FNDP, which is permutation test based on a $ F $-type statistic; and (ii) the global version of the method proposed by \cite{pini2017interval}, which is the two-way extension of the method of \cite{zhang2014one}, referred to as TGPF.
Both for the FNDP and TGPF methods, the distribution of the test statistic is approximated by using the Manly's scheme \citep{manly2006randomization} with 1000 random permutations.
Moreover, also the TRPbon and TRPfdr method (Section \ref{sec_oneway}) are considered with 30 random projections.
For each triplet (C$m$,$ a$,$ b $), with $ m=0,\dots,6 $, and $ a,b\in \lbrace 0,0.05,0.10,0.25,0.50\rbrace $, the five proposes and the four competing methods are applied $ N=500 $ times to the generated functional sample to test, at level $ \alpha=0.05 $, $ H_{0,A} $, $ H_{0,B} $ and $ H_{0,AB} $ against $ H_{1,A} $, $ H_{1,B} $ and $ H_{1,AB} $, respectively.
Then, for each triplet and test, the empirical size and empirical power of the test were computed as the fraction of rejections out of $ N $ replications (also in this case, with maximum standard deviation equal to 0.0224). The former is considered when $ a=b=0 $, for $ H_{0,A} $, $ H_{0,B} $ against $ H_{1,A} $, $ H_{1,B} $, and when $ a<0.25 $ for $H_{0,AB} $ against $ H_{1,AB} $; whereas the latter is considered when $ a\neq 0$ or $b\neq0 $, for $ H_{0,A} $, $ H_{0,B} $ against $ H_{1,A} $, $ H_{1,B} $, and $ a\geq0.25 $ for $H_{0,AB} $ against $ H_{1,AB} $.
For the sake of conciseness, we summarize the results for cases that are statistically equivalent. For instance, when analyzing the null hypothesis $ H_{0,A} $ (resp., $ H_{0,B} $), for each value of $ a $ (resp., $ b $), the five values corresponding to $ b=\lbrace 0,0.05,0.10,0.25,0.50\rbrace $ (resp., $ a=\lbrace 0,0.05,0.10,0.25,0.50\rbrace $) are summarized through their median. Similarly, when analyzing $ H_{0,AB} $, the values corresponding to $ a<0.25 $ are substituted by their median for each value of $ b $.
Fig. \ref{fig_twoa} shows the empirical size ($ a=0 $) and power ($ a\neq0 $) of all tests for $ H_{0,A} $ against $ H_{1,A} $ as a function of $ a $. When $ a $ increases, the performance of all the methods in rejecting $ H_{0,A} $ enhances. In terms of empirical size (i.e., when $ a=0 $), the results are quite satisfactory for all the methods in case of no contamination (C0), symmetric magnitude contamination (C1-2) and both symmetric (C5) and asymmetric (C6) shape contamination.
However, in case of asymmetric magnitude contamination (C3-4), only the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT tests are able to control the level $ \alpha $, being approximately less than or equal to 0.05.
This behavior is analogous to that achieved in Scenario 1 of Section \ref{sec_oneway}.
In terms of empirical power ($ a\neq0 $), the proposed RoFANOVA test has comparable performance when there are no outliers (C0); whereas it is far better than the competitors for the contamination models C1-6. Note that, for asymmetric magnitude contamination (C3-4), only the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT tests should be considered, being the only ones able to control the level $ \alpha $.
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}}
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_a_C6_p2M3-eps-converted-to.pdf}\\
\end{tabular}}
\caption{\label{fig_twoa}Empirical size ($ a=0 $) and power ($ a\neq0 $) of all tests for $ H_{0,A} $ against $ H_{1,A} $ (at level $ \alpha=0.05 $) as a function of $ a $, for different contamination models (C0-6) in Scenario 2. The proposed and competing tests are displayed as black and grey lines, respectively.}
\end{figure}
In Fig. \ref{fig_twob}, the empirical size ($ b=0 $) and the empirical power ($ b\neq0 $) of all tests for $ H_{0,B} $ against $ H_{1,B} $ (at level $ \alpha=0.05 $) are displayed as a function of $ b $. Also in this case, the proposed tests outperform the competitors, in terms of power, for contamination models C1-6. They simultaneously have, in fact, comparable performance in absence of contamination (C0). Moreover, differently from Scenario 1, all the tests are able to approximately control the level $ \alpha $, even for the contamination models C3-4. This is expected in this case, because the asymmetry in the contamination affects the main effect $ f_i $, only, and not $ g_i $.
Among the proposed tests, the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT tend to perform better than the ones based on monotonic functional $ M $-estimator, viz., the RoFANOVA-MED and RoFANOVA-HUB tests.
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}}
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_b_C6_p2M3-eps-converted-to.pdf}\\
\end{tabular}}
\caption{\label{fig_twob}Empirical size ($ b=0 $) and power ($ b\neq0 $) of all tests for $ H_{0,B} $ against $ H_{1,B} $ (at level $ \alpha=0.05 $) as a function of $ b $, for different contamination models (C0-6) in Scenario 2. The proposed and competing tests are displayed as black and grey lines, respectively.}
\end{figure}
To test $ H_{0,AB} $ against $ H_{1,AB} $, results are presented in the Supplementary Material where it is shown that the empirical power of the proposed tests is much larger than that of the competitors, for all the contamination model C1-6. Moreover, and that among the RoFANOVA tests, the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT achieve the best performance.
\section{Real-case study: analysis of variance of applied to the analysis of spatter behaviour in laser powder bed fusion}
\label{sec_real}
To demonstrate the potential of the proposed approach, this Section presents the real-case study in additive manufacturing. In L-PBF, spatters are process by-products that can be ejected either by the melt pool, i.e., the region when the thin layer of powder is locally melted by the laser, in the form of hot and liquid droplets or by the powder bed regions surrounding the melt pool \citep{young2020types,ly2017metal,bidare2018fluid}. In the latter case, spatters consist of powder particles entrained by convective motions above and around the melt pool. For more details about the spatter generation mechanism, the reader is referred to \cite{young2020types,ly2017metal,bidare2018fluid}, and the literature cited therein.
The analysis of spatters in the L-PBF process has gathered an increasing interest in the last years because they can drive relevant information about the process state and the final quality of the manufactured part.
Studying the effect of controllable process factors and other operating conditions on the spatter behaviour allows getting a deeper comprehension of underlying physical phenomena. Such knowledge may be used to tune the process condition and enhance the quality and mechanical performances of the products, or to design in-line and real-time process monitoring methodologies \citep{colosimo2020machine}.
Hot spatters ejected as a consequence of the laser-material interaction can be observed by means of high-speed cameras installed into the L-PBF machine or placed outside its viewports. The mainstream literature devoted to spatter analysis and monitoring in L-PBF relies on video image processing methods to compute synthetic indices that capture salient aspects of the spatter behaviour, e.g., the number of ejected spatters in each video frame, their size, speed, etc. \citep{grasso2017phase,everton2016review}. In the real-case study presented in this section, instead of treating synthetic descriptors of the spatter ejections as univariate or multivariate variables, the spatter behaviour is translated into a functional format by means of the so-called spatter intensity function introduced in Section 1. Such function captures the spatial spread of ejected spatters and can be estimated for each manufactured layer and for each test treatment.
Section 4.1 presents the main experimental settings, whereas the results of the analysis and the comparison against benchmark methods are reported in Section 4.2.
\subsection{Experimental setting and data preprocessing}
The case study involves the production of specimens of size 5 x 5 x 12 mm via L-PBF of 18Ni(300) maraging steel powder, a steel alloy commonly used for tooling applications, with average particle size between 25 and $35 \mu$m. An industrial L-PBF system, namely a Renishaw AM250, was used, with a high-speed camera in the visible range placed outside the front viewport of the machine as shown in Fig. \ref{fig_setup}, left panel. Videos were recorded during the production of six layers with a sampling rate of 1000 fps (frames per second) and a spatial resolution of about $200 \mu$m/pixel. Specimens are placed as shown in Fig. \ref{fig_setup}, right panel, and produced by varying the energy density provided by the laser to the material. Process parameters corresponding to the energy density levels are reported in the Supplementary Material.
The laser was displaced by a scanner along a predefined path consisting of parallel scan lines, whose orientation changed layer by layer, with a default rotation of about $67^{\circ}$ every layer.
Details about locations and orientations of the six analysed layers are provided in the Supplementary Material.
Along each scan line, the laser melts the material with a pulsed mode, i.e., by exposing points equispaced apart of a quantity $d$ along each scan line with a point exposure duration $t$. The energy density was varied by varying $t$ and $d$.
Within the build chamber, where the L-PBF process takes place, a laminar flow of inert gas, called shielding gas, is used to prevent ejected spatters from falling on the build area, with consequent potential contamination effects, and vaporized material from depositing on the laser window leading to possible attenuation of the laser beam \citep{anwar2018study}.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/Figura_setup_camera_posizione-eps-converted-to.pdf}
\caption{\label{fig_setup} Setup of the high-speed camera in front of the Renishaw AM250 machine's viewport (left panel) and placement of manufactured specimens in the build area, within the camera's field of view (right panel): numbers shown in the specimens correspond to the energy density level, from 1 to 6, applied during the process.}
\end{figure}
The functional response variable is the spatter intensity and was estimated by applying the video image pre-processing method presented in \cite{repossini2017use}. Thanks to this approach, the centroid of each spatter in the video frame was computed and used to determine the spatial coordinates $(s,t)$ of each detected spatter. All details about the video image pre-processing steps can be found in \cite{repossini2017use}.
In order to spatially map the amount of spatters ejected during the production of each specimen in each layer, three additional pre-processing operations were performed. First, the location of spatters was referred to a spatial domain centered in the center of the scanned area of each specimen, to allow comparing the functional response variables for specimens produced in different locations. Second, the spatial domain was discretized into 60 by 80 adjacent squared cells, in order to count the number of spatters ejected in each layer within each cell.
Based on these pre-processing steps, the spatial spread of the spatters, in each layer and for each specimen, could be summarized into the function $ Y_{i,j,k}\left(s,t\right) $ defined on the bi-dimensional domain $ \mathcal{T}=\left[0,1\right]\times \left[0,1\right] $, where indices $ i=1\dots,6 $, $ j=1,\dots 6 $ and $ k=1,\dots,n_{ij} $ indicate the energy density level, the layer, and the number of replicates (specimens) for each treatment, respectively.
The spatter intensity function $ Y_{i,j,k}\left(s,t\right) $ is a smoothed version of the actual amount of spatters counted in every location of the spatial domain.
The number of replicates $ n_{ij} $ is fixed and equal to 3, as three specimens were produced for each energy density level, except for $ i=6 $ and $ j= 1$ where $ n_{ij}=2 $, due to a delamination occurred in initial layers prevented from producing one of the three specimens with the lowest energy density level.
The $ Y_{i,j,k} $ are obtained by means of a smoothing phase based on tensor product bases of cubic splines, with second derivative penalty as marginal smooths. The marginal basis dimensions, set equal to 30, and the smoothing parameters were chosen by using restricted maximum likelihood (REML) \citep{wood2017generalized}. The smoothing phase was performed by using the R package \texttt{mgcv} \citep{wood2017generalized}.
Then, in order to reduce phase variability, a registration phase was performed \citep{ramsay2005functional}. It consists in the shifting of each $ Y_{i,j,k}$ along the $ s $ and $ t $ axes to minimize the $ L^2 $ distance with respect to the reference curve, which was chosen such that the mean of pairwise distances among the aligned curves is minimum.
The functional observations $ Y_{i,j,k}$, $ i=1\dots,6 $, $ j=1,\dots 6 $ and $ k=1,\dots,n_{ij} $, for $ t=0.75 $ and $ s=0.5 $ are represented in Fig. \ref{fig_X} at different energy density levels and in different layers.
The graphical representation of cross-sections of the spatter intensity function in Fig. \ref{fig_X} was adopted to aid the superimposition and direct comparison of functional patterns corresponding to different experimental treatments.
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{cccc}
\includegraphics[width=0.25\textwidth]{Figures/Case-study_X_s_t075_lev-eps-converted-to.pdf} & \includegraphics[width=.25\textwidth]{Figures/Case-study_X_t_s050_lev-eps-converted-to.pdf}& \includegraphics[width=.25\textwidth]{Figures/Case-study_X_s_t075_lay-eps-converted-to.pdf} & \includegraphics[width=.25\textwidth]{Figures/Case-study_X_t_s050_lay-eps-converted-to.pdf}\\
(a) & (b) &(c) & (d)
\end{tabular}
}
\caption{\label{fig_X}The functional observations $ Y_{i,j,k}$ for $ t=0.75 $, (a) and (c), and $ s=0.5 $, (b) and (d), in the real-case study, for different energy density levels ((a) and (b)) and different scan strategies ((c) and (d)).}
\end{figure}
\subsection{Results}
\label{sec_cas_res}
The spatter intensity functions $ Y_{i,j,k}$ ($ i=1\dots,6 $, $ j=1,\dots 6 $ and $ k=1,\dots,n_{ij} $) are modeled according to equation \eqref{eq_modanova}, where
$ f_i $ is the energy density functional effect, $ g_i $ is the layer functional effect, $ h_{ij} $ is the interaction term between the energy density and the layer.
The equivariant functional M-estimators (Section \ref{subsec_mestimator}) are shown in the Supplementary Material.
The aim of the analysis is therefore to test the energy density effect $ H_{0,Flu}=H_{0,A} $ \eqref{eq_H0A}, the layer effect $ H_{0,Lay}=H_{0,B} $ \eqref{eq_H0B} (mainly related to the layer by layer variation of the laser scan direction) and their interaction effect $ H_{0,FluLay}=H_{0,AB} $ against the alternatives $ H_{1,Flu}=H_{1,A} $ \eqref{eq_H0A}, $ H_{1,Lay}=H_{1,B} $ \eqref{eq_H0B} and $ H_{1,FluLay}=H_{1,AB} $.
In particular, Fig. \ref{fig_res} shows (a) the residuals of the fitted model for $t=0.75$ (the approximate $t$ value of the spatter intensity peak), obtained by using the RoFANOVA-BIS test as implemented in Section \ref{sec_sim}, and (b) the boxplot of their $L^1$ norms, defined as $ ||f||_1=\int_{\mathcal{T}}|f\left(t\right)| dt $, for $f\in L^2\left(\mathcal{T}\right)$. Because $\mathcal{T}=\left[0,1\right]\times \left[0,1\right] $, the $L^1$ norm can be interpreted as the average value of the function over its domain.
It is clear from Fig. \ref{fig_res} that some outliers are present in this real-case study. However, except from a few residuals that plot far from the bulk of the data, there are some points that could not be easily labeled as outliers. As mentioned in the introduction, the L-PBF process is characterized by complex dynamics with many transient and local phenomena that not only affect the natural variability of the measured quantities, but could lead also to outlying patterns. Determining whether an experimental point is an outlier or not, and identifying its root causes can be a difficult task, which makes the diagnostic approach hardly applicable in the absence of additional data and information.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.3\textwidth]{Figures/Case-study_res_t075-eps-converted-to.pdf} & \includegraphics[width=.3\textwidth]{Figures/Case-study_res_boxplot.pdf} \\
(a) & (b)\\
\end{tabular}
\caption{\label{fig_res}(a) Residuals of the fitted model for $t=0.75$, obtained by using the RoFANOVA-BIS test as implemented in Section \ref{sec_sim}, and (b) boxplot of their $L^1$ norms.}
\end{figure}
Therefore, we applied the RoFANOVA test described in Section \ref{sec_sim}, viz, RoFANOVA-MED, RoFANOVA-HUB, RoFANOVA-BIS, RoFANOVA-HAM, and RoFANOVA-OPT, specifically adapted for bi-dimensional functional data. As in the Monte Carlo simulation study (Section \ref{sec_sim}), the tuning constants are chosen such that the $ 95\%$ asymptotic efficiency is achieved, the number of permutations $ B $ are set equal to $ 1000 $. The functional sample mean is used as starting value to compute the robust equivariant functional $ M $-estimators (Section \ref{subsec_mestimator}).
The results are shown in Table \ref{tab_pvalue}. All the tests agree in considering significant the interaction between the energy density and the layer.
\begin{table}
\caption{\label{tab_pvalue}p-values of the RoFANOVA tests for $ H_{0,Flu} $, $ H_{0,Lay} $ and $ H_{0,FluLay}$ against $ H_{1,Flu} $, $ H_{1,Lay} $ and $ H_{1,FluLay}$.}
\centering
\scriptsize
\resizebox{1\textwidth}{!}{
\begin{tabular}{cccccccc}\hline
&RoFANOVA-MED & RoFANOVA-HUB & RoFANOVA-BIS&RoFANOVA-HAM&RoFANOVA-OPT\\\hline
$ H_{0,FluLay}$&0.00&0.01 &0.00&0.00&0.00\\
$ H_{0,Flu}$&0.00&0.00&0.00&0.00&0.00\\
$ H_{0,Lay}$&0.00&0.00&0.00&0.00&0.00\\ \hline
\end{tabular}
}
\end{table}
When an the interaction effect is present, it is well-known that an interpretation of the main effects becomes less straightforward than if the interaction is not significant \citep{miller1997beyond}, because the layer effect upon the spatter intensity will differ depending on the energy density level.
In this case, the best way to interpret the results is through the interaction plot \citep{montgomery2017design}, which graphically represents the response means at different factor levels.
Fig. \ref{fig_int} shows an interaction plot adapted to deal with bi-dimensional data. In particular, the $ L^1 $ norms of the group means, corresponding to the RoFANOVA-BIS test, are plotted as a function of the energy density level and the layer. In this case, if an interaction is present, the trace of the average response across the levels of one factor, which is plotted separately for each level of the other factor, will not be parallel \citep{montgomery2017design}.
Fig. \ref{fig_int} shows that, as the energy density increases, the spatter intensity tends to increase as well. This is in agreement with the fact that a higher energy density generates a larger and hotter melt pool with more intense convective and recoil motions, which translates into a more intense spatter ejection \citep{yang2020monitoring,repossini2017use,bidare2018fluid}. More interestingly, Fig. \ref{fig_int} shows different patterns corresponding to different layers. Indeed, in layers 1, 2, and 6, the spatter intensity is increasing with respect to the energy density. These three levels were characterized by very similar laser scan directions, with a low angle relative to the shielding gas flow (between $10^{\circ}$ and $40^{\circ}$). When the scan direction is parallel (or little angled) to the gas flow, more powder bed particles are pushed along the laser path and increase the occurrence of particles heated up by the hot metal vapour emission and ejected as hot spatters. Under these conditions, increasing the energy density increases the intensity of convective motions that entrap the powder particles into the hot vapour emission and hence the spatter intensity \citep{bidare2018fluid}.
A different influence of the energy density on the spatter intensity was observed in layers 3, 4 and 5. In these layers, the laser scan direction was almost perpendicular to the shielding gas flow direction, i.e., with angles in the range $80^{\circ}$ to $90^{\circ}$. Under these conditions, particles are dragged away from the scan path, and reduce the amount of powder particles ejected as hot spatters, and hence, the overall spatter intensity \citep{bidare2018fluid}. In addition, the analysis reveals that, when the laser scan direction was about perpendicular to the gas flow, there was a range of intermediate energy densities (from level 3 to level 5) at which the influence of the energy density itself on the spatter intensity reduced or even inverted. This can be interpreted as follows. Conversely, when the laser scan direction is parallel to the gas flow, an increase of the energy density causes an increase of convective motions and metal vapour emissions that result also in higher spatter intensity. When the laser scan direction is perpendicular to the gas flow, an increase of the energy density still causes an increase of convective motions and metal vapour emissions, but such vapour emission has little effect on the spatter intensity, which makes the influence of the energy density mainly evident at very low or very high energy density levels only. Such interaction between the energy density and laser scan direction on the spatter intensity was explored in a very few studies in the literature. Nevertheless, it is particularly relevant to understand the underlying spatter behaviour and to design either process optimization or process monitoring tools that rely on the in-line observation of such ejected particles.
Finally, we cannot confidently affirm that spatter intensity is affected by layer (i.e., by laser scan direction that changes layer by layer), because we cannot distinguish if differences among layers are due to interactions, only, or to a systematic laser scan direction effect too.
\begin{figure}
\centering
\includegraphics[width=0.3\textwidth]{Figures/Case-study_X_int-eps-converted-to.pdf}
\caption{\label{fig_int}Interaction plot as a function of the energy density level and the layer in the real-case study. }
\end{figure}
Even if the use of RoFANOVA is recommended in light of the results shown by the Monte Carlo simulation study (Section \ref{sec_sim}), for the sake of completeness, the bi-dimensional version of the FNDP and TGPF test have been applied. For the latter, the Manly's scheme \citep{manly2006randomization} with 1000 random permutations was used to approximate the test statistic distribution.
By comparing the additional results, which are shown in Table \ref{tab_pvalue2}, with the proposed tests (Table \ref{tab_pvalue}) we note that they disagree in considering as significant the interaction between energy density and layer. In particular, the FNDP and the TGPF tests do not reject the null hypothesis of no interaction (i.e., large p-values).
This may suggest, in accordance with the Monte Carlo simulation results achieved in the two-way FANOVA design case (Section \ref{sec_twoway}), that FNDP and TGPF tests may have not enough statistical power to detect a technologically relevant interaction among the main factors.
\begin{table}
\caption{\label{tab_pvalue2}p-values of the FNDP and TGPF tests for $ H_{0,Flu} $, $ H_{0,Lay} $ and $ H_{0,FluLay}$ against $ H_{1,Flu} $, $ H_{1,Lay} $ and $ H_{1,FluLay}$.}
\centering
\scriptsize
\begin{tabular}{ccc}\hline
&FNDP&TGPF\\\hline
$ H_{0,FluLay}$&0.72&0.23\\
$ H_{0,Flu}$&0.00&0.00\\
$ H_{0,Lay}$&0.00&0.00\\ \hline
\end{tabular}
\end{table}
\section{Conclusion}
\label{sec_con}
In this paper, we have proposed the RoFANOVA test for the functional analysis of variance problem. In particular, the proposed method has been designed to be robust against functional outliers, which are increasingly common in complex problems and, as it is well known, can severely bias the analyses. Robustness comes from the use of robust test statistics based on the functional equivariant $ M $-estimator and the functional normalized median absolute deviation, which are the extensions of the classical $ M $-estimator and normalized median absolute deviation to functional data.
The test statistic is, then, incorporated in a permutation test, in order to solve the FANOVA problem in a nonparametric fashion.
The proposed approach is demonstrated to be flexible to different choices of the loss function, and, to be applicable to both one-dimensional and bi-dimensional functional data.
To the best of the authors' knowledge, this is the first example of a robust method for the FANOVA problem that is specifically designed to reduce the abnormal observation weights in the computation of the test statistic in comparisons with the standard least-squares loss function appeared in the literature, where attention has been mainly focused on non-robust methods.
The performance of the proposed method has been investigated by means of an extensive Monte Carlo simulation study, where the proposed RoFANOVA have been compared with other methods already present in the literature. The results have shown that the proposed tests clearly outperform the competitors in terms of both empirical size and empirical power when outlier contamination is present. Moreover, even in case of no outlier contamination the loss of power of the RoFANOVA tests with respect to competitors is negligible.
The proposed method was applied to a motivating real-case study in the field of additive manufacturing.
Apart from the known influence of the energy density on the spatter intensity, in agreement with previous studies, the RoFANOVA test revealed a statistically significant interaction between the energy density and the laser scan direction relative to the shielding gas flow.
The statistical significance of the interaction between these two factors was not identified by the other non-robust tests, which confirms the effectiveness of the proposed approach to applications where complex process dynamics may lead to outlying patterns that contaminate the experimental dataset. The validity of the proposed approach is naturally not limited to the case study here presented and, in general, to manufacturing applications.
In future research, the effects of heteroscedasticity on the RoFANOVA test should be investigated in order to be able to deal with a wider variety of settings. In addition, some efforts should be made to extend the proposed method to more complex FANOVA designs.
\bibliographystyle{chicago}
\section{Derivation of the constant $c$ in the $ \FuNMAD $ expression}
\label{app_1}
Following Theorem 3.4 of \cite{sinova2018m}, $\hat{\mu}_{s,med}$ is a strongly consistent estimator of $\tilde{\mu}_{s,med}=\argmin_{y\in L^2\left(\mathcal{T}\right) }\Ex\left[||X_i-y||\right]$.
If we assume $X$ is a Gaussian random process, then, by Proposition 3.2 of \cite{sinova2018m}, it follows that $\tilde{\mu}_{s,med}=\mu$, where $\mu$ is the mean function of the random element $X$.
So stated, from the population version of Equation \eqref{eq_funmad} and the definition of univariate population median, for each $t\in\mathcal{T}$ we have asymptotically
\begin{equation*}
0.5=Pr\left[|X\left(t\right)-\mu\left(t\right)|<c\FuNMAD\left(t\right)\right]=Pr\left[|Z|<c\frac{\FuNMAD\left(t\right)}{\sigma(t)}\right],
\end{equation*}
where $Z$ is a standard normal random variable.
Therefore, we have
\begin{equation*}
\Phi\left(c\frac{\FuNMAD\left(t\right)}{\sigma(t)}\right)-\Phi\left(-c\frac{\FuNMAD\left(t\right)}{\sigma(t)}\right)=0.5,
\end{equation*}
where $\Phi$ is the cumulative distribution function of the standard normal distribution.
Noticing that
\begin{equation*}
\Phi\left(-c\frac{\FuNMAD\left(t\right)}{\sigma(t)}\right)=1-\Phi\left(c\frac{\FuNMAD\left(t\right)}{\sigma(t)}\right),
\end{equation*}
then
\begin{equation}
c\frac{\FuNMAD\left(t\right)}{\sigma(t)}=\Phi^{-1}\left(3/4\right)=0.6745,
\end{equation}
and thus $c=0.6745$ makes $\FuNMAD$ an asymptotically pointwise consistent estimator of $\sigma$.
\section{Details on Data Generation}
In this section, the data generation process for Scenario 1 and Scenario 2 of the simulation study is described.
For Scenario 1, let $ \mathcal{T}=\left[0,1\right] $, then the three following different model with 3 level main effects $ f_i$ are considered
\begin{enumerate}[label=M\arabic*]
\item $ f_i\left(t\right)= t\left(1-t\right) $ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\item $ f_i\left(t\right)= t^i\left(1-t\right)^{6-i} $ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\item $ f_i\left(t\right)= t^{i/5}\left(1-t\right)^{6-i/5} $ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $.
\end{enumerate}
Model M1 corresponds to $ H_0 $: $ f_1=f_2=f_3$ true, whereas, M2 and M3 provide examples, with $ H_0 $ false, of monotone functions with different increasing patterns. In particular, M2 simulates $f_i$ differences that are smaller than M3, where $ f_i $ are quite separated.
To simulate different type of outlying curves, let $ B $ and $ U $ be two independent random variables following a Bernoulli (with parameter $ p $) and a discrete uniform (on $ \lbrace -1,1\rbrace $) distributions, respectively, and let $ T $ be a random number generated from a uniform distribution on $ \left(0,0.75\right) $.
Then, the following four contamination models $ C_i $ are considered
\begin{enumerate}[label=C\arabic*]
\setcounter{enumi}{-1}
\item $ C_i\left(t\right)= 0 $ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\item $ C_i\left(t\right)= BUM$ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\item $ C_i\left(t\right)= \begin{cases}
BUM & \text{if } t\geq T \\
0 & \text{if }t< T ,
\end{cases}$ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\item $ C_i\left(t\right)= \left(-1\right)^{i}BM$ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\item $ C_i\left(t\right)= \begin{cases}
\left(-1\right)^{i}BM & \text{if } t\geq T \\
0 & \text{if }t< T ,
\end{cases}$ for $ t\in \left[0,1\right] $ and $ i=1,2,3 $,
\end{enumerate}
with contamination size constant $ M =25 $ and $ p = 0.1 $.
The model C0 is representative of no contamination. C1-4 represents magnitude contaminations, i.e., curves are generated far from the center, with, in particular, C1-2 (resp., C3-4) representing symmetric and partial trajectory contamination models, that are independent (resp.,
dependent) from the level $i$ of the main effect.
Then, the curves $X_{ik}$ are generated, for $ i=1,2,3 $ and $ k=1,\dots,20 $, as
\begin{equation*}
X_{ik}\left(t\right)= f_i\left(t\right)+ C_i\left(t\right)+\varepsilon_{ik}\left(t\right) \quad t\in \left[0,1\right],
\end{equation*}
where the errors $ \varepsilon_{ik} $ are independent Gaussian processes with zero mean and covariance function $ \gamma\left(s,t\right)=\sigma^2 e^{\left(-|s-t|10^{-5}\right)}$.
In what follows, we consider two shape contamination models \citep{lopez2009concept,sinova2018m} that are both independent and dependent from the level $i$ of the main effect.
In this setting, the curves $X_{ik}$ are generated, for $ i=1,2,3 $ and $ k=1,\dots,20 $, as
\begin{equation*}
X_{ik}\left(t\right)= (1-B)Y_{ik}\left(t\right)+ BZ_{ik}\left(t\right) \quad t\in \left[0,1\right],
\end{equation*}
with
\begin{equation*}
Y_{ik}\left(t\right)= f_i\left(t\right)+\varepsilon_{ik}\left(t\right),
\quad \quad
Z_{ik}\left(t\right)= f_i\left(t\right)+\varepsilon_{ik,c}\left(t\right) \quad t\in \left[0,1\right],
\end{equation*}
where $ \varepsilon_{ik,c} $ are independent Gaussian processes with zero mean and covariance function $ \gamma_{i,c}\left(s,t\right)=\sigma^2 e^{\left(-|s-t|k_{\gamma_{c},i}10^{-5}\right)}$.
The following choices for $ k_{\gamma_{c},i} $ are considered
\begin{enumerate}[label=C\arabic*]
\setcounter{enumi}{4}
\item $ k_{\gamma_{c},i}=10^2 $ for $ i=1,2,3 $,
\item $ k_{\gamma_{c},i}=10^{2+i} $ for $ i=1,2,3 $.
\end{enumerate}
In all the cases considered, the curves $ X_{ik} $ are observed through $ 25 $ evenly spread discrete points and $ \sigma $ is equal to $ \sigma_{1}=1/25 $, $ \sigma_{2}=1.8/25 $, $ \sigma_{3}=2.6/25 $, $ \sigma_{4}=3.4/25 $, $ \sigma_{5}=4.2/25 $, $ \sigma_{6}=5/25 $.
Fig. \ref{fig_one_way ex} shows the realizations of the response curve for the three models M1, M2, and M3 in presence of no contamination (C0) for $\sigma=\sigma_{1}$.
\begin{figure}
\centering
\includegraphics[width=.3\textwidth]{Figures/One-way_mean_M1-eps-converted-to.pdf}
\includegraphics[width=.3\textwidth]{Figures/One-way_mean_M2-eps-converted-to.pdf}
\includegraphics[width=.3\textwidth]{Figures/One-way_mean_M3-eps-converted-to.pdf}
\caption{\label{fig_one_way ex}The response curve realizations for the three models M1, M2, and M3 in absence of contamination (C0) for $\sigma=\sigma_{1}$ in Scenario 1.}
\end{figure}
For Scenario 2, we assume $ \mathcal{T}=\left[0,1\right] $ and the functional response depending on a grand mean $ m $, 2 level main effects $ f_i$ and $ g_i $, and on an interaction term $ h_{ij} $ through two parameters $ a $ and $ b $ as follows
\begin{itemize}
\item $ m\left(t\right)=t(1-t) $ for $ t\in \left[0,1\right] $,
\item $ f_i\left(t\right)=a(-1)^i |\sin(4\pi t)| $ for $ t\in \left[0,1\right] $ and $ i=1,2 $,
\item $ g_j\left(t\right)=b(-1)^j I(t>0.5)$ for $ t\in \left[0,1\right] $ and $ j=1,2 $,
\item $ h_{ij}\left(t\right)=-f_i\left(t\right)g_j\left(t\right) I(a\geq0.25)$ for $ t\in \left[0,1\right] $ and $ i=1,2 $, $ j=1,2 $,
\end{itemize}
with $ a,b\in \lbrace 0,0.05,0.10,0.25,0.50\rbrace $.
For the contamination models C0-4 (Section \ref{sec_oneway}), the curves $X_{ijk}$ are generated, for $ i=1,2 $, $ j=1,2 $ and $ k=1,\dots,20 $, as
\begin{equation*}
X_{ijk}\left(t\right)= m\left(t\right)+f_i\left(t\right)+g_j\left(t\right)+h_{ij}\left(t\right)+C_i\left(t\right)+\varepsilon_{ijk}\left(t\right) \quad t\in \left[0,1\right],
\end{equation*}
where the errors $ \varepsilon_{ijk} $ are independent Gaussian processes with mean zero and covariance function $ \gamma\left(s,t\right)=\sigma^2 e^{\left(-|s-t|10^{-5}\right)}$.
The curves $ X_{ijk} $ are observed through $ 25 $ evenly spread discrete points.
Whereas, for the contamination models C5-6 (Section \ref{sec_oneway}), the curves $X_{ijk}$ are generated, for $ i=1,2 $, $ j=1,2 $ and $ k=1,\dots,20 $, as
\begin{equation*}
X_{ijk}\left(t\right)= (1-B)Y_{ijk}\left(t\right)+ BZ_{ijk}\left(t\right) \quad t\in \left[0,1\right],
\end{equation*}
with
\begin{equation*}
Y_{ijk}\left(t\right)= m\left(t\right)+f_i\left(t\right)+g_j\left(t\right)+h_{ij}\left(t\right)+\varepsilon_{ijk}\left(t\right),
\end{equation*}
\begin{equation*}
Z_{ijk}\left(t\right)= m\left(t\right)+f_i\left(t\right)+g_j\left(t\right)+h_{ij}\left(t\right)+\varepsilon_{ijk,c}\left(t\right),
\end{equation*}
for $ t\in \left[0,1\right] $, where $ \varepsilon_{ijk,c} $ are independent Gaussian process with mean zero and covariance function $ \gamma_{ij,c}\left(s,t\right)=\sigma^2 e^{\left(-|s-t|k_{\gamma_{c},i}10^{-5}\right)}$ with, as for Scenario 1, $ k_{\gamma_{c},i}=10^2 $ for C5 and $ k_{\gamma_{c},i}=10^{2+i} $ for C6. The random variable $ B $ follows a Bernoulli (with parameter $ p=0.1 $) distribution.
In this case, the curves $ X_{ik} $ are observed through $ 25 $ evenly spread discrete points with $ \sigma=0.3 $.
Fig. \ref{fig_two_way_ex} shows the realizations of the response curve for $a=b=0$, $a=0.5$ and $b=0$, and, $a=0$ and $b=0.5$, in absence of contamination (C0).
\begin{figure}
\centering
\includegraphics[width=.3\textwidth]{Figures/Two-way_a_1b_1-eps-converted-to.pdf}
\includegraphics[width=.3\textwidth]{Figures/Two-way_a_5b_1-eps-converted-to.pdf}
\includegraphics[width=.3\textwidth]{Figures/Two-way_a_1b_5-eps-converted-to.pdf}
\caption{\label{fig_two_way_ex}The response curve realizations for $a=b=0$, $a=0.5$ and $b=0$, and, $a=0$ and $b=0.5$ in presence of no contamination (C0) in Scenario 2. }
\end{figure}
\section{Additional results in the simulation study}
Fig. \ref{fig_twoint} shows the empirical size ($ a\leq 0.10 $ and $ b=0 $) and empirical power ($ a= 0.25,0.50 $ and $ b\neq0 $) of all tests for $ H_{0,AB} $ against $ H_{1,AB} $ (at level $ \alpha=0.05 $) as a function of $ b $, for different contamination models (C0-6).
In terms of empirical size ($ a\leq 0.10 $ and $ b=0 $), all the tests are able to approximately control the level $ \alpha $, except for the FNDP and TGPF tests for model C3 and for RoFANOVA-MED and RoFANOVA-HUB for model C4 at $ b=0.50 $.
For $ a= 0.25,0.50 $ and $ b\neq0 $, the empirical power of the proposed tests is much larger than that of the competitors, for all the contamination model C1-6. Moreover, in case of no contamination (C0), the power of the RoFANOVA tests is comparable to that of the competitors for $ a= 0.25,0.50 $.
Also in this case, among the RoFANOVA tests, the RoFANOVA-BIS, RoFANOVA-HAM and RoFANOVA-OPT are the best ones.
\begin{figure}
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}M{0.5\textwidth}}
\multicolumn{4}{c}{\huge $ \bm{a\leq 0.10} $}\\[0.2cm]
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a1_C6_p2M3-eps-converted-to.pdf}\\[3cm]
\multicolumn{4}{c}{\huge $ \bm{a=0.25} $}\\[0.2cm]
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a2_C6_p2M3-eps-converted-to.pdf}\\[3cm]
\multicolumn{4}{c}{\huge $ \bm{a=0.50} $}\\[0.2cm]
\multirow{2}{*}{\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C0_p2M3-eps-converted-to.pdf}}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C1_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C3_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C5_p2M3-eps-converted-to.pdf}\\
&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C2_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C4_p2M3-eps-converted-to.pdf}&\includegraphics[width=0.5\textwidth]{Figures/Two-way_int_a3_C6_p2M3-eps-converted-to.pdf}\\
\end{tabular}}
\caption{\label{fig_twoint}Empirical size ($ a\leq 0.10 $ and $ b=0 $) and power ($ a= 0.25,0.50 $ and $ b\neq0 $) of all tests for $ H_{0,AB} $ against $ H_{1,AB} $ (at level $ \alpha=0.05 $) as a function of $ b $, for different contamination models (C0-6) in Scenario 2. The proposed and competing tests are displayed as black and grey lines, respectively.}
\end{figure}
\section{Additional details about the real case-study}
Table \ref{TABLE A} presents the process parameters and corresponding energy density levels for the production of the specimens in the real case-study.
Table \ref{TABLE B} shows the locations of the six analysed layers along the specimen build direction (distance from the baseplate) and orientation of the laser scan direction relative to the shielding gas flow in each layer.
\begin{table}
\caption{\label{TABLE A} Process parameters and corresponding energy density levels.}
\centering
\scriptsize
\resizebox{1\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c}
\toprule
\multirow{5}{*}{\shortstack{ Energy\\ density level }} &\multirow{5}{*}{\shortstack{Laser exposure\\ time\\ $t$ ($\mu$s)} }&\multirow{5}{*}{\shortstack{Distance between\\ exposed points\\ along laser scan\\ track\\ $dp$ ($\mu$m)}}&\multirow{5}{*}{\shortstack{Distance between\\ parallell laser\\ scan tracks \\$dh$ ($\mu$m)}}&\multirow{5}{*}{\shortstack{Laser power\\ $P$ (W)}} &\multirow{5}{*}{\shortstack{Powder bed \\thickness\\ $z$ ($\mu$m )}}&\multirow{5}{*}{\shortstack{Energy density\\ $F$ (kJ/cm\textsuperscript{3})}}\\
&&&&&&\\
&&&&&&\\
&&&&&&\\
&&&&&&\\\midrule
1&39&65&80&200&50&30\\\midrule
2&85&85&80&200&50&50\\\midrule
3&104&65&80&200&50&80\\\midrule
4&125&62.5&80&200&50&100\\\midrule
5&115&50&80&200&50&115\\\midrule
6&104&40&80&200&50&130\\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}
\caption{\label{TABLE B} Location of analysed layers along the specimen build direction (distance from the baseplate)
and orientation of the laser scan direction relative to the shielding gas flow in each layer.}
\scriptsize
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{c|c|c}
\toprule
\multirow{4}{*}{\shortstack{Analysed layer
}} &\multirow{4}{*}{\shortstack{Layer height\\ along the build \\direction\\(mm)} }&\multirow{4}{*}{\shortstack{Laser scan angle\\ relative to the\\ shielding gas flow
}}\\
&&\\
&&\\
&&\\\midrule
1&31&$10^{\circ}$\\\midrule
2&56&$40^{\circ}$\\\midrule
3&83&$80^{\circ}$\\\midrule
4&110&$85^{\circ}$\\\midrule
5&137&$90^{\circ}$\\\midrule
6&163&$30^{\circ}$ \\
\bottomrule
\end{tabular}
}
\end{table}
To visually explore the effects of the energy density and the layer on the spatter intensity, Fig. \ref{fig_mat} shows the equivariant functional M-estimators (Section \ref{subsec_mestimator}) $ \bar{Y}_{r} $, $ \bar{Y}_{r,i\cdot} $, $ \bar{Y}_{r,\cdot j} $ and $ \bar{Y}_{r,ij} $ of the functional grand mean, and of the group means of $ \lbrace Y_{ijk} \rbrace_{k=1,\dots n_{ij},i=1,\dots 6 } $, $ \lbrace Y_{ijk} \rbrace_{k=1,\dots n_{ij},j=1,\dots 6 } $, and $ \lbrace Y_{ijk} \rbrace_{k=1,\dots n_{ij} } $, respectively.
\begin{sidewaysfigure}
\centering
\footnotesize
\begin{tabular}{M{0.14\textwidth}|M{0.090\textwidth}|M{0.090\textwidth}M{0.090\textwidth}M{0.090\textwidth}M{0.090\textwidth}M{0.090\textwidth}M{0.090\textwidth}}
\diagbox[height=2cm,width=1.4\textwidth,innerwidth=2.9cm]{ Fluency level}{ Scan strategy} & - &1&2&3&4&5&6 \\\hline&&&&&&&\\[-0.3cm]
-&\includegraphics[width=0.1\textwidth]{Figures/Case-study_grand_mean-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean2_1-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean2_2-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean2_3-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean2_4-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean2_5-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean2_6-eps-converted-to.pdf}\\\hline&&&&&&&\\[-0.3cm]
1&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean1_1-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_11-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_12-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_13-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_14-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_15-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_16-eps-converted-to.pdf}\\
2&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean1_2-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_21-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_22-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_23-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_24-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_25-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_26-eps-converted-to.pdf}\\
3&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean1_3-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_31-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_32-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_33-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_34-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_35-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_36-eps-converted-to.pdf}\\
4 &\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean1_4-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_41-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_42-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_43-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_44-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_45-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_46-eps-converted-to.pdf}\\
5&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean1_5-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_51-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_52-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_53-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_54-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_55-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_56-eps-converted-to.pdf}\\
6&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean1_6-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_61-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_62-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_63-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_64-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_65-eps-converted-to.pdf}&\includegraphics[width=0.1\textwidth]{Figures/Case-study_group_mean_66-eps-converted-to.pdf}\\
\end{tabular}
\caption{\label{fig_mat} Equivariant functional M-estimators $ \bar{Y}_{r} $, $ \bar{Y}_{r,i\cdot} $ (first column), $ \bar{Y}_{r,\cdot j} $ (first row) and $ \bar{Y}_{r,ij} $ of the functional grand mean, and of the group means of $ \lbrace Y_{ijk} \rbrace_{k=1,\dots n_{ij},i=1,\dots 6 } $, $ \lbrace Y_{ijk} \rbrace_{k=1,\dots n_{ij},j=1,\dots 6 } $, and $ \lbrace Y_{ijk} \rbrace_{k=1,\dots n_{ij} } $, respectively. }
\end{sidewaysfigure}
\bibliographystyle{chicago}
|
1,314,259,995,776 | arxiv | \section*{Rotationally invariant expansion}
The connectedness Ornstein-Zernike (cOZ) equation is given by
\begin{equation}\label{eq:cOZ}
P(1,2) = C^+(1,2) + \int \; \mathrm{d} 3 \rho(3) C^+(1,3) P(3,2),
\end{equation}
where $P(1,2)$ is the pair connectedness function, $C^+(1,2)$ the direct connectedness function, and $\rho(3)$ is the single-particle number density, which in the currently relevant isotropic case is $\rho(3) = \rho/2 \pi$ in two spatial dimensions. The numbers $1$, $2$ and $3$ are particle labels, and represent the orientational and positional degrees of freedom of the particles. As stated in the main text, this equation links the two unknown functions $C^+(1,2)$ and $P(1,2)$, and must be supplemented by a closure relation for $C^+(1,2)$. Here, we use the connectedness Percus-Yevick (cPY) closure, defined by
\begin{align}
P(1,2) &= 1 &&\text{if particles $1$ and $2$ overlap, and} \\
C^+(1,2) &= 0 &&\text{if particles $1$ and $2$ do not overlap.}
\end{align}
Since we solve the set of cOZ - cPY equations self-consistently, we cannot \textit{a priori} average out the orientational degrees of freedom. To properly include the orientational degrees of freedom, we use a rotationally invariant expansion.
We follow the approach by Ferreira and coworkers \cite{Ferreira_1991} for two-dimensional isotropic dispersions of anisometric particles, and extend it to connectedness percolation theory. We start by expanding the functions $P(1,2)$ and $C^+(1,2)$ as
\begin{equation}
P(1,2) = \sum_{m,n = -\infty}^{\infty} P^{mn}(\mathbf{r}) \Psi^{mn}(\vartheta_1,\vartheta_2),
\end{equation}
where $P^{mn}(\mathbf{r})$ are the projections of $P(1,2)$ on the rotationally invariant basis defined by $\Psi^{mn}(\vartheta_1,\vartheta_2)$, which, by definition, remain invariant under rotations of the whole system. Using the inter-molecular reference frame, which corresponds to describing the orientations of the particles with respect to their separation vector $\mathbf{r}$, we obtain
\begin{equation}
\Psi^{mn}(\vartheta_1,\vartheta_2) = \exp i (m \vartheta_1 + n \vartheta_2).
\end{equation}
Here, $\vartheta_1$ and $\vartheta_2$ are the angles between the separation vector $\mathbf{r}$ and the major axis of the particles $1$ and $2$, respectively. These basis functions are orthogonal, and therefore the projections $P^{mn}(\mathbf{r})$ are given by
\begin{equation}
P^{mn}(r) = \frac{1}{4 \pi^2}\int_0^{2\pi} \; \mathrm{d} \vartheta_1 \int_0^{2\pi}\; \mathrm{d} \vartheta_2 P(1,2)\Psi^{mn}(\vartheta_1,\vartheta_2).
\end{equation}
We can reduce the number of independent coefficients by noting that $P$ and $C^+$ are real functions, and using the symmetry of the particle interactions, which gives $P^{mn} = P^{-m-n} = P^{nm} = P^{-n-m}$. Secondly, the inversion symmetry of the particles demands that $m,n$ are even.
Since we solve the cOZ Equation~(\ref{eq:cOZ}) in Fourier space, we introduce the Fourier transforms of $P(1,2)$ and $C^+(1,2)$ as
\begin{equation}
\widehat{P}(\mathbf{q}, \vartheta_{1q}, \vartheta_{2q}) = \int \dint{\mathbf{r}} e^{i \mathbf{q}\cdot \mathbf{r}} P(1,2),
\end{equation}
where $\vartheta_{1q, 2q}$, are the angles between the symmetry axis of particles $1$ or $2$ and the direction of the wave vector $\mathbf{q}$. Hence, we can relate the Fourier-space projections and the real-space projections through the Fourier-Bessel transform
\begin{equation}
\widehat{P}^{mn}(q) = 2 \pi i^{m+n} \int \dint{r} \, r P^{mn}(r) J_{m+n}(qr),
\end{equation}
where $J_N$ is the Bessel function of order $N$ \cite{Ferreira_1991} and $q = |\mathbf{q}|$ the magnitude of the wave vector. Remaining in Fourier space, we use the convolution theorem to obtain a projection-coupled cOZ equation
\begin{equation}
\widehat{P}^{mn}(q) = \widehat{C}^{mn}(q) + \rho \sum_l \widehat{P}^{ml}(q)\widehat{C}^{-ln}(q),
\end{equation}
where we have written $C$ instead of $C^+$ for clarity. Here, the orientational degrees of freedom are only implicitly present in the projections.
We can follow the same approach for the (real-space) cPY closure, which can be written as
\begin{equation}
C^+(1,2) = f^+(1,2)\left[1 - P(1,2) + C^+(1,2)\right],
\end{equation}
where $f^+(1,2)$ is the contact function that is defined as $f^+(1,2) = 1$ if particles $1$ and $2$ overlap, \textit{i.e.}, are connected, and $f^+(1,2) = 0$ if particles $1$ and $2$ do not overlap. Using the same expansion, we obtain
\begin{equation}
\begin{split}
C^{mn}(r) = \sum_{m',n'= - \infty}^{\infty}\left[\delta_{n'0}\delta_{m'0}-\left(P^{m'n'}(r)-C^{m'n'}(r)\right)\right]\\ \times {f^+}_{n-n'}^{m-m'}(r),
\end{split}
\end{equation}
where ${f^+}_{n-n'}^{m-m'}(r)$ are the projections of $f^+(1,2)$, but using a sub- and superscript instead of double superscript notation for clarity. The projections of $f^+(1,2)$ can be calculated using the overlap criterion yielding
\begin{equation}
{f^+}_{n-n'}^{m-m'}(r) = \frac{1}{4 \pi^2} \int_{\mathrm{Overlap}} \dint\vartheta_1 \dint{\vartheta_2} \Psi^{mn}(\vartheta_1,\vartheta_2)\Psi^{m'n'}(\vartheta_1,\vartheta_2),
\end{equation}
and must generally be calculated numerically. These projections need only be calculated once for a given model, which simplifies the self-consistent calculation of the cOZ equation. The double integral ${f^+}^N_M(r)$ was simplified by Ferreira and co-workers \cite{Ferreira_1991} to
\begin{equation}
\begin{split}
{f^+}^M_N(r) = \frac{4}{\pi^2} \int_0^{\pi/2} \dint{\vartheta_1} \cos\left[M \vartheta_1 + \frac{N}{2}(\vartheta_{2\mathrm{a}} + \vartheta_{2\mathrm{b}}) \right]\\ \times \frac{\sin\left[\frac{N}{2}\left(\vartheta_{2\mathrm{a}} - \vartheta_{2\mathrm{b}}\right)\right]}{N},
\end{split}
\end{equation}
where $\vartheta_{2a}$ and $\vartheta_{2b}$ are defined as the bounds for the angles of particle $2$ between which the particles overlap for fixed separation distance $r$ and angle $\vartheta_1$. For line segments, these angles are given by
\begin{align}
\vartheta_{2\mathrm{a}} &= \begin{cases}
\arccos{\frac{\cos{\vartheta_1}-2x}{\sqrt{1+4x^2-4 x \cos{\vartheta_1}}}} \qquad &\text{for $\vartheta_1 \leq \arccos{x}$,}\\
\vartheta_1 +\arcsin{2 x \sin{\vartheta_1}} \qquad &\text{for $\vartheta_1 > \arccos{x}$,}
\end{cases} \\
\vartheta_{2\mathrm{b}} &= \pi + \vartheta_1 - \arcsin\left[2 x \sin\vartheta_1\right],
\end{align}
with $x = r/l$ \footnote{For line segments, these angles can be derived as follows: Note that the edges of the overlap region always correspond to the \textsl{tip} of one of the line segments touching the other line segment somewhere. Using this, the parametric representation of line segments and the inversion symmetry of particles, we find several solutions that are easily checked for consistency with the overlap criterion.}.
Finally, the mean cluster size as defined in the main text can be written as
\begin{equation}
S = 1 + \rho \lim\limits_{q \to 0} P^{00}(q) = \lim\limits_{q \to 0}\left(1- \rho \widehat{C}^{00}(q)\right)^{-1}
\end{equation}
in terms of the $m, n = 0$ projection only, which indeed corresponds to the isotropically averaged $P(1,2)$ and $C^+(1,2)$. Here, we have used the fact that for $q \to 0$, the projection-coupled cOZ equation can be shown to decouple \cite{Ferreira_1991}.
\section*{Ideal line segments}
We solve the set of equations given by the cOZ and cPY equation self-consistently using an iterative solver. This iterative solver treats the cOZ equation in Fourier space, while dealing with the cPY closure in real space. The Fourier-Bessel transforms are efficiently handled using a method based on a logarithmic grid spacing \cite{Talman_1978,Hamilton_2000}. While the rotationally invariant expansion is derived for an infinite number of basis functions $\Psi^{mn}(\vartheta_1,\vartheta_2)$, we limit it to a finite and preferably small number. Specifically, we truncate the expansion set at $|\mathrm{max}(m,n)| = n_\mathrm{max}$. Note that $n_\mathrm{max} = 0$ corresponds to a pre-averaged approximation, \textit{i.e.}, replacing $P(1,2)$, $C^+(1,2)$, and $f^+(1,2)$ by weighted disks. For detailed orientational information, we use a relatively large value of $n_\mathrm{max} = 20$, whereas for orientationally averaged quantities, such as the mean cluster size $S$, we can truncate it at a small value. To confirm that the mean cluster size $S$ is indeed only weakly dependent on $n_\mathrm{max}$, we plot it for various values for $n_\mathrm{max}$ in Fig.~\ref{fig:mcsrotinv}. While the deviation in the $n_\mathrm{max} = 0$ approach compared to a $n_\mathrm{max} = 20$ approach is visible, it relatively minor, approximately $10\%$ at $\rho l^2 = 14.1$. For $n_\mathrm{max} = 4$, the difference at the same density is only slightly more than one per cent. Due to the minor influence of $n_\mathrm{max}$ on the mean cluster size $S$, we argue that even the pre-averaged cPY approach ($n_\mathrm{max} = 0$) yields quantitative predictions within the cPY approximation. Fig.~\ref{fig:mcsrotinv} shows that the cPY approximation does not yield a reasonable percolation threshold as we discuss in the main text, and that this observation is not altered by the choice of $n_{\mathrm{max}}$.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\columnwidth]{Snmax-eps-converted-to.pdf}
\caption{The mean cluster size $S$ of ideal line segments of length $l$ as function of the reduced density $\rho l^2$, obtained within the cPY closure. We include the values where the rotational invariant expansion is truncated at a different max values $n_\mathrm{max} = 0$ (blue), $n_\mathrm{max} = 4$ (black, dotted) and $n_\mathrm{max} = 20$ (red).}
\label{fig:mcsrotinv}
\end{figure}
While the state of affairs is unsatisfying for determining the percolation threshold, we can still obtain valuable information on connection probabilities below the percolation threshold. We show the average pair connectedness functions for two densities for both cPY approximation and from Monte Carlo simulations in Fig.~\ref{fig:pcf},
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\columnwidth]{pcf25-eps-converted-to.pdf}
~
\includegraphics[width=0.48\columnwidth]{pcf50-eps-converted-to.pdf}
\caption{The orientationally averaged pair connectedness function $\langle P(r,\vartheta)\rangle$ as function of the scaled center-to-center distance $r/l$ of line segments of length $l$, for scaled densities $\rho l^2 = 2.5$ (left) and $\rho l^2 = 5.0$ (right). Here, we show both our theoretical prediction based on the cPY approximation, and the results from our Monte Carlo simulations.}
\label{fig:pcf}
\end{figure}
which show good agreement at low density, but less so at high density. Interestingly, evaluating cPY theory and Monte Carlo simulations at \textsl{different} densities does yield accurate results for the pair connectedness functions \footnote{This does not hold for the direct connectedness function, as it is still limited by the cPY approximation}. To highlight that this also holds for the non-averaged pair connectedness function, we plot the short and long range behaviour of the (normalized) pair connectedness function in Fig.~\ref{fig:pcftheta}, which for comparison with our Monte Carlo simulations is averaged over the angle-bin $36^\circ<\vartheta < 45^\circ$, where $\vartheta$ is now the angle between the major axes of the two particles. Here, we evaluate the cPY approximation at $\rho l^2 = 9.6$ with $n_\mathrm{max} = 20$ to properly capture the orientational information, whereas the Monte Carlo simulation results have been obtained for $\rho l^2 = 5$. Fig.~\ref{fig:pcftheta} is representative for all angles.
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\columnwidth]{shorttheta-eps-converted-to.pdf}
~
\includegraphics[width=0.48\columnwidth]{longtheta-eps-converted-to.pdf}
\caption{The normalized pair connectedness function $P(r,\vartheta)$ as function of the scaled center-to-center distance $r/l$. The pair connectedness function is averaged over the angle $\vartheta$ between the particles $36^\circ<\vartheta < 45^\circ$. Predictions due to cPY theory match Monte Carlo simulation results if evaluated at different densities, both at small and large separation.}
\label{fig:pcftheta}
\end{figure}
For the three-dimensional slender rod model, density renormalization schemes have been successfully used to extend the region of quantitative predictions of the second virial approximation from infinitely slender particles, to particles with an aspect ratio in the range of $L/D \sim \mathcal{O}(10^1)$ \cite{Schilling_2015}. Using the observation that the pair connectedness functions are in excellent agreement, as long as we evaluate our theoretical model at a different density, we propose an \textsl{ad-hoc} density renormalization approach as follows: (1) We match both the short-ranged and long-ranged behaviour of the pair connectedness function, (2) we assume that the deviation from the Monte Carlo results is due to it entering the critical region, \textit{i.e.}, the mean cluster size conforms to the scaling relation $S \sim |\rho - \rho_\mathrm{c}|^{-\gamma}$, and (3) we assume that the mean cluster size in cPY increases exponentially, which, from Fig.~\ref{fig:mcsrotinv}, seems to be a reasonable approximation. Under these considerations, we propose the following scaling relation
\begin{equation}\label{eq:fit}
\rho_\mathrm{PY} = A - B \ln\left(|\rho_\mathrm{MC} - \rho_\mathrm{c}|l^2\right),
\end{equation}
where $A$ and $B$ are fitting parameters and $\rho_\mathrm{c}$ is the percolation threshold. Since the critical exponent $\gamma = 43/18$ associated with the mean cluster size is known, the parameter $B$ could in principle be calculated. Treating $\rho_\mathrm{c}$ as known, we obtain $A = 7.69 \pm 0.01$, and $B = 4.26 \pm 0.02$, which, as shown in Fig.~\ref{fig:denstmap}, is in excellent agreement. Since the same (near) exponential growth is also observed for ideal disks, we believe that such a method should be applicable for all ideal two-dimensional particle geometries \cite{deBruijn_2020}.
While yielding excellent agreement with simulations, this density renormalization has two main drawbacks. Firstly, it relies on \textit{ad hoc} assumptions and requires direct input from simulations. Secondly, for non-ideal particles, \textit{e.g.}, obeying the cherry-pit model, this method cannot be straightforwardly applied. Practical application is therefore limited to simulation-assisted approaches.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\columnwidth]{DenstScaling-eps-converted-to.pdf}
\caption{The density mapping of the pair connectedness function obtained from Percus-Yevick theory $P_\mathrm{PY}(r)$ on the pair connectedness function obtained from our Monte Carlo simulations $P_\mathrm{MC}(r)$, for ideal line segments of length $l$ in a two-dimensional model. The scaled density ($\rho l^2)_\mathrm{PY}$ for Percus-Yevick theory is plotted as function of the equivalent scaled density of Monte Carlo simulations $(\rho l^2)_\mathrm{MC}$. The orange line indicates $(\rho l^2)_\mathrm{PY} = (\rho l^2)_\mathrm{MC}$, the black dots indicate the density mapping with an uncertainty of $\Delta(\rho l^2) = 0.1$. The blue curve is the proposed fit Eq.~\eqref{eq:fit} with parameters $A = 7.69 \pm 0.01$ and $B = 4.26 \pm 0.02$.}
\label{fig:denstmap}
\end{figure}
\section*{NNCPT Mean Cluster Size}
In the main text we derived
\begin{align}
P(r_{12}) &= f^+(r_{12}) + (1- f^+(r_{12})) \\ \label{eq:numericaleq}
&\times\int_{-\frac{1}{2}}^\frac{1}{2} \; \mathrm{d} t' \int \; \mathrm{d} \bs{r}_{3} \; \omega^+(\bs{r}_{3},l t') l_{\mathrm{eff}}(l |t'|) P(r_{32}) \; .
\end{align}
The integral over $\bs{r}_3$ is to be evaluated on the entire $\mathbb{R}^2$ plane and can thus be interchanged with the $t'$-integral. Defining
\begin{align}
C^+( \bs{r}_{3}) := \int_{-\frac{1}{2}}^\frac{1}{2} \; \mathrm{d} t' \; \omega^+(\bs{r}_{3},l t') l_{\mathrm{eff}}(l |t'|) \; ,
\end{align}
and recalling that we fixed line segment 1 in the origin we obtain
\begin{align}
P(|\bs{r}_{2}|) &= f^+(|\bs{r}_{2}|) + (1- f^+(|\bs{r}_{2}|)) \\ \label{eq:splitter}
&\times \int \; \mathrm{d} \bs{r}_{3} \; C^+(\bs{r}_{3}) P(|\bs{r}_2 - \bs{r}_3 |) \; .
\end{align}
The integral in the last equation is a two-dimensional convolution. The only important structural difference to the cOZ equation is the $|\bs{r}_{2}|$-dependent factor in front of the convolution. But, the support of $f^+(|\bs{r}_{2}|)$ is limited to a disk of radius $l$, so that for $|\bs{r}_{2}| > l$ the function in front of the convolution becomes unity. We split $P$ into two separate contributions:
\begin{align}
P(|\bs{r}_2|) = \underbrace{P(|\bs{r}_2|)\Theta(|\bs{r}_2|-l)}_{P_{<l}(|\bs{r}_2|)} + \underbrace{P(|\bs{r}_2|)\Theta(l-|\bs{r}_2|)}_{P_{>l}(|\bs{r}_2|)} \; ,
\end{align}
which applied to eq.~(\ref{eq:splitter}) yields two coupled integral equations:
\begin{eqnarray}
P_{<l}(|\bs{r}_2|) &=& f^+(|\bs{r}_{2}|) + (1- f^+(|\bs{r}_{2}|)) \\
&&\int \; \mathrm{d} \bs{r}_{3} \; C^+(\bs{r}_{3}) \nonumber
\left[P_{<l}(|\bs{r}_2 - \bs{r}_3 |) +
P_{>l}(|\bs{r}_2 - \bs{r}_3 |)\right] \\
P_{>l}(|\bs{r}_2|) &=& \nonumber
\int \; \mathrm{d} \bs{r}_{3} \; C^+(\bs{r}_{3})
\left[P_{<l}(|\bs{r}_2 - \bs{r}_3 |) +
P_{>l}(|\bs{r}_2 - \bs{r}_3 |)\right] \label{eq:Plarger}
\end{eqnarray}
As the support of $P_{<l}$ is limited and $P(\bs{r}_2|) \leq 1$ necessarily, the percolation threshold depends exclusively on the properties of $P_{>l}$. The impact of $P_{<l}$ in eq. (\ref{eq:Plarger}) can be accounted for by a new inhomogeneity $g$.
\begin{align}
P_{>l}(|\bs{r}_2|) = \underbrace{\int \; \mathrm{d} \bs{r}_{3} \; C^+(\bs{r}_{3}) P_{<l}(|\bs{r}_2 - \bs{r}_3 |) }_{g(|\bs{r}_2|)} \nonumber \\ + \int \; \mathrm{d} \bs{r}_{3} \; C^+(\bs{r}_{3})
P_{>l}(|\bs{r}_2 - \bs{r}_3 |) \label{eq:Plarger2}
\end{align}
The mean cluster size consists of the contributions of both $P_{>l}$ and $P_{<l}$, although the latter is necessarily bounded by $\pi l^2$, because $P$ is bounded by 1. Thus, we can in good approximation neglect this contribution. The remaining part solves Eq.~(\ref{eq:Plarger2}) which is of the standard convolution type and thus can be solved in Fourier space
\begin{align}
\hat{P}(\bs{q}) = \frac{\hat{g}(\bs{q})}{1-\hat{C}(\bs{q})}
\end{align}
$\hat{g}(\bs{0})$ describes the integral of $g$ over the real plane. However, $g$ is a convolution itself and thus measure preserving. So $\hat{g}(\bs{0}) = \hat{C}^+(\bs{0}) \hat{P}_{<l}(\bs{0})$. But since $f^+(\bs{r}) \leq P_{<l}(\bs{r}) \leq 1$ for any $\bs{r}$, we know
\begin{align}
\hat{f}^+(\bs{0}) \leq \hat{P}_{<l}(\bs{0}) \leq \pi l^2 \;.
\end{align}
Since $f^+$ does not depend on the density, $f^+(\bs{0})$ does only depend on $l$. Therefore, $\hat{P}_{<l}(\bs{0})$ is strongly constrained by density independent constants. As a consequence, for the sake of determining the mean cluster size the approximation $\hat{g}(\bs{q}) \approx \hat{C}^+(\bs{q})$ is well controlled. With this approximation eq.~(\ref{eq:Plarger2}) becomes an Ornstein-Zernike type equation, so that the mean cluster size can be computed via
\begin{align}
S = \lim_{\bs{q} \rightarrow \bs{0}} \frac{1}{1-C^+(\bs{q})} \; .
\end{align}
This is the approximation of the mean cluster size from NNCPT which is depicted in Fig.~1 of the main text.
\begin{figure}[h!]
\centering
\includegraphics[width=0.8 \columnwidth]{NND}
\caption{NNCPT closure: $C^+(\bs{r_3})$ for $\rho l^2=5$}
\label{fig:nncpt_kernel}
\end{figure}
|
1,314,259,995,777 | arxiv | \section{ \label{sec:intro} Introduction}
The behavior of systems under electrostatic forces is governed by the electric field $\mathbf{E}$,
which can be expressed as the gradient of a scalar potential $\Phi$:
\begin{equation}
\mathbf{E} = -\nabla \Phi \; .
\end{equation}
In the absence of free charges, the potential $\Phi$ is determined by the Laplace equation,
\begin{equation}
\nabla^{2} \Phi = 0
\end{equation}
for all points $\mathbf{x}$ in the simply connected domain $\Omega$. The Laplace equation
admits a unique solution for the field $\mathbf{E}$ when the conditions on the boundary of
the domain, $\partial \Omega$, are specified. The boundary conditions may be completely specified by
associating either a value for the potential $\Phi$ (Dirichlet), or the derivative of $\Phi$
with respect to the surface normal $\frac{\partial\Phi}{\partial n}$ (Neumann), for every point
on $\partial \Omega$.
One technique for numerically solving the Laplace equation is the boundary element method (BEM).
Compared to other popular methods designed to accomplish the same goal, such as Finite Element and Finite
Difference Methods \cite{poljak2005boundary}, the BEM method focuses on the boundaries of the system rather than
its domain, effectively reducing the dimensionality of the problem. BEM also facilitates the calculation of fields in regions that extend
out to infinity (rather than restricting computation to a finite region) \cite{szilagyi1988electron}.
When it is applicable these two features often make the BEM faster and more versatile than competing methods.
The basic underlying idea of the BEM involves reformulating the partial differential equation
as a Fredholm integral equation of the first or second type, defined respectively as,
\begin{equation}
f(\mathbf{x}) = \int \limits_{\partial\Omega} K(\mathbf{x},\mathbf{y}) \Phi(\mathbf{y}) d\mathbf{y}
\end{equation}
and
\begin{equation}
\Phi(\mathbf{x}) = f(\mathbf{x}) + \lambda \int \limits_{\partial\Omega} K(\mathbf{x},\mathbf{y}) \Phi(\mathbf{y}) d\mathbf{y} \;,
\end{equation}
where $K(\mathbf{x}, \mathbf{y})$ (known as the Fredholm kernel), and $f(\mathbf{x})$ are known, square-integrable
functions, $\lambda$ is a constant, and $\Phi(\mathbf{x})$ is the function for which a solution is sought.
Discretizing the boundary of the domain into $N$ elements and imposing the
boundary conditions on this integral equation through either a collocation,
Galerkin or Nystr\"{o}m scheme results in the formation of dense matrices which naively cost $\mathcal{O}(N^2)$
to compute and store and $\mathcal{O}(N^3)$ to solve \cite{liu2009fast}.
This scaling makes solving large problems (much more than $\sim 10^4$ elements)
impractical unless some underlying aspect of the equations involved can be exploited.
For example, for the Laplace equation there exist iterative methods,
such as Robin Hood \cite{lazic2008robin} \cite{formaggio2012solving},
which take advantage of non-local charge transfer allowed by the elliptic nature of the equation
to reduce the needed storage to $\mathcal{O}(N)$ and time of
convergence to $\mathcal{O}(N^{\alpha})$, with $1 < \alpha < 2$.
Another technique that has been used to accelerate the BEM solution to
the Laplace equation, and has also found wide applicability
in three dimensional electrostatic, elastostatic, acoustic, and other problems,
is the fast multipole method (FMM) \cite{liu2009fast}. The FMM was originally
developed by V. Rohklin and L. Greengard for the two dimensional Laplace boundary value problem
\cite{rokhlin1985rapid} and N-body simulation \cite{greengard1988rapid}. Fast multipole methods
are appropriate when the kernel of the equation is separable or approximately separable so that,
to within some acceptable error, it may be expressed as a series \cite{beatson1997short},
\begin{equation}
K(\mathbf{x},\mathbf{y}) \approx \sum \limits_{k=0}^{p} \psi_k(\mathbf{x}) \xi_{k}(\mathbf{y}) \;.
\end{equation}
In the case of the Laplace equation, the kernel is often approximated by an expansion in spherical coordinates,
with the functions $\psi_k(\mathbf{x})$ and $\xi_k(\mathbf{y})$ taking the form of
the regular and irregular solid harmonics \cite{epton1995multipole}, \cite{van1998shift}.
This expansion allows the far-field effects of a source to be represented in a compressed form by a set of
coefficients known as the \emph{multipole moments} of the source. The series is truncated to a maximum
degree of $p$ which is determined by the desired precision.
When applying BEM together with FMM (which we refer to as FMBEM) to solve the Laplace equation
over a complex geometry, it is necessary to determine the multipole moments of various subsets
of the surfaces involved. At the smallest spatial scale, this requires a means of computing
the individual multipole moments of each of the chosen basis functions (boundary elements). Geometrically,
these basis functions usually take the form of planar triangular and rectangular elements,
with the charge density on these elements either constant or interpolated between some
set of sample points. Since rectangular elements cannot necessarily discretize
an arbitrary curved surface without gaps or overlapping elements and can be decomposed into triangles,
we consider it sufficient to compute the multipole expansion of basis functions of the triangular type.
Once the solution of the Laplace equation is know for a specific geometry and boundary conditions, a common task
is to track of charge particles throughout the resultant electrostatic field. Evaluating the field
directly from all boundary elements of the geometry is costly. However, this process can be
significantly accelerated by constructing a local or remote multipole expansion of the source field in the region of interest.
The expansions can be precomputed with a time and memory cost which scales like $\mathcal{O}(N p^2)$, but result in
field evaluation which scales like $\mathcal{O}(p^2)$ instead of $\mathcal{O}(N)$ as per the direct method.
The usefulness of the multipole expansion in both FMBEM and charged particle tracking motivates us to find a method by which
to compute the multipole expansion of a triangle boundary element accurately and efficiently
\section{\label{sec:math-prelim} Mathematical Preliminaries}
For an arbitrary collection of charges bounded within a sphere of radius $R$ about the point $\mathbf{x}_{0}$, there
is a remote expansion for the potential $\Phi(\mathbf{x})$ given by \cite{jackson}, \cite{greengard1988rapid}:
\begin{equation}
\Phi(\mathbf{x}) = \sum \limits_{l=0}^{\infty} \sum \limits_{m=-l}^{l} \frac{ Q_{l}^{m} Y_{l}^{m}(\theta, \phi)}{r^{l+1}} \;.
\end{equation}
This approximation converges at all points $|\mathbf{x} - \mathbf{x}_{0} | > R$. The coefficients $Q_{l}^{m}$ are known as the multipole moments of the charge distribution.
The spherical harmonics $Y_{l}^{m}(\theta, \phi)$ are given by:
\begin{equation}
Y_{l}^{m}(\theta, \phi) = N_{l}^{m} P_{l}^{|m|}(\cos \theta) e^{i m \phi} \;,
\label{complexsphericalharmonic-def}
\end{equation}
where the coordinates $(r,\theta,\phi)$ are measured with respect to the origin $\mathbf{x}_{0}$, and
the function $P_{l}^{m}$ is the associated Legendre polynomial of the first kind.
Several normalization conventions exist for the spherical harmonics;
Throughout this paper we use the Schmidt semi-normalized convention where
$N_{l}^{m} = \sqrt{ (l - |m|)! / (l+ |m|)! }$. When the charge
distribution $\sigma(\mathbf{x'})$ is confined to a surface $\Sigma$,
the moments are given by the following integral:
\begin{equation}
Q^{m}_{l} = \int \limits_{\Sigma} \sigma(\mathbf{x}) \overline{Y_{l}^{m}} (\theta, \phi) r^{l} d \Sigma = \int \limits_{\Sigma} \sigma(\mathbf{x}) N_{l}^{m} P_{l}^{|m|}(\cos \theta) e^{-i m \phi}r^{l} d \Sigma \;.
\label{surfacemultipole}
\end{equation}
The integral given in equation (\ref{surfacemultipole}) can be addressed in a straightforward manner
through two dimensional Gaussian quadrature \cite{lether1976computation}. It can also be reduced to a
one dimensional Gaussian quadrature if one first computes an auxiliary vector field and applies Stokes'
theorem, as described by Mousa et al \cite{mousa2008toward}. However, for high-order expansions, accurate
evaluation of the numerical integration becomes progressively more expensive. It is therefore desirable to
obtain an analytic expression of the multipole moments.
\section{\label{sec:coord-sys} Coordinate system for integration}
In order to compute the multipole expansion of a triangle $\Sigma$ defined by points $\{\mathbf{P}_0, \mathbf{P}_1, \mathbf{P}_2 \}$,
we first must select the appropriate coordinate system to simplify the integration. Without loss
of generality, we choose a system so that the vertex $\mathbf{P}_0$ lies at the origin,
and the $\mathbf{\hat{e}}_1$ direction is parallel to the vector $\mathbf{P}_2 - \mathbf{P}_1$.
The plane defined by the triangle is then parameterized by the local coordinates $(u,v)$.
Formally, this local coordinate system $S$ can be defined with the following origin and basis vectors:
\begin{equation}
S : \left\{
\begin{array}{lr}
\mathcal{O} &= \mathbf{P}_{0} \\
\mathbf{\hat{e}}_0 &= \frac{\mathbf{Q} - \mathbf{P}_0}{|\mathbf{Q} - \mathbf{P}_0|} \\
\mathbf{\hat{e}}_1 &= \frac{\mathbf{P}_2 - \mathbf{P}_1}{|\mathbf{P}_2 - \mathbf{P}_1|} \\
\mathbf{\hat{e}}_2 &= \mathbf{\hat{e}}_0 \times \mathbf{\hat{e}}_1
\end{array}
\right. \;,
\label{coordinate_S}
\end{equation}
where $\{\mathbf{P}_0, \mathbf{P}_1, \mathbf{P}_2 \}$ are the points defining the triangle $\Sigma$ in the
original coordinate system. The point $\mathbf{Q}$ is the closest point to $\mathbf{P}_0$ lying on
the line joining $\mathbf{P}_1$ and $\mathbf{P}_2$. The position of $\mathbf{Q}$ in the $(u,v)$-plane is $(h,0)$ and
is given by:
\begin{equation}
\mathbf{Q} = \mathbf{P}_1 +
\left(\frac{ \left(\mathbf{P}_0 - \mathbf{P}_1 \right) \cdot \left(\mathbf{P}_2 - \mathbf{P}_1\right) }{\left|\mathbf{P}_2 - \mathbf{P}_1\right|^2}\right)
\left(\mathbf{P}_2 - \mathbf{P}_1\right) \;.
\end{equation}
Figure (\ref{triangle-coordinate-system}) shows the arrangement of this coordinate system.
\begin{figure}[h!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=5.cm]{./CoordinateSystem.pdf}
\subcaption{Triangle $\Sigma$ in global coordinate system.}
\label{fig:global_coord}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=4.cm]{./SpecialCoordinateSystem.pdf}
\subcaption{Triangle $\Sigma$ in local coordinate system $S$.}
\label{fig:special_coord}
\end{subfigure}
\caption{In (\ref{fig:global_coord}) the boundary element $\Sigma$ (shaded region)
is shown with arbitrary position and orientation in the global coordinate system.
A detailed view of the local coordinate system $S$, in which the integration is
performed, is shown in (\ref{fig:special_coord}), where the $w$ axis points out
of the page.}
\label{triangle-coordinate-system}
\end{figure}
\newpage
\section{\label{sec:integral-evaluation} Evaluation by recurrence}
For an arbitrary expansion origin and triangular surface element equation (\ref{surfacemultipole})
is very difficult to compute analytically, even for a constant charge density.
Additionally, the variety of schemes available for function interpolation over triangular domains,
such as the natural orthogonal polynomial basis put forth by
\cite{proriol1957famille},
\cite{dubiner1991spectral},
\cite{owens1998spectral} and
\cite{koornwinder1975two}, or the more commonly used variations on Lagrange and Hermite
interpolation \cite{wait1985finite}, \cite{taylor1972completeness}, \cite{barnhill1975}, \cite{chen1992boundary} complicates
any general approach. Therefore in order to proceed we choose a simplifying restriction on the general problem and
avoid these more advanced interpolation schemes in favor of a simpler but less well-conditioned
bivariate monomial basis, where the charge density on the triangle is expressed terms
of local orthogonal coordinates $(u,v)$ by:
\begin{equation}
\sigma(u,v) = \left\{
\begin{array}{lr}
\sum\limits_{a=0}^{N} \sum\limits_{b=0}^{N-a} s_{a,b} u^{a} v^{b} & : (u,v) \in \Sigma \\
0 & : (u,v) \notin \Sigma
\end{array}
\right. \;,
\label{chargedensity}
\end{equation}
where $N$ is the order of the interpolation, the variables
$(u,v)$ are as defined in figure (\ref{triangle-coordinate-system}),
and $s_{a,b}$ are the interpolation coefficients. Figure (\ref{fig:interp-orders}) shows an example
of the interpolated function for various $N$. It is possible to perform a change of
basis on the interpolating polynomials \cite{gander2005change} to compute the $s_{a,b}$
coefficients in terms of the coefficients of some other polynomial basis, however
we will defer discussion of this change of basis and its application to low-order Lagrange interpolation
to Appendix (\ref{sec:appendix-change-of-basis}).
\begin{figure}[b!]
\captionsetup[subfigure]{justification=centering}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics{./ZerothOrderTriangle.pdf}
\caption{Zero-th order, $N=0$.}
\label{fig:zero-order-basis-function}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics{./FirstOrderTriangle.pdf}
\caption{First order, $N=1$.}
\label{fig:first-order-basis-function}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics{./SecondOrderTriangle.pdf}
\caption{Second order, $N=2$.}
\label{fig:second-order-basis-function}
\end{subfigure}
\caption{Planar boundary elements with various orders of charge density interpolation.
Height above the element indicates the value of the local charge density.}\label{fig:basis_functions}
\label{fig:interp-orders}
\end{figure}
It is convenient to perform the integral in the spherical
coordinate system associated with $S$, since the $(u,v)$-plane is a surface of constant
$\theta$ where the differential surface element $d\Sigma = r \sin\theta dr d\phi$.
Since the local coordinates $(u,v)$ are
\begin{align}
& u(r,\phi) = r\cos\phi \\
& v(r,\phi) = r\sin\phi \;.
\label{localcoords}
\end{align}
The expression for the charge density becomes:
\begin{equation}
\sigma(r,\phi) = \left\{
\begin{array}{lr}
\sum\limits_{a=0}^{N} \sum\limits_{b=0}^{N-a} s_{a,b} (r\cos\phi)^{a} (r\sin\phi)^{b} & : (r,\phi) \in \Sigma \\
0 & : (r,\phi) \notin \Sigma
\end{array}
\right. \;.
\label{chargedensityspherical}
\end{equation}
Fixing $\theta=\pi/2$, inserting our expression for the charge density (\ref{chargedensityspherical})
into (\ref{surfacemultipole}) and then exchanging the order of integration and summation we find:
\begin{align}
Q^{m}_{l} &= \sum\limits_{a=0}^{N} \sum\limits_{b=0}^{N-a} s_{a,b}
N_{l}^{m} {P}^{|m|}_{l}(0) \int_{\phi_1}^{\phi_2} \int_{0}^{r(\phi)} (\cos\phi)^{a} (\sin\phi)^{b} e^{-i m \phi} r^{a+b+l+1} dr d\phi \;.
\label{surfacemultipole3}
\end{align}
As can be seen in figure (\ref{triangle-coordinate-system}) the upper limit on the $r$ integration is given by $r(\phi) = h/\cos\phi$.
Performing the integration over the $r$ coordinate leaves us with:
\begin{equation}
Q^{m}_{l} = \sum\limits_{a=0}^{N} \sum\limits_{b=0}^{N-a} \underbrace{ \left( \frac{ s_{a,b} h^{a+b+l+2}}{ a+b+l+2 } \right) N_{l}^{m} {P}^{m}_{l}(0) }_{\mathcal{K}_{l,m}^{a,b}}
\underbrace{ \int_{\phi_1}^{\phi_2}\frac{ (\sin\phi)^{b} e^{-i m \phi} }{ (\cos\phi)^{b+l+2}} d\phi }_{\mathcal{I}_{l,m}^{b}} \;.
\label{phiintegral_general}
\end{equation}
The prefactors $\mathcal{K}^{a,b}_{l,m}$ are easy to compute. To address integrals of the form $\mathcal{I}_{l,m}^{b}$
we split our integrand into imaginary and real components
$\mathcal{I}_{l,m}^{b} = \mathcal{A}_{l,m}^{b} - i \mathcal{B}_{l,m}^{b}$, where
\begin{align}
\mathcal{A}_{l,m}^{b} &= \int_{\phi_1}^{\phi_2} \frac{ (\sin\phi)^{b} \cos(m \phi) }{ (\cos\phi)^{b+l+2}} d\phi \label{alm} \\
\mathcal{B}_{l,m}^{b} &= \int_{\phi_1}^{\phi_2} \frac{ (\sin\phi)^{b} \sin(m \phi) }{ (\cos\phi)^{b+l+2}} d\phi \label{blm} \;.
\end{align}
Before evaluating these integrals, we pause to introduce the Chebyshev polynomials
\cite{abramowitz2012handbook}, \cite{mason2002chebyshev}.
The Chebyshev polynomials of the first kind $T_n(x)$ are defined recursively for $n\geq0$ through:
\begin{equation}
T_{n+1}(x) = 2x T_{n}(x) - T_{n-1}(x) \;. \label{firstkindrecursion}
\end{equation}
with $T_{0}(x) = 1$ and $T_{1}(x) = x$. Similarly, the Chebyshev polynomials of the
second kind, $U_{n}(x)$, are defined through:
\begin{equation}
U_{n+1}(x) = 2x U_{n}(x) - U_{n-1}(x) \;.
\end{equation}
with $U_{0}(x) = 1$ and $U_{1}(x) = 2x$.
These polynomials are noteworthy for our purposes because of the two following useful properties:
\begin{align}
T_{n}(\cos\phi) &= \cos(n\phi) \label{firstkindidentity} \\
U_{n}(\cos\phi) &= \frac{\sin((n+1)\phi)}{\sin \phi} \label{secondkindidentity} \;.
\end{align}
We can exploit these in order to evaluate $\mathcal{A}_{l,m}^{b}$ and $\mathcal{B}_{l,m}^{b}$ recursively.
We first address $\mathcal{A}_{l,m}^{b}$. Using (\ref{firstkindidentity}), we may rewrite (\ref{alm}) as
\begin{equation}
\mathcal{A}_{l,m}^{b} = \int_{\phi_1}^{\phi_2} \frac{ (\sin\phi)^{b} T_{m}(\cos\phi) }{ (\cos\phi)^{b+l+2}} d\phi \;.
\end{equation}
Expanding this using (\ref{firstkindrecursion}) gives
\begin{equation}
\mathcal{A}_{l,m}^{b} = 2 \int_{\phi_1}^{\phi_2} \frac{ (\sin\phi)^{b} T_{m-1}(\cos\phi) }{ (\cos\phi)^{b+l+1}} d\phi -
\int_{\phi_1}^{\phi_2} \frac{ (\sin\phi)^{b} T_{m-2}(\cos\phi) }{ (\cos\phi)^{b+l+2}} d\phi \;,
\end{equation}
which yields the recursion relationship for the $\mathcal{A}_{l,m}^{b}$:
\begin{equation}
\mathcal{A}_{l,m}^{b} = 2 A_{l-1,m-1}^{b} - \mathcal{A}_{l,m-2}^{b} \;.
\label{A-recursion_general}
\end{equation}
Similarly for the $\mathcal{B}_{l,m}^{b}$, we have:
\begin{equation}
\mathcal{B}_{l,m}^{b} = 2 \mathcal{B}_{l-1,m-1}^{b} - \mathcal{B}_{l,m-2}^{b} \;.
\label{B-recursion_general}
\end{equation}
\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.4\linewidth]{./RecursionScheme.pdf}
\caption{Graphical representation of recursion given in equation (\ref{A-recursion_general}) up to $l=3$.
Circles denote terms which must be computed as a base case, squares denote terms
which may be computed by recurrence. The arrows indicate dependence. Higher
order terms extend downwards and to the right, as denoted by the dotted lines and arrows. }
\label{reduction-graph}
\end{center}
\end{figure}
Given these recursion relationships, we can reduce
the integrals $\mathcal{A}_{l,m}^{b}$ and $\mathcal{B}_{l,m}^{b}$ of any
degree $0\leq l$ and order $0 \leq m \leq l$ into a series of terms, of which only the base cases
must be evaluated explicitly. Figure (\ref{reduction-graph}) shows a representation of the recursion
relationship. The base cases that are not further reducible through recurrence can
all be expressed in terms of single integral form $I_{p}^{q}$ where
\begin{equation}
I_{p}^{q} = \int_{\phi_1}^{\phi_2} \frac{(\sin\phi)^q}{(\cos\phi)^p} d\phi \;.
\label{Ipq}
\end{equation}
The base cases $\mathcal{A}_{l,0}^{b} = I_{b+l+2}^{b}$ and $\mathcal{A}_{l,1}^{b} = I_{b+l+1}^{b}$, while $\mathcal{B}_{l,1}^{b} = I_{b+l+2}^{b+1}$ and $\mathcal{B}_{l,0}^{b} = 0$.
The solutions to integrals of the form $I_{p}^{q}$ is addressed in Appendix (\ref{sec:appendix-integrals}).
It should be noted that during the process of computing the value of the moment $Q_l^m$ through recursion,
the real and imaginary parts of all moments with degree $\leq l$ and order $\leq m$ will be computed.
These values can be stored so that there is no need to repeat the recursion for each individual moment needed.
This is useful when determining the multipole expansion of a boundary element since all moments up to certain
maximal degree can be computed in one pass through the recurrence.
\section{\label{sec:moment-transform} Multipole moments under coordinate transformation}
We can make use of the results of the preceding section to compute the multipole expansion coefficients
of the boundary element $\Sigma$ with respect to an arbitrary origin and set of coordinate axes. Typically, we are most
interested in being able to construct the multipole moments $M_j^k$ of $\Sigma$ in the coordinate system
that has the canonical Cartesian coordinate axes, with an origin at an arbitrary point $\mathbf{S}_0$.
We denote this system as $S''$:
\begin{equation}
S'' : \left\{
\begin{array}{lr}
\mathcal{O} &= \mathbf{S}_0 \\
\hat{e}_0'' &= (1,0,0) \\
\hat{e}_1'' &= (0,1,0) \\
\hat{e}_2'' &= (0,0,1)
\end{array}
\right. \;.
\label{coordinateS}
\end{equation}
Therefore, we must first construct the coordinate transformation $A:S \rightarrow S''$, and then
determine how this coordinate transform operates on the coefficients $Q_{l}^{m}$ of the
multipole expansion given in $S$.
The rigid motion $A:S \rightarrow S''$ can be specified by a rotation $U:S \rightarrow S'$
followed by a translation $T:S' \rightarrow S''$. We can describe the translation by the displacement
$\mathbf{\Delta} = \mathbf{S}_0 - \mathbf{P}_0$, and the rotation $U$ by the Euler
angles $(\alpha,\beta,\gamma)$ following the $Z-Y'-Z''$ axis convention
of \cite{pinchon2007rotation} and \cite{gimbutas2009fast}. The Euler angles allow us to write
the rotation $U$ as the composition of three successive rotations
$U = U_{Z''}(\gamma) U_{Y'}(\beta) U_{Z}(\alpha)$. Explicitly,
$U$ is given by
\begin{equation}
U =
\begin{bmatrix}
\cos\gamma & -\sin\gamma & 0\\
\sin\gamma & \cos\gamma & 0\\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\cos\beta & 0 & -\sin\beta\\
0 & 1 & 0\\
\sin\beta & 0 & \cos\beta
\end{bmatrix}
\begin{bmatrix}
\cos\alpha & -\sin\alpha & 0\\
\sin\alpha & \cos\alpha & 0\\
0 & 0 & 1
\end{bmatrix}
\end{equation}
and can be related to the basis vectors of the coordinate system $S$ by:
\begin{equation}
U =
\left [ \begin{array}{ccc}
U_{00} & U_{01} & U_{02}\\
U_{10} & U_{11} & U_{12}\\
U_{20} & U_{21} & U_{22}
\end{array} \right]
=
\left[
\begin{array}{c}
\hat{e}_0\\
\hat{e}_1 \\
\hat{e}_2
\end{array}
\right]^{T} \;.
\end{equation}
It is well known that the Euler angles $(\alpha,\beta,\gamma)$ do not uniquely
describe an arbitrary rotation matrix $U$, however, a unique description is not necessary
for our purposes. A convenient set of choices is given in table (\ref{eulerangles}).
\begin{table}
\begin{center}
\begin{tabular}[width=11cm]{|c|c|c|c|} \hline
Angle & $U_{22} \neq \pm 1$ & $U_{22} = 1$ & $U_{22} = -1$ \\ \hline
$\alpha$ & $\mathrm{\texttt{atan2}}\left(\frac{-U_{21}}{\sin\beta}, \frac{-U_{20}}{\sin\beta} \right)$ & $0$ & $\pi$ \\ \hline
$\beta$ & $\mathrm{\texttt{acos}}(U_{22})$ & $\mathrm{\texttt{atan2}}(U_{10}, U_{00})$ & $\mathrm{\texttt{atan2}}(U_{01}, U_{11})$ \\ \hline
$\gamma$ & $\mathrm{\texttt{atan2}}\left(\frac{-U_{12}}{\sin\beta}, \frac{-U_{02}}{\sin\beta} \right)$ & $0$ & $0$ \\ \hline
\end{tabular}
\end{center}
\caption{Euler angles in terms of the elements of the matrix $U$ \label{eulerangles}}
\end{table}
With the transformation $A:S\rightarrow S''$ specified by the Euler angles $(\alpha,\beta,\gamma)$ and
the displacement $\mathbf{\Delta}$, we can determine the multipole moments of $\Sigma$ in $S''$ through
the application of theorems (\ref{wigner-rotation}) and (\ref{M2M-theorem}).
Theorem (\ref{wigner-rotation}), from Wigner \cite{wignergroup}, originates in quantum
mechanics \cite{edmonds}. It appears when needing to express the result of the action of the rotation
operator $\mathcal{D}^{l}(\alpha,\beta,\gamma)$ upon
a particular eigenstate $|l,m \rangle$ of total angular momentum $l$, which is associated with the spherical harmonic
$Y_{l}^{m}(\theta,\phi)$, in terms of the eigenstates of the rotated frame $|l',m' \rangle$. Note that since total angular
momentum is conserved, this rotation operator does not mix states with a distinct value of $l$ (thus $l=l'$).
Specifically, Wigner's theorem tells us the matrix elements of the rotation operator $\mathcal{D}^{l}(\alpha,\beta,\gamma)$,
which is a member of the $(2 l + 1)\times (2l +1)$ matrix representation of $SO(3)$.
A more succinct version of this theorem is given in \cite{gimbutas2009fast}, and is restated here in slightly a modified form.
\begin{theorem}
Assume there are two coordinate systems which share the same origin $S:(\mathcal{O}, \hat{e}_{0}, \hat{e}_{1} , \hat{e}_{2})$
and $S':(\mathcal{O}, \hat{e}_{0}', \hat{e}_{1}' , \hat{e}_{2}')$, that are related by
the rotation $U \in SO(3)$ specified by the Euler angles $\{\alpha,\beta,\gamma\}$ such
that $\hat{e}_i' = U \hat{e}_i$, for $i=0,\;1,\;2$. Furthermore assume that there is a function
$F(\theta,\phi)$ that can be expanded in terms of the
spherical harmonics $Y_{l}^{m}(\theta,\phi)$ such that:
\begin{equation}
F(\theta,\phi) = \sum\limits_{l=0}^{\infty} \sum\limits_{m=-l}^{l} Q_{l}^{m} Y_{l}^{m}(\theta,\phi)
\end{equation}
then there exists a function $f(\theta', \phi')$ such that
\begin{equation}
f(\theta', \phi') = F(\theta(\theta',\phi'), \phi(\theta',\phi') ) = \sum\limits_{l=0}^{\infty} \sum\limits_{m'=-l}^{l} q_{l}^{m'} Y_{l}^{m'}(\theta',\phi')
\end{equation}
where the coefficients $q_{l}^{m'}$ are given by:
\begin{equation}
q_{l}^{m'} = \sum\limits_{m=-l}^{l} \mathcal{D}^{l}_{m',m}(\alpha,\beta,\gamma) Q_{l}^{m}
\label{wigner-transform}
\end{equation}
where $\mathcal{D}^{l}_{m',m}(\alpha,\beta,\gamma)$ are elements of what is know as the Wigner D-matrix.
\label{wigner-rotation}
\end{theorem}
The direct evaluation of the coefficients $\mathcal{D}^{l}_{m',m}(\alpha,\beta,\gamma)$ through the use
of the expressions given by Wigner \cite{wignergroup}, \cite{edmonds} is beyond the scope of this paper. Regardless,
direct evaluation of (\ref{wigner-transform}) is known to be inefficient, as well as numerically unstable for
large values of $l$ and certain angles \cite{choi1999rapid}. However, given the wide applicability of spherical harmonics to quantum chemistry, fast multipole methods, and other areas, there
has recently been a large effort to develop efficient and stable methods to perform such rotations in both
real and complex spherical harmonic bases. The current state of the field of spherical harmonic rotation
is well summarized by \cite{lessig2012efficient}, with the algorithm developed by Pinchon et al. \cite{pinchon2007rotation}
being one of the fastest and most accurate. To avoid the need of complex matrix-vector multiplication, the method proposed by Pinchon et al. \cite{pinchon2007rotation} is
executed in the basis of real spherical harmonics $S_{l}^{m}(\theta, \phi)$ (with a different normalization convention).
To apply a rotation to the set of multipole moments $\{Q_{l}^{m}\}$ with $l$ fixed and $m$ ranging from $-l$ to $l$
we first must calculate the corresponding real basis $\{R_{l}^{m}\}$ coefficients. Then, to prepare this set of
moments $\{R_{l}^{m}\}$ for the rotation operator we arrange them to form the column vector $\mathbf{R}_{l}$:
\begin{equation}
\mathbf{R}_{l} = \left[ R_{l}^{-l},\; R_{l}^{-l+1},\; R_{l}^{-l+2},\; \ldots,\; R_{l}^{l-1},\; R_{l}^{l} \right ]^{T} \;.
\end{equation}
The application of the Wigner $\mathcal{D}^l$-matrix to this column vector produces the corresponding vector of rotated moments $\mathbf{r}_{l}$.
For efficiency, the $\mathcal{D}^l$-matrix is itself decomposed into several matrices, each of which may be applied to the vector $\mathbf{R}_{l}$ in succession:
\begin{equation}
\mathbf{r}_{l} = \mathcal{D}^{l}(\alpha,\beta,\gamma) \mathbf{R}_{l}
= \left[ X_{l}(\alpha) J_{l} X_{l}(\beta) J_{l} X_{l}(\gamma) \right] \mathbf{R}_{l}
\end{equation}
In this notation, the $X_{l}$ matrices effect a rotation about the $z$-axis,
while the $J_{l}$ matrices perform
an interchange of the $y$ and $z$ axes.
The advantage to this method is that the $X_{l}$ matrices have a simple sparse
form whose action on the vector $\mathbf{R}_{l}$ can be computed quickly,
as they consist only of non-zero diagonal and
anti-diagonal terms. The interchange matrices $J_{l}$, on the other hand, are completely independent of the rotation angles
and therefore only need to be computed once. While the computation of $J_{l}$ is beyond the scope
of this paper, there is an elegant recursive scheme to compute them up to any degree $l$ given by Pinchon et al. \cite{pinchon2007rotation}.
After the rotated moments $\mathbf{r}_l$ have been computed in the real basis, we need only convert them back to the complex basis to obtain the set of moments $\{ q_{l}^{m'} \}$.
Now that we have obtained the multipole moments $\{ q_{l}^{m'} \}$ in the coordinate system $S'$, we need to determine how they
are modified by a displacement of the expansion origin. This can be accomplished by the application of theorem (\ref{M2M-theorem}).
This theorem, presented by Greengard and Rohklin \cite{rokhlin1985rapid}, \cite{greengard1988rapid}, is a principle part of the fast multipole method, applied
during the operation of gathering the multipole expansions of smaller regions into larger collections, and
describes how a multipole expansion about one origin can be re-expressed as an expansion
about a different origin. Graphically, this is represented in figure \ref{fig:M2M}.
\begin{theorem}
\label{M2M-theorem}
Consider a multipole expansion with coefficients $\{O^{m}_{n}\}$ due to charges located within the sphere $D$ with radius $a$
centered about the point $\mathbf{P}_0$. This expansion converges for points outside of sphere $D$.
Now consider the point $\mathbf{S}_0 \notin D$ such that $\mathbf{\Delta} = \mathbf{S}_0 - \mathbf{P}_0
= (\rho, \alpha, \beta)$. We may form a new multipole expansion about the point $\mathbf{S}_0$ due
to the charges within $D$ which converges for points outside of the sphere
$D'$ which has its center at $\mathbf{S}_0$ and radius $a' = \rho + a$. The multipole moments of the new
expansion $\{M_{j}^{k}\}$ are given by:
\begin{equation}
M_{j}^{k} = \sum_{n=0}^{j} \sum_{m=-n}^{m=n} \frac{ O_{j-n}^{k-m} i^{|k| - |m| - |k-m| } A_{n}^{m} A_{j-n}^{k-m} \rho^{n} Y_{n}^{-m}(\alpha, \beta) }{A_{j}^{k}}
\end{equation}
where $ A_{n}^{m} = (-1)^{n}/\sqrt{(n-m)! (n+m)!} \label{norm}$.
\end{theorem}
Immediately applying this theorem to the set of moments $\{ q_{l}^{m'} \}$ results in the final objective of obtaining
the multipole moments of the boundary element $\Sigma$ in the coordinate system $S''$. However, the number of
arithmetic operations required by the application of theorem (\ref{M2M-theorem}) scales like $\mathcal{O}(p^4)$.
This high cost can be mitigated by the use of the special case of theorem (\ref{M2M-theorem}) along the $z$-axis. White et al. \cite{white1996rotating}
noted that it can be used to perform a multipole-to-multipole translation along any axis needed if a rotation is performed through
the use of theorem (\ref{wigner-rotation}) before and after the translation operation.
The first rotation applied aligns the $z$-axis with the vector $\mathbf{S}_0 - \mathbf{P}_0$, while
the second rotation is the inverse. The use of the rotation operator together with
the axial translation has a cost which scales like $\mathcal{O}(p^3)$, which
for high-degree expansions can provide useful acceleration when compared to the implementation of theorem (\ref{M2M-theorem}) alone.
\begin{figure}[width=11cm]
\begin{center}
\includegraphics{./M2MTranslation.pdf}
\caption{ Multipole to multipole translation. The solid shaded area indicates the
region where the original multipole expansion $\{O_{n}^{m}\}$ does not converge, the striped
area indicates the region where the new multipole expansion $\{M_{j}^{k}\}$ does not converge.}
\label{fig:M2M}
\end{center}
\end{figure}
The use of theorem (\ref{M2M-theorem}) to make the calculation of the multipole moments in the special
coordinate system $S$ centered on the vertex $\mathbf{P}_0$ generalizable to any arbitrary expansion center $\mathbf{S}_0$ puts
a constraint on the radius of convergence. The radius of convergence can be no less than $\rho + a$, where $\rho = |\mathbf{P}_0 - \mathbf{S}_0|$
and $a$ is the length of the longest side of the triangle $\Sigma$ that terminates on $\mathbf{P}_0$.
\section{\label{sec:results} Numerical Results}
In order to gain some understanding of the accuracy and efficiency of the algorithm presented
in this work, some numerical tests were performed with regard to the problem of evaluating
the electrostatic potential of a uniformly charged triangle (zero-th order interpolant).
All of the following tests were performed in double precision.
Since the integrals required to compute the multipole expansion of boundary elements are typically evaluated
using numerical quadrature, a straightforward two dimensional Gauss-Legendre quadrature method was used as a
benchmark against which to compare the speed and accuracy of the analytic algorithm.
It should be noted that this numerical integration routine has not been optimized, nor is
it the most efficient possible, it is only intended to provide a point of reference to a typically used means of computing
the multipole coefficients. There are several techniques to accelerate the numerical integration
over our benchmark implementation, such as adaptive quadrature \cite{Berntsen1991} or quadrature rules specifically
formulated for triangular domains such as Cowper \cite{cowper1973gaussian}. Cowper's rules require roughly three times
fewer function evaluations than the two-dimension Gauss-Legendre Gauss-Legendre with corresponding accuracy but are
only provided for a few different orders. The computation of the weights and abscissa for an arbitrary
order quadrature rule on a triangular domain is more complicated than the simple two-dimensional scheme, which are trivially generated
from the one dimensional Gauss-Legendre weights and abscissa. Though it is possible that these other methods
may be competitive, they were not implemented for this study, since is not the purpose of this
paper to survey the broad range of numerical integration methods available.
The benchmark numerical integration is performed by first converting the integral over the triangular domain given
by the points $\{\mathbf{P}_0, \mathbf{P}_1, \mathbf{P}_2\}$
to an integral over a rectangular domain through the use of a slightly modified version of the transform
described by Duffy \cite{duffy1982quadrature}. We can then write the surface integral given in equation (\ref{surfacemultipole}) as:
\begin{equation}
Q^{m}_{l} =
\int \limits_{0}^{L_1} \int \limits_{0}^{L_2} \sigma_0
\overline{Y_{l}^{m}}(\theta(\mathbf{r}), \phi(\mathbf{r}) ) |\mathbf{r}|^{l}
\left|\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right| dv du
= \int \limits_{0}^{L_1} \int \limits_{0}^{L_2} f(u,v) dv du \;,
\label{numerical_integral}
\end{equation}
where $\mathbf{r}(u,v) = (\mathbf{P}_0 + u\mathbf{\hat{n}}_1 + v (1 - u/L_1 )\mathbf{\hat{n}}_2 ) - \mathbf{x}_0$.
The point $\mathbf{x}_0$ is the origin of the expansion
and $L_i = |\mathbf{P}_i - \mathbf{P}_0|$ and $\mathbf{\hat{n}}_i = (\mathbf{P}_i - \mathbf{P}_0)/L_i$ for $i=1,2$.
The two dimensional integral over the $(u,v)$-plane is then performed using $m$-th order two dimensional
Gauss-Legendre quadrature \cite{abramowitz2012handbook}, given by:
\begin{equation}
Q^{m}_{l} = \frac{L_1 L_2}{4} \sum \limits_{i=1}^{m} \sum \limits_{j=1}^{m} w_i w_j f\left( \frac{L_1}{2}(x_i + 1) , \frac{L_2}{2}(x_j + 1) \right)
\end{equation}
where $w_i$ and $x_i$ are respectively, the one-dimensional Gauss-Legendre weights and abscissa as described by Golub et al. \cite{golub1969calculation}.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=11cm]{./relative-error.pdf}
\caption{Comparison of the accuracy of the multipole expansion against the direct method of evaluating the potential with
various degrees of the expansion. Coefficients of the multipole expansion are calculated using the analytic method described in this paper.
Relative error is shown as a function of the ratio $|\mathbf{x} - \mathbf{x}_0|/R_{\mathrm{source}}$, where $|\mathbf{x} - \mathbf{x}_0|$
is the distance of the evaluation point from the expansion origin, and $R_{\mathrm{source}}$ is the radius of the smallest sphere
enclosing the charge distribution.}
\label{analytic_accuracy}
\end{center}
\end{figure}
The first study consisted of $10^4$ triangles generated by randomly selecting
points on a sphere with arbitrary radius $R_{\mathrm{source}}$. These triangles where restricted to have an aspect ratio of less than 100.
For each triangle the multipole
expansion (for each degree up to $p=32$) about the origin $\mathbf{x}_0$ (the center of the sphere)
was calculated using the algorithm described in this work. For each triangle 100 random points $\mathbf{x}$ were selected in the volume
$R_{\mathrm{source}} < |\mathbf{x} - \mathbf{x}_0 | < 10^3 \times R_{\mathrm{source}}$, the angular coordinates of which
where uniformly distributed, while the radial coordinate followed a log uniform distribution
in order to provide enough statistics for points at small radius. At each test point the relative error between
the potential evaluated directly and the potential given by the multipole expansion was computed and histogrammed.
The relative error $\Phi_{\mathrm{error}} = |(\Phi_{\mathrm{multipole}} - \Phi_{\mathrm{direct}})/\Phi_{\mathrm{direct}}|$ on
the potential is plotted as a function relative distance from the expansion origin for various expansion degrees in figure (\ref{analytic_accuracy}).
The relative error on a $p=32$ degree expansion of the potential reaches approximately machine precision at roughly twice $R_{\mathrm{source}}$.
However, the constraint imposed by theorem (\ref{M2M-theorem}) on the radius of convergence in this particular
test geometry limits the minimum radius of convergence to approximately $2 \times R_{\mathrm{source}}$. Using a
higher degree expansion than $32$ does not result in a reduced radius of convergence for this geometry.
As a general rule, $\Phi_{\mathrm{error}}$ is a decreasing function
of distance until numerical roundoff starts to dominate near the level of machine precision.
However, this is only true so long as the method used to compute the multipole moments of the expansion respects the oscillatory
behavior of the spherical harmonics. For low degree expansions, numerical quadrature rules
with a small number of function evaluations can compute the the multipole moments
exactly to within machine precision. However, as the degree of the expansion
is increased the higher order spherical harmonics oscillate more rapidly and
progressively more expensive quadrature rules are needed to evaluate the coefficients to equivalent accuracy.
To explore this effect we repeated the previous study using our algorithm and the benchmark numerical quadrature method
with various orders $m =\{2,\;3,\;4,\;6,\;8,\;10\}$ and defined a quantity $R_{\mathrm{convergence}}$ (the radius of convergence)
as the minimum distance $|\mathbf{x} - \mathbf{x}_0|$ for which we have $\Phi_{\mathrm{error}}(\mathbf{x})$ less then some
threshold $t_{\mathrm{error}}$. Then for each method and expansion degree up to $p=32$ we computed
the radius of convergence at four thresholds $t_{\mathrm{error}} = \{10^{-5}, 10^{-8}, 10^{-11}, 10^{-14}\}$.
Figure (\ref{numerical_accuracy}) shows the behavior of $R_{\mathrm{convergence}}/R_{\mathrm{source}}$ as a function of expansion degree.
For example, from figure (\ref{numerical_accuracy}) one can see that up to an expansion degree of $p=8$,
the $4\times4$ Gauss-Legendre quadrature rule is sufficient to compute the multipole coefficients
to the same accuracy as our algorithm. However continuing to use the $4\times4$ Gauss-Legendre
quadrature rule while increasing the degree of the expansion up to $p=32$ does not result in a
more accurate evaluation of the potential. To obtain the full benefit of a high degree expansion
one must correspondingly increase the number of function evaluations used by numerical integration.
\begin{figure}[htp!]
\centering
\mbox{ \centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./convergence-10e-5.pdf}
\caption{Threshold of $10^{-5}$}
\label{quadrature_n4}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./convergence-10e-8.pdf}
\caption{Threshold of $10^{-8}$}
\label{quadrature_n6}
\end{subfigure}%
}
\\
\mbox{
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./convergence-10e-11.pdf}
\caption{Threshold of $10^{-11}$}
\label{quadrature_n8}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./convergence-10e-14.pdf}
\caption{Threshold of $10^{-14}$}
\label{quadrature_n10}
\end{subfigure
}
\caption{Relative radius of convergence as a function of the degree of the multipole expansion for various thresholds on the relative error and different
methods of calculating the multipole moments. For quadrature rules which compute the multipole moments with insufficient accuracy
the radius of convergence fails to decrease after reaching a certain degree. Note that up to $p=32$ the $10\times 10$ Gauss-Legendre quadrature rule
computes the multipole moments to equivalent accuracy as algorithm (\ref{multipole-moment-algo}).}
\label{numerical_accuracy}
\end{figure}
\begin{figure}[htp]
\begin{center}
\includegraphics[width=11cm]{./wallclock.pdf}
\caption{Wallclock time required to evaluate all of the multipole coefficients of a single triangle for the method detailed
in algorithm (\ref{multipole-moment-algo}) and various $m\times m$ point Gauss-Legendre quadrature. The dashed lines on the graphs
denote that for a fixed threshold $t_{\mathrm{error}}$ on the relative error in the potential, the corresponding $R_{\mathrm{convergence}}$
for that numerical quadrature rule was equivalent or less than $1.2 \times R_{\mathrm{convergence}}$ of the analytic method.}
\label{speed}
\end{center}
\end{figure}
To demonstrate the efficiency of this algorithm (at least in regard to the naive
two dimensional numerical integration using Gauss-Legendre quadrature), a comparison was made between the
time needed to compute all of the multipole expansion coefficients of a single triangle
(up to a certain degree) using the analytic algorithm and the time needed when
using numerical integration. This test was carried out on a computer with an Intel i7 processor
running at 1.9GHz, results are shown in figure (\ref{speed}). Individually the scaling of all methods is $\mathcal{O}(p^2)$ since this is approximately the number of moments to be computed.
However, beyond a certain maximal degree, a fixed order numerical quadrature rule will no longer compute
the multipole moments to a given threshold $t_{\mathrm{error}}$, and a higher order rule will be needed to retain
accuracy making the scaling of numerical integration effectively greater than $\mathcal{O}(p^2)$. This difference
in scaling can be seen figure (\ref{speed}) by noting how the position of the end of the solid line
(cut off for $t_{\mathrm{error}} = 10^{-14}$) has a larger slope than the analytic method. For all but the lowest degree $p \leq 4$ expansions,
the performance of the algorithm presented in this work is approximately an order of magnitude faster than
the lowest accuracy Gauss-Legendre quadrature rule considered, while for the highest degree tested ($p=32$)
it is nearly two orders of magnitude faster than the quadrature rule which obtains equivalent accuracy.
Unfortunately, the analytic method of computing the multipole moments is not applicable in all cases.
The first restriction is that the aspect ratio of the triangle must not be too large (exceeding 100). Since for a needle
like triangle the values of $\phi_1$ or $\phi_2$ can be very close to $\pi/2$ which causes the base case integrals (\ref{Ipq})
to diverge. This can however be easily avoided if the BEM mesh has been constructed with sufficient quality.
The second issue is that the use of theorem (\ref{M2M-theorem}) prevents convergence of the multipole expansion
within the sphere of radius $\rho + a$ centered on $\mathbf{S}_0$. This is typically unimportant since in most cases
where the a multipole expansion is useful the distance between the triangle and the expansion center
$\rho$ is usually much larger than the length of the triangle's longest side $a$.
However this restriction can be noticeable when the expansion origin and region of interest are very close to or on the triangle.
For example if $\mathbf{S}_0$ is one of the vertices opposite $\mathbf{P}_0$ then then minimum radius of
convergence would be $\sim2a$, whereas for a numerical method which requires no translation it would only be $a$.
Additionally, some numerical instability is expected to be encountered in the recursion relations
(\ref{A-recursion_general}) and ( \ref{B-recursion_general}) for high degree expansions where
the individual terms become much larger than their difference, however this does not appear to manifest itself until beyond $p=32$.
\section{\label{sec:conclusion}Conclusion}
We have presented a novel technique to evaluate the multipole expansion coefficients of a triangle.
This method evaluates the necessary integrals through recursion within the context
of a coordinate system with special orientation and placement. The results of the integration can then be generalized to
the case of an arbitrary system through the well known transformation properties of the spherical harmonics
under rotation and translation. A summary of the full method by which to compute the multipole moments of a triangle is detailed in algorithm (\ref{multipole-moment-algo}).
\begin{algorithm}
\caption{Computing the multipole moments of a triangular boundary element.}
\begin{algorithmic}[1]
\Require {Triangle $\Sigma:\{\mathbf{P}_0, \mathbf{P}_1, \mathbf{P}_2 \}$ and associated
charge density interpolation coefficients $\{s_{ab}\}$. }
\State Compute height $h$ and coordinate system $S$ for triangle $\Sigma$ according to equation (\ref{coordinate_S}).
\For {$l=0$ to $p$}
\For {$m=0$ to $l$}
\ForAll{ $s_{a,b} \neq 0$ }
\State Compute the prefactor $\mathcal{K}_{l,m}^{a,b}$ according to equation (\ref{phiintegral_general}).
\State Recursively compute the integral $\mathcal{I}_{l,m}^{b}$ according to
equations (\ref{A-recursion_general}) and (\ref{B-recursion_general}).
\EndFor
\State Compute the multipole moment
$Q_{l}^{m} = \sum \limits_{a} \sum \limits_{b} \mathcal{K}_{l,m}^{a,b} \mathcal{I}_{l,m}^{b}$
and $Q_{l}^{-m} = \overline{Q_{l}^{m}}$.
\EndFor
\EndFor
\State Compute the Euler angles $( \alpha, \beta, \gamma)$ of the rotation $U: S \rightarrow S'$ according to table (\ref{eulerangles}).
\State Compute the effect of the rotation $U$ on the set of moments; $\{Q_{l}^{m}\} \rightarrow \{q_{l}^{m'}\}$.
\State Compute the effect of the translation $\mathbf{\Delta}:S' \rightarrow S''$ on the moments; $\{q_{l}^{m'}\} \rightarrow \{q_{l}^{m}\}$.
\Ensure The multipole moments $\{q_{l}^{m}\}$ of the triangle $\Sigma$ in coordinate system $S''$.
\end{algorithmic}
\label{multipole-moment-algo}
\end{algorithm}
Furthermore we have demonstrated that the application of this method to the multipole expansion of triangles with uniformly constant charge density compares favorably
in terms of accuracy and speed to a simple numerical integration technique. This method can also be extended to the
case of non-uniform charge density, provided the interpolant can be represented as a sum over the bivariate monomials.
We expect this method may find use in solving the three dimensional Laplace equation
with the fast multipole boundary element method (FMBEM). In addition, this technique has
also been used for the accurate calculation of a electric fields needed for large scale charged
particle optics simulations. We speculate that other boundary integral equation (BIE) problems,
such as the Helmholtz equation in the low frequency limit $k \rightarrow 0$, might benefit from this approach
if the integrand in the multipole coefficient integrals can be expanded in
terms of the solid harmonics, and may warrant a future study.
\ack
The authors would like to thank Dr. Ferenc Gl\"{u}ck for valuable comments regarding
the preparation of this paper. This work was performed, in part, under DOE Contract DE-FG02-06ER-41420.
|
1,314,259,995,778 | arxiv | \section{Introduction}
Although much of this paper concerns dynamics, we begin with the following simple and very general non-dynamical construction.
Suppose that $f$ is a {transcendental entire function}, that $V \subsetneq {\mathbb{C}}$ is a simply-connected domain, and that $U$ is a connected component of $f^{-1}(V)$; it is a consequence of the Open Mapping Theorem that $U$ is also simply connected. Let $\phi \colon {\mathbb{D}} \to U$ and $\psi \colon {\mathbb{D}} \to V$ be Riemann maps, and set $g \defeq \psi^{-1} \circ f \circ \phi$; see Figure~\ref{fig:inner}. We begin with a result that summarises the properties of the map $g$. This is not entirely new but to the best of our knowledge it has not been stated in this generality before. Here an \emph{inner function} is a holomorphic self-map of ${\mathbb{D}}$ for which radial limits exist at almost all points of the unit circle, and belong to the unit circle. A particular class of inner functions is the class of \emph{Blaschke products}\footnote{Sometimes, the term ``Blaschke product'' is used more generally for a function of the form~\eqref{eq:Bdef} where some $a_n$ may also have $\lvert a_n\rvert> 1$, so that $B$ has poles in ${\mathbb{D}}$. These are not inner functions, and are not considered in this paper.}. These are functions of the form
\begin{equation}
\label{eq:Bdef}
B(z) \defeq e^{i\theta} \prod_{n = 1}^d \frac{|a_n|}{a_n} \frac{a_n-z}{1-\overline{a_n}z},
\end{equation}
where $\theta \in {\mathbb{R}}$, $d \in {\mathbb{N}} \cup \{\infty\}$, and $(a_n)_{1 \leq n \leq d}$ is a sequence of points of ${\mathbb{D}}$, which satisfies the condition $\sum (1-|a_n|) < \infty$. When $a_n = 0$ we interpret the term in the infinite product simply as $z$. If $d$ is finite then $B$ is called a \emph{finite Blaschke product of degree $d$}, and otherwise it is an \emph{infinite Blaschke product}.
\begin{proposition}
\label{prop:basics}
Suppose that $f$ is a {transcendental entire function}, that $V \subsetneq {\mathbb{C}}$ is a simply-connected domain, and that $U$ is a connected component of $f^{-1}(V)$. Let $\phi \colon {\mathbb{D}} \to U$ and $\psi \colon {\mathbb{D}} \to V$ be conformal, and set $g \defeq \psi^{-1} \circ f \circ \phi$. Then $g$ is an inner function, which, for an appropriate choice of $\phi$ and $\psi$, can be taken to be a Blaschke product. More precisely, exactly one of the following conditions holds.
\begin{enumerate}[(a)]
\item \emph{Finite valence:} $g$ is a finite Blaschke product of degree $d$, for some $d \in {\mathbb{N}}$, and $f|_U$ is of constant finite valence $d$.\label{theo:a}
\item \emph{Infinite valence:} $g$ is an infinite Blaschke product, and $U \cap f^{-1}(z)$ is infinite for all $z \in V$ with at most one exception.\label{theo:b}
\end{enumerate}
\end{proposition}
\begin{figure}
\begin{center}
\def.9\textwidth{.7\textwidth}
\input{inner-function-construction.pdf_tex}
\end{center}
\caption{\label{fig:inner}Construction of an associated inner function. Here, $f(z)=\lambda e^z$ is an exponential function
with real $\lambda<-e$, which has
an attracting periodic orbit of period $2$. The domains $U$ and $V$ shown are the two periodic Fatou components of $f$.}
\end{figure}
In the setting of Proposition~\ref{prop:basics}, we say that $g$ is an \emph{inner function associated to} $f \lvert _{U}$.
Such inner functions have been mostly considered in a dynamical setting where $U=V$ and $\phi=\psi$, see below. However,
they have also appeared in settings where $U\neq V$; see, for example, \cite[p.5]{Bishop}.
\begin{remark}
Note that Proposition~\ref{prop:basics} implies that there are many inner functions which cannot be associated to a transcendental entire function in the sense of this paper. For example, if $A$ is any closed subset of ${\mathbb{D}}$, of (logarithmic) capacity zero, then there is an inner function that omits all the points of $A$; see \cite{MR0481015}.
\end{remark}
In our first main result, which significantly generalises earlier results in a dynamical setting, we are interested in the singularities of the associated inner function; a point $\zeta \in \partial \mathbb{D}$ is called a \emph{singularity} of (an inner function) $g$, if $g$ cannot be extended holomorphically to any neighbourhood of $\zeta$ in $\mathbb{C}.$ For a transcendental entire function $f$, we denote by $S(f)$ the set of \emph{singular values} of $f$; in other words, the closure of the set of critical and finite asymptotic values of $f$. Our result is as follows.
\begin{theorem}
\label{theo:tracts}
Suppose that $f$ is a {transcendental entire function}, that $V \subsetneq {\mathbb{C}}$ is a simply-connected domain, and that
$U$ is a component of $f^{-1}(V)$ such that $f\colon U\to V$ is of infinite valence.
Suppose that $S(f)\cap V$ is compact, and let $D$ be
a bounded Jordan domain with $S(f)\cap V\subset D$ and $\overline{D} \subset V$.
Then the singularities of an associated inner function $g$
are in order-preserving one-to-one correspondence with the accesses to infinity in
$U \cap f^{-1}(D)$. In particular, the number of singularities of $g$ is equal to the number of components of $U \setminus f^{-1}(\overline{D})$.
\end{theorem}
\begin{remark}\mbox{}
\begin{enumerate}[(a)]
\item An \emph{access to infinity} in $U\cap f^{-1}(D)$ is a homotopy class
of curves to infinity in $U$; see Section~\ref{S.singularities}
\item By the final statement, we mean that the number of singularities and
the number of components are either both infinite, or both finite and equal.
We caution that, when infinite, the number of singularities may be
uncountable, while the number of components of
$U\setminus f^{-1}(\overline{D})$ is always countable.
\item In the case of finite valence, it follows from Proposition~\ref{prop:basics} that any associated inner product is a finite Blaschke product, and has no singularities.
\end{enumerate}
\end{remark}
We now consider associated inner functions in a dynamical setting. Let $f$ be a transcendental entire function, and denote by $f^n$ the $n$th iterate of $f$. The set of points for which the set of iterates $\{f^n\}_{n \in \mathbb{N}}$ form a normal family in some neighbourhood is the \emph{Fatou set} $F(f)$, and its complement in the complex plane is the \emph{Julia set} $J(f)$. The Fatou set is open, and so consists of connected components which are called \emph{Fatou components}. For an introduction to the properties of these sets see, for example, \cite{bergweiler93}.
In the case that $U$ is a simply-connected Fatou component, and $V$ is the Fatou component containing $f(U)$, then the conditions we discussed earlier all hold, and we can associate an inner function to $f|_U$.
A case of particular interest is when the Fatou component $U$ is \emph{forward invariant}, in other words such that $f(U) \subset U$. Note that it is well known that forward-invariant Fatou components are necessarily simply connected. In this case we have that $U = V$, we can set $\psi = \phi$, and the dynamics of $f$ on $U$ is \emph{conjugate} to the dynamics on ${\mathbb{D}}$ of the function $g \defeq \phi^{-1} \circ f \circ \phi$. Moreover, $g$ is unique in this respect, up to a conformal conjugacy. In this case we say that $g$ is an inner function \textit{dynamically associated to} $f|_U$. This construction appears to have first been considered
by T\"opfer~\cite[\S II]{topfer} in 1939. Its connection to inner functions appears to have first been
made by Kisaka~\cite{kisaka} and Baker and Dom\'inguez~\cite{BakerDominguez}. Compare also \cite{DevandG, fagella-henriksen, Baranski, BK, univalentbd, Bargmann, EFJS}.
Theorem~\ref{theo:tracts} has the following corollary, which is a significant generalisation of the main result of \cite{EFJS}. Here we consider the class $\mathcal B$ of {transcendental entire function}s for which $S(f)$ is bounded, and for a function $f \in \mathcal B$ a \emph{tract} is a component of $f^{-1}({\mathbb{C}} \setminus \overline{D'})$ where $D'$ is a bounded Jordan domain containing $S(f)$. It is well-known that the number of tracts is independent of the choice of $D'$.
\begin{cor}
\label{corr:tracts}
Suppose that $f \in \mathcal B$, and that ${S(f)} \subset F(f)$. Suppose also that $U$ is an unbounded forward-invariant Fatou component of $f$. Then the number of singularities of a dynamically associated inner function is at most equal to the number of tracts of $f$.
\end{cor}
This generalises \cite[Theorem 1.5]{EFJS}, in which the condition that $S(f)$ is a compact subset of the Fatou set was replaced by the condition that the \emph{postsingular set} defined by
\[
\mathcal{P}(f) \defeq \overline{\bigcup_{j \geq 0} f^j(S(f))},
\]
is a compact subset of the Fatou set; such functions are called \emph{hyperbolic}.
Lyubich (personal communication) has asked which inner functions $g$ arise as dynamically associated inner functions. Few authors have explicitly calculated $g$ for given functions
$f$. Indeed, we are aware of only three examples in the literature. First, T\"opfer \cite[{\S}V]{topfer}
considered the function $f(z)=\sin(z)$,
which has a triple fixed point at the origin, with two attracting directions, along the positive and negative real axis. Note that,
by symmetry, the same inner function $g$ can be dynamically associated to either immediate parabolic basin.
T\"opfer observes~\cite[p.~78]{topfer}
that $g$ can be taken to have the form
\begin{equation}\label{eqn:topfer} g(z) = \frac{z^2 + k}{kz^2+1} \end{equation}
for some $0<k<1$. He does not determine the correct value of $k$, which is $k=1/3$ since the function $g$ must have
a parabolic point at $z=1$ (see the remarks at the end of Section~\ref{sec:expo}).
Devaney and Goldberg \cite{DevandG} considered the Julia set of $f_\lambda(z) \defeq \lambda e^z$ for values of $\lambda$ such that $f$ has a completely invariant attracting basin $U$. They showed that these functions have a dynamically associated inner function of the form
\begin{equation}
\label{eq:DevandG}
g(z) = g_{\mu}(z) \defeq \exp\left(i \ \frac{\mu + \overline{\mu}z}{1+z}\right),
\end{equation}
where $\mu$ lies in the upper half-plane ${\mathbb{H}}$, and depends on $\lambda$. Note that $g$ is not an infinite Blaschke product -- indeed, the proof technique used in \cite{DevandG} depends on this fact -- but is conjugate to one. However,
Devaney and Goldberg did not determine which values of $\mu$ are realised.
The result of Devaney and Goldberg was generalised later by Schleicher. He considered the case that $f_\lambda$ has an attracting \emph{periodic} point; in this case $f_{\lambda}$ is \emph{hyperbolic}. He observes that the associated
inner functions can always be chosen to take a certain form, which is equivalent to~\eqref{eq:DevandG}; see \cite[Lemmas III 4.2, III 4.3]{Dierkhab} for details.
Finally, Baker and Dom\'inguez \cite{BakerDominguez} showed that the map $f(z) \defeq z + e^{-z}$ has an invariant Fatou component with the dynamically associated Blaschke product
\begin{equation}\label{eqn:parabolicblaschke}
g(z) \defeq \frac{3z^2 + 1}{3 + z^2},
\end{equation}
which is the same as T\"opfer's map~\eqref{eqn:topfer}. It is easy to see how to write $g$ in the form \eqref{eq:Bdef}.
In view of this, perhaps unexpected, dearth of specific examples, our next goal in this paper is to find classes of {transcendental entire function}s, $\mathcal{F}$, and classes of inner functions, $\mathcal{G}$, with the following properties;
\begin{enumerate}[(I)]
\item Each $f \in \mathcal{F}$ has a forward-invariant Fatou component $U$;\label{p1}
\item For each $f \in \mathcal{F}$ there is an inner function $g\in\mathcal{G}$ dynamically associated to~$f$;\label{p2}
\item For each $g \in \mathcal{G}$ there is an $f \in \mathcal{F}$ such that $g$ is dynamically associated to~$f$.\label{p3}
\end{enumerate}
We begin with a result of this form for finite Blaschke products. If such a Blaschke product
$g\colon{\mathbb{D}}\to{\mathbb{D}}$ is dynamically associated to an invariant attracting or parabolic Fatou component
of an entire function, then $g$ has either an attracting fixed point in
${\mathbb{D}}$, or a triple fixed point on $S^1=\partial {\mathbb{D}}$. Equivalently,
$J(g)=S^1$; see~\cite{fletcherblaschke}.
\begin{theorem}\label{theo:finiteblaschke}
Let $\mathcal{F}$ consist of all entire functions having a forward-invariant
attracting or parabolic Fatou component
$U$ that is a bounded Jordan domain. Let
$\mathcal{G}$ be the connectedness locus of finite Blaschke products;
that is, $\mathcal{G}$ consists of all finite Blaschke products of degree $d\geq 2$
with $J(g)= S^1$.
Then~\ref{p1}, \ref{p2} and \ref{p3} hold.
\end{theorem}
Observe that~\ref{p1} and~\ref{p2} hold by assumption and Proposition~\ref{prop:basics},
so the main content of the theorem is showing the existence of
an entire function realising a prescribed Blaschke product as its dynamically
associated inner function. This will be achieved by quasiconformal surgery.
Next, we give
a complete description of dynamical inner functions for exponential maps with attracting fixed points, thus completing
the work of Devaney and Goldberg.
Similarly as in \cite{Dierkhab}, we find it convenient to change coordinates from the unit disc to the upper half-plane and consider \textit{inner functions of the upper half-plane}
associated to $f|_U$; compare also \cite{Bargmann}. We use the
family of functions
\begin{equation}\label{eqn:tangent} g_{a,b}\colon {\mathbb{H}}\to{\mathbb{H}} \defeq a\tan(z) + b, \qquad a>0, b\in (-\pi/2,\pi/2]. \end{equation}
It is easy to check that $g_{a,b}$ is conjugate to $g_{\mu}$ as in~\eqref{eq:DevandG} for $\mu=2(b+ai)$. Thus the following result
also gives a complete description of the set of $\mu$ for which $g_{\mu}$
arises as an inner function of an exponential map. (Compare Figure~\ref{fig:tan_inner} in Section~\ref{sec:expo}.)
\begin{theorem}\label{theo:exp}
Set
\[ \mathcal{F} \defeq \{f_{\lambda}\colon f_{\lambda} \text{ has
an attracting fixed point} \} =
\{ f_{\tau\cdot e^{-\tau}}\colon \tau\in{\mathbb{D}}\setminus\{0\} \}, \]
where $f_{\lambda}(z)=\lambda e^z$. Also let
\begin{align} \label{eqn:expoG} \mathcal{G} \defeq& \{ g_{a,b}\colon g_{a,b}\text{ has an attracting fixed point in ${\mathbb{D}}$} \} \\
=& \left\{ g_{a,b}\colon a>1 \text{ or } \lvert b\rvert > \arccos(\sqrt{a}) - \sqrt{a}\cdot \sqrt{1-a} \right\}.\notag \end{align}
Then $\mathcal{F}$ and $\mathcal{G}$ satisfy~\ref{p1}, \ref{p2} and \ref{p3}. More precisely, for every $f_{\lambda}\in\mathcal{F}$, the family
$\mathcal{G}$ contains exactly one
dynamically associated inner function of $f$, and vice versa.
\end{theorem}
As in Schleicher's work, our theorem applies also to iterates of exponential maps. Indeed, more generally the following is true.
\begin{theorem}\label{theo:unisingular}
Let $\mathcal{G}$ be as above,
let $f$ be a transcendental entire function, and suppose that $U$ is a basin of attraction of period $n$ for $f$. If the cycle of $U$ contains only one singular value of $f$, and
$f^n\colon U\to U$ is of infinite valence, then the family $\mathcal{G}$ from~\eqref{eqn:expoG} contains exactly one
inner function dynamically associated to $f^n|_U$.
\end{theorem}
While \cite{DevandG} and \cite[Section~III.4]{Dierkhab} only treated
attracting dynamics, we can also consider parabolic basins,
thus completing the description of associated inner functions for periodic
Fatou components of exponential maps.
\begin{theorem}\label{theo:unisingularparabolic}
Let $f$ be a transcendental entire function, and suppose that $U$ is a parabolic basin of period $n$ for $f$. If the cycle of $U$ contains only one singular value of $f$, and
$f^n\colon U\to U$ is of infinite valence, then $\tan\colon {\mathbb{H}}\to{\mathbb{H}}$ is a dynamically associated inner function for $f^n$ on $U$.
\end{theorem}
We are able to give a similar description of dynamically associated inner functions in cases where $f^n\colon U\to U$ takes some value only finitely many times,
see Theorem~\ref{theo:finite}. In particular, this applies to many functions of the form
\[
f(z) \defeq \lambda P(z) e^{Q(z)};
\]
see Corollary~\ref{cor:exponentials}.
The case of Fatou components containing infinitely many critical points is more complicated. We begin with the following detailed example, which concerns sine functions with invariant Fatou components of infinite valence. (Recall that the example $f(z)=\sin(z)$ studied
by T\"opfer has invariant Fatou components of finite valence.)
\begin{theorem}
\label{theo.sine}
There is a homeomorphism $\psi \colon (0, 1) \to (1, \infty)$ with the following property. Let $\mathcal{F}$ be the family of {transcendental entire function}s
\begin{equation}
\label{sinfdef}
f_\lambda(z) \defeq \lambda \sin z, \quad\text{for } \lambda \in (0, 1).
\end{equation}
For $\tau > 1$ let
\[
a_n = a_n(\tau) \defeq \frac{\tau^n-1}{\tau^n+1}, \quad\text{for } n \in {\mathbb{N}},
\]
and let $\mathcal{G}$ be the family of infinite Blaschke products
\begin{equation}
\label{singdef}
g_\tau(z) \defeq z \prod_{n=1}^\infty \frac{a_n^2 - z^2}{1-a_n^2z^2}.
\end{equation}
Then \ref{p1}, \ref{p2} and \ref{p3} hold for these families, with $\tau = \psi(\lambda)$.
\end{theorem}
\begin{remark}\normalfont
The proof of Theorem~\ref{theo.sine} makes strong use of symmetries of the Julia sets of the functions involved. It is not easy to see, therefore, how one might extend these results to wider families.
\end{remark}
Let $g$ be an inner function dynamically associated to an invariant Fatou component $U$ of infinite valence for a transcendental entire function $f$.
Let $A$ denote the set of singularities of $g$. By the
Schwarz reflection principle, $g$ extends to a meromorphic function on $\hat{{\mathbb{C}}}\setminus A$, so we can think of $g$ as a global complex dynamical system.
As mentioned above, if $U$ is an attracting or parabolic basin, we have $J(g)=S^1$; see \cite[Lemma~2]{kisaka},
\cite[Lemmas~8 and~9]{BakerDominguez} and \cite[Theorem~2.24]{Bargmann}. Here
a point on the unit circle is in
the Julia set $J(g)$ if it has no neighbourhood on which the iterates
of $g$ are defined and normal \cite[Section~3]{BakerDominguez}. If $\# A=1$, then $g$ is (up to conformal conjugacy) a transcendental meromorphic function. When $A$ is countable,
$g$ belongs to a class of functions for which the theory of complex dynamics was developed by Bolsch~\cite{bolsch}. Similarly, if $f$ has only finitely many singular
values in the Fatou component $U$, then $g$ is a \emph{finite-type map} in the sense of Epstein~\cite{epsteinthesis}; see also~\cite{remperadial,epsteincheritat}. In particular, inner functions
allow us to construct many examples of functions in these classes for which the Julia set is a circle. A larger class of holomorphic functions for which Fatou-Julia iteration theory
has been extended is Epstein's theory of \emph{Ahlfors islands maps}; see \cite{remperadial,remperipponexotic}, and also~\cite{bakerdominguezherring} for the case
where $A\neq S^1$. It would be interesting to investigate when the inner function $g$ associated to $f$ satisfies this Ahlfors islands condition.
For families of entire functions with a finite number of singular values, it is plausible that the preceding observation about finite-type maps, together with surgery
techniques similar to Theorem~\ref{theo:finiteblaschke}, can lead to a
description of the associated inner functions. On the other hand, it appears to be very difficult
to develop general principles for inner functions associated to Fatou components where the singular values are allowed to accumulate on the boundary. It is perhaps
surprising that we can nonetheless give a very precise description in one particular case.
\begin{theorem}
\label{theo.Fatou}
Let $\mathcal{F}$ be the family of {transcendental entire function}s
\begin{equation}
\label{Fatoudef}
f_\lambda(z) \defeq \lambda + z + e^{-z}, \quad\text{for } \lambda > 0.
\end{equation}
Let $\mathcal{G}$ be the family of maps
\begin{equation}
\label{eq:gmudef}
g_{\lambda} \colon {\mathbb{H}} \to {\mathbb{H}}; \quad g_{\lambda}(z) \defeq z - \lambda \frac{\cot z}{2}, \quad\text{for } \lambda > 0.
\end{equation}
Then \ref{p1}, \ref{p2} and \ref{p3} hold for these families, with $g_{\lambda}$ being associated to the restriction of $f_{\lambda}$ to its single Fatou component.
\end{theorem}
\begin{remarks}
\mbox{ }
\begin{enumerate}
\item Again, the proof of this result makes strong use of symmetries of Julia sets, and does not extend to the case
where $\lambda$ is not real and positive.
\item The map $g_\lambda$ is conjugate to the map $h_\lambda(z) \defeq z + \lambda \tan z$ via the conjugation $z \mapsto \pi/2 - z$. We prefer the parameterisation $g_\lambda$ as it makes the proof slightly simpler. Note that the dynamics of the map $h_1$ was studied in \cite{Accesses}.
\item The dynamics of the maps $f_{\lambda}$ was studied in \cite{Fweb}, under the parameterisation
\[
h(z) \defeq z + a + be^{cz}, \quad b \ne 0, \ ac < 0.
\]
(Up to affine conjugacy, this is the same family as $\mathcal{F}$.)
\end{enumerate}
\end{remarks}
To conclude, let us return to the case of inner functions associated to $f\colon U\to V$ where $U\neq V$, and in particular to the case where $U$ and $V$ are
simply-connected \emph{wandering domains} of a transcendental entire function $f$. Similarly as in Theorem~\ref{theo:finiteblaschke}, we show that
every finite Blaschke product may arise in this manner.
\begin{theorem}\label{theo:wandering}
Let $g\colon {\mathbb{D}}\to{\mathbb{D}}$ be a Blaschke product of degree $d$, with $2\leq d<\infty$. Then there is a transcendental entire function $f$ having wandering
domains $U$ and $V=f(U)$ which are bounded Jordan domains and such that $g$ is an inner function associated to $f|_U$.
\end{theorem}
Moreover, using approximation theory we can construct a \emph{single} entire function $f$ with an orbit of
wandering domains whose associated inner functions approximate any desired
Blaschke product:
\begin{theorem}
\label{theo:finitecomplicated}
There is a {transcendental entire function} $f$ having a simply-connected wandering domain $U$ with the following property. Given a finite Blaschke product $B$ and $\epsilon > 0$, there is $n\geq 0$ and
a Blaschke product $g$ associated to $f\colon f^n(U)\to f^{n+1}(U)$, such that the following both hold.
\begin{enumerate}[(i)]
\item $|g(z) - B(z)| < \epsilon, \quad\text{for } z \in {\mathbb{D}}$.
\item $\operatorname{deg}(g)= \operatorname{deg}(B).$
\end{enumerate}
\end{theorem}
\emph{Acknowledgments:}
We would like to thank Dimitrios Betsakos and Misha Lyubich for their comments and observations which initiated this paper, Phil Rippon and Ian Short for useful discussions, and the referee for helpful comments.
\section{Proof of Proposition~\ref{prop:basics}}
\label{S.basics}
In this section, our goal is to prove Proposition~\ref{prop:basics}. To do this, we first need a little background on inner functions.
It is well-known that it is possible to factorise inner functions in a canonical way. First we define a \emph{singular inner function} as a function of the form
\begin{equation}
\label{eq:singinnerdef}
S(z) \defeq \exp \left(- \int \frac{e^{i\theta} + z}{e^{i\theta} - z} \ d\mu(\theta)\right),
\end{equation}
for some positive and singular measure $\mu$. We then have the following, which is due to Frostman \cite{Frostman}; see also \cite[p.72]{Garnett} together with \cite[Theorem 6.4]{Garnett}.
\begin{theorem}
\label{theo:inner}
If $g : {\mathbb{D}} \to {\mathbb{D}}$ is an inner function, then there is a Blaschke product $B$ and a singular inner function $S$ such that $g = B \cdot S$. Moreover, for all $\zeta \in {\mathbb{D}}$, except possibly for a set of capacity zero, the function
\begin{equation}
\label{eq:gzetadef}
g_{\zeta}(z) \defeq \frac{g(z) - \zeta}{1 - \overline{\zeta}g(z)},
\end{equation}
is a Blaschke product.
\end{theorem}
We will also use the following, which is a version of \cite[Theorem 4']{Heins}.
\begin{theorem}
\label{theo:heins}
Suppose that $f$ is a {transcendental entire function}, that $V$ is a domain, and that $U$ is a component of $f^{-1}(V)$. Then exactly one of the following holds.
\begin{enumerate}[(i)]
\item there exists $n \in {\mathbb{N}}$ such that $f$ assumes in $U$ every value of $V$ exactly $n$ times (counting multiplicities).
\item $f$ assumes in $U$ every value of $V$ infinitely often with at most one exception.
\end{enumerate}
\end{theorem}
We are now able to prove Proposition~\ref{prop:basics}.
\begin{proof}[Proof of Proposition~\ref{prop:basics}]
That $g$ is an inner function was shown in \cite{EFJS}, in a much less general context, and we repeat the argument for completeness. Suppose that $g$ was not inner. By Fatou's Theorem, there would exist a set $E \subset \partial \mathbb{D}$, of positive measure with respect to ${\mathbb{D}}$, on which $g$ had non-tangential limits of modulus strictly less than one, and on which $\phi$ had well-defined limits. It would follow that $$\phi(E\setminus \phi^{-1}(\{\infty\})) \subset \partial U$$ was a set of positive harmonic measure with respect to $U$ that was mapped by $f$ into $V$. This is a contradiction, since $f(\partial U) \subset \partial V$.
To see that $g$ can be taken to be a Blaschke product, choose $\zeta \in {\mathbb{D}}$ such that, by the second part of Theorem~\ref{theo:inner}, the function $g_{\zeta}(z) = \omega(g(z))$ is a Blaschke product, where
\[
\omega(z) \defeq \frac{z - \zeta}{1-\overline{\zeta}z}.
\]
Set $\tilde{\phi} \defeq \phi \circ \omega^{-1}$ and $\tilde{\psi} \defeq \psi \circ \omega^{-1}$, and so
\[
\tilde{g} \defeq \tilde{\psi}^{-1} \circ f \circ \tilde{\phi} = \omega \circ \psi^{-1} \circ f \circ \phi \circ \omega^{-1} = g_{\zeta} \circ \omega^{-1},
\]
is a Blaschke product which is associated to $f$.
If $g$ is a finite Blaschke product, of degree $d$ say, then it is easy to see that each point of ${\mathbb{D}}$ has exactly $d$ preimages up to multiplicity. The statement for $f$ is then immediate, and this gives the case~\ref{theo:a}.
Otherwise, $g$ is an infinite Blaschke product. In particular $g^{-1}(0)$ is infinite. It follows by Theorem~\ref{theo:heins} that $g^{-1}(\zeta)$ is infinite for all $\zeta \in {\mathbb{D}}$ except at most one point.
\end{proof}
\section{Singularities of inner functions}
\label{S.singularities}
The main goal of this section is to prove Theorem \ref{theo:tracts}.
Recall that by an \emph{access to infinity} within a domain $U$ we mean
a homotopy class of curves tending to infinity within $U$. Any
collection of pairwise disjoint curves to infinity comes equipped with
a natural \emph{cyclic order}, which records how these curves are
ordered around $\infty$ according to positive orientation.
If $U$ is simply connected, this in turn gives rise to a natural cyclic order
on accesses to infinity in $U$. Carath\'eodory's theory of \emph{prime ends},
see e.g.\ \cite[Chapter~2]{Pommerenke}, provides a natural
correspondence between accesses to infinity in $U$ and the set of points on
the unit circle where
a Riemann map has radial limit $\infty$. Indeed,
it follows from the definition that accesses to infinity are in one-to-one correspondence
with the prime ends represented by a sequence of cross-cuts tending to infinity,
and therefore the following result follows from
\cite[Corollary~2.17]{Pommerenke}.
Compare \cite[Correspondence Theorem]{Accesses} for details.
\begin{proposition}\label{prop:correspondence}
Let $U\subset{\mathbb{C}}$ be a simply-connected domain, and let $\phi\colon {\mathbb{D}}\to U$
be a conformal isomorphism. Set
\begin{equation}\label{eqn:infinityset}
\Theta \defeq \{\zeta\in S^1\colon \lim_{t\nearrow 1}\phi(t\zeta) = \infty \}
\end{equation}
For $\zeta\in \Theta$, let $\alpha(\zeta)$ denote the access to infinity in $U$
represented by $\phi([0,1)\cdot \zeta)$.
Then $\alpha$ is a cyclic-order-preserving
bijection between $\Theta$ and the set of accesses to infinity in $U$.
Moreover,
if $\gamma\colon [0,\infty)\to U$ is any curve to infinity in $U$ representing
an access $[\gamma]$, then $f^{-1}(\gamma(t))\to \alpha^{-1}([\gamma])$ as
$t\to\infty$.
\end{proposition}
\begin{figure}
\begin{center}
\def.9\textwidth{\textwidth}
\input{singularities-proof-2.pdf_tex}
\end{center}
\caption{\label{fig:singularities}Illustration of the proof of Theorem~\ref{theo:tracts}. Here $f$
is the fourth iterate of the exponential map from Figure~\ref{fig:inner}, and $U=V$ is an invariant Fatou component of $f$, whose
boundary is shown in black.
Observe that the set $\Theta' = \partial W_0\cap S^1$ of singularities of $g$ is infinite. Also note that $U$ has many more
accesses to infinity than $D_0$, so that $\Theta\supsetneq \Theta'$; in fact, $\Theta$ is dense in $S^1$.}
\end{figure}
Observe that Proposition~\ref{prop:correspondence} could also be used as a
definition of accesses to infinity in $U$ and their cyclic order.
The proof of Theorem~\ref{theo:tracts}
uses some ideas that were also used in the proof of \cite[Theorem 1.1]{EFJS};
for the reader's convenience, we shall give a largely self-contained account,
relying only on classical results on the boundary behaviour of
univalent functions.
\begin{proof}[Proof of Theorem~\ref{theo:tracts}]
Recall that $f$ is a {transcendental entire function}, $V \subsetneq {\mathbb{C}}$ is a simply-connected domain, and $U$ is a component of $f^{-1}(V)$ such that $f\colon U\to V$ has infinite valence.
Finally $D$ is a bounded Jordan domain containing $S(f) \cap V$, such that $\overline{D} \subset V$.
Set $D_0 \defeq f^{-1}(D)\cap U$. Then $f\colon U\setminus \overline{D_0}\to V\setminus \overline{D}$ is a covering map. Since $V\setminus \overline{D}$ is an annulus, and the map has infinite
degree, it follows that $f$ is a universal covering when restricted to
any connected component $T$
of $U\setminus \overline{D_0}$, and that consequently
$D_0$ is connected, simply connected
and unbounded. Compare \cite[Proposition~2.9]{BFR} for details.
Let $\mathcal{T}$ denote the set of components of $U\setminus \overline{D_0}$.
With a slight abuse of terminology we call these the \emph{tracts in $U$}.
Let $T\in\mathcal{T}$. By the above,
exactly one of the boundary components of $T$, $\Gamma(T)$ say, is a
preimage of $\partial D$, and so is an arc,
tending to infinity in both directions,
which is mapped by $f$ as an infinite-degree covering.
Now consider Riemann maps $\phi\colon {\mathbb{D}} \to U$ and $\psi\colon{\mathbb{D}} \to V$ and an inner function $g \defeq \psi^{-1} \circ f \circ \phi$ associated to $f|_U$.
Let $\Theta\subset S^1$ be defined as in~\eqref{eqn:infinityset}, and let $\Theta'\subset \Theta$
denote the subset corresponding to accesses to infinity in
$D_0$. Note that, by the F.~and M.~Riesz theorem
\cite[Theorem~1.7]{Pommerenke}, the set $\Theta$ has
zero Lebesgue measure and is therefore totally disconnected.
Let $X\subset S^1$ denote the set of singularities of $g$; note that $X$ is a compact
subset of $S^1$. We wish to show
that $X=\Theta'$, which will be achieved by studying the structure of
\[ W_0\defeq \phi^{-1}(D_0) = g^{-1}(W), \]
where $W=\psi^{-1}(D)$. (Compare Figure~\ref{fig:singularities}.)
The set $\partial W_0\cap {\mathbb{D}} = g^{-1}(\partial W)$ consists of the countably many
curves $\gamma(T) \defeq \phi^{-1}(\partial T)$ for $T\in\mathcal{T}$.
Each $\gamma(T)$ is an arc tending to $S^1$ in both directions.
By Proposition~\ref{prop:correspondence},
$\gamma(T)$ in fact has two end-points $a(T),b(T)\in \Theta'$ on the
unit circle. We may choose the labelling
such that $\gamma(T)$ separates
the arc $I(T)\defeq (a(T),b(T))$ of $S^1$, understood in positive orientation,
from $W_0$. This implies that
\begin{equation}\label{eqn:W0boundary}
\partial W_0 = \bigcup_{T\in\mathcal{T}} \gamma(T) \cup
\left(S^1\setminus \bigcup_{T\in\mathcal{T}} I(T)\right).
\end{equation}
\begin{claim}[Claim~1]
$a(T),b(T)\in X$ for all $T\in\mathcal{T}$.
\end{claim}
\begin{subproof}
The restriction $g\colon \gamma(T)\to \partial W$
is a universal covering. In particular, every point of $\partial W$ has infinitely
many preimages near $a(T)$ and $b(T)$, and these points must be
singularities of $g$.
\end{subproof}
\begin{claim}[Claim~2]
The map $g$ extends continuously to each $I(T)$
as an analytic universal covering $g\colon I(T)\to S^1$. In particular,
$I(T)\cap X = \emptyset$.
\end{claim}
\begin{subproof}
The map $g\colon \phi^{-1}(T)\to {\mathbb{D}}\setminus W$ is a universal covering map.
Since ${\mathbb{D}}\setminus W$ is an annulus, the restriction is
equivalent, up to analytic changes of coordinate in domain and range,
to the restriction of
the complex exponential map to a horizontal strip $S$, with
$a(T)$ and $b(T)$
corresponding to $-\infty$ and $+\infty$ on the boundary of $S$.
The conformal isomorphism between $\phi^{-1}(T)$ and $S$ extends
continuously to $I(T)$, and thus $g$ extends continuously to $I(T)$
as a universal covering of $S^1$. By the Schwarz
reflection principle, the extension is analytic.
\end{subproof}
\begin{claim}[Claim~3]
$W_0$ is a Jordan domain, and $\Theta' = \partial W_0\cap S^1=S^1\setminus\bigcup_T I(T)$.
\end{claim}
\begin{subproof}
Recall that the second equality holds by~\eqref{eqn:W0boundary}.
Define $\rho\colon S^1\to \partial W_0$ as follows. On each
$I(T)$, the map is a homeomorphism
$\rho\colon I(T) \to \gamma(T)$, fixing the endpoints $a(T)$ and $b(T)$.
Outside these intervals, i.e.\
on $\partial W_0\cap S^1$, $\rho$ agrees with the identity.
(Note that, by Claim~2, each $I(T)$ is
a non-degenerate interval, so such $\rho$ does indeed exist.)
Clearly $\rho$ is injective; it is surjective
by~\eqref{eqn:W0boundary}.
By definition, the restriction $\rho|_{\overline{I(T)}}$ is continuous for any $T\in\mathcal{T}$, as is $\rho|_{\partial W_0\cap S^1}$. So let
$(T_n)_{n=1}^{\infty}$ be a sequence of pairwise different elements of $\mathcal{T}$, and suppose that
$\zeta_n\in I(T_n)$ is a sequence converging to $\zeta \in S^1$. To establish continuity of $\rho$, we must prove that
$\rho(\zeta_n)\to \zeta$.
Since the $I(T_n)$ are pairwise disjoint, their lengths tend to zero and $a(T_n)\to \zeta$.
Let $\varepsilon>0$. By the Carath\'eodory prime end correspondence,
there is a cross-cut $C$ of ${\mathbb{D}}$, contained in the Euclidean disc
$D(\zeta,\varepsilon)$ of radius $\varepsilon$
around $\zeta$, such that $C$ separates $\zeta$ from $0$ and $\phi(C)$
is a cross-cut of $U$ with finite endpoints
\cite[Theorem~2.15]{Pommerenke}.
Since $\partial D_0\cap{\mathbb{C}}$ is locally an arc, and $\phi(C)$ is bounded,
$\partial T$ intersects $\phi(C)$
only for finitely many $T$. (Compare \cite[Lemma~2.1]{dreadlocks}.) So, for sufficiently large $n$, $\gamma(T_n)$ is disjoint from $C$, and
$C$ separates $a(T_n)$ from $\partial D(\zeta,\varepsilon)$ in ${\mathbb{D}}$. So $\rho(\zeta_n)\in \gamma(T_n) \subset D(\zeta,\varepsilon)$. We have proved that
$\rho(\zeta_n)\to \zeta$, and hence that $\rho$ is continuous. Thus
$\rho(S^1)=\partial W_0$ is a Jordan curve.
Clearly no $I(T)$ intersects $\Theta'$, and hence $\Theta'\subset \partial W_0\cap S^1$. On the
other hand, every $\zeta\in \partial W_0\cap S^1$ is accessible from
$W_0$ by the first part of the claim. If $\gamma \subset W_0$ is a curve tending
to $\zeta$, then $\phi(\zeta)$ must tend to
$\hat{\partial} U \cap \hat{\partial} D_0 = \{\infty\}$
(where $\hat{\partial}$ denotes
the boundary in the Riemann sphere $\hat{{\mathbb{C}}}$). Hence $\zeta\in \Theta'$
by Proposition~\ref{prop:correspondence}, as
required.
\end{subproof}
Since $\Theta'\subset \Theta$ is totally disconnected, it follows from Claim~3
that the set $\bigcup_T \{a(T),b(T)\}$ of endpoints of the intervals
$I(T)$ is dense in $\Theta'$. In particular, by Claim~1,
$\Theta'\subset X$, and $X\subset \Theta'$ by Claims~2 and~3.
We have established that $X=\Theta'$.
By Proposition~\ref{prop:correspondence}, the set
$\Theta'$ is in one-to-one cyclic-order-preserving correspondence with
the set of accesses to infinity in $D_0$. Thus we have proved the first
claim of the theorem. Moreover, clearly $\Theta' = S^1\setminus \bigcup_T I(T)$ is
finite if and only if $\mathcal{T}$ is finite. If this is the case,
the map $T\mapsto a(T)$ defines a bijection between $\mathcal{T}$ and
$\Theta'$. This completes the proof.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{corr:tracts}]
Suppose that $f \in \mathcal B$, that ${S(f)} \subset F(f)$, and that $U$ is an unbounded forward-invariant Fatou component of $f$. Choose a point $w \in U$, and let $D \subset U$ be a hyperbolic disc, centred at $w$, of sufficiently large hyperbolic radius that $S(f) \cap U \subset D$; this is possible since $S(f)$ is compact and does not meet $\partial U$.
It follows from Theorem~\ref{theo:tracts} that the number of singularities of an associated inner function is equal to the number of components of $U \setminus f^{-1}(\overline{D})$. Since $\overline{D} \subset U$, each component of $U \setminus f^{-1}(\overline{D})$ is contained in exactly one component of ${\mathbb{C}} \setminus (U \cap f^{-1}(\overline{D}))$, so it suffices to count the components of this latter set.
Now let $D' \supset D$ be a bounded Jordan domain containing $S(f)$, so that the tracts of $f$ are the components of ${\mathbb{C}} \setminus f^{-1}(\overline{D'})$. Since $D' \supset D$, no tract can meet more than one component of ${\mathbb{C}} \setminus (U \cap f^{-1}(\overline{D}))$. However, each component of ${\mathbb{C}} \setminus (U \cap f^{-1}(\overline{D}))$ meets at least one tract. This completes the proof.
\end{proof}
\section{Bounded-degree inner functions}
In this section, we prove Theorems~\ref{theo:finiteblaschke} and~\ref{theo:wandering}. The results are proved by a standard type of
\emph{quasiconformal surgery} \cite[Section~4.2]{brannerfagellasurgery},
which is analogous to the well-known proof of the straightening theorem
for polynomial-like mappings \cite{douadyhubbardpolynomiallike}.
Throughout this section, and only in this section, we shall use
without comment the standard notions and techniques of
quasiconformal surgery, as explained for example in~\cite{brannerfagellasurgery}.
We begin with Theorem~\ref{theo:wandering}, where the
surgery takes a particularly simple form. We shall require the following result,
which establishes the existence of a suitable function on which to perform the surgery.
\begin{figure}
\subfloat[$f(z)=-z^2 \cdot \exp\bigl(p(z)-c\bigr)$]{\fbox{\def.9\textwidth{.47\textwidth}%
\input{f-deg2-hyperbolic.pdf_tex}}}\hfill
\subfloat[$h(w) = 2w + p(\exp(w)) - c + \pi i$]{\fbox{\def.9\textwidth{.47\textwidth}%
\input{f-deg2-wandering.pdf_tex}}}
\caption{\label{fig:wandering}The function $f$ defined in~\eqref{eqn:Cstarmap} and its lift $h$,
illustrating Proposition~\ref{prop:wandering} in the case $d=2$. The Julia sets are shown in black and the basin of $-1$,
respectively its preimage under the exponential map, is shown in grey. Critical points are marked by asterisks.}
\end{figure}
\begin{proposition}\label{prop:wandering}
Let $d\geq 2$. Then there exists an entire function $h$ having a simply-connected wandering domain $W$ such that, for all $n\geq 0$,
$h^n(W)$ is a Jordan domain and
$h\colon h^n(W)\to h^{n+1}(W)$ is a proper map of degree $d$.
\end{proposition}
\begin{proof}
We use the well-known
method of obtaining wandering domains by lifting invariant components
of a self-map of ${\mathbb{C}}^* = {\mathbb{C}}\setminus\{0\}$. Compare \cite[p.~106]{hermanexemples}
and \cite[p.~414]{sullivanqcdynamicI} for early examples,
and \cite[p.~3]{classifyingwandering} for a general description of this method.
Define
\begin{align}
p(z) &\defeq 2\cdot \sum_{j=1}^{d-1} \binom{d-1}{j} \cdot \frac{z^{j}}{j} \quad\text{and}\notag \\
f(z) &\defeq -z^2 \cdot \exp\bigl(p(z)-c\bigr), \label{eqn:Cstarmap} \end{align}
where $c\defeq p(-1)$.
Then $f(0)=0$, $f(-1)=-1$, and
\begin{align*} f'(z) &= -\exp\bigl(p(z)-c\bigr)\cdot (2z + z^2p'(z)) \\ &=
-\exp\bigl(p(z)-c\bigr)\cdot 2z\cdot \left(1 + \sum_{j=1}^{d-1} \binom{d-1}{j} \cdot z^j\right) \\ &=
- \exp\bigl(p(z)-c\bigr)\cdot 2z\cdot(z+1)^{d-1}.\end{align*}
Hence $f$ has
super-attracting fixed points at $0$ and $-1$, and no other critical points. The super-attracting fixed point $0$ is also the
only asymptotic value of $f$.
Let $U$ be the Fatou component of $f$ containing $-1$. Since $U$ is a super-attracting basin containing no singular values
other than the fixed point, it follows that
$f\colon U\to U$ has degree $d$.
By~\cite[Theorem~1.10]{BFR}, $U$ is
a bounded Jordan domain (in fact, a quasidisc). See Figure~\ref{fig:wandering}.
Observe that $f^{-1}(0)=\{0\}$. Let $h$ be the lift of $f$ under $z=\exp(w)$ defined by
\[ h(w) \defeq 2w + p(\exp(w)) - c + \pi i. \]
Observe that $h\bigl((2k-1)\pi i\bigr) = (4k-1)\pi i$ for $k\in{\mathbb{Z}}$.
Let $W_k$ be the connected component of $\exp^{-1}(U)$
containing $(2^k-1)\pi i$. By \cite{waltersemiconjugacy}, we have
$\exp(J(h)) = J(f)$, so $W_k$ is a Fatou component of $h$, with
$h(W_k)\subset W_{k+1}$.
Since $U$ is a Jordan domain and is
mapped to itself as a proper map of degree $d$, it follows that
$W=W_1$ has the desired properties.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:wandering}]
Let $g$ be a finite Blaschke product of degree $d\geq 2$, and let $h$ and $W$ be as in
Proposition~\ref{prop:wandering}. Let $\tilde{g}$ be the Blaschke product
associated to $h\colon W\to f(W)$, say $\tilde{g} = \phi_1\circ h \circ \phi_0^{-1}$ with Riemann maps $\phi_0\colon W\to{\mathbb{D}}$ and $\phi_1\colon f(W)\to {\mathbb{D}}$.
Restricted to $S^1$, both $g$ and $\tilde{g}$ are analytic covering maps of degree $d$, and therefore there is an analytic map
$\theta\colon S^1\to S^1$ such that
\begin{equation}\label{eqn:gfunctionalrelation} g\circ \theta = \tilde{g}.
\end{equation} In particular, $\theta$ is quasisymmetric, and can therefore be extended to a quasiconformal
homeomorphism $\theta\colon {\mathbb{D}}\to{\mathbb{D}}$. Define a quasiregular map $\tilde{f}\colon{\mathbb{C}}\to{\mathbb{C}}$ by
\[ \tilde{f}(z) \defeq \begin{cases} h(z) & \text{if } z\notin W \\
\phi_1^{-1}(g(\theta(\phi_0(z)))) &\text{if }z\in W. \end{cases}. \]
Since $\partial W$ and $\partial f(W)$ are Jordan curves,
the maps $\phi_0$ and $\phi_1$ extend homeomorphically to the
boundary. By~\eqref{eqn:gfunctionalrelation}, $\tilde{f}$ is continuous at points
of $\partial W$. By the Bers gluing lemma, it follows that
$\tilde{f}$ is indeed quasiregular
on ${\mathbb{C}}$.
Let $\mu$ be the dilatation of
$\theta\circ\phi_0$, which is a Beltrami differential
on $U$. Extend $\mu$ to ${\mathbb{C}}$ as follows:
If $V$ is a component of the Fatou set of $h$ such that
$h^n(V)=W$ for $n>0$, let
$\mu|_V$ be the pull-back of $\mu|_{W}$ under $h^n$. On the
complement of the backward orbit of $W$, set $\mu=0$. Then $\mu$ is
invariant under $\tilde{f}$.
Apply the measurable Riemann mapping theorem to obtain
a quasiconformal map $\psi\colon{\mathbb{C}}\to{\mathbb{C}}$ whose dilatation is
$\mu$; then
\[ f\defeq \psi\circ \tilde{f}\circ \psi^{-1} \]
is an entire function. Moreover, consider $U\defeq \psi(W)$ and $V \defeq \psi(h(W))=f(U)$. Then
\begin{align*} \Phi_0 &\defeq \theta\circ \phi_0\circ \psi^{-1}
\colon W \to {\mathbb{D}} \qquad\text{and} \\
\Phi_1 &\defeq \phi_1\circ \psi^{-1} \colon V\to {\mathbb{D}} \end{align*}
are conformal isomorphisms and
\[ \Phi_1 \circ f = \phi_1 \circ \tilde{f}\circ \psi^{-1} = g \circ \theta \circ \phi_0 \circ \psi^{-1} = g \circ \Phi_0 \]
on $W$.
So $g$ is an associated inner function of $f\colon U\to V$.
\end{proof}
The proof of Theorem~\ref{theo:finiteblaschke} is similar. As with
Proposition~\ref{prop:wandering}, we should first establish
the existence of
suitable subjects for our surgery. (See Figure~\ref{fig:invariantfatou}.)
\begin{figure}
\subfloat[$\alpha_2$]{\fbox{\def.9\textwidth{.47\textwidth}%
\input{alpha2-hyperbolic.pdf_tex}}}\hfill
\subfloat[$\rho_2$]{\fbox{\def.9\textwidth{.47\textwidth}%
\input{rho2-parabolic.pdf_tex}}}
\caption{\label{fig:invariantfatou}The functions $\alpha_2$ and $\rho_2$ from Proposition~\ref{prop:invariantfatouexistence}. Julia sets are shown in black, and the critical points $0$ and $1$ are marked by
asterisks. The attracting basin of $0$ for $\alpha_2$ and the parabolic basin of $\rho_2$ are shown in grey.}
\end{figure}
\begin{proposition}\label{prop:invariantfatouexistence}
For every $d\geq 2$, there is an entire function $\alpha_d$
having an invariant super-attracting Fatou component $W$ which
is a bounded Jordan domain, and such that $f\colon W\to W$ is a
proper map of degree $d$.
Similarly, there is an entire function $\rho_d$ having an invariant parabolic Fatou
component $W$ which
is a bounded Jordan domain, and such that $f\colon W\to W$ is a
proper map of degree $d$.
\end{proposition}
\begin{proof}
A function $f$ with the properties required of $\alpha_d$ was already
described in~\eqref{eqn:Cstarmap}, but since we do not require
$\alpha_d$ to restrict to a self-map of ${\mathbb{C}}\setminus\{0\}$ here,
we can also give simpler formulae, such as
\[ \alpha_d\colon {\mathbb{C}}\to{\mathbb{C}}; \qquad z\mapsto \left(\frac{1-\cos(\pi\sqrt{z})}{2}\right)^d. \]
Then $S(\alpha_d)=\{0,1\}$, both $0$ and $1$ are super-attracting fixed points, and
$0$ is a degree $d$ critical point of $\alpha_d$.
Let $W$ be the connected component of $F(\alpha_d)$ containing $0$.
Then $W$ is simply connected, and
$\alpha_d\colon W\to W$ is a branched covering branched only over $0$.
Since both singular values of $\alpha_d$ belong to (super-)attracting basins,
the map $\alpha_d$ is hyperbolic in the sense of \cite{BFR}.
Again applying
\cite[Theorem~1.10]{BFR}, $U_0$ is a quasidisc.
For $\lambda>0$, define
\begin{equation}\label{eqn:flambda} f_{\lambda}(z) \defeq \frac{\alpha_d(z)+\lambda}{1+\lambda}.\end{equation}
Then $0$ and $1$ are still critical points of $f_{\lambda}$, with
critical values $\lambda/(1+\lambda)$ and $1$. Moreover,
$f_{\lambda}$ is increasing on $[0,1]$.
It is easy to see that there
is $\lambda_0>0$ such that the orbit of $0$ converges to an attracting
fixed point for $\lambda<\lambda_0$, to $1$ for $\lambda>\lambda_0$,
and to a parabolic fixed point for $\lambda=\lambda_0$. (See Figure~\ref{fig:bifurcation-flambda}.) Set
$\rho_d\defeq f_{\lambda_0}$, and let $W$ be the Fatou component
containing $0$. Then $\rho_d\colon W\to W$ is a degree $d$ proper map.
Moreover, since all singular values belong to attracting or parabolic basins,
$\rho_d$ is \emph{strongly geometrically finite} in the sense of
\cite{geometricallyfinite}. By \cite[Theorem~1.8]{geometricallyfinite},
$W$ is again a bounded Jordan domain.
\end{proof}
\begin{figure}
\subfloat[$\lambda = 0$]{\includegraphics[width=.3\textwidth]{alpha-hyperbolic-function-innerfunctions}}\hfill%
\subfloat[$\lambda = \lambda_0$]{\includegraphics[width=.3\textwidth]{parabolic-function-innerfunctions}}\hfill%
\subfloat[$\lambda < \lambda_0$]{\includegraphics[width=.3\textwidth]{flambda-after-bifurcation-innerfunctions}}
\caption{\label{fig:bifurcation-flambda}The graph of $f_{\lambda}$, as defined in~\eqref{eqn:flambda}, in the case $d=2$. We have $f_0 = \alpha_2$ and $f_{\lambda_0} = \rho_2$. Here
$\lambda_0\approx 0.0548$.}
\end{figure}
We now divide the connectivity locus $\mathcal{G}$ of finite Blaschke products
from Theorem~\ref{theo:finiteblaschke}
into subclasses as follows. For $d\geq 2$, denote by
$\mathcal{A}_{d}$ the Blaschke products of degree $d$ having an attracting
fixed point in ${\mathbb{D}}$, and by $\mathcal{P}_d$ those having a triple fixed point
on $S^1$. The elements of $\mathcal{A}_d$ are called
\emph{elliptic} Blaschke products, while those of $\mathcal{P}_d$ are said to be
parabolic with
\emph{zero hyperbolic step}; see \cite{fletcherblaschke}. Then
$\mathcal{G}=\bigcup_k \mathcal{A}_d\cup \mathcal{P}_d$.
Our goal is to extend the proof of Theorem~\ref{theo:wandering} to the case
of invariant attracting or parabolic components. To do so, we need to be able to
replace the (non-dynamical) quasisymmetric map $\theta$ by a conjugacy between
the two Blaschke products in question. This is possible by the following result.
(See also
\cite[Theorem~A]{clarkvanstrienrigidity} for a much more general, and extremely
deep result.)
\begin{proposition}\label{prop:conjugacy}
Let $d\geq 2$. Any two elements of $\mathcal{A}_d$ are quasisymmetrically
conjugate on $S^1$, and any two elements of $\mathcal{P}_d$ are quasisymmetrically
conjugate on $S^1$.
\end{proposition}
\begin{proof}
The maps in $\mathcal{A}_d$ are hyperbolic in the sense of rational dynamics,
and hence expanding on their
Julia sets. The result is well-known in this case; see \cite[Exercise~2.3 in Chapter~II]{demelovanstrien},
and compare \cite{petersenblaschke} for a more general theorem. See also
\cite[Section~4.2]{brannerfagellasurgery}.
For $\mathcal{P}_d$, the result
follows from \cite[Proposition~2.3]{lomonacopetersenshen}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:finiteblaschke}]
Let $g\in\mathcal{G}$, say of degree $d\geq 2$. If $g\in \mathcal{A}_d$, set
$h \defeq \alpha_d$ from Proposition~\ref{prop:invariantfatouexistence};
if $g\in \mathcal{P}_d$, set $h\defeq \rho_d$. Let
$W$ be the corresponding invariant Fatou component, let
$\phi\colon W\to {\mathbb{D}}$ be a Riemann map, and set $\tilde{g}\defeq \phi\circ h\circ \phi^{-1}$.
Then by Proposition~\ref{prop:conjugacy}, there is a quasisymmetric
homeomorphism $\theta\colon S^1\to S^1$ such that
$g\circ \theta = \theta\circ\tilde{g}$. Extend $\theta$ to a quasiconformal
map $\theta\colon {\mathbb{D}}\to{\mathbb{D}}$ and define
\[ \tilde{f}\colon {\mathbb{C}}\to{\mathbb{C}};\qquad z\mapsto \begin{cases} h(z) & \text{if } z\notin W \\
\phi^{-1}\bigl(\theta^{-1}\bigl(g(\theta(\phi(z)))\bigr)\bigr) &\text{if }z\in W. \end{cases}. \]
The argument now proceeds
exactly as in the proof of Theorem~\ref{theo:wandering}. The function
$\tilde{f}$ is
quasisymmetric. Straightening an invariant Beltrami differential
that extends the complex dilatation of $\theta\circ\phi$, we obtain
an entire function for which $g$ is a dynamically associated inner function.
\end{proof}
\begin{remarks}\mbox{}
\begin{enumerate}[(a)]
\item To carry out the surgery, we could have started with any function
$h$ having an invariant attracting or parabolic component $W$ of the
required degree $d$. (For simplicity, our proof used the fact that $W$ is
a Jordan domain, but it is easy to see that this is not essential.) In particular,
let $\tilde{g}$ be the dynamically associated inner function, and let
$g$ be a Blaschke product of the same degree and type (i.e., elliptic or parabolic
with zero hyperbolic step) as $\tilde{g}$. If additionally $g$ and $\tilde{g}$ are
\emph{quasiconformally equivalent}, i.e.\ differ only by pre- and post-composition with
quasiconformal
homeomorphisms of the disc, then there is an entire function $f$ quasiconformally
equivalent to $h$ for which $g$ is a dynamically associated inner function.
As mentioned in the introduction, it seems that a similar result holds for functions
with finitely many singular values~-- or, more generally, Fatou components $W$
for which $W\cap S(f)$ is compact. We shall leave this question for discussion in
a future paper.
\item Observe that we could also have deduced
Theorem~\ref{theo:wandering} from
(the proof of) Theorem~\ref{theo:finiteblaschke}, applying the surgery
for attracting basins
to the function $f$
defined in~\ref{eqn:Cstarmap}.
From this, we obtain another self-map of ${\mathbb{C}}\setminus\{0\}$ realising a desired
Blaschke product. Taking a lift of this second function, we obtain a wandering
domain with the desired property.
\item We have restricted to finite-valence
attracting and parabolic Fatou components,
where we obtained a complete description of the associated Blaschke
products. However, let us briefly comment
on the case of a finite-valence
\emph{Baker domain}, i.e.\ an invariant component of
the Fatou set on which
the iterates converge locally uniformly to infinity. Such a domain is
called \emph{hyperbolic}, \emph{simply parabolic} or \emph{doubly parabolic}, depending on whether the Denjoy-Wolff point of the
associated Blaschke product is attracting, a double fixed point, or
a triple fixed point. Compare \cite{fagella-henriksen}.
Doubly parabolic examples of every finite
degree $d$ exist~\cite[Section~4]{fagella-henriksen}. The corresponding
inner function belongs to $\mathcal{P}_d$, and we can apply our
surgery to see that every element of $\mathcal{P}_d$ is realised as
a dynamically associated inner function of an entire function with a
Baker domain.
An analogue of
Proposition~\ref{prop:conjugacy} also holds for Blaschke products
with $J(f)\neq S^1$
(again, this is a simple special
case of~\cite[Theorem~A]{clarkvanstrienrigidity}). Therefore, starting
with a function having a hyperbolic or simply-parabolic Baker domain
of finite degree, we can apply the same surgery technique. However,
as far as we are aware, the only known hyperbolic
and simply-parabolic Baker domains of finite valence
are univalent. Hence we cannot presently
answer the question which finite Blaschke products
arise as dynamically associated inner functions of entire functions with
Baker domains.
\end{enumerate}
\end{remarks}
\section{Inner functions of exponential maps}\label{sec:expo}
We begin with the following well-known observation concerning
exponential maps $f_{\lambda}(z)=\lambda e^z$.
\begin{lemma}[{\cite[Lemma~1.1]{DevandG}}]\label{lem:attractingexp}
$f_{\lambda}$ has a fixed point of multiplier $\tau\in{\mathbb{C}}\setminus\{0\}$
if and only if $\lambda = \tau\cdot e^{-\tau}$.
\end{lemma}
\begin{proof}
If $z$ is a fixed point of multiplier $\tau$, then
$z=f_{\lambda}(z) =f_{\lambda}'(z)=\tau$. So
$\tau = \lambda e^{\tau}$, as claimed.
\end{proof}
The following shows that unisingular inner functions with an attracting
fixed point are determined by their degree and their multiplier.
\begin{lemma}\label{lem:unisingular}
Let $\tau\in{\mathbb{D}}\setminus\{0\}$, and let $2\leq d\leq \infty$. Then, up to conjugacy by a M\"obius automorphism of ${\mathbb{D}}$, there exists a unique inner function
$g\colon{\mathbb{D}}\to{\mathbb{D}}$ of degree $d$ such that $g$ has an attracting fixed point of multiplier $\tau$ in ${\mathbb{D}}$, and such that $g$ has only one singular value in ${\mathbb{D}}$.
\end{lemma}
\begin{proof}
To prove existence, it is enough to exhibit the existence of a polynomial
or entire function
having an attracting fixed point of multiplier $\tau$, and having only
one singular value. Indeed, then the dynamically associated inner function
is of the stated form.
For $d<\infty$, such a function is given by the polynomial
\[ z\mapsto \frac{\tau}{d} \cdot ( (z+1)^d-1); \]
for $d=\infty$ the function $f_{\tau\cdot e^{-\tau}}$ has the
desired properties by Lemma~\ref{lem:attractingexp}.
So it remains to prove uniqueness. Suppose that $g$ and $\tilde{g}$ are both functions with the stated properties. Let $z_0$ and $\tilde{z}_0$ be the corresponding fixed points,
and $s$ and $\tilde{s}$ the singular values. By the K{\oe}nigs linearisation theorem, there is a simply-connected domain $U_0\supset \{z_0,s\}$ and a
conformal isomorphism
\[ \phi \colon U_0 \to B(0,\lvert\tau\rvert^{-1}) \]
such that $\phi(g(z)) = \tau \phi(z)$ and $\phi(s)=1$. An analogous function $\tilde{\phi}$ on a domain $\tilde{U}_0$ exists also for $\tilde{g}$. Set
\[ \psi_0\defeq \tilde{\phi}^{-1}\circ \phi \colon U_0 \to \tilde{U}_0. \]
Then $\psi_0$ conjugates $g$ to $\tilde{g}$ on $U_0$, with $\psi_0(z_0)=\tilde{z}_0$ and $\psi_0(s) =\tilde{s}$.
Now set $U_1\defeq g^{-1}(U_0) \supset U_0$. Then $g\colon U_1\to U_0$ is either a branched covering of degree $d$, branched only over $s$ (if $d<\infty$), or
a universal covering (otherwise); see \cite[Proposition 2.8]{BFR}.
The same is true for $\tilde{g}$. It follows that we can lift $\psi_0$ to a map
$\psi_1\colon U_1\to \tilde{U}_1$ such that $\psi_0(g(z)) = \tilde{g}(\psi_1(z))$ and
$\psi_1(z_0)=\tilde{z}_0$. We have $\psi_1(z)=\psi_0(z)$ near $z_0$, and hence by the identity theorem on all of $U_0$. In particular,
$\psi_1(s)=\tilde{s}$, and we can continue inductively.
In this manner, we obtain a conformal conjugacy $\psi$ between $g$ and $\tilde{g}$ on
\[ \bigcup g^{-n}(U_0) = {\mathbb{D}}. \]
In other words, $g$ and $\tilde{g}$ are M\"obius conjugate, as claimed.
\end{proof}
We now study the family of maps $g_{a,b}$ as in~\eqref{eqn:tangent}. (See Figure~\ref{fig:tan_inner}.)
\begin{figure}
\begin{center}
\def.9\textwidth{.9\textwidth}
\input{tangent_inner_labelled.pdf_tex}
\end{center}
\caption{\label{fig:tan_inner}The family $g_{a,b}$, for $-\pi/2<b<\pi/2$ and $0<a<\pi$. For parameters in the grey region,
$g_{a,b}$ has an attracting fixed point in ${\mathbb{H}}$. The curve
$\lvert b\rvert = \arccos(\sqrt{a})-\sqrt{a}\cdot \sqrt{1-a}$ is shown in black. Note that $g_{a,b}$ has a multiple
fixed point in ${\mathbb{R}}$ for each parameter on this curve, but only $\tan = g_{1,0}$ has a triple fixed point.
The strong dotted lines are curves of fixed argument for the
multiplier of the attracting fixed point of $g_{a,b}$.}
\end{figure}
\begin{proposition}\label{prop:G}
No two different maps $g_{a,b}$ are conformally conjugate. Moreover, $g_{a,b}$ has an attracting fixed point in ${\mathbb{H}}$ if and only if $a>1$ or $a\leq1$ and
$\lvert b\rvert > \arccos(\sqrt{a})-\sqrt{a}\cdot \sqrt{1-a}$.
\end{proposition}
\begin{proof}
If $g_{a,b}$ and $g_{\tilde{a},\tilde{b}}$ are conformally conjugate, then the conjugacy $\psi$ must preserve the set
$g_{a,b}^{-1}(\infty) = g_{\tilde{a},\tilde{b}}^{-1}(\infty)$, which consists of the odd multiples of $\pi/2$. So $\psi$
is a translation by an integer multiple of $\pi$. Since it must
also map singular values to singular values, we have $\psi(ai+b) = \tilde{a}i + \tilde{b}$. So $a=\tilde{a}$ and $b-\tilde{b}\in \pi {\mathbb{Z}}$. As $b,\tilde{b} \in (-\pi/2,\pi/2]$, we see that
$ b = \tilde{b}$ as required.
Recall that, by the Denjoy-Wolff theorem, for every $a$ and $b$
there is a point $\zeta_0\in\overline{{\mathbb{H}}}\cup\{\infty\}$ such that
$g_{a,b}^n\to \zeta_0$ locally uniformly on ${\mathbb{H}}$. We claim that
$\zeta_0\neq \infty$. Indeed, recall that $g_{a,b}$ is $\pi$-periodic,
and that $g_{a,b}(z)\to b + ai\in{\mathbb{H}}$ as $\operatorname{Im} z\to +\infty$.
Hence, if $z_n\defeq g_{a,b}^n(z_0)\to\infty$ for some
$z_0\in\overline{{\mathbb{H}}}$, we must have $\operatorname{Im} z_n\to 0$.
On the other hand, within any horizontal strip of bounded height we have
\[ \lvert \tan'(z)\rvert = \frac{1}{\lvert \cos(z)\rvert^2} \geq
C\cdot \lvert \tan(z)\rvert^2 \]
for some constant $C$. So, in particular,
$\lvert g_{a,b}'(z_n)\rvert \to \infty$. It follows that
\[ \operatorname{Im} z_{n+1} = \operatorname{dist}(z_{n+1},{\mathbb{R}}) \geq \operatorname{dist}(z_n,{\mathbb{R}}) = \operatorname{Im} z_n \]
for sufficiently large $n$. Since $\operatorname{Im} z_n\to 0$ and ${\mathbb{R}}$ is completely invariant,
this is possible only if $z_0\in{\mathbb{R}}$.
Hence $\infty$ cannot be the
Denjoy-Wolff point of $g_{a,b}$.
In particular,
$g_{a,b}$ has an attracting fixed point in ${\mathbb{H}}$ if and only if it does not
have an attracting or parabolic fixed point in ${\mathbb{R}}$.
Now, if $g_{a,b}$ has a fixed point of multiplier $\tau>0$ at $\alpha\in{\mathbb{R}}$, then $\alpha$ is not an odd multiple of $\pi/2$, and
\[ \tau = g_{a,b}'(\alpha) = \frac{a}{(\cos \alpha)^2}. \]
So $a=a_{\tau}(\alpha) = \tau\cdot (\cos \alpha)^2$, and in particular $a\leq \tau$. But also
\[ \alpha = g_{a,b}(\alpha) = \tau \cdot (\cos \alpha)^2\cdot \tan \alpha+b= \tau\cdot \cos \alpha\cdot \sin\alpha + b, \]
so
\[ b = b_{\tau}(\alpha) = \alpha - \tau\cdot \cos\alpha \cdot \sin\alpha. \]
Note that $a(\alpha)$ is a strictly decreasing function of $\alpha$ on $[0,\pi/2)$, and it is an easy exercise to see that, for $\tau\leq 1$,
$b_{\tau}(\alpha)$ is a strictly increasing function of $\alpha$.
To prove the claim, let us restrict to the case $b\in [0,\pi/2]$, which we can do by symmetry. If $g_{a,b}$ has a parabolic fixed point $\alpha$
in ${\mathbb{R}}$, then $a \leq 1$ and $\alpha\in[0,\pi/2]$. Therefore
\[ b = \alpha - \cos \alpha \cdot \sin\alpha = \arccos\sqrt{a} - \sqrt{a}\cdot \sqrt{1-a} =: \theta(a). \]
Moreover, for fixed $\alpha$, $a_{\tau}(\alpha)$ is an increasing function of $\tau$, while $b_{\tau}(\alpha)$ is a decreasing function of $\tau$.
It follows that $g_{a,b}$ has an attracting fixed point in ${\mathbb{R}}$ if and only if $a<1$ and $b<\theta(a)$, as claimed.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:unisingular}]
Let $f$ and $U$ be as in the theorem, and let $g\colon{\mathbb{H}}\to{\mathbb{H}}$ be an inner function dynamically associated to $f^n$ on $U$.
Then $g$ has an attracting fixed point in ${\mathbb{H}}$, and a single singular value in ${\mathbb{H}}$, which we may assume to be placed at $i$. Then
$g\colon {\mathbb{H}}\to {\mathbb{H}}\setminus \{i\}$ is a universal covering map. So is $\tan$, and the two maps agree up to pre-composition by a M\"obius transformation of the half-plane.
Applying a suitable M\"obius conjugacy to $g$, we see that it can be chosen of the form $g_{a,b}\in\mathcal{G}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:exp}]
The characterisation of $\mathcal{G}$ is in Proposition~\ref{prop:G}, and
claim~\ref{p1} holds by assumption. If $f\in\mathcal{F}$, then by Theorem~\ref{theo:unisingular}, there is an inner function
$g\in\mathcal{G}$ dynamically associated to $f$, and this function is unique by the first part of Proposition~\ref{prop:G}.
Finally, let $g\in\mathcal{G}$ have an attracting fixed point of multiplier $\tau\in{\mathbb{D}}\setminus\{0\}$ in ${\mathbb{D}}$. There is a unique $\lambda\in{\mathbb{C}}\setminus\{0\}$ such that
$f_{\lambda}$ has a fixed point
of multiplier $\tau$, namely $\lambda = \tau\cdot e^{-\tau}$. As we have just proved, there is a dynamically associated inner function $g_{a,b}\in\mathcal{G}$, and by
Lemma~\ref{lem:unisingular} and the first part of Proposition~\ref{prop:G}, we have $g=g_{a,b}$, as required.
\end{proof}
For the parabolic case, we use the following version of Lemma~\ref{lem:unisingular}.
\begin{lemma}\label{lem:unisingularparabolic}
Up to conformal conjugacy, $\tan\colon{\mathbb{H}}\to{\mathbb{H}}$ is the only inner
function of infinite valence
that has a unique singular value in ${\mathbb{H}}$ and that has
a fixed point of multiplicity $3$ in ${\mathbb{R}}$.
\end{lemma}
\begin{proof}
The proof is similar to Lemma~\ref{lem:unisingular}: Given two
functions $g$ and $\tilde{g}$ with the stated properties, we can use
Fatou coordinates to construct petals $U_0,\tilde{U}_0\subset {\mathbb{H}}$ of
$g$ and $\tilde{g}$ and a conformal isomorphism
$\psi_0\colon U_0\to\tilde{U}_0$ with $\psi_0\circ g = \tilde{g}\circ \psi_0$,
and such that $\psi_0(s) = \tilde{s}$. (Here $s$ and $\tilde{s}$ are
again the singular values of $g$ and $\tilde{g}$.)
We can again lift $\psi_0$ to a map $\psi_1$ on $g^{-1}(U_0)$,
chosen such that $\psi_1(g(s))=\tilde{g}(\tilde{s})$, and it follows as
before that $\psi_0 = \psi_1$ on $U_0$. The proof now proceeds as before,
and we conclude that $g$ and $\tilde{g}$ are M\"obius conjugate.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:unisingularparabolic}]
Suppose that $f$ and $U$ satisfy the hypotheses of the theorem.
Let $\phi\colon U\to{\mathbb{H}}$ be a conformal isomorphism, and let
$g \defeq \phi\circ f^n\circ \phi^{-1}$ be the dynamically associated inner
function, where $n$ is the period of the
parabolic Fatou component $U$.
Recall that $g$ is of infinite valence and
has only one singular value $\alpha\in{\mathbb{H}}$.
Consequently $g\colon {\mathbb{H}}\to {\mathbb{H}}\setminus\{\alpha\}$ is a universal
covering map. Applying a suitable M\"obius conjugacy,
$g$ can be taken of the form $g=g_{a,b}$
for unique choices of $a>0$ and $b\in (-\pi/2,\pi/2]$.
Since $f^n$ has no fixed point in $U$, the inner function
$g$ has no fixed point in ${\mathbb{H}}$, so its Denjoy-Wolff point $\zeta_0$
must lie on the boundary.
As noted in the proof of Proposition~\ref{prop:G}, we have
$\zeta_0\neq \infty$. The point $\zeta_0$ is thus either an attracting fixed point,
a double fixed point (with a single attracting direction along ${\mathbb{R}}$)
or a triple fixed point (with two repelling directions along the real axis).
As mentioned previously, the final case holds if and only if
$g$ has \emph{zero hyperbolic step}; i.e.\
\[ \operatorname{dist}_{{\mathbb{H}}}(g^k(z),g^{k+1}(z))\to 0 \]
as $k\to\infty$, for all $z\in{\mathbb{H}}$.
Here $\operatorname{dist}_{{\mathbb{H}}}$ denotes hyperbolic distance.
It is well-known that
\[ \operatorname{dist}_U(f^{kn}(z),f^{(k+1)n}(z)) \to 0 \]
for $z\in U$.
Indeed, the proof of the existence
of Fatou coordinates, see \cite[Chapter~10]{milnor}, shows that
all $z\in U$ eventually enter an \emph{attracting petal} $P\subset U$ on
which $f^n$ is conformally conjugate to the map $z\mapsto z+1$ on the
upper half-plane. So $f^{kn}(z)\in P$ for large enough $k$, and
\[ \operatorname{dist}_U(f^{kn}(z),f^{(k+1)n}(z)) \leq \operatorname{dist}_P(f^{kn}(z),f^{(k+1)n}(z) =
O(1/k). \]
Since $g$ is conformally conjugate to $f^n|_U$, we see that $g$ does indeed
have zero hyperbolic step, and hence a triple fixed point at $\zeta_0$.
By Lemma~\ref{lem:unisingularparabolic}, we have $g=\tan$, as claimed.
\end{proof}
\begin{remark}
In the specific case of the parabolic exponential map $f(z)=e^{z-1}$, we could
proceed somewhat more directly, using the inherent symmetry of the Julia set.
Indeed, the parabolic basin $U$ intersects the real axis in the interval
$(-\infty,1)$. This interval is a hyperbolic geodesic of $U$ by symmetry,
and contains the singular value $0$. For the inner function
$h=h_{a,b}\colon {\mathbb{H}}\to{\mathbb{H}}$, it follows that the hyperbolic
geodesic connecting the Denjoy-Wolff point to $\infty$ contains the
singular value. From this, we easily conclude that $b=0$, so that
$h= a\cdot \tan(z)$, with $a\leq 1$. We have
\[ \operatorname{dist}_{{\mathbb{H}}}(z,a\cdot \tan(z)) \asymp \operatorname{dist}_{{\mathbb{H}}}(z,a\cdot z) = \log 1/a \]
as $z\to 0$ in ${\mathbb{H}}$. Hence we must have $a=1$, as claimed.
\end{remark}
We note that similar results to Theorems~\ref{theo:unisingular}
and~\ref{theo:unisingularparabolic} hold when the cycle of $U$ contains only one
singular value and and $f^n\colon U\to U$ is proper of degree $d$.
Indeed, in this case the associated inner function $g$ is a finite-degree
\emph{unicritical} Blaschke product having connected Julia set, and the
connectedness locus of unicritical Blaschke products has been
described in detail in~\cite{fletcherblaschke,fletcheretalblaschke}. When
$d=2$, an elliptic Blaschke product fixing zero with multiplier $\lambda$ is given by
\[ z\mapsto z\cdot \frac{z+\lambda}{1+\overline{\lambda}z}, \]
see \cite[Section~4.2.1]{brannerfagellasurgery}, and a Blaschke product
with a parabolic fixed point is given by the function~\eqref{eqn:parabolicblaschke}.
Hence, if $U$ is a periodic Fatou component of period $n$ whose orbit
contains just one singular value, which is a critical value of degree $2$,
then one of the above Blaschke products is dynamically associated to
$f^n\colon U\to U$.
\section{A generalisation of exponential maps}
\label{S.exponentials}
We now generalise our considerations for exponential maps as follows.
\begin{theorem}\label{theo:finite}
Suppose that $f$ is an entire function and $U$ is an unbounded forward-invariant Fatou component on which $f$ has infinite valence, but such that
$f^{-1}(a)\cap U$ contains exactly $p$ points, counting multiplicity, for some $a\in U$ and $p\geq 0$.
Assume that an
inner function dynamically associated to $f|_U$ has a finite number $q\geq 1$ of singularities on $\partial {\mathbb{D}}$.
Then $f$ has a dynamically associated inner function of the form
\begin{equation}
\label{eq:Sdef}
g\colon {\mathbb{D}}\to{\mathbb{D}}; \qquad z \mapsto B(z) \exp\left(- \sum_{j=1}^q \left(c_j \frac{e^{i\theta_j}+z}{e^{i\theta_j} - z}\right)\right),
\end{equation}
for some finite Blaschke product $B$ of degree $p$, real numbers $\theta_1, \ldots, \theta_q$, and positive real numbers $c_1, \ldots, c_q$.
\end{theorem}
Before we prove the theorem, let us note a special case.
\begin{cor}
\label{cor:exponentials}
Suppose that $P, Q$ are polynomials of degree
$\deg P \geq 0$ and $\deg Q \geq 1$. Suppose also that the function
\[
f(z) \defeq P(z) e^{Q(z)},
\]
has an unbounded forward-invariant Fatou component, $U$, containing the origin, on which $f$ has infinite valence.
Then $f$ has a dynamically associated inner function of the form~\eqref{eq:Sdef}, with $q\leq \deg Q$ and $p\leq \deg P$.
\end{cor}
\begin{remark}\normalfont
If $\lambda$ is sufficiently small, then the conditions of this corollary hold for $\lambda f$; see, for example, \cite[Lemma 7.1]{DaveSurvey}.
\end{remark}
\begin{proof}[Proof of Corollary~\ref{cor:exponentials} using Theorem~\ref{theo:finite}]
There are at most $\deg P$ preimages of $0$ under $f$ in $U$
(counting multiplicity), and any associated inner function has at most
$\deg Q$ singularities on
$\partial{\mathbb{D}}$ by Theorem~\ref{theo:tracts} (note that $S(f)$ is finite, and hence $U\cap S(f)$ is compact). Hence the hypotheses of Theorem~\ref{theo:finite} are satisfied.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:finite}]
Let $g$ be an inner function dynamically associated to $f$. We shall assume that the Riemann map $\phi \colon {\mathbb{D}} \to U$ is chosen so that $\phi(0) = a$. By Theorem~\ref{theo:inner}, set $g = B \cdot S$, where $B$ is a Blaschke product and $S$ is a singular inner function of the form \eqref{eq:singinnerdef}.
Since $S$ is never zero, and $a$ has exactly $p$ preimages under $f$, counting multiplicity, it follows that $B$ must be a finite Blaschke product of degree $p$.
Note that this implies that $B$ has no singularities in the boundary of the disc.
It is easy to see, for example, by \cite[Theorem 6.2]{Garnett}, that the singularities of $g$ correspond exactly to the support of $\mu$. Since $g$ has only $q$ singularities, there exist real numbers $\theta_1, \ldots, \theta_q$ and positive real numbers $c_1, \ldots, c_q$ such that $\mu$ is equal to $q$ point masses, each of mass $c_j$, at the points $e^{i\theta_j}$. The result follows.
\end{proof}
We now give three applications of this result. Our first example
notes that, for the functions studied in Theorems~\ref{theo:exp}
and~\ref{theo:unisingularparabolic}, we recover the family
of maps $g_{a,b}$ from~\eqref{eqn:tangent}, up to conformal conjugacy.
\begin{example}
\label{ex:exp}
Suppose that $f$ has a Fatou component $U$
of period $n$ such that $U$ contains only one singular value
of $f$, and such that $f^n\colon U\to U$ is of infinite valence. Then
this restriction is a universal covering over a single point $a\in U$, and
therefore Theorem~\ref{theo:finite} applies with $p=0$ and $q=1$.
In particular, suppose that
$f(z) = \lambda e^z$, for some $\lambda \ne 0$, such that
$f$ has a forward-invariant attracting or parabolic
Fatou component, $U$. Then $U$ must contain the
origin, which is the only singular value of $f$, and
Corollary~\ref{cor:exponentials} applies with
$P \equiv \lambda$, $p=0$, $Q \equiv \operatorname{Id}$ and $q=1$.
The only Blaschke product of order zero is the rotation. Hence there exist
$\sigma\in (-\pi,\pi]$, $c>0$, and $\theta \in (-\pi, \pi]$ such that
\begin{equation}
\label{e1}
g(z) = e^{i\sigma} \exp\left(c \frac{z+e^{i\theta}}{z-e^{i\theta}}\right).
\end{equation}
Conjugating $g$ with a rotation if necessary, we can assume that $\theta = \pi$, in which case
\begin{equation}
\label{e2}
g(z) = \exp\left(i\sigma + c \frac{z-1}{z+1}\right).
\end{equation}
This is equivalent to~\eqref{eq:DevandG} and~\eqref{eqn:tangent}, with
$\mu = \sigma + ic$ and $(a,b)=(c,\sigma)/2$, respectively.
In particular, when $U$ is a parabolic basin, and
in particular for $f(z)=e^{z-1}$, we have $c=2$ and $\sigma=0$
by Theorem~\ref{theo:unisingularparabolic}.
So here $g$ takes the form
\[
g(w) = \exp\left(\frac{2(w-1)}{w+1}\right).
\]
\end{example}
\begin{example}
\label{ex:zexp}
Suppose we are in the family $f(z) = \lambda z e^z$, so that $P \equiv \lambda \operatorname{Id}$, $p=1$, $Q \equiv \operatorname{Id}$ and $q=1$. Suppose that $f$ has a forward-invariant Fatou component, $U$, of infinite valence, that contains the origin. Then the hypotheses of Corollary~\ref{cor:exponentials} hold.
The fact that $q=1$ means, again, that after the right choice of $\arg \phi'(0)$ we can take, for some positive $c$,
\[
S(z) = \exp\left(c \frac{z-1}{z+1}\right).
\]
For $B$, the fact that $p=1$ means that $B$ is a Blaschke product of degree one. However, we also know that $f(0) = 0$ and so $g(0) = 0$. Thus, for some $\sigma>0$, we have
\[
g(z) = e^{i\sigma} z \exp\left(c \frac{z-1}{z+1}\right).
\]
\end{example}
\begin{example}
\label{ex:powerexp}
Suppose we are in the family $f(z) = \lambda e^{z^q}$, for some $q \in {\mathbb{N}}$, so that $P \equiv \lambda$ and $Q(z) = z^q$. Suppose that the hypotheses of Corollary~\ref{cor:exponentials} hold. Since $f$ omits the origin, we get that the Blaschke product is just a constant. We also get, by obvious symmetry considerations, that the $\theta_j$ in \eqref{eq:Sdef} can be taken to be the $q$-th roots of unity, and the $c_j$ are all the same. Set $\omega = e^{2\pi i/q}$. Then there exists $\sigma>0$ and $c>0$ such that
\[
g(z) = e^{i\sigma} \exp\left(-c\sum_{j=1}^n \left(\frac{\omega^j+z}{\omega^j - z}\right)\right).
\]
\end{example}
\section{Proof of Theorem~\ref{theo.sine}}
\label{S.sine}
\begin{proof}[Proof of Theorem~\ref{theo.sine}]
It is easy to see that if $\lambda \in (0, 1)$, then
both critical values $\pm \lambda$ are in the immediate basin
of attraction $U$ of the attracting fixed point at $0$. It follows
easily that $U=F(f)$. (This was already observed by Fatou~\cite{fatou}.)
In particular,~\ref{p1} holds.
For simplicity write $f$ for $f_\lambda$. Choose the Riemann map $\phi\colon {\mathbb{D}} \to U$ so that $\phi(0) = 0$ and $\phi'(0) > 0$. Then $\phi$ maps points on the real line to the real line, because of the obvious symmetry of $U$. Let $g$ be the inner function $g \defeq \phi^{-1} \circ f \circ \phi$, and let
\[
g = B \cdot S,
\]
where $B$ is a Blaschke product and $S$ is a singular inner function, as usual.
Clearly $0$ is a simple zero of $f$, and so of $g$, and so of $B$. Notice that $f$ is $2 \pi$-periodic. Notice also that $f^2$ (the second iterate) is $\pi$-periodic. Hence $J(f)$ is $\pi$-periodic. The zeros of $f$ are the points $\pm\zeta_n$ where $$\zeta_n = n\pi, \quad\text{for } n \in {\mathbb{N}}.$$ Hence the other zeros of $g$, and so of $B$, are all of the form $\pm a_n$, for some increasing sequence with
$a_n \rightarrow 1$ as $n\rightarrow \infty$. Thus we can write $B$ as
\begin{equation}\label{eqn:B}
B(z) = z \cdot \prod_{n \geq 1} \frac{a_n^2 - z^2}{1 - a_n^2 z^2}.
\end{equation}
We claim that $S$ is, in fact, absent. To prove this claim, suppose otherwise.
If $D\subset U$ is a Jordan domain containing $[-\lambda,\lambda]$, then
$f^{-1}(U\setminus D)$ has exactly two components. Indeed,
by the elementary mapping properties of $\sin$, the set
$f^{-1}(\partial D)$ consists of two curves tending to $\pm\infty$, symmetrically
with respect to the real axis.
By Theorem~\ref{theo:tracts}, the map $g$ has two singularities on $\partial {\mathbb{D}}$. Since $\pm 1$ are singularities of $B$, this means that $S$ has at most two singularities, which would need to be positioned at $\pm 1$.
Since $U$ is symmetric on reflection in the imaginary axis, and since $f$ is an odd function, our choice of $\phi$ implies that $g$ is also an odd function. Hence $S$ is generated by two equal masses, each at $\pm 1$. In particular, by a calculation
from~\eqref{eq:singinnerdef}, there exists $c>0$ such that
\[
S(z) = \exp \left(c \frac{z^2 + 1}{z^2 - 1}\right).
\]
It follows that as $x \rightarrow 1$, we have that $S(x) \rightarrow 0$, and so $g(x) \rightarrow 0$. This is impossible, as $f(x)$ does not have a limit as $x \rightarrow \infty$. This contradiction proves our claim, and we have
$g=B$ with $B$ as in~\eqref{eqn:B}.
Next we seek to find a formula for the $a_n$. Because of the periodicity of $J(f)$ the hyperbolic distance in $U$ from $\zeta_n$ to $\zeta_{n+1}$ is constant, and in fact equal to the hyperbolic distance in $U$ from $0$ to $\zeta_1$. Call this distance $d$. By symmetry, the real axis is a hyperbolic geodesic of $U$. It
follows that $\operatorname{dist}(0,\zeta_n)=n\cdot d$.
Then the hyperbolic distance in ${\mathbb{D}}$ from $0$ to $a_n$ is also $n\cdot d$, and so
\[
\log \frac{1+a_n}{1-a_n} = n\cdot d,
\]
from which we calculate
\[
a_n = \frac{e^{n\cdot d} - 1}{e^{n\cdot d} + 1}= \frac{\tau^{n} - 1}{\tau^{n} + 1}
\]
where $\tau = e^d\in (1,\infty)$.
Clearly $d$, and hence $\tau$, depends on $\lambda$. Write $\tau = \tau(\lambda)$; it remains to show that $\tau\colon (0,1)\to (1,\infty)$ is a homeomorphism. Since $\phi(0) = g(0) = f(0) = 0$,
\[
\lambda = f'(0) = g'(0) = \prod_{n \geq 1} a_n^2 = \prod_{n \geq 1} \left(\frac{\tau^{n} - 1}{\tau^{n} + 1}\right)^2.
\]
In particular, $\lambda$ is uniquely determined by $\tau$.
The function $x\mapsto (x-1)/(1+x)$ is strictly increasing on
$[1,\infty)$. So $\lambda$ is a strictly increasing continuous
function of $\tau$. Moreover, it is easy to see that
$\lambda\to 0$ as $\tau\to 1$, and $\lambda\to 1$ as $\tau\to\infty$.
\end{proof}
\section{Fatou components with infinitely many critical values}
\label{S.infinite}
\begin{proof}[Proof of Theorem~\ref{theo.Fatou}]
It is easy to show that $f_\lambda$ has a completely invariant Fatou component, $U_\lambda$, which contains a right half-plane. Hence \ref{p1} holds.
For simplicity write $f$ for $f_\lambda$ and $U$ for $U_\lambda$. Note that $0 \in U$, and indeed (by a calculation) ${\mathbb{R}} \subset U$. Note also that $f(\overline{z}) = \overline{f(z)}$, and so $U$ is symmetric on reflection on the real line.
Let $\alpha > 0$, and choose the Riemann map $\phi : {\mathbb{H}} \to U$ so that $\phi(i\alpha) = 0$ and $i\phi'(i\alpha) > 0$. (We will choose $\alpha$ later). Then $\phi^{-1}$ maps points on the real axis to the positive imaginary axis, because of the symmetry of $U$. Let $h \defeq \phi^{-1} \circ f \circ \phi$ be an inner function of the upper half-plane.
Note that $f(w + 2\pi i) = f(w) + 2\pi i$, for $w \in {\mathbb{C}}$. This means that $U$ is periodic under translation of $2\pi i$. We can deduce that there exists $\kappa > 0$ such that $$\phi^{-1}(w + 2 \pi i) = \phi^{-1}(w) - \kappa, \quad\text{for } w \in {\mathbb{C}}.$$ It then follows that $h(z - \kappa) = h(z) - \kappa$, for $z \in {\mathbb{H}}$.
We claim that $h$ has one singularity, and this is at infinity.
(Observe that we cannot apply Theorem~\ref{theo:tracts}, as the
singular values of $f$ are not compactly contained in $U$.) Suppose that $\zeta$ is such that $|\zeta|$ is small. It can be shown by a calculation that the preimages under $f$ of $\zeta$ that are of large modulus are close to the points $-\log |y_n| + iy_n$, where
\[
y_n =
\begin{cases}
\frac{4n+1}{2}\pi, \text{ for } n \in {\mathbb{N}}, \\
\frac{4n+3}{2}\pi, \text{ for } -n \in {\mathbb{N}}.
\end{cases}
\]
These points can be connected to infinity by two curves in $U$ (one containing the points of positive imaginary part, and the other containing the points of negative imaginary part) that are each homotopic to $(0, +\infty)$. This establishes the fact that $h$ has only one singularity, since by transferring everything to the unit disc via a M\"obius map we can deduce that $g$ has exactly one singularity on $\partial {\mathbb{D}}$, the point where all preimages of almost every $z \in {\mathbb{D}}$ accumulate. Moreover, this singularity of $h$ is at $\lim_{x \to +\infty} \phi^{-1}(x) = i\infty$. This completes the proof of the claim.
Since $h$ has no singularities in $\overline{{\mathbb{H}}}$, by Schwarz reflection it
extends to a transcendental meromorphic map of the whole plane, which we continue to call $h$, and which maps ${\mathbb{H}}$ to itself. For simplicity we now write
\[
h(z) \defeq z + G(z) \defeq z + \frac{G_1(z)}{G_2(z)},
\]
where $G_1$ and $G_2$ are entire. Then
\[
G(z + \kappa) = G(z),
\]
in other words, $G$ is $\kappa$-periodic. Note that $\kappa$ depends linearly on $\alpha$; this can easily be seen by pre-composing $\phi$ with a map $z \mapsto cz$, for $c > 0$. Hence we can assume that $\alpha$ is chosen so that $\kappa = \pi$; in other words, $G$ is $\pi$-periodic.
Next we locate the poles and fixed points of $h$. We have $\phi(it)\in{\mathbb{R}}$ and $\phi(it)\to -\infty$ as $t\to 0$; hence $\phi(h(it))=f(\phi(it))\to +\infty$. So $h(it)\to\infty$ as $t\to 0$, and $0$ is a pole of
$h$. Since $h$ commutes with translation by $\kappa=\pi$, all the integer multiples of $\pi$, $\zeta_n=\pi n$, are also poles of $h$. We claim that there are no other poles. Indeed,
every pole of $h$ is the landing point of some piece of $h^{-1}(i\cdot [\alpha,\infty)) = \phi^{-1}(f^{-1}([0,\infty)))$. But
$f^{-1}([0,\infty))$ consists of countably many curves $\gamma_n$, each tending to $-\infty$ asymptotically at imaginary part $2n\pi$.
It follows easily that $\phi^{-1}(\gamma_n)$ lands at $\zeta_{-n}$, and these are the only poles of $h$.
Now we locate the fixed points of $h$. The fixed points of $f$ are the points $z_n \defeq -\log\lambda + (2n+1)\pi i$, for $n \in {\mathbb{Z}}$. These points are accessible boundary points of $U$, since, for each $n \in {\mathbb{N}}$, the set $\{ z_n + x : x > 0 \}$ lies in $U$. Since the set of fixed points is $2 \pi i$-periodic, the corresponding fixed points $w_n \defeq \phi^{-1}(z_n)$ of $h$ are a $\pi$-periodic set of real numbers; this follows from our choice of $\alpha$ above. Since the fixed points of $f$ are simple, all these are
simple fixed points of $h$.
It is easy to see that the poles $\zeta_n$ are simple, as otherwise $h$ could not preserve the upper half-plane. Similarly, $h$ cannot have any critical points on the real line. For, if $x \in {\mathbb{R}}$ and $h'(x) = 0$, then close to $x$ the map $h$ behaves like a power map and so cannot preserve the upper half-plane.
It follows that there is exactly one fixed point between every two poles. We deduce that the points $z_n$ are the only fixed points of $h$. Moreover, since $z_0$ and $z_{-1}$ are
symmetrically placed with respect to the real axis, the same must be true of $w_0$ and $w_{-1}$. Since $w_{n}-w_{n+1} = \pi$, the $w_n$ are at the \emph{odd} multiples of $\pi/2$.
Since the poles of $h$ are exactly the zeros of the sine function, it follows that there is an entire function $h_2$ such that
\[
G_2(z) = e^{h_2(z)} \sin z.
\]
Similarly, the fixed points of $h$ (which are all simple) are the zeros of $G_1$. Hence there is an entire function $h_1$ such that
\[
G_1(z) = e^{h_1(z)} \cos z.
\]
We have now concluded that there is an entire function $H$ such that
\[
h(z) = z + e^{H(z)} \cot z.
\]
Note that $G$, and hence $e^{H(z)}$, is $\pi$-periodic.
Now, if $x > 0$ is large, then $f(x + iy) \approx x + iy + \lambda$. We can deduce that if $y > 0$ is large, then $h(x + iy) = x + iy + i\nu(x, y)$, where $\nu(x, y)$ is small. It follows that $e^{H(z)}$ is bounded, and so must be constant. Since $f$ maps the real line to itself and maps large values of $x$ close to $x + \lambda$, $h$ maps the positive imaginary axis to itself and, when $y$ is large, maps $iy$ close to $i(y + \nu)$ for some positive $\nu$. Thus $e^{H(z)}$ is the constant $-\nu$, and we have obtained that
\[
h(z) = z - \nu \cot z.
\]
It remains to show that $\nu = \lambda/2$. Fix $\delta > \max(\pi,\nu)$. For $t>\delta$, we can apply Koebe's distortion theorem
to $\phi$, restricted to the disc $D(it,t)\subset{\mathbb{H}}$ centred at $it$, to obtain distortion estimates on the smaller disc $D(it,\delta)$. More precisely, setting
$r\defeq \delta/t<1$, we have
\[ \frac{(1-r)^2}{(1+r)^2} \leq \frac{\lvert \phi(z) - \phi(it)\rvert }{\lvert z-it\rvert }\cdot \frac{1}{\lvert \phi'(it)\rvert} \leq \frac{(1+r)^2}{(1-r)^2} \]
for all $z\in D(it,\delta)$. For $z=it + \pi$, we have $\phi(z)=\phi(it)+2\pi$, so $\lvert \phi'(it)\rvert \to 2$ as $t\to\infty$. Similarly, for
$z= h(it)$ and sufficiently large $t$, we have $z\in D(it,\delta)$ and
\[ \frac{\lvert \phi(z) - \phi(it)\rvert}{\lvert z-it\rvert } = \frac{\lvert f(\phi(it)) - \phi(it)\rvert }{\lvert h(it)-it\rvert } \to \frac{\lambda}{\nu}, \]
so $\phi'(it) \to \lambda/\nu$. We have shown $\nu = \lambda/2$, as required.
\end{proof}
\section{Proof of Theorem \ref{theo:finitecomplicated}}
\label{sec:approximation}
We first give a simple result about uniform convergence of finite Blaschke products in the unit disc, which we use in the proof of Theorem \ref{theo:finitecomplicated}.
\begin{proposition}
\label{prop:conv}
Suppose that $(B_n)_{n \in {\mathbb{N}}}$ is a sequence of finite Blaschke products of degree $d$, which converge locally uniformly on ${\mathbb{D}}$ to a finite Blaschke product, $B$, of degree $d$. Then the convergence is, in fact, uniform on ${\mathbb{D}}$.
\end{proposition}
\begin{proof}
In general we denote the open disc with centre $w \in {\mathbb{C}}$ and radius $r>0$ by
\[
D(w, r) \defeq \{ z \in {\mathbb{C}} : |z-w| < r \}.
\]
Let $\rho \in (0, 1)$ be such that all the zeros of $B$ lie in the disc $D(0, \rho)$. Set
\[
t \defeq \min \{ |B(z)| : |z| = \rho \} > 0.
\]
It follows from locally uniform convergence that for all sufficiently large values of $n$, we have
\[
|B(z) - B_n(z)| < t \leq |B(z)|, \quad\text{for } |z| = \rho.
\]
It then follows from Rouch\'e's theorem that, for all sufficiently large values of $n$, all the zeros of $B_n$ lie in $D(0, \rho)$. Hence there exists $r \in (0, 1)$ such that all the zeros of all the $B_n$ lie in $D(0, r)$.
By an easy calculation, we can deduce that there exists $r' > 1$ such that, with $D \defeq D(0, r')$, each $B_n$ is analytic in $D$, and the family $(B_n)_{n \in {\mathbb{N}}}$ is uniformly bounded in $D$. It then follows by the Vitali-Porter theorem, see, for example, \cite{Schiff}, that the $B_n$ converge locally uniformly to $B$ in $D$. The result follows, as $\overline{{\mathbb{D}}}$ is a compact subset of $D$.
\end{proof}
Now we give the proof of Theorem~\ref{theo:finitecomplicated}.
\begin{proof}[Proof of Theorem~\ref{theo:finitecomplicated}]
Let $(B_n)_{n \in {\mathbb{N}}}$ be a sequence of finite Blaschke products that is dense in the space of finite Blaschke products (in the topology of uniform convergence). Such a sequence exists, for example, by choosing functions of the form \eqref{eq:Bdef} with $d$ finite and all the variables $\theta, \Re a_1, \Re a_2, \ldots, \Re a_d$, and $\Im a_1 \Im a_2, \ldots, \Im a_d$ rational.
For each $n \in {\mathbb{N}}$ let $T_n$ be the translation $T_n(z) \defeq z +4n$, and let $D_n$ be the disc $D_n \defeq D(4n, 1)$. It follows by \cite[Theorem 5.3]{classifyingwandering} that there exists a transcendental entire function $f$ having an orbit of bounded,
simply-connected, escaping, wandering domains $(U_n)_{n \in {\mathbb{N}}}$ such that the following all hold.
\begin{enumerate}[(A)]
\item $\Delta_n' \defeq D(4n, r_n) \subset U_n \subset D(4n,R_n) \defeq \Delta_n,$ where $0 < r_n < 1 < R_n$ and $r_n,R_n \to 1$ as $n\to \infty$.\label{thesets}
\item $f_{n} \defeq T_{n+1}\circ B_n \circ T_n^{-1}$ is analytic on $\overline{\Delta_n}$, and $|f(z) - f_{n}(z)| \to 0$ as $n \to \infty$
uniformly on $\overline{\Delta_n}$; by ``uniformly'' we mean that for each $\epsilon > 0$ there exists $N \in {\mathbb{N}}$ such that $|f(z) - f_n(z)| < \epsilon$, for $z \in \Delta_n$ and $n \geq N$.\label{convs}
\item $f:U_n \to U_{n+1}$ has the same degree as $B_n$.\label{degree}
\end{enumerate}
This completes the definition of the function $f$. It remains to show that the Fatou components of $f$ have dynamically associated inner functions with the claimed properties. Suppose that $n \in {\mathbb{N}}$. Let $\phi_n:\mathbb{D} \to U_n$ be the Riemann map such that $\phi_n(0) = 4n$ and $\phi_n'(0) > 0$. Then
\begin{equation}
\label{eq:g}
g_{n}= \phi_{n+1}^{-1} \circ f \circ \phi_n
\end{equation}
is an inner function dynamically associated to $f|_{U_n}$.
We need to be able to approximate the Riemann maps in \eqref{eq:g}, and we claim that $\phi_n - T_n \to 0$ locally uniformly on ${\mathbb{D}}$ as $n \to \infty$. To prove this claim, we first consider translated copies of $U_n$, defined by
\[
U_n' \defeq T_n^{-1}(U_n), \quad\text{for } n \in {\mathbb{N}}.
\]
Note that, for each $n\in{\mathbb{N}}$, we have that $D(0, r_n) \subset U_n' \subset D(0, R_n)$. Suppose that $w_0 \in {\mathbb{D}}$. Then there exists a neighbourhood of $w_0$ that is contained in $U_n'$ for all sufficiently large $n \in \mathbb{N}$. Also, suppose that $w \in \partial {\mathbb{D}}$. Then it follows by \eqref{thesets} that there exists a sequence of points $w_n \in \partial U_n'$ such that $w_n \to w$ as $n \to \infty$. Hence $U_n' \to {\mathbb{D}}$ in the sense of kernel convergence; see \cite[p.13]{Pommerenke}. Set
\[
\phi_n' \defeq T_n^{-1} \circ \phi_n, \quad\text{for } n \in {\mathbb{N}}.
\]
It then follows from the Carath\'eodory kernel theorem (\cite[Theorem 1.8]{Pommerenke}) that $\phi_n' \to \operatorname{Id}$ locally uniformly in the unit disc as $n \to \infty$, where $\operatorname{Id}(z) \defeq z$. The claim above follows.
Suppose that $B$ is a given Blaschke product, and suppose that $B$ has degree $d$. Let $(n_p)_{p \in {\mathbb{N}}}$ be a sequence of integers such that the subsequence $(B_{n_p})_{p \in {\mathbb{N}}}$ converges uniformly to $B$ on ${\mathbb{D}}$ as $p\rightarrow\infty$. We can assume that each $B_{n_p}$ has degree $d$.
Note that it follows by \eqref{degree} that the degree of each $g_{n_p}$ is equal to $d$. Observe that the theorem requires that the sequence of functions $(g_{n_p})_{p \in {\mathbb{N}}}$ converges uniformly on ${\mathbb{D}}$ to the function $B$ as $p \rightarrow \infty$. We shall prove first that this sequence converges locally uniformly to $B$. We will then use Proposition~\ref{prop:conv} to deduce that this convergence is in fact uniform.
To prove locally uniform convergence, suppose that $K \subset {\mathbb{D}}$ is a given compact set. Choose $r \in (0, 1)$ sufficiently close to $1$ that $K \subset \Delta$ where $\Delta \defeq D(0, r)$. It follows by the claim above that
\begin{equation}
\label{eq:phi}
\tilde{\epsilon}_n(z) \defeq \phi_n(z) - T_n(z), \quad\text{for } z \in {\mathbb{D}},
\end{equation}
is analytic in ${\mathbb{D}}$, and that $\sup_{z \in \Delta} |\tilde{\epsilon}_n(z)| \to 0$ as $n \to \infty$. Similarly
\begin{equation}
\label{eq:phiminus1}
\hat{\epsilon}_{n}(z) \defeq \phi_n^{-1}(z) - T_{n}^{-1}(z), \quad\text{for } z \in U_n,
\end{equation}
is analytic in $U_n$, and that $\sup_{z \in T_n(\Delta)} |\hat{\epsilon}_n(z)| \to 0$ as $n \to \infty$. Finally, it follows by \eqref{convs} above that, for all sufficiently large $n\in {\mathbb{N}}$, we have that
\begin{equation}
\label{eq:f}
\epsilon_{n}(z) \defeq f(z) - f_{n}(z), \quad\text{for } z \in \Delta_n,
\end{equation}
is analytic in $\Delta_n$, and such that $\sup_{z \in \phi_n(\Delta)} |\epsilon_n(z)| \to 0$ as $n \to \infty$.
Note that, by \eqref{convs}, \eqref{eq:phi}, and \eqref{eq:f}, if $n \in {\mathbb{N}}$ is sufficiently large, then
\begin{align}
f(\phi_n(z)) &= f_n(\phi_n(z)) + \epsilon_n(\phi_n(z)) \nonumber \\
&= T_{n+1}(B_n(T_n^{-1}(T_n(z) + \tilde{\epsilon}_n(z)))) + \epsilon_n(\phi_n(z)) \nonumber \\
&= B_n(z + \tilde{\epsilon}_n(z)) + 4(n+1) + \epsilon_n(\phi_n(z)), \quad\text{for } z \in K.\label{eq1}
\end{align}
Since the sequence $(B_{n_p})_{p\in{\mathbb{N}}}$ converges uniformly to $B$ in ${\mathbb{D}}$, as $p$ tends to infinity,~\eqref{eq1} gives
\begin{equation} T_{n_p+1}^{-1}(f(\phi_{n_p}(z))) \to B(z) \label{eqn:eq2} \end{equation}
uniformly for $z\in K$.
In particular, if $\tilde{\Delta} = D(0,\tilde{r})$ is a disc containing $B(K)$, then $f(\phi_{n_p}(K))\subset T_{n_p}(\tilde{\Delta})$ for sufficiently large $p$.
It then follows, by \eqref{eq:g}, \eqref{eq:phiminus1}, and \eqref{eqn:eq2}, that, for $z \in K$,
\begin{align*}
g_{n_p}(z) =&\ \phi_{n_p+1}^{-1}\bigl(f(\phi_{n_p}(z))\bigr) \\
=&\ T_{n_p+1}^{-1}\bigl (f(\phi_{n_p}(z))\bigr) + \hat{\epsilon}_{n_p+1}\bigl(f(\phi_{n_p}(z))\bigr) \to B(z)
\end{align*}
as $p\to\infty$.
We have, therefore, established that the subsequence $g_{n_p}$ converges locally uniformly to $B$ in ${\mathbb{D}}$, as $p$ tends to infinity. Uniform convergence then follows by Proposition~\ref{prop:conv}. This completes the proof of Theorem~\ref{theo:finitecomplicated}. For, if $g_{n_p} \rightarrow B$ uniformly on ${\mathbb{D}}$, then, given $\epsilon > 0$, we can set $U= U_{n_p}$ and $V = U_{n_p+1}$ for a sufficiently large value of $p$. Note that $U$ and $V$ are then successive wandering domains in the orbit of $U_0$.
\end{proof}
\newcommand{\etalchar}[1]{$^{#1}$}
\newcommand{\noopsort}[1]{}
|
1,314,259,995,779 | arxiv | \section{Introduction}
To date, the correspondence between string theory in five-dimensional
anti-de Sitter (AdS) space and four-dimensional QCD has enjoyed a
number of successes (see
\cite{Erdmenger:2007cm,CasalderreySolana:2011us,Costa:2012fw,deTeramond:2012rt} and
references therein).
In this letter, we demonstrate another
success by showing that parameter-free AdS/QCD wavefunctions for the $\rho$
meson lead to predictions for the rate of diffractive $\rho$ meson
production ($\gamma^* p \to \rho p$) that agree with
the data collected at the HERA $ep$ collider.
In previous papers \cite{Forshaw:2010py,Forshaw:2011yj}, we took a
phenomenological approach and extracted the light-front wavefunctions
of the $\rho$ meson using the HERA data. We follow the same formalism
here, except that we now use the AdS/QCD wavefunctions predicted in \cite{Brodsky:2007hb,Brodsky:2008pg}.
\section{The AdS/QCD wavefunction}
Brodsky and de T\'eramond have recently shown that, in what they call a
first semiclassical approximation to light-front QCD \cite{deTeramond:2008ht}, the meson
wavefunction can be written in the
following factorized form
\begin{equation}
\phi(x,\zeta, \varphi)=\frac{\Phi(\zeta)}{\sqrt{2\pi \zeta}} f(x) \mathrm{e}^{i L \varphi}
\label{factorized-lc}
\end{equation}
where $L$ is the orbital quantum number and $\zeta=\sqrt{x(1-x)} b$ ($x$
is the light-front longitudinal momentum fraction of the quark
and $b$ the quark-antiquark transverse separation). The function $\Phi(\zeta)$ satisfies a
Schr\"odinger-like wave equation
\begin{equation}
\left(-\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2} - \frac{1-4L^2}{4 \zeta^2} + U(\zeta) \right) \Phi (\zeta)=M^2 \Phi (\zeta) \;,
\label{LFeigenvalue}
\end{equation}
where $U(\zeta)$ is the confining potential defined at equal light-front time. After identifying $\zeta$ with
the co-ordinate in the fifth dimension, Eq.~\eqref{LFeigenvalue}
describes the propagation of spin-$J$ string modes, in which
case $U(\zeta)$ is determined by the choice for the dilaton
field. We shall use the soft-wall
model~\cite{Karch:2006pv}, in which
\begin{equation}
U(\zeta)=\kappa^4 \zeta^2 + 2\kappa^2(J-1) \;.
\label{quadratic-dilaton}
\end{equation}
This potential encodes the
confinement dynamics of QCD and the challenge remains to derive it from
first-principles QCD.
Solving Eq.~(\ref{LFeigenvalue}) with this potential results in eigenvalues
\begin{equation}
M^2=4\kappa^2(n+J/2+L/2) \; ,
\label{mass-spectrum}
\end{equation}
which reproduces the correct meson mass spectrum. In particular, it
predicts a massless pion ($S=0, n=0, L=0$) and $M_\rho^2=2\kappa^2$ for the $\rho$ meson
($S=1, n=0, L=0$). The corresponding
eigenfunctions are \cite{Vega:2009zb}
\begin{equation}
\Phi (\zeta)= \kappa \sqrt{2\zeta} \exp \left(-\frac{\kappa^2 \zeta^2}{2}\right) \;.
\end{equation}
It remains to
specify the function $f(x)$ in Eq.~\eqref{factorized-lc}. This can be done by
comparing the expressions for the pion EM form factor
obtained in the light-front formalism and in AdS
space \cite{Brodsky:2007hb} and it results in
\begin{equation}
f(x) = {\cal{N}}\sqrt{x(1-x)} \;.
\end{equation}
The resulting wavefunction is thus
\begin{equation}
\phi(x,\zeta)= \mathcal{N} \frac{\kappa}{\sqrt{\pi}}\sqrt{x(1-x)} \exp \left(-\frac{\kappa^2 \zeta^2}{2}\right)~,
\label{lcwf-massless-quarks}
\end{equation}
where $\mathcal{N}$ is a normalisation constant. Assuming the meson is
dominated by its leading $q\bar{q}$ Fock component, $\mathcal{N}$ is fixed by
\begin{equation}
\int {\mathrm{d}}^2{\mathbf{b}} \; \mathrm{d} x \; |\phi(x,\zeta)|^2 = \int_0^1
\frac{\mathrm{d} x}{x(1-x)} f^2(x) = 1~. \label{eq:norm0}
\end{equation}
Brodsky and de
T\'eramond also have a prescription to account for non-zero quark
masses \cite{Brodsky:2008pg}: A Fourier transform to $k$-space gives
\begin{equation}
\tilde{\phi}(x,k) \propto \frac{1}{\sqrt{x(1-x)}} \exp \left(-\frac{M^2_{q\bar{q}}}{2\kappa^2} \right) ~,
\end{equation}
where the invariant mass squared of the $q\bar{q}$ pair is
$M^2_{q\bar{q}}=k^2/(x(1-x))$. For massive quarks, the invariant mass should rather be
$M^2_{q\bar{q}}=(k^2 + m_f^2 )(x(1-x))$. After substituting this into
the wavefunction and Fourier
transforming back to transverse position space, one obtains the final
form of the AdS/QCD wavefunction:
\begin{equation}
\phi(x,\zeta)= N \frac{\kappa}{\sqrt{\pi}}\sqrt{x(1-x)} \exp \left(-\frac{\kappa^2 \zeta^2}{2}\right) \exp\left(-\frac{m_f^2}{2\kappa^2 x (1-x)} \right)~,
\label{lcwf-massive-quarks}
\end{equation}
and $N$ is fixed by the generalization of Eq.~(\ref{eq:norm0}):
\begin{equation}
\int_0^1 \frac{\mathrm{d} x}{x(1-x)}
f^2(x)\exp\left(-\frac{m_f^2}{\kappa^2x(1-x)} \right) = 1~. \label{eq:norm}
\end{equation}
This is rather similar to the original Boosted Gaussian (BG)
wavefunction discussed in \cite{Nemchik:1996cw,Forshaw:2003ki}
\begin{equation}
\phi^{{\mathrm{BG}}} (x,\zeta) \propto x(1-x) \;
\exp \left(\frac{m_f^{2}R^{2}}{2}\right)
\exp \left(-\frac{m_f^{2}R^{2}}{8 x(1-x)}\right) \; \exp \left(-\frac{2 \zeta^{2}}{{R}^{2}}\right) \;.
\label{original-boosted-gaussian}
\end{equation}
If $R^2=4/\kappa^2$ then
the two wavefunctions differ only by a factor of $\sqrt{x(1-x)}$,
which is not surprising given that
in both cases confinement is modelled by a harmonic oscillator.
In what follows we shall consider a parameterization that accommodates
both the AdS/QCD and the BG wavefunctions:
\begin{equation}
\phi(x,\zeta) \propto [x(1-x)]^\beta \exp \left(-\frac{\kappa^2 \zeta^2}{2}\right) \exp\left(-\frac{m_f^2}{2\kappa^2 x (1-x)} \right)~.
\label{lcwf-massive-quarks-fit}
\end{equation}
\section{Comparing to data, QCD Sum Rules and the lattice}
To compute the cross-section for $\gamma^* p \to \rho p$ we use the
dipole model of high-energy scattering
\cite{Nikolaev:1990ja,Nikolaev:1991et,Mueller:1993rr,Mueller:1994jq}. In
this approach, the scattering amplitude is a convolution of the photon
and vector meson $q\bar{q}$ wavefunctions with the total cross-section
to scatter a $q\bar{q}$ dipole off a proton. QED is used to determine
the photon wavefunction and the dipole cross-section can be extracted
from the precise data on the deep-inelastic structure
function $F_2$. The details of this procedure can be found in
\cite{Forshaw:2011yj,Forshaw:2003ki}. All that remains is to specify the wavefunction of the meson.
The meson's light-front wavefunctions can be written in terms of the
AdS/QCD wavefunction $\phi(x,\zeta)$ \cite{Forshaw:2011yj}. For longitudinally
polarized mesons:
\begin{equation}
\Psi^{L}_{h,\bar{h}}(b,x) = \frac{1}{2\sqrt{2}}
\delta_{h,-\bar{h}}
\left( 1 + \frac{ m_{f}^{2} - \nabla^{2}}{M_{\rho}^2\; x(1-x)}\right) \phi_L(x,\zeta) ~,
\label{nnpz_L}
\end{equation}
where $\nabla^2 \equiv \frac{1}{b} \partial_b + \partial^2_b$ and
$h$ ($\bar{h}$) are the helicities of the quark (anti-quark).
For transversely polarized mesons:
\begin{equation}
\Psi^{T=\pm}_{h,\bar{h}}(b, x) = \pm [i e^{\pm i\theta}
( x \delta_{h\pm,\bar{h}\mp} - (1-x) \delta_{h\mp,\bar{h}\pm})
\partial_{b}+ m_{f}\delta_{h\pm,\bar{h}\pm}] \frac{\phi_T(x,\zeta)}{2x(1-x)}~,
\label{nnpz_T}
\end{equation}
where $be^{i\theta}$ is the complex form of the transverse separation,
$\mathbf{b}$. Rather than using Eq.~(\ref{eq:norm}) to fix the
normalization of $\phi$ we impose
\begin{equation}
\sum_{h,\bar{h}}\int \mathrm{d}^{2}{\mathbf{b}} \, \mathrm{d} x \,
|\Psi^{\lambda}_{h,\bar{h}}(b, x)|^{2} = 1 ~,
\label{normalisationTL}
\end{equation}
where $\lambda=L,T$. This means we allow for a polarization dependent
normalization (hence the subscripts on $\phi_{L,T}$). For longitudinal
polarization, the difference between the two normalization
prescriptions leads only to sub-leading
differences $\sim M_\rho^2/Q^2$ or $m_f^2/M_\rho^2$ in the
electroproduction scattering
amplitude, where $Q^2$ is the photon virtuality.
For transverse polarization, the two prescriptions lead to
a slightly different normalization but this can be attributed to the
ignoring of higher Fock
components in the wavefunction (since all of the normalization
integrals are in any case only unity up to corrections due to higher Fock components).
To compare with the HERA data we have to fix a value for the quark
mass. We take $m_f = 140$~MeV, which is the value used in the
fits to the deep-inelastic structure function, $F_2(x,Q^2)$ \cite{Forshaw:2011yj}.
We use the CGC[0.74] dipole model \cite{Soyez:2007kg,Watt:2007nr} (see \cite{Forshaw:2011yj} for details),
although the predicitions do not
vary much if we use other models that fit the HERA $F_2$ data
\cite{Forshaw:2004vv,Kowalski:2006hc}.
We also take $\kappa=M_\rho/\sqrt{2}=0.55$~GeV, which is the AdS/QCD prediction.
Figures~\ref{fig:xsecW} and \ref{fig:ratio} compare the AdS/QCD predictions, shown as the solid blue curves, with the HERA
data. The agreement is
good and the disagreement at high $Q^2$ is not
unexpected. This is the region where perturbative evolution of the
wavefunction will be relevant and the AdS/QCD wavefunction we use is
clearly not able to describe that. It should be stressed that these AdS/QCD
predictions are parameter-free.
We are also able to compute the electronic decay width
$\Gamma_{e^+e^-}$, which is related to the decay constant via
$$ f_\rho = \left( \frac{3\Gamma_{e^{+}e^{-}} M_\rho}{4 \pi
\alpha_{\mathrm{em}}^2} \right)^{1/2}.$$
Using
\begin{equation}
f_\rho = \frac{1}{2}
\left(\frac{N_c}{\pi}\right)^{1/2} \int_0^1 \mathrm{d} x \left(1 + \frac{m_{f}^{2}-\nabla^{2}}{M_\rho^2x(1-x)}\right) \phi_L(x,\zeta=0)~,
\label{longdecayB}
\end{equation}
we obtain $\Gamma_{e^{+}e^{-}}=6.66$~keV, which is to be compared with the measured value $\Gamma_{e^{+}e^{-}}=7.04 \pm 0.06~\mathrm{keV}$
\cite{Nakamura:2010zzi}.
\begin{figure}
\centering
\subfigure[~H1]{\includegraphics[width=0.85\textwidth]{H1xsec.pdf} }
\subfigure[~ZEUS]{\includegraphics[width=0.85\textwidth]{ZEUSxsec.pdf} }
\caption{Comparison the HERA cross-section data
\cite{Chekanov:2007zr,Collaboration:2009xp}. Solid blue curve is the
AdS/QCD prediction and the dashed red curve is the best fit.}
\label{fig:xsecW}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{HERAratio.pdf}
\caption{Comparison to the HERA data on the longitudinal to transverse cross-section ratio
\cite{Chekanov:2007zr,Collaboration:2009xp}. Solid blue curve is
the AdS/QCD prediction and the dashed red curve is the best fit.}
\label{fig:ratio}
\end{figure}
Figure \ref{fig:contour} shows the $\chi^2$ per data point in the
$(\beta,\kappa)$ parameter space (see
Eq. (\ref{lcwf-massive-quarks-fit})) \footnote{We include the
electroproduction data and also the decay constant $f_\rho$ in the fit.}. It confirms that the AdS/QCD
prediction lies impressively close to the minimum in $\chi^2$.
The best fit has a $\chi^2$ per data point equal to $114/76$ and is
achieved with $\kappa=0.56$ GeV and $\beta=0.47$. Better fits to the data are
possible, e.g. if one allows the parameters $\beta$ and $\kappa$ to
depend on the polarization of the meson. However, given that we have
not attempted to quantify the theory uncertainty we regard these as
good fits. The
curves resulting from the best fit are shown as the red dashed curves
in Figures~\ref{fig:xsecW} and \ref{fig:ratio}.
We have previously shown that the twist-$2$
Distribution Amplitude (DA) can be related to $\phi_L(x,\zeta)$
according to
\begin{equation}
\varphi(x,\mu) = \left(\frac{N_c}{\pi}\right)^{1/2} \frac{1}{2f_\rho}
\int \mathrm{d}
b \; \mu
J_1(\mu b) \left(1 + \frac{m_f^2 -\nabla^2}{M_\rho^2x(1-x)} \right) \phi_L(x,\zeta)~.
\label{tw2DAB}
\end{equation}
We note that $\int \mathrm{d} x \; \varphi(x,\mu \to \infty)=1$ recovers the decay
constant constraint. To compare to predictions
using QCD Sum Rules \cite{Ball:2007zt} and from the lattice \cite{Boyle:2008nj}, we
can also compute the moment:
\begin{equation}
\int_0^1 \mathrm{d} x \; (2x-1)^2 \varphi(x,\mu)~.
\end{equation}
We obtain a value of $0.228$ for the AdS/QCD wavefunction, which is to be compared with the
Sum Rule result of $0.24 \pm 0.02$ at $\mu = 3$~GeV \cite{Ball:2007zt} and the lattice
result of $0.24\pm 0.04$ at $\mu = 2$~GeV \cite{Boyle:2008nj}. The AdS/QCD
wavefunction neglects the perturbatively known evolution with the
scale $\mu$ and should be viewed as a parametrization of the DA at
some low scale $\mu \sim 1$ GeV. Viewed as such, the agreement is
good.
\begin{figure}
\includegraphics[width=0.8\textwidth]{contour.pdf}
\caption{The $\chi^2$ distribution in the $(\beta,\kappa)$ parameter space. The AdS/QCD prediction is the white star.}
\label{fig:contour}
\end{figure}
\section{Acknowledgements}
We thank Stan Brodsky, Mike Seymour and Guy de T\'eramond for their
helpful comments and suggestions.
The work of JRF is supported by the Lancaster-Manchester-Sheffield
Consortium for Fundamental Physics under STFC grant ST/J000418/1. R.S thanks the University of Manchester and the Institute for Nuclear Theory at the University of Washington for hospitality and financial support.
|
1,314,259,995,780 | arxiv | \section{Introduction}
As the number of optimization methods, and implementations of those methods, has increased, researchers have pursued comparative studies to evaluate their performance. When done well, such studies can be of great value in helping end-users choose the most suitable optimization method for their problems. Such studies are generally referred to as optimization benchmarking.
In the most general sense, benchmarking is the comparison of one or more products to an industrial standard product over a series of performance metrics. In the case of benchmarking optimization algorithms, the products are the specific implementations of given algorithms, and the performance metrics are generated by running the implementations on a series of test problems. This framework presents a certain clarity in benchmarking optimization algorithms, as there is at least some agreement on what constitutes ``better''. If one algorithm runs faster, uses less memory, and returns a better final function value, on all possible problems, then it can be considered better than the alternative. Of course, in practice such a clear conclusion seldom arises. Thus, interpreting the conclusions of algorithmic comparisons can be tricky.
Nonetheless, when done well, benchmarking optimization algorithms can have great practical value. It can reveal both strengths and weaknesses of an algorithm, which allows for better research focus. It can aid in determining if a new version of optimization software is performing up to expectations. And, it can help guide end-users in selecting a good choice of algorithm for a particular real-world problem.
However, when done poorly, benchmarking optimization algorithms can also be misleading. It can hide algorithm's weaknesses (or strengths), report improvements that do not exist, or suggest the incorrect algorithmic choice for a given situation.
In optimization benchmarking many subjective choices are made, such as the test set to solve, the computing environment to use, the performance criteria to measure, etc. Our primary objective in this paper is to help researchers to design a proper benchmarking approach that is more comprehensive, less biased, and less subject to variations within a particular software or hardware environment. Our secondary objective is to provide a comprehensive review of the benchmarking literature for optimization algorithms.
In pursuing these objectives, we focus on single-objective optimization algorithms that run in serial (i.e., that do not use parallel processing). Comparing algorithms for multi-objective optimization, or optimization algorithms that use parallel processing, introduce new levels of complexity to the benchmarking process. While we provide some comments on the challenges for benchmarking some algorithms in the conclusions, we consider these issues outside of the scope of this paper.
We also note that much of the presentation within this paper discusses algorithms as if the underlying optimization problem is a continuous unconstrained problem. This is for ease of presentation, and in most cases translating the ideas to other styles of optimization problems is clear. As such, we limit ourselves to discussing specific styles of optimization problems only when the translation is not straight-forward.
\subsection{Historical overview of benchmarking in optimization}
We begin with a brief historical overview of optimization benchmarking.
One of the very first studies in benchmarking of algorithms was given by Hoffman et al., in 1953 \cite{117}, in which three different methods for linear programming were compared. Although this computational experiment was performed early in the development of computers, when there existed almost no compiler and programming environment, the reported techniques have been used for a long time and can be considered as the foundation of the current comparison techniques. They include such ideas as, using test sets to compare algorithms, employing performance measures (accuracy, CPU time, number of iterations, and convergence rate), and paying attention to the impact of coding on the performance of algorithms.
Another early contribution to the field of benchmarking is Box's work from 1966 \cite{26}. In this work, Box evaluates the performances of eight algorithms for unconstrained optimization using a collection of 5 test problems with up to 20 variables. He considered the number of function evaluations, the importance of model size and the generality of the optimization algorithms.
In the late 1960's, optimization benchmarking research began to expand rapidly. Comparative studies have been performed throughout the optimization literature, for example in unconstrained optimization \cite{59, 53, 23}, constrained optimization \cite{60, 56, FamularoPuglieseSergeyev2002}, nonlinear least squares \cite{54, 55, 19, 109}, linear programming \cite{104, 145, 47}, nonlinear programming \cite{103, 54, 62, 51, 106, 107, 108, 105, 18, 19, 17, 21, 75, 102, 70}, geometric programming \cite{64, 58}, global optimization \cite{88, 78, 123, 80, 73, 22, SS00, ZZ08, KM16}, derivative-free optimization \cite{82, 95, 99, SK06, PSKZ14}, and other areas of optimization \cite{59, 53, 61, 120, 11,148, 68} -- amongst many more.
In addition, a few researchers have focused on improving the (optimization) benchmarking process. In 1979, Crowder et al.~\cite{38}, presented the first study that attempted to provide standards and guidelines on how to benchmark mathematical algorithms. It includes a detailed discussion of experimental design and noted the necessity of {\em a priori} experimental design. The authors paid attention to reproducibility of the results and provided a method for reporting the results. In 1990, similar research conducted by Jackson et al.\ \cite{9} delivered an updated set of guidelines. In 2002, Dolan and Mor\'e introduced performance profiles \cite{2}, which have rapidly become a gold standard in benchmarking of optimization algorithms with more recent work pointing out its limitations~\cite{GOULD-16}. In this paper, we attempt to provide a modern picture of best-practice in the optimization benchmarking process.
\subsection{Paper framework}
We now provide a general framework for benchmarking optimization algorithms, which we also use to structure discussions in the paper.
\begin{enumerate}
\item \textbf{Clarify the reason for benchmarking.} In Section \ref{sec:reason}, we discuss some of the common reasons to compare optimization algorithms, and some of the pitfalls that arise when the purpose of benchmarking is unclear.
\item \textbf{Select the test set.} In Section \ref{sec2}, a review of test sets for various problem categories is presented, the challenges related to test sets are discussed, and some guidelines are provided for assembling an appropriate test set.
\item \textbf{Perform the experiments}. In Section \ref{sec3}, we review and discuss various considerations related to the critical task of designing experiments, including performance measures, tuning parameters, repeatability of the experiments, and ensuring comparable computational environments.
\item \textbf{Analyze and report the results}. Section \ref{sec4} contains a review of different reporting methods for optimization algorithms, including tabular methods, trajectory plots, and ratio-based plots (such as performance and data profiles).
\end{enumerate}
In addition to the aforementioned sections, Section \ref{sec5} contains a review of recent advances in the field of automated benchmarking and Section \ref{sec6} presents some concluding thoughts.
\section{Reason for benchmarking}\label{sec:reason}
Having a clear understanding of the purpose of a numerical comparison is a crucial step that guides the rest of the benchmarking process. While seemingly self-evident, it is surprisingly easy to neglect this step. Optimization benchmarking has been motivated by a variety of objectives. For example:
\begin{enumerate}
\item To help select the best algorithm for working with a real-world problem.
\item To show the value of a novel algorithm, when compared to a more classical method.
\item To compare a new version of optimization software with earlier releases.
\item To evaluate the performance of an optimization algorithm when different option settings are used.
\end{enumerate}
In a practical sense, all of these work towards gathering information in order to rank optimization algorithms within a certain context. However, the context can, and should, play a major role in guiding the rest of the benchmarking process.
For example, if the goal is to select the best algorithm for a particular real-world application, then the test problems (Section \ref{sec2}) should come from examples of that application.
Alternately, if the goal is to show the value of a new optimization algorithm, then it is valuable to think about exactly where the algorithm differs from previous methods. Many new algorithms are actually improvements on how a classical method deals with some aspect of an optimization problem. For example, in \cite{c14} the authors develop a new method to deal with nonconvexity when applying a {\em proximal-bundle method} to a {\em nonsmooth optimization problem}. As such, to see the value of the method, the authors compare it against other proximal-bundle methods on a collection of nonconvex nonsmooth optimization problem. If they had compared their method against a quasi-Newton method on smooth convex optimization problems, then very little insight would have been gained.
Regardless of the reason, another question researchers must consider is what aspect of the algorithm is most important. Is a fast algorithm that returns infeasible solutions acceptable? Is it more important that an algorithms solves every problem, or that its average performance is very good? Is the goal to find a global minimizer, or a highly accurate local minimizer? The answers to these questions should guide the choice of performance metrics that need to be collected (Section \ref{sec3}) and how they should be analyzed (Section \ref{sec4}). Answering these types of questions before running the experiments is time well spent.
\section{Test sets} \label{sec2}
A test problem contains a test function together with some further criteria such as the constraint set, feasible domain, starting points, etc. A test set is a collection of test problems. Obviously, benchmarking only yields meaningful results when competing algorithms are evaluated on the same test set with the same performance measures.
The selection of the appropriate test sets to benchmark the performance of optimization algorithms is a widely noticed issue among researchers \cite{72, 8, 9, SSL13, ZZ08}. Generally, there are three sources for collecting test problems: real-world problems, pre-generated problems, and randomly-generated problems. Real-world problems can be found through instances of specific applications, and pre-generated problems exist in common test set libraries, see Table \ref{Table2}. Conversely, randomly-generated test problems are often more {\em ad hoc} in nature, with researchers creating methods to randomly generate test problems that are only used in a single paper \cite{13, c14, c19} (among many others). However, some researchers have gone to the effort to study methods to randomly-generate test problems for a given area; some examples appear in Table \ref{Table2b}.
While the real-world test sets provide specialized information about the performance of the optimization algorithms within a specific application, the results may be difficult to generalize. The difficulties lie in the facts that real-world test sets are often small and the problems are often application-specific. Nonetheless, if the goal is to determine the best algorithm to use for a particular real-world application, then a real-world test set focused on that application is usually the best option.
On the other hand, the artificial and randomly-generated test sets can provide useful information about the algorithmic characteristics of optimization algorithms. Artificial and randomly-generated test sets can be extremely large in size, thereby providing an enormous amount of comparative data. However, it can be difficult to rationalize their connection to the real-world performance of optimization algorithms. If the goal is to compare a collection of algorithms across a very wide spectrum, then artificial and randomly-generated test sets are usually the better option.
When selecting a test set, it is always important to keep the particular goal of the comparison in mind. Regardless of the goal, an appropriate test set should generally seek to avoid the following deficiencies.
\begin{enumerate}
\item[i.] \textbf{Too few problems.} Having more problems in the test set makes the experiment more reliable and helps the results reveal more information about the strengths or weaknesses of the evaluated algorithms.
\item[ii.] \textbf{Too little variety in problem difficulty.} A test set containing only simple problems is not enough to identify the strengths and weaknesses of algorithms. In contrast, a test set which only has problems that are so difficult that no algorithm can solve them, clearly, does not provide useful information on the relative performance of algorithms.
\item[iii.] \textbf{Problems with unknown solutions.} When possible, it is better to use test problems where the solution is known. Depending on the analysis performed (see section \ref{sec4}), the ``solution'' could be interpreted as the minimum function value, or the set of global minimizers. Having access to the solution greatly improves the ability to evaluate the quality of the algorithmic output. However, when the test set is comprised of real-world test problems, then a lack of known solutions may need to be accepted as inevitable.
\item[iv.] \textbf{Biased starting points.} Allowing different algorithms to use different starting-points will obviously bias the result. However, more subtle problems can also exist. For example, if a starting point lies on the boundary of a constraint set, then an interior point method will be severely disadvantaged. Another example comes from considering the Beale test function, which has a global minimizer at $(3,0.5)$ \cite{23}. If a {\em compass search} (see, e.g., \cite{KoldaLewisTorczon2003}) with an initial step length of $1$ is started at $(0.5, 0.5)$, then it will converges to the exact minimizer in just 4 iterations. However, if a starting point of $(0.51, 0.51)$ is used, then the exact same algorithm will use $63$ iterations to reach a point within $10^{-2}$ of the global minimizer.\footnote{Note that this example is artificially constructed to emphasize the results; the recommended starting point for the Beale test problem is $(1,1)$.}
\item[v.] \textbf{Hidden structures.} Many test sets have some structure that is not realistic in real-world problems. For example, about 50\% of the problems in the test set \cite{23} have solution points that occur at integer valued coordinates. An algorithm that employs some form of rounding may perform better than usual on these problems.
\end{enumerate}
Thus, when choosing test sets for the benchmarking task the following considerations should be taken into account as much as possible.
\begin{enumerate}
\item[i.] If the test set contains only few problems, then the experiment should be referred to as a \textit{case study} or a \textit{proof of concept}, but not benchmarking. While there is no fixed number that determines what is ``enough problems to be considered benchmarking'', we recommend that in order to achieve a reliable conclusion about the performance, an experiment should contain at least 20 test problems (preferably more). In the specific case of comparing a new version of an optimization algorithm with a previous version, the number of test problems should be significantly greater -- in the order of 100 or more. In all cases, the more problems tested, the more reliable the conclusions.
\item[ii.] When possible, a test set should include at least two groups of problems: an \textit{easy group} which consists of the problems that are easy to solve within a reasonable time on a regular contemporary computer using all the optimization algorithms tested, and a \textit{hard group} that contains the problems which are solvable but computationally expensive and may require a specific optimization algorithm.
\item[iii.] Whenever possible, ensure at least a portion of the test set includes problems with known solutions.
\item[iv.] For test sets that include starting points, new starting points can be generated by introducing a small (random) perturbations to the given starting points. For other test sets, randomly-generated starting points can be created from scratch. In either case, all starting points should be created for each problem, and then every algorithm should be provided the same starting point for testing. This approach can be further used to increase result reliability, by repeating tests on the same function with a variety of starting points.
\item[v.] Examine the test set with a critical eye and try to determine any hidden structure. Some structures can be removed through methods similar to the random perturbation of starting points in (iv). One quick test is to set an algorithm to minimize $f(x)$ starting at $x^0$ and then set the algorithm to minimize $\hat{f}(x) = f(x - p)$ starting from $\hat{x}^0 = x^0 - p$ (where $p$ is any random point). Constraints can then be shifted in a similar manner, effectively shifting the entire problem horizontally by the vector $p$. While it relocates the origin, and moves any constraints away from special integer values, it has no theoretical effect on the geometry of the problem. As such, the results of both tests should be extremely similar (theoretically they should be identical, but numerical errors may cause some deviation). If the results of both tests differ, then perhaps some hidden structure is being exploited by the algorithm, or perhaps some hidden constraints are causing issues. Regardless of the reason, the researcher should recognize the issue and consider a wider test set.
\end{enumerate}
Using suitable standard test sets is usually a good option when benchmarking optimization algorithms. In particular, it is usually easier to compare results across research groups when standard tests are employed, although even within a specific research field there is generally no consensus on the appropriate test set to draw specific conclusions. Many interesting and diverse test sets have been reported in the literature, see Tables \ref{Table2} and \ref{Table2b}.
\begin {table}[H]
\captionof{table}{Some test sets reported in the literature.\label{Table2}}
\begin{center}
\begin{tabularx} {\textwidth}{ss}
\toprule[1.5pt]
\head{Collection type} & \head{Resources} \\
\toprule[1.5pt]
Unconstrained optimization problems & \cite{4, 23, 63, 118} \\ \midrule
Global optimization & GAMS \cite{72}, COCONUT \cite{66}, and other collections
\cite{98, 90, 89, 65, 94, 96, FamularoPuglieseSergeyev2002} \\ \midrule
Linear programming & \cite{c20} \\ \midrule
Local optimization &\cite{90, 98} \\ \midrule
Nonlinear optimization problems & CUTEr \cite{85,86}, CUTEst \cite{111}, COPS \cite{6,8}, and collections \cite{52, 83, 84, 57, 142} \\ \midrule
Mixed integer linear programming & MIPLIB \cite{PEKO2002, 49} \\
\bottomrule[1.5pt]
\end{tabularx}
\end{center}
\end{table}
Producing random test sets using test problem generators has its own drawbacks. Are the generated problems representative or difficult? Is there any hidden structure in the problems? Some papers that use random test problem generators are listed in Table~\ref{Table2b}.
\begin {table}[H]
\captionof{table}{Some test problem generators reported in the literature.\label{Table2b}}
\begin{center}
\begin{tabularx} {\textwidth}{ss}
\toprule[1.5pt]
\head{Test problem generators} & \\
\toprule[1.5pt]
Network programming & \cite{157}\\\midrule
Nonlinear optimization problems & \cite{51,91} \\ \midrule
Combinatorial problems & \cite{131, 157} \\
\midrule
Quadratic Programming & \cite{156} \\
\midrule
Global Optimization & \cite{89, 93, 94, NL14} \\
\bottomrule[1.5pt]
\end{tabularx}
\end{center}
\end{table}
Figure \ref{Fig06TestSet} shows a decision tree that summarizes the fundamental decisions required for assembling an appropriate test set for benchmarking of optimization algorithms.
\begin{figure}[ht]
\centering
\def\svgwidth{\columnwidth}
\footnotesize
\input{flowchart1.pdf_tex}
\caption{Test set decision tree.}
\label{Fig06TestSet}
\end{figure}
\section{Performing the experiments}\label{sec3}
The performance of algorithms is influenced by two general types of factors: \textit{environmental factors} and \textit{algorithmic factors}.
\textit{Environmental factors} refer to factors that are out of the algorithm scope and usually beyond the control of the researcher. A common example is the computer environment used to test the algorithms, which includes processor speed, operating system, computer memory, etc. Environmental factors may also include the programmer's skill and the programming language/compiler used. This is particularly evident when multiple pieces of software by a variety of programmers are being compared. In essence, if the benchmarking process is repeated by another researcher elsewhere, then environmental factors are unlikely to remain constant, and so the benchmarking results are expected to change.
\textit{Algorithmic factors} are related to the algorithm itself. These are factors that are considered global across a variety of computing platforms. If the software is programmed by the researcher, then it is assumed these factors are independent from the implementation aspects of the algorithm.
\emph{Optimization benchmarking} seeks to measure the algorithmic factors, and proceeds under the key \emph{assumption} that, while environmental factors are expected to change the results, the algorithmic factors are sufficiently strong that the general ranking of algorithms should remain constant under the specific ranges of parameters under consideration.
To compare algorithms, it is necessary to collect data that measures the overall performance of each algorithm. This is done by running each algorithm on the test set (discussed in Section \ref{sec2}), and collecting data on the results. The data collection and the selection of performance measures is based on the research questions motivating the experimental study. In general, performance measures fall into 3 categories: efficiency, reliability, and quality of algorithmic output. We discuss these performance categories in Subsections \ref{ss:efficiency}, \ref{ss:reliability}, and \ref{ss:qualityofsolution}. Table \ref{Table3} provides a classification of the comparative measures for optimization algorithms based partly on the guidelines provided by Hoffman and Jackson \cite{124}.
\begin {table}[H]
\caption{Comparative measures.}
\centering
\begin{tabularx} {\textwidth}{lX}
\toprule[1.5pt]
\head{Performance Category} & \head{Example Criteria} \\
\toprule[1.5pt]
& 1. Number of fundamental evaluations \\
Efficiency & 2. Running time \\
& 3. Memory usage \\
\toprule[1.5pt]
& 1. Success rate \\
Reliability & 2. Number of constraint violations \\
& 3. Percentage of global solutions found \\
\toprule[1.5pt]
& 1. Fixed-cost solution result \\
Quality of Solution & 2. Fixed-target solve time \\
& 3. Computational accuracy \\
\toprule[1.5pt]
\end{tabularx}
\label{Table3}
\end{table}
In order to allow for maximal data analysis (and thereby the best understanding of overall performance), it is recommended to collect at least some data from every performance category.
\subsection{Efficiency}\label{ss:efficiency}
The efficiency of an optimization algorithm refers to the computational effort required to obtain a solution. In mathematical programming, there are two primary measures of efficiency, the number of fundamental evaluations and the running time. A third, less common, measure is memory usage.
\textbf{Number of fundamental evaluations}: The term fundamental evaluation is used to refer to any subroutine that is called by the algorithm in order to gain fundamental information about the optimization problem. The most obvious example is an objective function evaluation but the evaluation may involve complex simulation algorithms. Other fundamental evaluations could include gradient evaluations, Hessian evaluations, or constraint function evaluations. The number of fundamental evaluations can be used as a standard unit of time, and is often assumed to be platform independent. In many situations, the number of fundamental evaluations is a particularly important measure, as for real-world problems these evaluations often dominate over the internal workings of the algorithm
\cite{27, 38, c3, 7, 25, 13, 36,53, 61, 62, 66, 73, 105, 129, E85, PSKZ14, KS15}.
Note however that this measure is unreasonable when fundamental evaluations do not dominate the internal workings of the algorithm \cite{4}.
\textbf{Running Time}: Running time, as a measure for optimization benchmarking, is usually measured by either CPU time or wall clock time.\footnote{\textit{Wall clock time} refers to the amount of time the human tester has to wait to get an answer from the computer. Conversely, \textit{CPU time} is the amount of time the CPU spends on the algorithm, at the exclusion of operating system tasks and other processes.} Wall clock time contains CPU time, and has been argued to be more useful in real-world setting \cite{109}. However, Wall clock time is not reproducible or verifiable since it is tied to a specific hardware platform and software combination. CPU time is considerably more stable, as it is independent of background operations of the computer. Moreover, CPU time is more-or-less consistent for the same version of an operating systems running on the same computer architectures.
It should be noted that, due to the wide variety and complexity of modern computing architectures, the number of situations in which time is dominated by memory access costs is increasing, hence the precision of CPU timers has been reduced. To improve the precision of CPU timers, tools such as cache and memory access tracers can help obtaining a more accurate CPU time performance. For a more detailed discussion on these techniques we refer to \cite{c1, 112}.
Another issue with CPU time is the increasing prevalence of multi-core machines. Thorough reporting would require indicating the number of cores, the CPU time for each core, but also how efficiently the different levels of memory were used and cache hits/misses. Since such measurements are not straightforward to obtain for multi-core machines, the wall-clock time along with the hardware specifications are usually reported. (Unless the new algorithm contribution focuses specifically on optimizing computation for a multi-core architecture, in which case more precise measures are warranted.) Eventually, the onus is on the researchers to explain how simplified measurements support the conclusions drawn; this is especially true for multi-core machines.
Regardless of whether wall clock time or CPU time is used, in order to maximize the reliability of the data, it is important to ensure that any background operations of the computer are kept to a minimum. Furthermore, any manuscript regarding the benchmarking should clearly state which form of running time was collected.
\textbf{Other measures:} In addition to the categorization presented above, in some specific cases, there is another issue that influences the choice of an appropriate measure for running time: \textit{the type of algorithm}. For example, to evaluate the running time for branch-and-bound based algorithms, the number of branch-and-bound nodes is a common criterion, while for simplex and interior point based algorithms, the number of iterations is often used. Therefore, when deciding on the choice of a suitable efficiency measure, the type of algorithms to be evaluated should also be taken into account.
\subsection{Reliability}\label{ss:reliability}
The reliability and robustness of an optimization algorithm is defined as the ability of the algorithm to ``perform well'' over a wide range of optimization problems \cite{23, 81}. The most common performance measure to evaluate the reliability is success rate \cite{c4, 58, 105, SS00}. Success rate is gauged by counting the number of test problems that are successfully solved within a pre-selected tolerance. This can be done using objective function value, or distance of the solution point from a minimizer. In convex optimization these two approaches are largely, but not perfectly, interchangeable. However, if the objective function has multiple local minimizers, then the researcher must decide whether good local solutions are acceptable outcomes, or if the algorithm must converge to a global minimizer \cite{51, 55}. In addition to the success rate, the average objective function values and the average constraint violation values have also been reported to measure reliability \cite{51}.
When studying reliability, the researcher should consider whether the algorithms are deterministic, or non-deterministic, and repeat tests multiple times if the algorithm is non-deterministic. Reliability can be based on a fixed starting point (if one is given with the test set), but it is often better to use multiple starting points.
In deterministic optimization algorithms, reliability can be interpreted as the number of problems in the given test set that are solved by the optimization algorithm. When dealing with non-deterministic algorithms, it is important to repeat each test multiple times, to make sure that reliability is measured in aggregate, and not skewed by a single lucky (or unlucky) algorithmic run.
Using multiple repeats of each test raises the issue of how to aggregate the results. One option is to consider each algorithmic run as a separate test problem and then compare solvers across this expanded test set. This allows comparisons based on worst-case or best-case scenarios. Another option is to use averaged data, for example, average runtime, average solution accuracy, average reliability, etc. If averaging is used, then it is important to also include standard deviations of the data. In either case, data collection is best performed by considering each algorithmic run as a separate test problem, as average values can easily be extracted from this data, while reconstructing the full test data from averaged values is not possible.
It should be noted that, in some cases multiple repeats of a non-deterministic method is impractical due to the time it takes to solve a single problem.
\textbf{Multiple starting points:} As mentioned in Section \ref{sec2}, many academic test problems come with suggested starting points. While algorithms should always be tested using these starting points, it is often enlightening to test the algorithm using other starting points as well. Most deterministic algorithms should show little change in performance if a starting point is perturbed by a small random vector -- provided the new starting point retains whatever feasibility properties the algorithm requires in a starting point.
Hillstrom \cite{15} is one of the first to consider testing optimization algorithms at nonstandard starting points. He proposed using random starting points chosen from a box surrounding the standard starting point. In another approach to this problem, in \cite{23} the authors present a large collection of test functions along with some procedures and starting points to assess the reliability and robustness of unconstrained optimization algorithms. In some cases, prior knowledge is available about the solution of a test problem. Some methods use such information to construct a starting point close to the optimal solution \cite{150}.
Regardless of how starting points are selected, fair benchmarking requires all the algorithms to use the same starting point for each test. Therefore, starting points should be generated and stored outside of the testing process.
\subsection{Quality of algorithmic output}\label{ss:qualityofsolution}
The quality of the algorithmic output is obviously important when comparing optimization algorithms. Measuring quality falls into two easily separated categories: a known solution is available, and no known solutions are available.
\textbf{Known solution available:} When the expected solution for a problem is available, two methods can be employed to measure the quality of an algorithmic output: fixed-target and fixed-cost \cite{1, 41, 95}.
In the \textit{fixed-target} method, the required time (function calls, iterations, etc) to find a solution at an accuracy target $\varepsilon_{target}$ is evaluated. The main problem with fixed-target methods is that some algorithms may fail to solve a test problem. Therefore, the termination criterion cannot rely only on accuracy, but should also include some safety breaks such as the maximum computational budget. If the algorithm successfully reaches the desired accuracy, then the time to achieve the accuracy can be used to measure the quality of the algorithm on that test problem. If the algorithm terminates before reaching the desired accuracy, then it should be considered unsuccessful on that test problem.
Let $x^0$ be the initial point from a test run, $\bar{x} \in \mathbb{R}^n$ be the termination point obtained from the test run, and $x^* \in \mathbb{R}^n$ be the known solution for the problem. In the \textit{fixed-cost} approach, the final optimization error $f(\bar{x})-f(x^*)$ is checked after running the algorithm for a certain period of time, number of function calls, number of iterations, or some other fixed measurement of cost. Then, the smaller the final optimization error is, the better the quality of the algorithmic output.
The fixed-target versus fixed-cost decision can be seen as a multiobjective problem. It is analogous in engineering to minimizing cost, subject to constraints on performance versus maximizing performance, subject to a constraint on cost.
If a fixed-cost approach is used, then there are many options on how to quantify the accuracy of the algorithmic output. We need to determine whether $\bar{x}$ approximates $x^*$ or not. For example, this can be done using the function value or the distance from the solution:
$$f_{\tt acc} = f(\bar{x}) - f(x^*), \quad \mbox{and} \quad x_{\tt acc} = \|\bar{x} - x^*\| ~~\mbox{respectively}.$$
It is often valuable to ``normalize'' these quantities by dividing by the starting accuracy:
$$f_{\tt acc}^n = \frac{ f(\bar{x}) - f(x^*)}{f(x^0) - f(x^*)}, \quad \mbox{and} \quad x_{\tt acc}^n = \frac{\|\bar{x} - x^*\|}{\|x^0 - x^*\|}.$$
Finally, to improve readability, and reduce floating point errors, many researchers take a base-10 logarithm:
$$\begin{array}{rcl}
f_{\tt acc}^l &=& \log_{10}(f(\bar{x}) - f(x^*)) - \log_{10}(f(x^0) - f(x^*)), \quad \mbox{and} \\
x_{\tt acc}^l &=& \log_{10}(\|\bar{x} - x^*\|) - \log_{10}(\|x^0 - x^*\|).\end{array}$$
The values $f_{\tt acc}^l$ and $x_{\tt acc}^l$ can be loosely interpreted as the negative of the number of new digits of accuracy obtained (measured on a continuous scale), thus making these values very useful for discussion. Finally, to avoid exact solutions making an algorithm look better than it is, one can select a ``maximal improvement value'' $M$ (typically about 16) and set
\begin{equation}
\gamma= \begin{cases}
-f_{\tt acc}^l, & \mbox{if}~ -f_{\tt acc}^l \leq M \\
M, & -f_{\tt acc}^l > M ~\mbox{or}~ f(\bar{x}) - f(x^*) = 0,
\end{cases}\label{eq:accuracymeasure}
\end{equation}
or the analogous equation using $x_{\tt acc}^l$. Note that we have multiplied $f_{\tt acc}^n$ by $-1$, so $\gamma$ can be interpreted as the number of new digits of accuracy obtained up to a maximal improvement value of $M$.
Similar measures can be used to quantify the amount of constraint violation for a test run. Considering $\min \{ f(x) ~:~ g_i(x) \leq 0, i = 1, 2, ..., m\}$,
$$\begin{array}{ll}
\displaystyle\sum_{i=1}^m \max\{0, g_i(\bar{x})\} &\mbox{gives the sum of violated constraints},\\
\displaystyle\sum_{i=1}^m (\max\{0, g_i(\bar{x})\})^2 &\mbox{gives the squared sum of violated constraints},\\
\displaystyle\frac{1}{m} \sum_{i=1}^m \max\{0, g_i(\bar{x})\} &\mbox{gives the mean constraint violation, and}\\
\displaystyle\prod_{i : g_{i}(\bar{x}) > 0} g_i(\bar{x}) &\mbox{amounts to the geometric mean of the violated constraints}.
\end{array}$$
The selection of the appropriate strategy among the variety of approaches depends on the objectives of the experimental research, the problem structure, and the type of optimization algorithms used. The researcher should also carefully select the success criteria, e.g., how to fairly compare a solution that barely satisfies the constraints versus a solution that barely violates the constraints but returns a much lower objective function value.
\textbf{No known solution available:} In many situations, the test set used will not have known solutions to all problems. This is particularly true if the test set includes instances of real-world applications. To evaluate the quality of an algorithmic output in this situation, some new considerations are required \cite{114,115}.
First, it should be immediately obvious that, if no known solution is available, then fixed-target approaches cannot be applied. Fixed-cost approaches are still applicable, but since $f(x^*)$ is not known, measuring the accuracy of the final algorithmic output's function value, $f(\bar{x})$, becomes difficult. Measuring the accuracy of the final algorithmic output's point, $\bar{x}$, becomes essentially impossible.
To measure the quality of the final algorithmic output's function value $f(\bar{x})$, the simplest approach is to replace $f(x^*)$ with the best known value for the problem. For any given test run, this guarantees that at least one algorithm gets the exact answer, so it is important to select a reasonable maximal improvement value. Another approach is to estimate the optimal solution using statistical techniques. For example, in combinatorial optimization problems, some researchers \cite{c11, c12} use a sample of algorithmic outputs to predict the location of the solution. In \cite{c2}, such an approach is explained in an evaluation of non-deterministic algorithms. Another strategy is to calculate a lower bound on the cost of an optimal solution, and to compare the algorithmic output cost with that lower bound. As an example, the total sum of the weight list in packing problems can be considered as a lower bound on the total number of bins used in a packing. Finally, one may abandon comparing the algorithmic output quality with the optimal solution, and only assess the quality of the algorithmic output with similar results published in the literature or other algorithms being tested.
\subsection{Parameter tuning and stopping conditions}
Additional parameters, such as stopping tolerances, population size, step sizes, or initial penalty parameters, are required for most optimization algorithms.
Among such parameters, stopping conditions play a highly notable role, as different stopping conditions can drastically change the output of an algorithm \cite{SS00, SSL13, ZZ08}. Moreover, if stopping tests are internalized within a method, it may not be possible to ensure all algorithms use the same stopping conditions \cite{SS00, SK06}. However, if a fixed-cost or fixed-target approach (see Subsection \ref{ss:qualityofsolution}) is employed, then other stopping conditions can be turned off, thereby ensuring all algorithms use the same stopping conditions. If it is not possible to ensure all algorithms use the same stopping conditions, then researchers should recognize this potential source of error when drawing conclusions from the results.
Other parameters, such as initial step length, can also have an impact on the performance of an optimization algorithm. Such parameters often require tuning in order to obtain a better performance. If different choices of input parameters are allowed in an algorithm, researchers should mention the parameter settings used and how they were selected. Different strategies used for tuning parameters affect the benchmarking process. Choosing appropriate parameter settings for an optimization algorithm is usually based on experiments and statistical analysis.
The tuning strategy should be chosen in conjunction with a specific algorithm and in a replicable manner \cite{115}. Any improvements obtained from hand-tuning can of course be reported, but separately from more systematic comparative experiments. In some studies, algorithmic methods are presented to automate the tuning procedure of parameters \cite{AudetOrban2006, AudetDangOrgan2014, 145, 147, 121, 76}. The major disadvantage of these tuning methods is that they require a considerable computational investment because they usually try many possible settings to find an appropriate one. Nonetheless, in recent years some studies have specifically focused on automatic tuning of parameters in optimization solvers. Examples of these efforts include the machine learning based method proposed in \cite{145}, CPLEX automatic tuning tool \cite{c13}, use of derivative-free optimization \cite{AudetOrban2006}, ParamILS \cite{147}, and the procedure proposed in \cite{146} for mixed integer programming solvers. Similarly, some of the tuning techniques for non-deterministic methods include sequential parameter optimization (SPO) \cite{143, 113}, relevance and calibration approach \cite{144}, and F-Race \cite{c9}.
In view of the considerable research on automatic tuning of optimization solvers, a more accurate approach for benchmarking of optimization solvers requires a pre-processing step in which an automatic tuning method is employed to find the suitable configuration settings for all the solvers \cite{82}. As this is not always practical, it is important to emphasize that tuning parameters can have a major impact on the performance of an algorithm, therefore it is not appropriate to tune the parameters of some methods while leaving other methods at their default settings.
\section{Analyzing and reporting the results} \label{sec4}
Many studies use basic statistics (e.g., average solving time) to report the experimental results. Basic statistics are a reasonable starting point, but provide little information about the overall performance of optimization methods. Reporting methods can be loosely broken down into three categories: numerical tables, graphics, and performance ratio methods (e.g., performance and data profiles).
\subsection{Tables}
Numerical tables provide the most complete method of reporting benchmarking results, so for the sake of completeness, we recommend making full tables of results readily available. However, such tables are cumbersome, so are often better included in an appendix or in additional online material linked to an article.
As full tables of results can easily overwhelm a reader, researchers have developed various techniques that provide easy-to-understand and compact methods for reporting the experimental results. Summary tables can be employed as a first step \cite{SK06}. For example, in \cite{5} optimization methods were rated by the percentage of problems for which a method's time is termed \textit{competitive} or \textit{very competitive}. The solving time of an algorithm was called competitive if $T_s\le 2 T_{min}$ in which $T_s$ is the solving time obtained by an algorithm on a particular problem and $T_{min}$ is the minimum solving time obtained among all the algorithms on that specific problem. Similarly, if $T_s\le \frac{4}{3} T_{min}$, then they call that method very competitive. Tables such as these provide good talking points for discussing benchmarking data, but fail to give a complete picture of the results. One critic for this particular approach is it does not explore how much the table would change if, for example, the cut-off for very competitive was changed from $\frac{4}{3} T_{min}$ to $\frac{5}{4} T_{min}$.
Many other forms of summary tables are present throughout optimization benchmarking literature, however all suffer from the same fundamental problem -- to be readable a summary table must distill the results down to a highly condensed format, thereby eliminating much of the benchmarking information.
\subsection{Graphics}
Well-conceived graphics can provide more information than some other data presentations. Simple graphical methods, such as histograms, box-plots, and trajectory plots, provide a next step in analysis, while more complete methods include performance profiles, data profiles, and accuracy profiles. Depending on the objectives of an experimental research, one or more of these techniques might be useful to report the results. In \cite{132, 133}, different types of plots are introduced, which are useful for data representation in general.
\begin{figure}[ht]
\captionof{figure}{A sample trajectory plot.}
\centering
\fbox{\includegraphics[width=13cm]{trajectoryPlot}}
\label{Fig00}
\end{figure}
A more specialized plot for optimization algorithms is the trajectory plot \cite{1, 22, 108, 105, 106, c8, SS00}. In a trajectory plot, the performance of an optimization algorithm on a given test problem is visualized by plotting a path that connects the points generated by each iteration of the algorithm. An example appears in Figure \ref{Fig00}, where the trajectories of two algorithms attempting to minimize the Rosenbrock function are plotted. Both algorithms begin at the point $(3, 3)$, and the first iteration moves both algorithms to the point $(0.2, 3.5)$. Algorithm 1 (represented by the solid line) proceeds to $(0.7, -0.2)$ and continues in a zig-zag path towards the minimizer. Algorithm 2 (represented by the dashed line) proceeds to $(1.1, 1.3)$ and then follows a fairly straight path towards the minimizer, albeit with very small step sizes. While trajectory plots are useful to build a better understand of how each algorithm behaves, they are not particularly good for benchmarking as they can only present the results for one test problem at a time. They are also limited to plots of functions of 2 or 3 variables, or to plotting projections onto subspaces for more than 3 variables.
Another specialized plot for optimization benchmarking is the {\em convergence plot}. In a convergence plot the performance of different optimization methods is visualized by plotting the best function value found against some measure of fundamental evaluation (Section \ref{ss:efficiency}). An example convergence plot is given in Figure \ref{Fig01}.
\begin {figure}[H]
\captionof{figure}{A sample convergence plot.}
\centering
\fbox{\includegraphics[scale=0.5]{ConvergencePlot}}
\label{Fig01}
\end {figure}
In Figure \ref{Fig01} the results of four optimization methods are plotted for a given test problem. In this example, method M1 starts well, but stalls after about 300 function evaluations, while method M2 shows steady decrease for about 800 function evaluations before stalling. Method M3 initially decreases the fastest, but stalls after about 350 function evaluations. Finally, method M4 starts very slowly, but ultimately finds the lowest value. Like trajectory plots, convergence plots are useful for discussing some specific behavior of the algorithm, but are poor for benchmarking as they can only be used to analyze one test problem at a time.
While trajectory and convergence plots are useful to visualize a method on one problem, their main drawback is that they represent the results for a single problem per plot. So if the test set contains a large number of problems then it will be difficult to evaluate the overall performance of these methods. Other types of plots can be found in the literature, but generally have the same limitations as trajectory and convergence plots \cite{17} \cite{22}.
For many optimization algorithms, researchers are interested in how the problem scales with the size of the input (e.g., dimension of the problem). For such research it can be valuable to produce a {\em runtime plot}. Runtime plots visualize the data by plotting the time to solve across a series of problem instances with different sizes. Runtime plots can suffer from a similar issues to trajectory and convergence plots, namely, they represent the results for a single series of problem instances. However, this problem can be somewhat mitigated by aggregating data from a collection of problems to create an ``average runtime'' plot.
\subsection{Performance profiles}
According to \cite{SKM16}, the idea of creating graphical comparison of optimization methods dates back to at least 1978 in the paper by Grishagin \cite{G78}.\footnote{We thank ``Mathematics Referee \#1'' for pointing out that reference.} In 2000, Strongin and Sergeyev presented the idea of {\em operational characteristics} for an algorithm: a graphical method to visualize the probability that an algorithm solves a problem within a set time-frame \cite{SS00}. However, it was not until the 2002 paper by Dolan and Mor\'e, \cite{2}, that the idea of graphically presenting benchmarking results became mainstream. Dolan and Mor\'e (apparently unaware of works of Grishagin or Strongin and Sergeyev) denoted their proposed graphs {\em performance profiles}.
Performance profiles provide interesting information such as efficiency, robustness, and probability of success in a graphically compact form \cite{2}. Their use has grown rapidly in optimization benchmarking, and should certainly be considered for any benchmarking optimization research.
Let $\mathcal{P}$ be a set of problems, $\mathcal{S}$ a set of optimization solvers, and $\mathcal{T}$ a convergence test. Assume proper data has been collected. Performance profiles are now defined in terms of a performance measure $t_{p,s} > 0$, obtained for each pair of $(p,s)\in P\times S$. This measure can be the computational time, the number of function evaluations, etc. A larger value of $t_{p,s}$ shows worse performance. For each problem $p$ and solver $s$, the performance ratio is defined as
\begin{equation}\label{05}
r_{p,s}=
\begin{cases}
\displaystyle \frac{t_{p,s}}{\min\{t_{p,s}:s \in \mathcal{S}\}} &\text{if convergence test passed,} \\
\displaystyle \infty &\text{if convergence test failed.}
\end{cases}
\end{equation}
for a specific problem $p$ and test $s$ (the best solver has $r_{p,s} = 1$). The \textit{performance profile} of a solver $s$ is defined as follows
\begin{equation} \label{06}
\rho_s({\tau})=\frac{1}{\lvert \mathcal{P} \rvert} \text {size} \{\mathit{p} \in \mathcal{P}: r_{p,s} \le \tau \},
\end{equation}
where $\lvert \mathcal{P} \rvert$ represents the cardinality of the test set $\mathcal{P}$. Then, $\rho_s({\tau})$ is the portion of the time that the performance ratio $r_{p,s}$ for solver $s \in \mathcal{S}$ is within a factor $\tau \in \mathbb{R}$ of the best possible performance ratio.
Note that $\rho_s({1})$ represents the percentage of problems for which solver $s \in \mathcal{S}$ has the best performance among all the other solvers. And for $\tau$ sufficiently large, $\rho_s({\tau})$ is the percentage of the test set that can be solved by $s \in \mathcal{S}$. Solvers with consistently high values for $\rho_s({\tau})$ are of interest.
Figure \ref{Fig02} shows a sample performance profile plot (created using data from \cite{c7}) for logarithmic values of $\tau$. The logarithmic values are employed to deal with smaller values for $\tau$. This will result in a more accurate plot which shows the long term behavior of the methods. To demonstrate the difference, Figure \ref{Fig02-2} shows the same performance profile using non-logarithmic values of $\tau$. Depending on the data collected, logarithmic or non-logarithmic values of $\tau$ may be more appropriate. Researchers should create both profiles, but it may be only necessary to provide one in the final manuscript.
\begin {figure}[H]
\caption{An example performance profile using logarithmic values of $\tau$.}
\centering
\fbox{\includegraphics[scale=0.5]{PprofileLog}}
\label{Fig02}
\end {figure}
\begin {figure}[H]
\caption{The performance profile from Figure \ref{Fig02} using non-logarithmic values of $\tau$.}
\centering
\fbox{\includegraphics[scale=0.5]{PprofileNormal}}
\label{Fig02-2}
\end {figure}
The performance profiles in Figure \ref{Fig02} compares four different optimization methods on a test set of 60 problems. The method M1 has the best performance (in terms of CPU time) for almost 93\% of the problems; meaning that M1 is able to solve 93\% of the problems as fast or faster than the other two approaches. M3 solves roughly 11\% of the problems as fast or faster than the other approaches. On the other hand, given enough time M1 solves about 92\% of all problems, while M3 solves about 94\% of all problems. The graphs of M1 and M3 cross at about $\log_2(\tau) \approx 3$ (i.e., $\tau \approx 8$), the two methods solve the same number of problems if time to solve is relaxed to be within a factor of $8$.
Since performance profiles compare different methods versus the best method, the interpretation of the results should be limited to comparison to the best method and no interpretation should be made between, e.g., the second best and third best method since a switching phenomenon may occur~\cite{GOULD-16}\footnote{We thank ``Engineer Referee \#3'' for pointing out that reference.}. To compare the second and third best method, a new performance profile should be drawn without the first method, see the explicit examples provided in~\cite{GOULD-16}.
Performance profile plots can be customized by substituting the standard performance measure \textit{time}. For example, in \cite{4, 134, 136}, the objective function value is used as the performance measure to compare the profiles. In particular, $t_{p,s}$ is replaced with
\begin{equation}
m_{p,s} = \frac{\hat{f}_{p,s}(\text{after $k$ function evaluations}) - f^*}{(f_w-f^*)},
\end{equation}
for problem $p$ and solver $s$, where $f_w$ is the largest (worst) function value obtained among all the algorithms, and $\hat{f}_{p,s}$ is the estimated function value after $k$ function evaluations. In another example, \cite{SK15} creates a performance measure based on proximity to optimal points.
The primary advantage of performance profiles is that they implicitly include both speed and success rate in the analysis. The value of $\rho_s({\alpha})$ gives a sense of how promising the algorithmic outputs are relative to the best solution found by all the optimization algorithms that are compared together.
One criticism of performance profiles is that the researcher must select a definition for the convergence test passing and failing. Changing this definition can substantially change the performance profile \cite{HareKochLucet2011}. Also note that if a fixed-cost approach is used to performing the benchmarking experiments, then performance profiles become inappropriate, as all algorithms will use the same ``time''. Another criticism is that the profile is only showing performance with respect to the best method and does not allow one to compare other methods with each other due to the appearance of a switching phenomenon~\cite{GOULD-16}.
Nonetheless, performance profiles have become a gold-standard in modern optimization benchmarking, and should be included in optimization benchmarking analysis whenever possible with an appropriate interpretation.
\subsection{Accuracy profiles}
Similar to performance profiles, \textit{accuracy profiles} provide a visualization of an entire optimization benchmarking test set. However, accuracy profiles are designed for fixed-cost data sets. They begin by defining, for each problem $p \in \mathcal{P}$ and solver $s \in \mathcal{S}$, an accuracy measure (similar to equation \eqref{eq:accuracymeasure}):
$$\gamma_{p, s} = \begin{cases}
-f_{\tt acc}^{p,s}, & \mbox{if}~ -f_{\tt acc}^{p,s} \leq M \\
M, & -f_{\tt acc}^{p,s} > M ~\mbox{or}~ f_{\tt acc}^{p,s} ~\mbox{is undefined},
\end{cases}$$
where $f_{\tt acc}^{p,s} = \log_{10}(f(\bar{x}_{p,s}) - f(x^*_p)) - \log_{10}(f(x^0_p) - f(x^*_p))$, $\bar{x}_{p,s}$ is the candidate solution point obtained by solver $s$ on problem $p$, $x^*_p$ is the optimal point for problem $p$, and $x^0_p$ is the initial point for problem $p$. The performance of the solver $s \in \mathcal{S}$ on the test set $\mathcal{P}$ is measured using the following function
$$
R_s (\tau)= \frac{1}{|\mathcal{P}|} \text {size} \{\gamma_{p,s} | \gamma_{p,s} \ge \tau, p \in \mathcal{P} \}.
$$
The accuracy profile $R_{s} (\tau)$ shows the proportion of problems such that the solver $s \in \mathcal{S}$ is able to obtain a solution within an accuracy of $\tau$ of the best solution. An example accuracy profiles (using data from \cite{Hare-Sagastizabal-2006}) appears in Figure \ref{Fig05}.
\begin {figure}[H]
\caption{An example accuracy profile.}
\centering
\fbox{\includegraphics[scale=0.5]{AccProf}}
\label{Fig05}
\end {figure}
In Figure \ref{Fig05}, we see 4 methods (M1, M2, M3, and M4) plotted against each other in an accuracy profile format. Examining the profile, notice that method M1 achieves 5 digits of accuracy on almost all test-problems, and 6 digits of accuracy on about 75\% of test problems. All other method achieve this level of accuracy on 50\% or less of test problems. Thus, if 5 or 6 digits is the desired level of accuracy, then M1 is a clear winner. However, if the particular application requires much higher accuracy, then M3 becomes a contender. Indeed, only M3 was able to achieve 12 digits of accuracy on any reasonable portion of the test problems. (In this particular test, accuracy was capped at 16 digits, but no method managed to achieve this on a significant portion of the test problems.)
Accuracy profiles do not provide as much information as performance profiles, but are suitable when fixed-cost data sets are collected. This is appropriate in cases where the cost of obtaining the exact solution exceeds the budget, so the optimization target is to find as good a solution as possible within a limited time.
\subsection{Data profiles}
Mor\'{e} and Wild \cite{3} proposed data profiles as an adjustment to performance profiles for derivative-free optimization algorithms. Data profiles try to answer the question: what percentage of problems (for a given tolerance $\tau$) can be solved within the budget of $k$ function evaluations? They assume the required number of function evaluations to satisfy the convergence test is likely to grow as the number of variables increases. The data profile of an optimization algorithm $s$ is defined using~\cite{3}
\begin{equation} \label{07}
d_s({k})=\frac{1}{\lvert \mathcal{P} \rvert} \text {size} \left\{\mathit{p} \in \mathcal{P}: \frac{t_{p,s}}{n_p+1} \le k \right\},
\end{equation}
in which $t_{p,s}$ shows the number of function evaluations required to satisfy the convergence test, $n_p$ is the number of variables in the problem $p \in \mathcal{P}$, and $d_s(k)$ is the percentage of problems that can be solved with $k(n_p + 1)$ function evaluations. The value $k(n_p + 1)$ is used since $n_p + 1$ is to the number of function evaluations required to compute a ``simplex gradient'' (a one-sided finite-difference estimate of the gradient).
It is worth noting that data profiles could easily be defined replacing $\frac{t_{p,s}}{n_p+1}$ by any other measure of fundamental evaluations used. Moreover, if $\frac{t_{p,s}}{n_p+1}$ is replaced by iterations, then data profiles become a slight variation of {\em operational characteristics} defined in \cite{SS00}.
\begin {figure}[H]
\caption{An example data profile.}
\centering
\fbox{\includegraphics[scale=0.5]{DataProfile}}
\label{Fig03}
\end {figure}
Figure \ref{Fig03} shows a typical data profile. Suppose the user has a budget limit of 100 simplex gradients, according to Figure~\ref{Fig03}, with this budget the method M4 has the best performance by solving roughly $22\%$ of the problems; while M3 has the worst performance among all the solvers since with this budget it does not solve any problem.
Like performance profiles, data profiles are cumulative distribution functions, and thus, monotone increasing step functions with a range in $[0, 1]$. Data profiles do not provide the number of function evaluations required to solve a specific problem, but instead provide a visualization of the aggregate data. Also note that the data profile for a given solver $s \in S$ is independent of other solvers; this is not the case for performance profiles.
Although the data profiles are useful for benchmarking, they have not received the same extensive attention as the performance profiles. This is partly because they are newer, but perhaps also because they are primarily used with derivative-free optimization algorithms. However, data profiles could be easily adjusted to a broader class of algorithms by replacing $t_{p,s}$ with any measure of time, and $n_p + 1$ by any dimensional normalization factor. For example, for a sub-gradient based method, $d_s(\alpha)$ could be redefined as
$$d_s({\alpha})=\frac{1}{\lvert \mathcal{P} \rvert} \text {size} \{\mathit{p} \in \mathcal{P}: g_{p,s} \le \alpha \},$$
where $g_{p,s}$ is the number of sub-gradient evaluations. This might make them an appropriate tool for benchmarking bundle methods \cite[\S XIV-XV]{HUL93}.
Table \ref{Table4-Report} summarizes the reporting methods discussed in this section.
\begin {sidewaystable}[H]
\captionof{table}{Reporting methods summarization.\label{Table4-Report}}
\begin{center}
\begin{tabular*}{\textwidth}{lllll}
\toprule
\head{Reporting method} & \head{Evaluates} & \head{Advantage} & \head{Drawback} & \head{Recommendation} \\
\midrule
\begin{tabular}{m{3.0cm}}Full Data Tables\-\end{tabular}
& \begin{tabular}{m{3.5cm}}---\end{tabular}
& \begin{tabular}{m{3.8cm}}Comprehensive \end{tabular}
& \begin{tabular}{m{3.5cm}}Overwhelming \end{tabular}
& \begin{tabular}{m{4.0cm}}Provide in appendix or online data set.\end{tabular} \\
\midrule
\begin{tabular}{m{3.0cm}}Summary Tables \\\\ Simple Graphs \end{tabular}
& \begin{tabular}{m{3.5cm}}Varies\end{tabular}
& \begin{tabular}{m{3.8cm}}Brief\end{tabular}
& \begin{tabular}{m{3.5cm}}Incomplete\end{tabular}
& \begin{tabular}{m{4.0cm}}Provide as talking point, \\ but include other forms of analysis.\end{tabular} \\
\midrule
\begin{tabular}{m{3.0cm}} Trajectory Plots \\\\ Convergence Plots\end{tabular}
& \begin{tabular}{m{3.5cm}} Speed and Accuracy \\\\ Efficiency \end{tabular}
& \begin{tabular}{m{3.8cm}} Clear \\\\ Precise \end{tabular}
& \begin{tabular}{m{3.5cm}} Examines one problem at a time.\end{tabular}
& \begin{tabular}{m{4.0cm}} Good for case-studies, but should include other forms of analysis for larger data sets\end{tabular} \\
\midrule
\begin{tabular}{m{3.5cm}} Performance Profiles\- \\\\ Accuracy Profiles\- \\\\ Data Profiles\end{tabular}
& \begin{tabular}{m{3.5cm}} Speed and Robustness \\\\Accuracy\\\\Speed and Robustness\end{tabular}
& \begin{tabular}{m{3.8cm}} Strong graphical representation that incorporates the entire dataset.\end{tabular}
& \begin{tabular}{m{3.5cm}} Cannot be used for fixed-cost data sets\end{tabular}
& \begin{tabular}{m{4.0cm}} Include at least one of these three profiles whenever possible\end{tabular} \\
\bottomrule
\end{tabular*}
\end{center}
\end{sidewaystable}
\section{Automated benchmarking}\label{sec5}
As we have seen, the benchmarking process of optimization algorithms is a complicated task that requires a lot of effort from data preparation and transformation to analysis and visualization of benchmarking data. Accordingly, some researchers have begun the development of software tools to facilitate and automate developing test sets, solving the problems using a variety of optimization algorithms, and carrying out performance analysis and visualization of benchmarking data.
The PAVER server \cite{33,45} is an online server that provides some tools for automated performance analysis, visualization, and processing of benchmarking data. An optimization engine, either a modeling environment, such as AMPL \cite{127} or GAMS \cite{126}, or a stand-alone solver, is required to obtain solution information such as objective function value, resource time, number of iterations, and the solver status. Then, the benchmark data obtained by running several solvers over a set of problems can be automatically analyzed via online submission to the PAVER server. PAVER returns a performance analysis report through e-mail in HTML format. The tools available in PAVER allow either direct comparisons between two solvers or comparisons of more than two solvers simultaneously in terms of efficiency, robustness, algorithmic output quality, or performance profiles.
The High-performance Algorithm Laboratory~\cite{44} (HAL) is a computational environment designed to facilitate empirical analysis and design of algorithms. It supports conducting large computational experiments and uses a database to handle data related to algorithms, test sets, and experimental results. It also supports distributed computation on a cluster of computers. Its major advantage over other tools is its aim to develop a general purpose tool that can handle different categories of problems, although the initial deployment of problems and algorithms is tricky.
The Optimization Test Environment \cite{74} is another tool that can be used for benchmarking the performance of different optimization algorithms. It provides some facilities to organize and solve large test sets, extract a specific subset of test sets using predefined measures, and perform statistical analysis on the benchmarking data. The results obtained by each optimization algorithm is verified in terms of feasibility and correctness. A variety of information is reported such as the number of global numerical solutions found (i.e., the best solution found among all optimization algorithms), number of local solutions found, number of wrong claims, etc. For problem representation, it uses Directed Acyclic Graphs (DAGs) from the Coconut Environment \cite{128}. This user-friendly environment analyzes results and automatically summarizes them before reporting them in an easy-to-use format such as \LaTeX, JPEG, and PDF.
Other software tools for automating benchmarking process include EDACC \cite{140}, LIBOPT \cite{46}, CUTEr \cite{87} and a testing environment reported in \cite{5}.
Using automated performance analysis tools has the potential to facilitate the benchmarking process. Moreover, the automation of the process may reduce the risk of biased comparison, by taking some of the comparison decisions away from the algorithm designer. However, automated benchmarking tools are not yet accepted by the research community due to their shortcomings. The major drawback of these tools is that the flexibility of a researcher to design experiments based on their research objectives is restricted to the tools' limitations and the way they view the benchmarking process. Moreover, so far all of these tools operate in expert mode, meaning that the usability aspect needs to be improved in terms of application and design of experiments. In most cases preparation of an experiment beyond the scope of default facilities of the benchmarking tools is nontrivial and involves some customization, e.g., scripting. Further research in this direction will create valuable tools for the optimization community, but the current status is not ready for wide-spread use.
\section{Conclusion} \label{sec6}
This article reviews the issue of benchmarking optimization algorithms. For the sake of having a careful, less-biased, explicitly-stated, and comprehensive evaluation of the optimization algorithms an {\em a priori} benchmarking design is required. To develop an appropriate experimental design, the first thing to consider is to clarify the questions that are to be answered by the experiment. This includes selecting a suitable test set and suitable performance measures based on the objectives of the research. The data must be analyzed and processed in a transparent, fair, and complete manner. Within this paper we discuss each of these topics, and present a review of the state-of-the-art for each of these steps. We include several tables and figures that summarize the process, and provide key advice designed to lead to a fair benchmarking process.
A final important point must be raised in regards to optimization benchmarking:
\begin{quote}{\bf as in all scientific research, benchmarking optimization algorithms should be reported in a manner that allows for reproducibility of the experiments.}\end{quote}
When reporting results, make sure to describe algorithms, parameters, test problems, the computational environment, and the statistical techniques employed with an acceptable level of details. It should be clarified that it is usually difficult to provide enough information in a published paper to enable the reader to rerun the stated experiments and replicate completely the reported results. Moreover, the pace of computational development is so high that it is virtually impossible to entirely reproduce a computational experiment, due to development and modifications in operating systems, computer architecture, programming languages, etc. However, the minimum standard for replication of the experiments is that at least the authors themselves should be able to replicate the experiments \cite{38}. Therefore, it is important that the researcher keep all the programs and data necessary to redo all the computations and recreate all graphs. Such programs should be made available whenever possible.
\subsection{Some final insights and remarks from the referees}
This paper provides a high-level perspective on benchmarking of optimization algorithms. While it does not aim to be all encompassing, it hopefully provides a baseline for best practices when benchmarking optimization algorithms. Many nuances exist when dealing with benchmarking specific genres of optimization algorithms. We end with some final discussion on some of these nuanced areas. Many of these final remarks were provided through the insights of $5$ excellent referees.
The state-of-the-art in optimization benchmarking currently has (at least) two major voids that require further research: how to properly benchmark optimization algorithms that make use of parallel processing, and how to properly benchmark multi-objective optimization algorithms.
Evaluating the performance of parallel optimization algorithms is different from traditional optimization methods in various aspects: performance measures such as time, the appropriate test sets, the new measures of merit involved in parallel processing such as the concept of speedup, efficiency, etc. All of these concerns together with the fast pace of technological advances in parallel computing motivate a research on the benchmarking of parallel optimization algorithms. A good start in this regard is the research paper by Barr and Hickman \cite{119}.
Benchmarking multi-objective optimization algorithms is similarly in its infancy. Appropriate test sets and performance measures have yet to surface. Multi-objective optimization is a rapidly advancing field, and research into proper benchmarking in this discipline would be highly valuable.
A benchmarking challenge that we have not addressed is how to compare optimization algorithms that are different in nature\footnote{We thank ``Mathematics Referee \#1'' for pointing out this challenge.}. For example, consider the comparison of a deterministic and a non-deterministic method \cite{GK17,KM17}. If the multiple repeats of the non-deterministic method are considered, is it fair to compare the average quality to the single run of the deterministic method. Some ideas on this, including a proposed method for comparing deterministic and a non-deterministic methods, can be found in \cite{SKM16}.
Another benchmarking challenge that has not been fully address is how to compare algorithms that approach the same problem from fundamentally different view points\footnote{We thank ``Engineering Referee \#3'' and ``Mathematics Referee \#2'' for pointing out this challenge.}. For example, when working with constrained optimization problems, some researchers have explored {\em infeasible point methods} while others have focused on {\em interior point methods}. Infeasible point methods typically take a two phase approach, where one phase aims for decrease in function value and the second phase aims to improve feasibility. Interior point methods assume a strictly feasible starting point and use some form of penalty function to maintain feasibility of all trial points. Comparing these two styles of algorithms is very challenging, and possibly meaningless, as one assumes an infeasible starting point and the other assumes a feasible starting point. Other algorithms adopt a hybrid approach by approximating the feasible set with some tolerance~\cite{REGIS-17}; in that case, the tolerance parameter could greatly influence the result of the comparison.
A source of debate in benchmarking global optimization algorithms is how to deal with {\em rescaling of the domain}\footnote{We thank ``Mathematics Referee \#2'' for pointing out this challenge.}. Many global optimization algorithms are designed with the baseline assumption that the optimization problem's constrained region is the unit hypercube $[0,1]^n$. Of course, in practical applications this is not always true. Some algorithms deal with this at the solver level, using the constraint set's diameter to select parameters like initial step lengths; while other algorithms deal with this at the problem level, assuming that the end-user will rescale the constraint set to be the unit hypercube (which is not always easy to do). Comparisons of algorithms that place fundamentally different assumptions on the problem structure may impact the selection of an appropriate test set and may limit the conclusions one can draw from the numerical results.
Another potential limitations on what conclusions can be drawn from a numerical study is the sensitivity analysis of the parameters\footnote{We thank ``Mathematics Referee \#1'' for pointing out this challenge.}. A robust study should investigate a range of parameters and report on their impact on the validity of the conclusions. We leave the complexity of how best to report such information to future research.
|
1,314,259,995,781 | arxiv |
\section{Introduction}\label{sec:1_introduction}
Recently, generative models have shown great success in open-domain conversation along with the development of large-scale language models, yielding fluent and informative responses \citep{roller2021recipes,adiwardana2020towards,NEURIPS2020_1457c0d6}.
However, generative models suffer from the challenges of latency and computational resources for building real-time conversation systems due to auto-regressive decoding for response generation and a large GPU memory footprint.
\input{Figures/1_latency_to_human}
Meanwhile, retrieval models such as Bi-encoder and Poly-encoder \citep{humeau2019poly} is able to build efficient open-domain conversation systems by pre-defining the response set and searching the most relevant response to the given context from the response set.
In addition, a Bi-encoder dramatically reduces the latency when adopting efficient Maximum Inner Product Search (MIPS) libraries, such as FAISS \citep{johnson2019billion} and ScaNN \citep{avq_2020}.
Despite the outstanding efficiency, retrieval models have shown some lack of conversational ability compared to generative models.
Retrieval models are known to return an erroneous response when the pre-defined response set does not contain the proper response to the given context, while generative models deal with these cases more flexibly \citep{weston2018retrieve}.
Exemplar-based generative models \citep{weston2018retrieve, wu2019response, gupta2021controlling} try to mitigate this problem by combining the advantages of the two approaches, whereas the inherent inefficiency of the generative models remains since exemplar-based generative models employ a generative model for response generation.
To make an efficient yet fluent open-domain conversation system, which is mandatory for real-world applications, we propose a novel training method for retrieval models called \textbf{\textit{Generative-to-Retrieval distillation}} (G2R).
G2R enables retrieval models to leverage the knowledge of large-scale generative models in both data-level and model-level.
First, \textbf{\textit{data-level G2R}} augments the original dialogue dataset with the responses produced by a large-scale generative model using contexts in the original dialogue dataset.
Then, the produced responses are also added to the pre-defined response set.
The augmented dialogue dataset and response set are utilized for training a retrieval model at the training phase and for returning responses at the inference phase, respectively.
Although data-level G2R enables retrieval model to utilize high-quality responses generated by the large-scale generative model, it does not transfer the fine-grained knowledge from the generative model about the quality of individual responses.
\textbf{\textit{Model-level G2R}} resolves this limitation by transferring the response quality scores assessed by the large-scale teacher generative model into the scores of the student retrieval model.
This method induces the retrieval model to select a better response in terms of the response quality.
We empirically demonstrate that a retrieval-based conversation system, which consists of the G2R-applied retrieval model and a MIPS library, shows a substantial conversational ability while showing fast inference speed, as shown in Figure \ref{fig:fig_2_human_evaluation}.
For instance, our retrieval-based conversation system shows about a 20x speedup compared to the Blender model (90M parameters) while exhibiting a comparable human evaluation result on conversational ability.
\section{Method}\label{sec:2_method}
\subsection{Preliminaries}\label{seubsec:2_1_reliminaries}
\textbf{Retrieval models for Open-domain Conversation.}
Let $D=\{(c_i,r_i) \mid 1 \le i \le n\}$ denote the dialogue dataset that contains $n$ context-response pairs, where $c_i$ and $r_i$ are a context and its corresponding gold response of the $i$-th example, respectively.
At the training phase, retrieval models are trained to maximize the score of the gold response $r_i$ for the given context $c_i$ compared to the scores of negative responses.
At the inference phase, retrieval models return the response with the highest score for the given context $c$ from the pre-defined response set $R=\{r_i \mid 1 \le i \le n\}$ constructed from the dialogue dataset $D$.
\newline
\textbf{Knowledge Distillation.}
Knowledge Distillation \cite{hinton2015distilling} transfers the knowledge of the teacher model into the student model by adding a loss that matches the logits of the student model $z_s$ with the logits of the teacher model $z_t$.
For classification task with $l$ classes, the knowledge distillation loss is defined by the cross-entropy between the softened output probability of the student model and the teacher model:
\begin{multline}
\small
\mathcal{L_{KD}} = -\sum_{x \in X} \sum_{i=1}^l p_t(y_i|x) \log p_s(y_i|x) \\
= -\sum_{x \in X}\sum_{i=1}^l \left[ \frac{\exp(z_t(x, y_i)/T)}{\sum_j{\exp(z_t(x, y_j)/T)}} \times \right. \\
\left. \log \frac{\exp(z_s(x, y_i)/T)}{\sum_j{\exp(z_s(x, y_j)}/T)} \right],
\label{eq:knowledge_distillation_loss}
\end{multline}
where $p(y|x)$ and $z(x, y)$ are the softened probability and logit value of the models for the input $x$ and class $y$, respectively, and $T$ is a temperature parameter for smoothing the logit values.
\subsection{Retrieval-based Conversation System}\label{sec:2_2_TwoStagePipeline}
Our goal is to create an efficient open-domain conversation system based on the retrieval model.
However, naively utilizing the retrieval model can lead to the low efficiency when the size of the response set $R$ is large since the retrieval model has to calculate scores for all response candidates.
To this end, we adopt the Bi-encoder \citep{humeau2019poly} model with an efficient MIPS library to select proper responses efficiently without calculating a score for all response candidates.
Bi-encoder encodes a context $c$ and response $r$ into the fixed-length embedding respectively with Transformer architecture \citep{vaswani2017attention}, and defines the relevance score between $c$ and $r$ as the dot-product of two embeddings.
Therefore, an efficient MIPS library, FAISS \citep{johnson2019billion} for our case, can be utilized for speeding up the search process.
\subsection{Data-level G2R}\label{subsec:2_3_data_level_g2r}
\input{Figures/2_architecture}
It is well-known that utilizing an additional high-quality dialogue dataset is helpful for improving the performance of the retrieval model, as shown in \citet{zhang2020dialogue}.
Moreover, enriching the pre-defined response set $R$ with more diverse responses can help the retrieval model to respond appropriately to a variety of input contexts since it widens the opportunity to select an appropriate response.
However, it is highly labor-intensive and costly to acquire such high-quality dialogue datasets or responses through human-in-the-loop annotation such as in \citet{zhang2018personalizing} or \citet{smith2020can}.
Meanwhile, previous studies \cite{adiwardana2020towards, roller2021recipes,NEURIPS2020_1457c0d6} show that well-tuned large-scale generative models are able to achieve near-human conversational ability.
From these observations, we are motivated to leverage the generation result of large-scale generative models to extend the response set as well as the dialogue dataset for training a retrieval model, as shown in Figure \ref{fig:fig_1_architecture}(a).
For each context $c_i$ in the dialogue dataset $D$, a large-scale generative model $\mathcal{G}$ generates $m$ responses, $\{r^{\mathcal{G}}_{i,j} \mid 1 \le j \le m \}$.
Considering the generated responses as a gold response of the given context $c_i$, they are added to the dialogue dataset $D$ and the pre-defined response set $R$ as follows:
$D^{\mathcal{G}}= D \cup \{(c_i, r^\mathcal{G}_{i,j}) \mid 1 \le i \le n, 1 \le j \le m\}$ and $R^{\mathcal{G}} = R \cup \{r^\mathcal{G}_{i,j} \mid 1 \le i \le n, 1 \le j \le m\}$.
$D^{\mathcal{G}}$ and $R^{\mathcal{G}}$ denote the augmented dialogue dataset and response set, respectively.
After the augmentation, a retrieval model $\mathcal{R}$ is trained by minimizing the cross-entropy loss $\mathcal{L}_{CE}$ which maximizes the probability of selecting the ground-truth response $r$ among the set of randomly sampled negative responses $R^-$:
\begin{equation}
\mathcal{L}_{CE} = -\sum_{(c,r) \in \mathcal{D}^\mathcal{G}} \log\frac{\exp(\mathcal{R}(c, r))}{\sum_{r^- \in \{r\} \cup R^-} \exp(\mathcal{R}(c, r^-))},
\label{eq:data_distillation_loss}
\end{equation}
where $\mathcal{R}(c, r)$ is the score computed by the retrieval model $\mathcal{R}$ for the given context $c$ and response $r$.
Note that $R^-$ is created differently for every iteration by randomly sampling responses from $R^\mathcal{G}$ without replacement.
We employ the largest open-domain conversation model available, Blender 9.4B \citep{roller2021recipes}, as the large-scale generative model $\mathcal{G}$.
We apply top-k sampling \citep{fan2018hierarchical} for the diversity of responses since beam search tends to generate similar responses within the same context \citep{adiwardana2020towards}.
In addition, we sample responses multiple times with different minimum length constraints to diversify the specificity and length of generated responses.
\subsection{Model-level G2R}\label{subsec:2_4_model_level_g2r}
While data-level G2R provides additional high-quality dialogue data and diverse responses, it does not transfer the fine-grained knowledge about the quality of the individual responses from the large-scale generative model $\mathcal{G}$.
Model-level G2R is designed to address this problem by transferring the individual response-level quality score, assessed by the large-scale teacher generative model $\mathcal{G}$, into the student retrieval model $\mathcal{R}$.
We first define the quality score of the response from the perspective of the teacher generative model $\mathcal{G}$, denoted as $\mathcal{G}(c, r)$.
Then, the student retrieval model is trained to match the score $\mathcal{R}(c, r)$ of the student retrieval model with the score $\mathcal{G}(c, r)$ of the teacher generative model, similar to the conventional knowledge distillation technique \citep{hinton2015distilling}.
Overall process of knowledge distillation is depicted in Figure \ref{fig:fig_1_architecture}(b).
We define the generator score $\mathcal{G}(c, r)$ as the log-likelihood normalized by the length of response:
\begin{equation}
\mathcal{G}(c, r) = (\log P_{\mathcal{G}}(r|c)) / |r|,
\label{eq:log_likelihood_score}
\end{equation}
where $P_\mathcal{G}(r|c)$ is the probability of the response $r$ for the given context $c$ of the generative model $\mathcal{G}$ and $|r|$ is the number of tokens in the response $r$.
Log-likelihood is normalized with the length of response to mitigate the problem of preferring shorter responses \citep{murray2018correcting}.
We can derive the distillation loss $\mathcal{L}_{KD}$ by regarding the generator quality score $\mathcal{G}(c, r)$ and retriever score $\mathcal{R}(c, r)$ as the logits of teacher and student model, respectively.
Eq. \ref{eq:knowledge_distillation_loss} then turns into:
\begin{equation}
\begin{gathered}
P_\mathcal{G}^{KD}(c_i, r) = \frac{\exp(\mathcal{G}(c_i, r) / T)}{\sum_{r' \in R_i \cup R^-} \exp(\mathcal{G}(c_i, r') / T)}, \\
P_\mathcal{R}^{KD}(c_i, r) = \frac{\exp(\mathcal{R}(c_i, r) / T)}{\sum_{r' \in R_i \cup R^-} \exp(\mathcal{R}(c_i, r') / T)}, \\
\mathcal{L}_{KD} = -\sum_{i=1}^n \sum_{r \in R_i \cup R^-} P_\mathcal{G}^{KD}(c_i, r) \log P_\mathcal{R}^{KD}(c_i, r),
\end{gathered}
\end{equation}
where $R_i = \{r_i, r_{i,1}^\mathcal{G}, \cdots, r_{i,m}^\mathcal{G}\}$ is a set of positive responses correspond to the context $c_i$ in $D^\mathcal{G}$.
Since calculating the generator quality score $\mathcal{G}(c_i, r^-)$ for negative responses requires heavy extra computation, we simplify the calculation by approximating $P_\mathcal{G}^{KD}(c_i, r^-) \approx 0$, $\exp (\mathcal{G}(c_i, r^-) / T) \approx 0$ for randomly sampled negative responses $r^- \in R^-$.
Our final loss for the model-level G2R is a sum of original cross-entropy loss in Equation \ref{eq:data_distillation_loss} and the knowledge distillation loss where hyperparameter $\alpha$ controls the weights of each term:
\begin{equation}
\mathcal{L} = \alpha \mathcal{L}_{CE} + (1 - \alpha) \mathcal{L}_{KD}.
\label{eq:model_distillation_loss}
\end{equation}
\section{Experiments}\label{sec:3_experiments}
\subsection{Dataset}
We conduct experiments on the open-domain conversation datasets which consist of Blended Skill Talk \citep{smith2020can}, ConvAI2 \citep{zhang2018personalizing}, Empathetic Dialogues \citep{rashkin2019towards} and Wizard of Wikipedia \citep{dinan2018wizard}.
Following \citet{roller2021recipes}, all four datasets are used together for the following experiments, and we refer to the merged dataset as BST+.
We follow the method of splitting train, validation, and test set from \citet{smith2020can}.
\subsection{Metrics}
\textbf{Human Evaluation.}
We conduct a human evaluation to assess the quality of model responses.
Human evaluation is carried out on 200 examples randomly sampled from the BST+ test dataset.
Human judges are asked to evaluate the quality of the generated response with two criteria on a 0-2 scale: (i) \textit{Appropriateness (Appr.)} for evaluating whether the generated response is fluent, logical, and appropriate to its given context, and (ii) \textit{Informativeness (Info.)} for evaluating whether the generated response has meaningful information relevant to its given context.
Each example is rated by at least three unique human judges, and all the human evaluation is performed via Amazon Mechanical Turk.
\newline
\textbf{Automated Metrics.}
We also report various kinds of automated metrics.
\textit{MaUdE} \cite{sinha2020learning} is an unreferenced dialogue response evaluation metric calculated by the model that is trained to score positive responses as 1 while scoring syntactically and semantically negative responses as 0, using the ConvAI2 dataset.
Since \textit{MaUdE} shows a high correlation with human judgments on fluency and interestingness of responses, we use \textit{MaUdE} as a proxy metric for evaluating the overall quality of responses produced by each model.
For measuring the lexical diversity of generated responses we utilize \textit{Dist-2} and \textit{Dist-3} \citep{li2016diversity}, where \textit{Dist-n} is a ratio of distinct n-grams to the total number of n-grams in all the responses generated by each model.
\textit{Length}, the average number of tokens in generated responses, is reported for reference.
Last but not least, we measure and report the \textit{Latency} for generating a response for a single input context to verify the efficiency of the model.
Although we report the latency measured on the GPU-enabled environment, the latency measured by using only the CPU is reported in the supplementary material.
\input{Tables/Table_main_result}
\subsection{Models and Baselines}
\textbf{Blender.}
Blender, the state-of-the-art model in open-domain conversation task, is adtoped with different number of parameters: \textit{Blender 90M}, \textit{Blender 2.7B}, and \textit{Blender 9.4B}.
For response generation, we follow the decoding hyperparameters suggested in the original work \citep{roller2021recipes}.
\newline
\textbf{Distilled Blender.}
A small Blender model distilled from a larger generative model is employed to compare our result with a generative model that also utilizes the knowledge distillation technique.
We use 400M parameters Blender model distilled from \textit{Blender 2.7B} with TinyBERT style distillation \citep{jiao2020tinybert}, denoted as \textit{Distilled Blender}.
\newline
\textbf{Bi-encoder \& Poly-encoder.}
\textit{Bi-encoder} and \textit{Poly-encoder} with 256M parameters \citep{humeau2019poly}, pre-trained with the Pushshift Reddit comment dataset \citep{baumgartner2020pushshift} and fine-tuned on the BST+ dataset, are the baselines for retrieval models.
The \textit{Bi-encoder} model integrated with MIPS library, as described in Section \ref{sec:2_2_TwoStagePipeline}, is denoted as \textit{Bi-encoder (w/ FAISS)}.
\newline
\textbf{RetNRef.}
\textit{RetNRef} \citep{weston2018retrieve} is an exemplar-based generative model which incorporates the response of retrieval models into the input of the generative model.
Contrary to G2R, \textit{RetNRef} exploits the retrieval model to make the generative model better, while G2R exploits the knowledge of the generative model to make the retrieval model better.
We use the dialogue retrieval model described in \citet{roller2021recipes} trained with the $\alpha$-blending technique.
\newline
\textbf{Human Response.}
\textit{Human response} refers to the ground-truth label annotated in the BST+ dataset.
\newline
\textbf{G2R.}
Our system is built upon the retrieval-based conversation system described in Section \ref{sec:2_2_TwoStagePipeline}, where the Bi-encoder $\mathcal{R}$ is trained with our proposed G2R using \textit{Blender 9.4B} as the teacher generative model $\mathcal{G}$.
\textit{G2R-DM} denotes our model trained with the data-level G2R and the model-level G2R.
For a comprehensive analysis, two variants are adopted:
\textit{G2R-D} is trained with the data-level G2R only, and \textit{G2R-D (w/o FAISS)} further removes the use of the MIPS library, FAISS, from \textit{G2R-D}.
\subsection{Implementation Details}
We provide the details on our implementation and the hyperparameter values in the supplementary material.
For reproducibility, we release the augmented dialogue dataset and the implementation of G2R models.\footnote{\url{https://github.com/hyperconnect/g2r}}
\section{Experimental Results}\label{sec:5_experimental_results}
\subsection{Result Analysis}
We present the human evaluation result and the automated metrics in Table \ref{tab:main_result}.
Overall, our system trained with G2R achieves a "sweet-spot" between conversational ability and efficiency.
Our system maintains the low latency of \textit{Bi-encoder (w/ FAISS)} while boosting up the human evaluation results significantly, achieving comparable or better human evaluation scores than the \textit{Blender 90M} and human responses, respectively.
Taking a closer look, the Blender generative models and the distilled variant show high human evaluation metric while showing relatively large latency along with the lack of diversity, as shown in the Dist-2 and Dist-3 scores.
Retrieval baselines (\textit{Bi-encoder} and \textit{Poly-encoder}) show an opposite trend, exhibiting much lower latency and relatively higher response diversity but showing relatively lower conversational ability in terms of human evaluation metric.
Unlike human evaluation results, the MaUdE scores of the \textit{Bi-encoder} and the \textit{Poly-Encoder} are unexpectedly high.
However, we suspect this is because the MaUdE metric is trained on the ConvAI2 dataset, which is a subset of the BST+ dataset, and with a similar training objective of these retrieval models as described in Section \ref{sec:3_experiments}.
G2R-based models achieve far better human evaluation results compared to their original model, \textit{Bi-encoder (w/ FAISS)}.
Applying data-level G2R only (\textit{G2R-D}) significantly boosts the performance, making the model perform comparable to gold human response in terms of human evaluation.
Using data-level G2R enlarges the number of responses in the pre-defined response set $R^\mathcal{G}$ more than ten times, therefore using Bi-encoder without FAISS (\textit{G2R-D (w/o FAISS)}) leads to increased latency.
Although using FAISS induces a latency overhead for the case where the size of the response set is small (\textit{Bi-encoder (w/ FAISS)}), using FAISS in a larger response set as in \textit{G2R-D} enables us to maintain the low latency, while having a slight degradation of response qualities compared to the version without FAISS.
Further application of model-level G2R additionally boosts the performance of the retrieval model.
\textit{G2R-DM} shows a higher human evaluation score and MaUdE score than \textit{G2R-D}, and exhibits a comparable human evaluation score to the \textit{Blender 90M} model while running much faster.
While \textit{G2R-DM} shows a somewhat deficient human evaluation score compared to the bigger Blender generative models, it shows substantially lower latency (23.0x speedup over \textit{Distilled Blender}, 44.7x speedup over \textit{Blender 2.7B}).
In addition, \textit{G2R-DM} exhibits a much higher response diversity compared to the Blender generative models.
The \textit{RetNRef} model shows worse performance and delivers much higher latency compared to our \textit{G2R-DM} model.
\subsection{Statistics of the Responses augmented by the Data-level G2R}
\input{Tables/Table_Data_Distillation_Stats}
\input{Tables/Table_Ablation_Data_Distillation}
Table \ref{tab:data_distillation_response_stats} shows the basic statistics of the original response set $R$ and the response set $R^\mathcal{G}$ created by data-level G2R.
After applying the data-level G2R, $R^\mathcal{G}$ has roughly 11 times more candidates compared to the original response set $R$.
To verify if responses in the new response set $R^\mathcal{G}$ show more diversity, we count the number of unique tokens and bi-gram/tri-grams appearing in each response set.
The augmented response set has much more unique tokens and bi-gram/tri-grams than the original response set, implying that it covers more diverse topics, entities and shows more diversity in terms of phrases and expressions.
\subsection{Ablation Studies}
\textbf{Breakdown analysis of Data-level G2R. } We conduct an ablation study to analyze in detail how the performance of the model changes depending on how we use responses generated in the data-level G2R method.
In data-level G2R, generated responses are utilized in two ways: for augmenting the training dialogue dataset $D^{\mathcal{G}}$ of the retrieval model $\mathcal{R}$, and for building the augmented response set $R^\mathcal{G}$.
We separate these two utilization methods and evaluate models that use only each method.
Table \ref{tab:ablation_data_distillation} shows the evaluation results of these ablation models.
Along with the human evaluation metrics and automated metrics, we also report \textit{Hits@1/K} and \textit{Hits@5/K} \citep{roller2021recipes} of trained Bi-encoder model on the BST+ test set, which are widely adopted to evaluate the performance of retrieval models.
As shown in the table, only utilizing one of the methods does not show better performance compared to the model that utilizes both methods.
Utilizing the generated responses for building $R^\mathcal{G}$ improves the appropriateness score of the model, which supports the hypothesis we have raised in Section \ref{sec:2_method} that using a diverse response set is helpful for the model to respond more appropriately.
The use of augmented dialogue $D^\mathcal{G}$ for training $\mathcal{R}$ is helpful for increasing a human evaluation score, for both appropriateness and informativeness metrics, meaning that the retrieval model learns to select relevant and rich responses that the generative model created.
In addition, training with augmented dialogue $D^\mathcal{G}$ considerably improves the Hits metric of the retrieval model.
Nonetheless, using both methods shows the best human evaluation performance among all ablation models, indicating that using new examples for both training a retrieval model and building a response set is crucial for inducing a good performance.
\newline
\textbf{Different Dialogue Augmentation Strategy. }
Here, we implement a simple baseline inspired by \citet{zhu2020data} and \citet{zhang2020dialogue}, which augments training dialogue by utilizing top-$m$ responses of a retrieval model that has already been trained.
In this experiment, we use the \textit{Bi-encoder} model for this augmentation process, and the augmented dialogue dataset generated by this method is denoted as $D^\mathcal{R}$.
Comparison of data-level G2R with this baseline will enable us to verify that our method with a large generative model produces better quality training dataset than simply using a retrieval model.
\input{Tables/Table_Ablation_MD_Loss}
The result of this ablation study is reported in Table \ref{tab:ablation_data_distillation}.
As shown in the table, using $D^\mathcal{R}$ as the training dataset does not lead to a significant performance gain for all metrics, contrary to the case of using $D^\mathcal{G}$ which improves both human evaluation score and Hits metric.
This result strongly indicates that utilizing a large-scale generative model for dialogue augmentation as in data-level G2R is a much more effective augmentation strategy than using retrieval models.
\newline
\textbf{Utilizing a Different Generator Quality Score for Model-level G2R. }
Although we employ the log-likelihood score (LL score) for defining the generator quality score $\mathcal{G}(c, r)$ in model-level G2R, there are other methods that can be utilized as well.
One example is a Mutual Information score (MI score) \citep{li2016diversity}, which is a point-wise mutual information between the given context $c$ and response $r$.
Details about calculating the MI score of response are described in the supplementary material.
MI score is known to assign lower values to generic responses while escalating the score of responses that are more specific to the given context, so we expect that using the MI score will produce more specific and diverse responses compared to the LL score while having a slightly higher risk of returning responses with inappropriate details with respect to input context.
Therefore, we evaluate the variant of model-level G2R that uses MI score as $\mathcal{G}(c, r)$ and compare the performance with the model that uses LL score.
The results are provided in Table \ref{tab:ablation_model_distillation}.
Using MI score for the model-level G2R exhibits a slightly lower human evaluation score than using LL score, especially for the appropriateness score, implying that using the MI score may be less capable of producing an appropriate and accurate answer.
However, in terms of automated metrics, the MI score shows a higher MaUdE score.
In addition, using MI score shows higher response diversity compared to LL score as expected, indicating that MI score could be employed for more diverse responses of the conversation system.
\subsection{Case Study}
\input{Tables/Table_Qualitative_Results}
Table~\ref{tab:qualitative_results} provides an example of responses returned by the baseline models and our G2R models.
In this example, \textit{Bi-encoder (w/ FAISS)} returns the irrelevant response to the given context.
Blender models' responses are logically appropriate, however, they just simply change the topic (\textit{Blender 90M}, \textit{Distilled Blender}) or relatively lack of specific details (\textit{Blender 2.7B}, \textit{Blender 9.4B}).
\textit{G2R-D} tries to respond with detail, but the response contains a somewhat irrelevant phrase about groceries.
In contrast, \textit{G2R-DM} respond appropriately along with specific details talking about a particular book title.
We provide additional response examples in the supplementary material.
\section{Related Work}\label{sec:2_related_work}
\subsection{Open-domain Conversation}
The task of open-domain conversation has been studied based on retrieval models, generation models, or using both.
While retrieval models~\citep{wang2013dataset,ji2014information,wang2015syntax,yan2016learning,wu2017sequential,zhou2018multi,tao2019one,humeau2019poly} search a response relevant to a given context from a pre-defined response set, generative models~\citep{shang2015neural,vinyals2015neural,li2020don,holtzman2019curious,welleck2019neural,roller2021recipes} produce a response based on the given context with auto-regressive decoding.
It is known that the retrieval and generative models have advantages in the efficiency of inference and quality of generated responses, respectively.
To take both advantages, several exemplar-based generative models~\citep{guu2018generating,wu2019response,weston2018retrieve,cai2019retrieval,gupta2021controlling} have recently been proposed by combining the retrieval and generative models.
The main difference between our proposed training method and the exemplar-based generative models is that exemplar-based generative models provide the knowledge of retrieval models to generative models, while our proposed training method transfers the knowledge of generative models to retrieval models to focus on the efficiency of open-domain conversation systems.
\subsection{Knowledge Transfer from Large Models}
Transferring the knowledge from larger-scale teacher neural networks into smaller-scale student neural networks has been implemented to improve the performance of the student model, including data augmentation and knowledge distillation.
In the data augmentation perspective, several works \citep{schick2021generating, chang2021jointly, kumar2020data, yang2020g} utilize the generation result of pre-trained language models as a labeled example for text classification tasks.
\citet{lin2020world} utilize the inference result of the retrieval model and the generative model as a semi-negative dataset for training a student retrieval model.
Meanwhile, Knowledge distillation \citep{hinton2015distilling} transfers the knowledge of the teacher model into the student model by matching the student logits with softened teacher logits.
Knowledge distillation especially designed for specific tasks or model architectures exists, such as sequence generation task \citep{kim2016sequence, lin2020autoregressive}, retrieval models \citep{lu2020twinbert, vakili2020distilling} and for transformer architectures \citep{jiao2020tinybert, wang2020minilm, sun2020mobilebert}.
The most related work to our paper is Dialogue Distillation \citep{zhang2020dialogue}, which also proposes a data-level and model-level distillation for open-domain conversation models.
Our research differs from this work in three ways.
First, Dialogue Distillation requires additional unpaired text corpus, which could be hard to be obtained in certain circumstances.
We instead focus on utilizing the knowledge of large-scale generative models for augmenting additional data.
In addition, Dialogue Distillation does not enrich the pre-defined response set, which is crucial for improving the performance of the retrieval models, as shown in our experiments.
Last but not least, while Dialogue Distillation only considers the distillation within the homogeneous architecture, Generative-to-Generative or Retrieval-to-Retrieval, we focus on the model-level distillation between heterogeneous architectures, especially Generative-to-Retrieval, to take advantages of each architecture.
\section{Conclusion}\label{sec:6_conclusion}
We present G2R, a novel training scheme of retrieval model for open-domain conversation by distilling the knowledge of large-scale generative models in both data-level and model-level.
G2R enables retrieval models to build a highly efficient conversation system that exhibits a substantial level of conversational ability.
We believe that our work will serve as a stepping stone for creating an efficient and real-time open-domain conversation system.
\section*{Ethical Considerations}
We train our models with the BST+ dataset, and the models we used for the pre-training (Pre-trained Bi-encoder weights from \citet{humeau2019poly}) and generating the augmented dataset (Blender 9.4B) are trained with the Pushshift Comment Dataset \citep{baumgartner2020pushshift} and the BST+ dataset.
Both the BST+ dataset and the Pushshift dataset are publicly available.
Texts included in these datasets may include potentially abusive contents and underlying biases, and these toxicities and biases could have been unintentionally encoded in our models.
Therefore, methods for reducing the toxicity of the open-domain dialogue system \citep{xu2020recipes, dinan2019build} or methods for mitigating the bias of the dialogue model \citep{liu2020mitigating, dinan2020queens} are recommended to be jointly used with our method when deploying our model in production.
Like any other open-domain conversational system, our system might provide false or misleading information.
Furthermore, our system has the potential to return a response that contains private information.
Since our model is a retrieval-based model and the pre-defined response set is fixed, an effort for filtering out the responses that potentially contain false information, private information, profanity, and inappropriate content should be preceded.
We acknowledge that it is possible to have biases in human evaluation through Amazon Mechanical Turk. To reduce potential biases, we set a maximum number of annotations per worker. We did not ask the user's identity; therefore, their personal information, including their gender, race, ethnicity, etc., is not revealed.
\subsection{Reference models \& Dataset}
\section{Introduction}\label{sec:1_introduction}
In recent years, with the development of large-scale language models \citep{devlin2019bert,brown2020language}, generative models \citep{sutskever2014sequence,vaswani2017attention,lewis2020bart} have been popular in open-domain conversation \citep{roller2020recipes,zhou2020design,adiwardana2020towards}.
However, existing generative models are too inefficient to be deployed in real-time open-domain conversation systems.
For example, a Blender 9B model \citep{roller2020recipes} requires \textcolor{red}{four GPUs} but could generate only one response per second (1 throughput per second, TPS).
To alleviate this inefficiency, many open-domain conversation systems have employed retrieval models \citep{yoon2018learning,humeau2019poly}, which retrieve responses relevant to the given context from the pre-defined response repository.
Replacing autoregressive decoding of generative models, retrieval models speed up the response inference time with fast search libraries such as FAISS \citep{johnson2019billion} and FALCON \citep{NIPS2015_2823f479}.
However, retrieval models have a lack of flexibility since they return completely erroneous responses when there are no responses in the pre-defined response repository for the specific context~[cite].
In this paper, to take advantage of each approach, we propose a new learning method of retrieval models by utilizing large-scale generative models from the point of view in knowledge distillation.
Our learning method consists of data- and model-level distillation.
As stated in a previous study~\citep{zhang2020dialoguedistillation}, the performance of retrieval models could be improved by learning additional high-quality dialogue data.
However, collecting high-quality dialogue data is not easy because it usually requires labor-intensive and time-consuming.
Therefore,
As data-level distillation,
\section{Engines}
To produce a PDF file, pdf\LaTeX{} is strongly recommended (over original \LaTeX{} plus dvips+ps2pdf or dvipdf). Xe\LaTeX{} also produces PDF files, and is especially suitable for text in non-Latin scripts.
\section{Preamble}
The first line of the file must be
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\documentclass[11pt]{article}
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To load the style file in the review version:
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\usepackage[review]{emnlp2021}
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To use Times Roman, put the following in the preamble:
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Please see the \LaTeX{} source of this document for comments on other packages that may be useful.
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\section{Document Body}
\subsection{Footnotes}
Footnotes are inserted with the \verb|\footnote| command.\footnote{This is a footnote.}
\subsection{Tables and figures}
See Table~\ref{tab:accents} for an example of a table and its caption.
\textbf{Do not override the default caption sizes.}
\begin{table}
\centering
\begin{tabular}{lc}
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\textbf{Command} & \textbf{Output}\\
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\verb|{\"a}| & {\"a} \\
\verb|{\^e}| & {\^e} \\
\verb|{\`i}| & {\`i} \\
\verb|{\.I}| & {\.I} \\
\verb|{\o}| & {\o} \\
\verb|{\'u}| & {\'u} \\
\verb|{\aa}| & {\aa} \\\hline
\end{tabular}
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\textbf{Command} & \textbf{Output}\\
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\verb|{\c c}| & {\c c} \\
\verb|{\u g}| & {\u g} \\
\verb|{\l}| & {\l} \\
\verb|{\~n}| & {\~n} \\
\verb|{\H o}| & {\H o} \\
\verb|{\v r}| & {\v r} \\
\verb|{\ss}| & {\ss} \\
\hline
\end{tabular}
\caption{Example commands for accented characters, to be used in, \emph{e.g.}, Bib\TeX{} entries.}
\label{tab:accents}
\end{table}
\subsection{Hyperlinks}
Users of older versions of \LaTeX{} may encounter the following error during compilation:
\begin{quote}
\tt\verb|\pdfendlink| ended up in different nesting level than \verb|\pdfstartlink|.
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This happens when pdf\LaTeX{} is used and a citation splits across a page boundary. The best way to fix this is to upgrade \LaTeX{} to 2018-12-01 or later.
\subsection{Citations}
\begin{table*}
\centering
\begin{tabular}{lll}
\hline
\textbf{Output} & \textbf{natbib command} & \textbf{Old ACL-style command}\\
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\citep{Gusfield:97} & \verb|\citep| & \verb|\cite| \\
\citealp{Gusfield:97} & \verb|\citealp| & no equivalent \\
\citet{Gusfield:97} & \verb|\citet| & \verb|\newcite| \\
\citeyearpar{Gusfield:97} & \verb|\citeyearpar| & \verb|\shortcite| \\
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\caption{\label{citation-guide}
Citation commands supported by the style file.
The style is based on the natbib package and supports all natbib citation commands.
It also supports commands defined in previous ACL style files for compatibility.
}
\end{table*}
Table~\ref{citation-guide} shows the syntax supported by the style files.
We encourage you to use the natbib styles.
You can use the command \verb|\citet| (cite in text) to get ``author (year)'' citations, like this citation to a paper by \citet{Gusfield:97}.
You can use the command \verb|\citep| (cite in parentheses) to get ``(author, year)'' citations \citep{Gusfield:97}.
You can use the command \verb|\citealp| (alternative cite without parentheses) to get ``author, year'' citations, which is useful for using citations within parentheses (e.g. \citealp{Gusfield:97}).
\subsection{References}
\nocite{Ando2005,borschinger-johnson-2011-particle,andrew2007scalable,rasooli-tetrault-2015,goodman-etal-2016-noise,harper-2014-learning}
The \LaTeX{} and Bib\TeX{} style files provided roughly follow the American Psychological Association format.
If your own bib file is named \texttt{custom.bib}, then placing the following before any appendices in your \LaTeX{} file will generate the references section for you:
\begin{quote}
\begin{verbatim}
\bibliographystyle{acl_natbib}
\section{Implementation Details}
\subsection{Baseline models}
\textbf{Blender Models.} For Blender models (\textit{Blender 90M}, \textit{Blender 2.7B}, \textit{Blender 9.4B}, \textit{Distilled Blender}), we use the pre-trained weights released from ParlAI (Miller et al., 2017).
For a generation, we follow the decoding hyperparameters suggested from the original work (Roller et al., 2021) - using beam search with beam size 10, minimum beam length 20, and tri-gram beam blocking on context and response blocks.
\textbf{Retrieval Models.}
We train the \textit{Bi-encoder} and the \textit{Poly-encoder} baseline model on BST+ dataset with pre-trained weights released in ParlAI (Miller et al., 2017), which is originally disclosed by Humeau et al. (2019).
Both models have a network parameter size of 256M.
We train the model with BST+ dataset, with the batch size 512 and the configuration of using other responses in batch as random negatives, initial learning rate of 1e-5, \textit{ReduceOnPleteau} learning rate schedule with decay rate 0.5 and patience 1.
The validation Hits@1/K metric is employed as a proxy metric.
Also, we utilize Adamax optimizer (Kingma and Ba, 2015) with gradient clip value 0.1 for our experiments.
Note that most of the hyperparameters follow the default implementation of f Humeau et al. (2019) implemented in the ParlAI library.
These learning hyperparameters were also used for training other retrieval models in this paper, unless stated.
\textbf{RetNRef.}
We train the RetNRef model with a 256M Bi-encoder model architecture as a retriever and 90M Blender generative model architecture as a generator. We follow the $\alpha$-blending training scheme of (Roller et al., 2021), using blending parameter $\alpha=0.5$.
The model was trained with a batch size of 32 and an initial learning rate of 7e-6, with \textit{ReduceOnPleateau} learning rate scheduler with validation PPL as a proxy metric (with decay rate 0.5, patience 1).
For inference, we use the same decoding hyperparameters as in Blender generative models except for the minimum beam length constraint parameter.
We used 0 for this value since using a larger value induced a severe repeating problem in the generated response and hurt the performance of the model.
\subsection{FAISS}
FAISS (Johnson et al., 2019) is employed as an efficient MIPS library for our retrieval-based conversation pipeline.
Hierarchical Navigable Small World approximation (Malkov and Yashunin, 2018) is used for building a FAISS index, which was empirically found to be fast and accurate.
We use \textit{HNSW32\_Flat} index with \textit{efSearch} parameter 256 whlie using FAISS throughout our implementation.
\subsection{Data-level G2R}
We use the Blender 9B model (Roller et al., 2021) as our large-scale generative model $\mathcal{G}$.
Throughout our experiments, we use the BST+ training dataset as the original dialogue dataset $D$, without using the meta-information such as the persona information from ConvAI2 (Zhang et al., 2018) and the Wikipedia topic information from WoW (Dinan et al., 2018).
We use top-k sampling with $k=20$ and tri-gram beam blocking on context and response blocks.
We sample 5 samples each from two configurations that use the beam min length hyperparameter of 10 and 20, respectively, sampling a total of 10 samples from a single context $c_i$.
We mainly used ParlAI (Miller et al., 2017) for our experiments.
For training Data-level G2R based retrieval model, we compose a mini-batch by randomly selecting 48 unique contexts and randomly selecting 10 responses connected to each context, resulting in a total of 480 (context, response) pairs in a single batch.
512 random negatives are uniformly sampled from response repository $R^\mathcal{G}$ and used as a shared random negative among the examples in the batch.
We use the Bi-encoder model trained for baseline retrieval model as initial weights and use the same learning configuration as in the baseline retrieval model except for the initial learning rate value of 5e-5.
We tested the initial learning rate value $lr \in \{1e-5, 5e-5\}$ and selected $5e-5$ since this value has shown faster convergence and higher validation Hits@1/K metric.
We trained the model until the convergence of validation Hits@1/K metric and chose the model with the best Hits@1/K metric along the training process.
Training takes about 16 to 24 hours in a single NVIDIA DGX Station A100 workstation.
\subsection{Model-level G2R}
For model-level G2R, we use hyperparameter of $\alpha=0.9$, and $T=1$.
We did not perform hyperparameter search on $T$, and tried $\alpha \in \{0.5, 0.9\}$ and selected 0.9 for $\alpha $ since 0.9 has shown higher validation Hits@1/K metric.
We use the same training configuration as we train the model in data-level G2R.
\section{Metrics Details}
\subsection{Human evaluation}
For accurate human evaluation, we only received an annotation from turkers that satisfies the following requirements: (1) HITs approval rate greater than 95\%, (2) Location is one of Australia, Canada, New Zealand, United Kingdom, and the United States, (3) Lifetime number of HITs approved greater than 1000, following Li et al. (2018).
The instruction for the human evaluation is provided below:
\begin{quote}
Given the dialogue context, you need to rate the quality of the given response in terms of \textbf{appropriateness} and \textbf{informativeness}.
Appropriateness is a metric for evaluating whether \textbf{the given response is fluent, logical, and appropriate to its given context}. Please rate appropriateness with the range of 0 to 2, where 0 represents bad, and 2 represents excellent. Assign a lower score to the response if the response seems off (illogical, out of context, confusing).
Informativeness is a metric for evaluating whether \textbf{the given response has meaningful information relevant to its given context}. Please rate informativeness with the range of 0 to 2, where 0 represents bad, and 2 represents excellent. Please assign a higher score if the response is rich and specific to the context and a lower score if the response is bland and generic.
\end{quote}
\subsection{Measuring Latency}
We use NVIDIA DGX Station A100 for measuring the latency of the model, with Pytorch 1.7.1, Cuda 11.0, CuDNN 8.0.
We only utilize a single GPU (NVIDIA A100 GPU, 40GB Memory) for measuring the latency.
Latency is measured as the average inference time of 200 response generations after having 3 warm-up steps.
\subsection{Details for Calculating Metrics}
For calculating the MaUdE (Sinha et al., 2020) metric, we used the code provided by the authors\footnote{\url{https://github.com/facebookresearch/online_dialog_eval}}.
For calculating the Dist-2, Dist-3 metrics, and Length, we tokenized the generated response with the \texttt{casual\_tokenize} method of the \texttt{nltk} library (Loper and Bird, 2002) and calculated the metric over 200 generated responses.
\section{CPU Latency}
\input{Tables/Table_CPU_Latency}
\input{Tables/Table_Hits_Suppl}
\input{Tables/Table_MD_Example}
In Table \ref{tab:cpu_latency}, we report the latency of various models measured by using only CPU.
While retrieval models, especially \textit{Bi-encoder} and \textit{G2R-DM}, show an acceptable latency under 200ms, generation models such as \textit{Blender 90M} and \textit{Distilled Blender} exhibit inordinately high latency over 1 second.
In particular, \textit{Distilled Blender} shows the latency of 9.3 seconds.
The immensely high latency of generative models makes it extremely difficult to employ these models to build real-time conversation agents in a situation where only CPU is available for inference.
For calculating CPU latency, we utilized a Ubuntu machine with 40 Intel Xeon Silver 4210 CPU (2.20GHz) and 250GB RAM, and measured latency as the average inference time of 50 response generations after having 3 warm-up steps.
\section{Human Evaluation Details}
\input{Tables/Table_Main_Result_Interval}
\input{Tables/Table_Human_Eval_Pvalue}
We provide additional statistics about the human evaluation result, including 95\% confidence interval and p-values for two-tailed t-test between the human evaluation scores of two models in Table \ref{tab:main_result_interval} and Table \ref{tab:main_result_pvalue}, respectively.
Since the number of annotations was relatively small (200 examples) due to the cost of the human evaluation, the majority of the comparison is not statistically significant ($p<0.05$).
However, we observed that the comparison between \textit{G2R-D vs. Bi-encoder (w/ FAISS)}, \textit{G2R-DM vs. Bi-encoder (w/ FAISS)}, \textit{G2R-DM (MI Score) vs. Bi-encoder (w/ FAISS)} and \textit{G2R-DM vs. Poly-encoder} shows a statistically significant difference in terms of Sum of human evaluation score and the Appropriateness human evaluation score, proving that our G2R methods improve the performance of the retrieval model.
Also, note that the trend of the \textit{Sum} human evaluation score within 90M, 2.7B, and 9.4B Blender models is similar to the trend of ACUTE-Eval Engagingness evaluation result reported in the original paper (Roller et al.,
2021), which adds more reliability to our human evaluation result.
\section{Dataset Details}
BST+ dataset is a concatenation of four English dialogue dataset (Blended Skill Talk (Smith et al.,
2020), ConvAI2 (Zhang et al., 2018), Empathetic Dialogues (Rashkin et al., 2019) and Wizard of Wikipedia (Dinan et al., 2018)).
We use the Blender 9.4B model to augment this dataset as described in the \textit{Data-level G2R} section, and the augmented dataset consists of total 3,070,033 context-response pairs on 274,233 unique contexts.
As described in the Experiments section, we release the augmented BST+ dataset in \url{https://github.com/hyperconnect/g2r}.
\section{Validation and Test Hits@1/K metrics}
For reference, we report the Hits@1/K and the Hits@5/K metrics of our retrieval models measured on the validation and the test split of BST+ in Table \ref{tab:valid_test_hits}.
\section{Details for Model-level G2R Ablation Study}
Here, we provide additional details for calculating the MI score in the ablation study for model-level G2R.
MI score is calculated with the \textit{MMI-bidi} equation in the original paper (Li et al., 2016), but additionally normalized by the length of response in the same way LL score is normalized:
\begin{equation}
\mathcal{G}_{MI}(c, r) = (\log P_{\mathcal{G}}(r|c) - log P_{\mathcal{G}}(r)) / |r|.
\label{eq:mutual_information_score}
\end{equation}
Since calculating the unconditional language probability term $P_\mathcal{G}(r)$ in Equation \ref{eq:mutual_information_score} is intractable, we approximate this term by taking the average of the likelihood values of $r$ given dummy input contexts, including \textit{"."}, \textit{"<PAD>"} and \textit{"<UNK>"}.
This trick enables us to avoid undesirable alternative options for calculating $P_\mathcal{G}(r)$ with high computational burden, such as training a separate unconditional language model or calculating an intractable marginal probability $\sum_c P_\mathcal{G}(r|c) P(c)$.
\section{Data-level and Model-level G2R Examples}
Table \ref{tab:model_level_g2r_example} shows the example responses generated by the data-level G2R, and LL and MI score calculated for each response.
Data-level G2R is able to generate high-quality responses that are appropriately related to the input context.
Model-level G2R helps the retrieval model to distinguish between low-quality and high-quality responses since both LL and MI scores assign a low value for an inappropriate response.
Comparing both scores in this example, the MI score assigns higher values to a relatively more specific response compared to the LL score.
In comparison, the LL score assigns higher values to a relatively more general response that can also be appropriately utilized for a different context.
\section{Additional Response Examples}
\input{Tables/Table_Qualitative_Suppl}
\input{Tables/Table_Qualitative_Results_Full}
We provide additional examples of responses generated by the baseline models and our G2R models in Table \ref{tab:qualitative_results_suppl}.
Also, the full dialogue context for the case study example in the main paper is provided in \ref{tab:qualitative_results_full}.
\end{document}
|
1,314,259,995,782 | arxiv | \section{Introduction and outline}
In statistical mechanics the study of systems that are far from
equilibrium continues to attract considerable attention both in the
physics and in the mathematics literature. As it turns out,
exclusion processes first introduced by Spitzer
\cite{Spitzer1970} and their generalisations provide an excellent
set of models that display rich and interesting nonequilibrium phenomena. Moreover,
these processes are intimately related to a number of different
models from statistical mechanics, combinatorics, probability and
random matrix theory and a fruitful interplay between these fields,
triggered by the seminal work of Baik, Deift and Johansson
\cite{Baik99}, continues to produce spectacular results that yield
very precise information on the behavior of such systems.
It is beyond the scope of this article to explain all or even a fair amount
of these results in detail. Instead, we remain faithful to the original goal
of the set of lectures on which this manuscript is based, namely to explain
the fascinating developments in this field to an audience of scientists working
in many different areas of mathematics and theoretical physics. We do not assume
any significant acquaintance with the concepts of statistical mechanics or
probability theory. In order to keep our presentation as elementary as possible
we shall always focus on the simplest cases. Readers who wish to obtain more
information on a specific topic will be referred to the literature.
Here we will make use of the numerous reviews that have appeared recently in this area.
In the remaining part of the introduction we provide a short outline
of the topics that will be treated in this paper.
In Sect.~\ref{Equilibrium} we begin our discussion by explaining in
general terms what physicists mean by the distinction between
equilibrium and nonequilibrium systems, and by describing different
types of nonequilibrium behavior. We then introduce in
Sect.~\ref{Exclusion} a class of stochastic models, known as
\textit{simple exclusion processes}. They describe the stochastic motion of
interacting particles on a lattice where the interaction is given by
the exclusion property, i.e. two particles may not occupy the same
site simultaneously. Interacting particle systems provide useful
models for various nonequilibrium phenomena. Technically, they are
(discrete or continuous time) Markov chains, and we will argue below
in Sect.~\ref{Markov} that a simple yet precise criterion for the
equilibrium vs. nonequilibrium character of a given system can be
formulated within the general theory of Markov chains
\cite{Zia2007}.
We then proceed to explain in more detail how one dimensional simple
exclusion processes can be analyzed, specializing to the case
where particles move only to
neighboring lattice sites. If the probabilities to
move right or left differ from each other one obtains the
\textit{asymmetric simple exclusion process} (ASEP). Mostly this paper
will be concerned with ASEP and its subcase TASEP (\textit{totally
asymmetric simple exclusion process}) where all motion is
unidirectional. The criterion of Sect.~\ref{Markov} identifies ASEP (and
consequently TASEP) as nonequilibrium systems. In fact, ASEP has
become a paradigmatic model for driven transport of a single
conserved quantity and most of our discussion is focused on this
class of models. For readers who are not familiar with ASEP and TASEP
it might be useful at this point to have a look at the precise
definition in Sect.~\ref{jsec3.1}.
We begin our exposition of the analysis of ASEP with a discussion of
stationary measures in Sect.~\ref{Measures} where it is explained
why the uniform distribution always provides a stationary measure in
the simplest case of periodic boundary conditions.
For a macroscopic description of the dynamics the notion of the
hydrodynamic limit is introduced. The time evolution of the
macroscopic particle density is then described by a hyperbolic PDE
that can be solved by the method of characteristics
(Sect.~\ref{Hydro}). After this has been established it is most
natural to ask how much the process fluctuates around this
macroscopic description. At this point it is useful to realize that
ASEP is equivalent to a specific random model for surface growth
which is known as 'corner growth' or 'single step' model
(Sect.~\ref{Growthmodel}). Kardar,
Parisi and Zhang (KPZ) conjectured in their seminal paper
\cite{Kardar1986} that the fluctuation properties of a large class
of (growth) models are universal. Assuming that KPZ universality can
be applied to this particular growth model, one may obtain
predictions for the scaling exponents of the fluctuations of the
height of the surface and of the scaling exponent of the (spatial)
correlation length. The KPZ conjecture and its implications for ASEP
will be explained in Sect.~\ref{KPZ_conjecture}.
The KPZ conjecture is based on the KPZ equation which essentially
adds stochastic driving to the hyperbolic PDE that describes the
hydrodynamic limit. Unfortunately, the KPZ equation is difficult to
analyze and it was only very recently that fluctuation results for
solutions of the KPZ equation became available (cf.
Sect.~\ref{T8.4}). Before that an alternative approach to studying fluctuation properties was taken that turned out to be very fruitful. Rather than dealing with the KPZ equation itself one analyzes various specific models that are believed to belong to the
KPZ universality class. A major breakthrough was achieved through a
spectacular discovery on Ulam's problem for random permutations.
In the early 60's Ulam raised the question to determine the asymptotics of the length of the longest increasing subsequence of a random permutation of the numbers $1, \ldots,
N$, where it is assumed that all $N!$ permutations are equally
likely. It took about 15 years to prove that the expected value of the
length of the longest increasing subsequence behaves like
$2\sqrt{N}$ as $N$ becomes large and there was strong numerical
evidence that the fluctuations around the mean are of order
$N^{1/6}$. In a remarkable paper Baik, Deift, and
Johansson \cite{Baik99} did not only prove that $N^{1/6}$ was indeed the correct
scaling, but they also identified the limiting distribution for the
appropriately rescaled fluctuations. It came as a surprise that
the limiting distribution coincides with the Tracy-Widom
distribution that describes the fluctuations of the largest
eigenvalue of matrices from the Gaussian Unitary Ensemble (GUE) as
the matrix size tends to infinity.
The article \cite{Baik99} became the starting signal for an explosion of
research activities that continue until today. It became immediately
clear that there is a variety of related combinatorial models (e.g.
growth models, last passage percolation models, tilings, directed
polymers in a random environment, tandem queues) that can be
analyzed at the same level of detail and where Random Matrix
distributions appear in the asymptotic description. For further
information we refer the reader to the surveys
\cite{Aldous1999,TTracyWidom02,TJohansson02,TKonig05,Spohn2006,Majumdar2007}
and the monograph \cite{TBDS}. Two results that are of particular relevance for our
discussion of KPZ universality were obtained independently by
Pr\"ahofer and Spohn \cite{Prahofer00b} and Johansson \cite{Jo1}.
Pr\"ahofer and Spohn used the results in \cite{Baik99}, and further
developments in \cite{TBaRa01}, to describe the height fluctuations
of the polynuclear growth model (which is somewhat different from
the corner growth/ single step model mentioned above, cf.
Sect.~\ref{Universality}). They obtained the scaling law predicted
by KPZ universality. Moreover, they were able to identify the
limiting distributions of the fluctuations. Again these are given by
the Tracy-Widom distributions of Random Matrix Theory. A distinction
needs to be made depending on the curvature of the surface. Flat
surfaces lead to statistics from the Gaussian Orthogonal Ensemble
(GOE), whereas the height-fluctuations of curved surfaces are
described by GUE statistics.
On the other hand, the results that Johansson presented in
\cite{Jo1} immediately apply to the totally asymmetric simple
exclusion process with step initial condition. One obtains the
scaling exponent and the limiting distribution for the fluctuations
of the particle flux, that are again given by the GUE Tracy-Widom
distribution. We devote Sects.~\ref{tsec:R} -- \ref{tsec:A} to
explain \cite{Jo1} in great detail. This part of our presentation can be viewed as an expanded and more self-contained version of Sects.~3 and 4 of the review \cite{Sas1} by T. Sasamoto.
According to the philosophy of our paper we explain the results of
Johansson in the simplest case. More precisely, we consider the
particle flux at the origin for a discrete time version of TASEP
(dTASEP, cf. Sect.~\ref{jsec3.1} (iii)) with step initial data. In
Sect.~\ref{tsec:R} we formulate Johansson's result in Theorem
\ref{tsatz:R.1} and discuss its relation to KPZ universality.
The proof of Theorem \ref{tsatz:R.1} naturally falls into two parts.
The first part is of combinatorial nature. Via a
representation by waiting times (Sect.~\ref{tsec:C.1}) the problem
is mapped to finding the longest subsequence in a list of
alphabetically ordered two-letter random words that is weakly
increasing in the second letter (Sect.~\ref{tsec:C.2}). By the
Robinson-Schensted-Knuth algorithm (Sect.~\ref{tsec:C.3}) one may
represent the random words by pairs of Semi Standard Young Tableaux
of the same shape (see Definition \ref{tdef:C.1}). The advantage of
this representation is twofold: On the one hand the length of the
longest weakly increasing subsequence is simply given by the length
of the first row of the corresponding Young Tableau. On the other
hand there exist explicit formulae for counting the number of
Semistandard Young Tableaux of a given shape, that can be derived
using Schur polynomials (Sect.~\ref{tsec:C.4}). The result of all
this reasoning is formula (\ref{teq:C.28}), where $\Delta$ denotes
the Vandermonde determinant (see also (\ref{teq:R.9}) and
Definitions \ref{tdef:R.1} and \ref{tdef:R.2}).
The second part of the proof of Theorem \ref{tsatz:R.1} is the
asymptotic analysis of (\ref{teq:C.28}). The key observation is that
the right hand side of (\ref{teq:C.28}) has exactly the same
structure as the formula for the distribution of the largest
eigenvalue of GUE matrices. In particular the method of orthogonal
polynomials (Sect.~\ref{tsec:A.1}) can be applied to complete the
proof of Theorem \ref{tsatz:R.1} in Sect.~\ref{tsec:A.2}. The
somewhat miraculous appearance of the Tracy-Widom distribution for
the fluctuation of the particle flux of dTASEP is now explained on a
technical level by the fact that Hermite polynomials (used for GUE)
and Meixner polynomials (used for dTASEP) look the same near their
respective largest zeros after appropriate rescaling. The similarity
of Hermite and Meixner polynomials is no coincidence. We briefly
discuss the universal behavior of orthogonal polynomials in
Sect.~\ref{tsec:A.3}.
As it was mentioned above the work of Johansson \cite{Jo1} and of
Pr\"ahofer and Spohn \cite{Prahofer00b} mark the beginning of a
broad stream of research activities that continues to produce new
and exciting results at a rapid pace. In Sect.~\ref{Universality} we
briefly sketch and summarize those directions of recent research
that are closely related to the question of KPZ-universality. The
first generalization beyond \cite{Jo1} that we describe concerns the initial conditions.
The results of Johansson apply for step initial conditions where
every site to the left of the origin is occupied whereas every site
to the right is empty. Based on their work on the polynuclear growth
model Pr\"ahofer and Spohn \cite{Prahofer01} formulated a conjecture
for the fluctuations of the flux for TASEP in the case of a general
initial step profile with arbitrary constant particle densities
$\rho_L$ and $\rho_R$ to the left resp. right of the origin. In
Sect.~\ref{T8.1} we explain this conjecture that has recently been
fully established by Ben Arous and Corwin \cite{TBeCo09}. Most
remarkably, in a series of papers
\cite{TTracyWidom08b}-\cite{TTracyWidom10} C. Tracy and H. Widom
were able to extend some of these results to general ASEP. It should
be pointed out that their proof is based on the Bethe Ansatz and
does not use any of the nice but very special combinatorial identities that were crucial in the argument of Johansson.
Furthermore we provide in
Sect.~\ref{Universality} pointers to the recent literature regarding spatio-temporal correlations for (T)ASEP (Sect.~\ref{T8.2}), interacting particle systems beyond (T)ASEP
(Sect.~\ref{T8.3}), fluctuation results for the KPZ equation
(Sect.~\ref{T8.4}), and physical experiments where KPZ behavior can
be observed (Sect.~\ref{T8.5}). We conclude the paper with a few
remarks on integrability and universality.
\section{Equilibrium and nonequilibrium states}
\label{Equilibrium}
The most fundamental concept of statistical physics is the distinction between
\textit{microstates} and \textit{macrostates} in the description of systems
with many degrees of freedom. To fix ideas, consider a classical
$N$-particle system (say, a gas in a box)
described by a Hamilton function $H(q,p)$ of position
variables $q = (q_1,...,q_{dN})$ and momenta $p = (p_1,...,p_{dN})$.
Particles move in a region $\Omega \subset \mathbb{R}^d$ of volume $V = \vert \Omega
\vert$. Then a \textit{microstate} is simply a point $(q,p)$ in phase space, whereas
a \textit{macrostate} will be defined for the purposes of these lectures as
a measure $P_X(q,p) dq \, dp$ parameterized by a set of \textit{macroscopic}
state variables (in short \textit{macrovariables}) $X$.
Here $P_X(q,p)$ is a function on phase space and $dq \, dp$
denotes the canonical Liouville measure.
Examples of macrovariables
are energy, density, temperature or pressure. The macrovariables
parametrizing the macrostate $P_X$ could have a dependence on space and time,
but to be useful they should be chosen such that they are slowly varying.
This singles out in particular
the conserved quantities of the underlying $N$-particle system as candidates for
macrovariables. The mapping from the microstate $(q,p)$ to the
macrovariables $X$ is many-to-one, and the measure
$P_X(q,p) dq \, dp$ gives the probability to find the system in a particular
set of microstates $(q,p)$ under the constraint that the macroscopic state is described
by $X$. In principle, the time dependence (if any) of $P_X(q,p)$ is induced by the classical
Hamiltonian dynamics of the microstate variables $(q,p)$, but in practice
well-chosen macrovariables are often found to satisfy autonomous evolution
laws, such as the equations of hydrodynamics. The derivation of macroscopic evolution
equations from microscopic Hamiltonian dynamics is the goal of \textit{kinetic theory}.
A (much simplified) version of this problem will be addressed below in
Sect.~\ref{Hydro}.
In this perspective, \textit{equilibrium states} are a subclass of macrostates
which are attained at long times by a system that is isolated or in contact
with a time-independent, spatially uniform environment. Characteristic properties
of equilibrium states are that
\begin{itemize}
\item the macrovariables $X$ are time-independent and spatially homogeneous, and
\item there are no macroscopic currents (e.g., of mass or energy).
\end{itemize}
The two most important examples of equilibrium states are the following:
\begin{itemize}
\item[a.)] In an \textit{isolated} system the energy $E$ is conserved,
the appropriate macrovariables are $X = (E, V, N)$ and the equilibrium
state is the measure induced by the Liouville measure
on the energy shell $\{(q,p): H(q,p) = E\}$.
This is known in physics as the \textit{microcanonical} measure.
\item[b.)] In a system at \textit{constant temperature} $T$ particles exchange energy
with the walls of the box $\Omega$ in such a way that the mean energy is fixed.
The appropriate macrovariables are then $X = (T, V, N)$ and the equilibrium state
is of the form
$$
P_{T,V,N} \sim \exp[-\beta H], \;\;\; \beta = 1/T,
$$
known as the \textit{canonical} measure.
\end{itemize}
Having roughly characterized
equilibrium states, we may say that \textit{nonequilibrium}
states arise whenever the conditions for the establishment of equilibrium
are not fulfilled. As such, this definition is about as useful as it would
be to define some area of biology as the study of non-elephants. We can be
somewhat more precise by making a distinction between
\begin{itemize}
\item[(i)] \textit{Systems approaching equilibrium.} By definition,
the macrostate of such a system is time-dependent. In addition, systems in
this class often become spatially inhomogeneous; an important and much studied
case are systems undergoing phase separation \cite{Bray1994}.
\item[(ii)] \textit{Nonequilibrium stationary states (NESS).} These
systems are kept out of equilibrium by external influences. They
are stationary, in the sense that macroscopic state variables are
time-independent, and they may or may not be spatially homogeneous.
In any case they are characterized by non-vanishing macroscopic currents.
\end{itemize}
Examples for NESS are
\begin{itemize}
\item \textit{Heat conduction.} In a system with boundaries
held at different temperatures there is a stationary
energy current proportional to the temperature gradient
(\textit{Fourier's law}).
\item \textit{Diffusion.} In a system coupled to particle reservoirs
held at different densities there is a mass current proportional
to the density gradient (\textit{Fick's law}).
\item \textit{Electric conduction.} Here particles are charged and
move under the influence of a constant electric field. The particle
current is proportional to the field strength (\textit{Ohm's law}).
\end{itemize}
Among these three examples,
the first two can be further characterized as \textit{boundary driven},
in the sense that the NESS is maintained by boundary conditions
on the quantity that is being transported (heat, mass),
whereas the last example illustrates a \textit{bulk-driven} NESS
maintained by an external field acting in the bulk of the system.
NESS are the simplest examples of nonequilibrium states.
Nevertheless, their description in the framework of classical Hamiltonian
mechanics is conceptually subtle and technically demanding (see, e.g., \cite{Vollmer2002}).
The main reason is that a Hamiltonian system under constant driving inevitably
accumulates energy. In order to allow for the establishment of a steady state,
dissipation has to be introduced through the coupling to an external
reservoir, that is, a system with an infinite number of degrees of freedom.
These difficulties can be avoided by starting from \textit{stochastic}
microscopic dynamics. While less realistic on the microscopic level,
stochastic models provide a versatile framework for addressing
fundamental questions associated with the behavior of many-particle
systems far from equilibrium. The class of models of interest here are known
in the probabilistic community as \textit{interacting particle systems}.
These are lattice models with a discrete (finite or infinite) set of states
associated with each lattice site and local interactions. We focus specifically
on exclusion processes, which are introduced in the next section.
It is worth pointing out that
the notion of equilibrium states in statistical physics, as outlined above,
is much more restrictive than the usage of the corresponding term in most areas of mathematics, where
an \textit{equilibrium} is commonly understood to be any time-independent solution
of some deterministic or stochastic time evolution. Thus NESS are equilibria in
the mathematical sense. In Sect.~\ref{Markov} we will give a precise definition of
what distinguishes physical equilibria from other time-independent
states in the context of continuous time Markov chains.
\section{An introduction to exclusion processes}
\label{Exclusion}
\subsection{Definition}
\label{jsec3.1}
The simple exclusion process was introduced in 1970 by Frank Spitzer
\cite{Spitzer1970}. Particles occupy the sites of a
$d$-dimensional lattice, which for the purposes of this discussion will be
taken to be a finite subset $\Omega \subset \mathbb{Z}^d$. The particles
are indistinguishable, which implies that a microstate or configuration
of the system is given by
$$
\eta = \{ \eta_x \}_{x \in \Omega} \in \{ 0,1 \}^\Omega,
$$
where $\eta_x = 0$ (1) if site $x$ is vacant (occupied). The dynamics
can be informally described as follows (for a detailed construction
see \cite{Spitzer1970,Liggett1999}):
\begin{itemize}
\item Each particle carries a clock which rings according to a Poisson
process with unit rate (i.e., the waiting times between rings are
exponentially distributed).
\item When the clock rings the particle at site $x$ selects a
target site $y$ with probability $q_{xy}(\eta)$ and attempts to jump there.
\item The jump is performed if the target site is vacant and discarded
otherwise; this step implements the \textit{exclusion interaction}
between particles and enforces the single occupancy constraint
$\eta_x = 0$ or 1.
\end{itemize}
Together these rules define the exclusion process as a continuous time Markov
chain on a finite state space; some general properties of such chains will
be discussed in the next section. \textit{Interactions} (beyond the exclusion
interaction) can be introduced through the dependence of the jump matrix
$q_{xy}$ on the configuration $\eta$. Similarly,
\textit{inhomogeneity} associated with sites or particles can be introduced
by letting the waiting times and the jump matrix depend
explicitly on the particle positions or the particle labels,
see \cite{Krug2000}.
We next restrict the discussion to the one-dimensional case
with nearest neighbor hopping and without inhomogeneities or explicit
interactions. Then
$$
q_{xy} = q \delta_{y,x+1} + (1-q) \delta_{y,x-1}.
$$
Informally, the particle attempts to jump to the right with
probability $q$ and to the left with probability $1-q$.
The following cases are of interest:
\begin{itemize}
\item[(i)] $q = 1/2$ defines the \textit{symmetric
simple exclusion process}
(SSEP). We will see below that this is really an equilibrium system.
However, when defined on a finite lattice of sites $x = 1,...,L$
and supplemented with boundary rates $\alpha, \beta, \gamma, \delta$
which govern the injection $(\alpha, \delta)$ and extraction
$(\gamma, \beta)$ of particles at the boundary sites $i = 1$ and
$i = L$, this model provides a nontrivial example for a boundary-driven
NESS \cite{Derrida2007}.
\item[(ii)] $q \neq 1/2$ defines the \textit{asymmetric
simple exclusion process} (ASEP). When considered on the one-dimensional
ring (a lattice with \textit{periodic boundary conditions}) the
system attains a bulk-driven NESS in which there is a non-vanishing
stationary mass current. This is the simplest realization
of a \textit{driven diffusive system} \cite{Schmittmann1995}.
Note that the boundary conditions
are crucial here. On a finite lattice with closed ends, which prevent particles
from entering or leaving the system, an \textit{equilibrium} state is established
in which the bias in the jump probability is compensated by a density gradient;
this is the discrete analog of a gas in a gravitational field, as
described by the barometric formula. Another possibility is to consider
a finite lattice with open ends at which particles are injected and extracted
at specified rates \cite{Krug1991}. This leads to a NESS with a surprisingly
complex structure, see \cite{Blythe2007} for review.
\item[(iii)] $q = 1$ (or 0) defines the \textit{totally asymmetric
simple exclusion process} (TASEP).
In contrast to the case of general $q$,
this process can also be formulated in discrete time \cite{Yaguchi1986}:
In one time step $t \to t+1$, all particles attempt to move to the right
(say) simultaneously and independently with probability $\pi \in (0,1]$;
moves to vacant sites are accepted and moves to occupied sites discarded.
Such a discrete time dynamics cannot be defined for $0 < q < 1$, because it would
lead to conflicts when different particles attempt to simultaneously
access the same vacant site.
For $\pi \to 0$ the discrete time TASEP (\textit{dTASEP}) reduces
to the continuous time process in
rescaled time $\pi t$, while for $\pi = 1$ it becomes a deterministic
cellular automaton which has number 184 in Wolfram's classification
\cite{Wolfram1983,Krug1988}.
The case of general $\pi$ has been studied mostly in the context of
vehicular traffic modeling \cite{Schreckenberg1995,Chowdhury2000}.
Note that in terms
of the waiting time picture sketched above, the discrete time dynamics
corresponds to replacing the exponential waiting time distribution
by a geometric distribution with support on integer times only.
The exponential and geometric waiting time distributions are the only ones
that encode \textit{Markovian} dynamics \cite{Krug1998}.
The waiting time representation will play an important role in the exact solution
of the dTASEP presented below in Sect.\ref{tsec:C}.
\end{itemize}
\subsection{Continuous time Markov chains}
\label{Markov}
Before discussing some specific properties of exclusion processes, we
outline the general setting of continuous time Markov chains
(see \cite{Resnick2002} for an introduction). Consider a Markov chain
with a finite number of states $i=1,\ldots,C$ and transition rates
$\Gamma_{ij}$. The rates define the dynamics in the following way:
\begin{itemize}
\item[]
When the chain is in state $i$ at time $t$, a transition to state $j \neq i$
occurs in the time interval $[t,t+dt]$ with probability $\Gamma_{ij} dt$.
\end{itemize}
The key quantity of interest is the transition probability
$$
P_{ki}(t) = \textrm{Prob}[\textrm{state} \; i \; \textrm{at} \; t \vert
\textrm{state} \; k \; \textrm{at} \; 0] \equiv P_i(t)
$$
where the initial state $k$ is included through the initial condition
$P_i(0) = \delta_{ik}$. The transition probability satisfies the
evolution equation
\begin{equation}
\label{master}
\frac{d}{dt} P_i = \sum_{j \neq i} \Gamma_{ji} P_j - \sum_{j \neq i} \Gamma_{ij} P_i
= \sum_j A_{ji} P_j,
\end{equation}
which is known as the \textit{master equation} in physics \cite{vanKampen2001}
and as the \textit{forward equation} in the theory of stochastic processes \cite{Resnick2002}.
Here the \textit{generator matrix}
$$
A_{ij} = \left\{ \begin{array}{l@{\quad:\quad}l}
\Gamma_{ij} & i \neq j \\
-\sum_{k \neq i} \Gamma_{ik} & i = j \end{array} \right.
$$
has been introduced. The master equation simply accounts for the balance
of probability currents going in and out of each state of the Markov chain.
To bring out this structure we rewrite (\ref{master}) in the form
\begin{equation}
\label{cont}
\frac{d}{dt} P_i = \sum_j K_{ij}, \;\;\;\;
K_{ij} = \Gamma_{ji} P_j - \Gamma_{ij} P_i,
\end{equation}
where $K_{ij}$ is the \textit{net probability current} between states $i$ and
$j$ \cite{Zia2007}. If the chain is \textit{irreducible}, in the sense that
every state can be reached from every other state through a connected
path of nonzero transition rates, the solution of (\ref{master}) approaches
at long times a unique, stationary invariant measure $P_i^\ast$ determined
by the condition
\begin{equation}
\label{eigen}
\sum_j A_{ji} P_j^\ast = 0.
\end{equation}
The invariant measure is the left eigenvector of the generator matrix, with
eigenvalue zero. Based on (\ref{cont}) we can rewrite (\ref{eigen}) as
\begin{equation}
\label{currents}
\sum_j K^\ast_{ji} = 0 \;\;\;
\textrm{with} \;\;\; K_{ji}^\ast = \Gamma_{ji} P_j^\ast - \Gamma_{ij} P_i^\ast.
\end{equation}
Two classes of Markov chains may now be distinguished depending on how the
stationarity condition (\ref{currents}) is realized:
\begin{itemize}
\item[(i)] $K_{ij}^\ast = 0 \; \forall \; i,j$. In this case the
probability currents cancel between any two states $i,j$,
\begin{equation}
\label{detbal}
\Gamma_{ij} P_i^\ast = \Gamma_{ji} P_j^\ast,
\end{equation}
a condition that is known in physics as \textit{detailed balance}.
In the mathematical literature Markov chains with this property are
called \textit{reversible}, because (\ref{detbal}) implies that the
weight of any trajectory (with respect to the invariant measure)
is equal to that of its image under time-reversal
\cite{Resnick2002,Kelly1979}. Detailed balance or, equivalently, reversibility
is a fundamental property that any stochastic model of a physical
system \textit{in equilibrium} must satisfy, because equilibrium
states are distinguished by invariance under time reversal
\item[(ii)] $K_{ij}^\ast \neq 0$ at least for some pairs of states
$i,j$. Such a Markov chain is irreversible and describes a system in a
NESS.
\end{itemize}
Examples for both kinds of situations will be encountered in the next section.
\subsection{Stationary measure of the exclusion process}
\label{Measures}
We consider the ASEP on a ring of $L$ sites
with a fixed number $N$ of particles. The total number of microstates
$\eta$ is then $C = {L \choose N}$ and the transition
rates are
\begin{equation}
\label{ASEPrates}
\Gamma(\eta \to \eta') = \left\{ \begin{array}{l@{\quad:\quad}l}
q & (...\bullet \circ ...) \to (...\circ \bullet ...) \\
1-q & (...\circ \bullet ...) \to (...\bullet \circ ...) \\
0 & \textrm{else}. \end{array} \right.
\end{equation}
Here $(...\bullet \circ ...)$ denotes a local configuration
with an occupied site $(\bullet)$ to the left of a vacant site
$(\circ)$, and it is understood that only configurations
$\eta, \eta'$ that differ by the exchange of a single particle-vacancy
pair are connected through nonzero transition rates. The stationary
measure $P^\ast(\eta)$ is determined by the condition
\begin{equation}
\label{statASEP}
\sum_{\eta'} \Gamma(\eta' \to \eta) P^\ast(\eta') =
\sum_{\eta'} \Gamma(\eta \to \eta') P^\ast(\eta)
\;\;\;\; \forall \; \eta.
\end{equation}
As the simplest possibility, let us assume that the invariant measure is
uniform on the state space,
\begin{equation}
\label{uni}
P^\ast(\eta) = {L \choose N}^{-1} \;\;\;
\Rightarrow \;\;\;
K^\ast(\eta, \eta') = [\Gamma(\eta' \to \eta) -
\Gamma(\eta \to \eta') ] {L \choose N}^{-1}.
\end{equation}
We discuss separately the symmetric and the asymmetric process.
\begin{itemize}
\item $q=1/2$ (SSEP). Here the rate $q = 1-q = 1/2$ for all allowed processes, and for each
allowed process the reverse process occurs at the same rate. We conclude that detailed
balance holds in this case, $K^\ast = 0$, and the SSEP is reversible as announced previously.
\item $q \neq 1/2$ (ASEP). Because for any allowed process with rate $q$ the reverse process
occurs at rate $1 - q \neq q$ and vice versa,
detailed balance is manifestly broken, $K^\ast \neq 0$,
and we are dealing with an irreversible NESS. However, we now show that the uniform
measure (\ref{uni}) is nevertheless invariant. To see this, consider the
total transition rates for all processes leading into or out of a given configuration
$\eta$. We have
$$
\Gamma_{\mathrm{tot}}^\mathrm{in}(\eta) = \sum_{\eta'} \Gamma(\eta' \to \eta) =
q {\cal{N}}_{\circ \bullet}(\eta) + (1 - q) {\cal{N}}_{\bullet \circ}(\eta)
$$
where ${\cal{N}}_{\circ \bullet}(\eta)$ denotes the number of pairs of sites with
a particle to the right of a vacancy in the configuration $\eta$. Similarly
$$
\Gamma_{\mathrm{tot}}^\mathrm{out}(\eta) = \sum_{\eta'} \Gamma(\eta \to \eta') =
q {\cal{N}}_{\bullet \circ}(\eta) + (1 - q) {\cal{N}}_{\circ \bullet}(\eta).
$$
A little thought reveals that ${\cal{N}}_{\bullet \circ}(\eta) = {\cal{N}}_{\circ \bullet}(\eta)$
for any configuration $\eta$. Hence $\Gamma_{\mathrm{tot}}^\mathrm{in}(\eta) =
\Gamma_{\mathrm{tot}}^\mathrm{out}(\eta)$ for any $q$, and the stationarity condition
(\ref{statASEP}) is satisfied for the uniform measure (\ref{uni}).
\end{itemize}
A few remarks are in order.
\begin{itemize}
\item[(i)] The invariance of the uniform measure (\ref{uni}),
and the fact that it is independent of the bias $q$, relies
crucially on the ring geometry. With open boundary conditions allowing for the injection and extraction
of particles both the SSEP and the ASEP display nontrivial invariant measures
characterized by long-ranged correlations and the possibility of
phase transitions \cite{Derrida2007,Blythe2007}. For example, for the SSEP with
boundary densities $\rho_L$ at $x=1$ and $\rho_R$ at $x=L$ one finds a linear
mean density profile, as expected from Fick's law, but in addition there are
long-ranged density-density correlations on the scale $L$,
which take the form \cite{Derrida2007,Spohn1983}
$$
\mathbb{E} (\eta_{L\xi} \eta_{L \xi'}) -
\mathbb{E} (\eta_{L\xi}) \mathbb{E}(\eta_{L \xi'}) =
-\frac{\xi (1 - \xi')}{L} (\rho_L - \rho_R)^2.
$$
Here $\xi, \xi^\prime \in [0,1]$ are scaled position variables with
$\xi < \xi^\prime$.
\item[(ii)] The invariant measure of the dTASEP on the ring is
\textit{not} uniform. Rather, one finds
a Gibbs measure with repulsive nearest-neighbor interactions between
the particles \cite{Yaguchi1986,Schadschneider1993,Schreckenberg1995}.
This means that the probability of a configuration can be written as
a product of pair probabilities,
\begin{equation}
\label{Ising}
P_\rho^\ast(\eta) \sim \prod_{x} p_\rho(\eta_x,\eta_{x+1}),
\end{equation}
where the limit $N, L \to \infty$ at fixed density
\begin{equation}
\label{density}
\rho = \mathbb{E}(\eta_x) = N/L
\end{equation}
is implied and
\begin{equation}
\label{DTASEP1}
p_\rho(0,1) = p_\rho(1,0) = \frac{1 - \sqrt{1 - 4 \pi \rho(1- \rho)}}{2\pi},
\end{equation}
\begin{equation}
\label{DTASEP2}
p_\rho(0,0) = 1 - \rho - p_\rho(1,0), \;\;
p_\rho(1,1) = \rho - p_\rho(1,0).
\end{equation}
For $\pi \to 0$ this reduces to a Bernoulli measure of independent particles
(see Sect.~\ref{Hydro}), whereas for $\pi \to 1$ we have $p_\rho(1,0) \to (1 - \vert 1 -
2 \rho \vert)/2$, which implies that $p_\rho(1,1) \to 0$ for $\rho < 1/2$
and $p_\rho(0,0) \to 0$ for $\rho > 1/2$. At $\pi = 1$ and mean density $\rho = 1/2$ the
measure is concentrated on the two configurations $\eta_x = [1 \pm (-1)^{x}]/2$.
\item[(iii)] The invariance of the uniform measure for the ASEP on the ring
is an example of \textit{pairwise balance}
\cite{Schuetz1996}, a property that generalizes the detailed
balance condition (\ref{detbal}) into the form
$$
\Gamma(\eta \to \eta') P^\ast(\eta) = \Gamma(\eta'' \to \eta) P^\ast(\eta'').
$$
This means that for each configuration $\eta'$ contributing to the outflux of
probability out of the state $\eta$ there is a configuration $\eta''$ whose influx
contribution precisely cancels that outflux. In other words, the terms in the sums
on the two sides of (\ref{statASEP}) cancel pairwise.
\end{itemize}
\subsection{Hydrodynamics}
\label{Hydro}
An important goal in the study of stochastic interacting particle systems is to understand
how deterministic evolution equations emerge from the stochastic microscopic dynamics
on large scales \cite{Spohn1991,Kipnis1999,Bertini2007}.
This is similar to the (much harder) problem
of deriving hydrodynamic equations from the Newtonian dynamics of molecules in a gas
or a fluid. The mathematical procedure involved in the derivation of macroscopic
evolution equations for systems with conserved quantitities is therefore referred
to as the \textit{hydrodynamic limit}. Here we give a heuristic sketch of hydrodynamics
for the ASEP.
The key input going into the hydrodynamic theory is the relationship between the particle
density $\rho$ and the stationary particle current $J$.
The particle current is defined as the net number
of particles jumping from a site $x$ to the neighboring site
$x+1$ per unit time, which is independent of $x$ in the stationary state.
From the definition of the ASEP we have
$$
J = q \mathbb{E}[\eta_x (1 - \eta_{x+1})] - (1-q) \mathbb{E}[\eta_{x+1} (1 - \eta_{x})]
$$
where expectations are taken with respect to the invariant
measure. Since all configurations of $N$ particles on the lattice of $L$ sites
are equally probable,
$$
\mathbb{E}[\eta_x (1 - \eta_{x+1})] = \mathbb{E}[\eta_{x+1} (1 - \eta_{x})]
= \frac{N}{L} \frac{(L-N)}{L-1}.
$$
This is just the probability of finding a filled site next to a vacant site,
which is obtained by first placing one out of $N$ particles in one of $L$ sites,
and then placing one out of $L-N$ vacancies in one of
the remaining $L-1$ sites. We conclude
that
$$
J = \frac{(2 q -1) \rho(1- \rho)}{1 - 1/L} \to (2 q -1) \rho(1 - \rho) \;\;
\textrm{for} \;\; L \to \infty,
$$
where the particle density (\ref{density}) is kept fixed. Similarly
\begin{equation}
\label{Bernouilli}
\mathbb{E}[\eta_x \eta_y ] = \frac{N}{L}\frac{N-1}{L-1} \to \rho^2 =
\mathbb{E}[\eta_x] \mathbb{E}[\eta_y]
\;\; \textrm{for} \;\; L \to \infty
\end{equation}
for any pair of sites $x \neq y$. This implies that in the invariant measure
on the ring, restricted to a fixed finite number of sites,
for $L \to \infty$ each site is occupied independently with
probability $\rho$ (Bernoulli measure).
We can now formulate the basic idea of the hydrodynamic limit \cite{Lebowitz88}.
Suppose that we start the ASEP at time $t = 0$ from a Bernoulli measure
with a slowly varying density $\rho(x,0)$. Here ``slowly varying'' means
that variations occur on a scale $\ell \gg 1$. Since the invariant measure
of the ASEP is a Bernoulli measure of \textit{constant} density,
it is plausible that, if $\ell$ is chosen large enough, the evolving measure
will remain close to a Bernoulli measure with a time and space dependent
density $\rho(x,t)$ at all times; and because the particle density is locally
conserved, the evolution equation for $\rho(x,t)$ must be of conservation
type,
\begin{equation}
\label{hydro1}
\frac{\partial}{\partial t} \rho(x,t) + \frac{\partial}{\partial x}
j(x,t) = 0.
\end{equation}
In the limit $\ell \to \infty$ we may expect, in the spirit of a law of large
numbers, that the local particle current $j(x,t)$ converges to the
stationary current associated with the local density $\rho(x,t)$,
$$
j(x,t) \to J(\rho(x,t)),
$$
such that (\ref{hydro1}) becomes an autonomous, deterministic hyperbolic conservation law
\begin{equation}
\label{hydro2}
\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x} J(\rho) = 0
\end{equation}
for the density profile $\rho(x,t)$. Equation (\ref{hydro2}) is known as the
\textit{Euler} equation for the ASEP, because similarly to the Euler equation
in fluid mechanics it lacks a second order ``viscosity'' term $\nu
\partial^2 \rho/\partial x^2$. It must be emphasized that such a term does
\textit{not} appear to leading order, when the hydrodynamic limit is carried out
at fixed $q \neq 1/2$. It is only present in the \textit{weakly asymmetric}
case, which implies that $q \to 1/2$ in the limit $\ell \to \infty$
such that $\ell (q - 1/2)$ is kept fixed \cite{DeMasi1989}.
The Euler equation (\ref{hydro2}) has been rigorously established for a wide
range of models, including cases in which the invariant measure and the
current-density relation $J(\rho)$ are not explicitly known \cite{Seppalainen1999}.
We conclude this section by a brief discussion of the properties of the nonlinear
PDE (\ref{hydro2}), assuming a general (but convex) current-density relation
with $J(0) = J(1) = 0$. This includes in particular the dTASEP for
which
\begin{equation}
\label{JTASEP}
J(\rho) = \pi p_\rho(1,0) = \frac{1}{2}[1 - \sqrt{1 - 4 \pi \rho(1 -\rho)}].
\end{equation}
\begin{itemize}
\item[(i)] \textit{Shock formation}. Hyperbolic conservation laws
of the form (\ref{hydro2}) can generally be solved by the
\textit{method of characteristics}. To this end we first
rewrite (\ref{hydro2}) in the form
\begin{equation}
\label{cEq}
\frac{\partial \rho}{\partial t} + c(\rho) \frac{\partial \rho}{\partial x} = 0,
\end{equation}
where
\begin{equation}
\label{speed}
c(\rho) = \frac{dJ}{d\rho}.
\end{equation}
A characteristic is a trajectory of a point of constant density, and the key observation
is that the characteristics of (\ref{cEq}) are straight lines. Denoting by $v_{\rho_0}(x,t)$ the
position of a point of density $\rho_0 = \rho(x,0)$ at time $t$, we have to
satisfy the condition
$$
\rho(v_{\rho_0}(x,t),t) = \rho_0 = \rho(x,0)
$$
at all times. Taking the time derivative of this relation and using (\ref{cEq}) we see that
the solution is
$$
v_{\rho_0}(x,t) = x + c(\rho_0)t,
$$
i.e. points of constant density travel at the \textit{kinematic wave speed} (\ref{speed}).
The convexity of the current-density relation implies that $c(\rho)$ is a
decreasing function of the density. As a consequence characteristics collide
in regions of increasing initial density, $d \rho(x,0)/dx > 0$, leading
to the formation of density discontinuities (\textit{shocks}) in finite time.
At this point the description by the PDE (\ref{hydro2}) breaks down,
but the speed $V$ of a shock separating regions of density $\rho_L$
on the left and $\rho_R > \rho_L$ on the right is easily inferred from
mass conservation to be given by
\begin{equation}
\label{shockspeed}
V = \frac{J(\rho_R)-J(\rho_L)}{\rho_R - \rho_L}.
\end{equation}
Note that $V \to c$ for $\rho_L \to \rho_R$.
On the microscopic level shocks are represented by
the \textit{shock measures} of the ASEP \cite{Ferrari1991,Ferrari1992}.
These are inhomogeneous invariant
measures on $\mathbb{Z}$ which approach Bernoulli measures with density
$\rho_L$ and $\rho_R$ for $x \to -\infty$ and $x \to \infty$, respectively.
The microscopic structure of shocks has been studied in considerable
detail \cite{Derrida1993}.
\item[(ii)] \textit{Rarefaction waves.} If the initial density profile is a step
function
\begin{equation}
\label{step}
\rho(x,0) = \left\{ \begin{array}{l@{\quad:\quad}l}
\rho_L & x < 0 \\
\rho_R & x > 0 \end{array} \right.
\end{equation}
with $\rho_L > \rho_R$, a diverging fan of characteristics forms leading
to a broadening, self-similar density profile
\begin{equation}
\label{wave}
\rho(x,t) = \left\{ \begin{array}{l@{\quad:\quad}l}
\rho_L & x < c(\rho_L) t \\
\rho_R & x > c(\rho_R) t \\
\phi(x/t) & c(\rho_L) < x/t < c(\rho_R), \end{array} \right.
\end{equation}
where the shape function $\phi(\xi)$ can be computed from the current-density
relation $J(\rho)$. Inserting the ansatz (\ref{wave}) into
(\ref{cEq}) we see that
\begin{equation}
\label{phic}
\phi(\xi) = c^{-1}(\xi).
\end{equation}
For the continuous time ASEP the interpolating shape is linear,
$$
\phi(\xi) = \frac{1}{2} \left(1 - \frac{\xi}{2q-1} \right).
$$
\end{itemize}
\subsection{Mapping to a growth model}
\label{Growthmodel}
The representation of the one-dimensional ASEP as a growth model
seems to have been formulated first by Rost \cite{Rost1981}.
It is illustrated in Fig.~\ref{Rostfigure} for a step initial condition,
where all sites to the left of the origin are occupied ($\eta_x = 1$ for
$x \leq 0$) and all sites to the right of the origin are empty ($\eta_x = 0$
for $x > 0$). This initial condition will also play a central role below in Sect.\ref{tsec:R}.
\begin{figure}
\begin{center}
\resizebox{12cm}{!}{\includegraphics{Figure1.eps}}
\end{center}
\caption{(Color online) Schematic of the mapping between a configuration of the ASEP and the corresponding height
configuration $h_{x+\frac{1}{2}}$ (bold blue line). Initially all sites $x \leq 0$ in the ASEP
are occupied, and all sites $x > 0$ are vacant (step initial condition). The ASEP occupation variables
determine the height differences according to (\ref{ASEPslope}). At the same time, the height
increment $h_{x+\frac{1}{2}}(t) - h_{x+\frac{1}{2}}(0)$ counts the net number of particles that have
crossed the bond $(x,x+1)$ from left to right up to time $t$. In the figure two particles
have crossed the origin, and $h_{\frac{1}{2}} = 2$. Rotating the height configuration
by 45$^\circ$ provides a representation of the net number of jumps a given particle has undergone
as a function of the particle label, counted backwards from $x=0$ (the $l$'th particle
is the one that was located at $x = - l$ at time $t=0$).}
\label{Rostfigure}
\end{figure}
The mapping assigns to every configuration $\eta = \{\eta_x \}$ of the ASEP
a configuration of height variable $\{h_{x+\frac{1}{2}} \}$,
where the shift of the index indicates that the height variable $h_{x+\frac{1}{2}}$
lives on the bond connecting the ASEP sites $x$ and $x+1$. After fixing the height at
a reference point, e.g. by setting $h_{\frac{1}{2}}$ = 0, the height configuration
is uniquely determined by the relation
\begin{equation}
\label{ASEPslope}
h_{x+\frac{1}{2}} - h_{x-\frac{1}{2}} = \frac{1}{2} - \eta_x.
\end{equation}
The ASEP occupation variable encode the local slopes of the height profile,
which take the values $h_{x+\frac{1}{2}} - h_{x-\frac{1}{2}} = \pm \frac{1}{2}$, hence the
name ``single step model'' in the physics literature \cite{Meakin86,Plischke87}.
The step initial condition corresponds to the initial height profile
$$h_{x+\frac{1}{2}}(0) = \frac{1}{2} \vert x \vert,$$
which accounts for the designation as a ``corner growth model'' in the
mathematical literature.
It can be seen from Fig.~\ref{Rostfigure} that a particle jumping
across a bond to the
right (left) increases (decreases) the corresponding height variable by
one unit. Thus the height $h_{x+\frac{1}{2}}$ is an odd (even) multiple of
$\frac{1}{2}$ for odd (even) values of $x$, and the height change
$h_{x+\frac{1}{2}}(t) - h_{x+\frac{1}{2}}(0)$ is equal to the net number of particles
that have jumped across the bond $(x, x+1)$ from left to right up to time
$t$ (jumps from right to left are counted with a negative sign). Finally,
for the special case of
the step initial condition, the net number of jumps (forward minus
backward) performed by a given particle
can also be read off from the height profile (see Fig.~\ref{Rostfigure}).
The mapping (\ref{ASEPslope}) is clearly not restricted to the step initial
condition. Of particular interest are translationally invariant initial
conditions, which can be constructed deterministically or stochastically.
For example, to generate a deterministic initial condition of density $\rho = 1/n$,
one simply places a particle at every $n$'th site of the lattice, and a
stochastic initial condition is obtained by occupying sites independently
with probability $\rho$. The two types
of initial conditions differ in the \textit{roughness} of the corresponding
height configuration, which is quantified by the \textit{height
difference correlation function}
\begin{equation}
\label{hdiff}
G(r) = \mathbb{E}[(h_{y+r} - h_{y})^2] - \mathbb{E}[h_{y+r} - h_y]^2.
\end{equation}
An ensemble of height configurations on $\mathbb{Z}$ is said to be \textit{smooth} if
$\lim_{r \to \infty} G(r) < \infty$ and \textit{rough} otherwise \cite{Krug1991b}. The deterministic
initial conditions described above are smooth in this sense, whereas for the
stochastic initial condition a simple computation using (\ref{ASEPslope}) and
(\ref{Bernouilli}) shows that
\begin{equation}
\label{ASEPrough}
G(r) = \rho (1 - \rho) \vert r \vert.
\end{equation}
\section{The KPZ conjecture}
\label{KPZ_conjecture}
The asymmetric exclusion process and the equivalent growth model
introduced in the preceding subsection are representatives of a large class of
models, which was brought to the forefront
of research in nonequilibrium statistical physics in 1986 by a
seminal paper of Kardar, Parisi and Zhang (KPZ) \cite{Kardar1986}. Working in the
framework of a phenomenological stochastic continuum description, they formulated
what may be called a \textit{universality hypothesis} encompassing the fluctuation
properties of a large class of different microscopic models\footnote{For an introduction
to the idea of universality from a mathematical perspective see \cite{TDeift07}.}. The classic period
of research in this area has been extensively reviewed in the literature
\cite{Krug1991b,HalpinHealy1995,Krug1997}. Here we
aim to give a concise and simple presentation of the KPZ conjecture, in order
to place the more recent developments (to be elaborated in the following sections)
into their proper context.
\subsection{The Kardar-Parisi-Zhang equation}
We start from the hydrodynamic equation (\ref{hydro2}) with a general current-density
relation $J(\rho)$. Since we are interested in fluctuations around a
state of constant mean density $\bar \rho$, we write $\rho(x,t) = \bar \rho + u(x,t)$
and expand to second order in $u$, which yields
\begin{equation}
\label{expansion}
\frac{\partial u}{\partial t} = - c(\bar \rho) \frac{\partial u}{\partial x}
- \lambda u \frac{\partial u}{\partial x},
\end{equation}
where
\begin{equation}
\label{lambda}
\lambda = \frac{d^2J}{d^2\rho}(\bar \rho).
\end{equation}
The linear drift term on the right hand side
can be eliminated by a Galilei transformation $x \to x - ct$, which leaves us with
what is known (for $\lambda = 1$) as the \textit{inviscid Burger equation}.
Now fluctuations
are introduced (in the spirit of fluctuating hydrodynamics \cite{Spohn1991})
by adding a random force to the right hand side of (\ref{expansion}). In order to
guarantee mass conservation, this term must take the form of a derivative
$-\partial \zeta/\partial x$ of a stochastic process $\zeta(x,t)$ in space and time.
This is assumed to be a stationary Gaussian process with zero mean and a covariance function
\begin{equation}
\label{covariance}
\mathbb{E}[\zeta(x,t) \zeta(x',t')] = a_x^{-1} a_t^{-1} G[(x - x')/a_x, (t - t')/a_t]
\end{equation}
which vanishes beyond a small correlation length $a_x$ and a short correlation time $a_t$.
Usually one takes formally\footnote{In the hydrodynamic context \cite{Forster1977}
it is also of interest to consider the solutions of (\ref{noisy}) on scales
\textit{small} compared to the spatial driving scale $a_x$.} $a_x, a_t \to 0$,
which reduces the right hand side of (\ref{covariance})
to a product of $\delta$-functions,
\begin{equation}
\label{white}
\mathbb{E}[\zeta(x,t) \zeta(x',t')] \to D \delta(x - x') \delta(t - t')
\end{equation}
and turns the process $\zeta(x,t)$ into spatio-temporal white noise of strength $D$.
This rather violent driving has to be compensated by a small viscosity term
$\nu \partial^2 u/ \partial x^2$ with $\nu > 0$. Putting all ingredients together we thus arrive at
the \textit{stochastic Burgers equation}
\begin{equation}
\label{noisy}
\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial x^2}
- \lambda u \frac{\partial u}{\partial x} + \frac{\partial \zeta}{\partial x}
\equiv - \frac{\partial}{\partial x} j(x,t),
\end{equation}
first introduced in the context of randomly stirred
fluids \cite{Forster1977} and subsequently applied to fluctuations in
the exclusion process by van Beijeren, Kutner and Spohn \cite{vanBeijeren1985}.
To establish the connection to growth models we proceed in analogy to
the discrete case discussed in Sect.~\ref{Growthmodel}. We introduce the
\textit{height function} $h(x,t)$ through the time-integrated
particle current,
\begin{equation}
\label{height}
h(x,t) = \int_0^t j(x,s) ds,
\end{equation}
Supplementing this with the initial condition $u(x,0) = 0$, it follows
from the conservation law for $u$ that
\begin{equation}
\label{hu}
\frac{\partial h}{\partial x} = -u,
\end{equation}
and therefore
\begin{equation}
\label{KPZ}
\frac{\partial h}{\partial t} = \nu \frac{\partial^2 h}{\partial x^2} +
\frac{\lambda}{2} \left( \frac{\partial h}{\partial x} \right)^2 - \zeta,
\end{equation}
which is precisely the KPZ-equation \cite{Kardar1986}. In general there is also
a constant term on the right hand side of (\ref{KPZ}) which has been set to zero.
The defining feature of the equation is the quadratic nonlinearity on the right hand side, which is present whenever
$\lambda \neq 0$, that is, when the current is a (generic) nonlinear
function of the density\footnote{Note that it is possible to have
$\lambda \neq 0$ even if the current itself vanishes at the specific
mean density under consideration (see \cite{TBoFe08} for an example).}
[compare to (\ref{lambda})].
It is important to clearly understand the relation between
the stochastic PDE's (\ref{noisy},\ref{KPZ}) and the underlying discrete particle systems.
The coefficient $\lambda$ in (\ref{KPZ}) is defined through the current-density
relation according to (\ref{lambda}), but the viscosity $\nu$ and the noise strength $D$
in (\ref{white}) do not directly appear on the discrete level. To give these coefficients
a consistent interpretation, we start from the observation \cite{Forster1977,Huse1985} that
the invariant measure of (\ref{noisy}) with spatio-temporal
white-noise driving is spatial white noise\footnote{This remains true
for certain discretizations of (\ref{KPZ}) \cite{Krug1991b,TSaSp09}.} with
strength $D/2 \nu$. This is easy to check for the linearized equation ($\lambda = 0$) but
it remains true also for $\lambda \neq 0$, somewhat analogous to the invariance of
the uniform measure for the ASEP discussed in Sect.~\ref{Measures}. As a consequence, the
spatial statistics of $h(x,t)$ for long times is that of a Wiener process with
''diffusion constant'' $D/4 \nu$ \textit{in space}\footnote{An immediate consequence of (\ref{heightdiff}) is that
typical configurations of $h(x,t)$ are non-differentiable. This is the origin of the ill-posedness of the KPZ equation.} ,
\begin{equation}
\label{heightdiff}
\lim_{t \to \infty} \mathbb{E}[(h(x,t) - h(x',t))^2] = \frac{D}{2\nu} \vert x - x' \vert
\equiv A \vert x - x' \vert.
\end{equation}
This relation holds also on the discrete level, provided $\vert x - x' \vert$ is large
compared to the correlation length of the particle system, and it identifies
the ratio $A = D/2 \nu$ as a property of the invariant measure of the latter;
for the continuous time ASEP we read off the relation $A = \bar \rho
(1 - \bar \rho)$ from (\ref{ASEPrough}), and for the
discrete time TASEP $A$ can be computed from the transition probabilities
(\ref{DTASEP1},\ref{DTASEP2}) [see (\ref{ADTASEP})]. It can be seen from the relation (\ref{hu})
[or its discrete analogue (\ref{hdiff})] that
the height difference correlation function
is a measure of the fluctuations in the
particle number in the interval between $x$ and $x'$. For this reason $A$ has
been referred to as a (nonequilibrium) compressibility \cite{Hager2001}.
We note for later reference that the KPZ-equation (\ref{KPZ}) can be linearized using the Hopf-Cole transformation
\begin{equation}
\label{HopfCole}
Z(x,t) = \exp \left[ - \frac{\lambda}{2 \nu} h(x,t) \right],
\end{equation}
which was originally applied to the deterministic Burgers equation \cite{Hopf1950,Cole1951} and rediscovered
in the context of (\ref{KPZ}) by Huse, Henley and Fisher \cite{Huse1985}. Indeed, using (\ref{KPZ})
we see that $Z(x,t)$ evolves according to a heat equation with a multiplicative stochastic force,
\begin{equation}
\label{Z(x,t)}
\frac{\partial Z}{\partial t} = \nu \nabla^2 Z + \frac{\lambda}{2 \nu} \zeta(x,t) Z(x,t).
\end{equation}
The formal solution of (\ref{Z(x,t)}) is a Wiener path integral
describing the weight $Z(x,t)$ of a Brownian path (or ``directed polymer''
\cite{Kardar1986}) subject to the random space-time potential $\frac{\lambda}{2 \nu} \zeta(x,t)$.
\subsection{The universality hypothesis}
The considerations in the preceding subsection suggest that the details of the underlying particle system
enter the large scale fluctuations properties only through the two parameters $\lambda$
and $A$. These parameters define characteristic scales of height, length and time,
which can be used to non-dimensionalize any correlation function of interest. In the
non-dimensional variables the correlation functions are then conjectured
to be \textit{universal}, i.e.
independent of the specific microscopic model. This is the essence of the
universality hypothesis.
As an illustration, consider the probability distribution
of the height $h(x,t)$ at a given point $x$, corresponding to the
time-integrated current through a fixed bond in the exclusion process.
Because of translational invariance,
this cannot depend on $x$, and we have to find a combination of $\lambda$, $A$
and $t$ that has the dimension of $h$. Denoting the dimension of a quantity
$X$ by $[X]$, we read off from (\ref{KPZ}) that
$$
[\lambda] = \frac{[x]^2}{[h] [t]}
$$
and from (\ref{heightdiff}) that
$$
[A] = \frac{[h]^2}{[x]}.
$$
The unique combination with the dimension $[h]$ is
$(A^2 \vert \lambda \vert t)^{1/3}$, and hence we expect
that the rescaled height fluctuation
\begin{equation}
\label{heightscaled}
\tilde h = \frac{h}{(A^2 \vert \lambda \vert t)^{1/3}}
\end{equation}
should have a universal distribution. For example, the variance of the height
is predicted to be of the form \cite{Krug1992}
\begin{equation}
\label{variance}
\mathbb{E}[h(x,t)^2] - (\mathbb{E}[h(x,t)])^2 = c_2 (A^2 \vert \lambda \vert t)^{2/3}
\end{equation}
with a universal constant $c_2$ which is independent of the specific model
or of model parameters such as the update probability $\pi$ in the dTASEP
(see Remark \ref{KPZ_remark}). Similarly, the unique combination of $\lambda$, $A$ and $t$
that has the dimension $[x]$ of length is
\begin{equation}
\label{correlation_length}
\ell(t) = (A \lambda^2)^{1/3} t^{2/3},
\end{equation}
which defines the \textit{correlation length} of fluctuations; note that correlations spread
\textit{superdiffusively}, that is, faster than
$t^{1/2}$ \cite{vanBeijeren1985}. This is in contrast to
the case $\lambda = 0$, where a straightforward solution of
(\ref{KPZ}) shows that the correlation length grows diffusively, and
height fluctuations are Gaussian and of order $t^{1/4}$ \cite{Krug1997}.
This behavior has been explicitly demonstrated for interacting particle systems
in which the current is a linear (or constant) function of the
density\footnote{It is also possible to construct
situations where the leading order nonlinearity in the expansion of the current is of cubic or higher order
in the density fluctuations, so that $\lambda = 0$ but the problem remains
nonlinear \cite{Derrida1991}.
Non-rigorous analysis indicates that such nonlinearities are irrelevant (in the sense that
the diffusive behavior is preserved) when of quartic order or higher, while in the cubic
case fluctuations spread weakly superdiffusively as $t^{1/2} (\ln
t)^{1/4}$ \cite{Devillard1992,Binder1994}.} \cite{Ferrari1998,Krug2000a,Balazs2006}.
As an illustration of these considerations, and for later reference, we compute the scale factor
$A^2 \vert \lambda \vert$ for the dTASEP
at density $\bar \rho = 1/2$. Taking two
derivatives of the current function (\ref{JTASEP}) we find
$$
\lambda_{\mathrm{dTASEP}}(1/2) = - \frac{2 \pi}{\sqrt{1 - \pi}}.
$$
To determine the compressibility $A$ we appeal to the equivalence of the
invariant measure (\ref{Ising}) to the equilibrium state of
the one-dimensional Ising chain\footnote{The one-dimensional Ising
chain is treated in most textbooks on statistical physics, see
e.g. \cite{Plischke2006}.}.
Ising spins $\sigma_i$ are canonically related to the occupation variables
$\eta_i$ by $\sigma_i = 1 - 2 \eta_i = \pm 1$. The transition probabilities
(\ref{DTASEP1},\ref{DTASEP2}) make up the \textit{transfer matrix}
of the Ising chain, with the
density $\rho$ playing the role of the magnetic field (which vanishes when
$\rho = 1/2$) and the update probability $\pi$ controlling the nearest
neighbor coupling; since $p_{1/2}(0,1) > p_{1/2}(1,1)$ the coupling
is \textit{antiferromagnetic} for $\pi > 0$. Particle number fluctuations
translate to fluctuations of the magnetization, and hence the
compressibility is proportional to the magnetic susceptibility of the Ising
chain. This can be computed from the free energy per spin, which is proportional to the
logarithm of the largest eigenvalue of the transfer matrix, by taking
two derivatives with respect to the magnetic field. The final result is
\begin{equation}
\label{ADTASEP}
A_{\mathrm{dTASEP}}(1/2) = \frac{1}{4} \;
\frac{p_{1/2}(1,1)}{p_{1/2}(0,1)} = \frac{1}{4}
\sqrt{1 - \pi},
\end{equation}
and we conclude that
\begin{equation}
\label{scaleDTASEP}
(A^2 \vert \lambda \vert)_{\mathrm{dTASEP}} = \frac{1}{8} \pi \sqrt{1 - \pi}.
\end{equation}
The early work on KPZ-type processes was mostly concerned with establishing the
universality of the $t^{2/3}$-scaling of the variance (\ref{variance}) which,
once the role of $A$ and $\lambda$ has been recognized, is essentially a
consequence of dimensional analysis \cite{Krug1997}. Numerical evidence
of universality in a more refined sense, which encompasses universal
\textit{amplitudes} like $c_2$ in (\ref{variance}),
was presented in \cite{Krug1992}, where it was also pointed
out that different \textit{universality classes} characterized by the same
$t^{2/3}$ scaling but different amplitudes may arise
from different initial and boundary conditions. Specifically, three cases
were identified:
\begin{itemize}
\item[I.] \textit{Growth from a flat surface without fluctuations}. In the language
of exclusion processes, this corresponds to an \textit{ordered}
initial condition of constant density; for example, the case $\bar \rho = 1/2$
is realized by occupying all odd or all even sites of the lattice.
\item[II.] \textit{Growth from a flat surface with stationary roughness}, in the sense
of (\ref{heightdiff}). This corresponds to starting the exclusion process
in a configuration generated from the invariant measure, e.g. a Bernoulli
initial condition of density $\rho$ for the continuous time ASEP. In this
case the universal fluctuations of interest are visible only if the
density is chosen such that the kinematic wave speed
$c(\rho) = 0$; otherwise they will be masked by
the fluctuations in the initial condition which drift across the observation
point. The drift can be eliminated by moving the observation point
at the kinematic wave speed $c(\rho)$.
\item[III.] \textit{Growth of a cluster from a seed.} For the exclusion process
this corresponds to a step initial condition of the form (\ref{step}) with
$\rho_L > \rho_R$. When $\rho_L > 1/2 > \rho_R$ the relation (\ref{phic}) ensures
that the density at the origin $x = 0$ remains at $\phi = c^{-1}(0) = 1/2$
at all times. As in case II., current fluctuations at other values of $\rho$ can be studied
by moving the observation point along a general characteristic $x/t = \xi$
with $\phi(\xi) = \rho$.
\end{itemize}
Early attempts to derive refined universal information, such as amplitudes and scaling functions, directly
from the KPZ equation met with limited success \cite{Krug1992,Amar1992,Frey1996}. A full understanding
of the universality classes of the one-dimensional KPZ equation became available only through the spectacular
developments that were triggered a decade ago by the paper of Baik, Deift and Johansson \cite{Baik99}.
In the next three sections we explain the key steps of this development along the lines of the work of
Johansson \cite{Jo1}, and return to the broader issue
of KPZ universality in Sect.~\ref{Universality}.
\section{An exactly solvable model: dTASEP with step initial conditions}
\label{tsec:R}
In this section we begin our discussion of Johansson's result \cite{Jo1}
on the fluctuations of the particle flux for discrete time TASEP (dTASEP) with step initial data. We formulate the result in Theorem \ref{tsatz:R.1} below and we compare it with the predictions of KPZ theory described in the previous section.
Let us first recall the dTASEP model that has been introduced in Sect.~\ref{jsec3.1} (iii).
We denote the infinitely many particles of the system by integers $j = 0, 1, 2, \ldots$ and their respective
positions at integer times $t = 0, 1, 2, \ldots$ by $x_j(t) \in \mathbb Z$. We assume step initial conditions
$x_j(0)=-j$. Jumps to the right $x_j(t+1)=x_j(t)+1$ are attempted at every time step $t \ge 0$ by all particles
$j \ge 0$ independently with probability $\pi$, but have to be discarded by the exclusion property if at time $t$ the receiving site $x_j(t)+1$ is
occupied by another particle of the system, i.e. if $x_{j-1}(t)=x_j(t)+1$. In this case, particle $j$ remains on its site, $x_j(t+1)=x_j(t)$.
\begin{definition}\label{tdef:R.1}
We denote by $\mathbb P_\pi$ the probability measure on the (total) motion of the particle system that is induced by the
stochastic process described above.
\end{definition}
Let us first look at an example and compute the probability that the motion depicted
in Figure \ref{aa} occurs. To do this we only need to count for each particle $j = 0, 1, 2, 3$ how many times it had a choice to jump and how often it actually jumped.
\begin{figure}
\begin{center}
\resizebox{7.cm}{!} {\includegraphics{Figure2.eps}}
\end{center}
\caption{Sample path for dTASEP where only the motion of the four rightmost particles $j=0, 1, 2, 3$ is displayed up to some time $t\le 15$.}
\label{aa}
\end{figure}
\begin{center}
\begin{tabular}{rccc}
& $\sharp$ choices & $\sharp$ jumps & $\sharp$ stays\\
$j=0$ & $10$ & $5$ & $5$\\
$j=1$ & $11$ & $5$ & $6$\\
$j=2$ & $10$ & $4$ & $6$\\
$j=3$ & $9$ & $4$ & $5$\\
total & $40$ & $18$ & $22$
\end{tabular}
\end{center}
By the assumed stochastic independence of all jumps we have
\[
\mathbb P_\pi \ \textrm{(Figure \ref{aa} occurs)}\ =\pi^{18}(1-\pi)^{22}\;.
\]
Next we turn to the flux which is the quantity that we want to analyze.
\begin{definition}\label{tdef:R.2}
For $r \in \mathbb Z$, $t \in \mathbb N$ we denote the
total flux through the bond between sites $r$ and $r+1$
up to time $t$ by
\[
F_r(t):= \sharp \{ j\in\mathbb N\colon x_j(t)>r\}
-\sharp \{ j\in\mathbb N\colon x_j(0)>r\}\;,
\]
i.e. the total number of particles that have crossed from site $r$ to $r+1$ during the time interval $[0,t]$.
\end{definition}
For example, in the particular situation displayed in Figure \ref{aa} we have
\begin{center}
\begin{tabular}{l|c|c|c|c|}
$t$ & $3$ & $6$ & $9$ & $12$ \\ \hline
$F_{-1}(t)$ & $0$ & $1$ & $2$ & $2$ \\
$F_0(t)$ & $1$ & $2$ & $2$ & $3$ \\
$F_1(t)$ & $0$ & $1$ & $2$ & $2$
\end{tabular}
\end{center}
From now on we will only consider the flux $F_0(t)$ through the bond between sites $0$ and $1$
in order to keep the presentation as simple as possible.
Let us first recall what the discussion on the hydrodynamic limit presented in Sect.~\ref{Hydro}
implies for the current at $x=0$.
We are exactly in the situation of the rarefaction wave (see (ii) of
Sect.~\ref{Hydro}) with $\rho_L = 1$ and $\rho_R =0$ and with $J(\rho)$ given by (\ref{JTASEP}).
Since $c(1/2) = J'(1/2) = 0$ we learn from (\ref{wave}) and (\ref{phic}) that
$\rho(0, t) = \phi(0) = c^{-1}(0) = 1/2$ and again by (\ref{JTASEP}) it follows that the current $j(0, t)$
is given by
\begin{eqnarray}\label{teq:R.3}
j(0, t) = J(\rho(0, t)) = \frac{1}{2} (1 - \sqrt{1 - \pi}) =: J_{\pi}\, .
\end{eqnarray}
We therefore expect that $F_0(t)$ is approximately given by $J_{\pi} t$. Indeed, it is a corollary of
Theorem \ref{tsatz:R.1} below that $F_0(t)/t$ converges with probability 1 to $J_{\pi}$ as $t \to \infty$.
Now we turn to the more detailed predictions of KPZ theory. As it was explained in Sect.~\ref{Growthmodel}, the flux $F_0(t)$ corresponds to the height $h(1/2, t)$. The dimensional analysis of Sect.~\ref{KPZ_conjecture} suggests that the fluctuations of $F_0(t) - J_{\pi}t$
are of order $t^{1/3}$ (see (\ref{heightscaled})). Moreover, taking
(\ref{variance}) and (\ref{scaleDTASEP}) into account, it is predicted for large values of $t$ that the standard deviation of the centered and rescaled variable
$Z_{\pi}(t) := t^{-1/3}(F_0(t) - J_{\pi}t)$ is given by
\begin{eqnarray*}
\sqrt{\mbox{Var} (Z_{\pi}(t))} = C \pi^{1/3} (1-\pi)^{1/6}, \quad t \mbox{ large, }
\end{eqnarray*}
with $C$ independent of $\pi$. The result of Johansson forcefully reaffirms the KPZ conjecture for dTASEP. Even more is true: In the limit $t \to \infty$, the random variables $Z_{\pi}(t)/(\pi^{1/3} (1-\pi)^{1/6})$ do not only have the same second moments for all $0 < \pi < 1$, but they converge to exactly the same probability distribution.
One may think of this result in analogy to the Central Limit Theorem. There one considers
independent, identically distributed random variables $X_i$. The quantities for which we draw the
analogy to the fluxes $F_0(n)$ are the partial sums $S_n = X_1 + \ldots + X_n$. Under some weak assumptions
on the distribution of the $X_i$'s one has with probability 1 that $S_n/n$ converges to the expectation
$\mu := \mathbb E(X_1)$ for $n \to \infty$ (law of large numbers) and that the rescaled random variables
$n^{-1/2}(S_n - n \mu)$ tend to a Gaussian distribution (Central Limit Theorem).
In contrast to the Central Limit Theorem the rescaled variables $Z_{\pi}$ do not converge to a Gaussian distribution. Indeed, and quite surprisingly, the limiting distribution is given by the Tracy-Widom distribution of random matrix theory.
\begin{reminder}\label{trem:C.1}
(Tracy-Widom distribution)
\noindent
{\em
The Gaussian Unitary Ensemble GUE is defined as a sequence $\mathbb P_N$ of Gaussian probability
measures on $N\times N$ Hermitean matrices of the form
\[
d\mathbb P_N(M)=\frac{1}{Z_N}e^{-\mathrm{tr}(M^2)}dM
\]
where $Z_N$ denotes the norming constant and $dM$ abbreviates the product Lebesgue measure on the real and imaginary parts of the entries of $M$ respecting the Hermitean symmetry, i.e. $dM = \Pi_i \,dM_{ii} \,\,
\Pi_{i<j} \,d\mbox{Re}(M_{ij})\, d\mbox{Im}(M_{ij})$. Denote
by $\lambda_1(M)$ the largest eigenvalue of $M$ which is a random object. $\lambda_1(M)$ is expected to be at $\sqrt{2N}$ and the fluctuations are of order $N^{-1/6}$. One can prove that the appropriately rescaled largest eigenvalue converges to some distribution that is now called the Tracy-Widom distribution, i.e. for
$s\in \mathbb R$ we have
\[
\mathbb P_N\left(\frac{\lambda_1(M)-\sqrt{2N}}{(8N)^{-1/6}}\le s\right)\longrightarrow
TW_2(s) \qquad \mbox{as }N\to\infty\, .
\]
The function $TW_2$ can be expressed in terms of the Hastings--McLeod
solution of the Painlev\'e II equation \cite{TTracyWidom94} or by Fredholm determinants of integral operators
with Airy kernel (see Sect.~\ref{tsec:A.2} for more details). Note that the subindex $2$ of $TW_2$ is related
to the fact that GUE is a $\beta$-random matrix ensemble with $\beta=2$. Roughly speaking a $\beta$-ensemble
is an ensemble where the joint distribution of eigenvalues is of the form
\[
d\mathbb P_N(\lambda_1, \ldots, \lambda_N)= \frac{1}{\hat{Z}_N} |\Delta(\lambda)|^{\beta} \prod_{j=1}^N w_N(\lambda_j) d\lambda_j \, ,
\]
and $\Delta$ denotes the Vandermonde determinant (cf. Sect.~\ref{tsec:C.4} below).
See \cite{TMehta} for a general reference on Random Matrix Theory.
The densities of the Tracy-Widom distributions $TW_1$ and $TW_2$ are displayed in Fig.~\ref{3functions}(a) of Sect.~\ref{Universality}.
}
\end{reminder}
For our model, dTASEP with step initial data, the theorem of Johansson \cite{Jo1} implies:
\begin{theorem} \label{tsatz:R.1}
Let $0 < \pi < 1$. Set $J:=\frac{1}{2}(1-\sqrt{1-\pi})$ and $V:= 2^{-4/3} \pi^{1/3}(1-\pi)^{1/6}$. Then, for all $s \in
\mathbb R$ we have
\begin{eqnarray}\label{tN2.5}
\lim_{t \to \infty} \mathbb P_\pi \left(\frac{F_0(t)-Jt}{Vt^{1/3}}\le s\right) =
1-TW_2(-s)
\end{eqnarray}
\end{theorem}
\begin{remark}
\label{KPZ_remark}
As it was noted above Theorem \ref{tsatz:R.1} affirms and strengthens the KPZ predictions. In particular, the scaling of the flux is precisely that expected from KPZ theory. Comparison with
(\ref{scaleDTASEP}) shows that $V = (A^2 \vert \lambda \vert/2)^{1/3}$ in the notation of Sect.~\ref{KPZ_conjecture}.
\end{remark}
The results of Johansson in \cite{Jo1} are more general than stated above. Mean and fluctuations of the particle flux are described not only at the origin and continuous (in time) TASEP is also considered
by letting $\pi$ tend to $0$ and by rescaling time in an appropriate manner.
We would like to emphasize that Johansson's proof of Theorem \ref{tsatz:R.1} does not make any use of the
considerations regarding the hydrodynamic limit and the KPZ conjecture as presented above. Instead,
the problem treated in Theorem \ref{tsatz:R.1} should be viewed as a very special one within the class of models considered in Sects.~\ref{Exclusion}
and \ref{KPZ_conjecture}. This problem has the attractive feature that it is exactly solvable by a series of
beautiful and non-obvious observations which will be described in the following two sections.
\section{Proof of Theorem \ref{tsatz:R.1} -- part I: Combinatorics}
\label{tsec:C}
We begin our discussion of the proof of Theorem \ref{tsatz:R.1} by relating the flux
$F_0(t)$ to another random variable. For $j,k\in \mathbb N$ denote
\[
T(j,k):= \min \{ t\in\mathbb N\colon x_j(t)=k+1-j\}\;,
\]
that is the time by which particle $j$, that starts at site $x_j(0)=-j$, has just completed its
$(k+1)$-st jump. Observe, that at time $T_k:=T(k,k)$ we have
\[
x_0(T_k)>x_1(T_k)>\ldots >x_k(T_k)=1>0\ge x_{k+1} (T_k)>\ldots\;.
\]
Thus, at time $T_k$ exactly the first $k+1$ particles $0,1,\ldots ,k$ have already jumped from
site $0$ to site $1$ and $F_0(T_k)=k+1$. Moreover, for times $t<T_k$ we have
$F_0(t)\le k$. This implies the relation
\begin{equation}\label{teq:R.9}
\mathbb P_\pi (F_0(t)\le k)=\mathbb P_\pi (T_k>t)=1-\mathbb P_\pi (T(k,k)\le t)\;.
\end{equation}
In the present section we outline how the explicit formula (\ref{teq:C.28}) of Lemma \ref{tlemma:C.1} below for the probability
distribution of $T(k,k)$ can be derived. By a series of bijections we map our combinatorial model
via waiting times and random words to Semi Standard
Young Tableaux, a classical object of combinatorics and representation theory where explicit formulas
for counting are available. The asymptotic analysis of formula
(\ref{teq:C.28}) for $\mathbb P_\pi (T(k,k)\le t)$
is discussed in Sec.~\ref{tsec:A}.
\subsection{From discrete TASEP to waiting times}
\label{tsec:C.1}
We introduce an equivalent description of the dynamics of the particle system by a table of waiting times.
For $j,l\in\mathbb N$ we denote
\begin{eqnarray*}
w_{j,l} &:=& \mbox{ number of times particle $j$ decides to stay on site $l-j$ } \\
&& \mbox{ after it becomes possible to jump to site $l-j+1$. }
\end{eqnarray*}
For example, in the case of Figure \ref{aa} we can determine the following entries of the matrix $(w_{j,l})$ of waiting times.
\begin{equation}\label{teq:C.0}
\left(
\begin{array}{cccccc}
0&3&1&0&1&\ldots \\
2&0&0&1&3&\ldots \\
1&2&0&2&?&\ldots \\
2&0&2&1&?&\ldots \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{array}
\right)
\end{equation}
The key observation for computing $T(k,k)$ from the table of waiting times is the
following recursion for $T(j,k)$.
\begin{eqnarray}\label{teq:C.1}
\hspace{-30pt}
T(j,k)=1+w_{j,k}+\left\{\begin{array}{lcl}
0&,&\textrm{if}\ j=k=0\\
T(j,k-1)&,&\textrm{if}\ j=0, k>0\\
T(j-1,k)&,&\textrm{if}\ j>0, k=0\\
\max (T(j-1,k), T(j,k-1))&,&\textrm{if}\ j,k>0
\end{array}\right.
\end{eqnarray}
Indeed, to compute the time it takes the $j$-th particle to complete its $(k+1)$-st jump one needs to add
$1+w_{j,k}$ to the time when this jump became possible. For this jump to become possible, particle $j$ has to be on site $k-j$ (happens at
$T(j, k-1)$) and particle $j-1$ must have emptied neighboring site $k-j+1$ (happens at time $T(j-1,k)$). It is obvious from (\ref{teq:C.1}) that in order to compute $T(k, k)$ one only needs to know the $(k+1)\times (k+1)$ topleft section of the table of waiting times
$(w_{j,l})_{0\le j,l \le k}$.
Relation (\ref{teq:C.1}) allows to prove the following formula for $T(j,k)$ by induction on $(j+k)$:
\begin{eqnarray}\label{teq:C.2}
T(j,k)=j+k+1+\max\limits_{\mathcal P\in \Pi_{j,k}}\left(\sum\limits_{s
\scriptsize{\mbox{ on }} \mathcal P}w_s\right)\;.
\end{eqnarray}
Here $\Pi_{j,k}$ denotes the set of paths $\mathcal P$ in the table of waiting times that connect the
$(0,0)$-entry with the $(j,k)$-entry and satisfy the additional condition that only steps to the right-neighbor and to the
neighbor downstairs are permitted. More formally, we may write
\begin{eqnarray*}
\Pi_{j,k}=\left\{ (s_0,\ldots ,s_{j+k})\in(\mathbb N\times\mathbb N)^{j+k+1}\colon
s_0=(0,0), s_{j+k}=(j,k)\ \textrm{and}\right.\\
\left. s_i-s_{i-1} \in \{(1,0),(0,1)\}\ \textrm{for all}\ 1\le i\le j+k\right\}\;.
\end{eqnarray*}
For $\mathcal P =(s_0,\ldots ,s_{j+k})\in \Pi_{j,k}$ we understand
\[
\sum\limits_{s \scriptsize{\mbox{ on }} \mathcal P}w_s:= \sum\limits^{j+k}_{i=0}w_{s_i}
\]
We illustrate formula (\ref{teq:C.2}) with our running example. The corresponding table of waiting times
displayed in (\ref{teq:C.0}) has nine paths in $\Pi_{3,3}$
that maximize the sum of waiting times. Two of them are
\begin{eqnarray}\label{teq:C.N1}
\mathcal P_1 &\colon & (0,0)\to (0,1)\to (0,2)\to (0,3)\to (1,3)\to (2,3)\to (3,3)\\
\mathcal P_2&\colon & (0,0)\to (1,0)\to (2,0)\to (2,1)\to (2,2)\to (3,2)\to (3,3)
\end{eqnarray}
and we have
\[
\sum\limits_{s \scriptsize{\mbox{ on }} \mathcal P_1}w_s=\sum\limits_{s \scriptsize{\mbox{ on }} \mathcal P_2}w_s=8\;.
\]
Formula (\ref{teq:C.2}) then yields for the time when the fourth particle $j=3$ has just completed its fourth jump $T(3,3)=3+3+1+8=15$ which is easily verified from Figure \ref{aa}.
\begin{remark}
The probabilistic model we have arrived at, i.e. to search for right- and downward
paths that maximize the total waiting time, is also known as the last passage
percolation problem and that is precisely the model studied in the paper
\cite{Jo1} of Johansson. Interpreting $w_{j,l}$ as potential energies this can also be considered as the problem
of zero-temperature directed polymers in a random medium \cite{Kardar1987,Krug1992,Krug1994,HalpinHealy1995,Krug1998}.
\end{remark}
We are now ready to compute $P_{\pi} (T(k,k) \le t)$ in terms of the table of waiting times. Recall from (\ref{teq:C.2}) that $T(k, k)$ is completely determined by the topleft $(k+1) \times (k+1)$ corner of the table. Moreover, and again by (\ref{teq:C.2}), a table of waiting times corresponds to a particle dynamics with $T(k,k) \le t$ if and only if the topleft corner belongs to the set $W(k, t)$
which we define to be the set of $(k+1) \times (k+1)$ matrices $(w_{j,l})$ with entries that are non-negative integers
and which have the additional property
\begin{eqnarray}\label{teq:C.4}
\max_{\mathcal P\in \Pi_{k,k}}\left(\sum_{s \scriptsize{\mbox{ on }} \mathcal P}w_s\right) \le t - 2k - 1 \; .
\end{eqnarray}
It is straightforward to determine the probability that the topleft corner of the table of waiting times agrees with any given element $Q$ of $W(k, t)$. Indeed, we only need to count the total number of decisions
to either jump (always equals $(k+1)^2$) or to stay (equals the sum of all entries of $Q$ which we denote by $|Q|_1$) whenever a jump is not prohibited by the exclusion property. In summary
we obtain the following formula
\begin{eqnarray}\label{teq:C.3}
\mathbb P_{\pi} (T(k,k) \le t) = \sum_{Q \in W(k,t)} \pi^{(k+1)^2} (1-\pi)^{|Q|_1}.
\end{eqnarray}
\subsection{From waiting times to random words}
\label{tsec:C.2}
We associate with any $(k+1)\times (k+1)$ matrix $Q = (w_{j,l})$ of waiting times the sequence of pairs
$(j,l)_{0 \le j, l \le k}$, listed in lexicographical order, where the value of $w_{j,l}$ determines how often the index
$(j, l)$ appears in this list. In the case of $Q$ being the top left $4 \times 4$ submatrix in (\ref{teq:C.0}) the corresponding sequence of pairs reads
\begin{eqnarray}\label{teq:C.10}
\begin{array}{ccccccccccccccccc}
0&0&0&0&1&1&1&2&2&2&2&2&3&3&3&3&3\\
1&1&1&2&0&0&3&0&1&1&3&3&0&0&2&2&3
\end{array}
\end{eqnarray}
We may consider this list of pairs as a list of 17 two-letter words from the alphabet $\{0, 1, 2, 3\}$ in lexicographical order. This
explains the term ``random words'' often used in this context. Observe that any right-downward path $\mathcal P\in \Pi_{3,3}$ corresponds to a subsequence in this list of 17 two-letter words where both the first and the second row are weakly increasing.
A little more thought shows that the quantity $\max_{\mathcal P\in \Pi_{3,3}}\left(\sum_{s \scriptsize{\mbox{ on }} \mathcal P}w_s\right)$
is given by the length of the longest subsequence that is
weakly increasing in both rows. The sequence (\ref{teq:C.10}) has nine such subsequences of maximal length 8. The subsequences corresponding to the paths $\mathcal P_1$ and $\mathcal P_2$ of (\ref{teq:C.N1}) read
\begin{eqnarray*}
\begin{array}{cccccccc}
0&0&0&0&1&2&2&3\\
1&1&1&2&3&3&3&3
\end{array}, \qquad \mbox{ and } \qquad
\begin{array}{cccccccc}
1&1&2&2&2&3&3&3\\
0&0&0&1&1&2&2&3
\end{array} .
\end{eqnarray*}
Formula (\ref{teq:C.3})
translates to
\begin{eqnarray}\label{teq:C.12}
\mathbb P_{\pi} (T(k,k) \le t) = \sum_{\phi \in D(k,t)} \pi^{(k+1)^2} (1-\pi)^{\mbox{length of } \phi},
\end{eqnarray}
where $D(k, t)$ is the set of finite sequences $\phi$ of lexicographically ordered two-letter words from the
alphabet $\{0, 1, \ldots, k \}$ and for which the length of the longest subsequence of $\phi$ that increases weakly in both letters
is at most $t-2k-1$. By the Robinson--Schensted--Knuth correspondence we may enumerate the set $D(k, t)$
conveniently in terms of Semi Standard Young Tableaux. This is the content of the next section.
\subsection{From random words to Semi Standard Young Tableaux}
\label{tsec:C.3}
The Robinson--Schensted correspondence provides a bijection between permutations and Standard Young Tableaux that
is well known in combinatorics and in the representation theory of the permutation group. We now describe the extension
of this algorithm to random words which was introduced by Knuth \cite{Knu}. The basic algorithm that needs to be understood first
is the row insertion process. Suppose we have a weakly increasing sequence of integers, e.g. $0 \; 0 \; 1 \; 1 \; 1 \; 3$.
We insert an integer $r$ into this row by the following set of rules. If $r \ge 3$ we simply append $r$ at the end of the row.
In the case $r < 3$ we replace the unique number $s$ in the row that is strictly bigger than $r$ such that after the replacement
the sequence is still weakly increasing. We say that we have inserted $r$ by bumping $s$. For the sequence
$0 \; 0 \; 1 \; 1 \; 1 \; 3$ insertion of $r$ leads to
\begin{center}
\begin{tabular}{|l|l|l|}
$r$ & sequence after insertion of $r$ & bumped number \\ \hline
$0$ & $0 \; 0 \; 0 \; 1 \; 1 \; 3$ &$1$ \\
$1$ & $0 \; 0 \; 1 \; 1 \; 1 \; 1$ & $3$ \\
$2$ & $0 \; 0 \; 1 \; 1 \; 1 \; 2$ & $3$ \\
$3$ & $0 \; 0 \; 1 \; 1 \; 1 \; 3 \; 3$ & no number bumped
\end{tabular}
\end{center}
We now describe the procedure how a lexicographically ordered list of two-letter random words is transformed
into a pair of tableaux. In a first step, one only considers the sequence of the second letters of the random words
in order to build the first tableau. To this end
one inserts (see row insertion process described above) the second letters of the words, one after the other, into the first row of the tableau
that has been created before. In case a number is bumped, the bumped number is inserted into the second row of the tableau.
In case bumping occurs again in the second row, we insert the newly bumped number into the third row of the tableau.
This process is repeated until the bumping ends. In our example (\ref{teq:C.10}) this process leads to the following
sequence of tableaux.
\begin{equation*}
\hspace{-20pt}
\begin{array}{c}
1
\end{array}
\stackrel{1}{\longrightarrow}
\begin{array}{cc}
1&1
\end{array}
\stackrel{1}{\longrightarrow}
\begin{array}{ccc}
1&1&1
\end{array}
\stackrel{2}{\longrightarrow}
\begin{array}{cccc}
1&1&1&2
\end{array}
\stackrel{0}{\longrightarrow}
\begin{array}{cccc}
0&1&1&2\\
1&&&
\end{array}
\stackrel{0}{\longrightarrow}
\end{equation*}
\begin{equation*}
\hspace{-20pt}
\begin{array}{cccc}
0&0&1&2\\
1&1&&
\end{array}
\stackrel{3}{\longrightarrow}
\begin{array}{ccccc}
0&0&1&2&3\\
1&1&&&
\end{array}
\stackrel{0}{\longrightarrow}
\begin{array}{ccccc}
0&0&0&2&3\\
1&1&1&&
\end{array}
\stackrel{1}{\longrightarrow}
\end{equation*}
\begin{equation*}
\hspace{-20pt}
\begin{array}{ccccc}
0&0&0&1&3\\
1&1&1&2&
\end{array}
\stackrel{1}{\longrightarrow}
\begin{array}{ccccc}
0&0&0&1&1\\
1&1&1&2&3
\end{array}
\stackrel{3}{\longrightarrow}
\begin{array}{cccccc}
0&0&0&1&1&3\\
1&1&1&2&3&
\end{array}
\stackrel{3}{\longrightarrow}
\end{equation*}
\begin{equation*}
\hspace{-20pt}
\begin{array}{ccccccc}
0&0&0&1&1&3&3\\
1&1&1&2&3&&
\end{array}
\stackrel{0}{\longrightarrow}
\begin{array}{ccccccc}
0&0&0&0&1&3&3\\
1&1&1&1&3&&\\
2&&&&&&
\end{array}
\stackrel{0}{\longrightarrow}
\end{equation*}
\begin{equation*}
\hspace{-20pt}
\begin{array}{ccccccc}
0&0&0&0&0&3&3\\
1&1&1&1&1&&\\
2&3&&&&&
\end{array}
\stackrel{2}{\longrightarrow}
\begin{array}{ccccccc}
0&0&0&0&0&2&3\\
1&1&1&1&1&3&\\
2&3&&&&&
\end{array}
\stackrel{2}{\longrightarrow}
\end{equation*}
\begin{equation*}
\hspace{-20pt}
\begin{array}{ccccccc}
0&0&0&0&0&2&2\\
1&1&1&1&1&3&3\\
2&3&&&&&
\end{array}
\stackrel{3}{\longrightarrow}
\begin{array}{cccccccc}
0&0&0&0&0&2&2&3\\
1&1&1&1&1&3&3&\\
2&3&&&&&&
\end{array}
\end{equation*}
Using this procedure we have obtained a Semi Standard Young Tableau with 17 entries.
\begin{definition}\label{tdef:C.1}
By a Semi Standard Young Tableau (SSYT) we understand a tableau $\cal T$ of a finite number of integers
that are weakly increasing in each row and strictly increasing in each column. The shape $\lambda$ = sh$(\cal T)$
of $\cal T$ is denoted by the sequence of row lengths $(\lambda_0, \lambda_1, \ldots)$ that is required to be
a weakly decreasing sequence of non-negative integers. Furthermore, we set $|\lambda|:= \sum_i \lambda_i$ to be
the total number of cells in the tableau.
\end{definition}
In order to see that the procedure described above always leads to a SSYT one only needs to convince oneself that adding one number to
a SSYT will result in a SSYT with one more cell. The key observation here is that in the bumping procedure a number can only move downwards or left-downwards.
In our running example the final SSYT $\cal T^*$ has shape
$(8,7,2,0,0,\ldots)$ and it is no coincidence but a theorem that the length of the first row $\lambda_0$ equals the length of the longest weakly increasing subsequence that we have seen to be 8. This fact can be proven in general by showing inductively that at every step of the procedure the length of the longest weakly increasing subsequence (in both letters) up to some word in the list is given by the position at which the second letter of the word is inserted in the first row of the tableau. In our example (\ref{teq:C.10}) the length of the longest weakly increasing subsequence up to the 15-th word $(3,2)$ is six and correspondingly $2$ is inserted at the sixth position of the first row.
Note that the sequence (\ref{teq:C.10})
is not the only one that leads to the final tableau $\cal T^*$. For example, the sequence
\[
\begin{array}{ccccccccccccccccc}
0&0&0&0&1&1&1&2&2&2&2&2&3&3&3&3&3\\
1&1&1&2&3&3&3&0&0&1&1&3&0&0&0&2&2
\end{array}
\]
leads to the same $\cal T^*$. However, and this is the central message of the Robinson-Schensted-Knuth correspondence, one may encode the sequence of random words (\ref{teq:C.10}) in a unique way
if one records in addition how the tableau grows and if one remembers the first letters of the random words that have so far been neglegted.
This information is all encoded in the second tableau. We now demonstrate how to build this second tableau
with our running example. As a first step we record for the above described procedure which cell has been
added to the SSYT at which step.
\begin{equation}\label{teq:C.16}
\begin{array}{cccccccc}
1&2&3&4&7&11&12&17\\
5&6&8&9&10&15&16&\\
13&14&&&&&&
\end{array}
\end{equation}
Since we also need to remember the first letters of our 17 random words it is natural to replace
the entries in (\ref{teq:C.16}) in the following way. We note that the first 4 words in (\ref{teq:C.10}) have first letter 0 and we therefore
replace $1$, $2$, $3$, $4$ each by $0$. The next three words have first letter 1 and we replace $5$, $6$, $7$ each by $1$. Then there are five words starting with letter $2$, leading us to replace $8$, $9$, $10$, $11$, $12$ each by $2$.
The remaining five entries $13$, $14$, $15$, $16$, $17$ are each replaced by $3$. This leads to the tableau
\begin{equation}\label{teq:C.18}
\begin{array}{cccccccc}
0&0&0&0&1&2&2&3\\
1&1&2&2&2&3&3&\\
3&3&&&&&&
\end{array}\, .
\end{equation}
We have arrived at a
SSYT $\cal U^*$ that clearly has
the same shape as $\cal T^*$. In order to prove that the second tableau always yields a SSYT there is only one non-obvious property to verify, which is the strict increase in each column. To see this recall that for words $(a_1, a_2)$,
$(b_1, b_2)$ the number $b_2$ can only bump $a_2$ if $a_2 > b_2$ which implies by the lexicographical ordering of the list that $b_1 > a_1$. Therefore, in the second tableau, $b_1$ will be located below or left-below $a_1$ and by the weak increase within each row this suffices.
In summary we have mapped the sequence (\ref{teq:C.10}) of random words to the pair of SSYT's
$(\cal T^*, \cal U^*)$
\begin{equation*}
\begin{array}{cccccccc}
0&0&0&0&0&2&2&3\\
1&1&1&1&1&3&3&\\
2&3&&&&&&
\end{array}
\;, \qquad \qquad
\begin{array}{cccccccc}
0&0&0&0&1&2&2&3\\
1&1&2&2&2&3&3&\\
3&3&&&&&&
\end{array}
\end{equation*}
of equal shape $\lambda$. It is an
instructive exercise to reconstruct sequence (\ref{teq:C.10}) from the pair of SSYT's. In fact,
the proof that the above described procedure maps lexicographically ordered two letter words bijectively to pairs of SSYT's of equal shape
can be given by an explicit description of the inverse map. This bijection has two more features that are of interest to us.
Firstly, $|\lambda|$ equals the number of words in our list (= 17 in our running example). Secondly,
the length of a longest weakly increasing subsequence is exactly given by the length of the first row $\lambda_0$
(= 8 in our example) as we have already observed above (see \cite{Knu} for details).
This implies that the set $D(k, t)$ (cf. (\ref{teq:C.12})) is bijectively mapped onto the set of all pairs
$(\cal T, \cal U)$ of SSYT's of equal shape $\lambda$ satisfying
\[
t-2k-1 \ge \lambda_0 \ge \lambda_1 \ge \ldots
\]
with entries from $\{ 0, 1, \ldots, k \}$. Note that we have $\lambda_{k+1}=0$ because entries in each column are strictly increasing. We therefore arrive at
\begin{equation}\label{teq:C.20}
\mathbb P_{\pi} (T(k,k) \le t) = \sum_{t-2k-1 \ge \lambda_0 \ge \ldots \ge \lambda_{k} \ge 0} \pi^{(k+1)^2} (1-\pi)^{| \lambda |} L(\lambda, k)^2,
\end{equation}
where $L(\lambda, k)$ denotes the number of SSYT's of shape $\lambda = (\lambda_0, \ldots, \lambda_{k}, 0, \ldots)$
and with entries from $\{0, 1, \ldots, k \}$. We have now derived a representation for $\mathbb P_{\pi} (T(k,k) \le t)$
involving the combinatorial quantity $L(\lambda, k)$ that can be computed explicitly.
\subsection{Schur polynomials and an explicit formula for the distribution of $T(k, k)$}
\label{tsec:C.4}
There is a beautiful argument using Schur polynomials $s_{\lambda}$
that provides a formula for $L(\lambda, k)$. This argument is
explained in the appendix of \cite{Sas1} and we do not repeat it
here (see in addition \cite[Cor. 4.6.2]{TSagan01} for a derivation
of the representation \cite[(A.5)]{Sas1} of $s_{\lambda}$ by
determinants). The result is
\begin{equation}\label{teq:C.22}
L(\lambda, k) = \prod_{0\le i < j \le k} \frac{\lambda_i - \lambda_j + j - i}{j-i}.
\end{equation}
Introducing the new variables $y_i := \lambda_i -i +k$ and denoting the Vandermonde determinant by $\Delta(y) = \prod_{0\le i < j \le k}
(y_j-y_i)$ we obtain
\begin{equation}\label{teq:C.24}
\hspace{-30pt}
\mathbb P_{\pi} (T(k,k) \le t) = C_{\pi, k} \sum_{t-k-1 \ge y_0 > \ldots > y_{k} \ge 0}
\Delta(y)^2 \prod_{i=0}^k (1-\pi)^{y_i}, \quad \mbox{ where }
\end{equation}
\begin{equation} \label{teq:C.26}
C_{\pi, k} := \pi^{(k+1)^2} (1-\pi)^{-k(k+1)/2}
\prod_{0\le i < j \le k} \frac{1}{(j-i)^2}
\end{equation}
Observe that each term in the sum is a symmetric function in $y$ that vanishes if two components agree.
This leads to the final formula in this section for the probability distribution of $T(k, k)$ which is closely related to the distribution of the flux $F_0(t)$ via (\ref{teq:R.9}).
\begin{lemma}\label{tlemma:C.1}
\begin{eqnarray}\label{teq:C.28}
\mathbb P_{\pi} (T(k,k) \le t) &=& \frac{C_{\pi, k}}{(k+1)!}
\sum_{\scriptsize{\begin{array}{c} y \in \mathbb Z^{k+1} \\ 0 \le y_i \le t-k-1\end{array}}}
\Delta(y)^2 \prod_{i=0}^k (1 - \pi)^{y_i}\; .
\end{eqnarray}
\end{lemma}
This formula should be compared with the formula for the distribution of the largest eigenvalue of
the Gaussian Unitary Ensemble (cf. Reminder \ref{trem:C.1})
\begin{eqnarray}\label{teq:N.20}
\mathbb P_N (\lambda_1(M) \le \Lambda) = \frac{1}{\hat{Z}_N} \int_{(-\infty, \Lambda]^N} \Delta(y)^2
\prod_{j=1}^N e^{-y_j^2} \ dy
\end{eqnarray}
with some appropriate norming constant $\hat{Z}_N$. Observe that
this formula has exactly the same structure as (\ref{teq:C.28}). The
role played by the measure $e^{-x^2}dx$ for GUE is taken by the
discrete measure $\sum_{j=0}^{\infty} (1-\pi)^j \delta_j$ supported
on $\mathbb N$ for dTASEP. Here $\delta_j$ denotes the $\delta$-distribution concentrated at $j$. In the next section we discuss how the
method of orthogonal polynomials can be used to analyze the
asymptotics of such types of high-dimensional integrals.
\section{Proof of Theorem \ref{tsatz:R.1} -- part II: Asymptotic analysis}
\label{tsec:A}
In the previous section we have derived formulae (\ref{teq:C.28}) and (\ref{teq:R.9}) for the probability distribution of the flux $F_0(t)$.
We now need to analyze this formula asymptotically in a regime where $t$ and $k$ both become large and $k \approx J t + V t^{1/3} s$ with $s$ being an arbitrary but fixed real number and $J$, $V$ being defined as in Theorem \ref{tsatz:R.1}. The key to the analysis is the observation that the right hand side of (\ref{teq:C.28}) is structurally
the same as the standard formula for the probability distribution of the largest eigenvalue of GUE (\ref{teq:N.20}) and
the method of orthogonal polynomials (see Sect.~\ref{tsec:A.1}) can be applied. The role played by Hermite polynomials for GUE will
be taken by Meixner polynomials in our model. In both cases it is the behavior of the
orthogonal polynomials of large degree in a vicinity of their respective largest zero that matters in the
asymptotic analysis.
After appropriate rescaling this behavior can be described in terms of Airy functions for
both Hermite and Meixner polynomials (see Sect.~\ref{tsec:A.2}). On a technical level this explains the occurence of the Tracy-Widom distribution $TW_2$ for GUE as well as for dTASEP with step initial conditions. We include in Sect.~\ref{tsec:A.3} a brief discussion of the universal behavior of orthogonal polynomials.
\subsection{The method of orthogonal polynomials following an approach of Tracy and Widom}
\label{tsec:A.1}
Almost 50 years ago, Gaudin and Mehta have introduced orthogonal polynomials to random matrix theory in order to study local eigenvalue statistics. A very transparent version of the now so called \emph{method of orthogonal polynomials} is due to
Tracy and Widom \cite{TW2} and we will briefly outline their approach for our situation. See also the recent book of Deift and Gioev \cite[Chapt. 4]{TDeiftGioev09} for a self contained presentation of the method of orthogonal polynomials with some remarks on the history of the method. For us the method allows to express the right hand side of (\ref{teq:C.28}) in terms of a Fredholm determinant of an operator where the (discrete) integral kernel is given in terms of Meixner orthogonal polynomials.
We begin our discussion by introducing the discrete weight
\[
w_{\pi}(x) := \left\{
\begin{array}{ll}
0&, \mbox{ if } x < 0 \\ (1-\pi)^x &, \mbox{ if } x \ge 0
\end{array}
\right. , \quad x \in \mathbb Z\; .
\]
For a moment we let $(q_l)_{l\ge 0}$ be any sequence of polynomials with $q_l$ being of degree $l$ with (non-zero) leading coefficient
$\gamma_l$. Setting $\varphi_l (x) := q_l (x) \sqrt{w_{\pi}(x)}$ and using the definition of the Vandermonde determinant together with some basic properties of determinants we have for $y \in \mathbb N^{k+1}$ that
\[
\left[ \det (\varphi_l(y_i))_{0 \le i, l \le k} \right]^2 =
(\gamma_0 \ldots \gamma_k)^2 \Delta(y)^2 \prod_{i=0}^k (1 - \pi)^{y_i} \; .
\]
Furthermore, we set $I_s := [s, \infty)$ and denote by ${\bf 1}_{I_s}$ its characteristic function that
takes the value 1 on $I_s$ and 0 on $\mathbb R \setminus I_s$. We may then rewrite (\ref{teq:C.28}) in the form
\begin{eqnarray*}
\hspace{-30pt}
\mathbb P_{\pi} (T(k,k) \le t) = \frac{C_{\pi, k}}{(\gamma_0 \ldots \gamma_k)^2(k+1)!}
\sum_{y \in \mathbb Z^{k+1}} \left[ \det (\varphi_l(y_i)) \right]^2 \prod_{i=0}^k
\left( 1 - {\bf 1}_{I_{t-k}}(y_i)
\right).
\end{eqnarray*}
An identity due to Andr\'eief (see e.g. \cite[(3.3)]{TDeiftGioev09}) adapted to our context reads
\begin{eqnarray*}
\hspace{-40pt}
\sum_{y \in \mathbb Z^{k+1}} \left[ \det (\varphi_j(y_i)) \right]
\left[ \det (\varphi_l(y_i)) \right] \prod_{i=0}^k
f(y_i) = (k+1)! \det \left( \sum_{x \in \mathbb Z} \varphi_j(x) \varphi_l(x) f(x)
\right).
\end{eqnarray*}
One may prove this formula using the Leibniz sum for determinants. This allows to write the distribution of $T(k, k)$ as an determinant
\[
\mathbb P_{\pi} (T(k,k) \le t) =
\frac{C_{\pi, k}}{(\gamma_0 \ldots \gamma_k)^2} \det S \; ,
\]
where $S$ denotes the $(k+1) \times (k+1)$ matrix with entries
\[
S_{j, l} = \sum_{x \in \mathbb Z} \varphi_j(x) \varphi_l(x)
\left( 1 - {\bf 1}_{I_{t-k}}(x)
\right) \; ,
\qquad 0 \le j, l \le k \; .
\]
So far the choice of the polynomials $q_l$ of degree $l$ was arbitrary. Now we choose $(q_l)_l$ to be the sequence of
normalized orthogonal polynomials with respect to the discrete measure $\sum_{x \in \mathbb Z} w_{\pi}(x) \delta_x$
which belongs to the class of Meixner polynomials. We have
\[
\sum_{x \in \mathbb Z} \varphi_j(x) \varphi_l(x) = \sum_{x \in \mathbb Z} q_j(x) q_l(x) w(x) = \delta_{j, l}
\]
for $j, l \in \mathbb N$. Hence $S = I - R(t-k)$ with
\[
R(s)_{j, l} = \sum_{x \in \mathbb Z} \varphi_j(x) \varphi_l(x) {\bf 1}_{I_s}(x) =
\sum_{x \ge s} \varphi_j(x) \varphi_l(x) \; .
\]
In summary, we have so far derived
\begin{equation}\label{teq:A.10}
\mathbb P_{\pi} (T(k,k) \le t) = \frac{C_{\pi, k}}{(\gamma_0 \ldots \gamma_k)^2} \det (I - R(t-k)) .
\end{equation}
The prefactor
$C_{\pi, k}(\gamma_0 \ldots \gamma_k)^{-2}$ can be seen to equal $1$ by considering equation (\ref{teq:A.10}) for fixed $k$ in the limit $t \to \infty$. Thus
\begin{equation}\label{teq:AN.10}
\mathbb P_{\pi} (T(k,k) \le t) = \det (I - R(t-k)).
\end{equation}
The final idea in the argument of Tracy-Widom is to write $R(s)$ -- considered as a linear map
$\mathbb R^{k+1} \to \mathbb R^{k+1}$ -- as a product $R(s)= A(s)B(s)$, with
\begin{eqnarray*}
B(s): \mathbb R^{k+1} \to \ell_2(\mathbb Z \cap I_s)&,\qquad&(u_j)_{0 \le j \le k} \mapsto \sum_{j=0}^k u_j \varphi_j|_{I_s} \\
A(s): \ell_2(\mathbb Z \cap I_s) \to \mathbb R^{k+1}&,\qquad&f \mapsto
\left( \sum_{x \ge s} f(x) \varphi_{l}(x) \right)_{0 \le l \le k}
\end{eqnarray*}
Applying the formula $\det(I-AB)=\det(I-BA)$ that holds in great generality (see e.g. \cite[(3.1)]{TDeiftGioev09}) we have derived the
following Fredholm determinant formula for $\mathbb P_{\pi} (T(k,k) \le t)$.
\begin{lemma}\label{tlem:A.1}
$\mathbb P_{\pi} (T(k,k) \le t) = \det(I-\Sigma_k(t-k))$, where
\[
\Sigma_k(s) \colon \ell_2(\mathbb Z \cap I_s) \to \ell_2(\mathbb Z \cap I_s)\;, \quad f \mapsto
\left( \sum_{y \ge s} \sigma_k(x, y) f(y) \right)_{x \ge s}
\]
and $\sigma_k$ denotes the reproducing kernel $\sigma_k(x, y) := \sum_{j=0}^k \varphi_j(x) \varphi_j (y)$ with respect
to the Meixner polynomials.
\end{lemma}
It may seem somewhat strange to convert (\ref{teq:A.10}) that involves a determinant of some finite size matrix $I-R$
into a formula that involves the computation of a Fredholm determinant of an operator acting on the infinite dimensional space
$\ell_2(\mathbb Z \cap I_s)$.
However, one has to keep in mind that we are interested in an asymptotic result with $k \to \infty$.
Hence the size of $I-R$ goes to infinity and it is not at all clear how to perform the asymptotic analysis of the determinants.
In contrast, the operator
$I - \Sigma_k$ acts on the same space $\ell_2(\mathbb Z \cap I_s)$ for all $k$ and it is only the reproducing kernels $\sigma_k$ that dependend on $k$. As will be discussed below the kernels $\sigma_k$ are amenable to asymptotic analysis. In fact, due to the
Christoffel-Darboux formula for orthogonal polynomials we may express $\sigma_k$ just in terms of $\varphi_{k}$ and
$\varphi_{k+1}$. For large values of $k$ the behavior of these functions is rather well understood.
For example, if $x$ is somewhat larger than the largest zero of $\varphi_k$, then $|\varphi_k(x)|$ is very close to zero.
This implies that for values of $t-k$ that are somewhat larger than the
largest zeros of $\varphi_k$ and $\varphi_{k+1}$ the operator $\Sigma_k(t-k)$ is negligible and
thus $\mathbb P_{\pi} (T(k,k) \le t)$ is very close to 1.
If one reduces the value of $t-k$ to lie in a vicinity of the largest zero of $\varphi_k$ (which is also close to the
largest zero of $\varphi_{k+1}$) then the functions $\varphi_k$ and $\varphi_{k+1}$, appropriately rescaled, are described to leading order by Airy functions. In Sect.~\ref{tsec:A.2} we will use the just mentioned properties of Meixner
polynomials to complete the proof of Theorem \ref{tsatz:R.1}.
As it was noted in the last paragraph of Sect.~\ref{tsec:C.4}, the formula for the distribution of the
largest eigenvalue of GUE (\ref{teq:N.20}) is structurally the same as formula (\ref{teq:C.28}) for the distribution of $T(k, k)$
and the arguments described in the present section can
be applied in an analogous way. The only difference is that we need to use Hermite polynomials instead of
Meixner polynomials and that the summation operator $\Sigma_k$ is to be replaced by an integral operator with a
kernel that is given by the reproducing kernel for Hermite polynomials up to degree $N-1$ ($N$ as in $\mathbb P_N$,
cf. Reminder \ref{trem:C.1}). As in the Meixner case, the leading order behavior of Hermite polynomials near their largest zero is described by Airy functions. On a technical level this is the reason why the fluctuation of
the flux in dTASEP with step initial conditions follows asymptotically the same distribution as the fluctuation of the largest eigenvalue of
GUE. It is no coincidence that Meixner polynomials and Hermite polynomials of large degree look locally the same when rescaled appropriately. In fact, large classes of orthogonal polynomials display the same local behavior. We will comment on this universality property of orthogonal polynomials in Sect.~\ref{tsec:A.3}.
\subsection{Completing the proof of Theorem \ref{tsatz:R.1}}
\label{tsec:A.2}
We have argued above that in order to complete the proof of Theorem \ref{tsatz:R.1} we need to determine the asymptotic behavior of the reproducing kernel $\sigma_k$ (see Lemma \ref{tlem:A.1}) in a vicinity of the largest zero of $\varphi_k$. This requires some detailed analysis that we are not going to present here and we refer the reader to \cite[Sect. 5]{Jo1}. See also Sect.~\ref{tsec:A.3} for a few general remarks on the asymptotic analysis of orthogonal polynomials.
We start with some notation. The Airy function can be defined for $x \in \mathbb R$ by
\begin{eqnarray}\label{tAi.5}
\textrm{Ai}(x) := \frac{1}{2 \pi} \int_{\mathbb R} \exp \big( i[x(t+is)+(t+is)^3/3]\big) dt
\end{eqnarray}
with an arbitrary choice of $s>0$. The Airy kernel is given by
\begin{eqnarray*}
A(x, y) := \frac{\textrm{Ai}(x) \textrm{Ai}'(y) - \textrm{Ai}'(x) \textrm{Ai}(y)}{x-y} \; ,\quad x, y \in \mathbb R\; ,
\end{eqnarray*}
with the obvious interpretation on the diagonal $x=y$. The
Tracy-Widom distribution for $\beta=2$ can then be expressed as a Fredholm determinant
\begin{eqnarray*}
\textrm{TW}_2(s) := \det (I - A)|_{L^2[s, \infty)} \; , \quad s \in \mathbb R ,
\end{eqnarray*}
where $A$ denotes the integral operator associated with the Airy kernel. Note that one may derive a differential equation for $TW_2$ leading to another representation \cite{TTracyWidom94}
\[
\textrm{TW}_2 (s) = \exp \left( - \int_s^{\infty} (x-s) u(x)^2 dx \right) \; ,
\]
where $u$ denotes the Hastings-McLeod solution of the Painlev\'e II equation $u''= 2u^3 +xu$ that is singled out from all solutions of this ordinary differential equation by the asymptotic condition $u(x) \sim -$Ai$(x)$ for $x \to \infty$.
Observe that the Airy function solves the linearized Painlev\'e II equation $u''=x u$ with asymptotics Ai$(x) \sim
\frac{\exp \left(-(2/3)x^{3/2}\right)}{2 \sqrt{\pi} x^{1/4}}$ as $x \to \infty$.
In an interesting paper \cite{TBornemann09} Bornemann demonstrates that it is advantageous for numerical evaluations of $TW_2$ to start from the Fredholm determinant formula rather than using the Hastings-McLeod function.
The result in \cite[Lemma 3.2]{Jo1} on the reproducing kernel $\sigma_k$ for Meixner polynomials reads
\begin{equation}\label{teq:A.20}
c k^{1/3} \sigma_k( b k + c k^{1/3} \xi, b k + c k^{1/3} \eta) \to A(\xi, \eta) \quad \mbox{ for } k \to \infty,
\end{equation}
where $b= \pi^{-1}(1+\sqrt{1-\pi})^2$ and $c=\pi^{-1} (1-\pi)^{1/6} (1+\sqrt{1-\pi})^{4/3}$. We can now derive (\ref{tN2.5}) formally. From (\ref{teq:R.9}), Lemma \ref{tlem:A.1}, and (\ref{teq:A.20}) we learn that we should have
\begin{eqnarray}\label{tN2.10}
t-k = bk + ck^{1/3}(-s)(1 + o(1))\; ,
\end{eqnarray}
for $t \to \infty$ and with $k = Jt + Vst^{1/3}$. A straight forward calculation shows that
this can only be achieved if
\[
(b+1)J = 1 \quad \mbox{and} \quad V(b+1) = c J^{1/3}
\]
leading to the formulae for $J$ and $V$ as presented in Theorem \ref{tsatz:R.1}. Moreover, it is apparent that the remainder $o(1)$ in (\ref{tN2.10}) is of order ${\cal O}\left(t^{-2/3}\right)$.
Clearly, the argument just made does not fully prove Theorem \ref{tsatz:R.1} since additional estimates are needed to deduce the convergence of Fredholm determinants from the convergence of the kernels. For details see \cite[Sect. 3]{Jo1}.
Finally we observe that the linear growth of the mean and the $t^{1/3}$-scaling of the fluctuations of the flux $F_0(t)$ follow directly from condition (\ref{tN2.10}), i.e. from (\ref{teq:R.9}), Lemma \ref{tlem:A.1}, and (\ref{teq:A.20}) without any reference to KPZ theory.
\subsection{Remarks on the universal behavior of orthogonal polynomials}
\label{tsec:A.3}
We have discussed two examples for fluctuations that can be
described by the Tracy-Widom distribution: The largest eigenvalue of
GUE and the particle flux at the origin of dTASEP with step initial
conditions. Both systems can be analyzed by the method of orthogonal
polynomials. The fact that the limiting distribution for the
fluctuations agree in both cases is then a consequence of the fact
that the corresponding sets of orthogonal polynomials, Hermite and
Meixner, have the same asymptotic behavior in a vicinity of their
respective largest zeros after appropriate rescaling. This is no
coincidence. During the past ten years many detailed results on
various types of orthogonal polynomials have become available that
show universal behavior of orthogonal polynomials of large degree on
a local scale. In this section we give a rough description of this
universal behavior and outline a few approaches how such results can
be proved.
Let us assume that the support of the measure of orthogonality $\alpha$ is contained in $\mathbb R$. Then the normalized orthogonal polynomial $q_n$ of degree $n$ with positive leading coefficient, that is defined through
\[
\int\limits_{\mathbb R} q_n(x)q_m(x)\, d\alpha (x) =\delta_{n,m}\; ,
\]
has $n$ simple real roots which we denote by $x_i^{(n)}$. For many measures of orthogonality $\alpha$, and in particular in the case of the so called varying weights which we do not discuss here any further, there exists some natural scaling $x \to \hat{x}$ such that the
counting measures $\frac{1}{n}\sum^n_{i=1} \delta_{\hat{x}_i^{(n)}}$ associated with the rescaled zeros $\hat{x}_i^{(n)}$
converge for $n \to \infty$ to some measure $\mu$ of total mass 1.
The support $S$ of $\mu$ is compact and is always contained in the support of $\alpha$. For example, in the case of Hermite polynomials $d\alpha = e^{-x^2}\, dx$, the scaling is given by $\hat{x} = x/n^{1/2}$ and the limiting measure of zeros is given by $d\mu = \pi^{-1} \sqrt{2-x^2} \, {\bf 1}_{S}(x) \, dx$ with $S=[-\sqrt{2}, \sqrt{2}]$.
One should note that the scaling $x \to \hat{x}$ as well as the measure of zeros $\mu$ do depend on the measure of orthogonality $\alpha$. In order to explain what is universal about orthogonal polynomials we restrict ourselves to the common case that $S$, the support of $\mu$, consists of a single interval or a finite union of disjoint intervals. It is convenient to describe the behavior of $q_n$ by considering the corresponding functions $\varphi_n$ that are orthonormal with respect to Lebesgue measure. For example, in the case $d\alpha = w(x)dx$ we have $\varphi_n = q_n \sqrt w$. In the situation described above the large $n$ behavior of $\varphi_n(x)$ is generically the following:
\noindent
{\em For $\hat{x}$ outside $S$}: $\varphi_n(x)$ decays at an exponential rate to zero as $n\to\infty$.
\noindent
{\em For $\hat{x}$ in the interior of $S$}: $\varphi_n(x)$ is oscillating rapidly (in $\hat{x}$) and can be described to leading order by the cosinus function with slowly varying amplitude and frequency, which only depend on $\mu$.
\noindent
{\em For $\hat{x}$ close to a boundary point $b$ of $S$}: To leading order $\varphi_n(x)$ can be expressed in terms of special functions. One distinguishes between soft edges ($b$ lies in the interior of the support of $\alpha$) and hard edges ($b$ is also a boundary point of the support of $\alpha$).
In the first case the density of $\mu$ usually vanishes like a square root at $b$ and then the leading order of $\varphi_n(x)$ is described by the Airy function. This is in particular the case for Hermite and Meixner polynomials.
In the case of a hard edge the situation is a bit more complicated. In many cases Bessel functions can be used for asymptotic formulas for $\varphi_n(x)$ (see e.g. \cite{TKuijVanl03, TKMVV04}).
We conclude this section by mentioning a few methods how such asymptotics for $\varphi_n$ can be proved. We will in particular remark on the appearance of the Airy function.
\noindent
{\em I. Differential equations of second order}
We again discuss Hermite polynomials $d\alpha = e^{-x^2}dx$ as a typical example. The corresponding functions
$\varphi_n$ satisfy the second order differential equations
\[
\varphi''_n(x)+(2n+1-x^2)\varphi_n(x)=0\; .
\]
WKB analysis of these differential equations shows that the
oscillatory region $|x| < \sqrt{2n+1}$ is connected with the
exponentially decaying region $|x| > \sqrt{2n+1}$ by Airy functions.
This approach can be applied to a number of classical orthogonal
polynomials that are known to solve linear differential equations of
second order with nice coefficients.
\noindent
{\em II. Representation by contour integrals}
Such representations are known for a number of classical orthogonal polynomials (e.g. for Meixner polynomials, see e.g. \cite[Sect. 5]{Jo1})
and can be analyzed using the method of steepest descent. The appearance of the Airy function can be seen from its integral representation (\ref{tAi.5}) which generically provides a normal form at critical points of higher degeneracy.
\noindent
{\em III. Riemann-Hilbert problems}
The characterization of orthogonal polynomials as unique solutions of certain matrix Riemann-Hilbert problems (see \cite{TDeift99} and references therein)
works in principle for all types of weights and opens in particular the way to analyze non-classical orthogonal polynomials. Here the limiting measure of the zeros $\mu$, which can also be defined as the unique minimizer of a variational problem, yields the key to the asymptotic analysis. In the neighborhoods of boundary points of the support of $\mu$ at which the density
vanishes like a square root -- this is the generic case for a soft edge --
Airy functions arise naturally. This method for the asymptotic analysis of orthogonal polynomials was first
carried out in \cite{TDKMVZ}.
The method works best in the class of analytic weights, but progress has recently been made for weights
that have only a finite number of derivatives \cite{TMcLaughlinMiller08}. Orthogonal polynomials with respect to discrete measures
have been analyzed by Riemann-Hilbert techniques in \cite{TBKMM}.
\noindent
{\em IV. Reproducing kernels}
In recent years universality results for orthogonal polynomials, in particular results on the reproducing kernel (cf. Lemma \ref{tlem:A.1}), have been substantially generalized (see \cite{TLubinsky09}
and references therein). A very nice view on universality has been introduced in \cite{TLubinsky08} where classical results on reproducing kernels for entire functions of exponential type are being used (see \cite{TLevinLubinsky09} for the Airy kernel).
\section{KPZ-universality revisited}
\setcounter{footnote}{0}
\label{Universality}
In view of the universality conjecture formulated in
Sect.~\ref{KPZ_conjecture}, one expects that the results
derived in the preceding sections for a very special case --
the dTASEP with step initial conditions -- should carry over,
in a quantitative sense, to a much broader class of models.
The first explicit demonstration of this idea
was presented by Pr\"ahofer and Spohn in a series of papers
\cite{Prahofer00b,Prahofer00a,Prahofer01}, where an
alternative and independent route linking Ulam's problem to growth
models was established\footnote{Another link between the two classes
of problems was found by Majumdar and Nechaev \cite{Majumdar2004,Majumdar2007}. }.
The starting point is the one-dimensional polynuclear growth model (PNG),
an interacting particle system on the real line, in which particles
(antiparticles) move deterministically at unit speed to the right (left),
annihilate upon colliding, and are created in pairs according to a
two-dimensional Poisson process in space and time \cite{Krug89}.
Via the random set of particle creation events the model can be mapped
onto the problem of the longest increasing subsequence of a random
permutation, which in turn provides a link to the
Tracy-Widom distribution \cite{Baik99}.
For the case of a droplet growing from a seed (case III of
Sect.~\ref{KPZ_conjecture}), Pr\"ahofer and Spohn show that the resulting
fluctuation distribution is
identical (under the rescaling prescribed by KPZ theory) to that obtained
by Johansson for the dTASEP.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth,angle=-90]{Figure3.ps}
\end{center}
\caption{(a) The densities of the three universal distribution functions
$TW_2$, $TW_1$ and $F_0$ (from left to right).
Discrete points show simulation results
for the PNG model. Reprinted with permission from
\cite{Prahofer00a}. Copyright 2000 by the American Physical Society. (b) Phase
diagram for the distribution of current fluctuations in the TASEP with
Bernoulli step initial conditions. Here G denotes the Gaussian distribution and
G$^2$ (GOE$^2$) is the distribution of the maximum of two
independent Gaussian ($TW_1$) random variables.
Initial particle densities are $\rho_L = \rho_-$ to the left and
$\rho_R = \rho_+$
to the right of the origin.
Reprinted from \cite{Prahofer01}
with kind permission of Springer Science and Business Media.}
\label{3functions}
\end{figure}
Moreover, by imposing suitable boundary conditions \cite{Baik00} and
symmetry relations \cite{TBaRa01} on the set of Poisson points,
the cases of flat and rough initial conditions (case I and II of
Sect.~\ref{KPZ_conjecture}) can be handled as well \cite{Prahofer00a}.
For the flat initial condition (case I) the fluctuations are governed
by the GOE distribution $TW_1$, while for the rough
initial condition (case II) a new distribution $F_0$ emerges which so
far does not have an interpretation in terms of random matrix theory
\cite{Baik00}. The three distributions are depicted in
Fig.~\ref{3functions}(a).
Since the fundamental works of Johansson, Pr\"ahofer and Spohn the field
has developed rapidly, and it is impossible to do
justice to the new results in the framework of these
lecture notes. In the following subsections we therefore restrict ourselves
to briefly outlining the most important directions of research,
providing the interested reader with a few key references along which
recent advances can be traced.
\subsection{The Pr\"ahofer-Spohn conjecture and the ASEP}
\label{T8.1}
Based on the universality hypothesis, Pr\"ahofer and Spohn
translated the results obtained for the PNG model into a conjecture
for the fluctuations of the particle current through the origin for
the TASEP with a general step initial condition (\ref{step}), where particles are placed to the left (right)
of the origin according to a Bernoulli measure with density $\rho_L$ ($\rho_R$)
\cite{Prahofer01}. The fluctuation phase diagram in the plane of
the boundary densities is shown
in Fig.~\ref{3functions}(b). The overall features of
the diagram can be understood from hydrodynamics. First, the
Johansson result obtained at $\rho_L = 1$, $\rho_R = 0$ is seen to
extend throughout the region $\rho_L > 1/2 > \rho_R$. As explained
in Sect.~\ref{KPZ_conjecture}, this reflects the fact that the
density profile near the origin is independent of the boundary
densities in this case. For $\rho_L < 1/2$ and $\rho_R > 1/2$ the
application of the hydrodynamic formulae (\ref{wave}) [for $\rho_L >
\rho_R$] and (\ref{shockspeed}) [for $\rho_L < \rho_R$] show that
the density at the origin becomes $\rho_L$ and $\rho_R$,
respectively. In these cases the intrinsic current fluctuations are
masked by the initial fluctuations drifting across the origin,
leading to simple Gaussian statistics (regions marked G in the
diagram). The line $\rho_R + \rho_L = 1$, $\rho_L < \rho_R$, is
special, because there the shock speed (\ref{shockspeed}) vanishes
and the density at the origin shifts randomly between $\rho_L$ and
$\rho_R$. As a consequence, the current is distributed as the
maximum of two independent Gaussian random variables (denoted by
G$^2$ in the figure). Similarly, along the lines $\rho_L = 1/2$,
$\rho_R < 1/2$, and $\rho_R = 1/2$, $\rho_L > 1/2$, the distribution
is that of the maximum of two independent variables drawn from
$TW_1$. Finally, at the point $\rho_L = \rho_R = 1/2$ we have case
II behavior governed by the distribution $F_0$.
A proof of the Pr\"ahofer-Spohn conjecture for the TASEP was
recently presented by Ben Arous and Corwin \cite{TBeCo09} (see also
\cite{TNaSa04,TBaBePe05,Ferrari2006} for earlier partial results). Moreover, in a
remarkable series of papers Tracy and Widom have been able to
generalize these results to the (partially) asymmetric exclusion
process \cite{TTracyWidom08a,TTracyWidom08b,TTracyWidom09a,TTracyWidom09b,TTracyWidom09c,TTracyWidom09d,TTracyWidom10}.
The generalization is highly nontrivial, because the ASEP for
general $q$ is not a determinental process \cite{TTracyWidom09c}, and it requires a
novel set of techniques based on the Bethe ansatz \cite{TSchutz97,TRaSchu05,Gollinelli2006}.
\subsection{Spatio-temporal scaling}
\label{T8.2}
We have seen in Sect.~\ref{KPZ_conjecture} that the essence of the
KPZ conjecture is the universality of height fluctuations when
viewed on the appropriate scales defined by the height rescaling
(\ref{heightscaled}) and the correlation length
(\ref{correlation_length}). In other words, once the average growth
shape has been subtracted, one expects that the rescaled
fluctuations
$$
\bar{h}_t(y) \equiv (A^2 \vert \lambda \vert t)^{-1/3} h(y (A \lambda^2 t^2)^{1/3} ,t)
$$
converge for $t \to \infty$ to a universal stochastic process
${\cal{A}}(y)$, whose single-point distribution is one of the random
matrix distributions discussed above. The process ${\cal{A}}(y)$
was first explicitly characterized by Pr\"ahofer and Spohn for the
PNG-model in the droplet geometry (case III of
Sect.~\ref{KPZ_conjecture}), who named it the \textit{Airy process}
\cite{TPrSp02}. Subsequently Sasamoto identified the analogous
process for the case of flat initial conditions (case I)
\cite{TSasamoto05}. In line with the nomenclature used to
designate the corresponding single-point distributions ($TW_2$ for
case III and $TW_1$ for case I), the two processes are now called
Airy$_2$ and Airy$_1$ processes, respectively \cite{Borodin07,TBoFeSa08,TBoFe08,TFerrari08b}.
Whereas the Airy$_2$ process has a natural interpretation in the
random matrix context as the motion of the largest eigenvalue
in GUE matrix diffusion (Dyson's Brownian motion), the
corresponding relation does not hold for the Airy$_1$ process
\cite{TBoFePr08}. The corresponding process for initial
conditions with stationary roughness (case II) was studied in
\cite{Baik2010} (see also \cite{Imamura2004}).
In recent work Ferrari and collaborators have extended the analysis
to include correlations between height fluctuations at different
times $t$ and $t'>t$. As for the scaling of the height fluctuations
themselves, the characteristics of the hydrodynamic equation play a
special role for the decay of correlations. Whereas the decay along
generic space-time directions is governed by the correlation length
(\ref{correlation_length}), which is of order $t^{2/3}$, along the
characteristics the decorrelation time at time $t$ is set by $t$
itself, which implies a much slower decay
\cite{TFerrari08a,TCoFePe10a,TCoFePe10b}. This is in accordance with
KPZ phenomenology, which predicts that such correlations should
decay as $(t/t')^{\bar{\lambda}}$ with a universal autocorrelation
exponent $\bar{\lambda}$ \cite{Krech1997,Kallabis1999}. In contrast
to the scaling exponents of single-point height fluctuations
introduced in Sect.~\ref{KPZ_conjecture}, the autocorrelation
exponent depends explicitly on the growth geometry: For a flat
initial condition (case I) $\bar{\lambda} = 1$, whereas for a curved
cluster (case III) $\bar{\lambda} = 1/3$ \cite{Singha2005}.
\subsection{KPZ-scaling at large}
\label{T8.3}
The results described so far in this section were based on a small
set of exactly solvable models, the (T)ASEP's and the PNG model. On
the other hand, KPZ universality is expected to hold for a much
broader class of interacting particle systems and growth models
which is limited only by the requirement of local, stochastic
transition rules and a nonlinear dependence of the particle current
(or growth rate) on the particle density (or surface slope) [see
Sect.~\ref{KPZ_conjecture}]. It is therefore gratifying that the
class of models for which KPZ universality has been rigorously
established -- mostly in the sense of finding the exact scaling
exponents governing the order of fluctuations -- has been greatly
expanded in recent years. This has required the development of new
techniques that are purely probabilistic in nature and do not rely
on the specific analytic structure of the exactly solvable models.
The first result of this type was obtained in \cite{TCaGr06} for the
Hammersley process, an exlusion-type process in continuous space
which is closely related to the Ulam problem. Similar methods were
subsequently applied to a variety of interacting particle systems
\cite{TQuVa07,TQuVa08,TBaSe,TBaSe09}, including a class of zero
range processes\footnote{Zero range processes were introduced by
Spitzer \cite{Spitzer1970} and have been extensively studied in the
physics literature, see \cite{Evans2005}.} with general jump rates
that have a non-decreasing, concave dependence on the number of
particles \cite{TBaKoSe}. This constitutes a major step on the way
to proving KPZ universality in the broadest sense.
\subsection{The universality class of the KPZ equation}
\label{T8.4}
Ironically, although tremendous advances in the analysis of different representatives of the KPZ universality class
were achieved over the past decade, the one-dimensional KPZ equation (\ref{KPZ}) itself remained rather poorly understood.
We noted already
that the KPZ equation is mathematically ill-posed because of the highly
singular white noise term, and some regularization is needed to make it amenable to rigorous analysis. This can be
done by spatial discretization or by constructing the equation through a scaling limit from an asymmetric
exclusion process with weak asymmetry \cite{TBeGi97,TSaSp10a}. Both approaches have recently been used to prove the
correct order of fluctuations in the KPZ equation \cite{TSaSp09,TBaQuSe10} as well as refined
universality in the sense of the Tracy-Widom distribution \cite{TSaSp10b,TSaSp10c,Amir2010}. Thus it has finally been
established, as it were, that the KPZ equation belongs to its own universality class.
In another line of work an
independent, non-rigorous approach to establishing Tracy-Widom
universality has been developed which is based on applying the replica
method to the path weight $Z(x,t)$ in the Hopf-Cole
transformed equation (\ref{Z(x,t)}). In this approach one computes
moments $\mathbb{E}(Z^n)$ with respect to the stochastic force
$\zeta(x,t)$, which results in a
problem of $n$ bosonic, quantum-mechanical particles
interacting through an attractive $\delta$-function potential
\cite{Kardar1987b,HalpinHealy1995}. While the ground-state energy and
wave function of this quantum system
been known for a long time, the recent works
\cite{Calabrese2010,Dotsenko2010}
have succeeded in summing
over the full spectrum of excited states of the many-body Hamiltonian.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{Figure4.eps}
\end{center}
\caption{(Color online) Experimental demonstration of KPZ universality in a thin film
of turbulent liquid crystal. (a) Outlines of a growing turbulent
droplet at time intervals of 5 s. (b) Measured distribution of
shape fluctuations in comparison to the GUE prediction. Reprinted with permission from
\cite{Takeuchi2010}. Copyright 2010 by the American Physical Society.}
\label{Kazumasa}
\end{figure}
\subsection{Experiments}
\label{T8.5}
Despite the wide applicability of the KPZ theory to a large class of stochastic growth
models, experimental signatures of KPZ scaling in the real physical
world have proven to be surprisingly elusive. Early efforts focused on the investigation of
growth-induced (two-dimensional) surface roughness of crystals and thin solid films
\cite{Krim1995}. However, detailed consideration of the
physical processes governing such growth experiments has revealed that
they typically operate in regimes where KPZ asymptotics is practically
out of reach \cite{Krug1997,Michely2004}.
To date, the most thoroughly
studied experimental system that displays KPZ scaling is the slow, flameless
combustion of paper. By imaging and analyzing the one-dimensional smoldering front,
the $t^{1/3}$-scaling of fluctuations \cite{Maunuksela1997} as well as
the non-Gaussian shape of the corresponding probability distribution \cite{Miettinen2005}
were demonstrated.
Very recently, a new experimental system involving different phases
of driven, turbulent liquid crystal films became available, in which
the refined universality predictions of the theory can be tested
with unprecedented precision \cite{Takeuchi2010}. In Fig.~\ref{Kazumasa} we show a
series of snapshots of the growing turbulent droplet along with the
experimentally determined distribution of shape fluctuations. Under
the rescaling prescribed by KPZ theory, the predicted universal GUE
distribution is seen to emerge from the data without any adjustable
parameters.
\section{Integrability and Universality}
The central model discussed in our paper is the totally asymmetric
simple exclusion process (TASEP). This model has two features that
motivate our choice. Firstly, it belongs to the class of stochastic
interacting particle systems that are useful in the study of
transport phenomena in nonequilibrium systems. We are particularly
interested in the description of fluctuations around the mean
behavior which is governed by a deterministic evolution equation. In
general it is difficult to obtain such detailed information, mainly
because the interactions between the particles destroy stochastic
independence and the classical central limit theorem cannot be
applied. However, and this is the second feature, TASEP is an
exactly solvable model and the fluctuations can be analyzed by a
series of beautiful and non-obvious observations. We have seen by
explicit calculation that the fluctuations of the flux are described
by a formula that is structurally the same as the formula for the
fluctuations of the largest eigenvalue of matrices from the Gaussian
Unitary Ensemble (GUE). This provides one explanation for the much
celebrated link to Random Matrix Theory. There are also links
to the theory of integrable systems (see e.g. \cite{TDeift99, TMoerbeke02, TMoerbeke08}) that we have not explained at all and that we
mention here only in order to give an indication why TASEP is sometimes called an integrable model, even though there is no differential equation to integrate in sight.
A common feature of integrable systems is that their delicate
mathematical structure immediately breaks down if the model is
changed ever so slightly. One may think that this limits the
interest in the corresponding results. However, the recurring
experience with integrable systems has been, that even though the
method of proof is not applicable if the model is changed, the results may
persist. In our context a nice example of this principle of
universality is the recent work of Tracy and Widom where it is shown
by very different methods that a number of results for TASEP also
hold for ASEP. Universality results are also available for large
classes of matrix ensembles (see \cite{TDeiftGioev09, TErdos10} and references therein).
Despite these results, the question of universality is a subtle one
as can be seen for example from the
correlations of the fluctuations (cf. \cite{TBoFePr08} and Sect.~\ref{T8.2}). It remains
a great challenge to understand the realm of validity of the various
laws that have been established for the specific integrable models.
\section*{Acknowledgements}
This work was supported and inspired by SFB/TR 12 \textit{Symmetries
and universality in mesoscopic systems}. We are grateful to H. Spohn,
P.L. Ferrari, and to an unknown referee for valuable remarks, and to
M. Pr\"ahofer and K.A. Takeuchi for providing figures.
\section*{References}
|
1,314,259,995,783 | arxiv | \section{Introduction}
It has been known for a while that every finite (quasi)simple group
is determined by its complex group algebra, i.e., by the set of
character degrees with multiplicities \cite{BNOT15}. (Recall that,
by the Wedderburn theorem, the multiset of character degrees
counting multiplicities of $G$ determines and is determined by its
complex group algebra $\mathbb{C}{G}\cong \bigoplus_{\chi\in\mathrm{Irr}(G)}
\mathrm{Mat}_{\chi(1)}(\mathbb{C})$. Here, $\mathrm{Irr}(G)$ is the set of
all irreducible characters of $G$ and $\mathrm{Mat}_n(\mathbb{C})$
is the $\mathbb{C}$-algebra of $n\times n$ matrices.) The version
without multiplicity (and therefore stronger) is a conjecture
proposed by B. Huppert in the late nineties \cite{Huppert}. This
conjecture has been extensively studied over the past two decades,
and progress has been made on a case-by-case basis using the
classification of finite simple groups
\cite{Huppert06,nt,Bessenrodt}. Further discussion on this and other
isomorphism problems of similar nature can be found in the recent
survey \cite{Margolis}.
This paper is concerned with \emph{the codegree isomorphism problem}
for finite simple groups. For a character $\chi$ of a finite group
$G$, the \emph{codegree} of $\chi$ is $\normalfont{\mbox{cod}}(\chi):=
|G:\operatorname{ker}(\chi)|/\chi(1)$. This notion was first introduced and
studied (in a slightly different form) by D. Chillag and M. Herzog
\cite{Chillag1} and D. Chillag, A. Mann, and O. Manz
\cite{Chillag-Mann}. It was later developed into the current form
used nowadays by G. Qian, Y. Wang, and H. Wei \cite{Qian} and has
been proved to have remarkable connections with the structure of
finite groups \cite{Qian,i,dl,Moreto,q21, cn}.
Recently, there has been great interest in the codegree analogue of
Huppert's conjecture (Problem~20.79 of the Kourovka Notebook
\cite{Khukhro}), to which we will refer to as \emph{the codegree
isomorphism conjecture}. From now on $\normalfont{\mbox{cod}}(G)$ denotes the set of
all the codegrees of irreducible characters of $G$.
\begin{equation}
\tag{CIC}\label{eq:HCC}
\parbox{\dimexpr\linewidth-12em}{%
\strut
\emph{Let $S$ be a finite nonabelian simple group and $G$ a
finite group. Then $\normalfont{\mbox{cod}}(G)=\normalfont{\mbox{cod}}(S)$ if and only if $G\cong S$.}
\strut
}
\end{equation}
The approach so far to this problem is more or less similar to
Huppert's original method, and therefore, unfortunately, is still
case-by-case \cite{BahriAkh,Ahanjideh,gkl, gzy, LY22}.
Let $\normalfont{\mbox{cod}}(G)=\{c_1<c_2<...<c_k\}$ and $m_G(c_i)$ be the number of
irreducible characters of $G$ with codegree $c_i$. The multiset
$$C(G):=\{(c_i,m_G(c_i)):1\leq i\leq k\}$$ is called the
{group pseudo-algebra} of $G$, which can be viewed as the codegree
counterpart of the aforementioned complex group algebra $\mathbb{C}
G$. A natural weaker version of (\ref{eq:HCC}) asks whether $G$ and
$S$ must be isomorphic if $C(G)=C(S)$. We confirm this in our first
main result.
\begin{mainthm}\label{thm:main1}
Let $S$ be a finite simple group and $G$ a finite group. Then
$$C(G)=C(S) \text{ if and only if } G\cong S.$$
\end{mainthm}
The main novelty of this paper is a more uniform approach to these
codegree problems with as little case-by-case analysis as possible.
Our proof of Theorem \ref{thm:main1}, somewhat surprisingly, only
relies on the classification via the so-called \emph{simple order
theorem} (also known as the Artin-Tits theorem
\cite{Artin55,Kimmerle-et-al}), which states that two non-isomorphic
finite simple groups have the same order if and only if they are
either $PSL_4(2)$ and $PSL_3(4)$ or $\Omega_{2n+1}(q)$ and
$PSp_{2n}(q)$ for some $n\geq 3$ and odd $q$. This is perhaps the
first time that a result of this type is proved uniformly for all
simple groups.
There are two key ingredients in the proof of Theorem
\ref{thm:main1}. We find it remarkable that they admit strikingly
elementary proofs. The first provides a characterization of perfect
groups in terms of codegrees, see Theorem \ref{thm-char}. (Recall
that a group is perfect if it coincides with its derived subgroup.)
The second is an order-divisibility property involving character
codegrees of finite simple groups, see Theorem
\ref{thm-|S|divides|G|}.
In general the set of codegrees $\normalfont{\mbox{cod}}(G)$ of a finite group $G$ does
not determine its set of character degrees. However, this is not the
case when $G=S$ is simple. To see why, consider a prime divisor $p$
of $|S|$. Then $S$ has some nontrivial irreducible character of
$p'$-degree (by the squares of the degrees equation). As a result,
$S$ has some codegree divisible by $|S|_p$ -- the $p$-part of $|S|$,
and this is the largest power of $p$ dividing any codegree of $S$.
Therefore both the order $|S|$ and all the character degrees of $S$
are indeed determined by $\normalfont{\mbox{cod}}(S)$. This argument also proves that
if two simple groups have the same set of codegrees, then they have
the same order. Our next main result is a far stronger statement but
the proof requires the classification.
\begin{mainthm}\label{thm:main4}
Let $S$ and $H$ be finite simple groups such that $\normalfont{\mbox{cod}}(S)\subseteq
\normalfont{\mbox{cod}}(H)$. Then $S\cong H$.
\end{mainthm}
Theorem \ref{thm:main4} in fact is the first step in proving the
codegree isomorphism conjecture (\ref{eq:HCC}). Let $G$ be any
finite group and $H$ a simple group such that $\normalfont{\mbox{cod}}(G)=\normalfont{\mbox{cod}}(H)$, and
$N$ be a maximal normal subgroup of $G$ so that $S:=G/N$ is simple.
In order to prove $G\cong H$, one would first need to establish
$S\cong H$, under the assumption $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(G)=\normalfont{\mbox{cod}}(H)$.
This is precisely what we do in Theorem~\ref{thm:main4} (see Theorem
\ref{thm:last}).
\begin{remark}
Using Theorem \ref{thm:main4}, we will prove in a subsequent paper
\cite{hmt} that (\ref{eq:HCC}) holds for all sporadic groups,
alternating groups, groups of Lie type of low rank, and, for the
first time in the degree/codegree isomorphism problem, groups of Lie
type of arbitrary rank over a field of prime order. Furthermore,
perhaps unexpectedly, we reduce (\ref{eq:HCC}) to a problem on
$p$-groups.
\end{remark}
\begin{remark}
The proof of Theorem \ref{thm:main4} is fairly complicated and
combines several techniques. In particular, it essentially utilizes
some deep results on the representation theory of finite simple
groups, including the classification of prime-power-degree
representations \cite{Malle-Zalesskii,BBOO01}, lower/upper bounds
for the largest degree of irreducible representations
\cite{VK85,LMT13}, and the existence of $p$-defect zero characters
\cite{Michler,Willems,Granville-Ono}. Along the way we prove an
effective and \emph{explicit} upper bound for the largest character
degree $b(S)$ of an exceptional group $S$ of Lie type (Theorem
\ref{lem:exceptional-b(S)}). (See Lemma~\ref{lem:bounds-for-b(S)}
for previous related work on symmetric and alternating groups
\cite{VK85} and classical groups \cite{LMT13}.)
\end{remark}
\begin{remark}
We need bounds for the largest character degree $b(S)$ in order to
control the behavior of $f(S):=|S|/b(S)$ -- the smallest nontrivial
codegree of $S$. While the relevance of the smallest (or low-degree
in general) characters of (quasi/almost)simple groups is well-known
in group representation theory (see \cite{MM21} for the latest
results), the smallest codegree had not been studied much before.
This invariant arises naturally in the proof of Theorem
\ref{thm:main4} (see Lemma~\ref{lem:fS geq fH}) and measures the
relative growth of $b(S)$ compared to $|S|$. Our proof would be much
simpler if one can show that $f$ is \emph{divisibly increasing}
among nonabelian simple groups, by which we mean that, if $S$ and
$H$ are nonabelian simple of different orders such that $|S|$
divides $|H|$, then $f(S)<f(H)$. We indeed confirm this phenomenon
in many cases, particularly when one of the two groups involved is
alternating (see Propositions \ref{prop:mixed case I},
\ref{prop:mixed case II}, and \ref{prop:alternating}).
\end{remark}
The layout of the paper is as follows. In Section
\ref{sec:prime-power-codegrees}, we prove some results on prime
character codegrees and provide a short proof of a theorem of Riese
and Schmid on prime-power codegrees. In Section
\ref{sec:codegrees-simple-gps} we discuss the order-divisibility
property and its consequences involving character codegrees of
finite simple groups. Using the results in the preceding sections,
we prove Theorem \ref{thm:main1} in Section \ref{sec:theoremA}.
Results on bounding the largest character degree are presented in
Section \ref{sec:largest-degree}. Finally, the proof of Theorem
\ref{thm:main4} is carried out in Sections \ref{sec:theoremD-Lie},
\ref{sec:theoremD-mixed-case}, and
\ref{sec:theoremD-alternating-sporadic}.
\section{Prime-power codegrees}\label{sec:prime-power-codegrees}
In this section we prove some results on prime-power character
codegrees. These results show that, in contrast to character
degrees, there are significant restrictions on the structure of
groups with faithful irreducible characters of prime/prime-power
codegree.
We mainly follow the notation from \cite{Isaacs} for character
theory and \cite{Conway,Carter85} for finite simple groups.
Throughout, for a positive integer $n$ and a prime $p$, we write
$n_p$ to denote the maximal $p$-power divisor of $n$ and
$n_{p'}:=n/n_p$ to denote the maximal divisor not divisible by $p$
of $n$. Let $N\trianglelefteq\, G$ and $\theta\in\mathrm{Irr}(N)$. We write
$\mathrm{Irr}(G|\theta)$ for the set of irreducible constituents of
$\theta^G$ and $\mathrm{Irr}(G|N)$ for the set of irreducible characters of
$G$ whose kernels do not contain $N$. If $G$ is a group, $\pi(G)$ is
the set of primes that divide $|G|$. If $n$ is an integer, $\pi(n)$
is the set of primes that divide $n$, and if $\mathcal{S}$ is a set
of integers, then $\pi(\mathcal{S})$ is the set of primes that
divide some member of $\mathcal{S}$. As usual,
$\normalfont{\mbox{cd}}(G):=\{\chi(1):\chi\in\mathrm{Irr}(G)\}$ is the set of all irreducible
character degrees of $G$. Other notation will be recalled or defined
when necessary.
We begin by collecting some known facts on character codegrees that
we will use without explicit mention.
\begin{lem}\label{lem:1}
Let $G$ be a finite group and $\chi\in{{\operatorname{Irr}}}(G)$. The following hold:
\begin{enumerate}[\rm(i)]
\item
If $\chi$ is not the principal character, then $\normalfont{\mbox{cod}}(\chi)>\chi(1)$.
\item
If $N\trianglelefteq G$ and $N\leq\operatorname{ker}(\chi)$, then the codegree of
$\chi$ as a character of $G$ coincides with the codegree of $\chi$
viewed as a character of $G/N$.
\item
If $N\trianglelefteq\trianglelefteq G$ and $\theta\in{{\operatorname{Irr}}}(N)$ lies
under $\chi$, then $\normalfont{\mbox{cod}}(\theta)$ divides $\normalfont{\mbox{cod}}(\chi)$.
\item
$\pi(G)=\pi(\normalfont{\mbox{cod}}(G))$.
\item
If $G$ is abelian, then $\normalfont{\mbox{cod}}(G)=o(G)$, where $o(G)$ is the set of
orders of the elements of $G$.
\end{enumerate}
\end{lem}
\begin{proof}
Part (i) is \cite[Lem. 2.1]{dl}. Parts (ii) and (iii) are contained
in \cite[Lem. 2.1]{Qian}, and part (iv) is \cite[Lem. 2.4]{Qian}.
Now, we prove part (v). The inclusion $o(G)\subseteq\normalfont{\mbox{cod}}(G)$ follows
from \cite[Lem. 2.2]{dl}. Conversely, if $d\in\normalfont{\mbox{cod}}(G)$ there exists
$\chi\in{{\operatorname{Irr}}}(G)$ such that $d=|G:\operatorname{ker}(\chi)|$ (note that since $G$
is abelian, $\chi$ is linear). Since $G/\operatorname{ker}(\chi)$ is cyclic, we
conclude that $G$ has elements of order $d$.
\end{proof}
\subsection{Prime codegrees: characterizing perfect groups}
The goal in this subsection is to provide a characterization of
perfect groups in terms of the absence of prime codegrees.
\begin{thm}
\label{lem-p} Let $G$ be a finite group. Suppose that there exists
$\chi\in{{\operatorname{Irr}}}(G)$ faithful such that $\normalfont{\mbox{cod}}(\chi)=p$ is a prime
number. Then $G$ is either the cyclic group of order $p$ or a
Frobenius group with Frobenius kernel of order $p$.
\end{thm}
\begin{proof}
We argue by induction on $|G|$. Let $N\trianglelefteq G$ be minimal
such that there exists $\theta\in{{\operatorname{Irr}}}(N)$ lying under $\chi$ with
$\normalfont{\mbox{cod}}(\theta)=p$. Then
$$
\frac{|G|}{\chi(1)}=p=\normalfont{\mbox{cod}}(\theta)=\frac{[N:\operatorname{ker}(\theta)]}{\theta(1)},
$$
and we deduce that
$$
p=\normalfont{\mbox{cod}}(\chi)>\chi(1)=[G:N]|\operatorname{ker}(\theta)|\theta(1).
$$
In particular, $p$ does not divide any of the three factors in the
right hand side.
Suppose first that $N<G$. By the inductive hypothesis,
$N/\operatorname{ker}(\theta)$ is cyclic of order $p$ or a Frobenius group with
Frobenius kernel $K/\operatorname{ker}(\theta)$ of order $p$. In the latter case,
if $\lambda\in{{\operatorname{Irr}}}(K/\operatorname{ker}(\theta))$ lies under $\theta$ then
$\normalfont{\mbox{cod}}(\lambda)=p$. This contradicts the choice of $N$. (Note that
$K$ is normal in $G$ because $K$ is characteristic in $N$.)
Hence, we may assume that $N/\operatorname{ker}(\theta)$ is cyclic of order $p$.
By Clifford's theorem the faithful character $\chi_N$ is a sum of
$G$-conjugates of $\theta$. Let $T$ be a complete set of
representatives in $G$ for these conjugates. By \cite[Lem.
2.21]{Isaacs}, the intersection of $\operatorname{ker}(\theta^g)$, where $g$ runs
over $T$, is trivial. We conclude that $N$ embeds into the direct
product \[\prod_{g\in T} N/\operatorname{ker}(\theta^g).\] Each of the direct
factors has order $p$, and so $N$ is an elementary abelian
$p$-group. Since $p$ does not divide $|\operatorname{ker}(\theta)|$, we conclude
that $N$ is cyclic of order $p$. As $\theta$ is linear and
$\chi(1)=|G|/p$, we now have $\chi=\theta^G$. It follows that $G$ is
a Frobenius group with kernel $N$, as desired.
Now, we consider the case $N=G$. Let $M$ be a maximal normal
subgroup of $G$. Since $N=G$ the codegree of any irreducible
character of $N$ lying under $\chi$ is $1$. This means that
$\chi_M=\chi(1)\mathbf{1}_M$. But $\chi$ is faithful, so we deduce
that $M=1$ and $G$ is simple. If $G$ is abelian, then it is the
cyclic group of order $p$. If $G$ is not abelian, then
$|G|/p=\chi(1)<\sqrt{|G|}$ and it follows that $|G|<p^2$. By
Sylow's theorems, it follows that $G$ has a normal Sylow
$p$-subgroup. This contradiction completes the proof.
\end{proof}
The following consequence of Theorem \ref{lem-p} is already
mentioned in the introduction.
\begin{thm}
\label{thm-char} A finite nontrivial group $G$ is perfect if and
only if $G$ does not have any prime character codegree.
\end{thm}
\begin{proof}
By Lemma \ref{lem-p}, if $G$ has an irreducible character $\chi$ of
prime codegree then $G/\operatorname{ker}(\chi)$ is solvable. In particular, $G$ is
not perfect. Conversely, if $G$ is not perfect, then the abelian
group $G/G'$ has some irreducible character of prime codegree.
\end{proof}
\subsection{Prime power codegrees: the Riese-Schmid theorem}
Chillag and Herzog proved in \cite[Thm. 1]{Chillag1} that a simple
group does not possess nontrivial irreducible characters of prime
power codegree. The proof relied on a case by case analysis of the
simple groups, using the fact that, most of the times, they have
$p$-blocks of defect zero. This was generalized by Riese and Schmid
in \cite[Cor. 3]{RS} to quasisimple groups, using also block theory
and the classification. We offer a short proof of this result that
only depends on an easy consequence of the classification, which is
due to W. Kimmerle, R. Lyons, R. Sandling, and D.\,N. Teague
\cite[Thm. 3.6]{Kimmerle-et-al}:
\begin{lem}
\label{lem:aa} For every finite simple group $S$ and prime $p$,
$|S|<(|S|_{p'})^2$.
\end{lem}
The following is a restatement of \cite[Lem. 1]{RS} in the language
of codegrees.
\begin{lem}
\label{lem-rs} Let $p$ be a prime. Let $G$ be a finite group and
$\chi\in{{\operatorname{Irr}}}(G)$ faithful. Then $\normalfont{\mbox{cod}}(\chi)$ is a power of $p$ if
and only if $\chi$ is induced from a Sylow $p$-subgroup of $G$.
\end{lem}
\begin{thm}
\label{thm-quasi}
A quasisimple group $G$ does not possess nonprincipal characters of
prime power codegree.
\end{thm}
\begin{proof}
Suppose that there exists $\mathbf{1}_G\neq\chi\in{{\operatorname{Irr}}}(G)$ of
$p$-power codegree. Let $K:=\operatorname{ker}(\chi)$ and note that $K\leq{\mathbf Z}(G)$.
By Lemma \ref{lem-rs}, $\chi$ is induced from a Sylow $p$-subgroup
of $G/K$. Therefore, $$\chi(1)\geq|G/K|_{p'}.$$
By Lemma \ref{lem:aa}, we know that
$$|G/{\mathbf Z}(G)|<(|G/{\mathbf Z}(G)|_{p'})^2\leq(|G/K|_{p'})^2.$$ Hence, by
\cite[Cor. 2.30]{Isaacs},
$$
\chi(1)\leq|G:{\mathbf Z}(G)|^{1/2} <|G/K|_{p'},
$$
which violates the inequality above.
\end{proof}
The next result is a
restatement in terms of character codegrees of Theorem~B of
\cite{RS}. The proof in \cite{RS} uses Brauer's first and third main
theorems.
Recall that if a group $G$ has trivial $p'$-core
${\mathbf{O}}_{p'}(G)$, then it is defined to be $p$-constrained if the
$p$-core ${\mathbf{O}}_p(G)$ contains its centralizer.
\begin{thm}[Riese-Schmid]
\label{thm-rs} Let $G$ be a finite group and let $p$ be a prime.
Suppose that $\chi\in{{\operatorname{Irr}}}(G)$ is faithful of $p$-power codegree.
Then ${\mathbf{O}}_{p'}(G)=1$ and $G$ is $p$-constrained.
\end{thm}
\begin{proof}
By Theorem \ref{thm-quasi}, we know that $G$ is not simple. Let $N$
be a minimal normal subgroup of $G$ and let $\theta\in{{\operatorname{Irr}}}(N)$ lying
under $\chi$. Since $\theta$ has $p$-power codegree (by Lemma
\ref{lem:1}(iii)) and $\normalfont{\mbox{cod}}(\theta)>1$ (note that since $\chi$ is
faithful, $\theta\neq\mathbf{1}_N$), we deduce that $N$ is either an
elementary abelian $p$-subgroup or a direct product of nonabelian
simple groups of order divisible by $p$. In particular,
${\mathbf{O}}_{p'}(G)=1$.
We claim that $N$ is an elementary abelian $p$-group. Suppose that
$N=S_1\times\cdots\times S_t$, with $S_i\cong S$ for some nonabelian
simple group $S$ of order divisible by $p$. We wish to reach a
contradiction. Since $\theta\neq\mathbf{1}_N$, there exists a
nonprincipal $\psi\in{{\operatorname{Irr}}}(S_i)$ lying under $\theta$ for some $i$.
Note that $\normalfont{\mbox{cod}}(\psi)$ is a power of $p$ and this contradicts
Theorem \ref{thm-quasi}. The claim follows.
Write $P:={\mathbf{O}}_p(G)$ and $C:={\mathbf{C}}_G(P)$. Note that $C\cap P={\mathbf Z}(P)$
and ${\mathbf{O}}_{p'}(C)=1$ (because ${\mathbf{O}}_{p'}(G)=1$). We want to see that
$C\leq P$. Assume not. Take $K$ subnormal in $G$ such that
${\mathbf Z}(P)\leq K\leq C$ and $K/{\mathbf Z}(P)$ is simple. Since ${\mathbf{O}}_{p'}(C)=1$
and $G$ does not have nonabelian minimal normal subgroups, we
conclude that $K'$ is quasisimple.
Now, take $\gamma\in{{\operatorname{Irr}}}(K')$ lying under $\chi$. Again, we have that
$\gamma$ is not principal and $\normalfont{\mbox{cod}}(\gamma)$ is a $p$-power. This
contradicts Theorem \ref{thm-quasi}.
\end{proof}
We end this section with a variation of Theorem \ref{thm-quasi}.
\begin{thm}
\label{lem-cf} Let $G$ be a finite group. Suppose that $p$ is a
prime and $\chi\in{{\operatorname{Irr}}}(G)$ is faithful of $p$-power codegree. Then
$\normalfont{\mbox{cod}}(\chi)$ exceeds the $p$-part of the product of the orders of
the nonabelian composition factors in a composition series of $G$.
In particular, if $K/L$ is a non-abelian chief factor of $G$, then
$\normalfont{\mbox{cod}}(\chi)>|K/L|_p$.
\end{thm}
\begin{proof}
Let $n$ be the product of the orders of the non-abelian composition
factors in a composition series of $G$. Using again that $\chi$ is
induced from a Sylow $p$-subgroup and \cite{Kimmerle-et-al}, we
have
$$
\normalfont{\mbox{cod}}(\chi)>\chi(1)\geq|G|_{p'}\geq n_{p'}>n_p,
$$
as wanted.
\end{proof}
\section{An order-divisibility result for codegrees}\label{sec:codegrees-simple-gps}
The following order-divisibility result will be crucial in the
proofs of our main theorems.
\begin{thm}\label{thm-|S|divides|G|}
Suppose that $S$ is a finite simple group and $G$ a finite group
such that $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(G)$. Then $|S|$ divides $|G|$.
\end{thm}
\begin{proof}
Let $d_1,...,d_k$ be all the degrees of nontrivial irreducible
characters of $S$, and let $m_i$ ($1\leq i\leq k$) be the number of
those characters of degree $d_i$. By the assumption, for each $i$,
there exists $\chi_i\in\mathrm{Irr}(G)$ such that
\[
\frac{|S|}{d_i}=\frac{[G:\operatorname{ker}(\chi_i)]}{\chi_i(1)}=\frac{|G|}{\chi_i(1)|\operatorname{ker}(\chi_i)|}.
\]
It follows that
\[
\sum_{i=1}^k \frac{m_i d_i^2}{|S|^2}= \sum_{i=1}^k \frac{m_i
\chi_i(1)^2|\operatorname{ker}(\chi_i)|^2}{|G|^2},
\]
and thus
\[
\frac{\sum_{i=1}^k m_i
\chi_i(1)^2|\operatorname{ker}(\chi_i)|^2}{|G|^2}=\frac{\sum_{i=1}^k m_i
d_i^2}{|S|^2}=\frac{|S|-1}{|S|^2}.
\]
Therefore $|S|^2$ divides $|G|^2(|S|-1)$, and the theorem follows.
\end{proof}
We record some consequences of Theorem~\ref{thm-|S|divides|G|} that
will be needed in subsequent sections.
\begin{cor}\label{thm-|S|divides|G|2}
Suppose that $S$ and $H$ are finite simple groups such that $\normalfont{\mbox{cod}}(S)=
\normalfont{\mbox{cod}}(H)$. Then $|S|=|H|$.
\end{cor}
\begin{proof}
This directly follows from Theorem \ref{thm-|S|divides|G|}. (Or, alternatively, from the comment in the paragraph that precedes the statement of Theorem \ref{thm:main4}.)
\end{proof}
\begin{lem}\label{lem-x=H/S}
Suppose that $S$ and $H$ are finite nonabelian simple groups such
that $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(H)$. Let $x:=|H|/|S|$. Then $x\in
\mathbb{N}$ and $dx\in \normalfont{\mbox{cd}}(H)$ for every $1\neq d\in\normalfont{\mbox{cd}}(S)$.
\end{lem}
\begin{proof}
We know that $x\in \mathbb{N}$ by Theorem~\ref{thm-|S|divides|G|}. For
each $1\neq d\in\normalfont{\mbox{cd}}(S)$, we have $|S|/d\in \normalfont{\mbox{cod}}(H)$, and thus there
exists some $\chi\in\mathrm{Irr}(H)$ such that $|S|/d=|H|/\chi(1)$, implying
that $\chi(1)=dx$, as claimed.
\end{proof}
\begin{lem}\label{lem-chi-p-power}
Let $S$ and $H$ be finite simple groups of Lie type. Suppose that
the defining characteristic of $H$ is $p$ and $\normalfont{\mbox{cod}}(S)\subseteq
\normalfont{\mbox{cod}}(H)$. Then $|S|_{p'}=|H|_{p'}$ and there exists $\chi\in\mathrm{Irr}(S)$
such that $\chi(1)=|S|_p$.
\end{lem}
\begin{proof}
We first observe that $|S|$ is divisible by $p$ because otherwise
every codegree of $S$ is not divisible by $p$ but the only
nontrivial codegree of $H$ not divisible by $p$ is
$|H|_{p'}=|H|/\mathrm{St}_H(1)$.
By \cite{Michler,Willems}, $S$ has an irreducible character, say
$\chi$, of $p$-defect $0$, so that $\normalfont{\mbox{cod}}(\chi)$ is coprime to $p$.
Therefore we have
\[
|S|/\chi(1)= |H|_{p'}.
\]
It follows from Theorem~\ref{thm-|S|divides|G|} that
$\chi(1)|H|_{p'}=|S|$ divides $|H|$, implying that
$|S|_{p'}=|H|_{p'}$ and $\chi(1)=|S|_p$.
\end{proof}
Remark that when $p\geq 5$, the above result holds for all $S$. This
is because, in such case, irreducible $p$-defect zero characters
still exist in alternating groups (by \cite{Granville-Ono}) and in
sporadic simple groups \cite{Conway}.
\begin{lem}\label{lem-chi-p-power-2}
Let $S$ be a finite simple group of Lie type and $H$ a finite
nonabelian simple group. Suppose that $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(H)$ and
there are primes $p\neq r$ such that $H$ has a unique character
degree divisible by each $|H|_{p}$ and $|H|_r$. Then $|S|=|H|$ and
$\normalfont{\mbox{cd}}(S)\subseteq \normalfont{\mbox{cd}}(H)$.
\end{lem}
\begin{proof}
Repeating the arguments in the proof of Lemma~\ref{lem-chi-p-power},
we have $|S|_{p'}=|H|_{p'}$ and $|S|_{r'}=|H|_{r'}$, implying that
$|S|=|H|$ and $\normalfont{\mbox{cd}}(S)\subseteq \normalfont{\mbox{cd}}(H)$.
\end{proof}
\section{Group pseudo-algebras of simple groups: Theorem
\ref{thm:main1}}\label{sec:theoremA}
In this section we prove Theorem \ref{thm:main1}, using the results
in the preceding sections.
Theorem \ref{thm-|S|divides|G|} is useful in proving results
concerning codegrees of finite simple groups. One of them is the
next theorem, whose proof makes use of the simple order theorem.
Recall that the simple order theorem asserts that two
non-isomorphic finite simple groups have the same order if and only
if they are either $PSL_4(2)$ and $PSL_3(4)$ (of order $20,160$) or
$\Omega_{2n+1}(q)$ and $PSp_{2n}(q)$ (odd-dimensional orthogonal and
symplectic groups, of order $(1/2)q^{n^2}\prod_{i=1}^n (q^{2i}-1)$)
for some $n\geq 3$ and odd $q$ (see \cite[Thm. 5.1]{Kimmerle-et-al}
for instance). It was proved by E. Artin \cite{Artin551,Artin55} in
the fifties for known families of simple groups at the time, and
completed by J. Tits for the remaining families discovered later on
(see \cite{Kimmerle-et-al}). Artin's method is to consider certain
invariants associated to (orders of) simple groups that can be
computed explicitly and are able to distinguish the groups easily.
Therefore, the simple order theorem currently relies on the
classification of finite simple groups.
\begin{thm}\label{cor-cod(S)=cod(H)}
Suppose that $S$ and $H$ are finite simple groups such that
$\normalfont{\mbox{cod}}(S)= \normalfont{\mbox{cod}}(H)$. Then $S\cong H$.
\end{thm}
\begin{proof} The statement is trivial when both of $S$ and $H$ are abelian.
If one of the two groups is nonabelian, then, by Corollary
\ref{thm-char}, so is the other. So assume that both $S$ and $H$ are
nonabelian. By Theorem \ref{thm-|S|divides|G|2}, we have $|S|=|H|$
and hence it follows that $\normalfont{\mbox{cd}}(S)=\normalfont{\mbox{cd}}(H)$. Assume to the contrary
that $S$ is not isomorphic to $H$.
By the simple order theorem, we have
$$\{S,H\}=\{PSL_4(2),PSL_3(4)\}$$ or
$$\{S,H\}=\{\Omega_{2n+1}(q),PSp_{2n}(q)\}$$ for some odd prime power
$q=p^\ell$ and $n\geq 3$. The former case is eliminated using
\cite{Conway}, so we just need to show that
$\normalfont{\mbox{cd}}(\Omega_{2n+1}(q))\neq \normalfont{\mbox{cd}}(PSp_{2n}(q))$ for indicated $n$ and
$q$.
By Lusztig's classification of ordinary characters of finite groups
of Lie type (see \cite[Chapter 13]{DM}), irreducible characters of
$G:=Sp_{2n}(q)$ are parameterized by pairs $((s),\psi)$ where $(s)$
is the conjugacy class of a semisimple element $s\in
G^*:=SO_{2n+1}(q)$ and $\psi$ is a unipotent character of the
centralizer $C:=\mathbf{C}_{G^\ast}(s)$. Moreover, the degree of the
character $\chi_{((s),\psi)}$ associated to $((s),\psi)$ is
\[
\chi_{((s),\psi)}(1)=[G^\ast:C]_{p'}\psi(1).
\]
Let $\alpha\in\{\pm 1\}$ such that $4\mid (q^m-\alpha)$ and consider
a semisimple element $s\in G^*$ with spectrum $\{1,-1,...,-1\}$ such
that $C\cong GO_{2n}^\alpha$ (see \cite[Lem. 2.2]{Nguyen10}). Such
$s$ will then belong to $\Omega_{2n+1}(q)=[G^*,G^*]$, implying that
the semisimple character $\chi_{((s),\mathbf{1}_C)}$ associated to
the pair $((s),\mathbf{1}_C)$ is trivial on
$\mathbf{Z}(Sp_{2n}(q))$, by \cite[Lem. 4.4]{Navarro-Tiep13}. We
therefore have an irreducible character of $PSp_{2n}(q)$ of degree
\[\chi_{(s)}(1)=(|SO_{2n+1}(q)|/|GO_{2n}^\alpha|)_{p'}=(q^n+\alpha)/2.\]
To see that $\normalfont{\mbox{cd}}(PSp_{2n}(q)) \neq \normalfont{\mbox{cd}}(\Omega_{2n+1}(q))$ (for odd
$q$ and $n\geq 3$), it is enough to show that $(q^n+\alpha)/2$ is
not character degree of $\Omega_{2n+1}(q)$. By \cite[Thm.
6.1]{Tiep-Zalesskii96}, under our assumptions on $n$ and $q$ and the
additional condition $(n,q)\neq (3,3)$, the minimal (nontrivial)
irreducible character of $Spin_{2n+1}(q)$ has degree
\[
d(Spin_{2n+1}(q))=\left\{\begin{array}{ll} (q^{2n}-1)/(q^2-1)& \mathrm{ if }\ q\geq 5 \\
(q^n-1)(q^n-q)/2(q+1) & \mathrm{if}\ q=3,\end{array} \right.
\]
(For the definition of classical groups of various isogeny types,
including the odd-dimensional spin groups, we refer the reader to
\cite[p. 40]{Carter85}.) Note that $Spin_{2n+1}(q)$ is a central
extension of $\Omega_{2n+1}(q)$ and so every character degree of
$\Omega_{2n+1}(q)$ is one of $Spin_{2n+1}(q)$. It is now easy to
check that $d(Spin_{2n+1}(q))>(q^n+\alpha)/2$ for $n\geq 3$. For the
remaining case $(n,q)=(3,3)$, we note that $d(Spin_{2n+1}(q))=27$,
which is still greater than $(q^n+\alpha)/2=13$.
\end{proof}
Certainly, one has to do much more work to relax the hypothesis in
Theorem \ref{cor-cod(S)=cod(H)} to $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(H)$; that
is, to obtain Theorem \ref{thm:main4}.
\begin{lem}\label{lem:O2n+1andPSp}
Let $n\geq 3$ and $q$ be an odd prime power. Then
$\normalfont{\mbox{cd}}(\Omega_{2n+1}(q))\nsubseteq \normalfont{\mbox{cd}}(PSp_{2n}(q))$ and
$\normalfont{\mbox{cd}}(PSp_{2n}(q)) \nsubseteq \normalfont{\mbox{cd}}(\Omega_{2n+1}(q))$.
\end{lem}
\begin{proof}
We have seen in the proof of Theorem \ref{cor-cod(S)=cod(H)} that
$PSp_{2n}(q)$ possesses an irreducible character of degree
$(q^n+\alpha)/2$, where $\alpha\in\{\pm 1\}$ such that $4 \mid
(q^n-\alpha)$, and furthermore $(q^n+\alpha)/2\notin
\normalfont{\mbox{cd}}(\Omega_{2n+1}(q))$. Therefore, it suffices to show
$cd(\Omega_{2n+1}(q))\nsubseteq \normalfont{\mbox{cd}}(PSp_{2n}(q))$.
We claim that both \[(q^{2n}-1)/(q^2-1) \text{ and }
q(q^{2n}-1)/(q^2-1)\] are elements of $\normalfont{\mbox{cd}}(\Omega_{2n+1}(q))$ for
$q$ odd. Let $G:=Spin_{2n+1}(q)$, the universal cover of
$\Omega_{2n+1}(q)$. The dual group $G^\ast$ of $G$ (in the sense of
\cite[Def. 13.10]{DM}) is the projective conformal symplectic group
$PCSp_{2n}(q)$, which is the quotient of $\widetilde{G}=CSp_{2n}(q)$
by its center ${\mathbf Z}(\widetilde{G})\simeq C_{q-1}$. Consider a
semisimple element $s\in \widetilde{G}$ with spectrum
$Spec(s)=\{-1,-1,1,...,1\}$ and
\[\mathbf{C}_{\widetilde{G}}(s)\cong (Sp_{2}(q)\times
Sp_{2n-2}(q))\cdot C_{q-1}\] (see \cite[Lem. 2.4]{Nguyen10}). Let
$s^\ast$ be the image of $s$ under the natural homomorphism from
$\widetilde{G}$ to $G^\ast$. It is easy to see that, by the choice
of $s$, $\mathbf{C}_{\widetilde{G}}(s)$ is the complete inverse
image of $\mathbf{C}_{G^\ast}(s^\ast)$ under this homomorphism, and
thus
$\mathbf{C}_{G^\ast}(s^\ast)=\mathbf{C}_{\widetilde{G}}(s)/\mathbf{Z}(\widetilde{G})$
and
\[[G^\ast:\mathbf{C}_{G^\ast}(s^\ast)]_{p'}=[\widetilde{G}:\mathbf{C}_{\widetilde{G}}(s)]_{p'}
=\frac{|Sp_{2n}(q)|_{p'}}{|Sp_2(q)|_{p'}|Sp_{2n-2}(q)|_{p'}}=\frac{q^{2n}-1}{q^2-1},\]
where $p$ is the defining characteristic of $G$.
Consider the canonical homomorphism $f: Sp_2(q)\times
Sp_{2n-2}(q)\hookrightarrow \mathbf{C}_{\widetilde{G}}(s)
\rightarrow \mathbf{C}_{G^\ast}(s^\ast)$. Using \cite[Prop.
13.20]{DM}, we know that unipotent characters of $Sp_2(q)\times
Sp_{2n-2}(q)$ are of the form $\theta \circ f$ where $\theta$ runs
over the unipotent characters of $\mathbf{C}_{G^\ast}(s^\ast)$. In
particular, as $Sp_{2}(q)\cong SL_2(q)$ has unipotent characters of
degrees 1 and $q$, $\mathbf{C}_{G^\ast}(s^\ast)$ has two unipotent
characters of degree $1$ and $q$ as well. By the conclusion of the
previous paragraph, the Lusztig series $\mathcal{E}(G,(s^\ast))$ of
$G$ associated to the conjugacy class $(s^\ast)$ of $G^\ast$
contains two characters of degrees $(q^{2n}-1)/(q^2-1)$ and
$q(q^{2n}-1)/(q^2-1)$.
Note that $s^\ast \in PSp_{2n}(q)=[G^\ast,G^\ast]$ and
$|\mathbf{Z}(G)|=|G^\ast/(G^\ast)'|=2$, and therefore every
character in the Lusztig series $\mathcal{E}(G,(s^\ast))$ restricts
trivially to $\mathbf{Z}(G)$ (see \cite[Lem. 4.4]{Navarro-Tiep13}),
and so can be viewed as a character of
$G/\mathbf{Z}(G)=\Omega_{2n+1}(q)$. The claim is completely proved.
Suppose first that $q>3$ and assume to the contrary that
$\normalfont{\mbox{cd}}(\Omega_{2n+1}(q))\subseteq \normalfont{\mbox{cd}}(PSp_{2n}(q))$. Then we have
$(q^{2n}-1)/(q^2-1)\in \normalfont{\mbox{cd}}(PSp_{2n}(q))$, so that
$(q^{2n}-1)/(q^2-1)\in \normalfont{\mbox{cd}}(Sp_{2n}(q))$. Let
$\chi\in\mathrm{Irr}(Sp_{2n}(q))$ such that $\chi(1)=(q^{2n}-1)/(q^2-1)$.
Now, as $q>3$, we have $(q^{2n}-1)/(q^2-1)<(q^{2n}-1)/2(q+1)$, and
therefore, by the classification of irreducible characters of
$Sp_{2n}(q)$ of degrees up to $(q^{2n}-1)2(q+1)$ (\cite[Thm.
5.2]{Tiep-Zalesskii96}), we deduce that $\chi$ must be one of the
Weyl characters of degree $(q^n\pm 1)/2$ or the minimal unipotent
one of degree $(q^n-1)(q^n-q)/2(q+1)$. A simple check reveals that
none of these degrees matches the degree of $\chi$.
Now we suppose $q=3$. (In such case, $(q^{2n}-1)/(q^2-1)$ is indeed
a character degree of $PSp_{2n}(q)$.) As the case of $\Omega_7(3)$
and $PSp_6(3)$ can be checked directly, we suppose furthermore that
$n\geq 4$. We then have
$q(q^{2n}-1)/(q^2-1)<(q^{2n}-1)(q^{n-1}-q)/2(q^2-1)$. Examining the
degrees up to $(q^{2n}-1)(q^{n-1}-q)/2(q^2-1)$ of irreducible
characters of $Sp_{2n}(q)$ available in \cite[Cor. 4.2]{Nguyen10},
we observe that none of them is equal to $q(q^{2n}-1)/(q^2-1)$. We
have shown that $q(q^{2n}-1)/(q^2-1)\notin \normalfont{\mbox{cd}}(Sp_{2n}(q))$, and
therefore, by the above claim, $\normalfont{\mbox{cd}}(\Omega_{2n+1}(q))\nsubseteq
\normalfont{\mbox{cd}}(PSp_{2n}(q))$, as desired.
\end{proof}
We can now prove our first main Theorem \ref{thm:main1}, which in
fact follows from the following slightly stronger result. If $G$ and
$H$ are groups we say that $C(G)\subseteq C(H)$ if
$\normalfont{\mbox{cod}}(G)\subseteq\normalfont{\mbox{cod}}(H)$ and $m_G(c)\leq m_H(c)$ for every
$c\in\normalfont{\mbox{cod}}(G)$.
\begin{thm}\label{thm:main1repeated}
Let $H$ be a finite simple group and $G$ a nontrivial finite group
such that $C(G)\subseteq C(H)$. Then $G\cong H$.
\end{thm}
\begin{proof}
Suppose first that $H$ is abelian of prime order $p$. Then
$\normalfont{\mbox{cod}}(G)\subseteq\normalfont{\mbox{cod}}(H)=\{1,p\}$. Therefore, by Lemma 2.4 of \cite{dl}, $G$ is an elementary
abelian $p$-group and since $k(G)\leq p$, we conclude that $G$ is
cyclic of order $p$, as wanted.
So we may assume that $H$ is nonabelian. By Corollary \ref{thm-char}
and the assumption $C(G)\subseteq C(H)$, we have that $G$ is
perfect. Let $N\trianglelefteq\, G$ such that $S:=G/N$ is nonabelian simple. Now
$\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(G)\subseteq \normalfont{\mbox{cod}}(H)$ and therefore, by
Theorem \ref{thm-|S|divides|G|}, we have $|S|$ divides $|H|$.
Note that $C(S)\subseteq C(G)\subseteq C(H)$. Therefore there exists a subset
$I\subseteq \mathrm{Irr}(H)\backslash \{\mathbf{1}_H\}$ and a bijection
$f:\mathrm{Irr}(S)\backslash \{\mathbf{1}_S\} \rightarrow I$ such that
\[
\frac{|S|}{\chi(1)}=\frac{|H|}{f(\chi)(1)}
\]
for every $\chi\in \mathrm{Irr}(S)\backslash \{\mathbf{1}_S\}$. It follows
that
\[
\sum_{\chi\in \mathrm{Irr}(S)\backslash \{\mathbf{1}_S\}}
\frac{\chi(1)^2}{|S|^2} = \sum_{\psi\in I} \frac{\psi(1)^2}{|H|^2},
\]
and thus
\[
\frac{|S|-1}{|S|^2}\leq \frac{|H|-1}{|H|^2}.
\]
As the function $(x-1)/x^2$ is decreasing on $[2,\infty)$, we deduce
that $|S|\geq |H|$.
The conclusions of the last two paragraphs show that $|S|=|H|$. If
$S\cong H$ then $C(G/N)=C(S)=C(H)\supseteq C(G)$ and so $G/N$ and $G$ have
the same number of conjugacy classes, which is possible only when
$N=1$, and we are done.
So assume by contradiction that $S\ncong H$. Using again the simple
order theorem, we have $\{S,H\}=\{PSL_4(2),PSL_3(4)\}$ or
$\{S,H\}=\{\Omega_{2n+1}(q), PSp_{2n}(q)\}$ for some $n\geq 3$ and
odd $q$. For the former pair, using the character tables of both
$PSL_4(2)$ and $PSL_3(4)$ available in \cite{Conway}, one observes
that $7\in\normalfont{\mbox{cd}}(PSL_4(2)) \backslash \normalfont{\mbox{cd}}(PSL_3(4))$ and $63\in
\normalfont{\mbox{cd}}(PSL_3(4)) \backslash \normalfont{\mbox{cd}}(PSL_4(2))$, implying that none of
$\normalfont{\mbox{cod}}(PSL_3(4))$ and $\normalfont{\mbox{cod}}(PSL_2(4))$ contains the other, and this
violates the fact that $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(H)$. The latter pair
was already handled in Lemma \ref{lem:O2n+1andPSp}.
\end{proof}
\section{The largest character degree of finite simple
groups}\label{sec:largest-degree}
Let $b(G)$ denote the largest degree of an irreducible character of
a finite group $G$. Recall from the Introduction that if $S$ is a
simple group, then $f(S):=|S|/b(S)$ is the smallest nontrivial
character codegree of $S $. The following elementary fact explains
the relevance of $f(S)$, and therefore $b(S)$.
\begin{lem}\label{lem:fS geq fH}
Let $S$ and $H$ be finite simple groups such that $\normalfont{\mbox{cod}}(S)\subseteq
\normalfont{\mbox{cod}}(H)$. Then $f(S)\geq f(H)$.
\end{lem}
\begin{proof}
The hypothesis implies that $f(S)\in\normalfont{\mbox{cod}}(S)\subseteq\normalfont{\mbox{cod}}(H)$. Since $f(H)$ is the smallest nontrivial member of $\normalfont{\mbox{cod}}(H)$, it follows that $f(S)\geq
f(H)$.
\end{proof}
Under the hypothesis of Lemma \ref{lem:fS geq fH}, we showed in
Theorem \ref{thm-|S|divides|G|} that $|S|$ divides $|H|$. We will see in
later sections that, in many cases, the two conditions $f(S)\geq
f(H)$ and $|S|$ divides $|H|$ are enough to force $S\cong H$, as
stated in Theorem \ref{thm:main4}.
Browsing through character tables of small-order simple groups in
\cite{Conway}, one notices that if $|H|$ is a multiple of $|S|$ and $|H|>|S|$, then $b(H)>b(S)$. However, it seems that the largest
character degree grows slower than the order -- that is,
$f(H)>f(S)$. This is not so easy to prove generally, but we do
confirm it in several cases, particularly when either $S$ or $H$ is
an alternating group (see Sections \ref{sec:theoremD-mixed-case} and
\ref{sec:theoremD-alternating-sporadic}).
We shall need effective (both lower and upper) bounds for the
largest degree of an irreducible character of simple groups. For
symmetric groups, asymptotic and explicit bounds were obtained by
A.\,M. Vershik and S.\,V. Kerov in \cite{VK85} which can be used to
derive the corresponding bounds for alternating groups. For a group
$S$ of Lie type in characteristic $p$, an obvious (and in fact very
tight!) lower bound for $b(S)$ is the degree ${\mathsf {St}}_S(1)=|S|_p$ of the
Steinberg character ${\mathsf {St}}_S$. When $S$ is of classical type, explicit
upper bounds have been worked out by M. Larsen, G. Malle, and P.\,H.
Tiep in \cite{LMT13}. Unfortunately, upper bounds for exceptional
groups achieved in \cite{LMT13} are only asymptotic and its proof
does not allow one to obtain an explicit bound. We obtain Theorem
\ref{lem:exceptional-b(S)} below that we believe will be useful in
other applications.
\begin{lem}\label{lem:order-bound}
Let ${\mathbf{G}}$ be a simple algebraic group over the algebraic closure of
a finite field of order $q$ in characteristic $p$, $F:
{\mathbf{G}}\rightarrow {\mathbf{G}}$ a Steinberg endomorphism, and $G:= {\mathbf{G}}^F$ be the
corresponding finite group of Lie type. Let $r$ be the rank of
${\mathbf{G}}$. Then
\[
(q-1)^r\cdot|G|_p\leq |G|_{p'}\leq q^r\cdot|G|_p.
\]
\end{lem}
\begin{proof}
Note that finite groups ${\mathbf{G}}^F$ of the same isogeny type have the
same order, so we may work with ${\mathbf{G}}$ being of simply-connected
type. The inequalities are then straightforward to verify using the
order formulas for finite groups of Lie type available in \cite[p.
xvi]{Conway}.
\end{proof}
\begin{thm}\label{lem:exceptional-b(S)}
Let $S$ be a simple exceptional group of Lie type defined over a
field of order $q$ in characteristic $p$. Then the following hold:
\begin{enumerate}[\rm(i)]
\item $b(S)<256|S|_p$.
\item If $q>2$, then $b(S)<26|S|_p$.
\end{enumerate}
\end{thm}
\begin{proof}
The Tits group can be verified easily, so we assume that $S\neq
{}^2F_4(2)'$. We then may find a simple algebraic group $\mathbf{G}$
of adjoint type and a Steinberg endomorphism
$F:\mathbf{G}\rightarrow \mathbf{G}$ such that $S=[G,G]$ where
$G:=\mathbf{G}^F$ (see \cite[Prop. 24.21]{malletesterman}). Clearly
it suffices to show that $b(G)<256 {\mathsf {St}}_G(1)$.
Let $(\mathbf{G}^\ast,F^\ast)$ be the dual pair of $(\mathbf{G},F)$,
so $\mathbf{G}^\ast$ will be the corresponding simple algebraic
group of simply connected type, and set
$G^\ast:=({\mathbf{G}}^\ast)^{F^\ast}$. As mentioned before, Lusztig's
classification on complex characters of finite groups of Lie type
implies that the set of irreducible complex characters of $G$ is
partitioned into Lusztig series $\mathcal{E}(G,(s))$ associated to
various conjugacy classes $(s)$ of semisimple elements of $G^\ast$.
Furthermore, there is a bijection $\chi\mapsto\psi$ from
$\mathcal{E}(G,(s))$ to $\mathcal{E}({\mathbf{C}}_{G^\ast}(s),(1))$ such that
\begin{equation}\label{lusztig}
\chi(1)=\frac{|G|_{p'}}{|{\mathbf{C}}_{G^\ast}(s)|_{p'}}\psi(1).
\end{equation}
The detailed structure of centralizers of semisimple elements in a
finite exceptional groups of Lie type was determined by Carter
\cite{Carter78}, Deriziotis \cite{Deriziotis}, and
Deriziotis-Liebeck \cite{Deriziotis-Liebeck}. A well-known result of
Steinberg states that the centralizer ${\mathbf{C}}_{{\mathbf{G}}^\ast} (s)$ of a
semisimple element $s$ is a connected reductive subgroup of maximal
rank in ${\mathbf{G}}^\ast$. Such connected subgroup has a decomposition
${\mathbf{C}}_{{\mathbf{G}}^\ast} (s)=\mathbf{S}\mathbf{T}$, where $\mathbf{S}$ is a
semisimple subgroup, $\mathbf{T}$ is a central torus,
$\mathbf{S}\cap \mathbf{T}$ is finite, and
$|({\mathbf{C}}_{G^\ast}(s))^{F^\ast}|=|\mathbf{S}^\ast||\mathbf{T}^\ast|$
(see \cite[p. 48]{Deriziotis-Liebeck}). When $s$ is in $G^\ast$, the
centralizer ${\mathbf{C}}_{{\mathbf{G}}^\ast} (s)$ is $F^\ast$-stable and
${\mathbf{C}}_{G^\ast}(s)=({\mathbf{C}}_{{\mathbf{G}}^\ast} (s))^{F^\ast}$; and so
\[|{\mathbf{C}}_{G^\ast}(s)|=|S||T|\] where $S:=\mathbf{S}^{F^\ast}$ and
$T:=\mathbf{T}^{F^\ast}$. Let $r$ be the semisimple rank of
${\mathbf{G}}^\ast$ and $q$ (that will be a power of $p$) the absolute value
of all eigenvalues of $F$ on the character group of an $F$-stable
maximal torus of ${\mathbf{G}}$. Possible values for $|S|$ and $|T|$ are
available in \cite{Deriziotis,Deriziotis-Liebeck}. In particular, we
have
\[
|S|=\prod_i |L_{r_i}(q^{a_i})|
\]
and
\[
|T|=\prod_j \Phi_j(q),
\]
where $L_{r_i}(q^{a_i})$s are finite groups of Lie type (of
simply-connected type) of rank $r_i$ defined over a field of order
$q^{a_i}$ and $\Phi_j(q)$s are cyclotomic polynomials (and also
polynomials of the forms $q^2\pm \sqrt{2}q+1$, $q^2\pm \sqrt{3}q+1$,
or $q^4\pm \sqrt{2}q^3+q^2\pm \sqrt{2}q+1$ for Suzuki and Ree
groups) evaluated at $q$. As ${\mathbf{C}}_{{\mathbf{G}}^\ast} (s)$ has maximal rank,
we furthermore have
\begin{equation}\label{eq2}
\sum_i a_ir_i + \sum_j \deg(\Phi_j)=r.
\end{equation}
Now formula (\ref{lusztig}) implies that the typical degree of an
irreducible character of $G$ is of the form
\[
\chi(1)=\frac{|G|_{p'}}{\prod_i |L_{r_i}(q^{a_i})|_{p'} \prod_j
\Phi_j(q)} \psi(1),
\]
where $\psi\in \mathcal{E}({\mathbf{C}}_{G^\ast}(s),(1))$, a unipotent
character of ${\mathbf{C}}_{G^\ast}(s)$. By \cite[Thm. 3.1]{LMT13}, for any
finite group of Lie type $\mathbf{G}^F$, where ${\mathbf{G}}$ is a simple
algebraic group in characteristic $p$ and $F$ a Steinberg
endomorphism on ${\mathbf{G}}$, the degree ${\mathsf {St}}(1)=|\mathbf{G}^F|_p$ of the
Steinberg character ${\mathsf {St}}$ of $\mathbf{G}^F$ is strictly larger than
the degree of any other unipotent character. Therefore, the degrees
of unipotent characters of ${\mathbf{C}}_{G^\ast}(s)$, which are in fact the
same as those of the semisimple group $S$, are bounded by $\prod_j
|L_{r_i}(q^{a_i})|_{p}$. It follows that
\[
b(G)\leq \frac{|G|_{p'}}{\prod_i |L_{r_i}(q^{a_i})|_{p'} \prod_j
\Phi_j(q)} \prod_j |L_{r_i}(q^{a_i})|_{p},
\]
By Lemma \ref{lem:order-bound},
\[
\frac{|L_{r_i}(q^{a_i})|_{p}}{|L_{r_i}(q^{a_i})|_{p'}} \leq
\frac{1}{(q^{a_i}-1)^{r_i}}\leq \frac{1}{(q-1)^{a_ir_i}}.
\]
Also, it is easy to see that \[ \Phi_j(q)\geq (q-1)^{\deg \Phi_j}.
\]
We therefore deduce that
\[
b(G)\leq \frac{|G|_{p'}}{(q-1)^{\sum_i a_ir_i + \sum_j
\deg(\Phi_j)}}= \frac{|G|_{p'}}{(q-1)^r}.
\]
On the other hand, we have $|G|_{p'}\leq |G|_pq^r$ by again Lemma
\ref{lem:order-bound}, and it follows that
\[
\frac{b(G)}{|G|_p}\leq \frac{q^r}{(q-1)^r}.
\]
As the rank $r$ is at most $8$ for exceptional groups, the desired
inequalities follow.
\end{proof}
Bounds for $b(S)$ of alternating groups and classical groups are
collected in the following.
\begin{lem}\label{lem:bounds-for-b(S)}
Let $S$ be a finite simple group, $n$ a positive integer, and $q$ a
prime power.
\begin{enumerate}[\rm(i)]
\item For $S=\textup{\textsf{A}}_n$ with $n\geq 5$, \[\frac{1}{2}e^{-1.28255\sqrt{n}}\sqrt{n!} \leq b(S)\leq
e^{-0.11565\sqrt{n}}\sqrt{n!}.\] In particular,
\[
b(S)>\frac{1}{2}e^{-1.28255\sqrt{n}} (2\pi n)^{1/4}(n/e)^{n/2}
\]
\item For $S=\textup{\textsf{A}}_n$ with $n\geq 5$, \[b(\textup{\textsf{A}}_{n+1})\geq \frac{2(n+1)}{\sqrt{8n+1}+3}b(\textup{\textsf{A}}_n).\]
\item For $S=PSL_n(q)$ with $n\geq 2$,
\[{b(S)}<13(1+\log_q(n+1))^{2.54}{\mathsf {St}}_S(1).\]
\item For $S=PSU_n(q)$ with $n\geq 3$,
\[ {b(S)}<2(2+\log_q(n+1)^{1.27}{\mathsf {St}}_S(1).\]
\item For $S\in\{\Omega_{2n+1}(q), PSp_{2n}(q),P\Omega_{2n}^\pm(q)\}$ with $n\geq 2$ and $q$ odd,
\[ {b(S)}<38(1+\log_q(2n+1))^{1.27}{\mathsf {St}}_S(1).\]
\item For $S\in \{\Omega_{2n+1}(q), PSp_{2n}(q),P\Omega_{2n}^\pm(q)\}$ with $n\geq 2$ and $q$ even,
\[ {b(S)}<8(1+\log_q(2n+1))^{1.27}{\mathsf {St}}_S(1).\]
\end{enumerate}
\end{lem}
\begin{proof}
Part (i) follows from \cite[Thm. 1]{VK85} and Parts (iii)-(vi)
follow from \cite[Thm. 5.1, 5.2, and 5.3]{LMT13}. Let us prove Part
(ii).
Let $\chi\in\mathrm{Irr}(\textup{\textsf{S}}_{n})$ such that $\chi(1)=b(\textup{\textsf{S}}_{n})$. Let
$\lambda$ be the partition of $n$ corresponding to $\chi$ and
$Y_\lambda$ be the Young diagram associated to $\lambda$. By the
well-known branching rule, the induction $\chi^{\textup{\textsf{S}}_{n+1}}$ of
$\chi$ from $\textup{\textsf{S}}_n$ to $\textup{\textsf{S}}_{n+1}$ is the sum of irreducible
characters corresponding to the partitions of $n+1$ whose associated
Young diagrams are obtained from $Y_\lambda$ by adding a suitable
node. The number of those suitable nodes is at most
$(\sqrt{8n+1}+1)/2$ (see \cite[p. 1950]{HHN16}) and at most one of
the resulting Young diagrams is symmetric. We deduce that
\[
(n+1)b(\textup{\textsf{A}}_n)\leq (n+1)\chi(1)=\chi^{\textup{\textsf{S}}_{n+1}}(1)\leq
\frac{\sqrt{8n+1}+3}{2} b(\textup{\textsf{A}}_{n+1}),
\]
and the result follows.
\end{proof}
\section{Theorem~\ref{thm:main4}: Groups of Lie
type}\label{sec:theoremD-Lie}
In this section we prove Theorem \ref{thm:main4} when the groups
involved are of Lie type.
In the following, for simplicity, we say that two groups have the
same defining characteristic if there is a common characteristic
over which the groups can be defined.
\begin{prop}\label{prop:same-char}
Let $S$ and $H$ be finite simple groups of Lie type. Suppose that
$\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(H)$. Then $S$ and $H$ have the same defining
characteristic.
\end{prop}
\begin{proof} Suppose that the defining characteristic of $H$ is
$p$. By Lemma~\ref{lem-chi-p-power}, $|S|_{p'}=|H|_{p'}$ and there
is $\chi\in\mathrm{Irr}(S)$ such that $\chi(1)=|S|_p$. By
Lemma~\ref{lem-x=H/S}, it follows that
\begin{equation}\label{eq1}
d\cdot\frac{|H|_p}{\chi(1)}\in \normalfont{\mbox{cd}}(H) \text{ for every } d\in\normalfont{\mbox{cd}}(S).
\end{equation}
Certainly if $\chi$ is the Steinberg character of $S$ then we are
done. So we assume otherwise and aim to find a contradiction or end
up with a case where $H$ can be defined in another characteristic
not equal to $p$. By the classification of prime-power-degree
representations of quasi-simple groups \cite[Thm.
1.1]{Malle-Zalesskii}, we arrive at the following possibilities of
$S$ and $\chi(1)$.
\medskip
(i) $S=PSL_2(q)$, $\chi(1)\in\{q\pm 1\}$ or $q$ is odd and
$\chi(1)\in \{(q\pm 1)/2\}$. Observe that $\chi(1)$ cannot be $(q\pm
1)/2$ because otherwise, by taking $d=2\chi(1)$, we would have
$2|H|_p\in\normalfont{\mbox{cd}}(H)$, which is impossible. So $\chi(1)=q+\alpha=p^x$
for some $\alpha\in\{\pm 1\}$ and $x\in \mathbb{N}$. Suppose first
that $q=2^t$ for some $t\geq 2$. Then $2^t+\alpha=p^x$. By
Mihailescu's theorem \cite{Mih04} (previously known as Catalan's
conjecture), either $x=1$ so that $2^t+\alpha$ is a (Mersenne or
Fermat) prime or $\alpha=1$ and $t=3$. In the latter case, $p=3$ and
$|H|_{3'}=|S|_{3'}=56$, forcing $H$ to be ${}^2G_2(3)'$, which turns
out to be isomorphic to $S=PSL_2(8)$, as desired. So it remains to
consider the former case: $q+\alpha=2^t+\alpha=p$ is the defining
characteristic of $H$. Now $|H|_{p'}=|S|_{p'}=q(q-\alpha)$. Let
$p^a$ be the order of the underlying field of $H$. It is clear from
the order formulas of simple groups of Lie type (see \cite[p.
xvi]{Conway}) that $|H|_p< |H|_{p'}/(p^a-1)\leq |H|_{p'}/(p-1)$. We
therefore deduce that
\[|H|_p<\frac{q(q-\alpha)}{q+\alpha-1}.\]
Thus we must have $\alpha=-1$ and $|H|_p=p$. Now $H$ is a simple
group of Lie type in characteristic $p$ such that $|H|=p(p+1)(p+2)$.
This is impossible as for such a group $H$, one can check from the
order formula that $|H|_{p'}<(|H|_p)^2$.
Next we suppose $q\geq 5$ is odd. Then $p=2$ and $q+\alpha =|S|_2$.
Again by Mihailescu's theorem, either $q$ is a prime or $\alpha=-1$
and $q=9$. The case $q=9$ is eliminated in the same way as before.
So assume that $|H|_{2'}=q(q-\alpha)/2$ and $q$ is a prime. Note
that when $H$ is not of type $A_1$, every prime divisor of
$|H|_{p'}$ is smaller than $\sqrt{|H|_{p'}}$. Therefore our group
$H$ must be $PSL_2(q_1)$ for some $2$-power $q_1$, implying that
$q(q-\alpha)/2=q_1^2-1$. This, however, returns no relevant
solutions.
\medskip
(ii) For the remaining possibilities of $S$ and $\chi$, the
character $\chi$ has a decent small degree and we are able to
produce an irreducible character of $S$ whose degree is a proper
multiple of $\chi(1)$. Condition (\ref{eq1}) then implies that a
proper multiple of $|H|_p$ is a character degree of $H$, which is
impossible. This required character turns out to be chosen as a
unipotent character in most cases. We refer the reader to \cite[\S
13.8]{Carter85} for the description of unipotent characters of
finite classical groups.
The next possibility of $S$ and $\chi(1)$ is $S=PSL_n(q)$, $q>2$,
$n$ is an odd prime, $(n,q-1)=1$, and $\chi(1)=(q^n-1)/(q-1)$. If
$n=3$ then $SL_3(q)$ has an irreducible character of degree $q^3-1$
(see \cite{Simpson-Frame73}), which is a proper multiple of
$\chi(1)=q^2+q+1$. For $n\geq 5$, the unipotent character
parameterized by the partition $(2,n-2)$ with degree
\[
\chi^{(2,n-2)}(1)=\frac{(q^n-1)(q^{n-1}-q^2)}{(q-1)(q^2-1)}
\]
fulfills our requirement.
Another possibility is $S=PSU_n(q)$, $n$ is an odd prime,
$(n,q+1)=1$, and $\chi(1)=(q^n+1)/(q+1)$. This case is handled
similarly as in the previous one. Here we note that, when $n\geq 5$
is odd, the unipotent character parameterized by the partition
$(2,n-2)$ has degree
\[
\chi^{(2,n-2)}(1)=\frac{(q^n+1)(q^{n-1}-q^2)}{(q+1)(q^2-1)}.
\]
The next case is $S=PSp_{2n}(q)$, $n\geq 2$, $q=\ell^t$ with $\ell$
an odd prime, $tn$ is a $2$-power, and $\chi(1)=(q^n+1)/2$. Now the
unipotent character parameterized by the symbol ${0\hspace{6pt} 1
\choose n}$ has required degree
\[
\chi^{{0\hspace{6pt} 1 \choose
n}}(1)=\frac{(q^n+1)(q^{n}+q)}{2(q+1)}.
\]
The last possibility involving a family of groups is
$S=PSp_{2n}(3)$, $n\geq 3$ is a prime, and $\chi(1)=(3^n-1)/2$. Then
$S$ has a unipotent character with degree
\[
\chi^{{0\hspace{6pt} 1 \hspace{6pt} n\choose
-}}(1)=\frac{(3^n-1)(3^{n}-3)}{8}.
\]
\medskip
(iii) $(S,\chi(1))\in\{(Sp_6(2),7)$, $(Sp_6(2), 27)$, $({}^2F_4(2)',
27)$, $(G_2(2)', 7)$, $(G_2(2)', 27)$, $(G_2(3), 64)\}$. First
assume that $S=G_2(2)'$ and $\chi(1)=27$. Then $p=3$ and
$|H|_{3'}=|S|_{3'}=224$. It is easy to check that there is no such
simple group $S$ of Lie type in characteristic 3 with $27\mid
|S|_3$. In all other cases one can find a character $\psi\in\mathrm{Irr}(S)$
such that $\chi(1) \mid \psi(1)$ but $\chi(1)<\psi(1)$. Therefore
$\psi(1){|H|_p}/\chi(1)$ is a proper multiple of $|H|_p$ and thus
cannot be a character degree of $H$, violating condition
(\ref{eq1}).
\end{proof}
As seen in Lemma \ref{lem-chi-p-power} and Proposition
\ref{prop:same-char}, we face the situation where two simple groups
$S$ and $H$ of Lie type have the same characteristics $p$ and
$|S|_{p'}=|H|_{p'}$. It is surprising to us that this turns out to
happen only when $|S|=|H|$ (see Proposition \ref{prop:|S|_p'=|H|_p'}
below), and therefore they are among the coincidences appeared in
the simple order theorem.
We slightly modify two of the invariants in Artin's proof of the
simple order theorem (for classical groups) to prove our results.
\begin{define}
Let $S$ be a finite group of Lie type in characteristic $p$. Let
$\omega=\omega(S)$ and $\psi=\psi(S)$ respectively denote the
largest and the second largest orders of $p$ modulo a prime divisor
of $|S|_{p'}$. We will refer to $\omega(S)$ and $\psi(S)$ as the
Artin invariants of $S$.
\end{define}
In fact, when $p$ is the dominant prime of $|S|$, these $\omega(S)$
and $\psi(S)$ coincide with Artin's invariants defined in
\cite{Artin55}. We remark that there are only a few cases involving
Mersenne and Fermat primes where $p$ is not dominant in $|S|$, and
they are listed in \cite[Thm. 3.3]{Kimmerle-et-al}.
Assume for now that $S$ is not one of $G_2(2)'$, ${}^2G_2(3)'$, and
${}^2F_4(2)'$. (Note that $S_1=G_2(2)'\cong PSU_3(9)$ and
$S_2={}^2G_2(3)'\cong PSL_2(8)$ and, even though we do not allow
$S_1$ (or $S_2$) to be viewed as a Lie type group over a field of
order 2 (or 3), we do allow it to be viewed as one over the field of
order 9 (or 8).) Let $q=p^t$ be the order of the underlying field of
$S$. It is well-known that the order $|S|$ then has the standard
cyclotomic factorization in terms of $q$ as
\[
|S|=\frac{1}{d}q^k \prod_i\Phi_i(q)^{e_i},
\]
where $d$, $k$, $e_i$s depend on $S$ and can be found in \cite[Table
6]{Conway} and \cite[Tables C1, C2, and C3]{Kimmerle-et-al} for
instance, and $\Phi_i(q)$ is the value of the $i$th cyclotomic
polynomial evaluated at $q$. Replacing $q$ by $p^n$ and factorizing
each $\Phi_i(x^t)$ further into cyclotomic polynomials of $x$, one
has
\[
|S|=\frac{1}{d}p^{kt} \prod_i\Phi_i(p)^{f_i}
\]
for certain positive integers $f_i$s depending on $S$.
Using Zsigmondy's theorem, it is not difficult to see that, except
for some `small' cases, the invariants $\omega(S)$ and $\psi(S)$ are
precisely the largest and second largest, respectively, index $i$
such that $\Phi_i(p)$ appears in the cyclotomic factorization of
$|S|$ (see \cite[Lem. 4.6]{Kimmerle-et-al}). We refer the reader to
\cite[Tables A1, A2 and A3]{Kimmerle-et-al} for the list of
exceptions and the values of their Artin's invariants, including the
groups $G_2(2)'$, ${}^2G_2(3)'$, and ${}^2F_4(2)'$ we excluded
earlier. We reproduce in Table \ref{Artin invariants} the values of
$\omega(S)$ and $\psi(S)$ for the generic case only.
\begin{table}[ht]
\caption{$\omega(S)$ and $\psi(S)$ for simple groups of Lie type:
generic case.\label{Artin invariants}}
\begin{center}
\begin{tabular}{llll}
\hline \begin{tabular}{l} $S$\\ ($q=p^t$)\end{tabular} &
\begin{tabular}{l} Conditions\\ ($p$ a Mersenne prime)\end{tabular}&
$\omega(S)$& $\psi(S)$ \\ \hline
$PSL_n(q), n\geq 2$& \begin{tabular}{l} $(n,q)\neq (2,2^6),(3,2^2),(3,2^3)$,\\
$(4,2^2), (6,2),(7,2),(2,p^2),(3,p)$\end{tabular} & $nt$ & $(n-1)t$ \\
$PSU_4(q)$& & $6t$ & $4t$ \\
$PSU_n(q), n\geq 3$ odd& $(n,q)\neq (3,2^3),(5,2),(3,p)$ & $2nt$ & $2(n-2)t$ \\
$PSU_n(q), n\geq 6$ even& $(n,q)\neq (6,2)$ & $2(n+1)t$ & $2(n-1)t$ \\
$\Omega_{2n+1}(q), n\geq 2$& $(n,q)\neq (2,2^8),(3,2),(4,2),(2,p)$ & $2nt$ & $2(n-1)t$ \\
$PSp_{2n}(q), n\geq 3$& & $2nt$ & $2(n-1)t$ \\
$P\Omega^+_{2n}(q), n\geq 4$& $(n,q)\neq (4,2),(5,2)$ & $2(n-1)t$ & $2(n-2)t$ \\
$P\Omega^-_{2n}(q), n\geq 4$& $(n,q)\neq (4,2)$ & $2nt$ & $2(n-1)t$ \\
${}^2B_2(2^t), t\geq 3$ odd& $t\equiv 3(\bmod 6)$ & $4t$ & $4t/3$ \\
${}^2B_2(2^t), t\geq 3$ odd& $t\equiv \pm1(\bmod 6)$ & $4t$ & $t$ \\
$G_2(q), q\geq 3$& $q\neq 4$ & $6t$ & $3t$ \\
${}^2G_2(3^t), t\geq 3$ odd& & $6t$ & $2t$ \\
${}^3D_4(q)$& $q\neq 2$ & $12t$ & $6t$ \\
$F_4(q)$& & $12t$ & $8t$ \\
${}^2F_4(2^t), t\geq 3$ odd& & $12t$ & $6t$ \\
$E_6(q)$& & $12t$ & $9t$ \\
${}^2E_6(q)$& & $18t$ & $12t$ \\
$E_7(q)$& & $18t$ & $14t$ \\
$E_8(q)$& & $30t$ & $24t$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{prop}\label{prop:|S|_p'=|H|_p'}
Let $p$ be a prime. Suppose that $S$ and $H$ are two non-isomorphic
simple groups of Lie type in characteristic $p$ and
$|S|_{p'}=|H|_{p'}$. Then $\{S,H\}=\{PSL_4(2),PSL_3(4)\}$ or
$\{S,H\}=\{\Omega_{2n+1}(q), PSp_{2n}(q)\}$ for some $n\geq 3$ and
odd $q$. In particular, $|S|=|H|$.
\end{prop}
\begin{proof}
By the assumptions, we have $\omega(S)=\omega(H)$ and
$\psi(S)=\psi(H)$. First we consider the case where both $S$ and $H$
are generic so that their invariants $\omega$ and $\psi$ are
available in Table \ref{Artin invariants}. Comparing those values,
we can find all the collections of groups with equal values of
$\omega$ and $\psi$. We list these collections in Table \ref{Artin
invariants2} (each row in the table is one such collection). Now one
simply compare the $p'$-parts of orders of groups in each
collection. It turns out that the only pair with the same $p'$-parts
of orders is $\{\Omega_{2n+1}(q), PSp_{2n}(q)\}$ with some $n\geq 3$
and odd $q$.
Assume now that at least one of the two groups, say $S$, is
non-generic. That is, $S$ is among the exceptions listed in the
second column of Table \ref{Artin invariants}. The values of the
invariants $\omega$ and $\psi$ of these groups are available in
\cite[Tables A2 and A3]{Kimmerle-et-al}. The analysis is basically
the same as in the generic case, but more tedious. We first find all
the possible groups $H$ with $\omega(S)=\omega(H)$ and
$\psi(S)=\psi(H)$, and then compare $|S|_{p'}$ and $|H|_{p'}$, where
$p$ is the defining characteristic of $S$ and $H$.
Let us demonstrate the case $S=PSL_3(4)$ as an example. Then
$\omega(S)=4$ and $\psi(S)=3$. But there are only two other simple
groups of Lie type with the same values of $\omega$ and $\psi$,
namely $PSL_4(2)$ and $P\Omega^+_8(2)$. However,
$|P\Omega^+_8(2)|_{2'}\neq |PSL_3(4)|_{2'}=|PSL_4(2)|_{2'}$, and so
we come up with another possible pair for $\{S,H\}$, namely
$\{PSL_4(2),PSL_3(4)\}$, as stated in the theorem.
\end{proof}
\begin{table}[ht]
\caption{Simple groups of Lie type with the same values of $\omega$
and $\psi$: generic case.\label{Artin invariants2}}
\begin{center}
\begin{tabular}{l}
\hline
$PSL_n(p^{2s})$, $\Omega_{2n+1}(p^s)$, $PSp_{2n}(p^s)$, $P\Omega^+_{2(n+1)}(p^s)$, $P\Omega^-_{2n}(p^s)$ \\
$PSL_3(p^{2s})$, $PSU_4(p^s)$, $\Omega_{7}(p^s)$,
$PSp_{6}(p^s)$, $P\Omega^+_{8}(p^s)$\\
$PSL_2(p^{6s})$, $\Omega_5(p^{3s})$, $G_2(p^{2s})$, ${}^3D_4(p^s)$\\
$PSL_2(p^{3s})$, $G_2(p^s)$\\
$PSL_3(p^{4s})$, $PSU_4(p^{2s})$, $\Omega_{7}(p^{2s})$,
$PSp_{6}(p^{2s})$, $P\Omega^+_{8}(p^{2s})$, $F_4(p^s)$\\
$PSL_2(2^{6s})$, $\Omega_5(2^{3s})$, $G_2(2^{2s})$, ${}^3D_4(2^s)$, ${}^2F_4(2^s)$, $s\geq 3$ odd\\
$PSL_4(p^{3s})$, $E_6(p^s)$\\
$PSL_4(p^{6s})$, $\Omega_9(p^{3s})$, $P\Omega_{10}^+(p^{3s})$,
$P\Omega_{8}^-(p^{3s})$, $E_6(p^{2s})$\\
$PSL_3(p^{6s})$, $PSU_4(p^{3s})$, $\Omega_{7}(p^{3s})$,
$PSp_{6}(p^{3s})$, $P\Omega^+_{8}(p^{3s})$, ${}^2E_6(p^s)$\\
$PSL_3(p^{12s})$, $\Omega_{7}(p^{6s})$, $PSp_{6}(p^{6s})$, $P\Omega^+_{8}(p^{6s})$, $F_4(p^{3s})$, ${}^2E_6(p^{2s})$\\
$PSL_5(p^{6s})$, $\Omega_{11}(p^{3s})$, $PSp_{10}(p^{3s})$, $P\Omega_{12}^+(p^{3s})$, $P\Omega_{10}^-(p^{3s})$, $E_8(p^s)$\\
$PSU_n(q)$, $PSU_{n-1}(q)$, $n\geq 7$ odd\\
$PSU_3(2^{2s})$, ${}^2B_2(2^{3s})$, $s$ odd\\
$PSU_3(3^{s})$, ${}^2G_2(3^{s})$, $s\geq 3$ odd\\
$PSU_9(p^s)$, $PSU_8(p^s)$, $E_7(p^s)$\\
$\Omega_{2n+1}(q)$, $PSp_{2n}(q)$, $n\geq 3$, $q$ odd\\
\hline
\end{tabular}
\end{center}
\end{table}
The next theorem improves Theorem \ref{cor-cod(S)=cod(H)} when the
relevant groups are of Lie type.
\begin{thm}\label{thm:Lie type}
Let $S$ and $H$ be finite simple groups of Lie type such that
$\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(H)$. Then $S\cong H$.
\end{thm}
\begin{proof}
By Lemma \ref{lem-chi-p-power} and Propositions \ref{prop:same-char}
and \ref{prop:|S|_p'=|H|_p'}, we have that $S$ and $H$ fall into one
of two pairs of groups concluded in Proposition
\ref{prop:|S|_p'=|H|_p'}. The result now follows by Lemma
\ref{lem:O2n+1andPSp}.
\end{proof}
\section{Theorem~\ref{thm:main4}: The mixed case of alternating groups and groups of Lie
type}\label{sec:theoremD-mixed-case}
In this section, we prove Theorem~\ref{thm:main4} in the mixed
situation where the set of codegrees of an alternating group $S$ is
contained in that of a simple group $H$ of Lie type, or vice versa.
In the following proposition we remark that the condition on $m$ is
necessary, due to the coincidences of isomorphic simple groups:
$A_5\cong PSL_2(4)\cong PSL_2(5)$, $A_6\cong PSL_2(9)$, and
$A_8\cong PSL_4(2)$. We also recall that $f(X):=|X|/b(X)$, where
$b(X)$ is the largest character degree of $X$.
\begin{prop}\label{prop:mixed case I}
Suppose $m=7$ or $m\geq 9$. Let $H$ be a simple group of Lie type.
If $|\textup{\textsf{A}}_m|$ divides $|H|$, then $f(H)>f(\textup{\textsf{A}}_m)$. As a consequence,
$\normalfont{\mbox{cod}}(\textup{\textsf{A}}_m)\nsubseteq \normalfont{\mbox{cod}}(H)$.
\end{prop}
\begin{proof}
Let $p$ be the defining characteristic and $q$, a power of $p$, the
order of the underlying field of $H$. Consider first the case $H$
being of exceptional type. Using Lemma \ref{lem:exceptional-b(S)},
we have
\[
f(H)> \frac{|H|}{256|H|_p}=\frac{1}{256}|H|_{p'}.
\]
As $|\textup{\textsf{A}}_m|$ divides $|H|$, it follows that $ f(H)>
(1/256)|\textup{\textsf{A}}_m|_{p'}.$ Therefore, to prove the theorem, it suffices
to show \[b(\textup{\textsf{A}}_m)\geq 256|\textup{\textsf{A}}_m|_p.\] Let us assume for now that
that $m\geq 10$. In particular, the dominant prime in $|\textup{\textsf{A}}_m|=m!/2$
is $2$. We therefore just need to show $b(A_m)\geq 256|\textup{\textsf{A}}_m|_2$. As
$|\textup{\textsf{A}}_m|_2\leq 2^{m-2}$, for this we want to show
\[
b(\textup{\textsf{A}}_m)\geq 64\cdot2^m.
\]
Note that $b(\textup{\textsf{A}}_{19})=64,664,600>64\cdot 2^{19}$ (see \cite{McK86}
for the degree of the largest irreducible characters and associated
partitions of symmetric groups of degree up to 75, from which one
can deduce the exact value or a good bound for the one of
corresponding alternating groups). Now one just inducts on $m$ with
the help of Lemma \ref{lem:bounds-for-b(S)}(ii) to achieve the
desired bound for $m\geq 19$.
Suppose that $m\leq 18$ and recall that we are still dealing with
exceptional groups. When $q=2$, the proposition can be verified
directly, so assume that $q\geq 3$. In such case, $b(H)<26|H|_p$ by
Lemma \ref{lem:exceptional-b(S)}, and whence the above estimate can
be refined so that we only need to prove $b(\textup{\textsf{A}}_m)\geq 26|\textup{\textsf{A}}_m|_p$,
which turns out to be true for all $18\geq n\geq 13$. For the
remaining values $m\leq 12$, the arguments go as follows. First we
are done if $f(\textup{\textsf{A}}_m)\leq \sqrt{|H|}$, as $f(H)>\sqrt{|H|}$, so we
may assume that $|H|<f(\textup{\textsf{A}}_m)^2$. For each $m\leq 12$, there are
indeed no possibilities for $H$ satisfying $|H|<f(\textup{\textsf{A}}_m)^2$ and
$|A_m|$ divides $|H|$.
Following the same idea as in the case of exceptional groups, but
using Lemma~\ref{lem:bounds-for-b(S)} instead, we can show that in
fact $f(H)>f(\textup{\textsf{A}}_m)$ for every $H$ of classical type and $m\geq 19$.
Let us present the details for only the linear groups.
Consider $H=PSL_n(q)$ for some $n\geq 2$ and $q$ a prime power. By
Lemma \ref{lem:bounds-for-b(S)}(i), we have
\[f(\textup{\textsf{A}}_m)=\frac{m!}{2b(\textup{\textsf{A}}_m)}\leq e^{1.28255\sqrt{m}}(m!)^{1/2}.\]
Thus, if $|H|\geq e^{2.5651\sqrt{m}}m!$ then $f(H)>\sqrt{|H|}\geq
e^{1.28255\sqrt{m}}(m!)^{1/2}\geq f(\textup{\textsf{A}}_m)$ and we would be done. We
therefore can assume that $|H|< e^{2.5651\sqrt{m}}m!$, which in
particular implies that $n<m$. Using Lemma
\ref{lem:bounds-for-b(S)}(iii), we see that, as before, it is enough
to show that $b(\textup{\textsf{A}}_m)\geq 13(1+\log_q(n+1))^{2.54}|\textup{\textsf{A}}_m|_p$. Since
$m\geq n+1$ and $|\textup{\textsf{A}}_m|_p\leq 2^{m-2}$, for this it is sufficient
to show that
\[
b(\textup{\textsf{A}}_m)\geq 13(1+\log_2m)^{2.54}2^{m-2}.
\]
This last inequality is indeed true for $m=20$, and therefore is
true for all $m\geq 20$, by induction and Lemma
\ref{lem:bounds-for-b(S)}(ii). Checking directly, we see that the
inequality $b(\textup{\textsf{A}}_m)\geq 13(1+\log_q(n+1))^{2.54}|\textup{\textsf{A}}_m|_2$ is still
valid for $n=19$.
As for the exceptional types, we are left to consider the small
cases $m\leq 18$. Again we are done if $f(\textup{\textsf{A}}_m)\leq \sqrt{|H|}$, so
we may assume that $|H|<f(\textup{\textsf{A}}_m)^2$. For each $m$, we search for
relevant $H$ satisfying $|H|<f(\textup{\textsf{A}}_m)^2$ and $|A_m|$ divides $|H|$
and find that, for such an $H$, the inequality $f(H)>f(\textup{\textsf{A}}_m)$
always holds true.
\end{proof}
We shall need the following result on $2$-defect zero and $3$-defect
zero characters of alternating groups, which easily follows from
earlier work of F. Garvan, D. Kim and D. Stanton \cite{GKS90} on the
so-called \emph{$p$-core partitions}. They are partitions having no
hook lengths divisible by $p$. Using Garvan-Kim-Stanton's result, A.
Granville and K. Ono \cite{Granville-Ono} proved the existence of
$p$-defect zero characters with $p\geq 5$ in symmetric and
alternating groups.
\begin{lem}\label{lem:defect zero Am}
Let $m$ be a positive integer.
\begin{enumerate}[\rm(i)]
\item $\textup{\textsf{A}}_m$ has a $2$-defect zero irreducible character if and only
if $m=2k^2+k$ or $m=2k^2+k+2$ for some $k\in\mathbb{N}$.
\item $\textup{\textsf{A}}_m$ has a $3$-defect zero irreducible character if and only
if there is a prime $\ell\equiv 2(\bmod 3)$ such that the the exact
power of $\ell$ dividing $3m+ 1$ is odd.
\end{enumerate}
\end{lem}
\begin{proof}
See the discussion in \cite[pp. 333-334]{Granville-Ono}.
\end{proof}
\begin{prop}\label{prop:mixed case II}
Let $S$ be a simple group of Lie type and $8\neq m\geq 7$ an
integer. Then $\normalfont{\mbox{cod}}(S)\nsubseteq \normalfont{\mbox{cod}}(\textup{\textsf{A}}_m)$. In fact, if $|S|$
divides $|\textup{\textsf{A}}_m|$ and $m\geq 44$, then $f(S)<f(\textup{\textsf{A}}_m)$.
\end{prop}
\begin{proof}
Assume by contradiction that $\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(\textup{\textsf{A}}_m)$. By
Lemma~\ref{lem:fS geq fH}, we then have $f(S)\geq f(\textup{\textsf{A}}_m)$.
Suppose that the defining characteristic of $S$ is $p$. Observe that
$f(S)\leq |S|/{\mathsf {St}}_S(1)=|S|_{p'}$. Furthermore, $|S|_{p'}<|S|_p^2$
(see \cite[Proof of Thm. 12]{Cossey}) and $|S|_p\leq |\textup{\textsf{A}}_m|_p$ by
Theorem~\ref{thm-|S|divides|G|}. Therefore we have $f(S)<
(|\textup{\textsf{A}}_m|_p)^2$. Assume for a moment that $m\geq 10$ so that
$|\textup{\textsf{A}}_m|_p\leq |\textup{\textsf{A}}_m|_2\leq 2^{m-2}$. We now have $f(S)< 2^{2m-4}$.
On the other hand, it is clear that $f(\textup{\textsf{A}}_m)>\sqrt{m!/2}$.
Therefore, we would be done if $m!\geq 2^{4m-7}$. By the well-known
estimate $m!>\sqrt{2\pi m}(m/e)^m$, this is certainly true when
$m\geq44$. So we may now suppose that $m\leq 43$.
As mentioned above, every simple group of Lie type, and therefore
$S$ in particular, has a $2$-defect zero irreducible character,
which means that $S$ has an odd codegree and so does $\textup{\textsf{A}}_m$ as
$\normalfont{\mbox{cod}}(S)\subseteq \normalfont{\mbox{cod}}(\textup{\textsf{A}}_m)$. It follows that $m=2k^2+k$ or
$m=2k^2+k+2$ for some $k\in\mathbb{N}$, by Lemma~\ref{lem:defect zero
Am}(i). This forces $m$ to be one of $10, 12, 21, 23, 36$, or $38$.
By the same reason, $\textup{\textsf{A}}_m$ has a codegree not divisible by $3$ and
so Lemma~\ref{lem:defect zero Am}(ii) further narrows down the
choices for $m$: $m\in\{10,12,21,36\}$. In fact, when $m=21$ or
$36$, we still have $f(\textup{\textsf{A}}_m)> |\textup{\textsf{A}}_m|_2^2$, and since
$|\textup{\textsf{A}}_m|_2^2>f(S)$, it follows that $f(\textup{\textsf{A}}_m)>f(S)$, which is a
contradiction.
Suppose $m=10$. The inequality $f(\textup{\textsf{A}}_m)<(|\textup{\textsf{A}}_m|_p)^2$ then forces
$p=2$ or $3$. If $p=2$ then $|S|_{2'}=|\textup{\textsf{A}}_{10}|/\chi(1)$, where
$\chi\in\mathrm{Irr}(\textup{\textsf{A}}_{10})$ is one of the two $2$-defect zero
irreducible characters of equal degree $384$, implying
$|S|_{2'}=10!/(2\cdot 384)=4725$. It is easy to see from
\cite{Conway} that there is no such group of Lie type in
characteristic $2$. If $p=3$ then $|S|_{3'}=10!/(2\cdot 567)=3200$
since $\textup{\textsf{A}}_{10}$ has a unique $3$-defect zero character of degree
$567$, which again leads to a contradiction as there is no such
group in characteristic $3$. The case $m=12$ is treated similarly
and we skip the details.
\end{proof}
\section{Theorem~\ref{thm:main4}: Alternating and sporadic
groups}\label{sec:theoremD-alternating-sporadic}
\begin{prop}\label{prop:alternating}
Let $m<n$ be positive integers. Then $f(\textup{\textsf{A}}_m)<f(\textup{\textsf{A}}_n)$.
Consequently, $\normalfont{\mbox{cod}}(\textup{\textsf{A}}_m)\nsubseteq \normalfont{\mbox{cod}}(\textup{\textsf{A}}_{m+1})$.
\end{prop}
\begin{proof}
It suffices to show that $b(\textup{\textsf{A}}_{m+1})<(m+1)b(\textup{\textsf{A}}_m)$. Let
$\chi\in\mathrm{Irr}(\textup{\textsf{A}}_{m+1})$ such that $\chi(1)=b(\textup{\textsf{A}}_{m+1})$. As shown
in \cite[p. 1956]{HHN16}, such $\chi$ must be the restriction of an
irreducible character, say $\psi$, of $\textup{\textsf{S}}_{m+1}$ whose associated
partition, say $\lambda$, is not self-conjugate. In particular,
$\chi(1)=\psi(1)$. As in the proof on Lemma
\ref{lem:bounds-for-b(S)}(ii), let $Y_\lambda$ be the Young diagram
associated to $\lambda$. The restriction $\psi_{\textup{\textsf{S}}_m}$ of $\psi$ to
$\textup{\textsf{S}}_n$ is the sum of irreducible characters corresponding to the
partitions of $n$ whose associated Young diagrams are obtained from
$Y_\lambda$ by removing a suitable node. The number of those
suitable nodes is at most $(\sqrt{8m+9}-1)/2$, so
\[
b(\textup{\textsf{A}}_{m+1})=\psi(1)\leq \frac{\sqrt{8m+9}-1}{2} b(\textup{\textsf{S}}_m).
\]
Since $b(\textup{\textsf{S}}_m)<2b(\textup{\textsf{A}}_m)$ as already mentioned above, it follows
that
\[
b(\textup{\textsf{A}}_{m+1})< (\sqrt{8m+9}-1) b(\textup{\textsf{A}}_m),
\]
which implies our desired inequality $b(\textup{\textsf{A}}_{m+1})<(m+1)b(\textup{\textsf{A}}_m)$
for $m\geq 5$. The result is easily checked for smaller $m$.
\end{proof}
\begin{prop}\label{prop:sporadic}
Theorem \ref{thm:main4} is true when either $S$ or $H$ is a sporadic
simple group.
\end{prop}
\begin{proof}
The case where both $S$ and $H$ are sporadic simple groups can be
verified by using the available data in \cite{Conway}.
Suppose that $S$ is a sporadic group and $H=\textup{\textsf{A}}_m$ for some $m\geq
5$. Let $p_S$ be the largest prime divisor of $|S|$. By Theorem
\ref{thm-|S|divides|G|}, we have $|S|$ divides $|\textup{\textsf{A}}_m|$, so
$p_S\leq m$. By Lemma \ref{lem:fS geq fH}, we have $f(S)\geq
f(\textup{\textsf{A}}_m)>\sqrt{m!/2}$. It follows that $f(S)\geq \sqrt{p_S!/2}$.
Again using \cite{Conway}, it can be checked that this can never
happen.
Next we assume that $S$ is a sporadic group and $H$ is a simple
group of Lie type in characteristic $p$. Suppose first that $S$ has
an irreducible character, say $\chi$, of $p$-defect zero. Then, as
argued in the proof of Lemma \ref{lem-chi-p-power}, we have
$|S|_{p'}=|H|_{p'}$ and $\chi(1)=|S|_{p}$. In particular, $\chi(1)$
is a prime power, and therefore, \cite[Thm. 1.1]{Malle-Zalesskii}
yields
\[
(S,p,\chi(1))\in\{(M_{11}/M_{12}, 11,11), (M_{11},2,16),
(M_{24}/Co_{2}/Co_{3},23,23)\}
\]
(We note that $M_{12}$ has another irreducible character of prime
power degree, namely $16$, but the character is not of $2$-defect
zero and thus does not fit our situation.) However, for each of
these possibilities, there is no simple group of Lie type $H$ in
characteristic $p$ such that $|H|_{p'}=|S|_{p'}$. Next, we suppose
that $S$ has no characters of $p$-defect zero. By
\cite[Cor.~2]{Granville-Ono}, $$p\in\{2,3\} \text{ and }
S\in\{M_{12}, M_{22}, M_{24}, J_2, HS, Suz, Ru, Co_1, Co_3,BM\}.$$
Now we just apply Lemma~\ref{lem:bounds-for-b(S)} and argue
similarly as in the proof of Proposition~\ref{prop:mixed case I},
with $S$ in place of $\textup{\textsf{A}}_m$, to arrive at $f(H)>f(S)$, and thus it
follows from Lemma \ref{lem:fS geq fH} that
$\normalfont{\mbox{cod}}(S)\nsubseteq\normalfont{\mbox{cod}}(H)$.
Now we consider the case where $S=\textup{\textsf{A}}_m$ for some $m\geq 5$ and $H$
a sporadic simple group. Using Theorem \ref{thm-|S|divides|G|}, we
have $m!/2$ divides $|H|$ and so $m$ is at most $\overline{p}_H-1$,
where $\overline{p}_H$ is the smallest prime not dividing $|H|$.
This constraint is enough to ensure that $f(A_m)<f(H)$, and thus
$\normalfont{\mbox{cod}}(\textup{\textsf{A}}_m)\nsubseteq \normalfont{\mbox{cod}}(H)$ by Lemma \ref{lem:fS geq fH}.
Finally we consider the case where $S$ is a simple group of Lie type
and $H$ a sporadic simple group. As in the proof of
Proposition~\ref{prop:mixed case II} , we have $f(H)\leq (|H|_p)^2$,
where $p$ is the defining characteristic of $S$. The only possible
$p$ satisfying such condition is $p=2$. Now $|S|_{2'}$ is an odd
codegree of $S$, and hence of $H$, and so $|S|_{2'}=|H|/\chi(1)$ for
some $2$-defect zero character $\chi\in\mathrm{Irr}(H)$. There are in fact
only $16$ sporadic simple groups having a $2$-defect zero
irreducible character. For such a group and such a character, there
are no $S$ satisfying the indicated condition. This concludes the
proof.
\end{proof}
Theorem \ref{thm:main4} follows from Theorem \ref{thm:Lie type} and
Propositions \ref{prop:mixed case I}, \ref{prop:mixed case II},
\ref{prop:alternating}, and \ref{prop:sporadic}.
For future work on the codegree isomorphism conjecture
(\ref{eq:HCC}), we record the following immediate consequence of
Theorem \ref{thm:main4}.
\begin{thm}\label{thm:last}
Let $S$ be a finite nonabelian simple group. Let $G$ be a minimal
counterexample to (\ref{eq:HCC}) with respect to $S$ -- that is, $G$
is minimal subject to the conditions $\normalfont{\mbox{cod}}(G)=\normalfont{\mbox{cod}}(S)$ and $G \ncong
S$. Then $G$ has a unique minimal normal subgroup $N$ and $G/N\cong
S$.
\end{thm}
\begin{proof}
Let $N$ be a maximal normal subgroup of $G$. Since
$\normalfont{\mbox{cod}}(G/N)\subseteq\normalfont{\mbox{cod}}(G)=\normalfont{\mbox{cod}}(S)$ and $G/N$ is simple, it follows
from Theorem \ref{thm:main4} that $G/N\cong S$. Furthermore, by the
minimality of $G$ as a counterexample, we have that $N$ is a minimal
normal subgroup of $G$. (If $G$ has a normal subgroup $M$ such that
$M<N$, then $\normalfont{\mbox{cod}}(G/N)\subseteq \normalfont{\mbox{cod}}(G/M)\subseteq \normalfont{\mbox{cod}}(G)$, forcing
$\normalfont{\mbox{cod}}(G/M)=\normalfont{\mbox{cod}}(S)$.) Also, $N$ is the unique minimal normal
subgroup of $G$ since, otherwise, $G=S\times S$, which violates the
assumption $\normalfont{\mbox{cod}}(G)=\normalfont{\mbox{cod}}(S)$.
\end{proof}
We conclude the paper with a couple of remarks. First, the group
pseudo-algebra $C(G)$ seems to better distinguish finite groups than
the usual complex group algebra $\mathbb{C}G$. For instance, while
any two abelian groups $A$ and $B$ of the same order have the same
complex group algebra $\mathbb{C}A=\mathbb{C}B$, it was shown in
\cite{mor23} that $A\cong B$ if and only if $C(A)=C(B)$. It has even
been speculated that a finite group $G$ and an abelian group $A$ are
isomorphic if and only if $C(G)= C(A)$. This, if true, would
indicate that abelian groups have very distinctive character
codegrees (counting multiplicities). Theorem \ref{thm:main1} shows
that simple groups indeed have very distinctive codegrees.
Our results are likely to remain true for quasi and/or almost simple
groups. However, at the time of this writing, we do not see yet a
uniform proof for these larger families of groups as the one
presented in this paper.
|
1,314,259,995,784 | arxiv | \section{Introduction}
Let $S=G/P$ be a rational homogeneous manifold associated to a simple root $\alpha_k$.
The identity component $\operatorname{Aut}_0(S)$ of the automorphism group of $S=G/P$
is equal to $G$ excepting the cases where
$(G, \{\alpha_k\})$ is $(B_{\ell}, \{\alpha_{\ell}\})$,
$(C_{\ell}, \{\alpha_1\})$ or $(G_2, \{\alpha_1\})$.
In these cases, we will think of $S=G/P$ as a rational homogeneous manifold $G'/P'$ with $\operatorname{Aut}_0(S)=G' \supsetneq G$.
The ample generator of the Picard group of $S$ induces a $G$-equivariant embedding of $S$ into a projective space.
Under the action of a Borel subgroup of $G$, $S$ has only finitely many orbits.
These orbits give rise to a cell decomposition of $S$, so that the homology space of $S$ is generated freely by the homology classes of their closures, Schubert varieties. In particular, the homology class of a (complex) subvariety of $S$ is a linear combination of the homology classes of Schubert varieties with nonnegative coefficients.
Homogeneous submanifolds associated to subdiagrams of the marked Dynkin diagram of $S$ are smooth Schubert varieties of $S$, and these are all smooth Schubert varieties when $S$ is associated to a long root (Proposition 3.7 of \cite{HoM}). They are rigid except for certain linear spaces $S_0$ in a rational homogeneous manifold $S$ associated to a short root.
\begin{theorem} [Theorem 1.1 of \cite{HoM}] \label{known result I} Let $S=G/P$ be a rational homogeneous manifold associated to a simple root and let $S_0=G_0/P_0$ be a homogeneous submanifold associated to a subdiagram $\mathcal D(S_0)$ of the marked Dynkin diagram $\mathcal D(S)$ of $S$.
Then any subvariety of $S$ having the same homology class as $S_0$ is induced by the action of $\operatorname{Aut}_0(S)$,
excepting when $(S, S_0)$ is given by
\begin{enumerate}
\item [\rm (a)] $S=(C_{n}, \alpha_k)$, $S_0=\mathbb P^{b-k }$, $ \Lambda=\{ \alpha_{k-1}, \alpha_b \}$, $ 2 \leq k < b \leq n ${\rm ;}
\item [\rm (b)] $S=(F_4, \alpha_3)$, $S_0=\mathbb P^3$ or $\mathbb P^1$, $\Lambda=\{ \alpha_1, \alpha_4\}$ or $ \{ \alpha_2, \alpha_4\}${\rm ;}
\item [\rm (c)] $S=(F_4, \alpha_4)$, $ S_0=\mathbb P^2 $ or $\mathbb P^1$, $\Lambda =\{\alpha_2\}$ or $ \{ \alpha_3 \}$
\end{enumerate}
\noindent
where $\Lambda$ denotes the set of simple roots in $\mathcal D(S) \backslash \mathcal D(S_0)$ which are adjacent to the subdiagram $\mathcal D(S_0)$.
\end{theorem}
On the other hand there are non-homogeneous smooth Schubert varieties when $S$ is associated to a short root. For example, an odd symplectic Grassmannian in the symplectic Grassmannian $Gr_{\omega}(k,V)$, which was introduced in (\cite{Mi}), is a smooth Schubert variety but is not homogeneous.
Here, $(V, \omega)$ is a complex vector space of dimension $2n$ with a symplectic form $\omega$ and $Gr_{\omega}(k,V)$ is the variety consisting of $\omega$-isotropic $k$-subspaces of $V$.
Fix an isotropic flag $F_{\bullet}:F_0 \subsetneq F_1 \subsetneq \dots \subsetneq F_{2n} =V$. The subvariety $Gr_{\omega}(k,V;F_a, F_{2n-1-a})$ of $Gr_{\omega}(k,V)$ consisting of $\omega$-isotropic subspaces of $V$, which contain $F_a$ and which are contained in $F_{2n-1-a}$, is called an odd symplectic Grassmannian. A smooth Schubert variety of the symplectic Grassmannian is either a homogeneous submanifold associated to a subdaigram of the marked Dynkin diagram of $Gr_{\omega}(k,V)$, an odd symplectic Grassmannian, or a linear space (Theorem 1.2 of \cite{HoC}). Furthermore, an odd symplectic Grassmannian $S_0=Gr_{\omega}(k,V; F_a, F_{2n-1-a})$ for
$0 \leq a \leq k-2$ is rigid, in the same sense as in Theorem \ref{known result I}, that is, any subvariety of $S=Gr_{\omega}(k,V)$ having the same homology class as $S_0$ is induced by the action of $\operatorname{Aut}_0(S)=\mathbb P Sp (V, \omega)$. (Theorem 1.2 of \cite{HoM}). \\
In this paper we will extend these results to other pair $(S,S_0)$ consisting of a rational homogeneous manifold $S$ associated to a short root and a smooth Schubert variety $S_0$ of $S$. For the history and background of this kind of rigidity problem, see \cite{HoM}. Linear spaces of $S$ are classified in (\cite{LM}): a connected component of the space of linear spaces in $S$ corresponds to a linear Schubert variety of $S$. Some connected components have more than one $G$-orbits, i.e., for some linear Schubert varieties of $S$ there is a deformation in $S$ which is not obtained by the action of $G$ (For details see \cite{LM}).
From now on, we will focus on non-linear smooth Schubert varieties.
\begin{theorem} \label{theorem main}
Let $S=G/P$ be a rational homogeneous manifold of type $(F_4, \alpha_3)$ or of type $(F_4, \alpha_4)$. Then a non-linear smooth Schubert variety $S_0$ of $S$ is either a homogenous submanifold associated to a subdiagram of the Dynkin diagram of $S$ or a horospherical variety embedded into $S$ of the following form:
\begin{enumerate}
\item[\rm(1)] $S_0=(C_2, \alpha_2, \alpha_1)$ and $S=(F_4, \alpha_3)$;
\item[\rm(2)] $S_0 =(B_3, \alpha_2, \alpha_3)$ and $S=(F_4, \alpha_3)$.
\end{enumerate}
Furthermore,
any subvariety of $S$ having the same homology class as $S_0$ is induced by the action of $\operatorname{Aut}_0(S)$.
\end{theorem}
Together with Theorem 1.2 of \cite{HoC} and Theorem 1.2 of \cite{HoM} for the case where $S$ is the symplectic Grassmannian $Gr_{\omega}(k,V)$, which are explained in the above, we get the following result.
\begin{theorem} \label{Theorem 2} Let $S=G/P$ be a rational homogeneous manifold associated to a short root. Then a non-linear smooth Schubert variety $S_0$ of $S$ is either a homogenous submanifold associated to a subdiagram of the Dynkin diagram of $S$ or a horospherical variety embedded into $S$ of the following form:
\begin{enumerate}
\item[\rm(1)] $S_0=(C_m, \alpha_{i+1}, \alpha_i)$ and $S=(C_n, \alpha_k)$, $2 \leq m \leq n$ and $1 \leq i \leq m-1$ and $n-k=m-i$;
\item[\rm(2)] $S_0=(C_2, \alpha_2, \alpha_1)$ and $S=(F_4, \alpha_3)$;
\item[\rm(3)] $S_0 =(B_3, \alpha_2, \alpha_3)$ and $S=(F_4, \alpha_3)$.
\end{enumerate}
\noindent
In particular, any smooth Schubert varieties of $S$ is linear when $S$ is of type $(F_4, \alpha_4)$.
\end{theorem}
\begin{theorem} \label{Theorem 1} Let $S=G/P$ be a rational homogeneous manifold associated to a short simple root and let $S_0$ be a non-linear smooth Schubert variety of $S$. Then any subvariety of $S$ having the same homology class as $S_0$ is induced by the action of $\operatorname{Aut}_0(S)$.
\end{theorem}
For notations, see Section \ref{Section horospherical}. For example, $(C_m, \alpha_{i+1}, \alpha_i)$ denotes the odd symplectic Grassmannian consisting of isotropic $(i+1)$-subspaces of $(\mathbb C^{2m+1}, \omega)$ and $(C_n, \alpha_k)$ denotes the symplectic Grassmannian $Gr_{\omega}(k, \mathbb C^{2n})$ consisting of isotropic $k$-subspaces of $(\mathbb C^{2n}, \omega)$. \\
We remark that Richmond-Slofstra \cite{RiSl} obtained the same classification of smooth Schubert varieties of rational homogeneous manifold of Picard number one by using a combinatorial method developed by \cite{BiPo} (Grassmnnian Schubert varieties in \cite{RiSl} are Schubert varieties in rational homogeneous manifolds of Picard number one in our paper). We reprove it by a geometric method (Proposition 3.7 of \cite{HoM} and Theorem \ref{Theorem 2}). One advantage of our geometric method is that it gives not just classification but also their rigidity (Theorem \ref{Theorem 1}) at the same time. Moreover, we describe smooth Schubert varieties of rational homogeneous manifolds of Picard number one geometrically: it is either a homogeneous submanifold associated to a subdiagram of the Dynkin diagram of $S$, a linear space, or a horospherical variety. This is not true for rational homogeneous manifolds of higher Picard number. For example, odd symplectic flag manifolds (\cite{Mi}) are smooth Schubert varieties of symplectic flag manifolds but they are not horospherical. \\
The remainder of this paper is organized as follows. In Section 2 we give basic definitions and properties of Schubert varieties, horospherical varieties. We also explain our main tool, varieties of minimal rational tangents.
We will restrict ourselves to the case when $S$ is of type $(F_4, \alpha_3)$ or of type $(F_4, \alpha_4)$. In Section 3 we classify smooth Schubert varieties of the rational homogeneous manifold of type $(F_4, \alpha_3)$ and prove their rigidity, and we complete the proof of Theorem \ref{theorem main} in the last section by showing that any smooth Schubert varieties of the rational homogeneous manifold of type $(F_4, \alpha_4)$ is linear.
\section{Preliminaries}
\subsection{Schubert varieties}
Let $G$ be a connected semisimple algebraic group over $\mathbb C$.
Take a Borel subgroup $B$ of $G$ and a maximal torus $T$ in $B$. Denote by $\Delta^+$ the system of positive roots of $G$ and by $\Phi=\{ \alpha_1, \cdots, \alpha_{\ell}\}$ the system of simple roots of $G$. For a root $\alpha$, write $\alpha = \sum_{i=1}^{\ell} n_i(\alpha)\alpha_i$.
Let $\frak t$ be the Lie algebra of $T$.
To each simple root $\alpha_k$ we associate a parabolic subgroup $P$ of $G$, whose Lie algebra $\mathfrak p$ is given by $\mathfrak p = \mathfrak t + \sum_{n_k(\alpha) \geq 0} \frak g_{\alpha}$.
The reductive part of $\frak p$ is given by $ \mathfrak t + \sum_{n_k(\alpha)=0}\frak g_{\alpha}$ and the nilpotent part of $\frak p$ is given by $ \sum_{n_k(\alpha) >0}\frak g_{\alpha}$.
The homogeneous manifold $S=G/P$ is called the rational homogeneous manifold associated to $\alpha_k$. We will denote it by $(G, \alpha_k)$.
Let $\mathcal W$ be the Weyl group of $G$. For $w \in \mathcal W$, set $\Delta(w)=\{ \beta \in \Delta^+ : w(\beta) \in -\Delta^+\}$.
Define a subset $\mathcal W^P$ of $\mathcal W$ by
$\mathcal W^P:=\{ w \in \mathcal W: \Delta(w) \subset \Delta(U_P)\},$ where $\Delta(U_P)=\{
\alpha \in \Delta^+: n_{\alpha_k}(\alpha) >0\}$. Then we have a cell decomposition
$S=\coprod _{w \in \mathcal W^P} B.x_w $,
where $x_w=wP, w \in \mathcal W^P$ are $T$-fixed points in $S$.
For each $w \in \mathcal W^P$, the closure $S(w)$ of $B. x_w$ is called {\it the Schubert variety of type} $w$.
\subsection{Horospherical varieties}
\label{Section horospherical}
Let $L$ be a connected reductive algebraic group.
Let $H$ be a closed subgroup of $L$.
A homogeneous space $L/H$ is said to be {\it horospherical} if $H$ contains the unipotent radical of a Borel subgroup of $L$. In this case, the normalizer $N_L(H)$ of $H$ in $L$ is a parabolic subgroup $P$ of $L$ and $P/H$ is a torus $(\mathbb C^{\times})^r$. Thus there is a $(\mathbb C^{\times})^r$-bundle structure on $L/H$ over $L/P$. A normal $L$-variety is called {\it horospherical} if it contains an open dense $L$-orbit isomorphic to a horospherical homogeneous space $L/H$.
For a dominant weight $\varpi$ of $L$ let $V_L(\varpi)$ denote the irreducible representation space of $L$ with highest weight $\varpi$. Fix a Borel subgroup of $L$.
Let $\{\alpha_1, \cdots, \alpha_n\}$ be the system of simple roots of $L$ and let $\{ \varpi_1, \cdots, \varpi_{n}\}$ be the system of fundamental weights of $L$. Take a highest weight vector $v_i$ in $V_L(\varpi_i)$ for $i=1, \cdots n$. Then the $L$-orbit of $[v_i]$ in $\mathbb P(V_L(\varpi_i))$ is the rational homogeneous variety of type $(L, \alpha_i)$.
For $i \not=j$, the closure of the $L$-orbit of $[v_i+v_j]$ in $\mathbb P(V_L(\varpi_i) \oplus V_L(\varpi_j))$ is a horospherial $L$-variety (Proposition 2.1 of \cite{HoH}).
We will denote the closure of $L.[v_i+v_j]$ in $\mathbb P(V_L(\varpi_i) \oplus V_L(\varpi_j))$ by
$(L, \alpha_i, \alpha_j)$.
It has three $G$-orbits: one open orbit $L.[v_i + v_j]$ and two closed orbits, $ L.[v_i]$ and $ L.[v_j]$. For more details on horospherical varieties see \cite{Pa}.
\begin{proposition} [Proposition 1.8 and Proposition 1.9 and Proposition 1.10 of \cite{Pa}] \label{homogeneous horospherical varieties} $\,$
\begin{enumerate}
\item The horospherical variety $(A_n, \alpha_1, \alpha_n)$ with $n \geq 2$ is isomorphic to the rational homogeneous manifold of type $ (D_{n+1}, \alpha_1)$.
\item The horospherical variety $(A_n, \alpha_i, \alpha_{i+1})$ with $n \geq 3$ and $1 \leq i \leq n-1$ is isomorphic to the rational homogeneous manifold of type $ (A_{n+1}, \alpha_{i+1})$.
\item The horospherical variety $(D_n, \alpha_{n-1}, \alpha_{n})$ with $n \geq 4$ is isomorphic to the rational homogeneous manifold of type $(D_{n+1}, \alpha_n)=(B_n, \alpha_n)$.
\end{enumerate}
\end{proposition}
\begin{proposition}[Theorem 0.1 and Theorem 1.7 of \cite{Pa}] \label{classification nonhomogeneous}
Let $L$ be a connected reductive algebraic group. Let $X$ be a smooth projective horospherical $L$-variety of Picard number one.
Then $X$ is either homogeneous or one of the following.
\begin{enumerate}
\item [{\rm(1)}] $(B_n, \alpha_{n-1}, \alpha_n)$, $n \geq 3${\rm;}
\item [{\rm(2)}] $(B_3, \alpha_1, \alpha_3)${\rm;}
\item [{\rm(3)}] $(C_n, \alpha_{i+1}, \alpha_i)$, $n\geq 2$ and $i \in \{ 1,2, \cdots, n-1\}${\rm;}
\item [{\rm(4)}] $(F_4, \alpha_2, \alpha_3)${\rm;}
\item [{\rm(5)}] $(G_2, \alpha_2, \alpha_1)$.
\end{enumerate}
\end{proposition}
In Proposition 4.1 of \cite{HoH}, we describe an equivariant embedding of a smooth horospherical variety of Picard number one into a rational homogeneous manifold of Picard number one as a linear section. Among them $(C_m, \alpha_{i+1}, \alpha_{i})$ is a smooth Schubert variety of $(C_{m+1}, \alpha_{i+1})$. We have two more smooth Schubert varieties as follows.
\begin{proposition} \label{linear sections} Let $S$ be a rational homogeneous manifold of type $(F_4, \alpha_3)$ and let $S_0$ be one of the following horospherical varieties:
\begin{enumerate}
\item $S_0=(B_3, \alpha_{2}, \alpha_3)${\rm;}
\item $S_0=(C_2, \alpha_{2}, \alpha_1)${\rm;}
\end{enumerate}
\noindent Then there is an embedding of $S_0$ into $S$ as a smooth Schubert variety.
\end{proposition}
\begin{proof}
We recall how to embed $X=(B_3, \alpha_2, \alpha_3)$ into $S=(F_4, \alpha_3)$. For details see \cite{HoH}. The rational homogeneous manifold $\mathbb S=(B_3, \alpha_3)$ can be embedded into the variety $\mathcal C_x(\mathcal S)$ of minimal rational tangents of $\mathcal S=(F_4, \alpha_4)$ at $x \in \mathcal S$, and the isotropy group $\mathcal P$ of $\mathcal G=Aut(\mathcal S)$ at $x$ acts transitively on $\mathbb S$. Thus the cone $\widehat{\mathbb S}$ over $\mathbb S$ with vertex $x$ can be embedded into $\mathcal S$ as a linear section and $\mathcal P$ stabilizes $\widehat{\mathbb S}$. Furthermore, $X$ is the Fano variety $F_1(\widehat{\mathbb S})$ of lines lying on the cone $\widehat{\mathbb S}$ over $\mathbb S$, and $S$ can be embedded into the Fano variety $F_1(\mathcal S)$ of lines lying on $\mathcal S$. The embedding of $X$ into $S $ is induced by the embedding of $\widehat{\mathbb S}$ into $\mathcal S$. Therefore, $\mathcal P$ stabilizes $X$.
Since the stabilizer of $X$ in $G=\operatorname{Aut}(S)$ contains a Borel subgroup of $G$ and $X$ is irreducible, $X$ is a Schubert variety (Proposition 2.1 in \cite{HoM}). This completes the proof for the case (1).
For (2) just embed $S_0$ into a rational homogeneous manifold $S_1$ of type $(C_3, \alpha_2)$ and consider the embedding of $S_1$ into $S$ as a homogeneous submanifold associated to a subdiagram of the marked Dynkin diagram of $S$.
\end{proof}
\subsection{Varieties of minimal rational tangents} \label{section vmrt of Schubert varieties}
Let $X$ be a uniruled projective manifold with an ample line bundle $\mathcal L$. By a (parameterized) {\it rational curve} on $X$ we mean a nonconstant holomorphic map $f: \mathbb P^1 \rightarrow X$.
A rational curve $f$ is said to be {\it free} if the pull-back $f^*TX$ of the tangent bundle $TX$ of $X$ on $\mathbb P^1$ is semipositive. A free rational curve $f$ such that the degree $f^*\mathcal L$ is minimum among all free rational curves is called a {\it minimal rational curve}.
Let $\mathcal H$ be a connected component of $\text{Hom}(\mathbb P^1, X)$ containing a minimal rational curve and let $\mathcal H^0$ be the subset consisting of free rational curves. The quotient space $\mathcal K=\mathcal H^{0}/\operatorname{Aut}(\mathbb P^1)$ of (unparameterized) minimal rational curves is called
a {\it minimal rational component}.
Fix a minimal rational component $\mathcal K$. When we say a minimal rational curve we mean a rational curve belonging to $\mathcal K$. For a general $x \in X$ the space $\mathcal K_x$ of minimal rational curves passing through $x$ is a projective manifold. Define a rational map $\Psi$ from $\mathcal K_x$ to $\mathbb P(T_xX)$ by sending a minimal rational curve immersed at $x$ to the tangent line at $x$. The strict transformation $\mathcal C_x(X)$ of $\Psi$ is called the {\it variety of minimal rational tangents of} $X$ {\it at} $x$. The union of $\mathcal C_x(X)$ over general $x \in X$ forms a fiber bundle $\mathcal C(X)$ over $X$. The variety of minimal rational tangents was introduced in \cite{HwM99} to study geometric structures on uniruled projective manifolds. For more details on the variety of minimal rational tangents and its applications to the study of geometric structures on uniruled projective manifolds, see \cite{Mk16}, the most recent survey. \\
Let $S=G/P$ be a rational homogeneous manifold associated to a simple root. Then the Picard number of $S$ is one and the ample generator $\mathcal L$ of the Picard group defines a $G$-equivariant embedding of $S$ into the projective space $\mathbb P(H^0(S, \mathcal L)^*)=\mathbb P^N$. Lines $\mathbb P^1$ in $\mathbb P^N$ lying on $S$ are minimal rational curves, and we will choose the family $\mathcal K$ of lines lying on $S$ as our minimal rational component, so that the variety $\mathcal C_x(S)$ of minimal rational tangents of $S$ at any $x$ in $S$ is defined by the space of all tangent directions of lines lying on $S$ passing through $x$. If $S$ is associated to a long root, then $G$ acts on $\mathcal K$ transitively. If $S$ is associated to a short root, then $\mathcal K$ has two $G$-orbits. In any case, by a general line we mean a line corresponding to a point in the open $G$-orbit in $\mathcal K$, and by a general point in $\mathcal C_x(S)$ we mean the tangent direction of a general line.
Let $\mathcal C_x(S)^{gen}$ denote the subvariety of $\mathcal C_x(S)$ consisting of the tangent directions of general lines in $S$. For an explicit description of the variety $\mathcal C_x(S)$ of minimal rational tangents of $S$ and its application to the deformation rigidity of $S$, see \cite{HwM02}, \cite{HwM04b}, and \cite{HwM05}.
Let $S_0$ be a Schubert variety of $S$. By Proposition 3.1 of \cite{HoM}, $S_0$ is covered by lines of $S$ lying on $S_0$ and is of Picard number one (the same arguments in the proof work for the case when $S_0$ is singular). Consider the family $\mathcal K_0$ of all lines lying on $S_0$.
The stabilizer $Stab_G(S_0)$ of $S_0$ in $G$ is a parabolic subgroup of $G$. By a {\it general} point in $S_0$ we mean a point $x$ in the open orbit of $Stab_G(S_0)$ in $S_0$. In particular, the base point of $S_0$ is a general point.
For a general point $x$ of $S_0$, define the variety $\mathcal C_x(S_0)$ of minimal rational tangents of $S_0$ at $x$ by the set of tangents directions of lines lying on $S_0$ passing through $x$. Then $\mathcal C_x(S_0)=\mathcal C_x(S) \cap \mathbb P(T_xS_0)$ (Proposition 3.1 of \cite{HoM}). By a {\it general} point of $\mathcal C_x(S_0)$ we mean a point in $\mathcal C_x(S_0) \cap \mathcal C_x(S)^{gen}$. \\
Let $x_0$ be the base point of $S$ at which the isotropy group of $G$ is $P$.
Let $B$ be a Borel subgroup of $G$ contained in $P$ and $T$ be a maximal torus of $B$. Let $L$ be the reductive part of $P$ containing $T$.
\begin{proposition} [p.352 of \cite{HoM}, Proposition 4.1 of \cite{HoC}] \label{necessary conditions} Let $S=G/P$ be a rational homogeneous manifold associated to a simple root and let $S_0$ be a Schubert variety. Let $x=gx_0$ be a general point of $S_0$ and
let $(L \cap B)_x$ denote the conjugate $g(L \cap B)$ of the Borel subgroup $L \cap B$ of $L$. Then
\begin{enumerate}
\item $\mathcal C_x(S_0)$ is invariant under the action of $(L \cap B)_x$.
\item If $S_0$ is smooth, then $\mathcal C_x(S_0)$ is smooth and is the closure of a $(L \cap B)_x$-orbit in $\mathcal C_x(S)$.
\end{enumerate}
\end{proposition}
\begin{proof} For the base point $x_w=w.x_0$ of $S_0$, (1) and (2) follows from the arguments in p.352 of \cite{HoM} or Proposition 4.1 of \cite{HoC}.
It remains to show (1) for a general point $x$ of $S_0$, i.e., for any point in the orbit of $Stab_G(S_0)$ of $x_w $. By arguments in p.352 of \cite{HoM}, $\mathcal C_{x_w}(S_0)$ is invariant under the action of $(L\cap B)_{x_w} =w(L \cap B)$. Then $\mathcal C_{gx_w}( gS_0)$ is invariant under the action of $gw(L \cap B)$ for any $g \in G$. In particular, for $b \in Stab_G(S_0)$, $\mathcal C_{bx_w}( S_0)=\mathcal C_{bx_w}(b S_0)$ is invariant under the action of $bw(L \cap B)$. Therefore, for a general point $x=bx_w$ of $S_0$, $\mathcal C_x(S_0)$ is invariant under the action of $(L \cap B)_x=bw(L \cap B)$.
\end{proof}
We will consider the following two conditions (I), (II) on the variety $ \mathcal C_x(S_0)$ of minimal rational tangent of the `model' Schubert variety $S_0$:
\begin{enumerate}
\item [\rm(I)] at a general point $\alpha \in \mathcal C_x(S_0)$, for any $h \in P_x$ sufficiently
close to the identity element $e\in P_x$ and satisfying
$T_{\alpha}\left(h\mathcal C_x(S_0) \right) =
T_{\alpha}\left(\mathcal C_x(S_0) \right)$ we must have $h\mathcal C_x(S_0) = \mathcal C_x(S_0)$;
\item [\rm(II)] any local deformation of $ \mathcal C_x(S_0)$ in $\mathcal C_x( S )$ is induced by the action of $P_x$.
\end{enumerate}
\begin{proposition} [Proposition 3.2 of \cite{HoM}] \label{general case - inductive step}
Let $S=G/P$ be a rational homogeneous manifold associated to a simple root, and $S_0 $
be a smooth Schubert variety of $S$.
Assume that $\mathcal C_x(S_0)$ satisfies {\rm(I)} and {\rm(II)}
at a general point $x \in S_0$.
Then, the following holds true.
\begin{enumerate}
\item[\rm (1)]
If a smooth subvariety $Z$ of $S$ is uniruled by lines of $S$ lying on $Z$ and contains $x$ as a general point with $\mathcal C_x(Z) = \mathcal C_x( S_0)$, then $S_0$ is contained in $Z$.
\item[\rm (2)]
Any local deformation of $S_0$ in $S$ is induced by the action of $G$.
\end{enumerate}
\end{proposition}
\section{$(F_4, \alpha_3)$-case}
Let $S=G/P$ be the rational homogeneous manifold of type $(F_4, \alpha_3)$.
Let $o \in S$ be the base point. Then
$\mathcal C_o(S)$ is the projectivization of the cone
$$\{ e \otimes q + (f\wedge f') \otimes q^2 : e \wedge f \wedge f'=0, e, f, f' \in E, q \in Q \} $$
in $(E \otimes Q) \oplus (\wedge^2 E \otimes S^2 Q)$, where $E$ is a complex vector space of dimension 3 and $Q$ is a complex vector space of dimension 2 (see \cite{HwM04b}).
Via the map $[e \otimes q + (f\wedge f') \otimes q^2] \in \mathcal C_0(S) \mapsto [q] \in \mathbb P(Q)$, $\mathcal C_o(S)$ can be think of as a fiber bundle over $\mathbb P(Q)=\mathbb P^1$ with fiber isomorphic to the smooth quadric $\mathbb Q^4 \subset \mathbb P(E \oplus \wedge^2E)$.
Let $\rho: P \rightarrow GL(T_{o}S)$ be the isotropy representation.
Then $\rho(P)$ is $(SL(E) \times SL(Q))\ltimes (E^* \otimes Q^*)$, where $E^* \otimes Q^*$ acts on $E \otimes Q$ trivially and maps $\wedge^2 E \otimes S^2 Q$ to $E \otimes Q$
If $S_0$ is the homogeneous submanifold associated to the subdiagram of of type $(C_3, \alpha_2)$ of $S$, then $\mathcal C_x(S_0)$ is the linear section of $\mathcal C_x(S)$ by $\mathbb P((F_2 \otimes Q) \oplus (F_2^{\perp} \otimes S^2Q))$, where $F_2$ is a subspace of $E$ of dimension $2$, and is isomorphic to $\mathbb P(\mathcal O(-1)^2 \oplus \mathcal O(-2))$.
If $S_0$ is the horospherical variety $(B_3, \alpha_2, \alpha_3)$ in $S$, then $\mathcal C_x(S_0)$ is the linear section of $\mathcal C_x(S)$ by $\mathbb P((F_1 \otimes Q) \oplus (F_1^{\perp} \otimes S^2Q))$, where $F_1$ is a subspace of $E$ of dimension $1$, and is isomorphic to $\mathbb P(\mathcal O(-1) \oplus \mathcal O(-2)^2)$.
If $S_0$ is the horospherical variety $(C_2, \alpha_{2}, \alpha_1)$ in $S$, then $\mathcal C_x(S_0)$ is the linear section of $\mathcal C_x(S)$ by $\mathbb P((e \otimes Q) \oplus (f^* \otimes S^2Q) )$, where $e \in E$ and $f^* \in E^*$ be such that $\langle e, f^* \rangle =0$, and is isomorphic to $\mathbb P(\mathcal O(-1) \oplus \mathcal O(-2))$.
\begin{lemma} \label{classification in Q4}
Let $B^1$ be a Borel subgroup of $SL(E)$.
The smooth closures of $B^1$-orbits in $\mathbb Q^4 \subset \mathbb P(E \oplus \wedge^2E) \simeq \mathbb P(E \oplus E^*)$ intersecting the open $SL(E)$-orbit are given by
$$\mathbb Q^4, \mathbb P(F_1 \oplus F_1^{\perp}),\, \mathbb P(F_2 \oplus F_2^{\perp}),\, \mathbb P(V_1 \oplus W_1)$$
where $F_i$ ($i=1,2$) is a subspace of $E$ of dimension $i$ and $F_i^{\perp}$ is the annihilator of $F_i$, and $V_1$ is a subspace of $E$ of dimension one and $W_1$ is a subspace of $V_1^{\perp}$ of dimension one.
\end{lemma}
\begin{proof}
Take a basis $\{e_1, e_2, e_3 \}$ of $E$ compatible with $B^1$.
Let $\tilde E$ be a vector space of dimension 4 containing $E$. Extend $\{e_1, e_2, e_3\}$ to a basis $\{e_1, e_2, e_3, e_4\}$ of $\tilde E$.
Recall that the isomorphism $E \oplus \wedge^2 E \rightarrow \wedge^2 \tilde E$ is given by $e + f \wedge f' \mapsto e\wedge e_4 + f \wedge f'$ and, under this isomorphism, the closure of $SL(E).[e_1 + e_1 \wedge e_2]$ in $\mathbb P(E \oplus (\wedge ^2E))$ is isomorphic to $G(2,4) \simeq \mathbb Q^4 \subset \mathbb P(\wedge^2 \tilde E) \simeq \mathbb P^5$ (Proposition \ref{homogeneous horospherical varieties}). Identifying $\wedge^2 E$ with $E^*$ and considering quadratic form on $E \oplus E^*$, we can see that the closure of $SL(E).[e_1 +e_3^*]$ in $\mathbb P(E \oplus E^*)$ is $\mathbb Q^4 \subset \mathbb P^5$.
Now $\mathbb Q^4$ has three $SL(E)$-orbits, $\mathbb P(E)$, $\mathbb P( E^*)$ and the open orbit $\mathcal O$.
The closures of $B^1$-orbits in $\mathbb Q^4$ which intersect the open orbit $\mathcal O$ are
\begin{enumerate}
\item[(a)] $ \operatorname{cl}(B_1.(e_1+e_3^*))=\mathbb Q^4$
\item[(b)] $\operatorname{cl}(B_1.(e_1+ e_2^*))$, $\operatorname{cl}(B_1.(e_2+e_3^*))$ (3-dimensional and singular)
\item[(c)] $\operatorname{cl}(B_1.(e_2 +e_1^*))=\mathbb P^2$, $\operatorname{cl}(B_1.(e_3+e_2^*))=\mathbb P^2$
\item[(d)] $\operatorname{cl}(B_1.(e_3+e_1^*))=\mathbb P^1$
\end{enumerate}
We may express $\mathbb P^2$'s in (c) as $\mathbb P(F_2 \oplus F_2^{\perp})$ and $\mathbb P(F_1 \oplus F_1^{\perp})$, where $F_i $ is a subspace of $E$ of dimension $i$ and $F_i^{\perp}$ is the annihilator of $F_i$ for $i=1,2$.
\end{proof}
The space $\mathcal K$ of $\mathbb P^2$'s in $\mathbb Q^4 \subset \mathbb P(E \oplus \wedge^2E)\simeq \mathbb P(E \oplus E^*)$ has two connected components, $\mathcal K_1$ and $\mathcal K_2$, each of which is isomorphic to $\mathbb P^3$. One of them contains $\mathbb P(E)$, and the other contains $\mathbb P(E^*)$.
\begin{lemma} \label{deformtion of P2 in Q4}
Each connected component of the space of $\mathbb P^2$'s in $\mathbb Q^4 \subset \mathbb P(E \oplus E^*)$ has two $SL(E) \ltimes E^*$-orbits: one is closed and the other is open.
\end{lemma}
\begin{proof}
Let $\mathbb P(F)$ be a $\mathbb P^2$ contained in $\mathbb Q^4$ in the same connected component as $\mathbb P(E)$. If $\mathbb P(F) \not= \mathbb P(E)$, then we have $\operatorname{dim} (\mathbb P(F) \cap \mathbb P(E))=0$, and ($\mathbb P(F) \cap \mathbb P(E^*)$ has dimension 1 or is empty). In the first case, we have
$$F=F_1 \oplus F_1^{\perp}$$ for some subspace $F_1 \subset E$ of dimension 1.
In the second case, there is a linear map $\varphi: E \rightarrow \wedge^2 E$ such that $\operatorname{dim} \text{Ker}\, \varphi =1$ and $F=F_{\varphi}$, where $F_{\varphi} \subset E$ is the graph of $\varphi$. Let $\varphi_1:E \rightarrow \wedge^2 E$ be a linear map defined by $\varphi_1(e) = e_1 \wedge e$, where $e_1$ is a basis of $\text{Ker}\, \varphi $. The condition $e \wedge \varphi(e) =0$ for any $e \in E$ implies that $\varphi $ is $\lambda \varphi_1$ for some $\lambda \in \mathbb C^{\times} =\mathbb C -\{0\}$. To see this, extend $\{e_1\}$ to a basis $\{e_1, e_2, e_3\}$ of $E$ and write
\begin{eqnarray*}
\varphi(e_2)&=& \varphi^2_{12}e_1 \wedge e_2 + \varphi^2_{23}e_2 \wedge e_3 + \varphi^2_{31} e_3 \wedge e_1 \\
\varphi(e_3)&=& \varphi^3_{12}e_1 \wedge e_2 + \varphi^3_{23}e_2 \wedge e_3 + \varphi^3_{31} e_3 \wedge e_1.
\end{eqnarray*}
From $0=e_2 \wedge \varphi(e_2)=e_3 \wedge \varphi(e_3)$ it follows that $\varphi^2_{31}=\varphi^3_{12}=0$. From $0=(e_1+ e_2) \wedge(\varphi(e_1) + \varphi(e_2))=e_1 \wedge \varphi(e_2) + e_2 \wedge \varphi(e_2) = e_1 \wedge \varphi(e_2)$
it follows that $\varphi^2_{23}=0$. Similarly, $\varphi^3_{23}=0$.
%
From $0=(e_2 + e_3) \wedge (\varphi(e_2) + \varphi(e_3))=e_2 \wedge \varphi(e_3) + e_3 \wedge \varphi(e_2)$, it follows that $e_2 \wedge (\varphi^3_{31}e_3 \wedge e_1) + e_3 \wedge (\varphi^2_{12} e_1 \wedge e_2)=0$ and thus $\varphi^3_{31}=-\varphi^2_{12}$. Put $\lambda:=\varphi^2_{12}$. Then $\varphi = \lambda \varphi_1$ and
$F_{\varphi}$ is spanned by $$e_1,\,\, e_2 + \lambda e_1 \wedge e_2,\,\, e_3 -\lambda e_3 \wedge e_1.$$
%
We remark that $\lim_{\lambda \rightarrow 0}F_{\lambda \varphi_1} =E$ and $\lim_{\lambda \rightarrow \infty} F_{\lambda \varphi_1} = F_1 \oplus F_1^{\perp}$, where $F_1 =\text{Ker}\, \varphi_1$.
The action of $E^*$ on $E \oplus \wedge^2 E$ is given by zero on $E$ and by the interior product on $\wedge^2E$. For example, $ce_1^*.(e_1 \wedge e_2) = ce_2 + e_1 \wedge e_2$ and $ce_1^*.(e_3 \wedge e_1) = - ce_3 + e_3 \wedge e_1$, where $c \in \mathbb C$. Hence, there is an element $e \in E^*$ which maps $\mathbb P(F_1 \oplus F_1^{\perp})$ to $\mathbb P(F_{\varphi})$, while $\mathbb P(E)$ is fixed by the action of $SL(E) \ltimes E^*$. Therefore, the connected component of the space of $\mathbb P^2$'s in $\mathbb Q^4$ containing $\mathbb P(E)$ has two $SL(E) \ltimes E^*$-orbits, the orbit of $\mathbb P(E)$ (which is a one point set $\{\mathbb P(E) \}$) and the orbit of $\mathbb P(F_1 \oplus F_1^{\perp})$, where $F_1$ is a subspace of $E$ of dimension one. The first one is closed and the second one is open. \\
Let $\mathbb P(F)$ be a $\mathbb P^2$ contained in $\mathbb Q^4$ in the same connected component as $\mathbb P(E^*)$. If $\mathbb P(F) \not=\mathbb P(E^*)$, by the same arguments as in the previous case, $F$ is either $F_2 \oplus F_2^{\perp}$ for some subspace $F_2$ of $E$ of dimension $2$ or is spanned by
$$e_2 \wedge e_3 + \lambda e_2,\,\, e_3 \wedge e_1 -\lambda e_1,\,\, e_1 \wedge e_2$$
for some basis $\{e_1, e_2, e_3\}$ of $E$.
Subspaces $F$ of the first form are fixed by the action of $E^*$.
For each subspace $F$ of the second form, there is an element $e^* \in E^*$ which maps $E^* $ to $\langle e_2 \wedge e_3 + \lambda e_2, e_3 \wedge e_1 -\lambda e_1, e_1 \wedge e_2 \rangle$ (just think of $E^*$ as $\langle e_2\wedge e_3, e_3\wedge e_1, e_1 \wedge e_2 \rangle$). Therefore, the connected component of the space of $\mathbb P^2$'s in $\mathbb Q^4$ containing $\mathbb P(E^*)$ has two $SL(E) \ltimes E^*$-orbits, the orbit of $\mathbb P(E^*)$ and the orbit of $\mathbb P(F_2 \oplus F_2^{\perp})$, where $F_2$ is a subspace of $E$ of dimension two. The first one is open and the second one is closed.
\end{proof}
\noindent {\bf Remark.} Let $\mathcal Y_0 =\mathbb P(V_1 \oplus W_1)$ where $V_1 \subset E$ is a subspace of dimension one and $W_1 \subset V_1^{\perp}$ is a subspace of dimension one. Since any line $\mathbb P^1$ in $\mathbb Q^4$ is the intersection of two $\mathbb P^2$'s, contained in different connected components of $\mathcal K$, any local deformation $\mathcal Y_t$ of $\mathcal Y_0$ is the intersection $\mathcal X_{1,t} \cap \mathcal X_{2,t}$, where $\mathcal X_{i,t}$ belongs to in $\mathcal K_i$ for $i=1,2$. In the proof of Proposition \ref{deformtion of P2 in Q4}, we prove that, up to the action of $SL(E) \ltimes E^*$, $\mathcal X_{1,t}=\mathbb P(V_1 \oplus V_1^{\perp})$. Since $\mathcal Y_t=\mathcal X_{1,t} \cap \mathcal X_{2,t}$ is $\mathbb P^1$, $\mathcal X_{2,t}$ is of the form $\mathbb P(F_{2,t} \oplus F_{2,t}^{\perp})$, where $F_{2,t}$ is a subspace of $E$ of dimension two, and $\mathcal Y_t$ is of the form $\mathbb P(V_1 \oplus W_{1,t})$, where $W_{1,t} $ is a subspace of $ V_1^{\perp}$ of dimension one. Therefore, up to the action of $SL(E) \ltimes E^*$ again, $\mathcal Y_t =\mathcal Y_0$. \\
\begin{proposition} \label{classification of vmrt}
Let $S=G/P$ be the rational homogeneous manifold of type $(F_4, \alpha_3)$.
Varieties of minimal rational tangents of smooth Schubert varieties of $S$ are of the following forms: \\
\begin{tabular} {|c|c|}
\hline
$\mathcal C_o(S) \cap \mathbb P((E\otimes q) \oplus (E^* \otimes q^2))$ \,\,&\,\, $\mathcal C_o(S)$ \\ [3pt]
\hline
$\mathbb P(E \otimes q)$ \,\,&\,\, $\mathcal C_o(S) \cap \mathbb P((F_1 \otimes Q) \oplus (F_1^{\perp}\otimes S^2Q))$ \\[3pt]
$\mathbb P((F_2 \otimes q) \oplus (F_2^{\perp}\otimes q^2))$ & $\mathcal C_o(S) \cap \mathbb P((F_2 \otimes Q)\oplus (F_2^{\perp}\otimes S^2Q))$ \\[3pt]
\hline
$\mathbb P(F_2 \otimes q)$ \,\,&\,\, $\mathcal C_o(S) \cap \mathbb P((V_1 \otimes Q) \oplus (W_1\otimes S^2Q))$ \\[3pt]
\hline
$\mathbb P(F_1 \otimes q)$ \,\,& \,\,$\mathbb P(e \otimes Q)$ \\[3pt]
\hline
\end{tabular}
%
\vskip 10 pt \noindent
where $F_i$ is a subspace of $E$ of dimension $i$ for $i=1,2$ and $V_1$ is a subspace of $E$ of dimension one and $W_1$ is a subspace of $V_1^{\perp}$ of dimension one and $e \in E$ and $q \in Q$.
The corresponding smooth Schubert varieties are \\
\begin{tabular}{|c|c|}
\hline
$(B_3, \alpha_3)$ \,\,&\,\, $S$ \\[3pt]
\hline
$\mathbb P^3_{A_3}$ \,\,&\,\, $(C_3,\alpha_2)$ \\[3pt]
$ (B_2, \alpha_2)$ \,\,&\,\, $(B_3, \alpha_2, \alpha_3)$ \\[3pt]
\hline
$\mathbb P^2_{A_2}$ \,\,&\,\, $(C_2, \alpha_2, \alpha_1)$\\[3pt]
\hline
$ (A_1, \alpha_1)$ \,\,&\,\, $ (A_2, \alpha_1)$ \\[3pt]
\hline
\end{tabular}
\vskip 10 pt \noindent
where $(L, \alpha_i)$ denotes the homogeneous submanifold of $S$ associated to a subdiagram of type $(L, \alpha_i)$, and $(L, \alpha_i, \alpha_j)$ denotes the horospherical variety embedded in $S$ as in Proposition \ref{linear sections}, and $\mathbb P_{A_3}^3$ and $\mathbb P_{A_2}^2$ denote $\mathbb P^3$ and $\mathbb P^2$ in $(B_3, \alpha_3)$ which are not associated to a subdiagram of the Dynkin diagram of $S$.
\end{proposition}
\begin{proof} Recall that the variety
$\mathcal C_o(S)$ of minimal rational tangents of $S=G/P$ is the projectivization of the cone
$$\{ e \otimes q + f^* \otimes q^2 : \langle e , f^*\rangle=0, e \in E, f^* \in E^*, q \in Q \} $$
in $(E \otimes Q) \oplus ( E^* \otimes S^2 Q)$, where $E$ is a complex vector space of dimension 3 and $Q$ is a complex vector space of dimension 2, and that the fiber over $[q] \in \mathbb P(Q)$ is $\{ e \otimes q + f^* \otimes q^2 : \langle e ,f^*\rangle=0, e \in E, f^* \in E^*\} \simeq \mathbb Q^4$.
The semisimple part $L = L^1 \times L^2$ of $P$ is $SL(E) \times SL(Q)$ and $L \cap B$ is $B^1 \times B^2$, where $B^1$ is a Borel subgroup of $SL(E)$ and $B^2$ is a Borel subgroup of $SL(Q)$.
Let $S_0$ be a smooth Schubert variety of $S $ and $w \in \mathcal W^P$ be the element corresponding to $S_0$, i.e., $S_0$ is the closure of the $B$-orbit $B.x$ at $x=w.o$. By Proposition \ref{necessary conditions} $\mathcal C_o(w^{-1}S_0)$ is the closure of a $B^1 \times B^2$-orbit $B^1 \times B^2 (e \otimes q + f^* \otimes q^2)$, where $(e, f^*) \in E \oplus E^*$ satisfies $\langle e, f^* \rangle=0$ and $q\in Q$. \\
\noindent {\bf Case 1.} If $S_0$ does not have a general line, then $\mathcal C_o(w^{-1}S_0)$ is contained in $\mathcal C_o(S) \backslash \mathcal C_0(S)^{gen}$, and thus it is contained in $\mathbb P (E \otimes Q)$. Therefore, $\mathcal C_o(w^{-1}S_0)$ is one of the followings: $\mathbb P(E \otimes q)$, $\mathbb P(F_2 \otimes q)$, $\mathbb P(F_1 \otimes q)$, $\mathbb P(e \otimes Q)$. \\
\noindent {\bf Case 2.} If $S_0$ has a general line, then $\mathcal C_o(w^{-1}S_0)$ intersects $\mathcal C_o(S)^{gen}$ nontrivially.
By Lemma \ref{classification in Q4} the smooth closure of a $B^1 \times B^2$-orbit in $\mathcal C_o(S)$ which is a linear section of $\mathcal C_o(S)$ and intersects $\mathcal C_o(S)^{gen}$, is one of the followings: \\
\begin{tabular} {|c|c|}
\hline
$\mathcal C_o(S) \cap \mathbb P((E\otimes q) \oplus (E^* \otimes q^2))$ \,\,&\,\, $\mathcal C_o(S)$ \\ [3pt]
\hline
$\mathbb P((F_1 \otimes q) \oplus (F_1^{\perp} \otimes q^2))$ \,\,&\,\, $\mathcal C_o(S) \cap \mathbb P((F_1 \otimes Q) \oplus (F_1^{\perp}\otimes S^2Q))$ \\[3pt]
$\mathbb P((F_2 \otimes q) \oplus (F_2^{\perp}\otimes q^2))$ & $\mathcal C_o(S) \cap \mathbb P((F_2 \otimes Q)\oplus (F_2^{\perp}\otimes S^2Q))$ \\[3pt]
\hline
$\mathbb P((V_1 \otimes q)\oplus (W_1 \otimes q^2))$ \,\,&\,\, $\mathcal C_o(S) \cap \mathbb P((V_1 \otimes Q) \oplus (W_1\otimes S^2Q))$ \\[3pt]
\hline
\end{tabular}
\vskip 10 pt \noindent
where $F_i$ is a subspace of $E$ of dimension $i$ for $i=1,2$ and $V_1$ is a subspace of $E$ of dimension one and $W_1$ is a subspace of $V_1^{\perp}$ of dimension one and $e \in E$ and $q \in Q$.
(Note that $\mathcal C_o(w^{-1}S_0)$ cannot be contained in $\mathbb P(E^* \otimes S^2Q)$.)
Among them, the $P$-orbits of
$\mathbb P((F_1 \otimes q) \oplus (F_1^{\perp} \otimes q^2))$ and $\mathbb P((V_1 \otimes q)\oplus (W_1 \otimes q^2))$ are not closed (see the proof of Proposition \ref{deformtion of P2 in Q4}), so that they cannot be the varieties of minimal rational tangents of Schubert varieties. \\
Combining lists in Case 1 and in Case 2, we get the desired list.
\end{proof}
\begin{proposition} \label{F4alpha3} Let $S$ be the rational homogeneous manifold of type $(F_4, \alpha_3)$ and let $S_0$ be either $(C_2, \alpha_2, \alpha_1)$ or $(B_3, \alpha_2, \alpha_3)$. Then $\mathcal C_x(S_0)$ at a general point $x \in S_0$ satisfies {\rm(I)} and {\rm(II)} in Proposition \ref{general case - inductive step}.
\end{proposition}
\begin{proof} We will use the same notations as in Proposition \ref{classification of vmrt}.
Assume that $S_0$ is $(B_3, \alpha_2, \alpha_3)$. Then $\mathcal Z_0:=\mathcal C_x(S_0)$ is the linear section of $\mathcal Z:=\mathcal C_x(S)$ by $\mathbb P((F_1 \otimes Q) \oplus (F_1^{\perp} \otimes S^2Q)$ and thus $\mathcal Z_0$ is the projectivization $\mathbb P(\mathcal F)$ of the vector bundle $\mathcal F$ of rank 3 over $\mathbb P(Q)$, whose fiber over $[q] \in \mathbb P(Q)$ is $(F_1 \otimes q) \oplus (F_1^{\perp} \otimes q^2)$. Hence $\mathcal F$ is isomorphic to $\mathcal O(-1) \oplus \mathcal O(-2)^2$ over $\mathbb P^1$.
Any local deformation $\mathbb P(\mathcal F_t)$ of $\mathbb P(\mathcal F)$ is also isomorphic to $\mathbb P(\mathcal O(-1) \oplus \mathcal O(-2)^2)$, so that there is a subbundle $\mathcal F_{1,t} \subset \mathcal F_t$ such that $\mathcal F_{1,t} \otimes \mathcal O(1)$ is a trivial vector bundle of rank one. Then there is a subspace $F_{1,t} \subset E$ of dimension one such that the fiber of $\mathcal F_{1,t}$ at $[q] \in \mathbb P(Q)$ is $F_{1,t} \otimes q$. By acting an element of $SL(E)$, we may assume that $F_{1,t}=F_1$.
By the proof of Lemma \ref{deformtion of P2 in Q4}, the fiber of $\mathcal F_{t}$ at $[q] \in \mathbb P(Q)$ is the graph $F_{\lambda \varphi_1}$ of $\lambda \varphi_1:E \rightarrow E^*$ for some $\lambda \not=0\in \mathbb C \cup \{\infty\}$, depending on $[q]$. Since the assignment $[q] \in \mathbb P(Q) \mapsto \lambda \in (\mathbb C -\{0\})\cup \{\infty\}$ is holomorphic, it is constant. Consequently, $\mathcal F_t$ is $\mathcal F$ up to the action of $(SL(E) \times SL(Q))\ltimes (E^* \otimes Q^*)$.
For $h \in (SL(E) \times SL(Q))\ltimes (E^* \otimes Q^*)$ having nontrivial factor in $E^* \otimes Q^*$, there is a nonzero linear function $\varphi:E \rightarrow E^*$ such that $h\mathbb P(\mathcal F)=\mathbb P(\mathcal F_{\varphi})$, where $\mathcal F_{\varphi}$ is the vector bundle over $\mathbb P(Q)$ whose fiber at $[q] \in \mathbb P(Q)$ is $\{e \otimes q + \varphi(e) \otimes q^*: e \in E\}$. Then any point in $ \mathbb P(\mathcal F) \cap h\mathbb P(\mathcal F)$ is non-generic. If $h$ is in $SL(E) \times SL(Q)$ and $ \mathbb P(\mathcal F)$ is tangent to $h\mathbb P(\mathcal F))$ at $\alpha \in \mathbb P(\mathcal F) \cap h\mathbb P(\mathcal F))$, then $h\mathbb P(\mathcal F) = \mathbb P(\mathcal F)$.
By a similar arguments we get the desired results when $S_0$ is $(C_2, \alpha_2, \alpha_1)$.
\end{proof}
\begin{proposition} \label{classification F4 alpha3} Let $S=G/P$ be the rational homogeneous manifold of type $(F_4, \alpha_3)$. Then a smooth Schubert variety of $S$ is one of the following:
\begin{enumerate}
\item a homogeneous submanifold associated to a subdiagram of the marked Dynkin diagram of $S$;
\item a linear space;
\item $(B_3, \alpha_2, \alpha_3)$ or $(C_2, \alpha_1, \alpha_2)$, embedded as in Proposition \ref{linear sections}.
\end{enumerate}
\end{proposition}
\begin{proof} Proposition \ref{classification of vmrt} and
Proposition \ref{F4alpha3} and Proposition \ref{general case - inductive step}.
\end{proof}
\begin{proposition} \label{rigidity F4 alpha3}
Let $S$ be the rational homogeneous manifold of type $(F_4, \alpha_3)$ and let $S_0$ be either $(C_2, \alpha_2, \alpha_1)$ or $(B_3, \alpha_2, \alpha_3)$. Then any local deformation of $S_0$ in $S$ is induced by the action of $G$.
\end{proposition}
\begin{proof}
By Proposition \ref{general case - inductive step} and Proposition \ref{F4alpha3}, any local deformation of $S_0$ in $S$ is induced by the action of $G$.
\end{proof}
\section{$(F_4, \alpha_4)$-case}
In this section we will consider the case when $S$ is the rational homogeneous manifold of type $(F_4, \alpha_4)$ and prove that any smooth Schubert variety of rational homogeneous manifold $S$ of type $(F_4, \alpha_4)$ is linear (Proposition \ref{classification F4 alpha4}). We will use that $S$ is a hyperplane section of another rational homogenoeus manifold $S'$ of Picard number one, which is associated to a long simple roots, and that any smooth Schubert variety of $S'$ is a homogeneous submanifold associated to a subdiagram of the marked Dynkin diagram of $S'$ (Proposition 3.7 of \cite{HoM}). \\
Let $G$ be the simple group of type $F_4$ and let $W$ be the irreducible $G$-representation space of highest weight $\varpi_4$ and let $w_4$ be a highest weight vector in $W$. Then the $G$-orbit of $x_0:=[w_4]$ in $\mathbb P(W)$ is the rational homogeneous manifold $S=G/P$ of type $(F_4, \alpha_4)$.
Let $G'$ be the simple Lie group of type $E_6$ and
let $W'$ be the irreducible $E_6$-representation space of highest weight $\varpi_6$ and let $w_6'$ be a highest weight vector in $W'$. Then the $G'$-orbit of $x_0':=[w'_6] \in \mathbb P(W')$ is the rational homogeneous manifold $S'=G'/P'$ of type $(E_6, \alpha_6)$.
$W$ can be embedded into $W'$ equivariantly as a hyperplane with $x_0=x_0'$ and $S=G/P$ is the hyperplane section of $S'=G'/P'$ by $\mathbb P(W)$. Here, we use the same notation for the fundamental weights $\varpi_1, \dots, \varpi_4$ of $G$ of type $F_4$ and the fundamental weights $\varpi_1, \dots, \varpi_6$ of $G'$ of type $E_6$, for the simplicity of notations. We will adapt the same convention afterwards as long as it does not make any confusion.
For $w \in \mathcal W^P$, let $S(w)$ be the closure of $B$-orbit $B.x_w$ in $S$, and
for $w' \in \mathcal W^{P'}$, let $S'(w')$ be the closure of $B'$-orbit $B'.x_{w'}$ in $S'$, where $x_{w'}:=w'.x_0$.
The inclusion map $\mathcal W_G \hookrightarrow \mathcal W_{G'}$ from the Weyl group of $G$ to the Weyl group of $G'$ induces an injective map $$\mathcal W^P=\mathcal W_G/\mathcal W_P \hookrightarrow \mathcal W^{P'}=\mathcal W_{G'}/\mathcal W_{P'}$$
\noindent (Figure A and Figure B).
Thus for $w \in \mathcal W^{P}$, $B.x_w $ is contained in $B'.x_{w'}$ for a unique $w' \in \mathcal W^{P'}$. Then we have either $B.x_w =B'.x_{w'} \subset \mathbb P(W)$ or $B.x_w \subsetneq B'.x_{w'} $ and $B.x_w = B'.x_{w'} \cap \mathbb P(W)$, so that we have either $S(w) = S'(w') \subset \mathbb P(W)$ or $S(w) \subsetneq S(w')$ and $S(w) = S'(w') \cap \mathbb P(W)$. In any case we have $S(w) =S(w') \cap \mathbb P(W)$.
\begin{figure}[p]
\begin{subfigure}[b]{.4\textwidth}
\begin{tikzpicture}
\dynkinnode{0}{0}{$\tilde{S}_1$}
\dynkinarrow{0}{0.4}{0}{0.8}
\dynkinnode{-0.5}{0.6}{$s_{\tilde{\alpha}_1}$}
\dynkinnode{0}{1.2}{$\tilde{S}_2$}
\dynkinarrow{0}{1.6}{0}{2}
\dynkinnode{-0.5}{1.8}{$s_{\tilde{\alpha}_3}$}
\dynkinnode{0}{2.4}{$\tilde{S}_3$}
\dynkinarrow{0}{2.8}{0}{3.2}
\dynkinnode{-0.5}{3}{$s_{\tilde{\alpha}_4}$}
\dynkinnode{0}{3.6}{$\tilde{S}_4$}
\dynkinarrow{-0.2}{4}{-0.8}{4.4}
\dynkinarrow{0.2}{4}{0.8}{4.4}
\dynkinnode{-0.8}{4}{$s_{\tilde{\alpha}_2}$}
\dynkinnode{0.8}{4}{$s_{\tilde{\alpha}_5}$}
\dynkinnode{-1}{4.8}{$\tilde{S}_5$}
\dynkinnode{1}{4.8}{$\tilde{S}_6$}
\dynkinarrow{-1}{5.2}{-1}{5.6}
\dynkinarrow{1}{5.2}{1}{5.6}
\dynkinarrow{0.6}{5}{-0.6}{5.8}
\dynkinnode{-1.5}{5.4}{$s_{\tilde{\alpha}_5}$}
\dynkinnode{1.5}{5.4}{$s_{\tilde{s}_6}$}
\dynkinnode{0}{5}{$s_{\tilde{\alpha}_2}$}
\dynkinnode{-1}{6}{$\tilde{S}_7$}
\dynkinnode{1}{6}{$\tilde{S}_8$}
\dynkinarrow{-1}{6.4}{-1}{6.8}
\dynkinarrow{1}{6.4}{1}{6.8}
\dynkinarrow{-0.6}{6.2}{0.6}{7}
\dynkinnode{-1.5}{6.6}{$s_{\tilde{\alpha}_4}$}
\dynkinnode{1.5}{6.6}{$s_{\tilde{\alpha}_2}$}
\dynkinnode{0}{6.2}{$s_{\tilde{s}_6}$}
\dynkinnode{-1}{7.2}{$\tilde{S}_9$}
\dynkinnode{1}{7.2}{$\tilde{S}_{10}$}
\dynkinarrow{-1}{7.6}{-1}{8}
\dynkinarrow{1}{7.6}{1}{8}
\dynkinarrow{-0.6}{7.4}{0.6}{8.2}
\dynkinnode{-1.5}{7.8}{$s_{\tilde{\alpha}_3}$}
\dynkinnode{1.5}{7.8}{$s_{\tilde{\alpha}_4}$}
\dynkinnode{0}{7.4}{$s_{\tilde{s}_6}$}
\dynkinnode{-1}{8.4}{$\tilde{S}_{11}$}
\dynkinnode{1}{8.4}{$\tilde{S}_{12}$}
\dynkinarrow{-1.2}{8.8}{-1.8}{9.2}
\dynkinarrow{-0.8}{8.8}{-0.2}{9.2}
\dynkinarrow{0.8}{8.8}{0.2}{9.2}
\dynkinarrow{1.2}{8.8}{1.8}{9.2}
\dynkinnode{-1.8}{8.8}{$s_{\tilde{\alpha}_1}$}
\dynkinnode{-0.4}{8.8}{$s_{\tilde{s}_6}$}
\dynkinnode{0.4}{8.8}{$s_{\tilde{\alpha}_3}$}
\dynkinnode{1.8}{8.8}{$s_{\tilde{\alpha}_5}$}
\dynkinnode{-2}{9.6}{$\tilde{S}_{13}$}
\dynkinnode{0}{9.6}{$\tilde{S}_{14}$}
\dynkinnode{2}{9.6}{$\tilde{S}_{15}$}
\dynkinarrow{-1.8}{10}{-1.2}{10.4}
\dynkinarrow{-0.2}{10}{-0.8}{10.4}
\dynkinarrow{0.2}{10}{0.8}{10.4}
\dynkinarrow{1.8}{10}{1.2}{10.4}
\dynkinnode{-1.8}{10.4}{$s_{\tilde{s}_6}$}
\dynkinnode{-0.4}{10.4}{$s_{\tilde{\alpha}_1}$}
\dynkinnode{0.4}{10.4}{$s_{\tilde{\alpha}_5}$}
\dynkinnode{1.8}{10.4}{$s_{\tilde{\alpha}_3}$}
\dynkinnode{-1}{10.8}{$\tilde{S}_{16}$}
\dynkinnode{1}{10.8}{$\tilde{S}_{17}$}
\dynkinarrow{-1}{11.2}{-1}{11.6}
\dynkinarrow{1}{11.2}{1}{11.6}
\dynkinarrow{0.6}{11}{-0.6}{11.8}
\dynkinnode{-1.5}{11.4}{$s_{\tilde{\alpha}_5}$}
\dynkinnode{1.5}{11.4}{$s_{\tilde{\alpha}_4}$}
\dynkinnode{0}{11}{$s_{\tilde{\alpha}_1}$}
\dynkinnode{-1}{12}{$\tilde{S}_{18}$}
\dynkinnode{1}{12}{$\tilde{S}_{19}$}
\dynkinarrow{-1}{12.4}{-1}{12.8}
\dynkinarrow{1}{12.4}{1}{12.8}
\dynkinarrow{0.6}{12.2}{-0.6}{13}
\dynkinnode{-1.5}{12.6}{$s_{\tilde{\alpha}_4}$}
\dynkinnode{1.5}{12.6}{$s_{\tilde{\alpha}_2}$}
\dynkinnode{0}{12.2}{$s_{\tilde{\alpha}_1}$}
\dynkinnode{-1}{13.2}{$\tilde{S}_{20}$}
\dynkinnode{1}{13.2}{$\tilde{S}_{21}$}
\dynkinarrow{-1}{13.6}{-1}{14}
\dynkinarrow{1}{13.6}{1}{14}
\dynkinarrow{-0.6}{13.4}{0.6}{14.2}
\dynkinnode{-1.5}{13.8}{$s_{\tilde{\alpha}_3}$}
\dynkinnode{1.5}{13.8}{$s_{\tilde{\alpha}_1}$}
\dynkinnode{0}{13.4}{$s_{\tilde{\alpha}_2}$}
\dynkinnode{-1}{14.4}{$\tilde{S}_{23}$}
\dynkinnode{1}{14.4}{$\tilde{S}_{22}$}
\dynkinarrow{-0.8}{14.8}{-0.2}{15.2}
\dynkinarrow{0.8}{14.8}{0.2}{15.2}
\dynkinnode{-0.8}{15.2}{$s_{\tilde{\alpha}_2}$}
\dynkinnode{0.8}{15.2}{$s_{\tilde{\alpha}_3}$}
\dynkinnode{0}{15.6}{$\tilde{S}_{24}$}
\dynkinarrow{0}{16}{0}{16.4}
\dynkinnode{-0.5}{16.2}{$s_{\tilde{\alpha}_4}$}
\dynkinnode{0}{16.8}{$\tilde{S}_{25}$}
\dynkinarrow{0}{17.2}{0}{17.6}
\dynkinnode{-0.5}{17.4}{$s_{\tilde{\alpha}_5}$}
\dynkinnode{0}{18}{$\tilde{S}_{26}$}
\dynkinarrow{0}{18.4}{0}{18.8}
\dynkinnode{-0.5}{18.6}{$s_{\tilde{s}_6}$}
\dynkinnode{0}{19.2}{$\tilde{S}_{27}$}
\end{tikzpicture}
\caption{[ Figure A : Hasse diagram of $S'$ ]}
\end{subfigure}
\hspace{2cm}
\begin{subfigure}[b]{.4\textwidth}
\begin{tikzpicture}
\dynkinnode{0}{0}{$S_1$}
\dynkinarrow{0}{0.4}{0}{0.8}
\dynkinnode{-0.5}{0.6}{$s_{\alpha_4}$}
\dynkinnode{0}{1.2}{$S_2$}
\dynkinarrow{0}{1.6}{0}{2}
\dynkinnode{-0.5}{1.8}{$s_{\alpha_3}$}
\dynkinnode{0}{2.4}{$S_3$}
\dynkinarrow{0}{2.8}{0}{3.2}
\dynkinnode{-0.5}{3}{$s_{\alpha_2}$}
\dynkinnode{0}{3.6}{$S_4$}
\dynkinarrow{-0.2}{4}{-0.8}{4.4}
\dynkinarrow{0.2}{4}{0.8}{4.4}
\dynkinnode{-0.8}{4}{$s_{\alpha_1}$}
\dynkinnode{0.8}{4}{$s_{\alpha_3}$}
\dynkinnode{-1}{4.8}{$S_5$}
\dynkinnode{1}{4.8}{$S_6$}
\dynkinarrow{-1}{5.2}{-1}{5.6}
\dynkinarrow{1}{5.2}{1}{5.6}
\dynkinarrow{0.6}{5}{-0.6}{5.8}
\dynkinnode{-1.5}{5.4}{$s_{\alpha_3}$}
\dynkinnode{1.5}{5.4}{$s_{\alpha_4}$}
\dynkinnode{0}{5}{$s_{\alpha_1}$}
\dynkinnode{-1}{6}{$S_7$}
\dynkinnode{1}{6}{$S_8$}
\dynkinarrow{-1}{6.4}{-1}{6.8}
\dynkinarrow{1}{6.4}{1}{6.8}
\dynkinarrow{-0.6}{6.2}{0.6}{7}
\dynkinnode{-1.5}{6.6}{$s_{\alpha_2}$}
\dynkinnode{1.5}{6.6}{$s_{\alpha_1}$}
\dynkinnode{0}{6.2}{$s_{\alpha_4}$}
\dynkinnode{-1}{7.2}{$S_9$}
\dynkinnode{1}{7.2}{$S_{10}$}
\dynkinarrow{-1}{7.6}{-1}{8}
\dynkinarrow{1}{7.6}{1}{8}
\dynkinarrow{-0.6}{7.4}{0.6}{8.2}
\dynkinnode{-1.5}{7.8}{$s_{\alpha_3}$}
\dynkinnode{1.5}{7.8}{$s_{\alpha_2}$}
\dynkinnode{0}{7.4}{$s_{\alpha_4}$}
\dynkinnode{-1}{8.4}{$S_{11}$}
\dynkinnode{1}{8.4}{$S_{12}$}
\dynkinarrow{-1}{8.8}{-1}{9.2}
\dynkinarrow{1}{8.8}{1}{9.2}
\dynkinnode{-1.5}{9}{$s_{\alpha_4}$}
\dynkinnode{1.5}{9}{$s_{\alpha_3}$}
\dynkinnode{-1}{9.6}{$S_{16}$}
\dynkinnode{1}{9.6}{$S_{17}$}
\dynkinarrow{-1}{10}{-1}{10.4}
\dynkinarrow{1}{10}{1}{10.4}
\dynkinarrow{0.6}{9.8}{-0.6}{10.6}
\dynkinnode{-1.5}{10.2}{$s_{\alpha_3}$}
\dynkinnode{1.5}{10.2}{$s_{\alpha_2}$}
\dynkinnode{0}{9.8}{$s_{\alpha_4}$}
\dynkinnode{-1}{10.8}{$S_{18}$}
\dynkinnode{1}{10.8}{$S_{19}$}
\dynkinarrow{-1}{11.2}{-1}{11.6}
\dynkinarrow{1}{11.2}{1}{11.6}
\dynkinarrow{0.6}{11}{-0.6}{11.8}
\dynkinnode{-1.5}{11.4}{$s_{\alpha_2}$}
\dynkinnode{1.5}{11.4}{$s_{\alpha_1}$}
\dynkinnode{0}{11}{$s_{\alpha_4}$}
\dynkinnode{-1}{12}{$S_{20}$}
\dynkinnode{1}{12}{$S_{21}$}
\dynkinarrow{-1}{12.4}{-1}{12.8}
\dynkinarrow{1}{12.4}{1}{12.8}
\dynkinarrow{-0.6}{12.2}{0.6}{13}
\dynkinnode{-1.5}{12.6}{$s_{\alpha_3}$}
\dynkinnode{1.5}{12.6}{$s_{\alpha_4}$}
\dynkinnode{0}{12.2}{$s_{\alpha_1}$}
\dynkinnode{-1}{13.2}{$S_{23}$}
\dynkinnode{1}{13.2}{$S_{22}$}
\dynkinarrow{-0.8}{13.6}{-0.2}{14}
\dynkinarrow{0.8}{13.6}{0.2}{14}
\dynkinnode{-0.8}{14}{$s_{\alpha_1}$}
\dynkinnode{0.8}{14}{$s_{\alpha_3}$}
\dynkinnode{0}{14.4}{$S_{24}$}
\dynkinarrow{0}{14.8}{0}{15.2}
\dynkinnode{-0.5}{15}{$s_{\alpha_2}$}
\dynkinnode{0}{15.6}{$S_{25}$}
\dynkinarrow{0}{16}{0}{16.4}
\dynkinnode{-0.5}{16.2}{$s_{\alpha_3}$}
\dynkinnode{0}{16.8}{$S_{26}$}
\dynkinarrow{0}{17.2}{0}{17.6}
\dynkinnode{-0.5}{17.4}{$s_{\alpha_4}$}
\dynkinnode{0}{18}{$S_{27}$}
\end{tikzpicture}
\caption{[ Figure B : Hasse diagram of $S$ ]}
\end{subfigure}
\end{figure}
By using this relation between $\mathcal W^{P}$ and $\mathcal W^{P'}$ and the description of the Zariski tangent space $T_{x_0}S(w)$ of the Schubert variety $S(w)$ at the base point $x_0$ (Theorem 3.2 of \cite{Po}) we can show that the dimension of $T_{x_0}S(w)$ is greater than the length of $w$ unless $S(w)$ is a linear space, so that there is no smooth Schubert variety other than linear spaces in $S$. Instead of doing this, we apply the theory of the variety of minimal rational tangents again as in the previous section for the unity of the method. \\
The semisimple part of the reductive part $L$ of $P$ is of type $B_3$ and
the variety $\mathcal Z:=\mathcal C_{x_0}(S)$ of minimal rational tangents of $S$ at $x_0$
is the closure of $L$-orbit of $[v_1+v_3]$ in $\mathbb P(V)$, where $V$ is the direct sum $V(\varpi_1) \oplus V(\varpi_3)$, where $V(\varpi_i)$ is the $B_3$-representation space of highest weight $\varpi_i$ for $i=1,2,3$ (see \cite{HwM05}). $\mathcal Z$ is smooth and is of Picard number one and is uniruled by lines lying on $\mathcal Z$.
Let $z_0:=[v_1] \in \mathbb P(V)$. Then the $P$-orbit of $z_0$ is open in $\mathcal Z$ and the $L$-orbit of $z_0$ is closed. Let $Q$ denote the isotropy group of $L$ at $z_0$. Then the semisimple part of the reductive part $H$ of $Q$ is of type $B_2$ and the variety $\mathcal A:=\mathcal C_{z_0}(\mathcal Z)$ of minimal rational tangents of $\mathcal Z$ at $z_0$ is the closure of $H$-orbit of $[u_1+u_2]$, where $u_i$ is a highest weight vector of $B_2$-representation space $U(\varpi_i)$ of highest weight $\varpi_i$ for $i=1,2$.
Let $\mathcal X$ be the closure of a $H \cap B$-orbit in $\mathcal Z$. As in the case of Schubert varieties, for a point $x$ in the open $H\cap B$-orbit in $\mathcal X$, we define the variety $\mathcal C_{x}(\mathcal X)$ of minimal rational tangents by the set of tangent directions of lines lying on $\mathcal X$ passing through $x$.
\begin{proposition} \label{classification F4 alpha4} Let $S=G/P$ be the rational homogeneous manifold of type $(F_4, \alpha_4)$. Then any smooth Schubert variety of $S$ other than $S$ itself is linear.
\end{proposition}
\begin{proof}
Let $S_0$ be a Schubert variety of type $w$, i.e., the closure of $B$-orbit of $x_w:=wx_0$, where $w \in \mathcal W^P$.
By Proposition \ref{necessary conditions}, $\mathcal C_{x_w}(S_0)$ is invariant under the action of the Borel subgroup $w(L \cap B)$ of $w(L)$.
Thus $\mathcal Z_0:=\mathcal C_{x_0}(w^{-1}S_0)$ is invariant under the action of $L \cap B$.
Assume that $S_0$ is smooth. Then $\mathcal Z_0$ is smooth and is the closure of an $L \cap B$-orbit in $\mathcal Z$ (Proposition \ref{necessary conditions}). It suffices to show that $\mathcal Z_0$ is linear.
As in the case when $S$ is of type $(F_4, \alpha_3)$, we may be able to classify $L \cap B$-orbits in $\mathcal Z$ and to determine which closures are smooth. Instead of doing this, we will prove that the variety $\mathcal C_z(\mathcal Z_0)$ of minimal rational tangents of $\mathcal Z_0$ at a general point $z \in \mathcal Z_0$ is linear, by showing that it is the closure of a $(H \cap B)_z$-orbit in $\mathcal C_z(\mathcal Z)$ and by using that any smooth closure of $(H \cap B)$-orbit in $\mathcal A=\mathcal C_{z_0}(\mathcal Z)$ is linear.
If $S_0$ does not contain a general line, then $\mathcal Z_0$ is contained in $ \mathcal Z \cap \mathbb P(V(\varpi_3))$ which is a rational homogeneous manifold of type $(B_3, \alpha_3)$, and thus $\mathcal Z_0$ is linear because any smooth Schubert variety of the rational homogeneous manifold of type $(B_3, \alpha_3)$ is linear.
From now on, we will assume that $S_0$ contains a general line, i.e., $\mathcal Z_0$ intersects $\mathcal Z^{gen}=\mathcal Z - \mathcal Z \cap \mathbb P(V(\omega_3))$ nontrivially.
$\mathcal Z_0$ is uniruled by lines in $\mathcal Z$
because $L \cap B$ has an open orbit in $\mathcal Z_0$.
Let $z=g z_0 $, where $g\in P$, be a point in the open $L \cap B$-orbit in $\mathcal Z_0$.
By the same arguments as in the proof of Proposition 3.1 of \cite{HoM}, the variety $\mathcal A_0:=\mathcal C_{z_0}(g^{-1}\mathcal Z_0)$ of minimal rational tangents of $g^{-1}\mathcal Z_0$ at $z_0 $ is a smooth linear section of $\mathcal A$. However, it is not obvious that $\mathcal A_0$ is invariant under the action of the Borel subgroup $H \cap B$ of $H$ (the same arguments in the proof of Proposition \ref{necessary conditions} do not apply because $\mathcal Z$ is no longer a rational homogeneous manifold).
\begin{lemma} \label{necessary conditions inductive step}
$\mathcal A_0$ is invariant under the action of $H \cap B$.
\end{lemma}
Together with the fact that $\mathcal A_0$ is smooth, we get that $\mathcal A_0$ is the closure of an $H\cap B$-orbit in $\mathcal A$.
Now $\mathcal A=(B_2, \alpha_1, \alpha_2)=(C_2, \alpha_2, \alpha_1)$ is the odd symplectic Grassmannian $Gr_{\omega}(2, \mathbb C^5)$, smooth orbit closures of a Borel subgroup of $H$ (of $B_2$-type) in $\mathcal A $ other than $\mathcal A$ itself are linear. Therefore, $\mathcal A_0$ is linear and hence $\mathcal Z_0$ is linear. Consequently, $S_0$ is linear. This completes the proof of Proposition \ref{classification F4 alpha4}.
\end{proof}
In the remaining section we will prove Lemma \ref{necessary conditions inductive step}. We will consider $S=G/P$ as a hyperplane section of a rational homogeneous manifold $S'=G'/P'$ associated to a long simple root, whose variety of minimal rational tangent is again a rational homogeneous manifold of Picard number one. \\
The semisimple part of the reductive part $L'$ of $P'$ is of type $D_5$
and the variety $\mathcal Z'$ of minimal rational tangents of $S'$ at $x_0$
is the $L'$-orbit of $z_0':=[v'_5]$ in $\mathbb P(V')$, where $V'$ is the $D_5$-representation space of highest weight $\varpi_5$ and $v'_5$ is a highest weight vector in $V'$.
Since $S$ is a hyperplane section of $S'$, $\mathcal Z$ is a hyperplane section of $\mathcal Z'$, too. The reason why we introduce $S'$ is that its variety $\mathcal Z'$ of minimal rational tangents is a rational homogeneous manifold, so that we can apply arguments in Section \ref{section vmrt of Schubert varieties} to the closures of $L'\cap B'$-orbits in $\mathcal Z'$, while the variety $\mathcal Z$ of minimal rational tangents of $S$ is not.
We will identify $z_0$ with $z_0'$ so that $\mathcal Z$ is the hyperplane section of $\mathcal Z'$ by $\mathbb P(V)$ as follows. As a representation space of $D_4$, $V'$ is the direct sum $V''(\varpi_3) \oplus V''(\varpi_4)$, where $V''(\varpi_i)$ is the $D_4$-representation space of highest weight $\varpi_i$ for $i=1, \dots, 4$, and as a $D_4$-variety, $\mathcal Z'$ is isomorphic to the closure of $L''$-orbit of $[v''_4+v''_3]$ in $\mathbb P(V''(\varpi_4) \oplus V''(\varpi_3))$, where $v''_i$ is a highest weight vector in $V''(\varpi_i)$ for $i=1,\dots, 4$ (Proposition \ref{homogeneous horospherical varieties}). Since $\mathcal Z'$ is homogeneous, we may identify $z_0'=[v_5']$ with $[v_4'']$. If we identify $z_0 $ with $z_0' $,
$V$ is a hyperplane of $V'$ and $\mathcal Z$ is the hyperplane section of $\mathcal Z'$ by $\mathbb P(V)$. The embedding of $\mathcal Z$ into $\mathcal Z'$ is that of $(B_3, \alpha_1, \alpha_3)$ into $(D_4, \alpha_4,\alpha_3)=(D_5, \alpha_5)$.
Let $Q'$ be the isotropy group of $L'$ at $z_0'$.
The semisimple part $H'$ of the reductive part of $Q'$ is of type $A_4$ and
the variety $\mathcal A'$ of minimal rational tangents of $\mathcal Z'$ at $z_0'$ is the $H'$-orbit of $[u_2']$, where $u_2'$ is a highest weight vector of $A_4$-representation space $U'$ of highest weight $\omega_2$.
The semisimple part $H''$ of the reductive part of the isotropy group of $L''$ at $z_0'$ is of type $D_3=A_3$.
As an $A_3$-representation space $U'$ is the direct sum $U''(\varpi_1) \oplus U''(\varpi_2)$, where $U''(\varpi_i)$ is the $A_3$-representation space of highest weight $\varpi_i$ for $i=1,2,3$, and as an $A_3$-variety $\mathcal A'$ is isomorphic to the closure of $A_3$-orbit of $[u''_1+u''_2]$, where $u''_i$ is a highest weight vector of $U''(\varpi_i)$ for $i=1,2,3$ (Proposition \ref{homogeneous horospherical varieties}). As before, if we identify $[u_1]$ with $[u''_1]$, $U$ is a hyperplane of $U'$ and $\mathcal A$ is the hyperplane section of $\mathcal A'$ by $\mathbb P(U)$.
The embedding of $\mathcal A$ into $\mathcal A'$ is the embedding of $(B_2, \alpha_1, \alpha_2)$ into $(D_3, \alpha_3, \alpha_2)=(A_3, \alpha_1, \alpha_2)=(A_4, \alpha_2)$. \\
\noindent {\it Proof of Lemma \ref{necessary conditions inductive step}.}
Let $S_0' =S'(w')$, $w' \in \mathcal W^{P'}$, be the Schubert variety of $S'$ corresponding to $S_0$.
From $S_0 = S_0'\cap \mathbb P(W) $ it follows that
$\mathcal Z_0 =\mathcal Z_0' \cap \mathbb P(T_{x_0}w^{-1}S_0) =\mathcal Z_0' \cap \mathbb P(T_{x_0}S)$.
$\mathcal Z_0'$ may have more than one irreducible components, but, since $\mathcal Z_0$ is smooth, there is an irreducible component ${\mathcal Z'_0}^0$ of $\mathcal Z_0'$ such that $\mathcal Z_0 ={\mathcal Z'_0}^0 \cap \mathbb P(T_{x_0}S)$.
By the invariance of $\mathcal Z_0' $ under the action of $L' \cap B'$ (Proposition \ref{necessary conditions}), ${\mathcal Z'_0}^0$ is the closure of an $L'\cap B'$-orbit in $\mathcal Z'$, i.e., a Schubert variety of $\mathcal Z'$. By Proposition \ref{necessary conditions} again, for a general point $g' z_0'$ in ${\mathcal Z'_0}^0$, $\mathcal C_{g' z_o'}( {\mathcal Z'_0}^0)$ is invariant under the action of $g'(H' \cap B')$ and thus ${\mathcal A_0'}^0:=\mathcal C_{ z_o'}( {g'}^{-1}{\mathcal Z'_0}^0)$ is invariant under the action of $ H' \cap B' $. Now
$\mathcal Z_0
=\mathcal Z_0' \cap \mathbb P(T_{x_0}S)$, we have $\mathcal A_0
= {\mathcal A_0'}^0 \cap \mathbb P(T_{z_0} \mathcal Z ) $. Since ${\mathcal A_0'}^0$ is invariant under the action of $H' \cap B'$, $\mathcal A_0$ is invariant under the action of $H \cap B =(H' \cap B') \cap H$.
\qed \\
\vskip 10 pt
\noindent {\it Proof of Theorem \ref{theorem main}.} By Proposition \ref{classification F4 alpha4} any smooth Schubert variety of $S$ of type $(F_4, \alpha_4)$ is linear.
Now the first statement follows from Proposition \ref{classification F4 alpha3}, and the second statement follows from Proposition \ref{rigidity F4 alpha3}.
\qed \\
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\end{document}
|
1,314,259,995,785 | arxiv | \section{Introduction}
Optimal filtering theory defines different optimality criteria, such
as minimizing the conditional mean square estimation error (MSEE),
given the measurements \cite{Jazwinsky}, maximizing the a posteriori
probability (MAP) density function (pdf) of the signal, given the
measurements \cite{VanTrees}, Bellman's criterion of minimum noise
energy (MNE) \cite{Bellman}, \cite{Lee}, \cite{Lee1}, and more. In
problems of phase estimation, that lead to loss of lock and cycle
slips, an important optimality criterion is maximizing the mean time
to lose lock (MTLL) or to exit a given region, which is also a well
known control problem \cite{Fleming}, \cite{ZZ}, \cite{ADS},
\cite{MR}. Approximation methods for finding the various optimal
filters have been devised for problems with small noise, including
large deviations and WKB solutions of Zakai's equation, the extended
Kalman filter (EKF) \cite{Katzur}, \cite{Picard86}\cite{Picard91},
\cite{Hijab} and others. The EKF and WKB approximations produce explicit suboptimal
finite-dimensional filters, which in case of phase estimation are
the well known phase trackers, such as the phase locked loop (PLL),
delay locked loop (DLL), angle tracking loops, and so on
\cite{Stensby}. The MSEE in these phase trackers is asymptotically
optimal \cite{Katzur}, \cite{Hijab}.
The suboptimal phase trackers are known to lose lock (or slip
cycles) \cite{Stensby}. The MTLL in these filters is simply the mean
first passage time (MFPT) of the estimation error to the boundary of
the lock region. The MFPT from an attractor of a dynamical system
driven by small noise has been calculated by large deviations and
singular perturbation methods \cite{FW}, \cite{MS}, \cite{book}, and
in particular, for the PLL \cite{BS}. The MTLL in particle filters
for phase estimation was found in \cite{FB}. It has been found
recently that minimizing the MNE leads to a finite, yet much longer
MTLL than in the above mentioned phase estimators \cite{Doron},
\cite{Ehud}. This raises the question of designing a causal (or
noncausal) phase estimator with maximal MTLL.
The MTLL is the fundamental performance criterion in phase tracking
and synchronization systems. Thus, for example, a phase tracking
system is considered locked, as long as the estimation error
$e(t)=x(t)-\hat x(t)$ is in $(-\pi,\pi)$. When the error exceeds
these limits, the estimation is said to be unlocked, and the system
relocks on an erroneous equilibrium point, with a deviation of
$2\pi$. Another example is an automatic sight of a cannon. The sight
is said to be locked on target if the positioning error is somewhere
in between certain limits. Similar problems, in which the
maximization of exit time is an optimality criterion, were
considered by several authors \cite{ZZ}, \cite{ADS}, \cite{MR}. In
\cite{ZZ}, a simpler filtering problem is considered, in which the
error $e(t)$ is measured, rather than the state variable $x(t)$. It
is solved under the further assumption of a linear measurement
inside a domain. In \cite{ADS}, \cite{MR} the state process is
controlled through its drift, rendering it a control rather than a
filtering problem.
In this paper we show that for small noise the maximum MTLL filter
is Bellman's MNE filter \cite{Lee}. It follows that the result of
\cite{Ehud} for the MTLL of the optimal MNE phase filter, is
asymptotically an upper bound for any other filtering scheme. In
view of the results of \cite{Ehud}, the potential gain of the
optimal MNE filter over the first order EKF-PLL is $12$ dB.
\section{Formulation}\label{sec:Form}
An important class of filtering problems with small
measurements noise can be reduced to the model of a diffusion
process
\begin{eqnarray}\label{proc_eq}
dx(t)&=&m(x,t)\,dt+\varepsilon\sigma\,dw(t),
\end{eqnarray}
measured in a noisy channel
\begin{eqnarray}\label{meas_eq}
dy(t)&=&h(x,t)\,dt+\varepsilon\rho\,d\nu(t),
\end{eqnarray}
where $m(x,t)$ and $h(x,t)$ are possibly nonlinear, continuous
functions. The processes $w(t)$ and $\nu(t)$ are independent
standard Brownian motions, and $\varepsilon$ is a small parameter. If
$m(x,t)$ is a linear function and the noise in (\ref{proc_eq}) is
not small, an appropriate scaling of time and dependent variables
scales the small measurements noise into the diffusion equation as
well, giving the canonic system (\ref{proc_eq}), (\ref{meas_eq})
\cite{Doron}. The optimal filtering problem is to find a causal
estimator $\hat x(t)$ of $x(t)$, given the measurements
$y_0^t=\{y(s)\,:\, 0\leq s\leq t\}$, such that the mean first time
the error signal,
\begin{eqnarray}
e(t)=x(t)-\hat x(t),\label{exhat}
\end{eqnarray}
leaves a given lock domain $L\subset\hbox{\bb R}$, is maximal. More
specifically, for any adapted function $\hat x(t)\in{\cal C}(\hbox{\bb R}^+)$
(measurable with respect to the filtration generated by $y(t)$), we
define an error process by (\ref{exhat}) and the first time to lose
lock by
\begin{eqnarray}
\tau=\inf\left\{t\,:\,e(t)\in\partial L\right\}.\label{tauL}
\end{eqnarray}
The optimal filtering problem is to maximize $E[\tau\,|\,y_0^\tau]$
(see definition (\ref{Etauy}) below) with respect to all
adapted continuous functions $\hat x(t)$. For example, if
$h(x,t)=\sin x$ in a phase estimation problem, then $L=(-\pi,\pi)$
and lock is lost when $e(t)=\pm\pi$.
We can rewrite the model equations (\ref{proc_eq}), (\ref{meas_eq})
in terms of the error process $e(t)$ as
\begin{eqnarray}
de(t)&=&M_{\hat x}(e(t),t)\,dt+\varepsilon\sigma\,dw(t)\label{err_eq}\\
&&\nonumber\\
dy(t)&=&H_{\hat x}(e(t),t)\,dt+\varepsilon\rho\,d\nu(t)\label{err_meas}
\end{eqnarray}
where
\begin{eqnarray*} M_{\hat x}(e(t),t)&=&m(\hat x(t)+e(t))-\dot{\hat x}(t)\\
&&\\
H_{\hat x}(e(t),t)&=&h(\hat x(t)+e(t)),\end{eqnarray*}
and the filtering problem is to find $\hat x(t)$, such that
$E[\tau\,|\,y_0^\tau]$ is maximal.
The survival probability of a trajectory $(e(t),y(t))$ of
(\ref{err_eq}) with absorption at $\partial L$ and (\ref{err_meas}) can be
expressed in terms of the pdf $p(e,y,t\,\big|\,\xi,\eta,s)$ of the
two-dimensional process with an absorbing boundary condition on $\partial
L$. It is the solution of the Fokker-Planck equation (FPE)
\begin{eqnarray}
\frac{\partial p(e,y,t\,\big|\,\xi,\eta,s)}{\partial t}&=&-\frac{\partial
M_{\hat x}(e,t)p(e,y,t\,\big|\,\xi,\eta,s)}{\partial e}-\frac{\partial
H_{\hat x}(e,t)p(e,y,t\,\big|\,\xi,\eta,s)}{\partial y}+\nonumber\\
&&\nonumber\\
&& \frac{\varepsilon^2\sigma^2}{2}\frac{\partial^2
p(e,y,t\,\big|\,\xi,\eta,s)}{\partial
e^2}+\frac{\varepsilon^2\rho^2}{2}\frac{\partial^2
p(e,y,t\,\big|\,\xi,\eta,s)}{\partial y^2}\label{2DFPE}
\end{eqnarray}
for $e,\xi\in L,\ y,\eta\in\hbox{\bb R}$, with the boundary and initial
conditions
\begin{eqnarray}
p(e,y,t\,|\,\xi,\eta,s)&=&0\qquad\qquad\qquad\quad\mbox{for}\quad e\in\partial L,\
y\in\hbox{\bb R},\ \xi\in L,\ \eta\in\hbox{\bb R}\label{BC}\\
&&\nonumber\\
p(e,y,s\,|\,\xi,\eta,s)&=&\delta(e-\xi,y-\eta)\ \quad\mbox{for}\quad
e\in L,\ y\in\hbox{\bb R},\ \xi\in L,\ \eta\in\hbox{\bb R}\label{IC}.
\end{eqnarray}
The pdf is actually the joint density and probability function
$p(e,y,t\,|\,\xi,\eta,s)=\Pr\{e(t)=e,y(t)=y,\tau>t\,|\,\xi,\eta,s\}$
and thus the survival probability is
\begin{eqnarray}
\Pr\{\tau>t\,|\,\xi,\eta,s\}=S_{e(\cdot),y(\cdot)}(t)&=&
\int_{L}\int_{\hbox{\bb R}} p(e,y,t\,|\,\xi,\eta,s)\,de\,dy,\label{Sxy}
\end{eqnarray}
and it decays in time.
\section{Simulation with particles}
To simulate the filtering problem on a finite interval $0\leq t\leq
T$, we discretize (\ref{proc_eq}), (\ref{meas_eq}) on a sequence of
grids
$$\left\{t_i=i\Delta t,\quad i=0,1,\ldots,N,\quad\Delta t=\frac{T}{N}\right\},$$
and define discrete trajectories by the Euler scheme
\begin{eqnarray}
x_N(t_{i+1})&=&x_N(t_i)+\Delta t\,m(x_N(t_i),t_i)+
\varepsilon\sigma\,\Delta w(t_i)
\label{dxN}\\ &&\nonumber\\
y_N(t_{i+1})&=&y_N(t_i)+\Delta t\,h\left(x_N(t_i),t_i\right)
+\varepsilon\rho\,\Delta\nu(t_i), \label{dyN}
\end{eqnarray}
for $i=0,1,\ldots,N-1$, where $\Delta w(t_i)$ and $\Delta
\nu(t_i)$ are independent zero mean Gaussian random variables with
variance $\Delta t$. The discretized version of (\ref{err_eq}),
(\ref{err_meas}) is
\begin{eqnarray}
e_N(t_{i+1})&=&e_N(t_i)+\Delta t\,M_{\hat x}\left(e_N(t_i),t_i\right)+
\varepsilon\sigma\,\Delta w(t_i)
\label{de}\\ &&\nonumber\\
y_N(t_{i+1})&=&y_N(t_i)+\Delta t\,H_{\hat x}\left(e_N(t_i),t_i\right)
+\varepsilon\rho\,\Delta\nu(t_i).
\label{dy}
\end{eqnarray}
Given an observed trajectory $\{y_{N}(t_i)\}_{i=0}^N$, we
sample $n$ trajectories
$\left\{\{x_{j,N}(t_i)\}_{i=0}^N\right\}_{j=1}^n$, according to the
scheme (\ref{dxN}), which produce error trajectories
$\left\{\{e_{j,N}(t_i)\}_{i=0}^N\right\}_{j=1}^n$, and determine
their first exit times from $L$, denoted
$\left\{\tau_{j,N}\right\}_{j=1}^n$ (we set $\tau_{j,N}=T$ if
$\{e_{j,N}(t_i)\}_{i=0}^N$ does not exit $L$ by time $T$)
\cite{Crisan2003}, \cite{Amblard}, \cite{Fischler},
\cite{Arulampalam}, \cite{elMoral_Guionnet}. The conditional MTLL is
defined on the ensemble by
\begin{eqnarray}
&&E\left[\tau_N\wedge T\,|\,\left\{y_N(t_i),\,i=0,1,\ldots,\displaystyle\left[\frac{\tau_{N}}
{\Delta t}\right]\wedge N\right\}\right]=\label{EtauN}\\
&&\nonumber\\
&&\frac{\displaystyle\sum_{j=1}^n(\tau_{j,N}\wedge T)\exp
\left\{\displaystyle\frac{1}{\varepsilon^2\rho^2}\displaystyle\sum_{k=0}^{\left[\displaystyle\frac{\tau_{j,N}\wedge T}{\Delta t}\right]}\left[H(e_{j,N}(t_{k-1}),t_{k-1})\Delta y_{k,N}-\frac12
H^2(e_{j,N}(t_{k-1}),t_{k-1})\Delta t\right]\right\}}{\displaystyle\sum_{j=1}^n\exp
\left\{\displaystyle\frac{1}{\varepsilon^2\rho^2}\displaystyle\sum_{k=0}^{\left[\displaystyle\frac{\tau_{j,N}\wedge T}{\Delta t}\right]}\left[H(e_{j,N}(t_{k-1}),t_{k-1})\Delta y_{k,N}-\frac12
H^2(e_{j,N}(t_{k-1}),t_{k-1})\Delta t\right]\right\}}.\nonumber
\end{eqnarray}
We define
\begin{eqnarray}
E[\tau\,|\,y_0^{\tau}]=
\lim_{T\to\infty}\lim_{n\to\infty}\lim_{N\to\infty}
E\left[\tau_N\wedge T\,|\,\left\{y_N(t_i),\,i=0,1,\ldots,\displaystyle\left[\frac{\tau_{N}}
{\Delta t}\right]\wedge N\right\}\right].\label{Etauy}
\end{eqnarray}
The conditional MTLL $E[\tau\,|\,y_0^{\tau}]$ is a random variable
on the $\sigma$-algebra ${\cal F}_\infty=\displaystyle\bigcup_{t>0}{\cal
F}_t$, where ${\cal F}_t$ is the $\sigma$-algebra generated by the
measurements process $y(\cdot)$ up to time $t$. Our purpose is to
find $\hat x(t)$ that maximizes $E[\tau\,|\,y_0^\tau]$ in the class
of continuous adapted functions.
\section{The joint pdf of the discrete process}
The pdf of a trajectory of $(e_N(t),y_N(t))$ is the Gaussian
\begin{eqnarray}
&&p_N(e_1,e_2,\ldots,e_N;y_1,y_2,\ldots,y_N;t_1,t_2,\dots,t_N)=
\prod_{k=1}^N\left[\frac{\displaystyle\exp\left\{-\frac{{\cal
B}_k(\mbox{\boldmath$x$}_k,\mbox{\boldmath$x$}_{k-1})}{2\varepsilon^2\Delta t }\right\}}
{2\pi\varepsilon^2\rho\sigma\Delta t }\right],\label{peNyN}
\end{eqnarray}
where the exponent is the quadratic form
\begin{eqnarray*}
{\cal B}_k(\mbox{\boldmath$x$}_k,\mbox{\boldmath$x$}_{k-1})&=&\left[\mbox{\boldmath$x$}_k-\mbox{\boldmath$x$}_{k-1}-\Delta t\mb{a}_{k-1}\right]^T
\mb{B}\left[\mbox{\boldmath$x$}_k-\mbox{\boldmath$x$}_{k-1}-\Delta t\mb{a}_{k-1}\right],
\end{eqnarray*}
such that
\begin{eqnarray*}
\mbox{\boldmath$x$}_k=\left[\begin{array}{l}e_k\\mbox{\boldmath$y$}_k\end{array}\right],\quad
\mb{a}_k = \left[\begin{array}{l}M_{\hat x}(e_k,t_k)\\H_{\hat x}(e_{k},t_k)\end{array}\right],\quad\mb{B}=
\begin{bmatrix} \sigma^{-2} & 0 \\ 0 & \rho^{-2} \\
\end{bmatrix}.
\end{eqnarray*}
The Wiener path integral \cite{Schulman}, \cite{Freidlin},
\cite{Kleinert}, \cite{unidirect}, \cite{ZakaiUstunel}
\begin{eqnarray}
&&p(e,y,t\,|\,\xi,\eta,s)=\label{pxyt}\\
&&\nonumber\\
&&\lim_{N\to\infty} \underbrace{\int_{L}de_1\int_{L}de_2\cdots
\int_{L}de_{N-1}}_{N-1}\underbrace{\int_{\hbox{\bb R}}dy_1\int_{\hbox{\bb R}}dy_2\cdots
\int_{\hbox{\bb R}}dy_{N-1}}_{N-1}\times\nonumber \\
&&\nonumber\\
&&\prod_{k=1}^N\left[\frac{\displaystyle\exp\left\{-\frac{{\cal B}_k(\mbox{\boldmath$x$}_k,\mbox{\boldmath$x$}_{k-1})}{2\varepsilon^2\Delta t }\right\}}
{2\pi\varepsilon^2\rho\sigma\Delta t }\right],\nonumber
\end{eqnarray}
with $e_N=e,\ y_N=y,\ e_0=\xi,\ y_0=\eta$, is the solution of the
FPE (\ref{2DFPE}) with the boundary and initial conditions
(\ref{BC}) and (\ref{IC}).
The pdf (\ref{peNyN}) can be written as
\begin{eqnarray}
&&p_N(e_1,e_2,\ldots,e_N;y_1,y_2,\ldots,y_N;t_1,t_2,\dots,t_N)=\label{break}\\
&&\nonumber\\
&&\prod_{k=1}^N\left[\frac{1}{{\sqrt{2\pi\Delta t }}\,\varepsilon\sigma}\exp\left\{
-\frac{[e_k-e_{k-1}-\Delta t M_{\hat x}(e_{k-1},t_{k-1})\,]^2}
{2\varepsilon^2\sigma^2\Delta t }\right\}\times\right.\nonumber\\
&&\nonumber\\
&&\left. \exp\left\{\frac{1}{\varepsilon^2\rho^2}
H_{\hat x}(e_{k-1},t_{k-1})(y_k-y_{k-1})-\frac{1}{2\varepsilon^2\rho^2}
H_{\hat x}^2(e_{k-1},t_{k-1})\Delta t \right\}\right]\times\nonumber\\
&&\nonumber\\
&& \left[\prod_{k=1}^N
\frac{\exp\left\{-\displaystyle\frac{(y_k-y_{k-1})^2}{2\varepsilon^2\rho^2\Delta t
}\right\}} {{\sqrt{2\pi\Delta t }}\,\varepsilon\rho}\right],\nonumber
\end{eqnarray}
where, by the Feynman-Kac formula \cite{Schulman}, \cite{Freidlin},
\cite{Kleinert}, \cite{unidirect}, \cite{ZakaiUstunel}, the first
product gives in the limit the function
\begin{eqnarray*}
&&\varphi(e,t,\rho)=\\
&&\\
&&\lim_{N\to\infty} \underbrace{\int_{L}de_1\int_{L}de_2\cdots
\int_{L}de_{N-1}}_{N-1}\prod_{k=1}^N\left[\frac{1}{{\sqrt{2\pi\Delta
t }}\,\varepsilon\sigma}
\times\right.\\
&&\\
&&\left.\exp\left\{-\frac{[e_k-e_{k-1}-\Delta t M_{\hat x}
(e_{k-1},t_{k-1})\,]^2}{2\varepsilon^2\sigma^2\Delta t }\right\}\times\right.\nonumber\\
&&\nonumber\\
&&\left. \exp\left\{\frac{1}{\varepsilon^2\rho^2}
H_{\hat x}(e_{k-1},t_{k-1})(y_k-y_{k-1})-\frac{1}{2\varepsilon^2\rho^2}
H_{\hat x}^2(e_{k-1},t_{k-1})\Delta t \right\}\right],
\end{eqnarray*}
which is the solution of the Zakai's equation in Stratonovich form
\cite{WZ}
\begin{eqnarray}
d_S\varphi(e,t,\rho)&=&\left\{-[\,M_{\hat
x}(e,t)\varphi(e,t)\,]_e+\frac{1}{2}[\,\varepsilon^2\sigma^2
\varphi(e,t\,]_{ee}-\frac{\varphi(e,t)H_{\hat x}^2(e,t)}{2\varepsilon^2\rho^2}\right\}\,dt +\nonumber\\
&&\nonumber\\
&& \frac{\varphi(e,t)H_{\hat x}(e,t)}{\varepsilon^2\rho^2}
\,d_Sy(t),\label{ZakaiStr1}
\end{eqnarray}
with the boundary conditions
\begin{eqnarray}
\varphi(e,t,\rho)=0\quad\mbox{for}\quad e\in\partial L.\label{ZBC}
\end{eqnarray}
Therefore the joint density
\begin{eqnarray*}
&&p_N(e_N,t_N;\,y_1,y_2,\ldots,y_N)=\\
&&\\
&&
\Pr\{e_N(t_N)=e_N,\tau>t;y_N(t_1)=y_1,y_N(t_2)=y_2,\ldots,y_N(t_N)=y_N\}
\end{eqnarray*}
can be written at $t=t_N, e_N=e$ as
\begin{eqnarray}
p_N(e,t;\,y_1,y_2,\ldots,y_N)&=&\left[\varphi(e,t,\rho)+o(1)\right]
\prod_{k=1}^N\frac{1}{\sqrt{2\pi\Delta t}\varepsilon\rho}
\exp\left\{-\frac{(y_k-y_{k-1})^2}{2\varepsilon^2\rho^2\Delta t}\right\},
\label{jointN}
\end{eqnarray}
where $o(1)\to0$ as $N\to\infty$. Equivalently,
\begin{eqnarray}
\varphi(e,t,\rho)=\frac{p_N(e,t;y_1,y_2,\ldots,y_N)}
{\displaystyle\prod_{k=1}^N\frac{1}{\sqrt{2\pi\Delta t}\varepsilon\rho}
\exp\left\{-\frac{(y_k-y_{k-1})^2}{2\varepsilon^2\rho^2\Delta t}\right\}}+o(1),
\label{aposteriori}
\end{eqnarray}
which can be interpreted as follows: $\varphi(e,t,\rho)$ is the
joint conditional density of $e_N(t)$ and $\tau>t$, given the entire
trajectory $\{y_{N}(t_i)\}_{i=0}^N$, however, the probability
density of the trajectories $\{y_{N}(t_i)\}_{i=0}^N$,
\begin{eqnarray*}
p_N^B(y_0^t)=\prod_{k=1}^N\left[\frac{\exp\left\{-\displaystyle\frac{(y_k-y_{k-1})^2}
{2\varepsilon^2\rho^2\Delta t}\right\}}{\sqrt{2\pi\Delta
t}\varepsilon\rho}\right],
\end{eqnarray*}
is Brownian, rather than the a priori density imposed by
(\ref{err_eq}), (\ref{err_meas}).
Now,
\begin{eqnarray*}
&&\Pr\{\tau>t_N,y_N(t_1)=y_1,y_N(t_2)=y_2,\ldots,y_N(t_N)=y_N\}=\\
&&\\
&&
\Pr\{\tau>t_N\,|\,y_N(t_1)=y_1,y_N(t_2)=y_2,\ldots,y_N(t_N)\}\times\\
&&\\
&& \Pr\{y_N(t_1)=y_1,y_N(t_2)=y_2,\ldots,y_N(t_N)=y_N\},
\end{eqnarray*}
which we abbreviate to
\begin{eqnarray}
\Pr\{\tau>t,y_0^t\}=\Pr\{\tau>t\,|\,y_0^t\}p_N(y_0^t),\label{Abbreviated}
\end{eqnarray}
where the density
$p_N(y_0^t)=\Pr\{y_N(t_1)=y_1,y_N(t_2)=y_2,\ldots,y_N(t_N)=y_N\}$ is
defined by the system (\ref{dxN}), (\ref{dyN}), independently of
$\hat x(t)$.
We now use the abbreviated notation (\ref{Abbreviated}) to write
\begin{eqnarray}
\Pr\{\tau>t\,|\,y_0^t\}&=&
\frac{\Pr\{\tau>t,y_N(t_1)=y_1,y_N(t_2)=y_2,\ldots,y_N(t_N)=y_N\}}{p_N(y_0^t)}
\nonumber\\
&&\nonumber\\
&=&\int_{L}\frac{p_N(e,t;y_1,y_2,\ldots,y_N)}{p_N(y_0^t)}
\,de\nonumber\\
&&\nonumber\\
&=&\frac{p_N^B(y_0^t)}{p_N(y_0^t)}\int_{L}\left\{\varphi(e,t,\rho)+o(1)\right\}\,de.
\label{Survival}
\end{eqnarray}
As $N\to\infty$, both sides of eq.(\ref{Survival}) converge to a
finite limit, which we write as
\begin{eqnarray}
\Pr\{\tau>t\,|\,y_0^t\}=\alpha(t)\int_L\varphi(e,t)\,de,\label{SPHI}
\end{eqnarray}
where
$$\alpha(t)=\lim_{N\to\infty}\frac{p_N^B(y_0^t)}{p_N(y_0^t)},$$
is a function independent of $\hat x(t)$.
Next, we show that $E[\tau\,|\,y_0^\tau]$, as defined in
(\ref{EtauN}), (\ref{Etauy}), is given by
\begin{eqnarray}
E[\tau\,|\,y_0^\tau]=\int_0^\infty\Pr\{\tau>t\,|\,y_0^t\}\,dt.\label{EPtau}
\end{eqnarray}
Indeed, since $\Pr\{\tau>t\,|\,y_0^t\}\to0$ exponentially fast as
$t\to\infty$, we can write
\begin{eqnarray*}
\int_0^\infty\Pr\{\tau>t\,|\,y_0^t\}\,dt=\lim_{T\to\infty}\int_0^T
td\Pr\{\tau <t\,|\,y_0^t\}
\end{eqnarray*}
and
\begin{eqnarray*}
\int_0^T td\Pr\{\tau
<t\,|\,y_0^t\}=\lim_{N\to\infty}\sum_{i=1}^Ni\Delta t\Delta
\Pr\{\tau <i\Delta t\,|\,y_0^{i\Delta t}\},
\end{eqnarray*}
where
\begin{eqnarray*}
\Delta \Pr\{\tau <i\Delta t\,|\,y_0^{i\Delta t}\}= \Pr\{\tau
<i\Delta t\,|\,y_0^{i\Delta t}\}-\Pr\{\tau<(i-1)\Delta
t\,|\,y_0^{(i-1)\Delta t}\}.
\end{eqnarray*}
Now, we renumber the sampled
trajectories $e_{j,N}(t_i)$ in the numerator in (\ref{EtauN})
according to increasing $\tau_{i,N}$, so that in the new enumeration
$\tau_{i,N}=i\Delta t$. Then we group together the terms in the sum
that have the same $\tau_{i,N}$ and denote their sums $m_{i,N}$, so
that (\ref{EtauN}) becomes
\begin{eqnarray}
E\left[\tau_N\wedge T\,|\,\left\{y_N(t_i),\,i=0,1,\ldots,\displaystyle\left[\frac{\tau_{N}}
{\Delta t}\right]\wedge N\right\}\right]&=&\frac{\displaystyle\sum_{i=1}^Ni\Delta t\:m_{i,N}}
{\displaystyle\sum_{i=1}^N m_{i,N}}.
\label{rearranged}
\end{eqnarray}
Finally, we identify
\begin{eqnarray*}
\Delta \Pr\{\tau <i\Delta t\,|\,y_0^{i\Delta
t}\}=\frac{m_{i,N}}{\displaystyle\sum_{i=1}^{N}m_{i,N}}\left(1+o(1)\right)
\end{eqnarray*}
where $o(1)\to0$ as $N\to\infty$. Hence (\ref{EPtau}) follows.
Finally, we identify
\begin{eqnarray*}
\Delta \Pr\{\tau <i\Delta t\,|\,y_0^{i\Delta
t}\}=\frac{m_{i,N}}{\displaystyle\sum_{i=1}^Nm_{i,N}}\left(1+o(1)\right)
\end{eqnarray*}
where $o(1)\to0$ as $N\to\infty$. Hence (\ref{EPtau}) follows.
\subsection{Asymptotic solution of Zakai's equation and the optimal filter}
For small $\varepsilon$ the solution of (\ref{ZakaiStr1}) with the
boundary conditions (\ref{ZBC}) is constructed by the method of
matched asymptotics \cite{Bender}, \cite{MS}, \cite{book}. The outer
solution is given by large deviations theory \cite{Hijab},
\cite{Freidlin}, \cite{Stroock}, \cite{DZ} as
\begin{eqnarray*}
\varphi_{\mbox{\footnotesize { outer}}}(e,t)&=&\exp\left\{-\frac{\psi(e,t)}{\varepsilon^2}\right\},
\end{eqnarray*}
where
\begin{eqnarray}\label{psi}
\psi(e,t)=\inf_{\displaystyle e(\cdot)\in{\cal C}_e^1([0,t])}\int_0^{t\wedge\tau}\left\{\left[\frac{\dot
e(s)-M_{\hat x}(e(s),s)}{\sigma}\right]^2+\left[\frac{\dot
y(s)-H_{\hat x}(e(s),s)}{\rho}\right]^2\right\}\,ds,
\end{eqnarray}
and
\begin{eqnarray*}
{\cal C}_e^1([0,t])&=&\left\{e(\cdot)\in{\cal
C}^1([0,t])\,:e(0)=e\right\}.
\end{eqnarray*}
We denote by $\tilde e(t)$ the minimizer of the integral on the
right hand side of eq.(\ref{psi}). The outer solution
$\varphi_{\mbox{\footnotesize { outer}}}(e,t)$ does not satisfy the
boundary conditions (\ref{ZBC}), so a boundary layer correction
$k(e,t,\varepsilon)$ is needed to obtain a uniform asymptotic
approximation,
\begin{eqnarray}
\varphi(e,t)\sim
\varphi_{\mbox{\footnotesize{uniform}}}(e,t)=\varphi_{\mbox{\footnotesize{outer}}}(e,t,\rho)k(e,t,\varepsilon)=
\exp\left\{-\frac{\psi(e,t)}{\varepsilon^2}\right\}k(e,t,\varepsilon).\label{uniform}
\end{eqnarray}
The boundary layer function has to satisfy the boundary and matching
conditions
\begin{eqnarray}
k(e,t,\varepsilon)=0\quad\mbox{for}\quad e\in\partial L,\quad
\lim_{\varepsilon\to0}k(e,t,\varepsilon)=1\quad\mbox{for}\quad e\in
L,\label{match}
\end{eqnarray}
uniformly on compact subsets of the interior of $L$.
Since the survival probability is
\begin{eqnarray*}
\Pr\left\{\tau>t\,|\,y_0^t\right\
=\int_{L}\alpha(t)\exp\left\{-\frac{\psi(e,t)}{\varepsilon^2}\right\}k(e,t,\varepsilon)\,de,
\end{eqnarray*}
the MTLL, according to (\ref{EPtau}), is given by
\begin{eqnarray}\label{ETAU}
E[\tau\,|\,y_0^\tau]=
\int_0^\infty\int_{L}\alpha(t)\exp\left\{-\frac{\psi(e,t)}{\varepsilon^2}\right\}
k(e,t,\varepsilon)\,de\,dt.
\end{eqnarray}
In view of (\ref{exhat}), the minimizer $\tilde e(t)$ of the
integral on the right hand side of (\ref{psi}) can be represented as
$\tilde e(t)=\tilde x(t)-\hat x(t)$, where $\tilde x(t)$ is the
minimizer of the integral
\begin{eqnarray}
\Psi(x,t)=\inf_{\displaystyle x(\cdot)\in{\cal
C}_x^1([0,t])}\int_0^{t\wedge\tilde\tau}\left\{\left[\frac{\dot
x(s)-m(x(s),s)}{\sigma}\right]^2+\left[\frac{\dot
y(s)-h(x(s),s)}{\rho}\right]^2\right\}\,ds,\label{xtilde}
\end{eqnarray}
where $\tilde\tau=\inf\{t\,:\,\tilde x(t)-\hat x(t)\in\partial L\}$
and
\begin{eqnarray*}
{\cal C}_x^1([0,t])=\left\{x(\cdot)\in{\cal
C}^1([0,t])\,:x(0)=x\right\}.
\end{eqnarray*}
Writing $\psi(e,t)=\Psi(x,t)$ and
$k(e,t,\varepsilon)=K(x,t,\varepsilon)$, we rewrite (\ref{ETAU}) as
\begin{eqnarray}
E[\tau\,|\,y_0^\tau]= \int_0^\infty\int_{L+\hat
x(t)}\alpha(t)\exp\left\{-\frac{\Psi(x,t)}{\varepsilon^2}\right\}
K(x,t,\varepsilon)\,dx\,dt.\label{ETAU2}
\end{eqnarray}
The integral in (\ref{ETAU2}) is evaluated for small $\varepsilon$
by the Laplace method, in which the integrand is approximated by a
Gaussian density with mean $\tilde x(t)$ and variance proportional
to $\varepsilon^2$. It is obviously maximized over the functions
$\hat x(t)$ by choosing $\hat x(t)$ so that the domain of
integration covers as much as possible of the area under the
Gaussian bell. If $L$ is an interval, then the choice $\hat
x(t)=\tilde x(t)$ is optimal. We conclude that for small noise, the
minimum noise energy filter $\tilde x(t)$ is asymptotically the
maximum MTLL filter.
\section{Discussion}
The main result of this paper is a proof that for small noise,
the minimum noise energy filter maximizes the mean time the
estimation error stays within a given region, e.g., maximizes the
mean time to lose lock in problems of phase tracking and
synchronization. The MNE filter is not finite-dimensional, however
finite discrete approximations, such as Viterbi-type algorithms
\cite{Unger}, \cite{ViterbiCoding}, can give arbitrary accuracy. The
practical aspects of finding the true MNE filter, or otherwise
adequate approximations for it, was partially dealt with in
\cite{Ehud} and still remains an interesting issue for further
studies.
Katzur {\em et. al.} \cite {KBS}, and subsequently Picard
\cite{Picard86}\cite{Picard91}, have shown that for nonlinear, but
monotone measurement functions, the MNE filter is to leading order
identical to the extended Kalman filter. However, for measurement
functions which are non-monotone, this is apparently not the case.
Ezri \cite{Doron} and Fischler \cite{Ehud} have considered the
problem of phase filtering and smoothing respectively, in which the
stochastic phase process $x(t)$ is measured in a low noise channel
by the vector function $\mbox{\boldmath$h$}(x) =
\left[\:\sin(x),\,\cos(x)\:\right]^T$. They show that there is a
huge gap between the MTLLs of the extended Kalman filter (smoother)
or particle filter, and the MNE filter (smoother), respectively.
The great advantage of the MNE filter in the case of phase
estimation is explained by the observation that finite-dimensional
approximations to the MAP or minimal MSEE filters (the EKF or the
finite dimensional filters of Katzur \cite{KBS}), do not capture
large deviations of the signal or of the measurements noise. They
are optimal only near local maxima of the a posteriori probability
density. The MNE filter, in contrast, is a \textit{global} MAP
estimator and can track large deviations. Thus, it is less
vulnerable to loss of lock phenomena, relative to the above
mentioned filters.\\
\noindent {\bf Acknowledgment:} The authors thank B.Z. Bobrovsky, O.
Zeitouni, D. Ezri, B. Nadler and A. Taflia for useful discussions.
|
1,314,259,995,786 | arxiv | \section{INTRODUCTION}
Model-free reinforcement learning has been making significant progress in complex sensorimotor control problems, particularly when optimizing end-to-end vision-based policies \cite{nvidia_driving}. The lack of need for an explicit dynamics model has nevertheless incurred a significant cost in the form of long training times, large number of interactions with the environment, and in many cases, uninformed exploration. These drawbacks often make model-free reinforcement learning impractical and unsafe to apply to real robotic systems.
We propose a method that combines reinforcement learning (RL) with demonstrations and imitation learning (IL) in order to address these issues and accelerate the policy optimization process. Our method consists of two phases: (a) offline training of a generative model to be used as a state-action potential function $\Phi(s,a)$ for reward shaping, and (b) online RL that uses the learned potential to shape the sparse task reward, making the learning problem easier. Although generally applicable, in our case we use normalizing flows~\cite{pmlr-v37-rezende15} and Generative Adversarial Networks~\cite{NIPS2014_5423, ho_gail} learned from a small number of demonstrations for (a) and TD3~\cite{Fujimoto2018AddressingFA} for (b),
Our method provides an alternative to existing methods that combine RL with demonstrations, because it gracefully handles the case of suboptimal and noisy demonstrations. It does this by shaping the reward function to incorporate user demonstrations in the form of \textit{advice}~\cite{Wiewiora:2003:PMA:3041838.3041938} that biases the optimization process towards areas of the state-action space that the demonstrator deems high-value, without biasing the learned policy away from the optimal solution.
\textbf{Drawbacks of existing approaches:} The majority of existing works that combine RL with demonstrations~\cite{DBLP:conf/aaai/HesterVPLSPHQSO18, Vecerk2017LeveragingDF, Zhu-RSS-18} implicitly assume optimality of demonstrations, or lack of bias in the off-policy data. If the demonstration dataset is $\mathcal{D}=\{(s_i, a_i), i=1...N\}$ these methods typically solve a variant of the following problem:
\begin{eqnarray}
\underset{\theta}{\text{max}} \; \mathcal{V}^{\pi_{\theta}}(s_0) \;\; \text{subject to} \;\; \pi_{\theta}(s_i)=a_i \;\; \forall i
\label{eqn:rl_il_1}
\end{eqnarray}
where $\mathcal{V}^{\pi_{\theta}}(s_0) = \mathbb{E}_{\pi_{\theta}} \left[ \sum_{t=0}^{\infty} \gamma^t r(s_t, a_t) \; | \; s_0\right]$ is the value function corresponding to the policy $\pi_{\theta}$ and the fixed starting state $s_0$. This problem ends up being converted to one that instead has a soft regularization term for the demonstrations:
\begin{eqnarray}
\underset{\theta}{\text{max}} \; \mathcal{V}^{\pi_{\theta}}(s_0) - \beta \sum_{(s_i, a_i) \in \mathcal{D}}(\pi_{\theta}(s_i) - a_i)^2
\label{eqn:rl_il_2}
\vspace{-0.3cm}
\end{eqnarray}
There are a number of drawbacks to this formulation:
(a) It assumes optimal demonstrations, which is often not a realistic assumption, particularly when suboptimal actions occur at states near which optimal trajectories need to pass through. In another instance of this same issue, a number of recent works, for example~\cite{DBLP:conf/aaai/HesterVPLSPHQSO18, Vecerk2017LeveragingDF}, include the demonstrations permanently in the replay buffer for off-policy RL methods, which again assumes optimality of the demonstrations. Suboptimality could be a result of the demonstrator not optimizing the same underlying reward as the RL problem, or not being an expert. Both of these possibilities are unaccounted for by the formulations in Eqns.~\ref{eqn:rl_il_1} and ~\ref{eqn:rl_il_2} and can bias the learned policy away from the optimal policy.
(b) A number of recent papers, for example~\cite{rajeswaran}, address (a) by reducing the effect of the demonstrations over time, by replacing $\beta$ with a decreasing sequence $\beta_t$ such that $\lim_{t\to\infty} \beta_t = 0$. While this addresses the issue of suboptimality and eventually forgets the demonstrations, it introduces another design parameter, namely the speed at which the demonstrations will be forgotten.
(c) The formulations in Eqns.~\ref{eqn:rl_il_1} and ~\ref{eqn:rl_il_2} cannot gracefully handle multi-modal action distributions at a given state. If the dataset includes $(s_i, a_i)$ and $(s_i, a_j)$ then the policy is forced to compromise by selecting the average action, which might be neither desirable nor safe. Multi-modal policies avoid this issue, but deterministic or unimodal policies do not.
\textbf{Our contributions:} Our work extends that of Brys, Harutunyan et al.~\cite{Brys:2015:RLD:2832581.2832716}, who incorporate demonstrations via shaping potentials that, unlike the generative models that we make use of here, are not suited for handling high-dimensional structured state spaces. Our method addresses the issues above and brings the following advantages over formulations like the one in Eqn.~\ref{eqn:rl_il_2}:
\begin{itemize}
\item It does not make any assumptions about optimality of the demonstrations, and it does not allow the demonstrations to introduce bias to the learned policy.
\item It does not require a forgetting schedule for the demonstrations.
\item It can handle multi-modal demonstrations gracefully.
\end{itemize}
\noindent We demonstrate these properties via an extensive range of simulations as well as via real robot experiments on the Franka Emika 7DOF compliant arm.
\section{RELATED WORK}
There is a large number of methods augmenting RL with demonstrations, many of them in the realm of discrete MDPs and game playing, which we cannot cover here, but we include the main ideas from continuous control tasks.
\textbf{RL + Shaping:} Our work builds upon Wiewiora et al.~\cite{Wiewiora:2003:PMA:3041838.3041938}, who showed that a state-action potential biases the $Q$-function of the original MDP, by the exact amount of the shaping potential. They introduce the notion of \textit{advice} for an RL agent. The class of shaping potentials they considered, however, was limited to discrete action and state spaces, and was not applicable to robotics, or high-dimensional systems. Our work addresses this setting by using shaping potentials that are directly trained from demonstration data via generative models. Also related is the seminal work of Ng et al.~\cite{ng1999policy}, that introduced the notion of reward shaping and the conditions under which policy invariance holds when rewards are modified.
\textbf{RL + Optimal Demonstrations:} Deep Q-Learning from Demonstrations~\cite{DBLP:conf/aaai/HesterVPLSPHQSO18}, DDPG from Demonstrations~\cite{Vecerk2017LeveragingDF}, as well as \cite{Zhu-RSS-18}, implicitly assume optimality of the demonstrations data, unlike our work herein. In fact,~\cite{Vecerk2017LeveragingDF} assumes that the demonstration data are in the form $(s,a,r,s')$, which constrains the type of interactions that the demonstrator can have with the robot. Having access to the reward, in addition to states and actions, is difficult in scenarios where the robot is not aware of the task that the demonstrator is executing. It is also restrictive in the sense that it does not easily allow extensions, where only the states are given but not the actions. We therefore avoid assuming imitation data in that form and opt for tuples $(s,a)$ as the demonstration dataset. The notion of \textit{advice} assumes other forms, for example high-level Linear Temporal Logic formulas that guide the (discrete) RL process, as was done in~\cite{icarteusing}. SQIL~\cite{reddy} incorporates demonstrations in the replay buffer and assigns a reward of +1 to them.
\textbf{RL + Suboptimal Demonstrations:} In~\cite{rajeswaran} a schedule is followed for forgetting the demonstrations. Optimization eventually focuses on the RL objective, but only after the policy has been initialized with behavioral cloning. Abrupt transitioning was also studied in ~\cite{Cheng-UAI-18}. A shaping approach to combine imitation and RL objective is described in~\cite{sun2018truncated}, but using state-based potentials. AC-Teach~\cite{kurenkov2019acteach} handles the case of suboptimal demonstrations using an ensemble of demonstrators and Bayesian actor-critic methods. Nair et al.~\cite{nair_exploration} provide another way of not assuming optimal demonstrations, called \emph{Q-Filtering}, whereby they only keep the terms of the behavioral cloning loss for which the demonstrated action has higher $Q$ value than the action returned by the policy. A similar idea appears in~\cite{Ning2020ReinforcementLW}, as well as in~\cite{drop} where demonstrations are dropped when outside of a confidence interval. In~\cite{Jing2020ReinforcementLF} imperfect demonstrations are handled in terms of a soft constraint, using constrained RL. \cite{rlid} provides a unified reinforcement learning objective that also handles imperfect demonstrations through soft optimality. Our method differs from this paper in that we only assume $(s,a)$ pairs, while it also assumes access to the reward. \cite{nair2020accelerating} presents a method for doing policy search while keeping it close to the behavior policy that has been learned from an offline dataset. In~\cite{Grollman2012} the case of failed demonstrations is considered, enabling the robot to learn from both successful and unsuccessful demonstrations. TAMER~\cite{tamer} and other interactive RL approaches assume a continuous user feedback mechanism that provides the RL reward, which is a significant burden on the user.
\textbf{Batch/Offline RL:} Most works in this area ~\cite{pmlr-v97-fujimoto19a, stabilizing_q_learning, levine2020offline} aim to constrain the effect of poorly estimating $Q$ away from the collected data. Importance sampling is a commonly used way to do this, as long as one has access to the behavioral policy or can approximate it~\cite{Precup:2001, DBLP:journals/corr/MunosSHB16, DBLP:journals/corr/JiangL15, wu2019behavior, kumar2020discor}, which may lead to lack of scalability in high dimensions. Alternatively, one could force the Q-estimation to be conservative~\cite{kumar2020conservative}, but it remains unclear whether the learned policy can be easily refined via online RL. Challenges in refining a policy that has been initialized through offline RL are described in~\cite{nair2020accelerating}.
\textbf{Residual RL:} Methods in this family~\cite{residual_rl_1, rpl, residual_rl_controller} decompose the policy into a sum of two parts, one representing prior knowledge, namely trained from demonstration data, and one residual policy that is learned through RL.
\section{METHODOLOGY}
\subsection{State-Action, Potential-Based Reward Shaping}
Given a Markov Decision Process (MDP) $\mathcal{M}=(\mathcal{S}, \mathcal{A},\mathcal{T}, r, \gamma)$, reward shaping, as introduced in the seminal work by Ng et al \cite{ng1999policy} refers to modifying the (often sparse) reward function in order to solve another MDP $\mathcal{\widetilde{M}}=(\mathcal{S}, \mathcal{A},\mathcal{T}, \widetilde{r}, \gamma)$ such that:
\begin{equation}
\widetilde{r}_t = r(s_t,a_t, s_{t+1}) + \gamma \Phi(s_{t+1}) - \Phi(s_t)
\end{equation}
\noindent The function $\Phi$ is called a \textit{shaping potential}, and it is meant to make sparse reward functions more dense by providing more reward signal for the recursive computation of the state-action value function $Q(s,a)$. Ng et al. showed that the optimal value functions between the original and modified MDPs satisfies the following equation:
\begin{equation}
\widetilde{Q}^*(s,a) + \Phi(s) = Q^*(s,a)
\end{equation}
\noindent Every optimal policy corresponding to these state-action value functions will satisfy $\pi^*(s) = \text{argmax}_a Q^*(s,a) = \text{argmax}_a \widetilde{Q}^*(s,a)=\widetilde{\pi}^*(s)$. In other words, every optimal policy for $\mathcal{M}$ will be optimal for $\mathcal{\widetilde{M}}$ and vice versa, so the optimal behavior is not affected by the shaping function, even though the value function is.
Wiewiora et al. \cite{wiewiora2003potential} showed that the shaping potential did not need to depend only on states, it could also depend on actions. The modified reward then becomes:
\begin{equation}
\widetilde{r}_t = r(s_t,a_t, s_{t+1}) + \gamma \Phi(s_{t+1}, a_{t+1}) - \Phi(s_t, a_t)
\label{eqn:state_action_potential}
\end{equation}
\noindent which gives rise to the following state-action value function:
\begin{equation}
\widetilde{Q}^*(s,a) + \Phi(s,a) = Q^*(s,a)
\end{equation}
\noindent In this case of state-action shaping potentials, there are no guarantees about the preservation of the optimal policy of the original MDP to the modified MDP. In fact, the optimal policy of the original MDP is
\begin{equation}
\pi^*(s) = \text{argmax}_a \left[ \widetilde{Q}^*(s,a) + \Phi(s,a)\right]
\label{eqn:advice_policy}
\end{equation}
\noindent while the optimal policy of the modified MDP is $\widetilde{\pi}^*(s) = \text{argmax}_a \left[ \widetilde{Q}^*(s,a)\right]$, which is in general different.
Wiewiora et al. \cite{wiewiora2003potential} demonstrated potential functions for discrete state and action spaces, that were constrained to low-dimensional discrete planning problems, which are not applicable to robotics. Our paper analyzes the case where the state and action space is high-dimensional and continuous, and the shaping potential is trained via generative models, in order to support many types of demonstration data and improve the convergence properties of imitation-guided model-free RL. As long as we are able to optimally solve Eqn.~\ref{eqn:advice_policy} and $\widetilde{Q}^*(s,a)$ is well estimated, the learned policy incorporates \textit{advice} $\Phi(s,a)$, without imposing the demonstrations as hard constraints, and without introducing bias compared to the optimal policy.
\subsection{Potentials Based On Normalizing Flows}
One of the types of state-action shaping potentials that we consider herein is a trained function ${\Phi_{\psi, c}(s,a)=c \; \text{log} \; p_{\psi}(s,a)}$ on demonstration data $\mathcal{D}=\{(s_i, a_i), i=1...N\}$. One class of generative models that have emerged in the last few years, that is able to directly optimize this log-density objective on a given dataset are \textit{normalizing flows}.
The main idea behind this class of models is that we can use the change-of-variables formula for probabilistic distributions to transform a normal distribution (a distribution that is easy to sample) to an arbitrary distribution (from which it is difficult to sample). Given a random variable $z_0 \in \mathbb{R}^d$, such that $z_0 \sim p_0(z_0)=\mathcal{N}(0, I_d)$, and an invertible, smooth function $f: \mathbb{R}^d \rightarrow \mathbb{R}^d$ with $z_1=f(z_0)$, the change of variables formula for distributions is:
\begin{eqnarray}
p_1(z_1) & = & p_0(z_0)\; \bigg |\text{det}\left(J_f(z_0)\right) \bigg|^{-1}
\end{eqnarray}
\noindent Rezende and Mohamed \cite{pmlr-v37-rezende15} chained multiple of these bijective transformations to create a normalizing flow:
\begin{eqnarray}
z_0 \sim p_0(z_0),
z_K & = & f_K \circ f_{K-1} \circ ... \circ f_1(z_0) \\
p_K(z_K) & = & p_0(z_0) \prod_{k=1}^K \; \bigg |\text{det} \left( J_{f_k}(z_{k-1}) \right) \bigg|^{-1}
\end{eqnarray}
\noindent where $\circ$ denotes function composition. The vast majority of the recent literature on normalizing flows concerns itself with different ways to parameterize bijective functions $f_{\psi_i}(z)$ in a way that chaining multiple of them results in an expressive enough output distribution. We follow Papamakarios et al \cite{papamakarios2017masked} and we use the same bijective transformation as Masked Autoregressive Flow (MAF):
\begin{eqnarray}
z_k^{(1)} & = & \mu_{w_{k_1}} + \text{exp}(\alpha_{v_{k_1}}) z_{k-1}^{(1)} \nonumber \\
z_k^{(i)} & = & \mu_{w_{k_i}}(z_k^{(1:i-1)}) + \text{exp}(\alpha_{v_{k_i}}(z_k^{(1:i-1)})) z_{k-1}^{(i)} \quad \quad
\label{eqn:maf_transform}
\end{eqnarray}
\noindent Here, the superscript $i \leq d$ indexes the dimensions of the random variable $z_k \in \mathbb{R}^d$, and makes the $i^{\text{th}}$ entry of the output variable depend only on entries $1...i$ of the input variable. This preserves the triangular structure of the Jacobian matrix, so the determinant remains easy to compute. The parameters of the transform $f_{\psi_k}(z_{k-1})$ described in Eqn. ~\ref{eqn:maf_transform} are $\psi_k=(w_{k_1}, v_{k_1}, ..., w_{k_d}, v_{k_d})$. The exponential term for the scaling factor is meant to ensure the positivity of standard deviation.
Training a normalizing flow is typically done via maximum likelihood estimation, by optimizing the parameters $\psi=(\psi_1, \psi_2,...,\psi_K)$, so that the log likelihood of the points in the sample dataset is maximized. In our case, we treat $z_K=(s,a)$, since we assume access to states and not high-dimensional image data. The log-likelihood objective we want to maximize is:
\begin{equation}
\mathcal{L}(\psi, \mathcal{D}) = -\sum_{(s_i, a_i) \in \mathcal{D}} \; \sum_{k=1}^K \; \text{log} \; \bigg|\text{det}\left(J_{f_{\psi_k}}(z_{k-1})\right)\bigg|
\end{equation}
\noindent In order to avoid learning density functions $p_K(z_K)$ that exhibit large changes whenever $z_K=(s,a)$ changes slightly, we regularize the Jacobian of the learned density with respect to its input $z_K$. Our final training cost for the shaping potential based on normalizing flows is:
\begin{equation}
\mathcal{L}_{\text{flow}}(\psi, \mathcal{D}) = \mathcal{L}(\psi, \mathcal{D}) + \eta || \nabla_{z_K} \text{log} \; p_K(z_K) ||^2
\end{equation}
\noindent Once the optimal parameters $\psi^*$ are identified from the training process, we use the following shaping potential:
\begin{eqnarray}
\Phi_{\psi^*, c}(s,a)=c \; \text{log} \; ( p_{\psi^*}(s,a) + \epsilon)
\label{eqn:nf_potential}
\end{eqnarray}
\noindent with $z_K=(s,a)$, $c \in \mathbb{R}^{+}$ a hyperparameter, and $\epsilon$ is a small constant to prevent numerical issues and the log probability from going to negative infinity.
\textbf{Scalability:} We note that if we had chosen to make the policy input be high-dimensional, for example image-based, our current model with $z_K=(s,a)$ would be very slow to train due to the cost of evaluating the Jacobian in Eqn. \ref{eqn:nf_potential} and the autoregressive structure of the flow transform in Eqn.~\ref{eqn:maf_transform}. That said, as we will see in the experimental results section, we have used normalizing flow shaping potentials with dimension of $s,a$ being around 30 without any issues.
\subsection{Potentials Based On Generative Adversarial Networks}
The second type of state-action shaping potentials that we consider in this paper are functions $\Phi_{\psi, c}(s,a)=c \; D_{\psi}(s,a)$, trained on demonstration data $\mathcal{D}=\{(s_i, a_i), i=1...N\}$, where $D_{\psi}(s,a)$ is the discriminator of a Generative Adversarial Network (GAN) \cite{NIPS2014_5423}. These models also include a generative model $G_{\phi}(z)=\widetilde{x}$ that accepts a noise input $z \sim \mathcal{N}(0, I_d)$ and transforms it into a more structured random variable $\widetilde{x} \in \mathbb{R}^d$.
Training the generator and the discriminator is not done via maximum likelihood in this case, but through a minimax optimization problem. Let $p_r(x)$ be the real distribution of the random variable $x$ and $p_{\phi}(x)$ is the distribution induced by the generator. The end goal of the training process is to optimize the parameters of the generator, so that the distance between the real distribution and the generated distribution is minimized. The discriminator parameters are optimized so that its output is high on real samples and low on (fake) generated samples.
We follow Arjovsky et al. \cite{pmlr-v70-arjovsky17a} and Gulrajani et al \cite{Gulrajani:2017} to estimate the Wasserstein-1, or Earth Mover's distance, in order to evaluate the cost of the optimal transport plan between two probability distributions $p_r$ and $p_{\phi}$:
\begin{equation}
W(p_r, p_{\phi})=\underset{\gamma \in \Pi(p_r, p_{\phi})}{\text{inf}} \; \mathbb{E}_{(x,y) \sim \gamma(x,y)} \left[ ||x-y||\right] \quad
\end{equation}
\noindent where $\gamma(x,y)$ indicates how much mass needs to be transported from $x$ to $y$ to transform the distribution $p_r$ to $p_{\phi}$, and $\Pi(p_r, p_{\phi})$ is the set of all joint distributions, whose marginals are $p_r$ and $p_{\phi}$. Given a fixed generator $G_{\phi}$, the intractable definition above is equivalent to the more tractable one:
\begin{equation}
W(p_r, p_{\phi})=\underset{D \in \mathcal{F}}{\text{sup}} \; \left[ \mathbb{E}_{x \sim p_r} \left[D(x) \right] - \mathbb{E}_{\widetilde{x} \sim p_{\phi}} \left[D(\widetilde{x}) \right] \right]
\end{equation}
\noindent where $\mathcal{F}=\{D: \mathbb{R}^d \rightarrow \mathbb{R} \; \text{such that} \; ||D||_L \leq 1\}$ is the set of discriminator functions with Lipschitz constant 1. Sampling from $p_{\phi}$ is done by $z \sim \mathcal{N}(0, I_d)$ and $\widetilde{x}=G_{\phi}(z)$. To impose the Lipschitz constant of 1 on the discriminator we follow WGAN-GP in Gulrajani et al. \cite{Gulrajani:2017}, and impose a soft constraint to its gradient. The approximate Wasserstein distance can be computed this way:
\begin{eqnarray}
L_1(\psi, \phi) & = & \mathbb{E}_{x \sim p_r} \left[D_{\psi}(x) \right] - \mathbb{E}_{\widetilde{x} \sim p_{\phi}} \left[D_{\psi}(\widetilde{x}) \right] \\
L_2(\psi, \phi) & = & \mathbb{E}_{\widehat{x} \sim p_{\phi}} \left[ (||\nabla_{\widehat{x}} D_{\psi}(\widehat{x})|| - 1)^2\right] \\
\widetilde{W}(p_r, p_{\phi}) & = & \underset{\psi}{\text{max}} \; L_1(\psi, \phi) - \alpha L_2(\psi, \phi) \label{eqn:approx_wasserstein}
\end{eqnarray}
\noindent where $\widehat{x} = \epsilon x + (1 - \epsilon) \widetilde{x}$ with $\epsilon \sim U[0,1], x \sim p_r, \widetilde{x} \sim p_{\phi}$ is used to enforce the Lipschitz constraint on samples between the real distribution and the generated distribution, since the Lipschitz constant needs to be 1 for every possible input to the discriminator.
The approximate Wasserstein distance in Eqn.~\ref{eqn:approx_wasserstein} corresponds to a fixed generator. For the generator to improve and minimize this distance, we solve the following problem:
\begin{eqnarray}
\psi^*, \phi^* = \text{arg} \underset{\phi}{\text{min}}\underset{\psi}{\text{max}} \; L_1(\psi, \phi) - \alpha L_2(\psi, \phi)
\end{eqnarray}
\noindent The potential then becomes $\Phi_{\psi^*, c}(s,a)=c \; D_{\psi^*}(s,a) \label{eqn:gan_potential}$
\noindent \textbf{Scalability:} Training the potential on high-dimensional demonstration data is scalable as GAN training has been demonstrated to produce realistic images of faces at high resolution~\cite{karras2018progressive}.
\subsection{Combining Reinforcement and Imitation Learning via Shaping}
\begin{algorithm}
\caption{TD3 with Demonstrations via Reward Shaping}\label{alg1}
\begin{algorithmic}[1]
\Statex \textbf{Offline pre-training}
\State $\text{Collect demonstrations} \; \mathcal{D} = \{(s_i, a_i), i=1...N\} $
\State $\text{Train shaping potential} \; \Phi_{\psi^*, c}(s,a) \; \text{from Eqn.}~\ref{eqn:nf_potential} \; \text{or} \; \text{GAN}$
\Statex \hrulefill
\State $\text{Given MDP} \; \mathcal{M}=(\mathcal{S}, \mathcal{A},\mathcal{T}, r, \gamma)$
\State $\text{Consider MDP} \; \mathcal{\widetilde{M}}=(\mathcal{S}, \mathcal{A},\mathcal{T}, \widetilde{r}, \gamma) \; \text{from Eqn.}~\ref{eqn:state_action_potential} \; \text{with}$
\Statex $\widetilde{r}_t = r(s_t,a_t, s_{t+1}) + \gamma \Phi_{\psi^*,c}(s_{t+1}, a_{t+1}) - \Phi_{\psi^*, c}(s_t,a_t) $
\Statex \hrulefill
\Statex \textbf{TD3 training with reward shaping}
\State $\text{Initialize two critic networks for} \; \widetilde{M}: \widetilde{Q}_{\theta_1}, \widetilde{Q}_{\theta_2}$
\State $\text{Initialize actor network} \; \pi_{\phi}$
\State $\text{Initialize target networks} \; \theta_1^{'} \leftarrow \theta_1, \theta_2^{'} \leftarrow \theta_2, \phi^{'} \leftarrow \phi$
\State $\text{Initialize replay buffer} \; \mathcal{B} \; \text{to empty}$
\While{not converged}
\For{$\text{episode}\ e = {1...E}$}
\For{$\text{step}\ t = {1...T}$}
\State $\text{Apply action} \; a = \pi_{\phi}(s) + \epsilon, \epsilon \sim \mathcal{N}(0, \sigma)$
\State $\text{Observe reward} \; r \; \text{and new state} \; s' \; \text{from} \; \mathcal{M}$
\State $\text{Store transition tuple} \; (s, a, r, s') \; \text{in} \; \mathcal{B}$
\EndFor
\EndFor
\For {$\text{batch}\ b = {1...B}$}
\State $\text{Sample mini-batch} \; \mathcal{B}_b \; \text{of} \; (s, a, r, s') \; \text{from} \; \mathcal{B}$
\State $\text{Sample mini-batch} \; \mathcal{D}_b \; \text{of} \; (s_d, a_d) \; \text{from} \; \mathcal{D}$
\State $a' \leftarrow \pi_{\phi'} (s') + \epsilon, \epsilon \sim \text{clip}(\mathcal{N}(0, \sigma'), -\delta, \delta)$
\State $\text{Target value}\;$
\Statex $\quad \quad \quad y = \widetilde{r} + \gamma \text{min}\{\widetilde{Q}_{\theta_1'}(s',a'), \widetilde{Q}_{\theta_2'}(s',a')\}$
\State $\text{Update critics} \; \theta_i \leftarrow \text{argmin}_{\theta_i} \sum (y - \widetilde{Q}_{\theta_i}(s,a))^2$
\If {$b \; \text{mod} \; d$}
\State $\text{Update policy}$
\State $\phi \leftarrow \text{argmax}_{\phi} \sum_{s \in \mathcal{B}_b \cup \mathcal{D}_b}[ \widetilde{Q}_{\theta_1}(s,\pi_{\phi}(s)) +$
\Statex $\quad \quad \quad \quad \quad \quad \quad \Phi_{\psi^*,c}(s,\pi_{\phi}(s)) ]$
\EndIf
\EndFor
\State $\text{Update target networks}$
\State $\theta'_i \leftarrow \tau \theta_i + (1-\tau) \theta'_i$
\State $\phi' \leftarrow \tau \phi + (1-\tau) \phi'$
\EndWhile
\end{algorithmic}
\end{algorithm}
We now show how to integrate the learned shaping potentials in a model-free reinforcement learning method. We use Twin Delayed Deterministic Policy Gradient (TD3) \cite{Fujimoto2018AddressingFA} since it is one of the best performing model-free RL methods at the time of writing. TD3 is an actor-critic method that maintains two critic networks for the $Q$ function and one actor network for the deterministic policy. The use of the double-$Q$ networks helps by reducing overestimation bias in the $Q$-function, which leads to suboptimality in the learned policy.
\section{EVALUATION}
We evaluate our method both in simulation and on a real robot. Our aim is to clarify the following questions:
\begin{itemize}
\item Does our method exceed the performance of (a) behavioral cloning and (b) pure RL?
\item Is our method robust to random seeds?
\item Is our method robust to suboptimal demonstrations? In particular, does it do better than RL with behavioral cloning, as formulated in Eqn. ~\ref{eqn:rl_il_2}?
\item Is our method practical on a real robot?
\end{itemize}
\noindent We answer all these questions in the affirmative and we analyse our experimental results below.
\subsection{Robustness to Random Seeds}
\begin{figure}[t!]
\centering
\includegraphics[width=0.23\textwidth]{figures/RobustToSeedsRL_fig1.pdf}
\includegraphics[width=0.22\textwidth]{figures/RobustToSeedsRL_fig2.pdf}
\includegraphics[width=0.22\textwidth]{figures/RobustToSeedsRL_legend.pdf}
\caption{GAN and Normalizing Flow (NF) shaping and baseline results for \textit{peg insertion} and \textit{pick and place} tasks on the Fetch environment adopted from OpenAI Gym. The initial position of the gripper is selected randomly but at a certain distance away from the hole, and demonstrations are near-optimal. In both cases, both RL $+$ shaping methods converge to the optimal policy. TD3 fail to converge due to insufficient exploration, i.e. it never finds the goal state. Behavioral Cloning (BC) only succeeds when the arm is initialized to certain states. The empirical mean has been computed from 5 random seeds, and the error bars represent 1$\sigma$ standard deviation.}
\label{fig:robusttoseedRL}
\vspace{-0.3cm}
\end{figure}
The issue of robustness of policies learned via reinforcement is intricately linked to the choice of random seeds, which determine the sequence of pseudorandom number generation that will drive the exploration process, as well as the random dynamics of the environment. Henderson et al \cite{henderson2018deep} showed that many recent deep RL methods are extremely sensitive to the selection of random seeds.
We evaluated our method on complex manipulation tasks of \textit{pick and place} and \textit{peg insertion} in simulation.
(1) \textit{peg insertion}: the end effector of the robot arm is initialized at a random location that is at a certain distance away from the hole, holding the peg. The location and orientation of the hole is fixed. A reward of $0$ is given when more than half of the peg is inserted in the hole and $-1$ otherwise.
(2) \textit{pick and place}: the object is placed at a random location that is at a certain distance away from both the end effector of the arm and the goal. The initial pose of the robot arm and the goal location are fixed. A reward of $0$ is given when the object is within a small threshold around the goal location and $-1$ otherwise.
For both environments, the episode length is set to $40$, and the environment does not terminate early.
Fig.~\ref{fig:robusttoseedRL} shows our method and baseline results for \textit{peg insertion} and \textit{pick and place} tasks. We consider two baselines for all experiments, namely Behavioral Cloning (BC), which is pure supervised learning, and pure model-free RL (TD3) without demonstrations. All empirical results are presented with empirical mean and a single standard deviation across $5$ random seeds. The demonstration data for Fig.~\ref{fig:robusttoseedRL} are near optimal with no additional noise having been added.
Fig.~\ref{fig:robusttoseedRL} shows that while the two RL with shaping methods converge to goal, the Behavioral Cloning and pure RL method fail to explore sufficiently to find the goal area. The GAN shaping method converged to $4/5$ seeds, so the lower mean and the higher variance is due to that.
\subsection{Robustness to Suboptimal Demonstrations}
To illustrate the sensitivity of TD3$+$BC to noise, we simplified the \textit{peg insertion} task by fixing the initial pose of the robot arm and limiting the state space to a 2D plane as shown in Figure~\ref{fig:robusttonoisetrajectory}. We provided suboptimal demonstration data that encourages the agent to lift the peg to a high location and then perform the insertion, shown as red arrows. In addition, we also included demonstration data that pushes the learned policy away from the optimal trajectory, shown as green arrows. More crucially, these suboptimal actions are given in areas of the state space where the optimal trajectory passes through, so the imitation objective is directly clashing with the RL objective.
\vspace{-0.2cm}
\begin{figure}[h!]
\centering
\includegraphics[width=0.23\textwidth]{figures/RobustToNoiseTrajectory_fig1.pdf}
\includegraphics[width=0.23\textwidth]{figures/RobustToNoiseTrajectory_fig2.pdf}
\vspace{-0.2cm}
\caption{Illustration of our method's robustness to noisy demonstration data. The left figure shows the provided demonstration data to all $3$ methods on the right figure: TD3 with GAN shaping and $\lambda$TD3 + BC for $\lambda=0.0001$ and $\lambda=0.01$, which refers to the relative weight of the RL objective compared to the Behavioral Cloning (BC) objective. In this dataset, suboptimality in the demonstration data (red action vectors) is introduced by exaggerating the lift of the peg. Green action vectors on the left figure force this. Crucially, the suboptimal demonstrations are in an area of the state space where the optimal trajectory needs to pass through, so the two objectives will clash. The green curve shows the performance of the policy trained with one choice of $\lambda$ which in turns puts more emphasis on the demonstration data, which leads to convergence to a suboptimal policy. With careful tuning of $\lambda$, TD3 $+$ BC achieves suboptimal performance, whereas with GAN Shaping, the policy performs optimally. }
\label{fig:robusttonoisetrajectory}
\end{figure}
The effect of these suboptimal demonstrations is shown in Fig.~\ref{fig:robusttonoiseavgreturn}. In particular, we see that the RL + shaping methods converge to the optimal policy, regardless of whether the demonstration data is near-optimal or not. On the other hand, the RL + Behavioral Cloning (with constraints, such as in Eqn.~\ref{eqn:rl_il_2}), is sensitive to the relative weight of the RL vs the imitation objective. When the role of the RL objective gets reduced by $\lambda=0.0001$ compared to the imitation objective, the learned policy does not manage to find the optimal solution, while for other such settings (e.g. $\lambda=0.01$) it does. This sensitivity to the relative weighting of the RL and imitation objective is undesirable, as it will affect any demonstrator forgetting schedule that adjusts this weight over time. GAIL~\cite{ho_gail}, one of the leading imitation learning methods, is unable to solve the task in the presence of imperfect demonstrations.
\textbf{Limitations}: We note that, in general, including the potential function in the policy optimization requires tuning the regularization parameters of the generative model that change how sharply peaked the density model or the discriminator is around observed data points. In practice, this could make the optimization landscape such that it is easy to get stuck in local minima.
\begin{figure}[t!]
\centering
\includegraphics[width=0.23\textwidth]{new_figures/RobustToNoiseOptimalDemoFullSize.pdf}
\includegraphics[width=0.223\textwidth]{new_figures/RobustToNoiseSuboptimalDemo_fig.pdf}
\includegraphics[width=0.23\textwidth]{new_figures/RobustToNoiseSuboptimalDemo_legend.pdf}
\caption{2D Peg insertion task. Comparison of our method that involves TD3 with shaping vs $\lambda$TD3+BC with various $\lambda$ weights, in the case of \textit{optimal demonstrations} (top) and \textit{suboptimal demonstrations} (bottom). The latter are shown in Fig.~\ref{fig:robusttonoisetrajectory}. The hyperparameter $\lambda$ refers to the relative weighting of the RL objective vs the behavioral cloning objective. These results show that $\lambda$TD3+BC is sensitive to this hyperparameter. For example, $\lambda=0.0001$ does not find the optimal policy, whereas the other methods do. Both shaping methods outperform $\lambda$TD3+BC. GAIL, an imitation learning method, is not able to solve the task, neither with optimal nor with suboptimal demonstrations.}
\label{fig:robusttonoiseavgreturn}
\vspace{-0.3cm}
\end{figure}
\subsection{Real Robot Experiments}
For real robot experiments, we use a Franka Emika Panda 7DOF arm to perform the \textit{peg insertion} task similar to the experiments in simulation as discussed in section $A$. The learned policy controls the end effector of the robot in Cartesian velocity control mode. In order to encourage faster convergence, we fixed the initial pose of the arm and modified the reward structure such that a reward of $0$ is given when the peg is inside the hole, $-0.5$ is given when the agent is holding the peg above the hole, and $-1$ otherwise. During training, the number of episode steps is set to $100$ and episodes do not terminate early.
\setlength{\tabcolsep}{0.1em}
\def0.5{0.5}
\begin{figure}
\begin{center}
\begin{tabular}[ht!]{ccc}
\includegraphics[width=0.16\textwidth]{figures/rl_0} &
\includegraphics[width=0.16\textwidth]{figures/rl_1} &
\includegraphics[width=0.16\textwidth]{figures/rl_4} \\
\includegraphics[width=0.16\textwidth]{figures/gan_4} &
\includegraphics[width=0.16\textwidth]{figures/gan_1} &
\includegraphics[width=0.16\textwidth]{figures/gan_3} \\
\end{tabular}
\end{center}
\label{table:sample_from_video}
\caption{Snapshots of the trained peg in hole policy after 60, 120, and 300 epochs, at the end of policy execution. On the top row, the policy has been trained using RL. On the bottom row, the policy has been trained through our method, RL and reward shaping.}
\vspace{-0.5cm}
\end{figure}
The success rates of our method and the baselines on the \textit{peg insertion} task on the real robot arm are presented in Fig.~\ref{fig:realrobotexps2smooth}, where we compare pure RL and RL with GAN shaping. The failure of TD3 to discover the goal area inside the peg holder is not surprising given the long horizon and sparse rewards involved in the task. To generate demonstration data, a near-optimal predefined trajectory was used.
Fig.~\ref{fig:realrobotexps2smooth} shows the average return from 5 episodes. Since the episode length is set to $100$, and the agent receives $-1$ when the peg is not above or in the hole, an average reward of $-100$ means the robot received no reward throughout the entire episode. We can see that with our method, RL with GAN Shaping, the robot is able to collect rewards in $20$ steps. Note that the agent does not have access to this cumulative, dense reward during training. This dense return is used here for evaluation purposes only.
\vspace{-0.2cm}
\begin{figure}[h!]
\centering
\includegraphics[width=0.4\textwidth]{new_figures/RealRobotExpFullSize.pdf}
\vspace{-0.4cm}
\caption{Comparison of TD3 with TD3 with shaping from a GAN potential, trained from demonstrations on a \textit{peg insertion} task. Our method finds a good policy in about 200 episodes. The performance reduction after that is due to the RL objective trying to optimize the trajectory of the peg so as to minimize time to arrive to the bottom of the hole. As it tries to optimize and straighten the trajectory, the peg starts hitting the holder more frequently, which delays the learning. To address this the reward can be modified to penalize contact with the holder.}
\label{fig:realrobotexps2smooth}
\end{figure}
\vspace{-0.5cm}
\section{CONCLUSION}
We addressed the problem of combining reinforcement learning with suboptimal demonstrations, using results from reward shaping and state-action potentials in order to model the demonstration data as \textit{advice}, and not as a set of \textit{constraints}, which is a popular method currently used in practice. We modeled the demonstration data as deep generative models, based on normalizing flows and Generative Adversarial Networks, and we showed that RL with generative model potentials is typically more robust than RL with behavioral cloning constraints, even in the presence of suboptimal data. We showed that our method is practical on a real robot arm, in addition to validating our method in simulation.
|
1,314,259,995,787 | arxiv |
\section{Introduction}
\label{introduction}
ASR is a long-lasting problem in audio processing and has been extensively studied for decades. Recently, end-to-end deep neural network based methods are becoming one of the most promising approaches and have dominated numerous ASR leader boards. An end-to-end ASR model often integrates acoustic model, pronunciation model and language model coherently. There are several widely used structures for end-to-end ASR modelling such as connectionist temporal classification (CTC)~\cite{dahl2011context,mohri2008speech,graves2006connectionist}, attention-based encoder–decoder model (AED)~\cite{miao2015eesen,chorowski2014end,chorowski2015attention,zhang2017very}, recurrent neutral network transducer (RNN-T)~\cite{graves2012sequence,graves2013speech,rao2017exploring} and Transformer~\cite{vaswani2017attention}. In particular, Conformer~\cite{gulati2020conformer} is a variant of Transformer and the most advanced structure for ASR due to its high performance.
Attention is the key component in Conformer which aims to capture global dependencies. Compared with conventional encoders such as the long short-term memory (LSTM)~\cite{graves2013speech} which implicitly encodes temporal information in hidden layers, the attention-based encoders explicitly store the global temporal dependencies through a relevance weighted matrix and use it to weight the input sequence. Although previous methods demonstrate the effectiveness of the attention mechanism, it suffers two crucial drawbacks: 1). It is less effective to capture local dependencies; 2). It requires quadratic space-time computational complexity with respect to the sequence length. Numerous approaches have been proposed to address these limitations.
To emphasize the local dependencies, ContextNet~\cite{han2020contextnet} proposes using convolution module to capture local dependencies and leveraging a squeeze-and-excitation module to summarize global context from input sequence. Conformer integrates the convolution module and self-attention network (SAN) module in a transformer block in a sequential manner to augment local dependencies and obtains the state-of-the-art performance. In this paper, we propose a new locality-biased attention to blend local dependencies in a SAN module as well and achieves a higher performance.
For the quadratic computational complexity, a common solution in ASR is to restrict the input speech length or truncate the long form speech into several smaller speeches. However, the model's performance may degrade due to the lack of long-range information dependencies. This solution does not touch the essence of the limitation as the network still suffers the quadratic complexity. Efficient transformers are then proposed to reduce the quadratic space-time complexities directly. Sparse transformers often use local windows~\cite{beltagy2020longformer}, learnable sparse attention matrix~\cite{tay2020sparse}, locality-sensitive hashing attention mechanism~\cite{kitaev2020reformer}, and Kullback-Leibler divergence based sparsity measurement~\cite{wang2021efficient} to skip some attention weight's computation and thus reduce the computational complexities. Linear transformers, on the other hand, reduce the complexities to linear with some kernel tricks. For example, the Gaussian-kernalized self-attention~\cite{kashiwagi2021gaussian} uses a variant of the Gaussian kernel to represent attention weight and replace dot-product self-attention.
Random Feature Attention (RFA)~\cite{peng2021random} proposes a linear time and space attention that uses random feature methods to approximate the softmax function and applies in Transformer. Performers~\cite{choromanski2020rethinking} estimates vanilla attention with comparable accuracy using a novel fast attention via positive orthogonal random features approach, which has linear space-time complexities. Since they involved softmax approximation in the attention process, and these errors may accumulated through layers, these methods often suffer from degenerated performances. For the ASR task, as it is trying to map the continuous speech features into discrete transcripts, it is more sensitive to these approximation errors and will cause severer performance drop.
In this paper, we solve these two drawbacks with one stone: a new locality-biased linear attention that not only ensures neighbouring tokens to have a higher weights than the distanced tokens but also enjoys linear space-time computational complexity. Our LBLA consists of two components: a kernel function that allows us to use kernel tricks to reduce the computational complexity to linear, and a locality-biased re-weighting mechanism to impose the locality bias. It is worth noting that not all locality biased re-weighting matrix can be applied here as it also needs to satisfy the linear attention requirements, so that we can use the kernel tricks on the re-weighting matrix as well. The experiments are conducted on the LibriSpeech corpus~\cite{panayotov2015librispeech}. They show that compared with the vanilla Conformer model, the proposed method increases the inference speed by 6\%$\sim$22\% and obtains the comparable word error rate (WER) performance. After finetuning with model initialization, the proposed method obtains WER reduced by relatively 2.4\%$\sim$5.7\%.
The paper is organized as follows: Section \ref{rnnt} gives a brief introduction about conformer encoder architecture. The proposed conformer-based LBLA model is described in Section \ref{sec_hmm_free}, followed by experiments and discussions in Section \ref{src_exp}.
\section{Preliminaries}
\label{rnnt}
\subsection{Softmax Attention}
End-to-end ASR model maps the arbitrary input feature sequence $X$=$\{x_{1},x_{2},..x_{t}\}$ of length $T$ into output sequence $Y$=$\{y_{1},y_{2},...y_{u}\}$ of length $U$ directly. Usually, the length of label $U$ is much smaller than length of speech frames $T$. Transformer is proposed in~\cite{vaswani2017attention}, which is a stack of transformer layer consists of self-attention network (SAN), feed forward network (FFN), layernorm and residual connection. Self-attention mechanism in SAN can be represented below:
\begin{equation}
\label{eq: attention}
{O} =\text { Attention }(Q, K, V)= \operatorname{Softmax}\left(\frac{Q K^{T}}{\sqrt{d_{k}}}\right)V
\end{equation}
where $d_{k}$ denotes the dimension of $K$, $\frac{1}{\sqrt{d_{k}}}$ is scaling factor, $Q$, $K$ and $V$ refer to the query, key and value that are projected to high-level vectors using parameter matrices $W_{q}$,$W_{k}$ and $W_{v}$:
\begin{equation}
Q = XW_{q},K = XW_{k},V = XW_{v}
\end{equation}
Multi-head attention (MHA) mechanism also proposed in~\cite{vaswani2017attention} projects query, key and value to several separately linear vector subspace. Then all of value embedding concatenated and project again with $W_{o}$:
\begin{equation}
\text { $head_{i}$ }=\operatorname{Attention}(Q_{i}W_{k}^{i},K_{i}W_{k}^{i},V_{i}W_{k}^{i})
\end{equation}
\begin{equation}
{O} =\text { MHA }(Q, K, V)=\operatorname{concat}({head_{1},..., head_{h}})W_{o}
\end{equation}
Where $h$ is the number of heads and $W_{q}^{i}$,$W_{k}^{i}$ and $W_{v}^{i}$
are parameter matrices for i-th attention head.
\subsection{Linear Attention}
As shown in Eq.~\ref{eq: attention}, query $Q \!\in\! \mathbb{R}^{T\times d}$ and key $K \!\in\! \mathbb{R}^{T\times d}$ are utilized for the attention weight matrix to represent the proximity between them, where $T$ is the length of input sequence and $d$ is the embedding length. Self-attention mechanism calculates attention weight matrix $QK^{T}\!\in\! \mathbb{R}^{T\times T}$ followed by softmax function that is applied to obtain normalization.
Since the complexity of calculating attention weight matrix is $\mathcal{O}(T^{2})$, which has quadratic time and memory bottleneck with respect to the input sequence length. In order to address this issue, we approximate the attention weight matrix and rewrite self-attention function as:
\begin{equation}
\quad {O}_{i}=\sum_{j} \frac{{exp}\left(Q_{i}K_{j}^T\right) V_{j}}{\sum_{j} {exp}\left(Q_{i}K_{j}^T\right)},
\end{equation}
\begin{equation}
{O}=\left[{O}_{1}, \ldots,{O}_{T}\right]^{T}
\end{equation}
Denote operator $\exp(Q_{i}K_{j}^T)$ as the proximity between i-th query and j-th key, the proximity measure function $\mathcal{P}(\cdot)$ can be defined as:
\begin{equation}
\quad \mathcal{P}(Q_{i},K_{j})=\operatorname{exp}(Q_{i}K_{j}^T)
\end{equation}
Then self-attention can be rewrite as:
\begin{equation}
\quad {O}_{i}=\sum_{j} \frac{\mathcal{P}\left(Q_{i}, K_{j}\right) V_{j}}{\sum_{j} \mathcal{P}\left(Q_{i}, K_{j}\right)}
\end{equation}
The proximity measure function $\mathcal{P}(\cdot)$ need to be calculated $T^2$ times. To maintain a linear computation complexity, we adopt a decomposable proximity function below:
\begin{equation}
\mathcal{P}(Q_{i}, K_{j})=\psi(Q_{i})\psi(K_{j})^T
\label{eq: decomp}
\end{equation}
where $\psi(\cdot)$ is a kernel function that maps the queries and keys to hidden representations. Consequently, we could rewrite self-attention in the form of kernel functions as:
\begin{equation}
\quad {O}_{i}=\sum_{j}\frac{\psi(Q_{i})(\psi(K_{j})^TV_{j})}{\sum_{j} \psi(Q_{i})\psi(K_{j})^T}
\label{eq: trick}
\end{equation}
Instead of explicitly calculating the attention weight matrix $QK^{T}$$\in$$\mathbb{R}^{T\times T}$, key and value are calculated first as $\psi(K_{j})^TV_{j}$$\in$$\mathbb{R}^{d\times d}$. Note that the dimension of head $d$ is always much smaller than the input speech sequence length of $T$. By using this trick, the computational complexity of calculating matrix can be reduce to $\mathcal{O}(Td^2)\approx \mathcal{O}(T)$ when $T\gg d$.
\section{Locality-Biased Linear Attention}
\label{sec_hmm_free}
In this section, we first provide a detailed description of the two components in our LBLA module, \emph{i.e.,} the kernel functions and the locality-biased re-weighting matrix, then illustrate how to integrate the LBLA module to a Conformer block.
\subsection{The Kernel Function}
To linearize the attention, the kernel functions have to satisfy the Eq.~\ref{eq: decomp}, then we can apply the kernel trick in Eq.~\ref{eq: trick} to reduce the computational complexities. How to find the right kernel function for the ASR task is a non-trivial task. Previous methods in natural language processing~\cite{katharopoulos2020transformers, zhen2021cosformer} suggest that the kernel function needs to project the tokens to a non-negative space such that the negatively-correlated information can be excluded in contextual information aggregation. Weak-attention~\cite{shi2020weak} suggests that ASR will likely benefit from sparse and localized attention weight. Following these suggestions, we explore three commonly used non-negative activation functions as our kernel functions: the ReLU function, the Exponential function, and the Sigmoid function. We defined these functions as below:
\begin{align}
\psi(x)&=\operatorname{ReLU}(x), \\
\psi(x)_{exp}&=\operatorname{exp}(x),\\
\psi(x)_{sig}&=\operatorname{Sigmoid}(x).
\end{align}
We use dot product as our proximity measure function $\mathcal{P}(\cdot)$ so that the requirement of Eq.~\ref{eq: decomp} fulfilled.
In the experiment section, we extensively ablate these selections and find that the Sigmoid one achieves the most accurate results. Therefore, in our LBLA, we use the Sigmoid function as our kernel function.
\subsection{Locality-biased Re-weighting Mechanism}
The Eq.~\ref{eq: decomp} computes the proximity measurements without any re-weighting mechanism. In Eq.~\ref{eq: reweight}, we show a locality-biased proximity measurement with a re-weighting matrix $\omega(\cdot)$.
\begin{equation}
\mathcal{P}_{local}(Q_{i}, K_{j})=\psi(Q_{i})\psi(K_{j})^T\omega(i-j)
\label{eq: reweight}
\end{equation}
where the $\omega(\cdot)$ needs to satisfy the Eq.~\ref{eq: linear} for linearization.
\begin{equation}
\textstyle
\omega(i-j) = \sum_1^n f(i)g(j)
\label{eq: linear}
\end{equation}
where $f(\cdot), g(\cdot)$ are decomposed functions of $\omega$ and $n$ is the number of decomposed terms.
Not many re-weighting matrices meet this requirement. Here, we use a cosine re-weighting matrix which can be decomposed with the Ptolemy's theorem as below:
\begin{equation}
\begin{scriptsize}
\begin{aligned}
\omega_{\operatorname{cos}}(i-j) &=
\operatorname{cos}(\frac{\pi}{2}\!\times\!\frac{i\!-\!j}{T})\\ &= \cos (\frac{\pi i}{2 T})\cos (\frac{\pi j}{2 T})\!+\!\sin (\frac{\pi i}{2 T}) \sin (\frac{\pi j}{2 T})
\end{aligned}
\end{scriptsize}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth,height=3cm]{locality.png}
\caption{Visualization of the locality-biased re-weighting patterns. The black cross denotes the query position and the mask denotes the corresponding locality weights.}
\label{fig:locality}
\end{figure}
Figure~\ref{fig:locality} illustrates our locality-biased re-weighting matrix, where the neighbouring tokens are assigned with higher weights and the distanced ones are with lower weights.
We now achieve the overall expression of our LBLA:
\begin{equation}
\textstyle
{O}= \frac{\mathcal{P}_{local}(Q, K) V}{\sum \mathcal{P}_{local}(Q, K)}= \frac{Q^{\cos }(K^{\cos } V)+Q^{\sin }(K^{\sin } V)}{\sum Q^{\cos }K^{\cos } +\sum Q^{\cos }K^{\cos } }
\end{equation}
where \scalebox{0.9}{$Q_{i}^{cos}\!=\!\widetilde{Q}_{i}\cos (\frac{\pi i}{2 M}), Q_{i}^{sin}\!=\!\widetilde{Q}_{i}\sin (\frac{\pi i}{2 M}),K_{j}^{cos}\!=\!\widetilde{K}_{j}\cos (\frac{\pi j}{2 M})$}, \scalebox{0.9}{$K_{j}^{sin}\!=\!\widetilde{K}_{j}\sin (\frac{\pi j}{2 M})$, $\widetilde{Q}_{i}=\psi(Q_{i}), \widetilde{K}_{j}=\psi(K_{j})$}.
Compared with vanilla softmax attention, our LBLA has a linear computational complexity with locality-biased re-weighting mechanism.
\subsection{Locality-biased Conformer Block}
Conformer has obtained state-of-the-art performance in ASR~\cite{gulati2020conformer,guo2021recent,li2021better,ng2021pushing,chung2021w2v,zhang2021bigssl}. As a variant of Transformer, Conformer uses convolution module to augment local dependencies. A single vanilla conformer block consists of convolution module, multi-head self-attention module and two independent feed forward networks. In our locality-biased Conformer block, we replace the multi-head self-attention module with our LBLA module and keep the rest modules unchanged. As illustrated in Figure~\ref{fig:architecture}, our block starts with a FFN, then the LBLA module to capture locality-biased global context followed by a convolution module that captures local context, another FFN is deployed as the last module. Especially, FFN is employed half-step residual weights in Conformer. The convolution module consists of pointwise convolution and gated linear unit, followed by one dimension convolution module and batchnorm, the last in swish activation function and another pointwise convolution.
\begin{figure}[t]
\centering
\vspace{-0mm}
\includegraphics[width=\linewidth]{architecture2.pdf}
\vspace{-8mm}
\caption{The structrue of LBLA based conformer model. The encoder contains locality-biased self-attention module, macaron-like feed-forward module and convolution module, followed by a post layernorm}
\label{fig:architecture}
\end{figure}
\section{Experiments}
\label{src_exp}
\subsection{Experimental Setup}
The experiments are conducted using LibriSpeech corpus, which has 960-hour audios for training. WER is evaluated on LibriSpeech test-clean and test-other test sets respectively. The test-clean contains simple and clean audios, test-other contains complex and noisy audios. All the neural networks are trained using Wenet toolkit~\cite{yao2021wenet}.
Models are trained with 3727 sub-word units, which generated from training transcripts of LibriSpeech using SentencePiece~\cite{kudo2018sentencepiece}. Besides, the features embedding fed into the encoder is a 80-dimensional log filter-bank feature with 10ms frame shift and 25ms window size followed by Spec-Augment~\cite{park2019specaugment}. Two convolution layers are added with ReLU~\cite{agarap2018deep} activation function to downsample the frame rate to 4 with absolute position embedding layer.
The encoder has 12 layers, each conformer layer has convolution module with kernel size 31, model dimension 256 and two feed forward layers with dimension 2048. Dropout~\cite{srivastava2014dropout} rate is set to 0.1 for all layers. The decoder is a standard transformer decoder with 6 layers, each transformer layer has a FFN layer with dimension 2048, 4 attention heads and dropout is also set to 0.1 for all layers.
CTC criterion is utilized to optimize the CTC loss as an auxiliary loss. Models are trained with the Adam~\cite{kingma2014adam} optimizer for about 150000 steps, peak learning rate is $4\times10^{-3}$ and warm up is 25000 steps. In inference, CTC greedy search and attention rescoring are applied to evaluate the performance of the model while beam size is set to 10. The amount of model parameter is about 49M.
Besides, We trained our models on one machine with 8 NVIDIA Tesla A100 GPUs.
\subsection{Results}
All models are trained on LibriSpeech from scratch. The proposed model use kernel functions ReLU, Sigmoid~\cite{mount2011equivalence} and Exponential for comparison. We have observed that it's beneficial to change the number of attention head. Table~\ref{exp_ctcg} shows the result of CTC greedy search and attention rescoring respectively.
\begin{table}[th]
\caption{WER(\%) of LBLA on LibriSpeech task using CTC greedy search decoding. B0 indicates the aforementioned Conformer baseline. Our LBLA model with the Sigmoid achieves better performance than the vanilla Conformer in every aspects.}
\label{exp_ctcg}
\scriptsize
\centering
\setlength{\tabcolsep}{1.5mm}{
\begin{tabular}{lcccccc}
\toprule
\multirow{4}{*}{model} & \multirow{4}{*}{activation}& \multirow{4}{*}{\makecell[c]{attention\\head}} &\multicolumn{2}{c}{test clean} & \multicolumn{2}{c}{test other} \\
\cline{4-7}
{}& {}& {}&\makecell[c]{ ctc\\ greedy\\search}&\makecell[c]{ attention\\ rescoring}&\makecell[c]{ ctc\\ greedy\\search}&\makecell[c]{ attention\\ rescoring}\\
\midrule
B0 & - & 4 & 3.71 &3.2 & 9.67 & 8.74 \\
\midrule
R0 & ReLU & 1 & 3.75 & 3.27 &10.15 & 9.16 \\
R1 & ReLU & 4 & 3.78 & 3.34 &9.81 & 8.87 \\
R2 & ReLU & 8 & 3.82 & 3.4 &9.97 & 8.98 \\
\midrule
E0 & Exponential & 1 &{3.7} & 3.28 & 9.68 & 8.88 \\
E1 & Exponential & 4 & 3.73 & 3.35 &{9.56} & 8.82 \\
E2 & Exponential & 8 & 3.71 & 3.25 & 9.73 & {8.74}\\
\midrule
S0 & Sigmoid & 1 & 3.70 & 3.29 & 9.55 & 8.74 \\
S1 & Sigmoid & 4 & 3.70 & 3.27 & 9.59 & 8.83 \\
S2 & Sigmoid & 8 &\textbf{3.58} &\textbf{3.16} &\textbf{9.50} &\textbf{8.61} \\
\bottomrule
\end{tabular}
\vspace{-2.0em}
}
\end{table}
Firstly, we evaluate the performance of models that use CTC greedy search. As illustrated in Table~\ref{exp_ctcg}, Conformer baseline achieves a WER 3.71/9.67 on test-clean/test-other. Compared with baseline, $R$0-$R$2 perform slightly worse and achieve a relative degradation of WER by 1\%$\sim$2.9\% on test-clean and 1.4\%$\sim$4.9\% on test-other. The reason causing the performance degradation may be that $ReLU()$ forces negative value to be zero leading to loss some information of relevance weighted matrix. Compared with baseline, $E$0-$E$2 obtain the comparable performance . $E1$ even achieves a relative reduction of WER by 1.2\% on test-other.
$S$0-$S$2 perform better and $S$2 achieves a relative reduction of WER by 3.5\% on test-clean and 1.5\% on test-other.
Experiments show that the proposed LBLA model achieves the comparable performance compared with vanilla Confomer, and exponential-based kernel function such as Sigmoid function and Exponential function could enlarge the distance among the proximity $\mathcal{P}(\cdot)$ to force the attention matrix to be more sparse.
\iffalse
\begin{table}[th]
\caption{WER(\%) of LBLA on LibriSpeech task using attention rescoring decoding.}
\label{exp_rescore}
\footnotesize
\centering
\begin{tabular}{lcccc}
\toprule
model & activation& {\makecell[c]{ attention\\ head}} & {test clean} & {test other}\\
\hline
Conformer & - & 4 & 3.2 & 8.74 \\
\midrule
R0 & ReLU & 1 & 3.27 & 9.16 \\
R1 & ReLU & 4 & 3.34 & 8.87 \\
R2 & ReLU & 8 & 3.4 & 8.98 \\
\midrule
S0 & Sigmoid & 1 & 3.9 & 11.58 \\
S1 & Sigmoid & 4 & 3.9 & 11.58 \\
S2 & Sigmoid & 8 & 3.9 & 11.58 \\
\midrule
E0 & exp & 1 & 3.28 & 8.88 \\
E1 & exp & 4 & 3.35 & 8.82 \\
E2 & exp & 8 & 3.25 & \textbf{8.74} \\
\bottomrule
\end{tabular}
\end{table}
\fi
We also evaluate the performance of models using attention rescoring. Conformer baseline achieves a WER 3.2/8.74 on test-clean/test-other. $R$0-$R$2 are slightly worse than baseline and achieve a relative degradation of WER by 2.1\%$\sim$6\% on test-clean and 1.5\%$\sim$4.8\% on test-other.
$E$0-$E$2 achieve a relative degradation of WER by 2.5\%$\sim$4.7\% on test-clean and 0\%$\sim$1.6\% on test-other. However, $S$0-$S$2 obtain the best performance in our experiments and achieve a relative reduction of WER by 1.2\% on test-clean and 1.5\% on test-other. The proposed LBLA model also achieves better performance.
In our experiments, we obverse that validation loss is increasing while training last several epochs. It seems that the larger learning rate cannot make the LBLA model converge well in the late training stage. Consequently, We change the learning rate for late training stage while the rest strategies keep same by resume the model from middle of the training stage, all models' number of training steps keep same. As illustrated in Table~\ref{exp_finetune}. F1 outperforms the baseline trained from scratch by relatively 2.4\%$\sim$5.7\%.
\begin{table}[th]
\caption{WER(\%) of LBLA with initialization on LibriSpeech task. Models half the learning rate for late training stage.}
\label{exp_finetune}
\footnotesize
\centering
\setlength{\tabcolsep}{1.7mm}{
\begin{tabular}{lccccc}
\toprule
\multirow{4}{*}{model} & \multirow{4}{*}{initialize} &
\multicolumn{2}{c}{test clean} & \multicolumn{2}{c}{test other}\\
\cline{3-6}
{}& {} &\makecell[c]{ ctc\\ greedy\\search}&\makecell[c]{ attention\\ rescoring}&\makecell[c]{ ctc\\ greedy\\search}&\makecell[c]{ attention\\ rescoring}\\
\midrule
{Conformer} & scratch & 3.71 & 3.2 & 9.67 & 8.74 \\
\midrule
F0 & E0 & {3.6} & {3.19} & 9.62 & 8.78 \\
F1 & S2 & \textbf{3.50} & \textbf{3.08} & \textbf{9.43} & \textbf{8.53} \\
\bottomrule
\end{tabular}
}
\vspace{-0.5em}
\end{table}
\begin{table}[th]
\caption{Inference speed of LBLA on LibriSpeech. It's defined as decoding utterance length per second. Three sub sets is used to evaluate the inference speed, which test-set1 contains all data in test-clean, test-set2 contains medium length speech longer than 10s in test-clean and test-set3 contains long speech longer than 20s in test-clean. All experiments are evaluated on a single thread of CPU.}
\label{exp_speed}
\footnotesize
\centering
\setlength{\tabcolsep}{5.0mm}{
\begin{tabular}{lccc}
\toprule
model & test-set1 & {test-set2} & {test-set3}\\
\midrule
Conformer & 22.8 & 22.0 & 20.7 \\
\midrule
R0 & 22.9 &23.6 & 23.0 \\
R1 & \textbf{24.3} & \textbf{25.6} & 25.2 \\
R2 & 24.1 & 25.0 & 24.2 \\
\midrule
E0 & 23.7 & 24.3 & 23.8 \\
E1 & 23.7 & 24.5 & 25.0 \\
E2 & 24.0 & 24.8 & 25.0 \\
\midrule
S0 & 23.7 & 24.4 & 23.7 \\
S1 & 23.8 & 24.7 & 24.1 \\
S2 & 24.1 & 25.0 & \textbf{25.3} \\
\bottomrule
\end{tabular}
}
\vspace{-1em}
\end{table}
\begin{table}[th]
\caption{Ablation study of LBLA, we remove its features: (1) removing cosine re-weight without additional relative positional embedding; (2) removing kernel function; (3) removing normalization for attention. NC represents no convergence.
}
\label{exp_ablation}
\footnotesize
\centering
\setlength{\tabcolsep}{2.1mm}{
\begin{tabular}{lcccc}
\toprule
\multirow{4}{*}{model} &
\multicolumn{2}{c}{test clean} & \multicolumn{2}{c}{test other}\\
{} &\makecell[c]{ ctc\\ greedy\\search}&\makecell[c]{ attention\\ rescoring}&\makecell[c]{ ctc\\ greedy\\search}&\makecell[c]{ attention\\ rescoring}\\
\midrule
S2 &3.58 &3.16 &9.50 &8.61 \\
\ \ $-$cosine re-weight & {3.84} & {3.37} & 9.61 & 8.83 \\
\ \ $-$Sigmoid & {$NC$} & {$NC$} & $NC$ & $NC$ \\
\ \ \ \ \ $+$ReLU & 3.82 & 3.4 &9.97 & 8.98 \\
\ \ $-$normalization & {$NC$} & {$NC$} & $NC$ & $NC$ \\
\bottomrule
\end{tabular}
\vspace{-2.0em}
}
\end{table}
To evaluate the time consumption of inference, we conduct the experiments on three test sets that contain different duration of speech. As illustrated in Table~\ref{exp_speed}, our proposed LBLA model obtains the better performance on time consumption. Compared with baseline, $R1$ increases the speed of inference by relatively 6\% on test-clean and $S2$ increases relatively 22\% on long speech sub set. Our proposed model will be more efficient with respect to the input sequence length.
LBLA model differs from a Conformer in some ways, specially, the inclusion of a kernel function and locality-biased re-weight mechanism. Normalization is applied similar to vanilla attention mechanism. We study these effects of these differences by removing features. Table~\ref{exp_ablation} shows the impact to the LBLA model. Kernel function and normalization are important to stabilize training until convergence. Locality-biased re-weight is an efficient way to obtain shift-invariant positional information and better performance.
\section{Conclusions}
\label{sec_con}
In this paper, we proposed a LBLA model which introduces a locality-biased linear mechanism in Conformer block to enjoy linear space-time complexities. The LBLA contains a kernel function to ensure linear computational complexity and a cosine reweighing matrix to impose more weight on neighbouring tokens. We evaluated it extensively on the LibriSpeech corpus. Compared with the vanilla Conformer, our proposed model enjoys linear computational complexity, obtains comparable WER performance and increases the inference speed by 6\%$\sim$22\%.
\bibliographystyle{IEEEtran}
|
1,314,259,995,788 | arxiv | \section{Introduction}\label{sec:introduction
Modern sensors are collecting
high-dimensional data at unprecedented volume and speed; human {analysts}
cannot keep pace.
For instance, many sources of intelligence data must be translated by human experts before they can be widely accessible to analysts and actionable; the translation step is a significant bottleneck \cite{sept11}.
Typical NASA
missions collect terabytes of data every day
\cite{JPLBigData,WiredSKA,LSSTWeb,LSST}. {Incredibly,} the Large
Hadron Collider (LHC) at CERN ``generates so much data that scientists
must discard the overwhelming majority of it---hoping hard they've
not thrown away anything useful.'' \cite{LHC} There is a pressing need
to help analysts {prioritize} data {\em accurately and efficiently} from
a storage medium or a data stream. This task is complicated by the
fact that, typically, the data is neither thoroughly annotated nor
meaningfully catalogued. Failure to extract relevant data could lead
to incorrect conclusions in the analysis, while extraction of
irrelevant data could overwhelm and frustrate human analysts,
throttling the discovery process.
{This paper focuses on} {\em scalable online data processing
algorithms that can winnow large datasets to produce smaller subsets
of the most important or informative data for human analysts}.
{This process is described as ``data thinning.''} Often, the data thinning process
involves flagging observations which are inconsistent with previous
observations from a specified class or category of interest, or are
ranked highly according to a learned ranking function. Typically we
are interested in methods which can perform these assessments from
streaming data, as batch algorithms are inefficient on very large
datasets.
{One} generic approach to the problem of data thinning for large
quantities of (possibly streaming) high-dimensional data {requires
estimating and tracking} a probability distribution $f_t$ underlying the
stream of observations $x_t$, and {flagging} an observation as anomalous
whenever $\widehat{f}_t(x_t) < \tau$ for some small threshold
$\tau>0$, as demonstrated in past work
\cite{raginsky_OCP,onlineSocialAnomalies}. Ultimately, {the} goal is
to ensure that the flagged data is salient to human analysts on
the receiving end without being buried in an avalanche of irrelevant
data. Within this general framework, {there are} three key
challenges:
\begin{itemize}
\item {\bf Dynamic environments:} The data may not be from a
stationary distribution. For {example}, it may exhibit diurnal,
location- or weather-dependent patterns. Effective data thinning
methods must adapt to those dynamics and sources of bias. Global
summary statistics and naive online learning algorithms will fail in
this context.
\item {\bf High-dimensional data:} Individual data points $x_t$ may be
high-dimensional, resulting in the classical ``curse of
dimensionality'' \cite{bellman1961adaptive,hastie2009element}. While
large quantities of data {may be} available, the
combination of high-dimensional data and a non-stationary
environment still results in an ill-posed estimation problem.
\item {\bf Real-time processing:} In applications {like those
with NASA and CERN}, large quantities of streaming data preclude
computationally intensive or batch processing.
\end{itemize}
\subsection{Data thinning for wide-area motion imagery}
While our approach is not restricted to imaging data, one important application of our data thinning approach is real-time video analysis.
Recent advances in optical engineering have led to the advent of new
imaging sensors that collect data at an unprecedented rate and scale;
these data often cannot be transmitted efficiently or analyzed by
humans due to their sheer volume. For example, the ARGUS system
developed by BAE Systems is reported to collect video-rate gigapixel
imagery \cite{argus,argus2}, and even higher data rates are anticipated
soon \cite{mosaic3,mosaic4,mosaic2}. This type of data is often
referred to as wide-area motion imagery (WAMI). Currently WAMI streams
are used primarily in a forensic context -- after a significant event
occurs ({\em e.g.,~} a security breach), the data immediately preceding the
event are analyzed {\em reactively} to piece together what led to that
event. However, there is a strong need for predictive analysis which
can be used to help {\em anticipate} or detect negative events in real
time.
Unfortunately, the latter form of analysis is often infeasible for two
reasons: {(1)} the data acquisition rate exceeds the capacity of
many sensor platforms' downlinks; and {(2)} size, weight, and power
constraints limit processing capabilities on airborne sensor
platforms. Thus an {\em emerging and fundamental challenge is
efficiently downloading salient information to ground-based analysts
over a limited-bandwidth channel}. While data compression has a long
history, conventional compression methods may distort information
particularly relevant to analysts. In particular, standard motion
imagery compression techniques typically focus on optimizing peak
signal-to-noise ratio or psycho-visual metrics which apply globally to
an entire video and are often unrelated to any specific task.
{Instead, a} better solution would be to identify unique objects or regions of
WAMI, and transmit only features of these objects. This concept is
illustrated in Fig.~\ref{fig:concept}. Ideally, {this} method will
identify regions and features of a data stream most critical to a
given task, and prioritize these features when preparing data for
storage or transmission. This task is clearly related to ``visual
saliency detection'' ({\em cf.,~}
\cite{SaliencyItti,SaliencySR,SaliencyGBVS,rao2010using}); we describe
the connections between the proposed work and saliency detection in
Section~\ref{sec:RelatedWork}.
Note that in this setting a key challenge is that the sensor may be
placed on a vibrating platform that introduces significant jitter into
the data and precludes direct comparison of successive frames. While
real-time video stabilization has been considered in the video
processing literature ({\em cf.,~}
\cite{erturk2002real,hansen1994real,ratakonda1998real,chang2006robust,battiato2010robust}),
such methods are often robust for small motions associated with a
hand-held device and break down with large motions associated with
mechanical vibrations. More robust methods capable of processing
larger degrees of jitter can be computationally prohibitive on
energy-constrained platforms
\begin{figure}
\centerline{\includegraphics[width=.6\textwidth]{Figures/ThinningIllustration.pdf}}
\caption{{\small Conceptual illustration of proposed objectives. An airborne
platform collects wide-area motion imagery (WAMI), identifies
task-specific salient patches, and transmits only those
patches. The ground-based receiver can then perform more
sophisticated processing, including registration, geolocation, and
activity analysis.}}
\label{fig:concept}
\end{figure}
\subsection{Problem formulation and approach}
\label{sec:formulation}
Suppose we are given a sequence of data $x_1, x_2, \ldots,$ and for
$t = 1, 2, \ldots$, $x_t \in \mathbb{R}^{p}$, where $p$ denotes the
{\em ambient dimension}. Assume that $x_t$ {comes} from some unknown
distribution, {\em i.e.,~} there exists some sequence of distributions $P_t$
such that
$$
x_t\sim P_t \quad t=1,2,\ldots
$$
where $P_t$ evolves over time, and its distribution density function
is denoted by $f_t$. The goal is to find the $x_t$ that are unusual
or anomalous. In particular, we assign each observation $x_t$ {an} {\em
anomalousness score} proportional to its negative log likelihood
under the estimated model---{\em i.e.,~} $-\log f_t(x_t)$. Observations with a
low anomalousness score can then either be directed to a human
{analyst} or flagged for further processing and analysis.
The key challenge here is two-fold: (a) the dimension of the signal,
$p$, can be quite large, and (b) $f_t$ may evolve rapidly over
time. The combination of these factors means that our problem is
ill-posed, because we are unlikely to gather enough samples to
reliably learn the density $f_t$.
{This paper proposes} a method for estimating and tracking the
time-series of density functions $f_t$ over $\mathbb{R}^p$. In
stationary, low-dimensional settings, we might consider a Gaussian
mixture model that could be estimated, for instance, using an online
expectation-maximization (EM) algorithm
\cite{same2007online}. However, the non-stationary setting and high
dimensions make that approach unviable, as we demonstrate
experimentally later in the paper. {The proposed approach, by contrast,
considers} a
constrained class of Gaussian mixture models in which the Gaussian
covariance matrices (each in the positive-semidefinite cone
$\mathcal{S}^{p}_{+}$) are low-rank. This model is equivalent to
assuming most $x_t$ lie near a union of low-dimensional
subspaces. While this union of subspaces is unknown {\em a priori}, we
may leverage recent advances in subspace tracking ({\em cf.,~}
\cite{BalzanoNowakRecht2010,ChiJournal2012,ChiEldarCalderbank2012,roseta})
and subspace clustering ({\em cf.,~} \cite{aggarwal2000finding, bradley2000k,
bohm2004computing, achtert2006mining, achtert2007robust,
achtert2007exploring}) to yield an accurate sequence of density
estimates $\hat f_t$, and mitigate the curse of
dimensionality
In addition, we consider certain computational {and} statistical
tradeoffs associated with the data thinning problem. In particular,
{there are two ways to} reduce the computational complexity associated with computing
anomalousness scores. {First,} we can reduce the frequency with which we update our
model. {Second, we can} subsample the elements of each $x_t$ and leverage missing
data models for fast calculations and updates. We demonstrate that
these methods, which are not amenable to standard stochastic filtering
methods, can yield significant computational speedups with only small
decreases in thinning accuracy.
\subsection{Contributions and paper layout}
{This paper presents} a data thinning method for high-dimensional
streaming data in a dynamic environment. The algorithm adapts to
changing environments using tracking methods for union of
subspaces. {As shown by} both synthetic and real-data experiments,
{the} algorithm (a) efficiently tracks the subspaces in which most
observation lie and hence precisely detects observations that occur with low
probability, and (b) can be applied to a variety of real-world
applications and tasks.
{Section~\ref{sec:RelatedWork} describes related work.} Section~\ref{sec:model} explains {the}
probability density model based on unions of subspaces, and
Section~\ref{sec:tracking} presents the algorithm for tracking such
densities. Section~\ref{sec:computation} describes the computational
and statistical tradeoffs associated with the proposed approach.
Section~\ref{sec:syncexp} {reports} synthetic experiments
{which demonstrates} the ability of {the} algorithm to precisely track the
density and detect anomalous signals within a changing environment.
Section~\ref{sec:WAMIexp}, {tests the} algorithm on the wide-area
motion imagery (WAMI) videos {to detect} salient objects, while
Section~\ref{sec:Enronexp} {tests the} algorithm on the Enron email
database to detect major events.
\section{Related work}
\label{sec:RelatedWork}
While data thinning is an emerging concept associated with modern
high-dimensional, high-velocity data streams, the formulation described
in Section~\ref{sec:formulation} is closely related to anomaly
detection, visual saliency detection, and subspace clustering and
tracking.
\subsection{Anomaly detection}
The study of anomaly detection has a long and rich history, where the
earliest papers can date back to the 19th century
\cite{edgeworth1887xli}. Despite the long history of the study of
anomaly detection, most existing detection methods {do not work well with high
dimensional data}, and often do not work online. A 2009 survey on
anomaly detection \cite{chandola2009anomaly} categorizes the available
methods into classification-based methods, nearest neighbor-based
methods, cluster-based methods, information theoretic methods,
statistical anomaly detection methods, and spectral methods.
Among the six categories, classification based methods ({\em cf.,~}
\cite{de2000reject, barbara2001detecting, scholkopf2001estimating,
roth2004outlier, roth2006kernel}) require a large training pool with
labeled data that is typically unavailable in the settings of
interest. Also, {the classification based methods depend highly} on the training data,
{and} do not {effectively} adapt to changing environments. Nearest neighbor ({\em cf.,~}
\cite{zhang2006detecting, otey2006fast, ghoting2008fast,
tao2006mining, wu2006outlier}) and cluster-based methods ({\em cf.,~}
\cite{ertoz2004finding, yu2002findout, budalakoti2006anomaly,
pires2005using}) can both be extended to work online, but the
computational costs are usually high, scaling with the amount of
data. Furthermore, the performance of the nearest neighbor and
cluster-based methods highly depend on the distance measure, and the
optimal distance measure is highly problem-dependent.
Certain statistical methods ({\em cf.,~} \cite{solberg2005detection,
aggarwal2008outlier, chen2005simultaneous, agarwal2007detecting})
assume that the data are drawn from some standard or predetermined
distribution, and determines outliers by computing the likelihood of
the signal coming from such distributions. These methods can often
work online, and do not rely on a big training set, but estimating the
distribution of high-dimensional data is a non-trivial task, and the
statistical assumptions {do} not always hold true, especially for
high-dimensional data where there could be spatial correlations.
Information theoretic techniques ({\em cf.,~} \cite{he2005optimization,
ando2007clustering, keogh2004towards}) identify the anomalies by
trying to find a small subset such that removing the subset will
greatly reduce the complexity of the whole set. The approach requires
no supervision, and does not make assumptions about the underlying
statistical distribution of the data. However, they usually have
exponential time complexity and are batch methods. Additionally, it is
difficult to assign anomalousness scores to a single data point.
Spectral methods ({\em cf.,~} \cite{agovic20086, dutta2007distributed,
ide2004eigenspace, shyu2003novel}) assume that data can be embedded
into a lower dimensional subspace, and detect anomalies over the
embedded space rather than the original space. Because {spectral methods}
essentially operate on a reduced-dimensional representation of the
data, {they} are well-suited to high-dimensional
data. Spectral methods can also be integrated with other methods, and
are thus highly versatile. However, spectral methods can incur high
computational costs; even online anomaly detection algorithms ({\em cf.,~}
\cite{breunig2000lof}, \cite{kriegel2008angle} and
\cite{ahmed2009online}) face this challenge. Furthermore, the
subspace model underlying spectral methods is less flexible than the
union of subspace model underlying {this paper's proposed} method.
\subsection{Visual saliency detection}
In the special case of imagery or video data, data thinning is closely
related to visual saliency detection. Like anomaly detection,
saliency detection has been widely studied over the last few decades.
A standard benchmark for comparison in image saliency detection is
proposed by Itti et al. in \cite{SaliencyItti}. {This} paper attempts to
explain human visual search strategies, using biologically motivated
algorithms. However, this algorithm is too slow to apply to real time
videos. Hou and Zhang in \cite{SaliencySR} use spectral analysis to
detect salient objects for faster speed. However, {the analysis} breaks down when
multiple types of salient objects are present in the scene.
Graph-based methods ({\em cf.,~} \cite{SaliencyGBVS}) work well even when
there is no central object in the scene, which is often difficult for
other methods to handle, but suffers from high computational
complexity. Rao et al. proposed a cluster-based algorithm in
\cite{rao2010using}, where the salient object is identified by first
clustering all the pixels according to their local features, and then
finding the group of pixels that contains the most salient
information. It works better than \cite{SaliencyItti}, but not as well
as the graph-based algorithms. The information theoretic model based
algorithm proposed in \cite{SaliencyAIM} claims to work as well as
\cite{SaliencyItti}, but requires less tuning. This work is
improved in \cite{zhang2008sun} for faster speed and better
performance.
Methods for image saliency detection have been extended to video
saliency detection, but those methods assume a stable imaging platform
and video stream free of jitter. In the WAMI application described
above, however, sensors can be placed on vibrating platforms that
preclude most video saliency detection methods.
\subsection{Subspace clustering and tracking}
{The proposed method} is also closely related to the subspace clustering and
tracking algorithms. Subspace clustering is a relatively new, but
vibrant field of study. These methods cluster observations into
low-dimensional subspaces to mitigate the curse of dimensionality,
which often make nearest-neighbors-based methods inaccurate
\cite{beyer1999nearest}. Early works in the field can only identify
subspaces that are parallel to the axes, which is not useful when the
data is not sparse, but lives on an arbitrarily oriented
hyperplane. Newer methods \cite{aggarwal2000finding, bradley2000k,
bohm2004computing, achtert2006mining, achtert2007robust,
achtert2007exploring}, which are also called correlation clustering
methods, can identify multiple arbitrarily angled subspaces at the
same time, but all share the same problem of high computational
cost. Even \cite{achtert2007exploring}, which is shown to beat other
methods in speed, still has an overall complexity of $O(p^2T^2)$,
where $p$ is the dimension of the problem, and $T$ is the total number
of data points. More recent methods based on sparse modeling ({\em cf.,~}
\cite{elhamifar2009sparse,elhamifar2013sparse,wang2016noisy,
vidal2010tutorial,groupsparse_ssp}) require solving convex
optimization problems that can be inefficient in high-dimensional
settings. Thus, the high complexity of the algorithms make them less
than ideal candidates for an efficient online algorithm.
Subspace tracking is a classical problem that experienced recent
attention with the development of algorithms that are robust to
missing and outlier elements of the data points $x_t$. For example,
the Grassmannian Rank-One Update Subspace Estimation (GROUSE)
\cite{BalzanoNowakRecht2010}, Parallel Estimation and Tracking by
REcursive Least Squares (PETRELS)
\cite{ChiJournal2012,ChiEldarCalderbank2012}, and Robust Online
Subspace Estimation and Tracking Algorithm (ROSETA) \cite{roseta}
effectively track a single subspace using incomplete data
vectors. These algorithms are capable of tracking and adapting to
changing environments. The subspace model used in these methods,
however, is inherently strong, whereas a plethora of empirical studies
have demonstrated that high-dimensional data often lie near manifolds
with non-negligible curvature
\cite{allard2012multi,roweis2000nonlinear,belkin2003laplacian}.
In contrast, the non-parametric mixture of factor analyzers
\cite{ChenSilvaPaisley2012} uses a mixture of low-dimensional
approximations to fit to unknown and spatially-varying (but static)
curvatures. The Multiscale Online Union of SubSpaces Estimation
(MOUSSE) method developed by Xie et al. \cite{xie2013change} employs
union of subspaces tracking for change point detection in
high-dimensional streaming data. Thanks to the adoption of the
state-of-the-art subspace tracking techniques, the algorithm is both
accurate and efficient (with complexity linear in $p$). However,
MOUSSE cannot be directly applied for our data thinning task for a few
reasons. First, MOUSSE is designed for change-point detection and does
not have a probabilistic model. Thus observations in a rare subspace
would still be treated as typical, which makes it difficult to
discover the rare observations. Second, MOUSSE can only process one
observation at a time, {\em i.e.,~} it does not allow for mini-batch updates
that can be helpful in data thinning applications, where data could
arrive in blocks. Last but not least, although MOUSSE is able to deal
with missing data, \cite{xie2013change} does not explore the
computational-statistical tradeoffs that are important for time- or
power-sensitive applications. {This paper presents} a method
that is designed for the data thinning task, has a specific
statistical model, and allows for mini-batch updates which increases
the algorithm's efficiency. Also, we will explore the
computational-statistical tradeoffs in Section~\ref{sec:computation}.
\section{Data thinning via tracking union of subspaces}
\label{sec:tracking}
\subsection{Union of subspaces model}
\label{sec:model}
Recall from Section~\ref{sec:formulation} that each $x_t \in \mathbb{R}^p$
is assumed {to be} drawn from a distribution with density $f_t$, and that
$f_t$ is modeled as a mixture of Gaussians where each Gaussian's
covariance matrix is the sum of a rank-$r$ matrix (for $r<p$) and a
scaled identity matrix. We refer to this as a {\em dynamic low-rank GMM}. In
particular, the $j^{\rm th}$ Gaussian mixture component is modeled as
$$\mathcal{N}\left(\mu_{j,t},\Sigma_{j,t}\right)$$
where $\mu_{j,t} \in \mathbb{R}^p$ is the mean and
$$\Sigma_{j,t} = V_{j,t} \Lambda_{j,t} V_{j,t}^T + \sigma_j^2 I.$$
Here $V_{j,t} \in \mathbb{R}^{p \times r}$ is assumed to have orthonormal
columns, and $\Lambda_{j,t} \in \mathbb{R}^{r \times r}$ is a diagonal
matrix with positive diagonal entries.
If $\sigma_j = 0$, then $\Sigma_{j,t}$ would be rank-$r$ and any point
drawn from that Gaussian would lie within the subspace spanned by the
columns of $V_{j,t}$ -- shifted by $\mu_{j,t}$. By allowing $\sigma_j
> 0$ we model points drawn from this Gaussian lying near
that $r$-dimensional shifted subspace.
Overall, we model
\begin{equation}
f_t = \sum_{j=1}^{K_t} q_{j,t} \mathcal{N}\left(
\mu_{j,t}, V_{j,t} \Lambda_{j,t} V_{j,t}^T + \sigma_j^2 I
\right)
\label{eq:GMMmodel}
\end{equation}
where $K_t$ is the number of mixture components in the model at time
$t$ and $q_{j,t}$ is the probability of $x_t$ coming from mixture component $j$.
To better understand this model, we can think of each observation
$x_t$ as having the form $v_t + w_t$, where $v_t$ lies in a union of
subspaces (or more precisely, because of the Gaussian means, a union
of hyperplanes) defined by the $V_{j,t}$s and within ellipsoids
embedded in those hyperplanes, where the ellipsoid axis lengths are
determined by the $\Lambda_{j,t}$s.
Fig.~\ref{fig:subspaceill} illustrates the union of subspaces
model. Fig.~\ref{fig:biker} shows a sample image where one person is
walking on a road with trees on both sides \cite{sampleimage}. In such
a situation, we would want to be able to learn from a sequence of such
images that the trees, grass and the road which occupy most of the
pixels are typical {of} the background, and label the person as salient
because it is uncommon in the scene. Fig.~\ref{fig:bikerill}
illustrates the union of subspaces model. When we divide the image
into patches, the vast majority of patches are plant, and road
patches, and only a few patches contain the person. Thus, the plant
and road patches live on a union of subspaces as illustrated and can
be thinned, leaving anomalous patches for further analysis.
\begin{figure}
\centering
\subfloat[Image of a pedestrian walking on a road with trees on the sides]{\includegraphics[width=.45\textwidth]{Figures/sampleimage.pdf}\label{fig:biker}} ~
\subfloat[Illustration of the union of subspaces
idea]{\includegraphics[width=.45\textwidth]{Figures/USB.pdf}\label{fig:bikerill}}
\caption{{Illustration of the union of subspaces
idea. Fig.~\ref{fig:biker} shows a pedestrian walking on a road
with trees on the sides \cite{sampleimage}. The road and the
plants occupy most of the pixels, and they can be considered
living in a union of subspaces. The person on the road
would be considered as an outlier. }}
\label{fig:subspaceill}
\end{figure}
\subsection{Algorithm highlights}
{This section explains how the proposed method estimates} the evolving Gaussian
mixture model using the techniques from the union of subspaces
tracking algorithms.
These steps are summarized in in Fig.~\ref{fig:flowchart}.
As seen, {this} data thinning method shares some features with the online EM
algorithm for GMM estimation. However, there are a few key differences
{which are elaborated} below:
\begin{itemize}
\item We constrain covariances to lie in a {\em union of subspaces},
which significantly reduces the problem size for estimating the
covariance matrices. This constraint improves the accuracy of the
algorithm, and also makes our method much stabler when the
environment is changing rapidly relative to the data
availability. This constraint also reduces computation time. (More
details of computational complexity are discussed in
Section~\ref{sec:complexity}.)
\item In some settings, such as when working with WAMI data, we
receive groups of $x_t$'s simultaneously and can perform model
updates more efficiently using {\em mini-batch techniques}. (The
mini-batch approach is discussed in Section~\ref{sec:Multi}.)
\item For large, high-velocity data streams, real-time processing
{is} paramount. Even evaluating the likelihood of each new observation
can be time consuming. We explore {\em subsampling-based
approximations} which reduce computational burden yet still yield
accurate results. (Accuracy and computational complexity tradeoffs
are discussed in Section~\ref{sec:computation}.)
\item For the online EM algorithm for GMM estimation, the number of
mixture components is selected {\em a priori}, and does not change for the
duration of the task. This would work when the environment does not
change over time, but is inappropriate for applications that work in
dynamic environments. {The proposed} method {\em adapts to changing numbers of
mixture components}, which allows the mixture model to better
track the environmental dynamics. {The method} adapts the number of mixture
components using a multiscale representation of a hierarchy of
subspaces, which allows us to reduce the model order using a simple
complexity regularization criterion. {The method} also tracks hidden subspaces
which are then used to increase the model order when data
calls for it. (More details about the multi-scale model is discussed
in Section~\ref{sec:algorithm}.)
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{Figures/flow_chart.pdf}
\caption{Flow chart of the main steps in the data thinning method.}
\label{fig:flowchart}
\end{figure}
\subsection{The Online Thinning algorithm}
\label{sec:algorithm}
This section describes the updates of the parameters associated with
{the proposed} dynamic low-rank GMM in \eqref{eq:GMMmodel}. The updates of the
mixture component weights ($q_{j,t}$) and means ($\mu_{j,t}$) are
computed using stochastic gradient descent. The updates of the
covariance matrices are more sophisticated and leverage subspace
tracking methods. In particular, we focus on methods which admit
observations $x_t$ with missing elements; this will allow us to
subsample $x_t$ for computational speedups. These updates are detailed
below.
The biggest challenge is updating $K_t$, the number of mixture
components. In real-life applications, the number of mixture
components is in general (a) not known {\em a priori}, and (b) can
change with $t$. Thus a mechanism for adaptively choosing the number
of subspaces is needed. Reducing model order is slightly less
challenging because it is relatively simple to merge two nearby
mixture components. However, increasing model order is a much more
complex issue, especially in an online setting.
To address these challenges, we organize these mixture components
using a tree structure, as illustrated in Fig.~\ref{fig:tree}. The
idea for a multiscale tree structure stems from the multiscale
harmonic analysis literature \cite{donoho97cart} and online updates of
such models are introduced in \cite{xie2013change}. In our setting, at
time $t$, the $j^{\mbox{th}}$ node is associated with a Gaussian
distribution parameterized by its mean vector $\mu_{j,t}$, low-rank
covariance matrix parameters $V_{j,t}, \Lambda_{j,t}$, and weight
$q_{j,t}$. Most of the probability mass associated with each Gaussian
is an ellipsoid centered at $\mu_{j,t}$, where $V_{j,t}$ and
$\Lambda_{j,t}$ characterize the principle axes and principal axis
lengths, respectively, of the ellipsoid. Finally, $q_{j,t}$ is
approximately the probability of an observation falling inside this
ellipsoid.
In the tree structure, we denote the set of leaf nodes as
$\mathcal{J}_t\triangleq\{j:j^{\mbox{th}} \mbox{ node is a leaf node at time } t\}$
and have $K_t \triangleq |\mathcal{J}_t|$. The leaves of the tree correspond to the
Gaussian mixture components in the model shown in
Eq.~\eqref{eq:GMMmodel}. Each parent node corresponds to a single
Gaussian which approximates the weighted sum of the Gaussians
associated with its two children, where the weights correspond to the
children's $q$ parameters. Each of the tree leaves is also associated
with two {\em virtual} children nodes. The virtual children nodes
correspond to their own Gaussian distributions that can be used to
grow the tree. The decision of pruning and growing are made based on
(a) the accuracy of the Gaussian mixture model, {\em i.e.,~} the cumulative
(with a forgetting factor) anomalousness score, and (b) the size of
the mixture model, {\em i.e.,~} the total number of leaf nodes at time $t$.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{Figures/MousseIllustrationSuper3.pdf}
\caption{Multiscale representation of low-rank Gaussian mixture
model. Consider a density with its mass concentrated along the black
dashed curve. Each successive level in the multiscale representation
has more Gaussian mixture components (depicted via contour plots)
with covariance matrices corresponding to more compact ellipsoids,
and hence yields a more accurate approximation of the underlying
density. Given a particular binary tree representation of a GMM, the
approximation error can be allowed to increase or decrease by
pruning or growing the binary tree connecting the different
scales. The ellipsoids are all very compact along some axes because
they correspond to covariance matrices that are the sum of a
low-rank matrix and a scaled identity matrix.}
\label{fig:tree}
\end{figure}
\subsubsection{Computation of the Gaussian mixture likelihood (and
anomalousness score)}
{The proposed} algorithm uses the negative log-likelihood of {the} Gaussian mixture
model give the data point as its anomalousness score.
The likelihood of $x_t$ under the Gaussian associated with
node $j$ is given by {(recall $\Sigma_{j,t} = V_{j,t} \Lambda_{j,t} V_{j,t}^T + \sigma_j^2 I$)}
\begin{equation}
p_{j,t}(x_t)=\frac{1}{(2\pi)^{p/2}|\Sigma_{j,t}|^{1/2}}e^{-\frac{1}{2}(x_{t}-\mu_{j,t})^T{\Sigma}_{j,t}^{-1}(x_{t}-\mu_{j,t})}.
\label{eq:likelihood}
\end{equation}
Using {the} model
in Eq.~\eqref{eq:GMMmodel}, the Gaussian mixture negative
log-likelihood function (and hence anomalousness score) for any $x_{t}\in\mathbb{R}^{p}$ is:
\begin{equation}
\begin{aligned}
s_t(x_t) =& -\log f_t(x_t)\\
=& -\log \left(\sum_{j\in \mathcal{J}_t}q_{j,t} p_{j,t}(x_{t})\right).
\label{eq:score}
\end{aligned}
\end{equation}
\subsubsection{Selective update}
With the observation of each $x_t$, {the algorithm} first compute the likelihood of
$x_t$ under each of the Gaussian mixture components, and then assign
$x_t$ to the component that maximizes the likelihood. Specifically,
after the likelihood computations above, $x_t$ is assigned to the
mixture component
\begin{equation}
j^*_{t}\triangleq\arg\max_{j\in\mathcal{J}_t}\{p_{j,t}(x_{t})\}.
\label{eq:j_star}
\end{equation}
Note that {the weights $q_{j,t}$ are not used} here in order to avoid biasing towards
components with large weights. {This} assignment is made in order to
reduce the computational complexity of the parameter update step: with
each $x_t$, instead of updating all the parameters of the entire tree,
{the algorithm} only updates the tree branch associated leaf node $j_t^*$. That is,
{the algorithm} updates the parameters of node $j_t^*$, all of its ancestors, and
one of node $j_t^*$'s virtual children (the one under which $x_t$ is
more likely). This approach significantly reduces the time complexity
of the updates, especially when the model is complex ({\em i.e.,~} when the
number of leaf nodes is large).
\subsubsection{Mini-batch update}
\label{sec:Multi}
In previous sections, we have always assumed that we have one
observation $x_t\in\mathbb{R}^p$ arriving at each time $t$. However, in
many applications, multiple observations can arrive
simultaneously. For example, in WAMI settings, hundreds of image
patches in a single frame arrive at the same time. One way to deal
with this is simply treat each patch as arriving at a different time,
and update the model parameters separately with each
observation. However, when the number of patches is large (for HD
videos, {there can be} thousands of patches per frame), this sequential
processing can be extremely time-consuming.
To reduce the computation cost, we can instead update the mixture
model in mini-batches, {\em i.e.,~} when multiple observations {are received} at
the same time, we first compute the anomalousness score of each
observation, and assign them to their own mixture component. The
collection of observations assigned to a given mixture component then
form a mini-batch. {The} mixture model and tree
structure {are then updated} only once for each mini-batch. When the size of mini-batches
is much larger than 1 ({\em e.g.,~} hundreds of image patches assigned to a
tree of size $K_t=10$), this approach significantly reduces the number
of times {needed} to update the mixture component parameters and tree
structures, and thus saves computation time. Note that this mini-batch
processing does not affect the computation of the anomalousness score
and component assignment, where each observation {is processed}
sequentially as if they arrive separately.
Thus, now instead of assuming a single vector $x_t$ arrives at time
$t$, we assume that we receive a collection of observations stored in
matrix $X_t = [x_{t,1},\ldots,x_{t,N_t}]\in\mathbb{R}^{p\times N_t}$ at
time $t$, where $x_{t,i} \in\mathbb{R}^p$ for all $i=1,\ldots,N_t$. A
special case of this is $N_t = 1$, which is the sequential update
without mini-batches. After assigning each column in $X_t$ to the
$K_t$ leaf nodes in the hierarchical tree structure based on their
distance to the corresponding mixture components, we can rewrite $X_t$
into mini-batches, $X_t= [X_{j_{1},t},\cdots, X_{j_{K_t},t}]$, where
$\{j_1,\ldots,j_{K_t}\} \subseteq \mathcal{J}_t$. Here each
$X_{j_i,t}\in\mathbb{R}^{p\times n_{j,t}}, i=1,\ldots,K_t$ is a block of
$n_{j,t}$ data points that are assigned to the $j_i^{\mbox{th}}$ node
in the tree (must be a leaf node). Note that
$\sum_{j\in\mathcal{J}_t} n_{j,t} = N_t$.
Our update equations are based on a ``forgetting factor''
$\alpha \in (0,1)$ that places more weight on more recent
observations; this quantity affects how quickly a changing
distribution is tracked and is considered a tuning parameter to be set
by the end user. Then for each leaf node $j$ that needs updates ({\em i.e.,~}
with assigned observations), the weights $q_{j,t}$ are then updated by
\begin{equation}
q_{j,t+1} = \alpha q_{j,t} + (1-\alpha)\frac{n_{j,t}}{N_{t}}.
\label{eq:q}
\end{equation}
Note that for the leaf nodes the weights need to add to 1, {\em i.e.,~}
$\sum_{j\in\mathcal{J}_t} q_{j,t} = 1$ for all $t$. If we initialize $q_{j,1}$
s.t. $\sum_{i\in\mathcal{J}_1} q_{j,1}=1$ , and the weight of any parent node
is the sum of the weights of its two children, then this update
preserves $\sum_{i\in\mathcal{J}_t} q_{j,t}=1$ for all $t$. The mixture
component means $\mu_{j,t}$ are updated by
\begin{equation}
\mu_{j,t+1} = \alpha \mu_{j,t} + \frac{(1-\alpha)}{n_{j,t}}X_{j,t}\mathbbm{1}_{n_{j,t}\times 1}.
\label{eq:mu}
\end{equation}
The diagonal matrix
\[{\Lambda}_{j, t} \triangleq \mbox{diag}\{\lambda_{j, t}^{(1)}, \ldots, \lambda_{j, t}^{(r)}\} \in \mathbb{R}^{r\times r},\]
with $\lambda_{j, t}^{(1)}, \ldots, \lambda_{j, t}^{(r)}\ge 0$, contains eigenvalues of the
covariance matrix of the projected data onto each subspace. Let
\begin{equation}
M_{j,t} = [\mu_{j,t},\ldots,\mu_{j,t}] \in\mathbb{R}^{p\times n_{j,t}}
\label{eq:M}
\end{equation}
be a means matrix computed by concatenating $n_{j,t}$ copies of $\mu_{j,t}$ together. Let
\begin{equation}
B_{j,t} = V_{j,t}^{\#} (X_{j,t} - M_{j,t}),
\label{eq:B}
\end{equation}
be the residual signal, {where the superscript $^{\#}$ denotes the pseudo-inverse of a matrix (for orthonormal $V_{j,t}$, the pseudo-inverse is its transpose).} Denote its $m^{\mbox{th}}$ row as
$B_{j,t}^{(m)}$. Then we can update
\begin{equation}
\lambda_{j, t+1}^{(m)}=
\alpha\lambda_{j, t}^{(m)} + (1-\alpha) \|B_{j,t}^{(m)}\|_2^2, m = 1,
\ldots, r.
\label{eq:Lambda}
\end{equation}
The subspace matrices $V_{j,t}$ are updated using
Algorithm.~\ref{alg:mpetrels}. The updates of $V_{j,t}$ and
$\Lambda_{j,t}$ are a mini-batch
extension of the PETRELS \cite{ChiEldarCalderbank2012,ChiJournal2012} update equations,
with an added step of orthonormalization of $V_{j,t+1}$ since PETRELS does not guarantee the orthogonality of $V_{j,t+1}$.
For the ancestors of each leaf node that got updated, we
combine all the mini-batches assigned to its children, and update
the node with the same formulae as above using the combined
mini-batches. For the virtual children of leaf nodes that got
updated, we divide each mini-batch into two sub-mini-batches based
on the likelihood of each observation under the Gaussian of the
virtual node, and update each virtual node with its assigned
sub-mini-batch.
\begin{algorithm}
\caption{Mini-Batch Update of Covariance Parameters}\label{alg:mpetrels}
\begin{algorithmic}[1]
\State {\bf Initialize:} $V_{j,1}$ (with training data), $R_{j,1}=c\mathbbm{1}_{r\times r}, c\ll1$
\State {\bf input:} $X_{j,t}, V_{j,t}, R_{j,t}, M_{j,t}$
\State $B_{j,t} = V_{j,t}^{\#} (X_{j,t} - M_{j,t})$
\State $R_{j,t+1} = \alpha R_{j,t} + B_{j,t}B_{j,t}^T$
\State $\wt{V}_{j,t+1} = {V}_{j,t} + \left( (X_{j,t} - M_{j,t})B_{j,t}^T - V_{j,t}B_{j,t}B_{j,t}^T \right) R_{j,t+1}^{\#}$
\State {\bf Orthonormalization}
\Statex \quad ${V}_{j,t+1} = \wt{V}_{j,t+1}\left(\wt{V}_{j,t+1}^T\wt{V}_{j,t+1}\right)^{-\frac{1}{2}}$
\State {\bf Output:} $V_{j,t+1}, R_{j,t+1}$
\end{algorithmic}
\end{algorithm}
\subsubsection{Tree structure update}
The growing (splitting nodes) and pruning (merging nodes) of the tree
structure allow the complexity of the GMM to adapt to the diversity of
the observed data. The number of nodes in the tree controls the
tradeoff between the model accuracy and complexity. {The} proposed
method determines whether to grow or prune the tree by greedily
minimizing a cost function consisting of the weighted cumulative
anomalousness score (with weights corresponding to the forgetting
factor $\alpha$ described above) and the model complexity ($|\mathcal{J}_t|$).
Define $\epsilon_t$ as the cumulative anomalousness score where
$\epsilon_0 = 0$, and
$$\epsilon_{t+1} = \alpha\epsilon_t + \frac{1}{N_t}\sum_{i=1}^{N_t}s_t(x_{t,i}).$$
For each node $j$ (including virtual children), a similar
cumulative score $e_{j,t}$ {is kept} based only on the mini-batches assigned to
that node. Let
$\mathcal{I}_{j,t}\triangleq\{i:x_{t,i} \mbox{ assigned to } j^{\mbox{th}}
\mbox{ node or its children}\}$
(for virtual nodes this set is the indices of its sub-mini-batch), {initialize}
$e_{j,0}=0$, and $e_{j,t}$ is updated by
$$
e_{j,t+1} = \alpha e_{j,t}
+ \frac{1}{|\mathcal{I}_{j,t}|}
\sum_{i\in\mathcal{I}_{j,t}}-\log\left(p_{j,t}(x_{t,i})\right).
$$
Let ${\mbox{\sc tol}}$ be a pre-set error tolerance. For each leaf node
$j_1\in\mathcal{J}_t$ that is assigned new observations, let $j_0$ be its
parent, $j_2$ be its sibling, and $j_{1,1}, j_{1,2}$ be its virtual
children. Let $\gamma$ be a positive constant. Split node
$j_1$ if
\begin{equation}
\epsilon_{t+1} \le {\mbox{\sc tol}},
\label{eq:split1}
\end{equation}
and
\begin{equation}
e_{j_1,t} + \gamma K_t > \frac{q_{j_{1,1},t}e_{j_{1,1},t} + q_{j_{1,2},t}e_{j_{1,2},t}}{q_{j_{1,1},t}+q_{j_{1,2},t}} + \gamma (K_t+1).
\label{eq:split}
\end{equation}
Note the left side of Ineq.~\eqref{eq:split} is the penalized
cumulative score of node $j_1$ (where the penalty is proportional to
the number of nodes in the tree), while the right side of
Eq.~\eqref{eq:split} is the average penalized cumulative score of node
$j_1$'s two virtual children. We split node $j_1$ if the average
penalized cumulative score is smaller at the virtual children level.
Similarly, merge nodes $j_1$ and $j_2$ if
\begin{equation}
\epsilon_{t+1} \ge {\mbox{\sc tol}}
\label{eq:merge1}
\end{equation}
and
\begin{equation}
e_{j_0,t} + \gamma (K_t-1) < \frac{q_{j_1,t}e_{j_1,t} + q_{j_2,t}e_{j_2,t}}{q_{j_1,t}+q_{j_2,t}} + \gamma K_t,
\label{eq:merge}
\end{equation}
Note the left side of Ineq.~\eqref{eq:merge} is the penalized (with
tree size) cumulative score of node $j_1$'s parent $j_0$, while the
right side of Eq.~\eqref{eq:split} is the average penalized cumulative
score of node $j_1$ and its sibling $j_2$. We merge $j_1$ and $j_2$ if
the average penalized cumulative score of $j_1$ and $j_1$ is larger
than the penalized score of their parent. The use of these penalized
scores to choose a tree which is both (a) a good fit to the observed
data and (b) a small as possible to avoid overfitting is common in
classification and regression trees
\cite{CART,willett:tmi03,willett:jsac04,WillettNowak2005,scott2006minimax,willett:density}.
The splitting and merging operations are detailed in
Algorithm~\ref{alg:split} and Algorithm~\ref{alg:merge}. The complete
Online Thinning algorithm is summarized in Algorithm~\ref{alg:OTM}.
\begin{algorithm}[h!]
\caption{Grow tree}
\begin{algorithmic}[1]
\State {\bf Input:} Node $j$ with virtual children nodes $k$ and $\ell$
\State Update $\mathcal{J}_{t+1} = \mathcal{J}_{t}\bigcup \{k,\ell\} \backslash \{j\} $
\State Create new virtual children: $k_1, k_2$ for new leaf node $k$, and $j_{1,1}, j_{1,2}$ for new leaf node $\ell$
\State Let $v_{i,t}^{(1)}$ be the first column of $V_{i,t}, i\in\{k,\ell\}$
\State Initialize virtual
nodes $k_1, k_2, j_{1,1}$ and $j_{1,2}$:
\Statex \quad for $i\in \{k,\ell\}$
\begin{align*}
\mu_{i_1, t+1} &= \mu_{i, t} + \sqrt{\lambda_{i,
t}^{(1)}} v^{(1)}_{i, t}/2 \\
\mu_{i_2, t+1} &= \mu_{i, t} - \sqrt{\lambda_{i,
t}^{(1)}} v^{(1)}_{i, t}/2\\
V_{i_1, t+1} &= V_{i, t} \\
V_{i_2, t+1} &= V_{i, t} \\
\lambda_{i_1, t+1}^{(1)} & = \lambda_{i, t}^{(1)}/2 \\
\lambda_{i_2, t+1}^{(1)} &= \lambda_{i, t}^{(1)}/2 \\
\lambda_{i_1, t+1}^{(m)} &=\lambda_{j, t}^{(m)}, \quad
m = 2, \ldots, r \\
\lambda_{i_2, t+1}^{(m)} &= \lambda_{j, t}^{(m)}, \quad
m = 2, \ldots, r \\
q_{i_1, t+1} &= q_{j,t} / 2\\
q_{i_2, t+1} &= q_{j,t} / 2
\end{align*}
\end{algorithmic}
\label{alg:split}
\end{algorithm}
\begin{algorithm}[h!]
\caption{Prune tree}
\begin{algorithmic}[1]
\State {\bf Input}: Node $j$ with children nodes $j_1$ and $j_2$ to be
merged
\State Delete all four virtual children nodes of $j_1$ and $j_2$
\State Update $\mathcal{J}_{t+1} = \mathcal{J}_t\bigcup\{j\}\backslash \{j_1, j_2\}$
\State Define $j_1$, $j_2$ as the virtual children nodes of the
new leaf node $j$
\end{algorithmic}
\label{alg:merge}
\end{algorithm}
\begin{algorithm}[h!]
\caption{Online Thinning with Mini-Batch Updates}
\begin{algorithmic}[1]
\State {\bf input:} error tolerance ${\mbox{\sc tol}}>0$, threshold $\tau>0$, forgetting factor $\alpha\in(0,1)$
\State {\bf initialize:} tree structure, set initial error
$\epsilon_1 = 0$
\For{$t = 1, 2, \ldots$}
\State receive new data
$X_{t}\in\mathbb{R}^{p\times N_t}$
\For{$i = 1, 2, \ldots, N_t$}
\State let $x_{t,i}$ be the $i^{\mbox{th}}$ column of $X_{t}$
\State for all $j\in\mathcal{J}_t$, compute likelihood of $x_{t,i}$ under
node $j$:
$$ p_{j,t}(x_{t,i})=\frac{1}{(2\pi)^{p/2}|\Sigma_{j,t}|^{1/2}}e^{-\frac{1}{2}(x_{t,i}-\mu_{j,t})^T{\Sigma}_{j,t}^{-1}(x_{t,i}-\mu_{j,t})}$$
\State compute anomalousness score $s_t(x_{t,i})$:
$$s_t(x_{t,i}) = -\log \left(\sum_{j\in \mathcal{J}_t}q_{j,t}
p_{j,t}(x_{t,i})\right)$$
\State assign $x_{t,i}$ to leaf node $j^*_{t}\triangleq\arg\max_{j\in\mathcal{J}_t}\{p_{j,t}(x_{t,i})\}.$
\State compute the likelihood of $x_{t,i}$ under $j^*$'s two virtual
children nodes, and also assign $x_{t,i}$ to the virtual child with
higher likelihood
\EndFor
\State update $\epsilon_{t+1} = \alpha\epsilon_t + \frac{1}{N_t}\sum_{i=1}^{N_t}s_t(x_{t,i})$
\For{ all nodes $j$ in the tree}
\State set $\mathcal{I}_{j,t}\triangleq\{i:x_{t,i} \mbox{ assigned to } j^{\mbox{th}}
\mbox{ node or its children}\}$
\If {$\mathcal{I}_{j,t}$ is not empty}
\State {denote all data assigned to node $j$ or its children as $
X_{j,t} = [x_1,\ldots,x_{n_{j,t}}]$}
\State update $e_{j,t+1} = \alpha e_{j,t}
+ \frac{1}{|\mathcal{I}_{j,t}|}
\sum_{i\in\mathcal{I}_{j,t}}-\log\left(p_{j,t}(x_{t,i})\right)$
\State update $q_{j,t+1} = \alpha q_{j,t} + (1-\alpha)\frac{n_{j,t}}{N_{t}}$
\State update $\mu_{j,t+1} = \alpha \mu_{j,t} +
\frac{(1-\alpha)}{n_{j,t}}X_{j,t}\mathbbm{1}_{n_{j,t}\times 1}$
\State set $M_{j,t} = [\mu_{j,t},\ldots,\mu_{j,t}] \in\mathbb{R}^{p\times n_{j,t}}$
\State set $B_{j,t} = V_{j,t}^\# (X_{j,t} - M_{j,t})$
\For{$m = 1,\ldots, r$}
\State update
$\lambda_{j, t+1}^{(m)}=
\alpha\lambda_{j, t}^{(m)} + (1-\alpha) \|B_{j,t}^{(m)}\|_2^2$
\EndFor
\State update $V_{j,t}$ by calling Algorithm~\ref{alg:mpetrels}
\If{$\epsilon_{t+1} \le {\mbox{\sc tol}}$ and $e_{j_1,t} + \gamma K_t > \frac{q_{j_{1,1},t}e_{j_{1,1},t} + q_{j_{1,2},t}e_{j_{1,2},t}}{q_{j_{1,1},t}+q_{j_{1,2},t}} + \gamma (K_t+1)$}
\State call Algorithm~\ref{alg:split}
\ElsIf{$\epsilon_{t+1} \ge {\mbox{\sc tol}}$ and $e_{j_0,t} + \gamma (K_t-1) < \frac{q_{j_1,t}e_{j_1,t} + q_{j_2,t}e_{j_2,t}}{q_{j_1,t}+q_{j_2,t}} + \gamma K_t$}
\State call Algorithm~\ref{alg:merge}
\EndIf
\Else { update $q_{j,t+1} = \alpha q_{j,t}$}
\EndIf
\EndFor
\State $\mathcal{X}_t = \{x_{t,i}:s_t(x_{t,i}) > \tau\}$
\EndFor
\State {\bf output:}
sequence of thinned data $\mathcal{X}_1,\ldots,\mathcal{X}_T$
\end{algorithmic}
\label{alg:OTM}
\end{algorithm}
\subsection{Subsampling observations}
When $p$ is large and computation time is critical, we can subsample
the elements of each $X_t$ and leverage missing data models for fast
calculations and updates. Algorithm~\ref{alg:mpetrels} is a modified
version of PETRELS \cite{ChiJournal2012,ChiEldarCalderbank2012} with
mini-batches. Note that PETRELS was specifically designed to work with
missing entries, where \cite{ChiJournal2012,ChiEldarCalderbank2012}
thoroughly investigated the effect of missing data in subspace
tracking algorithms.
Specifically, to modify our Online Thinning algorithm for $X_t$ with
subsampled entries, we define $\Omega_t\subseteq \{1,\ldots,p\}$ be
the subset of entries {used} at time $t$. Assume all $\Omega_t$ have
the same size and define $|\Omega|\triangleq|\Omega_t|, \forall t$. Define
an operator $P_{\Omega_t}(\cdot)$ that selects the rows indexed by
$\Omega_t$. {Then, for the likelihood and score computation, denote
$\Sigma_{j,t,\Omega_t} \triangleq P_{\Omega_t}(V_{j,t}) \Lambda_{j,t} P_{\Omega_t}(V_{j,t})^T + \sigma_j^2 I_{|\Omega_t|}$, and compute
\begin{equation}
\begin{aligned}
p_{j,t}(x_t)&=\frac{1}{(2\pi)^{p/2}|\Sigma_{j,t}|^{1/2}}\exp\left\{-\frac{1}{2}\left[P_{\Omega_t}(x_{t,i})\right.\right. \\
&\left.\left.-P_{\Omega_t}(\mu_{j,t})\right]^T
{\Sigma}_{j,t,\Omega_t}^{-1}\left[P_{\Omega_t}(x_{t,i})-P_{\Omega_t}(\mu_{j,t})\right]\right\}
\end{aligned}
\notag
\end{equation}
as the likelihood.
For the mini-batch update step, replace $X_{j,t}, \mu_{j,t},$ and $V_{j,t}$ with
$P_{\Omega_t}(X_{j,t}), P_{\Omega_t}(\mu_{j,t}),$
and $P_{\Omega_t}(V_{j,t})$, respectively in Eq.~\eqref{eq:mu},\eqref{eq:M}, and~\eqref{eq:B}, and
use Algorithm~\ref{alg:mpetrels2} instead of Algorithm~\ref{alg:mpetrels}.}
\begin{algorithm}
\caption{Mini-Batch Update of Covariance Parameters with Subsampling}\label{alg:mpetrels2}
\begin{algorithmic}[1]
\State {\bf Initialize:} $V_{j,1}$ (with training data), $R_{j,1}=c\mathbbm{1}_{r\times r}, c\ll1$
\State {\bf input:} $\Omega_t, P_{\Omega_t}(X_{j,t}), V_{j,t}, R_{j,t}, P_{\Omega_t}(M_{j,t})$
\State $B_{j,t} = P_{\Omega_t}(V_{j,t})^{\#} \left[P_{\Omega_t}(X_{j,t}) - P_{\Omega_t}(M_{j,t})\right]$
\State $R_{j,t+1} = \alpha R_{j,t} + B_{j,t}B_{j,t}^T$
\State $\wt{V}_{j,t+1} = {V}_{j,t}$
\State $P_{\Omega_t}(\wt{V}_{j,t+1}) = P_{\Omega_t}(\wt{V}_{j,t+1}) + \left[ (P_{\Omega_t}(X_{j,t})\right.$
\Statex $\left. - P_{\Omega_t}(M_{j,t}))B_{j,t}^T - P_{\Omega_t}(V_{j,t})B_{j,t}B_{j,t}^T \right] R_{j,t+1}^{\#}$
\State {\bf Orthonormalization}
\Statex \quad ${V}_{j,t+1} = \wt{V}_{j,t+1}\left(\wt{V}_{j,t+1}^T\wt{V}_{j,t+1}\right)^{-\frac{1}{2}}$
\State {\bf Output:} $V_{j,t+1}, R_{j,t+1}$
\end{algorithmic}
\end{algorithm}
\subsection{Computational complexity}
\label{sec:complexity}
As discussed in Section~\ref{sec:tracking}, the union of subspaces
assumption significantly reduces the problem size for estimating the
covariance matrices. This not only improves the algorithm accuracy and
stability for high dimensional problems, but also reduces computation
time.
{
Take the computation of $(x-\mu_{j,t})^T\Sigma_{j,t}^{-1}(x-\mu_{j,t})$ as an example.
$\Sigma_{j,t}$ is the covariance matrix of the
$j^{\mbox{th}}$ mixture component at time $t$. For a full-rank GMM
model, computing the $\Sigma_{j,t}^{-1}$ takes $O(p^3)$
operations, and computing $(x-\mu_{j,t})^T\Sigma_{j,t}^{-1}(x-\mu_{j,t})$ given $\Sigma_{j,t}^{-1}$ takes $O(p^2)$ operations. Thus the total complexity is $O(p^3)$.
However, with the low-rank assumption we have
$\Sigma_{j,t}= V_{j,t}\Lambda_{j,t}V_{j,t}^T + \sigma^2 I$, and with
the Woodbury matrix identity \cite{woodbury1950inverting}, we can compute
\begin{equation}
\Sigma_{j,t}^{-1} = \sigma^{-2} I + \sigma^{-4} V_{j,t} (\Lambda_{j,t}^{-1} + V_{j,t}^T V_{j,t})^{-1} V_{j,t}^T,
\notag
\end{equation}
and
\begin{equation}
\begin{aligned}
&(x-\mu_{j,t})^T\Sigma_{j,t}^{-1}(x-\mu_{j,t}) = \\
&\sigma^{-2} (x-\mu_{j,t})^T(x-\mu_{j,t}) \\
&+ \sigma^{-4} (x-\mu_{j,t})^TV_{j,t} (\Lambda_{j,t}^{-1} + V_{j,t}^T V_{j,t})^{-1} V_{j,t}^T(x-\mu_{j,t}).
\end{aligned}
\notag
\end{equation}
Note that computing $(\Lambda_{j,t}^{-1} + V_{j,t}^T V_{j,t})^{-1}$ is easy because (a) $V_{j,t}^T V_{j,t} = I_{r}$ since the columns of $V_{j,t}$ are orthonormal, and (b) $\Lambda_{j,t}$ is diagonal.
Computing the whole equation takes $O(pr+ r^2)=O(pr)$ operations. Thus, by using the low-dimensional structure, we reduced the computation complexity from $O(p^3)$ to $O(pr)$.
}
{
Another example is the computation of the determinant of $\Sigma_{j,t}$. For a full-rank GMM
model, computing $|\Sigma_{j,t}|$ takes $O(p^3)$ operations. For our low-rank model with $\Sigma_{j,t}= V_{j,t}\Lambda_{j,t}V_{j,t}^T + \sigma^2 I$, we can use the matrix determinant lemma \cite{harville1998matrix} and compute
\begin{equation}
\begin{aligned}
|\Sigma_{j,t}| &= \sigma^2\left|\Lambda_{j,t}\right| \left|\Lambda_{j,t}^{-1}+\sigma^{-2}V_{j,t}^TV_{j,t}\right|.
\end{aligned}
\notag
\end{equation}
The number of operation needed is O(r) since $V_{j,t}^T V_{j,t} = I_{r}$ and $\Lambda_{j,t}$ is diagonal.
}
{
\subsubsection{Computational complexity without subsampling}
{
Below we summarize the {computational} complexity of each major steps of the Online Thinning algorithm.{Let} $T$ {be} the total number of time steps. {For} simplicity, we assume the
mini-batch sizes $N_t$ are the same for all $t$. Let $N \triangleq T N_t$ be
the total number of observations {received}, and $K_{\mbox{max}}$ be the
maximum number of leaf nodes in the tree.
}
\begin{itemize}
\item {Likelihood and score computation} has a complexity of
$$O(NK_{\mbox{max}} pr),$$
where each likelihood computation takes $O(pr)$ operations, and this is computed $K_{\mbox{max}}+2$ times per observation ($K_{\mbox{max}}$ leaf nodes plus two virtual children of the assigned leaf node) for all $N$ observations. The computation of the anomalousness score is computationally inexpensive since it is a weighted sum of pre-computed likelihoods.
\item Mini-batch updates {have complexities of} at most
$$O\left(N K_{\mbox{max}} pr + \frac{N}{N_t} K_{\mbox{max}} pr^2\right),$$
where the first term $N K_{\mbox{max}} pr$ comes from the calculation of $B_{j,t}$ and $\wt{V}_{j,t+1}$, and the second term comes from the orthonormalization of $\wt{V}_{j,t+1}$. The term $\frac{N}{N_t}$ is the number of batches {received}
\item Tree structure updates {have complexities of} at most $$O(N K_{\mbox{max}} pr+\frac{N}{N_t}K_{\mbox{max}} pr),$$
where the first term is an upper bound of complexity for updating the cumulative likelihood $e_{j,t}$ of the parents of leaf nodes (for leaf nodes and their virtual children, the likelihood is computed when at the score computing stage). In the second term, the term $pr$ comes from the number of operations needed to copy the subspace when splitting nodes, and the maximum number of splitting at each time $t$ is bounded by the maximum number of leaf nodes $K_{\mbox{max}}$ (merging nodes is computational inexpensive).
\end{itemize}
Adding three steps together, the Online Thinning algorithm has a complexity {of} at most
$$O\left(N K_{\mbox{max}} pr + \frac{N}{N_t} K_{\mbox{max}} pr^2\right).$$
}
{
\subsubsection{Computational complexity with subsampling}
Let $|\Omega|$ be the number of entries {observed} after subsampling, then the complexity of each major step {is} as follows.
\begin{itemize}
\item {Likelihood and score computation} has a complexity of
$$O(NK_{\mbox{max}} |\Omega|r),$$
which scales with $|\Omega|$.
\item Mini-batch updates {have complexities of} at most
$$O\left(N K_{\mbox{max}} (|\Omega|r+r^2) + \frac{N}{N_t} K_{\mbox{max}} pr^2\right),$$
where the first term $N K_{\mbox{max}} (|\Omega|r+r^2)$ comes from the calculation of $B_{j,t}$ and $\wt{V}_{j,t+1}$, and is affected by subsampling. Note the extra $NK_{\mbox{max}} r^2$ comes from the added complexity from computing $P_{\Omega_t}(V_{j,t})^{\#}$. When no subsampling is done, $V_{j,t}^{\#}=V_{j,t}^{T}$. However, this is {generally} not true when we perform subsampling, and this extra step adds complexity. However, in general $|\Omega|$ scales with $p$ and is much larger than $r$.
The second term comes from the orthonormalization of $\wt{V}_{j,t+1}$, which is not affected by subsampling.
\item Tree structure updates {have complexities of} at most
$$O(N K_{\mbox{max}} |\Omega|r+\frac{N}{N_t}K_{\mbox{max}} pr),$$
where the first term comes from the likelihood computation, and scales with $|\Omega|$. The second comes from the tree splitting, which is not affected by subsampling.
\end{itemize}
Adding three steps together, the Online Thinning algorithm has a complexity {of} at most
$$O\left(N K_{\mbox{max}} (|\Omega|r + r^2) + \frac{N}{N_t} K_{\mbox{max}} pr^2\right).$$
}
\subsubsection{Remarks}
{
Subsampling changes the first term of {the} complexity. Higher subsampling rates (smaller $|\Omega|$)
{reduce} the complexity of computing the likelihood, and affect some steps in the mini-batch update. However, {subsampling} does not affect the orthonormalization of $V_{j,t}$, or the splitting and merging of tree structures. Additionally, subsampling makes the computation of $P_{\Omega_t}(V_{j,t})^{\#}$ difficult, since in general $P_{\Omega_t}(V_{j,t})^{\#}\neq P_{\Omega_t}(V_{j,t})^{T}$. Still, the effect of this added complexity in computing $P_{\Omega_t}(V_{j,t})^{\#}$ is generally small since $|\Omega|$ is usually much larger than $r$.
}
{
Changing the size of mini-batches
$N_t$ changes $\frac{N}{N_t}$, and thus the second term of {the} complexity. Specifically, in the algorithm,
changing $N_t$ changes (a) the number of times {needed} to update the tree
structure, (b) the number of times {needed to} update $R_{j,t}$ and its pseudo-inverse (see
Algorithm~\ref{alg:mpetrels}), and (c) the
number of time {needed} to perform the orthonormalization of $V_{j,t}$
(see Algorithm~\ref{alg:mpetrels}).
When $N_t$ has the same order of (or larger than) $K_{\mbox{max}} r$, and without subsampling, the algorithm's complexity is linear to the total number of observations $N$, the observation dimension $p$, the tree size $K_{\mbox{max}}$, and the subspace dimension $r$.
}
\section{Computational and statistical tradeoffs}
\label{sec:computation}
Different systems have different delay allowances and precision
requirements, and it is natural to ask how much performance we
sacrifice by trying to reduce the computation time. Understanding such
tradeoffs is crucial for applications where real-time processing is
required, yet the computational power is limited, as with many mobile
surveillance systems. {This section explores} the tradeoff between
processing time and detection accuracy for the Online Thinning
algorithm.
There are two primary ways to reduce the computational complexity of
the data thinning: {(1)} by randomly subsampling the entries of $x_t$,
{\em i.e.,~} we only use partially observed data to update the dynamic low-rank GMM
model parameters and
estimate the anomalousness score; and {(2)} by varying the size
of the mini-batches. Note that these are made possible because, as
discussed in Section~\ref{sec:tracking}, data thinning (a) is robust
to unobserved entries, and (b) can process data in mini-batches,
respectively.
To explore this further, two experiments {are conducted}---one in which we
vary the mini-batch size, and one in which we vary the subsampling
rate. For these experiments, {the data is generated} as follows: The ambient
dimension is $p=100$. We first generate points in $\mathbb{R}^p$ in a
union of three (shifted) subspaces of dimension ten; in which 95\% of
the points lie in the union of the first two subspaces. The other 5\%
of the points lie in a third subspace that is orthogonal to the other
two. All three subspaces have shifts close to 0. We then add white
Gaussian noise with variance $\sigma^2 = 0.1$ to these points to
generate our observations. The two subspaces where the 95\% of
observations come from are dynamic, where {the subspaces rotate at a}
speed $\delta>0$. For $j=1,2$, we have
$$
V_{j,t+1} = V_{j,t} + \delta\frac{B}{\|B\|_F}V_{j,t},
$$
where $B$ is a $p\times p$ skew-symmetric matrix. Denote the set of
$x_t$'s coming from each of the three subspaces as
$\mathcal{X}_j, j=1,2,3$, respectively. The goal is to identify the
5\% of the observations that come from $\mathcal{X}_3$.
The experiment streams in four thousand observations in total. An
initial model is estimated using the first one thousand samples, and
the models are then updated in an online fashion for the remaining
three thousand samples. The anomalousness score is calculated as the
negative log-likelihood of each data point according to the estimated
model. We then select observations $x_t$ for which $s_t(x_t) > \tau$,
and compute the detection rate and false alarm rate
$$P_D(\tau) = \frac{|\{t:x_t \in\mathcal{X}_3, s_t(x_t) > \tau\}|}{|\{t:x_t \in \mathcal{X}_3\}|},$$
$$P_F(\tau) = \frac{|\{t:x_t \in\mathcal{X}_1\bigcup\mathcal{X}_2, s_t(x_t) > \tau\}|}{|\{t:x_t \in \mathcal{X}_1\bigcup\mathcal{X}_2\}|}.$$
The threshold $\tau$ is tuned to minimize the detection error
$1-P_D(\tau)+P_F(\tau)$. Each experiment is averaged over ten random
realizations.
{The first experiment varies} the percentage of entries {observed} in
each $x_t$. Subsampling reduces the dimension of $x_t$,
which saves time in many of the operations in the algorithm. With more
observed entries, {the} estimates of the likelihoods
under each mixture component {are more accurate}, and hence the thinning
performance {is better}. However, the computation of likelihoods and updates of
the dynamic low-rank GMM parameters will also be slower.
Fig.~\ref{fig:timeaccuracy1} shows the detection error of our approach
as a function of subsampling rate ($|\Omega|/p$). The two curves
correspond to different subspace rotation speed ($\delta$). We vary
the subsampling rate from 25\% to 100\%. The detection error is kept
at less than 5\% even at a subsampling rate of 55\%.
\begin{figure}[!t]
\centering
{\includegraphics[width=0.5\textwidth]{Figures/new_time_accu_1005.pdf}}
\caption{Detection error as a function of subsampling rate. The
two curves correspond to different subspace rotation speed
($\delta$). A subsampling rate at 55\% still keeps the detection
error less than 5\%. }
\label{fig:timeaccuracy1}
\end{figure}
{The second experiment varies} $N_t$, the size of the mini-batches.
{The batch size $N_t$ varies} from 10
to 1000. Fig.~\ref{fig:timeaccuracy2} displays the detection error as
a function of $N_t$. The three curves correspond to different subspace
rotation speed ($\delta$). The detection error increases slightly as
$N_t$ increase, since reducing $N_t$ in general improves the ability
of the algorithm to follow the changing subspaces. For all three
values of $\delta$, the change in detection error relative to $N_t$ is
less than 2\%.
\begin{figure}[!t]
\centering
{\includegraphics[width=0.5\textwidth]{Figures/timevsacc_2_2.pdf}}
\caption{Detection error as a function of mini-batch size $N_t$. The three curves correspond to different
subspace changing speeds ($\delta$). The detection error increases as
$N_t$ increases. For all three $\delta$ values, the change in
detection error relative to $N_t$ is less than 2\%.}
\label{fig:timeaccuracy2}
\end{figure}
\section{Synthetic data experiments}
\label{sec:syncexp}
{This section compares the} Online Thinning approach based on
tracking a dynamic low-rank GMM with (a) a classical full-rank static GMM and
(b) an online GMM estimation algorithm. Neither
of these comparators has the low-rank structure {exploited by the Online Thinning algorithm}. The
synthetic data is generated according to the same model as in Section~\ref{sec:computation}.
The experiment streams in four thousand observations in total. For
Online Thinning and the classical online GMM, an initial model is
estimated using the first one thousand samples, and the models are
then updated in an online fashion for the remaining three thousand
samples. The anomalousness score is calculated as the negative
log-likelihood of each data point according to the estimated
model. For the classical static GMM algorithm, we estimate a GMM model
on the entire four thousand data points (after all samples come in) at
once, and assign an anomalousness score to each sample proportional to
the negative log likelihood of the data point coming from the
estimated GMM model. Fig.~\ref{fig:sync} compares the detection
accuracy (in ROC curves) of Online Thinning and the two comparator
algorithms in two settings, where in \ref{fig:sync1}, the true
subspaces used to generate the data are kept static throughout the
experiment, and in \ref{fig:sync2} and \ref{fig:sync3}, the true
subspaces rotate at a small rate ($5\times10^{-3}$ and
$2\times10^{-2}$, respectively) at each time step. Each plotted
experiment is averaged over thirty random realizations. As seen in the
plots, Online Thinning using the dynamic low-rank GMM outperforms the
online and static algorithms based on a classical full-rank GMM model
in all cases, especially when the subspaces change over time.
The reasons behind the performance gap when the subspaces
change over time can be explained by the underlying models of the three
algorithms. Both the batch GMM and online GMM algorithms rely on full-rank
GMM models, which make the problem ill-posed, and, {therefore,} estimating the
covariance matrices becomes difficult. Furthermore, the batch GMM algorithm
relies on a static model, which introduces bias when the environment is dynamic.
On the other hand, Online Thinning is based on a
dynamic low-rank GMM model, and thus faces a much less ill-posed problem
by having a union of subspace assumption (which significantly reduces
the number of unknowns in the covariance matrices). At the same time,
Online Thinning focuses on the most recent samples by weighing down
the past samples, and can thus follow the changes in the subspaces.
\begin{figure*}[!t]
\centering
\subfloat[Static subspaces]{\includegraphics[width=0.33\textwidth]{Figures/compare3_0.pdf}\label{fig:sync1}}~
\subfloat[Subspaces changing at rate $5\times10^{-3}$]{\includegraphics[width=0.33\textwidth]{Figures/compare3_5e-3.pdf}\label{fig:sync2}}~
\subfloat[Subspaces changing at rate $2\times10^{-2}$]{\includegraphics[width=0.33\textwidth]{Figures/compare3_2e-2.pdf}\label{fig:sync3}}
\caption{Comparison between Online Thinning using a dynamic low-rank
GMM, a classical online GMM, and a classical static batch GMM assuming
true subspace rank (ten) is {known}. \ref{fig:sync1}
shows the comparison between Online Thinning and GMM when the subspaces
are static. \ref{fig:sync2} and \ref{fig:sync3} show the comparison
between Online Thinning and GMM when the subspaces change at a rate of
$5\times10^{-3}$ and $2\times10^{-2}$, respectively. Online Thinning
outperforms both the online and batch GMM algorithms in all cases,
especially when the subspaces change over time. }
\label{fig:sync}
\end{figure*}
In Fig.~\ref{fig:sync}, the true subspace
rank {is assumed known}. However, the real rank of the subspaces is not always known {\em a
priori}. To further assess the performance of Online Thinning in such
situations, we repeat the above experiment but compute rank-six and rank-eight
approximations of the rank-ten subspaces; the results are displayed in
Fig.~\ref{fig:sync_r7}. Note that the classical (full-rank) GMM
algorithms are not affected by the rank assumption. As seen in the
plots, the performance of Online Thinning
slightly degrades when the rank of the subspace is given incorrectly
to the algorithm. However, Online Thinning still outperforms the classical batch GMM and online GMM
algorithms when the subspaces rotates at a rate of $\delta=1\times 10^{-2}$, even when the rank of the subspaces {is significantly under-estimated}.
\begin{figure}[!t]
\centering
{\includegraphics[width=0.5\textwidth]{Figures/compare_all_rank.pdf}}
\caption{Comparison between Online Thinning, online GMM and batch GMM assuming the subspace rank {is under-estimated} at six and eight for the Online Thinning algorithm. We show the comparison between Online Thinning, online GMM and regular GMM algorithms when the subspaces change at a rate of $1\times10^{-2}$. Even when {the rank is incorrectly estimated at six (correct rank is ten)}, Online Thinning outperforms both classical batch GMM and online GMM algorithms.}
\label{fig:sync_r7}
\end{figure}
\section{Wide-area motion imagery experiments}
\label{sec:WAMIexp}
{This experiment compares} Online Thinning with the SUN (Saliency
Using Natural statistics) algorithm proposed by Zhang {\em et al}
in~\cite{zhang2008sun}. The SUN algorithm is representative of the
state-of-the-art saliency detection algorithms~\cite{borji2013state},
provides a general framework for many models, performs as well as or
better than previous models, and is computationally
efficient~\cite{zhang2008sun}.
We perform this comparison on a real surveillance video capturing an
empty field near a highway. In the video, a car is parked on the lot,
and two people can be seen walking in and out of the scene on the
field. We use this video because it is clear that the car and the
people are most salient in the scene. The original video can be found
at \url{https://youtu.be/mX1TtGdGFMU}. For the Online Thinning
algorithm, we use SIFT (scale-invariant feature transform) features
\cite{lowe1999object} of frame $t$ as our observation $X_t$ at time
$t$. Specifically, we use the package from \cite{vedaldi08vlfeat} to
compute the dense SIFT features ({\em i.e.,~} SIFT features computed over a
pre-set grid of points on each frame) as features. Each frame of the video is of size
$960\times540$, and the {grid is placed} so that one SIFT feature is
computed for each $25\times 25$ patch. {Each frame have} roughly
eight hundred SIFT feature vectors. The dimension of each
SIFT feature vector is 128.
Fig.~\ref{fig:pklot} shows the result of Online Thinning and the SUN
algorithms on this surveillance video at frames 50 and
100. Figures~\ref{fig:pklot11} and~\ref{fig:pklot12} show the original
frames, while in~\ref{fig:pklot21} and~\ref{fig:pklot22}, we flag the
top 5\% patches with the highest anomalousness or saliency scores by
the Online Thinning and SUN algorithms. In the results, green patches
are flagged by both methods, blue patches are only flagged by Online
Thinning, and red patches are only flagged by SUN. Note that in both
frames, the people in the scene are mostly labeled by blue, {\em i.e.,~} they
are only flagged by Online Thinning. {The} Online Thinning
outperforms the SUN algorithm by more consistently flagging small rare
patches such as the people; this is in part due to the adaptivity of
{Online Thinning} to dynamic environments. The result video can be found at
\url{https://www.youtube.com/watch?v=DyLJThawgi0}.
\begin{figure}[!t]
\centering
\subfloat[Original, frame 50]{\includegraphics[width=0.4\textwidth]{Figures/gray50.jpg}\label{fig:pklot11}}~
\subfloat[Original, frame 100]{\includegraphics[width=0.4\textwidth]{Figures/gray100.jpg}\label{fig:pklot12}}\\
\subfloat[5\% most salient patches, frame 50]{\includegraphics[width=0.4\textwidth]{Figures/frame50.jpg}\label{fig:pklot21}}~
\subfloat[5\% most salient patches, frame 100]{\includegraphics[width=0.4\textwidth]{Figures/frame100.jpg}\label{fig:pklot22}}
\caption{Data thinning result using Online Thinning and SUN algorithms on the surveillance video at frames 50 and 100. The first row shows the original video, and the second row shows the data thinning results. In the results, green patches are flagged by both methods, blue patches are only flagged by Online Thinning, and red patches are only flagged by SUN. Online Thinning outperforms the SUN algorithm by consistently flagging the people, which are sometimes missed by the SUN algorithm.}
\label{fig:pklot}
\end{figure}
Motion imagery taken from a moving camera ({\em e.g.,~} video taken from an
unmanned arial vehicle) is often jittery due to mechanical vibrations
in the camera platform. Such jittering often poses difficulty to the
data thinning task. The magnitude of the vibrations precludes
standard video stabilization techniques used, for instance, for
handheld video cameras. {This experiment demonstrates} that the
proposed method can robustly flag salient objects from a jittery
video; the flagged patches can then be processed off-line (as
discussed in the introduction), and software video-stabilization
methods can be applied to these frames {alone} to co-register them.
Specifically, to demonstrate the effect of jittering, we artificially
add random rotations and small shifting to each of the frames before
processing. The jittered video can be found at
\url{https://youtu.be/oKzIOryxR0s}. Then, we flag and extract patches
with high anomalousness scores using {the} proposed Online Thinning
algorithm. Finally, we use a feature-matching-based approach {\em on
only the flagged patches} to generate a stabilized, thinned
video~\cite{lee2009video,matsushita2005full}. Fig.~\ref{fig:stable2}
shows the original jittered video frames
(left column) and corresponding stabilized detection results (right
column). Note that despite the rotation and shifting of the original
frames, the stabilized result is consistently showing the car
and the people without significant shifting or shaking. The
result video can be found at \url{https://youtu.be/DyLJThawgi0}.
\begin{figure}[!t]
\centering
\subfloat[Jittering video frame 60]{\includegraphics[width=0.33\textwidth]{Figures/n11.jpg}\label{fig:stb11}}~
\subfloat[Stabilized detection frame 60]{\includegraphics[width=0.33\textwidth]{Figures/n12.jpg}\label{fig:stb12}} \\
\subfloat[Jittering video frame 61]{\includegraphics[width=0.33\textwidth]{Figures/n21.jpg}\label{fig:stb21}}~
\subfloat[Stabilized detection frame 61]{\includegraphics[width=0.33\textwidth]{Figures/n22.jpg}\label{fig:stb22}} \\
\subfloat[Jittering video frame 120]{\includegraphics[width=0.33\textwidth]{Figures/n51.jpg}\label{fig:stb31}}~
\subfloat[Stabilized detection frame 120]{\includegraphics[width=0.33\textwidth]{Figures/n52.jpg}\label{fig:stb32}} \\
\subfloat[Jittering video frame 121]{\includegraphics[width=0.33\textwidth]{Figures/n61.jpg}\label{fig:stb41}}~
\subfloat[Stabilized detection frame 121]{\includegraphics[width=0.33\textwidth]{Figures/n62.jpg}\label{fig:stb42}}
\caption{The original, jittered video frames (left columns) and
corresponding thinning results after stabilization (right
columns). The thinning result consistently shows the car and the
people without significant shifting or shaking, even though the
stabilization was performed using {\em only} the flagged patches. }
\label{fig:stable2}
\end{figure}
Under different situations, the meaning of ``anomalousness'' and ``saliency'' can also be
different. For example, a moving car on a busy street during daytime
may be seen as normal, while the same car should be considered
anomalous or salient if it appears in some vacant lot when no other cars are
around. Conventional non-adaptive saliency detection algorithms often
lack the flexibility of changing the definition of saliency over time,
while Online Thinning, as an online algorithm, has the ability to
learn the environment over time and adapt to new needs.
{A third experiment compares} Online Thinning with the classical
batch and online GMM algorithm with a real-life parking lot
surveillance video data. The video is a time-lapse of a parking lot
where cars arrive and gradually fill up the entire lot. For Online
Thinning and the classical online GMM, an initial model is estimated
using the first frame, and the models are then updated frame by
frame. The anomalousness score is calculated as the negative
log-likelihood of each data point according to the estimated
model. For the classical static GMM algorithm, we estimate a GMM model
on the first twenty frames, and assign an anomalousness score to each
data point proportional to the negative log likelihood of the data
point coming from the estimated GMM model. {The batch
GMM is trained only with the patches from} the first twenty frames to simulate a setting
in which a probability model is learned in one set of environmental
conditions and does not adapt to a changing environment.
For all three algorithms, dense SIFT features from each frame
$t$ {are used} as the observation $X_t$. Each frame of the video is of size
$960\times540$, and the {grid is placed} so that one SIFT feature is
computed for each $25\times 25$ patch. {Each frame have} roughly
eight hundred SIFT feature vectors. The dimension of each
SIFT feature vector is 128.
Fig.~\ref{fig:pklot2} shows the result of Online Thinning
(Alg.~\ref{alg:OTM}) and both classical batch and online GMM
algorithms on the surveillance video at frames 21 and 232. Red-colored
patches are flagged as having high anomalousness scores.
Figures~\ref{fig:pklot211},~~\ref{fig:pklot212} and~\ref{fig:pklot213}
show the result on frame 21, where the lot is still relatively empty,
and all three algorithms flagged similar items in the scene (incoming
car, people in the
lot). Figures~\ref{fig:pklot221},~\ref{fig:pklot222}
and~\ref{fig:pklot223} show the result on frame 232, when the lot is
about half full. At frame 232, the Online Thinning algorithm has
learned that cars are common objects in the video, and has thus
adapted to assigning lower anomalousness scores to most cars.
Instead, the Online Thinning algorithm assigns higher anomalousness
scores to relatively uncommon objects like black pole, building
windows, and cars parked differently from others. The batch GMM
algorithm does not adapt to the video, and assigns most cars with high
anomalousness scores. The online GMM algorithm flags fewer patches
than the batch GMM algorithm when the parking lot is filled
up. However, online GMM still flags significantly more cars than the
Online Thinning algorithm. Note that in the video, the parking lot is
filled up gradually, and most of the cars in the parking lot at frame
232 has shown up in the scene for a long time. However, at frame 232,
the Online GMM algorithm still flags a significant amount of cars in
the parking lot, while the Online Thinning algorithm has stopped
flagging cars that have been in the scene for a long time. This
suggests that the online GMM algorithm adapts to the environment at a
slower rate than the Online Thinning algorithm.
\begin{figure}[!t]
\centering
\subfloat[Online Thinning, frame 21]{\includegraphics[width=0.33\textwidth]{Figures/MOUSSE1.jpg}\label{fig:pklot211}}~
\subfloat[Online Thinning, frame 232]{\includegraphics[width=0.33\textwidth]{Figures/MOUSSE2.jpg}\label{fig:pklot221}}\\
\subfloat[Batch GMM, frame 21]{\includegraphics[width=0.33\textwidth]{Figures/GMM1.jpg}\label{fig:pklot212}} ~
\subfloat[Batch GMM, frame 232]{\includegraphics[width=0.33\textwidth]{Figures/GMM2.jpg}\label{fig:pklot222}} \\
\subfloat[Online GMM, frame 21]{\includegraphics[width=0.33\textwidth]{Figures/CEM1.jpg}\label{fig:pklot213}} ~
\subfloat[Online GMM, frame 232]{\includegraphics[width=0.33\textwidth]{Figures/CEM2.jpg}\label{fig:pklot223}}
\caption{Result of Online Thinning and classical batch and online GMM
algorithms on the surveillance video at frames 21 and
232. Red-colored patches are flagged as salient according to the
different probability
models. \ref{fig:pklot211},~\ref{fig:pklot212},
and~\ref{fig:pklot213} show the result on frame 21, where the lot is
still relatively empty, and all three algorithms flag similar items
in the scene (incoming car, people in the
lot). \ref{fig:pklot221},~\ref{fig:pklot222}, and~\ref{fig:pklot223}
show the result on frame 232, when the lot is about half full. The
Online Thinning algorithm has learned that cars are common objects
in the video, and has thus adapted to assigning lower anomalousness
scores to most cars. The batch GMM algorithm does not adapt to the
video, and assigns most cars with high anomalousness scores. The
online GMM algorithm flags less cars with high anomalousness scores
than batch GMM algorithm, but still flags more cars than the Online
Learning algorithm. This suggests the online GMM algorithm adapts to
the environment slower than the Online Learning algorithm.}
\label{fig:pklot2}
\end{figure}
\section{Enron email experiments}
\label{sec:Enronexp}
Data thinning can also be applied to text documents to find anomalous
texts and topics. The development of latent Dirichlet allocation
(LDA)~\cite{blei2003latent} for text document topic modeling and other
methods have allowed us to analyze the topics of a collection of
documents. The Enron data is a collection of about fifty thousand
emails within the Enron corporation between the year 1998 and
2002. The dataset has been explored in the context of social network
analysis \cite{diesner2005communication} and event detection
\cite{aggarwal2012event, raginsky2012sequential, horn2011online}. In
\cite{raginsky2012sequential, horn2011online}, the authors used the
email addresses and time stamps and successfully predicted major
events in the company by finding days during which email
correspondence shows abnormal patterns. In our work, we also try to
detect significant events in the company's history by using the Enron
database. However, we approach the problem by using the count of
``topic words'' found in the emails, instead of the contact
information which does not reflect the content of the emails.
The challenge here is that the count data cannot be modeled as
Gaussian, and {pre-}processing {is needed} before applying the method. We
see each of the word-count of topic words in the email as an
independent Poisson realization of some underlying rate. By using the
Anscombe transformation \cite{anscombe1948transformation}, we can
approximate the normalized data as arising from a Gaussian mixture
model, and thus apply the Online Thinning
Algorithm~\ref{alg:OTM}.
{This experiment applies} the Online Thinning algorithm to the
Enron email dataset for event detection. To process the Enron emails,
we first generate a five-hundred-word topic list using LDA
\cite{darling2011theoretical}, where the list includes fifty
topics, and each topic has ten associated keywords. For each
email, the number of times each keyword appears {is counted and recorded}
in a fifty-dimensional vector $y_t\in\mathbb{N}^{50}$ where each entry
$[y_{t}]_i$ corresponds to how many times the keywords in topic $i$
appears in this email. Here $[\cdot]_i$ indicates the $i^{\mbox{th}}$
element of a vector. The feature vectors {are then normalized} using the
Anscombe transform \cite{anscombe} by setting
$[x_t]_i = 2\sqrt{[y_t]_i + \frac{3}{8}};$
note that $[x_t]_i$ is asymptotically normal with mean
$2\sqrt{[y_t]_i + \frac{3}{8}} + \frac{1}{4\sqrt{[y_t]_i}}$ and unit
variance. Online Thinning is then applied to the transformed data
data (the $x_t$'s), and we flag emails by thresholding the
anomalousness score assigned by the Online Thinning algorithm.
Fig.~\ref{fig:Enron} shows the number of selected emails versus time
(date). The major peaks in the plot correspond to the following time
and events:
\begin{enumerate}
\item December 13, 2000: Enron announces that president and chief
operating officer Jeffrey Skilling will take over as chief executive
in February. \cite{Dec13}
\item May 9, 2001: ``California utility says prices of gas were
inflated'' by Enron collaborator, and blackouts affect 167,00 Enron
customers. \cite{May09,May08}
\item October 24, 2001: Enron ousts its chief financial officer Andrew
S. Fastow, and the shares of Enron fell to the lowest price since
early 1995 \cite{ousts}.
\item November 28, 2001: Enron shares plunge below \$1. \cite{Nov28}
\item January 30, 2001: Stephen Cooper takes over as Enron CEO, and
Enron Metals is sold to a unit of Sempra Energy. \cite{Jan29, Jan30}
\end{enumerate}
As seen, the flagged dates cluster around the time when significant
events happen in the Enron company.
\begin{figure}[!t]
\centering
{\includegraphics[width=0.6\textwidth]{Figures/enron.pdf}}
\caption{number of selected emails versus time (date). The large peaks in the plot all correspond to major events in the history of the company. The red curve is smoothed using a Gaussian filter.}
\label{fig:Enron}
\end{figure}
\section{Conclusion}
{This paper} proposed an online data thinning method for
high-dimensional data with changing environment. At the heart of the proposed
algorithm is a union of subspaces tracking algorithm,
which allows for fast and accurate data thinning in a variety of
applications with both subsampled data and mini-batch updates.
The core idea of the proposed approach is to track a Gaussian mixture
model whose covariance matrices each are dominated by a low-rank
component. Under this model, most observations are concentrated in a
union of subspaces, a model growing in popularity in image, video, and
text analysis because of its flexibility and robustness to
over-fittings. Unlike traditional GMMs, the low-rank structure
proposed here mitigates the curse of dimensionality and facilitates
efficient tracking in dynamic environments. Furthermore, by
leveraging the recent advances in subspace tracking and subspace
clustering techniques, the proposed method is able to accurately
estimate the mixture density without adding a significant
computational burden. Another important feature of the proposed method
is the ability to track an arbitrary number of mixture components. The
adoption of a tree-like hierarchical structure for the union of
subspaces model allows the method to adaptively choose the number of
subspaces needed at each time stamp, and thus greatly {improves} the
flexibility of the method and accuracy when tracking highly dynamic
densities.
|
1,314,259,995,789 | arxiv | \section{Introduction}
\label{sec:I}
Neutron stars, which are stellar remnants of core-collapse supernova
explosions that occur at the last moment of massive stars, are
composed of matter under extreme conditions, namely, such high
density and large neutron excess as to be very difficult to realize on earth.
In fact, the density inside the star can significantly exceed the
normal nuclear density under strong gravitational field,
while the neutron excess can become extremely large under charge neutral
and beta equilibrium conditions \citep{NS}. Thus, observations of
neutron stars are expected to help us to probe the physics under such
extreme conditions, particularly given the difficulty in terrestrial
laboratories in obtaining relevant information about matter in
neutron stars. Even at the present time when information from two solar
mass neutron stars and a binary neutron star merger is available \citep{A2018},
however, the equation of state (EOS) of neutron star matter and
hence the neutron star structure are still uncertain.
In spite of the uncertainties in the EOS of neutron star matter,
a schematic view of neutron star structure can be drawn
as follows. Under the envelop composed mostly of a liquid
metal, the matter is considered to have a lattice structure via the
inter-ionic Coulomb interaction. This crystalline region is
referred to as a crust. In the deepest region of the crust,
below which the matter becomes uniform and constitutes a core, nuclei present
are so closely packed that the nuclear shape, which is normally roughly
spherical, could turn to cylinder (``spaghetti''), slab (``lasagna''), tube or
cylindrical hole (``ani-spaghetti''), and bubble or spherical hole (``Swiss
cheese'') as the density increases.
Such exotic shapes are often called nuclear pasta \citep{LRP1993,O1993}.
This nuclear pasta is embedded in a gas of dripped neutrons and thus
can be viewed as a liquid-gas mixed phase of nuclear matter. Since the
crystalline order of the phases of cylindrical nuclei, slab-like nuclei,
and tubes is low-dimensional, furthermore, these phases are liquid
crystals \citep{PP1998}. Interestingly, it is known that the appearance
of pasta structures depends strongly on the slope parameter of nuclear
symmetry energy \citep{OI2007}, of which the determination is one of
the important problems in nuclear physics \citep{L2017}.
Observational evidence for the presence of nuclear pasta would thus be
highly desired.
To extract information of neutron star interiors from observations,
asteroseismology is a very powerful technique, just like the
seismology for the Earth and the helioseismology for the Sun.
That is, since the characteristic frequencies observed from
neutron stars may well be more or less related to the interior
properties, one could obtain the interior information by somehow
observing such frequencies, identifying them as eigenfrequencies of
some global oscillation modes, and then solving an inverse problem.
Such frequencies could be obtained from gravitational waves that
would radiate from the interiors and reach us due to the strong permeability.
In fact, many possibilities of extracting the
neutron star properties via direct detection of the gravitational waves
have been proposed (e.g., \cite{AK1996,STM2001,SKH2004,SYMT2011,DGKK2013}).
Study in this direction is so promising as to make us expect to obtain
important information on the neutron star interiors in the near future.
As long as neutron star asteroseismology is concerned,
quasi-periodic oscillations (QPOs) in X-rays have been only known
electromagnetic signals of global oscillations. Up to now, three
giant flares have been observed from different soft gamma repeaters (SGRs).
In two of them, namely, SGR 1900+14 and SGR 1806-20, several QPOs have
been found in the X-ray afterglow following the respective
giant flare, where the observed QPO frequencies are in the range of tens of Hz
up to kHz \citep{I2005,SW2005,SW2006}. In SGR 1806-20, another QPO,
i.e., the 57 Hz QPO, was also found from the shorter and less energetic
recurrent 30 bursts \citep{QPO2}. Since the central object in the SGR is
considered to be a strongly magnetized neutron star, the observed QPOs
may well come from the global oscillations of the
neutron star. Given that typically, the frequencies induced by
acoustic oscillations in the star are around kHz \citep{VH1995}, one
has difficulty in identifying the QPOs of frequency lower than
$\sim 100$ Hz as the acoustic oscillations. In practice, it is
generally accepted that the mechanisms for explaining such lower QPO
frequencies are either the crustal torsional oscillations, the
magnetic oscillations, or the coupled modes (magneto-elastic oscillations).
However, calculations of the magnetic oscillation frequencies suffer
several uncertainties. The geometry and strength distribution of the
internal magnetic fields are poorly known, although the magnetic
oscillations depend strongly on such magnetic structure
\citep{GCFMS2013}. In addition, one has to take into account the
uncertain core EOS if the magnetic fields penetrate into
the core region. To avoid such uncertainties, in this study we focus on the
crustal torsional oscillations by ignoring the coupling to the magnetic
oscillations. Note that even in the absence of this coupling, the calculated
eigenfrequencies of the crustal torsional oscillations are still controlled by
several physical parameters that are relatively well-known but not yet
determined, i.e., such crustal properties as the shear modulus and the
superfluid density, as well as the star's radius $R$ and mass $M$. By
identifying the observed QPO frequencies as such eigenfrequencies,
therefore, one can manage to constrain the crustal properties \citep{SA2007,SW2009,GNJL2011,SNIO2012,PA2012,SNIO2013a,SNIO2013b,S2014,S2016,SIO2016,SIO2017a,SIO2018}.
In most of these earlier studies of the crustal torsional
oscillations, it was assumed that only the phase of spherical nuclei
oscillates quasiperiodically, while the pasta phases remain free from
oscillations. Since most of the pasta phases are liquid crystals, however,
their elastic properties could be responsible for global oscillations. In
contrast to a naive view that the shear modulus decreases continuously
in the pasta phases and eventually vanishes at the crust-core boundary,
which was adopted in \cite{S2011,PP2016}, we have recently attempted
to introduce a more realistic effect of the pasta elasticity into
the torsional oscillations \citep{SIO2017a,SIO2018}. In this attempt,
it was noted that for slab-like nuclei, the transverse shear response
vanishes for long-wavelength perturbations \citep{dGP1993,PP1998}.
That is, within the linear analysis, the phase of slab-like nuclei
behaves as a fluid. This indicates that the torsional oscillations
that could be excited within the phases of spherical and cylindrical
nuclei would be separable from those within the phases of
tubes and bubbles.
In our recent study \citep{SIO2018}, we calculated eigenfrequencies
of the torsional oscillations that occur inside the phases of
spherical and cylindrical nuclei and showed that the QPO frequencies observed
in SGR 1806-20 and SGR 1900+14, except for the 26 Hz QPO observed in SGR
1806-20, can be explained in terms of such oscillations. Additionally,
since the torsional oscillations are supposed to be confined
within a layer composed of spherical and cylindrical nuclei, we
discussed the overtone torsional oscillations, which have radial
nodes in such a manner that is dependent on the thickness of the layer. By
identifying the kHz QPO observed in SGR 1806-20 as the 1st overtone
torsional oscillation, we attempted to constrain the incompressibility
of symmetric nuclear matter for given $M$ and $R$. By combining the
resultant constraint with the constraint from empirical data for nuclear
giant monopole resonances, furthermore, not only did we manage to
constrain $M$ and $R$, but we obtained a more severe constraint on the
slope parameter $L$ of nuclear symmetry energy.
Even before our previous work \citep{SIO2018}, we suggested the
possibility that the 26 Hz QPO in SGR 1806-20 stems from torsional
oscillations that occur only in a deeper layer of the crust than the
slab phase, i.e., in a layer of composed of tubes and bubbles. As a first
step \citep{SIO2017a}, we focused on the torsional oscillations in the
bubble phase alone by simply assuming zero elasticity in the tube phase.
It was noted that the lowest fundamental frequency in the bubble
phase could explain the 26 Hz QPO because the enthalpy density is
relatively small in the bubble phase. In this work, by taking into
account the effect of the tube phase, we will give a more realistic
evaluation of the eigenfrequencies of torsional oscillations that occur
in the tube-bubble layer and thereby examine whether one could still
explain the 26 Hz QPO. Within the same framework, moreover,
we will discuss possible identification of newly found QPOs in SGR
1806-20 by a Bayesian procedure \citep{MCS18}.
In Sec.\ \ref{sec:II}, we summarize a model for the neutron star crust
that is constructed in such a way as to depend on the EOS of
nuclear matter. Section \ref{sec:III} is devoted to description of the
shear modulus that is consistent with the crust model summarized in Sec.\
\ref{sec:II}. In Sec.\ \ref{sec:IV}, we calculate the eigenfrequencies of
fundamental shear torsional oscillations in two elastic layers within the
crust and compare them with the low-lying QPO frequencies observed from
SGR 1806--20. Finally, concluding remarks and details of such comparison
are given in Sec.\ \ref{sec:V} and Appendix \ref{sec:appendix_1}, respectively.
Throughout the text, we use units in which $c=G=1$, where $c$ and $G$ denote
the speed of light and the gravitational constant, respectively.
\section{Model for neutron star crust}
\label{sec:II}
We start with construction of a neutron star crust in a spherically
symmetric configuration. This is because for neutron stars observed as SGRs
the magnetic and rotational energies are much smaller than the
gravitational binding energy \citep{K1998,H1999}. Then, the crust can be
constructed by integrating the Tolman-Oppenheimer-Volkoff (TOV) equation
together with the EOS of matter in the crust. Correspondingly, the
metric is given in spherical polar coordinates as
\begin{equation}
ds^2 = -e^{2\Phi(r)} dt^2 + e^{2\Lambda(r)} dr^2 + r^2 d \theta^2 + r^2\sin^2\theta d\phi^2,
\end{equation}
where $\Lambda(r)$ is directly connected to the mass function, $m(r)$, via
$\exp(-2\Lambda)=1-2m/r$.
It is advantageous that we dispense with the core EOS, which is
significantly uncertain. In integrating the TOV equation, therefore,
we set the values of $R$ and $M$ and then go inward from the star's
surface down to the crust-core boundary \citep{IS1997}.
To construct the crust in equilibrium, one has to prepare the EOS of
crustal matter that is in beta equilibrium and globally charge neutral.
This EOS can in turn be constructed in such a way that is dependent on
the bulk energy of zero temperature nuclear matter per baryon, which can
generally be expanded in the vicinity of the saturation point of
symmetric nuclear matter with respect to the baryon number density
($n_{\rm b}$) and the neutron excess ($\alpha$) (see \cite{L1981}):
\begin{equation}
w(n_{\rm b}, \alpha) = w_0 + \frac{K_0}{18n_0^2}(n_{\rm b} - n_0)^2 + \left[S_0 + \frac{L}{3n_0}(n_{\rm b} - n_0)\right]\alpha^2. \label{eq:w}
\end{equation}
Here $w_0$ and $K_0$ are the bulk energy and the incompressibility of
the symmetric nuclear matter at the saturation density of $n_{\rm b}=n_0$,
while $S_0$ and $L$ are the parameters that characterize
the nuclear symmetry energy, $S(n_{\rm b})$, i.e., $S_0=S(n_0)$ and
$L=3n_0(dS/dn_{\rm b})$ at $n_{\rm b}=n_0$. Among these five saturation
parameters, $n_0$, $w_0$, and $S_0$ are fairly well
constrained from empirical data for masses and charge radii of stable
nuclei. On the other hand, the constraint on the remaining two parameters,
$K_0$ and $L$, are relatively more difficult to obtain, because
these are related to the density change from $n_{\rm b}=n_0$. In this
study, therefore, we adopt the phenomenological EOSs of crustal
matter that were constructed by \cite{OI2003,OI2007} in such as way as to
depend on $K_0$ and $L$ (hereafter refereed to as OI-EOSs). These
EOSs allow us to systematically examine the dependence of the crustal
oscillations on $K_0$ and $L$.
Let us briefly summarize the OI-EOSs. The expression for the
energy of bulk nuclear matter used in the OI-EOSs was
constructed in a Pade form with respect to the density and in a
parabolic approximation with respect to the neutron excess, and fitted to
empirical data for masses and charge radii of stable nuclei within the
Thomas-Fermi approach \citep{OI2003}. Consequently, the saturation
parameters in Eq.\ (\ref{eq:w}) were given for more than 200 sets
of $K_0$ and $y\equiv -K_0S_0/(3n_0L)$. Then, within the Wigner-Seitz
approximation for five nuclear shapes, i.e., sphere, cylinder, slab,
tube, and bubble, the equilibrium nuclear shape and size in the crust
were determined as a function of $n_{\rm b}$ by optimizing the
energy density functional in the presence of a neutralizing
uniform electron gas and a gas of dripped neutrons~\citep{OI2007}.
In this study we confine ourselves to several sets of the saturation
parameters, which cover not only typical but also extreme cases as in
Table~\ref{tab:EOS}. We remark that the typical values are
empirically deduced as, e.g., $30\, \, \raisebox{-0.8ex}{$\stackrel{\textstyle <}{\sim}$ } L \, \, \raisebox{-0.8ex}{$\stackrel{\textstyle <}{\sim}$ } 80$ MeV \citep{Newton2014}
and $K_0=230\pm 40$ MeV~\citep{KM2013} or $250 \, \, \raisebox{-0.8ex}{$\stackrel{\textstyle <}{\sim}$ } K_0 \, \, \raisebox{-0.8ex}{$\stackrel{\textstyle <}{\sim}$ } 315$
MeV~\citep{SSM2014}.
Since we focus on the torsional oscillations that are trapped inside
the phases of tubes and bubbles in this study, we also show the
transition densities from the slab to the tube phase (S-CH),
from the tube to the bubble phase (CH-SH), and from the bubble to
the uniform phase (SH-U) in Table~\ref{tab:EOS}. As already
predicted by \cite{OI2007}, the pasta structure is more
difficult to appear for larger $L$. In fact, some of the
pasta structures are predicted to disappear for the cases with
$L=76.4$ and 146.1 MeV, which are denoted by the asterisk
in the column of $K_0$ in Table~\ref{tab:EOS}. We remark that
the thickness of each pasta phase strongly depends on not only $K_0$
and $L$ but also the stellar compactness ($M/R$) \citep{SIO2017b}.
We also remark that the transition densities tabulated in
Table~\ref{tab:EOS} are not obtained at constant pressure; in a
real situation, the density jumps at the transition pressures, but this
jump is tiny because the transitions are of weak first order.
\begin{table}
\centering
\begin{minipage}{100mm}
\caption{
The transition densities at the S-CH, CH-SH, and SH-U boundaries are shown for
several sets of the OI-EOSs characterized by $K_0$ and $L$. In the
cases in which the asterisk is affixed to the value of $K_0$, some
of the pasta phases are not predicted to appear. The values with $*1$
and $*2$ denote the transition densities from the cylindrical-hole to the
uniform phase and from the phase with spherical nuclei to the
uniform phase, respectively.
}
\begin{tabular}{cc|cccc}
\hline\hline
$K_0$ (MeV) & $L$ (MeV) & S-CH (fm$^{-3}$) & CH-SH (fm$^{-3}$) & SH-U (fm$^{-3}$) \\
\hline
180 & 17.5 & 0.09811 & 0.10206 & 0.10321 \\
180 & 31.0 & 0.08739 & 0.09000 & 0.09068 \\
180 & 52.2 & 0.07733 & 0.07885 & 0.07899 \\
230 & 23.7 & 0.09515 & 0.09817 & 0.09866 \\
230 & 42.6 & 0.08411 & 0.08604 & 0.08637 \\
230 & 73.4 & 0.07284 & 0.07344 & 0.07345 \\
360 & 40.9 & 0.09197 & 0.09379 & 0.09414 \\
$^*$360 & 76.4 & 0.07890 & --- & 0.07918$^{*1}$ \\
$^*$360 & 146.1 & --- & --- & 0.06680$^{*2}$ \\
\hline\hline
\end{tabular}
\label{tab:EOS}
\end{minipage}
\end{table}
In considering the torsional oscillations, furthermore, the
effective enthalpy, $\tilde{H}$, that participates in the oscillations is
another important factor, because the shear velocity $v_s$ is given by
$v_s^2=\mu/\tilde{H}$, where $\mu$ is the shear modulus to be discussed
in the next section, and because the fundamental frequency of the
torsional oscillations is proportional to $v_s$ \citep{HC1980}. In practice,
for the torsional oscillations in the tube and bubble phases, the effective
enthalpy can be expressed as
\begin{equation}
\tilde{H} = \frac{N_i + {\cal R}(A - N_i)}{A}H, \label{eq:H}
\end{equation}
where $N_i$ denotes the number of neutrons inside a single tube or
bubble, $A$ is the total nucleon number in a Wigner-Seitz cell, and $H$
denotes the local enthalpy given by $H=\varepsilon + p$ with the energy density
($\varepsilon$) and pressure ($p$). The coefficient ${\cal R}$ is a parameter
that characterizes a participant ratio, i.e., how much ratio of nucleons
outside the tube or bubble comove with it non-dissipatively via
entrainment, namely, Bragg scattering off the corresponding lattice. Note
that the non-participant nucleons behave as a superfluid. There are two
extreme cases: All the nucleons inside the Wigner-Seitz cell contribute
to the effective enthalpy for ${\cal R}=1$ (maximum enthalpy), while
no nucleons outside the tube or bubble do so for ${\cal R}=0$ (minimum
enthalpy). In general, ${\cal R}$ has an intermediate value that depends
on the band structure and pairing gap for the nucleons outside the tube or
bubble and hence changes with density. In this study, we simply
consider only the extreme cases in which ${\cal R}$ is constant at
1 or 0 in the whole region of the tube and bubble phases. We remark that the
value of ${\cal R}$ in the bubble phase is predicted to be
$\sim 0.34-0.38$ at $n_{\rm b}=0.08$ fm$^{-3}$, according to the band
calculations by \cite{Chamel2012}. Incidentally, we naively assume that
all the $N_i$ neutrons comove with the interface of the tube or bubble, just
like bubbles in boiled water. This might not be always the case with the
superfluid system considered here in which a non-dissipative hydrodynamic flow
could arise in such a way that some neutrons go across the interface
\citep{MU2016}.
\section{Shear modulus}
\label{sec:III}
Let us now proceed to the shear modulus, $\mu$, which is
associated with the distortion energy to be produced
by adding long-wavelength transverse shear deformation
of each of the five phases of inhomogeneous nuclear matter.
The distortion energy comes mainly from the change of the Coulomb
energy due to the deformation, and a particular form of
the corresponding shear modulus was adopted in our previous
studies except for the tube phase.
In the case of a bcc Coulomb lattice composed of
spherical nuclei, the effective shear modulus was
originally derived as
\begin{equation}
\mu_{\rm sp} = 0.1194\frac{n_i(Ze)^2}{a}, \label{eq:musp}
\end{equation}
where $n_i$, $Z$, and $a$ denote the ion number density,
the charge number of the nuclei, and the radius of the
Wigner-Seitz cell, i.e., $n_i=(4\pi a^3/3)^{-1}$
\citep{OI1990,SHOII1991}. Note that this $\mu_{\rm sp}$
was obtained via Monte Carlo method by averaging over all
directions of the wave vector of the distortion
with the assumption that each nucleus is a point particle.
Recently, this shear modulus has been
modified by taking into account the effect of electron
screening \citep{KP2013} and the effect of polycrystalline
nature \citep{KP2015}. In this study, however,
we adopt the traditional formula given by Eq.\ (\ref{eq:musp})
for simplicity.
The elastic properties in the rod and slab phases
have been discussed by \cite{dGP1993,PP1998}. The shear
modulus in the phase of cylindrical nuclei was
derived through the deformation energy to be produced
by adding a two-dimensional displacement perpendicular
to the elongated direction of the equilibrium
configuration of cylindrical nuclei. In practice, it can
be effectively expressed as
\begin{equation}
\mu_{\rm cy} = \frac{2}{3}E_{\rm Coul} \times 10^{2.1(w_2-0.3)}, \label{eq:mucy}
\end{equation}
where $E_{\rm Coul}$ and $w_2$ denote the Coulomb energy
per volume of a Wigner-Seitz cell and the volume
fraction of cylindrical nuclei, respectively, and
the coefficient of $2/3$ comes from the average over
all directions between the wave-vector of the distortion
and the elongated direction under the assumption that
crystallites of cylindrical nuclei randomly point.
We remark that
in the liquid drop model $E_{\rm Coul}$ is given by
\begin{equation}
E_{\rm Coul} = \frac{\pi}{2} (\rho_p R_p)^2 w_2\left[\ln\left(\frac{1}{w_2}\right)-1+w_2\right],
\label{eq:E_coul}
\end{equation}
where $\rho_p$ and $R_p$ are the proton charge density and the proton radius
of a cylindrical liquid drop \citep{RPW1983}.
By following a similar line of argument,
it was shown that the deformation energy in the phase of
slab-like nuclei becomes of higher order with respect
to the displacement. That is, this phase behaves as a
fluid within the linear response. This is the reason
why one can consider the torsional oscillations inside the
phases of spherical and cylindrical nuclei separately
from those inside the phases of tubes and bubbles.
The shear modulus in the tube (bubble) phase, i.e., $\mu_{\rm ch}$
($\mu_{\rm sh}$), can be derived in a similar fashion to that
in the phase of cylindrical (spherical) nuclei, because the liquid
crystalline structure of tubes (bubbles) is the same as that in the phase
of cylindrical (spherical) nuclei. In this study, therefore, we adopt
Eq.\ (\ref{eq:mucy}) for the tube phase and Eq.\ (\ref{eq:musp}) for the bubble
phase by properly replacing the relevant quantities in these
formulae: In the tube phase, $w_2$ in Eq.\ (\ref{eq:mucy})
(including $E_{\rm Coul}$) is replaced
by the volume fraction of a gas of dripped neutrons, while in the
bubble phase $n_i$ and $Z$ are replaced by the number density of
bubbles and the effective charge number $Z_{\rm bubble}$ of a bubble,
respectively \citep{SIO2017a}. In practice, $Z_{\rm bubble}$ is given
by $Z_{\rm bubble}=n_QV_{\rm bubble}$, with the volume of the bubble,
$V_{\rm bubble}$, and the effective charge number density of the bubble,
$n_Q$, defined by the difference of the charge number density
inside the bubble from that outside the bubble, i.e.,
$n_Q=-n_{\rm e}-(n_{\rm p}-n_{\rm e})=-n_{\rm p}$ with the proton number density
outside the bubble ($n_{\rm p}$) and the number density of a uniform electron
gas ($n_{\rm e}$).
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.5]{mu-K180} &
\includegraphics[scale=0.5]{mu}
\end{tabular}
\end{center}
\caption
(Color online)
Left: Profile of the shear modulus in the tube phase (thin lines) and bubble phase
(thick lines), calculated for the neutron star models with
$M=1.4M_\odot$ and $R=12$ km. Here, $K_0$ is fixed at 180 MeV,
while $L$ takes the value as labeled in the unit of MeV.
Right: For the neutron star model with $M=1.4M_\odot$ and
$R=12$ km constructed with $K_0=180$ MeV and $L=55.2$ MeV,
the profile of the shear modulus in
the phase of spherical nuclei (Sp) and in the phase of cylindrical nuclei (Cy)
is shown as well as that in the tube (CH) phase and the bubble (SH) phase.
\label{fig:mu}
\end{figure*}
In Fig.~\ref{fig:mu}, we illustrate the profile of the shear modulus inside the tube
and bubble phases for neutron star models constructed from the first
three sets of the OI-EOSs listed in Table~\ref{tab:EOS}. From this figure,
one can observe that the shear modulus becomes discontinuous at the transition
between the tube and bubble phases, which is similar to the case of the
transition between the phases of spherical and cylindrical nuclei
\citep{SIO2018}. In addition, it is to be noted that the shear modulus
in the tube phase can decrease as the density increases and that this
tendency becomes stronger for larger $L$. This tendency may well
come from the decrease of the volume fraction of a gas of dripped
neutrons with density (e.g., \cite{WI2003}).
\section{Torsional oscillation frequencies and comparison with QPOs}
\label{sec:IV}
We now turn to evaluations of the eigenfrequencies of fundamental
torsional oscillations in the sphere-cylinder and tube-bubble layers of the
crust of a neutron star. To this end, we start with the perturbation
equation in a spherical coordinate system, which is given by linearizing
the relativistic equation of motion that determines the torsional
oscillations \citep{ST1983,Sotani2007} as
\begin{equation}
{\cal Y}'' + \left[\left(\frac{4}{r} + \Phi' - \Lambda'\right)+\frac{\mu'}{\mu}\right]{\cal Y}'
+ \left[\frac{\tilde{H}}{\mu}\omega^2e^{-2\Phi} - \frac{(\ell+2)(\ell-1)}{r^2}\right]e^{2\Lambda}{\cal Y}=0,
\label{eq:perturbation}
\end{equation}
where ${\cal Y}$ denotes the Lagrangian displacement in the $\phi$ direction,
while $\tilde{H}$ is the effective enthalpy given by Eq.\ (\ref{eq:H}).
With the appropriate boundary conditions, the problem to solve becomes an
eigenvalue problem, where $\omega$ is the eigenvalue. Then, the
eigenfrequency of the torsional oscillations $f$ is given by $f=\omega/(2\pi)$.
As for the boundary conditions relevant to the torsional
oscillations that are excited inside the tube and bubble phases,
there are two boundaries, namely, the boundary between the bubble phase
and uniform matter, which corresponds to the inner boundary, and the boundary
between the slab and tube phases, which corresponds to the outer
boundary. In practice, one has to impose the zero-traction conditions
at the inner and outer boundaries, i.e., ${\cal Y}'=0$. In addition, one has
to impose the junction condition at the boundary between the tube and bubble
phases, where the traction should be continuous, i.e.,
\begin{equation}
\mu_{\rm ch}{\cal Y}' = \mu_{\rm sh}{\cal Y}'.
\end{equation}
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.5]{t2B1-M14R12a} &
\includegraphics[scale=0.5]{t2B0-M14R12a}
\end{tabular}
\end{center}
\caption
(Color online)
Fundamental frequencies of the $\ell=2$ torsional oscillations in the tube
and bubble phases, as obtained for the neutron star models with
$M=1.4M_\odot$ and $R=12$ km as well as with various sets of $L$
and $K_0$. Here, the left and right panels correspond to the results
for the maximum and minimum enthalpies that participate in the
oscillations, i.e., ${\cal R}=1$ and 0, respectively (see the text for
details). The solid line denotes the fitting given by Eq.\ (\ref{eq:fitting}).
\label{fig:t2L2-M14R12}
\end{figure*}
In Fig.~\ref{fig:t2L2-M14R12}, we plot the $\ell=2$ fundamental
frequencies of torsional oscillations in the tube and bubble phases
that are calculated for the neutron star models with $M=1.4M_\odot$ and
$R=12$ km, with various EOS parameter sets shown in
Table~\ref{tab:EOS}, and with the maximum and minimum enthalpies
(${\cal R}=1$ and 0). From this figure, one can observe that the frequency
depends only weakly on $K_0$, but shows a significant sensitivity
to $L$. In fact, we find that the $\ell=2$ fundamental frequencies can
be well fitted as a function of $L$ via
\begin{equation}
{}_0t_2 = c_2^{(0)} + c_2^{(1)}\sqrt{L} + c_2^{(2)}L, \label{eq:fitting}
\end{equation}
where $c_2^{(0)}$, $c_2^{(1)}$, and $c_2^{(2)}$ are the adjustable
parameters that depend on $M$ and $R$ as well as the value of
${\cal R}$. The obtained fitting [Eq.\ (\ref{eq:fitting})] is also
shown in Fig.~\ref{fig:t2L2-M14R12}. We remark that this fitting has
a different functional form from that obtained for the fundamental
frequencies of crustal torsional oscillations in the phases of
spherical and cylindrical nuclei \citep{SIO2018}. We also remark that the
fundamental frequency in the tube and bubble phases can be smaller than
that in the phases of spherical and cylindrical nuclei and that
the fundamental frequencies with general values of $\ell$ can
also be well fitted in the same functional form:
\begin{equation}
{}_0t_\ell = c_\ell^{(0)} + c_\ell^{(1)}\sqrt{L} + c_\ell^{(2)}L,
\label{eq:fitting1}
\end{equation}
with a different set of the adjustable parameters $c_\ell^{(0)}$,
$c_\ell^{(1)}$, and $c_\ell^{(2)}$. Hereafter we will thus attempt to
identify the observed QPO frequencies by using the fitting formula
(\ref{eq:fitting1}) for the tube-bubble layer, in addition to
the formula given in \cite{SIO2018} for the sphere-cylinder layer.
Note that the obtained fundamental frequencies in the tube-bubble
layer are generally lower than those obtained in our earlier
analysis by assuming that only the bubble phase is oscillating
\citep{SIO2017a}. This tendency is more significant for larger values
of $L$. This is partly because for larger $L$, the bubble phase is
less likely to appear, as shown in Table~\ref{tab:EOS}, and partly
because the shear modulus and hence the shear velocity is relatively
small in the tube phase, as shown in Fig.\ \ref{fig:mu}. As we shall
see later, therefore, the 26 Hz QPO observed from SGR 1806-20 is
identified as the $\ell=4$ fundamental torsional oscillation in the
tube-bubble layer, in contrast to our earlier analysis \citep{SIO2017a}
in which it was identified as the $\ell=2$ fundamental torsional
oscillation.
Up to now, we have already done many trials to identify the QPOs
observed in SGR 1806-20 and SGR 1900+14 as the crustal torsional
oscillations. As long as we adopt the QPO frequencies derived in the
conventional non-Bayesian analyses of RXTE data for the X-ray afterglows
of the giant flares and the recurrent X-ray outbursts, such identification
has worked out relatively well
\citep{SNIO2012,SNIO2013a,SNIO2013b,SIO2016,SIO2017a,SIO2018}.
In fact, the observed QPOs, except for the 26 Hz QPO in SGR 1806-20,
can be identified as the torsional oscillations inside the phases of
spherical and cylindrical nuclei in such a way that the QPOs of
frequencies lower than $\sim 200$ Hz correspond to the fundamental
oscillations with various values of $\ell$, while the higher QPOs observed
in SGR 1806-20 correspond to the overtones \citep{SIO2018}.
In this case, since it is still uncertain how much fraction of dripped
neutrons in the phase of cylindrical nuclei would be locked to the
motion of protons in the nuclei, we introduced a parameter $N_s/N_d$,
where $N_d$ and $N_s$ respectively denote the number of dripped neutrons
outside the cylindrical nuclei and the number of a superfluid part
of the dripped neutrons that behave independently of the oscillations,
and examined the extreme cases with $N_s/N_d=0$ and 1. We remark that
all (none) of the dripped neutrons outside the cylindrical nuclei
participate in the oscillations for $N_s/N_d=0$ $(1)$. We also
remark that for the corresponding value of $N_s/N_d$ in the phase of
spherical nuclei, we adopt the results by \cite{Chamel2012}, which are
based on the band theory.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{1806-L-M14R12Ns10a}
\end{center}
\caption
(Color online)
The $\ell=2$, 3, and 4 fundamental frequencies (painted regions)
of the torsional oscillations in the layer of the tube and bubble
phases, calculated as a function of $L$ for the neutron star
models with $M=1.4M_\odot$ and $R=12$ km. The lower and upper
boundaries of the painted regions correspond to the results obtained
for the maximum enthalpy (${\cal R}=1$) and the minimum enthalpy
(${\cal R}=0$), respectively. For reference, the low-lying QPO
frequencies derived by the conventional non-Bayesian analysis for
SGR 1806-20 are shown by horizontal lines. The QPO frequencies
except 26 Hz can be interpreted as manifestations of the
$\ell=2$, 3, 6, and 10 fundamental torsional oscillations that are
excited in the layer composed of spherical and cylindrical nuclei
\citep{SIO2018}, as illustrated by the solid lines that
denote the corresponding eigenfrequencies obtained by assuming that
the dripped neutron outside the cylindrical nuclei do not
participate in the oscillations (minimum enthalpy, i.e.,
$N_s/N_d=1$). The vertical thick line, i.e., $L=73.4$ MeV, denotes
the optimal value of $L$ for explaining the observed QPOs except
the 26 Hz QPO in terms of the torsional oscillations in the
sphere-cylinder layer with minimum enthalpy, while the vertical thin line,
i.e., $L=70.4$ MeV, denotes the corresponding value of $L$
in the case of maximum enthalpy, i.e., $N_s/N_d=0$ \citep{SIO2018}.
\label{fig:M14R12}
\end{figure}
Let us now illustrate how the newly examined torsional
oscillations in the tube-bubble layer could be accommodated
to the QPO observations of frequencies lower than 100 Hz,
including 26 Hz, for typical $M$-$R$ sets of the stellar models.
For $M=1.4M_\odot$ and $R=12$ km, such illustration can be seen
from Fig.~\ref{fig:M14R12}, in which the 18, 29, 57, and
92.5 Hz QPOs in SGR 1806-20 are as usual identified as
the $\ell=2$, 3, 6, and 10 fundamental frequencies in the
sphere-cylinder layer, whereas the 26 Hz QPO, which is difficult
to explain in terms of the oscillation in the sphere-cylinder layer,
can reasonably be identified as the $\ell=4$ fundamental
frequency in the tube-bubble layer. We remark that the
optimal value of $L$ for explaining the observed
low-lying QPOs ranges between 70.4 and 73.4 MeV
for neutron stars with $M=1.4M_\odot$ and $R=12$ km.
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.5]{1806-L-M13R13Ns10a} &
\includegraphics[scale=0.5]{1806-L-M18R12Ns10a}
\end{tabular}
\end{center}
\caption
(Color online)
Same as Fig.~\ref{fig:M14R12}, but for the neutron star models
with $M=1.3M_\odot$ and $R=13$ km in the left panel and
with $M=1.8M_\odot$ and $R=12$ km in the right panel.
The optimal values of $L$ denoted by the vertical
thick and thin lines are $L=70.8$ and 67.5 MeV,
respectively, in the left panel and $L=63.5$ and 59.6 MeV,
respectively, in the right panel.
\label{fig:M13M18}
\end{figure*}
We then examine whether or not the above-mentioned identification,
which strictly holds for $(M, R)=(1.4M_\odot,12 {\rm km})$, still works
out for other sets of $(M, R)$. For neutron star models with
$(M, R)=(1.3M_\odot,13 {\rm km})$ and $(1.8M_\odot,12{\rm km})$,
we again calculate the eigenfrequencies of the double-layer torsional
oscillations, as shown in Fig.~\ref{fig:M13M18}. We find that
the 18, 29, 57, and 92.5 Hz QPOs in SGR 1806-20 can be still consistent
with the $\ell=2,3,6$, and 10 fundamental frequencies in the sphere-cylinder
layer for such a range of the optimal $L$ as 67.5--70.8 MeV and
59.6--63.5 MeV, respectively. This shift of the optimal $L$ could
open up an opportunity of selecting $M$ and/or $R$ because
the $L$ dependence of the fundamental frequencies
in the sphere-cylinder layer is different from that
in the tube-bubble layer.
In fact, one can observe the tendency that the more
massive neutron star, the more difficult to explain
the 26 Hz QPO in terms of the $\ell=4$ fundamental
oscillation in the tube-bubble layer,
as long as we adopt the optimal value of $L$ that enables us to identify
the 18, 29, 57, and 92.5 Hz QPOs as the oscillations in the
sphere-cylinder layer. Note that the
fundamental frequencies scale as $R^{-1}$ both in the tube-bubble
and sphere-cylinder layers, implying that $R$ is not constrained
in the present approach. We can thus conclude that
light neutron star models are favored over
heavy ones in our identification. Incidentally, the
$(M, R)=(1.3M_\odot,13 {\rm km})$ case is consistent with
the neutron star models considered to be relevant
as a result of the comparison of the constraint on $K_0$,
which is obtained by assuming that the lowest
overtone frequency in the sphere-cylinder layer is
equal to the QPO frequency of 626.5 Hz observed from
SGR 1806-20, with the terrestrial constraint on $K_0$
\citep{SIO2018}. Furthermore, the
$(M, R)=(1.3M_\odot,13 {\rm km})$ case is consistent with
the mass and radius formulas for low-mass neutron
stars \citep{SIOO2014}, given the optimal
value of $L\sim 70$ MeV, and also with the constraint
on the mass and radius of each of the merging binary
neutron stars \citep{A2018}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{1806-L-M13R13Ns10c}
\end{center}
\caption
(Color online) Relations between the newly found
QPOs of 51.4, 97.3, and 157 Hz in SGR 1806-20
\citep{MCS18}, which are shown by horizontal solid lines,
and a selected set of the crustal torsional oscillations
for the neutron star models with $M=1.3M_\odot$ and $R=13$
km. The 51.4 and 97.3 Hz QPOs are identifiable as the
$\ell=8$ and 15 fundamental torsional oscillations in the
tube-bubble layer, while the 157 Hz QPO is identifiable as
the $\ell=17$ fundamental torsional oscillations in the
sphere-cylinder layer. The dashed and dotted lines
denote the originally discovered QPOs, which except for the 26 Hz QPO
have already been identified by us as manifestations of the fundamental
torsional oscillations in the sphere-cylinder layer, while the 26 Hz QPO
is identified as the $\ell=4$ oscillation in the tube-bubble layer as mentioned in text.
\label{fig:M13R13c}
\end{figure}
Thanks to the smaller shear modulus in the tube phase,
which leads to the smaller fundamental frequencies in the
tube-bubble layer than those in the sphere-cylinder
layer, we have a chance to explain not only the originally
discovered QPOs but also the QPOs newly found by
a Bayesian procedure, e.g., the 51.4, 97.3, and 157 Hz
QPOs in SGR 1806-20 \citep{MCS18}\footnote{In appendix
\ref{sec:appendix_1}, we tabulate a possible
correspondence between the crustal torsional oscillations and
all the 26 QPOs shown in Table 1 in \cite{MCS18}.}.
In practice, we illustrate the identification of
these three QPOs for the neutron star models with
$M=1.3M_\odot$ and $R=13$ km in Fig.~\ref{fig:M13R13c}.
As already shown in \cite{SIO2018}, the frequencies of 18, 29,
57, 92.5, and 150 Hz can be identified as the
$\ell=2$, 3, 6, 10, and 16 fundamental frequencies in
the sphere-cylinder layer. In a similar way, we find
that the newly found QPO of 157 Hz can also be
identified as the $\ell=17$ fundamental frequency, while
the newly found QPOs of 51.4 and 97.3 Hz
can be identified as the fundamental oscillations in
the tube-bubble layer, as is the case with
the 26 Hz QPO, in such a way that 26, 51.4, and 97.3 Hz
correspond to the $\ell=4$, 8, and 15 fundamental oscillations
in the tube-bubble layer.
\section{Conclusion}
\label{sec:V}
We have calculated the eigenfrequencies of the torsional oscillations in
the tube-bubble layer, in contrast to our previous work in which we
calculated those only in the bubble layer, and successfully identified
the newly found QPOS as the fundamental oscillations either in
the tube-bubble or sphere-cylinder layer. In the course of the
calculations, we find that the shear modulus, which characterizes the
torsional oscillations, decreases in the tube phase as the slope
parameter $L$ increases. As a result, the fundamental frequencies in the
tube-bubble layer can become smaller than those in the
sphere-cylinder layer. We also find that the fundamental frequencies
in the tube-bubble layer can be parameterized as a function of $L$,
and that the dependence on $L$ is different from that obtained for
the fundamental frequencies in the sphere-cylinder layer.
Remarkably, such a different dependence on $L$ helps us to explain
not only the QPO frequencies originally discovered in SGR 1806-20 but
also those newly found in the same object by a Bayesian procedure
in terms of the eigenfrequencies of the fundamental torsional oscillations
either in the tube-bubble or sphere-cylinder layer of a relatively low
mass neutron star constructed from the EOS of $L\sim 70$ MeV.
We also remark that such a neutron star model and the suitable value
of $L$ are consistent with the mass and radius formulas of low-mass
neutron stars and the constraint from the gravitational waves from the
neutron star binary merger, GW170817.
As a possible extension of this study, it would be of interest to
analyze the QPO widths, which could give us information of the internal
magnetic structure via possible coupling of the crustal torsional
oscillations with the Alfv\'{e}n continuum in the core \citep{MCS18}.
Generally, magnetars are considered to have a toroidal field that is by
an order of magnitude higher than the poloidal field. The question of
whether or not this picture is relevant might be possibly answered.
This work was supported in part by Grants-in-Aid for Scientific Research
through Grant Nos.\ 17K05458, 18H01211, and 18H05406
provided by Japan Society for the Promotion of Science (JSPS).
|
1,314,259,995,790 | arxiv | \section{Introduction}
In the last decade the fabrication of self-assembled quantum dots (QDs) has been intensively studied. The interest has been, and still is, stimulated by applications of QDs in optoelectronic devices. From previous studies it is well known that the optical and electronic properties of QDs are strongly affected by their size, shape, and material composition. Despite years of intense studies, control over these properties remains difficult. One major problem is the change in QD morphology during the growth of the capping layer. Traditionally control over the QD height, one aspect of the change in morphology, can be achieved with monolayer precision by the double capping method \cite{Paranthoen2001} or the so-called indium flush method \cite{Wasilewski1999}, a variation on the former technique. In the latter technique the growth of the capping layer is interrupted, at which point the temperature is raised to remove any surface resident indium. This effectively locks the height of the QD and prevents any further In segregation \cite{Keizer2010}. Another approach in shape control of QDs is the use of surfactants. Recently, antimony has received a great deal of attention in its role during the capping process due to its surfactant properties. It has been shown that Sb reduces the surface diffusion of other atoms but without getting incorporated itself \cite{Harmand2004}, allowing the achievement of fully pyramidal shaped QDs \cite{Ulloa2007a}. Yet another approach to gain control over the erosion of QDs during overgrowth and thus over the shape of the QD is the removal of the driving force: lattice strain. This can be achieved in lattice matched QDs grown by droplet epitaxy. First reported by Koguchi \textit{et al.} \cite{Koguchi1991}, this technique involves low temperature growth of unstrained group III-element droplets that are subsequently crystallized into QDs by incorporation of group V-elements. It has been shown that this technique can be used to grow nearly pure nanostructures \cite{Keizer2010a} with a typical size distribution of 10--20\% \cite{Mano2005}.
In this paper the techniques of indium flush, antimony capping, and droplet epitaxy are studied by means of cross-sectional scanning tunneling microscopy (X-STM). We first investigate the degree of control that can be achieved over the height of InGaAs/GaAs QDs and the wetting layer (WL) by means of an indium flush. We then go on by showing that antimony capping can be employed to prevent QD erosion during the capping process of InAs/InP QDs. Finally, the intermixing in, and the shape of, GaAs/AlGaAs QDs grown by droplet epitaxy is examined in detail.
\section{Experimental setup}
All X-STM measurements were performed at room temperature under UHV ($p<6\times10^{-11}$\,mbar) conditions with an Omicron STM-1, TS2 Scanner. The STM was operated in constant current mode on {\it in situ} cleaved (110)-surfaces. Electrochemically etched tungsten tips were used. The QD layers were grown by molecular beam epitaxy (MBE). The details of the growth procedure for the different material systems will be described separately in their corresponding sections.
\section{Indium flush}
The material system used to investigate the indium flush technique consists of InGaAs QD layers grown by MBE on an \textit{n}-type GaAs (001) orientated substrate. An undoped GaAs buffer layer of 420\,nm was grown at 690\,$^{\circ}$C, followed by a growth interruption of approximately 2\,min that allowed the temperature to be lowered to 600\,$^{\circ}$C, the nominal growth temperature of the QD layers. Following this, three sequences consisting of four QD layers of 1.98\,nm (7\,ML) In$_{0.5}$Ga$_{0.5}$As were deposited. During the whole growth process the As flux was kept constant at a pressure of $1.26\times10^{-5}$\,mbar. Three out of the four QD layers were grown with the indium flush method which consists of the following procedure. First the QD layers are partially capped with a GaAs layer of which the thickness was varied. Next, the temperature is raised to 650\,$^{\circ}$C for 30\,s and lowered again to the nominal growth temperature after which a second GaAs capping layer is deposited. In total, the annealing step takes place over a time window of $\approx180$\,s. The total structure was capped with 200\,nm GaAs.
\begin{figure}[b]
\includegraphics[width=8.6cm]{figure_1.pdf}
\caption{\label{figure_1}X-STM images of the InGaAs WL as a function of the capping layer thickness. (a) 2\,nm (b) 3\,nm (c) 6\,nm first capping layer thickness and (d) conventionally grown capping layer.}
\end{figure}
\begin{figure}
\includegraphics[width=8.6cm]{figure_2.pdf}
\caption{\label{figure_2}In segregation as a function of first capping layer thickness and bilayer position from the start of the WL.}
\end{figure}
We begin by analysing the WL thickness and composition. In figure~\ref{figure_1}, four typical X-STM images of WLs grown with different capping layer thicknesses are depicted. Even without any statistical analysis it is evident that the height of the WL can be controlled by varying the height of the capping layer. In addition, the In segregation appears to terminate abruptly in case of WLs that underwent an indium flush. This is a clear indication that most of the surface resident In is removed during the flush step, preventing further segregation. In order to make our analysis more quantitative we counted and marked the bilayer position from the start of the WL for approximately 3000 In atoms. An $\approx400$\,nm cross-sectional region of each WL present in the sample was analysed in this manner. In figure~\ref{figure_2}, the result of our statistical analysis is shown. The conventionally grown WL exhibits the expected exponential decay of the In concentration and In segregation length ($\approx25$\,nm) \cite{Offermans2005b}. In contrast, the WLs grown with the indium flush procedure show a stronger decay and shorter segregation length. This implies that In segregates out of the WLs and leaves the surface during the indium flush step. This additional loss of already buried In is strongest in case of the thinnest capping layer. The total amount of In that remains after flush-off is thus strongly dependent on the capping layer thickness due to desorption and additional segregation. The thickness of the final WL is found to be 6, 8, 10, 12 bilayers for 2, 3, 4, 6\,nm thick first capping layers, respectively. Note, that the statistical analysis presented in figure~\ref{figure_2} reveals that the WL extends further than is expected from figure~\ref{figure_1}. It is reported that the critical WL thickness for In$_{0.5}$Ga$_{0.5}$As QD formation is $\approx5$\,ML (1.4\,nm) \cite{Snyder1992}. We assume that all the In that is deposited after reaching this critical thickness goes into the formation of QDs. If we add the thickness of the critical layer ($d_{\mathrm{crit}}$) to the thickness of the first capping layer ($d_{\mathrm{cap}}$) and compare the resulting sum with the experimentally found thickness of the final WL we find good agreement. This is depicted quantitatively by the dotted red line and open red boxes in figure~\ref{figure_4}, that show $d_{\mathrm{crit}}+d_{\mathrm{cap}}$ and the experimentally determined average WL thickness, respectively. This result shows that In segregation beyond the position of the flush is completely suppressed; In is absent in the final GaAs capping layer.
\begin{figure}[t]
\includegraphics[width=8.6cm]{figure_3.pdf}
\caption{\label{figure_3}X-STM image of one conventionally grown QD and three QDs grown with an indium flush step incorporated in the growth process. The thickness of the first capping layer was varied.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=8.6cm]{figure_4.pdf}
\caption{\label{figure_4}QD height (black points) as a function of the thickness of the first capping layer. The black line is a linear fit. The dotted red line represents the sum of the critical layer thickness (5\,ML) and the first capping layer thickness (dashed blue line). The experimentally determined average thickness of the final WL is given by the open red boxes.}
\end{figure}
In order to determine the influence of the indium flush step on the structural properties of the QDs we determined the width and height of a total of 48 cleaved QDs. The width of the QDs ranged up to 100\,nm. The height of the conventionally grown QDs was found to vary between 7 and 10\,nm. The QD layers were found to be weakly coupled, as one would expect with the GaAs spacer layer being 30\,nm thick and slightly strained InGaAs QDs \cite{Bruls2003}, resulting in occasional stacking of the QDs. Figure~\ref{figure_3} shows one of the sequences consisting of four QD layers where the QDs are stacked. The thickness of the first capping layer was varied in the first three QD layers from 2, 3 to 6\,nm. The last layer is a conventionally grown QD layer, i.e. without the application of an indium flush step. As can be seen, the application of an indium flush step results in lowering of the QD height as compared to the conventionally grown QDs. The shape of the conventionally grown QD is lens like as expected for typical InGaAs QDs \cite{Wang2004}. The heights of all the observed QDs as a function of the first capping layer thickness are plotted in figure~\ref{figure_4}. Since, the lateral width of all the observed QDs was found to be of the order of 60\,nm, we can assume that none of the QDs is cleaved through their edge and that figure~\ref{figure_4} represents the spread in the height distribution of the QDs due to the growth process. We found a linear relation between the QD height and the first capping layer thickness up to $\approx 7$\,nm, indicated by the black line. Since increasing the first capping layer beyond a height of 7\,nm would make the growth procedure resemble conventional growth, we expect the QD height to saturate at this value. This is indicated in figure~\ref{figure_4} by the black horizontal line which represents the average height of the conventionally grown QDs. Note, that the average QD height in the absence of the first capping layer intersects at an offset. Moreover, the QDs are found to be higher than the final WL, see the dashed red line, open red boxes, and inset of figure~\ref{figure_4}. From this we conclude that the performed indium flush is incomplete.
\section{Antimony capping}
In the previous section we have shown that the indium flush technique can be used to lower the height of InGaAs QDs. We continue with an investigation of antimony capping, a technique that can be employed to prevent QD erosion during capping. Four InAs QD layers separated by 30 nm of InP were grown on an \textit{n}-type (311)B oriented InP substrate by solid source MBE. The growth temperature was set at 450$^\circ$C. The QDs were formed by the deposition of 2.1\,ML (001) equivalent monolayers. After QD formation, a 30\,s growth interrupt (GI) under As pressure was performed for all layers. Previously, it has been shown that As/P exchange is limited under such GI conditions \cite{ulloa2007c}. The first QD layer was overgrown with an InP capping layer. This first QD layer will be considered as the reference layer. For the second QD layer a growth interrupt under a Sb beam equivalent pressure of 2.7\,$\times\,10^{-7}$\,Torr (GISb) was performed during 30\,s before the growth of the InP capping layer. For the third and fourth layers, respectively a 1 nm and 2 nm GaAs$_{0.51}$Sb$_{0.49}$ (lattice matched to InP) thick layer was deposited after a 5\,s GISb.
\begin{figure}
\includegraphics[width=8.6cm]{figure_5.pdf}
\caption{\label{figure_5}Two 60\,nm$\,\times$\,15\,nm X-STM images. (a) InAs QD capped with InP after a 30\,s GI. (b) InAs QD capped with InP after a 30\,s GI $+$ 30\,s GISb. The bright spots correspond to Sb atoms.}
\end{figure}
In figure~\ref{figure_5}a, an X-STM image of a typical QD in the reference layer is shown. These QDs are found to have a flat top facet. The homogeneity of the contrast within the QD indicates that it consist of almost pure InAs. No digging in of the WL in the underlying material as in \cite{ulloa2007c} was observed. The intermixing at the corners is minimal, like in the case of InAs QDs in AlAs \cite{Offermans2005}. The average height and width estimated from 20 individually observed QDs in the reference layer are found to be 2.0 nm and 25\,nm, respectively. Before capping, the QDs have an asymmetric pyramidal shape, bounded by low-index facets \{001\}, \{111\}B, and \{110\} \cite{Lacombe1999}. Height histograms of the uncapped QDs deduced from AFM analysis, and height histograms of the capped QDs in the reference layer as observed by X-STM are shown in figure~\ref{figure_6}a-b. For the uncapped QDs, a Gaussian distribution centered around 3.3 nm is found, whereas after InP capping the height distribution is truncated at 2.4 nm. As demonstrated previously \cite{ulloa2007c}, the truncated distribution and the flat top facet of InAs/InP QDs are to a large extend the consequence of QD decomposition. This decomposition is driven by the strain mismatch between the InP capping layer and the InAs QDs.
\begin{figure}
\includegraphics[width=8.6cm]{figure_6.pdf}
\caption{\label{figure_6}QD height distribution of a) uncapped InAs QDs, (b) InP capped InAs QDs, (c) InP capped InAs QDs after 30\,s GISb, and d) GaAsSb capped InAs QDs after 5\,s GISb.}
\end{figure}
Figure~\ref{figure_5}b, shows InAs QDs for which a 30\,s GISb has been performed before the InP capping layer was grown. The bright spots correspond to Sb atoms remaining in the InP capping layer and in the InAs QDs after the GISb and the succeeding growth of the capping layer. Given the total amount of Sb supplied to the surface and the observed amount of Sb after capping, we conclude that a large part is desorbed during overgrowth. Segregation of the small fraction of Sb that gets incorporated in the InP capping layer is clearly shown. Within the QDs the back diffusion of Sb is negligible and a preferential incorporation of Sb is observed at the outermost layers of the QDs. Again, the InAs QD corners appear well defined with minimal intermixing and formation of an InAsP alloy, just as is the case with the QDs in the reference layer. The presence of Sb on the surface induces changes on the QD shape; the mean height is now 3.5\,nm (see figure~\ref{figure_6}c) and the mean diameter 21\,nm, corresponding to the dimensions of the uncapped QDs. We can explain the observed shape preservation by the well documented surfactant effect of Sb atoms \cite{Aivaliotis2007,Harmand2004}. An Sb surfactant can limit the in-plane diffusion of atoms on the surface. Accordingly, the InAs diffusion from the QD apex to the periphery should be reduced due to the presence of Sb atoms on the surface. This freezing of the mass transport on the growth front results in the preservation of the shape of the uncapped QDs.
\begin{figure}
\includegraphics[width=8.6cm]{figure_7.pdf}
\caption{\label{figure_7}Two 100\,nm$\,\times$\,20\,nm X-STM images of InAs QDs and the WL capped with (a) 1\,nm and (b) 2\,nm GaAsSb after a 30\,s GI $+$ 5\,s GISb. A single step edge is visible at the left side in both images.}
\end{figure}
X-STM images of the third and fourth QD layers are shown in figure~\ref{figure_7}a-b. These layers were, after 30\,s GI $+$ 5\,s GISb, capped with a thin layer of GaAsSb (lattice matched to InP). As was the case with the 30\,s GISb, the QDs in these layers are taller than those in the reference layer; for both layers, an average height of 3.2 nm (see figure~\ref{figure_6})d and a base diameter of 21 nm are deduced, corresponding to the dimensions of the uncapped QDs. Again, the intermixing in the QDs is negligible. Similar shape conservation has been reported when InGaAs or GaAsSb strained capping layers are grown on InAs/GaAs QDs \cite{ulloa2007c}. In that case a phase separation is observed in the ternary capping layer on top of the QDs. In our case, the GaAsSb layer is lattice matched to the InP substrate and the observed conformal growth of GaAsSb on InAs/InP QD might be related as previously to a low group III atoms migration when Sb atoms are present on surface.
\section{Droplet Epitaxy}
Having shown that the indium flush technique and the surfactant properties of Sb allow control over the shape and height of SK-grown QDs, we now turn our attention to QDs in a lattice matched materials system. More specifically, a GaAs/AlGaAs QD layer grown on an \textit{n}-type (001) oriented GaAs substrate by droplet epitaxy. The sample was grown in the following manner. First an AlGaAs buffer layer is grown at 580\,$^{\circ}$C. Next, the sample is cooled down to 200\,$^{\circ}$C, the As flux switched off, and the As evacuated from the growth chamber. The result is an As-stabilized \textit{c}(4$\times$4) surface. Subsequently, 3.75\,ML Ga, of which the first 1.75\,ML changes the excess As into a two-dimensional GaAs layer \cite{Sanguinetti2003}, is deposited at a rate of 0.5\,ML/s. The remainder of 2\,ML will form liquid Ga droplets on the surface. Next, these droplets are crystallized into a GaAs QDs by supply of an As$_4$ flux ($2\,\times\,10^{-4}$\,Torr beam equivalent pressure). Still under As$_4$ flux, the sample is then annealed at 350\,$^{\circ}$C for 10 minutes. Subsequently the structures are capped with 50\,nm AlGaAs deposited at 350\,$^{\circ}$C, followed by a second annealing step at 650,$^{\circ}$C under As$_4$ flux for 5 minutes. This last anneal step is inserted into the growth procedure to ensure that the next layer is grown on a defect free surface. Next, another capping layer of 40\,nm is grown at 580\,$^{\circ}$C. The total structure was capped with 600\,nm GaAs. A post growth anneal step, which is usually performed to improve the optical properties of the QDs was not performed on this sample. All the images presented in this section are recorded with high negative bias ($\approx\,$-3.2\,V) between sample and STM tip. At these tunneling conditions and with the color scaling used, dark regions represent AlAs rich regions while bright regions represent GaAs rich regions.
\begin{figure}[t]
\includegraphics[width=8.6cm]{figure_8.pdf}
\caption{\label{figure_8}40\,nm\,$\times$\,34\,nm topographic image of a typical GaAs/AlGaAs QD (top) and an average cross-sectional profile (top graph) and separation between bilayers (bottom graph) along the line in the top figure.}
\end{figure}
A total of 11 QDs where observed by X-STM. A typical QD is shown in figure~\ref{figure_8}. As can be seen in this topographic image, the QDs are sharply defined by abrupt interfaces. The thickness of the WL was found to be less than 1\,bilayer~\cite{Keizer2010a}, as expected. The bow tie feature is most likely a foreign atom and is of no interest in the current study. Since AlAs and GaAs are lattice matched materials, the QDs are expected to be strain free. This is checked by taking a cross-sectional profile of the QD in figure~\ref{figure_8}. Three distinct regions can be observed. From left to right: an AlAs rich region, the GaAs QD, and the AlGaAs matrix. The height difference between these regions is due to electronic contrast. More importantly, all the regions are flat, there is no outward relaxation as observed in QDs grown with lattice-mismatched systems \cite{Offermans2005b}. To further illustrate that the GaAs QD is strain free, the distance between adjacent bilayers along the cross-sectional profile was measured. For this analysis the STM piezo elements were calibrated by performing a 2D FFT on the AlGaAs matrix. The result is shown in the bottom graph of figure~\ref{figure_8}. As can be seen, there is little deviation from the expected value of 0.565\,nm (dashed line), indicating that the QD is indeed strain free. Note that there is an Al rich region on top of the QD. This can be explained by the difference in mobility of Al and Ga atoms; the Ga atoms are more mobile and will migrate along the side of the QD during capping while the Al atoms, which are less mobile, are more likely to remain on top of the QD. The driving force behind the migration of the incoming adatoms away from the top of the QD is the convex curvature of the growth front at the position of the QDs \cite{Xie1994}. Note that this different from the SK-grown QDs of the previous sections were strain induced by the lattice mismatch is the driving process.
\begin{figure}[t]
\includegraphics[width=8.6cm]{figure_9.pdf}
\caption{\label{figure_9}30\,nm\,$\times$\,60\,nm topographic image (left) of two QDs. An atomic grid is overlain on top of a close up of the QD dot (right). Al and Ga atoms in the QD are indicate by respectively red and yellow squares.}
\end{figure}
Whether intermixing of Al is a factor of importance in the formation of GaAs/AlGaAs QDs grown by droplet epitaxy is a question frequently raised in the literature \cite{Sanguinetti2002,Mantovani2004,Heyn2007}. In all QDs imaged we have observed some degree of intermixing. In figure~\ref{figure_9} left panel, two typical QDs are shown. Even without further analysis it is evident that some intermixing of Al has taken place, see dark spots inside the QDs. To make a more quantitative analysis we overlaid a grid with atomic dimensions on top of a close up of the QD that showed the strongest intermixing, see figure~\ref{figure_9} right panels. On this grid, the positions of the Al and Ga atoms are marked with respectively red and yellow squares. The concentration of Al in this particular QD is determined to be 6\%. Here we would like to point out that the observed Al intermixing varied strongly from dot to dot, see for example the QD depicted in figure~\ref{figure_8} were the degree of intermixing is considerably lower, and that the 6\% can be considered as an upper limit of intermixing in these QDs.
\begin{figure}
\includegraphics[width=8.6cm]{figure_10.pdf}
\caption{\label{figure_10}Profile of three QDs extracted from the X-STM data (open circles). A Gaussian function is fitted to the largest QD (red line). The other two QDs (green and blue line) are assumed to have the same 3D-structure as the largest QD but cleaved off center. The projection of the (111)-direction on the cleavage plane is given by the dashed black line.}
\end{figure}
Concerning the shape of the QDs, we notice that the side facets of the observed QDs are not exactly straight. The maximum side facet angles were found to be in the range 34--55$^\circ$ per QD, were the upper limit corresponds to a \{111\} facet (54.7$^\circ$). If we assume that (1) all the QDs are approximately of equal height and (2) the observed height difference is due to the position of the cleavage plane relative to the center of the QD, this result excludes QD shapes with constant facet angles like rectangular (truncated) pyramids \cite{Bruls2002}. Since it has been reported that uncapped AlGaAs/GaAs QDs have \{111\} facets \cite{Lee1998,Mano2008}, we conclude that the shape of the QDs is somewhat changed during capping. Figure~\ref{figure_8} shows the highest QD we found. Since it is the highest, we assume that this QD is cleaved directly through its center. Consequently, we used the profile of this QD to generate a 3D-profile by fitting a Gaussian function, see figure~\ref{figure_10} (red line), and rotating it around the symmetry axis along the growth direction. Next, we checked whether the profile of all other observed QDs (illustrated for two exemplary QDs by the green and blue lines) correspond to profiles obtained by cleaving the obtained 3D-profile at specific distances from the center. As can be seen in figure~\ref{figure_10}, this is the case. From this we conclude that the observed QDs are Gaussian shaped QDs of approximately the same height but cleaved at different position from their center.
\section{Conclusion}
To summarize, we have investigated three techniques that can be used to gain control over the shape of QDs. The indium flush technique allows control over the height of the WL and the InGaAs QDs. The resulting QDs have a flatted top facet. We have shown that not only surface resident In but also buried In that segregates out of the WL is desorbed during the indium flush. Concerning the technique of antimony capping, we have shown that a growth interrupt under Sb flux prior to capping preserves the shape of the uncapped QDs. The same could be achieved by the growth of a GaAsSb capping layer. This capping layer was found to conformally cover the growth front. In both case the preservation of QD shape is attributed to the surfactant properties of Sb. In QD layers grown by droplet epitaxy the WL was found to be less than 1\,bilayer thick. As expected in lattice-matched systems, we found no strain present in the GaAs/AlGaAs QDs. Without strain there is no driving force for QD erosion during overgrowth, resulting in high QD of which the shape was found to be Gaussian. We conclude that indium flush, antimony capping, and droplet epitaxy can all be used as a tool to both shape QDs and WL.
|
1,314,259,995,791 | arxiv | \section{Introduction} \label{sec:Intro}
Many important questions in solar physics concern phenomena that take place in the low-$\beta$ environment of the corona, such as flares and coronal mass ejections (CMEs), active region (AR) evolution and dynamics, heating of the corona and sources of the solar wind. For studying these, it is often useful to have a model of the 3D magnetic field in the corona as it still cannot be observed and mapped directly. Such models can be potential \citep{Jiang12a}, linear force-free field \citep[LFFF; constant-$\alpha$; e.g.][]{Demoulin94, Abramenko96, Jiang12a}, or non-linear force-free field \citep[NLFFF; $\alpha(r)$; e.g.][]{vanBallegooijen04, Wiegelmann04, Valori05, Wheatland06, Schrijver06, Malanushenko12, Inoue12, Jiang12b}. Potential field source surface models have been in use for a long time and although well representative of the structure of the field at large heights in the corona, they by definition lack currents, and hence free energy, which is important for powering solar eruptions. An alternative are NLFFFs, which have gained significant popularity in recent years with the advent of numerous methods for their computation, which use either line-of-sight or vector photospheric magnetograms to produce a model of coronal magnetic fields or to extrapolate the observed photospheric magnetic field to the corona. However, these 3D magnetic fields are intrinsically complicated and although basic topological features, such as null points (NPs), fan-spine surfaces and flux ropes (FRs) can often be approximately identified just by inspecting field line plots, often, there is a need of quantitative topological analysis in order to make sense of the complicated 3D coronal magnetic field structure, its dynamics, and evolution.
Topological features in 2D and 2.5D, such as NPs \citep[e.g.][]{GorbatchevSomov88, Parnell10}, separatrices \citep{GorbatchevSomov88}, separator field lines, and null lines have been explored in solar physics context since the 80s. They are known to separate the field in connectivity domains. However, in the mid-90s a new topological term arose, namely quasi-separatrix layers \citep[QSLs;][]{Priest95, Demoulin96b}, which are the 3D generalizations of the above-mentioned features, now separating the field into quasi-connectivity domains. While the linkage of magnetic field lines over separatrices and NPs is discontinuous, across QSLs it is continuous but drastically changes.
In the early description of \cite{Demoulin96a}, the strength of QSLs, i.e. the amount of the change in field line linkage, is measured by the norm of the Jacobian of the mapping of neighboring field lines \citep{1994ApJ...437..851L} from one end of the photosphere to the other. However, this quantity is not invariant with respect to the direction of tracing of the field lines. Consequently, \cite{Titov07} came up with an alternative covariant quantity quantifying QSL strengths, called the squashing factor, $Q$.
QSLs (as well as NPs and separatrices) are preferential sites for build-up of current sheets in the presence of footpoint motions, and hence are preferential sites where reconnection can take place \citep{Aulanier05b}. Although, there is still no quantitative relationship \citep[possibly because it depends on the exact field configuration and footpoint motions; see][]{Galsgaard03b}, it has been suggested that the higher the value of $Q$, the thinner the current layer at that particular QSL \citep{Aulanier05b}. This makes these topological features very important for studies of storage and release of magnetic free energy in the process of reconnection at all scales.
Quantitative studies of topology by deriving QSL maps in 2D from potential, LFFFs, and NLFFFs have been used over the past decade to tackle many problems in solar physics. The existence of a QSL wrapping around the flux rope and crossing itself at a very high-$Q$ topological feature, a hyperbolic flux tube \citep[HFT;][]{Titov07, Savcheva12a, Savcheva12b, Zhao14, Liu14}, has become the basic feature in the standard flare model in 3D \citep{Aulanier12, Janvier13}, confirmed by observations \citep{Janvier14, Savcheva15, Savcheva16a, Janvier16, Zhao16}. In this picture, tether-cutting reconnection happens at the HFT under the FR between J-shaped oppositely directed field lines, which slip \citep{Aulanier06b} over the photospheric traces of the HFT \citep{Janvier13} and produce S-shape field lines that feed the FR and post-flare arcade. This scenario was put forward supported by data-constrained NLFFF models and MHD simulations by \cite{Savcheva12b}. In this picture, the photospheric traces of the HFT are 2J-shaped \citep{Titov07, Aulanier10} and they match the 2J-shaped flare ribbons of classical two-ribbon flares \citep{Chandra09,Schrijver11}. The match between the shapes of QSLs and flare ribbons has been achieved recently by \cite{Liu14}, \cite{Savcheva15}, and \cite{Zhao16}. These QSLs have been shown to move together with the flare ribbons in direction perpendicular to the polarity inversion line (PIL) \citep{Savcheva16a, Janvier16}.
The QSLs derived in \cite{Savcheva16a} and \cite{Janvier16} have been derived based on NLFFFs constrained only by pre-flare observations (magnetograms, and EUV and X-ray images), but have managed to reproduce the flaring topology and its evolution to a large extent. That indicates that these kinds of studies have potential predictive power as the use of topology analysis can show us the likely sites of flare reconnection a few hours before the event, as shown in \cite{Savcheva12a}. One could imagine going further and using the flare ribbon information and QSLs to work backwards and improve the initial NLFFF, thus providing better initial conditions for global data-driven MHD simulations of CME initiation and propagation \citep{Savcheva16b}.
Further studies show the evolution of QSL maps of solar ARs over several days, noting the effects of: flux cancellation on building sigmoidal flux ropes (\cite{Savcheva12a}, who showed the transition from bald-patch separatrix surfaces \citep[BPSS;][]{Titov93} to a HFT); quadrupolar topology on the possible breakout scenario \citep{Zhao14}; flux emergence on the development of a fan-spine NP topology \citep{Jiang16}. The global topology of active regions before eruption has been shown to be important for the characteristics of the dynamics, be it an eruption or just loop reconfiguration \citep{Janvier16, Jiang16, Pontin16, Chintzoglou16}.
Knowing the locations, extent, shape, and connections between connectivity domains, and the features they contain or border, can prove vital for understanding links between seemingly unconnected faraway regions on the Sun that erupt sequentially or almost simultaneously, i.e. sympathetic eruptions. A detailed study of one such event (1-2 August, 2010) was conducted by \cite{Titov12}, who showed that filaments embedded in neighboring pseudostreamers are activated sequentially after the first filament erupts and destabilizes the system \citep{Torok11}. Even if it is a single CME, the potential of the CME to have a large longitudinal extent or to present with a significant energetic particle signature at any point in the heliosphere is most probably dependent on the specifics of the global 3D topology in the corona and heliosphere as the CME evolves and propagates \citep{Masson13}. As a related phenomenon, the propagation of EUV dimmings may also turn out to be dependent on the global solar topology, neighboring the directly related AR \citep{Downs16}.
On a smaller scale, reconnection at QSLs have been potentially found important for the heating of the solar corona \citep{Schrijver10}. Reconnection in loop braiding has been theoretically and numerically explored for this purpose as well \citep{WilmotSmith09a, Pontin15}. QSLs in the outskirts of ARs have been shown to drive plasma outflows \citep{Baker09} as evidenced by blueshifts in Hinode/EIS velocity maps of ARs, which could be important for understanding the outflow of plasma from the corona that contributes to the slow solar wind. Potential solar wind sources can be further derived my means of the S-web model of \cite{Archontis09}, which utilizes QSLs at the source surface and below to look at the connectivity domains surrounding active regions and coronal holes, as well as the connections between them.
Ultimately, with the speed-up and automation of NLFFF codes and QSL computation methods, we will be able to implement 3D QSL analysis in space weather predictive operations aimed at identifying the next likely region to erupt, studying the effect of the propagation of the CME ejecta and its particles, and predicting the direction and sign of the CME magnetic field when it reaches the Earth's magnetosphere. One step on this path is obtaining a fast, reliable 3D QSL code that can work on the whole Sun or in an AR in great detail. Such codes have been developed and used before for analyzing potential coronal magnetic field models \citep[][]{0004-637X-806-2-171}, as well as experimental flux rope configuration \citep[][]{PhysRevLett.103.105002}, yet they were never made public.
In this paper, we introduce a fast, freely-available, open-source code, \qsl, aimed at calculating 3D QSL maps, whose development was motivated by several potential uses, such as:
\begin{itemize}
\item Studying large resolution QSL physics and its application to reconnection theory.
\item Exploring large parameter spaces of possible topologies.
\item 3D studies of active region evolution, CME initiation and propagation.
\item Obtaining the evolution of topology over large periods of time with high cadence from data-driven or idealized MHD simulations at a wide range of scales.
\end{itemize}
The paper is organized as follows. In Section~\ref{over} we give an overview of the code. In Section~\ref{algo} we give details about the algorithm used in \qsl. We show illustrative results in Section~\ref{ex} and give our concluding remarks in Section~\ref{summary}.
\section{Code Overview}\label{over}
\qsl is written in C++ and depends on the Boost\footnote{\url{http://www.boost.org/}} and VexCL\footnote{\url{https://github.com/ddemidov/vexcl}} \citep{vexcl} libraries, on a working OpenCL\footnote{\url{https://www.khronos.org/opencl/}} implementation, as well as on their respective dependencies. The visualization scripts require Python with SciPy \citep{scipy} and PyEVTK\footnote{\url{https://www.python.org/}, \url{https://www.scipy.org/}, \url{https://bitbucket.org/pauloh/pyevtk}}. The code is intended to be run on a graphics processing unit (GPU). However, it can be run multithreaded on a CPU if one uses the POCL\footnote{\url{http://portablecl.org/}} \citep{pocl} OpenCL implementation.
The input for \qsl is 3D cubes containing the values of the magnetic field components sampled on a rectilinear grid in either Cartesian or spherical coordinates. For the exact file structure, we refer the reader to the manual distributed with the code.
The output of the code can be 2D or 3D arrays of $Q$ values, depending on whether the code is run in 2D or 3D mode to produce slices or data cubes, respectively. For slices, the output can be rendered as an image using the provided Python script. The output of 3-dimensional calculations is exported to VTK format,
which can then be visualized using Paraview, VisIt or Mayavi among many\footnote{\url{http://www.vtk.org/}, \url{http://www.paraview.org/}, \url{https://wci.llnl.gov/simulation/computer-codes/visit/}, \url{http://docs.enthought.com/mayavi/mayavi/}}.
When the desired output is a slice through the volume of interest, the slices can have two types of geometry: planar or spherical. The code supports constructing planar slices of arbitrary orientation. In this case, one needs to specify the orientation, center, and axes span of the slice. Spherical slices are slices at a specified fixed radius, spanning a given range in latitude and longitude.
\section{Algorithm}\label{algo}
Below we write down the equations solved by \qsl. The code can work in both Cartesian and spherical coordinates. Whenever we find it useful, we quote the explicit equations solved by the code for spherical coordinates.
\subsection{Integrating field lines}
The magnetic field lines ${\bm{x}}(\lambda)$ (where $\lambda$ is an affine parameter) are calculated as the integral curves of the unit magnetic field, ${\hat{\bm{B}}}$. Thus, in Cartesian coordinates, we trivially have:
\begin{eqnarray}\label{fl}
\partial_{\lambda}{\bm{x}}(\lambda)={\hat{\bm{B}}}({\bm{x}}(\lambda))\ .
\end{eqnarray}
In spherical coordinates, the field lines are given as solutions to (keeping the $\lambda$ dependence explicit):
\begin{eqnarray}\label{fl_sph}
r(\lambda)\cos\Big(\theta(\lambda)\Big)\partial_\lambda \phi(\lambda)={{\hat{B}}}_\phi\Big(\phi(\lambda),\theta(\lambda),r(\lambda)\Big)\nonumber\\
r(\lambda)\partial_\lambda \theta(\lambda)={{\hat{B}}}_\theta\Big(\phi(\lambda),\theta(\lambda),r(\lambda)\Big)\nonumber\\
\partial_\lambda r(\lambda)={{\hat{B}}}_r\Big(\phi(\lambda),\theta(\lambda),r(\lambda)\Big)\ .
\end{eqnarray}
Here we used the fact that the magnetic field components are written in the spherical orthonormal basis $\hat \phi,\ \hat \theta,\ \hat r$, which correspond to longitude, latitude, and radius, respectively.
\subsection{Interpolation schemes}
\qsl allows one to use different interpolation schemes when calculating the values of the magnetic field vectors on the right-hand side of the above system of equations. This capability can be used to test the robustness of QSL maps on the interpolation order. The available interpolation schemes are
trilinear, triquadratic \citep{quadratic_interp} and tricubic \citep{cubic_interp}.
To be able to write down the interpolation schemes explicitly, we need to introduce some notation first. The input magnetic field data cubes specify the values $\bm{B}^{{\bm{v}}}$ of the magnetic field at points with coordinates ${\bm{x}}_{{\bm{v}}}$, sampled on a rectilinear grid. Thus, ${\bm{v}}$ is a 3-vector, with each of its components running over the indices of the input 3d magnetic field array. Next, we would like to write down the value of ${\hat{\bm{B}}}$ at some arbitrary position ${\bm{x}}$. To do that, we need to identify the cell within the input array within which ${\bm{x}}$ lies. The interpolation kernels cover 8, 27, or 64 vertices neighboring that cell for trilinear, triquadratic and tricubic interpolation, respectively. Of the 8 vertices in the immediate neighborhood of ${\bm{x}}$, let us denote by ${\bm{c}}$ that vertex, which lies closest to the origin of the array. Therefore, for any of the above interpolation schemes, we can write:
\begin{eqnarray}\label{interp}
{\hat{\bm{B}}}({\bm{x}})=\sum\limits_{{\bm{v}} \in \hbox{stencil}}{{\hat{\bm{B}}}}^{{\bm{v}}+{\bm{c}}}f_{{\bm{v}}}\Big({\bm{x}}-{\bm{x}}_{{\bm{c}}};\bm{h}_{{\bm{c}}}\Big)\ ,
\end{eqnarray}
where the sum runs over the 8, 27, or 64 vertices which span the respective interpolation stencil around ${\bm{x}}$. The physical dimensions of the array cell containing ${\bm{x}}$ are specified by $\bm{h}_{\bm{c}}$. The interpolation kernels $f_{\bm{v}}$ depend on the selected interpolation order.
As an example, for trilinear interpolation, the interpolation kernels are given by:
\begin{eqnarray}
f_{0,0,0}({\bm{x}};{\bm{h}})&=&\left(1-\frac{x_0}{h_0}\right)\left(1-\frac{x_1}{h_1}\right)\left(1-\frac{x_2}{h_2}\right)\\
f_{0,0,1}({\bm{x}};{\bm{h}})&=&\left(1-\frac{x_0}{h_0}\right)\left(1-\frac{x_1}{h_1}\right)\left(\frac{x_2}{h_2}\right)\nonumber\\
f_{0,1,1}({\bm{x}};{\bm{h}})&=&\left(1-\frac{x_0}{h_0}\right)\left(\frac{x_1}{h_1}\right)\left(\frac{x_2}{h_2}\right)\nonumber\\
&\cdots&\nonumber
\end{eqnarray}
When running \qsl for spherical geometry, the input magnetic field is sampled on a rectilinear grid in spherical coordinates. Thus, in the above equation, we have $x_0=\phi-\phi_{\bm{c}}$, $x_1=\theta-\theta_{\bm{c}}$, $x_2=r-r_{\bm{c}}$, while $h_0,\ h_1,\ h_2$ give the grid spacing in longitude, latitude and radius, respectively, for the cell containing ${\bm{x}}$.
\newpage
\subsection{Field-line deviation using linearization}
The squashing factor $Q$ quantifies how neighboring field lines deviate from one another. The procedure described in \cite{Pariat12} for solving for $Q$ involves explicitly integrating three closely spaced field lines, after which one takes the finite differences in position of the footpoints of those field lines. Those differences in turn enter in the calculation of the squashing factor. However, calculating those field-line deviations using such a finite difference scheme puts severe constraints on the precision with which one should follow neighboring field lines. \cite{Pariat12} quote a fractional precision of $10^{-8}$ for their calculation, which results in severe speed penalties.
In \qsl, we alleviate that problem by calculating field-line deviations by linearizing the deviation equation as follows. The deviation between two neighboring field-lines ${\bm{x}}(\lambda)$ and ${\bm{x}}'(\lambda)$ is quantified by the difference in their positions: $\delta {\bm{x}}(\lambda)={\bm{x}}'(\lambda)-{\bm{x}}(\lambda)$. Here we assume that when $\lambda=0$, the positions along the two field lines are infinitesimally apart. Thus, the field-line deviation can be calculated as follows:
\begin{eqnarray}\label{dev}
\partial_\lambda \delta {\bm{x}}(\lambda)={\hat{\bm{B}}}\Big({\bm{x}}'(\lambda)\Big)-{\hat{\bm{B}}}\Big({\bm{x}}(\lambda)\Big)\approx \bigg(\delta {\bm{x}}(\lambda)\cdot\bm{\nabla}_{{\bm{x}}} \bigg){\hat{\bm{B}}}\Big({\bm{x}}(\lambda)\Big)\ ,
\end{eqnarray}
where we used perturbation theory to linearize the equation in the second equality.
In order to integrate the above equation, we need to be able to take the gradient of ${\hat{\bm{B}}}$. Recalling that ${\hat{\bm{B}}}({\bm{x}})$ is calculated using (\ref{interp}), taking the gradient is straightforward as it acts only on the interpolation kernels. In spherical coordinates, the field-line deviation is given explicitly below for reference:
\begin{eqnarray}\label{dev_sph}
r(\lambda)\cos\Big(\theta(\lambda)\Big)\partial_\lambda \delta \phi(\lambda)&=&\sum\limits_{{\bm{v}} \in \hbox{stencil}}
{{\hat{B}}}^{{\bm{v}}+{\bm{c}}}_{\phi}\times\delta f_{{\bm{v}}}\Big(\delta{\bm{x}}(\lambda);{\bm{x}}(\lambda)-{\bm{x}}_{\bm{c}};{\bm{h}}_{\bm{c}}\Big)
\nonumber\\
r(\lambda)\partial_\lambda \delta \theta(\lambda)&=&\sum\limits_{{\bm{v}} \in \hbox{stencil}}
{{\hat{B}}}^{{\bm{v}}+{\bm{c}}}_{\theta}\times\delta f_{{\bm{v}}}\Big(\delta{\bm{x}}(\lambda);{\bm{x}}(\lambda)-{\bm{x}}_{\bm{c}};{\bm{h}}_{\bm{c}}\Big)
\nonumber\\
\partial_\lambda \delta r(\lambda)&=&\sum\limits_{{\bm{v}} \in \hbox{stencil}}
{{\hat{B}}}^{{\bm{v}}+{\bm{c}}}_r\times\delta f_{{\bm{v}}}\Big(\delta{\bm{x}}(\lambda);{\bm{x}}(\lambda)-{\bm{x}}_{\bm{c}};{\bm{h}}_{\bm{c}}\Big)\ ,
\end{eqnarray}
where $\delta f$ is given by (after suppressing its arguments):
\begin{eqnarray}
\delta f_{\bm{v}}\equiv\bigg(\delta\phi(\lambda)\partial_\phi +\delta\theta(\lambda)\partial_\theta+\delta r(\lambda)\partial_r\bigg)f_{{\bm{v}}}\Big(\phi,\, \theta,\, r;\, \bm{h}_{\bm{c}}\Big)\Bigg|_{\phi=\phi(\lambda)-\phi_{\bm{c}},\, \theta=\theta(\lambda)-\theta_{\bm{c}},\, r=r(\lambda)-r_{\bm{c}}}\ ,
\end{eqnarray}
where the derivatives are taken analytically in the code for each interpolation kernel.
\subsection{Integration methods}
\qsl offers a choice between two integration schemes for integrating the field lines and field-line deviation vectors. One can use either an explicit Euler scheme, or
an adaptive Runge-Kutta Cash-Karp method \citep{Cash:1990:VOR:79505.79507} provided by the Boost \verb|runge_kutta_cash_karp54| stepper algorithm. The latter method can easily be substituted with any of the other integration methods offered by the Boost library.
The field line deviation vectors $\delta {\bm{x}}$ needed for the squashing factor calculation, are solved by \qsl using eq.~(\ref{dev}) (or for spherical geometry, using (\ref{dev_sph})) instead of the finite difference scheme of \cite{Pariat12}. This allows us to relax the precision and accuracy tolerances by many orders of magnitude. For the adaptive stepper, we have found a value of $10^{-2}$ to be more than sufficient for the real world example explored in this paper (see below). This tolerance is six orders of magnitude larger than the one quoted by \cite{Pariat12}. Choosing an explicit fixed-step Euler scheme with roughly 5 samplings per grid spacing gives about an order of magnitude speed-up relative to the adaptive stepper implementation that is incorporated in \qsl. One trades accuracy for such a speed-up. However, in our experiments, we have not encountered cases where using the adaptive stepper was beneficial. We still consider the adaptive stepper useful as it can be used for testing the convergence properties of the Euler scheme for the particular problem at hand.
\subsection{Squashing factor calculation}
Having introduced the basic equations allowing us to integrate the magnetic field lines and field-line deviations, we proceed to describe the calculation of the squashing factor.
To calculate the squashing factor, we use Method 3 of \cite{Pariat12}. That requires projecting the components ($\delta {\bm{x}}_\perp$) of $\delta {\bm{x}}$ perpendicular to the field line tangent, given by ${\hat{\bm{B}}}$. In spherical coordinates, we can do that if we write both ${\hat{\bm{B}}}$ and $\delta {\bm{x}}$ in the spherical orthonormal basis spanned by $\hat \phi,\ \hat\theta, \ \hat r$. The magnetic field is given by interpolating the input magnetic field using (\ref{interp}), while the components of the field line deviation vector are given by:
\begin{eqnarray}
\delta x_\phi=r(\lambda)\cos\Big(\theta(\lambda)\Big) \delta \phi(\lambda)\ , \ \ \
\delta x_\theta=r(\lambda) \delta \theta(\lambda)\ , \ \ \
\delta x_r=\delta r(\lambda)\ .
\end{eqnarray}
Since our choice of basis is orthonormal, calculating\footnote{Notice that normalizing a vector field and then interpolating it is not equivalent to interpolating a vector field and then normalizing it. It is up to the user to supply arrays of the magnetic field that are sampled finely enough to make this difference unimportant. To speed up the code, we first normalize the magnetic field data cubes, and only then interpolate them. Thus, ${\hat{\bm{B}}}({\bm{x}})$ as calculated from (\ref{interp}) is not guaranteed to be a unit vector. Therefore, when extracting $\delta {\bm{x}}_\perp$ the code explicitly normalizes ${\hat{\bm{B}}}({\bm{x}})$ beforehand.} $\delta {\bm{x}}_\perp=\delta {\bm{x}}-{\hat{\bm{B}}}\left({\hat{\bm{B}}}\cdot\delta{\bm{x}}\right)$ numerically is trivial. Here ${\hat{\bm{B}}}$ is evaluated at ${\bm{x}}(\lambda)$.
To calculate $Q$ at a position ${\bm{x}}_0$, we need to integrate $\delta {\bm{x}}(\lambda)$ for two sets of initial conditions (denoted with superscripts):
\begin{eqnarray}\label{ic}
\delta {\bm{x}}^{(1)}(\lambda=0)= \hat{\bm{a}}\ , \ \mathrm{and} \ \ \ \delta {\bm{x}}^{(2)}(\lambda=0)= \hat{\bm{b}}\ ,
\end{eqnarray}
where $\lambda=0$ corresponds to the initial condition for the field line ${\bm{x}}(\lambda=0)={\bm{x}}_0$, which passes through ${\bm{x}}_0$. The only restriction on vectors $\hat{\bm{a}}$ and $\hat{\bm{b}}$ is that they, combined with ${\hat{\bm{B}}}\Big({\bm{x}}_0\Big)$, form an orthonormal basis at ${\bm{x}}_0$. Integrating the field line ${\bm{x}}(\lambda)$ along with $\delta {\bm{x}}^{(1)}(\lambda)$ and $\delta {\bm{x}}^{(2)}(\lambda)$ involves solving equations (\ref{fl}) and (\ref{dev}) (which in spherical coordinates, correspond to (\ref{fl_sph}) and (\ref{dev_sph})), forwards to $\lambda_{\mathrm{F}}$, and then backwards to $\lambda_{\mathrm{B}}$, subject to the initial conditions (\ref{ic}). Those values of $\lambda$ correspond to parameter values for which ${\bm{x}}(\lambda)$ reaches the boundary of the region spanned by the input magnetic field\footnote{The standard squashing factor is calculated by knowing how much neighbouring field lines deviate from one field line footpoint to the other. However, one can envision applications where more localized $Q$ values may also be of interest. Thus, \qsl allows calculating $Q$ by integrating the field lines to $\lambda_F$ and $\lambda_B$ spanning a fixed maximum field line length. Note that choosing to enable that option gives a $Q$ which is no longer a global quantity, and is no longer constant along the length of a field line.}.
Let us define the solutions at the endpoints as:
\begin{eqnarray}
{\bm{a}}_{\mathrm{F}}\equiv\delta {\bm{x}}^{(1)}_\perp(\lambda_{\mathrm{F}})\ , \ \ \ {\bm{a}}_{\mathrm{B}}\equiv\delta {\bm{x}}^{(1)}_\perp(\lambda_{\mathrm{B}})\ ,\nonumber\\
{\bm{b}}_{\mathrm{F}}\equiv\delta {\bm{x}}^{(2)}_\perp(\lambda_{\mathrm{F}})\ , \ \ \ {\bm{b}}_{\mathrm{B}}\equiv\delta {\bm{x}}^{(2)}_\perp(\lambda_{\mathrm{B}})\ .
\end{eqnarray}
With these definitions, after a bit of algebra, one can show that $Q$ as calculated using Method 3 of \cite{Pariat12} can be reduced to the following expression, which is straightforward to implement numerically:
\begin{eqnarray}
Q=\frac{B_{\mathrm{F}} B_{\mathrm{B}}}{B^2_0}
\left[
a_{\mathrm{F}}^2 b_{\mathrm{B}}^2+
a_{\mathrm{B}}^2 b_{\mathrm{F}}^2-
2({\bm{a}}_{\mathrm{B}}\cdot {\bm{b}}_{\mathrm{B}})({\bm{a}}_{\mathrm{F}}\cdot {\bm{b}}_{\mathrm{F}})
\right]\ ,
\end{eqnarray}
where $B_{\mathrm{F}}\equiv B({\bm{x}}(\lambda_{\mathrm{F}}))$ is the unnormalized magnetic field magnitude corresponding to one of the endpoints, and similarly for $B_{\mathrm{B}}$ and $B_0$.
\subsection{Adaptive refinements}
In order to be able to identify QSLs, one needs to resolve high-$Q$ regions, which correspond to thin surfaces, separating the quasi-connectivity domains in 3D. Thus, a proper QSL code needs to perform adaptive refinements around those regions. One way to do that is to refine in regions where $Q$ (or its second derivative, for example) is larger than a predefined threshold. In \qsl, we employ an alternative method, which identifies those domain boundaries much more robustly. The method relies on using Field-line Length Edge (FLEDGE) maps which we introduce next.
We define a FLEDGE map to be \textit{any} map of the changes of the length of neighbouring field lines. As an example, such changes can be mapped out using the gradient magnitude from the Sobel operator applied to a 2D or 3D map of the length of field lines in a section or a volume. Examples of FLEDGE maps are shown in the last row of Figure~\ref{TDslices} (discussed further in the next section), where a Sobel filter was applied to the 2D maps of the Field-Line Length (FLL) shown in the middle row of that figure.
\begin{figure}[t!]
\plotone{TD_slices.pdf}\caption{Horizontal and vertical planar slices of a TD flux rope. The first column shows horizontal sections taken below the HFT; the second column is a horizontal section through the HFT; and the third is taken above the HFT at the $z$ locations identified above each column (the units of $x$, $y$ and $z$ are irrelevant). The rightmost column shows a vertical cross-section, similar to the one shown in Fig.\,\ref{3DTD}. The three rows correspond to: a QSL map, quantified by the logarithm of the squashing factor, $\log_{10}{Q}$; a field-line length (FLL) map (labelled by ``$\mathrm{length}$''); and a FLEDGE map (labelled by $S$), realized as the gradient magnitude from the Sobel operator applied to the FLL map. Note the correspondence between QSLs and large FLL jumps in the FLEDGE maps. Regions with open field lines around the flux rope are indicated with dark blue in the QSL map.}\label{TDslices}
\end{figure}
Having introduced FLEDGE maps, let us move on to describe the way adaptive refinements are handled by \qsl. That is done by first filling the slice or volume of interest with a Hilbert curve. This allows one to map the region of interest onto a one-dimensional curve. Thus, the array holding the $Q$ and FLL values in the code can be rendered one-dimensional, with successive elements of that array ordered according to position along the Hilbert curve.
Hilbert curves carry the useful property that neighboring points on the Hilbert curve are necessarily close together in real space, although the opposite does not necessarily hold. In the spirit of the FLEDGE maps described above, our criterion for refining the sampling in a region of interest is checking whether the jump in the FLLs between two successive samples along the Hilbert curve surpasses a certain threshold. One can envision many other possible choices, but we found this rule quite robust, converging on domain boundaries about an order of magnitude faster than using a threshold in $Q$ (or its second derivative along the Hilbert curve) as a refinement criterion. If the FLL jump threshold is surpassed, then the code calculates $Q$ (and the respective field-line length) halfway along the Hilbert curve between those two neighboring samples.
For convenience, \qsl includes a code which takes the $Q$ (or FLL) array sampled along the Hilbert curve, and converts it into a 2D or 3D array of $Q$ values sampled on a rectilinear grid spanning the respective slice or volume of interest. If more than one $Q$ value is found in a cell around a grid point, the value the array converter assigns to that point is the maximum $Q$ value in that cell. If there are no samples in a grid cell, then the array converter interpolates the $\log(Q)$ values along the Hilbert curve to fill in the gap.
The Hilbert curve refinements can miss a point that lies in between samples that are not close along the Hilbert curve, but are close in real space. To alleviate that problem, after each refinement step, we shift the Hilbert curve by a small amount in real space; then reorder the $Q$ and FLL arrays along that new Hilbert curve; perform the refinement step again; and then shift back the Hilbert curve to its original position, reordering the arrays along that original curve. We have found that applying this shifting technique nearly eliminates such misses, and makes any artifacts irrelevant.
The benefit of the Hilbert curve refinements (as opposed to using more sophisticated adaptive-mesh techniques) is that the code performing the adaptive refinements is about 50 lines long and requires no special book-keeping other than keeping track of the Hilbert coordinate of each point for which a $Q$ value is known. The functions responsible for the calculation of the $Q$ values are independent on the choice of refinement scheme. Thus, incorporating any other type of adaptive refinement in \qsl should be a straightforward coding exercise.
\section{Illustrative examples}\label{ex}
\subsection {Titov \& D{\'e}moulin flux rope}
In order to illustrate the capabilities of \qsl, in this section we show several 2D and 3D QSL and FLEDGE maps obtained with the code. Originally, the theory of QSLs has been developed for the Cartesian analytical model of a flux rope following the construction of \citet[TD;][]{TD99}, which has served as an analytical case study for QSL calculation methods \citep{Pariat12}. Thus, the first results from \qsl we include are obtained for the numerical implementation of the TD flux rope model as given by \cite{vanBallegooijen08}. Those are shown in 2D in Fig.~\ref{TDslices} and in 3D in Fig.~\ref{3DTD}.
\begin{figure}[t!]
\plotone{TD_3d.pdf}\caption{A 3d Cartesian rendering of the TD flux rope. The bottom grayscale section shows the vertical component of the magnetic field. The left column shows a 3D QSL map (quantified by $Q$), while the right column shows the corresponding 3D FLEDGE map (quantified by the gradient magnitude from the 3D Sobel operator applied to the FLL map). A cut through the middle of the flux rope is shown as well highlighting the boundary of the flux rope and the HFT underneath, which is clearly visible in the QSL map, but not in the FLEDGE map. The QSL associated with the tranisition to open field lines is made visible in the 2D FLEDGE section as well, although it is filtered out from the 3D rendering. Note the similarities between the QSL and FLEDGE maps.}\label{3DTD}
\end{figure}
In Fig.\,\ref{TDslices}, we have shown horizontal slices through the FR at 3 different heights (first three columns), as well as vertical slices through the flux rope (fourth column). The three rows in the figure correspond to: a QSL map, quantified by the squashing factor, $Q$; a map of the field-line length(FLL); as well as a FLEDGE map, quantified by the gradient magnitude from a Sobel filter applied to the FLL map.
The height of the sections in the first column ($z=0.01$) is taken below the peak of the HFT, so that the QSLs have a 2J shape with the J's facing away from each other \citep[for a cartoon illustration see Fig.\,8 of][]{Savcheva12a, TD99}. For the chosen value of flux rope twist, the horizontal maps display QSL hooks that are almost closed on each other. The effect of twist on the hooks is discussed in detail by \cite{Savcheva12b} and \cite{Zhao16}.
The second column ($z=0.97$) shows a single S-shaped QSL because the cut passes through the HFT \citep[see Fig.\,8][]{Savcheva12a}. The third column ($z=2$) shows the QSL that encircles the flux rope when the cut is taken above the HFT. The curve is almost closed due to the large amount of twist in the rope. The vertical section clearly shows the HFT under the flux rope core as the location where the QSLs that wraps around the flux rope intersects with itself.
From Fig.\,\ref{TDslices}, one can see that large values in the FLEDGE maps (locations where FLL jumps) correspond to quasi-domain boundaries, characterized by QSLs. Note that the FLEDGE maps also capture the boundary between open and closed field lines. The close correspondence between the QSL and FLEDGE maps is investigated further below.
\begin{figure}[t!]
\plotone{SOL_Q_3d_v2.pdf}\caption{Shown are different views of the 3D QSL (left column) and FLEDGE (right column) maps for region SOL2010-04-08. The geometry of the modelled region is that of a spherical wedge. The grayscale spherical slice shows the HMI magnetogram used in generating the NLFFF model of the region. The field lines show: the core of the flux rope (magenta); the overlying ``potential'' arcade (cyan); 2J field lines (yellow); S-shaped field line (blue); the flare arcade (green); and the HFT (red). The 3D-rendered surface of $Q$ and the Sobel gradient magnitude outlines the cavity of the flux rope. In the bottom row we show a vertical planar section through the spherical domain in the middle of the flux rope. An animated version of this figure can be found at \url{https://bitbucket.org/tassev/qsl_squasher/downloads/3D_QSL.m4v}.}\label{100408slice}
\end{figure}
In Fig.\,\ref{3DTD}, the 3D QSL (left column) and FLEDGE (right column) maps are rendered using Paraview. One can clearly see the QSL surface that wraps around the flux rope. In the top panel, a 2D section is added in the center of the rope showing the outline of the rope and the crossing of the QSL with itself at the HFT (the reddest part of the volume). A semi-transparent reddish feature from the 3D rendering can be seen to pass through the saddle point of the HFT in the 2D map. Note that 2D sections can be computed separately by \qsl, or can be extracted from the 3D volume of $Q$ in a visualization software. In the bottom panel of the Fig.\,\ref{3DTD} we have shown some sample field lines that belong to the flux rope and are contained within the 3D-rendered surface of the TD flux rope QSL. As in Fig.\,\ref{TDslices}, in Fig.\,\ref{3DTD} one can clearly see the close resemblance between the QSL and FLEDGE maps in 3D. The most apparent difference is the fact that the HFT is not prominent in the FLEDGE map. Thus, while we can use the locations of the largest $Q$ values as a proxy for the location of the HFT, there is no such correspondence between values in the FLEDGE map and HFT's.
\begin{figure}[t!]
\plotone{SOL_slices.pdf}\caption{The figure shows slices at 6\,Mm above the photosphere. The top-left panel shows the squashing factor, $Q$; the top-right panel shows a map of the field-line length (FLL); while the bottom two panels show (two different scalings of) FLEDGE maps of the region as quantified by the gradient magnitude from the Sobel operator applied to the FLL map. The dark-blue region in the QSL map corresponds to open field lines. Note the qualitative match between the QSL and FLEDGE maps for this real-world example.}\label{1004083D}
\end{figure}
\subsection {SOL2010-04-08 sigmoidal region}
Next, we demonstrate the capabilities of \qsl for a data-constrained unstable magnetic field model in spherical coordinates, produced with the flux rope insertion method \citep{vanBallegooijen04,Savcheva09}. The model is of SOL2010-04-08 sigmoidal region that produced a B-class flare and a CME on 08 April 2010 and its stability has been studied in detail in \cite{Su11}. The unstable model analyzed here is produced by addition of axial flux to the best-fit marginally stable model, so that a residual Lorentz force exists, which prevents the field from reaching a non-linear force-free equilibrium during the magnetifrictional relaxation of the field \citep{Savcheva15}. Such an unstable model has been used by \cite{Kliem13} and \cite{Savcheva16b} to produce an MHD eruption from this region. The lower boundary for the QSL calculation is set at 2\,Mm since lower than that the magnetic field models contain many low-lying bold patches \citep[BPs;][]{Titov93} which contain infinite values of $Q$, yet the field lines that pass through these BPs do not propagate to large heights. The introduction of that boundary significantly speeds up the computation as discussed in \cite{Savcheva12b}.
Our 3D results for SOL2010-04-08 are shown in Fig.~\ref{100408slice}. The left column shows the QSL map of the region, while the right column, the FLEDGE map, computed in the same way as for the TD flux rope.
The first two rows of Fig.~\ref{100408slice} show a top and side view of the spherical wedge domain of the magnetic field computation, including field lines sampling different quasi-connectivity domains, including the core of the flux rope (magenta) and the overlying arcade (cyan). The bottom row shows a planar vertical slice through the spherical domain in the middle of the flux rope. One can see the outline of the flux rope and the HFT (red field line) underneath. Notice that the HFT has already reached a significant height in the shown iteration of the magnetofrictional evolution \citep[see Fig.~10 in][]{Savcheva16a} and the erupting flux rope is in the process to turn into a CME.
As was the case with the TD flux rope, in this realistic example, the FLEDGE map recovers the general flux rope structure seen in the QSL map, including the intersection of the QSL surfaces at the location of the HFT. Yet, the HFT itself is not readily identified in the FLEDGE map.
However, calculating the 3D FLEDGE maps for SOL2010-04-08 took about 3\,min on a consumer workstation GPU (AMD W8100). The calculation of the 3D QSL map of the same region took 2 orders of magnitude more time. The main reason for this enormous difference is the fact that one does not need to perform adaptive refinements when computing FLEDGE maps as jumps in FLL are readily identified (see top-right panel of Fig.~\ref{1004083D}) even at low resolution, unlike local spikes in $Q$. Thus, we argue that FLEDGE maps offer a computationally cheap substitute of QSL maps that can be especially useful in the preliminary stages of any (quasi-)topological studies.
In Fig.\,\ref{1004083D} we have shown a spherical surface slice below the peak of the HFT. This can be recognized by the QSL pattern around coordinates (-200\,Mm,\,300\,Mm) in the top-left panel, showing the 2D QSL map. That QSL pattern can be recognized as the 2J-pattern seen in the first column of Fig.~\ref{TDslices}, as well as in the cartoon shown in Fig.~8 of \cite{Savcheva12a}. The QSL map is certainly complicated due to the intrinsic complexity of the observed HMI magnetic field (no smoothing has been applied). This effect of real magnetic fields on the complexity of QSL maps has been discussed in detail in \cite{Savcheva12a,Savcheva12b}. Yet, from the vertical slice in Fig.~\ref{100408slice}, one can see that most of the complex structures are contained at low heights above the photosphere, and do not interfere substantially with one's ability to read the flux-rope structure from the 3D QSL maps, thus highlighting the importance of calculating QSL maps in three dimensions.
For comparison, in the other three panels of Fig.\,\ref{1004083D}, we show the corresponding FLL map (top-right), as well as the FLEDGE map with two different scalings and color-codings in the bottom two panels. The similarities between the QSL and FLEDGE maps on the top-left and bottom-left are clear.
\section{Summary}\label{summary}
In this paper we presented \qsl: a free, publicly available, open-source code for fast calculation of Quasi-Separatrix Layer maps in two or three dimensions. It requires an input magnetic field sampled on a rectilinear grid in Cartesian or spherical coordinates.
We benchmarked the code by calculating 3D QSL maps for a model of the SOL2010-04-08 sigmoidal region on a consumer workstation GPU (AMD W8100). We found that the code achieves large processing speeds for three main reasons, each of which results in an order-of-magnitude speed-up:
\begin{itemize}
\item Running the code on the GPU as opposed to the workstation CPU results in about an order of magnitude speed-up.
\item Compared to previous studies \citep[e.g. ][]{Pariat12}, we drastically relax the precision requirements for the QSL calculation. We do that by applying perturbation theory when calculating field-line deviations, which are necessary for calculating the squashing factor, quantifying the QSL strength.
\item We use a new boundary detection criterion between quasi-connectivity domains, which quickly identifies possible QSL locations which need to be finely sampled by the code. That boundary detection criterion relies on finding the locations of abrupt field-line length changes. A map of these jumps in field-line length we dub a FLEDGE map. We find that using such FLL jumps as a refinement criterion, instead of a threshold in $Q$ (or its second derivative), results in an order of magnitude speed-up of the code.
\end{itemize}
For the realistic model discussed above, we clocked \qsl at several million $Q$ values per minute, which implies that a representative 3D QSL map can be obtained within a few hours.
We also presented a quick-and-dirty alternative to QSL maps: FLEDGE maps, which can be optionally output by \qsl. We show that, for the most part, FLEDGE maps and QSL maps identify similar topological features. Constructing high-resolution 3D FLEDGE maps with \qsl can be completed in minutes -- two orders of magnitude faster than calculating the corresponding 3D QSL maps. The main reason for this difference is the fact that one does not need to perform adaptive refinements when computing FLEDGE maps as jumps in field-line length are readily identified even at low resolution, unlike local spikes in $Q$. Thus, we argue that FLEDGE maps offer a computationally cheap substitute of QSL maps that can be especially useful in the preliminary stages of any (quasi-)topological studies.
The potential advantages to the solar physics community of having such freely-available, open-source codes are largely unexplored beyond published data-reduction pipelines. One of our goals in making \qsl public is stimulating others to get involved in a collaborative effort to produce codes open to inspection and verification. This has the benefit of avoiding the duplication of coding efforts and waste of public resources, as well as decoupling the scientific and coding efforts.
\qsl can be found at \url{https://bitbucket.org/tassev/qsl_squasher/}.
\bibliographystyle{apj}
|
1,314,259,995,792 | arxiv | \section{Adding matter : QCD with an arbitrary flavor number}
Usual QCD dynamics is characterized by spontaneous symmetry breaking
and dynamical mass generation, with the associated scale $\Lambda_{\mathrm{QCD}}$.
However, when the number of flavors exceeds a critical
number, an infra-red fixed point (IRFP) appears
and prevents the coupling from growing large
enough to break chiral symmetry.
The theory is then scale invariant - even conformal invariant.
In the intermediate region, the coupling `walks'
rather than runs
between two typical scales - this is the phenomena of scale separation for
which our results provide a preliminary evidence.
From a general field theory viewpoint, the analysis of the phase diagram
of strong interactions as a function of the number of flavor
adds to our knowledge of the theoretical basis of strong interactions and
their fundamental mechanisms. From a phenomenological viewpoint,
this study deals with a class of models which might play a relevant role
in model building beyond the standard model (BSM)
\cite{SCGT,Ph_review,Yamawaki:1985zg,Holdom:1984sk,Akiba:1985rr,Appelquist:1986an},
which explain the origin of mass using strong coupling mechanisms
realized in QCD with a large number of flavors.
All these topics are under active scrutiny
both theoretically and experimentally
\cite{Lat1,Lat2,Lat3,Lat4, Itou:2013ofa,
Appelquist:2011dp,Appelquist:Conformal,Appelquist:2009ka,Appelquist:2010xv,
Deuzeman:2008sc,Deuzeman:2009mh,Deuzeman:2012ee,Deuzeman:2012pv,Miura:2011mc,Miura:2012zqa,
Aoki:2014oha,Aoki:2013zsa,Aoki:2013xza,Aoki:2012eq,
Cheng:2014jba,Cheng:2013eu,Cheng:2011ic,
Hasenfratz:2011xn,Hasenfratz:2010fi,Hasenfratz:2009ea,
Fodor:2011tu,Fodor:2009wk,Ishikawa:2013tua,Ishikawa:2013wf,Iwasaki:2003de,
Finland:MWT,Svetitsky:sextet,Kogut:2010cz,Fodor:2009ar,Fodor:2012ty,
Catterall:MWT,Lucini:MWT,
DelDebbio:2010jy,DelDebbio:2010ze,deForcrand,
BraunGies,EKMJ,Alho:2012mh,Gursoy:2010fj,Alho:2013dka,Liao:2012tw,
Ryttov:2012nt,Ryttov:2007cx,Dietrich:2006cm,Antipin:2012sm,
Matsuzaki:2013eva,Matsuzaki:2012xx}
\subsection{Conformality}
Conformal invariance is anticipated to emerge
in the non-Abelian gauge theory with many species (flavors) of fermions
\cite{Caswell:1974gg,Banks:1981nn,Appelquist,Miransky:1997,Appelquist:1999hr}.
This is due to the IRFP
for $N_f > N_f^*$ at a coupling which is not
strong enough to break chiral symmetry:
a second zero of the two-loop
beta-function of a non-Abelian gauge theory
implies, at least perturbatively,
the appearance of IRFP conformal symmetry
\cite{Caswell:1974gg,Banks:1981nn}.
In color SU($3$) gauge
theory with $N_f$ massless fundamental fermions,
the second zero appears at $N_f\gtrsim 8.05$,
before the loss of asymptotic freedom (LAF) at
$N_f^{\mathrm{LAF}}=16.5$.
Analytic studies of the conformal transition of strong interactions
have produced a variety of predictions
for the conformal threshold:
the Schwinger-Dyson approach with
rainbow resummations
\cite{Appelquist,Miransky:1997,Appelquist:1999hr}
or the functional renormalization group method
\cite{BraunGies}
suggest the onset of conformal window around $N_f^* \sim 12$.
An all-order perturbative beta-function
\cite{Ryttov:2007cx}
inspired by the Novikov–Shifman–Vainshtein–Zakharov
beta-function of SQCD \cite{Novikov:1983uc}
leads to a bound $N_f^* > 8.25$.
Instanton studies at large $N_f$ \cite{Velkovsky:1997fe}
claimed a qualitative change of behaviour at $N_f=6$.
The $N_f^{*}$ has also been estimated
for different fermion representations \cite{Dietrich:2006cm}.
Holographic models for QCD in the Veneziano limit
find $3.7 < (N_f/N_c)^* < 4.2$ \cite{EKMJ}.
\subsection{Pre-conformality}
The direct inspection of theories at fixed $N_f$ is often inconclusive,
especially close to the expected threshold $N_f^*$.
An alternative approach to establish the existence of the
conformal window is to (try to)
observe directly the approach to conformality by monitoring
the evolution of the results obtained in the broken phase
as a function of $N_f$.
Moreover, the pre-conformal dynamics at flavor numbers just before
the onset of conformal invariance might serve as a paradigm
for the BSM model buildings that invokes non-perturbative
mechanisms of electroweak symmetry breaking
\cite{Yamawaki:1985zg,Holdom:1984sk,Akiba:1985rr,Appelquist:1986an}.
In such pre-conformal region,
the coupling should vary little --
should {\em walk} -- with the scale,
at variance with the familiar running of QCD.
One important question, of genuinely theoretical nature,
is to establish the existence and uncover the
properties of this new class of strongly interacting (quasi) conformal theories.
Because of this,
the sub--critical region,
when $N_f$ gets closer and closer to $N_f^*$,
is interesting per se.~\cite{Ph_review}.
In our study, such pre-conformal dynamics could manifest itself
either with a clear observation of a separation of scales,
or with a manifestation of a critical behaviour when approaching
$N_f^*$. One possibility is to observe the Miransky-Yamawaki essential
singularity~\cite{Miransky:1997}.
Alternatively, in an FRG approach~\cite{BraunGies},
the pseudo-critical line is almost linear with $N_f$
for small $N_f$, and displays a singular behaviour when
approaching $N_f^*$, which could be the only observable
effects, beyond Miransky scaling.
A {\em jumping} scenario in
which the change from a QCD dynamics to
the conformal window is abrupt is also a distinct possibility
\cite{Antipin:2012sm}.
\subsection{The high temperature path to conformality}
Chiral symmetry is restored at
high temperatures -- in the so-called quark-gluon plasma (QGP)
phase. Both physics intuition and phenomenological analysis
based on functional renormalization group~\cite{BraunGies}
and finite temperature holographic QCD \cite{Alho:2012mh}
indicate that the conformal phase of cold, many flavor QCD and
the high temperature chirally symmetric phase are continuously connected.
In particular, the onset of the conformal window coincides with
the vanishing of the transition temperature, and the conformal
window appears as a zero temperature limit of
a possibly strongly interacting QGP.
The analysis of the finite temperature phase transition
is a well-established
line of research within the lattice community. In our approach
we build on this experience and use the properties of a thermal
system to learn about general aspects of the phase diagram also
at zero temperature. According to the Pisarski-Wilczek
scenario~\cite{Pisarski:1983ms}, the most likely possibility for $N_f \ge 3$
is a first order chiral transition in the chiral limit, turning into a
crossover above a critical mass endpoint, and/or on lattices which are not
large enough. However it should be noted that closer to the conformal
window the dynamics of the light scalar mode might invalidate this simple
picture, and the nature of the thermal transitions poses specific issues.
We will identify the thermal crossover with confidence for a number of flavors
ranging from four to eight,
and we will complement these results with those
of the deconfinement transition in the quenched model.
Then, we study the approach to the conformal phase
in the light of the chiral phase transition
at finite temperature with variable number of flavors.
Further, we will argue that
even results in the bare lattice parameters
can be used directly to locate
the critical number of flavors,
thus generalising to finite temperature
the Miransky-Yamawaki phase diagram, Ref. \cite{Miransky:1997}.
\subsection{Setting the scale}
One ubiquitous problem in these studies is the setting of
a common scale among
theories which are essentially different.
We propose two alternative
possibilities to handle this problem,
one stemming from our own
work \cite{Miura:2011mc,Miura:2012zqa}, and the other
from a recent analysis \cite{Liao:2012tw}.
Interestingly, this latter approach analyses the dependence of
the confinement parameters on the matter content, and proposes
microscopic mechanisms for confinement
motivated by such $N_f$ dependence.
\subsection{Sketching the phase diagram}
The phase diagram of QCD emerging from these discussions is sketched in
Fig.~\ref{fig:Combined_phase}:
the axis is simply the number of light flavors. Ordinary QCD
-- two light flavors -- is marked by an arrow.
The conformal region is on the right hand side,
and is separated by the essential singularity for a critical number of flavor
(of about eleven according to the current estimates) from the hadronic phase.
The possibility of a first order transition has been discussed as well, and
we will get back to this later in this paper.
Clearly, as in any system undergoing a phase
transition, the nature and extent of the critical window are purely
dynamical questions whose answer cannot be guessed a priori.
Since the underlying dynamics is completely non-perturbative, lattice
calculations are the only tool to perform an ab initio, rigorous
study of these phenomena, and many lattice studies
have recently appeared~\cite{Lat1, Lat2, Lat3, Lat4}.
\begin{figure}
\includegraphics[width=8 truecm]{Combined_phase.pdf}
\caption{\label{Phaseplot} A projected view of the phase diagram of
QCD-like theories in the temperature ($T$), flavor number ($N_f$) and bare
coupling ($g$) space. In the T-$N_f$ plane, the critical line is a phase
boundary between the chirally broken hadronic phase and the chirally
symmetric quark gluon plasma, the zero temperature end point of which is
the onset of the conformal window. The zero temperature projected plane is
inspired by the scenario in Refs.~\protect\cite{Appelquist:1999hr,
Miransky:1997}.}
\label{fig:Combined_phase}
\end{figure}
We now turn to the presentation of our results. The interested reader
can find all the details in our published papers
\cite{Deuzeman:2008sc,Deuzeman:2009mh,Deuzeman:2012ee,Miura:2011mc,Miura:2012zqa},
and we here omitted many (sometimes important) details for the sake of a more
concise presentation.
Section \ref{sec:spectrum} is devoted to the strategies
we have used to obtain an (indirect) evidence of conformality
from a direct evidence of chiral symmetry restoration.
Some comments on the mass anomalous dimension associated with our
measurements are included as well.
In Sec.~\ref{sec:Tc},
we investigate the chiral phase transition at finite temperature
for various numbers of flavor,
and evaluate the onset of the conformal window $N_f^*$
via the vanishing of the transition temperature at large $N_f$.
The results are further exploited in Sec.\ref{sec:QGP}
to highlight a connection between the conformal window and
a {\em cold} QGP.
In Sec.~\ref{sec:scale},
we discuss the issues of scale separation and possible direct evidence,
which are still work in progress.
In Sec.~\ref{sec:summary},
we will summerize our review.
\section{The quest for conformality}\label{sec:spectrum}
In this
Section,
we discuss the existence of a conformal phase in $SU(N_c=3)$ gauge
theories in four dimensions.
In this lattice study, we explore the model in the bare parameter space,
varying the lattice coupling and bare fermion mass.
The analysis of the chiral order parameter and
the mass spectrum of the theory indicates the restoration of chiral symmetry
at zero temperature and the presence of a Coulomb-like phase,
depicting a scenario compatible with the existence of an
IRFP at nonzero coupling.
Following the T=0 plane of Fig.~\ref{fig:Combined_phase},
at a given $N_f>N^*_f$ and
increasing the gauge coupling from $g=0$, one crosses the line of the
IRFPs, going from a chirally symmetric
and asymptotically free phase (pre-conformal phase, shaded in the picture)
to a symmetric, but not asymptotically free one
(Coulomb-like or QED-like phase). A phase transition need not be associated with
the line of IRFPs, differently from what was originally speculated in
Ref.~\cite{Banks:1981nn}. At even larger couplings, a transition to a
strongly coupled chirally asymmetric
phase will always occur in the lattice regularized theory.
The latter is referred to as a bulk phase transition.
In the symmetric phases at nonzero coupling the conformal
symmetry is still broken by ordinary perturbative contributions.
They generate the running of the coupling constant which is different on the two
sides of the symmetric phase. See Ref.~\cite{Miransky:1997} for a
detailed discussion of this point. We emphasize that in the region
considered in this paper the conformal symmetry would still be broken by
Coulombic forces.
One lattice strategy to assess conformality was then the following:
first, it was demonstrated that the location of the transition from the chirally
symmetric to the broken phase is not sensitive to the physical temperature
and is therefore compatible with a bulk nature. Subsequently,
the bare fermion mass dependence of the chiral condensate on
the weak coupling side of the bulk transition clearly favored
a chiral symmetry restoration.
Finally, the behavior of the mass spectrum close to the
bulk transition will be studied, again confirming chiral symmetry restoration
without making use of detailed fits. The mass dependence of the spectrum
allowed the extraction of a candidate mass anomalous dimension.
These results are consistent with the scenario for conformality of
Fig.~\ref{fig:Combined_phase}.
In the following, we limited ourselves to the presentation of the spectrum
results which are probably those providing a cleanest visual evidence,
and which have been updated and expanded very recently.
All our simulations use
staggered fermions (Kogut-Susskind)
in the fundamental representation in color $SU(N_c=3)$.
Here we used a tree level Symanzik improved gauge action to suppress lattice
artifacts, and staggered fermions with the Naik
improvement scheme, that effectively extends the Symanzik improvement to
the matter content.
\subsection{Spectrum}
It has been noted in the past that
one can devise robust signatures of chiral symmetry based
on the analysis of the spectrum results.
One first significant spectrum observable is the ratio $m_\pi/m_\rho$,
between the mass of the lightest pseudoscalar state (pion) $m_\pi$
and the mass of the lightest vector state (rho) $m_\rho$.
In real-life QCD at zero temperature, chiral symmetry is spontaneously broken
and the pion is the (pseudo)Goldstone boson of the broken symmetry,
implying that its mass will behave as $m_\pi\sim \sqrt{m}$.
In contrast,
chiral symmetry is restored in the continuum limit in the conformal window.
At the IRFP and at infinite volume, the quark mass dependence of all hadron masses
in the spectrum is governed by conformal symmetry:
at leading order in the quark mass expansion all masses follow
a power-law with common exponent determined by the anomalous dimension
of the fermion mass operator at the IRFP.
Hence we expect a constant ratio. Away from the IRFP,
for sufficiently light quarks and finite lattice volumes,
the universal power-law dependence receives corrections,
due to the fact that the theory is interacting but no longer conformal.
The behaviour of the ratio is demonstrated in Fig. \ref{fig:mpimrhoRatio}:
a conformal scenario seems favoured in the range of masses we are
exploring. Note that the $m_\pi/m_\rho$
ratio should go to zero in the chiral limit in the broken phase, and
to a constant value if chiral symmetry is restored.
Analogous conclusions can be drawn from the inspection of
so-called an Edinburgh plot Fig.~\ref{fig:Edplot}.
The difference with the case of ordinary QCD is indeed striking.
The modest scattering of the data points
could be ascribed to the deviation from a perfect power law as discussed
above. It would then be of interest to repeat the same plot
for different couplings : at the IRFP it should indeed reduce to a point.
\begin{figure}
\begin{minipage}[htb]{\linewidth}
\begin{center}
\includegraphics[width=10cm]{mpimrhoRatio-vs-am-All.pdf}
\caption{Ratio $m_\pi/m_\rho$ as a function of the bare quark mass
for all existing data for $N_f=12$,
and $N_f=16$: $N_f=12$ data from \protect\cite{Fodor:2011tu} (red squares),
our $N_f=12$ data and $\beta_L=3.8,3.9,4.0$ (blue circles),
$N_f=16$ data from \protect\cite{Damgaard:1997ut} (magenta diamonds).}
\label{fig:mpimrhoRatio}
\end{center}
\end{minipage}
\hspace{0.2cm}
\begin{minipage}[htb]{\linewidth}
\begin{center}
\includegraphics[width=10cm]{EdPlot.png}
\caption{Edinburgh plot: $N_f=12$ data from \protect\cite{Fodor:2011tu} (red squares),
our $N_f=12$ data and $\beta_L=3.8,3.9$ (blue circles),
$N_f=16$ data from \protect\cite{Damgaard:1997ut} (magenta diamonds).
The QCD physical point (black star, leftmost) and
the heavy quark limit (free theory) point (black star, rightmost) are shown.}
\label{fig:Edplot}
\end{center}
\end{minipage}
\end{figure}
\subsection{Anomalous Dimension}
To see whether the theory has the anomalous dimension for $N_f = 12$,
we plot the pseudoscalar mass as a function of the
chiral condensate\cite{Kocic:1992is}, as in Fig. \ref{fig:pion_pbp_plot}.
The data are best fitted by a
simple power-law form
$(a m_\pi)^2 = A (a^3 \langle\bar\psi\psi\rangle)^{2\delta_\chi}$,
with $\delta_\chi = 0.64(1)$.
They clearly suggest that chiral symmetry is
restored and that the theory has anomalous dimensions \cite{Kocic:1992is}.
For comparison, in the symmetric
phase and in mean field we expect a linear dependence
with non negative intercept. The presence of anomalous dimensions is
responsible for negative curvature - noticeably opposite to what finite
volume effects would induce - and a zero intercept. The same graph in the
broken phase would show the opposite curvature and extrapolate with a
negative intercept.
\begin{figure}
\includegraphics[width=10truecm]{Pion_mass_vs_pbp.pdf}
\caption{\label{fig:pion_pbp_plot} { The relation between the
chiral condensate and the pion mass, for $6/g_L^2$ = 3.9 (blue) and 4.0
(red). The line represents a power law fit to the combined data }}
\end{figure}
Further,
in Fig.~\ref{fig:meson_mass_plot} we report on the measured values of
$m_\pi$ and $m_\rho$ as a function of the bare fermion mass from
our early work \cite{Deuzeman:2009mh}. Here the
lightest point at $am=0.025$ for the vector mass is absent, but a curvature
can still be appreciated.
Simulations were done on $16^3\times 24$ volumes,
while a set of measurements at larger volumes showed that finite volume
effects were under control.
The mass dependence
shown in Fig.~\ref{fig:meson_mass_plot} hints again at a few properties of
a chirally symmetric phase. We have fitted both the pion and the
rho mass with a power law ansatz
\begin{equation}
m_{\pi,\rho} = A_{\pi,\rho} m ^{\epsilon_{\pi,\rho}}
\end{equation}
and obtained the results
$A_\pi = 3.41(21)$, $\epsilon_\pi = 0.61(2)$,
$A_\rho = 4.47(61)$, $\epsilon_\rho = 0.66(5)$ at $6/g_L^2 = 3.9$, and
$A_\pi = 3.41(21)$, $\epsilon_\pi = 0.61(2)$,
$A_\rho = 4.29(11)$, $\epsilon_\rho = 0.66(1)$ at $6/g_L^2 = 4.0$.
The accuracies of these fits
are not comparable with those achieved by the fits to the chiral
condensate,
however they allow to draw a few conclusions.
First, the mass dependence of the vector and
pseudoscalar mesons is well fitted by a power-law.
Second, it is also relevant
that the exponents are not unity and $\epsilon_\pi \neq 1/2$.
The latter result
immediately tells that the pion seen here is not a Goldstone boson of a
broken chiral symmetry. In addition, both mesons have masses scaling with
roughly the same power, as it should be in a symmetric phase, and with
increasing degeneracy towards the chiral limit. The exponent of the power
law being not one, confirms that we are not in the heavy quark
regime. From the results for the exponent $\epsilon$ we can formally extract
a value for the anomalous dimension consistent with the other lattice results
as well as the analytic estimates \cite{Itou:2013ofa}.
Needless to say, a full control on the systematic and on the corrections to
scaling is needed before making such identification with confidence.
\begin{figure}
\includegraphics[width=10truecm]{Spectral_fit_vs_mass_2.pdf}
\caption{\label{fig:meson_mass_plot} {
The relation between
the bare quark mass and the masses of the pion (red) and rho meson (blue),
for $6/g_L^2$ = 3.6, 3.7, 3.8, 3.9 and 4.0 from the uppermost line
down. Power law fits to the separate values of beta are provided.}}
\end{figure}
\subsection{Inside the conformal phase : lattice matters at strong coupling!}
We have previously discussed
a strong coupling zero temperature transition -- a bulk transition --
within the conformal window.
However,
if we were to use a perfect action, the conformal phase discussed
above would extend all the way till the infinite coupling limit.
The role of improvement in this case
is really dramatic! A perfect action would destroy a phase transition.
No surprise, of course:
these are strong coupling phenomena taking place away
from the continuum limit, hence extra terms in the actions which are
irrelevant in the continuum might well become relevant.
But then, how would an ordinary improved action (as opposed
to a perfect action) affect the phase transition? The
evidence we have so far is in this case \cite{Deuzeman:2012ee}
the bulk transition moves
towards stronger coupling (consistently with the fact that
it will eventually disappear with a perfect action),
and a second transition develops. Among these two transitions
we have a phase with an unusual realization of chiral symmetry,
observed also in other studies \cite{Cheng:2011ic}.
From the perspective of the analysis of continuum many flavor QCD,
these observations are just due to a peculiar form of lattice artifacts.
Bulk transitions are however interesting for several
reasons including fundamental questions in the quantum field theory,
for example the existence of an non--trivial UV
fixed point in four dimension away from the perturbative domain
as well as modeling of condensed matter systems, such as graphene,
and the new phases discussed here might well be of interest in
these contexts.
\section{Near-conformal : continuum and lattice}\label{sec:Tc}
In this Section we discuss results for $N_f=0,~4,~6,~8$,
approaching the conformal window from below.
In this case the results have been obtained with
a fixed bare quark mass, and no attempt has been done to
extrapolate to the chiral limit.
In order to monitor the behaviour of these theories we had
to choose an observable, and we set for the (pseudo)critical
temperature. For each $N_f$
results are given for several values of $N_t$: this
is necessary
in order to control the approach to the continuum limit,
as we will show below.
\begin{table*}
\caption{
Summary of the (pseudo) critical lattice couplings $\beta_{\mathrm{L}}^{~\mathrm{c}}$
for the theories with $N_f=0,~4,~6,~8$, $am=0.02$
and varying $N_t=4,~6,~8,~12$
\protect\cite{Deuzeman:2008sc,Miura:2012zqa}.
}\label{Tab:bc}
\begin{center}
\begin{tabular}{c|cccc}
\hline\hline
$N_f\backslash N_t$ &
$4$&
$6$&
$8$&
$12$\\
\hline
$0$ &
$7.35\pm 0.05$&
$7.97\pm 0.07$&
$8.26\pm 0.06$&
$-$\\
$4$ &
$5.65\pm 0.05$&
$6.00\pm 0.05$&
$6.15\pm 0.15$&
$-$\\
$6$ &
$4.675\pm 0.05$&
$5.025\pm 0.05$&
$5.20\pm 0.05$&
$5.55\pm 0.1$\\
$8$ &
$-$&
$4.1125\pm 0.0125$&
$4.275\pm 0.05$&
$4.34\pm 0.04$\\
\hline\hline
\end{tabular}
\end{center}
\end{table*}
We have used a common bare fermion mass $ma = 0.02$
for all simulations at finite $N_f$.
Introducing a bare fermion mass,
any first order phase transition
will eventually turn into a crossover
for masses larger than some critical mass,
and any second order transition will immediately become
a crossover.
Since the chiral condensate looks smooth in our results,
we use the terminology of ``chiral crossover'' in the following.
In Table \ref{Tab:bc} we summarize
the (pseudo)critical lattice couplings $\beta_{\mathrm{L}}^{~\mathrm{c}}$ as a function
of $N_f$ and $N_t$
associated with the thermal crossover .
These are our raw data.
\subsection{IRFP from the lattice results}
Let us plot the lattice critical couplings
$g_{\mathrm{L}}^{~\mathrm{c}}(N_f,N_t) = \sqrt{10/\beta_{\mathrm{L}}^{~\mathrm{c}}(N_f,N_t)}$
(Table \ref{Tab:bc})
in the space spanned by the bare coupling $g_{\mathrm{L}}$
and the number of flavor $N_f$, and
consider the lines which connect
$g_{\mathrm{L}}^{~\mathrm{c}}$ with $N_t$ fixed: $g_{\mathrm{L}}^{~\mathrm{c}}(N_f)|_{N_t=\mathrm{fix}}$
(see Fig.~\ref{Fig:MY}).
These pseudo-critical thermal lines
separate a phase where chiral symmetry
(approximately) holds
from a phase where chiral symmetry is spontaneously broken
\footnote{
It would be of interest to study the interrelation of such lines
with the zero temperature first order transition line
observed in the conformal window
\cite{Deuzeman:2012ee,Deuzeman:2009mh,
Cheng:2011ic,Hasenfratz:2010fi,Hasenfratz:2011xn,Damgaard:1997ut,deForcrand}.}.
The resultant phase diagram may be seen
as an extension of the well-known
Miransky-Yamawaki phase diagram \cite{Miransky:1997}
to finite temperature.
We here argue that
the critical number of flavor $N_f^*$ can be read off
from the crossing point of thermal lines obtained for different $N_t$.
To see this,
we consider the well-known step-scaling function:
\begin{equation}
\Delta\beta_{\mathrm{L}}^s = \beta_{\mathrm{L}} - {\beta_{\mathrm{L}}}^{\prime}
\end{equation}
where $\beta_{\mathrm{L}}$ and $\beta_{\mathrm{L}}^{\prime}$
give the same physical scale $\xi$:
\begin{align}
\xi = a(\beta_{\mathrm{L}})\hat{\xi} = a({\beta_{\mathrm{L}}}^{\prime})\hat{\xi}^{\prime}
\ .\label{eq:unique_xi}
\end{align}
Here, $\hat{\xi}$ is the dimension-less
lattice correlation length, and
$\hat{\xi}/ \hat {\xi^{\prime}} = s$.
In our case, $\xi = T_c^{-1}$, $\hat{\xi} = N_t,
\hat{\xi}^{\prime} = N_t^{\prime}$,
and the above relation Eq.~(\ref{eq:unique_xi}) reads
\begin{align}
T_c^{-1} = N_t\ a(\beta_{\mathrm{L}}^{~\mathrm{c}}) = N_t^{\prime}\ a({\beta_{\mathrm{L}}^{~\mathrm{c}}}^{\prime})
\ .\label{eq:unique_Tc}
\end{align}
As discussed in the previous study Ref.~\cite{Hasenfratz:2011xn},
$\Delta \beta_{\mathrm{L}}^s = 0$ holds at the IRFP
regardless the scale factor $s$.
In principle,
we could then compute the step-scaling function from
our numerical results,
and try to see where it vanishes.
Alternatively,
we can look for the intersection
of pseudo-critical thermal lines:
obviously, $\Delta \beta_{\mathrm{L}}^s = 0$ holds at the intersection point
regardless the value of the scale factor $s$.
To demonstrate this procedure,
we consider
the pseudo-critical lines obtained for $N_t = 6$ and $N_t=12$
as shown in Fig.~\ref{Fig:MY}.
Note their positive slope:
the lattice critical coupling $g_{\mathrm{L}}^{~\mathrm{c}}$ is
an increasing function of $N_f$.
This is a consequence
of enhanced fermionic screening for a large number
of flavor, as noted first in Ref.~\cite{Kogut:1985pp}.
Interestingly, the slope decreases with increasing $N_t$,
which allows for a crossing point at a larger $N_f$.
Thus, we estimate the intersection at
$(g_{\mathrm{L}}^{~\mathrm{c}}, N_f^*) = (1.79\pm 0.12, 11.1\pm 1.6)$.
\begin{figure}
\includegraphics[width=10cm]{./MY_fit_select.pdf}
\caption{(Pseudo) critical values of the lattice coupling
$g_{\mathrm{L}}^{~\mathrm{c}}=\sqrt{10/\beta_{\mathrm{L}}^{~\mathrm{c}}}$ for theories with $N_f=0,~4,~6,~8$
and for several values of $N_t$
in the Miransky-Yamawaki phase diagram.
We have picked up $g_{\mathrm{L}}^{~\mathrm{c}}$ at $N_f = 6$ and $8$,
and considered ``constant $N_t$'' lines
with $N_t = 6,\ 12$.
{If the system is still described by
one parameter beta-function in this range of coupling,
the IRFP could be located at the intersection of the
fixed $N_t$ lines -- or equivalently, in the region where
the step-scaling function vanishes. To demonstrate the procedure
--as a preliminary example --
we have considered the intersection
of the $N_t = 12$ and $N_f = 6$ lines}.}
\label{Fig:MY}
\end{figure}
\subsection{Towards the continuum limit: estimating again the
conformal threshold}
Let us now fix $N_f$ and consider
the pseudo-critical temperatures $T_c$ in physical units:
\begin{align}
&T_c\equiv \frac{1}{a(\beta_{\mathrm{L}}^{~\mathrm{c}})\cdot N_t}\ .\label{eq:Tc}
\end{align}
We introduce the normalised critical temperature
$T_c/\Lambda_{\mathrm{L/E}}$
(see e.g. \cite{Gupta:2000hr})
where $\Lambda_{\mathrm{L}}$ ($\Lambda_{\mathrm{E}}$)
represents the lattice (E-scheme) Lambda-parameter
defined in the two-loop perturbation theory
with or without a renormalisation group inspired
improvement~\cite{CAllton}.
We consider the two-loop beta-function
\begin{align}
&\beta(g)
=-(b_0 {g}^3 + b_1 {g}^5)\ ,\label{eq:beta_func}\\
&b_0
=
\frac{1}{(4\pi)^2}
\Biggl(
\frac{11C_2[G]}{3}-\frac{4T[F]N_f}{3}
\Biggr)\ ,\label{eq:b0}\\
&b_1
=
\frac{1}{(4\pi)^4}
\Biggl(
\frac{34(C_2[G])^2}{3}
-\biggl(\frac{20C_2[G]}{3}+4C_2[F]\biggr)T[F]N_f
\Biggr)\ ,\label{eq:b1}
\end{align}
with $(C_2[G],\,C_2[F],\,T[F])=(N_c,\,(N_c^2-1)/(2N_c),\, 1/2)$.
The coupling $g$ can be either
the lattice bare coupling
$g_{\mathrm{L}} = \sqrt{10/\beta_{\mathrm{L}}}$ or the E-scheme renormalised coupling
$g_{\mathrm{E}} = \sqrt{3(1-\langle P \rangle(g_{\mathrm{L}}))}$,
where $\langle P\rangle(g_{\mathrm{L}})$ is the zero temperature
plaquette value.
If the one-loop perturbation theory exactly holds,
the E-scheme coincides the lattice scheme.
Integrating Eq.~(\ref{eq:beta_func}),
we obtain the well-known two-loop asymptotic scaling relation,
\begin{align}
R(g_{\mathrm{L/E}})\equiv
a(g_{\mathrm{L/E}})\Lambda_{\mathrm{L/E}}
= \bigl(b_0g_{\mathrm{L/E}}^2\bigr)^{-b_1/(2b_0^2)}
\exp\biggl[
\frac{-1}{2b_0g_{\mathrm{L/E}}}
\biggr]
\ ,\label{eq:RL}
\end{align}
where $\Lambda_{\mathrm{L}}$ ($\Lambda_{\mathrm{E}}$)
is the Lattice (E-scheme) Lambda-parameter.
To take into account higher order corrections,
we have also considered the renormalisation group inspired
improvement~\cite{CAllton}
\begin{align}
R^{\mathrm{imp}}(\beta_{\mathrm{L/E}})=
\Lambda_{\mathrm{L/E}}^{\mathrm{imp}}~a(\beta_{\mathrm{L/E}})
\equiv
\frac{R(\beta_{\mathrm{L/E}})}{1+h}
\times
\Biggl[
1 + h\
\frac{R^2(\beta_{\mathrm{L/E}})}{R^2(\beta_0)}
\Biggr]\ ,
\label{eq:RL_imp}
\end{align}
where $\beta_{\mathrm{L/E}} = 10/(g_{\mathrm{L/E}})^2$.
The coupling $\beta_0$ can be arbitrarily set
and the parameter $h$ is adjusted
so as to minimise the scaling violation.
Note that $h = 0$ reproduces the standard asymptotic scaling law
Eq.~(\ref{eq:RL}).
We now substitute $\beta_{\mathrm{L/E}}^{~\mathrm{c}}$
into the temperature definition Eq.~(\ref{eq:Tc}),
and insert the scale $\Lambda_{\mathrm{L/E}}$:
\begin{align}
\frac{1}{N_t}=\frac{T_c}{\Lambda_{\mathrm{L/E}}}
\times \Bigl(\Lambda_{\mathrm{L/E}}~a(\beta_{\mathrm{L/E}}^{~\mathrm{c}})\Bigr)
\ .\label{eq:T_Lam}
\end{align}
Eq.~(\ref{eq:T_Lam}) allows us
to define the (normalised) critical temperature
$T_c/\Lambda_{\mathrm{L/E}}$.
When we adopt the improvement Eq.~(\ref{eq:RL_imp}),
$T_c/\Lambda_{\mathrm{L/E}}$ is upgraded into
$T_c/\Lambda_{\mathrm{L/E}}^{\mathrm{imp}}$.
\begin{align}
\frac{T_c}{\Lambda_{\mathrm{L/E}}}
=\frac{R(g_{\mathrm{L/E}})}{N_t}
= \bigl(b_0g_{\mathrm{L/E}}^2\bigr)^{-b_1/(2b_0^2)}
\exp\biggl[
\frac{-1}{2b_0}
\biggr]
\ ,\label{eq:TcL}
\end{align}
where $g_{\mathrm{L/E}}$ denotes either the bare lattice coupling
or the coupling defined in the E scheme.
In addition, we consider the renormalisation group inspired
definition,
\begin{align}
\frac{T_c}{\Lambda_{\mathrm{L/E}}^{\mathrm{imp}}}
= \frac{R^{\mathrm{imp}}(g_{\mathrm{L/E}})}{N_t}
\ ,\label{eq:TcL_imp}
\end{align}
where $R^{\mathrm{imp}}$ is given by Eq.~(\ref{eq:RL_imp}).
The numerical results for
$T_c/\Lambda_{\mathrm{L/E}}$ and $T_c/\Lambda_{\mathrm{L/E}}^{\mathrm{imp}}$
are collected in
Table \ref{tab:TcL} and Table \ref{tab:TcLE}.
\begin{table*}
\caption{
Summary of
$T_c/\Lambda_\mathrm{L}$ and
$T_c/\Lambda_{\mathrm{L}}^{\mathrm{imp}}$ for various $(N_f,N_t)$.
The first (second) line at fixed $(N_f,N_t)$
shows the value of $T_c/\Lambda_\mathrm{L}$ ($T_c/\Lambda_{\mathrm{L}}^{\mathrm{imp}}$),
and the last two columns provide
the parameter $h$ and $\beta_0$ appeared
in the improved asymptotic scaling
Eq.~(\protect\ref{eq:RL_imp}).
}\label{tab:TcL}
\begin{center}
\begin{tabular}{c|cccc|cc}
\hline\hline
$N_f\backslash N_t$ &
$4$&
$6$&
$8$&
$12$&
$h$&
$\beta_0$\\
\hline
$0$ &
$18.11\pm 0.65$&
$18.21\pm 0.91$&
$16.56\pm 0.71$&
$-$&
$-$&
$-$\\
\quad &
$16.29\pm 0.75$&
$17.81\pm 1.02$&
$16.56\pm 0.78$&
$-$&
$0.05$&
$8.26$\\
\hline
$4$&
$21.99\pm 1.04$&
$19.98\pm 0.95$&
$17.12\pm 2.43$&
$-$&
$-$&
$-$\\
\quad &
$16.56\pm 1.44$&
$18.67\pm 1.38$&
$17.12\pm 3.41$&
$-$&
$0.30$&
$6.15$\\
\hline
$6$ &
$25.41\pm 1.43$&
$25.33\pm 1.43$&
$22.94\pm 1.29$&
$22.30\pm 2.52$&
$-$&
$-$\\
\quad &
$21.66\pm 1.64$&
$23.87\pm 1.58$&
$22.21\pm 1.40$&
$22.30\pm 2.66$&
$0.03$&
$5.55$\\
\hline
$8$ &
$-$&
$50.05\pm 0.87$&
$47.06\pm 3.28$&
$34.34\pm 1.91$&
$-$&
$-$\\
\quad &
$-$&
$34.32\pm 1.40$&
$42.67\pm 6.33$&
$34.34\pm 3.90$&
$1.08$&
$4.34$\\
\hline\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\caption{
Summary of $T_c/\Lambda_\mathrm{E}$
and $T_c/\Lambda_{\mathrm{L/E}}^{\mathrm{imp}}$ for $N_f = 6$ and $N_f = 8$.
The first (second) line at fixed $(N_f,N_t)$
shows the value of $T_c/\Lambda_\mathrm{E}$ ($T_c/\Lambda_{\mathrm{L/E}}^{\mathrm{imp}}$),
and the last two columns give
the parameter $h$ and $\beta_0$ appeared
in the improved asymptotic scaling Eq.~(\protect\ref{eq:RL_imp}).
For $N_f = 6$, the improvement was not necessary.
}\label{tab:TcLE}
\begin{center}
\begin{tabular}{c|cccc|cc}
\hline\hline
$N_f\backslash N_t$ &
$4$&
$6$&
$8$&
$12$&
$h$&
$\beta_0$\\
\hline
$6$ &
$74.22\pm 5.86$&
$75.47\pm 8.17$&
$74.56\pm 9.08$&
$75.13\pm 10.76$&
$-$&
$-$\\
\hline
$8$ &
$-$&
$422.54\pm 23.06$&
$422.61\pm 38.59$&
$316.03\pm 20.06$&
$-$&
$-$\\
\quad &
$-$&
$312.16\pm 33.13$&
$393.58\pm 60.01$&
$316.03\pm 31.52$&
$0.40$&
$4.34$\\
\hline\hline
\end{tabular}
\end{center}
\end{table*}
Let us know consider the results at fixed $N_f$:
for each $N_f$ , the
ratio in either Table approaches a constant by increasing $N_t$,
enabling us (with the due caveats) to interpret these asymptotic
values as continuum estimates.
Let us then take the values corresponding to the largest $N_t$
and consider their $N_f$ dependence : $T_c/\Lambda$ apparently
increases with $N_f$! How this can be reconciled with a vanishing
$T_c$ in the chiral limit? This is discussed below, and again in the
last Section.
\subsection{The critical number of flavor
and the vanishing critical temperature}
\label{subsec:TcM}
The apparent puzzle above immediately suggests that $\Lambda$
vanishes faster than $T_c$ when approaching $N_f$, i.e.
has a strong sensitivity to the IR dynamics affected by the conformal
threshold.
To observe the vanishing of $T_c$ we then need to properly define
a UV reference scale. Here we will review our first attempt
to do so which relies heavily on perturbation theory, while
in the last Section we will describe our ongoing work
on this subject.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{./u0_beta_all_II.pdf}
\caption{
The $\beta_{\mathrm{L}}$ dependence of the tadpole factor $u_0$
at zero temperature ($12^4$ lattice volume).
{At each $N_f$,
the dashed line represents the fit for data
with the ansatz $u_0 = 1 - A/(1 + B\cdot\beta_{\mathrm{L}}^2)$.}
We consider a constant $u_0$ ({\em e.g.} $u_0=0.8$ in figure),
and read off the corresponding lattice bare couplings $\beta_{\mathrm{L}}$,
which are used to define the scale $M$ at each
theory with $N_f$ flavors.}
\label{Fig:u0_const}
\end{center}
\end{figure}
Before going to details, we first explain the basic idea
which follows the FRG analysis by Braun and Gies~\cite{BraunGies}.
They used the $\tau$ lepton mass $m_{\tau} = 1.777$ (GeV)
as an $N_f$ independent UV reference scale for
theories with any number of flavors.
The initial condition of the renormalisation flow
has been specified via the strong coupling constant
in an $N_f$ independent way:
\begin{align}
\alpha_s(\mu = m_{\tau}) = 0.322\quad
\text{for}\quad {}^{\forall}N_f\ .\label{eq:ini_FRG}
\end{align}
Starting from the common initial condition Eq.~(\ref{eq:ini_FRG}),
the $N_f$ dependence of the critical temperature $T_c(N_f)$
emerges from
the
$N_f$ dependent
renormalisation flow
at the chiral phase transition scale
$\mu\sim\Lambda_{\mathrm{QCD}}\ll m_{\tau}$.
The $N_f$ dependence of $T_c$
as well as its novel non-analytic behaviour
in the pre-conformal region
becomes free from the choice of
the reference scale~\cite{BraunGies}
by using an $N_f$ independent UV reference scale
much larger than $T_c$.
In order to determine the reference coupling $g_{\mathrm{L}}^{~\mathrm{ref}}$
we utilise our plaquette results $\langle P \rangle$ (equivalently,
the tadpole factor $u_0 = \langle P \rangle^{1/4}$)
shown in Fig.~\ref{Fig:u0_const}.
Let us consider a constant $u_0$,
for instance $u_0 = 0.8$ in figure,
and read off the corresponding bare lattice couplings at each $N_f$.
The obtained $g_{\mathrm{L}}(N_f)$
is used as a reference coupling $g_{\mathrm{L}}^{~\mathrm{ref}}$
and the corresponding mass scale $M(g_{\mathrm{L}}^{~\mathrm{ref}})$
is again computed according to two loop scaling.
Some remarks on the aforementioned scale setting are in order:
First, we recall the scale setting procedure in the potential scheme,
where the measured normalised force $r^2F(r)$
is proportional to the renormalised coupling $\bar{g}$,
and the specification $\bar{g}^2\propto r_X^2F(r_X) = {}^\exists X$
sets a scale $r_X^{-1}$.
In short, we use our $u_0$ (or equivalently plaquettes)
to define $\bar{g}$,
and $u_0 = X$ is regarded as the analog of
the potential scheme scale setting.
Second,
in the leading order of the perturbative expansion,
the renormalised coupling is $N_f$ independent, and
proportional to the Wilson loop~\cite{Wong:2005jx} --
a property that we have already exploited in
the E-scheme calculation.
Hence the use of an $N_f$ independent $u_0$
approximately gives an $N_f$ independent scale setting,
similarly to
the FRG scale setting method Eq.~(\ref{eq:ini_FRG}).
And third, such an $N_f$ independent scale setting
can be performed in a sufficiently UV regime
$T_c(N_f) \ll M(g_{\mathrm{L}}^{~\mathrm{ref}})$
by adjusting the value of $u_0$ to satisfy
the condition $g_{\mathrm{L}}^{~\mathrm{ref}} \ll g_{\mathrm{T}}^{~\mathrm{c}}({}^{\forall}N_f)$.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{./Tc_M_Nf_2_1_II.pdf}
\caption{
Left:~
The $N_f$ dependence of $T_c/M$ where
$M$ is determined to be a UV scale corresponding to
$u_0=0.79$ (red box),
$0.80$ (blue $\bigcirc$), and
$0.81$ (magenta triangle)
at each theory with $N_f$.
The dashed lines represent fits for data
by assuming the expression Eq.~(\protect\ref{eq:BG_scaling})}
\label{Fig:TcM}
\end{center}
\end{figure}
Note that the
coupling at the lattice cutoff $a^{-1}(g_{\mathrm{L}}^{~\mathrm{c}})$
is $N_t\gg 1$ times larger than $T_c$.
Then, the scale hierarchy $T_c(N_f) \ll a^{-1}(g_{\mathrm{L}}^{~\mathrm{c}}(N_f))$
allows us to consider a reference scale
much larger than critical temperature
but smaller than the lattice cutoff
$T_c(N_f) \ll M(g_{\mathrm{L}}^{~\mathrm{ref}}) < a^{-1}(g_{\mathrm{L}}^{~\mathrm{c}}(N_f))$.
We find that $u_0\sim 0.8$ meets this requirement.
In summary,
the use of $g_{\mathrm{L}}^{~\mathrm{ref}}$ given by $u_0\sim 0.8$
is analogous to
the FRG scale setting method Eq.~(\ref{eq:ini_FRG}),
and is suitable for studying the vanishing
of the critical temperature
by utilising $T_c/M(g_{\mathrm{L}}^{~\mathrm{ref}})$.
Fig.~\ref{Fig:TcM}
displays the $N_f$ dependence of $T_c/M(g_{\mathrm{L}}^{~\mathrm{ref}})$
for $u_0 = 0.79$, $0.80$, and $0.81$.
Fitting the data points for $T_c/M(g_{\mathrm{L}}^{~\mathrm{ref}})$ at $N_f \geq 4$
by using the FRG motivated ansatz,
\begin{align}
T_c= K|N_f^* - N_f|^{(-2b_0^2/b_1)(N_f^*)}
\ ,\label{eq:BG_scaling}
\end{align}
where $b_{0,1}$ has been defined in Eqs.~(\ref{eq:b0}) and (\ref{eq:b1}),
the lower edge of the conformal window is estimated as:
$N_f^* = 10.48 \pm 1.01$ ($u_0 = 0.79$),
$N_f^* = 10.34 \pm 0.88$ ($u_0 = 0.80$),
$N_f^* = 10.23 \pm 0.80$ ($u_0 = 0.81$).
The error-bars involve both fit errors and
statistical errors of data.
We have further investigated the stability against
different choices of $u_0$:
$N_f^*$ is relatively stable
within the range $0.79\leq u_0\leq 0.94$.
The scale cannot be pushed further towards
the UV because of discritization errors.
On the other hand,
a small $u_0 \lesssim 0.7$ leads to $M(g_{\mathrm{L}}^{~\mathrm{ref}})\sim T_c$ or smaller.
In such a case,
the reference scale $M(g_{\mathrm{L}}^{~\mathrm{ref}})$
is affected by infra-red physics and cannot
be used to study the vanishing of $T_c$.
Despite these limitations,
the window of relative stability is however reasonably large,
and suffices to define
an average value for $N_f^*$.
We quote the average among
the three results obtained for $u_0=(0.79,0.80,0.81)$,
i.e. $N_f^* = 10.4 \pm 1.2$.
\section{Learning
about the Quark Gluon Plasma when studying the threshold
for conformality}\label{sec:QGP}
In this second subsection,
we will follow
the approach of a recent paper \cite{Liao:2012tw},
and compute the coupling $g_{\mathrm{T}}^{~\mathrm{c}} (N_f)$ at the scale of
the critical temperature for each $N_f$.
To obtain the coupling $g_{\mathrm{T}}^{~\mathrm{c}}$
at the scale of the temperature,
we evolve the coupling at the scale of the lattice spacing
$a$ up to the temperature inverse scale $N_t a$, still
making use of the two loop scaling, which, as we have seen,
is reasonably well satisfied.
The red ($\Box$) symbol in
Fig.~\ref{Fig:gTC} shows $g_{\mathrm{T}}^{~\mathrm{c}}$ as a function of $N_f$.
We superimpose a fit obtained by using the ansatz proposed in
Ref. \cite{Liao:2012tw}
\begin{align}
{N_f(g_{\mathrm{T}}^{~\mathrm{c}}) = A\cdot \log~
\bigl[B\cdot(g_{\mathrm{T}}^{~\mathrm{c}}- g_{\mathrm{T}}^{~\mathrm{c}}|_{N_f=0}) + 1\bigr]\ .\label{eq:gTC_fit}}
\end{align}
with $A$ and $B$ fit parameters, which describes well the data.
Since the critical temperature is
zero in the conformal phase,
the thermal critical coupling $g_{\mathrm{T}}^{~\mathrm{c}}$
should equal a {\em zero temperature} critical coupling $g^c$
when $N_f = N_f^*$.
Of course, $g^c$ is not known exactly
and we have to rely on approximations.
The first estimate is based on
the best available value $g_{\mathrm{SD}}^{\mathrm{c}}$
obtained by using
the two-loop Schwinger-Dyson equation \cite{Appelquist}.
In this case,
the lower edge of the conformal window $N_f^*$
is defined by the condition $g_{\mathrm{T}}^{~\mathrm{c}}(N_f^*) = g_{\mathrm{SD}}^{\mathrm{c}}(N_f^*)$.
In Fig.~\ref{Fig:gTC} $g_{\mathrm{SD}}^{\mathrm{c}}$ is plotted as a blue solid line.
We then estimate the intersection of $g_{\mathrm{T}}^{~\mathrm{c}}$ and $g_{\mathrm{SD}}^{\mathrm{c}}$ --
hence the onset of the conformal window
as well as the IRFP coupling at $N_f^*$ --
at $(g^*,N_f^*) = (2.79,13.2)\pm (0.13,0.6)$.
One second possibility for estimating $N_f^*$ is the following:
the conformal phase would emerge
when the coupling at IRFP
($g^{\mathrm{IRFP}}$) is not strong enough
to break chiral symmetry, {\em i.e.}
$g^{\mathrm{IRFP}} \leq g_{\mathrm{T}}^{~\mathrm{c}}$.
Here, we utilise the four-loop result for $g_{\mathrm{4l}}^{\mathrm{IRFP}}$
\cite{Ryttov:2012nt} as the best available.
In Fig.~\ref{Fig:gTC}, we show
$g_{\mathrm{4l}}^{\mathrm{IRFP}}$ as magenta $\bigcirc$,
with superimposed a linear interpolation.
In the plot,
we use the results for $g_{\mathrm{4l}}^{\mathrm{IRFP}}$ in the $\bar{\text{MS}}$ scheme.
The errors are estimated by considering the scheme
dependence \cite{Ryttov:2012nt}, which turns out to be rather mild
at four loops.
We can then locate the intersection of $g_{\mathrm{T}}^{~\mathrm{c}}$ and $g_{\mathrm{4l}}^{\mathrm{IRFP}}$
and obtain $(g^*,N_f^*) = (2.51,11.8)\pm (0.15,0.9)$.
Ideally,
the three lines in Fig. \ref{Fig:gTC}
should meet at a (single) IRFP fixed point, if all the
quoted results -- including the analytic ones -- were exact.
Indeed
the intersections we have estimated are consistent
within the largish errors.
We then quote the average of the above two estimates
as our final result from this analysis, $N_f^*\sim 12.5\pm 1.6$.
In addition,
we note that $g_{\mathrm{T}}^{~\mathrm{c}}$ is an increasing function
of $N_f$. This indicates that the quark-gluon plasma
is more strongly coupled at larger $N_f$,
as discussed in Ref.~\cite{Liao:2012tw}.
In turn, this observation might provide a clue into the
nature of the strongly interactive quark gluon plasma.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{./gT_2_1.pdf}
\caption{
The thermal critical coupling (red $\Box$)
and the fit for them
(dashed red line, with the ansatz Eq.~(\protect\ref{eq:gTC_fit}))
and the values of the zero temperature couplings in the conformal
phase from different estimates, see text for details.
At the critical
number of flavor the thermal critical coupling
should equal the critical
coupling associated with the IRFP.
The procedure is motivated by a recent
study by Shuryak in Ref.~\protect\cite{Liao:2012tw}.}
\label{Fig:gTC}
\end{center}
\end{figure}
\section{Two scales?}\label{sec:scale}
Let us elaborate on the circumstance that
$T_c/\Lambda$ computed using different schemes
($\Lambda = \Lambda_{\mathrm{L}}$ or $\Lambda_{\mathrm{E}}$)
consistently shows an increase with $N_f$, as initially
noted in \cite{Miura:2011mc}.
As discussed in \cite{Miura:2011mc} this indicates that
$\Lambda_{\mathrm{L/E}}$ vanishes faster than $T_c$
upon approaching the critical number of flavor. Within the various
uncertainties discussed here, this can be taken as a qualitative
indication of scale separation close to the critical
number of flavors.
In the Section above,
we have estimated the onset of the conformal phase $N^∗_f$ via the vanishing
of $T_c (N_f)/M$. As a next step, it is preferable to define
$T_c (N_f)/M$ without recourse to perturbation theory.
To this end, we have adopted the string tension $\sigma$ as a reference scale $M$,
and investigated $T_c / \sqrt{\sigma}$ (Fig.~\ref{Fig:Tc_s}).
The $\sigma$ is evaluated from the Wilson loop
measured on zero temperature lattices, for the
same set of pseudocritical couplings we have identified in our thermal study.
The $T_c/\sqrt{\sigma}$
remains stable in the error, again suggesting that our results are a reasonable
approximation of the continuum ones.
$T_c/\sqrt{\sigma}$ = 0.373(2)(+5,-6) (0.369(4)(+1,-5)) for $N_f = 6(8)$,
and the decreasing trend becomes less apparent with increasing
$N_f$ , and $T_c /\sqrt{\sigma}$ does not seem to
intercept the $N_f$ axis before the asymptotic freedom is lost ($N_f = 16.5$).
This may not be surprising. We find at least two reasons for the
non-vanishing $T_c / \sqrt{\sigma}$ : First, $\sigma$ would not
be a ``UV'' quantity and may also be vanishing when
a conformal phase sets in. In other
words, our result indicates that the regulator of
$T_c$ has to be more UV than $\sigma$ to elucidate the
vanishing of chiral symmetry breaking via $T_c$ . From
this point of view, a quantity $T_c w_0$ where $w_0$ is a
UV scale \cite{Borsanyi:2012zs} defined by the Wilson flow \cite{Luscher:2009eq}
may be a candidate which we are currently evaluating. Suppose
$T_c w_0$ displayed the expected hints of singularity at our
estimated $N_f^* $: our estimate of the critical number of
flavor would be confirmed, and we will have a significant evidence
of a scale separation - the two different scales bing $\sigma$ and $w_0$.
Again however one might argue
that a finite bare fermion mass breaks the conformality,
and both $T_c$ and $\sigma$
could be defined and finite even in the region $N_f \geq N^∗_f$.
Thus bare fermion mass
effects to $T_c /\sqrt\sigma$ should be further studied in future.
As indicated in Ref. \cite{Gursoy:2010fj},
the ratio $T_c /\sqrt{\sigma}$ is one of the input parameters to set a scale in
models based on the gauge/gravity duality at finite T.
Such inputs for the (would-be) walking
regime $N_f = 6$ and $N_f = 8$
are now available by the present study.
A final caveat concerns the occurrence of a small oscillatory behavior
in the effective mass of the Wilson loop
with smearing for $N_f=8$. It remains to be seen how these oscillations
relate to the bulk transition observed in the conformal window:
for instance these observations might confirm the original scenario in which
the bulk transition would still manifest itself in the QCD phase,
as a (pseudo)singularity unrelated with the chiral transition.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Tc_s.pdf}
\caption{
The $T_c/\sqrt{\sigma}$ as a function of $N_f$.
The symbol $\Box$ (red) represents the present results ($N_f = 6,8$).
For a comparison, we have quoted the $T_c/\sqrt{\sigma}$ from
\protect\cite{Laermann:1996xn} ($N_f = 0$),
\protect\cite{Karsch:2000kv} ($N_f = 2,3$),
\protect\cite{Engels:1996ag} ($N_f = 4$),
shown as $\bigcirc$ (blue) symbols.
}\label{Fig:Tc_s}
\end{center}
\end{figure}
\section{Summary}\label{sec:summary}
We have presented an overview of some of our results on the phases of QCD
at large number of flavor $N_f$, with some emphases on the scales of the theory, and on
the scale setting procedure: oversimplifying,
QCD generates one dynamically relevant scale for
a small number of flavors, becomes a multi-scale theory when approaching
the conformal window, and then looses its infra red scale.
We need consistent procedures of scale setting in order
to properly appreciate these phenomena. We are confident that we have taken
at least some steps towards this goal and
we hope that the strategies we have developed will help further sharpen
some of the still semi-quantitative estimates presented here.
For $N_f=12$, our measurements of the order parameter
and of the spectrum to our results provide evidence
towards the existence of a symmetric,
Coulomb-like phase on the weak coupling side of the lattice bulk transition.
In the scenario of Refs.~\cite{Appelquist,Miransky:1997},
such a Coulomb-like
region must be entangled to the presence of a conformal IRFP
for the theory with twelve flavors at a continuum limit.
We have then analyzed the spectrum results as a function of mass, and found
them to be well described by power-law fits with a mass anomalous dimension
consistent with other lattice results and
as well as the analytic estimates \cite{Itou:2013ofa}.
On the QCD side ($N_f < N_f^*$), we have investigated
the chiral phase transition/crossover
with $N_f = 0$ (quenched), $4$, $6$, and $8$.
We have discussed the possible implication for the
(pre-)conformal dynamics at large $N_f$,
and estimated, in a few independent ways,
the number of flavor $N_f^*$:
We have estimated the $N_f^*$ from the vanishing thermal scaling
by extrapolating our critical couplings $g_{\mathrm{L}}^{~\mathrm{c}}$ to
larger $N_f$ . This gives $N_f^*\sim 11.1\pm 1.6$.
We have extracted
a typical interaction strength $g_{\mathrm{T}}^{~\mathrm{c}}$
at the scale of critical temperature $T_c$
by utilising our $g_{\mathrm{L}}^{~\mathrm{c}}$ and the two-loop beta-function,
and compared $g_{\mathrm{T}}^{~\mathrm{c}}$ to
the zero temperature critical couplings ($g_{\mathrm{SD}}^{\mathrm{c}}$)
estimated by the two-loop Schwinger-Dyson equation
\cite{Appelquist}
as well as the IRFP position
($g_{\mathrm{4l}}^{\mathrm{IRFP}}$)
of the four-loop beta-function \cite{Ryttov:2012nt}.
The coincidence between $g_{\mathrm{T}}^{~\mathrm{c}}$ and $g_{\mathrm{SD}}^{\mathrm{c}}$ or
$g_{\mathrm{4l}}^{\mathrm{IRFP}}$ indicates
the vanishing critical temperature
with the emergence of the conformal phase.
Based on this reasoning,
we have estimated the onset of the conformal window
as $N_f^*\sim 12.5\pm 1.6$.
We have also confirmed
the increasing of $g_{\mathrm{T}}^{~\mathrm{c}}$ at larger $N_f$
which has been discussed in Ref.~\cite{Liao:2012tw}
and indicates
more strongly interacting non-Abelian plasma at larger $N_f$.
Further, we have examined the $N_f$ dependence of $T_c/M$
for a variety of choices for a reference scale $M$:
we have first considered a UV reference scale $M$
which is determined by utilising the tadpole factor $u_0$.
Then, $T_c/M$ turns out to be a decreasing function of $N_f$
consistently to the FRG observations~\cite{BraunGies},
and the vanishing $T_c/M$ indicates the emergence of the conformal window
around $N_f^* \sim 10.4 \pm 1.2$.
Then we have studied $T_c/\sqrt{\sigma}$ and we are currently extending
our study to $T_c w_0$ : the comparison among these different scale
setting procedures allows a controlled observation of the genuine
singularities - if any - associated with the onset of conformality,
and should highlight the emergence of different scales in the
pre-conformal window.
Last but not the least, we expect that
our {\em thermodynamic} lattice study for
the large $N_f$ non-Abelian gauge theory
plays an important role as a new connection
between the lattice and the Gauge/Gravity duality
\cite{Gursoy:2010fj,Alho:2012mh}.
\section{Acknowledgments}
MPL and KM were partially supported by the
PRIN `Frontiers of Strong Interactions' funded by the MIUR.
This work was in part based on the MILC Collaboration’s
public lattice gauge theory code~\cite{MILC}.
The numerical calculations for the chiral phase transition BG/P at CINECA in Italy
and the Hitachi SR-16000 at YITP, Kyoto University in Japan.
The numerical calculations for making the gauge
configurations at zero temperature were performed in BG/Q at CINECA in Italy. The Wilson loop
measurements were carried out on the high-performance computing system ϕ at KMI, Nagoya
University in Japan.
We wish to thank Marc Wagner for providing us the code for the
Wilson loop measurements.
\vskip 3 truecm
{\em Note added : After the acceptance of this work we have completed
our paper {\em On the particle spectrum and the conformal
window}\cite{Lombardo:2014pda},
which addresses the open issues of Section II providing a
nonperturbative
determination of the fermion mass anomalous dimension $\gamma =
0.235(46)$}.
|
1,314,259,995,793 | arxiv | \section{Introduction}
Generalized equivariant cohomology theories have received considerable attention in the modern research literature. Particular emphasis has been placed on cohomology computations in the presence of well-behaved equivariant stratifications. Indeed, Atiyah and Bott \cite{Yang-Mills} gave an inductive procedure for computing the ordinary equivariant cohomology of a manifold in terms of the cohomologies of the strata in an equivariant stratification. Kirwan \cite{KirwanBook} then applied related ideas to a Morse-type stratification arising from the norm-square of a moment map. A paper by Harada, Henriques, and Holm \cite{HHH2005} subsequently broadened this Atiyah-Bott-Kirwan framework to include generalized equivariant cohomology calculations via infinite stratifications. This work was partly motivated by a desire to develop a GKM-type theory for the partial flag varieties of Kac-Moody groups.
Our paper has two principal objectives. The first is to provide a straightforward, self-contained account of how to perform generalized torus-equivariant cohomology computations with a finite equivariant stratification of a smooth complex projective variety. While this is readily deducible from existing work, we believe it might serve as a convenient reference for other authors. More importantly, however, it provides the context for the second of our objectives-- a computation of the generalized torus-equivariant cohomology of a direct limit of smooth projective varieties with finitely many $T$-fixed points. More specifically, we will prove the following theorem.
\begin{theorem}\label{Main Theorem}
Suppose that $T$ is a compact torus with complexification $T_{\mathbb{C}}$, and let $E_T^*$ be one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. Let $X_0\subseteq X_1\subseteq X_2\subseteq\ldots$ be a sequence of equivariant closed embeddings of smooth complex projective $T_{\mathbb{C}}$-varieties, each with finitely many $T$-fixed points. If we define $X$ to be the direct limit of the varieties $X_n$ in their classical topologies, then $$E_T^*(X)\cong\prod_{x\in X^T}E_T^*(\text{pt})$$ as $E_T^*(\text{pt})$-modules.
\end{theorem}
While much of our work was inspired by \cite{HHH2005}, there are some important distinctions to be made. In \cite{HHH2005}, the authors first work in the context of a topological group $G$ and a fairly general stratified $G$-space $X$. Among other things, they provide some conditions on the stratification which explicitly determine the generalized $G$-equivariant cohomology of $X$ in terms of the cohomologies of the strata. By contrast, we deal with stratifications only in the context of a compact torus $T$ and a smooth complex projective $T_{\mathbb{C}}$-variety $Y$. We instead try to emphasize that the task of computing the generalized $T$-equivariant cohomology of $Y$ (and also direct limits of varieties $Y$) is especially simple.
Let us briefly outline the structure of this paper. Section \ref{sec:Generalized Equivariant Cohomology} begins with a brief overview of $T$-ring spectra and how they give rise to generalized $T$-equivariant cohomology theories. Recognizing that our arguments make extensive use of equivariant Euler classes, we include a short discussion of complex oriented theories. Also included in Section \ref{sec:Generalized Equivariant Cohomology} are brief descriptions of the three theories to which we will sometimes restrict our attention: ordinary equivariant cohomology $H_T^*$, (complex) equivariant K-theory $K_T^*$, and equivariant complex cobordism $MU_T^*$.
Section \ref{sec:Cohomology and Stratifications} is devoted to understanding the $E_T^*(\text{pt})$-module structure of $E_T^*(X)$, where $X$ is a $T$-space filtered by smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points. We begin with \ref{sub:Finite Stratifications}, in which Thom-Gysin sequences are used to compute the generalized $T$-equivariant cohomology of a finitely stratified smooth complex projective $T_{\mathbb{C}}$-variety. In \ref{sub:Finite Fixed Points}, we specialize to the case where our variety has finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*$, $K_T^*$, and $MU_T^*$. We conclude with \ref{sub:Direct Limits of Projective Varieties}, where we generalize to the case of direct limits of the varieties considered in \ref{sub:Finite Fixed Points}. This results in Theorem \ref{Main Theorem}.
In Section \ref{sec: Affine Grassmannian}, we give an example of a $T$-space satisfying the hypotheses of Theorem \ref{Main Theorem}, namely the affine Grassmannian of a simply-connected complex semisimple group.
\subsection*{Acknowledgements}
We gratefully acknowledge the support provided by Lisa Jeffrey and Paul Selick while this work was being prepared. We also wish to thank Steven Rayan for useful discussions and for his careful reading of this manuscript.
\section{Generalized Equivariant Cohomology}
\label{sec:Generalized Equivariant Cohomology}
\subsection{General Overview}
In the interest of clarity, we will begin with a brief overview of the pertinent parts of a generalized equivariant cohomology theory. Let $T$ denote a fixed compact torus, and define a $T$-space to be a compactly generated weak Hausdorff topological space $X$ endowed with a continuous action of $T$. These spaces form the objects of a category $\mathcal{C}_T$, whose morphisms are $T$-equivariant continuous maps. While this is precisely the category on which we would like to define our generalized $T$-equivariant cohomology theories, some of our arguments will be more transparent in the framework of the homotopy category of $T$-equivariant spectra.
Fix a complete $T$-universe, namely a real orthogonal $T$-representation $\mathcal{U}$ of countably infinite dimension, such that $\mathcal{U}$ contains infinitely many copies of each finite-dimensional $T$-representation. Recall that $T$-spectra indexed on\footnote{We will henceforth assume that all $T$-spectra are indexed on $\mathcal{U}$.} $\mathcal{U}$ (see Definition 9.4.1 of \cite{Axiomatic}) form a category, $TS^U$ \note{What are the maps, and can we comment more on what $\Sigma^\infty$ is? Right now it just plops out of nowhere}. Also, there is a suspension functor $\Sigma^{\infty}:(\mathcal{C}_T)_*\rightarrow TS^U$, where $(\mathcal{C}_T)_*$ is the category of based $T$-spaces. In this way, based $T$-spaces yield $T$-spectra, and we will sometimes make no distinction between a based $T$-space $X$ and its suspension spectrum $\Sigma^{\infty}(X)$.
The functor $\Sigma^\infty$ is one of a family of suspension functors $(\mathcal{C}_T)_*\rightarrow TS^U$ indexed by finite-dimensional real $T$-representations. Let $V$ be one such representation, and denote by $S^V$ its one-point compactification with base point at infinity. Note that the action of $T$ on $V$ extends to an action on $S^V$ that fixes the basepoint. Smashing against these spheres generalizes the usual suspension process, defining a functor $\Sigma^V:(\mathcal{C}_T)_*\rightarrow(\mathcal{C}_T)_*$ by
$$\Sigma^V(X):=S^V\wedge X.$$
If $V\subseteq W$ is an inclusion of finite dimensional $T$-representations, we define the relative suspension of a based $T$-space $X$ to be
$$(\Sigma^{\infty}_V(X))(W):=\Sigma^{V^{\perp}}(X),$$ where $V^{\perp}$ is the orthogonal complement of $V$ in $W$.
If $V$ does not include into $W$, we define $(\Sigma^{\infty}_V(X))(W)$ to be a point. The spaces $\{(\Sigma^{\infty}_V(X))(W)\}_W$ constitute a $T$-prespectrum and therefore determine a $T$-spectrum $\Sigma^{\infty}_V(X)$. Furthermore, $X \mapsto \Sigma^\infty_V(X)$ defines a functor
$$\Sigma^{\infty}_V:(\mathcal{C}_T)_*\rightarrow TS^U.$$ One may use this functor to define desuspensions of representation spheres:
$$S^{-V}:=\Sigma^{\infty}_V(S^0).$$
If $W$ is another finite-dimensional $T$-representation, we set
$$S^{W-V}:=S^W\wedge S^{-V}.$$
This gives us a $T$-spectrum $S^{\alpha}$ for each $\alpha$ in the representation ring $RO(T;U)$(see \cite{May}).
\subsubsection{Cohomology via spectra:}
We have developed the machinery necessary to explain how generalized $T$-equivariant cohomology theories arise from $T$-spectra. Denote by $\overline{h}TS^U$ the stable homotopy category of $T$-spectra obtained by inverting the weak equivalences in $TS^U$. Fix a $T$-spectrum $E$, and define a functor $\tilde E_T^0: \overline hTS^U \to \mathbb Z\text{-mod}$ by associating to each $T$-spectrum $F$ the abelian group $[F,E]:=\Hom(F,E)$ of morphisms in $\overline{h}TS^U$. One may extend $\tilde{E}_T^0$ to an $RO(T;U)$-graded functor by setting
\begin{equation}\label{eq:Grading}
\tilde{E}_T^{\alpha}(F):=[S^{-\alpha}\wedge F,E], \quad \alpha \in RO(T;U).
\end{equation}
We will be primarily interested in the underlying $\mathbb{Z}$-graded functor. More explicitly, if $n\in\mathbb{Z}$, then $\tilde{E}_T^n:\overline{h}TS^U\rightarrow\mathbb{Z}\text{-mod}$ is defined via \eqref{eq:Grading} by setting $\alpha$ equal to the appropriately signed $\vert n\vert$-dimensional trivial $T$-representation. The resulting $\mathbb{Z}$-graded functor $\tilde{E}_T^*$ then restricts to a reduced generalized $T$-equivariant cohomology theory on $(\mathcal{C}_T)_*$, with the associated unreduced theory $E_T^*$ on $\mathcal{C}_T$ given by
$$E_T^*(X):=\tilde{E}_T^*(X_+).$$
Here $X_+$ is the $T$ space formed by taking a disjoint union of $X$ and an additional base point.
If $E$ is additionally a commutative $T$-ring spectrum \cite[Chapter XII]{May}, then $E_T^*$ take values of the category $\text{CRing}_{\mathbb Z}$ of $\mathbb Z$-graded commutative rings.
We then have the following definition of a generalized $T$-equivariant cohomology theory suitable for our purposes.
\begin{definition}
A generalized $T$-equivariant cohomology theory is a $\mathbb{Z}$-graded functor $E_T^*:\mathcal{C}_T\rightarrow\text{CRing}_{\mathbb{Z}}$ resulting from a commutative ring $T$-spectrum $E$ as indicated above.
\end{definition}
\subsubsection{Additional Structure:}
Given a commutative $T$-ring spectrum $E$ and a $T$-space $X$, the map $X\rightarrow\text{pt}$ yields a morphism $E_T^*(\text{pt})\rightarrow E_T^*(X)$ of $\mathbb{Z}$-graded commutative rings. This map renders $E_T^*(X)$ a module over the ring $E_T^*(\text{pt})$.
A second consideration concerns equivariant Thom and Euler classes, and requires that we take $E_T^*$ to be a complex oriented theory \cite{CGK}. In more detail, suppose that $\xi$ is a $T$-equivariant complex vector bundle of rank $n$ over a $T$-space $X$, and let $Th(\xi)$ denote the associated Thom space. There exists a $T$-equivariant Thom class $u_T(\xi)\in\tilde{E}_T^{2n}(Th(\xi))$ which shares many of the properties of its non-equivariant counterpart, such as being natural under pullbacks and multiplicative over Whitney sums.
\note{I don't think that the properties of a such a class are known in general. Which things carry over from singular cohomology and which don't. Simply saying ``there is a Thom class'' is probably confusing.}
Associated to the Thom class is the Euler class, defined as follows: If $z: X_+ \to Th(\xi)$ is the zero section of the natural projection, define $e_T(\xi)\in E_T^{2n}(X)$ as
$$e_T(\xi):=z^*(u_T(\xi))\in\tilde{E}_T^{2n}(X_+)=E_T^{2n}(X).$$
\note{The approach here is motivated by \cite{CGK} as well as (What is this reference? It was hard coded) [6] in the bibliography of the former.}
Finally, one says that $E_T^*$ is a complex stable ring theory if for each finite-dimensional complex $T$-representation $V$, there exists a class $\alpha_{V}\in\tilde{E}_T^{\dim_{\mathbb{R}}(V)}(S^V)$ with the property that multiplication by $\alpha_V$ defines an isomorphism $\tilde{E}_T^*(X)\rightarrow \tilde{E}_T^*(S^V\wedge X)$ for all $T$-spaces $X$. Setting $X=S^0$ implies that $\tilde{E}_T^*(S^V)$ is freely generated by $\alpha_V$ as a module over $E_T^*(\text{pt})$.
We note that every complex oriented theory is a complex stable ring theory\cite{CGK}.
\subsection{Important Examples}
Despite having discussed generalized equivariant cohomology theories in the abstract, we will sometimes emphasize three important generalized $T$-equivariant cohomology theories: (ordinary) equivariant cohomology $H_T^*$, (complex) equivariant K-theory $K_T^*$, and equivariant complex cobordism $MU_T^*$. With this in mind, it will be prudent to recall the following proposition.
\begin{proposition}
Assume that $E_T^*$ is one of $H_T^*$, $K_T^*$, and $MU_T^*$. If $V$ is a finite-dimensional complex $T$-representation, then $E_T^*(S^V)$ is free and of rank one as a module over $E_T^*(\text{pt})$, and it vanishes in odd grading degrees.
\end{proposition}
We include a brief summary of those parts of each theory that will later prove relevant.
\subsubsection{Ordinary Equivariant Cohomology}
We denote by $ET\rightarrow ET/T=BT$ the universal principal $T$-bundle, characterized by the property that $ET$ is a contractible space on which $T$ acts freely. If $X$ is a $T$-space, then the product $X\times ET$ carries a $T$-action and we may form the Borel mixing space $$X_T:=(X\times ET)/T.$$ We then define the ordinary $T$-equivariant cohomology of $X$ (with integer coefficients) to be $$H_T^*(X):=H^*(X_T;\mathbb{Z}),$$ the integral cohomology of $X_T$. \note{Moved this up and changed how the reference is displayed} Of course, $H_T^*$ arises from the Eilenberg-MacLane $T$-spectrum \cite[Chapter XIII]{May}.
There is a natural ring isomorphism between the base ring $H_T^*(\text{pt})$ and $\Sym_{\mathbb{Z}}(X^*(T))$, the symmetric algebra of the weight lattice $X^*(T)$ of $T$. Indeed, a weight $\mu:T\rightarrow S^1$ yields an associated line bundle $$L(\mu):=\frac{ET\times\mathbb{C}}{(\alpha,z)\sim (t\alpha,\mu(t)z)}\rightarrow BT,$$ where $t\in T$ and $(\alpha,z)\in ET\times\mathbb{C}$. The ring isomorphism then associates to $\mu\in X^*(T)$ the first Chern class $c_1(L(\mu))\in H^2(BT;\mathbb{Z})=H^2_T(\text{pt})$.
\subsubsection{Equivariant K-Theory}
Our treatment follows that given in \cite{Segal}. Recall that for a compact $T$-space $X$, $K_T^0(X)$ is defined to be the Grothendieck group of the category of $T$-equivariant complex vector bundles {over} $X$. The operation of taking the tensor product of equivariant vector bundles renders $K_T^0(X)$ a commutative ring. One extends the definition of $K_T^0$ to a definition of $K_T^n(X)$ for $X$ locally compact and $n$ any integer. By virtue of Bott periodicity, there are natural $\mathbb{Z}$-module isomorphisms $K_T^n(X)\cong K_T^{n+2}(X)$, $n\in\mathbb{Z}$. In particular, if $n\in\mathbb{Z}$, then $K_T^{2n}(\text{pt})$ is naturally isomorphic to (the underlying abelian group of) the representation ring $R(T)$ of $T$. Note that $R(T)$ is freely generated over $\mathbb{Z}$ by $\{e^{\mu}:\mu\in X^*(T)\}$, where $e^{\mu}\in R(T)$ denotes the class of the one-dimensional complex $T$-representation of weight $\mu$. Furthermore, $K_T^{2n+1}(\text{pt})=K_T^{-1}(\text{pt})=0$. Hence, we shall identify $K_T^*(\text{pt})$ as a $\mathbb{Z}$-graded abelian group with $R(T)^{\oplus 2\mathbb{Z}}$. If we multiply elements in the grading components of the latter as elements of $R(T)$, then this becomes an isomorphism of $\mathbb{Z}$-graded commutative rings. \note{This last part should be verified.}
It will later be necessary to discuss the $T$-equivariant K-theory of spaces that are not locally compact. To encompass this larger class of spaces, we will define $T$-equivariant K-theory via its ring $T$-spectrum \cite[Chapter XIV]{May}.
\subsubsection{Equivariant Complex Cobordism}
\note{Need to write this section. The other two theories have brief intros, so it is asymmetric to leave out CC.}
Our discussion of the equivariant complex cobordism follows that of \cite{Sinha,May}.
As in Section \ref{sec:Generalized Equivariant Cohomology}, fix a complete $T$-universe $\mathcal{U}$ and let $BU^T(n)$ denote the Grassmannian of complex linear $n$-planes in $\mathcal{U}$. This Grassmannian comes equipped with a tautological line bundle $\xi_n^T \to BU^T(n)$, which is well known to serve as a model for the universal complex $n$-plane bundle. If $V$ is a finite-dimensional complex $T$-representation, let $\xi^T_V = \xi^T_{\dim_{\mathbb{C}}(V)}$. One then forms $Th(U)$, an $R(T)$-indexed pre-spectrum whose $V^{th}$ entry is $Th(\xi_V^T)$.
The spectrification of $Th(U)$ yields the spectrum $MU_T$.
\section{Cohomology and Stratifications}
\label{sec:Cohomology and Stratifications}
Herein we examine how to deduce the $E_T^*(\text{pt})$-module structure for spaces which admit equivariant stratifications. When there are only finitely many strata, the process amounts to inductively adding strata and will terminate after finitely many steps. We explore this case further in Section \ref{sub:Projective Variety Example}
using a natural stratification of a smooth projective $T_{\mathbb{C}}$-variety admitting finitely many $T$-fixed points.
Section \ref{sub:Direct Limits of Projective Varieties} then provides a generalization of Section \ref{sub:Projective Variety Example}, replacing smooth projective $T_{\mathbb{C}}$-varieties with direct limits thereof.
\note{Note to self: Add in some introduction here, then define $T$-equivariant stratification. Move original intro down.}
\subsection{Finite Stratifications}
\label{sub:Finite Stratifications}
Throughout this section let $T$ be a compact torus with complexification $T_{\mathbb{C}}$, and assume that $E_T^*$ is a complex oriented generalized equivariant cohomology theory.
\begin{definition}
\label{def:Finite Stratification}
Let $X$ be a smooth complex projective variety on which $T_{\mathbb C}$ acts algebraically. A \emph{$T$-equivariant stratification of $X$} consists of a finite partially ordered set $B$ and a collection $\{X_{\beta}\}_{\beta\in B}$ of pairwise disjoint smooth $T$-invariant locally closed subvarieties of $X$ satisfying
\begin{itemize}
\item[(i)] $X=\bigcup_{\beta\in B}X_{\beta}$, and
\item[(ii)] $\overline{X_{\beta}}=\bigcup_{\gamma\leq\beta}X_{\gamma}$ for all $\beta\in B$.
\end{itemize}
\end{definition}
\subsection*{Example} Examples of Definition \ref{def:Finite Stratification} include Bruhat cell decompositions of partial flag varieties. More precisely, suppose that $T_{\mathbb{C}}$ is a maximal torus of a connected, simply-connected complex semisimple group $G$. Suppose further that $T_{\mathbb{C}}\subseteq B\subseteq P$, where $B$ and $P$ are Borel and parabolic subgroups of $G$, respectively. Let $W$ denote the Weyl group and $W_P$ the subgroup of $W$ associated with $P$. One has the partial flag variety $X=G/P$, on which $G$ acts algebraically by left-multiplication. The $T$-fixed points of $X$ are naturally indexed by $W/W_P$. Also, each $B$-orbit contains a unique $T$-fixed point, giving the Bruhat decomposition $$X=\bigsqcup_{u\in W/W_P}BuP/P.$$ For each $u\in W/W_P$, set $X_u:=BuP/P$. Endowing $W/W_P$ with the Bruhat order, one has the closure relations $$\overline{X_{u}}=\bigsqcup_{v\leq u}X_v.$$ Hence, $\{X_u\}_{u\in W/W_P}$ is a $T$-equivariant stratification of $X$.\hfill $\square$ \\
Fix a smooth complex projective $T_{\mathbb C}$-variety $X$ and let $\{X_\beta \}_{\beta \in B}$ be a given equivariant stratification. For each fixed $\beta\in B$, let $N_{\beta}\rightarrow X_{\beta}$ denote the normal bundle of $X_{\beta}$ in $X$ and let $d(\beta)$ denote its rank. The bundle $N_{\beta}$ has a $T$-equivariant Thom class $u_T(\beta)\in \tilde E_T^{2d(\beta)}(Th(N_{\beta}))$ and an associated Euler class $e_T(\beta) \in E_T^{2d(\beta)}(X_\beta)$.
\begin{theorem}\label{thm:Finite Module Isomorphism}
Assume that for each $\beta\in B$, $E_T^*(X_{\beta})$ is a free module over $E_T^*(\text{pt})$, and that $e_T(\beta)$ is not a zero-divisor in $E_T^*(X_{\beta})$. There is an isomorphism
$$E_T^*(X)\cong\bigoplus_{\beta\in B}E_T^*(X_{\beta})$$
of $E_T^*(\text{pt})$-modules.
\end{theorem}
\begin{proof}
Following \cite{Yang-Mills}, we define a subset $J\subseteq B$ to be \textit{open} if whenever $\beta\in J$ and $\gamma\in B$ satisfy $\beta\leq\gamma$, we have $\gamma\in J$. This definition has the desirable property that if $J\subseteq B$ is open, then $$X_J:=\bigcup_{\beta\in J}X_{\beta}$$ is an open subset of $X$.
Choose a maximal element $\beta_1\in B$ and set $J_1:=\{\beta_1\}$, an open subset of $B$. We inductively define subsets $J_k\subseteq B$, $k\in\{2,\ldots,\vert B\vert\}$, by the condition that $J_k=\{\beta_1,\ldots,\beta_k\}$ with $\beta_k$ a maximal element of $B\setminus J_{k-1}$. By construction, $J_k$ is open for all $k$.
We have graded $E_T^*(\text{pt})$-module isomorphisms
\begin{equation}
\label{eq:Bundle}
E_T^*(X_{J_k},X_{J_{k-1}})\cong E_T^*(Th(N_{\beta_k}))\cong E_T^{*-2d(\beta_k)}(X_{\beta_k}),
\end{equation}
the second being the Thom Isomorphism (see \cite{May}, Theorem 9.2). Using \eqref{eq:Bundle}, the long exact sequence of the pair $(X_{J_k},X_{J_{k-1}})$ takes the form
\begin{equation}
\label{eq:Long-Exact}
\ldots\rightarrow E_T^{i-2d(\beta_k)}(X_{\beta_k})\xrightarrow\phi E_T^i(X_{J_k})\rightarrow E_T^i(X_{J_{k-1}})\rightarrow E_T^{i-2d(\beta_k)+1}(X_{\beta_k})\rightarrow\ldots.
\end{equation}
If $E_T^i(X_{J_k}) \to E_T^i(\beta_k)$ is the restriction map, the composition
$$E_T^{i-2d(\beta_k)}(X_{\beta_k}) \xrightarrow\phi E_T^i(X_{J_k}) \rightarrow E_T^i(X_{\beta_k})$$
is equivalent to multiplication by the equivariant Euler class $e_T(\beta_k)$. As $e_T(\beta_k)$ is not a zero divisor, the composition is injective, forcing $\phi$ to be injective. Hence \eqref{eq:Long-Exact} degenerates to the short exact sequence
\begin{equation}
\label{eq:Thom-Gysin}
0 \rightarrow E_T^{*-2d(\beta_k)}(X_{\beta_k})\rightarrow E_T^*(X_{J_k})\rightarrow E_T^*(X_{J_{k-1}})\rightarrow 0
\end{equation}
of $E_T^*(\text{pt})$-modules. Using \eqref{eq:Thom-Gysin} and induction, we will prove that
\begin{equation}
\label{eq:Induction Claim}
E_T^*(X_{J_k})\cong\bigoplus_{\ell\leq k}E_T^*(X_{\beta_\ell})
\end{equation} for all $k\in\{2,\ldots,\vert B\vert\}$, {from which the theorem will follow}.
In the base case $k=2$, our short exact sequence is $$0 \rightarrow E_T^{*-2d(\beta_2)}(X_{\beta_2})\rightarrow E_T^*(X_{J_2})\rightarrow E_T^*(X_{\beta_1})\rightarrow 0.$$ This sequence splits by virtue of the fact that $E_T^*(X_{\beta_1})$ is a free $E_T^*(\text{pt})$-module. Hence, $$E_T^*(X_{J_2})\cong E_T^*(X_{\beta_1})\oplus E_T^*(X_{\beta_2}).$$
Assume now that \eqref{eq:Induction Claim} holds for some $k\leq\vert B\vert -1$ and replace $k$ with $k+1$ in \eqref{eq:Thom-Gysin} to obtain the sequence
\begin{equation}
0\rightarrow E_T^{*-2d(\beta_{k+1})}(X_{\beta_{k+1}})\rightarrow E_T^*(X_{J_{k+1}})\rightarrow E_T^*(X_{J_{k}})\rightarrow 0.
\label{eq:SES}
\end{equation}
By assumption, $E_T^*(X_{J_k})$ is free, so \eqref{eq:SES} splits. Hence, \eqref{eq:Induction Claim} holds if we replace $k$ with $k+1$, and our induction is complete.
\end{proof}
\begin{remark}
The isomorphism in Theorem \ref{thm:Finite Module Isomorphism} does not respect the $\mathbb{Z}$-gradings of $E_T^*(X)$ and $\bigoplus_{\beta\in B}E_T^*(X_{\beta})$. To compensate for the degree-shift of $2d(\beta)$ appearing in \eqref{eq:Thom-Gysin}, one can identify $E_T^*(X_{\beta})$ as an $E_T^*(\text{pt})$-module with the principal ideal $\langle e_T(\beta)\rangle$ generated by $e_T(\beta)$. This gives us an isomorphism
\begin{equation}
\label{eq:Graded Isomorphism}
E_T^*(X)\cong\bigoplus_{\beta\in B}\langle e_T(\beta)\rangle
\end{equation}
on the level of both $E_T^*(\text{pt})$-modules and $\mathbb{Z}$-graded abelian groups.
\end{remark}
\note{I have moved this up from Section 2.2. Since this is the first section that talks about the Euler class not being a zero divisor, it seems like the conclusion of this section would be a good time to point out that the above Theorem applies to these cohomology theories. Also, add a part concerning $MU_T^*$. This may necessitate changing the lemma to say that $e_T(V)$ is not a zero divisor, for which the particular formulas (i) and (ii) are specifically mentioned in the proofs.}
\subsection{The Case of Finitely Many Fixed Points}\label{sub:Projective Variety Example}
\label{sub:Finite Fixed Points}
\note{I have completely changed the part of this section prior to the lemma.}
The approach outlined in Section \ref{sec:Cohomology and Stratifications} can be combined with a suitable Bia{\l}ynicki-Birula stratification to yield the $E_T^*$-module structure of a smooth complex projective $T_{\mathbb{C}}$-variety with finitely many $T$-fixed points. More explicitly, we will prove the following theorem:
\begin{theorem}\label{thm:Module Structure}
Suppose that $E_T^*$ is one of $H_T^*$, $K_T^*$, and $MU_T^*$. If $X$ is a smooth complex projective $T_{\mathbb{C}}$-variety with finitely many $T$-fixed points, then $E_T^*(X)$ is a free $E_T^*(\text{pt})$-module of rank $\vert X^T\vert$
\end{theorem}
For the duration of this section, we will assume that everything is as given in the statement of Theorem \ref{thm:Module Structure}.
\begin{lemma}\label{lemma:Coweight}
There exists a coweight $\lambda:\mathbb{C}^*\rightarrow T_{\mathbb{C}}$ with the property that the fixed points of the resulting $\mathbb{C}^*$-action on $X$ are precisely the $T$-fixed points.
\end{lemma}
\begin{proof}
Choose a coweight $\lambda$ such that for every $w\in X^T$ and weight $\mu:T_{\mathbb{C}}\rightarrow\mathbb{C}^*$ of the isotropy representation $T_wX$, the pairing $\langle\lambda,\mu\rangle$ is non-zero. This coweight yields an algebraic action of $\mathbb{C}^*$ on $X$, and we suppose that $Y$ is an irreducible component of $X^{\mathbb{C}^*}$. Note that $Y$ is a smooth closed $T_{\mathbb{C}}$-invariant subvariety of $X$.
By the Borel Fixed Point Theorem, $Y$ has a $T$-fixed point $y$. Since $T_yY$ is precisely the trivial weight space of the $\mathbb{C}^*$-representation on $T_yX$, our choice of $\lambda$ implies that $T_yY=\{0\}$. It follows that $Y=\{y\}$, giving the inclusion $X^{\mathbb{C}^*}\subseteq X^T$.
\end{proof}
Now, select $\lambda:\mathbb{C}^*\rightarrow T_{\mathbb{C}}$ as in Lemma \ref{lemma:Coweight}.
Given $w\in X^{\mathbb{C}^*}=X^T$, one has the smooth locally closed subvariety
\begin{equation}
\label{eq:BB definition}
X_w:=\left\{x\in X:\lim_{t\to 0}(\lambda(t)\cdot x)=w\right\}.
\end{equation}
The $X_w$ constitute a Bia{\l}ynicki-Birula stratification \cite{Bialynicki-Birula1973}, a $T$-equivariant stratification of $X$. Furthermore, $X_w$ is $T$-equivariantly homeomorphic to the $T$-submodule $(T_wX)^+$ of $T_wX$ spanned by the weight vectors whose weights have strictly positive pairing with $\lambda$. In particular, $X_w$ equivariantly retracts onto its $T$-fixed point $\{w\}$ and we have a ring isomorphism $r_w:E_T^*(X_w)\xrightarrow{\cong}E_T^*(\{w\})$. If $e_T(w)\in E_T^*(X_w)$ denotes the $T$-equivariant Euler class of the normal bundle of $X_w$ in $X$, then $r_w(e_T(w))$ is the $T$-equivariant Euler class of the quotient representation $T_{w}(X)/T_{w}X_w\rightarrow\{w\}$.
\begin{lemma}\label{lemma:Formula}
Let $V$ be a finite-dimensional complex $T$-representation such that $V^T = \{ 0\}$, viewed as a $T$-equivariant vector bundle over a point. If $E_T^*$ is $H_T^*, K_T^*$, or $MU_T^*$, then the $T$-equivariant Euler class $e_T(V) \in E_T^*(\text{pt})$ is not a zero divisor.
\end{lemma}
\begin{proof}
Note that $E_T^*(\text{pt})$ is an integral domain for each of the above three theories. By virtue of the Whitney sum formula, it therefore suffices to prove that $e_T(V)$ is non-zero when $V$ is one-dimensional.
Let $\mu\in X^*(T)$ be the (non-zero) weight of $V$. If $E_T^*=H_T^*$, then $e_T(V)$ is the ordinary Euler class of the associated bundle $ET\times_TV\rightarrow BT$. Under the usual ring isomorphism $H^*(BT;\mathbb{Z})\cong\Sym_{\mathbb{Z}}(X^*(T))$, this Euler class corresponds to the weight $\mu$.
When $E_T^*=K_T^*$, the equivariant Euler class of a complex $T$-representation is given by the alternating sum of its exterior powers in $K_T^*(\text{pt})$\cite[Chapter XIV, Theorem 3.2]{May}. Hence, $e_T(V)=1-[V]\in K_T^2(\text{pt})$, which is identified with $1-e^{\mu}$ under the isomorphism $K_T^2(\text{pt})\cong R(T)$. We thus see that $e_T(V)\neq 0$.
In the case of $MU_T^*$, we simply appeal to \cite{Sinha}.
\end{proof}
Since the $T$-fixed points in $X$ are isolated, zero is not a weight of the representation $T_{w}X/T_{w}X_w$. By Lemma \ref{lemma:Formula}, we conclude that $r_w(e_T(w))$ is not a zero-divisor in $E_T^*(\{w\})$, meaning that $e_T(w)$ is not a zero divisor. An application of Theorem \ref{thm:Finite Module Isomorphism} then yields an $E_T^*(\text{pt})$-module isomorphism
$$E_T^*(X)\cong\bigoplus_{w\in X^T}E_T^*(X_w).$$
In particular, $E_T^*(X)$ is free of rank $\vert X^T\vert$, proving Theorem \ref{thm:Module Structure}.
Theorem \ref{thm:Module Structure} will prove essential in extending our results to the case of direct limits of projective varieties. To realize the extension, we will require the following lemma.
\begin{proposition}\label{prop:Relative}
If $Y$ is a smooth closed $T_{\mathbb{C}}$-invariant subvariety of $X$, then
\begin{itemize}
\item[(i)] $E_T^*(X,Y)$ is a free $E_T^*(\text{pt})$-module of finite rank vanishing in odd grading degrees, and
\item[(ii)] the restriction map $E_T^*(X)\rightarrow E_T^*(Y)$ is surjective.
\end{itemize}
\end{proposition}
\begin{proof}
To prove (i), we will appeal to some general properties of model categories. \note{We might include an introductory reference.} Indeed, $T$-spaces form a model category in which the weak equivalences are the $T$-homotopy equivalences and the cofibrations are the morphisms with the $T$-homotopy extension property. Accordingly, we will begin by proving the following claim by induction: If $w_1,\ldots,w_n\in Y^T$ {and $X_{w_i}$ are the associated Bia\l ynicki-Birula strata}, then the inclusion
$$Y\rightarrow Y\cup\bigcup_{i=1}^nX_{w_i}$$
is an acyclic cofibration (ie. a cofibration that is also a weak equivalence). \note{This requires a reference. At the moment, the only reference is Peter May's response on Math Overflow.}
For the base case, let $Y_{w_1}\subseteq Y$ denote the Bia{\l}ynicki-Birula stratum of $Y$ associated with $w_1\in Y^T$. One has the pushout square
$$\xymatrix{
Y_{w_1} \ar[d] \ar[r] & X_{w_1} \ar[d] \\
Y \ar[r] & Y\cup X_{w_1}}$$
of inclusions. Note that $Y_{w_1}\rightarrow X_{w_1}$ is an acyclic cofibration. \note{This corresponds to an inclusion of finite-dimensional complex $T$-representations. Therefore, $(Y_w,X_w)$ is a $T$-NDR pair and $Y_w\rightarrow X_w$ is a $T$-cofibration. A reference is page 504 of "On $G$-ANR's and their $G$-Homotopy Types" by Murayama.}
Since the pushout of an acyclic cofibration is itself an acyclic cofibration, it follows that $Y\rightarrow Y\cup X_{w_1}$ is an acyclic cofibration. Now, assume that our claim holds for $\leq n$ points in $Y^T$. Given $w_1,\ldots,w_{n+1}\in Y^T$, we consider the pushout square $$\xymatrix{
Y \ar[d]^{i_2} \ar[r]^{i_1} & Y\cup\bigcup_{i=1}^nX_{w_i} \ar[d]^{j_2} \\
Y\cup X_{w_{n+1}} \ar[r]^{j_1} & Y\cup\bigcup_{i=1}^{n+1}X_{w_i}}$$ of inclusions. Noting that $i_1$ is an acyclic cofibration, the same is true of $j_1$. The inclusion $Y\rightarrow Y\cup\bigcup_{i=1}^{n+1}X_{w_i}$ is then a composition of the acyclic cofibrations $i_2$ and $j_1$, and so is itself an acyclic cofibration. This completes the induction. Setting $$Z:=\bigcup_{w\in Y^T}X_{w},$$ it follows that $Y\rightarrow Z$ is an acyclic cofibration. In particular, $E_T^*(Z,Y)=0$, and it just remains to prove that $E_T^*(X,Z)$ is free of finite rank and vanishes in odd degrees. \note{This would seem to follow from the G-Whitehead Theorem. A weak equivalence of G-CW complexes is a G-homotopy equivalence. Actually, we need not invoke this since our model-theoretic notion of a weak equivalence is precisely a $T$-homotopy equivalence.}
Recall that if $w\in X^T$, then $X_w$ is $T$-equivariantly homeomorphic to a finite-dimensional complex $T$-representation $V_w$. Choose an enumeration $\{w_1,\ldots,w_m\}$ of $X^T\setminus Y^T$ with the property that for all $k\in\{1,\ldots,m\}$, the quotient of $Z\cup\bigcup_{j=1}^kX_{w_j}$ by $Z\cup\bigcup_{j=1}^{k-1}X_{w_j}$ is $T$-equivariantly homeomorphic to the one-point compactification $S^{V_{w_k}}$. \note{Perhaps we must assume $X$ to be connected. Also, we should explain why this is possible.}
Using induction, we will prove that $E_T^*(Z\cup\bigcup_{j=1}^kX_{w_j},Z)$ is free of finite rank for all $k\in\{1,\ldots,m\}$, and that it vanishes in odd grading degrees.
Since $Z\cap X_{w_1}=\emptyset$, the inclusion $Z\rightarrow Z\cup X_{w_1}$ is a cofibration. \note{To see this, create the pushout square with $\emptyset$ in the upper left, $X_{w_1}$ and $Z$ on the off diagonal, and $Z\cup X_w$ on the bottom right}
Hence,
$$E_T^*\left(Z\cup X_{w_1},Z\right)\cong\tilde{E}_T^*\left((Z\cup X_{w_1})/Z\right)\cong\tilde{E}_T^*\left(S^{V_{w_k}}\right)$$
is free of finite rank, and vanishes in odd grading degrees. Now, assume that $E_T^*\left(Z\cup\bigcup_{j=1}^k X_{w_j},Z\right)$ vanishes in odd degrees and is free of finite rank. Since the inclusion $Z\cup\bigcup_{j=1}^kX_{w_j}\rightarrow Z\cup\bigcup_{j=1}^{k+1}X_{w_j}$ is a cofibration, we find that \note{For the next few lines, either the encompassing brackets need to be made bigger, or we must not use bigcup}
$$E_T^*\left(Z\cup\bigcup_{j=1}^{k+1}X_{w_j},Z\cup\bigcup_{j=1}^kX_{w_j}\right)\cong\tilde{E}_T^*\left(\bigg(Z\cup\bigcup_{j=1}^{k+1}X_{w_j}\bigg)\big/\bigg(Z\cup\bigcup_{j=1}^kX_{w_j}\bigg)\right)\cong\tilde{E}_T^*\left(S^{V_{w_{k+1}}}\right)$$
is also free of finite rank and vanishes in odd degrees. \note{Consider the pushout square with the empty set in the upper-left, $X_{w_{k+1}}$ and $Z\cup\bigcup_{j=1}^kX_{w_j}$ on the off-diagonal, and the union in the lower-right.}
Therefore, the long exact sequence of the pairs $(Z\cup\bigcup_{j=1}^{k+1}X_{w_j},Z\cup\bigcup_{j=1}^{k}X_{w_j})$, $(Z\cup\bigcup_{j=1}^{k+1}X_{w_j}, Z)$, $(Z\cup\bigcup_{j=1}^{k}X_{w_j}, Z)$ splits to give the short exact sequence
$$0\rightarrow E_T^*\left(Z\cup\bigcup_{j=1}^{k+1}X_{w_j},Z\cup\bigcup_{j=1}^{k}X_{w_j}\right)\rightarrow E_T^*\left(Z\cup\bigcup_{j=1}^{k+1}X_{w_j},Z\right)\rightarrow E_T^*\left(Z\cup\bigcup_{j=1}^{k}X_{w_j},Z\right)\rightarrow 0.$$
Since $E_T^*(Z\cup\bigcup_{j=1}^{k+1}X_{w_j},Z\cup\bigcup_{j=1}^{k}X_{w_j})$ and $E_T^*(Z\cup\bigcup_{j=1}^{k}X_{w_j},Z)$ are free of finite rank, the same is true of $E_T^*(Z\cup\bigcup_{j=1}^{k+1}X_{w_j},Z)$. We have therefore proved (i).
For (ii), we consider the long exact sequence of the pair $(X,Y)$. Indeed, (i) is then seen to imply that $E_T^n(X)\rightarrow E_T^n(Y)$ is surjective for even $n$. Furthermore, the isomorphism \eqref{eq:Graded Isomorphism} establishes that both $E_T^*(X)$ and $E_T^*(Y)$ vanish in odd grading degrees. The proof is therefore complete.
\end{proof}
\subsection{Direct Limits of Projective Varieties}
\label{sub:Direct Limits of Projective Varieties}
We now provide a generalization of our findings in Section \ref{sub:Projective Variety Example}, replacing projective varieties with direct limits thereof. As before, $T$ denotes a compact torus with complexification $T_{\mathbb{C}}$,
and $E_T^*$ is one of $H_T^*$, $K_T^*$, and $MU_T^*$. Suppose that $$X_0\subseteq X_1\subseteq X_2\subseteq\ldots\subseteq X_n\subseteq\ldots$$ is a sequence of equivariant closed embeddings of smooth complex projective $T_{\mathbb{C}}$-varieties with $(X_n)^T$ finite for each $n\geq 0$. Let $X$ be the topological direct limit of the $X_n$ in their analytic topologies, and endow $X$ with the induced direct limit topology. Note that $X$ then carries a continuous action of $T$. The following theorem then generalizes Theorem \ref{thm:Module Structure}:
\begin{theorem}\label{thm:Direct Limit}
Under the conditions stated above, there is an $E_T^*(\text{pt})$-module isomorphism $$E_T^*(X)\cong\prod_{x\in X^T}E_T^*(\text{pt}).$$
\end{theorem}
\begin{proof}
By Proposition \ref{prop:Relative}, each restriction map $E_T^*(X_{n+1})\rightarrow E_T^*(X_n)$ is surjective. Hence, the inverse system $\{E_T^*(X_n)\}_n$ of $E_T^*(\text{pt})$-modules has vanishing Milnor $\varprojlim^1$. It follows that the canonical map $E_T^*(X)\rightarrow\varprojlim_nE_T^*(X_n)$ is an isomorphism\cite{Axiomatic}.
It will therefore suffice to prove that $\{E_T^*(X_n)\}_n$ and $\{\bigoplus_{x\in (X_n)^T}E_T^*(\text{pt})\}_n$ are isomorphic as inverse systems of $E_T^*(\text{pt})$-modules, where the maps in the latter system are precisely the projection maps resulting from the inclusions $(X_n)^T\subseteq (X_{n+1})^T$. We will do this by inductively constructing $E_T^*(\text{pt})$-module isomorphisms $$\psi_n:E_T^*(X_n)\rightarrow\bigoplus_{x\in (X_n)^T}E_T^*(\text{pt})$$ making the diagrams
$$D_n:=\xymatrix{
E_T^*(X_{n+1}) \ar[d] \ar[r]^-{\psi_{n+1}} & \bigoplus_{x\in (X_{n+1})^T}E_T^*(\text{pt}) \ar[d]\\
E_T^*(X_n) \ar[r]^-{\psi_{n}} & \bigoplus_{x\in (X_n)^T}E_T^*(\text{pt})}$$
commute.
By Theorem \ref{thm:Module Structure}, we haves an $E_T^*(\text{pt})$-module isomorphism $\psi_0:E_T^*(X_0)\rightarrow\bigoplus_{x\in (X_0)^T}E_T^*(\text{pt})$.
Assume now that we have constructed isomorphisms $\psi_k:E_T^*(X_k)\rightarrow\bigoplus_{x\in (X_k)^T}E_T^*(\text{pt})$ for all $k\leq n$ so that the diagrams $D_0,\ldots,D_{n-1}$ commute. Since the restriction $\pi_n:E_T^*(X_{n+1})\rightarrow E_T^*(X_n)$ is surjective, the long exact sequence of the pair $(X_{n+1},X_n)$ degenerates to a short exact sequence
\begin{equation}
\label{eq:Short Exact Sequence}
0\rightarrow E_T^*(X_{n+1},X_n)\rightarrow E_T^*(X_{n+1})\xrightarrow{\pi_n} E_T^*(X_n)\rightarrow 0
\end{equation}
of $E_T^*(\text{pt})$-modules. Theorem \ref{thm:Module Structure} implies that $E_T^*(X_n)$ is free, so that \eqref{eq:Short Exact Sequence} admits a splitting $\varphi_n:E_T^*(X_{n+1})\rightarrow E_T^*(X_{n+1},X_n)$. Also, Proposition \ref{prop:Relative} implies that $E_T^*(X_{n+1},X_n)$ is free of rank $\vert (X_{n+1})^T\setminus (X_n)^T\vert$. We may therefore choose an $E_T^*(\text{pt})$-module isomorphism
$$\theta_n:E_T^*(X_{n+1},X_n)\xrightarrow{\cong}\bigoplus_{x\in (X_{n+1})^T\setminus (X_n)^T}E_T^*(\text{pt}).$$
The composite map
$$E_T^*(X_{n+1})\xrightarrow{(\pi_n,\varphi_n)} E_T^*(X_n)\oplus E_T^*(X_{n+1},X_n)\xrightarrow{\psi_n\oplus\theta_n}\bigoplus_{x\in (X_{n+1})^T}E_T^*(\text{pt})$$
is then an $E_T^*(\text{pt})$-module isomorphism that we shall call $\psi_{n+1}$. By construction, $D_n$ commutes for this choice of $\psi_{n+1}$, and this completes the proof.
\end{proof}
\note{I have moved the first two sections on the affine Grassmannian so that they now follow Sections 1 and 2.}
\section{The Affine Grassmannian}
\label{sec: Affine Grassmannian}
The affine Grassmannian $\mathcal Gr$ is a space of great interest to geometric representation theorists (see \cite{Kamnitzer,MV}, for instance). It is also very closely linked to the study of (algebraic) based loop groups (discussed in \cite{Mitchell,Magyar,PressleySegal}). Using the work done in the aforementioned papers, we can show that $\mathcal Gr$ is the perfect candidate for an application of Theorem \ref{thm:Direct Limit}.
\subsection{Definition and Filtration}
\label{sub:Definition and Filtration}
Let $G$ be a connected, simply-connected complex semisimple group. Fix a maximal torus $T_{\mathbb{C}}\subseteq G$ with compact real from $T_{\mathbb R}$, as well as a Borel subgroup $B$ containing $T_{\mathbb C}$. Take $W= N_G(T_{\mathbb C})/T_{\mathbb C}$ to be the associated Weyl group.
Let $X^*(T_{\mathbb{C}}):=\Hom(T_{\mathbb{C}},\mathbb{C}^*)$ and $X_*(T_{\mathbb{C}}):=\Hom(\mathbb{C}^*,T_{\mathbb C})$ be the weight and coweight lattices respectively, endowed with their usual pairing
$$\langle\cdot,\cdot\rangle:X_*(T_{\mathbb{C}})\otimes_{\mathbb{Z}}X^*(T_{\mathbb{C}})\rightarrow\mathbb{Z}.$$
The choice of Borel subgroup yields dominant weights $X^*(T_{\mathbb{C}})_+\subseteq X^*(T_{\mathbb{C}})$ and dominant coweights $X_*(T_{\mathbb{C}})_+\subseteq X_*(T_{\mathbb{C}})$. Take $\Delta\subseteq X^*(T_{\mathbb{C}})$ to be the collection of roots, and $\Pi\subseteq\Delta$ to be the subset of simple (positive) roots.
We shall assume that $G$ admits a finite-dimensional, faithful, irreducible representation $V(\alpha)$ of highest weight $\alpha\in X^*(T_{\mathbb{C}})_+$. This allows us to realize $G$ as a Zariski-closed subgroup of $GL(V(\alpha))$.
Consider the $\mathbb{C}$-algebras $\mathcal{O}:=\mathbb{C}[t]$ and $\mathcal{K}:=\mathbb{C}[t,t^{-1}]$, letting $G(\mathcal{O})$ and $G(\mathcal{K})$ denote the $\mathcal{O}$ and $\mathcal{K}$-valued points of $G$, respectively. Set-theoretically, the affine Grassmannian of $G$ is defined to be the coset space
$$\mathcal{G}r:=G(\mathcal{K})/G(\mathcal{O}).$$
Note that the $\mathbb{C}$-vector space $V(\alpha)\otimes\mathcal{K}$ admits the filtration
$$\ldots\subseteq V(\alpha)\otimes t^2\mathcal{O}\subseteq V(\alpha)\otimes t\mathcal{O}\subseteq V(\alpha)\otimes\mathcal{O}\subseteq V(\alpha)\otimes t^{-1}\mathcal{O}\subseteq V(\alpha)\otimes t^{-2}\mathcal{O}\subseteq\ldots.$$
We thus define a function $\text{val}:V(\alpha)\otimes\mathcal{K}\rightarrow\mathbb{Z}$ by
$$\text{val}(u):=\max\{k\in\mathbb{Z}:u\in V(\alpha)\otimes t^{k}\mathcal{O}\}.$$
As $G(\mathcal{K})$ acts on $V(\alpha)\otimes\mathcal{K}$ by virtue of the inclusion of $G$ into $\GL(V(\alpha))$, we may define $\text{Val}:G(\mathcal{K})\rightarrow\mathbb{Z}$ by
$$\text{Val}(g):=\min\{\text{val}(g\cdot v):v\in V(\alpha)\}.$$
Given $n\in\mathbb{Z}_{\geq 0}$, we set
$$G(\mathcal{K})_n:=\{g\in G(\mathcal{K}):\text{Val}(g)\geq -n\},$$
yielding a filtration
\begin{equation}
\label{eq:AffGeoFilt}
G(\mathcal{O})=G(\mathcal{K})_0\subseteq G(\mathcal{K})_1\subseteq G(\mathcal{K})_2\subseteq\ldots\subseteq G(\mathcal{K})
\end{equation}
of $G(\mathcal{K})$. Note that $G(\mathcal{K})_n$ is invariant under the right-multiplicative action of $G(\mathcal{O})$ on $G(\mathcal{K})$. Accordingly, we define
\begin{equation}
\label{eq:GrFilt}
\mathcal{G}r_n:=G(\mathcal{K})_n/G(\mathcal{O}),
\end{equation}
a smooth finite-dimensional projective scheme over $\mathbb{C}$. By exhibiting the affine Grassmannian as inductive limit of the schemes $\{\mathcal{G}r_n\}_{n\in\mathbb{Z}_{\geq 0}}$, we may realize $\mathcal{G}r$ as a projective ind-scheme. (For a treatment of ind-schemes, the reader might consult the appendix of \cite{FB}.)
Of course, we will endow $\mathcal{G}r$ with a topology other than the one it inherits as an ind-scheme. Namely, we will regard $\mathcal{G}r$ as the direct limit of the varieties $\{\mathcal{G}r_n\}_{n\in\mathbb{Z}_{\geq 0}}$ in their classical topologies.
\subsection{The Action of $\mathbb{C}^*$}
\label{sub:Action of C*}
There is a natural ``loop rotation'' action of $\mathbb{C}^*$ on $G(\mathcal{K})$. Indeed, the left-multiplicative action of $\mathbb{C}^*$ on itself yields a $\mathbb{C}^*$-action on $\Hom(\mathbb{C}^*,G)=G(\mathcal{K})$ by group automorphisms. More concretely, the inclusion $G\subseteq\GL(V(\alpha))$ associates to each point $\gamma\in G(\mathcal{K})$ an expansion $\gamma=\sum_{j\in\mathbb{Z}}\gamma_jt^j$, where $\gamma_j\in\End(V(\alpha))$ for all $j$. The action of $s\in\mathbb{C}^*$ on $\gamma$ is then given by $$s:\sum_{j\in\mathbb{Z}}\gamma_jt^j\mapsto\sum_{j\in\mathbb{Z}}\gamma_j(st)^j.$$ It follows that $G(\mathcal{K})_n$ is $\mathbb{C}^*$-invariant for $n\in\mathbb{Z}_{\geq 0}$. In particular, $\mathcal{G}r_0=G(\mathcal{O})$ is $\mathbb{C}^*$-invariant and the $\mathbb{C}^*$-action descends to an action on $\mathcal{G}r$ that preserves each subvariety $\mathcal{G}r_n$.
\subsection{The Generalized Torus-Equivariant Cohomology of $\mathcal{G}r$}
\label{sub:A Stratification}
Consider the compact torus $T:=T_{\mathbb{R}}\times S^1$, where $T_{\mathbb{R}}$ is the compact torus fixed in Section \ref{sub:Definition and Filtration}. As $T$ is a subgroup of $T_{\mathbb{C}}\times\mathbb{C}^*$, and the latter torus acts on $\mathcal{G}r$ via the commuting actions of $T_{\mathbb{C}}$ and $\mathbb{C}^*$, we obtain an action of $T$ on $\mathcal{G}r$.
Note that \eqref{eq:GrFilt} is thus a $T$-equivariant filtration. With Theorem \ref{thm:Direct Limit} in mind, it remains only to prove that $(\mathcal{G}r_n)^T$ is finite for all $n\geq 0$. To this end, let $\lambda\in X_*(T_{\mathbb{C}})$ be a coweight, and consider the point in $G(\mathcal{K})$ given by the composition
\begin{equation}
\label{eq:lambdaPoint}
\mathbb{C}^*\xrightarrow{\lambda} T_{\mathbb{C}}\hookrightarrow G,
\end{equation}
where $T_{\mathbb{C}}\hookrightarrow G$ is the inclusion. Let $t^{\lambda}\in\mathcal{G}r$ denote the class of \eqref{eq:lambdaPoint} in the affine Grassmannian. It turns out (see \cite{MV}) that the $T$-fixed points of $\mathcal{G}r$ are precisely the $t^{\lambda}$, for $\lambda\in X_*(T_{\mathbb{C}})$, leading us to prove the following lemma:
\begin{lemma}
For $n\geq 0$, $$(\mathcal{G}r_n)^T=\{t^{w\lambda}:w\in W, \text{ }\lambda\in X_*(T_{\mathbb{C}})_+, \langle\lambda,w_0\alpha\rangle\geq -n\},$$ where $w_0\in W$ is the longest element. In particular, $(\mathcal{G}r_n)^T$ is finite.
\end{lemma}
\begin{proof}
Since $\mathcal{G}r_n$ is $G$-invariant, one has an induced action of $W$ on $(\mathcal{G}r_n)^{T_{\mathbb{C}}}$. Because the actions of $G$ and $\mathbb{C}^*$ commute, the $W$-action leaves $(\mathcal{G}r_n)^{T_{\mathbb{C}}\times\mathbb{C}^*}=(\mathcal{G}r_n)^T$ invariant. Hence, if $\mu\in X_*(T_{\mathbb{C}})$ is $W$-conjugate to $\lambda\in X_*(T_{\mathbb{C}})_+$, then $t^{\mu}\in(\mathcal{G}r_n)^T$ if and only if $t^{\lambda}\in(\mathcal{G}r_n)^T$. Our task is therefore to prove that if $\lambda\in X_*(T_{\mathbb{C}})_+$, then $t^{\lambda}\in(\mathcal{G}r_n)^T$ if and only if $\langle\lambda,w_0\alpha\rangle\geq -n$.
Suppose that $\lambda\in X_*(T_{\mathbb{C}})_+$, and let $v\in V(\alpha)$ be a vector of weight $\xi\in X^*(T_{\mathbb{C}})$. Note that for all $t\in\mathbb{C}^*$, $$\lambda(t)\cdot v=\xi(\lambda(t))v=t^{\langle\lambda,\xi\rangle}v.$$ Hence, if we regard $\lambda$ as a point in $G(\mathcal{K})$, then $$\lambda\cdot v=v\otimes t^{\langle\lambda,\xi\rangle}\in V(\alpha)\otimes t^{\langle\lambda,\xi\rangle}\mathcal{O}.$$ Since $V(\alpha)$ has a basis of weight vectors, it follows that $\text{Val}(\lambda)$ is the minimum of $\langle\lambda,\xi\rangle$, where $\xi$ ranges over the weights of $V(\alpha)$. Noting that $w_0\alpha$ is the lowest weight of $V(\alpha)$, we conclude that $\text{Val}(\lambda)=\langle\lambda,w_0\alpha\rangle$. Therefore, $\lambda\in G(\mathcal{K})_n$ if and only if $\langle\lambda,w_0\alpha\rangle\geq -n$. This completes the proof.
\end{proof}
We may thus apply Theorem \ref{thm:Direct Limit} to compute the module structure of $E_T^*(\mathcal{G}r)$ for $E_T^*=H_T^*$, $K_T^*$, or $MU_T^*$. Indeed, we have $$E_T^*(\mathcal{G}r)\cong\prod_{\lambda\in X_*(T_{\mathbb{C}})}E_T^*(\text{pt})$$ as $E_T^*(\text{pt})$-modules.
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1,314,259,995,794 | arxiv | \section{Introduction}
\label{sec:intro}
Most models of dark matter (DM) assume that it is made up of point-like
particles, or at least can be treated as such in its interactions.
However, this is not necessarily the case. In the Standard Model (SM)
most of the SM-sector mass in the universe is in the form of composite states---atoms,
containing nuclei, made up of nucleons, which are themselves made up of
quarks and gluons. Particularly noteworthy is the remarkable
richness of SM-sector nuclear and atomic physics resulting from just a few basic conservation laws, most importantly
baryon-number and charge conservation, and a few relevant interactions, dominantly the short-range
strong nuclear binding force and long-range electromagnetic interaction.
The compositeness and variety of the states that result, atoms and nuclei, plays a vital role in many important physical
processes in the history of the universe, for example, in galaxy formation and
all of stellar physics. Given the ease with which bound states can arise, it is clearly important
to consider the possibility that DM may exist in the form of composite states too.
In this work, we will investigate a particular (and as we will
argue, very attractive) class of composite DM models---those
featuring cosmologically stable ``dark nuclei'' (DN) of large
dark ``nucleon'' number (DNN). In particular we focus on the case
where, similar to SM nuclei, there is a relatively short-range
strong ``nuclear'' binding force with a hard core repulsion,
and there is an approximately-conserved quantum number, DNN,
analogous to baryon number. Unlike the SM where both protons and
neutrons are relevant, for reasons of minimality and simplicity
we will take there to be just one kind of stable dark nucleon
(or more generally, the differences between different nucleon types
to be unimportant).
Importantly, we will in addition assume that {\em in the dark sector the
analogue of the long-range electromagnetic interaction between protons
is absent}, either because there is no massless ``dark photon''
or because the stable dark nucleon is uncharged.\footnote{Models
of composite DM previously considered in the literature include
``WIMPonium"~\cite{Pospelov2008,MarchRussell2008,Shepherd2009},
``dark atoms" \cite{Kaplan:2009de}, and most similarly to our
work, ``Q-balls"~\cite{Frieman1988, Kusenko1997} (non-topological
solitons of scalar fields). In addition the
DM candidates in technicolor-like theories are often composite
states~\cite{Nussinov:1985xr, Chivukula:1989qb}, though these typically
have small ``nucleon" number, and most importantly have constituents with
SM charges, unlike the cases we consider in this work.}
With these simple assumptions there exist a very broad range of stable
DN states of varying DNN, and with a binding energy per dark nucleon
that {\it saturates} at a finite value. This is similar to the SM where
there are stable nuclei at multiple baryon numbers with approximately
constant binding energy per nucleon, $\sim 8 {\, {\rm MeV}}$, over a range of
nuclear sizes. Unlike the SM, however, there is no longer a Coulomb
repulsion term, so that in the dark sector the binding energy per
dark nucleon truly does saturate, at least until gravitational effects become
important, and never turns over, unlike the Fe maximum in the SM, so
there are {\em stable DN up to very large DNN}. In terms of a simple
liquid-drop like model, analogous to the semi-empirical mass formula
in the SM, we take the binding energy per dark nucleon to behave
asymptotically as
\begin{equation}
\frac{B_k}{ k }= \alpha - \frac{\beta}{k^{1/3}} ,
\label{eq:semf}
\end{equation}
where $k\gg 1$ is the DNN, the volume energy term is $\alpha$, and the
surface tension term $\beta$. Roughly this set of assumptions was
considered in~\cite{Krnjaic2014}, which investigated the possibility that
they could result in the early-universe formation of large DN.~\footnote{Essentially
this holds for SM nuclei if Coulomb repulsion is absent (see \cite{Krnjaic2014} for such a model), and also for many
Q-ball models~\cite{Frieman1988,Frieman1989,Kusenko1997}. In the absence of short-range
hard-core repulsion, one obtains, in the fermionic case, states such
as those investigated in~\cite{Wise2014,Wise2014ii}. These are supported by degeneracy
pressure until their constituents become relativistic,
at which point the form of the attractive interactions becomes important---they
will generally not display the kind of scaling we consider in this paper.}
While the small bound states of such a model may have
highly variable properties,\footnote{See~\cite{Detmold2014,Detmold2014ii}
for a detailed investigation of a particular QCD-like model
focussing on the properties of the small bound states.} analogous
to the special properties of D or $^4$He, once the DN become large enough,
their properties, beyond just their binding energy, eq.(\ref{eq:semf}), can
obey simple scaling laws. Most importantly certain interaction cross
sections scale geometrically in the large DNN limit, at least when
suitably averaged over a range of DNN to eliminate ``magic number'' and resonant
effects which are of subdominant importance to the physics we study.
The main focus of this paper is the regime in which such scaling
properties can determine the final number distribution of DM;
namely, when larger states are built up via the
agglomeration of smaller ones at low temperatures, as per SM Big Bang Nucleosynthesis (BBN).
Like the BBN case, the simplest version of this process
applies when the DM is {\it asymmetric} \cite{Nussinov:1985xr,Gelmini:1986zz,Chivukula:1989qb,Barr:1990ca,Kaplan:1991ah,
Thomas:1995ze,Hooper:2004dc,Kitano:2004sv,Agashe:2004bm,Cosme:2005sb,Farrar:2005zd,Suematsu:2005kp,Tytgat:2006wy,Banks:2006xr,
Khlopov:2008ty,Kitano:2008tk,Kaplan:2009ag,Kohri:2009yn,Kribs:2009fy,Cohen:2009fz,An:2009vq,Khlopov:2010pq,Cohen:2010kn,Shelton:2010ta,
Davoudiasl:2010am,Haba:2010bm,Buckley:2010ui,Hall:2010jx,Dutta:2010va,Falkowski:2011xh,Frandsen:2011kt,Graesser:2011wi,Buckley:2011kk,
Iminniyaz:2011yp,MarchRussell:2011fi,MarchRussell2012,Hardy:2014dea,Unwin:2014poa}, { i.e.}} \newcommand{\etal}{{\it et al.}, with a particle--anti-particle
asymmetry that determines the final DM abundance, similarly to the baryon
asymmetry in the SM sector (though the symmetric case is also interesting).
The small-DN part of such a scenario was investigated in~\cite{Krnjaic2014}.
In the following we assume, unless specifically stated otherwise, the DM to be asymmetric
so the DNN will take positive integer values, and we denote the DN of various DNN as 1-DN, 2-DN, etc.
In the main body of this paper, Sections~\ref{sec:dn} and~\ref{sec:agg}, we will show that the
dark-sector analogue of SM BBN---which we call
``Big-Bang Dark Nucleosynthesis'' (BBDN)---can produce DN with DNN
up to or even exceeding $10^8$, resulting in a very wide variety of
striking changes to traditional DM phenomenology.
Specifically, we find the number distributions of DN resulting from BBDN
satisfy remarkably simple forms. For example, for asymmetric DM with
plausible underlying parameter values, DN with DNN $\gtrsim
10^8$ can be synthesised, with the number distribution taking one of two
characteristic scaling forms. If there is no substantial bottleneck
of BBDN at small DNN the distribution of DN sizes takes on a
peaked, in log space,
universal non-power-like functional form, independent of many details
of the initial distribution and interactions. This solution acts as an
attractor solution and we study how the distribution function of DN
approaches this universal scaling form. On the other hand, in the case
of a substantial bottleneck of BBDN at small DNN we find the surprising
result that DN of even larger DNN, $\gg 10^8$, are often synthesised,
again with a simple form of the number distribution. Such behaviours are
studied both via numerical solutions of the relevant equations, and from
physically motivated approximate analytical solutions.
\vspace{0.1in}
Such states can give rise to a variety of
interesting possibilities, including:
\begin{itemize}
\item For reasonable parameter values effectively very heavy ($\ge
10^8 {\, {\rm GeV}}$) DM can be produced by BBDN in the form of large DN (and
with one of two possible, essentially model independent
distribution functions). Such heavy DM is in contrast to the usual
unitarity limit for point-like DM in the case of thermal freeze-out
production \cite{Griest:1989wd},\footnote{Though thermal {\em freeze-in}
production of DM can produce superheavy elementary DM~\cite{Hall:2009bx} with an
energy density yield independent of the superheavy particle mass.}
and is not usually expected in asymmetric DM models, which seek to link the
DM and visible sector abundances.
\item Coherent enhancement of interactions: Processes that
do not probe the individual constituents will have amplitude
going as the DNN $k$, so scattering cross sections, in for example direct detection
experiments, can in principle be enhanced by $k^2\gtrsim 10^{16}$ compared to the case of a single
DM nucleon. Taking account of the reduction in the number density of such states, one still
finds an effective increase in direct detection interaction rates
scaling as $\sim k$, so effectively {\it reducing} expected collider signals
by $\sim 1/k$ for a given direct detection rate compared to the standard
point like case.
The physics of the coherent enhancement of direct detection signals, with
the accompanying possibility of novel form factors from the dark sector,
is discussed in detail in a companion paper \cite{NDMdirect}.
\item Inelastic processes at both ``high energy'', of order the DN
binding energy differences, and parametrically lower energy, from
long-wavelength collective excitations. Examples of the high energy
processes are ones that move between states of different DNNs---fusions and fissions,
but there is also the possibility of the extended structure of states
leading to a spectrum of excitations, as exists in SM nuclei and atoms
\cite{NDMdirect}.
\item States with large spin: for large composite states,
there is the possibility of interactions aligning the spins
of many of the constituent states, leading to DN with nuclear spin $\sim k$.
This is of potential interest in, for example, interactions with SM nuclei
in direct detection experiments, and capture in astrophysical objects.
\end{itemize}
Since our focus is the physics of BBDN, many of the above possibilities
are either only briefly touched upon in this work, or not treated at
all, and we leave their detailed study to a series of companion papers.
Finally, before turning to our analysis of DN and BBDN we emphasise that while there are many
specific models which can realise this kind of scenario, we will focus on regimes in which
the behaviour can be broadly model-independent in nature.
\section{Basics of dark nucleosynthesis}
\label{sec:dn}
While present-day DM may be composed of large bound states,
this is generically not the case in the hot early universe.
At large temperatures $T$, the entropy term in the free energy
dominates and the chemical equilibrium distribution
has almost all DM in small-number states.
For small $T$, compared to the binding energies, the energy term
dominates, and chemical equilibrium favours large bound states.
As we discuss in detail in later sections, starting from the situation at high temperatures,
large DN may be assembled by an aggregation process where fusions dominate dissociations and
fissions. Specifically, if interactions are not so weak
as to be frozen out by the time the equilibrium shifts to favour larger
states, then larger states will be built up until fusion reactions
freeze out due to a combination of falling velocity and a falling
number density from both Hubble expansion and the formation of the large DN themselves.
Since, for the DM masses we consider, the DM is dilute, if we are in kinetic equilibrium
then the transition from being kept in equilibrium by dissociations, to fusion
reactions dominating, generally happens fast enough to be only a small perturbation
to the subsequent fusion process (this technical point is discussed in detail in
Appendix~\ref{app:chemeq}). If thermalising interactions
are not sufficiently fast to reduce the energy of de-excitation
products before they hit another DN, then these may cause dissociations,
leading to very different behaviour from the fusions-only approximation
(c.f.\ SM recombination).\footnote{\label{fn:dex}In principle,
another exception to the statement that, late on, only fusions are important
is when the excited states produced by the fusion processes de-excite
by the emission of nucleon constituents, as occurs
in the SM~\cite{Bertulani2007} (many Q-ball models also de-excite by losing
charge, e.g.~\cite{Multamaki:2000qb}).
In general these emitted small-$k$ DN are either
quickly re-absorbed by larger DN and do not significantly alter
the dynamics of the aggregation process, or they act approximately
as an external bath with which the larger DN scatter. In the latter case,
as long as enough of the small-small fusion processes can occur without involving nucleon emission,
the mass fraction of small DN will be sub-dominant,
though they may dominate the number density.}
We will not consider this regime in the current
paper, assuming instead that the de-excitation products decay or thermalise
on fast enough timescales.\footnote{For example, if there is a bath
of states which a given de-excitation product interacts with more frequently
than it does with the DN, then the de-excitation products
will thermalise with that bath quickly
(cf.\ the SM electron bath during SM BBN).
Alternatively, the de-excitation products may decay quickly to lighter
hidden sector states (in particular, as discussed in Section~\ref{sec:lightdark},
if the de-excitation products are lighter than $\sim 100 {\, {\rm MeV}}$ then they generally
need to interact with lighter hidden sector states to reduce their
cosmological density).}
\subsection{Freeze-out of fusions}
\label{sec:fusionfo}
Given that fusions dominate at low $T$, we can obtain an estimate
of the maximum size of DN built up by the aggregation process
by investigating when fusion reactions freeze out.
First, let us suppose that the last fusions to freeze out
are those between large DN, and also that we end up with
a `peaked' mass distribution in which almost all of the mass is in $\sim A$-DN.
In this case, the rate of fusions for a single DN, per Hubble time,
is $\Gamma/H \sim \langle \sigma v \rangle n_A / H$,
where $\langle \sigma v \rangle$ is the thermally-averaged fusion
cross section, and $n_A$ is the number density of $A$-DN.
Since we have DNN conservation, $n_A = n_0/A$, where
$n_0$ is the total DNN density.
If the DN are non-relativistic, as required to be aggregating,
and in thermal equilibrium (e.g., with an external bath of light dark-sector particles as
we discuss later), then the DN velocity $v_A \simeq v_1 A^{-1/2}$.
For large $A$, saturation of the dark sector nuclear forces implies the internal mass-energy
density, $\rho_b$, of DN matter is roughly constant with size, and that fusion cross
sections scale approximately geometrically, $\sigma \simeq \sigma_1 A^{2/3}$, and so
\begin{equation}
\frac{\Gamma}{H}
\sim \frac{\sigma_1 v_1 n_0}{H} A^{2/3} A^{-1/2} A^{-1}
= \frac{\sigma_1 v_1 n_0}{H} A^{-5/6} \,.
\label{eq:foscaling}
\end{equation}
In general, the temperature, $T_b$, of the DN bath can differ from the temperature, $T$, which sets the
Hubble expansion rate in the radiation dominated era, and which we assume to be dominantly set by the
more numerous SM sector degrees of freedom, as the dark and SM sectors may be essentially decoupled
from each other.
With this in mind, and using $\sigma_1 \sim 4 \pi R_1^2$,
$v_1^2 \sim T_b / m_1$, and $4\pi R_1^3 \rho_b/3 \sim m_1$,
we then find
\begin{equation}
\frac{\sigma_1 v_1 n_0}{H} \sim
2 \times 10^7 \, \left(\frac{1 {\, {\rm GeV}} {\, {\rm fm}}^{-3}}{\rho_b}\right)^{2/3}
\left(\frac{g_\star(T)}{10.75}\right)^{1/2}
\left(\frac{m_1}{1 {\, {\rm GeV}}}\right)^{-5/6}
\left(\frac{T}{1 {\, {\rm MeV}}}\right)^{3/2}
\left(\frac{T_b}{T}\right)^{1/2} ,
\label{eq:fon1}
\end{equation}
where the parameters are normalised to SM values both for comparison and
because such a parameter region is naturally motivated by asymmetric
dark matter (ADM) models.
Thus, in this scenario, if dissociation stops being important
around $T = 1 {\, {\rm MeV}}$, then, from eq.(\ref{eq:foscaling}), the largest mass that could have been built up
is $(2 \times 10^7)^{6/5} m_1 \sim 5 \times 10^8 {\, {\rm GeV}}$,
corresponding to a radius of $\sim 480{\, {\rm fm}}$ for the fiducial parameter values.\footnote{Note that this does
not depend on the dark nucleon mass $m_1$. If some scaling other than $m_k\propto k$ between mass and
DNN held, e.g.\ as in the case of some Q-ball models~\cite{Frieman1989},
then eq.(\ref{eq:foscaling}) would still apply, but with $A$ replaced by the ratio of the final
mass to the mass for which $\sigma_1 v_1 n_0$ was evaluated.
Such models, in which binding energies were not a small fraction of the total mass,
have quite different properties.}
Beyond this size, the number density and velocity are too low for interactions
to occur. We will see that, for reasonable parameters, this bound
may be attained.
Note that, if we scale all
of our dimensional parameters to increase mass scales by a factor
$\lambda$ (i.e.\ $T \mapsto \lambda T$, $\rho_b \mapsto \lambda^4 \rho_b$, etc.),
then the freeze-out DN mass scales as $m_{{\rm fo}} \mapsto \lambda^{-7/5} m_{{\rm fo}}$.
Changing the mass scales of the constituents, e.g.\ by changing the confinement
scale in a strongly coupled theory, may be expected to have roughly this effect.
Thus, {\em decreasing} the mass scale of our constituents will tend to
{\em increase} the masses that could be built up, mainly since larger
geometric cross sections are available.
Going the other way, for $m_1 \gtrsim 100 {\, {\rm TeV}}$ (and corresponding
scalings of the other parameters), we have $\sigma_1 v_1 n_0 / H < 1$,
so there is no build-up in this regime, corresponding to
the usual unitarity bound for DM self-annihilations.
Alternatively, the last fusions to freeze out may be those
between `small' $+$ `large' DN.\footnote{The precise meaning of `small'
will depend on the aggregation dynamics, and as it becomes comparable
to the eventual sizes obtained, this scenario will approach the large $+$ large
case discussed above.}
Compared to large + large fusions,
while the possible number density of small DN is larger,
the number of separate fusion events onto a given DN needed for
the same increase in size is larger by the same factor.
However, since in thermal equilibrium the velocities
of small DN are larger, the overall rate of size
increase is {\em enhanced} by that factor.
To qualitatively understand this scenario, suppose that an order one proportion
of the dark nucleons are in small $k$-DN, and the rest in large $A$-DN.
Then, the rate at which an $A$-DN increases in size by $A$ is
given by
\begin{equation}
\Gamma \sim \langle \sigma v \rangle n_k \frac{k}{A}
\sim \delta \frac{1}{4} \sigma_1 v_1 n_0 k^{-1/2} A^{2/3} A^{-1} ~ ,
\label{eq:fobneck}
\end{equation}
where we used $\langle \sigma v\rangle_{A + k} = \frac{1}{4} \delta \sigma_1 v_1 k^{-1/2} A^{2/3}$,
with $\delta$ parameterising a possible suppression of small-large cross-sections
from the geometric value, e.g.\ from quantum reflection effects, as per
SM nucleon-nucleus interactions. So, the maximum attainable DNN of the DN is
\begin{equation}
A_{\rm fo} \sim 7 \times 10^{19} \left( \frac{\delta}{\sqrt{k}} \,
\frac{\sigma_1 v_1 n_0/H}{2 \times 10^7}\right)^3 ,
\end{equation}
corresponding to $m \sim 7 \times 10^{19} {\, {\rm GeV}}$ (and
scaling as $m_{{\rm fo}} \mapsto \lambda^{-5} m_{{\rm fo}}$) and
a radius of $3 \times 10^6 {\, {\rm fm}}$ for our fiducial parameter choices.
Similarly, the rate of fusions onto an $A$-DN for a given
$k$-DN is $\Gamma \sim \langle \sigma v \rangle n_A
\sim \frac{n_A}{n_k} \langle \sigma v \rangle n_k$,
so the largest DN that can absorb the entire population
of small DN also have approximately this same maximum size.
Thus, as long as $\delta$ is not too small,
and a large enough population of small DN exists
for long enough, there is the possibility
of producing much larger DN via this route,
than via an aggregation process in which the DN at any given
time are of approximately the same size.
Obtaining significant quantities of DN this large requires that a
population of small DN remains around until the end of the aggregation process,
i.e.\ that small $+$ small fusions are slow. If this is the case, but
small $+$ large fusion cross sections are roughly geometrical, and we
produce a number density of `seed' large DN $\sim n_0/A_{\rm fo}$, then,
as studied in Section~\ref{sec:bn}, we will end up with most of
the DM mass in DN of size $\sim A_{\rm fo}$. If we produce a larger seed density, the
maximum size will be reduced proportionally, up to a cross-over point at
which this size is lower than the freeze-out limit for large $+$ large
fusions, at which point we enter that regime. If a smaller
seed density is produced, then most of the DN never gets through the `bottleneck',
and remains small, with some sub-dominant population of large DN up to
$m_{\rm fo}$. Section~\ref{sec:bn} investigates these regimes in detail.
\subsection{Bottlenecks, and comparison to BBN}
Bottlenecks to nucleosynthesis, in the form of
`anomalously slow' small + small fusion rates for certain
channels, are important in Standard Model BBN.
There, only small nuclei are synthesised, with almost all
of the nucleons ending up in H and $^4$He, only small
amounts in the other $A\leq 7$ nuclei, and entirely negligible amounts
beyond this. The $B_D \sim 2 {\, {\rm MeV}}$ binding energy of
D means that, assuming only $A=1,2$ states are occupied,
$n_D/n_p \sim \eta (T/m_p)^{3/2} e^{B_D/T}$,
and since the baryon to photon ratio $\eta \simeq 6 \times 10^{-10}$,
this only becomes $\simeq 1$ at $T \simeq 0.06 {\, {\rm MeV}}$.
However, slightly before this temperature,
the D number density becomes large enough that $2+1$ and other processes
start occurring, and in fact their rates are $\gtrsim 10^4$
times the Hubble rate, as expected from a calculation along the lines of
eq.(\ref{eq:fon1}).
The real bottleneck preventing the synthesis of large
nuclei in the SM is the large binding energy per nucleon of $^4$He compared to
subsequent small nuclei. It is not until $^{12}$C that the binding energy per nucleon
exceeds $^4$He. In quasi-equilibrium among small-number nuclei, this large binding
energy means that $^4$He dominates the mass fraction along
with H, their ratio being set by the abundance of neutrons. The small number densities
of $A=7$ nuclei produced are nowhere near sufficient to make the rate of
production of energetically favoured $A \ge 12$ nuclei comparable
to the Hubble rate, and so we freeze out with
quasi-equilibrium abundances (for a textbook discussion see e.g.~\cite{Mukhanov2003}).
Note that this kind of bottleneck, involving a wrong-sign
binding energy difference rather than merely a small right-sign
difference as in the D example, gets worse rather than better
with decreasing temperature.\footnote{In the D case,
since $n_D$ increases exponentially
with falling temperature, the number of $A \ge 3$ states produced
beforehand is insufficient to bypass the bottleneck.
For this reason, it is somewhat difficult for the bypass process
to be dominant for right-sign binding energy differences.}
In summary, SM BBN provides an example of a nucleosynthesis
process where the presence of large binding energy differences among
small-number states creates a bottleneck. In fact, there are two
bottlenecks, the first of which, the D bottleneck, we get through well
before fusions have frozen out, but not the second, post-$^4$He.
A bottleneck in BBDN may similarly have the result of
preventing significant quantities of large DN from being formed.
However, there is also the possibility, as discussed in the previous section,
of a suitable bottleneck leading to the build-up of
even larger DN than would otherwise have been produced,
with a qualitatively different distribution of DN sizes.
Section~\ref{sec:bn} explores this scenario in detail.
\section{Aggregation process}
\label{sec:agg}
Recapping, the cross-over from the high-$T$ regime to that in which
dissociations are unimportant occurs in a sufficiently short time that only DN much smaller
than the freeze-out size are able to be built up in appreciable quantities during this period.
After this, there will be a period of effectively fusion-only aggregation
before we reach one of the limits discussed in Section~\ref{sec:fusionfo}.
This section investigates the details of this aggregation process.
Evolving with fusions only, the
Boltzmann equation for the $k$-DN number density, $n_k(t)$, is
\begin{equation}
\frac{d n_k(t)}{d t} + 3 H(t) n_k(t) = -\sum_{j = 1}^\infty \langle \sigma v\rangle_{j,k}n_j(t) n_k(t)
+ \frac{1}{2}\sum_{i + j = k}\langle \sigma v \rangle_{i,j} n_i (t) n_j(t)\,,
\end{equation}
where $\langle \sigma v\rangle_{i,j}$ is the velocity-averaged
cross section for $i + j \rightarrow (i+j)$.\footnote{When de-excitation
from some fusion events is via nucleon emission,
then, as mentioned in footnote~\ref{fn:dex}, the behaviour
may still approximately follow the fusions-only aggregation solution.
If large + large fusions are the dominant process,
as in the `scaling' regime discussed in Section~\ref{sec:scaling},
then the behaviour of small DN is not crucial.
If small + large fusions are most important, as in the
`bottlenecked' regime~\ref{sec:bn}, and these dominantly
de-excite via nucleon emission, this will just reduce their
net forward rate by some factor, and, in simple cases,
leaves the qualitative behaviour otherwise intact.}
This equation combines together the dilution of number densities
due to Hubble expansion, the change in cross sections due to
the decrease in DN velocities, and the change in the relative concentrations
of $k$-DN. We can separate these processes out by moving to different
variables. Writing the velocity-averaged cross sections as
$\langle \sigma v \rangle_{i,j} = \sigma_1 v_1 K_{i,j}$,
where the $K_{i,j}$ encode the relative rates of different fusion processes,
the Boltzmann equation in terms of the dimensionless yields $Y_k \equiv n_k/s$,
where $s(t)$ is the entropy density, is
\begin{equation}
\frac{d Y_k(t)}{dt} = \sigma_1 v_1(t) s(t) \left(
- Y_k(t) \sum_j K_{j,k}(t) Y_j(t) + \frac{1}{2} \sum_{i+j=k} K_{i,j}(t) Y_i(t) Y_j(t)\right) \,.
\end{equation}
Here we assume that entropy in conserved throughout the aggregation
process, so $s \propto a^{-3}$. This system of aggregation equations can be solved
in terms of relative concentrations, $y_k \equiv Y_k/Y_0$ (with $Y_0 = n_0/s$ the yield of dark
\emph{nucleons} irrespective of how they are distributed among DN of different sizes) and by
defining a new, dimensionless time $w$:
\begin{equation}
\frac{d y_k}{d w} = - y_k \sum_j K_{j,k} y_j + \frac{1}{2} \sum_{i+j=k} K_{i,j} y_i y_j \,,
\label{eq:agg}
\end{equation}
where
\begin{equation}
\frac{dw}{dt} = Y_0 \sigma_1 v_1(t) s(t) = n_0(t) \sigma_1 v_1(t) \, .
\label{eq:wt}
\end{equation}
Note that $Y_0$ is constant in time (assuming no entropy injection) by nucleon number conservation, and
the solution is normalised such that $\sum_k k \, y_k = 1$ throughout.
In words, the function $w(t)$ describes how quickly the number distribution of
DN changes, whereas the set of distributions moved through is determined
by the form of the `aggregation kernel' $K_{i,j}$ and the
corresponding solution to the aggregation system eq.(\ref{eq:agg}).
\subsection{Scaling regime}
\label{sec:scaling}
The system of aggregation equations, eq.(\ref{eq:agg}) for all $k$, is simplest to solve when the
$K_{i,j}$ do not depend on $w$, i.e.\ when we can absorb all of the time/temperature
dependence into the single quantity $v_1(t)$. The simplest case
in which this is true is when the DN are in kinetic equilibrium with each
other, at some temperature $T_b(t)$. $k$-DN then have mean-square
velocities $\langle v^2 \rangle = T_b/m_k = k^{-1} (T_b/m_1)$.
If fusion cross-sections scale approximately geometrically
for large DN, then a kernel of the form
\begin{equation}
K_{i,j} = (i^{2/3} + j^{2/3})\left(\frac{1}{i^{1/2}} + \frac{1}{j^{1/2}} \right) \,,
\label{eq:ker1}
\end{equation}
is a good approximation for large DN.\footnote{This specific form
matches cases investigated in the mathematical
literature~\cite{Leyvraz2006}.} In this expression we have, for
simplicity, replaced the relative velocity, which is non-relativistic in
the fusion regime, by an averaged velocity.
Figure~\ref{fig:ndist} shows, in both $w$ and $t$ space, numerical solutions for the
aggregation dynamics eq.(\ref{eq:agg}) with this kernel, starting from $y_1(0) = 1$
initial conditions. What is immediately noticeable is that the number distributions at different
times have almost the same shape, but shifted relative
to each other corresponding to an increase in average size. This arises
because our kernel is \emph{homogeneous}, $K_{b i, b j}
= b^\lambda K_{i,j}$ with, here, $\lambda = 1/6$. Physically,
if we scale up the DNN of all of the DN by some factor, this
just corresponds to an \emph{overall} scaling of the rate of fusions, and
doesn't change the relative rates of different-number processes.
It is known \cite{RednerBook} that, for such kernels, there is generally a `scaling solution'
such that almost any finitely-supported initial conditions eventually converge
to the form
\begin{equation}
y_k(w) = \frac{1}{\bar{k}(w)^2} f\left( \frac{k}{\bar{k}(w)}\right) \,,
\end{equation}
where $\bar{k}(w) \propto w^{1/(1-\lambda)}$ is the `characteristic size' of DN at time $w$,
$f$ is the kernel-dependent scaling solution, and the $1/\bar{k}^{2}$ factor ensures correct normalisation.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{ndist0.pdf}
\includegraphics[width=0.49\textwidth]{ndist1.pdf}
\caption{\emph{Left:}
Number distributions obtained by solving the system of equations
determining aggregation dynamics, eq.(\ref{eq:agg}) for all $k$,
from single-nucleon initial conditions, $y_k(0) = \delta_{k1}$, at equally-spaced
values in $\log w$ up to a maximum of $w = 75$. The first curve is a Kronecker delta
$y_1(0) = 1$, the second curve is the one with smallest intercept on the $k$ axis, etc.
Note that distribution of DN sizes very quickly broadens in $w$-time.
\emph{Right:} The same number distributions at half $e$-folding time
intervals, assuming DN temperature falling with scale factor and $w(t)$ evolving
as per eq.(\ref{eq:wTrel}), with $w_{\max} = 75$. As the physical time goes to infinity
the distribution is given by the $w=w_{\max}$ distribution shown by the thick solid curve.}
\label{fig:ndist}
\end{center}
\end{figure}
In our case, we can check whether the numerical solutions in
Figure~\ref{fig:ndist} obey this scaling behaviour by choosing
a plausible form for $\bar{k}$ (e.g.\ for a peaked number distribution,
we could take one over the total number of DN, $(\sum_k y_k)^{-1}$),
and plotting
$\bar{k}^2 y_k$ against $k/\bar{k}$, as shown
in Figure~\ref{fig:sdist}. We see that, for $y_1(0) = 1$ initial
conditions, the distribution converges very quickly to such a scaling solution, which
can then be used to extrapolate to larger $w$ values.
From the mathematical theory of aggregation developed in other contexts~\cite{Ernst1988,Leyvraz2006}, we expect
$f(x)$ to drop as $\sim x^{-\lambda} e^{-{\rm const.}\ x}$ for $x \gg 1$, and to have the form $f(x) \sim
x^\theta e^{-{\rm const.}/\sqrt{x}}$ for $x \ll 1$, with $\theta$ some constant, both of which match
sensibly onto the numerically obtained form. Also, the behaviour of
$\bar{k}(w)$ with $w$ matches, for larger $w$, the $\propto w^{6/5}$ form
predicted from $\lambda = 1/6$. This also holds for different choices of $\bar{k}$, as expected for
a peaked number distribution.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{scalingDist1.pdf}
\includegraphics[width=0.49\textwidth]{swscaling1.pdf}
\caption{Illustration of convergence of aggregation equation solutions
to `scaling' behaviour.
\emph{Left}: Solutions of aggregation equation after different
`times' $w$ (with $y_1(0) = 1$), plotted on scaled axes $\bar{k}^2 y_k$ versus
$k/\bar{k}$, where $\bar{k} = 1/\sum_k y_k$.
Green dashed line shows solution at $w = 3$, yellow at $w = 8$, red at $w = 25$,
and blue dashed at $w = 75$.
The almost complete overlap shows that convergence to the attractor
form of the `scaling solution' is rapid.
\emph{Right}: Plot of $\bar{k}$ behaviour with time---dots
are numerical $\bar{k}(w)$ values, lines are $w^{6/5}$ curves.
Blue are for $\bar{k}(w) \equiv 1/\sum_k y_k$, red are for $\bar{k}(w) \equiv \sum_k k^2 y_k$.
}
\label{fig:sdist}
\end{center}
\end{figure}
\subsection{(In-)Dependence on initial conditions}
\label{sec:initialcond}
As discussed in Section~\ref{sec:dn}, we do not expect to
start the fusions-only aggregation process with single-nucleon-only
initial conditions---instead, we will have whatever was produced
during the phase when dissociations were still important.
Furthermore, fusion cross sections between small DN
will probably not be well approximated by the geometrical
scaling appropriate to large DN; for example, SM cross
sections between small nuclei display very complicated behaviour.
Since, as we discuss in the next section, there is only a finite
$w$ time available for aggregation due to Hubble expansion, the question is whether
these initial conditions, and small-$k$ behaviour, converge to the scaling solution
quickly enough for appropriate measures of convergence.
Generally, initial conditions can be separated into a component for
which similar-size fusions are not frozen out, and one for which they are
(this component should be sub-dominant, otherwise aggregation does not significantly affect
the distribution at all).
Treating the latter case, in which the initial conditions include a subdominant large-$k$ `tail' for
which large~$+$~large fusions are frozen out (see the right panel
plot of Figure~\ref{fig:ictest} as an example), then as long as the aggregation
of smaller DN proceeds not much slower than the scaling behaviour,
only a small proportion of the small DN fuse with those in the tail.
To see this we can approximate the tail by purely $A$-DN, and use the fact that the
rate at which a given small $k$-DN fuses into the tail is $\Gamma_{k+A} \sim \delta \sigma_1 v_1 n_A A^{2/3} k^{-1/2}$.
So, writing the fraction of the DN in the tail as $\alpha = A n_A/n_0$,
\begin{equation}
\frac{d (1-\alpha)}{dt} \sim \Gamma_{k+a} (1-\alpha)
\quad \Rightarrow \quad
\frac{d\log(1-\alpha)}{dw} \sim - \delta \frac{n_A}{n_0} A^{2/3} k^{-1/2} ~.
\end{equation}
While $\alpha$ is small, so the small-$k$ build-up proceeds
like the scaling solution, we have $k \sim w^{6/5}$.
Since $A+A$ fusions are frozen out, $n_A$ is roughly constant throughout, so
\begin{equation}
\frac{d\alpha}{dw} \sim \delta \left(\frac{n_A}{n_0}\right)^{1/3} \alpha^{2/3} w^{-3/5}
\quad \Rightarrow \quad
\alpha_{\max} \sim \delta \frac{n_A}{n_0} w_{\max}^{6/5} \sim \delta \frac{n_A}{n_0} k_{\max} \,.
\end{equation}
Thus, either $\delta k_{\max} \gtrsim A$, in which case the tail is subsumed into
the scaling distribution, or else $\alpha$ is always $\ll 1$.
Note that this only occurs because $k$ increases sufficiently quickly with $w$.
As we shall see in Section~\ref{sec:bn}, if that does not happen,
then none of the DN may ever reach the scaling regime,
and $\alpha_{max} \sim \frac{n_A}{n_0} w_{\max}^3$ can become of order 1,
so all of the small DN can be absorbed by the frozen-out tail.
Dealing with the other case, suppose that the initial conditions have some component for
which similar-size fusions are not frozen out, e.g.\ the
left panel of Figure~\ref{fig:ictest}. If these have the same
average size as a monomer-only scaling solution after $\delta w$, then
the eventual distribution will be close to the monomer-only solution
after $w_{\max} + \delta w$. Since small $+$ large fusions are faster
than large $+$ large, the `memory' of the initial shape is erased fairly
efficiently.\footnote{Similarly, if the small-$k$ cross sections are larger than
those extrapolated down from large $k$, the effect is to start the
large-$k$ process slightly earlier, $w \rightarrow w + \delta w$, while if the
cross sections are smaller than the extrapolation then the solution
interpolates between the scaling and bottlenecked regimes---see
Section~\ref{sec:bn}.}
Figure~\ref{fig:ictest} illustrates these behaviours numerically,
with the left panel showing the convergence of non-frozen-out
initial conditions to a slightly-further-along scaling distribution,
and the right panel showing that a sub-dominant frozen-out tail
has little effect on the scaling solution obtained.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{expIC.pdf}
\includegraphics[width=0.49\textwidth]{tailIC.pdf}
\caption{Illustration of convergence of aggregation equation solutions
to `scaling' behaviour, for more complicated initial conditions.
\emph{Left}: Coloured curves are solutions of aggregation
dynamics, starting from $y_k(0) = e^{-k/30}$ initial conditions, at
equally-spaced $w$ values from 0 to 25. For comparison solid black curve shows
solution at $w=25$ for monomer-only initial conditions,
while black dashed curve is monomer-only solution at the later time $w=33.25$.
We see that despite the change in initial conditions we still end up
with the scaling distribution but now it is slightly shifted in $w$-evolution
compared to the monomer-only case.
\emph{Right}: Solutions of aggregation dynamics at
equally-spaced $w$ values from 0 to 25,
starting from initial conditions with most of the mass in single nucleons
($y_1(0) = 0.97$), but also a sub-dominant population
in a broad tail out to large $k$ values (as might arise
from e.g.\ build-up while most of the nucleons are trapped
behind a low-$k$ bottleneck).
The black curve is the solution at $w = 25$ for monomer-only initial conditions.
Since the initial state has only a small fraction of the nucleons in
the tail, and fusions between states in the tail are frozen out,
the tail has no significant effect on the majority of the scaling solution.
}
\label{fig:ictest}
\end{center}
\end{figure}
\subsection{Real-time behaviour}
The scaling distribution gives us the form of the $y_k(w)$ solution---to
obtain the real-time distribution $y_k(t)$, we need to know the
behaviour of $w(t)$ from solving eq.(\ref{eq:wt}). Generally, we are
most interested in the $t \rightarrow \infty$ limit, so want to find
$w(t\rightarrow \infty)$. To obtain this we in turn need to know the
behaviour of $v_1(t)$, which is simple in the case that the DN are in
\emph{kinetic} equilibrium throughout the aggregation process.
The simplest way for the DN to be in kinetic equilibrium
is for them to be in thermal contact with a bath of lighter particles
during the aggregation process. If this bath is relativistic,
so $T_b \propto 1/a$ (here ignoring possible changes in the number of species for simplicity),
and taking $v_1 = \sqrt{T_b/m_1}$, we obtain, assuming radiation domination,
\begin{equation}
\frac{dw}{dT} \simeq - \left. \frac{n_0 \sigma_1 v_1}{H}\right|_{T_0}
\left(\frac{T}{T_0}\right)^{1/2} \frac{1}{T_0} \,,
\end{equation}
where $T_0$ is the temperature when $w = 0$, i.e.\ at the start of the aggregation
process.
Solving this,
\begin{equation}
w(T) \simeq \frac{2}{3} \left. \frac{n_0 \sigma_1 v_1}{H}\right|_{T_0} \left(1 - \left(\frac{T}{T_0}\right)^{3/2}\right) \rightarrow
\frac{2}{3}\left. \frac{n_0 \sigma_1 v_1}{H}\right|_{T_0} \,,
\label{eq:wTrel}
\end{equation}
where the limit is $T \searrow 0$.
The right panel of Figure~\ref{fig:ndist} shows the solutions
obtained at different $t$ values, assuming this relationship between
$w$ and $T$, illustrating the convergence towards the $w = w_{\max}$
solution.
Alternatively, if the hidden sector bath is non-relativistic, so $T_b \propto a^{-2}$, we obtain
\begin{equation}
w(T) \simeq \frac{1}{2}\left. \frac{n_0 \sigma_1 v_1}{H}\right|_{T_0} \left(1 - \left(\frac{T}{T_0}\right)^{2}\right) \rightarrow \frac{1}{2}\left. \frac{n_0 \sigma_1 v_1}{H}\right|_{T_0} \,.
\label{eq:wTnr}
\end{equation}
Since $\bar{k} \sim w^{6/5}$, we have $\bar{k}(t \rightarrow \infty) \sim
\left(\left. \frac{n_0 \sigma_1 v_1}{H}\right|_{T_0}\right)^{6/5}$ in both cases.
This agrees with the approximate freeze-out calculation of
eq.(\ref{eq:foscaling}), which had $\Gamma/H \sim \frac{n_0 \sigma_1 v_1}{H} A^{-5/6}$.
Generally, since from eq.(\ref{eq:wt}) $\frac{dw}{d\log
a} = \frac{n_0 \sigma_1 v_1}{H}$, and during the radiation
era we have $H \propto a^{-2}$, $n_0 \propto a^{-3}$, and
writing $v_1 \propto a^{-\gamma}$, we have
$\frac{dw}{d\log a} \sim a^{-(1 + \gamma)}$, so,
$\Delta w \sim \Delta \left( a^{-(1 + \gamma)}\right)$
as we obtained in eqs.(\ref{eq:wTrel}) and~(\ref{eq:wTnr}).
Thus the {\it bulk of the aggregation process takes of order a Hubble time to
complete}. This is illustrated in the right panel of Figure~\ref{fig:ndist},
which shows solutions at half-$e$-folding-time intervals.
Such behaviour just comes from the freeze-out properties
of the interactions, so in the bottlenecked regime we take at
most this long as well---in fact, as we shall see in the next section,
that process may take much less than a Hubble time.
\subsection{Bottlenecked regime}
\label{sec:bn}
As illustrated in Section~\ref{sec:scaling}, if the fusion
rates, as parameterised by $w_{\max}$ and the $K_{i,j}$,
are not too small compared to the support of our initial conditions, then we reach the scaling
regime of the attractor solution. However, if some of the fusion rates are reduced far enough
to `trap' a proportion of the DN in a small-$k$ region
for long enough, then we will not reach the scaling regime.
Counter-intuitively, this can result in building
up \emph{larger} DN than would otherwise have been
the case. As roughly described by eq.(\ref{eq:fobneck}), this
occurs because small $+$ large fusions are less velocity-suppressed
in kinetic equilibrium than large $+$ large fusions, so,
if there is a bath of small DN present throughout the aggregation
process, build-up interactions may freeze out at higher $A$.
Figure~\ref{fig:scalingbn} shows a simple example
of moving between the scaling and bottlenecked regimes,
which may be helpful to keep in mind through the following.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{scaling_to_bn_linear.pdf}
\includegraphics[width=0.49\textwidth]{scaling_to_bn.pdf}
\caption{Illustration of transition between scaling regime and bottlenecked regime.
\emph{Left:} Mass distribution at $w = 25$, starting from initial conditions
of single nucleons, $y_k(0) = \delta_{k1}$, for the original kernel with $K_{1,1} = 4$ (purple),
and modified kernels with $K_{1,1} = 10^{-4}$ (red), and $K_{1,1} = 10^{-5}$ (yellow).
We see that, within this range, making $1+1$ fusions slower results in building up larger DN.
\emph{Right:} Number distribution at $w=25$, again starting from initial conditions of single nucleons,
for $K_{1,1} = (4, 1, 10^{-1}, 10^{-2}, \dots, 10^{-6})$. The dotted line aligned with
the $K_{1,1} = 10^{-6}$ (green) curve is $\propto k^{-2/3}$.
We transition from converging very quickly to the scaling solution,
to ending up with a power-law distribution cutting off at larger $k$.
Between $K_{1,1} = 10^{-5}$ and $10^{-6}$, the maximum $k$
reached hardly increases, with the overall number density in the power-law tail
just going down---we have reached the freeze-out limit for building up large DN.
}
\label{fig:scalingbn}
\end{center}
\end{figure}
Since small $+$ large fusions keep the number density
of large DN the same, they act to increase the rate of large $+$ large fusions,
which goes as $A^{2/3} A^{-1/2} = A^{1/6}$.
If $\Gamma/H$ becomes of order 1 or higher, then large-large
fusions will start operating and bring it down to $\sim 1$, establishing
a scaling distribution for the larger DN. Since the rate of fusions for
a single small DN is larger than that for large DN, this also
means that all of the small DN will be used up, clearing the bottleneck
and placing us in the scaling regime.
The more interesting case is when large-large interactions are always
frozen out. We can then model the aggregation process as a combination
of the slow creation of large post-bottleneck `seed' DN, and the fast
accretion of small pre-bottleneck DN onto these. Looking first
at the accretion process, from our previous assumptions the growth
rate for a large DN scales geometrically, $\propto R^2 \sim A^{2/3}$.
So, $dk/dw \sim k^{2/3} y_s$, where $y_s$ parameterises the concentration
(and size)
of small DN, and thus $k \sim \left( \int dw \, y_s \right)^3$.
If the bath of small DN is populated throughout most of the $w$ time, then
$k_{\max} \sim w_{\max}^3$, realising the freeze-out bound of eq.(\ref{eq:fobneck}).
As well as the maximum size, we are also interested in the number
distribution over $k$. If a seed DN is produced at time $w_{\rm inj}$,
then its eventual size will (for roughly constant $y_s$) be $k \sim
(w_{\max} - w_{\rm inj})^3$. More precisely, if we change time variable
to $z$, where $dz/dw = y_s$, we have $k \sim (z_{\max} - z_{\rm
inj})^3$. This means that the relationship between injection time
$z_{\rm inj}$ and final size $k$ is given by $-\frac{dz_{\rm inj}}{dk} \sim k^{-2/3}$.
Next, write $a_k(z)$ for the total concentration of seed $k$-DN injected by
time $z$. Since accretion only changes the $k$-number, and not the density,
of an injected seed population, we obtain
\begin{equation}
y_k = -\frac{d}{dk}\left(\sum_j a_j(z_{\rm inj}(j \rightarrow k))\right)
=
-\sum_j \frac{d a_j}{dz} \frac{dz_{\rm inj}(j \rightarrow k)}{dk}
\sim k^{-2/3} \sum_j c_j \frac{d a_j}{dz}
\end{equation}
For $k$ large compared to the $j$ which dominate the sum,
we can write this as $k^{-2/3} f_i(z_{\rm inj})$.
In particular, if for small $z$ the injection rate is roughly
constant (this is reasonable, since only a small fraction of the pre-bottleneck bath
has been used up, and the temperature, scale factor etc. change by only
a small amount), then for large $k$ we should have a power law
number distribution $y_k \sim k^{-2/3}$.
As we increase the injection rate $f_i$, we move through the
three different regimes described at the end of Section~\ref{sec:fusionfo}:
\begin{itemize}
\item If the injection rate is small enough, then most of the small
bath is not used, and most of the mass is stuck behind the bottleneck,
with a small proportion in a $y_k \sim k^{-2/3} f_i(z_{\rm inj})$ tail,
which extends to $k_{\max} \sim w_{\max}^3$. As an example, if we assume that
$f_i$ decreases with time, such that the nucleon number integral
$\int^{k_{\max}} dk\, k \, k^{-2/3} f_i$ is dominated by the upper
decade, $\sim k_{\max}^{4/3} f_i(0)$, then the upper limit of this regime
is given by $f_i(0) \sim k_{\max}^{-4/3} \sim w_{\max}^{-4}$.
\item For larger injection rates, we use up all of the small bath before
we reach $w_{\max}$, so we build up to correspondingly smaller $k_{\max}$.
Under the same assumptions on $f_i$, we have $k_{\max} \sim f_i(0)^{-3/4}$.
\item For sufficiently large injection rates, large-large fusions eventually become
important, and we enter the scaling regime. We expect this to happen roughly when
$k_{\max}$ for the addition process is comparable to the $k$ values of the scaling
peak. Since this has $\bar{k} \sim w^{6/5}$, under the above assumptions
on $f_i$ we expect the cross-over to be at around $f_i(0) \sim w_{\max}^{-8/5}$.
\end{itemize}
In the first regime, the proportion of DM in the pre-bottleneck bath stays
roughly constant, so $\frac{dz}{dw} = y_s \simeq {\rm const}$.
From the previous section, $\Delta w \sim \Delta (a^{-(1 + \gamma)})$,
so the bulk of the process again takes of order a Hubble time.
For the intermediate regime, we use up most of the small bath within
$\Delta w \simeq (3 k_{\rm max})^{1/3} < w_{\rm max}$,
a small fraction of a Hubble time. For the rest of $w$ time,
either large-large fusions are frozen out, in which case we only
make small modifications to the number distribution,
or we enter the scaling regime.
Figure~\ref{fig:scalingbn} shows the numerical solution of a particularly
simple bottlenecked example---using the geometrical kernel of
eq.(\ref{eq:ker1}), but reducing $K_{1,1}$. In this case,
the injection rate $f_i \propto y_1^2$, so for most of the addition
process it is constant, giving a power-law number distribution $y_k \sim k^{-2/3}$
for large $k$. The right panel of Figure~\ref{fig:scalingbn} illustrates how we move through
the three regimes identified above as we decrease $K_{1,1}$. We start
out converging quickly to the scaling distribution, but for
$K_{1,1} \lesssim 10^{-4}$ we never reach this regime, ending up with
a power law out to larger $k$. For $K_{1,1} \lesssim 10^{-5}$,
most of the $k=1$ population never makes it past the bottleneck,
and our power-law tail goes to smaller concentrations
rather than higher $k$ values.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{1plus_linear.pdf}
\includegraphics[width=0.49\textwidth]{1plus_comparison.pdf}
\caption{Further illustrations of behaviour in the bottlenecked regime.
\emph{Left:} Mass distribution at $w = 50$, starting
from single nucleons, for original kernel $K_{1,1} = 4$ (purple),
and modified kernels with
$K_{1,1} = 10^{-5}$ (yellow), $K_{1,1} = 10^{-6}$ (green) (latter two
in the `addition approximation', i.e. only taking into account
$1 + k \rightarrow (1+k)$ fusions). Comparing to
Figure~\ref{fig:scalingbn}, going to larger $w$ increases the difference
between the bottlenecked and scaling solutions.
\emph{Right:} Solid lines are number distributions
at $w = 25$, starting from single nucleons, for $K_{1,1} = 10^{-4}$ (red), $K_{1,1} = 10^{-5}$ (yellow), and
$K_{1,1} = 10^{-6}$ (green). Dashed lines are number distributions
in the addition approximation.
}
\label{fig:1plus}
\end{center}
\end{figure}
The right panel of Figure~\ref{fig:1plus} shows explicitly that,
for the bottlenecked solutions, we are close to being in the
addition-dominated regime, i.e.\ the dominant process is $1 + k \rightarrow (k+1)$.
Comparing the left panel to Figure~\ref{fig:scalingbn} illustrates that, as we increase $w_{\max}$,
the difference between the $k_{\max}$ attainable in the
scaling and bottlenecked regimes increases.
As was the case in the scaling regime, sufficiently small changes in the
initial conditions, or in the rates of individual processes, will not
make a major difference to the eventual number distribution. If we start
with a small mass fraction of the DN past the bottleneck, this will be
equivalent to an injection spike at $z=0$, and so a bump at the end
of the overall tail. If some of the rates for small $+$ large fusions
differ from the geometrical approximation, say for large DN of size
$k$, then this will affect all of the large DN that grow to sizes $> k$---as
long as the reduction in the rate is not large enough to cause a
further bottleneck, there will be little change in the qualitative form
of the final distribution.
\section{Aspects of dark sector phenomenology}
So far, we have investigated the BBDN process,
and the number distribution of DN that may result from it.
The DM self-interactions, and the extended
hidden sector needed to realise such models, mean that these
theories generically have the possibility of interesting hidden sector
phenomenology, including non-standard indirect detection signals,
modifications of halo properties, and early-universe signatures of extra species.
Also, though not required in these models, it is possible
that the DN have sufficiently strong interactions with
SM states to give signals in future direct detection experiments.
If that is the case, then such signals may differ from those
of usual WIMP DM, and have different relationships
to collider bounds \cite{NDMdirect}. Additionally, such interactions
may lead to the capture of DN in astrophysical objects,
and then self-interactions among captured DN could
become important.
In standard ADM models, annihilations with the relic
symmetric population can lead to astrophysical energy injections,
most detectably in the early universe,
putting a limit on how large this population can be. In nuclear DM models
there is additionally the energy injected by inelastic processes, particularly
the release of binding energy from fusions.
For simple ADM models with DM mass $\lesssim 100 {\, {\rm GeV}}$, annihilation of the symmetric
component generally needs to go into, or via, lighter hidden sector particles rather
than directly to the SM, to avoid constraints from
direct detection and collider experiments~\cite{MarchRussell2012}.
We will see below that in the case of DN,
the constraints from direct detection on the SM
couplings of individual DN constituents are often even tighter,
confirming the need to annihilate into lighter hidden sector particles.
In addition, if we want to build up DN via fusions from single nucleons,
we need some lighter state for at least the first few fusions to de-excite into, as
the limits on SM interactions are generally strong enough that
we cannot de-excite fast enough via those alone.
We also need some mediator particle to transmit the binding force.
If any of these additional states are sufficiently light, strongly-interacting,
or long-lived, then there may be astrophysical constraints
on their properties.
The amplitudes for sufficiently weak interactions between
DN and other states will, for interactions involving individual dark
nucleons, factorise into a form factor and a single-nucleon amplitude.
For momentum exchanges which are small compared to the
inverse radius of the DN, the form factor will
be $\propto A$, i.e.\ the interaction will be
coherence enhanced, and will fall off in some nuclear-structure-dependent
way for larger momenta \cite{NDMdirect} analogously to the scattering
of SM nuclei with WIMP DM.
Assuming that DN scatterings with SM nuclei in
DM direct experiments are in the small momentum transfer
regime, the coherence factor has the effect of
\emph{enhancing}, for a given SM-dark nucleon interaction
strength, the constraints from these experiments.
Taking the DN to be of a single size $A$, the number
density is $\propto 1/A$, but the scattering rate is coherence-enhanced
by $A^2$, so the overall event rate scales as $A$.
For the distributions discussed in the previous sections, both
the mass distribution $k y_k$ and the scattering-rate
distribution $k^2 y_k$ are peaked over a small logarithmic range
in $k$, so the $\propto A$ enhancement is a reasonable approximation.
The same is true, for simple injection profiles, in the bottlenecked case.
Since the rate for processes with only SM particles in the initial
state is generally set by the interaction strength
with individual dark nucleons, collider constraints,
and those from the annihilation of the symmetric
DM component in the early universe, are relatively less stringent
compared to direct detection.
In addition to these effects on overall rates, for intermediate momentum
transfers the effect of the momentum-dependent DN form factor may be
important. The possibility of `dark form factors' influencing the shape
of direct detection recoil spectrum has been investigated by a number of
authors, often in the context of suppressing low-momentum scatterings~\cite{Chang2009,Feldstein2009},
but also addressing soft form factors~\cite{Gelmini2002,McDermott:2011hx,Cherry2014}.
Also, for sufficiently large energy transfers, inelastic processes
involving excited DN states or fissions may be important.
In a forthcoming paper \cite{NDMdirect} we investigate the possible direct
detection phenomenology of DN in quantitative detail, as well as exploring
the possible consequences of its capture in astrophysical objects.
\subsection{Post-nucleosynthesis energetics in the dark sector}
\label{sec:selfint}
Since we have been considering DM with strong
self-interactions, an obvious question is whether these
self-interactions have late-time consequences.
For elastically-scattering DM with velocity-independent
self-scattering cross section, the observational
limit on the self-scattering cross section is
$\sigma_{XX}/m_X \lesssim 1 {\, {\rm barn}} / {\, {\rm GeV}}$ (see e.g.~\cite{Tulin2013}).
For a population of $A$-DN,
\begin{equation}
\frac{\sigma_{AA}}{m_A} \simeq \frac{0.05 {\, {\rm barn}}}{{\, {\rm GeV}}}
A^{-1/3}
\left(\frac{1 {\, {\rm GeV}}}{m_1}\right)^{1/3}
\left(\frac{1 {\, {\rm GeV}} {\, {\rm fm}}^{-3}} {\rho_b}\right)^{2/3} ,
\end{equation}
in the notation of eq.(\ref{eq:fon1}). From the results of
Section~\ref{sec:fusionfo}, if we scale
$m_1 \mapsto \lambda m_1$, $\rho_b \mapsto \lambda^4 \rho_b$ etc.,
then in the scaling regime, $A \mapsto \lambda^{-12/5} A$.
Thus, $\frac{\sigma_{AA}}{m_A} \mapsto \lambda^{-11/5}\frac{\sigma_{AA}}{m_A}$,
so we would hit the observational limit at $\sim 10 {\, {\rm MeV}}$ scales
for the constituents, assuming that we built up to the largest
allowed sizes.
Realistically, the situation can be more complicated than this.
Firstly, there would be a spectrum of DNN values. However,
for the scaling distribution, and simple forms of the bottlenecked
distributions, the area distribution $k^{2/3} n_k$ and the mass
distribution $k n_k$ both receive most of their contribution
from a single decade or so of nucleon numbers,
so the above calculations will hold approximately.
More interestingly, the fact that DN collisions may not be elastic
could have phenomenological consequences. Since, neglecting collisions,
DN of different sizes will have the same velocity distribution,
with e.g. $v \sim 10^{-3}$ in the Milky Way, large DN will have kinetic energies
much larger than the single-nucleon binding energy scale. Thus, it is possible
that DN-DN collisions will result in inelastic collisions,
including nuclear fragmentations, transitions to excited states,
and fusions.
Some types of collisions may be dissipative, reducing
the average kinetic energy (KE) per nucleon of the final state DN.
Examples include scattering to excited states, in which
some of the initial KE is lost into de-excitation products
(fusions with sufficiently small binding energy differences
also lose KE overall).
If these de-excitation products interact with the SM,
there may be indirect detection signals from these
processes.
Independently, the inelastic form of the scattering will
tend to lead to the contraction of DM mass distributions,
as compared to elastic scatterings.
It may be the case that, in regions of sufficiently high DM
density, there is the possibility of run-away contraction
(in particular, if the DM is self-gravitating, then
removing KE results in the distribution contracting
and also heating up, due to the negative heat capacity).
More generally, the effects and constraints will be
different from the usual simulations of elastic DM self-interactions.
On the other hand, the presumed binding-energy-based stability of
large DN means that there may be exothermic collisions, in which
the average KE of the DN is increased (c.f. fusions in stars, in the SM).
If the velocity kick given to the DN is large compared to the velocity
dispersion in the halo, exothermic collisions will have the effect of clearing out the
central high-density region, until the number density is low enough
that collisions occur once per particle per Hubble time.
Since smaller halos tend to have smaller velocity dispersions,
small velocity kicks may only modify structure on small scales
(where there are problems with $\Lambda$-CDM predictions for structure).
These general effects of inelastic DM self-interactions
are not specific to nuclear DM, and a number of models
have been investigated in the literature. In particular,
models in which the DM transforms under some approximate
symmetry can easily realise small mass splitting,
and the effects of illustrative cases corresponding to a Yukawa self-interaction
have been considered in~\cite{Loeb2010, Schutz2014}.
Turning to the energy released in de-excitations;
considering energy injections once the aggregation process has frozen
out, the injection rate is just set by the local DM velocities and number density
(which, in regions where only a small fraction of the DN undergo
self-interactions, will be the standard collisionless DM values).
Generally, if the rate is velocity-suppressed (as for large
DN-DN collisions), and freeze-out is before BBN, then present-day
cosmic ray (CR) constraints dominate those from earlier
times, if the injection is to sufficiently energetic SM
particles~\cite{Essig2013, Jedamzik2009}.
Velocity-independent processes with rate $\propto n^2$
have BBN, CMB and CR constraints within a few orders of magnitude of
each other, depending on the form of SM injections~\cite{Essig2013,
Madhavacheril2013, Jedamzik2009}.
In regions where only a small fraction of DN undergo fusions,
the proportion of the DM mass density represented
by the binding energy released in late-time fusions is
\begin{align}
\langle \sigma v \rangle n_A t_{\rm sys} \frac{\Delta BE}{M_A}
\sim \,
& 10^{-3} A^{-2/3}
\frac{\Delta BE}{A^{2/3} 0.01 m_1}
\Bigg\{\frac{\rho_{\rm DM}}{0.3 {\, {\rm GeV}} {\, {\rm cm}}^{-3}} \nonumber \\
&\left.\left(\frac{1 {\, {\rm GeV}} {\, {\rm fm}}^{-3}}{\rho_b}\right)^{2/3}
\left(\frac{1 {\, {\rm GeV}}}{m_1}\right)^{1/3}
\left(\frac{v}{10^{-3}}\right)
\left(\frac{t_{\rm sys}}{10 {\, {\rm Gyr}}}\right) \right\} \,,
\label{eq:latebe}
\end{align}
(where the $\Delta BE$ terms corresponds to the binding energy
difference in $A + A \rightarrow 2 A$ fusions scaling as $A^{2/3}
\beta$, in the notation of eq.(\ref{eq:semf}); $\beta = 0.01
m_1$ is around the SM value).
This corresponds to a proportion $\sim 0.1 A^{-1/3} \{\ \}$
of the DN undergoing collisions, where $\{\ \}$ indicates
the term in curly brackets in equation~\ref{eq:latebe}, so the total KE of the DN
involved is $\sim 7 \times 10^{-8} A^{-1/3} \{\ \}$ of the
DM mass density.
As an illustration of a comparable indirect detection signal, for
standard symmetric DM, the proportion of the DM mass density released in
$s$-wave annihilations is
\begin{equation}
\langle \sigma v\rangle n_X t_{\rm gal} \sim 3 \times 10^{-8} \left(\frac{100 {\, {\rm MeV}}}{m_X}\right)
\left(\frac{\langle \sigma v\rangle_X}{{\, {\rm pb}}}\right)\left(
\frac{\rho_{\rm DM}}{0.3 {\, {\rm GeV}} {\, {\rm cm}}^{-3}}\right) \,,
\end{equation}
where we take $t_{\rm gal} \sim 10 {\, {\rm Gyr}}$.
Since the detectability of CR signals from annihilating DM at $\sim
{\, {\rm pb}}$ cross-sections depends on the SM injection channel etc.~\cite{Cirelli2012},
the same applies for DN collisions within the $A$ ranges of interest.
Given the range of initial DN masses, and the possibility of
excited states of DN, such signals may have a richer structure than
the indirect detection signals usually considered.
Additionally, the geometric cross sections we have assumed give different
velocity dependence to the partial wave processes
usually considered.
Also, as considered above, for dissipative collisions in sufficiently
dense regions, there may be the possibility of (run-away) contraction
of the DN distribution, which could significantly affect signals from e.g.
the Galactic centre.
\subsection{Light dark sector states}
\label{sec:lightdark}
As mentioned above, DN models generally require additional lighter
hidden sector states. Since the assumption is that these do not make up
the bulk of the DM, they either need to never have a large abundance
(generally difficult to realise in thermal ADM histories),
be sufficiently light and weakly interacting with the SM to persist as
dark radiation, or their abundance needs to be reduced.
Reducing the yield of a species can occur by transferring its
energy density to other hidden sector species, or
to the SM. In the latter case, if this injection occurs during or
after BBN, there are constraints on its form and magnitude.
For injection to hadronic channels, the total energy density injected
after $T_\gamma \sim 1 {\, {\rm MeV}}$ must be significantly below that of the DM.
For electromagnetic ($e^\pm$ or $\gamma$) injection, the dominant effect at times before
thermalisation becomes inefficient, $\lesssim 10^4 {\, {\rm sec}}$, is
alteration of the photon-baryon ratio,\footnote{and also increasing the photon
temperature relative to that of neutrinos, so decreasing $N_{\rm eff}$
at later times compared to BBN.} which constrains the amount of energy injected
to $\lesssim 0.1 \rho_\gamma$~\cite{Simha2008}.
At later times, any energy injections, apart from those
to neutrinos, must again be significantly
sub-DM~\cite{Jedamzik2006, Hu1993,Slatyer2012}.
For hidden sector particles of mass $\gg 100 {\, {\rm MeV}}$,
their (symmetric) chemical equilibrium abundance (at the SM temperature)
by BBN times is significantly sub-DM. The dominant
constraints on their couplings to the SM generally come
from collider experiments, and permit decay times
of $\ll 1 \sec$ (e.g.~\cite{Lees2014}). So, in these cases, if such decays
are possible, then there are generally
consistent scenarios in which any initial abundance decays to
the SM before BBN.\footnote{As noted in previous sections,
obtaining sufficiently fast annihilations (directly to the SM) to reduce
the energy density to sub-DM levels is generally difficult
for $m \ll 100 {\, {\rm GeV}}$, without involving lighter states or non-minimal flavour structures.}
On the other hand, for hidden sector particles with $m \ll 100 {\, {\rm MeV}}$,
astrophysical, collider and other observations place strong constraints on their
SM couplings. These restrict their SM decay times to
$\gtrsim 1 \sec \left(\frac{100 {\, {\rm MeV}}}{m}\right)^{1 + 2k}$,
where $k \ge 0$ is set by the mass dimension of the decay operator
(e.g.~\cite{Raffelt2006, Kazanas2014}),
and mean that direct interactions with the SM are frozen out
below very high temperatures ($T_\gamma \gg m$).\footnote{Faster
decay times may be possible with more complex dark sectors,
but these generally involve additional lighter dark states.}
If $m \gtrsim 10 {\, {\rm keV}}$, purely SM decays occur
at $T_\gamma < m$, so the decay of a thermal abundance would transfer
$\gtrsim \rho_\gamma$ energy density to the SM.
For smaller $m$, the decay time is $\gtrsim 10^4 {\, {\rm sec}}$,
so the constraints on energy injection to the SM are much more
severe. In either case, limits on SM energy injection generally imply
that the majority of the initial thermal energy density needs
to be transferred to other, lighter hidden sector states.\footnote{It
may be possible to realise alternative scenarios in which energy is
transferred to the SM indirectly, via other hidden sector states. If the
hidden sector is at a slightly lower temperature
than the SM, and the transfer is before $\sim 10^4 {\, {\rm sec}}$,
this may be safe (sitting at the upper end of the mass range,
and transferring mostly before BBN, may also work).
This would require that number-changing interactions
with other hidden sector states are sufficiently fast
to keep the light state in thermal equilibrium during
the process. Three-body interactions with heavier
hidden sector states are suppressed by two factors
of the heavy state number density (which is small for
DM of SM mass or higher), and inelastic two-body collisions are only
relevant if such excitations are accessible at low temperatures,
imposing model-building constraints.
Alternatively, models in which SM energy injection is dominantly
to neutrinos may also be viable.} Considering these states in turn,
we eventually require that there be long-lived hidden sector states,
which will act as dark radiation, at least during the early universe.
Extra relativistic species are compatible with current observations~\cite{Ade2013}, and if the hidden sector is at a lower temperature
than the SM, its contribution to the effective number of such species
$N_{\rm eff}$ may be small.
Plausible candidates for these species within hidden sector models
include very light pseudo-Nambu-Goldstone bosons or $Z'$ states.
\section{Conclusions}
\label{sec:conclusions}
In this paper we have studied the ``Big Bang dark nucleosynthesis'' process by which
`nuclear' bound states of DM may be built up in the early universe. Specifically
we focussed on the case of asymmetric DM models where the nuclear binding energy per
dark nucleon saturates in the large nucleon number limit.
We find that, if fusions between small dark nuclei (DN)
happen sufficiently fast, and fusion cross-sections between
large DN scale on average geometrically, the resulting number distribution
generically takes on a universal form, illustrated in Figures~\ref{fig:ndist} and~\ref{fig:sdist}.
This result is broadly independent of small
changes to the initial conditions or fusion rates, assuming that the DN are in kinetic
equilibrium throughout the process, as discussed in Section \ref{sec:initialcond}. The
average mass of the DN built up during this process can be as large as $\sim 10^8 {\, {\rm GeV}}$.
If fusion reactions between small DN are not large enough to
reach this regime, but large-small fusions still occur at
an appreciable rate, then there is the counter-intuitive possibility
of building up even larger DN due to the higher velocities
of smaller particles, leading to larger fusion rates.
The resulting number distribution generally takes
the form of a power law multiplied by an `injection profile'
parameterising the change of small-small fusion rates with time.
Figures~\ref{fig:scalingbn} and~\ref{fig:1plus} illustrate this
behaviour.
Given this possibility of building up large dark nuclear bound states,
direct detection signals may be modified in a number of ways---by the
coherent enhancement of SM-DM interactions, by
dark form factors if the DN radius is significantly larger than SM nuclear radii,
and by inelastic interactions if there are sufficiently
low-lying excitations. We reserve detailed discussion
of such direct detection phenomenology to a forthcoming paper~\cite{NDMdirect}.
The possibility of inelastic collisions can also lead to interesting
astrophysical dark sector interactions as mentioned in Section~\ref{sec:selfint}.
These include annihilation-type indirect detection signals,
which are usually absent from asymmetric DM models
(and here may have a richer structure, corresponding to
the number distribution of DN). Inelastic collisions
may also modify the effects of DM self-interactions
on halo structure, especially at short distance scales.
In addition, as discussed in~\cite{NDMdirect}, there are a range
of possible consequences for the capture of DN by astrophysical objects
as well, from ejection by de-excitations, through to fusions leading to a very dense DN core.
Finally, the discussion in this paper has intentionally been as
model-independent as is practical, investigating idealised
versions of the behaviour that classes of models can display.
The physics is inspired by that observed in the Standard Model (SM)
sector, with the simple, but important, change that the Coulomb barrier
is removed. Although the example of the SM reassures us that a sufficiently
complicated model can realise the physics we discuss, it would be interesting to
investigate, along the lines of \cite{Hashimoto2011,Detmold2014ii,Krnjaic2014}, specific simple toy
models which realise all or some of the features discussed in this paper. As with the SM, we would
expect there to be additional features in the dark nuclear spectroscopy and interaction cross
sections, e.g.\ due to shell structure, on top of the general scaling behaviour of nuclear properties that
we have utilised. In addition the extra hidden sector states
required (to mediate binding forces, carry away de-excitation energy, etc)
beyond the dark nucleon itself can be relevant for astrophysical phenomenology
and constraints. In particular, if some of these states have masses $\lesssim 100 {\, {\rm MeV}}$,
there is likely to be a need for extra hidden sector states which
act as dark radiation. Another avenue for further work is the possibility
of symmetric DM models.
\section*{Acknowledgements}
We wish to thank Asimina Arvanitaki, Masha Baryakhtar, Peter Graham, Felix Kahlhoefer, and Surjeet Rajendran
for discussions. EH, RL and JMR thank the CERN Theory Group,
and SMW thanks the Oxford Physics Department, for hospitality
during completion of this work. This work was supported in part by ERC grant
BSMOXFORD no. 228169. EH and RL acknowledge support from STFC studentships, and
in addition, RL is grateful for support from the Buckee Scholarship at Merton College,
Oxford.
|
1,314,259,995,795 | arxiv | \section{Introduction}
Quantum computation provides an effective solution to certain problems, such as factoring large integers \cite{Shor} and searching unsorted data \cite{Grover}. The implementation of circuit-based quantum computation relies on the ability to realize a universal set of high-fidelity quantum gates, including arbitrary one-qubit gates and a nontrivial two-qubit gate \cite{Bremner}. However, there are two main obstacles to the realization of high-fidelity quantum gates. One is the control errors arising from inaccurate manipulation of quantum systems. The other one is the decoherence caused by the interaction between the quantum system and its environment. Geometric phases are only dependent on evolution paths of quantum systems but independent of the evolution details, and therefore quantum computation based on geometric phases is robust against control errors.
In 1984, Berry found that a quantum system in a nondegenerate eigenstate undergoing adiabatic and cyclic evolution can acquire a geometric phase in addition to a dynamical phase \cite{Berry}. The notion of geometric phases was then extended to quantum systems in degenerate eigenstates \cite{Wilczek}, in nonadiabatic evolution \cite{Aharonov,Anandan}, and in mixed states \cite{Uhlmann,Sjoqvist2000,Tong2004}. Until now, pure-state geometric phases have been used to realize quantum computation while mixed-state geometric phases have not.
The early proposals \cite{Zanardi,Jones,Duan} of geometric quantum computation are based on adiabatic Abelian geometric phases \cite{Berry} and adiabatic non-Abelian geometric phases \cite{Wilczek}. However, these proposals require quantum systems to undergo adiabatic evolution, which makes quantum systems evolve for a long time. To resolve this problem, nonadiabatic geometric quantum computation \cite{Wang,Zhu} based on nonadiabatic Abelian geometric phases \cite{Aharonov} and nonadiabatic holonomic quantum computation \cite{Sjoqvist,Xu} based on nonadiabatic non-Abelian geometric phases \cite{Anandan} were proposed.
Compared with nonadiabatic geometric quantum computation that uses the geometric phase as one parameter of a quantum gate, nonadiabatic holonomic quantum computation uses the holonomic matrix itself as a quantum gate. This makes nonadiabatic holonomic quantum computation possess a whole-geometric property. Due to the merits of both geometric robustness and high-speed implementation without the limit of adiabatic evolution, nonadiabatic holonomic quantum computation has received increasing attention.
The first protocol of nonadiabatic holonomic quantum computation is based on a three-level quantum system driven by two resonant laser pulses \cite{Sjoqvist}. It needs to combine two one-qubit gates to realize an arbitrary one-qubit gate. To simplify the operations, the single-shot protocol of nonadiabatic holonomic quantum computation \cite{Xu2015,Sjoqvist2016} and the single-loop protocol of nonadiabatic holonomic quantum computation \cite{Herterich} were proposed. The improved protocols allow us to realize an arbitrary one-qubit gate by a single-shot implementation and thus reduce the exposure time of nonadiabatic holonomic gates to error sources. To further shorten the exposure time of quantum gates to error sources, the path-shortening protocol of nonadiabatic holonomic quantum computation was put forward \cite{Xu2018}, where nonadiabatic holonomic gates can be realized based on a class of extended evolution paths that are shorter than the former ones.
The key to realizing nonadiabatic holonomic quantum computation based on these protocols is to find the Hamiltonians that make the quantum system satisfy both the cyclic evolution condition and the parallel transport condition.
Recently, a general approach of constructing Hamiltonians for nonadiabatic holonomic quantum computation was put forward \cite{Zhao}. By using this approach, one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. Up to now, a lot of works both in theories \cite{Johansson,Spiegelberg,Mousolou,Sjoqvist2015,Wang2016,Xu2017,XuGF,Mousolou2017,Mousolou2018,Zhao2017,Zhao2018,Xia,
Chen,Zhao2019,Ramberg,Xing} and experiments \cite{Feng,Li,Abdumalikov,Sun,Danilin,Zhang2019,Egger,Zu,Camejo,
Sekiguchi,Zhou,Nagata,Isida,Long} have contributed to nonadiabatic holonomic quantum computation.
While some protocols tried to reduce the influence of decoherence by shorting the exposure time of quantum gates, another line of protecting quantum gates against decoherence is to use decoherence-mitigation methods.
To make quantum gates robust against both control errors and decoherence, the combination of nonadiabatic holonomic quantum computation and decoherence-mitigation methods is a promising strategy. The first protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces was put forward in Ref. \cite{Xu}. Afterwards, a number of alternative protocols \cite{Liang,Zhang,ZhaoPZ,Wang2018} and physical implementation schemes \cite{Zhou2015,Xue,Xue2016,Liu2017,Zhu2019} were put forward. These proposals are based on the system-environment interaction being with some symmetry and they focus mainly on protecting nonadiabatic holonomic gates against collective decoherence, especially the collective dephasing.
In this paper, we propose a protocol of nonadiabatic holonomic quantum computation protected by dynamical decoupling. The decoherence-mitigation method used here is dynamical decoupling \cite{Viola}, and therefore there is no need to require system-environment interaction to have some symmetry. Due to the combination of nonadiabatic holonomic quantum computation and dynamical decoupling, our protocol not only possesses the intrinsic robustness against control errors but also protects quantum gates against environment-induced decoherence, regardless of collective decoherence or independent decoherence.
\section{Physical model}
Before proceeding further, we briefly review the basic idea of nonadiabatic holonomic quantum computation \cite{Sjoqvist,Xu}. Consider an $N$-dimensional quantum system exposed to the Hamiltonian $H(t)$. Assume there is
an $L-$dimensional subspace $\mathcal{S}(t)=\mathrm{Span}\{\ket{\phi_{k}(t)}\}^{L}_{k=1}$, where $\ket{\phi_{k}(t)}$ are orthonormal basis vectors and satisfy the Schr\"{o}dinger equation $i\ket{\dot{\phi}_{k}(t)}=H(t)\ket{\phi_{k}(t)}$.
If $\ket{\phi_{k}(t)}$ satisfy the following conditions
\begin{align}\label{eq}
&(\mathrm{i})~ \sum^{L}_{k=1}\ket{\phi_{k}(\tau)}\bra{\phi_{k}(\tau)}=
\sum^{L}_{k=1}\ket{\phi_{k}(0)}\bra{\phi_{k}(0)}, \notag\\
&(\mathrm{ii})~~\bra{\phi_{k}(t)}H(t)\ket{\phi_{l}(t)}=0,~~~~k,l=1,\cdot\cdot\cdot L,
\end{align}
then the unitary operator $U(\tau)$ with $\ket{\phi_{k}(\tau)}=U(\tau)\ket{\phi_{k}(0)}$ is a nonadiabatic holonomic gate acting on $\mathcal{S}(0)$. Here, $\tau$ is the evolution period.
Let us now elucidate our physical model. We consider a quantum system consisting of $N$ physical qubits, which interact though the well-known $XXZ$ coupling \cite{Yang,Johnson,Duan2003,Alcaraz,Canosa,Breunig}. The Hamiltonian reads
\begin{align}\label{eq1}
H=\sum_{k<l}\left[J^{x}_{kl}\left(\sigma^{x}_{k}\sigma^{x}_{l}+\sigma^{y}_{k}\sigma^{y}_{l}\right)
+J^{z}_{kl}\sigma^{z}_{k}\sigma^{z}_{l}\right],
\end{align}
where $J^{x}_{kl}$ and $J^{z}_{kl}$ are the real-valued controllable coupling parameters and $\sigma^{\alpha}_{m}$ represent the Pauli $\alpha$ operators ($\alpha=x,y,z$) acting on the $m$th qubit ($m=k,l$). For the quantum system considered here, we assume that each physical qubit interacts independently with its environment. The interaction Hamiltonian reads
\begin{align}\label{eq2}
H_{I}=\sum_{k,\alpha}\sigma^{\alpha}_{k}\otimes B^{\alpha}_{k},
\end{align}
where $B^{\alpha}_{k}$ is the environment operator corresponding to the system operator $\sigma^{\alpha}_{k}$.
If $B^{\alpha}_{k}$ is independent of the qubit index $k$, the environment-induced decoherence is reduced to collective decoherence. In particular, if the system operator is further taken as $\sigma^{z}_{k}$, then the collective decoherence yields collective dephasing. To protect nonadiabatic holonomic gates against collective dephasing,
nonadiabatic holonomic quantum computation in decoherence-free subspaces was proposed \cite{Xu}, and to protect nonadiabatic holonomic gates against collective decoherence, nonadiabatic holonomic quantum computation in noiseless systems was proposed \cite{Zhang}. For the more complicated decoherence induced by the interaction $H_{I}$ in Eq. (\ref{eq2}), dynamical decoupling provides an effective method to protect nonadiabatic holonomic gates against decoherence.
Dynamical decoupling operates by applying a periodic sequence of fast and strong symmetrizing pulses to quantum
systems to suppress the effect of undesired system-environment interaction. For the system-environment interaction in Eq. (\ref{eq2}), we can use a periodic sequence with the decoupling operations $\{\otimes^{N}_{k=1}I_{k},\otimes^{N}_{k=1}\sigma^{x}_{k},\otimes^{N}_{k=1}\sigma^{y}_{k},
\otimes^{N}_{k=1}\sigma^{z}_{k}\}$ to suppress its effect. The corresponding unitary operator over a period of time reads
\begin{widetext}
\begin{align}\label{req}
U_{I}=&\left[\left(\otimes^{N}_{k=1}\sigma^{z}_{k}\right)e^{-i H_{I}\tau} \left(\otimes^{N}_{k=1}\sigma^{z}_{k}\right)\right]
\left[\left(\otimes^{N}_{k=1}\sigma^{y}_{k}\right)e^{-i H_{I}\tau} \left(\otimes^{N}_{k=1}\sigma^{y}_{k}\right)\right]
\left[\left(\otimes^{N}_{k=1}\sigma^{x}_{k}\right)e^{-i H_{I}\tau} \left(\otimes^{N}_{k=1}\sigma^{x}_{k}\right)\right]
\left[\left(\otimes^{N}_{k=1}I_{k}\right)e^{-i H_{I}\tau} \left(\otimes^{N}_{k=1}I_{k}\right)\right]
\notag\\
=&e^{-i\left(\otimes^{N}_{k=1}\sigma^{z}_{k}\right)H_{I}\left(\otimes^{N}_{k=1}\sigma^{z}_{k}\right)\tau}
e^{-i\left(\otimes^{N}_{k=1}\sigma^{y}_{k}\right)H_{I}\left(\otimes^{N}_{k=1}\sigma^{y}_{k}\right)\tau}
e^{-i\left(\otimes^{N}_{k=1}\sigma^{x}_{k}\right)H_{I}\left(\otimes^{N}_{k=1}\sigma^{x}_{k}\right)\tau}
e^{-iH_{I}\tau}
\notag\\
=&e^{-i\left[\left(\otimes^{N}_{k=1}\sigma^{z}_{k}\right)H_{I}\left(\otimes^{N}_{k=1}\sigma^{z}_{k}\right)
+\left(\otimes^{N}_{k=1}\sigma^{y}_{k}\right)H_{I}\left(\otimes^{N}_{k=1}\sigma^{y}_{k}\right)
+\left(\otimes^{N}_{k=1}\sigma^{x}_{k}\right)H_{I}\left(\otimes^{N}_{k=1}\sigma^{x}_{k}\right)
+H_{I}\right]\tau}+O(\tau^2)
\notag\\
=&e^{-i\sum^{N}_{k=1}(-\sigma_{k}^{x}\otimes B_{k}^{x}-\sigma_{k}^{y}\otimes B_{k}^{y}+\sigma_{k}^{z}\otimes B_{k}^{z}
-\sigma_{k}^{x}\otimes B_{k}^{x}+\sigma_{k}^{y}\otimes B_{k}^{y}-\sigma_{k}^{z}\otimes B_{k}^{z}
+\sigma_{k}^{x}\otimes B_{k}^{x}-\sigma_{k}^{y}\otimes B_{k}^{y}-\sigma_{k}^{z}\otimes B_{k}^{z}
+\sigma_{k}^{x}\otimes B_{k}^{x}+\sigma_{k}^{y}\otimes B_{k}^{y}+\sigma_{k}^{z}\otimes B_{k}^{z})\tau}
+O(\tau^2)
\notag\\
=&\otimes^{N}_{k=1}I_{k}+O(\tau^2),
\end{align}
\end{widetext}
where $\tau$ is the duration time of pulse intervals and $I_{k}$ is the identity operator acting on the $k$th qubit.
This result indicates that up to the first-order term $O(\tau)$, the system-environment interaction can be completely eliminated by using a decoupling pulse sequence.
To realize dynamical-decoupling-protected nonadiabatic holonomic quantum computation, the decoupling pulse sequence needs to be inserted into the native dynamical evolution of the quantum system.
Therefore, we need to properly choose the Hamiltonian that not only makes the decoupling pulse sequence compatible with the dynamical evolution but also keeps the cyclic evolution condition as well as the parallel transport condition valid.
To this end, we choose the Hamiltonian in Eq. (\ref{eq1}), which commutes with the decoupling operations. In this case, we can realize the desired evolution protected by dynamical decoupling.
\section{Implementation}
To perform dynamical-decoupling-protected nonadiabatic holonomic quantum computation, we need to realize a universal set of quantum gates, including arbitrary one-qubit gates and a nontrivial two-qubit gate.
First, we realize an arbitrary one-qubit gate. To complete our realization, we utilize three physical qubits to encode a logical qubit. The specific encoding is
\begin{align}\label{req}
\ket{0}_{L}=\ket{001},~~\ket{1}_{L}=\ket{010}.
\end{align}
Meanwhile, we use $\ket{a}=\ket{100}$ as an auxiliary state.
In this case, we can apply the periodic sequence with decoupling operations $\{\otimes^{3}_{k=1}I_{k},\otimes^{3}_{k=1}\sigma^{x}_{k},
\otimes^{3}_{k=1}\sigma^{y}_{k},\otimes^{3}_{k=1}\sigma^{z}_{k}\}$ to the quantum system to protect quantum information against decoherence.
To realize nonadiabatic holonomic gates, we set the nonzero parameters of the Hamiltonian in Eq. (\ref{eq1}) to be
\begin{align}
&J^{x}_{12}=-\frac{J_{1}(t)}{2}\cos\phi_{1}\cos\frac{\theta_{1}}{2},~~
J^{x}_{13}=\frac{J_{1}(t)}{2}\cos\phi_{1}\sin\frac{\theta_{1}}{2},
\notag\\
&J^{z}_{23}=J_{1}(t)\sin\phi_{1},
\end{align}
where $J_{1}(t)$ is a time-dependent parameter, and $\phi_{1}$ and $\theta_{1}$ are time-independent parameters.
In this case, the Hamiltonian reads
\begin{align}\label{req1}
H_{1}(t)=&\frac{J_{1}(t)}{2}\cos\phi_{1}\bigg[-\cos\frac{\theta_{1}}{2}
\left(\sigma^{x}_{1}\sigma^{x}_{2}+\sigma^{y}_{1}\sigma^{y}_{2}\right)
\notag\\
&+\sin\frac{\theta_{1}}{2}\left(\sigma^{x}_{1}\sigma^{x}_{3}+\sigma^{y}_{1}\sigma^{y}_{3}\right)\bigg]
+J_{1}(t)\sin\phi_{1}\sigma^{z}_{2}\sigma^{z}_{3}.
\end{align}
By using the basis $\{\ket{0}_{L},\ket{1}_{L},\ket{a}\}$, this Hamiltonian can be recast as
\begin{align}
H_{1}(t)=&J_{1}(t)\cos\phi_{1}\left(\sin\frac{\theta_{1}}{2}\ket{a}_{L}\bra{0}-\cos\frac{\theta_{1}}{2}\ket{a}_{L}\bra{1}
+\mathrm{H.c.}\right)
\notag\\
&+J_{1}(t)\sin\phi_{1}(\ket{a}\bra{a}-\ket{0}_{LL}\bra{0}-\ket{1}_{LL}\bra{1}),
\end{align}
which can be further rewritten as
\begin{align}\label{eq3}
H_{1}(t)=&J_{1}(t)\cos\phi_{1}\left(\sin\frac{\theta_{1}}{2}\ket{a}_{L}\bra{0}-\cos\frac{\theta_{1}}{2}\ket{a}_{L}\bra{1}
+\mathrm{H.c.}\right)
\notag\\
&+2J_{1}(t)\sin\phi_{1}\ket{a}\bra{a}
\notag\\
&-J_{1}(t)\sin\phi_{1}(\ket{a}\bra{a}+\ket{0}_{LL}\bra{0}+\ket{1}_{LL}\bra{1}).
\end{align}
It is noteworthy that $\ket{a}\bra{a}+\ket{0}_{LL}\bra{0}+\ket{1}_{LL}\bra{1}$ is an identity operator and thus $-J_{1}(t)\sin\phi_{1}(\ket{a}\bra{a}+\ket{0}_{LL}\bra{0}+\ket{1}_{LL}\bra{1})$ can only generate a global phase
during evolution. This global phase does not affect the quantum gates and therefore the terms $-J_{1}(t)\sin\phi_{1}(\ket{a}\bra{a}+\ket{0}_{LL}\bra{0}+\ket{1}_{LL}\bra{1})$ in Eq. (\ref{eq3}) can be ignored. If we introduce two orthonormal states,
\begin{align}
\ket{d}&=\cos\frac{\theta_{1}}{2}\ket{0}_L+\sin\frac{\theta_{1}}{2}\ket{1}_L,
\notag\\
\ket{b}&=\sin\frac{\theta_{1}}{2}\ket{0}_L-\cos\frac{\theta_{1}}{2}\ket{1}_L,
\end{align}
the Hamiltonian then reads
\begin{align}
H_{1}(t)=&J_{1}(t)\cos\phi_{1}(\ket{a}\bra{b}+\ket{b}\bra{a})
+2J_{1}(t)\sin\phi_{1}\ket{a}\bra{a}.
\end{align}
The evolution operator corresponding to the above Hamiltonian can be written as $U_{1}(t)=\exp[-i\int^{t}_{0}H_{1}(t^{\prime})dt^{\prime}]$, which can be explicitly expressed as
\begin{align}\label{eq4}
U_{1}(t)=&\ket{d}\bra{d}+e^{-i\int^{t}_{0}J_{1}(t^{\prime})dt^{\prime}\sin\phi_{1}(\ket{a}\bra{a}+\ket{b}\bra{b})}
\notag\\
&\times e^{-i\int^{t}_{0}J_{1}(t^{\prime})dt^{\prime}[\cos\phi_{1}(\ket{a}\bra{b}+\ket{b}\bra{a})+
\sin\phi_{1}(\ket{a}\bra{a}-\ket{b}\bra{b})]}.
\end{align}
If the evolution time $T$ is taken to satisfy
\begin{align}
\int^{T}_{0}J_{1}(t)dt=\pi,
\end{align}
then the evolution operator is reduced to
\begin{align}\label{eq5}
U_{1}(T)=\ket{d}\bra{d}+e^{-i(\pi+\pi\sin\phi_{1})}\ket{b}\bra{b}+e^{-i(\pi+\pi\sin\phi_{1})}\ket{a}\bra{a}.
\end{align}
From Eqs. (\ref{eq4}) and (\ref{eq5}), we can see that a state initially prepared in the computational space $\mathcal{S}_{1}=\mathrm{Span}\{\ket{0}_{L},\ket{1}_{L}\}$ will evolve outside $\mathcal{S}_{1}$ during $t\in(0,T)$ and then return back to $\mathcal{S}_{1}$ at $t=T$. Thus, the cyclic evolution condition (i) is satisfied. By using the commutation relation $[H_{1}(t),U_{1}(t)]=0$, we can verify that $\bra{\phi(t)}H_{1}(t)\ket{\phi(t)}
=\bra{\phi(0)}U^{\dagger}_{1}(t)H_{1}(t)U_{1}(t)\ket{\phi(0)}
=\bra{\phi(0)}H_{1}(t)\ket{\phi(0)}=0$, where $\ket{\phi(t)}$ is an evolution state such that $\ket{\phi(t)}=U_{1}(t)\ket{\phi(0)}$ with $\ket{\phi(0)}\in\mathcal{S}_{1}$. It indicates that the parallel transport condition (ii) is satisfied. Therefore, $U_{1}(T)$ is a holonomic transformation. Acting on the computational space $\mathcal{S}_{1}$, the evolution operator $U_{1}(T)$
is equivalent to
\begin{align}
U_{1}=\ket{d}\bra{d}+e^{-i(\pi+\pi\sin\phi_{1})}\ket{b}\bra{b},
\end{align}
which plays the role of a nonadiabatic holonomic gate.
In the following, we demonstrate how to use dynamical decoupling to protect the dynamical evolution for the realization of nonadiabatic holonomic gates. We assume that the quantum system is coupled to its environment with the total Hamiltonian $\mathcal{H}(t)=H_{1}(t)+H_{E}+H_{I}$, where $H_{1}(t)$ is the system Hamiltonian in Eq. (\ref{req1}),
$H_{E}$ is the environment Hamiltonian, and $H_{I}$ is the interaction Hamiltonian in Eq. (\ref{eq2}). If the decoupling operations $\{\otimes^{3}_{k=1}I_{k},\otimes^{3}_{k=1}\sigma^{x}_{k},
\otimes^{3}_{k=1}\sigma^{y}_{k},\otimes^{3}_{k=1}\sigma^{z}_{k}\}$ are applied to the quantum system, the unitary operator reads
\begin{align}
\mathcal{U}(4\tau)=&\left[\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)e^{-i\int^{4\tau}_{3\tau}\mathcal{H}(t)dt} \left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)\right]
\notag\\
&\times
\left[\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)e^{-i\int^{3\tau}_{2\tau}\mathcal{H}(t)dt} \left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)\right]
\notag\\
&\times
\left[\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)e^{-i\int^{2\tau}_{\tau}\mathcal{H}(t)dt} \left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)\right]
\notag\\
&\times
\left[\left(\otimes^{3}_{k=1}I_{k}\right)e^{-i\int^{\tau}_{0}\mathcal{H}(t)dt} \left(\otimes^{3}_{k=1}I_{k}\right)\right]
\notag\\
=&e^{-i\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)\int^{4\tau}_{3\tau}\mathcal{H}(t)dt
\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)}
e^{-i\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)\int^{3\tau}_{2\tau}\mathcal{H}(t)dt
\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)}
\notag\\
&\times
e^{-i\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)\int^{2\tau}_{\tau}\mathcal{H}(t)dt
\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)}
e^{-i\int^{\tau}_{0}\mathcal{H}(t)dt}
\notag\\
=&e^{-i\left[\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)\int^{4\tau}_{3\tau}H_{1}(t)dt\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)
+\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)H_{I}\left(\otimes^{3}_{k=1}\sigma^{z}_{k}\right)\tau+H_{E}\tau\right]}
\notag\\
&\times
e^{-i\left[\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)\int^{3\tau}_{2\tau}H_{1}(t)dt\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)
+\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)H_{I}\left(\otimes^{3}_{k=1}\sigma^{y}_{k}\right)\tau+H_{E}\tau\right]}
\notag\\
&\times
e^{-i\left[\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)\int^{2\tau}_{\tau}H_{1}(t)dt\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)
+\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)H_{I}\left(\otimes^{3}_{k=1}\sigma^{x}_{k}\right)\tau+H_{E}\tau\right]}
\notag\\
&\times
e^{-i\left[\int^{\tau}_{0}H_{1}(t)dt
+H_{I}\tau+H_{E}\tau\right]}.
\end{align}
Since the chosen Hamiltonian $H_{1}(t)$ commutes with the decoupling operations, the unitary operator can be recast as
\begin{align}
\mathcal{U}(4\tau)=&e^{-i\left[\int^{4\tau}_{0}H_{1}(t)dt
+\sum_{\alpha=x,y,z}\left(\otimes^{3}_{k=1}\sigma^{\alpha}_{k}\right)H_{I}\left(\otimes^{3}_{k=1}\sigma^{\alpha}_{k}\right)\tau
+H_{I}\tau+4H_{E}\tau\right]}+O(\tau^2)
\notag\\
=&e^{-i\big[\int^{4\tau}_{0}H_{1}(t)dt+4H_{E}\tau\big]}+O(\tau^2)
\notag\\
=&e^{-i\int^{4\tau}_{0}H_{1}(t)dt}\otimes e^{-i4H_{E}\tau}+O(\tau^2)
\notag\\
=&U_{1}(4\tau)\otimes U_{E}(4\tau)+O(\tau^2),
\end{align}
where $U_{1}(4\tau)$ and $U_{E}(4\tau)$ are the evolution operators of the quantum system and its environment, and the relation $\sum_{\alpha=x,y,z}(\otimes^{3}_{k=1}\sigma^{\alpha}_{k})H_{I}(\otimes^{3}_{k=1}\sigma^{\alpha}_{k})
\tau+H_{I}\tau=0$ has been utilized to derive the second line. From the above result, one can see that up to the first-order term $O(\tau)$, the quantum system is completely decoupled from its environment at $t=4\tau$.
If we choose $\tau\ll T$, then we can repeat the above process over and over so that the effect of system-environment interaction will be suppressed in each interval of duration $4\tau$.
Therefore, we can use dynamical decoupling to protect nonadiabatic holonomic gates.
On the contrary, if we do not use dynamical decoupling to protect quantum gates, the unitary operator will be expressed as
\begin{align}\label{req2}
\mathcal{U}(4\tau)=&e^{-i\int^{4\tau}_{0}\mathcal{H}(t)dt}
\notag\\
=&e^{-i\left[\int^{4\tau}_{0}H_{1}(t)dt+4H_{I}\tau+4H_{E}\tau\right]}
\notag\\
=&\Big[e^{-i\int^{4\tau}_{0}H_{1}(t)dt}\otimes I_{E}\Big]
\left(e^{-i4H_{I}\tau}\right)
\left(I_{S}\otimes e^{-i4H_{E}\tau}\right)+O(\tau^2)
\notag\\
=&[U_{1}(4\tau)\otimes I_{E}]U_{I}(4\tau)[I_{S}\otimes U_{E}(\tau)]+O(\tau^2),
\end{align}
where $U_{I}(4\tau)$ is the evolution operator induced by the interaction Hamiltonian $H_{I}$, and $I_{S(E)}$ is the identity operator of the quantum system (environment).
From Eqs. (\ref{eq2}) and (\ref{req2}), we can obviously conclude that the system-environment interaction will affect the native dynamical evolution of the quantum system.
Finally, we demonstrate that an arbitrary one-qubit gate can be realized by using the nonadiabatic holonomic gate $U_{1}$. One can see that $U_{1}$ can be rewritten as
\begin{align}
U_{1}=e^{i\gamma_{1}/2}e^{-i\gamma_{1}(\sin\theta_{1}X+\cos\theta_{1}Z)/2},
\end{align}
where $\gamma_{1}=-(\pi+\pi\sin\phi_{1})$, and $X$ and $Z$ are the Pauli $x$ operator and Pauli $z$ operator acting on $\ket{0}_{L}$ and $\ket{1}_{L}$, respectively. Ignoring a trivial global phase, we can obviously see that $U_{1}$ is a quantum gate with a rotation axis in the $x-z$ plane and the rotation angle $\gamma_{1}$.
An arbitrary one-qubit gate can be realized by combining two such quantum gates about unparallel axes in the plane. For
example, $U_{1}$ is reduced to the quantum gate about the $x$ axis by setting $\theta_{1}=\pi/2$ and about the $z$ axis by setting $\theta_{1}=0$. By combining these noncommuting one-qubit gates, an arbitrary one-qubit gate can be realized.
Second, we realize a nontrivial two-qubit gate. To this end, we use six physical qubits to encode two logical qubits. To make the two-logical-qubit encoding compatible with the one-logical-qubit encoding, we encode two-logical-qubit states as
\begin{align}
\ket{00}_{L}&=\ket{001001},~\ket{01}_{L}=\ket{001010},
\notag\\
\ket{10}_{L}&=\ket{010001},~\ket{11}_{L}=\ket{010010}.
\end{align}
Meanwhile, we use $\ket{a_{1}}=\ket{011000}$ and $\ket{a_{2}}=\ket{000011}$ as auxiliary states.
In this case, we can utilize decoupling operations $\{\otimes^{6}_{k=1}I_{k},\otimes^{6}_{k=1}\sigma^{x}_{k},
\otimes^{6}_{k=1}\sigma^{y}_{k},\otimes^{6}_{k=1}\sigma^{z}_{k}\}$ to suppress the effect of the undesired system-environment interaction in Eq. (\ref{eq2}).
To realize nonadiabatic holonomic gates, we take the nonzero parameters of the Hamiltonian in Eq. (\ref{eq1}) as
\begin{align}
&J^{x}_{25}=-\frac{J_{2}(t)}{2}\cos\phi_{2}\cos\frac{\theta_{2}}{2},~~
J^{x}_{26}=\frac{J_{2}(t)}{2}\cos\phi_{2}\sin\frac{\theta_{2}}{2},
\notag\\
&J^{z}_{23}=J_{2}(t)\sin\phi_{2},
\end{align}
where $J_{2}(t)$ is time dependent, and $\phi_{2}$ and $\theta_{2}$ are time independent.
In this case, the Hamiltonian can be expressed as
\begin{align}
H_{2}(t)=&\frac{J_{2}(t)}{2}\cos\phi_{2}\bigg[-\cos\frac{\theta_{2}}{2}
\left(\sigma^{x}_{2}\sigma^{x}_{5}+\sigma^{y}_{2}\sigma^{y}_{5}\right)
\notag\\
&+\sin\frac{\theta_{2}}{2}\left(\sigma^{x}_{2}\sigma^{x}_{6}+\sigma^{y}_{2}\sigma^{y}_{6}\right)\bigg]
+J_{2}(t)\sin\phi_{2}\sigma^{z}_{2}\sigma^{z}_{3}.
\end{align}
By using the basis $\{\ket{00}_{L},\ket{01}_{L},\ket{10}_{L},\ket{11}_{L},\ket{a_{1}},\ket{a_{2}}\}$,
this Hamiltonian can be recast as
\begin{align}
H_{2}(t)=&J_{2}(t)\cos\phi_{2}\bigg[\left(\sin\frac{\theta_{2}}{2}\ket{a_{1}}_{L}\bra{00}
-\cos\frac{\theta_{2}}{2}\ket{a_{1}}_{L}\bra{01}+\mathrm{H.c.}\right)
\notag\\
&-\left(\cos\frac{\theta_{2}}{2}\ket{a_{2}}_{L}\bra{10}
-\sin\frac{\theta_{2}}{2}\ket{a_{2}}_{L}\bra{11}+\mathrm{H.c.}\right)\bigg]
\notag\\
&+J_{2}(t)\sin\phi_{2}[(\ket{a_{1}}\bra{a_{1}}-\ket{00}_{LL}\bra{00}-\ket{01}_{LL}\bra{01})
\notag\\
&+(\ket{a_{2}}\bra{a_{2}}-\ket{10}_{LL}\bra{10}-\ket{11}_{LL}\bra{11})],
\end{align}
which can be further recast as
\begin{align}\label{eq9}
H_{2}(t)=&J_{2}(t)\cos\phi_{2}\bigg[\left(\sin\frac{\theta_{2}}{2}\ket{a_{1}}_{L}\bra{00}
-\cos\frac{\theta_{2}}{2}\ket{a_{1}}_{L}\bra{01}+\mathrm{H.c.}\right)
\notag\\
&-\left(\cos\frac{\theta_{2}}{2}\ket{a_{2}}_{L}\bra{10}
-\sin\frac{\theta_{2}}{2}\ket{a_{2}}_{L}\bra{11}+\mathrm{H.c.}\right)\bigg]
\notag\\
&+2J_{2}(t)\sin\phi_{2}(\ket{a_{1}}\bra{a_{1}}+\ket{a_{2}}\bra{a_{2}})
\notag\\
&-J_{2}(t)\sin\phi_{2}(\ket{a_{1}}\bra{a_{1}}+\ket{a_{2}}\bra{a_{2}}+\ket{00}_{LL}\bra{00}
\notag\\
&+\ket{01}_{LL}\bra{01}+\ket{10}_{LL}\bra{10}+\ket{11}_{LL}\bra{11}).
\end{align}
It is noteworthy that $\ket{a_{1}}\bra{a_{1}}+\ket{a_{2}}\bra{a_{2}}+\ket{00}_{LL}\bra{00}
+\ket{01}_{LL}\bra{01}+\ket{10}_{LL}\bra{10}+\ket{11}_{LL}\bra{11}$ is an identity operator and thus $-J_{2}(t)\sin\phi_{2}(\ket{a_{1}}\bra{a_{1}}+\ket{a_{2}}\bra{a_{2}}+\ket{00}_{LL}\bra{00}
+\ket{01}_{LL}\bra{01}+\ket{10}_{LL}\bra{10}+\ket{11}_{LL}\bra{11})$ can only generate a global phase
during evolution. This global phase does not affect the quantum gates and therefore the terms $-J_{2}(t)\sin\phi_{2}(\ket{a_{1}}\bra{a_{1}}+\ket{a_{2}}\bra{a_{2}}+\ket{00}_{LL}\bra{00}
+\ket{01}_{LL}\bra{01}+\ket{10}_{LL}\bra{10}+\ket{11}_{LL}\bra{11})$ in Eq. (\ref{eq9}) can be ignored.
If we introduce four orthonormal states,
\begin{align}
\ket{d_{1}}&=\cos\frac{\theta_{2}}{2}\ket{00}_{L}+\sin\frac{\theta_{2}}{2}\ket{01}_{L},
\notag\\
\ket{b_{1}}&=\sin\frac{\theta_{2}}{2}\ket{00}_{L}-\cos\frac{\theta_{2}}{2}\ket{01}_{L},
\notag\\
\ket{d_{2}}&=\sin\frac{\theta_{2}}{2}\ket{10}_{L}+\cos\frac{\theta_{2}}{2}\ket{11}_{L},
\notag\\
\ket{b_{2}}&=\cos\frac{\theta_{2}}{2}\ket{10}_{L}-\sin\frac{\theta_{2}}{2}\ket{11}_{L},
\end{align}
the Hamiltonian can be further written as
\begin{align}
H_{2}(t)=&J_{2}(t)\cos\phi_{2}\left(\ket{a_{1}}\bra{b_{1}}+\ket{b_{1}}\bra{a_{1}}\right)
+2J_{2}(t)\sin_{2}\phi_{2}\ket{a_{1}}\bra{a_{1}}
\notag\\
&-J_{2}(t)\cos\phi_{2}\left(\ket{a_{2}}\bra{b_{2}}+\ket{b_{2}}\bra{a_{2}}\right)
\notag\\
&+2J_{2}(t)\sin\phi_{2}\ket{a_{2}}\bra{a_{2}}.
\end{align}
The evolution operator corresponding to this Hamiltonian then reads $U_{2}(t)=\exp[-i\int^{t}_{0}H_{2}(t^{\prime})dt^{\prime}]$, which can be explicitly expressed as
\begin{align}\label{eq7}
U_{2}(t)=&\ket{d_{1}}\bra{d_{1}}
+e^{-i\int^{t}_{0}J_{2}(t^{\prime})dt^{\prime}\sin\phi_{2}(\ket{a_{1}}\bra{a_{1}}+\ket{b_{1}}\bra{b_{1}})}
\notag\\
&\times e^{-i\int^{t}_{0}J_{2}(t^{\prime})dt^{\prime}[\cos\phi_{2}(\ket{a_{1}}\bra{b_{1}}+\ket{b_{1}}\bra{a_{1}})+
\sin\phi_{2}(\ket{a_{1}}\bra{a_{1}}-\ket{b_{1}}\bra{b_{1}})]}
\notag\\
&+\ket{d_{2}}\bra{d_{2}}+e^{-i\int^{t}_{0}J_{2}(t^{\prime})dt^{\prime}\sin\phi_{2}(\ket{a_{2}}\bra{a_{2}}+\ket{b_{2}}\bra{b_{2}})}
\notag\\
&\times e^{-i\int^{t}_{0}J_{2}(t^{\prime})dt^{\prime}[-\cos\phi_{2}(\ket{a_{2}}\bra{b_{2}}+\ket{b_{2}}\bra{a_{2}})+
\sin\phi_{2}(\ket{a_{2}}\bra{a_{2}}-\ket{b_{2}}\bra{b_{2}})]}.
\end{align}
If the evolution period $T$ is taken to satisfy
\begin{align}
\int^{T}_{0}J_{2}(t)dt=\pi,
\end{align}
the evolution operator is reduced to
\begin{align}\label{eq8}
U_{2}(T)=&\ket{d_{1}}\bra{d_{1}}
+e^{-i(\pi+\pi\sin\phi_{2})}\ket{b_{1}}\bra{b_{1}}
\notag\\
&+\ket{d_{2}}\bra{d_{2}}
+e^{-i(\pi+\pi\sin\phi_{2})}\ket{b_{2}}\bra{b_{2}}
\notag\\
&+e^{-i(\pi+\pi\sin\phi_{2})}\ket{a_{1}}\bra{a_{1}}+e^{-i(\pi+\pi\sin\phi_{2})}\ket{a_{2}}\bra{a_{2}}.
\end{align}
From Eqs. (\ref{eq7}) and (\ref{eq8}), we can see that a quantum state initially residing in the computation space $\mathcal{S}_{2}=\mathrm{Span}\{\ket{00}_{L},\ket{01}_{L},\ket{10}_{L},\ket{11}_{L}\}$ will evolve outside $\mathcal{S}_{2}$ during $t\in(0,T)$ and finally return back to $\mathcal{S}_{2}$ at $t=T$, i.e., the cyclic evolution condition (i) is satisfied. By using the commutation relation $[H_{2}(t),U_{2}(t)]=0$, we can verify that $\bra{\psi(t)}H_{2}(t)\ket{\psi(t)}
=\bra{\psi(0)}U^{\dagger}_{2}(t)H_{2}(t)U_{2}(t)\ket{\psi(0)}
=\bra{\psi(0)}H_{2}(t)\ket{\psi(0)}=0$, where $\ket{\psi(t)}$ is an evolution state such that $\ket{\psi(t)}=U_{2}(t)\ket{\psi(0)}$ with $\ket{\psi(0)}\in\mathcal{S}_{2}$. It means that the parallel transport condition (ii) is satisfied. Therefore, $U_{2}(T)$ is a holonomic transformation. When $U_{2}(T)$ acts on the computational space, it is equivalent to
\begin{align}
U_{2}=&\ket{d_{1}}\bra{d_{1}}
+e^{-i(\pi+\pi\sin\phi_{2})}\ket{b_{1}}\bra{b_{1}}
\notag\\
&+\ket{d_{2}}\bra{d_{2}}
+e^{-i(\pi+\pi\sin\phi_{2})}\ket{b_{2}}\bra{b_{2}},
\end{align}
which plays the role of a nonadiabatic holonomic gate. Similar to one-qubit gates, we can demonstrate that the two-qubit gate $U_{2}$ can be also protected by dynamical decoupling.
In the following, we demonstrate that $U_{2}$ is a nontrivial two-qubit gate.
We can see that $U_{2}$ can be rewritten as
\begin{align}
U_{2}=&\ket{0}_{LL}\bra{0}\otimes e^{-i\gamma_{2}(\sin\theta_{2}X+\cos\theta_{2}Z)/2}
\notag\\
&+\ket{1}_{LL}\bra{1}\otimes e^{-i\gamma_{2}(\sin\theta_{2}X-\cos\theta_{2}Z)/2}
\end{align}
with $\gamma_{2}=-(\pi+\pi\sin\phi_{2})$. Here, an unimportant global phase has been ignored.
Obviously, $U_{2}$ is a nontrivial two-qubit gate.
This two-qubit gate can realize a frequently used controlled phase gate when assisted by a one-qubit gate. Specifically, we first set $\theta_{2}=0$, and the two-qubit gate is reduced to $U_{2}=\ket{0}_{LL}\bra{0}\otimes\exp(-i\gamma_{2}Z/2)+\ket{1}_{LL}\bra{1}\otimes\exp(i\gamma_{2}Z/2)$. We then combine this gate and the one-qubit gate $\exp(i\gamma_{2}Z/2)$ acting on the second logical qubit, and the controlled phase gate $U_{C-P}=\ket{0}_{LL}\bra{0}+\ket{1}_{LL}\bra{1}\otimes\exp(i\gamma_{2}Z)$ can be realized.
\section{Conclusion}
In conclusion, we have put forward a protocol of nonadiabatic holonomic quantum computation protected by dynamical decoupling. A universal set of dynamical-decoupling-protected nonadiabatic holonomic gates is realized. Considering that geometric phases are only robust against control errors but cannot resist environment-induced decoherence, dynamical decoupling indeed provides an effective method to reduce the influence of the environment on nonadiabatic holonomic gates.
Our protocol can protect nonadiabatic holonomic gates against both collective decoherence and independent decoherence. Due to the combination of nonadiabatic holonomic quantum computation and dynamical decoupling, our protocol not only possesses the intrinsic robustness against control errors but also protects quantum gates against environment-induced decoherence.
\begin{acknowledgments}
This work was supported by the National Natural Science Foundation of China through Grants No. 11947221 and No. 11775129.
\end{acknowledgments}
|
1,314,259,995,796 | arxiv | \section{Introduction}
In a recent paper \cite{Ambrosio-Gigli-Savare11b} written jointly with Savar\'e, the first and second author introduced a notion of Riemannian Ricci lower
bound for metric measure spaces $(X,{\sf d},{\mbox{\boldmath$m$}})$, relying on the calculus tools they had developed in \cite{Ambrosio-Gigli-Savare11}.
This definition, in the spirit of the $CD(K,N)$ theory proposed by Lott-Villani \cite{Lott-Villani09} and
Sturm \cite{Sturm06I,Sturm06II} relies on optimal transportation tools and suitable convexity properties of the relative entropy functional
${\rm Ent}_{\mbox{\boldmath$m$}}$. In the framework of \cite{Ambrosio-Gigli-Savare11b}, these conditions are enforced
adding the assumption that the so-called Cheeger energy (playing here the role of the classical Dirichlet energy) is quadratic.\\
More precisely, the class of $RCD(K,\infty)$ spaces of \cite{Ambrosio-Gigli-Savare11b} can be defined in 3 equivalent
ways thanks to this equivalence result (see $\S\ref{ssevi}$ for the precise formulation of gradient flows involved here, in
the metric sense and in the $EVI_K$ sense):
\begin{theorem}\label{thm:main}
{\rm \cite{Ambrosio-Gigli-Savare11b} }Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ be a metric measure space with $(X,{\sf d})$ complete and separable, ${\mbox{\boldmath$m$}}(X)\in (0,\infty)$
and $\supp{\mbox{\boldmath$m$}}=X$. Then the following
are equivalent.
\begin{enumerate}
\item[(i)] $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a strict $CD(K,\infty)$ space and the $W_2$-gradient flow ${\mathscr H}_t$ of $\entv$ on $\ProbabilitiesTwo{X}$ is additive.
\item[(ii)] $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a strict $CD(K,\infty)$ space and $\mathbb{C}$ is a quadratic form on $L^2(X,{\mbox{\boldmath$m$}})$.
\item[(iii)] $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a length space and any $\mu \in \ProbabilitiesTwo{X}$ is the starting point of an $EVI_K$
gradient flow of $\entv$.
\end{enumerate}
\end{theorem}
This equivalence is crucial for the study of the spaces $RCD(K,\infty)$: for instance the fine properties of the heat
flow and the Bakry-Emery condition obtained in \cite{Ambrosio-Gigli-Savare11b} need (ii), while stability of $RCD(K,\infty)$ spaces under
Sturm's convergence \cite{Sturm06II} of metric measure spaces (a variant of measured Gromov-Hausdorff convergence)
depends in a crucial way on (iii) and on the stability properties of $EVI_K$ flows of \cite{Ambrosio-Gigli-Savare08}.
The aim of this paper is the extension of the theory of $RCD(K,\infty)$ spaces to a class of $\sigma$-finite metric measure
spaces. This extension includes fundamental examples such as the Lebesgue measure in $\mathbb{R}^n$,
noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with
lower Ricci curvature bounds studied by Cheeger and Colding \cite{Cheeger-Colding97I, Cheeger-Colding97II, Cheeger-Colding97III}. In our class of spaces
we obtain the perfect analogue of Theorem~\ref{thm:main} (see Theorem~\ref{thm:main1}).
Actually, even in the finite case we improve Theorem~\ref{thm:main}, replacing strict $CD(K,\infty)$ with $CD(K,\infty)$ in (i)
and (ii): this is possible mainly thanks to the fine results of Section~\ref{sec:Tapio}.
Let us now briefly and informally explain the terminology implicit in Theorem~\ref{thm:main} and the technical difficulties arising when one considers
$\sigma$-finite reference measures ${\mbox{\boldmath$m$}}$. Cheeger's energy $\mathbb{C}$ can be defined in $L^2(X,{\mbox{\boldmath$m$}})$ by a relaxation procedure
$$
\mathbb{C}(f):=\frac 12\inf\left\{\liminf_{h\to\infty}\int_X|Df_h|^2\,\d{\mbox{\boldmath$m$}}:\ \text{$f_h$ Lipschitz, $f_h\to f$ in $L^2(X,{\mbox{\boldmath$m$}})$}\right\},
$$
where $|Df|$ is the slope, see \eqref{eq:slope}. Instead of this direct construction,
we shall exclusively work in this paper with another equivalent one (equivalence follows by
Theorem~6.2 of \cite{Ambrosio-Gigli-Savare11}), based on the notion of weak upper gradient $\weakgrad{f}$, see Definition~\ref{def:wug}.
The weak upper gradient provides integral representation for $\mathbb{C}$, namely
$$
\mathbb{C}(f)=\frac12\int_X\weakgrad{f}^2\,\d{\mbox{\boldmath$m$}}\qquad\text{whenever $\mathbb{C}(f)<\infty$}.
$$
Since $\mathbb{C}$ is convex and lower semicontinuous on $L^2(X,{\mbox{\boldmath$m$}})$, its gradient flow ${\sf h}_tf$ is well defined starting from any initial
condition. One of the main results of \cite{Ambrosio-Gigli-Savare11} is the coincidence of ${\sf h}_t $ with the quadratic optimal transport distance semigroup
${\mathscr H}_t$ (the $W_2$ gradient flow of ${\rm Ent}_{\mbox{\boldmath$m$}}$) under the $CD(K,\infty)$ assumption: more precisely, if $f\in L^2(X,{\mbox{\boldmath$m$}})$ and
$\int f(x){\sf d}^2(x,x_0)\,\d{\mbox{\boldmath$m$}}(x)$ is finite, then ${\mathscr H}_t(f{\mbox{\boldmath$m$}})=({\sf h}_t f){\mbox{\boldmath$m$}}$, see Theorem~\ref{thm:heatgf}.
This explains the connection between (i) and (ii), where finiteness of ${\mbox{\boldmath$m$}}$ does not play any role. Passing to
the $EVI_K$ condition, deeply studied by the first two authors and Savar\'e in \cite{Ambrosio-Gigli-Savare08} and by
Daneri and Savar\'e in \cite{Daneri-Savare08}, it amounts (see Definition~\ref{def:EVIK}) to a family of differential inequalities indexed
by $\sigma\in\ProbabilitiesTwo{X}$:
\begin{equation}\label{eq:EVI}
\frac{\d}{\d t}\frac 12W_2^2(\mu_t,\sigma)\leq {\rm Ent}_{\mbox{\boldmath$m$}}(\sigma)-{\rm Ent}_{\mbox{\boldmath$m$}}(\mu_t)-\frac{K}{2}W_2^2(\mu_t,\sigma)
\quad\text{for a.e. $t\in (0,\infty)$.}
\end{equation}
Set $\mu_t=({\sf h}_t f){\mbox{\boldmath$m$}}$ and let $\varphi_t$ be Kantorovich potentials from $\mu_t$ to $\sigma$.
The analysis in \cite{Ambrosio-Gigli-Savare11b} shows that
\begin{equation}\label{eq:florence1}
\frac{\d}{{\d t}}\frac 12 W_2^2(\mu_t,\sigma)\leq \lim_{\eps\downarrow 0}\frac{\mathbb{C}(f_t-\eps\varphi_t)-\mathbb{C}(f_t)}{\eps}
\end{equation}
on the one hand, and that the $CD(K,\infty)$ condition gives
\begin{equation}\label{eq:florence2}
\lim_{\eps\downarrow 0}\frac{\mathbb{C}(\varphi_t-\eps f_t)-\mathbb{C}(\varphi_t)}{\eps}
\leq {\rm Ent}_{\mbox{\boldmath$m$}}(\sigma)-{\rm Ent}(\mu_t)-\frac{K}{2}W_2^2(\mu_t,\sigma)
\end{equation}
on the other hand.
If $\mathbb{C}$ is quadratic, then we can formally write that both the right hand side in \eqref{eq:florence1} and the left hand
side in \eqref{eq:florence2} coincide with $-\int_X D f_t\cdot D\varphi_t\,\d{\mbox{\boldmath$m$}}$, thus providing the connection from
(ii) to (iii). However, in the derivation of
\eqref{eq:florence2} a key role is played by the Sobolev regularity of $\log f_t$, that can be easily achieved if
$f_t\geq c>0$. But, this assumption is not compatible with the $\sigma$-finite case, since $f_t$ is a probability
density, and even local space-time lower bounds on $f_t$ can hardly be obtained in our framework, where no
finite dimensionality assumption on $(X,{\sf d},{\mbox{\boldmath$m$}})$ is made. It turns out that this derivation is still possible, but
only working in a time-dependent weighted Sobolev space: formally we write
$$
\int_X Df_t\cdot D\varphi_t\,\d{\mbox{\boldmath$m$}}=\int_X D\log f_t\cdot D\varphi_t\,\d(f_t{\mbox{\boldmath$m$}})
$$
and, thanks to the energy dissipation estimate
$$
{\rm Ent}_{\mbox{\boldmath$m$}}(f_T{\mbox{\boldmath$m$}})+\int_0^T\int_X\frac{\weakgrad{f_t}^2}{f_t}\,\d{\mbox{\boldmath$m$}}\,\d t\leq{\rm Ent}_{\mbox{\boldmath$m$}}(f{\mbox{\boldmath$m$}}),
$$
we know that $\log f_t$ belongs for a.e. $t$ to the Sobolev space with weight $f_t$. Then we prove that for a.e. $t>0$
the first inequality \eqref{eq:florence1} holds, when written in terms of weighted Sobolev spaces, for any choice of the Kantorovich potential
$\varphi_t$, while the second inequality \eqref{eq:florence2} holds for at least one. This suffices for the derivation of \eqref{eq:EVI}.
Besides the application to $\sigma$-finite $RCD(K,\infty)$ spaces, several results of this paper have an independent
interest and do not rely on curvature assumptions: see, for instance, Lemma~\ref{lem:GammaConvKant} which provides compactness properties
of Kantorovich potentials and Theorem~\ref{thm:change}, which analyzes the weighted Cheeger energies.
Also, it is worthwhile to mention
that existence of geodesics with $L^\infty$ bounds of Section~\ref{sec:Tapio} applies to $\sigma$-finite $CD(K,\infty)$ spaces,
i.e. no quadratic assumption
on $\mathbb{C}$ is needed for the results of the section. Also, since finiteness of ${\mbox{\boldmath$m$}}$ was used in \cite{Ambrosio-Gigli-Savare11b}
essentially only for the equivalence of Theorem~\ref{thm:main}, we describe in the last section the properties of $RCD(K,\infty)$ spaces
proved in \cite{Ambrosio-Gigli-Savare11b}, whose proof extends with no additional effort to the $\sigma$-finite case: among them we just mention the
Bakry-Emery condition
$$
\weakgrad {({\sf h}_t f)}^2\leq{\mathrm e}^{-2Kt}\weakgrad{f}^2\qquad\text{${\mbox{\boldmath$m$}}$-a.e. in $X$.}
$$
Further analysis of the Bakry-Emery condition will appear in the forthcoming paper \cite{AGSBaEm}.
The extension of the stability of the $RCD(K,\infty)$ condition under Sturm's metric measure convergence to
the $\sigma$-finite case is far from being trivial. We refer to \cite{AmbrosioGigliMondinoSavare} for the positive answer to this question.
The paper is organized as follows. In Section~\ref{sec:preliminaries} we gather a few facts on relative entropy and optimal
transportation, mostly stated without proofs (standard references are \cite{Ambrosio-Gigli11}, \cite{Ambrosio-Gigli-Savare08},
\cite{Villani09}); the only original contribution is a compactness result for Kantorovich potentials via De Giorgi's $\Gamma$-convergence stated in
Lemma~\ref{lem:GammaConvKant}.
In Section~\ref{sec:Cheeger} we recall the main results of the theory of weak gradients
as developed by the first two authors with Savar\'e in \cite{Ambrosio-Gigli-Savare11}, emphasizing also the connections with the points of view
developed by Cheeger in \cite{Cheeger00}, Koskela-MacManus in \cite{Koskela-MacManus} and
Shanmugalingam in \cite{Shanmugalingam00}. The main result of the section is Theorem~\ref{thm:change} which states that,
for probability densities $\rho=g{\mbox{\boldmath$m$}}$ with $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $\mathbb{C}(\sqrt{g})<\infty$, roughly speaking weak gradients w.r.t to ${\mbox{\boldmath$m$}}$ and
weak gradients with respect to $\rho$ are the same, even though no (local) lower bound on $g$ is assumed. Furthermore,
Cheeger's energy ${\mathbb{C}}_\rho$ induced by $\rho$ is quadratic if $\mathbb{C}$ is quadratic. Section~\ref{sec:Tapio} is crucial for the development
of (short time) $L^\infty$ estimates for displacement interpolation in $CD(K,\infty)$ spaces (see Theorem~\ref{thm:goodgeodesics} for a
precise statement) which are new in the situation when
$(X,{\sf d})$ is unbounded and ${\mbox{\boldmath$m$}}$ is not finite. These estimates,
which hold when the density of the first measure decays at least as $c_1{\mathrm e}^{-c_2{\sf d}^2(x,x_0)}$ for some $c_1,\,c_2>0$ and the second measure has
bounded density and support, are obtained combining carefully entropy minimization (an approach proposed by Sturm and
then developed by Rajala in \cite{R2011b,R2012}) and splitting of optimal geodesic plans. Section~\ref{sec:auxiliary}
is devoted to the proof of some auxiliary convergence results dealing with entropy, difference quotients of probability densities and Kantorovich
potentials, bilinear form ${\mathbb{C}}_\rho$ associated to a measure $\rho\in\ProbabilitiesTwo{X}$ as in Section~\ref{sec:Cheeger}. Section~\ref{sec:lastq}
contains the proof of Theorem~\ref{thm:main1}, which provides the equivalence result analogous to Theorem~\ref{thm:main}
in the present $\sigma$-finite setting.
\smallskip
\noindent {\bf Acknowledgement.} The authors warmly thank Giuseppe Savar\'e for his detailed and helpful comments
on a preliminary version of this paper and the reviewer for his constructive comments. The authors acknowledge the support
of the ERC ADG GeMeThNES. T.R. acknowledges the support of the Academy of Finland, project
no. 137528.
\section{Preliminaries}\label{sec:preliminaries}
In this section we introduce our notation, including the relative entropy functional ${\rm Ent}_{\mbox{\boldmath$n$}}$ in \eqref{eq:defRelentropy},
the slope $|Df|$ of a function $f$ in \eqref{eq:slope}, the one-sided slopes $|D^\pm f|$ in
\eqref{eq:onesidedslopes}, the class $AC^p(J;X)$ of absolutely continuous curves with metric derivative in $L^p(J)$,
the class of geodesics \eqref{defgeo} and the notions of geodesic and length space. We then review optimal transport,
prove the existence of special Lipschitz Kantorovich potentials (Proposition~\ref{prop:goodKant}) and prove a compactness theorem of
Kantorovich potentials (Lemma~\ref{lem:GammaConvKant}).
We assume throughout the paper that $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a metric measure space with $(X,{\sf d})$ complete and separable and ${\mbox{\boldmath$m$}}$ being a nonnegative
Borel measure finite on bounded sets and satisfying $\supp{\mbox{\boldmath$m$}}=X$.
We denote by $\Probabilities X$ the space of Borel probability measures on $(X,{\sf d})$ and
set \[
\ProbabilitiesTwo X :=\Big\{\mu\in\Probabilities X\ :\ \int_X{\sf d}^2(x_0,x)\,\d\mu(x)<\infty\,\,\,\text{for some (and hence all) $x_0\in X$}\Big\}.
\]
Given a nonnegative Borel measure ${\mbox{\boldmath$n$}}$, the \emph{relative entropy functional} ${\rm Ent}_{\mbox{\boldmath$n$}}:\ProbabilitiesTwo{X}\to [-\infty,\infty]$
with respect to ${\mbox{\boldmath$n$}}$ is defined as in Sturm's paper \cite{Sturm06I} by
\begin{equation}\label{eq:defRelentropy}
{\rm Ent}_{\mbox{\boldmath$n$}}(\mu):=
\begin{cases}
\lim\limits_{\epsilon\downarrow 0}\int_{\{\rho>\epsilon\}} \rho\log \rho\,\d{\mbox{\boldmath$n$}}& \text{if $\mu=\rho{\mbox{\boldmath$n$}}$}; \\
\infty & \text{otherwise.}
\end{cases}
\end{equation}
It coincides with $\int_{\{\rho>0\}}\rho\log\rho\,\d{\mbox{\boldmath$n$}}\in [-\infty,\infty)$ if the positive part of $\rho\log\rho$ is ${\mbox{\boldmath$n$}}$-integrable,
and it is equal to $\infty$ otherwise.
In the sequel we use the notation
\begin{equation}\label{def:DDDD}
D({\rm Ent}_{\mbox{\boldmath$n$}}):=\left\{\mu\in\ProbabilitiesTwo{X}:\ {\rm Ent}_{\mbox{\boldmath$n$}}(\mu)\in [-\infty,\infty)\right\}.
\end{equation}
By Jensen's inequality, ${\rm Ent}_{\mbox{\boldmath$n$}}$ is nonnegative when ${\mbox{\boldmath$n$}}\in\Probabilities{X}$. More generally,
we recall (see \cite[Lemma~7.2]{Ambrosio-Gigli-Savare11} for the simple proof) that when ${\mbox{\boldmath$n$}}$ satisfies the growth condition
\begin{equation}\label{eq:growthcond}
\int_X{\mathrm e}^{-{\sf c}{\sf d}^2(x_0,x)}\,\d{\mbox{\boldmath$n$}}(x) < \infty,
\end{equation}
for some $x_0\in X$ and ${\sf c}\in (0,\infty)$, then ${\rm Ent}_{\mbox{\boldmath$n$}}$ can bounded from below as follows. Letting $z=\int_X{\mathrm e}^{-c{\sf d}^2(x,x_0)}\,\d{\mbox{\boldmath$n$}}$ and
\begin{equation}\label{eq:grygorian2}
\tilde{\mbox{\boldmath$n$}}=\frac 1z {\mathrm e}^{-c{\sf d}^2(x,x_0)}{\mbox{\boldmath$n$}}\in\Probabilities{X},
\quad\qquadV(x)={\sf d}(x,x_0),
\end{equation}
and using the simple formula for the change of the reference measure
\begin{equation}\label{eq:changeentropy}
{\rm Ent}_{\mbox{\boldmath$n$}}(\mu)={\rm Ent}_{\tilde{\mbox{\boldmath$n$}}}(\mu)-c\int_XV^2\,\d\mu-\log z,\qquad\forall\mu\in\ProbabilitiesTwo X,
\end{equation}
we see that ${\rm Ent}_{\mbox{\boldmath$n$}}$ can be bounded from below in terms of the second moment of $\mu$. It is important to recall that if $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a $CD(K,\infty)$ space (see Definition~\ref{def:CD}), then the reference measure ${\mbox{\boldmath$m$}}$ always satisfies the growth condition \eqref{eq:growthcond},
as shown by Sturm in \cite[Theorem 4.24]{Sturm06I}.
\subsection{Metric structure}
We shall denote by ${\rm Lip}(X)$ the space
of Lipschitz functions $f:X\to\mathbb{R}$ and by ${\rm Lip}_b(X)$ the subspace of bounded
Lipschitz functions.
Given $f:X\to\mathbb{R}$ we define its slope $|Df|$ at $x$ by
\begin{equation}\label{eq:slope}
|Df|(x):=\limsup_{y\to x}\frac{|f(y)-f(x)|}{{\sf d}(y,x)}.
\end{equation}
We shall also use, in connection with Kantorovich potentials,
the one-sided counterparts of the slope, namely the ascending slope and descending slopes:
\begin{equation}\label{eq:onesidedslopes}
|D^+f|(x):=\limsup_{y\to x}\frac{[f(y)-f(x)]^+}{{\sf d}(y,x)},\qquad
|D^-f|(x):=\limsup_{y\to x}\frac{[f(y)-f(x)]^-}{{\sf d}(y,x)}.
\end{equation}
Given an open interval $J\subset\mathbb{R}$, an
exponent $p\in [1,\infty]$ and $\gamma:J\to X$, we say that
$\gamma$ belongs to $AC^p(J;X)$ if there exists $g\in L^p(J)$ satisfying
$$
{\sf d}(\gamma_s,\gamma_t)\leq\int_s^t g(r)\,\d r\qquad\forall s,\,t\in
J,\,\,s<t.
$$
The case $p=1$ corresponds to \emph{absolutely continuous} curves, denoted $AC(J;X)$.
It turns out that, if $\gamma$ belongs to $AC^p(J;X)$, there is a minimal function $g$ with this property, called
\emph{metric derivative} and given for a.e. $t\in J$ by
$$
|\dot\gamma_t|:=\lim_{s\to t}\frac{{\sf d}(\gamma_s,\gamma_t)}{|s-t|}.
$$
See \cite[Theorem~1.1.2]{Ambrosio-Gigli-Savare08} for the simple
proof. We say that an absolutely continuous curve $\gamma_t$ has
\emph{constant speed} if $|\dot\gamma_t|$ is (equivalent to) a
constant.
We call $(X,{\sf d})$ \emph{a geodesic space} if for any $x_0,\,x_1\in
X$ there exists
$\gamma:[0,1]\to X$ satisfying $\gamma_0=x_0$, $\gamma_1=x_1$ and
\begin{equation}\label{defgeo}
{\sf d}(\gamma_s,\gamma_t)=|t-s|{\sf d}(\gamma_0,\gamma_1)\qquad\forall s,\,t\in
[0,1].
\end{equation}
We will denote by $\geo(X)$ the space of all constant speed
geodesics $\gamma:[0,1]\to X$, namely $\gamma\in\geo(X)$ if
\eqref{defgeo} holds. Recall also that the weaker notion of \emph{length} space:
for all $x_0,\,x_1\in X$ and $\eps>0$ there exists $\gamma\in AC([0,1];X)$ such
that $\int_0^1|\dot\gamma_t|\,\d t<{\sf d}(x_0,x_1)+\eps$.
{F}rom the measure-theoretic point of view, when considering measures
on $AC^p(J;X)$ (resp. $\geo(X)$), we shall consider
them as measures on the Polish space $C(J;X)$ endowed with the sup
norm, concentrated on the Borel set $AC^p(J;X)$ (resp. closed set $\geo(X)$). We shall also use the
notation $\e_t:C(J;X)\to X$, $t\in J$, for the evaluation map at time $t$, namely
$\e_t(\gamma):=\gamma_t$.
\subsection{Optimal transport}
Given $\mu,\,\nu\in\ProbabilitiesTwo X$, we define the quadratic optimal transport distance $W_2$
between them as
\begin{equation}\label{eq:W2def}
W_2^2(\mu,\nu):=\inf\int_{X\times X} {\sf d}^2(x,y)\,\d{\mbox{\boldmath$\gamma$}}(x,y),
\end{equation}
where the infimum is taken among all Kantorovich transport plans, namely
probability measures ${\mbox{\boldmath$\gamma$}}$ on $X\times X$
such that
\[
\pi^1_\sharp{\mbox{\boldmath$\gamma$}}=\mu,\qquad\pi^2_\sharp{\mbox{\boldmath$\gamma$}}=\nu.
\]
Here, for $\mu\in \Probabilities X$, a topological space $Y$ and a $\mu$-measurable
map $T:X\to Y$, the push-forward measure $T_\sharp \mu\in \Probabilities Y$ is defined by
$T_\sharp\mu(B):=\mu(T^{-1}(B))$ for every Borel set $B\subset Y$.
Since $(X,{\sf d})$ is complete and separable,
the space $(\ProbabilitiesTwo X,W_2)$ is complete and separable. Since the cost ${\sf d}^2$ is lower semicontinuous,
the infimum in the definition \eqref{eq:W2def} of $W_2^2$ is attained.
All plans ${\mbox{\boldmath$\gamma$}}$ achieving the minimum will be called optimal.
For all $\mu,\,\nu\in\ProbabilitiesTwo{X}$ Kantorovich's duality formula holds:
\begin{equation}
\label{eq:dualitabase} \frac12 W_2^2(\mu,\nu)=\sup
\left\{\int_X\varphi\,\d\mu+\int_X\psi\,\d\nu:\
\varphi(x)+\psi(y)\leq \frac12 {\sf d}^2(x,y)\right\},
\end{equation}
where the supremum is taken among all functions $\varphi\in L^1(X,\mu)$ and $\psi\in L^1(X,\nu)$.
Recall that the \emph{$c$-transform} $\varphi^c$
of $\varphi:X\to\mathbb{R}\cup\{-\infty\}$ is defined by
\[
\varphi^c(y):=\inf\left\{\frac{{\sf d}^2(x,y)}2-\varphi(x):\ x\in X\right\}
\]
and that $\psi$ is said to be \emph{$c$-concave} if
$\psi=\varphi^c$ for some $\varphi$.
\begin{definition}[Kantorovich potential]\label{def:Kant}
We say that a map $\varphi:X\to\mathbb{R}\cup\{-\infty\}$ is a Kantorovich potential relative to $(\mu,\nu)$ if:
\begin{itemize}
\item[(i)] there exists a Borel map $\psi:X\to\mathbb{R}\cup\{-\infty\}$ such that $\psi\in L^1(X,\nu)$ and $\varphi=\psi^c$;
\item[(ii)] $\varphi\in L^1(X,\mu)$ and the pair $(\varphi,\psi)$ maximizes \eqref{eq:dualitabase}.
\end{itemize}
\end{definition}
Notice that the inequality $\varphi(x)+\psi(y)\leq\tfrac12{\sf d}^2(x,y)$, when integrated against an optimal plan
${\mbox{\boldmath$\gamma$}}$, forces the integrability of the positive part of $\varphi$. For this reason, in (ii) we may equivalently
require integrability of the negative part of $\varphi$ only. In the next proposition we illustrate
some key properties of Kantorovich potentials $\varphi$ and show how, in the special case when $\supp\nu$ is bounded,
a special choice of $\psi$ provides better properties of $\varphi=\psi^c$.
\begin{proposition}[Existence of Kantorovich potentials]\label{prop:goodKant}
If $\mu,\,\nu\in\ProbabilitiesTwo{X}$, then a Kantorovich potential $\varphi=\psi^c$ relative to $(\mu,\nu)$
exists and satisfies
\begin{equation}\label{eq:itforza1}
\varphi(x)+\psi(y)=\tfrac 12{\sf d}^2(x,y)\qquad\text{for ${\mbox{\boldmath$\gamma$}}$-a.e. in $(x,y)\in X\times X$}
\end{equation}
for any optimal Kantorovich plan ${\mbox{\boldmath$\gamma$}}$ and
\begin{equation}\label{eq:itforza}
|D^+\varphi|(x)\leq{\sf d}(x,y)\qquad\text{for ${\mbox{\boldmath$\gamma$}}$-a.e. $(x,y)$.}
\end{equation}
In addition, if $\supp\nu\subset\overline{B}_R(y_0)$ for some $R\geq 1$, then a locally Lipschitz
Kantorovich potential $\varphi=\psi^c$ exists
with $\psi\equiv -\infty$ on $X\setminus\supp\nu$, $\psi\leq R^2/2$ on $\supp\nu$ and
\begin{equation}\label{GrowthKant}
|D \varphi| (x) \leq R+{\sf d}(x,y_0), \qquad |\varphi(x)| \leq 2R^2 (1+{\sf d}^2(x,y_0)).
\end{equation}
\end{proposition}
\begin{proof} Since any complete and separable metric space can be isometrically embedded in a complete, separable and
geodesic metric space we can assume with no loss of generality that the space $(X,{\sf d})$ is geodesic.
The existence part is well known, so let us discuss briefly \eqref{eq:itforza}, the choice of gauge and
the regularity properties of $\varphi$ when $\nu$ has bounded support. From \eqref{eq:itforza1} and
the inequality $\varphi+\varphi^c\leq{\sf d}^2/2$ we get
$$
\varphi(z)-\varphi(x)\leq\frac{1}{2}\bigl({\sf d}^2(z,y)-{\sf d}^2(x,y)\bigr) \qquad\text{for all $z$}
$$
for ${\mbox{\boldmath$\gamma$}}$-a.e. $(x,y)$, so that $|D^+\varphi|(x)\leq{\sf d}(x,y)$ for $\gamma$-a.e. $(x,y)$.
Now, let us set
$$
\tilde\psi(x):=\begin{cases}
\psi(x) &\text{if $x\in\supp\nu$};\\
-\infty &\text{otherwise,}
\end{cases}
$$
and $\tilde{\varphi}:=(\tilde\psi)^c$.
Since $\tilde\varphi\geq\varphi$, it is obvious that its negative part is
$\mu$-integrable and that $(\tilde\varphi,\tilde\psi)$ is a maximizing pair, so that $\tilde\varphi$ is a Kantorovich
potential. From
$$
\tilde\varphi(x)=\inf_{y\in\supp\nu}\frac 12{\sf d}^2(x,y)-\tilde\psi(y)
$$
and the inclusion $\supp\nu\subset B_R(y_0)$ it is immediate to obtain the linear growth of $|D\tilde\varphi|$, in the form stated in \eqref{GrowthKant}.
Finally, possibly adding and subtracting the same constant to the potentials in the maximizing pair, we can assume
that $\tilde\varphi(y_0)=0$. Then, the inequality $\tilde\psi\leq \frac 12{\sf d}^2(y_0,\cdot)$ gives
$\tilde\psi\leq R^2/2$ on $\supp\nu$. The linear growth of $|D\tilde\varphi|$ gives the quadratic growth
of $|\varphi|$, since $(X,{\sf d})$ is geodesic. \end{proof}
In the proof of the next lemma we use De Giorgi's $\Gamma$-convergence. Strictly speaking, we use $\Gamma^-$-convergence,
the one designed for convergence of minimum problems. We recall the definition and the basic facts, referring to
Dal Maso's book \cite{DalMaso} for a full account of this theory. If $(Y,d)$ is a metric space
and $f_h:Y\to [-\infty,+\infty]$, $f:Y\to [-\infty,+\infty]$ are lower semicontinuous, we say that $(f_h)$ $\Gamma$-converges
to $f$ and write $f=\Gamma-\lim_h f_h$ if:
\begin{itemize}
\item[(a)] for any sequence $(y_h)\subset Y$ convergent to $y\in Y$, one has $\liminf_h f_h(y_h)\geq f(y)$;
\item[(b)] for all $y\in Y$ there exists $(y_h)\subset Y$ convergent to $y$ and satisfying $\limsup_h f_h(y_h)\leq f(y)$.
\end{itemize}
It is immediate to check that $\Gamma$-convergence is invariant by additive constant perturbations. In addition, (a)
yields that $f\mapsto\inf_A f$ is lower semicontinuous w.r.t. $\Gamma$-convergence for any open set $A\subset Y$, while
(b) yields that $f\mapsto\min_K f$ is upper semicontinuous w.r.t. $\Gamma$-convergence for any compact set $K\subset Y$.
If $Y$ is compact we can choose $A=K=Y$ to obtain
\begin{equation}\label{eq:vuoleilreferee}
\Gamma-\lim_{h\to\infty}f_h=f\qquad\Longrightarrow\qquad
\lim_{h\to\infty} \min_Y f_h=\min_Y f.
\end{equation}
We need one more property of $\Gamma$-convergence: if $Y$ is separable, then any sequence of lower semicontinuous
maps $f_h:Y\to [-\infty,+\infty]$ admits a $\Gamma$-convergent subsequence $f_{h(k)}$. To see this, let $\mathscr U$ be a
countable basis of open sets of $Y$ and extract with a diagonal argument a subsequence $h(k)$ such that $\inf_U f_{k(k)}$ has a limit in
$[-\infty,+\infty]$ for all $U\in\mathscr U$. Then, the function
$$
f(y):=\sup_{U\ni y,\,U\in{\mathscr U}}\lim_{k\to\infty}\inf_U f_{h(k)}\qquad y\in Y
$$
provides the $\Gamma$-limit of $f_{h(k)}$.
\begin{lemma}[Compactness of Kantorovich potentials]\label{lem:GammaConvKant}
Consider probability densities $\sigma,\,\eta=f{\mbox{\boldmath$m$}},\,\eta_n=f_n{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$ satisfying the following conditions:
\begin{itemize}
\item[(a)] $\sigma$ has compact support;
\item[(b)] $f_n \to f$ ${\mbox{\boldmath$m$}}$-a.e. in $X$ and
$\sup_n f_n (x) (1+{\sf d}^2(x, x_0)) \in L^1(X,{\mbox{\boldmath$m$}})$ for some $x_0 \in X$.
\end{itemize}
Suppose there exist $C>0$ and Kantorovich potentials $\varphi_n=\psi_n^c$ relative to $(\eta_n,\sigma)$ in the sense of Definition~\ref{def:Kant},
satisfying
\begin{equation}\label{eq:Assphin}
|\varphi_n(x)|\leq C (1+{\sf d}^2(x, x_0)) \quad\forall x\in X
\end{equation}
and
\begin{equation}\label{eq:Asspsin}
\psi_n\equiv -\infty \quad \text{on}\; X \setminus \supp \sigma\quad\text{and}\quad \psi_n (x)\leq C\,\,\forall x\in X.
\end{equation}
Then there exist a subsequence $n(k)$ and a Kantorovich potential $\varphi=\psi^c$ of the transportation problem relative to
$(\eta,\sigma)$ such that $\varphi_{n(k)} \to \varphi$ pointwise. In addition \eqref{eq:Assphin} is fulfilled by $\varphi$ and $\psi\leq C$.
\end{lemma}
\begin{proof}
Since $X$ is separable, by the compactness properties of $\Gamma$-convergence
we can assume with no loss of generality that $-\psi_n$ $\Gamma$-converges as $n\to\infty$, and we shall denote by $-\psi$ its
$\Gamma$-limit. Observe that, since by definition of $\Gamma$-convergence for every $x \in X$ there exists a sequence $x_n\to x$ such that
$-\psi_n(x_n)\to -\psi(x)$, $\psi$ still satisfies \eqref{eq:Asspsin}.
By the invariance of $\Gamma$-convergence under continuous additive perturbations we get
\begin{equation}
\left(\frac{1}{2} {\sf d}^2(x,\cdot)-\psi\right)= \Gamma-\lim_{n\to \infty} \left(\frac{1}{2} {\sf d}^2(x,\cdot)-\psi_n\right) \qquad \forall x \in X.
\end{equation}
Because of \eqref{eq:Asspsin} and of the compactness of $\supp\sigma$, we can use \eqref{eq:vuoleilreferee} to get
\begin{equation}\label{def:varphi}
\varphi_n(x)=\min_X \left(\frac{1}{2} {\sf d}^2(x,\cdot)- \psi_n\right) \to \min_X \left(\frac{1}{2}{\sf d}^2(x,\cdot)-\psi \right)=\varphi(x),
\end{equation}
where the last equality has to be understood as the definition of $\varphi(x)$. Obviously \eqref{eq:Assphin} is fulfilled
by $\varphi$, so that $\varphi\in L^1(X,f{\mbox{\boldmath$m$}})$. In connection with $\psi$, obviously its positive part is $\sigma$-integrable.
Now we claim that $\varphi=\psi^c$ is a Kantorovich potential for the limit transportation problem $(f{\mbox{\boldmath$m$}},\sigma)$; we have to prove that
\begin{equation}\label{claim:phiKant}
\int_X \varphi\, \d (f{\mbox{\boldmath$m$}})+\int_X \psi \,\d\sigma\geq \frac 1 2 W^2_2 (f {\mbox{\boldmath$m$}}, \sigma),
\end{equation}
since this inequality provides at the same time also integrability of the negative part of $\psi$.
Since by assumption $\varphi_n=\psi_n^c$ is a Kantorovich potential for $(f_n {\mbox{\boldmath$m$}}, \sigma)$, we already know that
\begin{equation}\label{eq:phinKant}
\int_X \varphi_n\, \d (f_n{\mbox{\boldmath$m$}})+\int_X \psi_n \,\d\sigma = \frac{1}{2} W^2_2 (f_n{\mbox{\boldmath$m$}}, \sigma).
\end{equation}
Using (b) it is immediate to check the weak convergence of $f_n {\mbox{\boldmath$m$}}$ to $f{\mbox{\boldmath$m$}}$, so that
(see for instance Proposition~2.5 in \cite{Ambrosio-Gigli11})
\begin{equation}\label{eq:LSCW22}
W^2_2(f{\mbox{\boldmath$m$}},\sigma) \leq \liminf_n W^2_2(f_n{\mbox{\boldmath$m$}},\sigma).
\end{equation}
Moreover, using (b) and \eqref{eq:Assphin}, the dominated convergence theorem gives
\begin{equation}\label{eq:convIntphin}
\int_X \varphi_n\, \d (f_n{\mbox{\boldmath$m$}}) \to \int_X \varphi \, \d (f{\mbox{\boldmath$m$}}).
\end{equation}
Finally, by the very definition of $\Gamma$-limit we have
\begin{equation}\nonumber\label{eq:PsiGammaL}
-\psi(x)=\inf\left\{\liminf_{n\to\infty}-\psi_n(x_n)|\,x_n \to x \right\}\leq \liminf_{n\to\infty} -\psi_n(x).
\end{equation}
Moreover, by assumption \eqref{eq:Asspsin}, $-\psi_n\geq -C$. Hence Fatou's lemma gives
\begin{equation}\label{eq:convIntphicn}
\limsup_{n\to \infty} \int_X \psi_n \,\d\sigma\leq \int_X \psi\, \d \sigma.
\end{equation}
Putting together \eqref{eq:phinKant}, \eqref{eq:LSCW22}, \eqref{eq:convIntphin} and \eqref{eq:convIntphicn} we get \eqref{claim:phiKant} as desired.
\end{proof}
Let us close this section by discussing the geodesic structure of $(\ProbabilitiesTwo{X},W_2)$, see
\cite[Theorem~2.10]{Ambrosio-Gigli11} or \cite{Lisini07}.
If $\mu_0, \,\mu_1\in\ProbabilitiesTwo X$ are connected by a constant speed geodesic $\mu_t$ in $(\ProbabilitiesTwo X, W_2)$,
then there exists ${\mbox{\boldmath$\pi$}} \in \Probabilities{\geo(X)}$ with $(\e_t)_\sharp{\mbox{\boldmath$\pi$}} = \mu_t$ for all $t\in [0,1]$
and
\[
W_2^2(\mu_s,\mu_t) = \int_{\geo(X)}{\sf d}^2(\gamma_s,\gamma_t)\,\d{\mbox{\boldmath$\pi$}}(\gamma)=
(s-t)^2\int_{\geo(X)}\ell^2(\gamma)\,\d{\mbox{\boldmath$\pi$}}(\gamma)\qquad\forall s,\,t\in [0,1],
\]
where $\ell(\gamma)={\sf d}(\gamma_0,\gamma_1)$ is the length of the geodesic $\gamma$.
The collection of all the measures ${\mbox{\boldmath$\pi$}}$ with the above properties is denoted by
$\gopt(\mu_0,\mu_1)$.
The measure ${\mbox{\boldmath$\pi$}}$ is not uniquely determined by $\mu_t$, unless $(X,{\sf d})$ is non-branching.
The relation between optimal geodesic plans and optimal Kantorovich plans is given by the fact that
$\gamma:=(\e_0,\e_1)_\sharp{\mbox{\boldmath$\pi$}}$ is optimal whenever ${\mbox{\boldmath$\pi$}}\in\gopt(\mu_0,\mu_1)$.
\subsection{Gradient flows}\label{ssevi}
In this section we review the notions of gradient flows in the metric sense, in the
$EVI_K$ sense and in the classical sense provided, in Hilbert spaces, by the theory of
monotone operators.
Let $(Y,{\sf d}_Y)$ be a complete and separable metric space and $K\in\mathbb{R}$.
We say that $E:Y\to\mathbb{R}\cup\{+\infty\}$ is
$K$-geodesically convex if for any $y_0,\,y_1\in D(E)$ there exists
$\gamma\in\geo(Y)$ satisfying $\gamma_0=y_0$, $\gamma_1=y_1$ and
\[
E(\gamma_t)\leq (1-t)E(y_0)+tE(y_1)-\frac K2t(1-t)
{\sf d}_Y^2(y_0,y_1)\qquad\text{for every } t\in[0,1].
\]
\begin{definition}[Metric formulation of gradient flow]\label{def:dissKconv}
Let $E:Y\to\mathbb{R}\cup\{+\infty\}$ be a $K$-geodesically convex and
l.s.c. functional. We say that a locally absolutely continuous curve
$[0,\infty)\ni t\mapsto y_t\in D(E)$ is a gradient flow of $E$
starting from $y_0\in D(E)$ if
\begin{equation}\label{eq:ede}
E(y_0)= E(y_t)+\int_0^t\frac12 |\dot y_r|^2+\frac12|D^- E|^2(y_r)\,\d r\qquad\forall t\geq 0.
\end{equation}
\end{definition}
Next we recall a stronger formulation of gradient flows, introduced and
extensively studied in \cite{Ambrosio-Gigli-Savare08},
\cite{Daneri-Savare08}.
\begin{definition}[Gradient flows in the $EVI_K$ sense]\label{def:EVIK}
Let $E:Y\to\mathbb{R}\cup\{+\infty\}$ be a lower semicontinuous functional,
$K\in\mathbb{R}$ and $(0,\infty)\ni t\mapsto y_t\in D(E)$ be a locally
absolutely continuous curve. We say that $(y_t)$ is a $K$-gradient
flow for $E$ in the Evolution Variational Inequalities sense (or,
simply, it is an $EVI_K$ gradient flow) if for any $z\in Y$ we have
\begin{equation}
\label{eq:defevi} \frac \d{\d t}\frac{{\sf d}_Y^2(y_t,z)}2+\frac
K2{\sf d}_Y^2(y_t,z)+E(y_t)\leq E(z)\qquad\text{for a.e.~$t\in
(0,\infty)$.}
\end{equation}
If $\lim\limits_{t\downarrow 0}y_t=y_0 \in \overline {D(E)}$, we say
that the gradient flow starts from $y_0$.
\end{definition}
Notice that the derivative in \eqref{eq:defevi} exists for
a.e.~$t>0$, since $t\mapsto{\sf d}_Y(y_t,z)$ is locally absolutely
continuous in $(0,\infty)$.
We recall some basic and useful properties of gradient flows in the
$EVI_K$ sense, see Proposition~2.22 in \cite{Ambrosio-Gigli-Savare11b};
we also refer to \cite[Chap.\ 4]{Ambrosio-Gigli-Savare08} for more results. In particular, we
emphasize that the maps ${\sf S}_t:y_0\mapsto y_t$ that at every $y_0$
associate the value at time $t\ge0$ of the unique $K$-gradient flow
starting from $y_0$ give raise to a continuous semigroup of
$K$-contractions according to \eqref{eq:21} in a closed (possibly
empty) subset of $Y$.
\begin{proposition}[Properties of gradient flows in the $EVI_K$ sense]\label{prop:evipropr}
Let $Y$, $E$, $K$, $y_t$ be as in Definition~\ref{def:EVIK}
and suppose that $(y_t)$ is an $EVI_K$ gradient flow of $E$ starting from
$y_0$. Then:
\begin{itemize}
\item[(i)] If $y_0\in D(E)$, then $y_t$ is also a metric gradient flow,
i.e.~\eqref{eq:ede} holds.
\item[(ii)] If $(\tilde y_t)$ is another $EVI_K$ gradient flow for $E$ starting
from $\tilde{y}_0$, then
\begin{equation}
{\sf d}_Y(y_t,\tilde y_t)\leq e^{-Kt}{\sf d}_Y(y_0,\tilde y_0).\label{eq:21}
\end{equation}
In particular, $EVI_K$ gradient flows uniquely depend on the
initial condition.
\item[(iii)]
Existence of $EVI_K$ gradient flows starting from any point in
$D\subset Y$ implies existence starting from any point in $\overline
D$.
\end{itemize}
\end{proposition}
If $(Y,{\sf d}_Y)$ is a Hilbert space with distance induced by the scalar product,
the gradient flow of a lower semicontinuous functional $E:Y\to \mathbb{R}\cup\{+\infty\}$
can also be defined as a locally absolutely continuous
map $y_t:(0,\infty)\to H$ satisfying
\begin{equation}\label{eq:defheat}
\frac{\d}{\d t}y_t\in -\partial^- E(y_t)\,\,\,\text{for a.e. $t>0$},
\qquad\lim_{t\downarrow 0}y_t=y\,\,\text{in $H$,}
\end{equation}
where the Frechet subdifferential $\partial^-E(y)$ is defined by
\begin{equation}\label{eq:subdiff}
\partial^- E(y):=\left\{\xi\in H:\ \liminf_{y'\to y}\frac{E(y')-E(y)-\langle\xi,y'-y\rangle}{{\sf d}_Y(y',y)}\geq 0\right\}.
\end{equation}
Under a $K$-convexity assumption the subdifferential can be equivalently defined
\begin{equation}\label{eq:subdiff1}
\partial^- E(y):=\left\{\xi\in H: E(y')\geq E(y)+\langle \xi,y'-y\rangle+\frac{K}{2}{\sf d}_Y^2(y',y)\,\,\,\text{for all $y'\in H$}\right\}.
\end{equation}
Differentiating the squared distance in \eqref{eq:defevi} yields that the $EVI_K$ formulation and \eqref{eq:defheat}
are equivalent in the Hilbert setting, for $K$-convex functionals.
\section{Weak gradients and weighted Cheeger energies}\label{sec:Cheeger}
In this section we recall the main results of the theory of weak gradients
as developed by the first two authors with Savar\'e in \cite{Ambrosio-Gigli-Savare11}, emphasizing the connections with the points of view
developed by Cheeger in \cite{Cheeger00}, Koskela-MacManus in \cite{Koskela-MacManus} and
Shanmugalingam in \cite{Shanmugalingam00}. We prove in Theorem~\ref{thm:change} the equivalence of weak gradients
defined with reference measures ${\mbox{\boldmath$n$}}$ and ${\mbox{\boldmath$m$}}$, under suitable assumptions on the density of ${\mbox{\boldmath$n$}}$ w.r.t. ${\mbox{\boldmath$m$}}$.
We introduce in \eqref{def:Cheeger} the weighted
Cheeger energy ${\mathbb{C}}_{\mbox{\boldmath$n$}}$ and show in Theorem~\ref{thm:weighted} that, under the assumptions of Theorem~\ref{thm:change},
${\mathbb{C}}_{\mbox{\boldmath$n$}}$ is quadratic whenever $\mathbb{C}$ is quadratic.
In the next two definitions we consider test plans and ``Sobolev" functions with respect to a reference nonnegative Borel measure
${\mbox{\boldmath$n$}}$ in $X$, finite on bounded sets. In the sequel we shall denote by ${\mathcal M}$ this class of measures, including both
probability measures and our reference measure ${\mbox{\boldmath$m$}}$.
\begin{definition}[Test plan]
We say that ${\mbox{\boldmath$\pi$}}\in\Probabilities{C([0,1];X)}$ is a 2-test plan relative to ${\mbox{\boldmath$n$}}\in{\mathcal M}$ if:
\begin{itemize}
\item[(i)] ${\mbox{\boldmath$\pi$}}$ is concentrated on $AC^2([0,1];X)$ and the $2$-action of ${\mbox{\boldmath$\pi$}}$ is finite:
$$
{\cal A}_2({\mbox{\boldmath$\pi$}}):=\int\int_0^1|\dot\gamma_t|^2\,\d t\,\d{\mbox{\boldmath$\pi$}}(\gamma)<\infty.
$$
\item[(ii)] There exists $C\geq 0$ such that $(\e_t)_\sharp{\mbox{\boldmath$\pi$}}\leq C{\mbox{\boldmath$n$}}$ for all $t\in [0,1]$.
\end{itemize}
\end{definition}
The following definition is inspired by the Heinonen-Koskela's concept \cite{Heinonen-Koskela98} of upper gradient, that we now illustrate.
A Borel function $G:X\to [0,\infty]$ is an upper gradient of a Borel function $f:X\to\mathbb{R}$ if
$$
|f(\gamma_b)-f(\gamma_a)|\leq \int_a^bG(\gamma_s)|\dot\gamma_s|\,\d s
$$
for any absolutely continuous curve $\gamma:[a,b]\to X$. Since the inequality is invariant under reparameterization one can also
reduce to curves defined in $[0,1]$.
Let $\Curvesnonpara{X}$ be the set of continuous parametric curves $C\subset X$ with finite length, where curves
equivalent under reparameterization are identified. Recall that any such curve $C$ can be written
as $\gamma([0,\ell])$, where $\ell$ is the length of $C$ and $\gamma:[0,\ell]\to X$ is Lipschitz
with $|\dot\gamma|=1$ a.e. in $[0,\ell]$. We shall denote by
$i:AC^2([0,1];X)\to\Curvesnonpara{X}$ the natural surjection.
Recall also that the the $2$-modulus of $\Gamma\subset\Curvesnonpara{X}$ is defined by
\begin{equation}\label{eq:Mod2}
{\rm Mod}_{2,{\mbox{\boldmath$n$}}}(\Gamma):=\inf\left\{\int_Xg^2\,\d{\mbox{\boldmath$n$}}:\
\text{$g:X\to [0,\infty]$ Borel, $\int_\gamma g\geq 1$ for all $\gamma\in\Gamma$}\right\}.
\end{equation}
Shanmugalingam proved in \cite{Shanmugalingam00} that functions with an upper gradient in $L^2(X,{\mbox{\boldmath$n$}})$
are absolutely continuous along ${\rm Mod}_{2,{\mbox{\boldmath$n$}}}$-a.e. curve in $\Curvesnonpara{X}$.
We also recall the following simple consequence of
\eqref{eq:Mod2}: for any ${\rm Mod}_{2,{\mbox{\boldmath$n$}}}$-negligible set $\Gamma$ there exist Borel functions $r_h:X\to [0,\infty]$
satisfying $\int_X r_h^2\,\d{\mbox{\boldmath$n$}}\to 0$ and $\int_\gamma r_h=\infty$ for all $\gamma\in\Gamma$. Also, the inequality
$$
{\rm Mod}_{2,{\mbox{\boldmath$n$}}}\bigl(\{\gamma:\ \int_\gamma g\geq t\}\bigr)\leq \frac{1}{t}\biggl(\int_Xg^2\,\d{\mbox{\boldmath$n$}}\biggr)^{1/2}\qquad t>0
$$
immediately yields that functions in $L^2(X,{\mbox{\boldmath$m$}})$ have a finite integral on $\gamma$ for ${\rm Mod}_{2,{\mbox{\boldmath$n$}}}$-a.e. $\gamma$.
\begin{definition}[The space ${\mathcal S}^2_{\mbox{\boldmath$n$}}$ and weak upper gradients]\label{def:wug}
Let $f:X\to\mathbb{R}$, $G:X\to [0,\infty]$ be Borel functions. We say that $G$ is a $2$-weak upper gradient relative to ${\mbox{\boldmath$n$}}$ of $f$ if
$$
|f(\gamma_1)-f(\gamma_0)|\leq \int_0^1G(\gamma_s)|\dot\gamma_s|\,\d s<\infty\qquad\text{for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$}
$$
for all $2$-test plans ${\mbox{\boldmath$\pi$}}$ relative to ${\mbox{\boldmath$n$}}$. \\
We write $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}$ if $f$ has a $2$-weak upper gradient in $L^2(X,{\mbox{\boldmath$n$}})$. The $2$-weak upper gradient
relative to ${\mbox{\boldmath$n$}}$ with minimal $L^2(X,{\mbox{\boldmath$n$}})$ norm (the so-called minimal $2$-weak upper gradient) will be denoted by $|D f|_{w,{\mbox{\boldmath$n$}}}$.
\end{definition}
\begin{remark}[Sobolev regularity along curves]\label{rem:charaweakgrad}{\rm
A consequence of ${\mathcal S}^2_{\mbox{\boldmath$n$}}$ regularity is (see Proposition~5.7 in \cite{Ambrosio-Gigli-Savare11}) the Sobolev property along curves, namely for any
$2$-test plan ${\mbox{\boldmath$\pi$}}$ relative to ${\mbox{\boldmath$n$}}$ the function $t\mapsto f(\gamma_t)$ belongs to the Sobolev space $W^{1,1}(0,1)$ and
$$
|\frac{\d}{\d t}f(\gamma_t)|\leq\weakgrad{f}(\gamma_t)|\dot\gamma_t|\qquad\text{a.e. in $(0,1)$}
$$
for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$. Conversely, assume that $g$ is Borel nonnegative, that for any $2$-test plan ${\mbox{\boldmath$\pi$}}$ the map
$t\mapsto f(\gamma_t)$ is $W^{1,1}(0,1)$ and that
$$
|\frac{\d}{\d t}f(\gamma_t)|\leq g(\gamma_t)|\dot\gamma_t|\qquad\text{a.e. in $(0,1)$}
$$
for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$. Then, the fundamental theorem of calculus in $W^{1,1}(0,1)$ gives that $g$ is a $2$-weak upper gradient of $f$.
}\fr\end{remark}
Because of the absolute continuity condition $({\mathrm e}_t)_\sharp{\mbox{\boldmath$\pi$}}\ll{\mbox{\boldmath$n$}}$ imposed on test plans, it is immediate to check that
the property of being in ${\mathcal S}^2_{\mbox{\boldmath$n$}}$, as well as $|D f|_{w,{\mbox{\boldmath$n$}}}$, are invariant under modifications of $f$ in ${\mbox{\boldmath$n$}}$-negligible sets.
Furthermore, these concepts are easily seen to be local with respect to ${\mbox{\boldmath$n$}}$ in the following sense: if $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}$ then
$f\in{\mathcal S}^2_{{\mbox{\boldmath$n$}}'}$ for all measures ${\mbox{\boldmath$n$}}'={\mbox{\boldmath$n$}}\res B$ with $B\subset X$ Borel, and $|Df|_{w,{\mbox{\boldmath$n$}}'}\leq |Df|_{w,{\mbox{\boldmath$n$}}}$ ${\mbox{\boldmath$n$}}'$-a.e. on $B$: this is due to the
fact that test plans relative to ${\mbox{\boldmath$n$}}'$ are test plans relative to ${\mbox{\boldmath$n$}}$. Conversely,
\begin{equation}\label{eq:localitynn}
\text{$f\in{\mathcal S}^2_{{\mbox{\boldmath$n$}}_R}$ with ${\mbox{\boldmath$n$}}_R:={\mbox{\boldmath$n$}}\res\overline{B}_R(x_0)$, $\sup_R\int_X|D f|_{w,{\mbox{\boldmath$n$}}_R}^2\,\d{\mbox{\boldmath$n$}}_R<\infty$}
\quad\Longrightarrow\quad f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}.
\end{equation}
This is due to the fact that any curve is bounded, hence any test plan ${\mbox{\boldmath$\pi$}}$ relative to ${\mbox{\boldmath$n$}}$ can be monotonically approximated
by test plans concentrated on curves contained in a bounded set.
Another property we shall need is the locality with respect to $f$, see \cite{AGSBaEm} for the simple proof.
\begin{proposition}[Locality]\label{prop:locality}
Let $f_1,\,f_2:X\to \mathbb{R}$ Borel and let $G_1,\,G_2\in L^2(X,{\mbox{\boldmath$n$}})$ be $2$-weak upper gradients of $f_1,\,f_2$ relative to
${\mbox{\boldmath$n$}}$ respectively.
Then
$$\tilde{G}_1:= \begin{cases}
G_1&\text{on $\{f_1\neq f_2\}$;}\\
\min\{G_1,G_2\}&\text{on $\{f_1=f_2\}$}
\end{cases}
$$
is a $2$-weak upper gradient of $f_1$. In particular, by minimality we get
\begin{equation}\label{eq:locality}
|D f_1|_{w,{\mbox{\boldmath$n$}}}=|D f_2|_{w,{\mbox{\boldmath$n$}}}\qquad\text{${\mbox{\boldmath$n$}}$-a.e. on $\{f_1=f_2\}$.}
\end{equation}
\end{proposition}
Weak gradients share with classical gradients many features, in particular the chain rule
\cite[Proposition~5.14]{Ambrosio-Gigli-Savare11}
\begin{equation}\label{eq:chainrule}
|D \phi(f)|_{w,{\mbox{\boldmath$n$}}}=\phi'(f)|D f|_{w,{\mbox{\boldmath$n$}}}\qquad\text{${\mbox{\boldmath$n$}}$-a.e. in $X$}
\end{equation}
for all $\phi:\mathbb{R}\to\mathbb{R}$ Lipschitz and nondecreasing on an interval containing the image of $f$.
By convention, as in the classical chain rule, $\phi'(f)$ is arbitrarily defined at all points
$x$ such that $\phi$ is not differentiable at $x$, taking into account the fact that
$|D f|_{w,{\mbox{\boldmath$n$}}}=0$ ${\mbox{\boldmath$n$}}$-a.e. on this set of points.
In the sequel we shall adopt the conventions
\begin{equation}\label{eq:convention}
\weakgrad{f}:=|D f|_{w,{\mbox{\boldmath$m$}}},\qquad\qquad{\mathcal S}^2:={\mathcal S}^2_{\mbox{\boldmath$m$}}.
\end{equation}
In Theorem~\ref{thm:equivalence} below we analyze in detail, the behaviour of $|D f|_{w,{\mbox{\boldmath$n$}}}$ and ${\mathcal S}^2_{\mbox{\boldmath$n$}}$
under modifications of the reference measure ${\mbox{\boldmath$n$}}$.
\begin{theorem}\label{thm:equivalence}
The following properties hold:
\begin{itemize}
\item[(a)] If ${\mbox{\boldmath$n$}}\in{\mathcal M}$ and $\Gamma\subset\Curvesnonpara{X}$ is ${\rm Mod}_{2,{\mbox{\boldmath$n$}}}$-negligible,
then any Borel set $\tilde\Gamma\subset AC^2([0,1];X)$ such that $i(\tilde\Gamma)\subset\Gamma$
is ${\mbox{\boldmath$\pi$}}$-negligible for any $2$-test plan ${\mbox{\boldmath$\pi$}}$ relative to ${\mbox{\boldmath$n$}}$. In addition, for any Borel and ${\mbox{\boldmath$n$}}$-negligible set
$N\subset X$ the following holds:
$$
{\rm Mod}_{2,{\mbox{\scriptsize\boldmath$n$}}}\bigl(\bigl\{\gamma\in\Curvesnonpara{X}:\ \int_{\gamma^{-1}(N)}|\dot\gamma|\,\d t>0\bigr\}\bigr)=0.
$$
\item[(b)] If either ${\mbox{\boldmath$n$}}\in\Probabilities{X}$ and $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}$, or ${\mbox{\boldmath$n$}}\in{\mathcal M}$ and $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}\cap L^1(X,{\mbox{\boldmath$n$}})$,
there exist $\phi_n\in {\rm Lip}_b(X)\cap L^2(X,{\mbox{\boldmath$n$}})$ satisfying $\phi_n\to f$ ${\mbox{\boldmath$n$}}$-a.e. in $X$ and $|D\phi_n|\to |D f|_{w,{\mbox{\boldmath$n$}}}$ in $L^2(X,{\mbox{\boldmath$n$}})$.
\item[(c)] If either ${\mbox{\boldmath$n$}}\in\Probabilities{X}$ and $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}$, or ${\mbox{\boldmath$n$}}\in{\mathcal M}$ and $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}\cap L^1(X,{\mbox{\boldmath$n$}})$,
then there exists a Borel function $\tilde f$ coinciding with $f$ out of an ${\mbox{\boldmath$n$}}$-negligible set and having an upper gradient in $L^2(X,{\mbox{\boldmath$n$}})$;
in addition, there exist upper gradients $G_n$ of $\tilde{f}$ converging to $|D f|_{w,{\mbox{\boldmath$n$}}}$ in $L^2(X,{\mbox{\boldmath$n$}})$.
\end{itemize}
\end{theorem}
\begin{proof}
(a) The first statement is a simple consequence of H\"older inequality, see \cite[Remark~5.3]{Ambrosio-Gigli-Savare11}. The second one
follows just by taking the function $g$ identically equal to $\infty$ on $N$ and null out of $N$ in \eqref{eq:Mod2}.
(b) Using the chain rule \eqref{eq:chainrule} we reduce the proof to the case of nonnegative functions $f$. If
$f$ belong to $L^2(X,{\mbox{\boldmath$n$}})$ the existence of $\phi_n$ is one of the main results of \cite{Ambrosio-Gigli-Savare11}, see Theorem~6.2 therein.
In the general case we approximate $f$ by the truncated functions $f_N=\min\{f,N\}$
and use the chain rule again to show $|D f_N|_{w,{\mbox{\boldmath$n$}}}\to |D f|_{w,{\mbox{\boldmath$n$}}}$ in $L^2(X,{\mbox{\boldmath$n$}})$.
Then, a diagonal argument provides the result.
(c) This is part of the theory developed by Koskela-MacManus in \cite{Koskela-MacManus} and
Shanmugalingam in \cite{Shanmugalingam00}: if $f_n\to f$ ${\mbox{\boldmath$n$}}$-a.e. and $G_n$ are upper gradients of
$f_n$ weakly convergent to $G$ in $L^2(X,{\mbox{\boldmath$n$}})$, then we can find a Borel function $\tilde{f}$ equal to $f$ ${\mbox{\boldmath$n$}}$-a.e.
and a Borel function $\tilde G$ equal to $G$ ${\mbox{\boldmath$n$}}$-a.e. such that $\tilde{G}$ satisfies the upper gradient property relative to
$\tilde{f}$ along ${\rm Mod}_{2,{\mbox{\boldmath$n$}}}$-almost every curve. In our case when $f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}$ we may apply statement (b)
with $G=|D f|_{w,{\mbox{\boldmath$n$}}}$ and choose $f_n=\phi_n$ to find $\tilde{f}$ and $\tilde{G}$. Then, denoting by $\Gamma$ the set of curves where
the upper gradient property fails and considering
$$
G_h:=\tilde{G}+r_h,
$$
where $r_h\in L^2(X,{\mbox{\boldmath$n$}})$ satisfy $\int_Xr_h^2\,\d{\mbox{\boldmath$n$}}\to 0$ and $\int_\gamma r_\epsilon=\infty$ for all $\gamma\in\Gamma$,
we obtain upper gradients $G_h$ of $\tilde{f}$ approximating $|D f|_{w,{\mbox{\boldmath$n$}}}$ in $L^2(X,{\mbox{\boldmath$n$}})$.
\end{proof}
\begin{theorem}[Change of reference measure]\label{thm:change}
Assume that $\rho=g{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$ with $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $\weakgrad{\sqrt{g}}\in L^2(X,{\mbox{\boldmath$m$}})$.
Then:
\begin{itemize}
\item[(a)] $f\in{\mathcal S}^2$ and $\weakgrad{f}\in
L^2(X,\rho)$ imply $f\in{\mathcal S}^2_\rho$ and $|D
f|_{w,\rho}=\weakgrad{f}$ $\rho$-a.e. in $X$;
\item[(b)] $\log g\in{\mathcal S}^2_\rho$ and $|D\log g|_{w,\rho}=\weakgrad g/g$ $\rho$-a.e. in $X$.
\end{itemize}
\end{theorem}
\begin{proof} (a) Thanks to the locality properties with respect to ${\mbox{\boldmath$m$}}$ stated after Definition~\ref{def:wug} (see in particular
\eqref{eq:localitynn}) we can reduce ourselves to the case when ${\mbox{\boldmath$m$}}(X)=1$.
Since the statement is invariant under
modification of $f$ and $g$ in ${\mbox{\boldmath$m$}}$-negligible sets,
by Theorem~\ref{thm:equivalence}(b) we can assume that $\sqrt{g}$ and $f$ are absolutely continuous along
${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-almost every curve in $\Curvesnonpara{X}$; even
more, we can assume that $f$ has an upper gradient
$H$ with $\int H^2\,\d{\mbox{\boldmath$m$}}<\infty$.
Let us prove first the inequality $|D f|_{w,\rho}\leq\weakgrad{f}$ $\rho$-a.e. in $X$. By a truncation argument
we can assume with no loss of generality that $f$ is bounded; under this assumption we can find
bounded Lipschitz functions $\phi_n$ with $|D\phi_n|\to \weakgrad{f}$ in $L^2(X,{\mbox{\boldmath$m$}})$. Since $g$ is bounded it
follows that $|D\phi_n|\to\weakgrad{f}$ in $L^2(X,\rho)$; we can now use the stability properties of weak upper
gradients \cite[Theorem~5.12]{Ambrosio-Gigli-Savare11} to obtain that $|D f|_{w,\rho}\leq\weakgrad{f}$ $\rho$-a.e. in $X$.
In order to prove the converse inequality $|D f|_{w,\rho}\geq\weakgrad{f}$ $\rho$-a.e. in $X$,
we consider a function $\tilde{f}$ coinciding with $f$ $\rho$-a.e. in $X$ and an upper gradient $L$ of $\tilde{f}$ with
$\int L^2\,\d\rho<\infty$. The converse inequality
follows by letting $L\to |Df|_{w,\rho}$ in $L^2(X,\rho)$, if we are able to show that
$$
L_1(x):=
\begin{cases}
H(x) &\text{if $g(x)=0$;}\\
\min\{H(x),L(x)\} &\text{if $g(x)>0$,}
\end{cases}
$$
is a $2$-weak upper gradient of $f$ relative to ${\mbox{\boldmath$m$}}$. More precisely, we will prove that the upper gradient inequality with $L_1$
in the right hand side holds along ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-almost every curve. We notice first that
$$
|\tilde{f}(\gamma_{\ell(\gamma)})-\tilde{f}(\gamma_0)|\leq\int_\gamma L
$$
along ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-a.e. curve $\gamma$ satisfying $\inf_\gamma g>0$ (here we are using
the invariance under reparameterization, selecting the
arclength one, with $\ell(\gamma)$ equal to the length of $\gamma$). Indeed, by definition of 2-modulus, the set
$$
\left\{\gamma\in\Curvesnonpara{X}:\ \inf_\gamma g>0,\,\,\int_\gamma L=\infty\right\}
$$
is not only ${\rm Mod}_{2,\rho}$-negligible, but also ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-negligible. If we write
the upper gradient inequality in averaged form
$$
\frac{1}{\epsilon\ell(\gamma)}\int_0^{\epsilon\ell(\gamma)}
|\tilde{f}(\gamma_{\ell(\gamma)-r})-\tilde{f}(\gamma_r)|\,\d r\leq\int_\gamma L
\quad\text{with}\quad\epsilon<\frac 12
$$
and use Theorem~\ref{thm:equivalence}(a) with the ${\mbox{\boldmath$m$}}$-negligible set $N=\{f\neq\tilde{f}\}\cap\{g>0\}$,
we may replace $\tilde{f}$ with $f$ in the previous inequality. Now we use the absolute continuity of $f$ along
${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-a.e. curve and pass to the limit along a sequence $\epsilon_k\downarrow 0$ to get
$$
|f(\gamma_b)-f(\gamma_a)|\leq\int_\gamma L
$$
along ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-a.e. curve $\gamma:[a,b]\to X$ with $\inf_\gamma g>0$.
The set of curves $\gamma\in\Curvesnonpara{X}$ containing a subcurve $\gamma':[a,b]\to X$ with $\inf_{\gamma'}g>0$
and $|f(\gamma_b')-f(\gamma_a')|>\int_{\gamma'} L$ is ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-negligible as well. If $\gamma$ does not
belong to this set and $f\circ\gamma$ is absolutely continuous, it is immediate to check
(recall that $g$ is continuous along ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-almost every curve) that its derivative
is bounded a.e. by $L_1\circ\gamma|\dot\gamma|$, whence the upper gradient inequality along $\gamma$ follows.
(b) We consider the functions $f_\eps=\log (g+\eps)$. Since $\weakgrad{g}^2/g^2\in L^1(X,\rho)$ it
is immediate to check that all functions $f_\eps$ satisfy the assumption in (a), hence $f_\eps\in{\mathcal S}^2_\rho$ and
$|D f_\eps|_{w,\rho}=|D f_\eps|_w=\weakgrad{g}/(g+\eps)$ $\rho$-a.e. in $X$. We can now pass to the limit
as $\eps\downarrow 0$ and use again the stability of weak upper gradients to get
$|D f|_{w,\rho}\leq\weakgrad{g}/g$ $\rho$-a.e. in $X$. The converse inequality follows by the chain rule \eqref{eq:chainrule}
with $\phi(s):=\log({\mathrm e}^s+1)$:
$$
\frac{\weakgrad{g}}{g+1}=|Df_1|_{w,\rho}=\phi'(f)|Df|_{w,\rho}=\frac{g}{g+1}|Df|_{w,\rho}.
$$
\end{proof}
\begin{remark}{\rm Notice that for the validity of (a) it suffices, as the proof shows, the existence
of a nonnegative function $\tilde{g}$ continuous along ${\rm Mod}_{2,{\mbox{\boldmath$m$}}}$-a.e. curve and
satisfying ${\mbox{\boldmath$m$}}(\{g\neq\tilde g\})=0$.
}\fr\end{remark}
We shall define $\mathbb{C}:L^1(X,{\mbox{\boldmath$m$}})\to [0,\infty]$, ${\mathbb{C}}_{\mbox{\boldmath$n$}}:L^1(X,{\mbox{\boldmath$n$}})\to [0,\infty]$ by
\begin{equation}\label{def:Cheeger}
\mathbb{C}(f):=\frac{1}{2}\int_X \weakgrad{f}^2\,\d{\mbox{\boldmath$m$}}\quad f\in{\mathcal S}^2,\qquad
{\mathbb{C}}_{\mbox{\boldmath$n$}}(f):=\frac{1}{2}\int_X |D f|_{w,{\mbox{\boldmath$n$}}}^2\,\d{\mbox{\boldmath$n$}}\quad f\in{\mathcal S}^2_{\mbox{\boldmath$n$}}
\end{equation}
with the conventions $\mathbb{C}(f)=\infty$ on $L^1(X,{\mbox{\boldmath$m$}})\setminus{\mathcal S}^2$, ${\mathbb{C}}_{\mbox{\boldmath$n$}}(f)=\infty$ on
$L^1(X,{\mbox{\boldmath$n$}})\setminus{\mathcal S}^2_{\mbox{\boldmath$n$}}$. We will choose ${\mbox{\boldmath$n$}}$, as explained in the introduction, to be
probability measures.
We shall also denote, whenever $\mathbb{C}$ (resp. $\mathbb{C}_{\mbox{\boldmath$n$}}$) is a quadratic form, by
\begin{equation}\label{eq:numeriamoanchequesta}
\mathcal E(f,g):=\frac{1}{2}\bigl(\mathbb{C}(f+g)-\mathbb{C}(f-g)\bigr)\qquad
\biggl(\text{resp. } \mathcal E_{\mbox{\boldmath$n$}}(f,g):=\frac{1}{2}\bigl({\mathbb{C}}_{\mbox{\boldmath$n$}}(f+g)-{\mathbb{C}}_{\mbox{\boldmath$n$}}(f-g)\bigr)\biggr)
\end{equation}
the associated symmetric bilinear form, defined on ${\mathcal S}^2\cap L^1(X,{\mbox{\boldmath$m$}})$ (resp. ${\mathcal S}^2_{\mbox{\boldmath$n$}}\cap L^1(X,{\mbox{\boldmath$n$}})$).
Still under the assumption that $\mathbb{C}$ is quadratic, as in \cite[Definition~4.13]{Ambrosio-Gigli-Savare11b}
(see also Gigli's work \cite{Gigli12} for a more general, non-quadratic framework) we can define
\begin{equation}\label{eq:defGamma}
\Gbil{f}{g}:=\lim_{\eps\downarrow 0}\frac{\weakgrad{(f+\eps g)}^2-\weakgrad{f}^2}{2\eps}
\qquad f,\,g\in{\mathcal S}^2,
\end{equation}
where the limit takes place in $L^1(X,{\mbox{\boldmath$m$}})$. Notice that $\Gbil{f}{f}=\weakgrad{f}^2$ ${\mbox{\boldmath$m$}}$-a.e. and that
$\Gbil{\cdot}{\cdot}$ provides integral representation
to $\mathcal E$, namely
$$
\mathcal E(f,g)=\int_X\Gbil{f}{g}\,\d{\mbox{\boldmath$m$}}.
$$
The inequality $\weakgrad{(f+\eps g)}^2\leq \bigl(\weakgrad{f}+\eps\weakgrad{g}\bigr)^2=
\weakgrad{f}^2+2\eps\weakgrad{f}\weakgrad{g}+\eps^2\weakgrad{g}^2$ provides the bound
\begin{equation}\label{eq:boundGamma}
\big|\Gbil{f}{g}\bigr|\le\weakgrad{f}\weakgrad{g}\qquad\text{${\mbox{\boldmath$m$}}$-a.e. in $X$.}
\end{equation}
Also, locality of weak gradients gives
\begin{equation}\label{eq:localityGamma}
\Gbil{f}{g}=\Gbil{f}{g'}\qquad\text{${\mbox{\boldmath$m$}}$-a.e. on $\{g=g'\}$.}
\end{equation}
We will need a chain rule with respect to the
second argument, see \cite[Lemma~4.7]{Ambrosio-Gigli-Savare11b} for the simple proof:
\begin{equation}\label{eq:chainriemannian}
\int_X\Gbil{f}{\phi(g)}\,\d{\mbox{\boldmath$m$}}=\int_X\phi'(g)\Gbil{f}{g}\,\,d\m
\end{equation}
for all $\phi:\mathbb{R}\to\mathbb{R}$ nondecreasing and Lipschitz on an interval containing the image of $g$,
with the same convention on the value of $\phi'(g)$ mentioned in \eqref{eq:chainrule}.
Finally, we will need the following lemma, whose proof is more delicate: it relies on the chain rule for
$\Gbil{\cdot}{\cdot}$ also with respect to the first factor and on the Leibniz rule with respect to the second factor
(see \cite{Ambrosio-Gigli-Savare11b} for finite measures and \cite[Proposition~4.20]{Gigli12} for the general case).
\begin{lemma}\label{lem:vabbeserve}
If $\mathbb{C}$ is quadratic, then $\Gbil{\cdot}{\cdot}$ is a symmetric bilinear form. In particular
$\int\weakgrad{f}^2 g\,\d{\mbox{\boldmath$m$}}=\int\Gbil{f}{f}g\,\d{\mbox{\boldmath$m$}}$ is a quadratic form for any nonnegative $g\in L^\infty(X,{\mbox{\boldmath$m$}})$.
\end{lemma}
\begin{theorem}[Weighted Cheeger energy]\label{thm:weighted}
Assume that $\rho=g{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$ with $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $\mathbb{C}(\sqrt{g})<\infty$. If $\mathbb{C}$ is a quadratic
form, then $\mathbb{C}_\rho$ is a quadratic form and
\begin{equation}\label{eq:transfer1}
\mathcal E_\rho(\log g,\varphi)=\mathcal E(g,\varphi)\qquad\text{
for all $\varphi:X\to\mathbb{R}$ Lipschitz with bounded support.}
\end{equation}
\end{theorem}
\begin{proof} By Theorem~\ref{thm:change}(a) and Lemma~\ref{lem:vabbeserve},
$\mathbb{C}_\rho$ is a quadratic form on bounded Lipschitz functions with bounded
support. By approximation $\mathbb{C}_\rho$ is a quadratic form
on bounded Lipschitz functions and eventually, taking
Theorem~\ref{thm:equivalence}(b) into account, on $L^2(X,\rho)$. \\
Let $f_\eps=\log(g+\eps)\in{\mathcal S}^2$. Then, using again the independence of weak gradients upon the reference measure given
by Theorem~\ref{thm:change}(a) and \eqref{eq:chainriemannian}, we get
\begin{eqnarray*}
\mathcal E_\rho(\varphi,f_\eps)&=&\lim_{\delta\downarrow 0} \frac{{\mathbb{C}}_\rho(\varphi+\delta f_\eps)-{\mathbb{C}}_\rho(\varphi)}{\delta}=
\lim_{\delta\downarrow 0}\int_X\frac{\weakgrad{(\varphi+\delta f_\eps)}^2-\weakgrad{\varphi}^2}{2\delta}\,\d\rho\\&=&
\int_X\Gbil{\varphi}{f_\eps}\,\d\rho=\int_X\Gbil{\varphi}{g}\frac{g}{g+\eps}\,\d{\mbox{\boldmath$m$}}.
\end{eqnarray*}
Passing to the limit as $\eps\downarrow 0$ provides the result, since convergence of the right hand sides is obvious, while
convergence of the left hand sides can be obtained working in the vector space
$H:=L^2(X,\rho')\cap{\mathcal S}^2_\rho$ endowed with the scalar product
$$
\langle h,h'\rangle:=\int_X hh'\,\d\rho'+\mathcal E_\rho(h,h')
\quad\text{with}\quad\rho':=\frac{1}{1+\log^2 g}\rho.
$$
This is indeed a Hilbert space because $\mathbb{C}_\rho$ is easily seen to be lower semicontinuous (since a truncation
argument allows the reduction to sequences uniformly bounded in $L^\infty(X,\rho)$) also w.r.t. $L^2(X,\rho')$
convergence; moreover, clearly $f_\eps\to f$ in $L^2(X,\tilde\rho)$ and since their norms are uniformly bounded we have
weak convergence in $H$. Finally $g\mapsto\mathcal E_\rho(\varphi,g)$ is continuous in $H$.
\end{proof}
\section{Existence of good geodesics}\label{sec:Tapio}
This section is devoted to the proof of the existence of geodesics
in $(\ProbabilitiesTwo{X},W_2)$ which are (at least for some initial time
interval) better than the ones given directly by the usual
$CD(K,\infty)$ inequality given by Lott and Villani \cite{Lott-Villani09} and Sturm \cite{Sturm06I}.
\begin{definition}\label{def:CD}
We say that $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a $CD(K,\infty)$ space if,
for all $\mu_0,\,\mu_1 \in D(\entv)$ (recall \eqref{def:DDDD}) there exists
a geodesic $(\mu_t) \in \geo(\ProbabilitiesTwo X)$ which
satisfies the convexity inequality
\begin{equation}\label{eq:CDdef}
\entv(\mu_t) \le (1-t)\entv(\mu_0) + t \entv(\mu_1)
- \frac{K}{2}t(1-t)W_2^2(\mu_0,\mu_1)\qquad\forall t\in [0,1].
\end{equation}
\end{definition}
The idea of constructing good geodesics in $CD(K,N)$ spaces was
recently used by Rajala in \cite{R2011b} to study $CD(K,N)$ spaces with branching
geodesics. There the initial motivation was to
obtain geodesics good enough so that the approach of \cite{R2011}
for proving local Poincar\'e inequalities could be adapted to these
spaces. Constructing geodesics by selecting midpoints is a standard approach,
see for example Gromov's proof that the GH limit of length spaces
is a length space \cite[Proposition 3.8]{Gromov07}
Here we modify some of Rajala's results \cite{R2011b} and \cite{R2012} to
the setting of this paper, repeating with some details the arguments
because on some occasions the adaptation is not trivial.
The version of these results which we will need in the later
sections is the following.
\begin{theorem}\label{thm:goodgeodesics}
Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ be a $CD(K,\infty)$ space and let
$\mu_0=\rho_0{\mbox{\boldmath$m$}},\,\mu_1=\rho_1{\mbox{\boldmath$m$}} \in D(\entv)$.
Assume in addition that $\mu_1$ has
bounded support and density and that the density $\rho_0$ satisfies the growth-bound
\begin{equation}\label{eq:decay}
\rho_0(x) \le c_1{\mathrm e}^{-c_2{\sf d}^2(x,x_0)}\qquad\forall x\in X
\end{equation}
for some $c_1,\,c_2>0$ and $x_0 \in X$.
Then there exist $t_0 \in (0,1)$ and a geodesic
$(\mu_t) \in \geo(\ProbabilitiesTwo X)$ between
$\mu_0$, $\mu_1$ satisfying the convexity inequality
\eqref{eq:CDdef} for all $t \in [0,1]$ and the density bound
\begin{equation}\label{eq:uniformbound}
\sup_{t \in [0,t_0]}||\rho_t||_{L^\infty(X,{\mbox{\boldmath$m$}})} < \infty.
\end{equation}
\end{theorem}
In $\S$\ref{ss41} we discuss the convexity of the entropy along intermediate measures formed using
an inductive process and prove existence of entropy minimizers. In $\S$\ref{ss42}
we review some result of Rajala in \cite{R2012} in $CD^*(K,N)$ spaces. In $\S$\ref{ss43}
we prove that the minimizers satisfy density bounds by adapting Rajala's result in \cite{R2011b}. Finally,
in $\S$\ref{ss44} we prove Theorem~\ref{thm:goodgeodesics} using these ingredients.
\subsection{Intermediate measures and the existence of minimizers}\label{ss41}
The measures with minimal entropy will be selected from the set of all
intermediate measures. Recall that for any two measures
$\mu_0,\, \mu_1 \in \ProbabilitiesTwo X$ the set of all intermediate
points (with a parameter $t \in (0,1)$), will be denoted by
\begin{align*}
\mathcal{I}_t(\mu_0,\mu_1) = \{\nu \in \ProbabilitiesTwo X
\,:\, W_2(\mu_0,\nu) = t W_2(\mu_0,\mu_1) \text{ and }
W_2(\mu_1,\nu) = (1-t) W_2(\mu_0,\mu_1)\}.
\end{align*}
It is not difficult to show that the set of $t$-intermediate points is a convex and closed subset
of $\ProbabilitiesTwo X$,
Even though the selection process is countable, it will define the whole
geodesic by completion. To get the convexity inequality \eqref{eq:CDdef}
for all times we will then need the lower semicontinuity of the entropy
w.r.t. $W_2$-convergence (a direct consequence of \eqref{eq:changeentropy} and of
the weak lower semicontinuity of ${\rm Ent}_{\mbox{\boldmath$n$}}$ in $\Probabilities{X}$ when ${\mbox{\boldmath$n$}}\in\Probabilities{X}$)
and tightness estimates.
Let us now indicate how the first property of the good geodesics follows
easily if we define the geodesic by taking any intermediate point where
\eqref{eq:CDdef} is satisfied.
\begin{proposition}\label{prop:combinedgeod}
Let $\mu_0, \,\mu_1 \in \ProbabilitiesTwo X$. Suppose that we have
selected inductively at step $(n+1)$ measures
$\mu_{t} \in \mathcal{I}_{\frac{t-s}{r-s}}(\mu_s,\mu_r)$ satisfying
\[
\entv(\mu_{t}) \le \frac{(r-t)}{(r-s)}\entv(\mu_s)
+ \frac{(t-s)}{(r-s)} \entv(\mu_r)
- \frac{K}{2}\frac{(t-s)}{(r-s)}\frac{(r-t)}{(r-s)}W_2^2(\mu_s,\mu_r),
\]
where $s < t < r$ and the times $s$ and $r$ are two consecutive
timepoints in the set of times where the measures have already been
selected at step $n$.
Then \eqref{eq:CDdef} holds for all $\mu_t$ chosen at the $(n+1)$-th step.
In particular, if the closure of the selected times is the whole interval $[0,1]$, defining
$\mu_t$ by completion, we have a geodesic between $\mu_0$ and $\mu_1$ along which
\eqref{eq:CDdef} holds.
\end{proposition}
\begin{proof} Suppose that we have selected a measure
$\mu_{t} \in \mathcal{I}_{t}(\mu_0,\mu_1)$ satisfying
\[
\entv(\mu_t) \le (1-t)\entv(\mu_0)
+ t \entv(\mu_1) - \frac{K}{2}t(1-t)W_2^2(\mu_0,\mu_1)
\]
and after it a measure $\mu_{ts} \in \mathcal{I}_{s}(\mu_0,\mu_t)$
satisfying
\[
\entv(\mu_{ts}) \le (1-s)\entv(\mu_0)
+ s \entv(\mu_t) - \frac{K}{2}s(1-s)W_2^2(\mu_0,\mu_t).
\]
Then for the measure $\mu_{ts}$ we also have
$\mu_{ts} \in \mathcal{I}_{ts}(\mu_0,\mu_1)$ and
\begin{align*}
\entv&(\mu_{ts}) \le (1-s)\entv(\mu_0) + s \entv(\mu_t) - \frac{K}{2}s(1-s)W_2^2(\mu_0,\mu_t) \\
\le\,& (1-s)\entv(\mu_0) + s \left((1-t)\entv(\mu_0) + t \entv(\mu_1) - \frac{K}{2}t(1-t)W_2^2(\mu_0,\mu_1) \right)\\
& - \frac{K}{2}s(1-s)W_2^2(\mu_0,\mu_t) \\
= \,& \left((1-s)+s(1-t)\right)\entv(\mu_0) + ts\entv(\mu_1) - \frac{K}{2}\left(ts(1-t)+t^2s(1-s)\right)W_2^2(\mu_0,\mu_1)\\
= \,& (1-ts)\entv(\mu_0) + ts\entv(\mu_1) - \frac{K}{2}ts(1-ts)W_2^2(\mu_0,\mu_1).
\end{align*}
Therefore the claim holds for all the points $t_i$. By the lower semicontinuity of the entropy it then holds also for the closure.
\end{proof}
Now that we know from Proposition~\ref{prop:combinedgeod} that
the first property of the geodesic in Theorem~\ref{thm:goodgeodesics}
is easily satisfied we turn to the more difficult part of obtaining
the density bound \eqref{eq:uniformbound}. To do this we will not
only select intermediate measures that satisfy \eqref{eq:CDdef},
but measures where the entropy is minimal. The obvious first step is
then to prove that there indeed exist such minimizers. In general the set
$\mathcal{I}_t(\mu_0,\mu_1)$, though closed, is not compact in
$(\ProbabilitiesTwo X, W_2)$. However, when we consider a subset of
$\mathcal{I}_t(\mu_0,\mu_1)$ with the entropy bounded from above, we
have compactness. In particular, we therefore have the existence of
minimizers.
\begin{lemma}\label{lma:minexists}
Let $\mu_0, \,\mu_1 \in \ProbabilitiesTwo{X}$.
Then for all $t\in[0,1]$ there exists a minimizer of the entropy in $\mathcal{I}_t(\mu_0,\mu_1)$.
\end{lemma}
\begin{proof}
Without loss of generality we can assume the existence of
$\nu \in \mathcal{I}_t(\mu_0,\mu_1)$ with $\entv(\nu)<\infty$.
We know that the entropy is lower
semicontinuous and that
$\mathcal{I}_t(\mu_0,\mu_1)$ is closed. The claim then follows
if we are able to show that the set
\[
\mathcal{K} = \{\mu \in \mathcal{I}_t(\mu_0,\mu_1)\,:\,
\entv(\mu) \le \entv(\nu)\} \subset \ProbabilitiesTwo X
\]
is relatively compact in $(\ProbabilitiesTwo X, W_2)$. It suffices
to prove that the set $\mathcal{K}$ is uniformly $2$-integrable
and tight, see \cite[Proposition 7.15]{Ambrosio-Gigli-Savare08}.
Let us first prove the uniform $2$-integrability of the set
$\mathcal{I}_t(\mu_0,\mu_1)$. This follows from the fact that for
any $\mu \in \mathcal{I}_t(\mu_0,\mu_1)$ we have
\[
\int_{X \setminus \overline{B}(x_0,k)} {\sf d}^2(x_0,x)\,\d\mu
\le \int_{X \setminus \overline{B}(x_0,k/2)} 4{\sf d}^2(x_0,x)\,\d(\mu_0+\mu_1)
\to 0, \quad \text{as }k \to \infty
\]
since $\mu_0,\,\mu_1 \in \ProbabilitiesTwo X$.
Let us next prove that $\mathcal{K}$ is tight. If $\tilde{\mbox{\boldmath$m$}}\in\Probabilities{X}$ is defined as in \eqref{eq:grygorian2},
\eqref{eq:changeentropy} shows that $\sup_{\mu\in\mathcal K}{\rm Ent}_{\tilde{\mbox{\boldmath$m$}}}(\mu)$ is finite. Then,
tightness of $\mathcal K$ is a simple consequence of the equi-integrability of the densities w.r.t. $\tilde{\mbox{\boldmath$m$}}$.
%
\end{proof}
As a technical tool we will need the excess mass functional
$\mathcal{F}_C \colon \ProbabilitiesTwo X \to [0,1]$ which is defined
for all thresholds $C \ge 0$ as
\begin{equation}\label{eq:excessmass}
\mathcal{F}_C(\mu) = \|(\rho-C)^+\|_{L^1(X,{\mbox{\boldmath$m$}})} + \mu^s(X),
\end{equation}
where $\mu = \rho {\mbox{\boldmath$m$}} + \mu^s$ with $\mu^s \perp {\mbox{\boldmath$m$}}$. This functional,
lower semicontinuous under weak convergence,
was used in \cite{R2011b} to obtain the first good geodesics in
$CD(K,N)$ spaces. The motivation for using the excess mass functional
is that its variations under perturbation of the minimizer are easier to estimate,
since one only cares about the amount of mass exceeding the threshold.
\subsection{Localization in transport distance}\label{ss42}
As we will later see, the task of finding the first good intermediate
measure between $\mu_0$ and $\mu_1$ is slightly more difficult than
finding the rest of the geodesic. This is due to the fact that after
some $\mu_t$ with $t \in (0,1)$ has been fixed we can consider the
transport distances to be essentially constant. This useful
observation was made by Rajala in \cite{R2012}. It follows from two simple
statements. First when one fixes an intermediate measure,
the length of the curves along which the transport is done gets
fixed. This is the content of the next proposition which was proved
in \cite[Proposition 1]{R2012}.
\begin{proposition}\label{prop:separation}
Let $\mu_0,\, \mu_1 \in \ProbabilitiesTwo X$ and $t_0 \in (0,1)$.
Suppose that there exist constants $0 \le C_1 \le C_2 < \infty$
and a measure ${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ with
\begin{equation}\label{eq:lengthbounds}
C_1 \le l(\gamma) \le C_2 \qquad\text{for }{\mbox{\boldmath$\pi$}}\text{-a.e. }\gamma \in \geo(X).
\end{equation}
Then the bounds in \eqref{eq:lengthbounds} hold $\tilde{\mbox{\boldmath$\pi$}}$-a.e.
for any $\tilde{\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ with
$(\e_{t_0})_\sharp\tilde{\mbox{\boldmath$\pi$}} = (\e_{t_0})_\sharp{\mbox{\boldmath$\pi$}}$.
\end{proposition}
In order to use the previous proposition we will need another
observation which is a simple consequence of cyclical monotonicity
(cf. Chapter 5 in Villani's survey \cite{Villani09} for a review of cyclical monotonicity).
Namely, when we work on a part of the transport with some bounds on
the lengths of the curves, this part will not get mixed with other
parts of the measure at any intermediate time. For the proof of this
fact see \cite[Lemma 2.5]{R2012}.
\begin{lemma}\label{lma:separation}
Take $0 \le C_1\le C_2 \le C_3 \le C_4 \le \infty$ and define
\[
A_1 = \{\gamma \in \geo(X) \,:\, C_1 \le l(\gamma) \le C_2 \}
\quad \text{and} \quad
A_2 = \{\gamma \in \geo(X) \,:\, C_3 < l(\gamma) \le C_4 \}.
\]
Then for any ${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ and any $t \in (0,1)$
there exists a Borel set $E \subset \geo(X)$ with ${\mbox{\boldmath$\pi$}}(E)=0$ such that
\[
\{(\gamma, \hat\gamma) \in (A_1\setminus E) \times (A_2\setminus E)\,:\,
\gamma_t = \hat\gamma_t\} = \emptyset.
\]
\end{lemma}
\subsection{Density bounds for the minimizers}\label{ss43}
The information from the minimizers of the entropy and of the excess
mass functional are obtained with a contradiction argument. First we
assume that there exists a minimizer which does not have the desired
density bound. After this we isolate the part of the minimizer where
the density bound is exceeded and redefine this part of the measure to
be something slightly better. If this new measure is again an
intermediate point and we have strictly decreased the energy we are minimizing (the entropy or the excess mass)
we obtain a contradiction, so that the minimizer must satisfy the density bound. To prove that we indeed get an intermediate point we use
the next lemma, whose proof relies on the joint convexity of $(\mu,\nu)\mapsto W_2^2(\mu,\nu)$,
which was again proved by Rajala in \cite[Lemma 3.5]{R2011b}.
\begin{lemma}\label{lma:combined}
Let $\mu_0,\, \mu_1 \in \ProbabilitiesTwo X$. Then for any $\lambda\in (0,1)$,
any ${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$, any Borel function
$f \colon \geo(X) \to [0,1]$ with $c = (f{\mbox{\boldmath$\pi$}})(\geo(X)) \in (0,1)$
and any
\[
\nu \in \mathcal{I}_\lambda\left(\frac1{c} (\e_{0})_\sharp\left(f{\mbox{\boldmath$\pi$}}\right),
\frac1{c} (\e_{1})_\sharp\left(f{\mbox{\boldmath$\pi$}}\right)\right)
\]
we have
\[
(\e_{\lambda})_\sharp\left((1-f){\mbox{\boldmath$\pi$}}\right) + c\nu \in \mathcal{I}_\lambda(\mu_0, \mu_1).
\]
\end{lemma}
The first step which uses the minimization of the excess mass functional $\mathcal{F}_C$
in \eqref{eq:excessmass} is the same one
that was taken in \cite[Proposition 3.11]{R2011b}. We repeat some key
points of the proof for the convenience of the reader. In \cite{R2011b}
the functionals $\mathcal{F}_C$ were minimized only in the bounded case.
A reduction to this case can be also made here and so the following
proposition which was proved in a slightly different form in
\cite[Proposition 3.9 and Proposition 3.11]{R2011b} will suffice.
\begin{proposition}\label{prop:excesszero}
Assume that $(X,{\sf d})$ is a bounded metric space with a finite measure ${\mbox{\boldmath$m$}}$.
Let $\nu_0, \,\nu_1 \in \ProbabilitiesTwo X$ and $t \in [0,1]$. Suppose
that there exists a constant $C> 0$ so that for any
${\mbox{\boldmath$\pi$}} \in \gopt(\nu_0,\nu_1)$ and $A\subset X$ Borel with
${\mbox{\boldmath$\pi$}}(\e_t^{-1}(A))> 0$ we have that for the measures
\begin{equation}\label{eq:hatdef}
\hat\nu_0 = \frac{1}{{\mbox{\boldmath$\pi$}}(\e_t^{-1}(A))}(\e_0)_\sharp\left({\mbox{\boldmath$\pi$}}\res\e_t^{-1}(A)\right), \qquad
\hat\nu_1 = \frac{1}{{\mbox{\boldmath$\pi$}}(\e_t^{-1}(A))}(\e_1)_\sharp\left({\mbox{\boldmath$\pi$}}\res\e_t^{-1}(A)\right)
\end{equation}
there exists a measure $\hat\nu \in \mathcal{I}_t(\hat\nu_0,\hat\nu_1)$
with
\begin{equation}\label{eq:entropybound1}
\entv(\hat\nu) \le \log\frac{C}{{\mbox{\boldmath$\pi$}}(\e_t^{-1}(A))}.
\end{equation}
Then there exists a minimizer $\mu_t$ of $\mathcal{F}_{C}$ in
$\mathcal{I}_t(\nu_0,\nu_1)$ and the minimum value is zero, so that
$\mu_t\ll{\mbox{\boldmath$m$}}$ and its density is less than $C$ ${\mbox{\boldmath$m$}}$-a.e. in $X$.
\end{proposition}
\begin{proof}
Take a threshold $C' > C$. It suffices to prove that the minimum of
$\mathcal{F}_{C'}$ in $\mathcal{I}_t(\nu_0,\nu_1)$ is zero and then
let $C' \downarrow C$. Without loss of generality we may assume that
all minimizers, whose existence is ensured by tightness of $\mathcal{I}_t(\nu_0,\nu_1)$
in $\Probabilities{X}$ and lower semicontinuity,
are absolutely continuous with respect to ${\mbox{\boldmath$m$}}$.
Indeed, suppose that there is a measure $\omega\in \mathcal{I}_t(\nu_0,\nu_1)$ with
a singular part. Let $A$ be an ${\mbox{\boldmath$m$}}$-negligible Borel set where
the singular part of $\omega$ is concentrated. By the assumption of the Proposition
together with Lemma~\ref{lma:combined} we can then redefine the part of
$\omega$ which is supported on $A$ to be a measure having finite
entropy. In particular it will be absolutely continuous with respect to
${\mbox{\boldmath$m$}}$. Since we are redefining only the singular part of $\omega$,
the value of the functional $\mathcal{F}_{C'}$ does not increase after
the redefinition.
Assume, contrary to the claim, that the infimum of $\mathcal{F}_{C'}$
in $\mathcal{I}_t(\nu_0,\nu_1)$ is positive. Denote by
$\mathcal{M}_\text{min} \subset \mathcal{I}_t(\nu_0,\nu_1)$ the set of
minimizers of $\mathcal{F}_{C'}$ in $\mathcal{I}_t(\nu_0,\nu_1)$.
Applying the proof of \cite[Proposition 3.9]{R2011b} we see that
the set $\mathcal{M}_\text{min}$ is always nonempty.
Take $\nu \in \mathcal{M}_\text{min}$ for which
\begin{equation}\label{eq:almostmaxpos}
{\mbox{\boldmath$m$}}(\{x \in X ~:~ \rho_\nu(x) > C'\}) \ge \left(\frac{C}{C'}\right)^{\frac14}
\sup_{\omega \in \mathcal{M}_\text{min}}{\mbox{\boldmath$m$}}(\{x \in X ~:~ \rho_\omega(x) > C'\}),
\end{equation}
where $\nu = \rho_\nu{\mbox{\boldmath$m$}}$ and $\omega = \rho_\omega {\mbox{\boldmath$m$}}$. Let
${\mbox{\boldmath$\pi$}} \in \gopt(\nu_0,\nu_1)$ be such that $(\e_t)_\sharp{\mbox{\boldmath$\pi$}} = \nu$.
There exists $\delta >0$ so that
\[
{\mbox{\boldmath$m$}}(A) > \left(\frac{C}{C'}\right)^{\frac12} {\mbox{\boldmath$m$}}(A')
\]
with
\begin{equation}\label{eq:deltabound}
A' = \{x \in X ~:~\rho_\nu(x) > C'\} \quad \text{ and }\quad
A = \{x \in A' ~:~ \rho_\nu(x) > C' + \delta\}.
\end{equation}
{F}rom the assumption of the proposition we know the existence of
a measure $\hat\nu=\hat{\rho}{\mbox{\boldmath$m$}}\in \mathcal{I}_t(\hat\nu_0,\hat\nu_1)$ with
$\entv(\hat\nu) \le \log ({C}/{\nu(A)})$,
where $\hat\nu_0$ and $\hat\nu_1$ are given by \eqref{eq:hatdef}.
By Jensen's inequality we then have
\begin{equation}\label{eq:bigsupport}
{\mbox{\boldmath$m$}}(\{\hat\rho>0\}) \ge \frac{\nu(A)}{C} \ge \frac{C'}{C} {\mbox{\boldmath$m$}}(A)
\ge \left(\frac{C'}{C}\right)^{\frac12} {\mbox{\boldmath$m$}}(A').
\end{equation}
We can now consider a new measure $\tilde\nu = \tilde\rho {\mbox{\boldmath$m$}}$ defined
as the combination
\begin{equation}\label{eq:tildenu}
\tilde\nu = \nu\res(X \setminus A) + \frac{C'}{C'+ \delta} \nu\res A
+ \frac{\delta}{C'+\delta}\nu(A) \hat\nu.
\end{equation}
By Lemma~\ref{lma:combined} and the convexity of $\mathcal{I}_t$ we have
$\tilde\nu \in \mathcal{I}_t(\nu_0,\nu_1)$. Due to the definition
\eqref{eq:deltabound} we only redistribute some of the mass above
the density $C'$ when we replace the measure $\nu$ by the measure
$\tilde\nu$, so that $\tilde\nu\in \mathcal{M}_{\text{min}}$. Let us calculate how much the excess mass functional
changes in this replacement:
\[
\mathcal{F}_{C'}(\nu) - \mathcal{F}_{C'}(\tilde\nu)
=\int_{\{\rho_\nu < C'\}}\min\left\{C'-\rho_\nu, \frac{\delta}{C'+\delta}\nu(A)\hat\rho\right\}\,\d{\mbox{\boldmath$m$}}.
\]
Because of the minimality of $\mathcal{F}_{C'}$ at $\nu$ this
integral must be zero. Therefore $\{\hat\rho>0\}\cap\{\rho_\nu<C'\}$
is ${\mbox{\boldmath$m$}}$-negligible. On the other hand, for any
$y\in \{\hat\rho>0\}\cap\{ \rho_\nu \ge C'\}$ we have
$\tilde\rho(y) > C'$ (if $y\in X\setminus A$ this is trivial, if $y\in A$ the second term
in \eqref{eq:tildenu} gives a contribution larger than $C'$). This, together with our choice
\eqref{eq:almostmaxpos} of $\nu$, leads to a contradiction:
$$
{\mbox{\boldmath$m$}}(\{\tilde\rho > C'\}) \geq {\mbox{\boldmath$m$}}(\{\hat\rho>0\})
\ge \left(\frac{C'}{C}\right)^{\frac12} {\mbox{\boldmath$m$}}(A')
\ge \left(\frac{C'}{C}\right)^{\frac14} \sup_{\omega \in \mathcal{M}_\text{min}}{\mbox{\boldmath$m$}}(\{\rho_\omega> C'\}).
$$
\end{proof}
Next we make another minimization. This time for the entropy itself.
A similar argument was used in \cite{R2012} to obtain good geodesics
in metric spaces satisfying the reduced curvature dimension
condition $CD^*(K,N)$.
\begin{proposition}\label{prop:spreadtolimit}
Let $\mu_0, \,\mu_1 \in \ProbabilitiesTwo X$ and $t \in [0,1]$.
Suppose that there exists a constant $C> 0$ so that for any
${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ and $A \subset X$ Borel with
${\mbox{\boldmath$\pi$}}(\e_t^{-1}(A))> 0$ we have that for the restricted measures
$\hat{\nu}_0,\,\hat{\nu}_1$ in \eqref{eq:hatdef}
there exists a measure
$\hat\nu \in \mathcal{I}_t(\hat\mu_0,\hat\mu_1)$ satisfying \eqref{eq:entropybound1}.
Then for any minimizer $\mu_\text{min}$ of the entropy in
$\mathcal{I}_t(\mu_0,\mu_1)$ we have $\mu_\text{min}\leq C{\mbox{\boldmath$m$}}$.
\end{proposition}
\begin{proof} Without loss of generality, we can assume $t\in (0,1)$.
Let $\nu=\rho{\mbox{\boldmath$m$}}$ be one of the minimizers of the entropy in
$\mathcal{I}_t(\mu_0,\mu_1)$, which by Lemma~\ref{lma:minexists} we
know to exist. By \eqref{eq:entropybound1} with $A=X$ we know that $\entv(\nu)<\infty$.
We need only to show that $\mathcal F_C(\nu)=0$.
Let ${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ be such that
$(\e_t)_\sharp{\mbox{\boldmath$\pi$}} = \nu$.
Suppose now by contradiction that $\mathcal{F}_C(\nu) > 0$, let
$\eta > 0$ be such that ${\mbox{\boldmath$m$}}(\{\rho > C + 2\eta\}) > 0$ and
define
\[
C_1 = \frac{1}{\eta}\left[{\mbox{\boldmath$m$}}(\{\rho > C + \eta\})
-{\mbox{\boldmath$m$}}(\{\rho > C + 2\eta\})\right] \ge 0.
\]
Since $\tau\mapsto g(\tau):={\mbox{\boldmath$m$}}(\{\rho \geq C + \tau\})$ is nonincreasing,
there exists $\delta \in (\eta,2\eta)$ such that $-g'(\delta)\leq C_1$. In particular, choosing $\delta$ in this way
and fixing $x_0\in X$, for $\phi\in (0,\eta/3)$ sufficiently small and $R=R(\phi)$ sufficiently large one has
${\mbox{\boldmath$m$}}(L') < {\mbox{\boldmath$m$}}(L) + (1+C_1)\phi$, where
\[
L = \{x \in B(x_0,R) \,:\, \rho(x) > C + \delta\}\qquad
\text{and}\qquad L' = \{x \in X \,:\, \rho(x) \ge C + \delta - 3\phi\}.
\]
Let $\Gamma \subset \geo(X)$ be a cyclically monotone set on which ${\mbox{\boldmath$\pi$}}$ is supported.
Fix $\bar\gamma \in \Gamma \cap {\mathrm e}_t^{-1}(L)$ and consider any $\gamma \in \Gamma \cap {\mathrm e}_t^{-1}(L)$.
Using cyclical monotonicity we get (similarly as in \cite[Theorem 8.22]{Villani09})
\begin{align*}
{\sf d}^2(\gamma_0, \gamma_1) & \le {\sf d}^2(\bar\gamma_0, \bar\gamma_1) + {\sf d}^2(\gamma_0, \gamma_1)
\le {\sf d}^2(\bar\gamma_0, \gamma_1) + {\sf d}^2(\gamma_0, \bar\gamma_1)\\
& \le \left({\sf d}(\gamma_t, \gamma_1) + \diam(L) + l(\bar\gamma)\right)^2 + \left({\sf d}(\gamma_0, \gamma_t) + \diam(L) + l(\bar\gamma)\right)^2\\
& = \left((1-t)^2 + t^2\right){\sf d}^2(\gamma_0, \gamma_1) + 2(\diam(L) + l(\bar\gamma)){\sf d}(\gamma_0, \gamma_1) + 2(\diam(L) + l(\bar\gamma))^2.
\end{align*}
Since $(1-t)^2 + t^2 = 1 - 2(1-t)t < 1$, the length of the geodesic $\gamma$ has a bound from above
given in terms of only $\diam(L)$ and $l(\bar\gamma)$.
Hence the measure ${\mbox{\boldmath$\pi$}}\res{\mathrm e}_t^{-1}(L)$ is supported in a uniformly bounded set of curves.
We can use Proposition~\ref{prop:excesszero} with $\nu_i=(\nu(L))^{-1}(\e_i)_\sharp{\mbox{\boldmath$\pi$}}\res\e_t^{-1}(L)$
to find a measure
\[
\tilde\nu = \tilde\rho {\mbox{\boldmath$m$}} \in \mathcal{I}_t\left(
\frac{(\e_0)_\sharp{\mbox{\boldmath$\pi$}}\res\e_t^{-1}(L)}{\nu(L)},
\frac{(\e_1)_\sharp{\mbox{\boldmath$\pi$}}\res\e_t^{-1}(L)}{\nu(L)}
\right)
\]
with $\tilde\rho\le {C}/{\nu(L)}$ ${\mbox{\boldmath$m$}}$-a.e. in $X$.
Now consider a new measure $\hat\nu = \hat\rho {\mbox{\boldmath$m$}}$ defined as the
combination
\[
\hat\nu = \nu\res (X \setminus L)
+ \frac{C + \delta-\phi}{C+ \delta} \nu\res L
+ \frac{\phi}{C+\delta}\nu(L) \tilde\nu.
\]
By Lemma~\ref{lma:combined} we have
$\hat\nu \in \mathcal{I}_t(\mu_0,\mu_1)$.
For $x \in L$ we have the estimates
\begin{align}
\hat\rho(x) & \le \frac{C + \delta -\phi}{C+ \delta}\rho(x)
+ \frac{\phi}{C+\delta}\nu(L)\tilde\rho(x)
\le \frac{(C + \delta-\phi)\rho(x) + C\phi}{C+\delta} \label{eq:tapio1}\\
& = \rho(x) + \frac{(C-\rho(x))\phi}{C + \delta}
< \rho(x) - \frac{\delta\phi}{C + \delta}\nonumber
\end{align}
and
\begin{equation}\label{eq:tapio2}
\hat\rho(x) \ge \frac{C + \delta - \phi}{C+ \delta}\rho(x)
> C + \delta - \phi.
\end{equation}
For $x \in L' \setminus L$ we have
\begin{equation}\label{eq:tapio3}
\hat\rho(x) \le \rho(x) + \frac{\phi}{C+\delta}\nu(L)\tilde\rho(x)
\le \rho(x) + \frac{C\phi}{C + \delta} < C + \delta + \phi
\end{equation}
and for $x \in X \setminus L'$ we get
\begin{equation}\label{eq:tapio4}
\hat\rho(x) \le \rho(x) + \frac{\phi}{C+\delta}\nu(L)\tilde\rho(x)
\le C + \delta - 3\phi + \frac{C\phi}{C + \delta}
< C + \delta - 2\phi.
\end{equation}
Write $C_2 = \frac{\delta}{C + \delta}{\mbox{\boldmath$m$}}(L)$. Let us estimate
the change in the entropy when we replace $\nu$ by $\hat\nu$:
using the convexity inequality $x\log x-y\log y\leq (x-y)(\log x+1)$
we can estimate from above $\entv(\hat\nu)-\entv(\nu)$ by
$$
\int_X(\hat\rho- \rho) (\log\hat\rho+1)\,\d{\mbox{\boldmath$m$}}
=\int_X(\hat\rho- \rho) \log\hat\rho\,\d{\mbox{\boldmath$m$}} .
$$
Now, we set $w:=\hat\rho-\rho$, split $X$ as $L\cup (X\setminus L')\cup (L'\setminus L)$
and use the fact that $w\leq 0$ on $L$ and $w\geq 0$ on $X \setminus L$, the inequalities
\eqref{eq:tapio1}, \eqref{eq:tapio2}, \eqref{eq:tapio3},
\eqref{eq:tapio4} and eventually the concavity of $\log$ to get
\begin{align*}
&~ \int_L w \log\left(C + \delta-\phi\right)\,\d{\mbox{\boldmath$m$}}
+ \int_{X \setminus L'} w\log\left(C + \delta-2\phi\right)\,\d{\mbox{\boldmath$m$}}
+ \int_{L' \setminus L} w\log\left(C + \delta+\phi\right)\,\d{\mbox{\boldmath$m$}} \\
= &~\left(\log\left(C + \delta-\phi\right) - \log\left(C + \delta - 2\phi\right)\right)\int_L w\,\d{\mbox{\boldmath$m$}}
+ \left(\log\left(C + \delta+\phi\right) - \log\left(C + \delta - 2\phi\right)\right) \int_{L'\setminus L} w\,\d{\mbox{\boldmath$m$}} \\
\le &~-\left(\log\left(C + \delta-\phi\right) - \log\left(C + \delta - 2\phi\right)\right) \frac{\delta\phi}{C + \delta}{\mbox{\boldmath$m$}}(L) \\
& + \left(\log\left(C + \delta+\phi\right) - \log\left(C + \delta - 2\phi\right)\right) \frac{C\phi}{C + \delta}{\mbox{\boldmath$m$}}(L'\setminus L) \\
< &~-\left(\log\left(C + \delta-\phi\right) - \log\left(C + \delta - 2\phi\right)\right) C_2\phi
+ \left(\log\left(C + \delta+\phi\right) - \log\left(C + \delta - 2\phi\right)\right) (1+C_1)\phi^2 \\
\le &- C_2\phi\frac{\phi}{C + \delta - 2\phi}+(1+C_1)\phi^2\frac{3\phi}{C + \delta - 2\phi} < 0
\end{align*}
for small enough $\phi\in (0,\eta/3)$. This contradicts the minimality of
the entropy at $\nu$.
\end{proof}
\subsection{Construction of the geodesic}\label{ss44}
\begin{proof}[Proof of Theorem \ref{thm:goodgeodesics}]
In this proof, to avoid a cumbersome notation, we switch to the ${\rm exp}$ notation
and set $C_1:=\|\rho_1\|_{L^\infty(X,{\mbox{\boldmath$m$}})}$.
Let $D > 0$ be such that $\supp(\mu_1) \subset B(x_0,D)$.
We will prove the claim with $$t_0 := \min\{\frac{c_2}{2K^-},\frac12\}.$$
The geodesic is constructed as follows. First we fix the
measure $\mu_{t_0} =\rho_{t_0}{\mbox{\boldmath$m$}}\in \mathcal{I}_{t_0}(\mu_0,\mu_1)$ to be a
minimizer of the entropy in $\mathcal{I}_{t_0}(\mu_0,\mu_1)$.
After this we define the rest of the geodesic for times
$t \in (0,t_0)$ inductively. Suppose that for some $n \in \mathbb{N}$
we have defined $\mu_{k2^{-n}t_0}$ for all $k = 0, 1, \ldots, 2^n$.
Then for all odd $k \in \mathbb{N}$ with $0 < k < 2^{n+1}$ we define
$\mu_{k2^{-n-1}t_0}$ to be a minimizer of the entropy in
$\mathcal{I}_{\frac12}(\mu_{(k-1)2^{-n-1}t_0},\mu_{(k+1)2^{-n-1}t_0})$.
We construct the geodesic on the interval $(t_0,1]$ in a similar way by
iteratively selecting the midpoints with minimal entropy.
The rest of the geodesic is given by completion. Let
${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ be such that $(\e_t)_\sharp{\mbox{\boldmath$\pi$}} = \mu_t$
for all $t \in [0,1]$.
Since we are selecting minimizers of the entropy among all the
possible intermediate measures in a $CD(K,\infty)$-space, the
selected measures satisfy the convexity inequality
\eqref{eq:CDdef} between the given endpoint measures.
Therefore, by Proposition~\ref{prop:combinedgeod} the inequality
\eqref{eq:CDdef} holds for all $t \in [0,1]$.
Let us then concentrate on the entropy estimates assumed in
Proposition~\ref{prop:excesszero} and Proposition~\ref{prop:spreadtolimit}.
Let ${\mbox{\boldmath$\pi$}} \in \gopt(\mu_0,\mu_1)$ and
$A\subset X$ Borel with $M := {\mbox{\boldmath$\pi$}}(\e_{t_0}^{-1}(A))>0$, write
\[
\hat\mu_0 = \hat\rho_0{\mbox{\boldmath$m$}} = \frac{1}{M}(\e_0)_\sharp\left({\mbox{\boldmath$\pi$}}\res\e_{t_0}^{-1}(A)\right) \quad\text{and}\quad
\hat\mu_1 = \hat\rho_1{\mbox{\boldmath$m$}} = \frac{1}{M}(\e_1)_\sharp\left({\mbox{\boldmath$\pi$}}\res\e_{t_0}^{-1}(A)\right),
\]
and take a measure $\nu \in \mathcal{I}_{t_0}\left(\hat\mu_0, \hat\mu_1\right)$
which satisfies the convexity inequality \eqref{eq:CDdef}
between these measures. Now, using \eqref{eq:decay},
we have the estimate (with $V(x)={\sf d}(x,x_0)$)
\begin{align*}
\entv(\nu)
\le~& (1-t_0)\entv\left(\hat\mu_0\right) + t_0 \entv\left(\hat\mu_1\right)
+ \frac{K^-}{2}t_0(1-t_0)W_2^2\left(\hat\mu_0, \hat\mu_1\right)\\
\le~& t_0\log\left(\frac{C_1}{M}\right)
+ (1-t_0)\int_X \hat\rho_0(x)\left(\log\hat\rho_0(x) + \frac{K^-}2t_0(D+V(x))^2\right)\,\d{\mbox{\boldmath$m$}}(x) \\
\le~& t_0\log\left(\frac{C_1}{M}\right)
+ (1-t_0)\int_X \hat\rho_0(x)\left(\log\left(\frac{c_1}M\right) - c_2V^2(x) + K^-t_0(D^2+V^2(x))\right)\,\d{\mbox{\boldmath$m$}}(x) \\
\le~& \log\left(\frac{\max\{C_1,c_1\}}{M}\right) + K^-D^2
= \log\left(\frac{\max\{C_1,c_1\}{\rm exp}[K^-D^2]}{M}\right),
\end{align*}
since $K^-t_0 \le c_2$ by the choice of $t_0$. By
Proposition~\ref{prop:spreadtolimit} we then have the estimate
\[
\|\rho_{t_0}\|_{L^\infty(X,{\mbox{\boldmath$m$}})} \le \max\{C_1,c_1\}{\rm exp}[K^-D^2]
\le \max\{C_1,c_1\}{\rm exp}[(2K^-+c_2)D^2]=: C.
\]
Next we prove that for all $t \in [0,t_0]$ we have
$\mu_t = \rho_t{\mbox{\boldmath$m$}}$ with the estimate
\begin{equation}\label{eq:decayestimate}
\rho_t(\gamma_t) \le C{\rm exp}\bigl[-\frac12(1-\frac{t}{t_0})(c_2 - K^-tt_0)\ell^2(\gamma)\bigr] \quad \text{for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma \in \geo(X)$.}
\end{equation}
First of all the estimate \eqref{eq:decayestimate} is true
for $t = t_0$. For $t = 0$ we have that, thanks to \eqref{eq:decay}, $\rho_0(\gamma_0)$ can be estimated from above by
\[
c_1{\rm exp}\bigl[-c_2{\sf d}^2(\gamma_0,x_0)\bigr] \le c_1{\rm exp}\bigl[-c_2([\ell(\gamma)-D]^+)^2\bigr]
\le c_1{\rm exp}\bigl[-\frac{c_2}2\ell^2(\gamma)+c_2D^2\bigr] \le C{\rm exp}(-\frac{c_2}2\ell^2(\gamma))
\]
and so \eqref{eq:decayestimate} holds also at $t=0$.
Suppose that for some $n \in \mathbb{N}$ the estimate
\eqref{eq:decayestimate} holds for all $t = k2^{-n}t_0$
with $k = 0,1,\dots, 2^n$. Take an odd integer $k$ with
$0 < k < 2^{n+1}$. Our aim is to prove \eqref{eq:decayestimate}
for $t = k2^{-n-1}t_0$.
Let $l \in (0,\infty)$ and $\epsilon > 0$ be such that we have
$\tilde M = {\mbox{\boldmath$\pi$}}(\{\gamma\,:\,l \le l(\gamma) \le l+\epsilon\})>0$.
Then by Proposition~\ref{prop:separation} we know that any measure
\[
\tilde{\mbox{\boldmath$\pi$}} \in \gopt\left(\frac1{\tilde M}(\e_0)_\sharp{\mbox{\boldmath$\pi$}}\res{\{\gamma\,:\,l \le \ell(\gamma) \le l+\epsilon\}},
\frac1{\tilde M}(\e_1)_\sharp{\mbox{\boldmath$\pi$}}\res{\{\gamma\,:\,l \le \ell(\gamma) \le l+\epsilon\}}\right)
\]
is concentrated on geodesics with lengths in the interval
$[l,l+\epsilon]$. On the other hand, by Lemma~\ref{lma:separation}
we know that
\[
(\e_{k2^{-n-1}t_0})_\sharp\tilde{\mbox{\boldmath$\pi$}} \perp (\e_{k2^{-n-1}t_0})_\sharp
{\mbox{\boldmath$\pi$}}\res\{\gamma\,:\, \ell(\gamma) \notin [l,l+\epsilon] \text{ and }\gamma_{k2^{-n-1}t_0} \in A\}.
\]
Therefore, in proving \eqref{eq:decayestimate} we may separately
deal with the parts of the measure where all the geodesics have
lengths in an interval $[l,l+\epsilon]$. Take now
a Borel set $A\subset X$ such that for the measure
$\hat{\mbox{\boldmath$\pi$}} = {\mbox{\boldmath$\pi$}}\res\{\gamma\,:\,l \le \ell(\gamma) \le l+\epsilon\text{ and }\gamma_{k2^{-n-1}t_0} \in A\}$
we have $\hat{M} = \hat{\mbox{\boldmath$\pi$}}(\geo(X))>0$.
Suppose that the measure
\[
\tilde\nu \in \mathcal{I}_\frac12\left(\frac1{\hat{M}}(\e_{(k-1)2^{-n-1}t_0})_\sharp\hat{\mbox{\boldmath$\pi$}},
\frac1{\hat{M}}(\e_{(k+1)2^{-n-1}t_0})_\sharp\hat{\mbox{\boldmath$\pi$}}\right)
\]
satisfies the convexity inequality \eqref{eq:CDdef}. Then
\begin{align*}
\entv(\tilde\nu) \le ~& \frac12\entv(\hat{M}^{-1}(\e_{(k-1)2^{-n-1}t_0})_\sharp\hat{\mbox{\boldmath$\pi$}}) + \frac12\entv(\hat{M}^{-1}(\e_{(k+1)2^{-n-1}t_0})_\sharp\hat{\mbox{\boldmath$\pi$}})\\
& + \frac{K^-}{8}W_2^2\left(\hat{M}^{-1}(\e_{(k-1)2^{-n-1}t_0})_\sharp\hat{\mbox{\boldmath$\pi$}},\hat{M}^{-1}(\e_{(k+1)2^{-n-1}t_0})_\sharp\hat{\mbox{\boldmath$\pi$}}\right)\\
\le ~& \frac12 \log{\frac{C}{\hat{M}}} -\frac14((1-(k-1)2^{-n-1})(c_2 - K^-(k-1)2^{-n-1}t_0^2)l^2) \\
&+ \frac12 \log{\frac{C}{\hat{M}}} -\frac14((1-(k+1)2^{-n-1})(c_2 - K^-(k+1)2^{-n-1}t_0^2)l^2) \\
& + \frac{K^-}{8}(2^{-n}t_0(l+\epsilon))^2\\
= ~& \log{\frac{C}{\hat{M}}} - \frac12((1-k2^{-n-1})(c_2 - K^-k2^{-n-1}t_0^2)l^2) + \frac{K^-}{8}2^{-2n}t_0^2(2l+\epsilon)\epsilon.
\end{align*}
Proposition \ref{prop:spreadtolimit} then gives
\[
\rho_t(\gamma_t) \le C{\rm exp}\bigl[-\frac12(1-\frac{t}{t_0})(c_2 - K^-tt_0)l^2 + \frac{K^-}{8}2^{-2n}t_0^2(2l+\epsilon)\epsilon\bigr]
\]
for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma \in \geo(X)$ with
$\ell(\gamma) \in [l,l+\epsilon]$. By letting $\epsilon \downarrow 0$ we then obtain
\eqref{eq:decayestimate} for $t = k2^{-n-1}t_0$.
Notice that the estimate \eqref{eq:decayestimate} gives
$ \rho_t(\gamma_t) \le C{\rm exp}\bigl[-\frac12(1-\frac{t}{t_0})(c_2 - K^-tt_0)\ell^2(\gamma)\bigr] \le C$
for all $t \in [0,t_0]$ for ${\mbox{\boldmath$\pi$}}$-a.e $\gamma \in \geo(X)$,
which is equivalent to \eqref{eq:uniformbound}.
\end{proof}
\section{Convergence results}\label{sec:auxiliary}
This section is devoted to the proof of some auxiliary convergence results. The first one deals with
entropy convergence. Recall the notation $V(x)={\sf d}(x,x_0)$.
\begin{lemma}\label{lem:ConvEnt}
Let $f_n{\mbox{\boldmath$m$}}$, $f{\mbox{\boldmath$m$}}$ be positive finite measures in $X$. If $f_n \uparrow f$ ${\mbox{\boldmath$m$}}$-a.e. and
$\int fV^2\,\d{\mbox{\boldmath$m$}}<\infty$, then
\begin{equation}\label{eq:ConvEnt}
\int_X f_n \log f_n \,\d {\mbox{\boldmath$m$}} \to \int_X f\log f\, \d {\mbox{\boldmath$m$}}.
\end{equation}
The same conclusion holds if $f_n \down f$ ${\mbox{\boldmath$m$}}$-a.e. and $\int f_1V^2\,\d{\mbox{\boldmath$m$}}<\infty$.
\end{lemma}
\begin{proof} Assume first that ${\mbox{\boldmath$m$}}$ is a finite measure. Let us first consider the case $f_n \uparrow f$.
Observe that the function $t\mapsto t \log t$ is decreasing on $[0,{\mathrm e}^{-1}]$ and increasing on $[{\mathrm e}^{-1},\infty)$; we write it as the difference
$\phi_1-\phi_2$, with
$$
\phi_1(t):=
\begin{cases}
-\frac{1}{{\mathrm e}} &\text{if $t\in [0,\frac{1}{{\mathrm e}}]$;}\\
t\log t & \text{if $t\geq\frac{1}{{\mathrm e}}$,}
\end{cases}\qquad\qquad
\phi_2(t):=
\begin{cases}
-\frac{1}{{\mathrm e}}-t\log t &\text{if $t\in [0,\frac{1}{{\mathrm e}}]$;}\\
0 & \text{if $t\geq\frac{1}{{\mathrm e}}$.}
\end{cases}
$$
Notice that $\phi_i$ are nondecreasing and bounded from below. Therefore
we can apply the monotone convergence theorem for
$\int\phi_i(f_n)\,\d{\mbox{\boldmath$m$}}$ to conclude. In the case $f_n\down f$ the argument is the same.
In the general $\sigma$-finite case we use
\eqref{eq:changeentropy} to reduce ourselves to the previous case, noticing that our assumptions
on $f_n$ imply $\int f_nV^2\,\d{\mbox{\boldmath$m$}}\to\int fV^2\,\d{\mbox{\boldmath$m$}}<\infty$.
\end{proof}
Recall that, according to Definition~\ref{def:wug} and \eqref{eq:convention},
the space ${\mathcal S}^2$ consists of ${\mbox{\boldmath$m$}}$-measurable functions having a weak upper gradient in $L^2(X,{\mbox{\boldmath$m$}})$.
\begin{lemma}\label{lem4.4revised}
Let $x_0\in X$, $\mu=f{\mbox{\boldmath$m$}},\,\sigma=g{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$ with $f(x)\leq c_1{\mathrm e}^{-c_2{\sf d}^2(x,x_0)}$ for some $c_1,\,c_2>0$,
$\inf_{B_R(x_0)}f>0$ for
all $R>0$ and $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ with bounded support.
Let ${\mbox{\boldmath$\pi$}}\in\gopt(\mu,\sigma)$ be a good geodesic given by Theorem~\ref{thm:goodgeodesics}. Then:
\begin{itemize}
\item[(1)] For $h\in{\mathcal S}^2$ satisfying $\weakgrad{h}\in L^2(X,\mu)$ and
\begin{equation}\label{assNablaf}
\weakgrad{h}^2(x) \leq C(1+ {\sf d}^2(x,x_0)) \quad \text{for any $x\in B^c_{R_*}(x_0)$}
\end{equation}
for some $C,\,R_*>0$, the following holds (understanding the integrals on $\geo(X)$)
\begin{equation}
\limsup_{t \down 0} \int\left| \frac{h(\gamma_t)-h(\gamma_0)}{{\sf d}(\gamma_t, \gamma_0)}\right|^2 \d {\mbox{\boldmath$\pi$}}(\gamma) \leq \int \weakgrad{h}^2(\gamma_0) \,
\d{\mbox{\boldmath$\pi$}}(\gamma).
\end{equation}
\item[(2)] For all Kantorovich potentials $\varphi$ relative to $(\mu,\sigma)$ with $|D \varphi|$ having linear growth
one has
\begin{equation}\label{derKant}
\lim_{t\downarrow 0} \frac{\varphi(\gamma_0)-\varphi(\gamma_t)}{{\sf d}(\gamma_0, \gamma_t)}= \lim_{t\downarrow 0} \frac{{\sf d}(\gamma_0,\gamma_t)}{t}=\weakgrad{\varphi}(\gamma_0) \quad \text{in } L^2(C([0,1];X), {\mbox{\boldmath$\pi$}}).
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof} (1) Call $f_t$ the density of $(\e_t)_\sharp {\mbox{\boldmath$\pi$}}$, i.e. $(\e_t)_\sharp {\mbox{\boldmath$\pi$}} = f_t{\mbox{\boldmath$m$}}$; we know that for $t>0$ sufficiently small,
say $t\in (0,t_0)$, $f_t$ exists and there exists a constant $C_*$ such that $f_t\leq C_*$ ${\mbox{\boldmath$m$}}$-a.e. in $X$ for all $t\in (0,t_0)$.
By definition of weak upper gradient, for any $t \in (0,t_0)$ and ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$ one has
\begin{equation} \nonumber
\left| \frac{h(\gamma_t)-h(\gamma_0)}{{\sf d}(\gamma_t,\gamma_0)} \right|^2 \leq
\frac{\left(\int_0^t \weakgrad{h}(\gamma_s)|\dot{\gamma}_s| \d s \right)^2 }
{{\sf d} ^2(\gamma_t,\gamma_0)} \leq \frac 1 t \int_0^t \weakgrad{h}^2(\gamma_s) \d s,
\end{equation}
therefore applying twice Fubini's theorem and using the identity $(\e_t)_\sharp{\mbox{\boldmath$\pi$}}=f_t{\mbox{\boldmath$m$}}$ we get
\begin{equation}
\int \left| \frac{h(\gamma_t)-h(\gamma_0)}{{\sf d}(\gamma_t,\gamma_0)} \right|^2 \d {\mbox{\boldmath$\pi$}}(\gamma) \leq
\int \left( \frac 1 t \int_0^t \weakgrad{h}^2(\gamma_s) \d s \right) \d {\mbox{\boldmath$\pi$}}(\gamma)
= \int_X \left(\frac{1}{t} \int_0^t f_s \d s \right) \weakgrad{h}^2 \,\d {\mbox{\boldmath$m$}}.
\end{equation}
The conclusion of the lemma follows once the following claim is proved:
\begin{equation}\label{claim:f}
\lim_{t\down 0} \int_X \left(\frac{1}{t} \int_0^t f_s \d s \right) \weakgrad{h}^2\, \d {\mbox{\boldmath$m$}}= \int_X \weakgrad{h}^2 f\,\d {\mbox{\boldmath$m$}}.
\end{equation}
In order to prove the claim we use both the uniform $L^\infty$ estimates on $f_t$ and the
$2$-uniform integrability of $V^2$ w.r.t. $f_t{\mbox{\boldmath$m$}}$. Notice first that the local boundedness of
$f^{-1}$ implies $\weakgrad{h}^2\in L^1(B_R(x_0),{\mbox{\boldmath$m$}})$ for all $R>0$; moreover
\begin{equation}\label{eq:sigmat}
\bar{f}_t:=\left(\frac{1}{t} \int_0^t f_s \d s \right)\to f \quad \text{in duality with } L^1(B_R(x_0),{\mbox{\boldmath$m$}}).
\end{equation}
Indeed the weak convergence $f_t {\mbox{\boldmath$m$}} \to f{\mbox{\boldmath$m$}}$ implies the weak convergence of $\bar{f}_t$ to $f$ in the duality with
$C_b(B_R(x_0))$; then \eqref{eq:sigmat} follows by the uniform $L^\infty$ bound on $\bar{f}_t$.
Second, observe that \eqref{assNablaf} gives
\begin{eqnarray}
\left|\int_X \bar{f}_t \weakgrad{h}^2 \,\d {\mbox{\boldmath$m$}}-\int_{B_R(x_0)} \bar{f}_t \weakgrad{h}^2\, \d {\mbox{\boldmath$m$}} \right| &\leq&
\frac{C}{t} \int_0^t\int_{B_R^c(x_0)} (1+{\sf d}^2(x,x_0)) f_s \,\d {\mbox{\boldmath$m$}}\d s \label{ClaimPart1}\\
&& \to 0 \quad \text{as } R\to \infty \text{ uniformly in } t \in (0,t_0)\nonumber;
\end{eqnarray}
the second line comes from the observation that the geodesic $(f_s {\mbox{\boldmath$m$}})_{s\in [0,1]}$ is a compact subset in $(\ProbabilitiesTwo X, W_2)$,
hence tight and 2-uniformly integrable (see \cite[Proposition~7.1.5]{Ambrosio-Gigli-Savare08}).
The claim \eqref{claim:f} follows then combining \eqref{ClaimPart1} and \eqref{eq:sigmat}.
\noindent
(2) Observe we are under the assumptions of the Metric Brenier Theorem 10.3 in \cite{Ambrosio-Gigli-Savare11}, therefore there exists
a Borel function $L$ satisfying
$L(\gamma_0):={\sf d}(\gamma_0,\gamma_1)$ for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma \in \geo(X)$ and, in addition,
\begin{equation}\label{eq:metrbren}
\weakgrad{\varphi}(x)=|D^+ \varphi| (x)= L(x) \quad \text{for ${\mbox{\boldmath$m$}}$-a.e. $x\in X$.}
\end{equation}
It trivially follows that for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma \in\geo(X)$
$$\weakgrad{\varphi}(\gamma_0)={\sf d}(\gamma_0,\gamma_1)=\frac{{\sf d}(\gamma_0,\gamma_t)}{t} \quad
\text{for every $t\in (0,1)$.}$$
The missing part is the $L^2$ convergence of difference quotients, proved and stated in \cite{Ambrosio-Gigli-Savare11} under
a different set of assumptions: we adapt the argument to our case, where $|D \varphi|$ has linear growth.
Since by optimality we have for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$ that
$$\varphi(\gamma_0)+\varphi^c(\gamma_1)=\frac{{\sf d}^2(\gamma_0,\gamma_1)}{2}, \qquad \varphi(\gamma_t)
+\varphi^c(\gamma_1)\leq \frac{{\sf d}^2(\gamma_t,\gamma_1)}{2} ,$$
we get with a subtraction that
$$\varphi(\gamma_0)-\varphi(\gamma_t)\geq \frac{1-(1-t)^2 }{2}{\sf d}^2(\gamma_0,\gamma_1)
=\frac{2t-t^2}{2} {\sf d}^2(\gamma_0,\gamma_1) \quad \text{for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$.}$$
Therefore, dividing both sides by ${\sf d}(\gamma_t,\gamma_0)=t{\sf d}(\gamma_1,\gamma_0)$, for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$ one has
\begin{equation}\label{liminfKant}
\liminf_{t \down 0} \frac{\varphi(\gamma_0)-\varphi(\gamma_t)}{{\sf d}(\gamma_0,\gamma_t)}\geq
{\sf d}(\gamma_0,\gamma_1)=\weakgrad{\varphi} (\gamma_0).
\end{equation}
On the other hand, by definition of ascending slope
\begin{equation}\label{limsupKant}
\limsup_{t\down 0} \frac{\varphi(\gamma_0)-\varphi(\gamma_t)}{{\sf d}(\gamma_0,\gamma_t)}\leq
|D^+ \varphi|(\gamma_0).
\end{equation}
So, combining \eqref{eq:metrbren} and \eqref{liminfKant} with \eqref{limsupKant} we get
\begin{equation}\label{eq:ConvKanta.e.}
\lim_{t \down 0} \frac{\varphi(\gamma_0)-\varphi(\gamma_t)}{{\sf d}(\gamma_0,\gamma_t)} = \weakgrad{\varphi}(\gamma_0) \quad
\text{for ${\mbox{\boldmath$\pi$}}$-a.e. $\gamma$.}
\end{equation}
Now we claim that
\begin{equation}\label{eq:weakConvKant}
\frac{\varphi(\gamma_0)-\varphi(\gamma_t)}{{\sf d}(\gamma_0,\gamma_t)} \rightharpoonup \weakgrad{\varphi}\circ\e_0
\qquad \text{weakly in } L^2(\geo(X),{\mbox{\boldmath$\pi$}}).
\end{equation}
Since by assumption $|D \varphi|$ has linear growth, by part (1) of the present lemma we have
\begin{equation}\label{eq:part1present}
\limsup_{t\down 0}\int \left |\frac{\varphi(\gamma_0)-\varphi(\gamma_t)}{{\sf d}(\gamma_0,\gamma_t)}\right |^2\,\d{\mbox{\boldmath$\pi$}}\leq
\int \weakgrad{\varphi}^2(\gamma_0)\,\d{\mbox{\boldmath$\pi$}}.
\end{equation}
If $\psi$ is a weak limit point of the difference quotients as $t\downarrow 0$, by Mazur's lemma a sequence of convex combinations of
these difference quotients strongly converges in $L^2(\geo(X),{\mbox{\boldmath$\pi$}})$ to $\psi$. Since a further subsequence converges ${\mbox{\boldmath$\pi$}}$-a.e., from
\eqref{eq:ConvKanta.e.} we obtain that $\psi=|D^+ \varphi|$. By weak compactness, the claim follows.
We conclude by observing that the lower semicontinuity of the norm under weak convergence together with \eqref{eq:part1present}
ensure convergence of the $L^2(\geo(X),{\mbox{\boldmath$\pi$}})$ norms.
Since in Hilbert spaces weak convergence and convergence of the norms give strong convergence, the lemma is proved.
\end{proof}
Our third result deals with weak convergence in the weighted Cheeger space: it will be applied to sequences
of Kantorovich potentials. In this and in the next lemma we assume that $\mathbb{C}$ is quadratic, so that
by Theorem~\ref{thm:weighted} $\mathbb{C}_\eta$ is quadratic whenever $\eta=g{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo X$ with $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ and with $\mathbb{C}(\sqrt{g})<\infty$.
Recall that $\mathcal E_\eta$ denotes, according to \eqref{eq:numeriamoanchequesta}, the bilinear form associated to
$\mathbb{C}_\eta$.
\begin{lemma}\label{lem:WeakConvK}
Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ have a quadratic Cheeger energy.
Let $\eta=g{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo X$ with $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $\mathbb{C}(\sqrt{g})<\infty$.
Consider a sequence $(f_n)\subset{\mathcal S}^2$ with
\begin{equation}\label{assfn}
\sup_{n \in \mathbb{N}} \int_X \weakgrad{f_n}^2\,\d\eta < \infty, \qquad \sup_{n\in\mathbb{N}} |f_n|(x)\leq C (1+{\sf d}^2(x,x_0)),
\end{equation}
and assume that $f_n\to f$ ${\mbox{\boldmath$m$}}$-a.e. in $X$. Then
\begin{equation}
\lim_{n\to\infty} \mathcal E_{\eta} (f_n,\log g)= \mathcal E_{\eta}(f,\log g).
\end{equation}
\end{lemma}
\begin{proof} We argue as in Theorem~\ref{thm:weighted}. Let us consider the weighted measure
$$
\tilde\eta:=\frac{1}{1+V^2}\eta
$$
and the corresponding weighted Sobolev space $H:=L^2(X,\tilde\eta)\cap{\mathcal S}^2_\eta$, endowed with the scalar product
$$\langle f,g\rangle_H:=\int_X fg \, \d\tilde\eta + \mathcal E_{\eta} (f,g).$$
Observe that, since $L^2(X,\tilde\eta)$ is a Hilbert space, in order to check the completeness of the norm $\|\cdot\|_H$ induced by this scalar product
it is enough to check the lower semicontinuity of $\|\cdot\|_H$ with respect to strong convergence in $L^2(X,\tilde\eta)$; but this is clear since ${\mathbb{C}} _\eta$ is lower semicontinuous with respect to $L^2(X,\eta)$ convergence and, on sequences uniformly bounded in $L^\infty(X,\eta)$,
the finiteness of $\eta$ turns $L^2(X,\tilde\eta)$ convergence into $L^2(X,\eta)$ convergence. By a truncation argument one obtains that ${\mathbb{C}}_\eta$ is
$L^2(X,\tilde\eta)$-lower semicontinuous. We conclude that $(H,\langle\cdot,\cdot\rangle_H)$ is a Hilbert space
(it is even separable, see \cite[Proposition~4.10]{Ambrosio-Gigli-Savare11b}, but we shall not need this fact in the sequel).
Now since $\eta\in \Probabilities X$, from the second assumption \eqref{assfn} and dominated convergence we have that
$f_n \to f$ strongly in $L^2(X,\tilde\eta)$. On the other hand, the first assumption in \eqref{assfn} implies that $\|f_n\|_H$ is bounded.
By reflexivity if follows that $f_n\to f$ weakly in $H$. The conclusion follows by noticing that, since $\mathbb{C}(\sqrt{g})<\infty$, the map
$$h\mapsto \mathcal E_\eta(h,\log g) $$
is linear and continuous from $H$ to $\mathbb{R}$.
\end{proof}
In this last result we estimate how much $\mathcal E_\rho(\log g,\varphi)$ changes under modifications of
the density $g$ of $\rho$.
\begin{lemma}\label{lem:essential} Let $\eta=g{\mbox{\boldmath$m$}},\,\eta'=g'{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$ with $g,\,g'\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $\mathbb{C}(\sqrt{g}),\,\mathbb{C}(\sqrt{g'})$
finite. Let
$\varphi:X\to\mathbb{R}$ be a locally Lipschitz function whose gradient has linear growth. Then, setting $E:=\{g\neq g'\}$, one has
\begin{eqnarray}\label{eq:essential}
&&|\mathcal E_\eta(\log g,\varphi)-\mathcal E_{\eta'}(\log g',\varphi)|\\ &\leq&\biggl(\int_E\weakgrad{\sqrt{g}}^2\,\d{\mbox{\boldmath$m$}}\biggr)^{1/2}
\biggl(\int_E \weakgrad{\varphi}^2\,\d\eta\biggr)^{1/2}+\biggl(\int_E\weakgrad{\sqrt{g'}}^2\,\d{\mbox{\boldmath$m$}}\biggr)^{1/2}
\biggl(\int_E\weakgrad{\varphi}^2\,\d\eta'\biggr)^{1/2}.\nonumber
\end{eqnarray}
\end{lemma}
\begin{proof} By Lemma~\ref{lem:WeakConvK} we can assume, by a simple approximation argument, that $\varphi$ has bounded support.
Under this assumption the quantity to be estimated reduces, thanks to \eqref{eq:transfer1} and \eqref{eq:localityGamma}, to
$$
\left|\int_X\Gbil{\varphi}{g}-\Gbil{\varphi}{g'}\,\d{\mbox{\boldmath$m$}}\right|=
\left|\int_E\Gbil{\varphi}{g}-\Gbil{\varphi}{g'}\,\d{\mbox{\boldmath$m$}}\right|\leq
\int_E\bigl(\weakgrad{g}\weakgrad{\varphi}+\weakgrad{g'}\weakgrad{\varphi}\bigr)\,\d{\mbox{\boldmath$m$}}
$$
and, after dividing and multiplying by $\sqrt{g}$ and $\sqrt{g'}$, we can use H\"older's inequality to provide the result.
\end{proof}
\section{Equivalence of the different formulations of $RCD(K,\infty)$}\label{sec:lastq}
In this section we prove the following result, extending Theorem~\ref{thm:main} to
$\sigma$-finite metric measure spaces.
\begin{theorem}\label{thm:main1}
Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ be a metric measure space with $(X,{\sf d})$ complete, separable, ${\mbox{\boldmath$m$}}$ finite on bounded sets
and with $\supp{\mbox{\boldmath$m$}}=X$. Then the following properties are equivalent.
\begin{enumerate}
\item[(i)] $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a $CD(K,\infty)$ space and the semigroup ${\mathscr H}_t$ on $\ProbabilitiesTwo{X}$ is additive.
\item[(ii)] $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a $CD(K,\infty)$ space and $\mathbb{C}$ is a quadratic form on $L^2(X,{\mbox{\boldmath$m$}})$.
\item[(iii)] $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a length space, \eqref{eq:growthcond} holds and any $\mu \in \ProbabilitiesTwo{X}$ is the starting point of an $EVI_K$
gradient flow of $\entv$.
\end{enumerate}
Any metric measure space $(X,{\sf d},{\mbox{\boldmath$m$}})$ satisfying these assumptions and one of the equivalent properties (i), (ii), (iii) will be
called ($\sigma$-finite) $RCD(K,\infty)$ space.
\end{theorem}
Here ${\mathscr H}_t$ is the
$W_2$-gradient flow of $\entv$, according to Definition~\ref{def:dissKconv} (which is known to exist and to be unique for any given initial datum in $D({\rm Ent}_{\mbox{\boldmath$m$}})$, see \cite{Gigli10} and \cite{Ambrosio-Gigli-Savare11}), while
${\sf h}_t$ stands for the gradient flow of $\mathbb{C}$ in $L^2(X,{\mbox{\boldmath$m$}})$ (or, equivalently, the $EVI_0$ gradient flow).
Note that the implications (i) to (ii) and (iii) to (i) were already proved by the first two
authors with Savar\'e in \cite{Ambrosio-Gigli-Savare11b}, because the same proof works in the $\sigma$-finite
setting. The key
implication from (ii) (or (i)) to (iii) is given by the derivative of quadratic optimal transport distance along the heat flow
and of the entropy
along a geodesic, estimated in the next two subsections. Consequently we shall always assume in this section
that $\mathbb{C}$ is quadratic.
We denote by $\Delta$ the infinitesimal generator of the linear semigroup ${\sf h}_t$, so that
$$
\frac{\d}{\d t}{\sf h}_tf=\Delta {\sf h}_t\qquad\text{for a.e. $t>0$.}
$$
Also, since $\mathbb{C}$ is quadratic, $\Delta$ is related to the bilinear form $\mathcal E$ in \eqref{eq:numeriamoanchequesta}
by
\begin{equation}\label{eq:intbypartsoo}
\int_X g\Delta f\,\d{\mbox{\boldmath$m$}}=\mathcal E(f,g)\qquad\forall g\in{\mathcal S}^2\cap L^2(X,{\mbox{\boldmath$m$}}),\,\,f\in D(\Delta).
\end{equation}
One of the main result of the work of the first two authors with Savar\'e \cite{Ambrosio-Gigli-Savare11} has been
the following identification theorem in $CD(K,\infty)$, see (8.5), Theorem~8.5 and
Theorem~9.3(iii) therein.
\begin{theorem}[The heat flow as gradient flow]\label{thm:heatgf}
Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ be a $CD(K,\infty)$ space and let $f\in
L^2(X,{\mbox{\boldmath$m$}})$ be such that $\mu=f{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$.
Then ${\mathscr H}_t\mu={\sf h}_tf{\mbox{\boldmath$m$}}$ for all $t\geq 0$, $t\mapsto\entv({\mathscr H}_t\mu)$ is
locally absolutely continuous in $[0,\infty)$, and
\begin{equation}\label{eq:edissrateflow}
-\frac{\d}{\d
t}\entv({\mathscr H}_t\mu)=|\dot{{\mathscr H}_t\mu}|^2=\int_{\{{\sf h}_tf>0\}}\frac{\weakgrad
{{\sf h}_tf}^2}{{\sf h}_tf}\,\d{\mbox{\boldmath$m$}}\qquad\text{for a.e.~$t>0$.}
\end{equation}
\end{theorem}
In other words, one can unambiguously define the heat flow on a
$CD(K,\infty)$ space either as the gradient flow of $\mathbb{C}$
in $L^2(X,{\mbox{\boldmath$m$}})$ or as the $W_2$-gradient flow of $\entv$.
\subsection{Derivative of $W_2^2(\cdot,\sigma)$ along the heat flow }
Notice that this result, whose proof is achieved by a duality argument, requires no curvature assumption.
We need only to assume that $\mathbb{C}$ is quadratic and that ${\mbox{\boldmath$m$}}$ satisfies the growth condition \eqref{eq:growthcond}.
\begin{theorem}
\label{thm:derw2} Let
$\mu=f{\mbox{\boldmath$m$}}\in D(\entv)$ and define $\mu_t:=({\sf h}_tf){\mbox{\boldmath$m$}}=f_t{\mbox{\boldmath$m$}}$.
Let $\sigma\in\ProbabilitiesTwo X$ with bounded support. Then, for a.e. $t>0$ the following property holds:
for any Kantorovich potential $\varphi_t$ relative to $(\mu_t,\sigma)$ whose
slope has linear growth, one has
\begin{equation}
\label{eq:derw2} \frac{\d}{\d t}\frac12W_2^2(\mu_t,\sigma)=
-\mathcal E_{\mu_t}(\varphi_t,\log f_t).
\end{equation}
\end{theorem}
\begin{proof} By the energy dissipation estimate \eqref{eq:edissrateflow},
we have $\int_0^\infty\mathbb{C}(\sqrt{f_t})\,{\d t}<\infty$. Furthermore, the maximum principle
proved in Theorem~4.20 of \cite{Ambrosio-Gigli-Savare11} shows that
$f_t\leq \|f\|_\infty$ ${\mbox{\boldmath$m$}}$-a.e. in $X$ for all
$t\geq 0$. Also, by Proposition~\ref{prop:goodKant} the potential $\varphi_t$ belongs to
$L^1(X,\nu)$ for all $\nu\in\ProbabilitiesTwo{X}$ and its slope has linear growth. Furthermore, the
$L^1$ estimate is uniform in $t$ and in bounded subsets of $\ProbabilitiesTwo
X$ and the estimate on the slope depends on $\sigma$ only.
Thanks to \eqref{eq:edissrateflow}, the map $t\mapsto f_t{\mbox{\boldmath$m$}}$ is a locally absolutely continuous curve
in $\ProbabilitiesTwo X$, hence the derivative on the left hand side of
\eqref{eq:derw2} exists for a.e.~$t>0$. Also, the derivative of
$t\mapsto f_t$ exists in $L^2(X,{\mbox{\boldmath$m$}})$ and coincides with $\Delta f_t$ for a.e.~$t>0$.
Fix $t_0>0$ where both properties hold, which is also
a Lebesgue point for $\mathbb{C}(\sqrt{f_t})$.
We now claim that
\begin{equation}\label{eq:tyuh}
\lim_{h\downarrow 0}\int_X\psi\frac{
f_{t_0}-f_{t_0-h}}{h}\,\d{\mbox{\boldmath$m$}}=-{\mathcal
E}_{\mu_{t_0}}(\psi,\log f_{t_0})
\end{equation}
for all locally Lipschitz functions $\psi$ whose gradient has linear growth.
The proof of \eqref{eq:tyuh} is easy if we assume, in addition, that
$\psi$ has bounded support. Indeed,
$h^{-1}(f_{t_0+h}-f_{t_0})\to\Delta f_{t_0}$ as $h\to 0$ in
$L^2(X,{\mbox{\boldmath$m$}})$, so that \eqref{eq:transfer1} and \eqref{eq:intbypartsoo} give
$$
\lim_{h\to 0}\int_X\psi\frac{
f_{t_0+h}-f_{t_0}}{h}\,\d{\mbox{\boldmath$m$}}= \int_X\psi\Delta f_{t_0}\,\d{\mbox{\boldmath$m$}}=
-{\mathcal E}(\psi,f_{t_0})=-{\mathcal
E}_{\mu_{t_0}}(\psi,\log f_{t_0}).
$$
For the general case, let $\chi_N:X\to [0,1]$ be satisfying ${\rm
Lip}(\chi_N)\leq 1$, $\chi_N\equiv 1$ on $B_N(x_0)$ and
$\chi_N\equiv 0$ on $X\setminus B_{2N}(x_0)$ and define
$\psi^N:=\psi\chi_N$. Applying Lemma~\ref{le:luigi} below with
$\varphi_N:=\psi-\psi^N$ we get
\[
\sup_{|h|<t_0/2}\left|\int
\varphi_N\frac{\rho_{t_0+h}-\rho_{t_0}}{h}\,\d{\mbox{\boldmath$m$}}\right|^2\\
\leq \sup_{|h|<t_0/2}\frac8h\int\limits_{t_0-|h|}^{t_0+|h|}
\mathbb{C}(\sqrt{f_s})\int_X\weakgrad{\varphi_N}^2\,\d\mu_s\,\d s.
\]
Hence (by our choice of $t_0$ and the
$2$-uniform integrability of $\mu_s$)
\[
\limsup_{N\to\infty}\sup_{|h|<t_0/2}\left|\int_X\varphi_N\frac{
f_{t_0+h}-f_{t_0}}{h}\,\d{\mbox{\boldmath$m$}}\right|=0,
\]
which, taking into account that ${\mathcal
E}_{\mu_{t_0}}(\psi^N,\log f_{t_0})\to {\mathcal
E}_{\mu_{t_0}}(\psi,\log f_{t_0})$ thanks to Lemma~\ref{lem:WeakConvK},
implies \eqref{eq:tyuh}.
Now, notice that since
$\varphi_{t_0}$ is a Kantorovich potential for $(\mu_{t_0},\sigma)$
one has
\[
\begin{split}
\frac 12 W_2^2(\mu_{t_0},\sigma)&=\int_X\varphi_{t_0}\,\d\mu_{t_0}+\int\varphi_{t_0}^c\,\d\sigma\\
\frac 12
W_2^2(\mu_{t_0-h},\sigma)&\geq\int_X\varphi_{t_0}\,\d\mu_{t_0-h}+
\int\varphi_{t_0}^c\,\d\sigma\qquad\text{for all $h$ such that
$t_0-h> 0$.}
\end{split}
\]
Taking the difference between the first identity and the second
inequality and using the claim with $\psi=\varphi_{t_0}$ we get
\[
\frac 12 W_2^2(\mu_{t_0+h},\sigma)-\frac 12 W_2^2(\mu_{t_0},\sigma)\geq -h\mathcal E_{\mu_{t_0}}(\log f_{t_0},\varphi_{t_0})+o(h).
\]
Since $t\mapsto W_2^2(\mu_t,\sigma)$ is differentiable at $t=t_0$ we conclude.
\end{proof}
\begin{lemma}\label{le:luigi} Let $\mu_s=f_s{\mbox{\boldmath$m$}}$ be as in the previous theorem and
let $\varphi:X\to\mathbb{R}$ be locally Lipschitz, with $|D \varphi|$ having linear
growth. Then, for $[s,t]\subset (0,\infty)$ one has
\begin{equation}
\label{eq:trucco} \left|\int
\varphi\frac{f_t-f_s}{t-s}\,\d{\mbox{\boldmath$m$}}\right|^2\leq \frac
8{t-s}\int_s^t \mathbb{C}(\sqrt{f_r})\biggl(\int\weakgrad
\varphi^2\,\d\mu_r\biggr)\,\d r.
\end{equation}
\end{lemma}
\begin{proof}
Assume first that $\varphi\in L^2(X,{\mbox{\boldmath$m$}})$. Then integrating by parts we
get
\[
\left|\int \varphi\Delta f_r\,\d{\mbox{\boldmath$m$}}\right|^2\leq\left(\int\weakgrad
\varphi\,\weakgrad{f_r}\,\d{\mbox{\boldmath$m$}}\right)^2\leq\int\weakgrad
\varphi^2\,\d\mu_r\,\int\frac{\weakgrad{f_r}^2}{f_r}\,\d{\mbox{\boldmath$m$}},
\]
for all $r>0$, and the thesis follows by integration in $(s,t)$. For
the general case, we approximate $\varphi$ by $\varphi\chi_N$, with $\chi_N$
chosen as in the proof of the previous theorem.
\end{proof}
\subsection{Derivative of the entropy along $\entv$-convex $L^\infty$-bounded geodesics}
The goal of this subsection is to prove the following theorem, where both the curvature condition and the fact
that $\mathbb{C}$ is quadratic play a role.
\begin{theorem}[Entropy inequality]\label{Thm:DerEntr}
Assume that $(X,{\sf d},{\mbox{\boldmath$m$}})$ is a $CD(K,\infty)$ space.
Let $\eta=f{\mbox{\boldmath$m$}},\,\sigma=g{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo X$ with $g$ uniformly bounded and having compact support, $f$ uniformly bounded with
$\mathbb{C}(\sqrt{f})<\infty$. Then there exists a Kantorovich potential $\varphi$ from $\eta$ to $\sigma$ such that $|D\varphi|$ has linear growth
and
\begin{equation}\label{eq:Step1}
\entv(\sigma)-\entv(\eta)- \frac{K}{2} W_2^2(\eta,\sigma) \geq -\mathcal E_{\eta}(\varphi,\log f).
\end{equation}
\end{theorem}
The proof of Theorem~\ref{Thm:DerEntr}, carried by approximation, is presented at the end of the subsection;
the first crucial step is the following proposition, whose proof relies on
Proposition~\ref{prop:goodKant} and Lemma \ref{lem4.4revised}.
\begin{proposition}\label{Prop1}
Under the assumptions of Theorem~\ref{Thm:DerEntr}, for $\delta>0$ call
\begin{equation}\label{eq:chietad}
f_{\delta,n}=c_{\delta,n}[(\chi_n^2) \eta \vee \delta{\mathrm e}^{-2cV^2}],
\end{equation}
where $c$ is strictly larger than the constant ${\sf c}$ in \eqref{eq:growthcond},
$c_{\delta,n}$ is the normalizing constant such that $f_{\delta,n} {\mbox{\boldmath$m$}}$ is a probability density,
$\chi_n$ is a $1$-Lipschitz cut-off function equal to $1$ on $B_n(x_0)$ and null outside $B_{2n}(x_0)$.\\
Then there exists a Kantorovich potential $\varphi_{\delta,n}$ from $\eta_{\delta,n}:=f_{\delta,n} {\mbox{\boldmath$m$}}$ to $\sigma$
satisfying the growth conditions
\begin{equation}\label{growthPhidn}
|\varphi_{\delta,n}(x)|\leq C(\sigma)(1+{\sf d}^2(x,x_0)),\quad\quad\quad |D\varphi_{\delta,n}|(x)\leq C(\sigma)(1+{\sf d}(x,x_0)),
\end{equation}
such that
\begin{equation}\label{eq:Prop1}
\entv(\sigma)-\entv(\eta_{\delta,n})- \frac{K}{2} W_2^2(\eta_{\delta,n},\sigma) \geq
-\mathcal E_{\eta_{\delta,n}}(\varphi_{\delta,n},\log f_{\delta,n}).
\end{equation}
\end{proposition}
\begin{proof} First of all we are under the assumptions of Theorem~\ref{thm:goodgeodesics}, so let ${\mbox{\boldmath$\pi$}}\in \gopt(\eta_{\delta,n},\sigma)$ and let
$(\e_t)_\sharp{\mbox{\boldmath$\pi$}}=\mu_t=f_t{\mbox{\boldmath$m$}}$, $t\in[0,1]$, be the associated good geodesic from $\eta_{\delta,n}$ to $\sigma$ with a
uniform $L^\infty$ bound on the density for $t\in (0,t_0)$ and the $K$-convexity of the entropy. Let also $\varphi$ be the Kantorovich potential,
given by Proposition~\ref{prop:goodKant}, with quadratic growth and whose slope has linear growth.
Let us now check that $f_{\delta,n}$ satisfies the assumptions of Lemma~\ref{lem4.4revised}. Indeed, $|D \log f_{\delta,n}| \leq C(1+{\sf d}(x,x_0))$
whenever ${\sf d}(x,x_0)>2n$, because in this set $f_{\delta_n}$ coincides with $c_{\delta,n}\delta{\mathrm e}^{-2cV^2}$; in addition, the locality of weak
gradients and the partition $X=\{\chi^2_n \eta>\delta{\mathrm e}^{-2cV^2} \} \cup \{\chi^2_n \eta\leq \delta{\mathrm e}^{-2cV^2}\}$ ensure
that $\weakgrad{\log f_{\delta,n}}\in L^2(X,\eta_{\delta,n})$ because the finiteness of $\mathbb{C}(\sqrt{f})$ ensures that $\weakgrad{\log f}\in L^2(X,\eta)$.
Observe that the convexity of $z\mapsto z \log z$ gives
\begin{equation}\label{eq:logpi}
\frac{\entv(\mu_t)-\entv(\eta_{\delta,n})}{t} \geq \int_X \log f_{\delta,n} \frac{f_t-f_{\delta,n}}{t} \,\d {\mbox{\boldmath$m$}} =
\int \frac{\log(f_{\delta,n}\circ\e_t)-\log(f_{\delta,n} \circ\e_0)}{t}\, \d {\mbox{\boldmath$\pi$}}.
\end{equation}
Define the functions $F_t,\,G_t: AC^2([0,1];X)\to \mathbb{R}$ as
\begin{equation}
F_t(\gamma):=\frac{\log(f_{\delta,n} \circ\e_0)-\log(f_{\delta,n} \circ \e_t)}{{\sf d}(\gamma_0, \gamma_t)} \label{def:Ft},\qquad
G_t(\gamma):=\frac{\varphi\circ\e_0-\varphi\circ\e_t}{{\sf d}(\gamma_0, \gamma_t)}.
\end{equation}
Multiplying and dividing the right hand side of \eqref{eq:logpi} by ${\sf d}(\gamma_0,\gamma_t)$ we obtain
\begin{equation}\label{QLastEq}
\liminf_{t \down 0} \frac{\entv(\mu_t)-\entv(\eta_{\delta,n} {\mbox{\boldmath$m$}})}{t}\geq
- \limsup_{t\down 0} \int F_t(\gamma) \frac{{\sf d}(\gamma_0,\gamma_t)}{t} \d {\mbox{\boldmath$\pi$}}(\gamma)
\end{equation}
Now we claim that
\begin{equation}\label{QLastEqbis}
- \limsup_{t\down 0} \int F_t(\gamma) \frac{{\sf d}(\gamma_0,\gamma_t)}{t} \d {\mbox{\boldmath$\pi$}}(\gamma)= -\limsup_{t \down 0} \int F_t G_t \d {\mbox{\boldmath$\pi$}}.
\end{equation}
The proof of \eqref{QLastEqbis} follows at once by
\begin{equation}\label{eq:chepalle}
\lim_{t\down 0 } \int\left| G_t(\gamma)-\frac{{\sf d}(\gamma_0,\gamma_t)}{t}\right |^2\,\d{\mbox{\boldmath$\pi$}}=0
\quad \text{and} \quad \sup_{t\leq t_0} \int |F_t|^2\,\d{\mbox{\boldmath$\pi$}}<\infty.
\end{equation}
The first fact in \eqref{eq:chepalle} is ensured by (2) of Lemma~\ref{lem4.4revised}, as well as the
identity
\begin{equation}\label{eq:logope}
\int \weakgrad{\varphi}^2\circ\e_0\,\d {\mbox{\boldmath$\pi$}}=\lim_{t\downarrow 0}\int |G_t|^2 \d{\mbox{\boldmath$\pi$}}.
\end{equation}
The second fact in \eqref{eq:chepalle} is ensured by (1) of the same lemma applied
to $h=\log f_{\delta,n}$. Combining \eqref{QLastEq} and \eqref{QLastEqbis} we get
\begin{equation}\label{QLastEqter}
\liminf_{t \down 0} \frac{\entv(\mu_t)-\entv(\eta_{\delta,n} {\mbox{\boldmath$m$}})}{t}\geq
-\limsup_{t \down 0} \int F_t G_t \d {\mbox{\boldmath$\pi$}}.
\end{equation}
Now, applying Lemma~\ref{lem4.4revised} to $h=\varphi+\epsilon \log f_{\delta,n}$ gives that
\begin{equation}
\int \weakgrad{(\varphi+\epsilon \log f_{\delta,n})}^2\circ\e_0 \d {\mbox{\boldmath$\pi$}} \geq
\limsup_{t\down 0} \int |G_t(\gamma)+\epsilon F_t(\gamma)|^2 \d {\mbox{\boldmath$\pi$}}(\gamma) \label{eq:HorVert}.
\end{equation}
Subtracting to \eqref{eq:HorVert} the equality \eqref{eq:logope} and dividing by $\epsilon$ gives
\begin{equation}\label{lastEq}
\limsup_{t\down 0} \int G_t F_t \,\d {\mbox{\boldmath$\pi$}} \leq \liminf_{\epsilon \down 0}
\int_X \frac{\weakgrad{(\varphi+\epsilon \log f_{\delta,n})}^2-\weakgrad{\varphi}^2}{2\epsilon} f_{\delta,n} \,\d {\mbox{\boldmath$m$}}
= \mathcal E_{\eta_{\delta,n}} (\log f_{\delta,n},\varphi),
\end{equation}
where we used again the uniform bound on the $L^2$ norm of $F_t$.
Combining \eqref{QLastEqter} and \eqref{lastEq} we obtain
\begin{equation}\label{eq:Last1}
\liminf_{t \down 0} \frac{\entv(\mu_t)-\entv(\eta_{\delta,n})}{t}\geq - \mathcal E_{\eta_{\delta,n}} (\log f_{\delta,n},\varphi).
\end{equation}
The conclusion follows by \eqref{eq:Last1} recalling that, by construction, the entropy is $K$-convex along the geodesic $(\mu_t)_{t\in [0,1]}$, see \eqref{eq:CDdef}.
\end{proof}
\noindent
{\bf Proof of Theorem~\ref{Thm:DerEntr}.} In this proof we denote for brevity $a\vee b=\max\{a,b\}$. For every $\delta\in (0,1)$ define the density
\begin{equation}\label{def:etadelta}
\tilde{f}_{\delta}:=f \vee (\delta {\mathrm e}^{-2cV^2}) \quad\text{and } f_{\delta}:=c_{\delta} \tilde{f}_{\delta} \text{ with $c_\delta\uparrow 1$
as $\delta\downarrow 0$}
\end{equation}
(here $c>0$ is the constant in \eqref{eq:growthcond}),
so that $\tilde{f}_\delta \geq f $ and $c_{\delta}$ are the normalizing constants. We need a further regularization of $f_\delta$; to this aim,
let $\chi_n$ be standard cut-off functions, namely
$0 \leq \chi_n \leq 1$, $\Lip (\chi_n) \leq 1$, $\chi_n\equiv 1$ on $B_n(x_0)$ and $\chi_n\equiv 0$ on $B^c_{2n}(x_0)$. Then,
for every $n>1,\,\delta>0$ we define the densities
\begin{equation}\label{def:etadeltan}
\tilde{f}_{\delta,n}:= (\chi^2_n f) \vee (\delta {\mathrm e}^{-2cV^2}) \quad\text{and } f_{\delta,n}:=c_{\delta,n} \tilde{f}_{\delta,n} \text{ with } \; c_{\delta,n}\downarrow c_\delta \text{ as } n\to\infty,
\end{equation}
so that $\tilde{f}_{\delta,n} \leq \tilde{f}_\delta $ and $c_{\delta,n}$ are the normalizing constants.
Of course $f_{\delta,n}$ is uniformly bounded and $\eta_{\delta,n}:=f_{\delta,n} {\mbox{\boldmath$m$}} \in \ProbabilitiesTwo X$, moreover $\mathbb{C}(\sqrt{f_{\delta,n}})$ is
finite. Indeed by the chain rule and the locality of the weak gradients we have that
\begin{eqnarray*}
\weakgrad{\sqrt{f_{\delta,n}}} & =& \sqrt{c_{\delta,n}}|D(\chi_n \sqrt{f})|_w\\
&\leq& \sqrt{c_{\delta,n}} \left( \chi_n \weakgrad{\sqrt{f}}+ \sqrt{f}\weakgrad{\chi_n} \right) \quad\text{if $\chi^2_nf \geq \delta {\mathrm e}^{-2cV^2}$}\\
\weakgrad{\sqrt{f_{\delta,n}}} & =& \sqrt{\delta\, c_{\delta,n} } \weakgrad{{\mathrm e}^{-2c V^2}} \\
& \leq& 4c \sqrt{\delta \,c_{\delta,n}}\, {\sf d}(\cdot,x_0) \,e^{-2c V^2} \quad\text{otherwise}.
\end{eqnarray*}
Since by assumption $\mathbb{C}(\sqrt{f}) < \infty$, it follows not only that $\weakgrad{\sqrt{f_{\delta,n}}}^2$ are uniformly
bounded in $L^1(X,{\mbox{\boldmath$m$}})$, but also that they are equi-integrable:
\begin{equation}\label{eq:equiBoundF}
\sup_{\delta\in (0,1),\, n \in \mathbb{N} } \mathbb{C}(\sqrt{f_{\delta,n}})< \infty\quad\text{and}\quad
E_j\downarrow\emptyset\,\,\Rightarrow\sup_{\delta\in (0,1),\,n\in\mathbb{N}}\int_{E_j}\weakgrad{\sqrt{f_{\delta,n}}}^2\,\d{\mbox{\boldmath$m$}}\to 0.
\end{equation}
Observe that $(\eta_{\delta,n},\sigma)$ has the structure described in Proposition~\ref{Prop1}, so there exists a Kantorovich potential
$\varphi_{\delta,n}$ from $\eta_{\delta,n}$ to $\sigma$ satisfying the growth conditions \eqref{growthPhidn}
and such that the entropy inequality holds:
\begin{equation}\label{eq:EntEstAppr}
\entv(\sigma)-\entv(\eta_{\delta,n})- \frac{K}{2} W_2^2(\eta_{\delta,n},\sigma)
\geq -\mathcal E_{\eta_{\delta,n}}(\varphi_{\delta,n},\log f_{\delta,n}).
\end{equation}
\noindent
{\bf Passage to the limit as $n\to\infty$.} Consider the transportation problem from $\eta_\delta:=f_\delta {\mbox{\boldmath$m$}}$ to $\sigma$.
We claim the existence of a Kantorovich potential $\varphi_\delta$ such that
\begin{equation}\label{eq:EntEstdelta}
\entv(\sigma)-\entv(\eta_{\delta})- \frac{K}{2} W_2^2(\eta_{\delta},\sigma) \geq -\mathcal E_{\eta_{\delta}}(\varphi_\delta,\log f_{\delta}).
\end{equation}
We would like to pass to the limit as $n\to\infty$ in \eqref{eq:EntEstAppr}. Let us start by considering the left hand side: applying
Lemma~\ref{lem:ConvEnt} to $\tilde{\eta}_{\delta,n}\uparrow \tilde{\eta}_{\delta}$ ${\mbox{\boldmath$m$}}$-a.e, and recalling that $c_{\delta,n}\down c_{\delta}$ as $n\to \infty$,
we get
\begin{equation}\label{eq:ConvEnteta}
\entv (\eta_{\delta,n}) \to \entv (\eta_\delta) \quad \text{as } n\to \infty.
\end{equation}
It is easy to check that $\eta_{\delta,n} $ weakly converge to $\eta_{\delta}$ and have uniformly integrable 2-moments, so by
\cite[Proposition~7.1.5]{Ambrosio-Gigli-Savare08} we have
\begin{equation}\label{eq:ConvW}
\lim_{n\to\infty} W_2^2(\eta_{\delta,n} ,\sigma) = W_2^2(\eta_\delta,\sigma).
\end{equation}
Now let us show the convergence of the right hand side of \eqref{eq:EntEstAppr}. To simplify the problem we prove first that
\begin{equation}\label{eq:etadnetad}
\lim_{n\to\infty} \bigl| \mathcal E_{\eta_{\delta,n}}(\varphi_{\delta,n},\log f_{\delta,n})-\frac{c_{\delta,n}}{c_{\delta}}
\mathcal E_{\eta_{\delta}}(\varphi_{\delta,n},\log f_{\delta}) \bigr|=0.
\end{equation}
Notice that, calling $A_\delta:=\{x\in X:\ f(x)\geq \delta{\mathrm e}^{-2cV^2(x)} \}$ we have $f_{\delta,n}=\frac{c_{\delta,n}}{c_\delta}f_\delta$
on the complement $(A_{\delta}\cap B_n(x_0))\cup A_\delta^c$
of $A_\delta\setminus B_n(x_0)$. Since $A_\delta\setminus B_n(x_0)\downarrow\emptyset$
we can use \eqref{eq:essential} of Lemma~\ref{lem:essential} to obtain \eqref{eq:etadnetad}, taking
\eqref{eq:equiBoundF} into account.
{F}rom \eqref{eq:etadnetad}, and taking into account that $c_{\delta,n}\to c_\delta$ as $n\to\infty$,
in order to prove the convergence of the right hand side of \eqref{eq:EntEstAppr}, it is enough to show the existence of
a Kantorovich potential $\varphi_\delta$ for $(\eta_\delta,\sigma)$ such that
\begin{equation}\label{reduct:convEdn}
\mathcal E_{\eta_\delta} (\varphi_{\delta,n}, \log f_{\delta}) \to \mathcal E_{\eta_\delta} (\varphi_{\delta}, \log f_{\delta})\quad \text{as } n\to \infty.
\end{equation}
Now we use in a crucial way Lemma~\ref{lem:GammaConvKant}, which ensures the existence of a Kantorovich potential $\varphi_\delta$
for $(\eta_\delta,\sigma)$ and of a subsequence $n(k)$ such that $\varphi_{\delta,n(k)} \to \varphi_{\delta}$ pointwise in $X$. Recalling
that $|\varphi_{\delta,n}|\leq C(1+V^2)$ and that $\int |D \varphi_{\delta,n}|_w^2 \,\d\eta_\delta$ is uniformly bounded,
we are in position to apply Lemma~\ref{lem:WeakConvK} and to conclude that \eqref{reduct:convEdn} holds.
Therefore we proved the convergence of all terms in \eqref{eq:EntEstAppr}, so that \eqref{eq:EntEstdelta} holds.
\noindent
{\bf Passage to the limit as $\delta\downarrow 0$.} The inequality \eqref{eq:EntEstdelta} passes to the limit as $\delta\down 0$: more precisely,
we claim the existence of a Kantorovich potential $\varphi$ from $f{\mbox{\boldmath$m$}}$ to $\sigma$ such that
\begin{equation}\label{eq:EntEstdel}
\entv(\sigma)-\entv(\eta)- \frac{K}{2} W_2^2(\eta,\sigma)\geq -\mathcal E_\eta(\varphi,\log f).
\end{equation}
As in the passage to the limit as $n\to\infty$, Lemma~\ref{lem:ConvEnt} easily implies that
$\entv(\eta_{\delta})\to\entv(\eta)$,
moreover it is easy to check that $\eta_\delta$ weakly converge to $\eta$ and have uniformly integrable 2-moments, so
\cite[Proposition~7.1.5]{Ambrosio-Gigli-Savare08} gives
$W_2(\eta_\delta,\sigma)\to W_2(\eta,\sigma)$.
In order to show the convergence of the right hand side of \eqref{eq:EntEstdel} we first prove that
\begin{equation}\label{eq:EdE}
\lim_{\delta\down 0}|\mathcal E_{\eta_\delta}(\varphi_\delta, \log f_\delta)-c_\delta\mathcal E_{\eta} (\varphi_\delta, \log f)|=0.
\end{equation}
First of all notice that, after calling $A_\delta:=\{x\in X: f(x)\geq \delta{\mathrm e}^{-2c V^2(x)} \}$, we have $f_\delta=c_\delta f$
on $A_\delta$. Since $X\setminus A_\delta\downarrow \{f=0\}$ as $\delta\downarrow 0$ and $\weakgrad{f}=0$
${\mbox{\boldmath$m$}}$-a.e. on $\{f=0\}$, we can use \eqref{eq:essential}
of Lemma~\ref{lem:essential} to show \eqref{eq:EdE}, taking \eqref{eq:equiBoundF} into account.
Now that \eqref{eq:EdE} is proved, taking into account that $c_\delta\to 1$ as $\delta\downarrow 0$,
it is enough to prove the existence of a Kantorovich potential $\varphi$ from $\eta$ to $\sigma$ such that
\begin{equation}\label{eq:QF}
\lim_{i\to\infty} \mathcal E_\eta (\varphi_{\delta_i}, \log f)=\mathcal E_\eta (\varphi, \log f).
\end{equation}
for some sequence $\delta_i\downarrow 0$.
Recall that $\varphi_\delta$ were constructed using Lemma~\ref{lem:GammaConvKant}, so they still satisfy the growth condition \eqref{growthPhidn};
applying again Lemma~\ref{lem:GammaConvKant} we get the existence of a Kantorovich potential $\varphi$ from $\eta$ to $\sigma$ and
$\delta_i\downarrow 0$ such that
$\varphi_{\delta_i}\to \varphi$ pointwise in $X$ as $i \to \infty$. Moreover, by \eqref{eq:itforza}
and $f\leq c^{-1}_\delta f_\delta\leq 2f_\delta$
for $\delta$ small enough, we have
$$\int_X \weakgrad{\varphi_{\delta_i}}^2\; f\,\d {\mbox{\boldmath$m$}} \leq 2\int_X \weakgrad{\varphi_{\delta_i}}^2 f_{\delta_i}\, \d {\mbox{\boldmath$m$}} \leq 2
W^2_2(\eta_{\delta_i},\sigma),$$
for $i$ large enough. Hence we can apply Lemma~\ref{lem:WeakConvK} and conclude that \eqref{eq:QF} holds.
Therefore \eqref{eq:EntEstdel} is proved and the proof of Theorem~\ref{Thm:DerEntr} is then complete.
\hfill$\Box$
\subsection{Proof of Theorem~\ref{thm:main1}.}
The implications from (i) to (ii) and from (iii) to (i) can be proven exactly
as in Theorem~5.1 of \cite{Ambrosio-Gigli-Savare11b} (as these proofs need no finiteness assumption on ${\mbox{\boldmath$m$}}$),
so let us focus on the implication from (ii) to (iii). Note that Sturm has proven in \cite{Sturm06I} (see Remark~4.6(iii) therein)
that $\supp{\mbox{\boldmath$m$}}$ is a length space for all $CD(K,\infty)$ spaces $(X,{\sf d},{\mbox{\boldmath$m$}})$ (his proof, based on an approximate
midpoint construction, does not use the local compactness).
It remains to show that the $EVI_K$-condition holds assuming
the $CD(K,\infty)$ condition and the fact that $\mathbb{C}$ is quadratic.
By the contractivity properties of $EVI_K$-gradient flows stated in Proposition~\ref{prop:evipropr} it is sufficient to
show that $\mu_t:=({\sf h}_tf){\mbox{\boldmath$m$}}$ is an $EVI_K$ gradient flow for $\entv$
for any initial measure $f{\mbox{\boldmath$m$}}\in\ProbabilitiesTwo{X}$ whose density $f$ is bounded and
satisfies $\mathbb{C}(\sqrt{f})<\infty$. By the maximum principle
proven in \cite{Ambrosio-Gigli-Savare11} (see Theorem~4.20 therein) one has
${\sf h}_tf\leq\|f\|_{L^\infty(X,{\mbox{\boldmath$m$}})}$ ${\mbox{\boldmath$m$}}$-a.e. in $X$ for all $t\geq 0$, furthermore
$\{\mu_t:\ t\in [0,T]\}$ is a bounded subset of $\ProbabilitiesTwo{X}$ for all $T>0$
and \eqref{eq:edissrateflow} gives
\begin{equation}
\int_0^\infty \mathbb{C}(\sqrt{{\sf h}_t f})\,{\d t}<\infty.
\end{equation}
By a simple density argument on the class of ``test'' measures $\sigma$ in \eqref{eq:EVI}
(see for instance \cite[Proposition~2.20]{Ambrosio-Gigli-Savare11b}), we can restrict ourselves to measures
$\sigma$ of the form $g{\mbox{\boldmath$m$}}$ with $g\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $\supp\sigma$ compact.
By \eqref{eq:derw2} of Theorem~\ref{thm:derw2} we get that for a.e. $t>0$, for any choice
of a Kantorovich potential $\varphi_t$ from $\mu_t$ to $\sigma$ whose slope has linear growth,
one has
\begin{equation}
\label{eq:derw22} \frac{\d}{\d t}\frac12W_2^2(\mu_t,\sigma)=
-\mathcal E_{\mu_t}(\varphi_t,\log {\sf h}_tf).
\end{equation}
Therefore, to conclude that \eqref{eq:EVI} holds, it suffices to show for a.e. $t>0$ the existence of
a Kantorovich potential $\varphi_t$ from $\mu_t$ to $\sigma$ whose slope has linear growth and
satisfies
\begin{equation}
-\mathcal E_{\mu_t}(\varphi_t,\log {\sf h}_t f)\leq \entv(\sigma)-\entv(\mu_t)- \frac{K}{2} W_2^2(\mu_t,\sigma).
\end{equation}
This is precisely the statement of Theorem~\ref{Thm:DerEntr} (with $\eta=\mu_t$)
and this concludes the proof. \hfill$\Box$
\section{Properties of $RCD(K,\infty)$ spaces}\label{sec:last}
In this section we state without proof some properties of $RCD(K,\infty)$ spaces whose proofs, given by the first two authors
and Savar\'e in \cite{Ambrosio-Gigli-Savare11b}. Their proofs do not rely on the finiteness assumption of ${\mbox{\boldmath$m$}}$. Refer to \cite{Ambrosio-Gigli-Savare11b}
for details of proofs and a more complete discussion.
\subsection{The heat semigroup and its regularizing properties}
In this section we describe more in detail the properties of the
$L^2$-semigroup ${\sf h}_t$ in a $RCD(K,\infty)$ space
and the additional information that one can
obtain from the identification with
$W_2$-semigroup ${\mathscr H}_t$.
By the definition of $RCD(K,\infty)$ spaces, we know that for any $x\in X$ there
exists a unique $EVI_K$ gradient flow ${\mathscr H}_t(\delta_x)$ of
$\entv$ starting from $\delta_x$, related to ${\sf h}_t$ by
\begin{equation}\label{eq:heatlheatw}
({\sf h}_t f){\mbox{\boldmath$m$}}=\int f(x)\,{\mathscr H}_t(\delta_x)\,\d{\mbox{\boldmath$m$}}(x)
\qquad\forall f\in L^2(X,{\mbox{\boldmath$m$}}).
\end{equation}
Since $\entv(\ke xt)<\infty$ for any $t>0$, one has $\ke
xt\ll{\mbox{\boldmath$m$}}$, so that $\ke xt$ has a density, that we shall denote by
$\ked xt$. The functions $\ked xt (y)$ are the so-called transition
probabilities of the semigroup. By standard measurable selection
arguments we can choose versions of these densities in such a way
that the map $(x,y)\mapsto\ked xt(y)$ is ${\mbox{\boldmath$m$}}\times{\mbox{\boldmath$m$}}$-measurable
for all $t>0$.
In the next theorem we prove additional properties of the flows. The
information on both benefits of the identification theorem: for
instance the symmetry property of transition probabilities is not at
all obvious when looking at ${\mathscr H}_t$ only from the optimal
transport point of view, and heavily relies on
\eqref{eq:heatlheatw}. On the other hand, the regularizing
properties of ${\sf h}_t$ are deduced by duality by those of
${\mathscr H}_t$, using in particular the contractivity estimate
\begin{equation}
\label{eq:contrw2} W_2({\mathscr H}_t(\mu),{\mathscr H}_t(\nu))\leq
e^{-Kt}W_2(\mu,\nu)\qquad t\geq0,\ \mu,\,\nu\in\ProbabilitiesTwo {X,{\mbox{\boldmath$m$}}}
\end{equation}
and the regularization estimates for the Entropy and its slope
\begin{equation}
\label{eq:24}
\mathrm I_K(t)\ent{{\mathscr H}_t(\mu)}+\frac {(\mathrm I_K(t))^2}2|D^- \entv|^2({\mathscr H}_t(\mu))
\le \frac 12 W_2^2(\mu,{\mbox{\boldmath$m$}})
\end{equation}
which are typical of $EVI_K$-solutions, with $\mathrm I_K(t):=\int_0^t {\mathrm e}^{K r}\,\d r$.
Notice also that \eqref{eq:contrw2} yields $
W_1({\mathscr H}_t(\delta_x),{\mathscr H}_t(\delta_y))\le{\mathrm e}^{-Kt}{\sf d}(x,y)$
for all $x,\,y\in X$ and $t\geq 0$. This implies that $RCD(K,\infty)$
spaces have Ricci curvature bounded from below by $K$
according to the $W_1$-contractivity property taken as definition in Ollivier \cite{Ollivier09} and Joulin \cite{Joulin07}.
\begin{theorem}[Regularizing properties of the heat flow]\label{thm:facili}
(Theorem~6.1 in {\rm \cite{Ambrosio-Gigli-Savare11b}}) Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ be a $RCD(K,\infty)$ space. Then:
\begin{itemize}
\item[(i)] The transition probability densities are symmetric
\begin{equation}\label{eq:symmetrictp}
\ked xt (y)=\ked yt (x)\qquad\text{${\mbox{\boldmath$m$}}\times{\mbox{\boldmath$m$}}$-a.e.~in $X\times
X$, for all $t>0,$}
\end{equation}
and satisfy for all $x\in X$ the Chapman-Kolmogorov formula:
\begin{equation}
\label{eq:chapman} \ked x{t+s}(y)=\int\ked xt (z)\ked zs
(y)\,\d{\mbox{\boldmath$m$}}(z)\qquad\text{for ${\mbox{\boldmath$m$}}$-a.e.~$y\in X$, for all $t,\,s\geq
0$.}
\end{equation}
\item[(ii)] The formula
\begin{equation}
\label{eq:l1} \tilde{\sf h}_tf(x):=\int f(y)\,\d\ke xt(y)\qquad
x\in X
\end{equation}
provides a version of ${\sf h}_t f$ for every $f\in L^2(X,{\mbox{\boldmath$m$}})$, an
extension of ${\sf h}_t$ to a continuous contraction semigroup in
$L^1(X,{\mbox{\boldmath$m$}})$ which is pointwise everywhere defined if $f\in
L^\infty(X,{\mbox{\boldmath$m$}})$.
\item[(iii)] The semigroup $\tilde{\sf h}_t$ maps contractively $L^\infty(X,{\mbox{\boldmath$m$}})$ in
$C_b(X)$ and, in addition, $\tilde{\sf h}_tf(x)$ belongs to $
C_b\bigl((0,\infty)\times X\bigr)$.
\item[(iv)] If $f:X\to\mathbb{R}$ is Lipschitz, then $\tilde{\sf h}_tf$ is Lipschitz on $X$
as well and $\Lip(\tilde{\sf h}_tf)\leq e^{-Kt}\Lip(f)$.
\end{itemize}
\end{theorem}
\begin{theorem}[Bakry-\'Emery in $RCD(K,\infty)$ spaces]\label{thm:bakryemery}
(Theorem~6.2 in {\rm \cite{Ambrosio-Gigli-Savare11b}}) For any $f\in L^2(X,{\mbox{\boldmath$m$}})\cap{\mathcal S}^2$ and $t>0$ we have
\begin{equation}\label{eq:bakryemery}
\weakgrad{({\sf h}_tf)}^2\leq
{\mathrm e}^{-2Kt}{\sf h}_t(\weakgrad{f}^2)\qquad\text{${\mbox{\boldmath$m$}}$-a.e.~in $X$.}
\end{equation}
In addition, if
$\weakgrad{f}\in L^\infty(X,{\mbox{\boldmath$m$}})$ and $t>0$, then
${\mathrm e}^{-Kt}\bigl(\tilde{\sf h}_t\weakgrad{f}^2\bigr)^{1/2}$ is an
upper gradient of $\tilde{\sf h}_tf$ on $X$, so that
\begin{equation}
\text{$|D^- \tilde{\sf h}_tf|\leq{\mathrm e}^{-Kt}
\bigl(\tilde{\sf h}_t\weakgrad{f}^2\bigr)^{1/2}$\quad pointwise in
$X$,} \label{eq:17}
\end{equation}
and $f$ has a Lipschitz version $\tilde{f}:X\to\mathbb{R}$,
with ${\rm Lip}(\tilde{f})\leq\|\weakgrad{f}\|_\infty$.
\end{theorem}
The regularization properties \eqref{eq:24} of $EVI_K$-flows provide
an $L\log L$ regularization of the semigroup ${\mathscr H}_t$
starting from arbitrary measures in $\ProbabilitiesTwo{X}$.
When $X$ is a $RCD(K,\infty)$-space with $K>0$, then
combining the slope inequality for $K$-geodesically convex functionals
\cite[Lemma 2.4.13]{Ambrosio-Gigli-Savare08}
\begin{displaymath}
\ent\mu\le \frac 1{2K}|D^-\entv|^2(\mu)
\end{displaymath}
with the identity $|D^-\entv|^2(f{\mbox{\boldmath$m$}})=\int\weakgrad{f}^2/f\,\d{\mbox{\boldmath$m$}}$ between slope and Fisher information,
we get
the Logarithmic-Sobolev inequality
\begin{equation}
\label{eq:+logsob}
\int_X f\log f\,\d{\mbox{\boldmath$m$}} \le \frac 1{2K} \int_{f>0} \frac{\weakgrad f^2}f\,\d{\mbox{\boldmath$m$}}
\quad\text{if }\sqrt f\in W^{1,2}(X,{\sf d},{\mbox{\boldmath$m$}}),\ f{\mbox{\boldmath$m$}}\in \Probabilities X,
\end{equation}
which in particular yields the hypercontractivity of ${\sf h}_t$,
see e.g.\ \cite{Aneetal00}. When ${\sf h}_t$ is ultracontractive, i.e.\
there exists $p>1$ such that
\begin{equation}\label{eq:stronger}
\|{\sf h}_t f\|_p \leq C(t)\|f\|_1\qquad\text{for every } f\in
L^2(X,{\mbox{\boldmath$m$}}),\,\,t>0,
\end{equation}
then one can also obtain global Lipschitz regularity
for the transition probabilities \cite[Proposition~6.4]{Ambrosio-Gigli-Savare11b}, see also \cite[Proposition~4.4]{GigliKuwadaOhta10}.
The stronger regularizing property \eqref{eq:stronger} is known to
be true, for instance, if doubling and Poincar\'e hold in
$(X,{\sf d},{\mbox{\boldmath$m$}})$, see \cite[Corollary~4.2]{Sturm96}.
We conclude this section with an example of application
of the Bakry-\'Emery estimate \eqref{thm:bakryemery}, which can be proven following
the $\Gamma$-calculus tools of Bakry \cite{Bakry06}, see
Theorem~6.5 in {\rm \cite{Ambrosio-Gigli-Savare11b}} for a detailed proof.
\begin{theorem}[Lipschitz regularization]\label{thm:lipreg}
If $f\in L^2(X,{\mbox{\boldmath$m$}})$ then ${\sf h}_t f\in {\mathcal S}^2$ for
every $t>0$ and
\begin{equation}
\label{eq:19}
2\, \mathrm I_{2K}(t)\weakgrad {{\sf h}_t f }^2\le {\sf h}_t
f^2\quad{\mbox{\boldmath$m$}}\text{-a.e.\ in $X$};
\end{equation}
in particular,
if $f\in L^\infty(X,{\mbox{\boldmath$m$}})$ then $\tilde{\sf h}_t f\in \Lip(X)$ for every
$t>0$ with
\begin{equation}
\label{eq:28}
\sqrt{2\,\mathrm I_{2K}(t)}\,{\rm Lip}(\tilde{\sf h}_t f)\le
\|f\|_\infty\quad\text{for every } t>0.
\end{equation}
\end{theorem}
\subsection{Connections with Dirichlet forms and Markov processes}
Since $\mathbb{C}$ is quadratic, lower semicontinuous in $L^2(X,{\mbox{\boldmath$m$}})$ and since
$\weakgrad{f}$ has strong locality properties, it turns out that the bilinear form
$\mathcal E$ associated to $\mathbb{C}$, whose domain is from now on restricted from $L^1(X,{\mbox{\boldmath$m$}})\cap{\mathcal S}^2$ to
$L^2(X,{\mbox{\boldmath$m$}})\cap{\mathcal S}^2$, is a local Dirichlet form. In the theory of
Dirichlet forms a canonical object is the induced distance, namely
\begin{equation}\label{eq:fuku2}
{\sf d}_{{\mathcal E}}(x,y):=\sup\left\{|\tilde g(x)-\tilde g(y)|:\
g\in D({\mathcal E}),\,\,[g]\leq{\mbox{\boldmath$m$}}\right\}\qquad\forall (x,y)\in
X\times X,
\end{equation}
where the function $\tilde{g}$ is the continuous representative in the
Lebesgue class of $g$, see Theorem~\ref{thm:bakryemery}). Another canonical object
is the local energy measure, namely the measure $[u]$ defined by
$$
[u](\varphi):=\mathcal E(u,u\varphi)-\frac{1}{2}\mathcal E(u^2,\varphi)
\qquad\varphi\in L^2(X,{\mbox{\boldmath$m$}})\cap{\mathcal S}^2.
$$
A consequence of Lemma~\ref{lem:vabbeserve} is that $[u]=\weakgrad{u}^2{\mbox{\boldmath$m$}}$ for all $u\in L^2(X,{\mbox{\boldmath$m$}})\cap{\mathcal S}^2$.
Also the distances can be identified:
\begin{theorem}[Identification of ${\sf d}_{{\mathcal E}}$ and ${\sf d}$]\label{thm:idistances}
(Theorem~6.10 of {\rm \cite{Ambrosio-Gigli-Savare11b}}) The function ${\sf d}_{{\mathcal E}}$ in \eqref{eq:fuku2} coincides
with ${\sf d}$ on $X \times X$.
\end{theorem}
Finally, using a tightness property of $\mathcal E$, the theory of Dirichlet forms can be applied
to obtain the representation of transition probabilities in terms of a continuous Markov process:
\begin{theorem}[Brownian motion]\label{thm:brownian}
(Theorem~6.8 of {\rm \cite{Ambrosio-Gigli-Savare11b}}) Let $(X,{\sf d},{\mbox{\boldmath$m$}})$ be a $RCD(K,\infty)$ space. There exists a unique
(in law) Markov process $\{{\mathbf X}_t\}_{\{t\geq 0\}}$ in
$(X,{\sf d})$ with continuous sample paths in $[0,\infty)$ and
transition probabilities ${\mathscr H}_t(\delta_x)$, i.e.
\begin{equation}\label{eq:transitionmp}
{\mathbf P}\bigl({\mathbf X}_{s+t}\in A\bigl|{\mathbf
X}_s=x\bigr)=\ke xt(A) \qquad\forall s,\,t\geq 0,\,\,\text{$A$
Borel}
\end{equation}
for ${\mbox{\boldmath$m$}}$-a.e. $x\in X$.
\end{theorem}
\subsection{Tensorization}
Recall that a metric space $(X,{\sf d})$ is said to be non branching if the map $(\e_0,\e_t):\geo(X)\to X^2$
is injective for all $t\in (0,1)$, i.e., geodesics do not split.
\begin{theorem}[Tensorization]\label{thm:tensor}
(Theorem~6.13 of {\rm \cite{Ambrosio-Gigli-Savare11b}}) Let $(X,{\sf d}_X,{\mbox{\boldmath$m$}}_X)$, $(Y,{\sf d}_Y,{\mbox{\boldmath$m$}}_Y)$ be metric measure spaces and define the
product space $(Z,{\sf d},{\mbox{\boldmath$m$}})$ as $Z:=X\times Y$,
${\mbox{\boldmath$m$}}:={\mbox{\boldmath$m$}}_X\times{\mbox{\boldmath$m$}}_Y$ and
\[
{\sf d}\big((x,y),(x',y')\big):=\sqrt{{\sf d}_X^2(x,x')+{\sf d}_Y^2(y,y')}.
\]
Assume that both $(X,{\sf d}_X,{\mbox{\boldmath$m$}}_X)$ and $(Y,{\sf d}_Y,{\mbox{\boldmath$m$}}_Y)$ are $RCD(K,\infty)$ and non branching.
Then $(Z,{\sf d},{\mbox{\boldmath$m$}})$ is $RCD(K,\infty)$ and
non branching as well.
\end{theorem}
In \cite{AGSBaEm} the first two authors in collaboration with Savar\'e proved that the tensorization property of $RCD(K,\infty)$
persists even when the non branching assumption on the base spaces is removed.
\def$'${$'$}
|
1,314,259,995,797 | arxiv | \section{Introduction\label{s:intro}}
During the last decades, the number of atomic and molecular species detected in the interstellar medium (ISM) has increased considerably thanks to (i) the improved sensitivity of facilities like the IRAM\,30m telescope in Spain or the Atacama Large Millimeter/submillimeter array (ALMA) in Chile, and (ii) new laboratory measurements of transitions of new species included in catalogues such as the Cologne Database for Molecular Spectroscopy (CDMS). Almost 200 different species have been found in Galactic/extragalactic environments such as cold dense cores, hot molecular cores, circumstellar disks, evolved stars or large diffuse molecular clouds. These 200 species do not consist only of simple molecules like the most abundant H$_2$ and CO, but also include complex species usually defined as molecules with 6 or more atoms (\citealt{HvD2009}; see the CDMS database\footnote{http://www.astro.uni-koeln.de/cdms/molecules} for a summary of detected species in space).
Molecular hydrogen, H$_2$, is by far the most abundant molecule in the Universe, followed by carbon monoxide, CO. Therefore, the intermolecular forces between these two species are of fundamental interest. If the CO-H$_2$ van der Waals complex\footnote{We note that this complex does not correspond to the formaldehyde molecule, H$_2$CO.} exists in measurable amounts in the ISM, it could be a sensitive indicator for low temperatures. The binding energy of the complex is so small --- typically 20~cm$^{-1}$ or 30~K --- that the relative abundance of the complex in the gas phase is expected to increase at lower temperatures.
There is an open debate about the feasibility of observing such weakly-bound species because their formation rates at the very low densities of interstellar molecular clouds (below 10$^{7}$~cm$^{-3}$) are low, due to the small probability of three-body collisions, which is the main formation route of van der Waals complexes in the laboratory. On the other hand, the large timescale on which these processes occur in interstellar space makes radiative association, which is usually a slow process, quite feasible (e.g.\ \citealt{Klem06}). Also non-equilibrium conditions in the ISM may strongly favor the formation and concentration of the CO-H$_2$ complex over time on the surfaces of dust grains in shielded regions at low temperatures, with release to the gas-phase occurring by localized heating processes such as turbulence or jets/outflows (e.g.\ \citealt{Allen97}). However, one also has to consider that CO tends to be frozen out onto dust grains in very cold, dense regions, and it seems difficult to release the CO-H$_2$ complex from grains without destroying it. On the other hand, this is completely unchartered territory, and even sensitive upper limits are useful. A detection of this complex would challenge many beliefs we have about the chemistry of dense molecular clouds.
There have been several attempts to observe complexes containing CO and H$_2$ molecules. The detection of the H$_2$ dimer, (H$_2$)$_2$, in the atmosphere of Jupiter has been reported by \citet{Mck88}, while searches for the CO dimer, (CO)$_2$ \citep{Van79}, and the CO-paraH$_2$ complex \citep{Allen97} in Galactic molecular clouds were not successful thus far. Laboratory data have clarified later a spectroscopic problem of these unsuccessful searches. The extensive millimeter-wave (MMW) studies of the CO dimer \citep{Su07} have shown that the radio astronomical search of this complex was based on frequencies which cannot be unambiguously attributed to (CO)$_2$. In the case of the CO-paraH$_2$ complex, the interstellar search was outside the correct frequency position of the most promising R(0) line, as later identified by the first MMW study of CO-paraH$_2$ \citep{Pak99}, and only the weaker Q(1) line was correctly tuned.
Recent laboratory studies of the CO-H$_2$ complex have provided precise MMW frequencies with uncertainties of about 50~kHz for the complex in different spin modifications and for its deuterated isotopologues: CO-paraH$_2$ \citep{Po09ApJ}, CO-orthoH$_2$ \citep{Jan13}, CO-orthoD$_2$ \citep{Po09OptSp} and CO-HD \citep{Po15}. Therefore, the availability of precise rest frequencies and modern astronomical receivers (with a sensitivity several times better than the old receivers used 20 years ago), combine in a great opportunity to detect for the first time a van der Waals complex in the ISM.
In this paper we present IRAM\,30m observations of a cold region in the Taurus molecular cloud in the search for the CO-H$_2$ van der Waals complex. In Sect.~\ref{s:obs} we describe the observations. In Sect.~\ref{s:res} we present the main results. Unfortunately, we do not have a detection of the CO-H$_2$ complex but we can set a new limit that can be used in future chemical modelling. In addition to the search for the CO-H$_2$ complex, the IRAM\,30m observations allowed us to conduct a spectral line survey of a very cold region ($\sim 10$~K), and we report the detection of complex organic molecules (COMs) as well as first time tentative detections of species in this object. In Sect.~\ref{s:disc} we discuss our results, and we end the paper with a summary of the main results in Sect.~\ref{s:summary}.
\begin{figure}[t]
\begin{center}
\begin{tabular}[b]{c}
\includegraphics[width=0.9\columnwidth]{CO-paraH2.pdf} \\
\includegraphics[width=0.9\columnwidth]{CO-orthoH2.pdf} \\
\end{tabular}
\end{center}
\caption{Energy level diagram for the CO-paraH$_2$ (top panel) and CO-orthoH$_2$ (bottom panel) van der Waals complex. \textit{Top panel}: The energy levels are labeled by quantum numbers $J$, $j_\mathrm{CO}$ and $l$ where $J$ is the total angular momentum, $j_\mathrm{CO}$ refers to the rotation of the CO sub-unit and $l$ refers to the end-over-end rotation of the complex. $K$ corresponds to the projection of $J$ onto the intermolecular axis. The labels $e$ and $f$ indicate the parity of the $J$ levels within a given stack. The parity of an even-$J$ level is `$+$' for stacks labeled by $e$ and `$-$' for $f$, while the parity of an odd-$J$ level is `$-$' for $e$ and `$+$' for stacks labeled $f$. The insert shows the approximate geometrical structure of the CO-H$_2$ complex (see \citealt{Po09ApJ} for details). \textit{Bottom panel}: The energy levels are labeled by quantum numbers $J$, parity $P$ and $n_J$,$P$, a consecutive number of the state for the given values of $J$ and $P$ (see \citealt{Jan13} for details). In both panels, the targeted transitions are indicated by arrows.}
\label{f:energylevels}
\end{figure}
\begin{table}
\caption{Frequencies of the brightest CO-H$_2$ targeted lines}
\label{t:transitions}
\centering
\begin{tabular}{l c c}
\hline\hline
&Transition
&Frequency (MHz)
\\
\hline
CO-paraH$_2$ &(1,1,0)--(0,0,0) & 108480.857 \\
&(1,1,1)--(1,0,1) & \phn91012.364 \\
CO-orthoH$_2$ &(2,f,2)--(1,f,1) & 101907.919 \\
&(3,e,2)--(2,e,1) & \phn93433.726 \\
CO-orthoD$_2$ &(1,1,0)--(0,0,0) & 102791.612 \\
&(1,1,1)--(1,0,1) & \phn89483.510 \\
CO-HD &(1,1,0)--(0,0,0) & 105636.\phnnn \\
\hline
\end{tabular}
\tablefoot{The uncertainty in the frequency measurements is about 50~kHz for all the transitions, except for CO-HD with a few tens of MHz. The labelling of the quantum numbers of the transitions are explained in detail in Fig.~\ref{f:energylevels}.}
\end{table}
\begin{figure*}[h!]
\begin{center}
\begin{tabular}[b]{c}
\includegraphics[width=0.95\textwidth]{plot_COH2.pdf} \\
\end{tabular}
\caption{\textit{Top panels}: Full spectrum observed with the IRAM\,30m telescope towards the cold dense core in TMC-1C. The mean rms noise level is $\sim 2$~mK. Most of the detected lines emit only in one channel (channel width 0.5--0.7~km~s$^{-1}$), suggesting that the linewidth of the different lines is $\le 0.7$~km~s$^{-1}$ (see Sect.~\ref{s:molecules}). \textit{Bottom panels}: Close-up views of the frequency ranges around the brightest transitions of the CO-H$_2$ van der Waals complex. The corresponding frequencies are listed in Table~\ref{t:transitions}, and are shown in the panels with a vertical dotted line. The expected linewidth is $\approx0.3$~km~s$^{-1}$, as measured in higher spectral resolution observations (e.g.\ \citealt{Spez13}).}
\label{f:specCOH2}
\end{center}
\end{figure*}
\section{Observations\label{s:obs}}
The observations were carried out from 2015 May 6 until May 9 at the IRAM\,30m telescope, located in Pico Veleta (Granada, Spain) under the project code 131-14. We have chosen to attempt the detection of the CO-H$_2$ complex in the nearest star forming region: the Taurus molecular cloud complex (e.g.\ \citealt{Ola88, Suz92, Rob00}), in particular towards a cold, dense condensation nearby TMC-1C which has measured low excitation temperatures of 3--7~K \citep{Spez13}, and for which a kinetic temperature of 10~K reproduce the observations presented by \citet{Spez16}. This object harbours the physical conditions (low temperature and high density, $\sim 4\times10^{4}$~cm$^{-3}$; \citealt{Schnee2007}) necessary to search for the CO-H$_2$ complex. We note that the density is still low enough to not have all the CO frozen out onto the dust grains\footnote{Referring to the work of \citet{Caselli1999}, a model in which CO is condensed out onto dust grains at densities above $10^5$~cm$^{-3}$ and has a roughly canonical abundance at lower hydrogen densities, is supported by the observations of gas-phase depletion in the L1544 cloud core.}. The coordinates used for the observations are $\alpha_{2000}$ = 04$^\mathrm{h}$41$^\mathrm{m}$16.$^\mathrm{s}$1 and $\delta_{2000}$ = +25\degr49\arcmin43\farcs8, coincident with the coordinates used in \citet{Spez13}.
We tuned the telescope to cover a number of transitions of the CO-H$_2$ complex and its deuterated isotopologues in the 3~mm band (E090) of the EMIR receiver. All four EMIR sub-bands were connected to the Fast Fourier Transform Spectrometers (FTS), with a spectral resolution of 200~kHz, which results in $\sim 0.5$--$0.7$~km~s$^{-1}$ at the corresponding frequencies. In Table~\ref{t:transitions}, we list the most intense transitions of the complex covered in our spectral setup. The frequency coverage was selected in order to optimize the simultaneous search of the strongest CO-paraH$_2$ and CO-orthoH$_2$ lines. The energy level diagrams for CO-paraH$_2$ and CO-orthoH$_2$ are shown in Fig.~\ref{f:energylevels}. In total, our observations cover an effective bandwidth of 16~GHz, ranging from 85.87 to 93.65~GHz in the lower sideband, and from 101.55 to 109.33~GHz in the upper sideband. The total observing time was 20~hours. We used the ON-OFF observation mode, with the reference position located at the offset (800\arcsec, 600\arcsec). The telescope pointing was checked every 1.5~hours and was found to be accurate to $\sim 5$\arcsec, i.e.\ only a fraction of the beam size of the telescope at these frequencies: $\sim 30$\arcsec. The weather conditions were stable during the observations with zenith opacities of 0.02--0.07 and system temperatures of 80--100~K. The observed spectra was calibrated following the standard procedures, and analyzed using the GILDAS\footnote{The GILDAS software package is developed by the IRAM and Observatoire de Grenoble, and can be downloaded at http://www.iram.fr/IRAMFR/GILDAS} software package. We converted the spectra to the main beam temperature scale, using a forward efficiency of 0.95, and a beam efficiency of 0.79 and 0.81 for the upper and the lower sidebands, respectively. The final spectrum has a noise level of $\sim 2$~mK.
\section{Results and analysis\label{s:res}}
\subsection{The CO-H$_2$ complex\label{s:coh2}}
In Fig.~\ref{f:specCOH2} we show, in the top panels, the full spectrum obtained with the IRAM\,30m telescope. A number of bright lines have been detected throughout the covered frequency range and they will be discussed in Sect.~\ref{s:molecules}. The bottom panels of Fig.~\ref{f:specCOH2} show a close-up view of the frequency ranges around the frequencies of the brightest CO-H$_2$ lines, corresponding to the CO-orthoH$_2$ and CO-paraH$_2$ transitions listed in Table~\ref{t:transitions}. No lines belonging to the CO-H$_2$ complex are detected at the corresponding frequencies (indicated in the figure with red dotted vertical lines). It is important to note that the noise at the high-frequency transitions is slightly larger than the average one (i.e.\ 2~mK). This larger noise is due to ripples in the baseline that were not possible to completely remove. However, since their wavelength is much larger than the expected linewidths, $\sim 0.3$~km~s$^{-1}$, they do not affect the search for the CO-H$_2$ transitions. None of the four transitions have been detected, and we therefore, set an upper limit of $\sim 6$~mK (corresponding to 3$\sigma$) for the intensities of these lines.
\begin{table}
\caption{Species, temperatures, column densities and transitions detected above $5\sigma$ towards TMC-1C (see Sect.~\ref{s:molecules} for details)}
\label{t:molecules}
\centering
\begin{tabular}{l c c c}
\hline\hline
Species
&$T$ (K)
&log($N$) (log[cm$^{-2}$])
&Transitions
\\
\hline
$^{13}$C$^{18}$O &7.0 &$13.30\pm0.50$ &1 \\
$^{13}$CS &7.0 &$11.65\pm0.26$ &1 \\
SO &$6.2\pm0.7$ &$13.25\pm0.28$ &2 \\
S$^{18}$O &6.2 &$11.81\pm0.26$ &1 \\
CCH &7.0 &$13.42\pm0.11$ &6 \\
CCS &$4.7\pm0.7$ &$12.95\pm0.38$ &4 \\
HCN &7.0 &$12.26\pm0.18$ &1 \\
H$^{13}$CN &7.0 &$11.51\pm0.29$ &1 \\
HC$^{15}$N &7.0 &$10.71\pm0.21$ &1 \\
HNC &7.0 &$12.92\pm1.01$ &1 \\
HN$^{13}$C &7.0 &$11.76\pm0.13$ &1 \\
H$^{15}$NC &7.0 &$11.51\pm0.30$ &1 \\
HCO &$6.1\pm1.8$ &$11.03\pm0.30$ &4 \\
HCO$^{+}$ &7.0 &$12.09\pm0.35$ &1 \\
H$^{13}$CO$^{+}$ &7.0 &$11.59\pm0.38$ &1 \\
HC$^{17}$O$^+$ &7.0 &\phn$9.97\pm1.50$ &1 \\
DCS$^{+}$ &7.0 &$10.88\pm0.24$ &1 \\
SO$_2$ &7.0 &$11.95\pm0.23$ &1 \\
N$_2$H$^+$ &7.0 &$12.42\pm0.19$ &1 \\
CCCS &$6.2\pm0.5$ &$12.67\pm0.31$ &2 \\
CCCO &7.0 &$11.26\pm0.23$ &1 \\
c-CCCH &$6.0\pm1.9$ &$12.14\pm0.35$ &6 \\
H$_2$CS &$14.8\pm2.1$ &$12.67\pm0.15$ &2 \\
HDCS &14.8 &$12.01\pm0.98$ &1 \\
H$_2$C$^{34}$S &14.8 &$11.43\pm1.59$ &1 \\
HOCO$^+$ &7.0 &$11.25\pm0.19$ &1 \\
HNCO &7.0 &$13.11\pm0.11$ &1 \\
NH$_2$D &7.0 &$12.59\pm1.00$ &1 \\
C$_4$H &$7.8\pm0.9$ &$13.12\pm0.26$ &2 \\
l-C$_3$H$_2$ &$7.2\pm0.7$ &$11.18\pm0.31$ &3 \\
c-C$_3$HD &$6.0\pm1.8$ &$12.14\pm0.35$ &4 \\
c-C$_3$D$_2$ &6.0 &$10.99\pm1.90$ &1 \\
CH$_2$CO &7.0 &$12.42\pm0.37$ &1 \\
HC$_3$N &$4.2\pm0.3$ &$14.62\pm0.45$ &2 \\
HCCNC &7.0 &$11.81\pm0.15$ &1 \\
HCOOH &7.0 &$11.64\pm0.13$ &1 \\
CH$_3$CN &$7.1\pm1.0$ &$11.80\pm0.45$ &2 \\
CH$_3$OH &7.0 &$12.86\pm1.27$ &1 \\
CH$_2$DOH &7.0 &$11.60\pm0.50$ &1 \\
CH$_3$CCH &$15.0\pm3.6$ &$13.31\pm0.13$ &3 \\
CH$_3$CCD &15.0 &$11.97\pm0.16$ &2 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}[t]
\begin{center}
\begin{tabular}[b]{c}
\includegraphics[width=0.8\textwidth]{spectra_01.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_02.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_03.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_04.pdf} \\
\end{tabular}
\end{center}
\caption{Spectral line survey towards TMC-1C. Each panel shows about 1~GHz of the total 16~GHz frequency band. The observed spectrum is shown in dark grey. Each identified transition is indicated with a blue dotted line and the name of the corresponding species. The green dotted lines correspond to ghost lines from the image sideband.}
\label{f:molecules}
\end{figure*}
\begin{figure*}[t]
\ContinuedFloat
\begin{center}
\begin{tabular}[b]{c}
\includegraphics[width=0.8\textwidth]{spectra_05.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_06.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_07.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_08.pdf} \\
\end{tabular}
\end{center}
\caption{Continued.}
\end{figure*}
\begin{figure*}[t]
\ContinuedFloat
\begin{center}
\begin{tabular}[b]{c}
\includegraphics[width=0.8\textwidth]{spectra_09.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_10.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_11.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_12.pdf} \\
\end{tabular}
\end{center}
\caption{Continued.}
\end{figure*}
\begin{figure*}[t]
\ContinuedFloat
\begin{center}
\begin{tabular}[b]{c}
\includegraphics[width=0.8\textwidth]{spectra_13.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_14.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_15.pdf} \\
\includegraphics[width=0.8\textwidth]{spectra_16.pdf} \\
\end{tabular}
\end{center}
\caption{Continued.}
\end{figure*}
\subsection{Spectral line survey\label{s:molecules}}
The broad frequency range covered with the IRAM\,30m telescope permits not only to study the CO-H$_2$ lines, but also to perform a spectral line survey of this cold dense condensation. It is worth mentioning that the high sensitivity achieved with our observations is adequate to search, e.g., for COMs (molecules with 6 or more atoms) in low-temperature environments. COMs have long been detected in the interstellar medium, especially in hot molecular cores associated with high-mass star forming regions (e.g.\ \citealt{Cum86, Bl87, SanchezMonge2013, SanchezMonge2014}). The advent of sensitive instruments has also revealed a chemical complexity associated with low-mass hot cores (or hot corinos; e.g.\ \citealt{Caz03}) and intermediate-mass hot cores (e.g.\ \citealt{SanchezMonge2010}). Despite their formation routes remain uncertain, both warm gas-phase and grain-surface reactions have been invoked to account for their presence in low-mass protostars. In this latter scheme, COMs result from radical-radical reactions on the grains as radicals become mobile when the nascent protostar warms up its surroundings and the resulting molecules are subsequently desorbed into the gas phase at higher temperatures or by shock events produced by winds/jets (e.g.\ \citealt{GnH06}). In the last years, the detection of COMs in cold environments ($T<30$~K; \citealt{Bac12, Vas14}) represents a challenge for the chemical models and an opportunity to improve and clarify the role of the grain-surface and gas-phase chemistry.
The top panels of Fig.~\ref{f:specCOH2} show the spectral line survey towards the condesantion in TMC-1C. We have identified 75 lines with an intensity $>5\sigma$. We note that the spectral resolution of only 0.5--0.7~km~s$^{-1}$ is not enough to resolve most of the lines, suggesting that all they are excited in an environment with a temperature $<30$~K, i.e\ in cold gas\footnote{Most of the lines are detected in one single channel. The exceptions are species such as HCN and N$_2$H$^+$ due to the hyperfine structure, and $^{13}$C$^{18}$O with a weak blue-shifted wing. The thermal linewidth for gas at 30~K is 0.23~km~s$^{-1}$, 0.18~km~s$^{-1}$ and 0.16~km~s$^{-1}$, for species with mean molecular weights of 25 (e.g.\ CCH), 40 (e.g.\ CH$_3$CCH) and 56 (e.g.\ CCS), respectively.}. The identification of the lines was done using the CDMS \citep{Mul05} and JPL \citep{Pic98} databases, and later on confirmed by creating synthetic spectra of each species using the XCLASS\footnote{The eXtended CASA Line Astronomy Software Suite (XCLASS) can be downloaded at https://www.astro.uni-koeln.de/projects/schilke/XCLASSInterface} software \citep{Moeller2015}. XCLASS is a toolbox for the Common Astronomy Software Applications (CASA) package containing new functions for modeling interferometric and single-dish data. The included myXCLASS program calculates synthetic spectra by solving the radiative transfer equation for an isothermal object in one dimension where the required molecular data are taken from an embedded database containing entries from the CDMS and JPL databases using the Virtual Atomic and Molecular Data Center (VAMDC) portal. The contribution of a molecule is defined by an user-defined number of components where each component is described by four main parameters: excitation temperature, column density, velocity width and velocity offset. In order to achieve a good description of the data we fit these parameters using the included model optimizer package MAGIX \citep{Moeller2013}. By performing an internal resampling, XCLASS makes sure that the line is sampled properly, even if the velocity resolution of the data is coarse.
In Table~\ref{t:molecules} we list the identified species with the corresponding excitation temperature (column~2), column density (column~3) and number of transitions above $5\sigma$ (column~4). The 75 detected lines come from 41 different species (including isotopologues) and four unidentified lines at the frequencies 90.593~GHz, 90.602~GHz, 92.872~GHz and 101.981~GHz with main beam temperatures of 18~mK, 18~mK, 30~mK and 15~mK, respectively. For those species with more than one transition we have fitted with XCLASS simultaneously the excitation temperature and column density\footnote{For species like C$_3$H$_2$ or CH$_3$OH, with different spin symmetries: ortho/para, the total column density is given.}. If only one transition is detected above $5\sigma$ for a given species, we fit only the column density and fix the excitation temperature to 7~K, which corresponds to the average temperature of the other transitions and is consistent with the value measured also in \citet{Spez13}. For the different detected isotopologues, we have fixed the excitation temperature to be the same of the main species, and we have fitted only the column density. In all cases we consider a linewidth of 0.3~km~s$^{-1}$. In Fig.~\ref{f:molecules} we present the whole spectral survey, indicating the location of the different detected transitions. A number of COMs have been detected in this cold dense core. We discuss the results in Sect.~\ref{s:moleculesDisc}.
\section{Discussion\label{s:disc}}
\subsection{Detectability of the CO-H$_2$ complex\label{s:coh2Disc}}
In Sect.~\ref{s:coh2} we report an upper limit of $\sim 6$~mK for the CO-H$_2$ lines. We follow the same approach as in \citet{Allen97} to establish an upper limit for the column density of the CO-paraH$_2$ complex. We use the dipole moment of CO and the total partition function of CO-paraH$_2$ calculated from the now known energy level scheme of the complex (see Fig.~\ref{f:energylevels}). Our calculations result in a value of $\sim 3\times10^{12}$~cm$^{-2}$ for the column density of the complex and a fractional abundance of the CO-paraH$_2$ dimer relative to CO of $\sim 5\times10^{-6}$, assuming the CO column density to be $6\times10^{17}$~cm$^{-2}$ (derived from the $^{13}$C$^{18}$O column density listed in Table~\ref{t:molecules}, and assuming standard $^{12}$C/$^{13}$C and $^{16}$O/$^{18}$O ratios of 60 and 500, respectively.
In the following, we estimate what number density of the CO-H$_2$ molecular complex do we expect under the ISM conditions, and compare it to the new upper limit. All the reaction rates used in the following are generic rates, which have not specifically been measured or calculated, and are taken from the review paper by \citet{vD14}. The given reactions are the basic types of reactions in space. Following \citet{vD14}, there are two basic processes by which molecular bonds can be formed in the interstellar molecular clouds: radiative association and formation on grain surface with subsequent release to the gas phase. In the radiative association process, the binding energy of a new molecule or molecular complex is carried out through the emission of a photon, and can be described as:
\begin{equation}
\mathrm{H}_2 + \mathrm{CO} \rightarrow \mathrm{CO}\mathrm{-}\mathrm{H}_2 + h\nu
\end{equation}
and proceeds at the rate of a radiative association reaction $k_1\approx10^{-17}$--$10^{-14}$~cm$^3$~s$^{-1}$. For the case of the formation on grain surfaces, a dust particle accommodates the released energy, and the process can be described as
\begin{equation}
\mathrm{H}_2 + \mathrm{CO}\mathrm{-}\mathrm{grain} \rightarrow \mathrm{CO}\mathrm{-}\mathrm{H}_2 + \mathrm{grain}\nonumber
\end{equation}
which proceeds at a rate of $k_2\approx10^{-17}$~cm$^3$~s$^{-1}$.
On the other side, there are three processes for the destruction of the complex: photodissociation, collisional dissociation and neutral-neutral bond rearrangement. The first one can be described by
\begin{equation}
\mathrm{CO}\mathrm{-}\mathrm{H}_2 + h\nu \rightarrow \mathrm{products}
\end{equation}
with a reaction rate of $k_3\approx10^{-10}$--$10^{-8}$~cm$^3$~s$^{-1}$. The second and third ones can be given by
\begin{equation}
\mathrm{CO}\mathrm{-}\mathrm{H}_2 + \mathrm{M} \rightarrow \mathrm{products}
\end{equation}
where M being a reaction partner, with rates for collisional dissociation of $k_4\approx10^{-26}$~cm$^3$~s$^{-1}$ and for bond rearrangement of $k_5\approx10^{-11}$--$10^{-9}$~cm$^3$~s$^{-1}$.
We consider a dense condensation with an H$_2$ density of [H$_2$] = $4\times10^{4}$~cm$^{-3}$, the CO density given by [CO] = [CO-grain] = 10$^{-4}$ [H$_2$], and assume [M] = [H$_2$]. Under these conditions, {the formation is dominated by radiative association, while the destruction mainly occurs by the bond rearrangement. As it is stated by \citet{vD14}, collisional dissociation of molecules is only important in regions of very high temperature ($>3000$~K) and density. Thus,} we determine the CO-H$_2$ abundance in the equilibrium as [CO-H$_2$] = ($k_1$[H$_2$][CO])/($k_5$[M]) = $4\times10^{-8}$--$4\times10^{-3}$~cm$^{-3}$ and [CO-H$_2$]/[CO] $\sim 10^{-8}$--$10^{-3}$. The obtained range for a possible abundance of CO-H$_2$ is quite wide. From the comparison of our estimated [CO-H$_2$]/[CO] abundance to the upper detection limit of [CO-H$_2$]/[CO]$\sim5\times10^{-6}$, we can conclude that the complex might be detected by observations with one or two orders higher sensitivity.
\subsection{Molecular inventory in cold regions\label{s:moleculesDisc}}
Table~\ref{t:molecules} and Figure~\ref{f:molecules} reveal a relatively rich chemistry in the cold dense core TMC-1C. Despite the average excitation temperature being of only 7~K, we are able to detect a number of species with 6 or more atoms: CH$_3$CN, CH$_3$OH and CH$_3$CCH. The column densities for these species are in the range $10^{11}$--$10^{13}$~cm$^{-2}$, which results in abundances of $10^{-12}$--$10^{-10}$ assuming a H$_2$ column density of 10$^{22}$~cm$^{-2}$ (e.g.\ \citealt{Schnee2005}). These abundances are about two orders of magnitude lower than the typical abundances found toward more massive hot molecular cores. We have searched for more complex species, such as methyl formate (CH$_3$OCHO) or dimethyl ether (CH$_3$OCH$_3$), but we have not detected them with an upper limit on the column density of about $10^{12}$~cm$^{-2}$, assuming an excitation temperature of 7~K. Similarly to the cold core L1689B studied by \citet{Bac12} we also detect ketene (CH$_2$CO), with a column density of $\sim 3\times10^{12}$~cm$^{-2}$ in complete agreement with the column densities determined for L1689B. In addition to the main isotopologues of the detected species, we also detect transitions of the deuterated counterparts CH$_3$CCD and CH$_2$DOH. The deuteration level is estimated to be about 0.045 for CH$_3$CCH and 0.055 for CH$_3$OH, however, this deuteration fractions should be better constraint with future observations of other transitions and with higher spectral resolution (necessary to resolve the lines). The uncertainty of the column density listed in Table~\ref{t:molecules} does not includes the uncertainty in the linewidth, which can not be measured in our coarse spectral resolution observations. The column densities can differ by 30\% if the linewidth is increased/decreased by 0.1~km~s$^{-1}$, or by 50\% if the variation is 0.2~km~s$^{-1}$. Therefore, the column densities reported in Table~\ref{t:molecules} have to be considered with caution. High-spectral resolution observations are necessary to improve the determination of the excitation temperature and column density. Another source of uncertainty in the column density determination is the excitation temperature: Observations of more transitions for the different molecules are required to better constraint the column density and to search for non-LTE effects.
In general, a number of deuterated compounds have been detected: DCS$^{+}$, HDCS, NH$_2$D, c-C$_3$HD, c-C$_3$D$_2$, CH$_2$DOH and CH$_3$CCD. The deuteration fraction is 0.2 for H$_2$CS, 0.07 for c-C$_2$HD, and about 0.05 for CH$_3$CCH and CH$_3$OH. It is worth noting that the column density measured for c-C$_3$HD and c-C$_3$D$_2$ is in agreement with the recent measurements of \citet{Spez13}.
Finally, in addition to the COMs discussed above, we highlight the detection of some species: (\textit{a}) HCS$^{+}$ has been observed in previous surveys towards Taurus molecular cores (e.g.\ \citealt{Oh98, Kai04}). Here, we present for the first time, a tentative detection of the deuterated counterpart DCS$^{+}$. A detailed study of different deuterated species may help to better understand the routes of deuteration, in particular for those more complex species, and to compare with similar studies conducted in high-mass star forming regions (e.g.\ \citealt{Fontani2011, Fontani2015}); (\textit{b}) Similarly, we report for the first time a tentative detection of HOCO$^{+}$ in this source, for which we determine a column density of $\sim 2\times10^{11}$~cm$^{-2}$; and (\textit{c}) The detection of HCO is common in photon-dominated regions (PDRs; e.g.\ \citealt{schilke2001}), where the chemistry is dominated by the presence of large amounts of far-UV photons. The Taurus molecular cloud is a low-mass star forming complex, and therefore there are no high-mass stars in the region able to produce enough UV photons. In this survey we report the detection of HCO in a cold, dense core, not associated with a PDR, with a column density of $\sim 10^{12}$~cm$^{-2}$. \citet{Bacmann2016} studied HCO in a number of cold prestellar cores, and related its abundance with that of other species such as H$_2$CO, CH$_3$O and finally CH$_3$OH. The authors determine the abundance ratios between the different species to be HCO:H$_2$CO:CH$_3$O:CH$_3$OH $\sim 10:100:1:100$, when the formation of methanol occurs via hydrogenation of CO on cold grain surfaces. The observed abundances of the intermediate species HCO and CH$_3$O suggest they are gas-phase products of the precursors H$_2$CO and CH$_3$OH, respectively. We measure an abundance ratio of HCO:CH$_3$OH $\sim 1:10$ for our cold, dense core (see Table~\ref{t:molecules}), consistent with the results reported by \citet{Bacmann2016}.
\section{Summary\label{s:summary}}
We have used the IRAM\,30m telescope to conduct sensitive observations of a cold, dense core in TMC-1C, with the goal of detecting the CO-H$_2$ van der Waals complex. We have not detected any transition of the CO-paraH$_2$ and CO-orthoH$_2$ compounds with a rms noise level of $\sim 2$~mK for a spectral resolution of 0.7~km~s$^{-1}$. This sets a new strong upper limit for the abundance of the complex: [CO-H$_2$]/[CO]~$\sim 5\times10^{-6}$. We estimate that the expected abundance of the complex, with respect to CO, in the ISM can be $\sim 10^{-8}$--$10^{-3}$, which suggest that more sensitive observations would be required to search for and detect for the first time the CO-H$_2$ complex in the ISM.
Our sensitive spectral line survey have revealed the detection of 75 different spectral lines coming from 41 different species (including isotopologues). The excitation temperature is $\sim 7$~K, consistent with previous estimates. We detect a number of complex organic molecules such as CH$_3$CN, CH$_3$OH, CH$_3$CCH and deuterated isotopologues. The detection of these species in a cold object is consistent with the similar findings in other objects (e.g.\ L1689B, \citealt{Bac12}). Future studies of these complex species to better constraint the physical parameters, as well as the study of more rare isotopologues, can help to improve the current understanding of the formation of complex species in the cold ISM.
\begin{acknowledgements}
We acknowledge the comments and suggestions of the anonymous referee that helped to improve the manuscript. AP would like to thank Nicolas Billot for his help with the observations and data processing and the IRAM team. This work was supported by Deutsche Forschungsgemeinschaft through grant SFB 956 (subprojects A6, B4 and C3).
\end{acknowledgements}
\bibliographystyle{aa}
|
1,314,259,995,798 | arxiv | \section{Introduction}
Reaction-diffusion systems are important models for various nonlinear processes in nonequilibrium physics and biological systems~\cite{rf:1}.
Turing first proposed a reaction-diffusion model to explain pattern formation in developmental biology~\cite{rf:2}. Various pattern formation has been studied in theoretical models~\cite{rf:3,rf:4}.
There is a characteristic wavelength in the Turing pattern. Kondo and Asai observed that the number of stripes in the skin pattern of a tropical fish increases to keep the characteristic wavelength constant, when the fish grows with time~\cite{rf:5}. In the skin pattern, the periodic pattern is constructed of pigment cells with different colors. The different types of pigment cells are mutually competitive.
On the other hand, there are several observations that various types of cells are differentiated during the developmental process, but the number ratio of the different cell types does not change very much as the body size increases. A typical example is the cellular slime mold Dictyostelium discoideum. The amoebic state changes to the fruiting body via the migrating slug state, if the breeding condition becomes worse. Prespore cells and prestalk cells are differentiated in the process. The number ratio is kept almost constant even if the body size is changed~\cite{rf:6}. The ratio control and the pattern formation are performed in two steps. In an early stage, the ratio of the two-type cells is regulated by DIF (Differentiation Inducing Factor), and later prestalk cells move into a tip region within the slug through a chemotaxis by the cAMP~\cite{rf:7,rf:8}. The two cell types, i.e., the prespore and the prestalk are competitive in this system. Another example is the differentiation of blood cells from the stem cells in the bone marrow. The number ratio of the red blood cells, the white blood cells, and the platelets is kept to be roughly constant.
Another typical rule of the cell differentiation is a cascade control of the differentiation process by complicated gene networks. In the segmentation process of the Drosophila, the differentiation proceeds from a large scale to a small scale in a hierarchical manner. A cascade network of genes and proteins such as bicoid protein, gap genes and pair-rule genes are identified in detail~\cite{rf:9}. Several competitive relations between two genes are known also in this process. For example, the gene engrailed (en) for the posterior compartmental specification and the gene wingless (wg) for the anterior compartment are mutually competitive. Several theoretical models of the hierarchical gene network were proposed for the segmentation process~\cite{rf:10,rf:11}.
The analyses of the specific gene network and the pattern formation for each system are important in the developmental biology. However, in this paper, we consider a very simple cascade model of competitive reaction-diffusion equations and propose a mechanism of the ratio control from a view point of the nonlinear dynamics.
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{fig1.eps}
\end{center}
\caption{(a) Eigenvalue $\lambda_k$ for Eq.~(2) at $a=0.4,c=2,d=1,D_X=1,D_Y=20$ and $b=5$ (solid curve), $b=4$ (dashed curve), and $b=3$ (dotted curve).
(b) Stationary profiles of $X_1$ (solid curve) and $X_2$ (dashed curve) at $a=0.4,b=5,c=2,d=1$ and $D_X=1$.
}
\label{f1}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=14cm]{fig2.eps}
\end{center}
\caption{(a) Eigenvalue $\lambda_k$ for Eq.~(4) at $a=0.4,c=2,d=1,D_X=1$ and $b=5$ (solid curve), $b=3$ (dashed curve), and $b=1$ (dotted curve).
(b) Stationary profiles of $X_1$ (solid curve) and $X_2$ (dashed curve) at $a=0.4,b=5,c=2,d=1,D_X=1$. (c) Time evolution of $X_1$ (solid curve) and $X_2$ (dashed curve) from $X_1=0.4+0.04\cos(2\pi x/L)+0.1\cos(4\pi x/L)$ and $X_2=0.4-0.1\cos(4\pi x/L)$.
}
\label{f2}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{fig3.eps}
\end{center}
\caption{(a) Stationary profiles of $X_1$ (solid curve) and $X_2$ (dashed curve) at $a=0.4,b=5,c=2,d_1=0.4,d_2=1$ and $D_X=1$ for Eq.~(6). (b) Size ratio of the $X_1$-dominant region as a function of $d_1$ for $a=0.4,b=5,c=2,d_1=0.4,d_2=1$ and $D_X=1$. The rhombi denote numerical results and the dashed curve is $d_1/(d_1+d_2)$.
}
\label{f3}
\end{figure}
\section{Ratio-control in a competitive reaction-diffusion system}
If two cell-types are mutually competitive, only one type of cells are locally activated by the lateral inhibition. The two cell-types are bistable, and a homogeneous state of one-type cell will appear. However, if some long-range enhancement of the competitive type is included in the system, two types of cells will appear in a large system. Meinhardt and Gierer proposed a model of the mutual induction of locally exclusive states, and generated a stripe-like pattern~\cite{rf:12}. We study first a simplified version of their model.
The model is written as coupled reaction-diffusion equations:
\begin{eqnarray}
\frac{\partial X_1}{\partial t}&=&\frac{cX_1+dY_2}{1+aX_1+bX_2}-X_1+D_X\nabla^2X_1,\nonumber\\
\frac{\partial X_2}{\partial t}&=&\frac{cX_2+dY_1}{1+aX_2+bX_1}-X_2+D_X\nabla^2X_2,\nonumber\\
\frac{\partial Y_1}{\partial t}&=&X_1-Y_1+D_Y\nabla^2Y_1,\nonumber\\
\frac{\partial Y_2}{\partial t}&=&X_2-Y_2+D_Y\nabla^2Y_2,
\end{eqnarray}
where $a,b,c,d$ are positive parameters, $X_1$ and $X_2$ obey the Hill type equation, $Y_1$ and $Y_2$ are produced respectively from $X_1$ and $X_2$. The Hill type equation is often used in the biochemistry and the gene regulatory network~\cite{rf:13,rf:14}. The enhancement and the suppression of the reaction are expressed in the numerator and the denominator in the first term of on the right-hand side of the Hill equation. That is, both $X_1$ and $X_2$ acts as activators for themselves by the terms of $cX_1$ or $cX_2$, and acts as inhibitors for the other by the term of $bX_2$ and $bX_1$. The two components $X_1$ and $X_2$ compete with each other when $b$ is sufficiently large. The terms $aX_1$ and $aX_2$ in the denominators express the effect of the saturation.
We assume that $X_1$ and $X_2$ correspond to proteins (morphogens) determining competitive cell types such as the pigment cells of different colors in the skin pattern, or the prestalk and the prespore in the slime mould. The term of $dY_2$ and $dY_1$ implies that $Y_2$ and $Y_1$ are activators respectively for $X_1$ and $X_2$. We assume that $D_Y\gg D_X$. It means that the morphogens $Y_1$ and $Y_2$ diffuse rapidly and work as long-range activators respectively for $X_2$ and $X_1$.
There is a uniform solution $X_1=X_2=Y_1=Y_2=X_0=(c+d-1)/(a+b)$ to Eq.~(1).
It is expected that $X_2$ is depressed when $X_1$ increases locally and vice versa, because $X_1$ and $X_2$ are mutually competitive.
If the perturbation of the form: $X_1=X_0+\delta x_1\cos(kx), X_2=X_0-\delta x_1\cos(kx), Y_1=X_0+\delta y_1\cos(kx), Y_2=X_0-\delta y_1\cos(kx)$ are assumed owing to the competitive relation, the deviations $\delta x_1$ and $\delta y_1$ satisfy
\begin{eqnarray}
\frac{d\delta x_1}{dt}&=&\frac{c\delta x_1-d\delta y_1}{1+(a+b)X_0}+\frac{(c+d)(b-a)X_0\delta x_1}{\{1+(a+b)X_0\}^2}-(1+D_Xk^2)\delta x_1,\nonumber\\
\frac{d\delta y_1}{dt}&=&\delta x_1-(1+D_Yk^2)\delta y_1.
\end{eqnarray}
The eigenvalue $\lambda_k$ can be calculated from Eq.~(2). Figure 1(a) displays $\lambda_k$ for $b=5$ (solid curve), $b=4$ (dashed curve) and $b=3$ (dotted curve) at $a=0.4,c=2,d=1,D_X=1$ and $D_Y=20$. (The parameter values $a,b,c,d,D_X$ and $D_y$ are not realistic biological ones, but we study the nonlinear system as just a model system in this paper.)
The Turing type instability is expected for a finite wavenumber. We have performed a one-dimensional numerical simulation. The system size $L=100$, and the periodic boundary conditions are imposed. Figure 1(b) shows stationary profiles of $X_1(x)$ (solid curve) and $X_2(x)$ (dashed curve) at $a=0.4,b=5,c=2,d=1,D_X=1$ and $D_Y=20$. A spatially periodic pattern appears, in which the $X_1$-dominant region and $X_2$-dominant region reciprocate spatially. This is a Turing pattern in the competitive reaction-diffusion systems. Note that the $Y$ component with large diffusivity acts as activators for the competitor, and it facilitates the competitor in spatially distant regions. It is different from the usual Turing pattern in usual activator-inhibitor systems, where the inhibitor with large diffusivity inhibits the activator. The mechansim of the pattern formation by the long-range activation of the competitor is essentially the same as the model proposed by Meinhardt and Gierer almost 30 years ago~\cite{rf:12}.
Kondo and Asai performed a numerical simulation of the usual type of activator-inhibitor system to explain the skin pattern of the tropical fish~\cite{rf:5}, but our model might be more plausible for the pattern formation, because the pigment cells are mutually competitive.
Next, we generalize the model equation (1) to a model equation in which the whole system is separated into two regions of $X_1$-dominant region and $X_2$-dominant region. If $D_Y$ is infinitely large and a steady state is obtained for the $Y$ component, Eq.~(1) is reduced to
\begin{eqnarray}
\frac{\partial X_1}{\partial t}&=&\frac{cX_1+d\langle X_2\rangle}{1+aX_1+bX_2}-X_1+D_X\nabla^2X_1,\nonumber\\
\frac{\partial X_2}{\partial t}&=&\frac{cX_2+d\langle X_1\rangle}{1+aX_2+bX_1}-X_2+D_X\nabla^2X_2,
\end{eqnarray}
where $\langle X\rangle$ is the spatial average of $X$, i.e., $\langle X\rangle=(1/L)\int_0^LXdx$ in a one-dimensional system and $\langle X\rangle=(1/L^2)\int_0^L\int_0^LXdxdy$ in a two-dimensional system. It is because $Y$ becomes uniform owing to the infinitely large diffusivity of $Y$ in Eq.~(1).
This is our original model. For this model equation, the small deviation $\delta x_1$ from the uniform solution satisfies
\begin{equation}
\frac{d\delta x_1}{dt}=\frac{c\delta x_1}{1+(a+b)X_0}+\frac{(c+d)(b-a)X_0\delta x_1}{\{1+(a+b)X_0\}^2}-(1+D_Xk^2)\delta x_1,
\end{equation}
for $k\ne 0$, because $\delta y_1$ is assumed to be zero owing to the uniformity of $Y$. On the other hand, for $k=0$,
\begin{equation}
\frac{d\delta x_1}{dt}=\frac{(c-d)\delta x_1}{1+(a+b)X_0}+\frac{(c+d)(b-a)X_0\delta x_1}{\{1+(a+b)X_0\}^2}-\delta x_1.
\end{equation}
Fugure 2(a) displays the eigenvalues $\lambda_k$ for $b=5$ (solid curve), $b=3$ (dashed curve) and $b=1$ (dotted curve) at $a=0.4,c=2,d=1,D_X=1$.
The eigenvalue $\lambda_k$ at $k=0$ is negative for these parameter values, therefore, the Fourier mode with $k=0$ is stable. The uniform solution is unstable for $b=5$ and $b=3$. The eigenvalue takes the largest value for the smallest wave number $k=2\pi/L$. Figure 2(b) shows a stationary solution for $b=5$ for the one-dimensional system with periodic boundary conditions of size $L=100$.
The initial condition was $X_1=0.5-0.1(x/L)$ and $X_2=0.4+0.1(x/L)$.
For this initial condition, the whole space is separated into the $X_1$-dominant region at $x<L/2$ and the $X_2$-dominant region at $x>L/2$.
There are domain walls between the two domains. However, even if the initial condition is random, the whole space is separated into the two regions with the same size, although the position of the $X_1$-dominant region becomes random. It is because $\lambda_k$ takes a maximum value at $k=2\pi/L$.
If the initial condition is $X_1=0.4+0.04\cos(2\pi x/L)+0.1\cos(4\pi x/L)$ and $X_2=0.4-0.1\cos(4\pi x/L)$, four domains, i.e., two $X_1$-dominant regions and two $X_2$-dominant regions appear initially, however, there is an attractive interaction between the domain walls and finally the two-domain structure is obtained as shown in Fig.2(c). The time evolution is similar to the coarsening process in the one-dimensional Ginzburg-Landau (GL) equation: $\partial u/\partial t=u-u^3+\partial^2u/\partial x^2$~\cite{rf:15}. However, it is different from the GL system in that the final state is a uniform state in the GL system and the final state is a two-domain state in our system. It is a unique point in our model that the two-domain structure of equal size appears naturally for any system size $L$.
Equation~(3) can be generalized into an asymmetric model:
\begin{eqnarray}
\frac{\partial X_1}{\partial t}&=&\frac{cX_1+d_1\langle X_2\rangle}{1+aX_1+bX_2}-X_1+D_X\nabla^2X_1,\nonumber\\
\frac{\partial X_2}{\partial t}&=&\frac{cX_2+d_2\langle X_1\rangle}{1+aX_2+bX_1}-X_2+D_X\nabla^2X_2,
\end{eqnarray}
where $d_1$ and $d_2$ are assumed to take different values.
Figure 3(a) diplays stationary profiles of $X_1$ (solid curve) and $X_2$ (dashed curve) at $a=0.4,b=5,c=2,d_1=0.4,d_2=1,$ and $D_X=1$ for $L=100$. The sizes of the $X_1$-dominant region and the $X_2$-dominant region are not the same in this asymmetric model. However, the maximum and the minimum values of $X_1$ and $X_2$ and the width of the domain walls are almost the same, which are expressed as $X_{max}$ and $X_{min}$. If $d_1\langle X_2\rangle$ and $d_2\langle X_1\rangle$ are assumed to takes constant values and $d_1\langle X_2\rangle<d_2\langle X_1\rangle$, a domain wall between a $X_1$-dominant region and a $X_2$-dominant region moves as the $X_1$-dominant region shrinks, which is similar to the motion of the domain wall in the asymmetric Ginzburg-Landau equation: $\partial u/\partial t=u+\epsilon u^2-u^3+\partial^2u/\partial x^2$. However, if the $X_1$-dominant region shrinks, $\langle X_1\rangle$ decreases and $\langle X_2\rangle$ increases in time in our system. This is due to a negative feedback effect involved in our system.
Finally, the domain walls become stationary, when $d_1\langle X_2\rangle=d_2\langle X_1\rangle$ is satisfied. If the system size $L$ is sufficiently large, the ratio $r=l/L$ of the domain size $l$ of the $X_1$-dominated region is evaluated from the condition $d_1\langle X_2\rangle=d_2\langle X_1\rangle$ as
\begin{equation}
d_1\{(1-r)X_{max}+rX_{min}\}=d_2\{rX_{max}+(1-r)X_{min}\},
\end{equation}
because $\langle X_2\rangle\sim (1-r)X_{max}+rX_{min}$ and $\langle X_1\rangle\sim rX_{max}+(1-r)X_{min}$, if the width of domain walls is not taken into consideration. If $X_{min}\ll X_{max}$, $r$ is approximated at $r=d_1/(d_1+d_2)$.
Thus, the size ratio is determined by the ratio of the parameters $d_1$ and $d_2$ and does not depend on the system size $L$. We studied the control of the domain size in the Ginzburg-Landau type equation in the previous work, which was the same mechanism as the present one~\cite{rf:16}.
Figure 3(b) displays numerically obtained size ratio $r$'s as a function of $d_1$ at $a=0.4,b=5,c=2,d_2=1$, and $D_X=1$ for $L=100$. The dashed curve denotes $r=1/(d_1+1)$. Fairly good agreement is seen.
The ratio control can be attained by the choice of the parameters $d_1$ and $d_2$. This is a mechanism of the ratio control of our differentiation model.
The feedback effect via the $Y$-variable with infinitely large diffusivity controls the ratio of the different cell types. In this asymmetric model, a two-domain structure appears naturally, and the ratio of the domain size is uniquely determined by the system parameters. It might be applicable to the ratio control in the early stage of the slime mold.
\begin{figure}[t]
\begin{center}
\includegraphics[width=14cm]{fig4.eps}
\end{center}
\caption{(a) Hierarchical network of fourteen elements in three layers.
The solid lines denote the active interaction and the dashed lines denote the competitive interaction.
(b) Stationary profiles of $X_i$ ($i=1,\cdots,14$) in the cascade model similar to Eq.~(8) in a one-dimensional space. The system is composed of fourteen elements in three layers. The system size is $L=96$. The parameter values are $a=0.4,b=5,c=2,d=1,\alpha=0.4$ and $D_X=0.1$. (c) Stationary profiles of $X_i$ ($i=1,\cdots,14$) at $a=0.4,b=5,c=2,d_1=0.8,d_2=1,\alpha=0.4$ and $D_X=0.1$.
}
\label{f4}
\end{figure}
\section{A cascade model of cell differentiation}
Next, we construct a new cascade model of the competitive reaction-diffusion equations as shown in Fig.~4(a) based on Eqs.~(3) and (6), although another cascade model based on a more complicated version of Eq.~(1) was proposed by Meinhardt to study the hierarchical subdivision into gap-gene, pair-rule gene and segment polarity gene in the Drosophila~\cite{rf:10}.
In Fig.~4(a), a system of fourteen elements in three layers is shown. Each element denotes a protein (morphogen) with number $i$ which determines the $i$th cell type. The horizontal dashed lines represent the competitive interaction between two elements as in the previous model, and the solid lines from the upper layer to the lower layer represent an active interaction from a upper-level element to a lower-level element.
A negative feedback from the lower-level to the upper level might be important, but the feedback effect is not considered in our model for the sake of simplicity. The active and competitive interactions are represented by the Hill type equations.
For example, a model equation for a system of six elements in two layers is written as
\begin{eqnarray}
\frac{\partial X_1}{\partial t}&=&\frac{cX_1+d_1\langle X_2\rangle}{1+aX_1+bX_2}-X_1+D_X\nabla^2X_1,\nonumber\\
\frac{\partial X_2}{\partial t}&=&\frac{cX_2+d_2\langle X_1\rangle}{1+aX_2+bX_1}-X_2+D_X\nabla^2X_2.\nonumber\\
\frac{\partial X_3}{\partial t}&=&\frac{\alpha X_1(cX_3+d_1\langle X_4\rangle)}{1+aX_3+bX_4}-X_3+D_X\nabla^2X_3,\nonumber\\
\frac{\partial X_4}{\partial t}&=&\frac{\alpha X_1(cX_4+d_2\langle X_3\rangle)}{1+aX_4+bX_3}-X_4+D_X\nabla^2X_4.\nonumber\\
\frac{\partial X_5}{\partial t}&=&\frac{\alpha X_2(cX_5+d_1\langle X_6\rangle)}{1+aX_5+bX_6}-X_5+D_X\nabla^2X_5,\nonumber\\
\frac{\partial X_6}{\partial t}&=&\frac{\alpha X_2(cX_6+d_2\langle X_5\rangle)}{1+aX_6+bX_5}-X_6+D_X\nabla^2X_6,
\end{eqnarray}
where $\alpha$ is the coupling constant of the active interaction from the upper layer to the lower layer. Owing to the competitive interaction, $X_2$ is almost zero in the $X_1$-dominant region. At the domain, $X_5$ and $X_6$ are also almost zero because $X_5$ and $X_6$ are activated by $X_2$. On the other hand, the $X_1$-dominant region is separated into the $X_3$-dominant region and the $X_4$-dominant region because of the competitive interaction between $X_3$ and $X_4$. The size ratio of the two regions are also determined by the ratio of $d_1$ and $d_2$. We have performed numerical simulations of a system of fourteen elements in the three layers. It is a one-dimensional system with system size $L=96$. Figure 4(b) displays stationary profiles of $X_i$ ($i=1,\cdots,14$) for the three layer system at $a=0.4,b=5,c=2,d=1,\alpha=0.4$ and $D_X=0.1$.
Firstly, the whole space is separated into the two domains: a $X_1$-dominant region and a $X_2$-dominant region. Next, the $X_1$-dominant region is separated into the two domains: a $X_3$-dominant region and a $X_4$-dominant region, and the $X_2$-dominant region is separated into the two domains: a $X_5$-dominant region and a $X_6$-dominant region. Similarly, the $X_3$-dominant region is separated into $X_7$-dominant and $X_8$-dominant regions, the $X_4$-dominant region is separated into $X_9$-dominant and $X_{10}$-dominant regions, the $X_5$-dominant region is separated into $X_{11}$-dominant and $X_{12}$-dominant regions, and, the $X_6$-dominant region is separated into $X_{13}$-dominant and $X_{14}$-dominant regions
As a result, a hierarchical pattern appears. The domain size decreases as $1/2,1/4$ and 1/8 as the layer is lowered. This is a kind of binary-tree decomposition of the whole space. If the parameter $\alpha$ is assumed to take a different value $\alpha_k$ for each layer, and is set to be $\alpha_k=(c+d/2)/\{X_{max}(c+d/2^k)\}$ for the $k$th layer, peak values of $X_i$ take almost the same value $X_{max}$ for each layer, because $\langle X_i\rangle=X_{max}/2^{k}$ in the $k$th layer. For the well tuned parameter values of $\alpha_k$, a self-similar binary decomposition of dominant regions will occur. If $d_1\ne d_2$, a multi-fractal-like pattern appears, because the size ratio is $r,1-r$ in the first layer, $r^2,r(1-r),(1-r)^2$ in the second layer, and $r^3,r^2(1-r),r(1-r)^2,(1-r)^3$ in the third layer. Figure 4(b) displays a stationary profiles of $X_i$ ($i=1,\cdots, 14$) for $d_1=0.8$ and $d_2=1$ at $a=0.4,b=5,c=2,d=1,\alpha=0.4$ and $D_X=0.1$. Inhomogeneous decomposition of dominant regions is clearly seen.
\begin{figure}[t]
\begin{center}
\includegraphics[width=14cm]{fig5.eps}
\end{center}
\caption{Hierarchical differentiation of the fourteen elements with the three layers in a two-dimensional space. The $X_i$-dominant regions ($i=1,\cdots, 14$) are depicted with different colors in a square of size $48\times 48$. The parameter values are $d_1=0.8$ and $d_2=1$ at $a=0.4,b=5,c=2,d_1=0.4,d_2=1,\alpha=0.4$ and $D_X=0.2$. A hierarchical structure is clearly seen. The colored regions imply that $X_i$ satisfies $X_i>1$. (a) $X_1$-dominant and $X_2$-dominant regions. (b) $X_3$, $X_4$, $X_5$, and $X_6$-dominant regions. (c) $X_7$, $X_8$, $X_9$, $X_{10}$, $X_{11}$, $X_{12}$, $X_{13}$, and $X_{14}$-dominant regions. The number $i$ indicates the cell type.
}
\label{f5}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=5cm]{fig6.eps}
\end{center}
\caption{Time evolution of the number ratio $r_i$ of $X_i$-dominant regions satisfying $X_i>1$ for $i=1,2,3,4,7$ and 8.
}
\label{f6}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{fig7.eps}
\end{center}
\caption{Three snapshot patterns of the $X_i$ dominant region satisfying $X_i>1$ for $i=3.4.5$ and 6 at (a) $t=50$, (b) $t=100$, and (c) $t=2000$. The snapshot at $t=20000$ is shown in Fig.~5(b). The number $i$ indicates the cell type.
}
\label{f7}
\end{figure}
Finally, we have performed a numerical simulation in two dimensions. The system size is $48\time 48$, and the periodic boundary conditions are imposed. The network of the interaction is the same as before in Fig.~4(a), that is, the number of elements are fourteen and the layer number is three. The parameter values are $d_1=0.8$ and $d_2=1$ at $a=0.4,b=5,c=2,d_1=0.4,d_2=1,\alpha=0.4$ and $D_X=0.2$.
The initial conditions are random between 0.5 and 0.6.
Figure 5(a),(b) and (c) display patterns at $t=20000$ which represent the dominant region satisfying $X_i>1$ for each $i$ using different color for (a) $i=1$ and 2, (b) $i=3,\cdots,6$, and (c) $i=7,\cdots,14$.
These states are final stationary states after the transient time evolution as shown later in Fig.~6 and Fig.~7. The whole region is divided into two domains: a $X_1$-dominant region and a $X_2$-dominant region. The $X_1$-dominant region is a circular blue region which is located near the four edges in Fig.~5(a). The area ratio of the $X_1$-dominant region and the $X_2$-dominant region is $0.42:0.58$, which is close to $d_1:d_2=0.44:0.56$. The $X_1$-dominant region is subdivided into two domains: a $X_3$-dominant region and a $X_4$-dominant region. The area ratio of the two regions is $0.44:0.56$, which is also close to the ratio $d_1:d_2=0.44:0.56$. Similarly, the $X_3$-dominant region is further subdivided into a $X_7$-dominant region and a $X_8$-dominant region. The area ratio of the two regions is $0.42:0.58$, which is also close to the ratio $d_1:d_2=0.44:0.56$. Thus, the hierarchical differentiation is clearly seen. And it is shown that the area ratio is determined by the ratio of $d_1$ and $d_2$.
Figure 6 displays the time evolution of the number ratio of the $X_i$-dominant regions for $i=1,2,3,4,7$ and 8. The number ratio becomes a stationary values at $t\sim 30$ for $i=1$ and 2, at $t\sim 70$ for $i=3$ and $4$, at $t\sim 1100$ for $i=7$ and 8. That is, the stationary state is attained sequentially from the upper layer to the lower layer. Figures 7(a), (b) and (c) show three snapshot patterns of the $X_3$-,$X_4$-, $X_5$- and $X_6$-dominant regions at $t=50,100$ and $2000$. The dominant regions are initially small and random, and the size and the location are determined gradually. Figure 7(c) is almost similar to Fig.~5(b), which means that the time evolution is almost stationary at $t=2000$.
\section{Summary}
Competitive interactions and hierarchical interactions are typical in gene networks. We have proposed a simple cascade model of competitive reaction-diffusion equations for the cell differentiation. We have first reconfirmed a spatially-periodic pattern in a simple one-dimensional competitive reaction diffusion equations, which was originally proposed by Meinhardt and Gierer. It is a kind of the Turing pattern, but the origin of the formation of the periodic pattern is the long-range enhancement of the competitor. If the diffusion constant for $Y$-variable is assumed to infinity, we get a new model, in which a two-domain structure appears naturally from an random initial condition, and the size ratio of the two domains can be controlled by system parameters owing to the negative feedback effect of the domain size. Next, we have generalized the model to a hierarchical model. The $X_i$-dominant regions appear in a cascade manner from the upper layer to the lower layer. The ratio of the $X_i$-dominant regions are well controlled by the system parameters. Our simple cascade model is not a realistic model based on biological experiments, however, it is a useful model to consider the Turing pattern, the ratio control, and the hierarchical differentiation in a unified manner. We expect that our model might be applicable to specific cell differentiation processes by changing the reaction network and modifying the system parameters suitably.
|
1,314,259,995,799 | arxiv | \section{Introduction}\label{introduction}
After the Epoch of Recombination around redshift $z\sim 1100$, the Universe entered the `Dark Ages' era during which it was completely neutral and devoid of any radiation sources. During this period, small perturbations in matter density grew under gravitational instability, and matter started to accumulate in localised over-density peaks. The formation of the first luminous objects (stars and galaxies) in these overdense regions marked the beginning of the so-called Cosmic Dawn (CD) era spanning the redshift range $30 > z > 12$ \citep{pritchard2007}. X-ray and Ultraviolet radiation from the first stars and galaxies began to heat and ionize the neutral hydrogen (HI hereafter) in the surrounding Inter-Galactic Medium (IGM), starting off the Epoch of Reionization (EoR) ($12 > z > 6$) during which HI in the IGM transitioned from being fully neutral to ionized \citep{madau1997}.
The redshifted 21-cm signal corresponding to the hyperfine transition of HI has been identified as an excellent probe of the HI distribution in the IGM during the CD and the EoR \citep{madau1997,shaver1999,furlanetto2006,pritchard2012,zaroubi2013}. A number of ongoing and upcoming experiments, such as the LOw Frequency ARray\footnote{\url{http://www.lofar.org/}}(LOFAR; \citealt{vanhaarlem2013}), the Giant Meterwave Radio Telescope\footnote{\url{http://gmrt.ncra.tifr.res.in/}}(GMRT; \citealt{paciga2011}), the Murchison Widefield Array\footnote{\url{http://www.mwatelescope.org/}}(MWA; \citealt{tingay2013,bowman2013}), the Precision Array for Probing the Epoch of Reionization\footnote{\url{http://eor.berkeley.edu/}}(PAPER; \citealt{parsons2010}), the Hydrogen Epoch of Reionization Array\footnote{\url{http://reionization.org/}}(HERA; \citealt{deboer2017}), NenuFAR\footnote{\url{https://nenufar.obs-nancay.fr/}}(New extension in Nan\c cay Upgrading loFAR; \citealt{zarka2012}), and the Square Kilometre Array\footnote{\url{http://skatelescope.org/}}(SKA; \citealt{mellema2013,koopmans2015}) are seeking to detect the brightness temperature fluctuations in the cosmological 21-cm signal using statistical methods e.g. the power spectrum. Complementary to these 21-cm power spectrum measurement experiments, several efforts such as the Experiment to Detect the Global Epoch of Reionization Signature (EDGES; \citealt{bowman2018}), the Large-aperture Experiment to Detect the Dark Ages (LEDA; \citealt{bernardi2016}), the Shaped Antenna measurement of the background RAdio Spectrum 2 (SARAS 2; \citealt{singh2017}), the Sonda Cosmol\'{o}gica de las Islas
para la Detecci\'{o}n de Hidr\'{o}geno Neutro (SCI-HI; \citealt{voytek2014}), the Probing Radio Intensity at high $z$ from Marion (PRIZM; \citealt{philip2018}), and the Netherlands-China Low frequency Explorer\footnote{\url{https://www.ru.nl/astrophysics/research/radboud-radio-lab-0/projects/netherlands-china-low-frequency-explorer-ncle/}}$^,$\footnote{\url{https://www.astron.nl/r-d-laboratory/ncle/netherlands-china-low-frequency-explorer-ncle}} (NCLE) are seeking to measure the sky-averaged spectrum of the 21-cm signal.
At present, several instruments targeting the EoR redshifts have placed upper limits on the brightness temperature power spectrum of the redshifted 21-cm signal. \cite{paciga2013} provided the first $2\sigma$ upper limit on the brightness temperature of $\Delta_{21}^2 < (248\,\text{mK})^2$ at wavenumber $k \approx 0.5\,h\,\text{cMpc}^{-1}$ at redshift $z=8.6$ using the GMRT. \cite{beardsley2016} used MWA to set a $2\sigma$ upper limit of $\Delta_{21}^2 < (164\,\text{mK})^2$ at $k \approx 0.27\,h\,\text{cMpc}^{-1}$ at $z=7.1$. The PAPER project also provided an upper limit of $\Delta_{21}^2 < (22\,\text{mK})^2$ in the wavenumber range $0.15 \leq k \leq 0.5\,h\,\text{cMpc}^{-1}$ at $z=8.4$ \citep{ali2015}, but have recently retracted their claim due to issues with their analysis strategy (see the erratum \citealt{ali2018}). The tightest $2\sigma$ upper limit on the 21-cm power spectrum yet is $\Delta_{21}^2 < (79.6\,\text{mK})^2$ at $k \approx 0.053\,h\,\text{cMpc}^{-1}$ in the redshift range $z= 9.6 -10.6$ and was provided by \cite{patil2017} using the LOFAR High Band Antenna (HBA) array. Instruments such as HERA, NenuFAR, and SKA-low which can potentially probe the CD redshifts are now in hardware roll-out stages (the latter is still in the development stage). \cite{ewall-wice2016} used low frequency MWA observations ($75-113$ MHz) to place an upper limit of $\Delta^2_{21} < (10^4\,\text{mK})^2$ at $k\approx 0.5$ on the power spectrum of the brightness temperature fluctuations of the 21-cm signal in the redshift range $12\lesssim z \lesssim 18$, which in most models corresponds to the epoch of X-ray heating during the CD (see e.g. \citealt{glover2003,fialkov2014,ross2017}).
In this work, we explore, for the first time, the possibility of observing the redshifted 21-cm signal from the CD era using the LOFAR-Low Band Antenna (LBA) array which observes in the $30-90$ MHz frequency range. We use LOFAR-LBA dual pointing observations of the North Celestial Pole (NCP field hereafter) and an adjacent field centred on the 3C220.3 radio galaxy (3C220 field hereafter), which is $\sim 7^{\circ}$ away from the NCP, to study the challenges (systematic biases) in CD studies with the LOFAR-LBA and to set the first upper limits on the 21-cm brightness temperature power spectrum in the redshift range $z = 19.8 - 25.2$. We also demonstrate the application of a novel dual-pointing calibration strategy to calibrate the LOFAR-LBA data and the application of Gaussian Process Regression (GPR) as a powerful foreground removal technique in CD experiments.
The paper is organised as follows: in Section \ref{sec:observations_preprocess}, we briefly describe the LOFAR-LBA system, the observational setup and preprocessing steps. In Section \ref{sec:calibration}, we describe the multi-beam calibration strategy to calibrate the LOFAR-LBA data. In Section \ref{sec:LBAnoise}, we assess the noise in the observed data and address the systematic biases, such as excess noise in Stokes I versus V using various statistical methods. We describe Gaussian Process Regression (GPR) in Section \ref{sec:GPR} and its application in removing residual foregrounds in LOFAR-LBA data. In Section \ref{sec:PSpec-results}, we determine the power spectra for both fields and derive upper limits on the 21-cm power spectrum in the redshift range $z = 19.8 - 25.2$. Finally, in Section \ref{sec:conclusions}, we summarise the work and discuss future prospects.
\section{Observations and preprocessing}\label{sec:observations_preprocess}
We used the LOFAR-LBA system with dual pointing setup to simultaneously observe the NCP field and the 3C220 field, which is $\sim 7^{\circ}$ away from the NCP. The NCP is the primary target field of the LOFAR-EoR KSP and has been used to set the first upper limits on the EoR power spectrum using LOFAR (see \citealt{patil2017}). The observational setup and preprocessing steps are described in the following subsections.
\subsection{LOFAR-Low Band Array}\label{subsec:lofar_lba}
The LOFAR-LBA consists of 38 stations spread across the Netherlands, providing shortest baseline lengths of $\sim 80$ m and longest baseline lengths of $\sim 100$ km. Out of these 38 stations, 24 stations (known as core stations) are spread within a core of 2 km radius, providing a densely sampled $uv$-plane. The remaining 14 stations (known as remote stations) are spread across the North-Eastern part of the Netherlands. Each LOFAR station consists of 96 low band dual-polarization dipole antennas randomly spread within an area of 81 m diameter. The voltages measured with the cross dipoles are digitised using a 200 MHz sampling clock covering the frequency range of 10-90 MHz. The digitised data is beam-formed to produce a digitally steerable station beam. At a given time, only 48 out of 96 dipoles can be combined in the beam-former. This allows a user to choose from three different station configurations in LOFAR-LBA mode viz: \texttt{LBA\_INNER} where the 48 innermost dipoles (array width $\sim 30$ m) are beam-formed, \texttt{LBA\_OUTER} where the 48 outermost dipoles (array width $\sim 81$ m) are beam-formed, and \texttt{LBA\_SPARSE} where half of the innermost 48 dipoles, plus half of the outermost 48 dipoles (array width $\sim 81$ m) are beam-formed. Each configuration results in a specific Field of View (FoV) as well as different sensitivity due to mutual coupling between the dipoles. The LOFAR-LBA system has an instantaneous bandwidth of 96 MHz. However, multiple pointings in the sky can be traded against the observable bandwidth depending on the number of pointings. In the case of two pointings, the bandwidth is reduced to 48 MHz per pointing. Readers may refer to \cite{vanhaarlem2013} for more information about the observation capabilities of LOFAR.
\subsection{Observations}\label{subsec:observations}
We use 14 hours of synthesis observation data of the NCP and the 3C220 fields, which were observed simultaneously with dual beam pointings using the \texttt{LBA\_OUTER} mode of the LOFAR-LBA system. The data were recorded during LOFAR observation Cycle 6 (ID: L557804, November 4-5, 2016). The observational details of the data are summarised in Table \ref{tab:obs_details}. The digitised data from beam-formed stations were correlated with 1 second time resolution and 3 kHz frequency resolution. The recorded data consists of 244 sub-bands for each field within the frequency range of 38-86 MHz. Each sub-band has a width of 195.3 kHz and consists of 64 channels. The recorded correlations (XX, XY, YX and YY) are stored in a Measurement Set (MS). The raw data volume for each field is $\sim18$ Terabytes and is preprocessed to reduce the data volume, as described in the next section.
\begin{table}
\centering
\caption{Observational details of the data.}
\label{tab:obs_details}
\begin{tabular}{ll}
\hline
\textbf{Parameter} & \textbf{value} \\
\hline
Telescope & LOFAR LBA \\
Observation cycle and ID & Cycle 6, L557804\\
Antenna configuration & \texttt{LBA\_OUTER} \\
Number of stations & 38 (NL stations) \\
Observation start time (UTC) & Nov 4, 2016; 16:21:44 \\
Number of pointings & 2 \\
Phase centre ($\alpha,\delta$; J2000): & \\
\quad NCP field & 00h00m00s, $+90^{\circ}00^{\prime}00^{\prime\prime}$ \\
\quad 3C220 field & 09h39m23s, $+83^{\circ}15^{\prime}26^{\prime\prime}$\\
Duration of observation & 14 hours \\
Minimum frequency & 38.08 MHz \\
Maximum frequency & 85.54 MHz \\
Target bandwidth & 48 MHz \\
Primary Beam FWHM & $3.88^{\circ}$ at 60 MHz \\
Field of View & 12 $\text{deg}^2$ at 60 MHz\\
SEFD & $\sim25$ kJy at 60 MHz\\
Polarisation & Linear X-Y \\
Time, frequency resolution: \\
\quad Raw Data & 1 s, 3 kHz \\
\quad After flagging step 1 & 2 s, 12 kHz (archived)\\
\quad After flagging step 2 & 2 s, 61 kHz \\
\hline
\end{tabular}
\end{table}
\subsection{Data selection and preprocessing}\label{subsec:preprocessing}
LOFAR-LBA has lower sensitivity and a relatively high RFI corruption level for frequencies above 70 MHz. Therefore, we used only 33 MHz bandwidth with the frequency range 39-72 MHz for preprocessing and further analysis. We used the standard LOFAR software pipeline (see e.g. LOFAR imaging cookbook \footnote{\url{https://www.astron.nl/radio-observatory/lofar/lofar-imaging-cookbook}}) for preprocessing the observed raw data. Processing steps include RFI-excision and averaging the data. Flagging of RFI corrupted data is performed on the highest resolution data (1 second, 3 kHz) to minimise information loss. We use the \texttt{AOFlagger} software \citep{offringa2010,offringa2012} to flag RFI corrupted data. Two channels on both edges of every sub-band were also discarded to avoid edge effects due to the polyphase filter. The remaining data were averaged in frequency and time to an intermediate-resolution of 12 kHz and 2 seconds, resulting in 15 channels per sub-band. This intermediate resolution data is archived for future use. To reduce the data volume further, it was averaged in frequency to 61 kHz and the auto-correlations were also flagged. The resulting data consists of 3 channels of 61 kHz each per sub-band and has a time resolution of 2 seconds. We flagged the remote station RS503LBA in all sub-bands for both fields because of its proximity to a windmill, which causes strong RFI in the visibilities of the station. We also observed that CS302LBA had poor gain upon inspecting the visibilities and flagged it for both fields. The flagging and averaging were performed separately on both 3C220 and NCP field datasets, although some correlation is obviously expected. Figure \ref{fig:uvcov} shows LOFAR-LBA $uv$-coverage (the inner region with $|\vect{u}|<600\lambda$) for the 3C220 field pointing for 14 hours track at 60 MHz after exclusion of flagged visibilities and the radial profile of the $uv$-coverage.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{LBA_uvcov_new.png}
\caption{\textit{Left panel:} The inner region ($|\vect{u}| < 600\lambda$) of LOFAR-LBA $uv$-coverage for the 3C220 field for 14 hour track at 60 MHz after excluding flagged visibilities. \textit{Right panel:} The radial $uv$-density $\frac{\textrm{d}N}{2\pi|\vect{u}| \textrm{d}|\vect{u}|}$ corresponding to the left panel.}
\label{fig:uvcov}
\end{figure*}
\section{Calibration Scheme}\label{sec:calibration}
The visibilities recorded by LOFAR are corrupted by the instrumental (complex station gains, primary beam, instrumental bandpass, clock-drift etc.) and environmental (ionosphere) factors. Calibration of the LOFAR-LBA system involves estimating the errors that corrupt the measured visibilities, and to obtain an accurate estimate of the true visibilities from observed data. Calibration of LOFAR-LBA data involves two major steps: (a) Direction Independent (DI) calibration and, (b) Direction Dependent (DD) calibration. DI calibration involves estimation of a single instrumental gain (represented by a complex $2\times 2$ Jones matrix) for each beam-formed station, and DD calibration accounts for the direction dependent errors arising from wave propagation effects through the ionosphere and the primary beam. We use \texttt{SAGECal-CO}\footnote{\url{http://sagecal.sourceforge.net/}} to perform the major calibration steps. \texttt{SAGECal-CO} performs calibration in a distributed way using consensus optimisation \citep{boyd2011}, which is an effective way to improve the quality of the calibration of radio interferometric data. In \texttt{SAGECal-CO}, the calibration problem is transformed into consensus optimisation by adding frequency smoothness of systematic errors as a constraint. It uses an Alternating Direction Method of Multipliers (ADMM) algorithm to reach convergence. Readers may refer to \cite{yatawatta2015,yatawatta2016,yatawatta2017,yatawatta2018} for a detailed description of the \texttt{SAGECal-CO} algorithm and its capabilities.
\\
Upon inspection of the raw visibilities, we observed that Cas\,A ($\sim 30^{\circ}$ away from NCP) and Cyg\,A ($\sim 50^{\circ}$ away from the NCP) superpose significant side-lobes onto both fields. It is crucial to subtract these sources before performing DI calibration to avoid errors due to these side-lobes. We use DD-calibration in \texttt{SAGECal-CO} to subtract Cas\,A and Cyg\,A. \cite{gehlot2018} (G18 hereafter) showed that the residuals after subtraction of bright sources such Cas\,A and Cyg\,A are significant as well as incoherent over timescales of a few minutes, depending on the strength of ionospheric scintillations. Therefore, we use a solution time and frequency interval of 30 seconds and 61 kHz to subtract Cas\,A and Cyg\,A, which is optimised to incorporate ionospheric effects while maintaining a decent signal-to-noise ratio ($\gtrsim 10$) for the given solution interval. We use the Cas\,A and Cyg\,A shapelet models \footnote{Cas\,A and Cyg\,A models were derived from wide-band LOFAR-LBA and HBA observations of Cas\,A and Cyg\,A. Each source has about 200 components (shapelets and point). See \cite{yatawatta2011} for more details.} as an input model for calibration and subtraction. The subtraction was performed individually on both fields.
The two fields, 3C220 and NCP, given their different pointings and gain solutions, have slightly different morphologies. The 3C220 field consists of a reasonably bright source located at the phase centre (the 3C220.3 radio galaxy with a flux of $\sim 38$ Jy at 74 MHz \citep{cohen2007}) which can be utilised as a bandpass calibrator, making calibration of the 3C220 field fairly straightforward. However, the NCP field does not have such relatively bright sources near the phase centre, which makes it more difficult to calibrate the field. Therefore, we adopt a calibration strategy where we calibrate the 3C220 field first and then use the output calibration products to calibrate the NCP field, given that the bandpass calibration solutions should be similar between the fields because of the same electronics, and that any effect of the beam should be spectrally smooth near the phase centre. A similar technique to calibrate the LOFAR-LBA data to study the ionospheric effects is shown in de Gasperin et al. (in preparation) and \cite{degasperin2018}. Similar types of calibration strategies are more common in radio survey experiments, although in those cases it is often required to switch between sources in time. The calibration settings (e.g. solution interval, frequency resolution, ADMM iterations, regularization factor) for the two fields are chosen to account for any rapidly varying effects in time and frequency such as the ionosphere while maintaining a reasonable signal to noise ratio per solution interval. Most of these settings are decided on the basis of the analysis and lessons learned in G18 as well as the analysis of the LOFAR-EoR data (see e.g. \citealt{patil2017}).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{continuum.png}
\caption{Left and right panels show Stokes $I$ continuum `dirty' images ($39-72$ MHz) of the 3C220 and NCP field respectively, after DI calibration. The images are not cleaned, and were produced using $\leq2000\lambda$ baselines with the Briggs -0.1 weighting scheme. The observed image rms is $\sigma_{\text{rms}}\sim 7$ mJy for the 3C220 field and $\sim 5.5$ mJy for the NCP field respectively. These values are still $\sim 10$ times higher than the expected rms ($\sim 0.7$ mJy) calculated using SEFD (System Equivalent Flux Density) estimates for LOFAR-LBA. The values of $\sigma_{\text{rms}}$ can be calculated from SEFD using the relation: $\sigma_{\text{rms}} = \text{SEFD}/\sqrt{2N(N-1)\Delta \nu \Delta t}$, where SEFD $\sim 30$ kJy at 55 MHz, $N = 29$ (corresponding to $2000\lambda$ baseline range), $\Delta\nu = 33$ MHz, $\Delta t = 0.9\times 14$ hours (assuming flagged data at $10\%$ level).}
\label{fig:continuum}
\end{figure*}
\subsection{Calibrating the 3C220 field}\label{subsec:3C220cal}
To calibrate the 3C220 field, we use a calibration strategy similar to that discussed in G18. The 3C220.3 radio galaxy is a double-lobed source of $\sim 8$ arcsec extent, but it is unresolved with the LOFAR-LBA array which has a maximum resolution of $\sim 15$ arcsec. Therefore, we use a single point source representing 3C220.3 with $38$ Jy flux at 74 MHz \citep{cohen2007} and a spectral index of -0.8 as a starting model for DI calibration. The major steps involved in the calibration of the 3C220 field are as follows:
\begin{enumerate}
\item Calibrate the raw visibilities using the 3C220.3 point source model in \texttt{NDPPP} \footnote{\url{http://www.lofar.org/operations/doku.php?id=public:user_software:ndppp}} to obtain the station gain solutions with 30 seconds and 61 kHz calibration solution intervals and subsequently apply them to the data. This step is performed separately for each sub-band (without consensus optimisation). We include the primary beam\footnote{Current LBA primary beam models are based on Electro-Magnetic (EM) simulations of the LOFAR-LBA dipoles (private communication with LOFAR Radio Observatory).} in the calibration step in \texttt{NDPPP}. Note that the LOFAR-LBA beam model has only been implemented in \texttt{NDPPP} at present. Hence, it is utilised for primary DI calibration for both fields. Note that we do not exclude any baselines during DI calibration steps for both fields.
\item Deconvolve (clean) and image the calibrated visibilities using the \texttt{WSClean} package \citep{offringa2014} with the following settings: cleaning threshold = $0.5\sigma$, weighting scheme = uniform, imaging baseline range = $0-5000\lambda$, 4th order linear polynomial\footnote{Using log polynomials to fit source spectra is unstable in \texttt{WSClean}. Therefore, we use an ordinary 4th order linear polynomial to fit source spectra. However, \texttt{SAGECal-CO} is only compatible with log polynomials. Therefore, we separately fit the source spectra with a 3rd order log-polynomial to make it compatible with \texttt{SAGECal-CO}.} for fitting the source spectrum over 15 points which correspond to averaged flux over 2.2 MHz bands spread within 33 MHz bandwidth. The cleaning parameters are chosen such that the modelled sources with lowest flux values are still a factor of few above the confusion limit at 60\,MHz ($\sigma_c \sim 10 \text{mJy/beam}^{-1}$, see \citealt{vanhaarlem2013} for calculation of $\sigma_c$). Since we do not apply the primary beam correction during imaging, the source fluxes are apparent and their spectra also possess the primary beam variations which are less smooth compared to the intrinsic source spectra. Using a 4th order polynomial for spectral fitting easily captures these beam variations compared to a lower order polynomial and facilitates better source subtraction. Step (i) is repeated once more using the clean model of 3C220.3 obtained in step (ii) and deconvolution is performed to obtain a more accurate 3C220.3 clean model. Further iterations were not required as the model converged.
\item Use \texttt{SAGECal-CO} to perform DI calibration of raw visibilities and subtract 3C220.3 using consensus optimisation (7 iterations and regularization factor of 5) over a 33 MHz frequency range. We provide the final clean model of 3C220.3 obtained after step (ii) as input to \texttt{SAGECal-CO} and use a calibration solution interval of 30 seconds and 183.1 kHz. The obtained gain solutions are subsequently applied to the residual visibilities.
\item Repeat the deconvolution with the same settings (but with lower clean-mask = $4\sigma$) in step (ii) to clean and image the residual visibilities after step (iii). The output clean model of the radio sources in the field contains 1270 components (points plus Gaussians) with flux $> 40$ mJy at 55 MHz. We repeated Step (iii) with this updated sky-model to perform DI-calibration and subtraction of 3C220.3 from the visibilities. Using a more complete sky-model in DI calibration allows for the mitigation of calibration errors due to unmodeled sources and produces accurately calibrated visibilities. The gain solutions obtained after this step are later utilised in the calibration of the NCP field.
\item Use DD-calibration with \texttt{SAGECal-CO} to subtract the clean-model obtained in step (iv). \texttt{SAGECal-CO} accounts for DD errors by obtaining the gain solutions in multiple directions. It subtracts the sources in each direction by multiplying the obtained gain solutions with the predicted visibilities and subtracting the product from the observed visibilities. We divide the 1270 components into 4 clusters using the weighted K-means clustering algorithm \citep{kazemi2013a} and use the centres of these clusters as four different directions. This algorithm operates on a unit sphere and the corresponding weights are determined by the source flux. The algorithm creates clusters such that the cluster size is minimized while maintaining similar net flux in each cluster to ensure sufficient signal to noise ratio for each cluster. We use a gain solution interval of 20 minutes and 183.1 kHz and 20 ADMM iterations for each gain solution while keeping the same regularization factor of $\rho=5$ \citep{yatawatta2016} as in DI calibration. We discard the baselines $\leq 200\lambda$ in the DD-calibration. These excluded baselines ($<200\lambda$) are used for further analyses and power spectrum estimation. Using a baseline cut avoids any bias due to unmodeled diffuse emission on shorter baselines excluded from calibration (see e.g. \citealt{patil2016,barry2016,ewall-wice2017,gehlot2018} for more details). It also mitigates the suppression of the 21-cm signal caused by the use of an incomplete sky-model in the calibration, as shown in \cite{patil2017} and \cite{sardarabadi2018}, and we will test this further in future. However, the exclusion of short baselines from the calibration also has a drawback that longer baselines are prone to calibration errors. These errors, when applied to excluded baselines, cause the foregrounds to leak into the ``EoR-window" on corresponding baselines \citep{barry2016,patil2016}. Using the smoothness of gain solutions as a constraint in the calibration largely mitigates this effect \citep{sardarabadi2018}. An optimal baseline selection criteria for calibration which accounts for these effects itself requires fairly rigorous analysis and is beyond the scope of this paper. The choice of $200\lambda$ baseline cut comes from the reason that the radial profile of the $uv$-coverage (see right panel of figure \ref{fig:uvcov}) is relatively flat within $20\lambda \leq |\vect{u}| \leq 200\lambda$ range and drops for longer baselines. This is an optimal choice for power spectrum estimation because of relatively uniform $uv$-coverage.
\item Image the residual visibilities in step (v) with \texttt{WSClean}. We used the following settings: weighting scheme = natural, pixel size = 3 arcmin, Image dimensions = $300\times 300$ pixels, imaging baselines = $15 - 200 \lambda$. Note that we do not deconvolve the final residual images. The output Stokes $I$, $V$ and Point Spread Function (PSF) images were stored for further analysis. The left panel of figure \ref{fig:continuum} shows the dirty continuum image of the 3C220 field after DI calibration where the 3C220.3 has been subtracted.
\end{enumerate}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Noise-PS_1ch12s_b_nu_new.png}
\caption{Stokes $I$ ($P_{\Delta_t I}(|\vect{u}|,\nu)$) and $V$ ($P_{\Delta_t V}(|\vect{u}|,\nu)$) noise spectra for the 3C220 field. Left and right panels correspond to Stokes $I$ and $V$ respectively.}
\label{fig:noise_PS_IV}
\end{figure*}
\subsection{Calibrating the NCP field}\label{subsec:NCPcal}
The absence of very bright sources makes the NCP field more difficult to calibrate using the strategy we employed for the 3C220 field. Therefore, we utilise a different approach. The NCP field consists of a moderately bright source (3C061.1) which lies at the edge of the primary beam causing the source to exhibit peculiar behaviour in its gain solutions. We, therefore, subtract 3C061.1 from the raw visibilities using DD-calibration with \texttt{SAGECal-CO} with the same settings as we employed for the Cas\,A and Cyg\,A subtractions. The 3C061.1 input model is adapted from the intrinsic model of 3C061.1 (points + shapelets, at 150 MHz) used in the LOFAR-EoR data processing pipeline (see e.g. \citealt{patil2017}). The fluxes in the model were scaled properly to match the flux values quoted in \cite{laing1980} and \cite{hales1995}. After subtraction of 3C061.1, visibilities were calibrated using the following steps:
\begin{enumerate}
\item Apply the DI gain solution amplitudes of the 3C220 field obtained in step (iv) in section \ref{subsec:3C220cal} to the NCP field visibilities to set the amplitude scale.
\item Deconvolve (clean) and image the resulting visibilities using \texttt{WSClean} with the following settings: cleaning threshold = $0.5\sigma$, weighting scheme = uniform, imaging baseline range = $0-2000\lambda$, 2nd order polynomial for fitting the source spectrum over 5 points which correspond to an averaged flux over 6.6 MHz bands spread over 33 MHz.
\item Perform DI calibration of the visibilities with \texttt{SAGECal-CO} using consensus optimisation (with same settings as in DI calibration of the 3C220 field) over the 33 MHz frequency range. The clean model obtained in step (ii) is provided as input. We use a calibration solution interval of 10 minutes and 183.1 kHz. The obtained gain solutions are subsequently applied to the visibilities. We repeat steps (ii) and (iii) in a self-cal loop with 3 iterations. The final clean model after 3 self-cal iterations contains 1470 components (points plus Gaussians) with flux $> 40$ mJy at 55 MHz.
\item Perform phase calibration using \texttt{NDPPP} on the visibilities obtained after step (i). We use the final clean model obtained after step (iii) as input and choose 30 seconds, 183.1 kHz as the calibration solution interval.
\item Use DD-calibration with \texttt{SAGECal-CO} to subtract the clean-model obtained in step (iii). We divide 1470 components into three clusters representing three directions (which represent three non-overlapping regions within the primary beam) using the weighted K-means clustering algorithm (same as in step (v) of section \ref{subsec:3C220cal}). We use a gain solution interval of 20 minutes and 183.1 kHz and 20 ADMM iterations for each gain solution. We discard the baselines $\leq 200\lambda$ to avoid errors due to unmodeled diffuse emission on shorter baselines and to avoid signal suppression.
\item Image the residual visibilities in step (v) with \texttt{WSClean} using the following settings: weighting scheme = natural, pixel-size = 3 arcmin, Image dimensions = $300\times 300$ pixels, imaging baselines = $15 - 200 \lambda$. The output Stokes $I$, $V$ and PSF images were stored for further analysis. The right panel of figure \ref{fig:continuum} shows the dirty continuum image of the NCP field after DI calibration.
\end{enumerate}
We only use the beam model during the DI-calibration and image deconvolution steps, and we do not correct the residual images for the primary beam. Also, we do not analyse the Stokes $Q$ and $U$ data. The latter is mainly because we do not include any polarised (compact or diffuse) emission in sky-model used for the calibration. Any unmodeled emission in Stokes $Q$ and $U$ can essentially bias the calibration. The currently utilised calibration scheme is defined such that only the Stokes $I$ and $V$ are constrained by the sky-model, whereas, the Stokes $Q$, $U$ have the freedom to rotate. Moreover, in the RM-synthesis analysis in G18, we did not observe any signature of the polarised emission in RM-space, suggesting the absence of significant polarised emission at these low frequencies. Because the Rotation Measure scales as $\lambda^2$, any polarised emission at low frequencies (40-70\,MHz) is essentially depolarised by the intervening magneto-ionic medium and the rapidly varying ionosphere. However, in G18, we observed strong polarization leakage of the bright off-axis source Cas\,A from Stokes $I$ to $Q$, $U$, and also in Stokes $V$ but at a much weaker level compared to Stokes $Q$ and $U$. This effect is mitigated by subtraction of Cas\,A and Cyg\,A during using DD-calibration at higher time resolution (30 seconds) and is already accounted for in the current analysis. The leakage of (partly) polarised foregrounds to Stokes $I$, if any, is expected to be significantly lower than the current noise level and currently does not affect our analysis.
At this point, we have residual data cubes that are DI calibrated and where the sky model has been subtracted using their DD gain solutions. These residual cubes form the input for subsequent analyses. In the following sections, we will discuss these analyses.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{ratio_1ch12s_b-nu.png}
\caption{The ratio $P_{\Delta_t I} / P_{\Delta_t V}$ of the noise spectra shown in figure \ref{fig:noise_PS_IV}. The ratio is flat except for a few outliers at shorter baselines.}
\label{fig:Ratio_noise_PS_IV}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Hist_combined-field.png}
\caption{The left and the right panels show normalised histograms of the distribution of the ratio values $P_{\Delta_t I} / P_{\Delta_t V}$ for the 3C220 and the NCP fields, respectively. The red and blue vertical lines represent the median and the mean of the distribution respectively. For the 3C220 field, the distribution has a median value of $1.46$ and a mean value of $1.52$. Similarly, the median and the mean values for the NCP field are $1.32$ and $1.38$ respectively.}
\label{fig:Hist_noise-ratio}
\end{figure*}
\section{Noise statistics in LOFAR-LBA}\label{sec:LBAnoise}
Current estimates of the average Signal Equivalent Flux Density (SEFD) per station of the LOFAR-LBA array are derived from the observations of bright sources at zenith. However, the SEFD of LOFAR varies as a function of angle from the zenith. Therefore, using zenith SEFD estimates to derive the noise on the visibilities and rms in the images (also noise power spectra) typically underestimates the SEFD for the fields away from the zenith. To avoid this bias, we estimate the noise and hence the noise spectrum (in baseline-frequency space) for the 3C220 field from the visibilities. A standard method to estimate the noise on visibilities is to subtract the un-gridded visibilities corresponding for two contiguous time-steps at the highest time resolution. However, this method is not feasible for large LOFAR-LBA datasets ($\sim18$ TB per dataset) because of a large number of baselines and time-steps. Therefore, we use an alternative approach where we estimate the noise spectrum from the gridded visibilities (see e.g \citealt{jacobs2016,beardsley2016,ewall-wice2016}). We expect that these two approaches become equivalent to each other for datasets with a large number of time-samples and leave additional comparison tests between the two approaches for future analyses.
We split the DI-calibrated visibilities of the 3C220 field into even and odd samplings with 12 seconds cadence such that these samplings are interleaved in time. Note that for the baseline range $20\lambda \leq |\vect{u}|\leq 200\lambda$ which we probe in our analysis, the sky and the PSF do not vary over a 12 second interval. Also, any sky leakage over 12 seconds will appear as a wedge in the cylindrically averaged power spectrum, which we do not observe in the analyses (shown in later sections). Moreover, we expect the system to be coherent over 12 seconds and only ionospheric effects are expected to change. We image these even and odd samplings using \texttt{WSClean} with the `natural' weighting scheme. We Fourier Transform (FT) the even and odd image cubes and properly scale visibilities in each $uv$-cell with corresponding sampling density to remove the effect of gridding weights during imaging. We calculate the azimuthally averaged (spatial) power spectrum of the difference as $P_{\Delta_t I}(|\vect{u}|,\nu) \equiv \langle \Delta_{t}\tilde{I} \rangle^2 = \langle \tilde{I}_{\text{even}} - \tilde{I}_{\text{odd}} \rangle^2 / 2$, where $\tilde{I}_{\text{even}}$ and $\tilde{I}_{\text{odd}}$ are the Fourier transforms of the even and odd Stokes $I$ image cubes respectively, $\vect{u} = (u,v)$ is the baseline vector (in units of wavelength) in the $uv$-plane and $|\vect{u}| = \sqrt{u^2 + v^2}$ and $\nu$ is the frequency. Similarly, $P_{\Delta_t V}(|\vect{u}|,\nu)\equiv \langle \Delta_{t}\tilde{V} \rangle^2$ is determined using the even and odd Stokes $V$ image cubes.
\subsection{Physical Excess Noise}\label{subsec:ExcessNoise}
Figure \ref{fig:noise_PS_IV} shows $P_{\Delta_t I}$ and $P_{\Delta_t V}$ for the $20-200 \lambda$ baseline range for the 3C220 field. We observe that both $P_{\Delta_t I}$ and $P_{\Delta_t V}$ spectra are relatively flat. The bright tilted streaks are a consequence of varying $uv$-density as a function of baseline length in LOFAR-LBA. We compare $P_{\Delta_t I}$ and $P_{\Delta_t V}$ by calculating their ratio. Figure \ref{fig:Ratio_noise_PS_IV} shows the ratio $P_{\Delta_t I} / P_{\Delta_t V}$ of the spectra shown in figure \ref{fig:noise_PS_IV}. We observe that the ratio is remarkably flat, except for a few outliers at shorter baselines ($\leq 40 \lambda$). Most of these outliers are also coincident with baselines where $uv$-density is relatively low. These outliers might arise due to imperfect calibration and slight differences in flagging of RFI affected baselines post calibration along with $uv$-density variations. Ideally, if the noise properties of Stokes $I$ and $V$ are statistically identical and if the sky and the PSF do not change over a 12 seconds interval, $P_{\Delta_{t} I}$ and $P_{\Delta_{t} V}$ are expected to be identical assuming that the sky has a negligible circular polarised emission component and Stokes $V$ is virtually empty. However, we observe excess power in Stokes $I$ compared to Stokes $V$, which is largely constant over the $20 - 200\lambda$ baseline range and over the 30 MHz bandwidth. Although the power in both Stokes $I$ and $V$ varies slightly with increasing baseline length, the ratio remains constant, suggesting that this slight variation is a result of varying $uv$-density. Most correlations that are spectrally smooth, e.g. due to intrinsic foregrounds, instrumental mode-mixing and ionosphere, appear as a ``wedge" in the two-dimensional power spectrum. Whereas systematic effects with specific spectral structure e.g. cable reflections may appear as a distinct feature above the ``wedge". However, only those effects that are non-smooth in frequency or possess noise-like behaviour affect most scales in the two-dimensional power spectrum. In later sections (see section \ref{subsec:2Dcompare_with_noise}) we show that the corresponding 2D power spectra for Stokes $I$ and $V$ noise estimates are featureless and devoid of any ``wedge" like structure or other distinct features corresponding to systematic effects. Hence this physical excess noise, for all practical purposes, is treated as additional white noise in Stokes $I$ that is seemingly uncorrelated in frequency and remains more or less the same for different baseline lengths.
The left panel of figure \ref{fig:Hist_noise-ratio} shows the normalised histogram of the distribution of $P_{\Delta_t I} / P_{\Delta_t V}$ values for the 3C220 field. The distribution has a median value of $1.46$ and a mean value of $1.54$, with most values lying within the range $1-2$. The noise spectra and their ratio for the NCP field also exhibit similar behaviour as the 3C220 field that the ratio $P_{\Delta_t I} / P_{\Delta_t V}$ is flat in frequency-baseline space. However, the distribution of the ratio values (see right panel of figure \ref{fig:Hist_noise-ratio}) has a slightly lower median and mean values of $1.32$ and $1.38$ respectively. The cause of this excess power in $P_{\Delta_t I}$ is still unknown, but it is higher for the 3C220 field which has a bright source at the centre, compared to the NCP field which is devoid of relatively bright sources. We are currently investigating the cause of this excess, but given that the excess is different for the two fields, ionospheric or interplanetary scintillation noise might be a probable reason for this excess, although the rapid decorrelation with frequency remains unexplained.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{dPS_IV_final_new.png}
\caption{\textit{Top panel:} The differential Stokes $I$ and $V$ power spectra calculated using residual images of the 3C220 field. The solid red curve corresponds to $P_{\Delta_{\nu} I}$ and the dashed red curve corresponds to $P_{\Delta_{\nu} V}$. The solid and dashed black curves correspond to $P_{\Delta_{t} I}$ and $P_{\Delta_{t} V}$ respectively, at $\nu = 59.95$ MHz. \textit{Bottom panel:} The ratio $P_{\Delta_{\nu} I}/P_{\Delta_{\nu} V}$ (red curve) and the ratio $P_{\Delta_{\nu} I}/P_{\Delta_{t} I}$ (blue curve). The dotted vertical line shows the location of the $200\lambda$ baseline cut.}
\label{fig:dPS_IV}
\end{figure}
\subsection{Variance at the inter-sub-band level}\label{subsec:excess_compare}
We use the azimuthally averaged power spectrum of the difference of Stokes $I$ and $V$ images between two contiguous sub-bands (differential power spectrum) to study the behaviour of variance at the inter-sub-band level (see e.g. \citealt{patil2016,gehlot2018}). This method is based on the assumption that Stokes $I$ images are composed of total sky signal convolved with the PSF plus additive noise. Assuming that the sky signal, which is smooth in frequency does not change\footnote{ For a spectral index of $-2.55$, sky brightness changes at $\sim 0.8\%$ level for 195 kHz frequency separation at 60 MHz, which has a negligible contribution to the difference.} between two consecutive sub-bands 195 kHz apart, and any contribution due to the sky signal should largely drop out in the differential Stokes $I$ images. Similarly, differential Stokes $V$ images should contain only noise. However, effects which are non-smooth at the sub-band level are expected to leave their signature in the differential Stokes images.
We use Stokes $I$ and $V$ residual images of the 3C220 field ($\nu_1 = 59.76$ MHz and $\nu_2 = 59.95$ MHz, located at the most sensitive part of the band) after DD-calibration to estimate the differential power spectra $P_{\Delta_{\nu} I}$ and $P_{\Delta_{\nu} V}$, and determine their ratio $P_{\Delta_{\nu} I}/P_{\Delta_{\nu} V}$. The top panel of figure \ref{fig:dPS_IV} shows $P_{\Delta_{\nu} I}$ (red solid curve) and $P_{\Delta_{\nu} V}$ (red dashed curve). We also show a slice of $P_{\Delta_{t} I}$ and $P_{\Delta_{t} V}$ at $59.95$ MHz in the same plot for comparison. We observe that the power spectra are more or less flat on baselines $|\vect{u}| > 200\lambda$ and increase rapidly for $|\vect{u}| > 200\lambda$. This can be attributed to variations in the $uv$-coverage of LOFAR-LBA. The variations in these power spectra also correlate well with the $uv$-coverage profile shown in figure \ref{fig:uvcov}.
The bottom panel of figure \ref{fig:dPS_IV} shows the ratio $P_{\Delta_{\nu} I}/P_{\Delta_{\nu} V}$ (red curve) and the ratio $P_{\Delta_{\nu} I}/P_{\Delta_{t} I}$ (blue curve). We also observe that the ratio $P_{\Delta_{\nu} I}/P_{\Delta_{\nu} V}$ is relatively flat and has value $\sim 2-3$ over the presented baseline range. This suggests that the rapid upturn in the power spectra shown in the top panel is due to $uv$-coverage variations and cancels out in the ratio. The excess variance in $P_{\Delta_{\nu} I}$ compared to $P_{\Delta_{\nu} V}$ is possibly due to random errors produced in calibration and/or erratic ionosphere. These random errors when applied to the data or the sky-model during subtraction, affect both Stokes $I$ and $V$. However, these errors lead to larger variance when applied to the emission in Stokes $I$ compared Stokes $V$ which lacks any emission (or below thermal noise, if any), resulting in excess noise at sub-band level.
The ratio we observe here is considerably smaller than that we observed in G18 ($P_{\Delta_{\nu} I}/P_{\Delta_{\nu} V} \gtrsim 10$). This lower Stokes $I$ noise is in part achieved because \texttt{SAGEcal-CO} enforces frequency smoothness of the gain solutions in the calibration process, and also because the ionospheric activity is more benign compared to the observation in G18 where frequency smoothness was not enforced in the calibration and the ionosphere behaved erratically. To quantify the ionospheric activity, we use ionospheric Rotation Measure (RM) estimates from the GPS data. The ionospheric RM levels for the current observation are of order $\sim0.1-0.15\,\text{rad\,m}^{-2}$ during 90\% of the observation which is $\gtrsim 4$ times lower than the ionospheric RM levels (RM $> 0.4\,\text{rad\,m}^{-2}$) for the observation in G18, suggesting benign ionospheric activity.
Furthermore, from comparison of $P_{\Delta_{\nu} I}$ with $P_{\Delta_{t} I}$, we observe that there is a sudden jump in the ratio at $|\vect{u}|\sim 200\lambda$. The ratio is $\gtrsim 2$ for $|\vect{u}| < 200$ and it continues to increase as the baseline length decreases, compared to the values ($\sim 1-2$) for $|\vect{u}| > 200$. We attribute this effect to the $200\lambda$ baseline cut used in the DD-calibration. The sky-model incompleteness or ionospheric effects can introduce random errors during the calibration step. These random errors on gain solutions when applied to the baselines excluded during the calibration step, increase the variance in Stokes $I$ compared to Stokes $V$ on excluded baselines \citep{patil2016,barry2016,ewall-wice2016,ewall-wice2017,gehlot2018,sardarabadi2018}.
\section{Gaussian Process Regression}\label{sec:GPR}
After subtracting the calibration sky-model using DD-calibration, any remaining foreground emission within the primary beam consists of unresolved sources, sources below the confusion noise, sources not included in the model, and diffuse emission. These foregrounds should vary slowly with frequency, making them separable from the 21-cm signal which has rapid spectral fluctuations. We use a Gaussian Process Regression (GPR) technique (see \citealt{mertens2018} for more details) to remove this remaining foreground emission and other spectral structures imparted on the data due to instrumental mode-mixing, such as instrumental chromaticity and imperfect calibration residuals. In this section, we briefly describe GPR and its application to LOFAR-LBA data.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{3C220-foregrounds.png}
\caption{The 3C220 field Stokes $I$ image cube slices (in brightness temperature units) across the centre of a spatial axis after different processing steps. \textit{Left panel:} A slice of the image cube after DD-calibration (data). \textit{Middle panel:} The GPR model of the smooth foregrounds (intrinsic + mode-mixing) corresponding to the data. \textit{Right panel:} The residuals after subtracting the GPR model from the data. The dashed black lines represent the frequency range ($54-68$ MHz) used for power spectrum estimation. The residuals after GPR are featureless except for a few outliers.}
\label{fig:3C220-foregrounds}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{NCP-foregrounds.png}
\caption{As figure \ref{fig:3C220-foregrounds} but for the NCP field. Similar to the 3C220 field, the residuals in the NCP field after GPR are featureless.}
\label{fig:NCP-foregrounds}
\end{figure*}
\subsection{Methodology}\label{subsec:GPR-method}
The visibilities observed by an interferometer ($\mathcal{V}_{\text{obs}}(\vect{u},\nu)$) can be written as a sum of different components viz. the signal of interest ($\mathcal{V}_{21}(\vect{u},\nu)$), the foreground contribution ($\mathcal{V}_{\text{sky}}(\vect{u},\nu)$), instrumental mode-mixing ($\mathcal{V}_{\text{mix}}(\vect{u},\nu)$) and noise ($\mathcal{V}_{\text{n}}(\vect{u},\nu)$), i.e.
\begin{equation}
\mathcal{V}_{\text{obs}}( \vect{u} ,\nu) = \mathcal{V}_{21}(\vect{u},\nu) + \mathcal{V}_{\text{sky}}(\vect{u},\nu) + \mathcal{V}_{\text{mix}}(\vect{u},\nu) + \mathcal{V}_{\text{n}}(\vect{u},\nu).
\end{equation}
Each of these components has a distinct spectral behaviour which is exploited by GPR to separate them from each other and eventually remove the foreground components from the observed visibilities, leaving residuals with the signal of interest buried below the noise \citep{mertens2018}. GPR models these different components with Gaussian Processes (GP). A GP ($f \sim \mathcal{GP} (m,\kappa)$) is the joint distribution of a collection of normally distributed random variables and is defined by its mean $m$ and covariance function $\kappa$. Values of $\kappa$ specify the covariance between pairs of points at different frequencies and determine the structure of the function (e.g. its smoothness in frequency) which can be modelled with a GP. GPs are often described by parameterized priors in GPR, and the GP prior is selected such that it maximises the Bayesian evidence, estimated by conditioning these priors to the observations (see \citealt{rasmussen2005} for an extensive review). The parameters of the covariance functions (also known as `hyper-parameters') can be estimated using standard optimisation or MCMC algorithms. The observed data ${\bf d}$, being a stacked set of gridded visibilities as a function of frequency, can be modelled as
\begin{equation}
{\bf d} = {\bf f}_{\text{fg}}(\nu) + {\bf f}_{21}(\nu) + {\bf n},
\end{equation}
where ${\bf f}_{\text{fg}}(\nu)$ corresponds to the foreground component, ${\bf f}_{21}(\nu)$ corresponds to the signal of interest and ${\bf n}$ is the noise. The 21-cm signal is expected to decorrelate over frequency scales > 1 MHz, whereas foregrounds are expected to be smooth on 1 MHz scales and decorrelate over a much larger frequency range. The covariance function $K\equiv \kappa$ for this model can be written as a sum of foreground covariance function $K_{\text{fg}}$ and a 21-cm signal covariance function $K_{21}$, i.e.
\begin{equation}
K = K_{\text{fg}} + K_{21}.
\end{equation}
$K_{\text{fg}}$ can further be decomposed into $K_{\text{int}}$, which corresponds to intrinsic foregrounds (large-scale correlation of $\sim 10 - 100$ MHz) and $K_{\text{mix}}$, which corresponds to instrumental mode mixing such as side-lobe noise (decorrelates within $\sim 1-5$ MHz). The joint probability distribution for the observed data ${\bf d}$ and function values ${\bf f}_{\text{fg}}$ of the foreground model at the same frequency $\nu$ is then given by
\begin{equation}
\begin{bmatrix} {\bf d} \\ {\bf f}_{\text{fg}} \end{bmatrix} \sim \mathcal{N} \left( \ \begin{bmatrix} 0 \\ 0 \end{bmatrix} , \begin{bmatrix} K_{\text{fg}} + K_{21} + \sigma_n^2 I & K_{\text{fg}} \\ K_{\text{fg}} & K_{\text{fg}} \end{bmatrix} \ \right),
\end{equation}
where $\sigma_n^2$ is the noise variance, $I$ is the identity matrix and $K\equiv K(\nu,\nu)$. After GPR, the foreground model can be retrieved as
\begin{gather}
E({\bf f}_{\text{fg}}) = K_{\text{fg}}\left[ K + \sigma_n^2 I \right]^{-1} {\bf d}, \\
\text{cov}({\bf f}_{\text{fg}}) = K_{\text{fg}} - K_{\text{fg}} \left[ K + \sigma_n^2 I \right]^{-1} K_{\text{fg}},
\end{gather}
where $E({\bf f}_{\text{fg}})$ and $\text{cov}({\bf f}_{\text{fg}})$ are the expectation values and covariance of the foregrounds respectively. The residuals ${\bf d}_{\text{res}}$ after foreground model subtraction are
\begin{equation}
{\bf d}_{\text{res}} = {\bf d} - E({\bf f}_{\text{fg}}).
\end{equation}
Readers may refer to \cite{mertens2018} for a detailed description of the GPR technique and its application as a novel method for foreground removal and 21-cm signal estimation.
\subsection{Application of GPR to the LOFAR-LBA data}\label{subsec:GPR-application}
We use GPR to remove remaining foreground emission from the residual visibilities after DD calibration. We test various covariance functions as kernels to model different components of the residual visibilities in GPR. We use a Bayesian framework to compare different covariance functions and select those models that maximise the marginal likelihood (or Bayesian evidence). We tested several GPR covariance kernels e.g. Radial Basis Functions, Rational Quadratic functions and different classes of Matern covariance functions to model foreground components and finally selected the ones with the maximum Bayesian evidence.
To model the intrinsic foreground emission (unmodeled sources and diffuse emission) we choose a RBF(Radial Basis Function) covariance function as kernel. The RBF kernel is essentially a square exponential or Gaussian function defined as:
\begin{equation}
\kappa_{\text{RBF}} (\nu_{\text{p}},\nu_{\text{q}}) = \exp \left( \dfrac{-|\nu_{\text{q}} - \nu_{\text{p}}|^2}{2l^2} \right)
\end{equation}
where $l$ is the characteristic coherence scale in frequency. We use $5-100$ MHz as the prior for the range of coherence scales of the intrinsic foregrounds. To model the mode-mixing component of the foregrounds, we choose a Rational Quadratic (RQ) covariance function defined as:
\begin{equation}
\kappa_{\text{RQ}} (\nu_{\text{p}},\nu_{\text{q}}) = \left( 1 + \dfrac{|\nu_{\text{q}} - \nu_{\text{p}}|^2}{2\alpha l} \right)^{-\alpha} \ ,
\end{equation}
where $l$ is the coherence scale and $\alpha$ is the so-called power-parameter. RQ functions can be understood as infinite sums of Gaussian covariance functions with characteristic coherence scales \citep{rasmussen2005}. We use $1-8$ MHz as prior values for the coherence scales and $\alpha = 0.1$ for the mode-mixing component. To account for the 21-cm signal, we use an Exponential covariance function, which is a special case of a Matern class covariance function \citep{stein1999} defined as:
\begin{equation}
\kappa_{\text{matern}} (\nu_{\text{p}},\nu_{\text{q}}) = \dfrac{2^{1-n}}{\Gamma (n)} \left[ \dfrac{\sqrt{2n}|\nu|}{l} \right]^{n} K_{n} \left( \dfrac{\sqrt{2n}|\nu|}{l} \right) \ ,
\end{equation}
where $|\nu| = |\nu_{\text{q}} - \nu_{\text{p}}|$ and $K_{n}$ is the modified Bessel function of the second kind (not to be confused with the covariance functions defined in section \ref{subsec:GPR-method}). The `hyper-parameter' $l$ represents the characteristic coherence scale. Special classes of Matern covariance functions can be obtained by choosing various values for $n$, e.g. choosing $n = 1/2$ gives an exponential kernel. We use a frequency coherence scale of $0.01-1.5$ MHz for the 21-cm signal with an initial value of 0.5 MHz. Using \texttt{21cmFAST} simulations \citep{mesinger2007,mesinger2011}, \cite{mertens2018} have shown that these coherence scales covers a wide range of possible 21-cm signal models.
We use the residual image-cubes obtained after DD-calibration for foreground removal. We perform GPR foreground removal on the inner $3.5^{\circ}\times 3.5^{\circ}$ region of the image cubes (which is slightly smaller than the primary beam FWHM $\sim 4^{\circ}$) to limit sky curvature and primary beam effects. We selected the $50-72$ MHz frequency range for GPR foreground removal, which is 8 MHz wider than the power spectrum estimation window, for better foreground fitting and removal. Figure \ref{fig:3C220-foregrounds} shows slices through the Stokes $I$ image cubes for the 3C220 field across the center of one of the two spatial axes before GPR (left panel), the GPR foreground fit (middle panel) and the residuals after subtracting the foreground model obtained with GPR from the data (right panel). Similarly, figure \ref{fig:NCP-foregrounds} shows the slices of Stokes $I$ image cubes for the NCP field. We observe that the Stokes $I$ residuals after foreground removal with GPR for both the 3C220 and NCP fields are featureless (except for a few outliers) and do not appear to have spatial or spectral structure. In the following section, we use these residuals after GPR to create cylindrically and spherically averaged power spectra for the 3C220 and the NCP fields.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{3C220_ps2d_IV-mod_new.png}
\caption{The cylindrically averaged Stokes $I$ and $V$ power spectra for the 3C220 field. \textit{Top row (left to right):} $P_{I}(k_{\perp},k_{\parallel})$ before foreground removal, GPR foreground model, and after foreground removal with. \textit{Bottom row (left to right):} Same as top row but for Stokes $V$. The solid grey lines correspond to a $5^{\circ}$ field of view which is slightly larger than the primary beam FWHM at 60 MHz. The dashed grey lines correspond to the instrumental horizon.}
\label{fig:3C220-ps2d_IV}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{3C220_ps2d_ratio.png}
\caption{The ratio of the cylindrically averaged Stokes $I$ and $V$ power spectra for the 3C220 field. \textit{Left panel:} $P_{I}/P_{V}$ before foreground removal with GPR. \textit{Right panel:} $P_{I}/P_{V}$ after foreground removal with GPR.}
\label{fig:3C220-ps2d_ratio}
\end{figure*}
\section{Power Spectra Results}\label{sec:PSpec-results}
In this section we present the cylindrically and spherically averaged power spectra for the 3C220 and NCP fields. Cylindrically averaged power spectra (or 2D cosmological power spectra) are widely used in 21-cm experiments to assess various 21-cm signal contaminants such as foregrounds, side-lobe noise and systematic biases. Cylindrically averaged power spectra ($P(k_{\perp},k_{\parallel})$) are defined as \citep{parsons2012,thyagarajan2015a}:
\begin{equation}
P(k_{\perp},k_{\parallel}) = \dfrac{X^2 Y}{\Omega_A B} \langle | \tilde{\mathcal{V}}(\vect{u},\eta) |^2 \rangle ,
\end{equation}
where $\tilde{\mathcal{V}}(\vect{u},\eta)$ is the FT of the visibilities in the frequency direction, $\Omega_A$ is the integral of the square of the primary beam over solid angle across the sky, and $B$ is the bandwidth of the visibility cube. $X$ and $Y$ are the conversion factors from angle and frequency to transverse co-moving distance ($D(z)$) and the co-moving depth along the line of sight ($\Delta D$), respectively. The wave numbers $k_{\perp}$ and $k_{\parallel}$ are related to baseline vector ($\vect{u} = (u,v)$ in units of wavelength) and the Fourier dual to frequency ($\eta$) as:
\begin{equation}
k_{\perp} = \dfrac{2\upi \left( |\vect{u}| \right)}{D(z)} , \ \ k_{\parallel} = \dfrac{2\upi \nu_{21} H_{\text{0}} E(z)}{c(1+z)^2} \eta \ ,
\end{equation}
where $\nu_{21}$ is the rest frame frequency of the 21-cm spin flip transition of HI, $z$ is the redshift corresponding to the observation frequency, $c$ is the speed of light, $H_0$ is the Hubble constant and $E(z) \equiv \left[ \Omega_M(1+z)^3 + \Omega_k(1+z)^2 + \Omega_{\Lambda}\right]^{1/2} $ is a function of the standard cosmological parameters. The spherically averaged dimensionless power spectrum can be estimated from $P(k_{\perp},k_{\parallel})$ as:
\begin{equation}
\Delta^2(k) = \dfrac{k^3}{2\pi^2} P(k) ,
\end{equation}
where $k = \sqrt{k_{\perp}^2 + k_{\parallel}^2}$. We determine $P(k_{\perp},k_{\parallel})$ for both the 3C220 and NCP fields using the gridded visibility cubes of the $3.5^{\circ}\times 3.5^{\circ}$ region of the image cubes with 14 MHz bandwidth (54-68 MHz), corresponding to the redshift range $z=19.8 - 25.2$. We weigh the $uv$-cells using an empirical weighting scheme where $uv$-cells in Stokes $I$ and $V$ with higher noise are down-weighted. In this scheme, the weights are scaled by the inverse of the per-visibility variance $\sigma^2[\mathcal{V}(u,v)]$ in each $uv$-cell estimated by computing robust variance statistics of Stokes $V$ along the frequency direction as:
\begin{equation}
\sigma^2[\mathcal{V}(u,v)] = \{ \hat{\sigma}_{\nu}[\sqrt{N_{\text{vis}}(u,v,\nu)} \times \mathcal{V}_{V}(u,v,\nu)]\} ^2
\end{equation}
where $\mathcal{V}_{V}(u,v)$ are the gridded Stokes $V$ visibilities, $N_{\text{vis}}(u,v,\nu)$ is the number of visibilities in a $uv$-cell and $\hat{\sigma}_{\nu}$ is a robust standard deviation estimator. The weights ($W$) are then given by:
\begin{equation}
W(u,v,\nu) = N_{\text{vis}}(u,v,\nu) \dfrac{\langle \sigma^2[\mathcal{V}(u,v)] \rangle}{\sigma^2[\mathcal{V}(u,v)]}.
\end{equation}
While the per-visibility variance should theoretically be identical in each $uv$-cell, it becomes different in the presence of systematics that can affect baselines differently. This empirical weighting scheme allows one to reduce the impact of those systematics. We emphasise that only Stokes $V$ is used in determining those weights.
To Fourier transform the visibilities along the frequency direction, we use a Least Square Spectral Analysis (LSSA) method (full least squares FT-matrix inversion, see e.g. \citealt{barning1963,lomb1976,stoica2009,trott2016}). We apply a `Hann'\footnote{The `Hann' window is defined as $W(n) = 0.5 -0.5\cos \left[\dfrac{2\pi n}{(M-1)} \right]$, where $0\leq n\leq M-1$ (see e.g. \citealt{blackman1958,harris1978}).} window function to the data prior to the frequency transform to control the side-lobes along the $\eta$ axis, however, this window function somewhat increases the noise. Although using a `Hann' window introduces minor correlations between different $k_{\parallel}$ modes, the residual spectra are similar to the ones produced using a Top-hat window. Therefore, we currently ignore this effect in our analysis. The resulting $\tilde{\mathcal{V}}(\vect{u},\eta)$ cubes after frequency transform are scaled accordingly with the conversion factors $X$ and $Y$ calculated using $\Lambda$CDM cosmological parameters that are consistent with the Planck 2016 results \citep{planck2016}, and then cylindrically and spherically averaged to obtain $P(k_{\perp},k_{\parallel})$ and $\Delta^2(k)$, respectively.
\subsection{The 3C220 field: cylindrical power spectra}\label{sec:PSpec_3C220}
In this section, we examine the cylindrical power spectra for the 3C220 field. The top row of figure \ref{fig:3C220-ps2d_IV} shows $P(k_{\perp},k_{\parallel})$ for Stokes $I$ before foreground removal, the GPR foreground model, and after GPR foreground removal. We observe that the lowest $k_{\parallel}$ bins in Stokes $I$ are dominated by smooth foreground emission (see top left panel) even after subtraction of the sky-model during DD-calibration. This foreground emission is modelled (shown in top middle panel) and effectively removed by the GPR foreground removal method (see top right panel). We also observe an excess power around the horizon line in Stokes $I$ prior to GPR, which is not removed during GPR, suggesting that this excess power has much lower spectral smoothness or decorrelates quickly over time between gridded visibilities and cannot be modelled with a GP with current settings. This structure is reminiscent of the `pitchfork' observed in G18 and is possibly caused by the residuals after Cas\,A and Cyg\,A subtraction. An inaccurate source model, ionospheric effects, the time variation of the primary beam, or a combination of these effects might explain these residuals. We expect the modelling errors to be negligible as their corresponding models are derived from high spatial resolution images and also because Cas\,A and Cyg\,A appear as compact sources on shorter baselines. Ionospheric effects, however, become stronger at lower elevations due to projection effects and subtraction of Cas\,A and Cyg\,A at 30 seconds and 61 kHz calibration resolution might not be sufficient to correct for ionospheric effects, especially on the shorter baselines. Also, the primary beam changes with time as the 3C220 field is tracked. Therefore, a combination of ionospheric effects, beam errors and time variation of the primary beam is likely capable of producing such an effect.
The bottom row of figure \ref{fig:3C220-ps2d_IV} shows $P(k_{\perp},k_{\parallel})$ for Stokes $V$ before foreground removal, the GPR foreground model, and after GPR foreground removal. We observe that the Stokes V power spectrum is featureless before and after foreground removal, which suggests that any foreground emission/leakage in Stokes $V$ is lower than the excess variance in the current data (see bottom middle panel). We also do not observe any visible signature of Cas\,A and Cyg\,A residuals. The vertical bands in Stokes $I$ and $V$ near $k_{\perp} \approx 0.08$ and $0.14$ are due to varying $uv$-density and drop out in the ratio $\frac{P_{I}}{P_{V}}$. The ratio after foreground removal, as shown in the right panel of figure \ref{fig:3C220-ps2d_ratio}, is relatively flat compared to the one before foreground removal, except for the above-mentioned region near the horizon. The ratio has a median value of $2.07$ which is higher than the median value ($\sim 1.46$) observed in the ratio $P_{\Delta_t I} / P_{\Delta_t V}$ for the 3C220 field (see section \ref{subsec:ExcessNoise}). However, it is consistent with the excess at the sub-band level (see section \ref{subsec:excess_compare}) caused by the use of a baseline cut in the DD-calibration.
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{NCP_ps2d_IV-mod_new.png}
\caption{The cylindrically averaged Stokes $I$ and $V$ power spectra for the NCP field. \textit{Top row (left to right):} $P_{I}(k_{\perp},k_{\parallel})$ before foreground removal, GPR foreground model, and after foreground removal with. \textit{Bottom row (left to right):} Same as top row but for Stokes $V$.}
\label{fig:NCP-ps2d_IV}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{NCP_ps2d_ratio.png}
\caption{The ratio of the cylindrically averaged Stokes $I$ and $V$ power spectra for the NCP field. \textit{Left panel:} $P_{I}/P_{V}$ before foreground removal with GPR. \textit{Right panel:} $P_{I}/P_{V}$ after foreground removal with GPR.}
\label{fig:NCP-ps2d_ratio}
\end{figure*}
\subsection{The NCP field: cylindrical power spectra}\label{sec:PSpec_NCP}
In this section, we assess the cylindrical power spectra for the NCP field. Figure \ref{fig:NCP-ps2d_IV} shows $P(k_{\perp},k_{\parallel})$ for Stokes $I$ and $V$. The top left panel shows the power spectrum after DD-calibration, where low $k_{\parallel}$ modes are dominated by the power due to foreground emission. Similar to the 3C220 field, this power is effectively removed with GPR (see top right panel). We do not observe a `pitchfork' in Stokes $I$ or $V$ (presumably) due to Cas\,A and Cyg\,A residuals opposed to the 3C220 field. This might be primarily because the NCP is stationary on the sky and therefore the beam does not change (only rotates) as the observation progresses. It is also likely that the Cas\,A and Cyg\,A are closer to the null for the NCP, causing the power on/around the structure to be below the noise. Similar to the 3C220 field, Stokes $V$ power spectra for the NCP field appear featureless before and after foreground removal (see figure \ref{fig:NCP-ps2d_IV}).
The behaviour of the ratio $\frac{P_{I}}{P_{V}}$ (see figure \ref{fig:NCP-ps2d_ratio}) is also similar to that of the 3C220 field. The ratio becomes relatively flat after foreground removal except for a few outliers at the small $k_{\perp}$ values. The ratio has a median value of $2.10$, which is almost equivalent to the median we observed for the 3C220 field, but higher than the median of the ratio $P_{\Delta_t I} / P_{\Delta_t V}$ for the NCP field (Section \ref{subsec:ExcessNoise}). This excess can also be attributed to the baseline cut in DD-calibration as discussed in Section \ref{subsec:excess_compare}, which we know causes excess power.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ps2d-noise_IV.png}
\caption{The cylindrically averaged Stokes $I$ and $V$ noise power spectra $P_{I}^N$ (left panel) and $P_{V}^N$ (right panel) for the 3C220 field determined from the difference cubes $\Delta_t\tilde{I}$ and $\Delta_t\tilde{V}$ respectively.}
\label{fig:ps2d-noise_IV}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{ps2d-noise_ratio.png}
\caption{The ratio $\frac{P_{I}^N}{P_{V}^N}$ for the 3C220 field. We observe that the ratio has a median value of $1.51$.}
\label{fig:ps2d-noise_ratio}
\end{figure}
\subsection{Comparison with noise power spectra}\label{subsec:2Dcompare_with_noise}
We determine the cylindrically averaged noise power spectra $P_{I}^N$ and $P_{V}^N$ corresponding to the Stokes $I$ and $V$ difference cubes $\Delta_t \tilde{I}$ and $\Delta_t \tilde{V}$ respectively for the 3C220 field (e.g. see section \ref{sec:LBAnoise}), by passing these cubes through the power spectrum estimation pipeline. Note that we do not perform foreground removal on these data-cubes because we expect the sky component to drop out on time scales of 12 seconds. Figure \ref{fig:ps2d-noise_IV} shows $P_{I}^N$ (left panel) and $P_{V}^N$ (right panel). We observe that power in both Stokes $I$ and $V$ is lower for small $k_{\perp}$ values and higher for larger $k_{\perp}$ because the $uv$-density of LOFAR-LBA decreases with increasing baseline length and drops out in the ratio $\frac{P_{I}^N}{P_{V}^N}$ shown in figure \ref{fig:ps2d-noise_ratio}. From comparison of $P_V(k_{\perp},k_{\parallel})$ for the 3C220 and NCP fields with $P_{V}^N$, we notice that $P_V(k_{\perp},k_{\parallel})$ deviates from $P_{V}^N$ for lower $k_{\perp}$ ($<0.1$) values. This deviation in $P_V(k_{\perp},k_{\parallel})$ compared to $P_{V}^N$ can be attributed to the baseline cut used in the DD-calibration, which increases the noise on the baselines excluded from the calibration.
Moreover, $\frac{P_{I}^N}{P_{V}^N}$ has a median value of $1.51$, which is consistent with the median value of $1.46$ for the ratio $P_{\Delta_t I} / P_{\Delta_t V}$. The NCP field, however, has a slightly lower median value of $1.3$. We observe that this excess power in Stokes $I$ for both the 3C220 and the NCP fields at 12 seconds level does not depend on the calibration, and is present at the same level throughout the analysis even after DD-calibration, foreground removal and also in the auto-correlations (results not shown here). This excess is different from the calibration cut induced noise and might have a physical origin. To account for this physical excess noise in the estimation of the spherically averaged power spectrum, we multiply the residual Stokes $V$ gridded visibilities after DD-calibration (since Stokes $V$ is an independent estimator of the thermal noise of the instrument) with the square-root of the median of the ratio $P_{\Delta_t I} / P_{\Delta_t V}$ (calculated in section \ref{subsec:ExcessNoise}) to obtain an estimate of the noise in Stokes $I$. We use the median instead of the mean because the median is a more robust representative of the central tendency of the distribution, whereas the mean is sensitive to outliers and becomes biased in the presence of strong outliers. This excess noise bias corrected Stokes $V$ is used as an estimator for the noise in the data in the foreground removal and spherically averaged power spectrum estimation steps.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{PS3D_IV_new-v2.png}
\caption{The spherically averaged Stokes $I$, $V$ and excess noise bias corrected Stokes $V$ power spectra. \textit{Left panel:} $\Delta_{I}^2$ and $\Delta_{V}^2$ for the 3C220 field before (blue and orange curves respectively) and after (red and purple curves respectively) foreground removal. \textit{Right panel}: $\Delta_{I}^2$ and $\Delta_{V}^2$ for the NCP field using the same colour scheme as in the left panel. The dashed grey and dashed black curves represent noise bias corrected Stokes $V$ power spectrum $\Delta_{I,n}^2$ and noise power spectrum estimate $\Delta_N^2$, respectively, for the corresponding fields. The errorbars represent the $2\sigma$ errors on the power spectra.}
\label{fig:ps3d_IV}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{PS3D_ulim_new-v2.png}
\caption{Noise bias corrected spherically averaged Stokes $I$ power spectra ($\Delta_I^2 - \Delta_{I,n}^2$) for the 3C220 and NCP fields. Blue circles represent the 3C220 field and red crosses represent the NCP field. The errorbars correspond to the $2\sigma$ errors on the power spectra.}
\label{fig:ps3d_ulim}
\end{figure}
\subsection{Spherically averaged power spectra}
We finally determine the Stokes $I$ and $V$ spherically averaged power spectra ($\Delta^2(k)$) for both the 3C220 and NCP fields before and after foreground removal for the redshift range $z = 19.8 - 25.2$. Figure \ref{fig:ps3d_IV} shows Stokes $I$ power spectra $\Delta_{I}^2$ and Stokes $V$ power spectra $\Delta_{V}^2$ for both the 3C220 (left panel) and NCP fields (right panel) before and after foreground removal. We use the physical excess noise bias corrected (using the median values from Section \ref{subsec:ExcessNoise}) Stokes $V$ visibilities as an estimator of the noise component in the Stokes $I$ power spectrum ($\Delta_{I,n}^2$), in order to account for the physical excess noise in Stokes $I$ compared to Stokes $V$. The dashed gray curves in figure \ref{fig:ps3d_IV} represent the excess bias corrected Stokes $V$ power spectra $\Delta_{I,n}^2$. For both fields, we observe that the power on smaller $k$ modes is dominated by large-scale foreground emission which comprises diffuse emission, unmodeled sources and sources below the confusion noise prior to foreground removal. A recent analysis of the wide-field AARTFAAC-12 HBA data (at 122 MHz) presented in \cite{gehlot2019} shows strong diffuse emission around the NCP on degree scales ($\vect{u} < 200$) which is well beyond the thermal noise. This emission becomes stronger at lower frequencies (50-70\,MHz) suggesting that the smallest $k$ modes are fully dominated by diffuse emission. Residual Stokes $I$ power on the smallest $k$ modes after foreground removal is an order of magnitude lower than the former. However, GPR does not remove any power from Stokes $V$, which means that any structure in Stokes $V$ is spatially and spectrally incoherent and behaves as uncorrelated noise. For the 3C220 field, Stokes $I$ residuals approach $\Delta_{I,n}^2$ at smaller $k$ modes, however these are still higher than $\Delta_{I,n}^2$ by $\sim 30\%$ on large $k$ modes. On the other hand, Stokes $I$ residuals for the NCP field are higher than $\Delta_{I,n}^2$ by $\sim50\%$ on most $k$ modes except for the lowest one. This remaining excess power, after correcting for the physical excess noise bias, is likely due to the baseline cut used during the DD-calibration.
Assuming that the physical noise properties of Stokes $I$ and $V$ are statistically identical, we use the post GPR excess noise bias corrected Stokes $V$ power spectrum ($\Delta_{I,n}^2$) to remove the noise component in the residual Stokes $I$ power spectrum. The noise bias corrected power spectrum $\Delta_I^2 - \Delta_{I,n}^2$ for the 3C220 (blue circles) and the NCP field (red crosses) are shown in figure \ref{fig:ps3d_ulim}. The dashed curves show thermal noise power spectrum estimate $\Delta^2_N$ estimated from $\Delta_t\tilde{V}$ for the 3C220 (`skyblue' coloured) and the NCP field (`coral' coloured). We observe that $\Delta_I^2 - \Delta_{I,n}^2$ for both fields are consistent with each other within the $2\sigma$ uncertainty for modes $k\lesssim 0.2\,h\,\text{cMpc}^{-1}$ and deviate slightly on $k\gtrsim 0.2\,h\,\text{cMpc}^{-1}$. This is possibly due to different morphologies of the two fields on small angular scales. The $\Delta_I^2 - \Delta_{I,n}^2$ for both fields, within $2\sigma$ uncertainty, agree with their respective noise estimate $\Delta_N^2$ (determined from $\Delta_t\tilde{V}$) which is a more accurate estimator of the thermal noise of the system. The $\Delta_N^2$ for both fields show power-law like behaviour and agree with each other on all $k$ modes. We find a $2\sigma$ upper limit of $\Delta_{21}^2 < (14561\,\text{mK})^2$ at $k\sim 0.038\,h\,\text{cMpc}^{-1}$ for the 3C220 field and $\Delta_{21}^2 < (14886\,\text{mK})^2$ at $k\sim 0.038\,h\,\text{cMpc}^{-1}$ for the NCP field in the redshift range $z = 19.8-25.2$. Both upper limits are consistent with each other within $2\%$. The upper limits $\Delta_I^2 - \Delta_{I,n}^2$ for the two fields are still dominated by systematics on most $k$ modes. A deeper understanding of systematics (and their mitigation) and a more accurate estimate of the noise bias is required to remove this additional bias. From our current analysis, we observed that it is harder to model the exact noise bias, which is crucial to obtain more robust upper limit. We are currently developing improved estimators of the incoherent noise power spectrum which might be thermal noise and also include incoherent random errors e.g. due to the ionosphere, calibration etc for noise bias subtraction. We are also exploring other methods (e.g. cross-correlating independent datasets) to estimate 21-cm power spectrum which circumvents several issues with noise bias subtraction and plan to incorporate them in future analyses.
\section{Summary and Outlook}\label{sec:conclusions}
In this work, we have explored the possibility of statistical measurement of the redshifted 21-cm signal of neutral hydrogen from the Cosmic Dawn using the LOFAR-Low Band Antenna system. We have presented the first upper limits on the power spectrum of the 21-cm signal in the high redshift range of $z = 19.8 - 25.2$ using LOFAR-LBA data with dual-pointing setup pointed at the NCP and the radio galaxy 3C220.3 simultaneously. Our main conclusions are:
\begin{enumerate}
\item For the 3C220 field, after 14 hours of integration, a $2\sigma$ upper limit of $\Delta_{21}^2 < (14561 \ \text{mK)}^2$ at $k = 0.038\,h\,\text{cMpc}^{-1}$ is reached on the power spectrum of 21-cm brightness temperature fluctuations. Similarly, for the NCP field, we reach a $2\sigma$ upper limit of $\Delta_{21}^2 < (14886 \ \text{mK)}^2$ at $k = 0.038 \,h\,\text{cMpc}^{-1}$ in the redshift range $z = 19.8 - 25.2$. Both upper limits are consistent with each other within $2\%$ level. Upper limits for both the 3C220 and the NCP fields are still dominated by the systematics.
\item We demonstrate the application of a multiple pointing method to calibrate LOFAR-LBA dual pointing observations.
\item We observe an excess of noise in the ratio of Stokes $I$ and $V$ noise spectra over short time-scales (12 seconds) in baseline-frequency space, derived from the Stokes $I$ and $V$ difference image-cubes created from even and odd visibility samplings at 12-second level. This excess is independent of frequency and baseline length and is also not affected by calibration. This excess noise is different from that introduced during calibration and already exists before DI and after DD calibration and does not change during those steps. The excess is different for the two fields and seems to have no clear origin. We suspect it to be caused by (diffractive) ionospheric scintillation noise, but we leave this analysis for the future.
\item We show that introducing frequency smoothness of instrumental gains as a constraint in both Direction Independent and Direction Dependent calibration of LOFAR-LBA data greatly reduces the calibration induced excess variance on the sub-band level in Stokes $I$ compared to Stokes $V$ in contrast to the un-constrained case presented in G18, where we found an excess by a factor $\sim 10$. However, an excess of $\sim 2 - 3$ still remains, which can be explained by the exclusion of short baselines during DD-calibration as shown for LOFAR-HBA data calibration as well in \cite{patil2016} and \cite{sardarabadi2018}.
\item After foreground removal using Gaussian Process Regression, the Stokes $I$ power spectrum is $\sim 2$ times that of Stokes $V$ for both fields and is featureless on most scales. However, we observe a `pitchfork' like structure in the 3C220 field at low $k_{\perp}$ near the horizon line. We expect this structure to be caused by Cas\,A and Cyg\,A residuals as seen by G18.
\end{enumerate}
\subsection{Outlook}
Detection of the redshifted 21-cm HI signal from Cosmic Dawn and the Epoch of Reionization promises to be an excellent probe to study the early phases of the evolution of the universe and has the potential to unveil exotic astrophysical phenomena. With the analysis shown in this work, a CD experiment with LOFAR-LBA will require $>10^4$ hours of integration (power spectrum sensitivity of $\sim(100\,\text{mK})^2$ in CD redshift range) in order to constrain the optimistic CD X-ray heating and baryon-Dark Matter scattering models (see e.g \citealt{fialkov2018,cohen2018}). We plan to improve the analysis in the future by improving the enforcement of spectral smoothness in calibration, better modelling of the instrument (improving beam models) and by using the new Image Domain Gridder (IDG) combined with \texttt{WSClean} (see e.g. \citealt{veenboer2017,vandertol2018}) to subtract off-axis sources. The upcoming LOFAR 2.0 upgrade will also increase the sensitivity of the LOFAR-LBA system. The combination of all these improvements will allow us to improve the CD 21-cm power spectrum sensitivity significantly.
Moreover, recently a deep absorption feature ($-0.5$ K deep) centred at $\sim78$ MHz ($z \sim 17 $) in the averaged sky spectrum was presented by \cite{bowman2018}, possibly being the sought-after 21-cm signal absorption feature seen against the Cosmic Microwave Background during the CD era. The suggested absorption feature is considerably ($\sim 2.5$ times) stronger and wider than predicted by standard astrophysical models \citep{barkana2018}. If confirmed, such a strong signal will lead to a large increase in the 21-cm brightness temperature fluctuations in the redshift range $z = 17-19$ corresponding to the deepest part of the absorption profile \citep{barkana2018,fialkov2018}, making it possible to detect its signal in a much shorter integration time compared to what was previously expected. Motivated by this, we have commenced a large scale program called the AARTFAAC Cosmic Explorer (ACE) which uses the Amsterdam-ASTRON Radio Transients Facility And Analysis Center (AARTFAAC) correlator based on LOFAR, to obtain wide-field data for statistical detection of the 21-cm brightness temperature fluctuations within the redshift range of the absorption feature. The techniques discussed in this paper and lessons learned here will be useful in understanding and mitigating the challenges in AARTFAAC data processing and analysis, as well as in the NenuFAR, the HERA and the upcoming SKA-low, which can also observe the similar redshift range. The SKA-low will also support multi-beam observations, and thus also can take advantage of the dual-beam calibration strategy we have demonstrated.
\section*{Acknowledgements}
BKG and LVEK acknowledge the financial support from a NOVA cross-network grant. FGM acknowledges support from a SKA-NL roadmap grant from the Dutch Ministry of OCW. LOFAR, the Low Frequency Array designed and constructed by ASTRON, has facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the International LOFAR Telescope (ILT) foundation under a joint scientific policy.
\bibliographystyle{mnras}
|
1,314,259,995,800 | arxiv | \section{Main Results}
To state our results we need to define the restriction operator.
Let $U\subset \Omega$ be domain in $\mathbb{C}^n$ and $R^{\Omega}_U:A^2(\Omega)\to A^2(U)$
denote the restriction operator. That is, $R^{\Omega}_Uf=f|_U$. Then
the adjoint $R^{\Omega *}_U:A^2(U)\to A^2(\Omega)$ of $R^{\Omega}_U$ is a bounded linear map
and one can show that (see, for example, \cite{ChakrabartiSahutogluPreprint})
\[R^{\Omega *}_Uf(z)=\int_UK^{\Omega}(z,w)f(w)dV(w),\]
where $dV$ is the Lebesgue measure in $\mathbb{C}^n$. We note that
if $\overline{U}\subset\Omega$, then Montel's Theorem implies
that $R^{\Omega}_U$ is compact. Also $R_U^*TR_U$ is a bounded linear operator
on $A^2(\Omega)$ whenever $T$ is a bounded linear map on $A^2(U)$.
Throughout this paper $Ef$ denotes the extension of $f$ onto $\mathbb{C}^n$
trivially by zero and $R_U$ will denote $R_U^{\Omega}$ when the domain
$\Omega$ is clear from the context. Then the formula for $R^{\Omega *}_U$
above is $R^{\Omega *}_U=P_{\Omega}E$.
For $z,$ $w\in \Omega$, let $K_z^{\Omega}(w)=K^{\Omega}(w,z)$.
Notice that the normalized Bergman kernel $k_z^{\Omega}$ is well-
defined whenever $K^{\Omega}(z,z)\neq 0$. In \cite{Englis07}, Engli\v{s}
observes that there are unbounded domains in $\mathbb{C}^{n}$ for
which the zero set $\mathcal{Z}$ of the Bergman kernel on the
diagonal $K^{\Omega}(z,z)$ is not empty. Namely, we denote
\[ \mathcal{Z}=\left\{ z\in \Omega :K^{\Omega}(z,z)=0\right\}.\]
\begin{definition}
A domain $\Omega$ in $\mathbb{C}^n$ is called a non-trivial Bergman domain if
$A^2(\Omega)\neq \{0\}$.
\end{definition}
We note that $\Omega$ is a non-trivial Bergman domain if and only if
$\mathcal{Z}\neq \Omega$. If $\Omega$ is bounded, then $\mathcal{Z}$ is
empty because the constant functions
belong to $A^2(\Omega)$ and $K^{\Omega}(z,z) \geq 1/\|1\|^2>0$ for all $z\in \Omega$.
Therefore, bounded domains are non-trivial Bergman domains as well.
The set $\mathcal{Z}$, if not empty and not equal to $\Omega$,
is a real-analytic variety in $\Omega$ with zero Lebesgue measure and it is a
relatively closed subset of $\Omega$. The normalized Bergman kernel
$k^{\Omega}_z$ is a well defined function in $A^2(\Omega)$ for $z\in \Omega\setminus \mathcal{Z}$.
In this paper we will always assume that $\Omega$ is a non-trivial Bergman domain.
In the example given in \cite{Englis07}, there exists a bounded function $\phi$ on
an unbounded pseudoconvex complete Reinhardt domain
$\Omega$ such that the Berezin transform $B_{\Omega}T_{\phi}$ of the (bounded) Toeplitz
operator on $\Omega$ has a singularity at a point in $\mathcal{Z}$. However, the map
$z\mapsto k^{\Omega}_z$ is continuous from $\Omega\backslash \mathcal{Z} $ to
$L^2(\Omega)$ since
\begin{align}\label{Eqn0}
\left\| k^{\Omega}_z-k^{\Omega}_w\right\|_{L^2(\Omega)}^2
=2-2\text{Re}\left\langle k^{\Omega}_z,k^{\Omega}_{w}\right\rangle
=2-2\text{Re}\frac{K^{\Omega}(w,z)}{\sqrt{K^{\Omega}(z,z)}\sqrt{K^{\Omega}(w,w)}}
\end{align}
and both $K^{\Omega}(w,z)$ and $K^{\Omega}(w,w)$ converge to
$K^{\Omega}(z,z)$ as $w$ converges to $z$ in $\Omega\backslash \mathcal{Z}$.
Hence, the Berezin transform $B_{\Omega}T$ of a bounded operator $T$ on $A^2(\Omega)$
is always a well-defined, bounded and continuous function,
on $\Omega\backslash \mathcal{Z}$. This can be seen from the
inequality $|B_{\Omega}T(z)|=|\langle Tk^{\Omega}_z,k^{\Omega}_z\rangle |\leq \|T\|$ and
\begin{align*}
\left|B_{\Omega}T(z)-B_{\Omega}T(w)\right|
\leq &\left|\langle Tk_z,k^{\Omega}_z-k^{\Omega}_w\rangle\right|
+\left|\langle T(k^{\Omega}_z-k^{\Omega}_w),k^{\Omega}_w\rangle\right| \\
\leq &\left\|T\right\| \left\|k^{\Omega}_z-k^{\Omega}_w\right\|_{L^2(\Omega)}
+\left\| T(k^{\Omega}_z-k^{\Omega}_w)\right\|_{L^2(\Omega)}
\end{align*}
for every $z$, $w\in\Omega\backslash \mathcal{Z}$.
Our first two results below can be seen as analogues of
Ramadanov's and Skwarczy\'nski's Theorems.
\begin{theorem}\label{ThmBounded}
Let $\{\Omega_j\}$ be a sequence of domains in $\mathbb{C}^n$ such that
$\Omega_j\subset \Omega_{j+1}$ for all $j$ and
$\Omega=\cup_{j=1}^{\infty}\Omega_j$ be a non-trivial Bergman domain.
Let $T$ be a bounded linear map on $A^2(\Omega)$. Then
$B_{\Omega_j}R_{\Omega_j}TR^*_{\Omega_j}\to B_{\Omega}T$ uniformly on compact
subsets of $\Omega\backslash \mathcal{Z}$ as $j\to \infty$.
Furthermore, if $\Omega$ is bounded, then
$EB_{\Omega_j}R_{\Omega_j}TR^*_{\Omega_j}\to B_{\Omega}T$ in $L^p(\Omega)$ as
$j\to \infty$ for all $0<p<\infty$.
\end{theorem}
\begin{theorem}\label{ThmDecreasing}
Let $\Omega$ be a non-trivial Bergman domain and $\{\Omega_j\}$ be a sequence of
domains in $\mathbb{C}^n$ such that $\Omega\subset\Omega_{j+1}\subset \Omega_j$
for all $j$. Assume $K^{\Omega_j}(z,z)\to K^{\Omega}(z,z)$ as $j\to\infty$
for every $z\in\Omega$. Let $T$ be a bounded linear map on $A^2(\Omega)$. Then
$B_{\Omega_j}(R^{\Omega_j}_{\Omega})^*TR^{\Omega_j}_{\Omega}\to B_{\Omega}T$ uniformly
on compact subsets of $\Omega\backslash \mathcal{Z}$ as $j\to \infty$.
Furthermore, if $\Omega$ is bounded, then
$B_{\Omega_j}(R^{\Omega_j}_{\Omega})^*TR^{\Omega_j}_{\Omega}\to B_{\Omega}T$ in $L^p(\Omega)$
as $j\to \infty$ for all $0<p<\infty$.
\end{theorem}
The next result describes the convergence of the Berezin
transforms when the symbols of Toeplitz operators are restricted
onto the subdomains. To clarify the notation below, $\phi|_U$ denotes
the restriction of $\phi$ onto $U, R_U\phi$.
\begin{theorem}\label{ThmLinfty}
Let $\{\Omega_j\}$ be a sequence of domains in $\mathbb{C}^n$ such that
$\Omega_j\subset \Omega_{j+1}$ for all $j$ and $\Omega=\cup_{j=1}^{\infty}\Omega_j$
be a non-trivial Bergman domain.
Assume that $T=\sum_{m=1}^lT_{\phi_{m,1}}\cdots T_{\phi_{m,k_m}}$
is a finite sum of finite products of Toeplitz operators with
bounded symbols on $\Omega$ and
$T^{\Omega_j} =\sum_{m=1}^lT_{\phi_{m,1}|_{\Omega_j}}\cdots T_{\phi_{m,k_m}|_{\Omega_j}}$
for each $j$. Then $B_{\Omega_j}T^{\Omega_j}\to B_{\Omega}T$ uniformly on compact
subsets of $\Omega\setminus \mathcal{Z}$ as $j\to \infty$. Furthermore,
if $\Omega$ is bounded, then $EB_{\Omega_j}T^{\Omega_j}\to B_{\Omega}T$ in $L^p(\Omega)$ as
$j\to \infty$ for all $0<p<\infty$.
\end{theorem}
\begin{remark}
We note that the $T^{\Omega_j}$ in the theorem above depends on the
symbols and hence representation of $T$. However, representation
of products of Toeplitz operators is not unique. For instance,
\c{C}elik and Zeytuncu in \cite{CelikZeytuncu16} showed that
there exists a Reinhardt domain $\Omega$ in $\mathbb{C}^2$ such that there
exists non-trivial nilpotent Toeplitz operators on $A^2(\Omega)$.
Hence the zero operator has multiple representations.
However, since the Berezin transform of $T$ is independent of
its representation, the Berezin transforms of $T^{\Omega_j}$ converge
to the same limit for any representation of $T$.
\end{remark}
For a function $\phi\in L^{q}(\Omega)$, assuming the Toeplitz operator $T_{\phi}$
is bounded on $A^2(\Omega)$, we define the Berezin transform $B_{\Omega}\phi$
of $\phi$ as $B_{\Omega}\phi(z)=B_{\Omega}T_{\phi}(z)$ for $z\in \Omega$. Hence
\begin{align*}
B_{\Omega}\phi(z) = \langle T_{\phi} k^{\Omega}_z,k^{\Omega}_z\rangle
=\langle P_{\Omega}\phi k^{\Omega}_z,k^{\Omega}_z\rangle
=\langle \phi k^{\Omega}_z,k^{\Omega}_z\rangle
=\int_{\Omega}\phi(w)|k^{\Omega}_z(w)|^2dV(w).
\end{align*}
As a consequence of Theorem \ref{ThmLinfty} and Dini's Theorem
we have the following corollary.
\begin{corollary}\label{CorLInfty}
Let $\{\Omega_j\}$ be a sequence of domains in $\mathbb{C}^n$ such that
$\Omega_j\subset \Omega_{j+1}$ for all $j$ and
$\Omega=\cup_{j=1}^{\infty}\Omega_j$ be a non-trivial Bergman domain.
Assume that $\phi\in L^q(\Omega)$ for some $0<q<\infty$ so that
$T_{\phi}$ is bounded on $A^2(\Omega)$. Then there exists a subsequence
$\{j_k\}$ and functions $\phi_k\in L^{\infty}(\Omega_{j_k})$
such that $B_{\Omega_{j_k}}\phi_k\to B_{\Omega}\phi$ uniformly
on compact subsets of $\Omega\backslash \mathcal{Z}$. If $\Omega$ is
bounded, then $EB_{\Omega_{j_k}}\phi_k\to B_{\Omega}\phi$ in $L^p(\Omega)$
as $k\to \infty$ for all $0<p<\infty$.
\end{corollary}
We note that, as Proposition \ref{PropLp} below shows, $\phi_k$ in the
corollary above might have to be different from $R_{\Omega_k}\phi$.
\begin{remark}
If the domain $\Omega$ is not bounded, then the Berezin transform
$B_{\Omega}T_{\phi}$ of the Toeplitz operator of a bounded symbol
$\phi$ does not have to be in $L^p(\Omega)$. For instance, let
$\phi(z)=\text{Re}(z)$ and $\Omega=\{z\in\mathbb{C}: 0<\text{Re}(z)<1\}.$
We note that $K^{\Omega}(z,z)\neq 0$ for any $z\in \Omega$ as $(z+1)^{-1}$
is square integrable on $\Omega$.
Since $\phi$ is bounded and harmonic, we conclude that
$B_{\Omega}T_{\phi}=\phi$ which is not in $L^p(\Omega)$ for any $0<p<\infty$.
\end{remark}
In the following proposition we compute the asymptotics of
the Berezin transform of $\log|z|$ on annuli that converge
to the punctured disc. Also it shows that the first conclusion in
Theorem \ref{ThmLinfty} is not true if we drop the assumption
that the symbol is bounded. The function
$\log|z|\in L^p(\mathbb{D}\setminus \{0\})$ for all
$0<p<\infty$ and, Lemma \ref{Lemma2} implies that,
\[B_{\mathbb{D}\setminus\{0\}}\log|z|
=B_{\mathbb{D}}\log|z|
=\frac{1}{2}(|z|^2-1).\]
\begin{proposition}\label{PropAsymp}
Let $A_r=\{z\in \mathbb{C}:r<|z|<1\}$ and $\phi(z)=\log|z|$. Then
\[B_{A_r}\phi(z)\to \frac{|z|^2}{4}-\frac{1}{4|z|^2}\]
uniformly on compact subsets of $\mathbb{D}\setminus\{0\}$
as $r\to 0^+$.
\end{proposition}
The following proposition shows that the last statement in
Theorem \ref{ThmLinfty} is not true in general for operators in
the Toeplitz algebra. One can argue as follows. Let
$\phi(z)=\log|z|$ be a symbol on $\mathbb{D}^*=\mathbb{D}\setminus \{0\}$.
One can show that $T_{\phi}$ is compact on $A^2(\mathbb{D}^*)$
(as $A^2(\mathbb{D}^*)=A^2(\mathbb{D})$ and $\phi=0$ on the unit circle).
However, compact operators are in the Toeplitz algebra
(see \cite[Theorem 6]{Englis92}). Hence $T_{\phi}$ is in
the Toeplitz algebra; yet, by Proposition \ref{PropLp} below,
$\{B_{A_r}T^{A_r}_{\phi}\}$ does not converge to $B_{\mathbb{D}^*}T_{\phi}$ in $L^p$.
\begin{proposition}\label{PropLp}
Let $A_r=\{z\in \mathbb{C}:r<|z|<1\}$, $\mathbb{D}^*=\mathbb{D}\backslash \{0\}$, and $\phi(z)=\log|z|$.
Then $T_{\phi}$ is a compact operator on $A^2(\mathbb{D}^*)$ and
\begin{eqnarray*}
\lim_{r\to 0^+}\|EB_{A_r}T^{A_r}_{\phi}\|_{L^p(\mathbb{D}^*)}= \infty,
\end{eqnarray*}
while $\|B_{\mathbb{D}^*}T_{\phi}\|_{L^p(\mathbb{D}^*)}<\infty$ for all $1\leq p\leq \infty$.
\end{proposition}
\section{Proofs of Theorems \ref{ThmBounded}, \ref{ThmDecreasing},
\ref{ThmLinfty} and Corollary \ref{CorLInfty}}
We start with a simple lemma.
\begin{lemma}\label{Lem:Adjoint}
Let $\Omega$ be a non-trivial Bergman domain in $\mathbb{C}^n$ and $U\subset \Omega$
be a subdomain. Then $R_U^*K_z^U=K_z^{\Omega}$ for $z\in U$.
\end{lemma}
\begin{proof} For $z\in U$ and $f\in A^2(\Omega)$ we have
\[ f(z)=\langle R_Uf,K_z^U\rangle_U =\langle f,R_U^*K_z^U\rangle_{\Omega}.\]
Because of the uniqueness of the Bergman kernel, we conclude
that $R_U^*K_z^U=K_z^{\Omega}$.
\end{proof}
We will need the following results of Ramadanov and Skwarczy\'nski
(see \cite[Theorem 12.1.23 and Theorem 12.1.24]{JarnickiPflugBook1Ed2}
and also \cite{Ramadanov67,Ramadanov83,IwinskiSkwarczynski75, SkwarczynskiThesis}).
\begin{theorem}[Ramadanov]\label{ThmRamadanov}
Let $\Omega_j$ be an increasing sequence of domains in $\mathbb{C}^n$ such that
$\Omega=\cup_{j=1}^{\infty} \Omega_j$. Then, $K^{\Omega_j}\to K^{\Omega}$ as $j\to\infty$ locally
uniformly on $\Omega \times \Omega$.
\end{theorem}
\begin{theorem}[Skwarczy\'nski]\label{ThmIwinskiSkwarczynski}
Let $\{\Omega_j\}$ be a sequence of domains in $\mathbb{C}^n$ such that
$\Omega\subset \Omega_{j+1}\subset \Omega_j$. Then, $K^{\Omega_j}\to K^{\Omega}$ as $j\to\infty$ locally
uniformly on $\Omega \times \Omega$ if and only if $K^{\Omega_j}(w,w)\to K^{\Omega}(w,w)$
as $j\to\infty$ for all $w\in \Omega$.
\end{theorem}
Let $U$ be a subdomain of a domain $\Omega$. Since
\[K^{\Omega}(z,z)=\sup\{|f(z)|^2:f\in A^2(\Omega)\text{ and } \|f\|=1\},\]
we have $0\leq K^{\Omega}(z,z) \leq K^{U}(z,z)$ for every $z\in U$. Hence, if
$K^{\Omega}(z,z)\not = 0$, then $K^{U}(z,z)\not = 0$.
\begin{lemma}\label{LemRestriction}
Let $\{\Omega_j\}$ be a sequence of domains in $\mathbb{C}^n$ such that
$\Omega_j\subset \Omega_{j+1}$ for all $j$ and
$\Omega=\cup_{j=1}^{\infty}\Omega_j$ be a non-trivial Bergman domain.
Then for each compact set $K\subset\Omega\backslash \mathcal{Z}$, we have
\[\lim_{j\to\infty}\sup_{z\in K}\|R^*_{\Omega_j}k_z^{\Omega_j}-k^{\Omega}_z\|_{L^2(\Omega)}=0.\]
\end{lemma}
\begin{proof}
First we note that $0\leq K^{\Omega}(z,z)\leq K^{\Omega_j}(z,z)$ for all $j$ and $z\in K$.
So since $K\subset \Omega\setminus \mathcal{Z}$ we have $K^{\Omega_j}(z,z)\neq 0$ for
all $j$ so that $K\subset \Omega_j$. Let $j_0$ be chosen such
that $K\subset \Omega_{j_0}$. Lemma \ref{Lem:Adjoint} implies that
$R^*_{\Omega_j}k_z^{\Omega_j}=K_z^{\Omega}/\sqrt{K^{\Omega_j}(z,z)}$ for $j\geq j_0$.
Then for $z\in K$ and $j\geq j_0$ we have
\begin{align*}
\|R^*_{\Omega_j}k_z^{\Omega_j}- k_z^{\Omega}\|_{L^2(\Omega)}
=& \left\|\frac{K_z^{\Omega}}{\sqrt{K^{\Omega_j}(z,z)}}
-\frac{K_z^{\Omega}}{\sqrt{K^{\Omega}(z,z)}}\right\|_{L^2(\Omega)} \\
=&\left\|k_z^{\Omega}\left(1-\sqrt{K^{\Omega}(z,z)}/\sqrt{K^{\Omega_j}(z,z)}\right)\right\|_{L^2(\Omega)} \\
= &\left|1-\sqrt{K^{\Omega}(z,z)}/\sqrt{K^{\Omega_j}(z,z)}\right|.
\end{align*}
Ramadanov's Theorem (Theorem \ref{ThmRamadanov}) implies that
$K^{\Omega}(z,z)/K^{\Omega_j}(z,z)\to 1$ uniformly on $K$ as $j\to \infty$.
Therefore,
$\sup_{z\in K}\|R^*_{\Omega_j}k_z^{\Omega_j}-k^{\Omega}_z\|_{L^2(\Omega)}\to 0$ as $j\to \infty$.
\end{proof}
The following Lemma, which is used in the proof of Theorem \ref{ThmBounded},
might be of interest on its own right.
\begin{lemma} \label{LemSeries}
Let $\Omega$ be a non-trivial Bergman domain in $\mathbb{C}^n$ and $U\subset \Omega$
be a subdomain. Let $T$ be a bounded operator on $A^2(\Omega)$. Then
\[\frac{B_{\Omega}T(z)}{B_U(R_UTR^*_U)(z)} = \frac{K^U(z,z)}{K^{\Omega}(z,z)}\]
for $z\in U\backslash \mathcal{Z}$.
\end{lemma}
\begin{proof}
For $z\in U\backslash \mathcal{Z}$, we use Lemma \ref{Lem:Adjoint} to get
\begin{align*}
B_U(R_UTR_U^*)(z)
=&\langle TR_U^*k_z^U,R_U^*k_z^U\rangle_{\Omega} \\
=&\frac{\langle TK_z^{\Omega},K_z^{\Omega}\rangle_{\Omega}}{K^U(z,z)} \\
=&\frac{K^{\Omega}(z,z)}{K^U(z,z)}B_{\Omega}T(z).
\end{align*}
Hence the proof of Lemma \ref{LemSeries} is complete.
\end{proof}
\begin{corollary}
Let $\Omega$ be a non-trivial Bergman domain in $\mathbb{C}^n, U\subset \Omega$
be a subdomain, and $T$ be a bounded linear
operator on $A^2(\Omega)$. Assume that $p\in \overline{U}$ and $1\leq\alpha<\infty$
such that $\frac{K^U(z,z)}{K^{\Omega}(z,z)}\to \alpha$ as $z\to p$, $z\in U\backslash \mathcal{Z}$. Then
$B_{\Omega}T$ is continuous at $p$ if and only if $B_U(R_UTR^*_U)$ is
continuous at $p$.
\end{corollary}
Now we are ready to prove Theorem \ref{ThmBounded}.
\begin{proof}[Proof of Theorem \ref{ThmBounded}]
The proof of locally uniform convergence is a result of Theorem
\ref{ThmRamadanov} together with Lemma \ref{LemSeries}. Indeed,
Theorem \ref{ThmRamadanov} implies that
\begin{eqnarray*}
K^{\Omega_j}(z,z)/K^{\Omega}(z,z)\to 1
\end{eqnarray*}
locally uniformly on $\Omega\times \Omega$ as $j\to \infty$. Then Lemma \ref{LemSeries}
implies that
\begin{eqnarray*}
B_{\Omega_j}R_{\Omega_j}TR^*_{\Omega_j}\to B_{\Omega}T
\end{eqnarray*}
locally uniformly on
$\Omega$ as $j\to\infty$.
To prove the second part we assume that $\Omega$ is bounded and $0<p<\infty$.
From the first part of the proof, we know that
$B_{\Omega_j}R_{\Omega_j}TR^*_{\Omega_j}\to B_{\Omega}T$ uniformly on compact sets as
$j\to \infty$. Furthermore,
$|B_{\Omega}T(z)|\leq \|T\|$ and $|EB_{\Omega_j}R_{\Omega_j}TR^*_{\Omega_j}(z)|\leq \|T\|$
for all $z\in \Omega$ and all $j$. Then, using the Lebesgue Dominated
Convergence Theorem, we conclude that
$EB_{\Omega_j}R_{\Omega_j}TR^*_{\Omega_j}\to B_{\Omega}T$ in $L^p(\Omega)$ as $j\to\infty$.
\end{proof}
\begin{lemma}\label{LemL2Conv}
Let $\Omega$ be a non-trivial Bergman domain and $\{\Omega_j\}$ be a sequence of
domains in $\mathbb{C}^n$ such that $\Omega\subset\Omega_{j+1}\subset \Omega_j$ for all $j$.
Assume that $K_{\Omega_j}(z,z)\to K_{\Omega}(z,z)$ as $j\to\infty$ for every $z\in\Omega$.
Then for each compact set $K\subset\Omega\backslash\mathcal{Z}$, we have
\[\lim_{j\to\infty}\sup_{z\in K}\|R^{\Omega_j}_{\Omega}k^{\Omega_{j}}_z-k^{\Omega}_z\|_{L^2(\Omega)}=0.\]
\end{lemma}
\begin{proof}
If $K^{\Omega}(z,z)>0$ for some $z\in\Omega$, then $K^{\Omega_j}(z,z)>0$ for large $j$
because $K^{\Omega_j}(z,z)$ increases to $K^{\Omega}(z,z)$ as $j\to\infty$. Furthermore,
there exists an open neighborhood of $z$ for which the normalized
Bergman kernels $k^{\Omega_{j}}$ and $k^{\Omega}$ are well-defined for $j$
large enough. Since $K\subset\Omega\backslash \mathcal{Z}$ is compact,
all of the functions in the statement are well-defined for large $j$,
and the limit makes sense.
Let $\varepsilon>0$ be given. For each $z\in K$, we choose a compact
$S_{z}\subset \Omega$ so that
$\left\| k^{\Omega}_z\right\|_{L^2(\Omega \backslash S_z)}<\varepsilon$.
Recall that the map $z\mapsto k^{\Omega}_{z}$ is continuous from
$\Omega\backslash \mathcal{Z}$ to $L^2(\Omega)$ (see \eqref{Eqn0}). For any
$z\in \Omega\setminus \mathcal{Z}$ we choose an open set
$U_z\subset \Omega\backslash \mathcal{Z}$ so that $z\in U_z$
and $\left\| k^{\Omega}_z-k^{\Omega}_w\right\|_{L^2(\Omega)}<\varepsilon$
when $w\in U_{z}$. Then
\[\left\| k^{\Omega}_w\right\|_{L^{2}(\Omega \backslash S_z)}
<\varepsilon+\left\| k^{\Omega}_z\right\|_{L^2(\Omega\backslash S_z)}
<2\varepsilon\]
for $w\in U_{z}$. Since $K$ is compact, there exist
$z_{1},\cdots ,z_{m}\in K$ so that $K\subset \cup _{j=1}^{m}U_{z_j}$.
The set $S=\cup _{j=1}^{m}S_{z_j}\subset \Omega$ is compact as well and
\begin{eqnarray*}
\sup_{w\in K}\left\| k^{\Omega}_w \right\|_{L^2(\Omega \backslash S)}<2\varepsilon.
\end{eqnarray*}
Using Theorem \ref{ThmIwinskiSkwarczynski}, we have
\begin{align}\label{Eqn1}
\sup_{z\in K, w\in S}\left |k^{\Omega_j}_z(w)-k^{\Omega}_z(w)\right |
<\frac{\varepsilon}{\sqrt{Vol(S)+1}}
\end{align}
and
\begin{align*}
\sup_{z\in K, w\in S}\left ||k^{\Omega_j}_z(w)|^2-|k^{\Omega}_z(w)|^2\right|
<\frac{\varepsilon^2}{Vol(S)+1}
\end{align*}
for large enough $j$. Then by integrating the above inequality
over $S$ and using
$\left\| k^{\Omega}_z\right\|_{L^2(\Omega \backslash S)}<2\varepsilon$
we get
\begin{align*}
\|k_z^{\Omega_j}\|^2_{L^2(S)}\geq \|k_z^{\Omega}\|^2_{L^2(S)}-\varepsilon^2
> 1-4\varepsilon^2-\varepsilon^2 =1-5\varepsilon^2,
\end{align*}
which implies that $\|k_z^{\Omega_j}\|_{L^2(\Omega\backslash S)}<\sqrt{5}\varepsilon$
when $j$ is large enough. Then using \eqref{Eqn1} we get
\begin{align*}
\|R^{\Omega_j}_{\Omega}k^{\Omega_{j}}_z-k^{\Omega}_z\|_{L^2(\Omega)}
&\leq \left\| k^{\Omega_{j}}_z-k^{\Omega}_z \right\|_{L^2(S)}
+\|k^{\Omega}_z\|_{L^2(\Omega\backslash S)}+\|k^{\Omega_j}_z\|_{L^2(\Omega\backslash S)} \\
&<(3+\sqrt{5})\varepsilon
\end{align*}
for $j$ large and $z\in K$. Hence,
\[\lim_{j\to\infty}\sup_{z\in K}\|R^{\Omega_j}_{\Omega}k^{\Omega_{j}}_z-k^{\Omega}_z\|_{L^2(\Omega)}=0.\]
The proof is finished.
\end{proof}
\begin{proof}[Proof of Theorem \ref{ThmDecreasing}]
For $z\in\Omega\backslash \mathcal{Z}$, we define $f(z)=B_{\Omega}T (z)$ and
\begin{align*}
f_j(z)=&B_{\Omega_j}(R^{\Omega_j}_{\Omega})^*TR^{\Omega_j}_{\Omega}(z)\\
g_j(z)=&\langle TR^{\Omega_j}_{\Omega}k^{\Omega_{j}}_z,k^{\Omega}_z \rangle_{L^2(\Omega)}
\end{align*}
for each $j$. Then
\[f_j(z)
=\langle (R^{\Omega_j}_{\Omega})^*TR^{\Omega_j}_{\Omega}k^{\Omega_{j}}_z,k^{\Omega_j}_z \rangle _{L^2(\Omega_j)}
= \langle TR^{\Omega_j}_{\Omega}k^{\Omega_{j}}_z,R^{\Omega_j}_{\Omega}k^{\Omega_j}_z \rangle_{L^2(\Omega)}.\]
Let $K\subset\Omega$ be a compact set. By Cauchy-Schwarz inequality we have
\begin{align*}
\sup_{z\in K}|g_j(z)-f(z)|
=&\sup_{z\in K}\left| \langle TR_{\Omega}^{\Omega_j}k_z^{\Omega_j}-Tk_z^{\Omega},
k_z^{\Omega}\rangle \right| \\
\leq& \sup_{z\in K}\left\| TR_{\Omega}^{\Omega_j}k_z^{\Omega_j}-Tk_z^{\Omega}\right\|_{L^2(\Omega)}\\
\leq &\left\| T\right\| \sup_{z\in K}
\left\| R_{\Omega}^{\Omega_j}k_z^{\Omega_j}-k_z^{\Omega}\right\|_{L^2(\Omega)}.
\end{align*}
The last term above converges to zero by Lemma \ref{LemL2Conv}.
Therefore, the sequence $\{g_j\}$ converges to $f$ uniformly on $K$.
Using Cauchy-Schwarz inequality again we have
\begin{align*}
|f_j(z)-g_j(z)| = \left|\langle TR^{\Omega_j}_{\Omega}k^{\Omega_j}_z,
R^{\Omega_j}_{\Omega}k^{\Omega_j}_z-k^{\Omega}_z \rangle \right|
\leq \|T\|\|R^{\Omega_j}_{\Omega}k^{\Omega_j}_z-k^{\Omega}_z\|_{L^2(\Omega)}.
\end{align*}
Lemma \ref{LemL2Conv} implies that the last term above converges
to zero uniformly on $K$. Hence, $|f_j-g_j|\to 0$ uniformly on
$K$ as $j\to \infty$. Therefore, $\{f_j\}$ converges to $f$ uniformly on $K$.
As in the proof of Theorem \ref{ThmBounded} we prove the second part
as follows. We assume that $\Omega$ is bounded. From the previous part of
this proof we know that $\{f_j\}$ converges to $f$ uniformly on compact
subset of $\Omega$. Furthermore, $\|f_j\|_{L^{\infty}(\Omega)}\leq \|T\|$ for all $j$.
Then using the Lebesgue Dominated Convergence Theorem, we conclude
that $\{f_j\}$ converges to $f$ in $L^p(\Omega)$ as $j\to\infty$ for all $0<p<\infty$.
\end{proof}
Now we are ready to prove Theorem \ref{ThmLinfty}.
\begin{proof}[Proof of Theorem \ref{ThmLinfty}]
It is enough to prove the result for finite product of Toeplitz
operators as it is easy to conclude the theorem for the finite sums
of such operators. So let $T=T_{\phi_m}\cdots T_{\phi_1}$ where
$\phi_1, \ldots, \phi_m\in L^{\infty}(\Omega)$. One can easily show
that $B_{\Omega}T \in L^{\infty}(\Omega)$ and
$B_{\Omega_j}T^{\Omega_j}\in L^{\infty}(\Omega_j)$ for all $j$. Furthermore,
one can show that
\[\max\{\|B_{\Omega_j}T^{\Omega_j}\|_{L^{\infty}(\Omega_j)},
\|B_{\Omega}T\|_{L^{\infty}(\Omega)}\}
\leq \|\phi_1\|_{L^{\infty}(\Omega)}\cdots \|\phi_m\|_{L^{\infty}(\Omega)}.\]
Let $f_j(z)=|B_{\Omega}T(z)-EB_{\Omega_j}T^{\Omega_j}(z)|$ for $z\in \Omega$. Then
\begin{align}\label{IneqBounded}
\|f_j\|_{L^{\infty}(\Omega)}
\leq 2\|\phi_1\|_{L^{\infty}(\Omega)}\cdots \|\phi_m\|_{L^{\infty}(\Omega)}
\end{align}
for all $j$.
We will use induction to prove that
\[\sup\{|T^{\Omega_j}k_z^{\Omega_j}(w)-Tk_z^{\Omega}(w)|:z,w\in K\}\to 0 \]
as $j\to \infty$.
So first let us assume that
$T=T_{\phi_1}$ is a Toeplitz operator. Let $K$ be a compact
set in $\Omega\backslash\mathcal{Z}$. As in the proof of Lemma \ref{LemL2Conv}
for a given $\varepsilon>0$, there exists a compact set $S\subset \Omega$ and
$j_0\in\mathbb{N}$ such that $K\Subset \Omega_j$,
$\|k^{\Omega}_z\|_{L^2(\Omega\setminus S)}<\varepsilon$ for all $z\in K$,
and $\|k^{\Omega_j}_z\|_{L^2(\Omega_j\setminus S)}<\varepsilon$ for all $z\in K$
and $j\geq j_0$. Let us consider the following equalities.
\begin{align*}
T_{\phi_1}k^{\Omega}_z(w)-T^{\Omega_j}_{\phi_1}k^{\Omega_j}_z(w)
=& \langle\phi_1 k^{\Omega}_z,K^{\Omega}(.,w)\rangle_{\Omega}
-\langle \phi_1 k^{\Omega_j}_z,K^{\Omega_j}(.,w)\rangle_{\Omega_j}\\
=&\langle\phi_1 k^{\Omega}_z,K^{\Omega}(.,w)\rangle_{S}
-\langle \phi_1 k^{\Omega_j}_z,K^{\Omega_j}(.,w)\rangle_{S}\\
&+\langle\phi_1 k^{\Omega}_z,K^{\Omega}(.,w)\rangle_{\Omega\setminus S}
-\langle \phi_1 k^{\Omega_j}_z,K^{\Omega_j}(.,w)\rangle_{\Omega_j\setminus S}.
\end{align*}
There exists $C_K>1$ such that $1/C_K\leq K^{\Omega_j}(w,w)\leq C_K$ for all $w\in K$
and all $j\geq j_0$ since by Theorem \ref{ThmRamadanov}, the continuous
functions $\{K^{\Omega_j}(w,w)\}$ converges to $K^{\Omega}(w,w)$ uniformly on $K$.
Without loss of generality we can assume that
\begin{align*}
\|k^{\Omega}_z\|_{L^2(\Omega\setminus S)}
&<\frac{\varepsilon}{\sqrt{K^{\Omega}(w,w)}}, \\
\|k^{\Omega_j}_z\|_{L^2(\Omega_j\setminus S)}
&<\frac{\varepsilon}{\sqrt{K^{\Omega_j}(w,w)}}
\end{align*}
for $j\geq j_0$ and all $z,w\in K$. Then
\[\left|\langle\phi_1 k^{\Omega}_z,K^{\Omega}(.,w)\rangle_{\Omega\setminus S}\right|
+ \left|\langle \phi_1 k^{\Omega_j}_z,K^{\Omega_j}(.,w)\rangle_{\Omega_j\setminus S}\right|
\leq 2\varepsilon \|\phi_1\|_{L^{\infty}(\Omega)}\]
for all $z,w\in K$. Also
\[\sup\left\{\left|\langle \phi_1 k^{\Omega}_z,K^{\Omega}(.,w)\rangle_{S}
- \langle\phi_1 k^{\Omega_j}_z,K^{\Omega_j}(.,w)\rangle_S\right|:z,w\in K\right\}\to 0\]
as $j\to \infty$ (a consequence of Theorem \ref{ThmRamadanov}). Then
\[\limsup_{j\to\infty} \sup\left\{\left|T_{\phi_1}k^{\Omega}_z(w)
-T^{\Omega_j}_{\phi_1}k^{\Omega_j}_z(w)\right|:z,w\in K\right\}
\leq 2\varepsilon \|\phi_1\|_{L^{\infty}(\Omega)}.\]
Since $\varepsilon$ is arbitrary, we conclude that
\[\sup\left\{\left|T^{\Omega}_{\phi_1}k^{\Omega}_z(w)
-T^{\Omega_j}_{\phi_1}k^{\Omega_j}_z(w)\right|:z,w\in K\right\}\to 0\]
as $j\to \infty$. We note that for $z\in \Omega_j$ we have
\begin{align}
\nonumber \left|B_{\Omega_j}T_{\phi_1}^{\Omega_j}(z)\right.-&\left.B_{\Omega}T_{\phi_1}(z)\right|
= \frac{\left| \sqrt{\frac{K^{\Omega}(z,z)}{K^{\Omega_j}(z,z)}}
\left\langle T^{\Omega_j}_{\phi_1}k_{z}^{\Omega_j},K_z^{\Omega_j}\right\rangle_{\Omega_j}
-\left\langle T_{\phi_1}k_z^{\Omega},K_z^{\Omega}\right\rangle_{\Omega }\right|}{\sqrt{K^{\Omega}(z,z)}} \\
\label{Eqn4}=&\frac{1}{\sqrt{K^{\Omega}(z,z)}}\left|\sqrt{\frac{K^{\Omega}(z,z)}{K^{\Omega_j}(z,z)}}
T^{\Omega_j}_{\phi_1}k_z^{\Omega_j}(z)-T_{\phi_1}k_{z}^{\Omega}(z)\right|.
\end{align}
Hence $B_{\Omega_j}T^{\Omega_j}_{\phi_1}\to B_{\Omega}T_{\phi_1}$ uniformly on compact
subsets of $\Omega\backslash \mathcal{Z}$ as $j\to \infty$.
Next we show the induction step. Let $\widetilde{T}=T_{\phi_{m-1}}\cdots T_{\phi_1}$
and $\widetilde{T}^{\Omega_j}=T_{\phi_{m-1}|_{\Omega_j}}\cdots T_{\phi_1|_{\Omega_j}}$.
As the induction hypothesis we assume that
$ \widetilde{T}^{\Omega_j}k^{\Omega_j}_z \to \widetilde{T}k^{\Omega}_z $ uniformly on
compact subsets as $j\to \infty$. Then
\begin{align*}
Tk^{\Omega}_z(w)-T^{\Omega_j}k^{\Omega_j}_z(w)
=& \langle\phi_m\widetilde{T}k_z^{\Omega}, K^{\Omega}(.,w)\rangle_{\Omega}
- \langle\phi_m\widetilde{T}^{\Omega_j}k_z^{\Omega_j}, K^{\Omega_j}(.,w)\rangle_{\Omega_j}\\
=&\langle\phi_m\widetilde{T}k_z^{\Omega}, K^{\Omega}(.,w)\rangle_{S}
- \langle\phi_m\widetilde{T}^{\Omega_j}k_z^{\Omega_j}, K^{\Omega_j}(.,w)\rangle_{S}\\
&+\langle\phi_m\widetilde{T}k_z^{\Omega}, K^{\Omega}(.,w)\rangle_{\Omega\setminus S} \\
&- \langle\phi_m\widetilde{T}^{\Omega_j}k_z^{\Omega_j}, K^{\Omega_j}(.,w)\rangle_{\Omega_j\setminus S}
\end{align*}
As in the previous case, we have
\begin{align*}
\left|\langle\phi_m\widetilde{T}k_z^{\Omega}, K^{\Omega}(.,w)\rangle_{\Omega\setminus S}\right|
\leq & \|\phi_m\|_{L^{\infty}(\Omega)}\|\widetilde{T}\|\|k_z^{\Omega}\|_{L^2(\Omega\setminus S)}\sqrt{K^{\Omega}(w,w)}\\
\leq &\varepsilon \|\phi_m\|_{L^{\infty}(\Omega)}\cdots \|\phi_1\|_{L^{\infty}(\Omega)}
\end{align*}
and
\begin{align*}
\left|\langle\phi_m\widetilde{T}^{\Omega_j}k_z^{\Omega_j},
K^{\Omega_j}(.,w)\rangle_{\Omega_j\setminus S}\right|
\leq &\|\phi_m\|_{L^{\infty}(\Omega)}\|\widetilde{T}^{\Omega_j}\|\|k_z^{\Omega_j}\|_{L^2(\Omega_j\setminus S)}\sqrt{K^{\Omega_j}(w,w)}\\
\leq &\varepsilon \|\phi_m\|_{L^{\infty}(\Omega)}\cdots \|\phi_1\|_{L^{\infty}(\Omega)}.
\end{align*}
Then
\begin{align*}
\left|\langle\phi_m\widetilde{T}k_z^{\Omega},
K^{\Omega}(.,w)\rangle_{\Omega\setminus S}\right|
&+\left|\langle\phi_m\widetilde{T}^{\Omega_j}k_z^{\Omega_j},
K^{\Omega_j}(.,w)\rangle_{\Omega_j\setminus S}\right|\\
&\leq 2\varepsilon \|\phi_m\|_{L^{\infty}(\Omega)}\cdots \|\phi_1\|_{L^{\infty}(\Omega)}
\end{align*}
for all $z,w\in K$. Furthermore, by induction hypothesis, we have
\[\sup\{|\widetilde{T}^{\Omega_j}k_z^{\Omega_j}(w)- \widetilde{T}k_z^{\Omega}(w)|:z,w\in K\}
\to 0\]
as $j\to \infty$. Then
\[\sup\left\{\langle\phi_m\widetilde{T}k_z^{\Omega}, K^{\Omega}(.,w)\rangle_{S}
- \langle\phi_m\widetilde{T}^{\Omega_j}k_z^{\Omega_j},
K^{\Omega_j}(.,w)\rangle_{S}:z,w\in K\right\}\to 0\]
as $j\to \infty$. Hence,
\[\sup\{|T^{\Omega_j}k_z^{\Omega_j}(w)- Tk_z^{\Omega}(w)|:z,w\in K\} \to 0\]
as $j\to \infty$. Similar to \eqref{Eqn4} one can show that
\begin{align*}\label{Eqn5}
\left|B_{\Omega_j}T^{\Omega_j}(z)-B_{\Omega}T(z)\right|
=\frac{1}{\sqrt{K^{\Omega}(z,z)}}\left|\sqrt{\frac{K^{\Omega}(z,z)}{K^{\Omega_j}(z,z)}}
T^{\Omega_j}k_z^{\Omega_j}(z)-Tk_{z}^{\Omega}(z)\right|.
\end{align*}
Therefore, $f_j\to 0$ uniformly on $K$ as $j\to\infty$.
To prove the second part we assume that $\Omega$ is bounded. Then
the Lebesgue Dominated Convergence Theorem together
with \eqref{IneqBounded} implies that $\int_{\Omega}|f_j(z)|^pdV(z)\to 0$
as $j\to \infty$. Hence, $EB_{\Omega_j}T^{\Omega_j} \to B_{\Omega}T$ in $L^p(\Omega)$
as $j\to \infty$.
\end{proof}
Using very similar arguments as in the proof of Theorem \ref{ThmLinfty}
one can prove the following corollary.
\begin{corollary}\label{CorLinftyDecreasing}
Let $\Omega$ be a non-trivial Bergman domain and $\{\Omega_j\}$ be a sequence of
domains in $\mathbb{C}^n$ such that $\Omega\subset\Omega_{j+1}\subset \Omega_j$ for all $j$.
Assume $K^{\Omega_j}(z,z)\to K^{\Omega}(z,z)$ as $j\to\infty$ for every $z\in\Omega$.
Let $T=\sum_{m=1}^lT_{\phi_{m,1}}\cdots T_{\phi_{m,k_m}}$
be a finite sum of finite products of Toeplitz operators with
bounded symbols on $\Omega_1$ and
$T^{\Omega_j} =\sum_{m=1}^lT_{\phi_{m,1}|_{\Omega_j}}\cdots T_{\phi_{m,k_m}|_{\Omega_j}}$
for each $j$. Then $B_{\Omega_j}T^{\Omega_j}\to B_{\Omega}T$ uniformly on compact
subsets of $\Omega\setminus \mathcal{Z}$ as $j\to \infty$.
Furthermore, if $\Omega_1$ is bounded, then $EB_{\Omega_j}T^{\Omega_j}\to B_{\Omega}T$
in $L^p(\Omega)$ as $j\to \infty$ for all $0<p<\infty$.
\end{corollary}
We finish this section with the proof of Corollary \ref{CorLInfty}.
\begin{proof}[Proof of Corollary \ref{CorLInfty}]
Let $\phi\in L^q(\Omega)$ and let $K\subset\Omega\backslash\mathcal{Z}$ be compact.
First assume that $\phi$ is real valued and $\phi\geq 0$ on $\Omega$.
For each $k\geq 1$ we define $\phi_k=\min\{\phi, k\}$.
Hence, $\phi_k\in L^{\infty}(\Omega)$ and $B_{\Omega}\phi_k(z)$ increases to
$B_{\Omega}\phi(z)$ for each $z\in\Omega$. By Dini's Theorem, $B_{\Omega}\phi_k$
converges uniformly to $B_{\Omega}\phi$ on $K$. By Theorem \ref{ThmLinfty},
for each $k\geq 1$ there exists $j_k$ so that
\[\sup_{z\in K}|EB_{\Omega_{j_k}}\phi_k(z)-B_{\Omega}\phi_k(z)|\leq \frac{1}{k}.\]
This means that $EB_{\Omega_{j_k}}\phi_k$ converges uniformly to
$B_{\Omega}\phi$ on $K$. If $\Omega$ is bounded and $p>0$, then by the
last statement of Theorem \ref{ThmLinfty}, we can find $j_k$ so that
$\|EB_{\Omega_{j_k}}\phi_k-B_{\Omega}\phi_k\|_{L^p(\Omega)}\leq 1/k$.
By Monotone Convergence Theorem, we conclude that
$\|B_{\Omega}\phi_k-B_{\Omega}\phi\|_{L^p(\Omega)}\to 0$ as $k\to\infty$.
Therefore, $\|EB_{\Omega_{j_k}}\phi_k-B_{\Omega}\phi\|_{L^p(\Omega)}\to 0$ as
$k\to\infty$. Now let $\phi\in L^q(\Omega)$ be real valued. Then we
write $\phi=\phi^+-\phi^-$ where $\phi^+, \phi^-\geq 0$ on $\Omega$.
Since $B_{\Omega}\phi=B_{\Omega}\phi^+-B_{\Omega}\phi^-$, we can apply the
first part of the proof to each term. Finally, if $\phi$ is complex
valued then we can apply the previous part of the proof to the
real and imaginary parts of $\phi$.
\end{proof}
\section{Proofs of Propositions \ref{PropAsymp} and \ref{PropLp}}
Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ be the unit disk in the complex plane.
The Poisson kernel (see, for instance, \cite[Definition 1.2.3]{RansfordBook})
on the unit disk is defined as
\[P(z,\zeta)=\text{Re}\left(\frac{\zeta+z}{\zeta-z}\right)
=\frac{1-|z|^2}{|\zeta-z|^2}\]
where $z\in\mathbb{D}$, $|\zeta|=1$.
\begin{lemma}\label{Lemma1}
Let $0<s<1$ and $z\in\mathbb{D}$. Then
\begin{align*}
\frac{1}{2\pi}\int_0^{2\pi} (P(sz,e^{it}))^2dt=\frac{1+s^2|z|^2}{1-s^2|z|^2}.
\end{align*}
\end{lemma}
\begin{proof}
Let us fix $z=\rho e^{i\theta}$. In \eqref{Eqn2},
we use the property that
\[P(s\rho e^{i\theta},e^{it})=P(s\rho e^{it},e^{i\theta});\]
and in \eqref{Eqn3} we use the facts that $P$, the Poisson
kernel, is the kernel of the integral operator that solves the
Dirichlet problem and $P(.,e^{it})$ is harmonic on $\mathbb{D}$
(see \cite{RansfordBook}).
\begin{align}
\nonumber \frac{1}{2\pi}\int_0^{2\pi} \left(P(sz,e^{it})\right)^2dt=&
\frac{1}{2\pi}\int_0^{2\pi} P(sz,e^{it})P(sz,e^{it})dt\\
\nonumber =&\frac{1}{2\pi}\int_0^{2\pi} P(s\rho e^{i\theta},e^{it})P(sz,e^{it})dt\\
\label{Eqn2}=& \frac{1}{2\pi}\int_0^{2\pi} P(s\rho e^{it},e^{i\theta})P(sz,e^{it})dt\\
\label{Eqn3} =& P(s^2\rho z,e^{i\theta})\\
\nonumber =&\frac{1-s^4|z|^4}{(1-s^2|z|^2)^2}\\
\nonumber =&\frac{1+s^2|z|^2}{1-s^2|z|^2}.
\end{align}
Hence, the proof of Lemma \ref{Lemma1} is complete.
\end{proof}
A function $u(z,w)$ in $\mathbb{D}^2$ is said to be separately subharmonic if
when one of the variables is fixed in $\mathbb{D}$, $u$ is subharmonic in the
other variable.
\begin{lemma}\label{Lemma2}
Let $\displaystyle G_a(z)=\log\left | \frac{a-z}{1-\overline{a} z}\right | $ be the Green's
function for $\mathbb{D}$ with pole at $a\in \mathbb{D}$. Then
\begin{align*}
B_{\mathbb{D}}G_a(z)=\frac{1}{2}\left (\left |\frac{a-z}{1-\overline{a} z}
\right|^2-1\right)
\end{align*}
and the function $u(z,a)=B_{\mathbb{D}}G_a(z)$, defined for
$(z,a)\in\mathbb{D} ^2$, is separately subharmonic on $\mathbb{D} ^2$.
\end{lemma}
\begin{proof}
First suppose that $a=0$. Using Lemma \ref{Lemma1} in the fourth
equality below we get
\begin{align*}
B_{\mathbb{D}}G_0(z)=&\frac{(1-|z|^2)^2}{\pi}
\int_{\mathbb{D}}\frac{\log|w|}{|1-\overline{w} z|^4}dV(w)\\
=&\frac{(1-|z|^2)^2}{\pi}\int_0^1 s\log s \int_0^{2\pi}
\frac{1}{|1-se^{-it} z|^4}dtds\\
=&2 (1-|z|^2)^2\int_0^1 \frac{s\log s}{(1-s^2|z|^2)^2}
\frac{1}{2\pi}\int_0^{2\pi}
\frac{(1-s^2|z|^2)}{|e^{it}-sz|^2}
\frac{(1-s^2|z|^2)}{|e^{it}-sz|^2}d
ds\\
=&2 (1-|z|^2)^2\int_0^1 \frac{s\log s}{(1-s^2|z|^2)^2}
\frac{1+s^2|z|^2}{(1-s^2|z|^2)}ds\\
=&2 (1-|z|^2)^2\int_0^1
\frac{s(1+s^2|z|^2)\log s}{(1-s^2|z|^2)^3}ds.
\end{align*}
One can show that
\[\int \frac{x(1+|z|^2x^2)\log x}{(1-|z|^2x^2)^3}dx\
=\frac{x^2\log x}{2(|z|^2x^2-1)^2}+\frac{1}{4|z|^2(|z|^2x^2-1)}+C.\]
Therefore,
\[2 (1-|z|^2)^2\int_0^1
\frac{s(1+s^2|z|^2)\log s}{(1-s^2|z|^2)^3}ds
=\frac{1}{2}(|z|^2-1).\]
Let $a\in\mathbb{D}\backslash\{0\}$. Let $\displaystyle \psi_a(w)=\frac{a-w}{1-\overline{a} w}$
be the M\"{o}bius transform on the disk. Then, using
\cite[Chapter 2]{HedenmalmKorenblumZhuBook}
(see also \cite[Section 6.3]{ZhuBook}) we have
\begin{align*}
B_{\mathbb{D}}G_a(z)=&\int_{\mathbb{D}}G_a(\psi_z(w))dV(w)\\
=&\int_{\mathbb{D}}G_0(\psi_a\circ\psi_z(w))dV(w) \\
=&\int_{\mathbb{D}}G_0(\psi_{\psi_{a}(z)}(w))dV(w)\\
=& B_{\mathbb{D}}G_0(\psi_{a}(z)) = \frac{1}{2}\left (\left
|\frac{a-z}{1-\overline{a} z}\right |^2-1\right ).
\end{align*}
Hence, the proof of Lemma \ref{Lemma2} is complete.
\end{proof}
\begin{proof}[Proof of Proposition \ref{PropAsymp}]
The Bergman kernel of the annulus $A_r$ is
(see \cite[Example 12.1.7 (c)]{JarnickiPflugBook1Ed2})
\[K^{A_r}(z,w)=-\frac{1}{2\pi z\overline{w} \log r}
+\frac{1}{\pi z\overline{w} } \sum_{k\neq 0} \frac{kz^k\overline{w}^k}{1-r^{2k}}.\]
Let $K$ be a compact subset of $\mathbb{D}\setminus\{0\}$. Then
for small enough $r>0$ the set $K$ is a compact subset of $A_r$.
Let us fix $z_0\in K\Subset A_r$ and let us
break down the function $K^{A_r}(z_0,w)$ into four pieces as
\[K^{A_r}(z_0,w)=\psi^0_{r,z_0}(w)+\psi^1_{r,z_0}(w)
+\psi^2_{r,z_0}(w) +\psi^3_{r,z_0}(w)\]
where
\begin{align*}
\psi^0_{r,z_0}(w)=&-\frac{1}{2\pi z_0\overline{w} \log r},\\
\psi^1_{r,z_0}(w)=&\frac{r^2}{(1-r^2)\pi z_0^2 \overline{w}^2},\\
\psi^2_{r,z_0}(w)=& \frac{1}{\pi z_0^2}
\sum_{k=2}^{\infty}\frac{k}{1-r^{2k}}
\left(\frac{r}{z_0}\right)^{k-1}\left(\frac{r}{\overline{w}}\right)^{k+1},\\
\psi^3_{r,z_0}(w)=&\frac{1}{\pi z_0\overline{w} } \sum_{k=1}^{\infty}
\frac{kz_0^k\overline{w}^k}{1-r^{2k}}.
\end{align*}
One can check that the $\sup\{|\psi^1_{r,z_0}(w)|:z_0\in K,w\in A_r\}$
and $\sup\{|\psi^3_{r,z_0}(w)|:z_0\in K,w\in A_r\}$ stay bounded as
$r\to 0^+$. Furthermore, $\sup\{|\psi^2_{r,z_0}(w)|:z_0\in K,w\in A_r\}$
converges to zero as $r\to 0^+$.
Now we will estimate the Berezin transform of $\phi(w)=\log|w|$
on $A_r$ at $z_0$. First we can write $|K^{A_r}(z_0,w)|^2$ as
\begin{align*}
|K^{A_r}(z_0,w)|^2 =\left|\psi^0_{r,z_0}(w)\right|^2
+\left|\psi^3_{r,z_0}(w)\right|^2 +\Psi_{r,z_0}(w)
\end{align*}
where
\begin{align*}
\Psi_{r,z_0}(w)=&2\text{Re}\left(\psi^0_{r,z_0}(w)\sum_{j=1}^3
\overline{\psi^j_{r,z_0 }(w)}
+\psi^1_{r,z_0}(w)\sum_{j=2}^3\overline{\psi^j_{r,z_0 } (w)}\right)\\
&+2\text{Re}\left(\psi^2_{r,z_0}(w)\overline{\psi^3_{r,z_0}(w)}\right)
+\left|\psi^1_{r,z_0}(w)\right|^2+\left|\psi^2_{r,z_0}(w)\right|^2.
\end{align*}
Now we will show that
$\sup\left\{\left|\int_{A_r}\phi(w)\Psi_{r,z_0}(w)dV(w)\right|
:z_0\in K\right\}\to 0$ as $r\to 0^+$.
Using polar coordinates we compute
\begin{align*}
\int_{A_r}|\phi(w)|\left|\psi^0_{r,z_0}(w)\right| dV(w)
=&\frac{1}{|z_0|\log r}\int_r^1 \log \rho d\rho \\
=&\frac{r-r\log r-1}{|z_0|\log r}\to 0
\end{align*}
uniformly on $K$ as $r\to 0^+$. Hence using the fact that
$\psi^1_{r,z_0},\psi^2_{r,z_0},\psi^3_{r,z_0}$ stay bounded uniformly
on $A_r$ for all $z_0\in K$ we conclude that
\[\int_{A_r}\phi(w)\psi^0_{r,z_0}(w)
\sum_{j=1}^3\overline{\psi^j_{r,z_0 } (w)} dV(w) \to 0\]
uniformly on $K$ as $r\to 0^+$. Similarly, we conclude that
\[\int_{A_r}\phi(w) \left|\psi^1_{r,z_0}(w)\right|^2 dV(w)\to 0\]
and
\[ \int_{A_r}\phi(w)\psi^1_{r,z_0}(w)\sum_{j=2}^3
\overline{\psi^j_{r,z_0 } (w)} dV(w) \to 0\]
uniformly on $K$ as $r\to 0^+$ because
$\psi^1_{r,z_0},\psi^2_{r,z_0},\psi^3_{r,z_0}$ stay
bounded uniformly on $A_r$ for all $z_0\in K$ and
\begin{align*}
\int_{A_r} |\phi(w)|\left|\psi^1_{r,z_0}(w)\right| dV(w)
= &-\frac{2r^2}{(1-r^2) |z_0|^2}\int_r^1\frac{\log\rho}{\rho} d\rho \\
=&\frac{r^2(\log r)^2}{(1-r^2)|z_0|^2} \to 0
\end{align*}
uniformly on $K$ as $r\to 0^+$. Finally, since
$\psi^3_{r,z_0}$ stays bounded uniformly on $A_r$ while
$\sup\{|\psi^2_{r,z_0}(w)|:z_0\in K,w\in A_r\}\to 0$
as $r\to 0^+$ we get
\[\int_{A_r}\phi(w) \left|\psi^2_{r,z_0}(w)\right|^2 dV(w)\to 0\]
and
\[ \int_{A_r}\phi(w)\psi^2_{r,z_0}(w) \overline{\psi^3_{r,z_0 } (w)} dV(w) \to 0\]
uniformly on $K$ as $r\to 0^+$. Therefore, we showed that
\[\sup\left\{\left|\int_{A_r}\phi(w)\Psi_{r,z_0}(w)dV(w)\right|:
z_0\in K\right\} \to 0 \text{ as } r\to 0^+.\]
Now we turn to $\int_{A_r}\phi(w)\left|\psi^0_{r,z_0}(w)\right|^2 dV(w)$.
\begin{align*}
\int_{A_r}\phi(w)\left|\psi^0_{r,z_0}(w)\right|^2 dV(w)
=\frac{1}{2\pi |z_0|^2(\log r)^2}\int_r^1 \frac{\log \rho}{\rho} d\rho
=-\frac{1}{4\pi |z_0|^2}.
\end{align*}
Finally,
\[K^{A_r}(z_0,z_0)\to K^{\mathbb{D}}(z_0,z_0)
=\frac{1}{\pi(1-|z_0|^2)^2}\]
uniformly for all $z_0\in K$ as $r\to 0^+$ and
\[\sup\left\{\left|\psi^3_{r,z_0}(w)\right|^2
- \left|K^{\mathbb{D}}(w,z_0)\right|^2:z_0\in K,
w\in \mathbb{D}\right\}\to 0\]
as $r\to 0^+$. Therefore, we have
\begin{align*}
B_{A_r}\phi(z_0)
=& \int_{A_r}\phi(w)\frac{|K^{A_r}(w,z_0)|^2}{K^{A_r}(z_0,z_0)}dV(w)\\
=& \int_{A_r}\phi(w)\frac{|\psi^0_{r,z_0}(w)|^2}{K^{A_r}(z_0,z_0)}dV(w)
+\int_{A_r}\phi(w)\frac{|\psi^3_{r,z_0}(w)|^2}{K^{A_r}(z_0,z_0)}dV(w)\\
&+\int_{A_r}\phi(w)\frac{\Psi_{r,z_0}(w)}{ K^{A_r}(z_0,z_0)}dV(w)
\end{align*}
and
\[B_{A_r}\phi(z_0)\to -\frac{(|z_0|^2-1)^2}{4|z_0|^2}+B_{\mathbb{D}}\phi(z_0)
=\frac{|z_0|^2}{4}-\frac{1}{4|z_0|^2}\]
uniformly on $K$ as $r\to 0^+$ because Lemma \ref{Lemma2} implies that
$B_{\mathbb{D}}\phi(z_0)=\frac{1}{2}(|z_0|^2-1)$. Therefore, we showed that
\[B_{A_r}\phi(z)\to \frac{|z|^2}{4}-\frac{1}{4|z|^2}\]
uniformly on compact subsets of $\mathbb{D}\setminus\{0\}$ as $r\to 0^+$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{PropLp}]
The functions $\{e_n:n=0,1,2,\ldots\}$ form an orthonormal basis
for $A^2(\mathbb{D}^*)$ where $e_n(z)=\sqrt{\frac{n+1}{\pi}}z^n$.
Using integration by parts, we compute
\[T_{\phi}e_n(z)
=\left(2(n+1)\int_0^1 r^{2n+1}\log rdr\right)z^n
=-\frac{z^n}{2n+2}
=-\frac{\sqrt{\pi}}{2(n+1)^{3/2}} e_n(z).\]
Hence, $T_{\phi}$ is a compact diagonal operator on $A^2(\mathbb{D}^*)$
and by \cite[Theorem 6]{Englis92} it is in the Toeplitz algebra.
Let $f(z)=\frac{|z|^2}{4}-\frac{1}{4|z|^2}$. Proposition \ref{PropAsymp}
implies that for any $\varepsilon>0$ and any compact set
$K\Subset \mathbb{D}\setminus \{0\}$ we can choose $r_0>0$
sufficiently small so that $K\Subset A_r$ and
\begin{align*}
\|EB_{A_r}T^{A_r}_{\phi}\|^p_{L^p(\mathbb{D}^*)}
&=\int_{A_r}|B_{A_r}\phi(z)|^pdV(z)
\geq \int_K|B_{A_r}\phi(z)|^pdV(z)\\
&\geq \int_K|f(z)|^pdV(z)-\varepsilon
\end{align*}
for all $0<r\leq r_0$. Then
\[\liminf_{r\to 0^+}\|EB_{A_r}T^{A_r}_{\phi}\|^p_{L^p(\mathbb{D}^*)}
\geq \|f\|_{L^p(K)}^p-\varepsilon.\]
Since $K$ and $\varepsilon$ are arbitrary, we conclude that
\[\liminf_{r\to 0^+}\|EB_{A_r}T^{A_r}_{\phi}\|^p_{L^p(\mathbb{D}^*)}\\
\geq \|f\|^p_{L^p(\mathbb{D}^*)}.\]
Furthermore, one can show that
$\|f\|_{L^p(\mathbb{D}^*)}=\infty$ if and only if $p\geq 1$.
Therefore,
\[\lim_{r\to 0^+}\|EB_{A_r}T^{A_r}_{\phi}\|_{L^p(\mathbb{D}^*)}=\infty.\]
Finally, $\|B_{\mathbb{D}^*}T_{\phi}\|_{L^p(\mathbb{D}^*)}<\infty$
for all $1\leq p\leq \infty$ because Lemma \ref{Lemma2} implies that
$B_{\mathbb{D}^*}T_{\phi}= (|z|^2-1)/2$.
\end{proof}
|
1,314,259,995,801 | arxiv | \section{Introduction} \label{sec:intro}
This paper is concerned with the algorithmic problem of listing all spanning trees of the fan graph. Applications of efficiently listing all spanning trees of general graphs are ubiquitous in computer science and also appear in many other scientific disciplines \cite{chakraborty}. In fact, one of the earliest known works on listing all spanning trees of a graph is due to the German physicist Wilhelm Feussner in 1902 who was motivated by an application to electrical networks \cite{1902}. In the 120 years since Feussner's work, many new algorithms have been developed, such as those in the following citations \cite{berger,Char,cummins,gabow,hakimi,holzmann,kamae,kapoor,kishi,matsui,mayeda,minty,Shioura1995,uno,smith1997generating,winter}.
For any application, it is desirable for spanning tree listing algorithms to have the asymptotically best possible running time, that is, $O(1)$-amortized running time. The algorithms due to Kapoor and Ramesh \cite{kapoor}, Matsui \cite{matsui}, Smith \cite{smith1997generating}, Shioura and Tamura \cite{Shioura1995} and Shioura et al. \cite{uno} all run in $O(1)$-amortized time. Another desirable property of such listings is to have the \emph{revolving-door} property, where successive spanning trees differ by the addition of one edge and the removal of another. Such listings where successive objects in a listing differ by a constant number of simple operations are more generally known as \textit{Gray codes}. The algorithms due to Smith \cite{smith1997generating}, Kamae \cite{kamae}, Kishi and Kajitani \cite{kishi}, Holzmann and Harary \cite{holzmann} and Cummins \cite{cummins} all produce Gray code listings of spanning trees for an arbitrary graph. Of all of these algorithms, Smith's is the only one that produces a Gray code listing in $O(1)$-amortized time. A stronger notion of a Gray code for spanning trees is where the revolving-door makes strictly local changes. More specifically, we would like the differing edges to share a common endpoint. Such a Gray code property, which we call a \textit{pivot Gray code}, is not given by any previously known algorithm. This leads to our first research question.
\begin{quote} \small
{\bf Research Question \#1} Given a graph $G$ (perhaps from a specific class), does there exist a pivot Gray code listing of all spanning trees of $G$? Furthermore, can the listing be generated in polynomial (ideally constant) time per tree using polynomial space?
\end{quote}
A related question that arises for any listing is how to \emph{rank}, that is, find the position of the object in the listing, and \emph{unrank}, that is, return the object at a specific rank. For spanning trees, an $O(n^3)$-time algorithm for ranking and unranking a spanning tree of a specific listing for an arbitrary graph is known~\cite{colbourn1989unranking}.
\begin{quote} \small
{\bf Research Question \#2} Given a graph $G$ (perhaps from a specific class), does there exist a (pivot Gray code) listing of all spanning trees of $G$ that can be ranked and unranked in $O(n^2)$ time or better?
\end{quote}
An algorithmic technique recently found to have success in the discovery of Gray codes is the greedy approach. An algorithm is said to be \emph{greedy} if it can prioritize allowable actions according to some criteria, and then choose the highest priority action that results in a unique object to obtain the next object in the listing.
When applying a greedy algorithm, there is no backtracking; once none of the valid actions lead to a new object in the set under consideration, the algorithm halts, even if the listing is not exhaustive. The work by Williams~\cite{williams2013greedy} notes that some very well-known combinatorial listings can be constructed greedily, including the binary reflected Gray code (BRGC) for binary strings, the plain change order for permutations, and the lexicographically smallest de Bruijn sequence.
Recently, a very powerful greedy algorithm on permutations (known as Algorithm J, where J stands for ``jump'') generalizes many known combinatorial Gray code listings including many related to permutation patterns, rectangulations, and elimination trees~\cite{MUTZE2020,MUTZEHoang2019,MUTZERectangulations2021}. However, no greedy algorithm was previously known to list the spanning trees of an arbitrary graph.
\begin{quote} \small
{\bf Research Question \#3} Given a graph $G$ (perhaps from a specific class), does there exist a greedy strategy to list all spanning trees of $G$? Moreover, does such a greedy strategy exist where the resulting listing is a pivot Gray code?
\end{quote}
\noindent
In most cases, a greedy algorithm requires exponential space to recall which objects have already been visited in a listing. Thus, answering this third question would satisfy only the first part of {\bf Research Question \#1}. However, in many cases, an underlying pattern can be found in a greedy listing which can result in space efficient algorithms~\cite{MUTZE2020,williams2013greedy}.
To address these three research questions, we applied a variety of greedy approaches to structured classes of graphs including the fan, wheel, $n$-cube, and the compete graph. From this study, we were able to affirmatively answer each of the research questions for the fan graph. It remains an open question to find similar results for other classes of graphs.
\subsection{New Results} \label{sec:results}
The \textit{fan graph} on $n$ vertices, denoted $F_n$, is obtained by joining a single vertex (which we label $v_\infty$) to the path on $n-1$ vertices (labeled $v_2, ... , v_n$) -- see Fig.~\ref{fig:F5}.
\begin{wrapfigure}[7]{r}{0.35\textwidth}
\begin{center}
\vspace*{-0.5cm}
\hspace{-0.6cm}
\includegraphics[scale=0.35, trim=0 0 0 0cm, clip]{Figures/F5.png}
\caption{$F_5$}
\vspace{-0.2cm}
\label{fig:F5}
\end{center}
\end{wrapfigure}
Note that we label the smallest vertex $v_2$ so that the largest non-infinity labeled vertex equals the total number of vertices. Let $\trees{n}$ denote the set of all spanning trees of $F_n$.
We discover a greedy strategy to generate $\trees{n}$ in a pivot Gray code order. We describe this greedy strategy in Section 2. The resulting listing is studied to find an $O(1)$-amortized time recursive algorithm that produces the same listing using only $O(n)$ space, which is presented in Section 3. We also show how to rank and unrank a spanning tree of the greedy listing in $O(n)$ time in Section 3, which is a significant improvement over the general $O(n^3)$-time ranking and unranking that is already known. We conclude with a summary in Section 4.
\section{A Greedy Generation for $\trees{n}$}
With our goal to discover a pivot Gray code listing of $\trees{n}$, we tested a variety of greedy approaches.
There are two important issues when considering a greedy approach to list spanning trees: (1) the labels on the vertices (or edges) and (2) the starting tree. For each of our approaches, we prioritized our operations by first considering which vertex $u$ to pivot on, followed by an ordering of the endpoints considered in the addition/removal. We call the vertex $u$ the \emph{pivot}.
Our initial attempts focused only on pivots that were leaves. As a specific example, we ordered the leaves (pivots) from smallest to largest. Since each leaf $u$ is attached to a unique vertex $v$ in the current spanning tree, we then considered the neighbours $w$ of $u$ in increasing order of label.
We restricted the labeling of the vertices to the most natural ones, such as the one presented in Section~\ref{sec:results}. For each strategy we tried all possible starting trees. Unfortunately, none of our attempts lead to exhaustive listings. Applying these strategies on the wheel, $n$-cube, and complete graph was also unsuccessful.
By allowing the pivot to be any arbitrary vertex, we experimentally discovered several exhaustive listings for $\trees{n}$ for $n$ up to 12 (testing every starting tree for $n=12$ took about eight hours). One listing stood out as having an easily defined starting tree as well as a nice pattern which we could study to construct the listing more efficiently. It applied the labeling of the vertices as described in Section~\ref{sec:results} with the following prioritization of pivots and their incident edges:
\begin{quote}
Prioritize the pivots $u$ from smallest to largest and then for each pivot, prioritize the edges $uv$ that can be removed from the current tree in increasing order of the label on $v$, and for each such $v$,
prioritize the edges $uw$ that can be added to the current tree in increasing order of the label on $w$.
\end{quote}
Since this is a greedy strategy, if an edge pivot results in a spanning tree that has already been generated or a graph that is not a spanning tree, then the next highest priority edge pivot is attempted. Let \textsc{Greedy}$(T)$ denote the listing that results from applying this greedy approach starting with the spanning tree $T$. The starting tree that produced a nice exhaustive listing was the path $v_\infty, v_2, v_3, \ldots, v_n$, denoted $P_n$ throughout the paper.
Fig.~\ref{fig:F2F3F4F5} shows the listings \textsc{Greedy}$(P_n)$ for $n=2,3,4,5$. The listing \textsc{Greedy}$(P_6)$ is illustrated in Fig.~\ref{fig:F6Generation}. It is worth noting that starting with the path $v_\infty,v_n, v_{n-1}, \ldots, v_2$ or the star (all edges incident to $v_\infty$) did not lead to an exhaustive listing of $\trees{n}$.
As an example of how the greedy algorithm proceeds, consider the listing \textsc{Greedy}$(P_5)$ in Fig.~\ref{fig:F2F3F4F5}. When the current tree $T$ is the 16th one in the listing (the one with edges $\{v_2v_\infty, v_2v_3, v_3v_4, v_5v_\infty\}$), the first pivot considered is $v_2$. Since both $v_2v_3$ and $v_2v_\infty$ are present in the tree, no valid move is available by pivoting on $v_2$. The next pivot considered is $v_3$. Both edges $v_3v_2$ and $v_3v_4$ are incident with $v_3$. First, we attempt to remove $v_3v_2$ and add $v_3v_\infty$, which results in a tree previously generated. Next, we attempt to remove $v_3v_4$ and add $v_3v_\infty$, which results in a cycle. So, the next pivot, $v_4$, is considered. The only edge incident to $v_4$ is $v_4v_3$. By removing $v_4v_3$ and adding $v_4v_5$ we obtain a new spanning tree, the next tree in the greedy listing.
To prove that \textsc{Greedy}$(P_n)$ does in fact contain all trees in $\trees{n}$, we
demonstrate it is equivalent to a recursively constructed listing that we obtain by studying the greedy listings. Before we describe this recursive construction we mention one rather remarkable property of \textsc{Greedy}$(P_n)$ that we will also prove in the next section: If $X_n$ is last tree in the listing \textsc{Greedy}$(P_n)$, then \textsc{Greedy}$(X_n)$ is precisely \textsc{Greedy}$(P_n)$ in reverse order.
\begin{figure}
\begin{center}
\includegraphics[scale=0.25]{Figures/F2F3F4F5.pdf}
\caption{\textsc{Greedy}$(P_n)$ for $n=2,3,4,5$. Read left to right, top to bottom.}
\label{fig:F2F3F4F5}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale = 0.28]{Figures/F6GenerationTest.pdf}
\caption{\textsc{Greedy}$(P_6)$ read from left to right, top to bottom. Observe that S1 is \textsc{Greedy}$(P_5)$ with $v_6 v_5$ added, S2 is the reverse of \textsc{Greedy}$(P_5)$ with $v_6 v_\infty$ added, S3 is \textsc{Greedy}$(P_4)$ with $v_6 v_5$ and $v_6 v_\infty$ added, except the edge $v_4 v_\infty$ is replaced by $v_4 v_5$, and S4 is the last five trees of \textsc{Greedy}$(P_4)$ in reverse order ($v_4 v_\infty$ is now present) with $v_6 v_5$ and $v_6 v_\infty$ added.}
\label{fig:F6Generation}
\end{center}
\end{figure}
\section{An $O(1)$-amortized time Pivot Gray Code Generation for $\trees{n}$}
In this section we develop an efficient recursive algorithm to construct the listing \textsc{Greedy}$(P_n)$. The construction generates some sub-lists in reverse order, similar to the recursive construction of the BRGC. The recursive properties allow us to provide efficient ranking and unranking algorithms for the listing based on counting the number of trees at each stage of the construction. Let \spt{n} denote the number of spanning trees of $F_n$. It is known that
\[\spt{n} = f_{2(n-1)} = 2 \frac{((3-\sqrt{5})/2)^{n}-((3+\sqrt{5})/2)^{n-2}}{5-3\sqrt{5}},\]
where $f_n$ is the $n$th number of the Fibonacci sequence with $f_1=f_2=1$ \cite{fanformula}.
By studying the order of the spanning trees in \textsc{Greedy}$(P_n)$,
we identified four distinct stages S1, S2, S3, S4 that are highlighted for \textsc{Greedy}$(P_6)$ in
Fig.~\ref{fig:F6Generation}. From this figure, and referring back to Fig.~\ref{fig:F2F3F4F5} to see the recursive properties, observe that:
\begin{itemize}
\item The trees in S1 are equivalent to \textsc{Greedy}$(P_{5})$ with the added edge $v_6 v_{5}$.
\item The trees in S2 are equivalent to the reversal of the trees in \textsc{Greedy}$(P_{5})$ with the added edge $v_6 v_{\infty}$.
\end{itemize}
\noindent
The trees in S3 and S4 have both edges $v_6 v_{5}$ and $v_6 v_{\infty}$ present.
\begin{itemize}
\item In S3, focusing only on the vertices $v_4,v_3,v_2,v_\infty$, the induced subgraphs correspond to \textsc{Greedy}$(P_{4})$, except whenever $v_4v_\infty$ is present, it is replaced with $v_4v_5$ (the last five trees).
\item In S4, focusing only on the vertices $v_4,v_3,v_2,v_\infty$, the induced subgraphs correspond to the trees in \textsc{Greedy}$(P_{4})$
where $v_4v_\infty$ is present, in reverse order.
\end{itemize}
Generalizing these observations for all $n \geq 2$ leads to the recursive procedure {\sc Gen}($k,s_1,\mathit{varEdge}$) given in Algorithm~\ref{alg: Gen}, which uses a global variable $T$ to store the current spanning tree with $n$ vertices. The parameter $k$ indicates the number of vertices under consideration; the parameter $s_1$ indicates whether or not to generate the trees in stage S1, as required by the trees for S4; and the parameter $\mathit{varEdge}$ indicates whether or not a variable edge needs to be added as required by the trees for S3. The procedure {\sc RevGen}($k,s_1,\mathit{varEdge}$), which is left out due to space constraints, simply performs the operations from {\sc Gen}($k,s_1,\mathit{varEdge}$) in reverse order.
For each algorithm the base cases correspond to the edge moves in the listings \textsc{Greedy}$(P_2)$ and \textsc{Greedy}$(P_3)$.
Let $\LIST{n}$ denote the listing obtained by initializing $T$ to $P_n$, printing $T$, and calling {\sc Gen}($n,1,0$). Let $L_n$ denote the last tree in this listing. Let $\REVLIST{n}$ denote the listing obtained by initializing $T$ to $L_n$, printing $T$, and calling {\sc RevGen}($n,1,0$). Thus, $\REVLIST{n}$ is the the listing $\LIST{n}$ in reverse order.
\begin{algorithm}
\footnotesize \caption{}
\label{alg: Gen}
\begin{algorithmic}[1]
\Procedure{Gen}{$k, s_1, varEdge$}
\If{$k = 2$} \Comment{$F_2$ base case}
\If{$varEdge$} $T \gets T - v_2 v_\infty + v_2 v_3$; \textsc{Print}$(T)$
\EndIf
\ElsIf{$k = 3$} \Comment{$F_3$ base case}
\If{$s_1$}
\If{$varEdge$} $T \gets T - v_3 v_2 + v_3 v_4$; \textsc{Print}$(T)$
\Else{ $T \gets T - v_3 v_2 + v_3 v_\infty$}; \textsc{Print}$(T)$
\EndIf
\EndIf
\State $T \gets T - v_2 v_\infty + v_2 v_3$; \textsc{Print}$(T)$
\Else
\If{$s_1$}
\State \textsc{Gen}{}$(k-1, 1, 0)$ \Comment{S1}
\If{$varEdge$} $T \gets T - v_k v_{k-1} + v_k v_{k+1}$; \textsc{Print}$(T)$
\Else{ $T \gets T - v_k v_{k-1} + v_k v_\infty$}; \textsc{Print}$(T)$
\EndIf
\EndIf
\State \textsc{RevGen}{}$(k-1, 1, 0)$ \Comment{S2}
\State $T \gets T - v_{k-1} v_{k-2} + v_{k-1} v_k$; \textsc{Print}$(T)$
\State \textsc{Gen}{}$(k-2, 1, 1)$ \Comment{S3}
\If{$k > 4$} $T \gets T - v_{k-2} v_{k-1} + v_{k-2} v_\infty$; \textsc{Print}$(T)$
\EndIf
\State \textsc{RevGen}{}$(k-2, 0, 0)$ \Comment{S4}
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
Our goal is to show that $\LIST{n}$ exhaustively lists all trees in $\trees{n}$ and moreover, the listing is equivalent to {\sc Greedy}($P_n$). We accomplish this in two steps: first we show that $\LIST{n}$ has the required size, then we show that $\LIST{n}$ is equivalent to {\sc Greedy}($P_n$). Before doing this, we first comment on some notation.
Let $T - v_i$ denote the tree obtained from $T$ by deleting the vertex $v_i$ along with all edges that have $v_i$ as an endpoint. Let $T + v_i v_j$ (resp. $T-v_i v_j$) denote the tree obtained from $T$ by adding (resp. deleting) the edge $v_i v_j$. For the remainder of this section, we will let $T_n$ denote the tree $T$ specified as a global variable for \textsc{Gen}{} and \textsc{RevGen}{}, and we let $T_{n-1}=T-v_n$ and $T_{n-2}=T-v_n-v_{n-1}$.
\begin{lemma} \label{EdgeMovesLemma}
For $n \geq 2$, $|\LIST{n}| = |\REVLIST{n}| = \spt{n}$.
\end{lemma}
\begin{proof}
This result applies the Fibonacci recurrence and straightforward induction by counting the number of trees recursively generated in each stage S1, S2, S3, S4 as described earlier in this section. The base cases for $n=2,3,4$ are easily verified by stepping through the algorithms. A formal proof is omitted due to space constraints.~\hfill $\Box$
\end{proof}
To prove the next result, we first detail some required terminology.
If $T \in \trees{n}$, then we say that the operation of deleting an edge $v_i v_j$ and adding an edge $v_i v_k$ is a \emph{valid} edge move of $T$ if the result is a tree in $\trees{n}$ that has not been generated yet. Conversely, if the result is not a tree in $\trees{n}$, or the result is a tree that has already been generated, then it is not a \emph{valid} edge move of $T$. We say an edge $v_i v_j$ is \emph{smaller} than edge $v_i v_k$ if $j<k$. An edge move $T_n - v_i v_j + v_i v_k$ is said to be \emph{smaller} than another edge move $T_n - v_x v_y + v_x v_z$ if $i<x$, if $i=x$ and $j<y$, or if $i=x$, $j=y$, and $k<z$.
\begin{lemma} \label{Gen=GreedyTheorem}
For $n\geq 2$, $\LIST{n} = \textsc{Greedy}(P_n)$ and $\REVLIST{n} = \textsc{Greedy}(L_n)$.
\end{lemma}
\begin{proof}
By induction on $n$. It is straightforward to verify that the result holds for $n=2,3,4$ by iterating through the algorithms. Assume $n>4$, and that $\LIST{j} = \textsc{Greedy}(P_j)$ and $\REVLIST{j} = \textsc{Greedy}(L_j)$ for $2\le j<n$. We begin by showing $\LIST{n} = \textsc{Greedy}(P_n)$, breaking the proof into each of the four stages for clarity. \\
\noindent \underline{S1:} Since $n>4$ and $s_1=1$, \textsc{Gen}{}$(n-1, 1, 0)$ is executed. By our inductive hypothesis, $\LIST{n-1} = \textsc{Greedy}(P_{n-1})$. These must be the first trees for \textsc{Greedy}$(P_n)$, as any edge move involving $v_n v_{n-1}$ or $v_n v_\infty$ is larger than any edge move made by \textsc{Greedy}$(P_{n-1})$. Since \textsc{Greedy}$(P_{n-1})$ halts, it must be that no edge move of $T_{n-1}$ is possible. So \textsc{Greedy}$(P_n)$ must make the next smallest edge move, which is $T_n - v_n v_{n-1} + v_n v_\infty$. Since $T_n$ is a spanning tree, it follows that $T_n - v_n v_{n-1} + v_n v_\infty$ is also a spanning tree (and has not been generated yet), and therefore the edge move is valid. At this point, \textsc{Gen}{}$(n, 1, 0)$ also makes this edge move, by line 13.\\
\begin{sloppypar}
\noindent \underline{S2:} \textsc{RevGen}{}$(n-1, 1, 0)$ ($T_{n-1} = L_{n-1}$) is then executed. By our inductive hypothesis, $\REVLIST{n} = \textsc{Greedy}(L_{n-1})$. Since \textsc{Greedy}$(L_{n-1})$ halts, it must be that no edge moves of $T_{n-1}$ are possible. At this point, $T_{n-1} = P_{n-1}$ because \textsc{RevGen}{}$(n-1, 1, 0)$ was executed. The smallest edge move now remaining is $T_n - v_{n-2} v_{n-1} + v_n v_{n-1}$. This results in $T_n = P_{n-2} + v_n v_{n-1} + v_n v_\infty$, which is a spanning tree that has not been generated. So, \textsc{Greedy}$(P_n)$ must make this move. \textsc{Gen}{}$(n, 1, 0)$ also makes this move, by line 15. So, $\LIST{n}$ must equal \textsc{Greedy}$(P_n)$ up to the end of S2.\\
\noindent \underline{S3:} Next, \textsc{Gen}{}$(n-2, 1, 1)$ starting with $T_{n-2} = P_{n-2}$ is executed. Since $varEdge = 1$, $v_{n-2} v_{n-1}$ is added instead of $v_{n-2} v_\infty$. \textsc{Greedy}$(P_n)$ also adds $v_{n-2} v_{n-1}$ instead of $v_{n-2} v_\infty$ since $v_{n-2} v_{n-1}$ is smaller than $v_{n-2} v_\infty$ and this edge move results in a tree not yet generated. Other than the difference in this one edge move, which occurs outside the scope of $T_{n-2}$, \textsc{Gen}{}$(n-2, 1, 0)$ and \textsc{Gen}{}$(n-2, 1, 1)$ (both starting with $T_{n-2}=P_{n-2}$) make the same edge moves. Since we also know that $\LIST{n-2} = \textsc{Greedy}(P_{n-2})$ by the inductive hypothesis, it follows that $\LIST{n}$ continues to equal \textsc{Greedy}$(P_n)$ after line 16 of \textsc{Gen}{}$(n,1,0)$ is executed. We know that $T_{n-2} = L_{n-2}$ after \textsc{Gen}{}$(n-2, 1, 0)$. However, $T_{n-2} = L_{n-2} - v_{n-2} v_\infty + v_{n-2} v_{n-1}$ instead because \textsc{Gen}{}$(n-2, 1, 1)$ was executed ($varEdge=1$). It must be that no edge moves of $T_{n-2}$ are possible because \textsc{Greedy}$(P_{n-2})$ (and \textsc{Gen}{}$(n-2, 1, 1)$) halted. The smallest edge move now remaining is $T_n - v_{n-2} v_{n-1} + v_{n-2} v_\infty$. This results in $T_{n-2} = L_{n-2}$. Also, $T_n = T_{n-2} + v_n v_{n-1} + v_n v_\infty$ is a spanning tree since $T_{n-2}$ is a spanning tree of $F_{n-2}$. So \textsc{Greedy}$(P_n)$ makes this move. \textsc{Gen}{}$(n, 1, 0)$ also makes this move, by line 17, and thus $\LIST{n} = \textsc{Greedy}(P_n)$ up to the end of S3. \\
\end{sloppypar}
\noindent \underline{S4:} Finally, \textsc{RevGen}{}$(n-2, 0, 0)$ starting with $T_{n-2} = L_{n-2}$ is executed. By our inductive hypothesis, $\REVLIST{n-2} = \textsc{Greedy}(L_{n-2})$. From the recursive definition of \textsc{RevGen}{}, it is clear that \textsc{RevGen}{}$(n-2, 0, 0)$ and \textsc{RevGen}{}$(n-2, 1, 0)$ make the same edge moves until \textsc{RevGen}{}$(n-2, 0, 0)$ finishes executing. So, by the inductive hypothesis, the listings produced by \textsc{RevGen}{}$(n-2, 0, 0)$ and \textsc{Greedy}$(L_{n-2})$ are the same until this point, which is where \textsc{Gen}{}$(n, 1, 0)$ finishes execution. By Lemma~\ref{EdgeMovesLemma} we have that $|\LIST{n}| = \spt{n}$. Therefore, \textsc{Greedy}$(P_n)$ has also produced this many trees, and each tree is unique. Thus, it must be that all $\spt{n}$ trees of $F_n$ have been generated. Thus, \textsc{Greedy}$(P_n)$ also halts.
Since $\LIST{n}$ and \textsc{Greedy}$(P_n)$ start with the same tree, produce the same trees in the same order, and halt at the same place, it follows that $\LIST{n} = \textsc{Greedy}(P_n)$. It is relatively straightforward to show that $\REVLIST{n} = \textsc{Greedy}(L_n)$ by using similar arguments as above. This proof is omitted due to space constraints.
\hfill $\Box$
\end{proof}
Since $\LIST{n}$ is the reversal of $\REVLIST{n}$, we immediately obtain the following corollary.
\begin{corollary}
For $n \geq 2$, \textsc{Greedy}$(P_n)$ is equivalent to \textsc{Greedy}$(L_n)$ in reverse order.
\end{corollary}
Because \textsc{Greedy}$(P_n)$ generates unique spanning trees of $F_n$, Lemma~\ref{EdgeMovesLemma} together with Lemma~\ref{Gen=GreedyTheorem} implies our first main result. This result answers {\bf Research Question \#3} and the first part of {\bf Research Question \#1} for fan graphs.
\begin{theorem}
For $n \geq 2$, $\LIST{n}$ = \textsc{Greedy}$(P_n)$ is a pivot Gray code listing of $\trees{n}$.
\end{theorem}
To efficiently store the global tree $T$, the algorithms \textsc{Gen}{} and \textsc{RevGen}{} can employ an adjacency list model where each edge $uv$ is associated only with the smallest labeled vertex $u$ or $v$. This means $v_\infty$ will never have any edges associated with it, and every other vertex will have at most 3 edges in its list. Thus the tree $T$ requires at most $O(n)$ space to store, and edge additions and deletions can be done in constant time. Our next result answers the second part of {\bf Research Question \#1} for fan graphs.
\begin{theorem}
For $n\geq 2$, $\LIST{n}$ and $\REVLIST{n}$ can be generated in $O(1)$-amortized time using $O(n)$ space.
\end{theorem}
\begin{proof}
For each call to \textsc{Gen}{}$(n, s_1, varEdge)$ where $n>3$, there are at most four recursive function calls, and at least two new spanning trees generated. Thus, the total number of recursive calls made is at most twice the number of spanning trees generated. Each edge addition and deletion can be done in constant time as noted earlier. Thus each recursive call requires a constant amount of work, and hence the overall algorithm will run in $O(1)$-amortized time. There is a constant amount of memory used at each recursive call and the recursive stack goes at most $n-3$ levels deep; this requires $O(n)$ space. As mentioned earlier, the global variable $T$ stored as adjacency lists also requires $O(n)$ space.
\hfill $\Box$
\end{proof}
\subsection{Ranking and Unranking}
We now provide ranking and unranking algorithms for the listing $\LIST{n}$ of all spanning trees for the fan graph $F_n$.
Given a tree $T$ in $\LIST{n}$, we calculate its rank by recursively determining which stage (recursive call) $T$ is generated. We can determine the stage by focusing on the presence/absence of the edges $v_n v_{n-1}$, $v_n v_\infty$, $v_{n-2} v_\infty$, and $v_{n-2} v_{n-1}$. Based on the discussion of the recursive algorithm, there are $t_{n-1}$ trees generated in S1, $t_{n-1}$ trees generated in S2, $t_{n-2}$ trees generated in S3, and $t_{n-2} - t_{n-3}$ trees generated in S4. S3 is partitioned into two cases based on whether $v_{n-2} v_{n-1}$ ($varEdge)$ is present. For the remainder of this section we will let $T_{n-1} = T - v_n$ and $T_{n-2} = T - v_n - v_{n-1}$.
For $n>1$, let $R_n(T)$ denote the rank of $T$ in the listing $\LIST{n}$. If $n=2,3,4$, then $R_n(T)$ can easily be derived from Fig.~\ref{fig:F2F3F4F5}. Based on the above discussion, for $n\geq 5$:
\begin{small}
\begin{equation*}
R_n(T) =
\begin{cases}
2 \spt{n-1} + 2 \spt{n-2} - R_{n-2}(T_{n-2}) + 1
& \text{if $e_1, e_2, e_3 \in T$} \\
2 \spt{n-1} + R_{n-2}(T_{n-2} + e_3)
& \text{if $e_1, e_2, e_4 \in T$, $e_3 \not \in T$} \\
2 \spt{n-1} + R_{n-2}(T_{n-2})
& \text{if $e_1, e_2 \in T$, $e_3, e_4 \not \in T$} \\
2 \spt{n-1} - R_{n-1}(T_{n-1}) + 1 & \text{if $e_2 \in T$, $e_1 \not \in T$} \\
R_{n-1}(T_{n-1}) & \text{if $e_1 \in T$, $e_2 \not \in T$}
\end{cases}
\end{equation*}
\end{small}
where $e_1 = v_n v_{n-1}$, $e_2 = v_n v_\infty$, $e_3 = v_{n-2} v_\infty$, and $e_4 = v_{n-2} v_{n-1}$.
Determining the tree $T$ at rank $r$ in the listing $\LIST{n}$ follows similar ideas by constructing $T$ starting from a set of $n$ isolated vertices one edge at a time. Let $U_n(T, r, e)$ return the tree $T$ at rank $r$ for the listing $\LIST{n}$. Initially, $T$ is the set of $n$ isolated vertices, $r$ is the specified rank, and $e = v_n v_\infty$.
If $n=2,3,4$, then $T$ is easily derived from Fig.~\ref{fig:F2F3F4F5}. For these cases, if the edge $v_n v_\infty$ is present, then it is replaced by the edge $e$ that is passed in.
\begin{footnotesize}
\begin{equation*}
U_n(T, r, e) =
\begin{cases}
U_{n-1}(T {+} e_1, r, v_{n-1} v_\infty)
& \text{if $0 < r \leq \spt{n-1}$,} \\
U_{n-1}(T {+} e, 2 \spt{n-1} {-} r {+} 1, v_{n-1} v_\infty)
& \text{if $\spt{n-1} < r \leq 2 \spt{n-1}$,} \\
U_{n-2}(T {+} e_1 {+} e, r {-} 2\spt{n-1}, e_4)
& \text{if $2 \spt{n-1} < r \leq 2 \spt{n-1} {+} \spt{n-2}$,} \\
U_{n-2}(T {+} e_1 {+} e, 2 \spt{n-1} {+} 2 \spt{n-2} {-} r {+} 1, e_3)
& \text{otherwise.}
\end{cases}
\end{equation*}
\end{footnotesize}
where $e_1 = v_n v_{n-1}$, $e_3 = v_{n-2} v_\infty$, and $e_4 = v_{n-2} v_{n-1}$.
Since the recursive formulae to perform the ranking and unranking operations each perform a constant number of operations and the recursion goes $O(n)$ levels deep, we arrive at the following result provided the first $2(n{-}2)$ Fibonacci numbers are precomputed. We note that the calculations are on
numbers up to size $t_{n-1}$.
\begin{theorem}
The listing $\LIST{n}$ can be ranked and unranked in $O(n)$ time using $O(n)$ space under the unit cost RAM model.
\end{theorem}
This answers {\bf Research Question \#2} for fan graphs.
\section{Conclusion}
We answer each of the three Research Questions outlined in Section~\ref{sec:intro} for the
fan graph, $F_n$. First, we discovered a greedy algorithm that exhaustively listed all spanning trees of $F_n$ experimentally for small $n$ with an easy to define starting tree.
We then studied this listings which led to a recursive construction producing the same listing that runs in $O(1)$-amortized time using $O(n)$ space. We also proved that the greedy algorithm does in fact exhaustively list all spanning trees of $F_n$ for all $n\geq 2$, by demonstrating the listing is equivalent to the aforementioned recursive algorithm. It is the first greedy algorithm known to exhaustively list all spanning trees for a non-trivial class of graphs. Finally, we provided an $O(n)$ time ranking and unranking algorithms for our listings, assuming the unit cost RAM model. It remains an interesting open problem to answer the research questions for other classes of graphs including the wheel, $n$-cube, and complete graph.
\bibliographystyle{splncs04}
|
1,314,259,995,802 | arxiv | \section{Introduction}
\noindent The increasing amount of astrophysical data from distant Ia supernovae \cite{riess}, \cite{perlmutter}, \cite{hicken}, cosmic microwave background anisotropy \cite{komatsu}, \cite{larson}, and large scale galaxy surveys \cite{abazajian1}, \cite{tegmark3}, all indicate that the Universe is currently undergoing a phase of accelerated expansion. This late time acceleration is believed to be caused by some kind of negative-pressure form of matter known as dark energy. The combined analysis of cosmological observations also suggests that the universe is spatially flat, and consists of about $30\%$
of dark matter, and $70\%$ of homogeneously distributed dark energy with negative pressure. Despite this high percentage of the dark energy component, its nature as well as its cosmological origin remains unknown and represents one of the fundamental problems of theoretical cosmology. The cold dark matter model with a cosmological constant ($\Lambda$CDM) provides an excellent explanation for the accelerated expansion of the universe and other existing observational data \cite{jassal}, \cite{wilson}, \cite{davis}, \cite{allen}. Nevertheless according to the amount of astrophysical data, it remains very probably that the dark energy density is weakly time dependent, raising the interest for the search of a dynamical origin of dark energy. At the present a wide variety of models have been proposed to explain the nature of the dark energy and the accelerated expansion (see \cite{copeland, sergeiod} for review).
Among the different models of dark energy, the holographic dark energy approach is quite interesting as it incorporates some concepts of the quantum gravity known as the holographic principle (\cite{bekenstein, thooft, bousso, cohen, susskind}). According to the holographic principle, the entropy of a system scales not with its volume, but with its surface area. In the cosmological context, the holographic principle will set an upper bound on the entropy of the universe \cite{fischler}. In the work \cite{cohen}, a relationship between the short distance cut-off $\Lambda$ and the infra-red cut-off $L$ was suggested in the frame of quantum field theory. This relationship was established by using the limit set by black hole formation, namely, if is the quantum zero-point energy density caused by a short distance (UV) cut-off, the total energy in a region of size $L$ should not exceed the mass of a black hole of the same size, thus $L^3\Lambda^4\leq LM_p^2$.
Applied to the dark energy issue, if we take the whole universe into account, then the vacuum energy related to this holographic principle is viewed as dark energy, usually called holographic dark energy \cite{cohen} \cite{hsu}, \cite{li}. The largest $L$ allowed is the one saturating this inequality so that the holographic dark energy density is defined by the equality $\rho_{\Lambda}=3c^2M_p^2L^{-2}$, where $c^2$ is a numerical constant and $M_p^{-2}=8\pi G$.\\
In the present work we propose a relation between the IR and UV cut-offs that obey the holographic bound $\rho_{\Lambda}\leq M_p^2L^{-2}$, without invoking the Planck scale. This relation can be established by proposing a dark energy density proportional to the Gauss-Bonnet (GB) 4-dimensional invariant and its modification. Besides his geometrical meaning, the GB invariant has the right dimension of energy density, and an energy density of the form $\rho\propto H^4$ can always obey the above mentioned bound for black hole formation.
The main motivation for the present model is to consider the possibility of a dark energy density that scales in a natural way as $\rho_{\Lambda}\propto L^{-4}$ (i.e. with its volume), and respect the physical bounds set by the holographic principle (as follows from the relation $M_p^2>>H^2$).
As will be shown, the dark energy density proportional to the GB invariant or its modification leads to interesting cosmological consequences. As we will see, this density gives the correct order of magnitude of the current critical density without involving the black hole bound, and could be considered as a non-saturated regime of the standard defined holographic principle.\\
The gauss-Bonnet term has been considered in many models of dark energy mainly as a term in the action, coupled to scalar field, or in modified theories including functions of the Gauss-Bonnet term $F({\cal G})$. The GB invariant is believed to give the leading-order correction to the low energy string gravity \cite{gross}, and its role on the dark energy issue was studied in \cite{odintsov5}, \cite{odintsov6}, \cite{odintsov7}, \cite{odintsov8}.
As it is well known, the infrared cut-off given by the future event horizon gives rise to accelerated expansion with an EoS parameter less than $-1/3$, but this model is based on non local quantities and faces problems with the causality \cite{li}. An holographic dark energy model which is based on local and non local quantities have been considered in \cite{sergei1, sergei2}. In \cite{gao}, \cite{granda}, \cite{granda1} an infrared cut-off for the holographic density was proposed, which describes the dark energy in good agreement with the astrophysical data, and may explain the cosmic coincidence.\\
The paper is organized as follows. In Sect. II we introduce the Gauss-Bonnet and Modified Gauss-Bonnet models and give a dynamical interpretation to the densities in terms of a tachyon scalar field. In sect. III we present an analysis of the cosmological dynamics, showing the evolution of the equation of state and the Hubble parameter. We also perform a statefinder diagnostic. Sect. IV is devoted to some discussion.
\section{The model}
\subsection*{Gauss Bonnet}
Guided by facts as that the 4 dimensional Gauss-Bonnet invariant is quadratic in curvature (and therefore has the dimension of energy density), that it appears in quantum corrections to low energy string gravity \cite{gross}, and considering a non saturated regime in the holographic principle, we introduce the dark energy density as follows
\begin{equation}\label{eq1}
\rho_\Lambda=\alpha{\cal G}
\end{equation}
where $\alpha$ is a positive dimensionless parameter and ${\cal G}$ is the 4-dimensional Gauss-Bonnet invariant ${\cal G}=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\eta\gamma}R^{\mu\nu\eta\gamma}$. According to \cite{horvat}, this proposal could be considered as a lower bound for the holographic dark energy. It is worth noting that the renormalized vacuum density for scalar field with conformal invariance in de Sitter space time is proportional to $H^4$ \cite{dowker}, \cite{davies}.
The IR cut-off related with this density is given by the ``length'' size of the GB invariant $L\sim ({\cal G})^{-1/4}$. In the flat FRW background $ds^2=-dt^2+a(t)^2\sum^{3}_{i=1}(dx^i)^2$, the Eq. (\ref{eq1}) takes the form
\begin{equation}\label{eq2}
\rho_{\Lambda}=24\alpha\left(H^4+H^2\dot{H}\right)
\end{equation}
In order to guarantee the observed current value of the DE density, $\rho_{DE}\sim M_p^2H_0^2$, it seems likely that we have to tune $\alpha$: according to (\ref{eq2}), $\rho_{\Lambda_0}\sim 24\alpha H_0^4$, which compared with $\rho_{DE}$ should give $24\alpha\sim M_p^2H_0^{-2}\sim 10^{122}$, but in fact the Friedmann equation together with the initial condition (i.e. the flatness condition in a flat FRW background) automatically take care about the correct magnitude of the dark energy density compatible with observations.\\
In absence of matter, the Friedmann equation with the energy density given by (\ref{eq2}), in the flat FRW background takes the form
\begin{equation}\label{eq3}
H^2=\frac{\kappa^2}{3}\rho_{\Lambda}=8\alpha \kappa^2 H^2\left(H^2+\dot{H}\right)
\end{equation}
where $\kappa^2=8\pi G=M_p^{-2}$. After simplifying
\begin{equation}\label{eq4}
\frac{dH}{dt}+H^2-\frac{1}{8\alpha\kappa^2}=0
\end{equation}
This solution gives the asymptotic behavior of the Hubble parameter at late times. Similar equation has been obtained in \cite{capozziello} for a model of scalar field with non-minimal derivative couplings. The Eq. (\ref{eq4}) has the solution
\begin{equation}\label{eq5}
H(t)=\frac{1}{(8\alpha\kappa^2)^{1/2}}\tanh\left[\frac{1}{\sqrt{8\alpha\kappa^2}}(t-t_0)\right]
\end{equation}
This asymptotic solution describes a bouncing universe that at far future approaches the de Sitter solution $H|_{t\rightarrow \infty}=(\frac{1}{8\alpha\kappa^2})^{1/2}$. In \cite{easson}, this solution was also studied in a scalar-tensor model with auxiliary scalar field. In terms of the variable $x=\log a$ and the scaled Hubble parameter $\tilde{H}=H/H_0$, the Eq. (\ref{eq5}) becomes
\begin{equation}\label{eq6}
\frac{1}{2}\frac{d\tilde{H}^2}{dx}+\tilde{H}^2-\tilde{A}=0
\end{equation}
where $\tilde{A}=1/(8\alpha\kappa^2 H_0^2)$. Solving this equation with the initial condition $\tilde{H}^2|_{x=0}=1$, gives the solution
\begin{equation}\label{eq7}
\tilde{H}^2=\tilde{A}+(1-\tilde{A}) e^{-2x}
\end{equation}
Note that the second term behaves as $a^{-2}$, producing an effect similar to that of the spatial curvature. Note also that $\tilde{A}$ should satisfy $\tilde{A}\leq 1$, since otherwise it would give unphysical results (i.e. $\tilde{H}^2<0$ in the past). Let's draw some conclusions from the Eq. (\ref{eq7}) and the parameter $\tilde{A}$. If we assume that $\tilde{A}\sim 1$ (it can be compared with the parameter of the energy density associated with the cosmological constant, which is about 0.7), then from the definition of $\tilde{A}$ follows
\begin{equation}\label{eq8}
\alpha=\frac{M_p^2}{8\tilde{A}H_0^2}\approx\frac{M_p^2}{8H_0^2}
\end{equation}
replacing back this value into the expression for the holographic density (\ref{eq2}), we give
\begin{equation}\label{eq9}
\rho_{\Lambda}\approx 3M_p^2 H_0^{-2}\left(H^4+H^2\dot{H}\right)
\end{equation}
giving the right magnitude of the current DE density $\rho_{\Lambda_0}\sim M_p^2 H_0^2$. So we don't need to involve from the beginning the Planck mass, and indeed the black hole bound, to obtain a probably interesting alternative for the holographic DE density. Compared with the usual saturated formula for the holographic density the present proposal could be interpreted as non saturated one. In this way the Friedmann equation and the initial condition take care about the current appropriate value for the dark energy density density.\\
The equation of state (EoS) parameter $w=-1-\frac{1}{3}\frac{d\tilde{H}^2/dx}{\tilde{H}^2}$, from (\ref{eq7}) is obtained as
\begin{equation}\label{eq10}
w=-1+\frac{2}{3}\frac{(1-\tilde{A})e^{-2x}}{\tilde{A}+(1-\tilde{A} e^{-2x}}=-1+\frac{2}{3}\frac{(1-\tilde{A})(1+z)^2}{\tilde{A}+(1-\tilde{A})(1+z)^2}
\end{equation}
in the last equality we used the redshift variable $1+z=e^{-x}$. This equation shows that $w<-1/3$, describing effectively evolving dark energy, with high redshift limit $w=-1/3$ at $z\rightarrow\infty$. At far future ($z\rightarrow -1$ or $x\rightarrow\infty$), the universe evolves toward de Sitter phase ($w\rightarrow-1$).
\subsection*{Modified Gauss Bonnet}
Let's consider the following modification of the GB dark energy density, that gives rise to interesting consequences.
\begin{equation}\label{eq11}
\rho_{\Lambda}=\gamma H^4+\delta H^2\dot{H}
\end{equation}
where $\gamma$ and $\delta$ are dimensionless constants. Replacing this density in the Friedmann equation gives
\begin{equation}\label{eq12}
\delta\frac{dH}{dt}+\gamma H^2-\frac{3}{\kappa^2}=0
\end{equation}
Integrating this equation gives the solution of the type (\ref{eq5}) with the same asymptotic behavior. In the $x$ variable and using the scaled quantities, the Eq. (\ref{eq12}) becomes
\begin{equation}\label{eq13}
\frac{1}{2}\frac{d\tilde{H}^2}{dx}+\frac{\gamma}{\delta}\tilde{H}^2-\frac{3}{\delta\kappa^2 H_0^2}=0
\end{equation}
After integration with the initial condition $\tilde{H}^2|_{x=0}=1$, leads to the solution
\begin{equation}\label{eq14}
\tilde{H}^2=\tilde{B}+(1-\tilde{B}) e^{-\frac{2\gamma}{\delta}x}
\end{equation}
where $\tilde{B}=3/(\gamma\kappa^2 H_0^2)$. An important characteristic of this solution is that coefficient of the exponent depends on the constants $\gamma$ and $\delta$, while the first term (corresponding to some density parameter) depends only on one constant $\gamma$, so we can handle them independently. According to this solution, if we consider $\gamma/\delta<0$, then the phantom behavior for dark energy is possible. In fact we can adjust these constants with the available observational data. For the same reasons discussed above, we can consider $\tilde{B}\sim 1$, which again reproduces the correct expression for the observed DE density $\rho_{\Lambda_0}\sim M_p^2 H_0^2$. The dark energy EoS parameter from (\ref{eq14}) is given by
\begin{equation}\label{eq15}
w=-1+\frac{2\gamma}{3\delta}\frac{(1-\tilde{B}) e^{-2(\gamma/\delta)x}}{\tilde{B}+(1-\tilde{B}) e^{-2(\gamma/\delta)x}}=-1+\frac{2\gamma}{3\delta}\frac{(1-\tilde{B}) (1+z)^{2(\gamma/\delta)}}{\tilde{B}+(1-\tilde{B}) (1+z)^{2(\gamma/\delta)}}
\end{equation}
for $\delta>0$ ($\gamma$ must always be positive), this EoS parameter runs between $w=-1+\frac{2\gamma}{3\delta}$ at $z\rightarrow \infty$ and $w=-1$ at $z\rightarrow -1$ (evolving towards de Sitter phase at future). Note that for $\gamma/\delta=3/2$ the EoS describes presureless matter-like behavior at $z\rightarrow \infty$.\\
Compared to the holographic model proposed in \cite{granda},\cite{granda1} in absence of matter, in that case we obtained power-law solution giving rise to constant EoS (see \cite{granda1}), and in the present model we obtained that even in absence of matter the EoS evolves dynamically.
\subsection*{Dynamical Interpretation}
In the present case, when we consider only the presence of dark energy fluid with density given by (\ref{eq2}) or (\ref{eq11}), we can find an exact expression for the scalar field and potential corresponding to this fluid.\\
Let's consider the scalar tachyon field with the action
\begin{equation}\label{eq11a}
S=-\int d^4x V(\phi)\sqrt{-\det\left[g_{\mu\nu}+\partial_{\mu}\phi\partial_{\nu}\phi\right]}
\end{equation}
In the flat Friedmann background, the energy density and pressure density of the tachyon scalar field are given by \cite{padmanabhan, copeland}
\begin{equation}\label{eq11b}
\rho_{\phi}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^2}},\,\,\,\, p_{\phi}=-V(\phi)\sqrt{1-\dot{\phi}^2}
\end{equation}
And the Einstein's equations take the form
\begin{equation}\label{eq11c}
H^2=\frac{\kappa^2}{3}\frac{V(\phi)}{\sqrt{1-\dot{\phi}^2}}
\end{equation}
and
\begin{equation}\label{eq11d}
3H^2+2\dot{H}=\kappa^2 V(\phi)\sqrt{1-\dot{\phi}^2}
\end{equation}
Combining Eqs. (\ref{eq11c}) and (\ref{eq11d}) one can express the tachyon field and the potential as follows
\begin{equation}\label{eq11e}
\dot{\phi}^2=-\frac{2}{3}\frac{\dot{H}}{H^2},
\end{equation}
and
\begin{equation}\label{eq11f}
V(\phi)=\frac{3H^2}{\kappa^2}\left[1+\frac{2\dot{H}}{3H^2}\right]^{1/2}
\end{equation}
Using the solution (\ref{eq14}) and solving the Eq. (\ref{eq11e}) in the $x$ variable, one finds
\begin{equation}\label{eq11g}
\phi=\sqrt{\frac{2\delta}{3\gamma \tilde{B}}}\frac{1}{H_0}\arctan\left[\sqrt{\frac{\tilde{B}}{1-\tilde{B}}}e^{\gamma x/\delta}\right]
\end{equation}
where $\delta>0$. And from (\ref{eq11f}) one can find the potential in terms of the scalar field as
\begin{equation}\label{eq11h}
V(\phi)=\frac{3\tilde{B}H_0^2}{\kappa^2}\frac{1}{\tan^2(\tilde{\phi})}\left[3\tan^4(\tilde{\phi})+2(3-\frac{\gamma}{\delta})\tan^2(\tilde{\phi})+3-\frac{2\gamma}{\delta}\right]^{1/2}
\end{equation}
where $\tilde{\phi}=\sqrt{\frac{3\gamma\tilde{B}}{2\delta}}\phi$. The GB case is obtained for $\gamma=\delta$. Is worth noting that the usual quintessence scalar field leads to inconsistencies (negative potential or complex $\phi$ for $x>0$) when trying to represent the solution (\ref{eq7}) or (\ref{eq14}) for $\delta>0$. So, the dynamics generated by the dark energy density (\ref{eq2}) or (\ref{eq11}) can be represented by an scalar field of the tachyon type with the potential (\ref{eq11h}).\\
\section{Adding matter content}
Here we consider the complete model with the matter component (usual barionic and dark matter). As we will see, adding the matter component causes strong changes in the differential equation, which becomes non-linear, and numerical calculation shows interesting results that deserve study. We will consider the general case of modified Gauss Bonnet model (MGB) and then analyze different cases, particularly the Gauss Bonnet model.
Adding the matter term $\rho_m=\rho_{m0}e^{-3x}$ to the Friedmann equation gives
\begin{equation}\label{eq16}
H^2=\frac{\kappa^2}{3}\left(\gamma H^4+\delta H^2\dot{H}+\rho_{m0} e^{-3x}\right)
\end{equation}
which in the $x$ variable and using the scaled magnitudes, takes the form
\begin{equation}\label{eq17}
\frac{\tilde{\delta}}{2}\tilde{H}^2\frac{d\tilde{H}^2}{dx}+\tilde{\gamma}\tilde{H}^4-\tilde{H}^2+\Omega_{m0}e^{-3x}=0
\end{equation}
where $\tilde{\gamma}=\kappa^2H_0^2\gamma/3$, $\tilde{\delta}=\kappa^2H_0^2\delta/3$ and $\Omega_{m0}=\kappa^2\rho_{m0}/(3H_0^2)$ ($\Omega_m=\kappa^2\rho_m/(3H^2)$). This equation should be solved with the initial condition
\begin{equation}\label{eq18}
\frac{d\tilde{H}^2}{dx}\Big|_{x=0}=1
\end{equation}
Note that this equation is more complicated that (\ref{eq4}) because in absence of matter we could cancel one $H^2$ factor, which considerably simplified the equation. This equation can not be solved analytically, but can be integrated numerically in a given redshift interval and for given values of the parameters $\tilde{\gamma}$ and $\tilde{\delta}$, taking into account the initial (flatness) condition (\ref{eq18}). Let's consider first the GB case which corresponds to $\tilde{\gamma}=\tilde{\delta}=8\kappa^2H_0^2\alpha=\tilde{\alpha}$ (see (\ref{eq3})).
Solving numerically (\ref{eq17}) in the redshift variable with $\Omega_{m0}=0.31$ and the initial condition (\ref{eq18}), we obtain the behavior of the EoS parameter as described in Fig. 1, for two values of $\alpha$.
\begin{center}
\includegraphics [scale=0.7]{wgbm.pdf}
\end{center}
\begin{center}
{fig. 1 \it The EoS parameter for the GB DE density with $\Omega_{m0}=0.31$, $\tilde{\alpha}=0.55$ (dashed line) and $\tilde{\alpha}=0.8$.}
\end{center}
And in Fig. 2 we show the evolution of $H(z)$ for the corresponding parameters of Fig. 1 against its observational values with error bars \cite{simon, stern}.\\
\begin{center}
\includegraphics [scale=0.7]{H-GB-m1.pdf}
\end{center}
\begin{center}
{Fig. 2 \it The observational $H(z)$ data with error bars and $H(z)$ form the solution of Eq. (\ref{eq17}) (for $\gamma=\delta$) with $\Omega_{m0}=0.31$, $\tilde{\alpha}=0.55$ (dashed line) and $\tilde{\alpha}=0.8$.}
\end{center}
The EoS parameter has been calculated according to
\begin{equation}\label{eq19}
w=-1-\frac{2}{3}\frac{\dot{H}}{H^2}=-1+\frac{1+z}{3\tilde{H}^2}\frac{d\tilde{H}^2}{dz}
\end{equation}
In the case $\tilde{\alpha}=0.8$, the current EoS parameter takes the value $w_0\sim -0.9$, and for $\tilde{\alpha}=0.55$ we have $w_0\sim -1.17$, so the model allows for quintom behavior. In both cases the EoS has a minimum value at future in the phantom region $w_{min}<-1$, and then evolves toward de Sitter phase.\\
For the MGB model, after numerical integration of Eq. (\ref{eq17}) with the initial condition (\ref{eq18}), $\Omega_{m0}=0.27$ and different choices of $\tilde{\gamma}$ and $\tilde{\delta}$ we found the EoS as showed in Fig. 3
\begin{center}
\includegraphics [scale=0.7]{wmgbm.pdf}
\end{center}
\begin{center}
{Fig. 3 \it The EoS parameter for the MGB DE density with $\Omega_{m0}=0.31$, and some representative values for the parameters $(\tilde{\gamma},\tilde{\delta})$ as shown in the graphic.}
\end{center}
In Fig. 4 we show evolution of $H(z)$ for the corresponding parameters of Fig. 3 against the observational values with error bars.
\begin{center}
\includegraphics [scale=0.7]{H-MGB-m1.pdf}
\end{center}
\begin{center}
{Fig. 4 \it The observational $H(z)$ data with error bars, and $H(z)$ from the solution of Eq. (\ref{eq17}) for the corresponding curves of Fig. 3}
\end{center}
Corresponding to each pair of parameters $(\tilde{\gamma},\tilde{\delta})$, the current values for EoS parameter are: $w_0\sim -0.77$ for $(0.9,0.6)$, $w_0\sim -0.8$ for $(0.9,0.7)$, $w_0\sim -0.99$ for $(0.7,0.5)$ and $w_0\sim -1.15$ for $(0.6,0.4)$. The last point gives currently quintom behavior, but all curves exhibit a minimum at future, bellow the phantom divide and then evolve towards the de Sitter phase. In all this analysis we have guided by the last data provided by the Planck Collaboration \cite{planck1,planck2}, namely $\Omega_m\sim 0.31$ and $H_0\sim 68 Km/s/Mpc$.
\subsection*{Statefinder Diagnostic}
An useful tool to compare different dark energy models uses derivatives of the scale factor beyond the second order \cite{sahni1, sahni2}. The diagnostic proposal called ``statefinder'' introduces new geometrical dimensionless parameters that characterize the properties of dark energy regardless of the model, as they depend on the observable Hubble parameter and its derivatives. The statefinder parameters $q$, $r$ and $s$ ($q$ is also known as the deceleration parameter) are defined as
\begin{equation}\label{eq20}
q=-1-\frac{\dot{H}}{H^2},\,\,\, r=\frac{\dddot{a}}{aH^3},\,\,\,\, s=\frac{r-1}{3(q-1/2)}
\end{equation}
For our analysis will be useful to write these parameters in terms of the Hubble parameter $H(z)$ and the redshift $z$ as
\begin{equation}\label{eq21}
\begin{aligned}
q=-1+\frac{1+z}{2H^2}\frac{dH^2}{dz}&,\,\,\, r=1+\frac{1}{2H^2}\left[(1+z)^2\frac{d^2H^2}{dz^2}-2(1+z)\frac{dH^2}{dz^2}\right]\\
& s=\frac{1}{3}\frac{(1+z)^2\frac{d^2H^2}{dz^2}-2(1+z)\frac{dH^2}{dz}}{(1+z)\frac{dH^2}{dz}-3H^2}
\end{aligned}
\end{equation}
An important characteristic of the pair ($r,s$) is that the spatially flat $\Lambda$CDM scenario corresponds to a fixed point ($1,0$) in the $s-r$ plane, with respect to which we can contrast the trajectories of other dark energy models. Though the relations between these statefinder parameters, namely $r(s)$ and
$r(q)$ can not be derived analytically for the present model, we give here a numerical analysis and plot a representative statefinder diagrams for the GB and the MGB models. In Fig. 5 we show the statefinder trajectories in the $s-r$ and $q-r$ planes for the GB model, taking $\tilde{\alpha}=0.8$ and Fig. 6 shows the statefinder trajectories for the MGB, taking $\tilde{\gamma}=0.9$ and $\tilde{\delta}=0.6$.\\
\begin{figure}[hbtp]
\begin{center}
\includegraphics [scale=0.55]{rs-GB.pdf}
\includegraphics [scale=0.55]{rq-GB.pdf}
\end{center}
\begin{center}
{Fig. 5. \it Evolution of the universe in the statefinder $s-r$ plane (left) and $q-r$ plane (right) for the GB DE model with $\tilde{\alpha}=0.8$, where the arrows along the curves denote the direction of evolution.}
\end{center}
\end{figure}
\noindent In the left graphic in Fig. 5, the pair ($r,s$) starts at the right of the $\Lambda$CDM fixed point, which is characteristic of quintessence behavior ($0<s<1$, $r<1$) and then evolves to the left of the $\Lambda$CDM (characteristic region of the Chaplygin gas model) \cite{sahni2} to asymptotically approach again the $\Lambda$CDM at late times. The trajectory in the $r-q$ plane starts in the region ($0<s<1$, $r<1$) and then crosses the $\Lambda$CDM line at some redshift in the past to evolve towards the de Sitter expansion at the future ($\Lambda$CDM$\rightarrow SS$ at $t\rightarrow \infty$). The trajectory $r(q)$ reaches a turning point in the phantom region ($q<-1$) and then evolves back to the SS point. The current values of the statefinder parameters are: $r_0\approx 1.92, s_0\approx -0.22, q_0\approx -0.86$.\\
\begin{figure}[hbtp]
\begin{center}
\includegraphics [scale=0.55]{rs-MGB.pdf}
\includegraphics [scale=0.55]{rq-MGB.pdf}
\end{center}
\begin{center}
{Fig. 6. \it Evolution of the universe in the statefinder $s-r$ plane (left) and $q-r$ plane (right) for the MGB DE model with $\tilde{\gamma}=0.9$ and $\tilde{\delta}=0.6$.}
\end{center}
\end{figure}
In Fig. 6 the parameter $r$ increases monotonically from unity to a maximum value and then decreases to unity. In this case the $r(s)$ trajectory approaches the steady state model (SS) (the de Sitter phase) asymptotically at late times ($\Lambda$CDM$\rightarrow SS$ at $t\rightarrow\infty$, $\Omega_m\rightarrow 0$). The evolution in the $r-q$ plane takes place in the upper half (over the $\Lambda$CDM line) and clearly shows that the evolution starts at the SCDM and ends at the SS model. The parameter $q$ reaches the minimum value in the future, in the region $q<-1$ (phantom phase) and turns back asymptotically to the de sitter expansion. The current values for the statefinder parameters are: $r_0\approx 2.19, s_0\approx -0.34, q_0\approx -0.65$.\\
According to the trajectories depicted above for both models, the expansion crosses the phantom divide at some point in the future, reaching a turning point in the region $q<-1$ and then evolving toward the de Sitter phase. The current value of $r$ is very close to the maximum value for both models. The $r(s)$ trajectory for the MGB model is a closed loop that starts in the early time limit of the $\Lambda$CDM fixed point (SCDM) and ends in the $t\rightarrow \infty$ limit of the $\Lambda$CDM (SS) with de Sitter expansion.\\
\section{Discussion}
We proposed an infrared cut-off for the DE density, based on the linear combination of $H^4$ and $H^2\dot{H}$ terms, that for particular values of the coefficients gives the GB 4-dimensional topological invariant. This proposal has the right dimension of density without introducing dimensional parameters and may be interpreted as a non-saturated regime in the conventional holographic principle, with the advantage that we don't need to resort to the limit imposed by the black hole formation. After solving the Friedamnn equation with the initial condition we get an expression for the dark energy density that incorporates the Planck mass, and behaves in a way compatible with observations. We have found a scalar tachyon field with potential given by Eq. (\ref{eq11h}), that represents the solutions (\ref{eq7}) and (\ref{eq14}), giving a dynamical interpretation to the models (\ref{eq2}) and (\ref{eq11}).
Unlike the holographic density proposed in \cite{granda}, \cite{granda1} in the absence of matter, the present model describes a dynamical varying EoS as stated by Eqs. (\ref{eq10}) and (\ref{eq15}), with $w$ running between $-1/3$ at $z\rightarrow\infty$ and $-1$ at $z\rightarrow-1$ for the GB DE and between $-1+\frac{2\gamma}{3\delta}$ at $z\rightarrow\infty$ and $-1$ at $z\rightarrow-1$ for the MGB DE. In absence of matter, according to (\ref{eq7}) the GB DE density at early times exhibits a behavior as $a^{-2}$, which was discussed in \cite{li}, but the energy balance is dominated by the matter behavior $a^{-3}$ for small $a$. The MGB model in absence of matter, and under some relation between the parameters reproduces a $\Lambda$CDM-like model.
The presence of matter drastically changes the behavior of the EoS, allowing the crossing of the phantom barrier at current and late times, presenting a minimum in the future bellow the phantom barrier, and then evolving toward the de sitter phase at $z\rightarrow -1$. This is due to the different nature of the Friedmann equation with the addition of matter content, that converts the Friedmann equation into a nonlinear differential equation. This could be interpreted as follows: the presence of dark matter enforces the effect of the DE at late times, but at early times is clear the dominance of the matter sector (especially in the MGB model), as seen in Figs. 1, 2 and in the statefinder diagrams given in Figs. 5 and 6. A remarkable aspect of this proposal is that independently of the values given to the parameters involved in the model, in all cases the universe evolves toward de Sitter phase in the far future ($z\rightarrow -1$).
Numerical calculations show that despite the fact that the EoS evolves toward the phantom region and ends in the de Sitter phase, the energy density and pressure are free of any finite time future singularities (i.e. $H$ and $\dot{H}$ are finite even in the region of phantom behavior shown in Figs. 1 and 2). This model also makes a concrete prediction about the final state of the universe, namely the universe ends in a de Sitter phase. We performed the statefinder diagnostic and have found the evolution paths in the $s-r$ and $q-r$ planes for the GB and MGB models. For the MGB model the $r(s)$ trajectory is a closed loop that starts at the early time limit of the $\Lambda$CDM fixed point and ends at the late time limit of the $\Lambda$CDM fixed point with de Sitter expansion.
It turns out that this choice of dark energy density (that scales as $L^{-4}$), which can be interpreted as a non-saturated variant of the holographic model, is a viable model which is able to explain the cosmic coincidence, it also leads to quintom behavior but remaining free of finite time future singularities, and is consistent with current observational data.
\section*{Acknowledgments}
This work was supported by Universidad del Valle under project CI 7890.
|
1,314,259,995,803 | arxiv | \section{Introduction}
\label{sec:intro}
\acf{PAI} is a comparatively young and rapidly emerging imaging modality that promises real-time, noninvasive, and radiation-free measurement of optical tissue properties\cite{wang2014photoacoustic}. In contrast to other optical imaging modalities, \ac{PAI} induces the emergence of acoustic signals to enable structural imaging of chromophores - molecular structures that absorb light - up to several centimeters deep into the tissue. This high depth penetration is possible because the acoustic scattering of the arising sound waves is orders of magnitude smaller than the optical scattering of the incident light in biological tissue. The underlying physical principle for signal generation is the \emph{PA effect}\cite{rosencwaig1976theory}. It is induced by extremely short light pulses that cause an initial pressure rise $p_0$ inside the tissue. The initial pressure $p_0 = \Gamma \cdot \mu_a \cdot \phi$ is proportional to the optical absorption coefficient $\mu_a$, the local light fluence $\phi$, and the temperature-dependent Gr\"uneisen parameter $\Gamma$. The deposited energy is released in form of sound waves that can be measured as time-series pressure data $p(t) = A(p_0, \theta)$ with appropriate acoustic detectors, such as ultrasound transducers. Here, acoustic forward operator $A$ operates on the initial pressure distribution $p_0$ taking into consideration the acoustic properties $\theta$ of the medium.\\
Due to its rapid development, \ac{PAI} has seen various clinical application attempts over the last few years. Among these, cancer research is a field where \ac{PAI} shows serious potential \cite{mallidi2011photoacoustic,li2019benign,zhang2018photoacoustic,quiros2018optoacoustics,oh2006three,weight2006photoacoustic,zhang2010subwavelength,zhang2010chronic,song2008noninvasive,erpelding2010sentinel,garcia2015dual}. In this use case, hemoglobin is the enabling endogenous chromophore, due to amplified and sustained angiogenesis \cite{hanahan2011hallmarks} being one of the hallmarks of cancer and due to the cancer cells' increased metabolism, which potentially induces a decrease in local blood oxygenation \cite{wang2012biomedical}. Furthermore, because inflammatory processes also change the hemodynamic behavior of tissue, \ac{PAI} is also used for imaging of inflamed joints \cite{wang2007noninvasive,rajian2012photoacoustic,jo2018detecting} or staging of patients with Crohn's disease \cite{knieling2017multispectral,waldner2016multispectral,lei2019characterizing}. To further increase the potential of \ac{PAI}, it is also applied in combination with other imaging modalities, especially ultrasound imaging \cite{niederhauser2005combined,aguirre2011potential,needles2013development,garcia2015dual,elbau2017quantitative,mandal2019multimodal}. \ac{PAI} is further used for brain imaging \cite{wang2003noninvasive,ku2005imaging,hu2009functional,yao2014photoacoustic,mohammadi2019skull} or surgical and interventional imaging applications, such as needle tracking \cite{kim2010handheld,su2010photoacoustic}.\\
The signal contrast of \ac{PAI} is caused by distinct wavelength-dependent absorption characteristics of the chromophores\cite{upputuri2016recent}. But to exploit information of $\mu_a$ for answering clinical questions, open research questions remain that can be categorized into four main areas. In the following, we explain these four major categories and summarize their principal ideas.\\
\textbf{Acoustic inverse problem.} The most pressing problem concerns the reconstruction of an image $S$ from recorded time-series data by estimating the initial pressure distribution $p_0$ from $p(t)$. This problem is referred to as the acoustic inverse problem. To this end, an inverse function $A^{-1}$ for the acoustic operator $A$ needs to be computed in order to reconstruct a signal image $S = A^{-1}(p(t)) \approx p_0 = \mu_a \cdot \phi \cdot \Gamma$ that is an approximation of $p_0$. Typical examples of algorithms to solve this problem are the universal back-projection\cite{xu2005universal}, delay-and-sum \cite{mozaffarzadeh2017double}, time reversal\cite{treeby2010k}, or iterative reconstruction schemes\cite{huang2013full}. While the acoustic inverse problem can be well-posed in certain scenarios (for example by using specific detection geometries) and thus can have a unique solution, several factors lead to considerable difficulties in solving it. These include wrong model assumptions\cite{cox2009challenges}, limited-view\cite{waibel_reconstruction_2018} and limited-bandwidth detectors\cite{buchmann2017characterization}, or device modeling\cite{sahlstrom2020modeling} and calibration errors\cite{buchmann2020quantitative}.\\
\textbf{Image post-processing.} \ac{PAI}, in theory, has exceptionally high contrast and spatial resolution\cite{xu2006photoacoustic}. Because the acoustic inverse problem is ill-posed in certain settings and because of the presence of noise, many reconstructed PA images suffer from distinct artifacts. This can cause the actual image quality of a PA image to fall short of its theoretical potential. To tackle these problems, image post-processing algorithms are being developed to mitigate the effects of artifacts and noise and thus improve overall image quality.\\
\textbf{Optical inverse problem.} Assuming that a sufficiently accurate reconstruction of $p_0$ from $p(t)$ has been achieved, the second principle problem that arises is the estimation of the underlying optical properties (most importantly the absorption coefficient $\mu_a$). It is an inverse problem and is referred to as the optical inverse problem. Furthermore, the problem has proven to be exceptionally involved, which can be derived by the fact that methods to solve the problem have not been successfully applied to \emph{in vivo} data yet. It belongs to the category of ill-posed inverse problems, as it does not necessarily possess a unique solution. Furthermore, several other factors make it hard to tackle, including wrong model assumptions\cite{cox2009challenges}, non-uniqueness and non-linearity of the problem\cite{shao2011estimating}, spectral coloring\cite{tzoumas2016eigenspectra}, and the presence of noise and artifacts\cite{kazakeviciute2016multispectral}. Quantification of the absorption coefficient has, for example, been attempted with iterative reconstruction approaches\cite{cox2006two}, via fluence estimation\cite{brochu2016towards}, or by using machine learning-based approaches\cite{kirchner2018context}.\\
\textbf{Semantic image annotation.} Based on the diagnostic power of optical absorption it is possible to generate semantic image annotations of multispectral PA images and a multitude of methods for it are being developed to specifically tackle questions of clinical relevance. To this end, algorithms are being developed that are able to classify and segment multispectral PA images into different tissue types and that can estimate clinically relevant parameters that are indicative of a patient's health status (such as blood oxygenation). Current approaches to create such semantic image annotations suffer from various shortcomings, such as long computation times or the lack of reproducibility in terms of accuracy and precision when being applied to different scenarios.\\
Simultaneously to the rapid developments in the field of \ac{PAI}, deep learning algorithms have become the \emph{de facto} state of the art in many areas of research\cite{shen2017deep} including medical image analysis. A substantial variety of medical applications include classical deep learning tasks such as disease detection \cite{liu2019comparison}, image segmentation \cite{tajbakhsh2020embracing}, and classification \cite{tandel2019review}. Recently, deep learning has also found entrance into the field of \ac{PAI}, as it promises unique advantages to solve the four listed problems, thus promoting clinical applicability of the developed methods. One further prominent advantage of deep learning is the extremely fast inference time, which enables real-time processing of measurement data.\\
This review paper summarizes the development of deep learning in \ac{PAI} from the emergence of the first PA applications in 2017 until today and evaluates progress in the field based on the defined task categories. In section \ref{sec:methods}, we outline the methods for our structured literature research. General findings of the literature review including the topical foci, data acquisition techniques, used simulation frameworks as well as network architectures are presented in section \ref{sec:general}. The reviewed literature is summarized according to the four principal categories in sections \ref{sec:reconstruction} to \ref{sec:clinical_translation}. Finally, the findings are discussed and summarized in section \ref{sec:discussion}.\\
\section{Methods of literature research}
\label{sec:methods}
Above, we described the \ac{PAI}-specific challenges that deep learning can be applied to and thus divided the topic into four major categories: \emph{I. Acoustic inverse problem}, \emph{II. Image post-processing}, \emph{III. Optical inverse problem}, and \emph{IV. Semantic image annotation}. We conducted a systematic literature review for the period between January 2017 and September 2020 and assigned each identified paper to the most suitable categories. For the search, we used several scientific search engines: \emph{Google Scholar}, \emph{IEEE Xplore}, \emph{Pubmed}, \emph{Microsoft Academic Search Engine}, and the \emph{arXiv} search function with the search string \textbf{("Deep learning” OR "Neural Network") AND ("photoacoustic" OR "optoacoustic")}. The search results were then refined in a multi-step process (see Fig. \ref{fig:search_algorithm}).\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/search_algorithm.pdf}
\caption{Overview of the literature review algorithm. First, potentially fitting papers are identified based on an initial search. The search results are complemented by adding additional papers found by other means than the search engines, and finally, non-relevant papers are excluded by removing duplicates and by abstract scanning.}
\label{fig:search_algorithm}
\end{figure}
First, potential candidates were identified based on an initial search using their title, as well as the overview presented by the search engine. The search results were complemented by adding additional papers found by means other than the named search engines. For this purpose, we specifically examined proceedings of relevant conferences, websites of key authors we identified, and websites of PA device vendors. Finally, non-relevant papers were excluded by removing Journal/Proceeding paper duplicates and by abstract scanning to determine whether the found papers match the scope of this review. Using the abstract, we excluded papers that did not apply deep learning, and those that did not match the scope of \emph{biomedical} \ac{PAI}. The remaining articles were systematically examined by the authors using a questionnaire to standardize the information that was to be extracted. While writing the paper, we continuously monitored the mentioned resources for new entries until the end of September 2020.\\
In total, applying the search algorithm as detailed above, we identified 66 relevant papers (excluding duplicates and related, but out-of-scope work) that have been published since 2017.
\section{General findings}
\label{sec:general}
The application of deep learning techniques to the field of \ac{PAI} has constantly been accelerating over the last three years and has simultaneously generated a noticeable impact on the field.\\
\textbf{Topical foci.} After categorization of the papers into the predetermined four areas, the papers were arranged into thematic subcategories (see Fig. \ref{fig:topics}). Papers related to the {Acoustic Inverse Problem} (section \ref{sec:reconstruction}) generally focus on the reconstruction of PA images from raw time-series pressure data but also related topics, such as dealing with limited-view or limited data settings, as well as the estimation of the speed of sound of tissue. The {Image Post-processing} (section \ref{sec:processing}) category entails papers that deal with image processing algorithms that are being applied after image reconstruction. The aim of such techniques usually is to improve the image quality, for example by noise reduction or artifact removal. The three papers assigned to the {Optical Inverse Problem} (section \ref{sec:quantification}) deal with the estimation of absolute chromophore concentrations from PA measurements. Finally, papers dealing with Semantic Image Annotation (section \ref{sec:clinical_translation}) tackle use cases, such as the segmentation and classification of tissue types or the estimation of functional tissue parameters, such as blood oxygenation.\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/topics.pdf}
\caption{Overview over the topical foci of current research towards applying deep learning algorithms to problems in biomedical \ac{PAI}.}
\label{fig:topics}
\end{figure}
\textbf{Data.} Data is key to successfully apply machine learning techniques to any given problem. We analyzed the usage of data in the reviewed papers and summarized the findings in Fig \ref{fig:data}.\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/training_and_test.pdf}
\caption{Analysis of the data used in the reviewed papers. (a) shows the distribution of the number of samples in the training data set, (b) shows the percentage of papers working with synthetic or experimental training data, (c) shows the percentage of papers that tested their approaches on multiple data sets including test data from a data distribution different than the training data and (d) shows how many papers tested their approach on real data.}
\label{fig:data}
\end{figure}
\emph{Training.} The number of training data ranged from 32 to 296.300 samples with a median number of training samples of 2.400. As evident from these findings, one of the core bottlenecks of the application of deep learning algorithms to \ac{PAI} is the lack of reliable experimental training data. This can in particular be caused by a lack of ground truth information on the underlying optical tissue properties or the underlying initial pressure distribution when acquiring experimental measurements. To address this issue, researchers make heavy use of simulated data and as a matter of fact, nearly 75\% of papers relied exclusively on these for training the neural network. Table \ref{tab:train_test} shows the distribution of papers that use experimental data. The table shows that the lack of experimental training data is particularly emphasized for the optical and acoustic inverse problem. In contrast to the other tasks, where manual image annotations can be used as a ground truth reference, the underlying initial pressure distribution or optical tissue properties are generally not known in experimental settings. We have identified three main strategies for generating synthetic training data in this review: random, model-based, and reference-based data generation:
\begin{enumerate}
\item \emph{Random data generation.} The first and simplest strategy generates data by creating completely random distributions of the optical and acoustic properties that are necessary for the simulation framework\cite{cai_end--end_2018}. Here, usually, a Gaussian distribution of the parameters in question is assumed and no dedicated structures are added to the data.
\item \emph{Model-based data generation.} Training data is created by defining geometrical structures that are assigned optical and acoustic properties according to a hand-crafted model\cite{grohl_estimation_2019}. Such a model might include literature references e.g. for the size, shape, and properties of typical absorbers in tissue. For the generation of training data, many different instances of the model are created that all yield different distributions of chromophores.
\item \emph{Reference-based data generation.} For the reference-based approach, reference images of different imaging modalities are taken as the basis for data generation\cite{hauptmann_model-based_2018}. They are processed in a way that allows for their direct usage to either create distinct segmentation patterns of, for example, vessels or as the initial pressure distribution for subsequent acoustic forward modeling.
\end{enumerate}
Naturally, researchers also utilized combinations of these approaches, including training on a large data set of simulated data and utilizing a smaller experimental data set to adjust the neural network to the experimental data distribution in a process called \emph{transfer learning}\cite{hauptmann_model-based_2018, pan2009survey}.\\
\begin{table}[h!tb]
\centering
\begin{tabular}{lccc}
\textbf{Problem} & \textbf{Exp. test data} & \textbf{Exp. train data} \\
\hline
\\
Acoustic inverse problem & 8 (40\%) & 1 (5\%)\\
Image post-processing & 14 (70\%) & 7 (35\%)\\
Optical inverse problem & 1 (33\%) & 1 (33\%)\\
Semantic image annotation & 12 (52\%) & 9 (39\%)\\
\\
\end{tabular}
\caption{Overview of the findings for training and test data used in the reviewed papers. The table shows the absolute and relative number of papers that use experimental data for testing or for training.}
\label{tab:train_test}
\end{table}
\emph{Testing.} In the field of medical imaging, only few prospective studies warrant reliable insights into the fidelity of deep learning methods \cite{liu2019comparison}. One of the major problems is that algorithms are not directly usable by clinicians due to technical or bureaucratic limitations \cite{panch2019inconvenient}. Given the fact that most approaches use simulated data to train their algorithms, there is a high probability that many of the presented algorithms - while yielding superb results on the publication data - could fail in a clinical scenario. This can be attributed to the fact that training data can suffer from several shortcomings compared to the data distribution in reality, such as a significant difference in the data (domain gap)\cite{ross2018exploiting}, an insufficient number of samples (sparsity)\cite{cciccek20163d}, or a selection bias\cite{kato2018learning}. Fig \ref{fig:data} shows that in \ac{PAI} 50\% of papers tested their deep learning approaches on multiple data sets that are significantly different from the training data distribution. Nearly all of these papers test their approaches on experimental data, and about 25\% of the examined papers test on \emph{in vivo} data.\\
\textbf{Simulation frameworks.} Given the necessity to create synthetic data sets for algorithm training, it is crucial to realistically simulate the physical processes behind \ac{PAI}. To this end, we have identified several eminent open-source or freely available frameworks that are being utilized in the field and briefly present five of them here:\\
1) The \emph{k-Wave}\cite{treeby2010k} toolbox is a third-party MATLAB toolbox for the simulation and reconstruction of PA wave fields. It is designed to facilitate realistic modeling of \ac{PAI} including the modeling of detection devices. As of today it is one of the most frequently used frameworks in the field and is based on a k-space pseudo-spectral time-domain solution to the PA equations.\\
2) The \emph{mcxyz} \cite{jacques2014coupling} simulation tool uses a Monte Carlo model of light transport to simulate the propagation of photons in heterogeneous tissue. With this method, the absorption and scattering properties of tissue are used to find probable paths of photons through the medium. The tool uses a \ac{3D} Cartesian grid of voxels and assigns a tissue type to each voxel, allowing to simulate arbitrary volumes.\\
3) The \emph{Monte Carlo eXtreme} \cite{fang2009monte} (MCX) tool is a Graphics Processing Units (GPU)-accelerated photon transport simulator. It is also based on the Monte Carlo model of light transport and supports the simulation of arbitrarily complex \ac{3D} volumes using a voxel domain, but is also capable of simulating photon transport for \ac{3D} mesh models (in the MMC version). Its main advantage is the support of GPU acceleration using a single or multiple GPUs.\\
4) The \emph{NIRFAST} \cite{dehghani2009near} modeling and reconstruction package was developed to model near-infrared light propagation through tissue. The framework is capable of single-wavelength and multi-wavelength optical or functional imaging from simulated and measured data. It recently integrated the NIRFAST optical computation engine into a customized version of 3DSlicer.\\
5) The \emph{Toast++}\cite{schweiger2014toast++} software suite consists of a set of libraries to simulate light propagation in highly scattering media with heterogeneous internal parameter distribution. Among others, it contains numerical solvers based on the finite-element method, the discontinuous Galerkin discretization scheme, as well as the boundary element method. \\
\textbf{Neural network Architectures.} \acp{CNN}\cite{gu2018recent} are currently the state-of-the-art method in deep learning-based \ac{PAI}. Here, especially the U-Net\cite{ronneberger2015u} architecture is highly popular and has been used in over 70\% of the reviewed papers. Other architectures such as residual neural networks or fully-connected networks only exhibit a prevalence of less than 10\% each. It should be noted that usually, slightly modified versions of the vanilla U-Net, especially fitting to the target application, have proven to yield the best performance.\\
\section{Acoustic Inverse Problem}
\label{sec:reconstruction}
The acoustic inverse problem refers to the task of reconstructing an image of the initial pressure distribution from measured time-series pressure data. Reconstructing a PA image from time-series data constitutes the main body of work, either by \emph{enhancing existing model-based approaches} (40\% of papers) or by performing \emph{direct image reconstruction} (35\% of papers). Furthermore, auxiliary tasks are being examined as well, such as the \emph{localization of wavefronts} from point sources (15\% of papers) and the \emph{estimation of the speed of sound} of the medium (10\% of papers). Information on these parameters is important to achieve optimal image reconstruction, thereby enhancing the image quality and improving the image's usefulness in clinical scenarios. Typical evaluation metrics that are used to assess the performance of reconstruction algorithms are the \ac{MSE}, \ac{MRE}, \ac{MAE}, \ac{PSNR}, \ac{SSIM} Coefficient, \ac{CC}, and \ac{SNR}. In total, we identified 20 papers that tackle the acoustic inverse problem, all of which use simulated PA data for training. Surprisingly, approximately 40\% of papers presented results on experimental data either using phantoms or \emph{in vivo} (animal or human) measurements. In the following, we summarize the literature partitioned into the already mentioned sub-topics: deep learning-enhanced model-based image reconstruction, direct image reconstruction, point source localization, and speed of sound estimation.\\
\subsection{Deep learning-enhanced model-based image reconstruction}
The central idea is to leverage the flexibility of deep learning to enhance already existing model-based reconstruction algorithms\cite{schwab_deep_2019,schwab_real-time_2018}, by introducing learnable components. To this end, Schwab \emph{et al.}\cite{schwab_learned_2019} proposed an extension of the weighted universal back-projection algorithm. The core idea is to add additional weights to the original algorithm, with the task of the learning algorithm then being to find optimal weights for the reconstruction formula. By learning the weights, the authors were able to reduce the error introduced from limited view and sparse sampling configurations by a factor of two. Furthermore, Antholzer \emph{et al.}\cite{antholzer_nett_2019} and Li \emph{et al.}\cite{li_nett_2020} leveraged neural networks to learn additional regularization terms for an iterative reconstruction scheme. Hauptmann \emph{et al.}\cite{hauptmann_model-based_2018} demonstrated the capability for \acp{CNN} to perform iterative reconstruction by training a separate network for each iteration step and integrating it into the reconstruction scheme. The authors showed that since their algorithm was trained on synthetic data, several data augmentation steps or the application of transfer learning techniques were necessary to achieve satisfactory results. Finally, Yang \emph{et al.}\cite{yang_accelerated_2019} demonstrated the possibility to share the network weights across iterations by using recurrent inference machines for image reconstruction.\\
\emph{Key Insights:} An interesting insight shared in one of the papers by Antholzer \emph{et al.}\cite{antholzer_nett_2019} was that model-based approaches seem to work better for "exact data", while deep learning-enhanced methods outperform purely model-based approaches on noisy data. This makes the application of deep learning techniques very promising for the typically noisier and artifact-fraught experimental data\cite{kazakeviciute2016multispectral,manwar2018photoacoustic}. On the other hand, currently employed deep learning models do not seem to generalize well from simulated to experimental data as evident from the fact that only 40\% of papers tested their method on experimental data (cf. table \ref{tab:train_test}). Ideally, the algorithms would have to be trained on experimental data.\\
\subsection{Direct image reconstruction}
The principal idea of direct image reconstruction with deep learning is to either completely replace the commonly used model-based methods or to integrate model-based methods as additional information for a deep learning-based solution. An overview of these approaches are summarized in Fig \ref{fig:aip:imageReconstruction}.\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/image_reconstruction.pdf}
\caption{Visualization of the principal approaches to deep learning-based PA image reconstruction. The time-series data is either given directly to a neural network, or after preprocessing steps, such as reference reconstructions or the calculation of hand-crafted feature maps. The goal of the reconstruction is to estimate the underlying initial pressure distribution.}
\label{fig:aip:imageReconstruction}
\end{figure}
The first approaches to direct image reconstruction with \acp{CNN} were proposed in 2018 by Waibel \emph{et al.}\cite{waibel_reconstruction_2018} and Anas \emph{et al.}\cite{anas_robust_2018}. Waibel \emph{et al.}\cite{waibel_reconstruction_2018} and Lan \emph{et al.}\cite{lan_reconstruct_2019} used modified U-Net architectures to estimate the initial pressure distribution directly from time-series pressure data, whereas Anas \emph{et al.}\cite{anas_robust_2018} used a \ac{CNN} architecture with dense blocks. Furthermore, Lan \emph{et al.} \cite{lan_ki-gan_2019,lan_hybrid_2019} proposed a method based on a generative adversarial network\cite{goodfellow2014generative} approach that - in addition to time-series data - also uses a reconstructed PA image as additional information to regularize the neural network. Guan \emph{et al.} \cite{guan_limited_2019} compared implementations of all these techniques to assess their merit in brain imaging within a neurological setting. They compared an algorithm that directly estimates the reconstructed image from time-series data, a post-processing approach, as well as a custom approach with hand-rafted feature vectors for the model. Their results show that adding additional information improves the quality of the reconstructed image, that iterative reconstruction generally worked best for their data, and that deep learning-based reconstruction was faster by 3 orders of magnitude.\\
\emph{Key Insights:} In contrast to deep learning-enhanced model-based reconstruction, direct deep learning reconstruction schemes are comparatively easy to train and most of the papers utilize the U-Net as their base architecture. In several works it was demonstrated that the \emph{infusion of knowledge} by regularizing the network with reference reconstructions or additional data from hand-crafted preprocessing steps led to very promising results\cite{lan_ki-gan_2019,lan_hybrid_2019}, generalizing them in a way that led to first successes on \emph{in vivo} data. Considering that deep learning-based image reconstruction outperforms iterative reconstruction techniques in terms of speed by orders of magnitude\cite{guan_fully_2019}, it is safe to say that these methods can be a promising avenue of further research. It has to be noted that, especially in terms of robustness and uncertainty-awareness, the field has much room for improvement. For example, Sahlstr\"om \emph{et al.} \cite{sahlstrom2020modeling} have modeled uncertainties of the assumed positions of the detection elements for model-based image reconstruction, but no comparable methods were applied in deep learning-based \ac{PAI} as of yet.\\
\subsection{Point source localization}
\begin{figure}[h!tb]
\centering
\includegraphics{figures/point_location_estimation.pdf}
\caption{Approaches for point source localization use time-series data as input to estimate either the pixel coordinates of the point of origin of the pressure wave or a heat map containing the probability of the source being in a certain location of the image.}
\label{fig:aip:pointSourceLocalization}
\end{figure}
The localization of the spatial position of point sources from time-series PA measurements was identified as a popular sub-task concerning PA image reconstruction. An algorithm for this could for example be used for the automatic detection and localization of point absorbers, such as needle tips, in a PA image. The general idea is to take time-series data to either regress numerical values for the pixel coordinates of the sources of the wavefronts or to output a two-dimensional map of the possible source locations (see Fig. \ref{fig:aip:pointSourceLocalization}).\\
To this end, Reiter \emph{et al.}\cite{reiter_machine_2017} presented an approach that uses a \ac{CNN} to transform the time-series data into an image that identifies the \ac{2D} point localization of the wavefront origin. They further use this approach to distinguish between signals and artifacts in time-series data. Johnstonbaugh \emph{et al.} \cite{johnstonbaugh_novel_2019} also use a \ac{CNN} in an encoder-decoder configuration to reconstruct the PA signal into an image containing a single point source. A similar architecture proposed by the same group \cite{johnstonbaugh_deep_2020} is also used to process the time-series data and output cartesian coordinates of the point source location.\\
\emph{Key Insights:} Similar to the deep learning-based direct reconstruction methods, methods for point source localization are exceptionally easy to train and can even be trained on \emph{in vitro} experimental data. This ease of accessibility made this task the first application of deep learning in \ac{PAI} \cite{reiter_machine_2017}. However, the integrability of these methods into clinical practice and their future impact beyond certain niche applications is questionable because \emph{in vivo} scenarios do typically not exclusively consist of point sources but comprise a very complex and heterogeneous distribution of chromophores.
\subsection{Speed of sound estimation}
A correct estimate of the speed of sound within the medium is an important constituent to successful image reconstruction. We identified two papers that explicitly incorporated the estimation of the speed of sound into their reconstruction. Shan \emph{et al.} \cite{shan_simultaneous_2019} used a \ac{CNN} to reconstruct the initial pressure distribution as well as the speed of sound simultaneously from the time-series data and Jeon \emph{et al.}\cite{jeon_deep_2020} trained a U-Net to account for the speed of sound aberrations that they artificially introduced to their time-series data.\\
\emph{Key Insights:} Automatically integrating estimates of the speed of sound into the image reconstruction algorithm can substantially enhance image quality and hence is an interesting direction of research. Nevertheless, the formulation of a corresponding optimization problem is inherently difficult, as it is not straightforward to assess the influence of a speed of sound mismatch on a reconstruction algorithm. Furthermore, the validation of these methods is difficult, as there typically is no \emph{in vivo} ground truth available.
\section{Image Post-Processing}
\label{sec:processing}
Being a comparatively young imaging modality, \ac{PAI} still suffers from distinct artifacts\cite{kazakeviciute2016multispectral}. These can have multiple origins and are primarily caused by hardware limitations such as light absorption in the transducer membrane or fluctuations in the pulse laser energy\cite{manwar2018photoacoustic}. Other issues can also lead to decreased image quality, such as under-sampling or limited-view artifacts, as well as other influences such as motion artifacts or artifacts specific to the reconstruction algorithm (see Fig \ref{fig:postprocessing}). Research in the field of using post-processing algorithms can broadly be divided into two main areas: the elimination of artifacts \ref{sec:processing:artifact} which mostly encompass systematic error sources and the enhancement of image quality \ref{sec:processing:quality} which is lost mainly through stochastic error sources.\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/post-processing.png}
\caption{Post-processing techniques are tasked to improve the image quality of a reconstructed PA image. The image quality can be reduced by many factors including under-sampling, limited-view artifacts, low laser energy, or the presence of motion during the measurement.}
\label{fig:postprocessing}
\end{figure}
\subsection{Artifact removal}
\label{sec:processing:artifact}
One principal approach to speed up image reconstruction is to use sparse data that only contains a fraction of the available time-series data. While this potentially leads to a significant increase in reconstruction speed, it comes with a cost in form of the deterioration of the image quality and the introduction of characteristic under-sampling artifacts. Several groups\cite{davoudi_deep_2019,antholzer_deep_2019,guan_fully_2019,farnia2020high} have shown that a large portion of these artifacts can be recovered using deep learning techniques. A core strength of such approaches is that experimental PA data can be utilized for training, by artificially undersampling the available channels and training the algorithm to predict the reconstructions from (1) full data, (2) sparse data, or (3) limited-view data \cite{antholzer_photoacoustic_2018,antholzer_deep_2018,deng_machine-learning_2019,zhang2020new,godefroy2020solving,tong2020domain}.\\
Reflection artifact can be introduced by the presence of acoustic reflectors in the medium (for example air). Allman \emph{et al.}\cite{allman_photoacoustic_2018} showed that deep learning can be used to distinguish between artifacts and true signals and Shan \emph{et al.}\cite{shan_accelerated_2019} demonstrated that the technology is also capable of removing such artifacts from the images. Furthermore, Chen \emph{et al.}\cite{chen_deep-learning-based_2019} introduced a deep learning-based motion correction approach for PA microscopy images that learns to eliminate motion-induced artifacts in an image.\\
\emph{Key Insights:} For limited-view or limited-data settings, experimental training data can easily be created by artificially constraining the available data, for example, by under-sampling the number of available time series data. On the other hand, for the task of artifact removal, it can be comparatively difficult to train models on \emph{in vivo} experimental settings for different sources of artifacts. This is, because artifacts can have various origins and are also dependent on the specific processing steps. Nevertheless, impressive results of the capability of learning algorithms to remove specific artifacts were demonstrated.
\subsection{Image quality enhancement}
\label{sec:processing:quality}
The quality and resolution of PA images are also limited by several other factors including the limited bandwidth of PA detectors, the influence of optical and acoustic scattering, the presence of noise due to the detection hardware, and fluctuations in the laser pulse-energy.\\
To remedy this, Gutta \emph{et al.}\cite{gutta_deep_2017} and Awasthi \emph{et al.}\cite{awasthi_deep_2020} proposed methods that aim to recover the full bandwidth of a measured signal. This is achieved by obtaining pairs of full bandwidth and limited bandwidth data using simulations that are used to train a neural network. Since experimental systems are always band-limited, the authors of these works rely on the presence of simulated data. On the other hand, more classical deep learning-based super-resolution algorithms were proposed by Zhao \emph{et al.}\cite{zhao_new_2020} to enhance the resolution of PA devices in the context of PA microscopy. For training of super-resolution approaches, the authors are theoretically not restricted by the domain of application and as such can also use data from sources unrelated to \ac{PAI}.\\
Several approaches have been proposed to enhance the image quality by improving the signal-to-noise ratio of image frames acquired with low energy illumination elements, such as LED-based systems. This has generally been done using \acp{CNN} to improve a single reconstructed image, for example by Vu \emph{et al.}\cite{vu_generative_2020}, Singh \emph{et al.}\cite{singh_deep_2020}, Anas \emph{et al.}\cite{anas_enabling_2018}, and Hariri \emph{et al.} \cite{hariri_deep_2020} or by using a neural network to fuse several different reconstructions into a higher-quality version, as proposed by Awasthi \emph{et al.}\cite{awasthi_pa-fuse_2019}.\\
\emph{Key Insights:} For the enhancement of image quality, common deep learning tasks from the field of computer vision\cite{voulodimos2018deep} can be translated to PA images relatively easily, as the algorithms are usually astonishingly straightforward to train and validate. We believe that applying well-established methods from fields adjacent to \ac{PAI} can be of excellent benefit to the entire field.
\section{Optical Inverse Problem}
\label{sec:quantification}
The optical inverse problem is concerned with estimating the optical tissue properties from the initial pressure distribution. The first method proposed to solve this inverse problem was an iterative reconstruction scheme to estimate the optical absorption coefficient\cite{cox2006two}. Over time, the iterative inversion schemes have become more involved\cite{cox2010quantitative} and Buchmann \emph{et al.}\cite{buchmann2020quantitative} achieved first successes towards experimental validation. Recently, data-driven approaches for the optical inverse problem have emerged, including classical machine learning\cite{kirchner2018context} as well as deep learning approaches. A tabulated summary of the identified papers can be found in Table \ref{tab:summary:oip} \\
\begin{table}[h!tb]
\centering
\begin{tabular}{lp{3cm}p{2cm}p{2cm}p{2cm}p{2cm}}
\textbf{Publication} & \textbf{Base \mbox{Architecture}} & \textbf{target $\mu_a$ $[cm^{-1}]$} & \textbf{background $\mu_a$ $[cm^{-1}]$} & \textbf{background $\mu_s'$ $[cm^{-1}]$} & \textbf{Validation Data}\\
\hline
\\
Cai \emph{et al.} \cite{cai_end--end_2018} & U-Net with residual blocks & N/A & $0.2 - 0.4$ & $5 - 10$ & \emph{in silico} \\
Gr{\"o}hl \emph{et al.} \cite{grohl_confidence_2018} & U-Net & $2 - 10$ & const. $0.1$ & const. $1.5 $ & \emph{in silico} \\
Chen \emph{et al.} \cite{chen_deep_2020} & U-Net & $0.1 - 2$ & $0.1 - 0.4$ & const. $10 $ & \emph{in vitro}\\
\\
\end{tabular}
\caption{Tabulated overview of the identified literature regarding the optical inverse problem. The table shows the publication, the base network architecture, the range of absorption and scattering parameters used for the training data, and the type of data that the approach was validated with.}
\label{tab:summary:oip}
\end{table}
For the identified deep learning-based approaches, the key objective is to estimate the optical absorption coefficients and subsequently the absolute concentrations of chromophores from the initial pressure distribution (see Fig. \ref{fig:oip:overview}).\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/absorption_estimation.pdf}
\caption{To solve the optical inverse problem, a neural network is tasked to estimate the underlying optical tissue parameters, primarily the optical absorption coefficient, from the initial pressure distribution $p_0$.}
\label{fig:oip:overview}
\end{figure}
Cai \emph{et al.} \cite{cai_end--end_2018} trained a U-Net with residual blocks to estimate the absolute concentration of \ac{ICG} alongside the relative ratio of $Hb$ and $HbO_2$. To this end, they simulated random smoothed maps of optical tissue properties for training and tested their approach on several simulated data sets, including one created from a digital mouse model\cite{dogdas2007digimouse}. Gr{\"o}hl \emph{et al.} \cite{grohl_confidence_2018} trained a total of four U-Net models to estimate fluence and optical absorption from the initial pressure distribution as well as directly from time-series pressure data. They also presented a method to estimate the expected error of the inversion, yielding an indicator for the model uncertainty. Their approach was trained and tested on simulated data, which contained tubular structures in a homogeneous background. Finally, Chen \emph{et al.}\cite{chen_deep_2020} trained a U-Net to directly estimate optical absorption from simulated images of initial pressure distribution. They trained and tested their model on synthetic data comprising geometric shapes in a homogeneous background, as well as another model on experimental data based on circular phantom measurements.\\
\emph{Key Insights:} Model-based methods to tackle the optical inverse problem suffer from the fact that many explicit assumptions have to be made that typically do not hold in complex scenarios\cite{kirchner2018context}. With data-driven approaches, many of these assumptions are only implicitly made within the data distribution, leaving room for a substantial improvement. Obtaining ground truth information on the underlying optical tissue properties \emph{in vivo} can be considered impossible and is exceptionally involved and error-prone even \emph{in vitro}\cite{grohl_confidence_2018}. As such, there has been no application yet to \emph{in vivo} data, leaving the optical inverse problem as one of the most challenging problems in the field of \ac{PAI}, which is reflected by the comparatively low amount of published research on this topic.\\
\section{Semantic Image Annotation}
\label{sec:clinical_translation}
While the topical areas until now have mostly considered PA data at a single wavelength, the power of \ac{PAI} for clinical use cases lies in its ability to discern various tissue properties through analysis of the changes in signal intensity over multiple wavelengths (see Fig \ref{fig:clinical_translation}). This allows for the estimation of functional tissue properties, especially blood oxygenation (section \ref{sec:clinical_translation:functional}), but also for the classification and segmentation (section \ref{sec:clinical_translation:classification}) of tissues and tissue types.\\
\begin{figure}[h!tb]
\centering
\includegraphics{figures/clinical_application.pdf}
\caption{For semantic tissue annotation (typically multispectral) PA measurements are used as the input and the algorithm is tasked to estimate the desired parameters, such as tissue oxygenation or segmentation maps of different tissue types. The black color in the oxygenation estimation denotes areas where oxygenation values cannot be computed.}
\label{fig:clinical_translation}
\end{figure}
\subsection{Functional property estimation}
\label{sec:clinical_translation:functional}
The estimation of local blood oxygenation $sO_2$ is one of the most promising applications of \ac{PAI}. In principle, information on the signal intensities of at least two wavelengths are needed to unmix the relative signal contributions of oxyhemoglobin $HbO_2$ and deoxyhemoglobin $Hb$: $sO_2 = HbO_2 / (HbO_2 + Hb)$. For \ac{PAI}, wavelengths around the isosbestic point ($\approx 800\,\text{nm}$) are commonly chosen for this task. Because linear unmixing should not directly be applied to the measured signal intensities due to the non-linear influence of the fluence, a lot of work has been conducted to investigate the applicability of neural networks to tackle this problem. Due to the unavailability of ground truth oxygenation information, the networks are currently being trained exclusively on simulated data. The problem was approached using fully-connected neural networks\cite{grohl_estimation_2019} as well as \acp{CNN}\cite{bench2020toward}.\\
The use of feed-forward fully-connected neural networks was demonstrated by Gr\"ohl \emph{et al.}\cite{grohl_estimation_2019} to be capable to yield accurate oxygenation estimations \emph{in silico} from single-pixel $p_0$ spectra. In addition, it was demonstrated that the application of the method to experimental \emph{in vitro} and \emph{in vivo} data yielded plausible results as well. Olefir \emph{et al.}\cite{olefir2020deep} demonstrated that introducing prior knowledge to the oxygenation estimation can improve performance. Specifically, the authors introduced two sources of information for regularization. On the one hand, they introduced fluence eigenspectra which they obtained from simulated data and on the other hand, they also estimated their results based on spectra from a larger patch of tissue thus introducing spatial regularization. They demonstrated the applicability of the method to \emph{in vitro} and \emph{in vivo} data in several experiments.\\
To make full use of the spatial context information, \acp{CNN} were employed to estimate blood oxygenation using the spectra of entire \ac{2D} images rather than single-pixel spectra. This was demonstrated by Yang \emph{et al.}\cite{yang_eda-net_2019,yang_quantitative_2019}, Luke \emph{et al.}\cite{luke_o-net_2019}, and Hoffer-Hawlik \emph{et al.}\cite{hoffer-hawlik_abso2luteu-net_2019}. Furthermore, Bench \emph{et al.}\cite{bench2020toward} showed the feasibility to estimate oxygenation from multispectral \ac{3D} images. It has to be noted that there exists a trade-off regarding the spatial extent and number of voxels of the input images and the number of training images that can feasibly be simulated for algorithm training. The approaches demonstrate the feasibility of using \acp{CNN} for estimating oxygenation with high accuracy (for reported values in the publication see Table \ref{tab:results_oxygenation_estimation}), however, a successful application of these methods \emph{in vitro} or \emph{in vivo} has not yet been shown, which is most probably caused by the large domain gap between simulated and experimental PA images.\\
\begin{table}[h!tb]
\centering
\begin{tabular}{ll}
\textbf{Publication} & \textbf{Reported sO$_2$ estimation error} \\
\hline
\\
Bench \emph{et al.}\cite{bench2020toward} & 4.4\% $\pm$ 4.5\% MAE\\
Cai \emph{et al.} \cite{cai_end--end_2018} & 0.8\% $\pm$ 0.2\% MRE\\
Gr\"ohl \emph{et al.}\cite{grohl_estimation_2019} & 6.1\% MedRE, IQR: [2.4\%, 18.7\%]\\
Hoffer-Hawlik \emph{et al.}\cite{hoffer-hawlik_abso2luteu-net_2019} & RSME consistently below 6\%\\
Luke \emph{et al.}\cite{luke_o-net_2019} & 5.1\% MedAE at 25dB SNR\\
Olefir \emph{et al.}\cite{olefir2020deep} & 0.9\%, IQR [0.3\%, 1.9\%] to 2.5\%, IQR [0.5\%, 3.5\%] MedAE*\\
Yang \emph{et al.}\cite{yang_quantitative_2019} & 1.4\% $\pm$ 0.2\% MRE \\
Yang and Gao \cite{yang_eda-net_2019} & 4.8\% $\pm$ 0.5\% MAE\\
\\
\end{tabular}
\caption{Overview of some of the reported errors on sO$_2$ estimation. Standard deviations or interquartile ranges (IQR) are shown if reported. It has to be noted that the used metrics as well as the conventions which pixels the error metrics are calculated on vary drastically between papers. As such the numbers are not directly comparable. For more detailed results, please refer to the linked papers. MedRE = Median Relative Error; MedAE = Median Absolute Error; MRE = Mean Relative Error; MAE = Mean Absolute Error; RSME = Root Mean Squared Error; *depending on dataset.}
\label{tab:results_oxygenation_estimation}
\end{table}
The estimation of other functional tissue properties has also been investigated, such as the detection of glucose by Ren \emph{et al.}\cite{ren_effects_2018}, the size of fat adipocytes by Ma \emph{et al.}\cite{ma_adipocyte_2018}, as well as the unmixing of arbitrary chromophores in an unsupervised manner by Durairaj \emph{et al.}\cite{durairaj_unsupervised_2020}.\\
\emph{Key Insights:} The estimation of functional tissue properties is closely related to the optical inverse problem, as functional properties can be derived from the optical properties of the tissue. But the direct estimation of the desired properties without quantification of the optical properties in an intermediate step is very popular. One reason for this is that there exist reference methods that can measure the functional properties and can be used to validate the results\cite{olefir2020deep}. This potentially also enables training an algorithm on experimental data, when using reference measurements as the ground truth. Taking the estimation of tissue oxygenation as an example showcases the potential rewards of comprehensively solving this family of problems, as it would enable a lot of promising applications, such as oxygen-level specific dosimetry in radiotherapy\cite{matsuo2014magnetic} or cancer type classification based on local variations in blood oxygenation in the tumor's microenvironment\cite{horsman1998measurement}.
\subsection{Tissue classification and segmentation}
\label{sec:clinical_translation:classification}
Multispectral image information can also be used to differentiate between different tissue types or to identify and detect tissue pathologies. In such cases, strategies for dataset curation differ depending on the use case but using experimental datasets is possible with manual data annotation. In the work of Moustakidis \emph{et al.} \cite{moustakidis_fully_2019} \emph{in vivo} images of a raster-scan optoacoustic mesoscopy (RSOM) system were utilized to automatically differentiate between different skin structures, while Lafci \emph{et al.} \cite{lafci_efficient_2020} used neural networks to segment the entire imaged object. Furthermore, Wu \emph{et al.}\cite{wu_multi-wavelength_2017} imaged \emph{ex vivo} tissue samples to monitor lesion formation during high-intensity focused ultrasound (HIFU) therapy, and Jnawali \emph{et al.}\cite{jnawali_deep_2019, jnawali_transfer_2019, jnawali_automatic_2020} also analyzed \emph{ex vivo} tissue to differentiate cancer tissue from normal tissue in pathology samples.\\
On the other hand, we identified several papers that used simulated data to train their networks. Typically, simulations of acoustic waves are conducted using pre-processed images of a different modality, such as CT images, and treating the intensity distribution as the initial pressure distribution. This was done by Zhou \emph{et al.}\cite{zhou_analysis_2019} to investigate the feasibility to differentiate healthy bone tissue from pathologies such as hyperosteogeny and osteoporosis. Further work by Zhang \emph{et al.}\cite{zhang_photoacoustic_2018} also examined the feasibility of DL-based breast cancer classification, Lin \emph{et al.}\cite{lin_computer-aided_2019} investigated the feasibility of endometrial cancer detection, and several groups including Zhang \emph{et al.}\cite{zhang_pathology_2018}, Luke \emph{et al.}\cite{luke_o-net_2019}, Chlis \emph{et al.}\cite{chlis_sparse_2019}, and Boink \emph{et al.}\cite{boink_partially-learned_2019} examined the segmentation of blood vessels. Finally, Allman et al.\cite{allman_deep_2019} conducted feasibility experiments that demonstrated the capability of neural networks to automatically segment needle tips in PA images.\\
\emph{Key Insights:} Semantic image annotation enables intuitive and fast interpretation of PA images. Given the number of potential applications of \ac{PAI}, we believe that semantic image annotation is a promising future research direction. Because the modality is comparatively young, high-quality reference data for algorithm training that are annotated by healthcare professionals are very rare. Furthermore, the cross-modality and inter-institutional performance of \ac{PAI} devices has to our knowledge not been examined as of yet. This makes validation of the proposed algorithms \emph{in vivo} difficult, as reflected by some of the presented work. As discussed throughout this review, the image quality of \ac{PA} images relies heavily on the solutions to the acoustic and optical inverse problems. This potentially introduces difficulties for manual data annotation and thus makes it more difficult to integrate developed methods into clinical practice.
\section{Discussion}
\label{sec:discussion}
The clinical translation of deep learning methods in \ac{PAI} is still in its infancy. Even though many classical image processing tasks, as well as PA-specific tasks, have already been tackled using deep learning techniques, vital limitations remain. For instance, many researchers resort to simulated data due to the lack of high-quality annotated experimental data. Accordingly, none of the proposed techniques were validated in a large-scale clinical \ac{PAI} study. In this section we will discuss the challenges for clinical translation of deep learning methods in \ac{PAI} and will conclude by summarizing the key findings of this review.\\
The challenges of clinical translation of spectroscopic optical imaging techniques have previously been extensively examined by Wilson \emph{et al.}\cite{wilson2018challenges}. While their work focused primarily on the general challenges and hurdles in translating biomedical imaging modalities into clinical practice, in this review, we focused on the extent to which the application of deep learning in particular could facilitate or complicate the clinical integration of \ac{PAI}. To this end, we have summarized important features that a learning algorithm should fulfill, based on the findings in other literature \cite{wilson2018challenges,miotto2018deep,kelly2019key}:\\
\textbf{Generalizability.} In general, training data \emph{must} be representative of the data encountered in clinical practice to avoid the presence of biases\cite{pannucci2010identifying} in the trained model. The acquisition of high-quality experimental data sets in \ac{PAI} is extremely problematic due to, for example, the high intra-patient signal variability caused by changes in local light fluence, the small amount of clinically approved PAI devices, and the lack of reliable methods to create ground truth annotations.\\
The lack of experimental data can be attributed to the comparative youth of \ac{PAI}, but also to the fact that semantic information for images is only available via elaborate reference measurement setups or manual data annotations, which are usually created by healthcare professionals. But even in commonplace imaging modalities like Computed Tomography (CT) or Magnetic Resonance Imaging (MRI), high-quality annotations are extremely costly, time-consuming and thus sparse compared to the total number of images that are taken. Finally, annotations of optical and acoustic properties are intrinsically not available \emph{in vivo}, as there currently exist no gold standard methods to non-invasively measure, for example, optical absorption, optical scattering, or the speed of sound.\\
To account for the lack of available experimental data, approximately 75\% of models were trained on simulated photoacoustic data. This practice has led to many insights into the general applicability and feasibility of deep learning-based methods. But methods trained purely on simulated data have shown poor performance when being applied to experimental data. Systematic differences between experimental PA images and those generated by computational forward models are very apparent. These differences are commonly referred to as the \textit{domain gap} and can cause methods to fail on \textit{in vivo} data despite thorough validation \textit{in silico}, since deep learning methods cannot easily generalize to data from different distributions. Closing this gap can make the \textit{in silico} validation of deep learning methods more meaningful. We see several approaches to tackle this problem:
\begin{enumerate}
\item Methods to create more realistic simulations. This comprises the implementation of digital twins of the respective PA devices or the simulation of anatomically realistic geometric variations of tissue.
\item Domain adaptation methods that are currently developed in the field of computer vision\cite{voulodimos2018deep} could help translate images from the synthetic to the real PA image domain.
\item Methods to refine the training process, such as extensive data augmentation specific for \ac{PAI}, the weighting of training data\cite{wirkert2017physiological}, content disentanglement\cite{ilse2020diva} or domain-specific architecture changes\cite{lan_ki-gan_2019}, as well as the tight integration of prior knowledge into the entire algorithm development pipeline\cite{hauptmann_model-based_2018}.
\end{enumerate}
A promising opportunity could lie in the field of \emph{life-long learning}\cite{ruvolo2013ella,chen2018lifelong}. This field strives to develop methods that have the ability to continuously learn over time by including new information while retaining prior knowledge\cite{parisi2019continual}. In this context, research is conducted towards developing algorithms that efficiently adapt to new learning tasks (\emph{meta-learning})\cite{finn2017model} and can be trained on various different but related tasks (\emph{multi-task learning})\cite{caruana1997multitask}. The goal is to create models that can continue to learn from data that becomes available after deployment. We strongly believe that the integration of such methods into the field of \ac{PAI} can have exceptional merit, as this young imaging modality can be expected to undergo frequent changes and innovations in the future.\\
\textbf{Uncertainty estimation.} We strongly believe that methods should be uncertainty-aware, since gaining insight into the confidence of deep learning models can serve to avoid blindly assuming estimates to be accurate\cite{what2017kendall,osawa2019practical}. The primary goal of uncertainty estimation methods is to provide the confidence interval for measurements, for example, by calculating the average and standard deviation over a multitude of estimation samples\cite{gal2016dropout}. On the other hand, such metrics might not be sufficient and a current direction of research is to recover the full posterior probability distribution for the estimate given the input, which for instance enables the automatic detection of multi-modal posteriors\cite{ardizzone2018analyzing}. Uncertainty estimation and Bayesian modeling of the inverse problems is an active field of research in \ac{PAI}\cite{tick2016image,tick2019modelling,sahlstrom2020modeling}. While a first simulation study\cite{grohl_confidence_2018} has demonstrated the benefits of exploiting uncertainty measures when using deep learning methods, the potential of this branch of research remains largely untapped.\\
\textbf{Out-of-distribution detection.} A major risk of employing deep learning-based methods in the context of medical imaging can be seen in their potentially undefined behavior when facing out-of-distribution (OOD) samples. In these situations, deep learning-based uncertainty metrics do not have to be indicative of the quality of the estimate\cite{kuleshov2018accurate} and methods that identify OOD situations should be employed to avoid trusting wrong estimations. In the field of multispectral imaging, OOD metrics were used to quantify domain gaps between data that a deep learning algorithm was trained on and newly acquired experimental data\cite{adler2019uncertainty,adler2019out}. We expect the investigation of well-calibrated methods for uncertainty estimation and the automatic detection of OOD scenarios to be integral towards the clinical translation of deep learning-based \ac{PAI} methodologies.\\
\textbf{Explainability.} The goal of the field of explainable deep learning is to provide a trace of the inference of developed algorithms \cite{holzinger2019causability}. The estimates of a deep learning algorithm should be comprehensible to domain experts in order to verify, improve, and learn from the system \cite{samek2017explainable}. In combination with techniques for uncertainty estimation and OOD detection, we believe that the explainability of algorithms will play an important role in the future of deep learning-based algorithms for PAI, especially in the biomedical context.\\
\textbf{Validation.} Thorough validation of methods is an integral part of clinical translation and as such plays a crucial role in the regulatory processes of medical device certification\cite{wilson2018challenges}. To this end, algorithms can be validated on several different levels, including in-distribution and out-of-distribution test data, as well as clinical validation in large-scale prospective studies\cite{steiner2018impact}. However, there is a systematic lack of prospective studies in the field of medical imaging with deep learning\cite{liu2019comparison}, and to our knowledge, there have been no such prospective deep learning studies in the field of \ac{PAI} yet. Some of the most impressive clinical trials in the field to date include the detection of Duchenne muscular dystrophy\cite{regensburger2019detection} and the assessment of disease activity in Crohn's disease\cite{knieling2017multispectral}. At least half of the reviewed papers have validated their methods on experimental data, but only approx. 20\% of papers have validated their methods on \emph{in vivo} data and even less on human measurements. This further demonstrates the vast differences in complexity within data obtained from phantoms versus living organisms. We expect that before deep learning methods for \ac{PAI} can reliably be used in a clinical context, much more pre-clinical work is needed to mature the proposed methodologies.\\
Another crucial aspect that we noticed during this review is the difficulty to compare many of the reported results. This is partly due to the fact that no standardized metrics or common data sets have so far been established in the field. Furthermore, the developed algorithms are tested only on in-house data sets that are usually not openly accessible. We have high hopes that these problems can be mitigated to a certain extent by the ongoing standardization efforts of the PA community, as promoted by the International Photoacoustic Standardisation Consortium (IPASC)\cite{bohndiek2019addressing}. Amongst other issues, this consortium is working on standardized methods to assess image quality and characterize \ac{PAI} device performance, on the organization of one of the first multi-centric studies in which PA phantoms are imaged all across the globe, as well as a standardized data format that facilitates the vendor-independent exchange of PA data.\\
\textbf{Computational efficiency.} Depending on the clinical use case, time can be of the essence (with stroke diagnosis being a prominent example\cite{fisher1995penumbra}) and the speed of the algorithm can be considered an important factor. \ac{PAI} is capable of real-time imaging\cite{gamelin2009real,kim2016programmable,kirchner2018signed} and the inference of estimates with deep learning can be exceptionally fast due to the massive parallelization capabilities of modern GPUs. The combination of these two factors can enable the real-time application of complex algorithms to \ac{PAI}. In the reviewed literature, it was demonstrated that entire high-resolution \ac{2D} and \ac{3D} images can be evaluated in a matter of milliseconds\cite{shang_two-step-training_2020}. In comparison to model-based methods, deep learning-based methods take a long time to train and fully optimize before they are ready to use. We believe that the drastic run-time performance increase could enable many time-critical applications of \ac{PAI} that might otherwise remain unfeasible.\\
\textbf{Clinical workflow integration.} Deep learning methods have already found success in several medical applications, especially in the field of radiology\cite{steiner2018impact,akkus2019survey,tonekaboni2019clinicians}. Nevertheless, we believe that the integrability of deep learning methods in \ac{PAI} heavily depends on the target clinical use case. The deep learning algorithm needs to have a clear impact on clinical practice, for example in terms of benefits for patients, personnel, or the hospital. Furthermore, the methods need to be easy to use for healthcare professionals, ideally being intuitive and introducing no significant time-burdens. \ac{PAI} is very promising for a multitude of clinical applications\cite{attia2019review}, which are mostly based on the differences in contrast based on local blood perfusion and blood oxygen saturation. To unleash to the full potential of \ac{PAI}, the inverse problems need to be solved to gain quantitative information on the underlying optical tissue properties. Deep learning can potentially enable an accurate, reliable, uncertainty-aware, and explainable estimation of the biomarkers of interest from the acquired PA measurements and thus provide unique opportunities towards the clinical translation of PAI. Nevertheless, thorough validation of the developed methods constitutes an essential first step in this direction.\\
\subsection{Conclusion}
This review has shown that deep learning methods possess unique advantages when applied to the field of \ac{PAI} and have the potential to facilitate its clinical translation in the long term. We analyzed the current state of the art of deep learning applications as pertaining to several open challenges in photoacoustic imaging: the acoustic and optical inverse problem, image post-processing, and semantic image annotation.\\
\textbf{Summary of findings:}
\begin{itemize}
\item Deep learning methods in \ac{PAI} are currently still in their infancy. While the initial results are promising and encouraging, prospective clinical validation studies of such techniques, an integral part of method validation, have not been conducted.
\item One of the core bottlenecks of the application of deep learning algorithms to \ac{PAI} is the lack of reliable, high-quality experimental training data. For this reason, about 75\% of deep learning papers in \ac{PAI} rely on simulated data for supervised algorithm training.
\item A commonly used workaround to create suitable experimental training data for image post-processing is to artificially introduce artifacts, for example, by deliberately using less information for image reconstruction.
\item Because the underlying optical tissue properties are inherently difficult to measure \emph{in vivo}, data-driven approaches towards the optical inverse problem have primarily relied on the presence of high-fidelity simulated data and have not yet successfully been applied \emph{in vivo}.
\item While direct image reconstruction with deep learning shows exceptional promise due to the drastic speed increases compared to model-reconstruction schemes, deep learning methods that utilize additional information such as reconstructions from reference methods or hand-crafted feature vectors have proven much more generalizable.
\item Approximately 50\% of papers test the presented methods on simulated data only and do not use multiple test sets that are significantly different from the training data distribution.
\item A successful application of oxygenation estimation methods using entire \ac{2D} or \ac{3D} images has not yet been shown \emph{in vitro} or \emph{in vivo}. This is most probably caused by the large domain gap between synthetic and experimental PA images.
\item Deep learning in \ac{PAI} has considerable room for improvement, for instance in terms of, generalizability, uncertainty estimation, out-of-distribution detection, or explainability.
\end{itemize}
|
1,314,259,995,804 | arxiv | \section{Introduction}
\label{intro}
The Sakai-Sugimoto model \cite{Sakai:2004cn,Sakai:2005yt} is a top-down realization of the gauge/string duality \cite{Maldacena:1997re}, which is, in a certain limit, dual to large-$N_c$ QCD.
Although only the opposite limit is accessible, that of large 't Hooft coupling $\lambda$, where the strict duality to QCD is lost, it has been successfully applied to meson, baryon, and
glueball properties in the vacuum \cite{Sakai:2004cn,Sakai:2005yt,Hata:2007mb,Imoto:2010ef,Brunner:2015yha}. The results suggest that the model can capture certain non-perturbative features that are very hard, if not impossible, to reproduce, unless the full theory is evaluated, for instance on the lattice.
Therefore, the model constitutes one of the most promising applications of the gauge/string duality in the context of QCD, although it always has to be kept in mind that we usually work in the limit of infinite coupling and
infinite number of colors, and that currently, at best, we can compute small corrections away from that limit.
A natural question is whether the Sakai-Sugimoto model can also be applied to medium physics, i.e., to QCD at finite temperature and density. Hot QCD, at zero or small baryon chemical potential, as encountered in relativistic
heavy-ion collisions, has been approached extensively with holographic methods \cite{casalderrey2014gauge}. Mostly, for this purpose, the original and best established version of the gauge/string correspondence has been employed, with ${\cal N}=4$ supersymmetric Yang-Mills theory being the dual field theory. Even though this theory is different from QCD, its application to the physics of heavy-ion collisions, or, more precisely,
the comparison of its predictions with the experimental results of the real-world quark gluon plasma, has turned out to be fruitful.
How about the complementary region, that of low temperature and large chemical potential?
This form of matter is interesting from a phenomenological point of view as well, because cold and dense matter exists inside compact stars, be it as nuclear matter
(neutrons and protons in the simplest form, probably including Cooper pairing) or as deconfined quark matter (three-flavor quark matter, also likely to form Cooper pairs) \cite{Schmitt:2010pn}.
Strongly coupled dense matter presents
an enormous theoretical challenge: it is currently out of reach for lattice QCD -- although there is progress in that direction \cite{Aarts:2015tyj},
and its phase structure is most likely to be very rich, not unlike ordinary condensed matter systems \cite{Alford:2007xm}. Moreover,
since compact stars have a density profile, they probe a wide region of the QCD phase diagram, possibly including both confined and deconfined matter. Even within conventional field-theoretical models that are already
simplified tremendously compared to full QCD, it is difficult to incorporate both nuclear and quark matter, and current efforts to compute the equation of state of ultra-dense matter often are restricted to
one form of matter or have to rely on a patchwork of different approaches, which is unsatisfying from a theoretical point of view.
Holographic approaches exist (although by far not as numerous as in the heavy-ion context), but they usually
rely on rudimentary descriptions of nuclear matter and do not contain quark matter at all \cite{Ghoroku:2013gja,Kim:2014pva},
or they use conventional, field-theoretical descriptions of nuclear matter and combine them with a holographic approach for (supersymmetric) quark matter \cite{Hoyos:2016zke}.
The Sakai-Sugimoto can potentially overcome some of these problems. Although, as argued above, we do not expect the model in its accessible limit to reflect real-world QCD in any rigorous way, it {\it does}
know about a chiral and deconfinement phase transition, about nucleons and quark matter at strong coupling. Therefore, it is worthwhile improving existing approximations, eventually constructing a
holographic equation of state which may be useful as an approach to dense matter complementary to more traditional methods.
\begin{figure}[t]
\centering
\includegraphics[width=.9\textwidth]{3cylinders}
\caption{Chirally broken phases with and without baryons, and chirally symmetric phase in the deconfined geometry of the Sakai-Sugimoto model, shown in the subspace spanned by the holographic direction $u\in[u_T,\infty]$,
where $u_T$ is given by the temperature (on the connected flavor branes this direction is, alternatively, parameterized by $z\in[-\infty,\infty]$),
and the compactified extra dimension $x_4$ with radius $M_{\rm KK}^{-1}$.
The black circles in the baryonic phase represent two instanton layers. In the pointlike approximation, with a single instanton layer located at the tip
of the connected branes $u_c$, the profile of the branes acquires a cusp at that point (hence the notation $u_c$ which is used here, even though the embedding is smooth in the case of finite-size instantons).
The calculations in the present work are performed in the decompactified limit, $L\ll \pi/M_{\rm KK}$.
}
\label{fig:3cylinders}
\end{figure}
Our calculation is done in the so-called decompactified limit of the model.
In this limit, the asymptotic separation of the D8- and $\overline{\rm D8}$-branes $L$ is small compared to the radius of the compactified extra dimension, for a schematic picture of the geometry of the model see
Fig.\ \ref{fig:3cylinders}. Moving the flavor branes asymptotically from their maximal separation (this is the "original" version of the model) very close together can be thought of as
going from large-$N_c$ QCD to a different dual theory
in which the gluon dynamics decouples, not unlike a Nambu--Jona-Lasinio model \cite{Antonyan:2006vw,Davis:2007ka,Preis:2012fh}. The reason for working in this limit is that now the chiral phase transition depends on the
baryon chemical potential, even without computing corrections away from the infinite-$N_c$ limit.
We employ various approximations for the baryonic phase.
Firstly, our approach is based on the flat-space instanton
solution for the non-abelian SU(2) gauge fields on the connected D8- and $\overline{\rm D8}$-branes, and we introduce a many-instanton system by adding up the squared field strengths of the single instantons.
Moreover, we average in position space over these configurations before we solve the equations of motion for the remaining fields.
For a single baryon, more sophisticated, but purely numerical solutions are known, and known to be more realistic when compared to real-world nucleons \cite{Cherman:2009gb,Bolognesi:2013nja,Rozali:2013fna}.
We give up on some degree of sophistication because we have to deal with additional complications, being interested in nuclear matter, not a single nucleon, and, since we work in the deconfined geometry, having to determine the embedding function of the connected flavor branes dynamically. The study of nuclear matter in the Sakai-Sugimoto
was initiated by approximating the instantons by delta functions \cite{Bergman:2007wp}, see phase diagram in Fig.\ \ref{fig:intro}, and the instanton approach has since then been further developed, including finite-size effects \cite{Ghoroku:2012am,Li:2015uea}. As an alternative approach, not further discussed here, one can use a homogeneous ansatz for the gauge fields which does not rely on any instanton
solution \cite{Rozali:2007rx,Li:2015uea,Elliot-Ripley:2016uwb}.
\begin{figure}[t]
\centering
\hbox{\includegraphics[width=.5\textwidth]{wobaryons}\includegraphics[width=.5\textwidth]{pointlike}}
\caption{$T$-$\mu$ phase diagrams in the decompactified limit of the Sakai-Sugimoto model, based on previous studies. Solid (dashed) lines are first-order (second-order)
phase transitions. Left panel: without baryons there is a chirally broken phase
with zero baryon number ("mesonic") and a chirally symmetric quark matter phase \cite{Horigome:2006xu}. Right panel: baryonic matter in the pointlike approximation shows a second-order baryon onset, and chiral
symmetry is, at low temperatures, not restored at any $\mu$ \cite{Bergman:2007wp,Preis:2011sp}. Here and in the following phase diagrams, we use a dimensionless temperature (the physical temperature is $T/L$) and a dimensionless
chemical potential (the physical chemical potential is $\mu \lambda /(4\pi M_{\rm KK} L^2)$).
The present work goes beyond the approximations used for these phase diagrams by allowing for finite-size -- and deformed -- instantons, and includes the possibility of more than one instanton layer.
}
\label{fig:intro}
\end{figure}
\section{Setup of the calculation}
\label{sec:setup}
Our goal is to determine the phase structure of the model
at nonzero $T$ and $\mu$, with an emphasis on the low-temperature regime, having in mind astrophysical applications. To this end, we focus on the three phases shown schematically in Fig.\ \ref{fig:3cylinders}, and
compare their free energies. The mesonic and quark matter phases are simply taken from the literature (see for instance appendix B of Ref.\ \cite{Li:2015uea} for a brief review of these phases, in the same notation as used below). The most difficult phase, where various approximations are necessary, is the baryonic phase. We shall discuss this phase in the following, explaining the setup and the calculation very briefly, all details can be found in Ref.\ \cite{Preis:2016fsp},
on which the results of these proceedings are based.
The Lagrangian we start from consists of a Dirac-Born-Infeld part and a Chern-Simons part,
\begin{equation} \label{Lag}
{\cal L} = u^{5/2} \sqrt{(1+u^3f_T x_4'^2-\hat{a}_0'^2+g_1)(1+g_2)} - n_I\hat{a}_0 q \, ,
\end{equation}
where $u$ is the holographic coordinate, prime denotes derivative with respect to $u$, $n_I$ is the instanton density, $f_T = 1-u_T^3/u^3$ with $u_T = \left(\frac{4\pi T}{3 M_{\rm KK}}\right)^2$, $\hat{a}_0$ is the
abelian gauge field with boundary condition $\hat{a}_0(\infty) = \mu$, and $x_4(u)$ is the embedding function of the flavor branes with boundary conditions $x_4(u_c)=0$, $x_4(\infty)=L M_{\rm KK}/2$.
We work exclusively with dimensionless quantities, for the corresponding dimensionful forms see Ref.\ \cite{Preis:2016fsp}.
The functions $g_1$, $g_2$, and $q$ arise from the non-abelian field strengths, $F_{iu}^2$, $F_{ij}^2$, and $\epsilon_{ijk} F_{ij}F_{ku}$, and they are based on the flat-space instanton approximation, averaged over position space. Explicitly, we have
\begin{equation}\label{g1g2sim}
g_1(u) \equiv \frac{f_T n_I}{3\gamma}\frac{\partial z}{\partial u}q(u) \, , \qquad g_2(u) \equiv \frac{\gamma n_I}{3u^3} \frac{\partial u}{\partial z} q(u) \, ,
\end{equation}
where the variables $z$ and $u$ are related by $u = (u_c^3+u_c z^2)^{1/3}$, and
\begin{equation}
q(u) \equiv 2\frac{\partial z}{\partial u} D(z) \, , \qquad \int_{u_c}^\infty du\, q(u) = 1 \, .
\end{equation}
The parameter $\gamma$ accounts for the anisotropy of the instantons. In flat space, corresponding to the \mbox{$\lambda=\infty$} limit, they are SO(4) symmetric in $(\vec{x},z)$
space, with $\vec{x}=(x_1,x_2,x_3)$. The deformation parameter $\gamma$ is a simple way -- keeping the functional
form of the flat-space solution -- to allow for oblate and prolate instantons. It is known from the full numerical solution that, upon going beyond the $\lambda=\infty$ limit, such a deformation does set in. In our simple parameterization, the instanton becomes elongated along the holographic direction for large $\gamma$ and along the spatial direction $x\equiv |\vec{x}|$ for small $\gamma$ (the instantons remain SO(3) symmetric in
position space). The function $D(z)$ can be understood as the instanton profile on the flavor branes. It includes $N_z$ many instanton layers that are allowed to spread along the $z$ direction,
\begin{equation} \label{Dz}
D(z) \equiv \int d^3 x \,D(x,z) \, , \qquad D(x,z) \equiv \frac{6}{\pi^2\gamma N_z}\sum_{n=0}^{N_z-1}\frac{(\rho/\gamma)^4}{[x^2+(z-z_n)^2/\gamma^2+(\rho/\gamma)^2]^4} \, ,
\end{equation}
where $\rho$ is the instanton width (giving, in combination with the deformation parameter $\gamma$, two different widths in holographic and spatial directions). We speak of instanton {\it layers} because, in general, they
also have a distribution in the spatial direction. The form of this distribution (for example a certain lattice structure) is irrelevant in our approximation because we neglect their interaction in position space and spatially average
over the field strengths squared. The locations of the instanton layers in the bulk is denoted by $z_n$,
and we assume them to be distributed equidistantly and symmetrically around $z=0$,
\begin{equation} \label{zn}
z_n= \left(1-\frac{2n}{N_z-1}\right) z_0 \, ,
\end{equation}
where the parameter $z_0$ (as well as the number $N_z$) has to be determined dynamically.
The free energy of the baryonic phase
\begin{equation}
\Omega = \int_{u_c}^\infty du\, {\cal L}
\end{equation}
is computed by first solving the equations of motion for $\hat{a}_0(u)$ and $x_4(u)$. We can solve them algebraically for $\hat{a}_0'(u)$ and $x_4'(u)$,
\begin{eqnarray}
\hat{a}_0'^2 &=& \frac{(n_IQ)^2}{u^5}\frac{1+g_1}{1+g_2-\frac{k^2}{u^8f_T}+\frac{(n_IQ)^2}{u^5}} \, , \qquad x_4'^2 = \frac{k^2}{u^{11}f_T^2}\frac{1+g_1}{1+g_2-\frac{k^2}{u^8f_T}+\frac{(n_IQ)^2}{u^5}} \, , \label{dx4}
\end{eqnarray}
where $k$ is an integration constant, and
\begin{equation}
Q(u) \equiv \int_{u_c}^u dv\,q(v)
\, .
\end{equation}
The solutions are inserted back into $\Omega$, and it remains to minimize the resulting expression with respect to the free parameters at given $T$ and $\mu$,
\begin{equation}
0 = \frac{\partial \Omega}{\partial k} =\frac{\partial \Omega}{\partial n_I} =\frac{\partial \Omega}{\partial u_c} =\frac{\partial \Omega}{\partial z_0} =\frac{\partial \Omega}{\partial \rho} =\frac{\partial \Omega}{\partial \gamma} \,,
\end{equation}
plus finding the integer $N_z$ that simultaneously minimizes $\Omega$.
If the baryonic phase is treated exactly as just outlined, the zero-temperature results are as follows: there is a second-order baryon onset (= $n_I$ continuous),
just like for the pointlike approximation (in fact, the instantons do become
pointlike, $\rho\to 0$ for $n_I\to 0$), and there is no chiral restoration at large $\mu$, which, again, was already seen in the pointlike approximation. Both features are unphysical in the sense that QCD behaves differently.
For any sensible equation of state, which accounts for the basic physical
properties of real-world baryons and which is capable to allow for hybrid stars, i.e., neutron stars with a quark matter core, a first-order baryon onset and chiral restoration at large $\mu$ are necessary.
It is not obvious whether the full solution of the Sakai-Sugimoto model does behave like QCD with regard to the baryon onset and chiral restoration. Therefore, a given approximation should not necessarily be judged
by these physical properties. More important from a theoretical point of view, the solution only has a single instanton layer, we have not found any solution for $N_z>1$.
The physical meaning of the instanton layers in the bulk is not obvious, but from this observation
we {\it can} conclude that our approximation is too simplistic; at least it is strongly suggested from other, complementary, approaches that the instanton distribution should spread out in the holographic direction
at large densities \cite{Rozali:2007rx,Elliot-Ripley:2016uwb,Kaplunovsky:2012gb}.
It turns out that the instantons do spread out dynamically if we improve our approximation by imposing the following constraints on the instanton width and deformation,
\begin{equation}\label{rhogamma}
\rho = \rho_0 u_c \, , \qquad \gamma = \frac{3}{2}\gamma_0u_c^{3/2} \, ,
\end{equation}
with $\rho_0$ and $\gamma_0$ now externally fixed, increasing the number of parameters of the model to five (besides $\lambda$, $M_{\rm KK}$ and $L$), and $u_c$ dynamically determined, as before. This somewhat
phenomenological approach accounts for instance for the fact that the instantons always have a nonzero width, as suggested from rigorous single-baryon results at finite $\lambda$. The particular scaling with $u_c$ is motivated by the
specific form of the stationarity equations, allowing for a complete elimination of $u_c$ from all equations but one. In the rest of these proceedings, we present the results from this approach.
\section{Results and conclusions}
Firstly, we discuss the structure of the solutions in the bulk, before we get to the phenomenologically more relevant properties of the solution. For specific values of $\rho_0$ and $\gamma_0$, we plot the instanton profiles
given in Eq.\ (\ref{Dz}) for three different chemical potentials, see Fig.\ \ref{fig:bumps}.
The figure shows the two-layer structure, with the distance between the two layers growing with increasing density. Additionally, the lower row of the figure shows the change in deformation:
the instanton width in the spatial (holographic) direction shrinks (grows) with increasing density. For all values of $\rho_0$ and $\gamma_0$ (which we have checked) and all densities, $N_z=2$ is the maximum number of
energetically preferred instanton layers.
We cannot exclude that this is an artifact of our approximation, for instance by restricting ourselves to equidistant layers in $z$. However, the same result was found in a completely different approximation
\cite{Elliot-Ripley:2016uwb}.
\begin{figure}[t]
\centering
\hbox{\includegraphics[width=.32\textwidth]{bumps1}\includegraphics[width=.33\textwidth]{bumps2}\includegraphics[width=.33\textwidth]{bumps3}}
\vspace{0.5cm}
\hbox{\includegraphics[width=.33\textwidth]{camel11}\includegraphics[width=.33\textwidth]{camel21}\includegraphics[width=.33\textwidth]{camel31}}
\caption{Instanton profiles in the holographic direction $z$ (upper row) and in the $(x,z)$ space (lower row) for three different chemical potentials and $T=0$, with parameters $\rho_0=2.5$, $\gamma_0=4$.
The thick (red) curves in the upper row
show the energetically preferred two-layer solution (at lower values of $\mu$, only one layer is favored, see right panel of Fig.\ \ref{fig:phases}).
For comparison, we also show the profiles for $N_z=1,3,\infty$. The lower row shows the two-layer solution for the same three chemical potentials. }
\label{fig:bumps}
\end{figure}
While we take the results of Fig.\ \ref{fig:bumps} as an indication of the improvement of our approximation through the constraints on $\rho$ and $\gamma$, let us now see whether we have also improved
the phenomenological properties of our baryonic matter. We start with the zero-temperature results, for two given values
of $\rho_0$, varying the second parameter $\gamma_0$ arbitrarily. The results are shown in Fig.\ \ref{fig:phases}. We see that there {\it is} a parameter range in which the baryon onset is first order, i.e., there is a binding energy
for nuclear matter. We also see that there is a (very small) region where, upon increasing $\mu$ at fixed $\rho_0$ and $\gamma_0$, the baryon onset is followed by a chiral phase transition to quark matter. In principle, one might proceed and match the 5 parameters of the model to physical data, for instance to saturation properties of nuclear matter. As a result, our model would predict the location of the chiral phase transition
to quark matter (or the absence thereof). This possibility is remarkable, we are not aware of any other model with a comparably small number of parameters, where low-density and high-density properties can be coupled in such a rigid way (usually, different models for the different density regimes have to be employed). Of course, our model has to be taken with a lot of care because of the above mentioned approximations, some of which are very crude.
In particular, it would be important to include the interaction of the instantons in position space, not only in the holographic direction, where we already have seen the repulsion of instanton layers.
Such an interaction would presumably give rise to an increased energy cost for the baryons to overlap, and thus possibly to a transition to quark matter at a lower chemical potential (or to the existence of a
such a transition in parameter regions where it does not exist in the present treatment). We are currently working on such an extension \cite{preparation}, and therefore defer a full-fledged evaluation of the model, including a matching of the parameters, to future work.
\begin{figure}[t]
\centering
\hbox{\includegraphics[width=.5\textwidth]{pdr15}\includegraphics[width=.5\textwidth]{pdr25}}
\caption{Zero-temperature phases for two slices through the phase space spanned by the parameters $\rho_0$, $\gamma_0$ (which set constraints on the instanton width and deformation) and the chemical potential
$\mu$. Solid (dashed) lines are first-order (second-order) phase transitions. The number of instanton layers $N_z$ in the baryonic phase is either one or two, and the transition between them can either be
continuous or discontinuous.}
\label{fig:phases}
\end{figure}
Here we continue with one aspect of a more quantitative evaluation, namely the strength of the first-order baryon onset, i.e., the size of the binding energy. We know that for symmetric
nuclear matter this binding energy is about 16 MeV, much less than the vacuum mass of the baryon $M_0 = 939\, {\rm MeV}$. In other words, the chemical potential for the baryon onset $\mu_0$ is only slightly smaller than the vacuum mass. Does our model allow for binding energies that are small in this sense, i.e., $\mu_0/M_0 \simeq 98\%$? Fig.\ \ref{fig:phases} suggests that it does because it contains a second-order
phase transition line (where the binding energy is zero) which terminates in two critical points and continues as first-order transition lines.
This implies that the jump in the baryon density becomes arbitrarily small close to the two
critical points, and consequently we should be able to find parameter values for $\rho_0$ and $\gamma_0$ for any given small binding energy. The dimensionless ratio $\mu_0/M_0$ is interesting also because all other parameters cancel, i.e., the condition of a given binding energy translates into a condition for $\rho_0$ and $\gamma_0$ alone. In our approach, we can calculate the vacuum mass of a baryon most easily by computing the chemical potential for vanishing density, $M_0 = N_c\mu(n_I\to 0)$ (the factor $N_c$ arises because $\mu$ is the {\it quark} chemical potential). We can thus, together with the calculation of the critical chemical potential
for the baryon onset $N_c\mu_0$,
determine a curve in the $\rho_0$-$\gamma_0$ parameter space on which the QCD binding energy is reproduced (more precisely, the {\it relative} binding energy, reproducing the correct {\it absolute} values
of $\mu_0$ and $M_0$ would require fitting one more parameter). The result is shown in Fig.\ \ref{fig:r0g0}.
\begin{figure}[t]
\centering
\includegraphics[width=.5\textwidth]{Er0g0}
\caption{Curve in the $\rho_0$-$\gamma_0$ parameter space on which the zero-temperature binding energy of symmetric nuclear matter is as small as in QCD, $\mu_0/M_0 = 98\%$
[(black) solid line].
In the shaded region there is no baryonic matter for any chemical potential, i.e., the curve that bounds this region is the location of the triple point in the phase diagrams of Fig.\ \ref{fig:phases}, where baryonic, mesonic, and chirally symmetric phases meet. For comparison, we show two curves [(red) dashed] with unrealistically large
relative binding energies, $\mu_0/M_0 = 73\%$ and 75\%. The (barely distinguishable) crosses on these two curves
mark the points that are chosen for the phase diagrams in Fig.\ \ref{fig:Tmuphases}. While they show a $T=0$ chiral phase transition at high density, the
points on the "QCD curve" have either no $T=0$ chiral phase transition at all or only at extremely large $\mu$ (possibly a consequence of our
simple approximation that appears to treat baryonic matter too favorable). }
\label{fig:r0g0}
\end{figure}
\begin{figure}[h]
\centering
\hbox{\includegraphics[width=.5\textwidth]{phases042}\includegraphics[width=.5\textwidth]{phases045}}
\caption{Phase diagram in the plane of temperature and chemical potential for a fixed $\rho_0$ and two values for $\gamma_0$.}
\label{fig:Tmuphases}
\end{figure}
Finally, we explore the full phase structure in the $T$-$\mu$ plane in Fig.\ \ref{fig:Tmuphases}. We have chosen the value of $\rho_0$ from the left panel of Fig.\ \ref{fig:phases}, and two values of $\gamma_0$
for which there is a zero-temperature phase transition to quark matter. As already obvious in Fig.\ \ref{fig:phases}, this requires some fine-tuning of the parameters, and the $T$-$\mu$ phase diagrams confirm that the
chiral phase transition is very sensitive to the choice of $\gamma_0$. Notice that the zero-temperature baryon onset
remains a transition from the mesonic phase to the baryonic phase at non-zero temperatures, i.e., the baryon number is always zero for chemical potentials below that transition. In QCD we expect a liquid/gas phase transition,
i.e., except for the exact $T=0$ limit, there is always a nonzero baryon density (albeit exponentially suppressed at small temperatures). Although our main focus has been on the zero-temperature properties, a better understanding of this $T>0$ behavior is another interesting problem for future studies.
{\it Acknowledgments.}
We acknowledge support from the {\mbox NewCompStar} network, COST Action MP1304.
A.S.\ is supported by the Science \& Technology Facilities Council (STFC) in the form of an Ernest Rutherford Fellowship.
|
1,314,259,995,805 | arxiv | \section{Introduction}
Crystal structure prediction with theoretical methods is a challenging
subject even for the simplest materials~\cite{Maddox1988,Needs2016}.
For complex materials with a large number of atoms in the unit cells,
$N$, finding the ground state crystal structure is particularly
difficult. The dimensionality of the search space grows as $3N+3$,
while the number of local minima increases exponentially with the
dimensionality, thus the effort required to find the global Gibbs free
energy minimum increases exponentially with $N$. This problem is
classified as NP-hard, and for large systems, searching all
configurations is unfeasible. However, significant progress has been
made with evolutionary algorithms~\cite{Oganov, Wang2010,Zurek2011},
random search techniques~\cite{AIRSS2006,AIRSS2011}, and others
methods including simulated annealing~\cite{WoodleyCatlow2008}, minima
hopping~\cite{Goedecker2004}, and metadynamics~\cite{Laio2002}. In
principle these methods do not require experimental input, however,
efficiency may be improved if it is incorporated. For example, powder
diffraction data has been used to restrict the search space to within
a known space group~\cite{Meredig2013}. In addition, knowledge
of energetically preferred structural elements like molecules and
functional groups may guide the structure generation process. One
could, for example, place entire H$_2$O molecules or NaCl pairs
instead of individual atoms. However, one must be aware that these
geometries may not be preserved at megabar pressures and thus
the implementation of such
constraints may eliminate the most favorable structures. While no
search method offers a rigorous path to finding the most stable structure,
they have all lead to many novel low-enthalpy candidate structures
and enriched our understanding of materials at high-pressure. Most
importantly, a number of theoretical predictions have later been
confirmed experimentally
~\cite{Oganov200695,Ono2007,MaLithium2008,Marques2011,Ma2009,Ma2011,Monserrat2016,Pickard2016,Needs2016,Dewaele2016,Struzhkin2016,Reilly2016}.
Knowledge of the globally stable structure allows one to calculate
physical properties under extreme conditions, where experimental
results are not yet obtainable.
For large unit cells, crystal structure prediction has remained a
challenge. When we applied the {\em ab initio} random structure search
technique~\cite{AIRSS2006}, with no symmetry constraints, to look for novel FeSiO${_3}$
structures~\cite{Zhang2014}, we found 74\% of the randomly generated
5-atom cells relaxed into symmetric structures, while the remaining
ones had no (or $P_1$) symmetry. During the relaxation of 10-atom
cells, the fraction of symmetric, non-$P_1$ structures decreased to
43\%. For 15-, 20-, 25- and 40-atom cells, the fraction of non-$P_1$
structures dropped 0.6\%, 0.9\%, 0.01\%, and 0.02\%,
respectively. This means more than 100 20-atom structures needed to be
relaxed in order to generate one symmetric structure that had a chance
of being the global enthalpy minimum. This argument adopts the common
assumption that the most stable structure has at least one symmetry
operation. The wealth of experimental data shows that most compounds
crystallize into symmetric structures at low temperature. This
tendency is expressed by Pauling's {\em rule of
parsimony}~\cite{Pauling1929} and supported by the energetics of
symmetry calculations~\cite{Wales1998}.
\subsection{Symmetry and Structure Prediction}
It has been recognized that symmetries are key to studying large
clusters~\cite{Wheeler2007,Lv2012,Oakley2013} and crystal
structures~\cite{Wang2010,AIRSS2011,Wang2012,Oganov2013}. In
Ref.~\cite{AIRSS2011}, this point is addressed by choosing $N_{op}$
specific symmetry operations. A subset of the atom positions are
chosen randomly and the remaining images are generated according to
symmetry. For the example of a mirror plane, the positions for half
of the atoms are chosen randomly while the other half are placed on
their mirror images. During the subsequent structural relaxation, a
more symmetric structure may emerge. With the added symmetry
operation, this structure can be derived from a supergroup of the
original group with $N_{op}$ operations. The evolutionary algorithm in
Refs.~\cite{Wang2010,Wang2012} relies on the particle swarm
optimization method to move from one generation to then next. The set of
230 space groups have been used to generate the initial set of
structures by selecting Wyckoff positions within a given space group
that are consistent with the chosen composition. Once a structure is
generated from a particular space group, the algorithm introduces a
penalty to prevent the generation of another structure from the same
space group. When this method was applied to the structural
optimization of TiO$_2$ with classical potentials~\cite{Wang2012} it
was shown in that the implementation of symmetry constraints
improved the efficiency of the search algorithm. With constraints,
more low energy structures were generated and approximately half as
many generations were needed to find the optimal structure. In
Ref.~\cite{Oganov2013} symmetry constraints are implemented by placing
atoms on the most general Wyckoff position with the option of merging
nearby atoms onto more symmetric Wyckoff positions, while allowing for
symmetry breaking in subsequent generations. It was shown
in~\cite{Oganov2013} that initializing the evolutionary algorithm with
symmetric structures improved efficiency while using classical
potentials to determine ground state structures of MgAl$_2$O$_4$.
In addition, implementation of symmetry constraints allowed the
determination of the ground state of
Mg$_{24}$Al$_{16}$Si$_{24}$O$_{96}$, a 160 atom unit cell structure
that was not found previously without symmetry constraints. In our
approach, we directly sample from the 230 space groups and all
associated 1,506 Wyckoff positions. This allows us to include all
Wyckoff positions consistently and select them with a high probability
without excluding any structure in principle. All space groups and
Wyckoff positions are treated with equal probability until we
eliminate structures in which atoms are very close. The first
water-salt structures that we generated with this method were reported
in Ref.~\cite{Domingos2016}. Independent of this work, a similar
approach was developed in Ref.~\cite{Zurek2017}. First a set of space
groups is selected. For every space group, a list of all possible
combinations of Wyckoff positions is assembled that are consistent
with the given composition. For large systems, this list may become exceptionally
large. For this reason, the size of this list was reduced by putting
similar Wyckoff positions into groups. This made the algorithm more
efficient but also changed the probability of how often certain
combinations of Wyckoff positions are selected. Conversely, in our algorithm, we
sample Wyckoff positions without generating such a list and
have thus no need to restrict its size. In
Ref.~\cite{Zurek2017}, 10 space groups were chosen when the initial
generation of TiO$_2$ structures were derived for the subsequent
evolutionary algorithm. Using classical potentials, it was shown that
symmetry constraints increase the probability of finding low energy
structures, but also the probability of generating high energy
structures, resulting in an increased average energy overall.
\subsection{High-pressure water-salt and carbon oxides}
Our goal is to design an efficient method to predict the crystal
structure of real materials at arbitrary pressures, without requiring
experimental input. Here we developed a symmetry-driven structure
search (SYDSS) technique to identify novel crystalline compounds at
high pressure. We applied our SYDSS algorithm to search for
(H$_2$O)$_n$-(NaCl)$_m$ and C$_n$O$_m$ compounds at megabar pressures. While salt
dissolves in water up to a maximum concentration, to our knowledge, no
stoichiometric H$_2$O-NaCl compound has been found in nature,
generated with laboratory experiments, or predicted theoretically.
However, at high pressure, the properties of materials change and
compounds that, while immiscible at ambient conditions, may form
stoichiometric compounds~\cite{FengHennig2008}. In
Ref.~\cite{Saitta2009}, a novel LiCl$\cdot$6H$_2$O structure was shown to
form at 2 GPa and it was suggested that other salt-ice compounds may
exists at higher pressure.
The discovery of novel high pressure compounds may improve our
understanding of the interior structure and dynamics of ice giant
planets~\cite{WilsonWongMilitzer2013}. If we assume, as an example,
NaCl were available in sufficient quantities, a separate H$_2$O-NaCl
layer~\cite{WilsonMilitzer2014} would form below the ice layer because
of its higher density. The density contrast of the two layers would
also introduce a convective barrier into the interior and potentially
prolong the cooling process of an ice giant planet.
The properties of carbon and oxygen are of high interest in planetary
science because together with hydrogen and nitrogen, they form the
planetary ices H$_2$O, CH$_4$, and NH$_3$ that make up the bulk of the
interiors of ice giant
planets~\cite{WilsonWongMilitzer2013}. Depending on the formation
conditions and the composition of the building materials, a variety of
planets and different interior structures are expected to
form~\cite{WilsonMilitzer2014, Bond2010,Madhusudhan2012}. Terrestrial
planets like Venus have thick and hot atmospheres that are rich in
CO$_2$. In the atmospheres of more massive exoplanets, we can expect
to find carbon-oxygen compounds that are exposed to yet higher
pressure. However, the properties of such compounds are not yet well
characterized at extreme conditions. With density functional molecular
dynamics simulations, Boates et al.~\cite{Boates2011} predicted CO$_2$
to exhibit a liquid-liquid phase transition at 0.5 Mbar. Leonhardi and
Militzer~\cite{Leonhardi2017} predicted a similar phase transition for
CO to occur between 0.1 and 0.2 Mbar. In the simulations, CO was also
observed to change phase from a molecular to a polymeric fluid. At yet
higher pressures, CO was found to spontaneously freeze into an
amorphous solid. Even though amorphous CO$_2$ structures have been
generated with high-pressure laboratory
experiments~\cite{Santoro2006}, one may expect that the amorphous CO
structures seen in the simulations do not correspond to the
thermodynamic ground state and that there is exists at least one
ordered solid CO structure with a lower free energy. Here we thus use
our SYDSS method to look for novel crystalline carbon-oxygen
structures with a carbon-to-oxygen ratio of 1:1 and variety of other
compositions.
In the original formulation, the {\em ab initio} random structure
search technique did not take advantage of crystal
symmetries and worked in the space group \textit{P}$_1$~\cite{AIRSS2006}.
Symmetric structures emerge, however, when atoms move onto Wyckoff
positions during the relaxation. If one wants to start with, and
maintain a certain set of crystal symmetries during the entire search
process, a special handling of the Wyckoff positions is unavoidable as
the following simple example of a mirror plane illustrates.
If an atom is placed exactly on the mirror plane then there exists
only one instance of it, otherwise there are two. Switching
continuously from one case to the other is difficult within the
context of \textit{ab initio} simulations because when one atom moves
closer to the mirror plane, the distance between the atoms becomes
small, repulsive forces become large and \textit{ab initio}
calculations with pseudopotentials typically do not converge. One
could, of course, remove one of the atoms if the distance between the
pair becomes too small but then the remaining structure would no
longer be symmetric. This means, for structural relaxation algorithms
that preserve the symmetry of the mirror plane, one needs to decide at
the very beginning whether the atom is on or off the mirror plane. In
both cases, the atom can still move in the subsequent relaxation and
occupy a more symmetric position.
While for a single mirror plane, only two cases need to be considered,
for a typical space group there exists a series of Wyckoff positions
that all need to be treated separately. Thus we decided to treat then
230 space groups and associated 1,506 Wyckoff
positions~\cite{IntTables,Bilbao} in a consistent fashion. This
means even screw axis symmetries are included and atoms, that are
far away from each other, have a higher chance of being placed on
symmetry positions. It is our goal to construct an algorithm that
does not exclude any structure but drastically increases the
probability that symmetric structures are generated successfully. To
prevent convergence issues in {\em ab initio} calculations, we
exclude, however, structures where atoms are unphysically close. For
this project, we conservatively chose the following minimum distances
between different Na, Cl, H, C and O species:
r$_{\rm NaNa}$ = 1.4,
r$_{\rm NaCl}$ = 1.2,
r$_{\rm ClCl}$ = 1.4,
r$_{\rm NaH}$ = 0.8
r$_{\rm ClH}$ = 0.8,
r$_{\rm HH}$ = 0.7,
r$_{\rm NaO}$ = 1.2,
r$_{\rm ClO}$ = 1.2,
r$_{\rm OH}$= 0.8,
r$_{\rm CO}$= 1.1,
r$_{\rm CC}$= 1.2,
and
r$_{\rm OO}$ = 1.2 \AA.
\section{Method}
Our SYDSS algorithm repeatedly steps through all 230 space groups
until a user-defined number of structures, $N_S$, have been generated
successfully. For a chosen space group, it selects lattice parameters
and angles at random, applies any constraints of the space group, and
scales the unit cell so that its volume matches a chosen target
volume. Then it builds a list of all atoms to be placed in the cell
(chemical composition times the number of formula units, $N_{FU}$). As
long as this list is not yet exhausted, our code loops over all
Wyckoff positions of the selected space group. Then it loops over all
atom types that have a sufficient number of atoms remaining to fill
all instances of the selected Wyckoff position. It chooses random
values for all free parameters of this position, generates the
coordinates, and checks whether the atoms satisfy all minimum distance
criteria~\cite{supercellEq}. If they do, our algorithm continues to
place the remaining atoms. If it fails to meet the distance criteria
at any point in this process, it discards the current configuration
and continues with the next space group until all $N_S$ initial
structures have been generated.
In our current implementation, the SYDSS algorithm has only a minimal
set of adjustable parameters: the atoms in the cell, the set of
minimum distances, and the target volume. The initial unit cell angles
are chosen between 40 and 140 degrees ~\cite{AIRSS2006} because the primitive cells of most structures can be represented in this way. This range could be broadened or one could sample from a smooth prior distribution that includes all angles. These choices, in
particular the distance criteria, imply that not all space groups
occur in the list of generated structures with equal probability.
Figure~\ref{Fig:initial_final} shows the probability distribution of
space groups in the set of initial structures of H$_2$O-NaCl
structures. Many space groups occur with very low or even zero
probability because an insufficient number of atoms remain to fill
all instances of a chosen Wyckoff position, or the inability to do so and satisfy
all minimum distance criteria. In particular, cubic systems (space groups
195-230) occur rarely among our generated H$_2$O-NaCl structures but they occur
frequently when we apply our algorithm to monatomic metals. Because of
its lack of symmetry constraints, space group \textit{P}$_1$ is still
among the space groups that are generated most often but its total
weight is now closer to 10\% compared to 100\% in ~\cite{AIRSS2006}.
If needed, additional biases could be introduced into the current
implementation of our SYDSS algorithm in order to reduce the
\textit{P}$_1$ probability further.
\begin{figure}[htb]
\includegraphics[width=.45\textwidth]{spg_hist07.eps}
\caption{Distribution of initial and final space groups among the
57,347 H$_2$O-NaCl structures that we generated with our SYDSS
algorithm. Both distributions are not identical because the space
group may change during the structural relaxation with DFT forces.
Some space groups have a low or zero occurrence probability. We set
their probabilities to 10$^{-3}$ to include them in this graph. }
\label{Fig:initial_final}
\end{figure}
Starting with these initial structures, structural relaxation at
constant pressure was carried out in the framework of density
functional theory, using the Perdew-Burke-Enzerhof
functional~\cite{Perdew1996} and the projector augmented wave method
~\cite{Kresse1996} as implemented in the Vienna \textit{ab initio}
simulation package~\cite{Kresse1999}. A basis-set cutoff energy of 980
eV used for the plane-wave expansion of the wave functions. For the
first round of relaxation, we used k-point grids of
4$\times$4$\times$4 for cells with less than 15 atoms and
2$\times$2$\times$2 for cells with greater than 15 atoms. The best of
these structures were then re-relaxed using higher density grids
(6$\times$6$\times$6 to 12$\times$12$\times$12) to ensure accurate
enthalpies. This method allowed us to search the structures more
efficiently by removing unlikely candidates early.
During relaxations, the symmetry of the initial space group was
preserved. This still allowed structures to attain higher symmetries
of a supergroup during the relaxation if the atoms move to more
symmetric positions while maintaining the symmetry operations of the
original space group. This means the space groups of the initial and
the relaxed structures may differ. Such transitions are illustrated in
Fig.~\ref{Fig:transitions}. Many transitions occurred between space
groups in the same crystal system but we also noticed transitions from
monoclinic (space groups 3-15) to orthorhombic (16-74), from
orthorhombic to tetragonal (75-142), and from trigonal (143-167) to
hexagonal (168-194) systems. In a few instances, a smaller primitive
unit cell emerged during the relaxation. Such a transition may plot
below the diagonal in Fig.~\ref{Fig:transitions} because the smaller
unit cell may have a lower space group number.
\begin{figure}[htb]
\includegraphics[width=.45\textwidth]{spg12.eps}
\caption{In this transition diagram, final versus initial space groups are plotted for the relaxation of 1 to 4 formula
unit structures. Transitions to higher space groups are much more
frequent because the structural relaxation often increases the symmetry.
The large circle indicates one possible pathway to our {\it Pnma} structure. The lines separate the 7 crystal systems.}
\label{Fig:transitions}
\end{figure}
\section{Results}
\subsection{Water-Salt Structure Search: (H$_2$O)$_n$-(NaCl)$_m$}
We generated and relaxed over 55,000 structures with compositions
(H$_2$O)$_n$-(NaCl)$_n$ having between 1 and 4 formula units at
pressures between 1 and 10 Mbar. We also explored additional
water-salt mixing ratios by relaxing over 11,000
(H$_2$O)$_n$-(NaCl)$_m$ structures with $n$:$m$=4:1, 3:1, 2:1, 3:2,
1:2, and 1:3. However, we were not able to find any thermodynamically
stable structures with $n \ne m$ and we will thus focus the following
discussion on structures with equal water-salt ratios where our
structure search was more successful.
Enthalpies of the computed (H$_2$O)$_n$-(NaCl)$_n$ structures were
then compared with the enthalpy of the H$_2$O and NaCl
endmembers. NaCl endmember enthalpies were calculated using the B2
structure which is stable in this pressure range~\cite{Bassett1968}.
H$_2$O endmember enthalpies were calculated using the high pressure
ice phases predicted at each pressure. At 1-2 Mbar, enthalpies were
calculated using the ice X structure~\cite{Polian1984}. For pressures
of 3-7 Mbar, enthalpies were calculated using the \textit{Pbcm}
structure~\cite{Benoit1996}. At 8 Mbar, the \textit{Pbca}
structure~\cite{Militzer2010} was used. At 9-10 Mbar, the
\textit{P}$3_1$21 structure ~\cite{Pickard2013} was assumed. To
further test our SYDSS method, we also applied it to pure water
ice. We relaxed 2,000 H$_2$O structures at 9 Mbar and reproduced the
\textit{P}$3_1$21 structure from Ref.~\cite{Pickard2013}.
After comparison with endmember data, three H$_2$O-NaCl structures
were found to have enthalpies lower than that of the combined
endmembers, suggesting a novel H$_2$O-NaCl structure would form at
high pressure. The enthalpy comparisons of the three best structures
with that of the endmembers is given in Fig.~\ref{Fig:enthalpy}. The
P$\bar{1}$ structure was found by relaxing a structure with one
formula unit of H$_2$O-NaCl, the P$2_1$ structure was found from two
formula unit structures, and the {\it Pnma} structure was found from
four formula unit structures. This enthalpy data predicts a novel
\textit{Pnma} symmetric H$_2$O-NaCl structure forming at 3.4 Mbar,
which is within the pressure range of diamond anvil cell
experiments~\cite{Lobanov2015}.
\begin{figure}[htb]
\includegraphics[width=.45\textwidth]{dH04.eps}
\caption{The difference in enthalpy, $H_{\rm H_2ONaCl} - H_{\rm
H_2O} - H_{\rm NaCl}$, per formula unit as function of
pressure. The arrow marks the pressure of 3.4 Mbar where the H$_2$O-NaCl
structure with {\em Pnma} symmetry is predicted to form from
H$_2$O and NaCl. The $P2_1$ and $P\bar 1$ structures were also
shown to have lower enthalpies than the endmembers at 5 Mbar but
the {\em Pnma} structure is energetically favored. }
\label{Fig:enthalpy}
\end{figure}
Out of 7,909 four-formula-unit H$_2$O-NaCl structures that were
successfully relaxed, 185 ($\sim$2\%) relaxed into the \textit{Pnma}
space group. Of these 185 \textit{Pnma} structures, 74 relaxed from
structures with P$2_12_12_1$ symmetry, 35 from \textit{Cc} symmetric
structures, and 76 initially started from \textit{Pnma} symmetric
structures. The fact that this structure was never generated from a
nonsymmetric initial structure (space group $P_1$) and the rate of
occurrence illustrates the advantages of implementing symmetry constraints in our algorithm.
The parameters of our novel orthorhombic H$_2$O-NaCl structure are
given in Tab.~\ref{table1} and two pictures are shown in
Fig.~\ref{Fig:images}. We verified that this
structure is dynamically stable by performing phonons calculations
with the Phonopy code~\cite{Togo2015} using 1x1x2, 1x2x1, and 2x1x1
supercells. The structure can be explained best by
analyzing the layering parallel to the $a$-$b$ planes. Layers with Cl$^-$ ions
alternate with layers of Na$^+$ and O$^{2-}$ ions. The Cl$^-$ ion
always occupy the same position in every layer. The Cl$^-$-Cl$^-$
distances are thus smaller than the separation between other ion pairs
of the same type. From layer to layer, the Na$^+$ and O$^{2-}$ ions
alternate between two positions, leading to unit cell with 20
atoms. The typical geometry of a H$_2$O molecule is well
preserved. The H$_2$O dipole moments lie in the $a$-$b$ planes and are
arranged in clockwise or anticlockwise direction around each column of
Cl$^-$ ion. Overall the charges are reasonably well balanced in this
structure.
\begin{table}
\begin{tabular}{c c c c c}
\hline
\hline
Atom& Wyckoff & x & y & z \\
\hline
Na& b & 0 & 0 & 1/2 \\
Cl& c & $-$0.476 & 1/4 & $-$0.370\\
H & d & ~~0.355 & $-$0.437 & $-$0.310 \\
O & c & ~~0.278 & 1/4 & ~~0.267 \\
\hline
\hline
\end{tabular}
\caption{Parameters of the orthorhombic H$_2$O-NaCl structure with {\it Pnma}
symmetry at 4 Mbar. The lattice parameters are $a$=3.942, $b$=3.849, and $c$=5.187 \AA.}
\label{table1}
\end{table}
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{Pnma_image_2_and_3_wide_b.eps}
\caption{ Novel orthorhombic NaCl-H$_2$O crystal structure with {\it
Pnma} symmetry. With decreasing size, the spheres denote the positions
of Cl, Na, O and H atoms. The structure has 20 atoms per unit
cell but has been doubled in $c$ direction in the right image.}
\label{Fig:images}
\end{figure}
C
We generated and relaxed over 700,000 C$_n$-O$_m$ structures with up
to 52 atoms per unit cells with ratios from n:m ranging from 1:7 to
6:1 at pressures between 1 and 50 Mbar. Enthalpies of the resulting
structures were compared to the C and O endmembers, which were
calculated using the stable carbon phases of diamond for 1-10 Mbar,
BC8 for 15-25 Mbar, and SC1 for 30-50 Mbar~\cite{Benedict2014}. For
oxygen endmembers $\zeta$-C2/m oxygen for 1-15
Mbar~\cite{Akahama1995,Ma2007} and the \textit{Cmcm} oxygen structure
above 20 Mbar~\cite{Sun2012} were used.
\begin{figure}[h]
\includegraphics[width=.45\textwidth]{CO_Figure_5.eps}
\caption{Enthalpy difference per formula unit as a function of composition
is plotted. (a) Representative enthalpy calculations for varying
compositions at 7 Mbar. (b) Enthalpy calculations at 50 Mbar suggest C$_4$O and
C$_2$O structures would be favorable in systems that contain more
carbon than CO$_2$.}
\label{Fig:COResults}
\end{figure}
In Fig.~\ref{Fig:COResults}, we plot the enthalpy difference per atom
between various C$_n$O$_m$ compounds that of the endmember phases,
$\Delta H = H_{\rm C_nO_m} - ( nH_{\rm C}+mH_{\rm O} )$. At every
pressure under consideration, the most stable CO$_2$ structures,
that we obtained, reproduced previous results~\cite{Lu2013}. The
convex hull at 7 Mbar in Fig.~\ref{Fig:COResults}a shows that no
stable structures are expected to exist besides CO$_2$ and the two
endmembers. However, convex hull diagram at 50 Mbar in
Fig.~\ref{Fig:COResults}b revealed the existence of two new stable
carbon-rich structures with C$_4$O and C$_2$O
compositions. Interestingly, no stable CO structures were found over
the entire pressure range. All CO structures, that we generated, were
found to have a higher enthalpy than a combination of carbon and
CO$_2$. Also none of our oxygen-rich compounds were found to be
stable. Only one structure with a C:O = 1:6 composition came close to
matching the combined enthalpies of pure oxygen and CO$_2$ but was not
found to be stable in the pressure range up to 50 Mbar.
The C$_4$O structure is monoclinic and has {\em C2/m} symmetry. It was
found from relaxing structures with 2 formula units (10 atoms). The
image in Fig. ~\ref{Fig:imageC4O} reveals a layered structure where
thin oxygen planes alternate with thick carbon layers. The oxygen
atoms form a 2D hexagonal lattice in planes spanned by the crystal
lattice vectors $b$ and $c$. The carbon atoms are arranged on four,
tightly stacked hexagonal layers in between.
\begin{figure}[h!]
\includegraphics[width=0.44\textwidth]{CONTCAR_C4O_45Mbar_222.eps}
\caption{Moniclinic C$_4$O crystal structure with {\it C2/m} symmetry at 45 Mbar. The
unit cell with 10 atoms as been doubled along every lattice vector
to better illustrate the C and O layers in the structure. The C
and O atoms are shown in dark and light color, respectively. }
\label{Fig:imageC4O}
\end{figure}
The C$_2$O structure can also be viewed as a layered structure but the
bonding is more complex and three dimensional. The structure is
orthorhombic and has {\em Pbca} symmetry. In Fig.~\ref{Fig:imageC2O},
the unit cell with 8 formula units (24 atoms) has been double in $b$
direction to illustrate the layers and 3D bonding. The shortest bonds
occur between the C and O atoms in the layers but C-O bond distances
vary considerably between 1.11 and 1.30 \AA~at 25 Mbar. The C-O layers
are connected by C-C bonds that are all between 1.17 and 1.18
\AA~long. Again, we verified the C$_2$O and C$_4$O structures structure
were dynamically stable by performing phonons calculations with the
Phonopy code~\cite{Togo2015} using 2x2x2 supercells.
\begin{figure}[h!]
\includegraphics[width=0.45\textwidth]{CONTCAR_C2O_25Mbar_122b.eps}
\caption{Orthorhombic C$_2$O crystal structure with {\it Pbca} symmetry at 25 Mbar. The
unit cell with 24 atoms as been doubled in $b$ direction. The C
and O atoms are shown in dark and light color, respectively.}
\label{Fig:imageC2O}
\end{figure}
We performed enthalpy calculations of these new C$_4$O and C$_2$O
structures in order to determine the pressure at which they are
favored over a decomposition into pure carbon and CO$_2$. In Fig.~\ref{Fig:enthalpyCO}, we choose to plot the resulting enthalpy
difference with respect to a mixture of pure carbon and C$_2$O
because this allows us to illustrate the C$_4$O and the C$_2$O
formulation pressures in a single diagram. We predict the C$_2$O
structure to form at 19.8 Mbar while the C$_4$O structure becomes
stable at 44.0 Mbar. The parameters of both structure given in
tables~\ref{tablePbca} and \ref{tableC2/m}. The formation pressures of
both structures are considerably larger than those that are typically
reached with diamond anvil cell experiments. This is not unexpected
because the diamond anvils would otherwise have reacted with the
samples in any experiment that contained sufficient amounts of free
oxygen. However, such pressures are accessible with dynamic
compression techniques that use ramp waves to compress the sample at
lower temperature than with standard shock wave
experiments~\cite{Eggert2014}.
\begin{figure}[h]
\includegraphics[width=.45\textwidth]{dH_CO_02}
\caption{The difference in enthalpy,
$H_{\rm C_nO_m} - \big[(n-2m) \times H_{\rm C} + m \times H_{\rm C_2O}\big]$, per atom as
function of pressure. The first arrow marks the pressure of 19.8 Mbar where
the C$_2$O structure with {\em Pbca} symmetry is predicted to
form. The second arrow marks the pressure 44.0 Mbar where the C$_4$O
structure with {\em C2/m} symmetry is predicted to form.}
\label{Fig:enthalpyCO}
\end{figure}
\begin{table}[h!]
\begin{tabular}{c c c c c}
\hline
\hline
Atom& Wyckoff & x & y & z \\
\hline
C & c & 0.848 & 0.497 & 0.850 \\
C & c & 0.325 & 0.115 & 0.531\\
O & c & 0.567 & 0.386 & 0.757 \\
\hline
\hline
\end{tabular}
\caption{Parameters of the orthorhombic C$_2$O structure with {\it Pbca} symmetry at 25 Mbar.
The lattice parameters are $a$=5.845, $b$=2.890, and $c$=2.899 \AA.}
\label{tablePbca}
\end{table}
\begin{table}[h]
\begin{tabular}{c c c c c}
\hline
\hline
Atom& Wyckoff & x & y & z \\
\hline
C & i & 0.347 & 0.000 & 0.220 \\
C & i & 0.893 & 0.000 & 0.605\\
O & b & 0.000 & 0.500 & 0.000 \\
\hline
\hline
\end{tabular}
\caption{Parameters of the monoclinic C$_4$O structure with {\it C2/m} symmetry at 45 Mbar. The
lattice parameters are $a$=2.960, $b$=1.434, $c$=3.916 \AA~and $\beta$ = 106.73$^\circ$.}
\label{tableC2/m}
\end{table}
\section{Conclusion}
Our SYDSS algorithm provides a systematic and consistent way to
generate symmetric candidate structures for relaxation with DFT forces
with the goal of predicting novel crystal structures at high
pressure. While no structure is excluded in principle, symmetric
structures are generated with high probability. This significantly
improves the efficiency of our structure search algorithm for large
unit cells with 20 atoms or more, if one adopts the common view that
ground state crystal structures are symmetric.
We applied our SYDSS technique to search for novel stoichiomtric
H$_2$O-NaCl compounds at high pressure because we assumed large unit
cells would be needed to accommodate atoms from both materials in an
optimal way. Indeed, our best structure has a comparatively
large primitive unit cell of 20 atoms. However, as with any random
search method, there is no guarantee there does not exist yet another
H$_2$O-NaCl structure with lower enthalpy unless our prediction is
confirmed with experiments. The predicted formation pressure of 3.4
Mbar is well within the reach of diamond anvil cell
experiments~\cite{Lobanov2015}. If indeed a yet more stable
H$_2$O-NaCl compounds exists, x-ray diffraction measurements should
reveal such a structure.
When we applied our SYDSS method to search for novel carbon-oxygen
compounds at megabar pressures, we identified two novel carbon-rich
but no oxygen-rich structures. At 19.8 Mbar, we predict an
orthorhombic C$_2$O structure to form from dense carbon and CO$_2$. At
44.0 Mbar, a novel monoclinic C$_4$O structure is expect to become
thermodynamically stable. Both transition pressures are beyond the
reach of static high pressure experiments but can in principle be
generated with dynamic compression techniques.
Also, we cannot completely rule out the existence of unknown
low-enthalpy structures of the H$_2$O, NaCl, carbon, and oxygen
endmembers even though one has looked for such structures carefully
with DFT methods carefully already. If a novel H$_2$O, NaCl, carbon,
or oxygen structure existed, the formation pressures of the predicted
novel compounds would be shifted to higher values than we have predict
here. However, in a diamond cell or ramp compression experiment one
would see such novel endmember structures. In either case, new
compounds or endmember structures are exprected to produced when
H$_2$O-NaCl and C-O mixtures are exposed to pressures of 3.4 and 19.8
Mbar, respectively.
\acknowledgments{The authors acknowledge support from the U.S. National Science Foundation (grant 1412646), the U.S. Department of Energy (grant DE-SC0010517), and University of California's lab fee program as well as encouraging discussions with R.
Caracas and other participants of the program ``Dynamics and
Evolution of Earth-like Planets'' at the Kavli Institute for
Theoretical Physics. In part, this work used the National Energy
Research Scientific Computing Center and the Extreme Science and
Engineering Discovery Environment. K. Driver provided comments on this
manuscript. }
|
1,314,259,995,806 | arxiv | \section*{Abstract}
\end{center}
In this paper, we consider three-dimensional dynamical
systems, as for example the Lorenz model. For these systems,
we introduce a method for obtaining families of
two-dimensional surfaces such that trajectories cross
each surface of the family in the same direction.
For obtaining these surfaces, we are guided by the
integrals of motion that exist for particular values
of the parameters of the system. Nonetheless families of
surfaces are obtained for arbitrary values of these
parameters.
Only a bounded region of the phase space is not filled by
these surfaces. The global attractor of the system must be
contained in this region.
In this way, we obtain information on the shape and location
of the global attractor. These results are more restrictive
than similar bounds that have been recently found by the
method of Lyapunov functions.\\
\\
\\
\\
{\bf Keywords} : Lorenz model/ Chaotic Dynamics/ Integrals of motion.
{\bf PACS numbers} : 05.45.+b / 02.30.Hq
}
\clearpage
The Lorenz equations (\ref{lo equa}) are one of the
classic models of nonlinear dynamics and chaos. These
equations were originally derived in a modal truncation
of the Boussinesq equations for thermal convection. They
read as follows~:
\begin{eqnarray}
\dot{x} & = & \sigma(y-x) \nonumber \\
\dot{y} & = & rx-y-xz \label{lo equa}\\
\dot{z} & = &xy-bz \nonumber
\end{eqnarray}
with $\sigma , b, r \geq 0 $.
There $\sigma$ corresponds to the Prandtl number, $b$ is
a geometric parameter and $r$ is the Rayleigh number in
units of the critical Rayleigh number.
These equations describe a dissipative dynamical system
for all values of $r$, $\sigma$ and $b$ because the
divergence of the flow field is always negative.
Hence 3-dimensional volumes in the phase space contract
to zero at a uniform exponential rate and the system's
attractor is necessarily of dimension less than three.
This model has become a classic in the area of nonlinear
dynamics. Its importance is not that it quantitatively
describes the hydrodynamics motion, but rather that it
illustrates how a simple model can produce very rich and
varied form of dynamics, depending on the value of a
parameter in the equations \cite{sparrow}.
In this paper, we are interested in the approximated
location in the phase space of the global attractor of
the system, which contains all dynamics evolving from
all initials conditions. The global attractor is the set
of points in phase space that can be arrived at from some
initial condition at an arbitrary long time in the past.
The two fundamental properties of global attractors
are \cite{doering navier}~:
\begin{itemize}
\item it is invariant under the evolution.
\item the distance of any solution from it vanishes as
$t \rightarrow + \infty $.
\end{itemize}
The last property is simply interpreted thus~: if the
solution starts initially outside the global attractor,
then it is attracted into it as $t \rightarrow + \infty $
and once inside it cannot escape. If it starts inside then
it stays inside.
The global attractor contains all the
asymptotic motion for the dynamical system. It is common
to talk of {\it multiple attractors} for a dynamical system,
and each of them may in its own right be considered as the
attractor for initial conditions within its own bassin
of attraction. The notion of global attractor
corresponds to the union of all possible such dynamically
invariant attracting sets. In particular, it contains all
possible structures such as fixed points, limit
cycles etc...
The global attractor is contained in an {\it absorbing ball} in phase
space, and we want to obtain analytic estimates about its
geometric shape. Moreover, this enables us to find good
estimates of its Lyapunov dimension. Estimates which give
the shape of the attractor are important as they lead to
a good upper bound on the dimension of the
Lorenz attractor \cite {doering et
gibbon}.
Until now, approximate locations of the
Lorenz's attractor in the phase space have been obtained
by the method of Lyapunov functions \cite{lorenz,sparrow,
doering et gibbon,strogatz,jackson}. Very recently, thanks
to this method, it has been shown that the global
attractor of the Lorenz equations is contained in a
volume bounded by a sphere, a cylinder, the volume
between two parabolic sheets, an ellipsoide and a cone
\cite{doering et gibbon}.\\
In this paper, we apply a different method for obtaining
analytic estimates for the location and shape of the
Lorenz attractor. The method is based on the determination
of families of 2-dimensional surfaces that are crossed by
the trajectories of the system only in one direction. In
the region filled by these surfaces, the dynamical
behaviour is very simple. The asymptotic {\it complex
behaviour} must be contained in the region of the
phase space that is not occupied by these surfaces.\\
For finding these families of surfaces, we will be guided
by the time-dependent integrals of motion that exist for
special values of the parameters of the system.
Integrals of motion for the Lorenz system have been
extensively studied in \cite{tabor,kus,tabor et weiss}.
The known integrals of motion are~:
\begin{itemize}
\item [{\bf a}] $I(x,y,z,t)=(x^2-2 \sigma z) e^{2 \sigma t}$
with $b=2 \sigma$ and $\sigma$ and $r$ arbitrary.
\item [{\bf b}] $I(x,y,z,t)=(y^2+z^2) e^{2t}$
with $b=1$, $r=0$ and $\sigma$ arbitrary.
\item [{\bf c}] $I(x,y,z,t)=(-r^2x^2+y^2+z^2) e^{2t}$
with $b=1$, $\sigma=1$ and $r$ arbitrary.
\item [{\bf d}] $I(x,y,z,t)=\left( \frac{(2 \sigma-1)^2}{\sigma}
x^2+
\sigma y^2-(4 \sigma-2)xy-\frac{1}{4 \sigma} x^4+x^2 z
\right)
e^{4 \sigma t}$ \\ with $b=6 \sigma-2$, $r=2 \sigma
-1$ and $\sigma$ arbitrary.
\item [{\bf e}] $I(x,y,z,t)=\left(
-rx^2-y^2+2xy+\frac{1}{4}x^4-x^2 z+
4(r-1)z
\right)
e^{4t}$ \\ with $b=4$, $\sigma=1$ and $r$
arbitrary.
\end{itemize}
For each of these integrals we have
$\frac{dI}{dt} \equiv 0$.
Let us consider case {\bf a} and let us define the family
of surfaces $x^2-2 \sigma z=k$, where k is an arbitrary
constant. The scalar product between the normal vector
$\vec{N}$ to this surface at a given point and the
tangent vector $\vec{T}$ to the trajectory of the Lorenz
system that goes through this point is given by~:
\begin{equation}
\vec{N} \cdot \vec{T}= (2x \vec{i} - 2 \sigma \vec{k})
\cdot \left( \sigma (y-x) \vec{i}+(rx-y-xz) \vec{j}+
(xy-bz) \vec{k} \right) =-bk \nonumber
\end{equation}
Therefore, for a given surface (i.e. for a given value of k)
this scalar product has the same sign for all the points of
the surface. Each surface of the family is crossed
in the same direction by the flow associated to the system.
This direction depends of the sign of the constant
k. Hence, for the case $b=2 \sigma$, the 3-d phase
space of the
Lorenz system is filled by two families of surfaces, the
families associated to positive and negative values of k.
The scalar product
$\vec{N} \cdot \vec{T}$ is positive (resp. negative) for
negative (resp. positive) values of k. It is clear that
the surface corresponding to k=0 plays a very special role.
All the trajectories of the system are attracted by this
surface. On this surface, the scalar product
$\vec{N} \cdot \vec{T}$ is zero. This surface is an
invariant manifold of the system, as can be seen
in fig. \ref{fig para m}.
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{para-m2.eps}
$$
\caption{The family of surfaces $x^2-2 \sigma z=k$ for
the case $b=2 \sigma$.
The bolded surface corresponds to $k=0$ and is the
attracting set of the system. Some trajectories
of the system are shown.}
\label{fig para m}
\end{figure}
It is clear that the existence of these
families of surfaces gives a lot of information about the
dynamics of the system. The behaviour of trajectories is
extremely simple in all the phase space with the
exception of the invariant surface $x^2-2 \sigma z=0$.
This surface contains the global attractor of the system for
the case $b=2 \sigma$. Here, the global attractor is contained
in a two-dimensional surface, as for the five cases
{\bf a}, {\bf b}, {\bf c}, {\bf d} and {\bf e}, that is when an
integral of motion
exists. The family of surfaces derivated above enables us to
characterize in a simple way this global attractor. The
determination of this family of surfaces follows immediatly
from the existence of the integral of motion {\bf a}, when
$b=2 \sigma$. Now, the natural question is~: if
$b \not= 2 \sigma$, is it possible to find similar
families of surfaces that the flow crosses in the
same direction at each point of the surface ? - in the
following, we will call this type of surface
{\it semipermeable} -
In this case, we do not have at our disposal an integral of
motion, and these surfaces can not fill the phase space
because, in the general case, the global attractor is not
contained in a two-dimensional set.\\
In order to find {\it semipermeable} surfaces in the general case
(when integrals of motion do not exist),
we will procede as follows~:\\
For the case of the integral of motion {\bf a},
we first propose, when $b \neq 2 \sigma$, a surface of the
same mathematical form as the integral of motion {\bf a}, but
with arbitrary coefficients~:
\begin{equation}
S=a_1 x^2+a_2z+a_3=0 \label{para}
\end{equation}
The scalar product $\vec{N} \cdot \vec{T}$ is now~:
\begin{eqnarray}
\vec{N} \cdot \vec{T} &=& 2 a_1 x \dot{x}+a_2 \dot{z}
\nonumber \\
&=&(2a_1 \sigma +a_2)xy-2 a_1 \sigma x^2 -a_2 b z
\label{ps para}
\end{eqnarray}
If we calculate this scalar product on the surface S, for the general
case $b \neq 2 \sigma$, we obtain ~:
\begin{equation}
\vec{N} \cdot \vec{T}/S=(2a_1 \sigma+a_2)xy+(b a_1-2 \sigma
a_1)x^2+b a_3 \label{ps para on}
\end{equation}
where we have replaced in (\ref{ps para}) $-a_2 z$ by
$a_3+a_1 x^2$. We now have an expression that depends only
on two variables~: $x$ and $y$. The problem of determining
the coefficients $a_1$, $a_2$ and $a_3$ in order for this
expression to have the same sign for arbitrary values of x
and y is considerably simpler than the analogous problem
in the three variables $x$,$y$ and $z$ that must be
solved in the method of the Lyapunov functionals.
To keep the same sign in (\ref{ps para on}) for arbitrary
values of $x$ and $y$, we must take $a_2=-2 \sigma a_1$. Then
we have~:
\begin{equation}
\vec{N} \cdot \vec{T}/S=a_1(b-2 \sigma)x^2+b a_3 \label{ps para fin}
\end{equation}
As $a_1$ must be nonzero, we can take $a_1=1$ without loss of
generality.
We now have two different cases~:
\begin{itemize}
\item [i)] $b>2\sigma $, we must take $a_3>0$ in
order to have a family of semipermeable surfaces.
We show this family, as well as
some trajectories of the system, in fig. \ref{fig para h}.
\item [ ii)] $b<2\sigma $, we must take $a_3<0$ in
order to have a family of semipermeable surfaces.
We show this family, as well as
some trajectories of the system, in fig. \ref{fig para b}.
\end{itemize}
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{para-h2.eps}
$$
\caption{The family of semipermeable surfaces (\ref{para})
for the case $b>2 \sigma$. The bolded surface is the {\it last}
surface of the family. Some trajectories of the system
are also shown. The critical points C+ and C-
are below these surfaces.}
\label{fig para h}
\end{figure}
\begin{figure}[hbtp]
$$
\epsfxsize=5cm
\epsfbox{para-b2.eps}
$$
\caption{The family of semipermeable surfaces(\ref{para})
for the case $b<2 \sigma$. The bolded surface is the {\it last}
surface of the family. Some trajectories of the system are also shown.
The chaotic attractor is above these surfaces.}
\label{fig para b}
\end{figure}
As we can see from the figures above, in the region filled by
the surfaces the dynamic of the system is very simple.
The {\it complex} behaviour can only occur in the region of
phase space that is not occupied by these surfaces.
In case ii), the global attractor of the system must be located in the
region $z>2 \sigma x^2$. For the case i), because of the
presence of the semipermeable surfaces, the flow cannot
enter the $z>2 \sigma x^2$ region upward and hence the
homoclinic trajectory cannot exist.
Therefore, motivated by the existence of the first
integral {\bf a}, valid in the case $b=2\sigma$, we have found
a family of semipermeable surfaces for arbitrary values
of the parameters of the system.
As we shall see, new families of semipermeable surfaces can be
found by using the other integrals of motion.\\
From the case {\bf b}, we deduce that the surfaces $y^2+z^2=k^2$ are
semipermeable for arbitrary values of k, when $b=1$ and
$r=0$. Guided by this result, we propose in the general
case a family of surfaces of the form~:
\begin{equation}
S=a_1(y-c_1)^2+a_2(z-c_2)^2-1=0
\label{cylindre}
\end{equation}
The scalar product $\vec{N} \cdot \vec{T}$ is given by~:
\begin{eqnarray}
\vec{N} \cdot \vec{T} & = &2 (
(a_2-a_1)xyz-a_2 b z^2-a_1 y^2+(a_1 r-a_2 c_2)xy+
a_1 c_1 xz \nonumber \\
& &+ a_2 c_2 b z+a_1 c_1 y-a_1 c_1 r x ) \label{ps cylindre}
\end{eqnarray}
The evaluation of $\vec{N} \cdot \vec{T}$ on the surface
(\ref{cylindre}) becomes far simpler if we take
$a_2=a_1$, $c_1=0$ and $c_2=r$. After this, (\ref{cylindre})
and (\ref{ps cylindre}) respectively become~:
\begin{equation}
S=a_1(y^2+(z-r)^2)-1=0
\end{equation}
\begin{equation}
\vec{N} \cdot \vec{T}=-2a_1(b z^2+y^2-rbz)
\end{equation}
The scalar product $\vec{N} \cdot \vec{T}$ calculated on
the surface $S$ is given by~:
\begin{equation}
\vec{N} \cdot \vec{T}/S=
a_1 \left( (1-b)z^2+r(b-2)z+r^2-\frac{1}{a_1} \right)
\label{ps cylindre on}
\end {equation}
Note that $\vec{N} \cdot \vec{T}/S$ is a function of
only one variable, as it is the case for expression
(\ref{ps para fin}) for the semipermeable parabolas.
In this case, the surface $S$ is not infinite in the $y$
and $z$ directions. In particular, the coordinate $z$
varies in the interval~:
$r-\frac{1}{\sqrt{a_1}} \leq z \leq r+\frac{1}{\sqrt{a_1}}$.
In consequence, the quadratic polynomial
(\ref {ps cylindre on}) must have the same sign only in
this interval and not for arbitrary values of
$z$. This condition determines the possible values
of $a_1$, that can be found by applying the Sturm's theorem.
The results are as follows~:
\begin{eqnarray}
b\leq 2 & \mbox{, }& \frac{1}{a_1}\geq r^2 \nonumber \\
b\geq 2 & \mbox{, }& \frac{1}{a_1}\geq
\frac{r^2 b^2}{4(b-1)} \label {cylindre min}
\end{eqnarray}
Hence, for arbitrary values of the parameters of the
system, we have found a family of semipermeable infinite
cylinders. The radius of these cylinders varies between
$+ \infty$ and the minimal values given in
(\ref{cylindre min}). The behaviour of some orbits with
respect to this family of surfaces is shown in
fig. \ref{fig cyl b}. The global attractor is
contained in the region not occupied by these surfaces.\\
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{cyl-b2.eps}
$$
\caption{Chaotic attractor stuck inside semipermeable cylinders.
The bolded circle corresponds to the {\it smallest} cylinder.}
\label{fig cyl b}
\end{figure}
From the integral of motion {\bf\ c}, we deduce the existence of
the family of semipermeable surfaces $z^2+y^2-rx^2-k=0$,
with $k$ arbitrary and $b=1$, $\sigma=1$.\\ Guided by this
result, we propose in the general case the family of surfaces~:
\begin{equation}
S=a_1 x^2+a_2 y^2+a_3 z^2-a_4=0 \; \mbox{ , with }
a_1 \, a_2 \, a_3 <0 \nonumber
\end{equation}
The scalar product $ \vec{N} \cdot \vec{T}$ is given by~:
\begin{equation}
\vec{N} \cdot \vec{T}=2 \left(
(a_3-a_2)xyz-a_3b z^2-a_2 y^2-a_1 \sigma x^2+(a_1 \sigma +a_2 r)xy
\right)
\end{equation}
In order to have the same sign for $\vec{N} \cdot \vec{T}$ on
the surface $S$, we take $a_3=a_2=1$. Then we have~:
\begin{equation}
S=a_1 x^2+ y^2+ z^2-a_4=0 \; \mbox{ , with } a_1<0 \label{cone}
\end{equation}
\begin{equation}
\vec{N} \cdot \vec{T}=2 \left(
-(y^2+b z^2)-a_1 \sigma x^2+(a_1 \sigma+r)xy
\right)
\end{equation}
This expression, calculated on the surface (\ref{cone}),
gives~:
\begin{equation}
\vec{N} \cdot \vec{T}/S=2 \left(
a_1(b- \sigma )x^2+(a_1 \sigma+r)xy+(b-1)y^2-b a_4
\right) \label {ps cone on}
\end{equation}
On the surface (\ref{cone}) the variables $x$ and $y$ vary
in such a way that the following inequality must be
satisfied~: $a_1 x^2+y^2 \leq a_4$. Therefore
(\ref{ps cone on}) must have the same sign for all values
of $x$ and $y$ that satisfy this inequality. The
solution of this algebraic problem is not simple. Hence
we do not give here the technical details of the
calculations.\\ This condition determines the possible
values of the coefficient $a_1$~:
\begin{eqnarray}
\frac{-2 \sigma r -(\sigma -1)^2-
\sqrt{(\sigma -1)^4+4 \sigma r (\sigma -1)^2}}
{2 \sigma ^2}
\leq a_1 \leq \nonumber \\
\frac{-2 \sigma r -(\sigma -1)^2+
\sqrt{(\sigma -1)^4+4 \sigma r (\sigma -1)^2}}
{2 \sigma ^2}
\end{eqnarray}
or
\begin{eqnarray}
\frac{2(b-1)(b-\sigma)-\sigma r
-2 \sqrt{(b-1)^2(b-\sigma)^2-\sigma r (b-1)(b- \sigma)}}
{\sigma ^2}
\leq a_1 \leq \nonumber \\
\frac{2(b-1)(b-\sigma)-\sigma r
+2 \sqrt{(b-1)^2(b-\sigma)^2-\sigma r (b-1)(b- \sigma)}}
{\sigma ^2}
\end{eqnarray}
The parameter $a_4$ is arbitrary, and the condition $a_1 < 0$ restricts the
possible values of the parameters $b$ and $\sigma$, which
are given in fig. \ref{fig hyper}.
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{hyper.eps}
$$
\caption{Parameters $\sigma$ and $b$ for which surfaces (\ref{cone})
are semipermeable.}
\label{fig hyper}
\end{figure}
The canonical values $r=28$, $\sigma=10$ and
$b=\frac{8}{3}$ lie in the region I. For parameters $\sigma$
and $b$ in region I of fig. \ref{fig hyper}, there are
two zones in phase space
that are filled with surfaces of the family. The behaviour
of some trajectories of the system, in relation to the
semipermeable surfaces is shown in fig. \ref{fig cone b}.
These results are more restrictive (they give a more precise
information about the location of the global attractor)
than similar results obtained recently in \cite {doering et
gibbon} by employing the method of Lyapunov functions.
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{cone-b2.eps}
$$
\caption{Lorenz attractor squeezed between semipermeable
hyperboloids (\ref{cone}). The bolded lines are the last
repelling cones. The thick dashed
line is the former bound obtained by Doering \& al.
\cite{doering et gibbon}. The parameters $\sigma$ and $b$
are in region I.}
\label{fig cone b}
\end{figure}
Therefore, our method can locate more
accurately the global attractor of the system in phase
space than before.
In region II of fig. \ref{fig hyper},
the scalar product $\vec{N} \cdot \vec{T}/S$ has opposite
sign with respect to region I and, in phase space, there are
two zones that are not filled by the surfaces of the
family. These two zones are not connected between them
and the critical points C- and C+ are contained in each
of the different regions. The behaviour of some
trajectories of the flow with respect to the
semipermeable surfaces and the position of the critical
points C- and C+ are shown in the fig. \ref{fig cone h}.
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{cone-h2.eps}
$$
\caption{Critical points C- and C+ separated by semipermeable
surfaces (\ref{cone}). The left (resp. right) zone is a part of the bassin
of attraction of C- (resp. C+). The parameters $\sigma$ and $b$
are in region II.}
\label{fig cone h}
\end{figure}
If one trajectory enters one of the two free regions in the
phase space, it cannot exit from it and hence
cannot pass in the other one. This restriction on the
behaviour of the orbits prevents the possibility of a
chaotic behaviour. The trajectories that evolve around
one of the critical points cannot go to the other free
region for evolving around the other critical point (we
make reference to the critical points C- and C+ only).
It is clear that the homoclinic bifurcation that precedes
the birth of the chaotic behaviour cannot occur in
region II of the parameter space.\\
There is still another result for region II of the parameter
space~: since the flow, once it has entered one of the two free
regions of the phase space, cannot escape from it,
it can only go to the critical point C- or C+ lying in this
region. So each one of the two free zones in
the phase space is a part of the attraction's basin of
C- or C+.\\
By applying the same method and guided by the
form of the integral of motion {\bf d}, we have found a
new family of
semipermeable surfaces. We do not give here the technical
details of the calculations. They are a little more
complicated than the calculations involved in the previous cases.
The results are as follows. The family of surfaces~:
$S=c_1+c_2 x^2-\frac{1}{4 \sigma} x^4+(2 \sigma-b) x y +
\sigma y^2+x^2 z=0 $ are semipermeable in the two following
cases~:
\begin{itemize}
\item[i)] $ b<6 \sigma-2$, $r<2 \sigma-1$, $c_1 \geq 0$
and $c_2 \in I_{c_2}$, where $I_{c_2}$ is the interval defined
by the two real roots of the quadratic polynomial in $c_2$~:
$4 \sigma^2 c_2^2+ 4 \sigma(b+ 2\sigma-b \sigma+2 r \sigma -
6 \sigma^2)c_2+b^2-4 b \sigma - 6 b^2 \sigma - 4 b r \sigma -
4 b^2 r \sigma + 4 \sigma ^2 +24 b \sigma ^2 +
9 b^2 \sigma ^2 + 8 r \sigma ^2 + 20 b r \sigma ^2 +
4 r^2 \sigma ^2 -24 \sigma^3-36 b \sigma^3 -
24 r \sigma^3 +36 \sigma ^4$. In this case, the sign of the
scalar product $\vec{N} \cdot \vec{T}/S$ is positive. The
canonical values of the Lorenz's parameters do not satisfy the
above two inequalities between $r$, $b$ and $\sigma$.
\item [ii)] $ b>6 \sigma-2$, $r>2 \sigma-1$, $c_1=0$
and $c_2 \in I_{c_2}$.\\ In this case, the sign of
$\vec{N} \cdot \vec{T}/S$ is negative. This family of surfaces
divides the phase space in three regions. Only one of them is
filled by the surfaces of the family. The two free regions are
disconnected and the critical points C+ and C- are located
in different regions. Here, as in one of the cases analysed above,
the homoclinic bifurcation, and hence the chaotic behaviour,
is not possible. The
critical points C+ and C- are stable and each of the two
free regions is part of the basin of attraction of each critical
point.
\end{itemize}
Finally, guided by the form of the first integral {\bf e}, we
have found another family of semipermeable surfaces.
Let us consider the family of surfaces~:
\begin{equation}
S=c_1+\frac{2-3b+b^2-c_2+2 \sigma -b \sigma -2 r \sigma}{2
\sigma}x^2-\frac{1}{4 \sigma}x^4+(2-b)xy+\sigma y^2+
(c_2+x^2)z=0 \label{fact4}
\end{equation}
These surfaces are semipermeable in four different cases~:
\begin{itemize}
\item [$\alpha$ ] $b>2(\sigma+1)$, $\sigma<1$, $c_1 c_2 \leq 0$,
$c_1<\frac{(c_2+k_3)(c_2k_2+b^2k_3)}
{64(\sigma-1)\sigma(\sigma+1)}$; \\ here
the flow is crosssing the surfaces downward.
\item [$\beta$ ] $b>2(\sigma+1)$, $\sigma<1$, $c_1 c_2
\leq 0$, $c_2>\frac{-bk_3}{b-4 \sigma +4}$,
$c_1<\frac{2-3b+b^2-c_2+2 \sigma-b \sigma-2 r \sigma}
{2 \sigma}$; \\ here
the flow is crossing the surfaces downward too.
\item [$\gamma$ ] $b<2(\sigma+1)$, $\sigma>1$, $c_1 c_2 \geq 0$,
$c_1>\frac{(c_2+k_3)(c_2k_2+b^2k_3)}
{64(\sigma-1)\sigma(\sigma+1)}$; \\ here
the flow is crossing the surfaces upward.
\item [$\delta$ ] $b<2(\sigma+1)$, $\sigma>1$, $c_1 c_2 \geq 0$,
$(b-4 \sigma+4)c_2<-bk_3$\\ $c_1>\frac{2-3b+b^2-c_2+2
\sigma-b \sigma-2 r \sigma}{2 \sigma}$; \\ here
the flow is crossing the surfaces upward too.
\item [] The quantities $k_2$ and $k_3$ are given by~:
$k_2=b^2+8b(\sigma-1)+16(1-\sigma^2)$ and
$k_3=2b-b^2-4 \sigma+2 b \sigma+4 r \sigma$.
\end{itemize}
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{fact4-h2.eps}
$$
\caption{Critical points separated by semipermeable
surfaces of type (\ref{fact4}) in case $\alpha$.
The far left (resp. right) zone is a part of the bassin of
attraction of C- (resp. C+).}
\label{fig fact4 h}
\end{figure}
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{fact4-b2.eps}
$$
\caption{Lorenz attractor enclosed by semipermeable surfaces
of type (\ref{fact4}) in case $\gamma$. The form of these
surfaces for $x^2<-c_2$
is a sink, what we cannot see in the figure, which is a projection.
The bolded curve is given by the equality in expression
(\ref{uppest}). The thick dashed curved
is the {\it last} parabola.}
\label{fig fact4 b}
\end{figure}
If $c_2>0$, each surface of the family is connected;
if $c_2<0$, all the surfaces are disconnected~: they are
divided in three parts.
In case $\alpha$, if we take
$r>\frac{1}{4 \sigma} (b-2)(b-2 \sigma)$, $c_2 \in
[-b^2 \frac{k_3}{k_2} \mbox{;} -k_3]<0$ and then $c_1>0$,
we then have disconnected semipermeable surfaces and the
critical points are under the
surfaces and separated by them. This is another configuration
where we know a part of the basin of attraction of each
critical point and where the homoclinic trajectory
cannot exists (see fig. \ref{fig fact4 h}).
The case $\gamma$ gives us information about the space extension
of the chaotic attractor ($r=28$, $\sigma=10$, $b=\frac{8}{3}$).
In this case, when $c_2<0$ and $c_1 \geq 0$, the surfaces
are disconnected and the flow crosses them upward
(see fig. \ref{fig fact4 b}).
The uppermost surface (for $c_1=0$ and $c_2=-k_3$)
is an additional bound for the Lorenz attractor. Hence it lies
entirely in the zone of the phase space where~:
\begin{equation}
{\large z \geq - \frac{\frac{2-3b+b^2-c_2+2 \sigma -b \sigma -2 r \sigma}
{2 \sigma}x^2-\frac{1}{4 \sigma}x^4+(2-b)xy+\sigma y^2}
{x^2-k_3} } \label {uppest}
\end{equation}
(for Lorenz's canonical values $k_3=1131.56$).\\
We have also found several semipermeable families of
ellipsoids. In fact, we have generalised results given in
\cite{doering et gibbon,sparrow,jackson}.
Surfaces like
\begin{equation}
S=\frac{c_3-r}{\sigma} x^2+y^2+(z-c_3)^2=R \label {ellipsoid}
\end{equation}
are semipermeable for the following cases~:
\begin{itemize}
\item $c_3 > r$ for arbitrary $\sigma$, $b$, $r$ and for values
of $R$ as in fig. \ref{fig elli}.
\begin{figure}[htbp]
$$
\epsfxsize=5cm
\epsfbox{elli.eps}
$$
\caption{Parameters for which surfaces (\ref{ellipsoid}) with $c_3 >r$
are semipermeable.}
\label{fig elli}
\end{figure}
The interest of having one free parameter (here $c_3$) in
addition to $R$ is that we may
lower the ellipsoids in phase space with $c_3$ and so restrict
more tightly
the region in which the chaotic attractor lies (with considering
the envelope of all the smallest (with R minimum) ellipsoids
when $c_3$ varies in $]r \mbox{; } + \infty[$). These results contain
known results about ellipsoids and several new ones.
\item if $c_3=r$ then $S$ is the cylinder
which we have studied above.
\item if $c_3<r$ then $S$ is an hyperboloid of
revolution. The revolution axis is $\{z=c_3,y=0\}$.
This surface is semipermeable for
$ b< \sigma \mbox{ and } \sigma >1 $
(a case which includes Lorenz's canonical values)
and $ R \leq \frac{b^2}{4 \sigma(b- \sigma)} {c_3}^2 $
For this case the scalar product is positive $\forall
\; \{x,y\} \in S $.
The flow crosses the last surface $\, \frac{c_3-r}
{\sigma}x^2+y^2+(z-c_3)^2=\frac{b^2}{4 \sigma(b-
\sigma)} {c_3}^2 $
outwards. This new information sharpens the bounding frontiers
of the attractor as we can see in fig. \ref{fig cone z c3}.
\end{itemize}
\begin{figure}[htbp]
$$
\epsfxsize=6cm
\epsffile{cone3d.eps}
$$
\caption{Lorenz's attractor and a semipermeable surface
of the family (\ref{ellipsoid}).
When we consider all the surfaces of the family, stronger
bounds on the location of the attractor are obtained.}
\label{fig cone z c3}
\end{figure}
Sparrow \cite{sparrow} has conjectured that all
trajectories of the Lorenz system eventually enter and
remain in the region $z \geq 0$ for all parameters values
$r$, $\sigma$ and $b$ (note that the plane $z=0$ is not
semipermeable). Sparrow proved this conjecture for the
case $b \leq \sigma +1$ by using the method of Lyapunov
functions.
The existence of semipermeable parabolas $z=\frac{x^2}
{2 \sigma}+a_0$ for
$a_0<0$ and $b<2 \sigma$ also proves the conjecture, but
for different values of parameters $\sigma$ and b. Indeed,
in this case the flow is crossing all
the parabolas upward. It doesn't mean that the $z=0$
plane itself is semipermeable, but the flow has to
end up with crossing upward
the last parabola $z=\frac{x^2}{2 \sigma}$. The
parabolas are in fact
driving the flow to the phase space zone where
$z>\frac{x^2}{2 \sigma} \geq 0$.
The cylinders family~: $y^2+(z-r)^2=R$ with $R \geq r^2$
and $b \leq 2$ also drives the flow inside the
{\it smallest}
cylinder ($R=r^2$) which lies in the $z \geq 0$
phase region.
The surfaces (\ref{fact4}), in case $\gamma$,
tell us also that the flow eventually crosses the uppermost surface
(given by the equality in (\ref{uppest})) upward. This
surface lies entirely in the $z \geq 0$ half space
(for $x^2 < -c_2$). The results of this work expand the
region of the parameter space for which the flow
eventually enters the zone of phase space with $z>0$.
This region is shown in fig. \ref{fig z plan}.\\
\begin{figure}[htbp]
$$
\epsffile{z-plan.eps}
$$
\caption{Range of parameters $\sigma$ and b for which the
flow eventually enters
the phase space zone $z>0$ ($\forall r$)}
\label{fig z plan}
\end{figure}
In conclusion, inspired by the integrals of motion that
exist for particular values of the parameters $b$, $\sigma$
and $r$, we were able to find several families of
surfaces, all crossed in the same direction by the
flow associated to the system.
From these results, we have deduced a rich quantity of
information about the geometrical location of the global
attractor of the system. This information is more restrictive
than similar results that had been found by the method
of Lyapunov functions. When compared to the Lyapunov technique,
we see that the fundamental advantage of this new method
is that one now has to study functions with one less variable.
Moreover, we have obtained information
about the spread of the basin of attraction of the
critical points C- and C+ when they are stable. We have
also determined regions of the parameter space where the
chaotic behaviour is not possible.
It is clear that the method used in this paper can be
applied to other 3-d dissipative dynamical systems that
the Lorenz one. We have choosen the latter owing to
the great importance that this system has played
in the study of chaotic dynamics.
\clearpage
|
1,314,259,995,807 | arxiv |
\section{Introduction}
For an integral polynomial $f$ and an odd prime $p$, we denote by $\mathfrak{R}_p(f)$ the image set of $f$ in $\mathbb{F}_p$, the finite field of order $p$. Many properties of $\mathfrak{R}_p(f)$ have been well-studied if $f$ is of small degree. For example, it is well-known that $|\mathfrak{R}_p(f)|=(p+1)/2$ if $f$ is quadratic. Following the work of von Sterneck \cite{vonsterneck} and Kantor \cite{kantor} in the early 20th century, the size of the image set of a cubic polynomial was determined: if $p>3$ is prime, then
\begin{align*}
\left| \mathfrak{R}_p(x^3 + ax^2 + bx + c) \right| &= \begin{cases}
p & a^2 - 3b=0,\ p\equiv -1 \pmod{3}; \\
\frac{p+2}{3} & a^2-3b = 0,\ p\equiv 1\pmod{3}; \\
\frac{2p-1}{3} & a^2-3b\ne 0,\ p\equiv -1 \pmod{3}; \\
\frac{2p+1}{3} & a^2-3b\ne 0,\ p\equiv 1\pmod{3}.
\end{cases}
\end{align*}
For quartic and higher degree polynomials, not as much is known. Sun \cite{sun} investigated $|\mathfrak{R}_p(f)|$ for quartic polynomials $f$ with no cubic term, Chou, Gomez-Calderon, and Mullen \cite{CGM} established $|\mathfrak{R}_p(f)|$ for Dickson polynomials $f$ (we will discuss this in greater detail later), and Cusick \cite{cusick} investigated an infinite family of polynomials over a finite field of characteristic $2$. Uchiyama \cite{uchiyama} provided sufficient conditions for a polynomial $f$ to satisfy the lower bound $|\mathfrak{R}_p(f)|>p/2$, but noted that this does not hold in general. Just a few years later, Birch and Swinnerton-Dyer made Uchiyama's bound more precise \cite{BSD}, and this was further improved upon by Voloch \cite{voloch}. Probabilistic methods over finite fields allow one to ask about the ``average'' value of $|\mathfrak{R}_p(f)|$, varying over polynomials of degree $n$, and this has proven to be a fruitful direction of research (see for example \cite{cohen,knopfmacher}), however ascertaining the value $|\mathfrak{R}_p(f)|$ for arbitrary polynomials still appears intractable at the time of writing.
Another interesting property of $\mathfrak{R}_p(f)$ is the \emph{residue sum}, denoted by $S_p(f)$, defined to be the sum of the elements of $\mathfrak{R}_p(f)$ in $\mathbb{F}_p$. Gauss \cite{Gauss} first proved that $S_p(x^2)=0$. Considering $f(x)=x^2$ as a special case of polygonal numbers, it is natural to investigate the residue sum of triangular numbers modulo $p$, which was shown to be $-16^{-1}$ in $\mathbb{F}_p$ by Stetson \cite{Stetson} in 1904. In other words, Stetson showed that
\[
S_p\left(\frac{x(x+1)}{2}\right)=-\frac{1}{16}.
\]
This result was later generalized by Gross, Harrington, and Minott \cite{GHM}, who computed for $a\not\equiv 0\pmod{p}$ that
\[
S_p\left(ax^2+bx+c\right)=-\frac{b^2-4ac}{8a}.
\]
We observe that the residue sums of quadratic polynomials are invariant for all odd primes $p$. Such is not true if $f$ has a higher degree. Finch-Smith, Harrington, and Wong \cite{FHW} showed that if $a\not\equiv 0\pmod{p}$ where $p>3$ is an odd prime, then
$$S_p\left(ax^3+bx^2+cx+d\right)=\begin{cases}
\dfrac{27a^2d-9abc+2b^3}{81a^2}\quad&\text{if }b^2\neq3ac\text{ and }p \equiv 1 \pmod{3};\vspace{3pt}\\
-\dfrac{27a^2d-9abc+2b^3}{81a^2}&\text{if }b^2\neq3ac\text{ and }p \equiv -1 \pmod{3};\vspace{3pt}\\
\dfrac{2\left(27a^2d-9abc+2b^3\right)}{81a^2}\quad&\text{if }b^2=3ac\text{ and }p \equiv 1 \pmod{3};\vspace{3pt}\\
0&\text{if }b^2=3ac\text{ and }p\equiv -1 \pmod{3}.
\end{cases}$$
While these results are interesting, the residue sums above have only been investigated for low-degree polynomials. In this article, we study $\mathfrak{R}_p(f)$ and $S_p(f)$ when $f$ is a \textit{Dickson polynomial}, which is an infinite family of polynomials with degrees that are arbitrarily large.
\begin{definition}
For a nonzero integer $a$, the Dickson polynomials $D_n(x,a)$ for $n\geq 0$ are defined recursively by $D_0(x,a)=2$, $D_1(x,a)=x$, and $D_n(x,a) = x D_{n-1}(x,a) - a D_{n-2}(x,a)$ for $n\ge 2$.
\end{definition}
The Dickson polynomials are ubiquitous in algebra and number theory. They are closely related to the \textit{Chebyshev polynomials} $T_n(x)$, and when $a=-1$, we recover the \textit{Lucas polynomials} $L_n(x)=D_n(x,-1)$. The Lucas polynomials are a ``polynomialization'' of the famous sequence of Lucas numbers, where the $n$th Lucas number can be obtained as $L_n(1)$.
As an illustrative example of residue sums, consider the Lucas polynomials at the prime $p=7$ as in Table~\ref{tab:table1}.
\begin{table}[h]\label{tab:table1}
\caption{Investigation of $S_7(L_n)$ for $1\le n\le 40$.}
\par\noindent\rule{0.7\textwidth}{0.4pt} \\
\vspace{0.2em}
\begin{tabular}{c | c}
$n$ & $S_7(L_n)$ \\
\midrule
1 & 0 \\
2 & 1 \\
3 & 0 \\
4 & 1 \\
5 & 0 \\
6 & 2 \\
7 & 0 \\
8 & 6
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(L_n)$ \\
\midrule
9 & 0 \\
10 & 1 \\
11 & 0 \\
12 & 2 \\
13 & 0 \\
14 & 1 \\
15 & 0 \\
16 & 1
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(L_n)$ \\
\midrule
17 & 0 \\
18 & 2 \\
19 & 0 \\
20 & 1 \\
21 & 0 \\
22 & 1 \\
23 & 0 \\
24 & 0
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(L_n)$ \\
\midrule
25 & 0 \\
26 & 1 \\
27 & 0 \\
28 & 1 \\
29 & 0 \\
30 & 2 \\
31 & 0 \\
32 & 1
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(L_n)$ \\
\midrule
33 & 0 \\
34 & 1 \\
35 & 0 \\
36 & 2 \\
37 & 0 \\
38 & 1 \\
39 & 0 \\
40 & 6
\end{tabular}
\centering
\end{table}
We remark that the residue sum $S_p(L_n)$ has a very limited number of possible values. Shockingly, this is not a property that is special to the case $p=7$. As a consequence of our study on the Dickson polynomials, we can provide a complete classification of $S_p(L_n)$ for all odd primes $p$ and positive integers $n$ which shows that $S_p(L_n)\in\{-1,0,1,2\}$. In fact, the following theorem implies that $S_p(D_n(x,a))\in \{-2a^{n/2},-a^{n/2},0,a^{n/2},2a^{n/2}\}$. As all our results hold for finite fields of odd characteristic, we state them in that generality where $q=p^k$ for $p$ an odd prime.
We denote by $\legendre{\cdot}{q}$ the quadratic character over $\mathbb{F}_q$, that is the multiplicative function defined by
\begin{equation}\label{eqn:quadratic-character}
\begin{aligned}
\legendre{a}{q} := a^{\frac{q-1}{2}}= \begin{cases}
0 & \mathrm{if\ }a=0\in\mathbb{F}_q; \\
1 & \mathrm{if\ }a\ \mathrm{is\ a\ quadratic\ residue\ in\ } \mathbb{F}_q; \\
-1 & \mathrm{if\ }a\ \mathrm{is\ not\ a\ quadratic\ residue\ in\ } \mathbb{F}_q.
\end{cases}
\end{aligned}
\end{equation}
This function generalizes the Legendre symbol over a field of prime order. Furthermore, it is natural to extend the definitions of $\mathfrak{R}_p$ and $S_p$ to the field of order $q$ and denote these by $\mathfrak{R}_q$ and $S_q$.
\begin{theorem}\label{thm:S-sum} Let $a$ be an integer, $n$ be a nonnegative integer, and $q$ be an odd prime power such that $a\neq 0 \in \mathbb{F}_q$. Let $d=\gcd(n,q-1)$ and $\delta = \gcd(n,q+1)$, and let $r$ be the highest power of 2 dividing $q^2-1$. Then the sum of the elements in the image of the Dickson polynomials is\footnote{When $(q-1)\mid n$ we remark that $\legendre{a}{q}^{n/d} a^{n/2} = a^n$, while when $(q+1)\mid n$ we have that $\legendre{a}{q}^{n/\delta}a^{n/2} = a^{n-n/\delta}$. We keep it in the stated form to highlight the symmetry.}
\begin{align*}
S_q(D_n(x,a)) &= \begin{cases}
0 & 2^{r-1}\mid n; \\
-\legendre{a}{q}^{\frac{n}{d}+\frac{n}{\delta}} a^{n/2} & \text{else},
\end{cases}
+ \begin{cases}
\legendre{a}{q}^{\frac{n}{d}}a^{n/2} & (q-1)\mid n; \\ 0 & \text{else},
\end{cases}
+ \begin{cases} \legendre{a}{q}^{\frac{n}{\delta}}a^{n/2} & (q+1)\mid n; \\
0 & \text{else}.
\end{cases}
\end{align*}
\end{theorem}
\begin{corollary} In the situation above, we have five possibilities for the residue sum, as $n$ and $a$ vary over all integers, and $q$ over all odd prime powers
\[
S_q(D_n(x,a)) \in \left\{ 0, \pm a^{n/2}, \pm 2a^{n/2} \right\}.
\]
\end{corollary}
This demonstrates the first non-trivial classification for $S_q(f)$ where $f$ varies over an infinite family of polynomials of unbounded degree. In the process of proving this theorem, we provide a complete characterization of the size of $\mathfrak{R}_q(D_n(x,a))$ for all $n$, $a$, and odd prime powers $q$, which is the main result of Chou, Gomez-Calderon, and Mullen \cite[Theorem 10]{CGM} for odd characteristic.
\begin{theorem}
Let $p$ be an odd prime power, $n$ an even natural number, $d=\gcd(q-1,n)$, $\delta=\gcd(2(q+1),n)$, and $2^r$ the highest power of 2. Then the size of the value set of the $n$th Dickson polynomial over $\mathbb{F}_q$ is
\begin{align*}
|\mathfrak{R}_q(D_n(x,a))|= \floor{\frac{q-1}{2d}}+\floor{\frac{q+1}{2\delta}}+1+\begin{cases}
1
& \mathrm{if\ }\legendre{a}{q}=-1 \mathrm{\ and\ } 2^{r-1}\mid\mid n; \\
0 & \mathrm{otherwise}.\end{cases}
\end{align*}
\end{theorem}
\subsection*{Acknowledgements}
The first named author is supported by an NSF Graduate Research Fellowship (DGE-1845298).
\section{Preliminaries}
In this section, we present some preliminary results and notation that will be useful in our investigation.
\begin{notation} Throughout this paper we will fix $p$ to be an odd prime, and $q$ to be some power of it, defining a finite field $\mathbb{F}_q$. We fix a primitive $(q^2-1)$st root of unity $\zeta_{q^2-1}$ to be a generator of the group of units $\mathbb{F}_{q^2}^\times$. For each positive factor $m$ of $q^2-1$, let $\zeta_m=\zeta_{q^2-1}^{(q^2-1)/m}$. In particular, $\zeta_{q-1}\in\mathbb{F}_q^\times$ is a primitive $(q-1)$st root of unity. When we consider a Dickson polynomial over $\mathbb{F}_q$ and $a$ nonzero in $\mathbb{F}_q$, let $A$ be the smallest positive integer such that $a=\zeta_{q-1}^A$. Finally, we denote by $S_q(f)$ the residue sum of an integral polynomial $f(x)$ over the finite field $\mathbb{F}_q$.
\end{notation}
\subsection{Dickson polynomials}
Using standard methods of solving recurrence relations, one can show that the Dickson polynomials admit a Binet formula expansion:
\begin{align}\label{eq:Ln}
D_n(x,a) = \omega(x,a)^n + \overline{\omega}(x,a)^n,
\end{align}
where
\begin{align*}
\omega(x,a) = \frac{x+\sqrt{x^2-4a}}{2}\quad\text{and}\quad\overline{\omega}(x,a) = \frac{x-\sqrt{x^2-4a}}{2}.
\end{align*}
Using the expressions for $\omega$ and $\overline{\omega}$, we make note of the following properties:
\begin{equation}\label{eqn:omega-properties}
\begin{aligned}
x &= \omega(x,a) + \overline{\omega}(x,a), \\
a &= \omega(x,a)\overline{\omega}(x,a).
\end{aligned}
\end{equation}
Since $a \neq 0$, from this expression we see $\overline{\omega}(x,a) = a\omega(x,a)^{-1}$.
\begin{example}
One may check that the small index Dickson polynomials are given by
\allowdisplaybreaks
\begin{align*}
D_0(x,a) &= 2 &D_4(x,a) &=x^4+4x^2a + 2a^2 \\
D_1(x,a) &= x &D_5(x,a) &= x^5 + 5x^3a + 5xa^2\\
D_2(x,a) &= x^2 - 2a &D_6(x,a) &= x^6 - 6x^4a +9x^2a^2 -2a^3\\
D_3(x,a) &= x^3 - 3xa &D_7(x,a) &=x^7-7x^5a+14x^3a^3-7xa^3.
\end{align*}
\end{example}
\begin{proposition}\label{prop:n-odd-case}
If $n$ is odd, then $S_q(D_n(x,a))=0$.
\end{proposition}
\begin{proof}
It follows from the recursive definition of $D_n(x,a)$ that if $n$ is odd, then $D_n(x,a)$ is an odd polynomial. Consequently, if $y\in\mathfrak{R}_q(D_n(x,a))$, then $-y\in\mathfrak{R}_q(D_n(x,a))$. Since $p$ is odd, $y\not\equiv -y$ in $\mathbb{F}_q$, and we deduce that $S_q(D_n(x,a))=0$.
\end{proof}
\begin{proposition}\label{prop:a0modp} Suppose that $a\equiv 0$ in $\mathbb{F}_q$. Then we have that
\begin{align*}
S_q(D_n(x,a)) &= \begin{cases} 1 & n=q-1; \\
0 & \text{else}. \end{cases}
\end{align*}
\end{proposition}
\begin{proof} Via the recursive relation of the Dickson polynomials, we have that $D_n(x,a) \equiv x^n$ whenever $a$ vanishes over $\mathbb{F}_q$. From this the problem reduces to summing $n$th powers over a finite field.
\end{proof}
\begin{assumption}\label{ass:ass} As \autoref{prop:n-odd-case} determines the residue sum $S_q(D_n(x,a))$ for all odd $n$, for the remainder of this paper, we will make the standing assumption that $n$ is even. Additionally \autoref{prop:a0modp} determines the case where $a\equiv 0 \in \mathbb{F}_q$, so we will assume without loss of generality that $a\not\equiv 0 \in \mathbb{F}_q$. Finally we will make the standing assumption that $q\ne 3$, however one may check by direct computation that \autoref{thm:S-sum} holds when $q=3$.
\end{assumption}
Using $\overline{\omega}(x,a)=a\omega(x,a)^{-1}$ , we simplify \autoref{eq:Ln} to
\begin{align*}
D_n(x,a) = \omega(x,a)^n + a^n\omega(x,a)^{-n},
\end{align*}
and we exploit this form of $D_n$ to prove the following proposition and other results throughout the paper.
\begin{proposition}\label{prop:dickson-poly-is-2-iff-omegan-is-1} Let $x,y\in\mathbb{F}_q$ be arbitrary. Then $D_n(x,a) = D_n(y,a)$ if and only if $\omega(x,a)^n = \omega(y,a)^{n}$ or $\omega(x,a)^n=\overline{\omega}(y,a)^n=a^n\omega(y,a)^{-n}$.
\end{proposition}
\begin{proof} Suppose that $D_n(x,a) = \omega(x,a)^n + a^n\omega(x,a)^{-n} = \omega(y,a)^n + a^n\omega(y,a)^{-n} = D_n(y,a)$. By multiplying both sides of the equation by $\omega(x,a)^n$ and rearranging the terms, we have that
\begin{align*}
\omega(x,a)^{2n} - \left(\omega(y,a)^n + a^n\omega(y,a)^{-n}\right)\omega(x,a)^n + a^n &= 0.
\end{align*}
That is, $\omega(x,a)^n$ is a solution of the polynomial $$t^2 - \left(\omega(y,a)^n + a^n\omega(y,a)^{-n}\right)t + a^n = \left(t-\omega(y,a)^n\right)\left(t-a^n\omega(y,a)^{-n}\right).$$
\end{proof}
\begin{proposition} \cite[Lemma~7]{CGM} Let $x\in \mathbb{F}_q$. Then we have that $\omega(x,a)^n = \bar{\omega}(x,a)^n$ if and only if $D_n(x,a) = \pm 2a^{n/2}$.
\end{proposition}
\begin{corollary}
For any $x\in \mathbb{F}_q$, we have that $D_n(x,a) = \pm2 a^{n/2}$ if and only if $\omega(x,a)^n = \pm a^{n/2}$.
\end{corollary}
\subsection{Hyperbolic, elliptic, and parabolic elements}
\begin{notation}
We partition $\mathbb{F}_q$ into three subsets, denoted by
\begin{align*}
\mathcal{H}_q(a) & = \left\{ x\in\mathbb{F}_q \colon \chi_q(x^2-4a)=1 \right\},\\
\mathcal{E}_q(a) &= \left\{x\in\mathbb{F}_p \colon \chi_q(x^2-4a)= -1 \right\},\\
\mathcal{P}_q(a) &= \left\{x\in\mathbb{F}_p \colon \chi_q(x^2-4a)=0 \right\}.
\end{align*}
We will refer to elements of $\mathcal{H}_q(a)$, $\mathcal{E}_q(a)$, and $\mathcal{P}_q(a)$ as \emph{hyperbolic}, \emph{elliptic}, and \emph{parabolic}, respectively. This terminology is inspired by the work of Bourgain, Gamburd, and Sarnak \cite{bgs} on showing the connectivity of the Markoff mod $p$ graphs.
\end{notation}
Our understanding of $\mathfrak{R}_q(D_n(x,a))$ will come from investigating the images of these three sets under the map $D_n$. We will denote by $D_n(\mathcal{H}_q(a),a)\subseteq\mathbb{F}_q$ the image set of the hyperbolic elements under the Dickson polynomial, and similarly for the elliptic and parabolic sets. In order to compute the residue sum $S_q(D_n(x,a))$, it will suffice to have a handle on these three image sets as well as their potential overlaps.
\begin{remark}\label{rmk:preliminary-observations}
We note here some preliminary observations about the quantity $\omega(x,a)$.
\begin{enumerate}
\item\label{rmk:preliminary-observations(1)} If $x\in \mathcal{H}_q(a)$, then $\omega(x,a)$ is an element of $\mathbb{F}_q$. In particular, $\omega(x,a)^{q-1} =1$, thus we may write $\omega(x,a) = \zeta_{q-1}^c$ for some $c$, where $\zeta_{q-1}$ is our fixed primitive $(q-1)$st root of unity.
\item If $x\in \mathcal{E}_q(a)$, then $\omega(x,a)$ is an element of $\mathbb{F}_{q^2}$ but not an element of $\mathbb{F}_q$. Thus we have that $\omega(x,a)^{q^2-1} = 1$.
\item Observe that $\mathcal{P}_q(a)$ is nonempty if and only if $\legendre{a}{q}=1$, where we recall that $\chi_q$ is the quadratic character as in \autoref{eqn:quadratic-character}. In this situation, if $x\in \mathcal{P}_q(a)$, then $\omega(x,a) = x/2$ is an element of $\mathbb{F}_q$. Moreover, we have that $x=\pm\sqrt{4a}=\pm2\sqrt{a}$. Now, since $n$ is even,
\begin{align*}
D_n(x,a) &= \omega(x,a)^n + \overline{\omega(x,a)}^{n} = \left( \frac{\pm 2\sqrt{a}}{2} \right)^n + \left( \frac{\pm 2\sqrt{a}}{2} \right)^{n} = 2 a^{n/2}.
\end{align*}
\end{enumerate}
\end{remark}
We now establish the following property of elliptic elements.
\begin{proposition}\label{prop:omega-properties-for-elliptic-elts} For all elliptic elements $x\in\mathcal{E}_q(a)$, we have that $\omega(x,a)^{q+1}= a \in \mathbb{F}_q$. In particular, we have that $\omega(x,a)=\zeta_{q^2-1}^{A+k(q-1)}$ in $\mathbb{F}_{q^2}$ for some integer $k$.
\end{proposition}
\begin{proof}
Via the freshman's dream, we may write $\omega(x,a)^q$ as
\begin{align*}
\omega(x,a)^q &= \frac{x + \left( \sqrt{x^2- 4a} \right)^q}{2},
\end{align*}
and we observe that
\begin{align*}
\overline{\omega}(x,a) &= \frac{x - \sqrt{x^2-4a}}{2}.
\end{align*}
As $x$ is not parabolic, the quantity $x^2 - 4a$ is nonvanishing, thus we have that $\left( x^2 - 4a \right)^{q-1} = 1$ in $\mathbb{F}_q$. As $\sqrt{x^2-4a}$ is not defined over $\mathbb{F}_q$, it is not fixed by the Frobenius endomorphism on $\mathbb{F}_{q^2}$.
This implies that $\left( x^2-4a \right)^{\frac{q-1}{2}} = -1$ in $\mathbb{F}_{q^2}$. Thus we see that
\begin{align*}
\omega(x,a)^q &= \frac{x + \left( \sqrt{x^2-4a} \right)^q}{2} = \frac{x + \left( x^2-4a \right)^{\frac{q-1}{2}} \sqrt{x^2 - 4a}}{2} = \frac{x - \sqrt{x^2-4a}}{2} = \overline{\omega}(x,a).
\end{align*}
Therefore $\omega(x,a)^{q+1} = \omega(x,a)\overline{\omega}(x,a)=a$.
By \autoref{rmk:preliminary-observations}, we have that $\omega(x,a) = \zeta_{q^2-1}^c$ for some $c$. From the observation that $\omega(x,a)^{q+1} = a = \zeta_{q-1}^A = \zeta_{q^2-1}^{(q+1)A}$, we must have that $c(q+1) \equiv A(q+1) \pmod{q^2-1}$. That is, $c = A + k(q-1)$ for some integer $k$, which we may assume to lie in the range $1\le k \le q+1$, since we only care about the residue of $c$ modulo $q^2-1$.
\end{proof}
We can now state explicitly what each set in the partition of $\mathbb{F}_q$ looks like:
\begin{proposition}\label{prop:hyp-ell-para-sets-description}
The hyperbolic, elliptic, and parabolic sets over the finite field $\mathbb{F}_q$ are given by:
\allowdisplaybreaks
\begin{align*}
\mathcal{H}_q(a) &= \OB{\zeta_{q-1}^c + \zeta_{q-1}^{A-c} \colon 1\le c\le q-1\ \mathrm{and }\ 2c\not\equiv A \pmod{q-1} }, \\
\mathcal{E}_q(a) &= \OB{\zeta_{q^2-1}^{A+k(q-1)}+\zeta_{q^2-1}^{Aq-k(q-1)} \colon 1\le k\le q+1\ \mathrm{and}\ 2k \not\equiv A \pmod{q+1} }, \\
\mathcal{P}_q(a) &= \begin{cases}
\OB{\pm 2 a^{1/2}} & \mathrm{if }\ \legendre{a}{q}=1; \\
\emptyset & \mathrm{if }\ \legendre{a}{q}=-1.
\end{cases}
\end{align*}
\end{proposition}
\begin{proof}
For $x$ hyperbolic, we know that $\omega(x,a) = \zeta_{q-1}^c$ for some $c$ by \autoref{rmk:preliminary-observations}. We should see for which $c$ we are getting hyperbolic elements. Since
\begin{align*}
\sqrt{x^2-4a} = \omega(x,a) - \bar{\omega}(x,a),
\end{align*}
we can check whether this quantity is defined over $\mathbb{F}_q$ (meaning that $x^2-4a$ is a residue). This is equivalent to checking that it is fixed under the Frobenius. Note that
\begin{align*}
\omega(x,a) - \bar{\omega}(x,a) = \zeta_{q-1}^c - \zeta_{q-1}^{A-c}.
\end{align*}
Applying the Frobenius, we see
\begin{align*}
\left( \zeta_{q-1}^c - \zeta_{q-1}^{A-c} \right)^q &= \zeta_{q-1}^{qc} - \zeta_{q-1}^{q(A-c)} = \zeta_{q-1}^{c} - \zeta_{q-1}^{A-c}.
\end{align*}
Thus for any $c$, we have that $\zeta_{q-1}^c + \zeta_{q-1}^{A-c}$ gives an element for which $\sqrt{x^2-4a} \in \mathbb{F}_q$. We should verify that it is not accidentally producing a parabolic element, i.e. that we are not accidentally getting $\sqrt{x^2-4a} = 0$. This would occur for some $c$ if
\begin{align*}
\sqrt{x^2-4a} &= \zeta_{q-1}^c - \zeta_{q-1}^{A-c} = 0,
\end{align*}
that is, if $2c\equiv A \pmod{q-1}$.
For elliptic elements, we want to verify that $\sqrt{x^2-4a}$ is not defined over $\mathbb{F}_q$, equivalently that it is not fixed under the Frobenius. So we want to throw out any $k$ for which
\begin{align*}
\left( \zeta_{q^2-1}^{A + k(q-1)} - \zeta_{q^2-1}^{Aq-k(q-1)} \right) &= \left( \zeta_{q^2-1}^{A + k(q-1)} - \zeta_{q^2-1}^{Aq-k(q-1)} \right)^q.
\end{align*}
This would give us the equality
\begin{align*}
\zeta_{q^2-1}^{A + k(q-1)} - \zeta_{q^2-1}^{Aq-k(q-1)} &= \zeta_{q^2-1}^{Aq + k(q^2-q)} - \zeta_{q^2-1}^{Aq^2-k(q^2-q)} \\
&= \zeta_{q^2-1}^{Aq + k(1-q)} - \zeta_{q^2-1}^{A - k(1-q)}.
\end{align*}
Rearranging, we see that this is the same as
\begin{align*}
2\zeta_{q^2-1}^{A+k(q-1)} &= 2\zeta_{q^2-1}^{Aq - k(q-1)}.
\end{align*}
Since $2$ is invertible in $\mathbb{F}_{q^2}$ we are left with the congruence
\begin{align*}
A + k(q-1) \equiv Aq - k(q-1) \pmod{q^2-1}.
\end{align*}
This is equivalent to $2k \equiv A \pmod{q+1}$. So we must omit these $k$'s out in order to ensure we are getting an elliptic element.
\end{proof}
We will be interested in the images of the hyperbolic, elliptic, and parabolic sets under the Dickson polynomial $D_n(x,a)$. In particular if we can understand the images over these sets, as well as their potential intersection, then we can completely understand $\text{im}(D_n(x,a))$.
\begin{lemma}\label{lem:CGM} \cite[Lemma~8]{CGM} Let $x,y\in \mathbb{F}_q^\times$, and let $x = u + a/u$ and $y = v + a/v$, where $u \in \mathbb{F}_q^\times$, and $v \in \mathbb{F}_{q^2}^\times$ so that $v^{q+1} = a$. Then if $u^n = v^n$ for some $n\ge 0$, this implies that
\begin{align*}
u^n = a^n/u^n = v^n = a^n/v^n.
\end{align*}
In particular they are all equal to $a^{n/2}$ or $-a^{n/2}$.
\end{lemma}
This result allows us to restrict the values of any possible overlap in the hyperbolic and elliptic images.
\begin{proposition}\label{prop:possible-hyp-ellip-overlap} There are only two possible values for the intersection of the hyperbolic and elliptic images, namely
\begin{align*}
D_n \left( \mathcal{H}_q(a),a \right) \cap D_n \left( \mathcal{E}_q(a) ,a\right) \subseteq \left\{ \pm 2a^{n/2} \right\}.
\end{align*}
\end{proposition}
\begin{proof}
Suppose that $x\in \mathcal{H}_q(a)$ and $y\in \mathcal{E}_q(a)$ so that $D_n(x,a) = D_n(y,a)$. Then there are some $c$ and $k$ for which
\begin{align*}
\omega(x,a)^n &= \zeta_{q-1}^{nc} \\
\omega(y,a)^n &= \zeta_{q^2-1}^{n(A+k(q-1)}.
\end{align*}
In order to have $D_n(x,a) = D_n(y,a)$, by \autoref{prop:dickson-poly-is-2-iff-omegan-is-1} we have that $\omega(x,a)^n = \omega(y,a)^n$ or $\omega(x,a)^n = \bar{\omega}(y,a)^n$.
In the first case, suppose that $\omega(x,a)^n = \omega(y,a)^n$. Since $y$ is elliptic, we have that $\omega(y,a)^{q+1} = a$ by \autoref{prop:omega-properties-for-elliptic-elts}. Therefore by invoking \autoref{lem:CGM} using $u = \omega(x,a)$ and $v = \omega(y,a)$, we have that $\omega(x,a)^n = \omega(y,a)^n =\pm a^{n/2}$. In particular this implies that
\begin{align*}
D_n(x,a) = D_n(y,a) = \pm 2a^{n/2}.
\end{align*}
In the latter case, if $\omega(x,a)^n = \bar{\omega}(x,a)^n$, we can observe that
\begin{align*}
\bar{\omega}(y,a)^{q+1} &= \frac{a^{q+1}}{\omega(y,a)^{q+1}} = \frac{a^{q+1}}{a} = a^q = a.
\end{align*}
Invoking \autoref{lem:CGM} with $v = \bar{\omega}(y,a)$, we have then that
\begin{align*}
\omega(x,a)^n = \bar{\omega}(y,a)^n = \pm a^{n/2},
\end{align*}
and therefore $D_n(x,a) = D_n(y,a) = \pm 2a^{n/2}$.
\end{proof}
\section{Evaluation of the residue sum}
As remarked earlier, our strategy for studying the residue sum $S_q(D_n(x,a))$ will be to investigate the sum over the hyperbolic, elliptic, and parabolic sets, as well as over their overlaps. To this end, we introduce some new notation: for any subset $B \subseteq \mathbb{F}_q$, we denote by $S_q^B(D_n(x,a))$ the sum over the distinct elements in $D_n(B,a)$. If $C \subseteq \mathbb{F}_q$ is another subset, we denote by $S_q^{B, C}(D_n(x,a))$ the sum over distinct elements of $D_n(B,a)\cap D_n(C,a)$, and we have similar notation for triple intersections. In this notation, the total sum will be computed as
\allowdisplaybreaks
\begin{align*}
S_q(D_n(x,a)) &= S_q^{\mathcal{H}_q(a)}(D_n(x,a)) + S_q^{\mathcal{E}_q(a)}(D_n(x,a)) + S_q^{\mathcal{P}_q(a)}(D_n(x,a)) \\
& \quad - S_q^{\mathcal{H}_q(a), \mathcal{E}_q(a)}(D_n(x,a)) - S_q^{\mathcal{H}_q(a), \mathcal{P}_q(a)}(D_n(x,a)) \\
& \quad - S_q^{\mathcal{E}_q(a), \mathcal{P}_q(a)}(D_n(x,a)) + S_q^{\mathcal{H}_q(a), \mathcal{E}_q(a), \mathcal{P}_q(a)}(D_n(x,a)).
\end{align*}
Our preliminary observations about the quantities $\omega(x,a)$ as $x$ varies over the hyperbolic and elliptic sets indicate that elements in $D_n(\mathcal{H}_q(a),a)$ and $D_n(\mathcal{E}_q(a),a)$ will be able to be characterized using roots of unity defined over $\mathbb{F}_q$ or its quadratic extension $\mathbb{F}_{q^2}$.
\begin{notation}\label{nota:d-delta} We will see that the residue sums $S_q(D_n(x,a))$ in \autoref{thm:S-sum} depend upon various properties of $n$, in particular the highest power of 2 dividing $n$ and the order of $n$ in $\mathbb{F}_q^\times$ and $\mathbb{F}_{q^2}^\times$ (which relates to divisors shared between $n$ and $q-1$ and $q+1$). To that end, we fix some notation:
\begin{align*}
d &:= \gcd(n,q-1) \qquad m := \frac{n}{d}\\
\delta &:= \gcd\left(n,q+1\right) \qquad \mu := \frac{n}{\delta}.
\end{align*}
We will also let $2^h$ denote the highest power of 2 dividing $q-1$, $2^\ell$ denote the highest power of $2$ dividing $q+1$, and $2^r$ the highest power of 2 dividing $q^2-1$.
\end{notation}
We remark the following relationship between $2^r$ and the divisors $d$ and $\delta$ which will come in handy throughout our computations.
\begin{proposition}\label{prop:r-d-delta-relationship} Let $d$, $\delta$, $h$, $\ell$, and $r$ be as in \autoref{nota:d-delta}
\begin{enumerate}
\item We have $\frac{q-1}{d}$ is odd if and only if $2^h\mid n$.
\item We have $\frac{q+1}{\delta}$ is odd if and only if $2^\ell \mid n$.
\item Both $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ are odd if and only if $2^{r-1} \mid n$.
\item Both $\frac{n}{d}$ and $\frac{n}{\delta}$ are even if and only if $2^r \mid n$.
\end{enumerate}
\end{proposition}
\begin{proof} (1) and (2) follow directly from the definition of $h$ and $\ell$.
As for (3), we notice that one of $\frac{q+1}{2}$ or $\frac{q-1}{2}$ will be odd, and therefore $h$ and $\ell$ cannot both be strictly greater than one. In particular, this tells us that $\max \left\{ h,\ell \right\} = h+\ell-1 = r-1$, from which the result follows.
For the forward direction of (4), let $2^s \mid\mid n$. Then $\frac{n}{d}$ even implies that $s>h$ and $\frac{n}{\delta}$ even implies that $s>\ell$. In particular $s>\max\{h,\ell\} = r-1$, and hence $s\ge r$.
For the backwards direction of (4), if $2^r\mid n$, then since $r = h+\ell$ and $h,\ell\ge 1$, we have that $2^{h+1}\mid n$ and $2^{\ell+1}\mid n$, implying that both $\frac{n}{d}$ and $\frac{n}{\delta}$ are even.
\end{proof}
It will also benefit us to record some parity constraints that can occur on these values. We will refer back to this result frequently.
\begin{proposition}\label{prop:parity-constraints} Continuing our notation from above:
\begin{enumerate}
\item Both $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ cannot be even.
\item If both $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ are odd, then we cannot have both $\frac{n}{d}$ and $\frac{n}{\delta}$ odd.
\item If $2^{r-1}\mid\mid n$, then $\frac{n}{d}$ and $\frac{n}{\delta}$ have opposite parity.
\end{enumerate}
\end{proposition}
\begin{proof} The first result follows from the fact that $q$ is an odd prime power, hence one of $\frac{q-1}{2}$ or $\frac{q+1}{2}$ must be odd. In particular since $2|d$ and $2\mid \delta$, one of $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ must be odd.
For the second observation, we remark that $4\mid (q-1)$ or $4\mid (q+1)$. This implies that either $4\mid d$ or $4\mid \delta$ (since we are assuming both $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ are odd), and therefore $4\mid n$. However, we must have that $2\mid\mid (q-1)$ or $2\mid\mid (q+1)$, and therefore $2\mid\mid d$ or $2\mid\mid \delta$. In particular there are more powers of $2$ dividing $n$ than divide one of $d$ or $\delta$, and therefore at least one of $\frac{n}{d}$ or $\frac{n}{\delta}$ must be even.
For the third observation, we have by \autoref{prop:r-d-delta-relationship} that $2^{r-1}\mid n$ is equivalent to both $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ being odd. However $2^r\nmid n$ means that $\frac{n}{d}$ and $\frac{n}{\delta}$ cannot both be even. Via observation (2) of this proposition, they cannot both be odd, therefore they must have opposite parity.
\end{proof}
\subsection{Summing over the hyperbolic and elliptic images}\label{subsec:hyp-sum}
Using the characterization of the hyperbolic and elliptic sets in \autoref{prop:hyp-ell-para-sets-description}, we can understand the hyperbolic and elliptic images, and therefore their sums.
We first treat the hyperbolic case. Via the Binet formula expansion, we may see that
\begin{equation}\label{eqn:hyperbolic-image}
\begin{aligned}
D_n(\mathcal{H}_p(a),a) &= \left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m(A-c)} \colon 1\le c \le q-1,\ 2c\not\equiv A \pmod{q-1} \right\}.
\end{aligned}
\end{equation}
We remark that the residue of $c$ modulo $\frac{q-1}{d}$ matters when recording elements in the hyperbolic image, however the condition $2c\not\equiv A \pmod{q-1}$ is not equivalent to the condition $2c\not\equiv A \pmod{\frac{q-1}{d}}$. So we can have elements $c$ so that $2c\equiv A \pmod{\frac{q-1}{d}}$, but $2c\not \equiv A \pmod{q-1}$. This is how elements like $\pm 2a^{n/2}$ can appear in the hyperbolic image. In order to deal with this, we can provide an alternative description of the hyperbolic image.
\begin{proposition} The hyperbolic image can be described as
\begin{align*}
&\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon 2c\not\equiv A \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} \cup \left\{ 2\zeta_{q-1}^{n c} \colon \substack{ 2c\equiv A \bmod{\frac{q-1}{d}} \text{ but} \\ 2c\not \equiv A \bmod{q-1} }\right\}_{c=1}^{\frac{q-1}{d}}
\end{align*}
\end{proposition}
Thus to characterize the hyperbolic image, it suffices to understand when these congruences can be solved. As we see in the following proposition, this depends upon the parity of $A$ and $\frac{q-1}{d}$, as well as whether or not $d=2$.
\begin{proposition}\label{prop:hyperbolic-image} We have that the hyperbolic image $D_n(\mathcal{H}_q(a),a)$ is equal to
\begin{align*}
\begin{cases}
\left\{2 \legendre{a}{q}^{n/d} a^{n/2}\right\} & d=q-1; \\
%
\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \right\}_{c=1}^{\frac{q-1}{d}} & A\text{ odd, } \frac{q-1}{d} \text{ even}; \\
%
\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{1}{2}\left( A + \frac{q-1}{d} \right) \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} \cup \left\{ 2(-1)^{n/d} a^{n/2} \right\} & A\text{ odd, } \frac{q-1}{d} \text{ odd}; \\
%
\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{A}{2} \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} & A\text{ even, } \frac{q-1}{d} \text{ odd, } d=2; \\
%
\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{A}{2} \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} \cup \left\{ 2 a^{n/2} \right\} & A\text{ even, } \frac{q-1}{d} \text{ odd, } d\ne 2; \\
%
\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{A}{2}, \frac{A}{2} + \frac{q-1}{2d} \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} \cup \left\{ -2a^{n/2} \right\} & A\text{ even, } \frac{q-1}{d} \text{ even, } d=2; \\
%
\left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{A}{2}, \frac{A}{2} + \frac{q-1}{2d} \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} \cup \left\{ 2a^{n/2},\ -2a^{n/2}\right\} & A\text{ even, } \frac{q-1}{d} \text{ even, } d\ne 2.
%
\end{cases}
\end{align*}
\end{proposition}
\begin{proof} We know that solutions to $2c\equiv A \pmod{\frac{q-1}{d}}$ exist if and only if $\gcd \left( 2, \frac{q-1}{d} \right)$ divides $A$, and in this setting there are precisely $\gcd \left( 2, \frac{q-1}{d} \right)$ such solutions.
\begin{enumerate}
\item In the case that $d=q-1$, we have that the hyperbolic image is simply $\{2\}$. However, since $d=q-1$, we can write $n = \frac{n}{d}(q-1)$, from which we can see that
\begin{align*}
a^{n/2} &= \left( a^{\frac{q-1}{2}} \right)^{\frac{n}{d}} = \legendre{a}{q}^{n/d}.
\end{align*}
Since these are both congruent to $\pm 1$, they square to 1, so we may rewrite $2 = 2 \legendre{a}{q}^{n/d} a^{n/2}$. When discussing potential overlap in the hyperbolic and elliptic images later, it will benefit us to characterize the hyperbolic image in this a priori more convoluted form.
\item In this case $\gcd \left( 2, \frac{q-1}{d} \right)$ is even, which does not divide $A$ since it is odd. Thus there are no solutions.
\item If $A$ is odd and $\frac{q-1}{d}$ is odd, then there is a unique solution of the form $c = \frac{1}{2}\left( A + \frac{q-1}{d} \right) + \ell \frac{q-1}{d}$ for some $\ell$. Multiplying this equality by $2$ we obtain
\begin{align*}
2c = A + \frac{q-1}{d} + 2 \ell\frac{q-1}{d} = A + \left( 2\ell + 1 \right) \frac{q-1}{d}.
\end{align*}
We note that $(2\ell + 1)$ is odd, while $d$ is always even. Therefore $\frac{2\ell + 1}{d}$ will never be an integer, and thus $2c\not \equiv A \pmod{q-1}$. Plugging in this $c$, we obtain
\begin{align*}
2\zeta_{\frac{q-1}{d}}^{\mu \left( \frac{1}{2}\left( A + \frac{q-1}{d} \right) + \ell \frac{q-1}{d} \right)} &= 2 \zeta_{\frac{q-1}{d}}^{\mu \frac{1}{2} \left( A + \frac{q-1}{d} \right)} \zeta_{\frac{q-1}{d}}^{\mu \ell \frac{q-1}{d}} = 2 \zeta_{q-1}^{\frac{n}{2} (A + \frac{q-1}{d})} \\
&= 2\zeta_{q-1}^{A \frac{n}{2}} \zeta_{q-1}^{\frac{q-1}{2} \frac{n}{d}} = 2(-1)^{n/d} a^{n/2}.
\end{align*}
\item If $A$ is even and $\frac{q-1}{d}$ is odd, then there is a unique solution, namely $c \equiv \frac{A}{2} \pmod{\frac{q-1}{d}}$. Any such solution will be an integer of the form $c = \frac{A}{2} + \ell \frac{q-1}{d}$ for some $\ell$, so we may multiply by 2 to obtain
\begin{align*}
2c = A + 2\ell \frac{q-1}{d}.
\end{align*}
If $d=2$, then this solution yields $2c\equiv A \pmod{q-1}$, so we must omit this value. In this case we see that
\begin{align*}
D_n(\mathcal{H}_q(a),a) &= \left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{A}{2} \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}}.
\end{align*}
If $d\ne 2$, then it is not the case that $c$ has to satisfy $2c\equiv A \pmod{q-1}$. This tells us that
\begin{align*}
D_n(\mathcal{H}_q(a),a) &= \left\{ \zeta_{\frac{q-1}{d}}^{m c} + \zeta_{\frac{q-1}{d}}^{m (A-c)} \colon c\not\equiv \frac{A}{2} \bmod{\frac{q-1}{d}} \right\}_{c=1}^{\frac{q-1}{d}} \cup \left\{ 2 \zeta_{\frac{q-1}{d}}^{m \frac{A}{2}} \right\}.
\end{align*}
Here we compute that $\zeta_{\frac{q-1}{d}}^{m \frac{A}{2}} = \zeta_{q-1}^{A \frac{n}{2}} = a^{n/2}$.
\item If $A$ is even and $\frac{q-1}{d}$ is even, then there are two solutions, namely $c \equiv \frac{A}{2} \bmod{\frac{q-1}{d}}$ and $c \equiv \frac{A}{2} + \frac{q-1}{2d} \bmod{\frac{q-1}{d}}$. Let's look at these two solutions individually.
\begin{enumerate}
\item For the case $c\equiv \frac{A}{2}$, we have that $c$ is an integer of the form $c = \frac{A}{2} + \ell \frac{q-1}{d}$. Multiplying by $2$ we obtain $2c = A + 2\ell \frac{q-1}{d}$. If $d=2$, we have that $2c\equiv A \pmod{q-1}$, so this $c$ yields a parabolic element.
\item For the case $c\equiv \frac{A} {2} + \frac{q-1}{2d}$, we have that $c = \frac{A}{2} + \frac{q-1}{2d} + \ell \frac{q-1}{d}$ for some $\ell$. Multiplying by $2$ yields
\begin{align*}
2c = A + (2\ell + 1)\frac{q-1}{d}.
\end{align*}
As $2\ell + 1$ is odd and $d$ is even, this choice of $c$ will never satisfy $2c \equiv A \pmod{q-1}$.
\end{enumerate}
\end{enumerate}
\end{proof}
\begin{corollary} The size of the hyperbolic set is:
\begin{align*}
|D_n(\mathcal{H}_q(a),a)| &= \floor{\frac{q-1}{2d}} + \begin{cases}
1 & A\cdot \frac{q-1}{d} \text{ odd}; \\
1 & A \text{ even, and } d\ne 2; \\
0 & \text{otherwise}. \end{cases}
\end{align*}
\end{corollary}
\begin{lemma}\label{lem:labelname} The hyperbolic sum is
\begin{align*}
S_q^{\mathcal{H}_q(a)}(D_n(x,a)) &= \begin{cases}
2 \legendre{a}{q}^{n/d} a^{n/2} & d = q-1; \\
0 & A\text{ odd, } \frac{q-1}{d} \text{ even}; \\
%
(-1)^{n/d} a^{n/2} & A\text{ odd, } \frac{q-1}{d} \text{ odd}; \\
-a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ odd, } d=2; \\
%
a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ odd, } d\ne 2; \\
%
-2a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ even, } d=2; \\
%
0 & A\text{ even, } \frac{q-1}{d} \text{ even, } d\ne 2.
%
\end{cases}
\end{align*}
\end{lemma}
\begin{proof} We may sum over the hyperbolic image as computed in \autoref{prop:hyperbolic-image} to obtain
\begin{align*}
S_q^{\mathcal{H}_q(a)}(D_n(x,a)) &= \begin{cases}
2 \legendre{a}{q}^{n/d} a^{n/2} & d = q-1; \\
0 & A\text{ odd, } \frac{q-1}{d} \text{ even}; \\
%
(-1)^{n/d} a^{n/2} & A\text{ odd, } \frac{q-1}{d} \text{ odd}; \\
-a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ odd, } d=2; \\
%
a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ odd, } d\ne 2; \\
%
-a^{n/2} +(-1)^{n/d} a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ even, } d=2 ; \\
%
a^{n/2} + (-1)^{n/d}a^{n/2} & A\text{ even, } \frac{q-1}{d} \text{ even, } d\ne 2.
%
\end{cases}
\end{align*}
In the latter two cases, $\frac{n}{d}$ is odd since $\frac{q-1}{d}$ is even, yielding the statement of the proposition.
\end{proof}
A similar analysis can be used to characterize the elliptic image. We observe via \autoref{prop:hyp-ell-para-sets-description} and the Binet formula that the elliptic image is
\allowdisplaybreaks
\begin{equation}\label{eqn:elliptic-image}
\begin{aligned}
&\quad\quad D_n(\mathcal{E}_p(a),a) = \left\{ \zeta_{q^2-1}^{n(A+k(q-1))}+\zeta_{q^2-1}^{n(Aq-k(q-1))}: 1\le k\le \frac{q+1}{\delta}\ \mathrm{and}\ 2k \not\equiv A \bmod{q+1} \right\}
\\
&=\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k}\colon 1\le k\le \frac{q+1}{\delta},\ 2k\not\equiv A \pmod{\frac{q+1}{\delta}} \right\} \cup \left\{ 2\zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} \colon \substack{ 2k\equiv A \bmod{\frac{q+1}{\delta}} \text{ but} \\ 2k\not \equiv A \bmod{q+1} } \right\}.
\end{aligned}
\end{equation}
Again we may better characterize this in various cases.
\begin{proposition}\label{prop:elliptic-image} We have that the elliptic image $D_n(\mathcal{E}_q(a),a)$ is equal to
\begin{align*}
\begin{cases}
\left\{ 2\legendre{a}{q}^{n/\delta} a^{n/2} \right\} & \delta=q+1; \\
%
\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k} \right\}_{k=1}^{\frac{q+1}{\delta}} & A\text{ odd, } \frac{q+1}{\delta}\text{ even}; \\
\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k}\colon k \ne \frac{1}{2}\left( A + \frac{q+1}{\delta} \right) \right\}_{k=1}^{\frac{q+1}{\delta}}\cup \left\{ 2(-1)^{n/\delta} a^{n/2}\right\} & A\text{ odd, } \frac{q+1}{\delta}\text{ odd}; \\
\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k}\colon k \ne \frac{A}{2} \right\}_{k=1}^{\frac{q+1}{\delta}} & A\text{ even, } \frac{q+1}{\delta}\text{ odd, } \delta=2; \\
\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k}\colon k\ne \frac{A}{2} \right\}_{k=1}^{\frac{q+1}{\delta}}\cup \left\{ 2a^{n/2} \right\} & A\text{ even, } \frac{q+1}{\delta}\text{ odd, } \delta\ne 2; \\
\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k}\colon k \ne \frac{A}{2},\ \frac{A}{2} + \frac{q+1}{2\delta} \right\}_{k=1}^{\frac{q+1}{\delta}}\cup \left\{ -2a^{n/2} \right\}
& A\text{ even, } \frac{q+1}{\delta}\text{ even, } \delta=2; \\
\left\{ \zeta_{q^2-1}^{nA} \zeta_{\frac{q+1}{\delta}}^{\mu k} + \zeta_{q^2-1}^{nAq} \zeta_{\frac{q+1}{\delta}}^{-\mu k}\colon k \ne \frac{A}{2},\ \frac{A}{2} + \frac{q+1}{2\delta} \right\}_{k=1}^{\frac{q+1}{\delta}}\cup \left\{ 2a^{n/2}, -2a^{n/2} \right\}
& A\text{ even, } \frac{q+1}{\delta}\text{ even, } \delta\ne 2. \\
%
\end{cases}
\end{align*}
\end{proposition}
\begin{proof} We can solve for the congruence $2k\equiv A \bmod{\frac{q+1}{\delta}}$.
\begin{enumerate}
\item When $\delta = q+1$, we can write $n = \frac{n}{\delta}(q+1)$, from which we see that any element in the elliptic image takes the form
\begin{align*}
\zeta_{q^2-1}^{n \left( A + k (q-1) \right)} + \zeta_{q^2-1}^{n \left( Aq - k(q-1) \right)} &= 2\zeta_{q^2-1}^{\frac{n}{\delta} A(q+1)} =2a^{n/\delta}.
\end{align*}
We remark that if $\delta = q+1$, we may write
\begin{align*}
a^{n/2} &= \left( a^{n/\delta} \right)^{\delta/2} = \left( a^{n/\delta} \right)^{\frac{q+1}{2}} = a^{n/\delta} \left( a^{n/\delta} \right)^{\frac{q-1}{2}} = a^{n/\delta} \legendre{a^{n/\delta}}{q}.
\end{align*}
We may verify that $\legendre{a^{n/\delta}}{q} = \legendre{a}{q}^{n/\delta}$, from which we compute
\begin{align*}
a^{n/\delta} = \legendre{a}{q}^{n/\delta} a^{n/2}.
\end{align*}
\item In this case there are no solutions to $2k\equiv A \pmod{\frac{q+1}{\delta}}$.
\item In this case, there is a unique solution, namely an integer of the form $k = \frac{1}{2} \left( A + \frac{q+1}{\delta} \right) + \ell \frac{q+1}{\delta}$ for some $\ell$. Multiplying by 2 we get
\begin{align*}
2k &= A + \left( 2\ell+1 \right)\frac{q+1}{\delta}.
\end{align*}
Since $\delta$ is even, we have that $\frac{2\ell+1}{\delta}$ will never be an integer, so such a $k$ will not satisfy $2k\equiv A \pmod{q+1}$.
\item There is a unique solution, $k \equiv \frac{A}{2} \pmod{\frac{q+1}{\delta}}$. This will be some integer of the form
\begin{align*}
k = \frac{A}{2} + \ell \frac{q+1}{\delta}.
\end{align*}
Multiplying by $2$, we have that
\begin{align*}
2k = A + 2\ell \frac{q+1}{\delta}.
\end{align*}
Thus we have to break into cases based on whether $\delta=2$ or $\delta\ne 2$.
\setcounter{enumi}{4}
\item We see that when $k = \frac{A}{2} + \frac{q+1}{2\delta}$, that the associated element in the elliptic image is
\begin{align*}
\zeta_{q^2-1}^{n \left( A + \left( \frac{A}{2} + \frac{q+1}{2\delta} \right)(q-1) \right)} + \zeta_{q^2-1}^{n \left( Aq - \left( \frac{A}{2} + \frac{q+1}{2\delta} \right)(q-1) \right)} &= \zeta_{q^2-1}^{n A \frac{q+1}{2}} \left( \zeta_{q^2-1}^{\frac{n}{\delta} \frac{q^2-1}{2}} + \zeta_{q^2-1}^{-\frac{n}{\delta} \frac{q^2-1}{2}} \right) \\
&= a^{n/2} \left( (-1)^{n/\delta} + (-1)^{-n/\delta} \right).
\end{align*}
Since $\frac{q+1}{\delta}$ is even, we have that $n/\delta$ is odd, so the above reduces to $-2a^{n/2}$.
\end{enumerate}
\end{proof}
\begin{corollary} The size of the elliptic set is:
\begin{align*}
|D_n(\mathcal{E}_q(a),a)| &= \floor{\frac{q+1}{2\delta}} + \begin{cases}
1 & A\cdot \frac{q+1}{\delta} \text{ odd}; \\
1 & A \text{ even, and } \delta\ne 2; \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
\end{corollary}
\begin{lemma}\label{lem:elliptic-sum} The elliptic sum is
\begin{align*}
S_q^{\mathcal{E}_q(a)}(D_n(x,a)) &= \begin{cases}
2\legendre{a}{q}^{n/\delta} a^{n/2} & \delta = q+1; \\
0
& A\text{ odd, } \frac{q+1}{\delta}\text{ even}; \\
(-1)^{n/\delta}a^{n/2}
& A\text{ odd, } \frac{q+1}{\delta}\text{ odd}; \\
- a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ odd, } \delta=2; \\
a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ odd, } \delta\ne 2; \\
-2 a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ even, } \delta=2; \\
0
& A\text{ even, } \frac{q+1}{\delta}\text{ even, } \delta\ne 2. \\
%
\end{cases}
\end{align*}
\end{lemma}
\begin{proof} We may sum over the elliptic image to get
\begin{align*}
S_q^{\mathcal{E}_q(a)}(D_n(x,a)) &= \begin{cases}
2\legendre{a}{q}^{n/\delta} a^{n/2} & \delta = q+1; \\
0
& A\text{ odd, } \frac{q+1}{\delta}\text{ even}; \\
(-1)^{n/\delta}a^{n/2}
& A\text{ odd, } \frac{q+1}{\delta}\text{ odd}; \\
- a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ odd, } \delta=2; \\
a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ odd, } \delta\ne 2; \\
- a^{n/2} + (-1)^{n/\delta}a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ even, } \delta=2; \\
(-1)^{n/\delta}a^{n/2} + a^{n/2}
& A\text{ even, } \frac{q+1}{\delta}\text{ even, } \delta\ne 2. \\
%
\end{cases}
\end{align*}
In the latter two cases, since $\frac{q+1}{\delta}$ is even and coprime to $n/\delta$, we have that $n/\delta$ is odd, which gives the statement of the lemma.
\end{proof}
In order to characterize potential overlaps in the images of the hyperbolic, elliptic, and parabolic sets, it will be easier to break into the case of $a$ a residue and non-residue.
\subsection{Overlaps in the non-residue case}
Via \autoref{prop:possible-hyp-ellip-overlap}, we know that any overlap in the hyperbolic and elliptic images must be a subset of $\left\{ \pm 2a^{n/2} \right\}$, and similarly we know the parabolic image to be $\left\{ 2a^{n/2} \right\}$ when $a$ is a residue, and empty otherwise. So it suffices to determine when either of $\pm 2a^{n/2}$ lie in the hyperbolic and elliptic image.
We remark however that these images will occur in the hyperbolic image precisely when the second set in \autoref{eqn:hyperbolic-image} is nonempty, therefore we can understand these images via the work already done in \autoref{prop:hyperbolic-image}. Similarly for the elliptic case we have solved for when $\pm 2a^{n/2}$ lies in the elliptic image in \autoref{prop:elliptic-image}. We can summarize these findings as follows.
\begin{proposition}\label{prop:nonresidue-EH-overlap} If $a$ is a non-residue, then the elliptic and hyperbolic overlap is the following:
\begin{align*}
D_n(\mathcal{H}_q(a),a) \cap D_n(\mathcal{E}_q(a),a) &= \begin{cases} \left\{ 2a^{n/2} \right\} & 2^r \mid n; \\
\emptyset & \text{otherwise.} \end{cases}
\end{align*}
\end{proposition}
\begin{proof} Via \autoref{prop:hyperbolic-image} when $A$ is odd, we will have $2(-1)^{n/d}a^{n/2}$ in the hyperbolic image if $\frac{q-1}{d}$ is odd (including when $\frac{q-1}{d} = 1$). Similarly via \autoref{prop:elliptic-image}, we will have $2(-1)^{n/\delta}a^{n/2}$ in the elliptic image when $\frac{q+1}{\delta}$ is odd (including when $\frac{q+1}{\delta} = 1$). Therefore in order to have any overlap we must have that the parities of $\frac{n}{d}$ and $\frac{n}{\delta}$ coincide. Since both $\frac{q-1}{d}$ and $\frac{q+1}{\delta}$ are odd, via \autoref{prop:parity-constraints} in order for the parities of $\frac{n}{d}$ and $\frac{n}{\delta}$ to agree, they must both be even. This condition is equivalent to $2^r\mid n$ by \autoref{prop:r-d-delta-relationship}.
\end{proof}
In the case where $A$ is odd, we can decompose the hyperbolic sum by making the $d=q-1$ case a separate condition as follows:
\allowdisplaybreaks
\begin{align*}
S_q^{\mathcal{H}_q(a)}(D_n(x,a)) &= \begin{cases}
2(-1)^{n/d}a^{n/2} & d = q-1; \\
0 & \frac{q-1}{d} \text{ even}; \\
%
(-1)^{n/d} a^{n/2} & \frac{q-1}{d} \text{ odd,}\ne 1, \\
\end{cases} \\
&= \begin{cases} (-1)^{n/d}a^{n/d} & \frac{q-1}{d}\text{ odd}; \\ 0 & \frac{q-1}{d}\text{ even}, \end{cases} + \begin{cases} (-1)^{n/d}a^{n/2} & d=q-1; \\ 0 & \text{else.} \end{cases}
\end{align*}
A similar argument shows that
\begin{align*}
S_q^{\mathcal{E}_q(a)}(D_n(x,a)) &= \begin{cases} (-1)^{n/\delta}a^{n/2} & \frac{q+1}{\delta}\text{ odd}; \\
0 & \frac{q+1}{\delta}\text{ even}, \end{cases} + \begin{cases}
(-1)^{n/\delta} a^{n/2} & \delta=q+1; \\
0 & \text{else}. \end{cases}
\end{align*}
\begin{lemma}\label{lem:A-odd-sum} Let $A$ be odd. Then the sum is given as
\begin{align*}
S_q(D_n(x,a)) &= \begin{cases}
0 & 2^{r-1}\mid n; \\
-(-1)^{\frac{n}{d}+\frac{n}{\delta}} a^{n/2} & 2^{r-1}\nmid n, \end{cases} + \begin{cases}
(-1)^{n/d}a^{n/2} & d=q-1; \\
0 & \text{else}, \end{cases} + \begin{cases}
(-1)^{n/\delta}a^{n/2} & \delta=q+1; \\
0 & \text{else}. \end{cases}
\end{align*}
\end{lemma}
\begin{proof} We can combine the three conditions:
\begin{align*}
\begin{cases} (-1)^{n/d}a^{n/d} & \frac{q-1}{d}\text{ odd}; \\
0 & \frac{q-1}{d}\text{ even}, \end{cases} + \begin{cases}
(-1)^{n/\delta}a^{n/2} & \frac{q+1}{\delta}\text{ odd}; \\
0 & \frac{q+1}{\delta}\text{ even}, \end{cases} - \begin{cases}
2a^{n/2} & \frac{n}{d},\ \frac{n}{\delta}\text{ even}; \\
0 & \text{else.} \end{cases}
\end{align*}
Rewriting these conditions using $h$, $\ell$, and $r = h+\ell$, we have
\begin{align*}
\begin{cases} (-1)^{n/d}a^{n/d} & 2^h\mid n; \\
0 & 2^h\nmid n,
\end{cases} + \begin{cases}
(-1)^{n/\delta}a^{n/2} & 2^\ell \mid n; \\
0 & 2^\ell \nmid n,
\end{cases} - \begin{cases}
2a^{n/2} & 2^{h+\ell}\mid n; \\
0 & 2^{h+\ell}\nmid n. \end{cases}
\end{align*}
Combining conditions, we can see that this simplifies to
\begin{align*}
\begin{cases}
0 & 2^r\mid n; \\
(-1)^{n/d}a^{n/2} + (-1)^{n/\delta}a^{n/2} & 2^{r-1}\mid\mid n; \\
(-1)^{n/d}a^{n/2}& 2^{r-1}\nmid n,\ \ell > h; \\
(-1)^{n/\delta}a^{n/2} & 2^{r-1}\nmid n,\ h>\ell.
\end{cases}
\end{align*}
Via \autoref{prop:parity-constraints}, when $2^{r-1}\mid\mid n$, we have that $\frac{n}{d}$ and $\frac{n}{\delta}$ have opposite parities, so we can merge the $2^{r-1}\mid\mid n$ and $2^r\mid n$ conditions.
When $\ell > h$, we have that $\ell= r-1$, so $2^{r-1}\nmid n$ is equivalent to $\frac{q+1}{\delta}$ being even, which implies $\frac{n}{\delta}$ is odd. Similarly in the last case, $\frac{n}{d}$ is odd, so we can merge the last two conditions. This gives
\begin{align*}
\begin{cases}
0 & 2^{r-1}\mid n; \\
-(-1)^{\frac{n}{d}+\frac{n}{\delta}} a^{n/2} & 2^{r-1}\nmid n. \end{cases}
\end{align*}
Adding back the $d=q-1$ and $\delta=q+1$ conditions, we obtain the desired statement.
\end{proof}
\subsection{Overlaps: the residue case}
In the residue setting, the parabolic image is precisely $\{2a^{n/2}\}$. Thus we need to study when $2a^{n/2}$ can lie in the hyperbolic and elliptic images in order to understand when they admit overlap with the parabolic image.
We can begin with the hyperbolic-elliptic overlap. We note that in almost all the cases in which they have overlap, the number at which they overlap is $2a^{n/2}$, which is parabolic. In particular, when this occurs, we will have
\[
S_q^{\mathcal{H}_q(a),\mathcal{E}_q(a)}(D_n(x,a)) - S_q^{\mathcal{H}_q(a),\mathcal{E}_q(a),\mathcal{P}_q(a)}(D_n(x,a)) = 0.
\]
We note that this difference will only ever be nonzero when $-2a^{n/2}$ lies in both the hyperbolic and elliptic images. However this cannot occur.
\begin{proposition}\label{prop:HE-HEP-difference} Let $a$ be a residue. Then
\begin{align*}
S_q^{\mathcal{H}_q(a),\mathcal{E}_q(a)}(D_n(x,a)) - S_q^{\mathcal{H}_q(a),\mathcal{E}_q(a),\mathcal{P}_q(a)}(D_n(x,a)) = 0.
\end{align*}
\end{proposition}
\begin{proof} We see that in the residue case, $\legendre{a}{q}=1$. So $-2a^{n/2}$ can only lie in the hyperbolic image when $\frac{q-1}{d}$ even, while $-2a^{n/2}$ can only lie in the elliptic image when $\frac{q+1}{\delta}$ is even. However these both can't simultaneously occur by \autoref{prop:parity-constraints}. Thus any hyperbolic-elliptic overlap occurs at $2a^{n/2}$ which therefore is parabolic as well.
\end{proof}
We can easily characterize the hyperbolic-parabolic and elliptic-parabolic overlaps by observing when $2a^{n/2}$ lies in the hyperbolic and elliptic images.
\begin{proposition}\label{prop:HP-overlap} Let $a$ be a residue. Then
\begin{align*}
S_q^{\mathcal{H}_q(a),\mathcal{P}_q(a)}(D_n(x,a)) &=%
\begin{cases}
2a^{n/2} & d\ne 2; \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
\end{proposition}
\begin{proposition}\label{prop:EP-overlap} Let $a$ be a residue. Then
\begin{align*}
S_q^{\mathcal{E}_q(a),\mathcal{P}_q(a)}(D_n(x,a)) &=%
\begin{cases}
2a^{n/2} & \delta\ne 2; \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
\end{proposition}
Now we can characterize the entire sum by combining these sums and their overlaps. Before doing so, we can begin to cancel some of the sums with others. First we can combine the elliptic sum with the elliptic-parabolic overlap:
\allowdisplaybreaks
\begin{align*}
S_q^{\mathcal{E}_q(a)}(D_n(x,a)) - S_q^{\mathcal{E}_q(a),\mathcal{P}_q(a)}(D_n(x,a)) &= \begin{cases}
2a^{n/2} & \delta = q+1; \\
%
- a^{n/2} & \frac{q+1}{\delta}\text{ odd, } \delta=2; \\
%
a^{n/2} & \frac{q+1}{\delta}\text{ odd, } \delta\ne 2; \\
%
-2 a^{n/2} & \frac{q+1}{\delta}\text{ even, } \delta=2; \\
%
0 & \frac{q+1}{\delta}\text{ even, } \delta\ne 2, \\
%
\end{cases} - \begin{cases}
2a^{n/2} & \delta\ne 2; \\
0 & \text{otherwise},
\end{cases} \\
&= \begin{cases}
0 & \delta = q+1; \\
-a^{n/2} & \frac{q+1}{\delta}\text{ odd, }\ne 1; \\
-2a^{n/2} & \frac{q+1}{\delta}\text{ even},
\end{cases} \\
&= \begin{cases}
-a^{n/2} & \frac{q+1}{\delta}\text{ odd}; \\
-2a^{n/2}& \frac{q+1}{\delta}\text{ even}, \end{cases} + \begin{cases}
a^{n/2} & \delta=q+1; \\ 0 & \text{else}. \end{cases}
\end{align*}
Combining the hyperbolic and the hyperbolic-parabolic overlap we see
\allowdisplaybreaks
\begin{align*}
S_q^{\mathcal{H}_q(a)}(D_n(x,a)) - S_q^{\mathcal{H}_q,(a)\mathcal{P}_q(a)}(D_n(x,a)) &=
\begin{cases}
2a^{n/2} & d = q-1; \\
-a^{n/2} & \frac{q-1}{d} \text{ odd, } d=2; \\
a^{n/2} & \frac{q-1}{d} \text{ odd, } d\ne 2; \\
-2a^{n/2} & \frac{q-1}{d} \text{ even, } d=2; \\
0 & \frac{q-1}{d} \text{ even, } d\ne 2,
\end{cases} - \begin{cases}
2a^{n/2} & d\ne 2; \\
0 & \text{otherwise},
\end{cases} \\
&= \begin{cases}
0 & d=q-1; \\
-a^{n/2} & \frac{q-1}{d}\text{ odd, }\ne 1; \\
-2a^{n/2} & \frac{q-1}{d}\text{ even},
\end{cases} \\
&= \begin{cases}
-a^{n/2} & \frac{q-1}{d}\text{ odd}; \\
-2a^{n/2} & \frac{q-1}{d}\text{ even},
\end{cases} + \begin{cases}
a^{n/2} & d=q-1; \\
0 & \text{else}. \end{cases}
\end{align*}
\begin{lemma}\label{lem:A-even-sum} Let $a$ be a residue. Then
\begin{align*}
S_q(D_n(x,a)) &= \begin{cases}
0 & 2^{r-1}\mid n; \\
-a^{n/2} & 2^{r-1}\nmid n,
\end{cases} + \begin{cases}
a^{n/2} & \delta=q+1; \\
0 & \text{else},
\end{cases} + \begin{cases}
a^{n/2} & d=q-1; \\
0 & \text{else}. \end{cases}
\end{align*}
\end{lemma}
\begin{proof} We first combine the conditions
\begin{align*}
\begin{cases}
-a^{n/2} & \frac{q+1}{\delta}\text{ odd}; \\
-2a^{n/2}& \frac{q+1}{\delta}\text{ even},
\end{cases} + \begin{cases}
-a^{n/2} & \frac{q-1}{d}\text{ odd}; \\
-2a^{n/2} & \frac{q-1}{d}\text{ even},
\end{cases} = \begin{cases}
-2a^{n/2} & 2^{r-1}\mid n; \\
-3a^{n/2} & \text{else}.
\end{cases}
\end{align*}
Adding back $2a^{n/2}$ from the parabolic sum and the $d=q-1$ and $\delta = q+1$ conditions yields the statement of the theorem.
\end{proof}
Combining \autoref{lem:A-odd-sum} and \autoref{lem:A-even-sum} yields the main theorem of the paper.
\subsection{Examples}
A priori, in order to totally characterize the sum of a family of Dickson polynomials, we must understand the quadratic character of $a$, whether $(q-1)$, $(q+1)$, or $2^{r-1}$ divide $n$, and the parities of $\frac{n}{d}$ and $\frac{n}{\delta}$. In the case of the Lucas polynomials $L_n(x) = D_n(x,a)$, many of these conditions coalesce --- for example the quadratic character of $-1$ is dependent upon the residue of our prime modulo four, which also determines possible parities of $\frac{n}{d}$ and $\frac{n}{\delta}$. In fact, modulo a fixed prime, knowledge of $d$ and $\delta$ alone determines the residue sum.
\begin{example} As in Figure~\ref{tab:table1}, consider when $p=7$. In this case there are a very limited number of possibilities for $d$ and for $\delta$. Since the values $d$ and $\delta$ completely determine $S_7(L_n)$, we provide the following table.
\begin{figure}[H]\label{fig:s7ln}
\caption{Possible values for $S_7(L_n)$.}
\par\noindent\rule{0.4\textwidth}{0.4pt} \\
\vspace{0.2em}
\begin{tabular}{c | c | c}
$d$ & $\delta$ & $S_7(L_n)$ \\
\midrule
2 & 2 & 1\\
2 & 4 & 1\\
2 & 8 & -1
\end{tabular} \quad
\begin{tabular}{c | c | c}
$d$ & $\delta$ & $S_7(L_n)$ \\
\midrule
6 & 2 & 2\\
6 & 4 & 2\\
6 & 8 & 0
\end{tabular}
\centering
\end{figure}
To provide an example when $p \equiv 1 \pmod{4}$, we can write an analogous table for $p=29$, although as expected it is much larger. Possible even values for $d = \gcd(n,28)$ are $d\in \left\{2,4,14,28\right\}$, while $\delta = \gcd(n,60)$ must be even as well, and thus lies in $\delta \in \left\{2,4,6,10,12,20,30,60\right\}$. However we remark that as 28 and 60 are both divisible by 4, we have that $4\mid d$ if and only if $4\mid \delta$, which gives us a restriction on the possible pairs that can show up. This yields the following table.
\begin{figure}[H]\label{fig:s29ln}
\caption{Possible values for $S_{29}(L_n)$.}
\par\noindent\rule{0.81\textwidth}{0.4pt} \\
\vspace{0.2em}
\begin{tabular}{c | c | c}
$d$ & $\delta$ & $S_{29}(L_n)$ \\
\midrule
2 & 2 & 1\\
2 & 6 & 1 \\
2 & 10 & 1 \\
2 & 30 & 0 \\
\end{tabular} \quad
\begin{tabular}{c | c | c}
$d$ & $\delta$ & $S_{29}(L_n)$ \\
\midrule
4 & 4 & 0\\
4 & 12 & 0 \\
4 & 20 & 0\\
\multicolumn{3}{c}{\textcolor{white}{.}}
\end{tabular} \quad
\begin{tabular}{c | c | c}
$d$ & $\delta$ & $S_{29}(L_n)$ \\
\midrule
14 & 2 & 1\\
14 & 6 & 1 \\
14 & 10 & 1 \\
14 & 30 & 0 \\
\end{tabular} \quad
\begin{tabular}{c | c | c}
$d$ & $\delta$ & $S_{29}(L_n)$ \\
\midrule
28 & 4 & 1\\
28 & 12 & 1 \\
28 & 20 & 1 \\
\multicolumn{3}{c}{\textcolor{white}{.}}
\end{tabular} \quad
\centering
\end{figure}
\end{example}
\begin{example} As another example, let $T_n(x) = \cos(n\arccos(x))$ denote the $n$th Chebyshev polynomial. It is well known that these are related to the Dickson polynomials for $a=1$ via the equality
\[
D_n(2x,1) = 2T_n(x).
\]
In particular this implies that for odd characteristic we have
\begin{align*}
S_q(T_n(x)) &= \frac{1}{2}S_q(D_n(x,1)).
\end{align*}
Invoking \autoref{thm:S-sum}, we may provide a characterization of this sum. As in the Lucas case, it admits an extremely constrained number of possible values. We may verify for all $n$ and $q$ that
\[
S_q(T_n(x)) \in \left\{\pm \frac{1}{2}, 0, 1\right\}.
\]
\end{example}
\section{Further Directions and Conclusion}
One natural direction to follow is to find other two step recurrences for which the above techniques can be employed. If one defines the polynomials $P_n(x)$ recursively by
\begin{align*}
P_n(x)=Ax\cdot P_{n-1}(x)+B\cdot P_{n-2}(x),
\end{align*}
given initial conditions
\begin{align*}
P_0(x)&=C \text{ and } P_1(x) = \frac{AC}{2}x,
\end{align*}
where $A,B \in \mathbb{Z}$ and $C$ is an even integer, then $P_n(x)$ shares many of the same properties with the Dickson polynomials $D_n(x,a)$. In particular, $P_n(x)$ is of degree $n$ for each $n$, is odd for $n$ odd and even for $n$ even, and admits the following Binet formula expansion:
\begin{align*}
&\qquad \quad P_n(x)=\frac{C}{2}
\left(\alpha(x)^n + \beta(x)^n\right),
\end{align*}
where $\alpha(x) = \frac{Ax+\sqrt{(Ax)^2+4B}}{2}$ and $\beta(x) = \frac{Ax-\sqrt{(Ax)^2+4B}}{2}$.
By studying the quadratic character of $A^2x^2+4B$, we obtain sets akin to the hyperbolic, elliptic, and parabolic from above. If we set $A=B=1$, then $P_n(x)=\frac{C}{2}L_n(x)$ and the values for $S_q(P_n)$ are in the set $\left\{\frac{-C}{2},0,\frac{C}{2},C\right\}$.
Another family of interest would be the Fibonacci polynomials, given by the initial conditions $F_1(x) = 1$, $F_2(x) = x$, and the recurrence relation
\[
F_n(x) = x\cdot F_{n-1}(x) + F_{n-2}(x).
\]
Due to the discrepancy between the indexing conventions on Dickson polynomials versus Fibonacci polynomials, the Fibonacci polynomials of even degree will always be odd, and hence $S_p(F_{2n}) = 0$ for all $p$ and $n$. An investigation of $S_p(F_n)$ at the prime 7 when $n$ is odd displays that sums over residues of Fibonacci polynomials are far less constrained than their Lucas counterparts.
\begin{figure}[H]\label{table3}
\caption{Investigation of $S_7(F_{2n-1})$ for $1\le n\le 40$.}
\par\noindent\rule{0.7\textwidth}{0.4pt} \\
\vspace{0.2em}
\begin{tabular}{c | c}
$n$ & $S_7(F_n)$ \\
\midrule
1 & 1 \\
3 & 4 \\
5 & 3 \\
7 & 0 \\
9 & 5 \\
11 & 3 \\
13 & 6 \\
15 & 6
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(F_n)$ \\
\midrule
17 & 1 \\
19 & 6 \\
21 & 3 \\
23 & 0 \\
25 & 0 \\
27 & 3 \\
29 & 6 \\
31 & 1
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(F_n)$ \\
\midrule
33 & 6 \\
35 & 6 \\
37 & 3 \\
39 & 5 \\
41 & 0 \\
43 & 3 \\
45 & 4 \\
47 & 1
\end{tabular}\quad
\begin{tabular}{c | c}
$n$ & $S_7(F_n)$ \\
\midrule
49 & 1 \\
51 & 4 \\
53 & 3 \\
55 & 0 \\
57 & 5 \\
59 & 3 \\
61 & 6 \\
63 & 6
\end{tabular}\quad \begin{tabular}{c | c}
$n$ & $S_7(F_n)$ \\
\midrule
65 & 1 \\
67 & 6 \\
69 & 3 \\
71 & 0 \\
73 & 0 \\
75 & 3 \\
77 & 6 \\
79 & 1
\end{tabular}
\centering
\end{figure}
An interesting direction of research would be to classify these sums in an analogous procedure to that presented in this paper, and begin to characterize the size of the image sets of Fibonacci polynomials modulo $p$. We observe that there is some $(p^2-1)$-fold periodicity in this table which is analogous to that observed for the Dickson polynomials. These values are also palindromic about $\frac{p^2-1}{2}$, which in the Dickson polynomials is explained by replacing $n$ by $p^2-1-n$ in Theorem~\ref{thm:S-sum}.
\color{black}
\bibliographystyle{amsalpha}
|
1,314,259,995,808 | arxiv | \section{Introduction}
\label{sec:intro}
The factorization theorem of Quantum Chromodynamics (QCD) allows to separate cross sections, $\sigma$,
into a hard-scattering matrix element, $\hat \sigma$, on one hand and parton distribution functions (PDFs) on the other.
While $\hat \sigma$, describing the short-distance structure of the reaction, is process-dependent but
perturbatively calculable, the PDFs, which account for the non-perturbative long-distance structure, are universal
but have to be determined from experimental data.
Beside the quark masses, the only free parameter of the QCD Lagrangian is the strong coupling constant, $\alpha_S$.
The renormalization group equation predicts the energy dependence of the strong coupling, i.e. a functional form for
$\alpha_S (Q)$, where $Q$ is the energy transferred in the reaction.
However, the absolute strength remains a free parameter of the theory,
which for convenience is usually compared at the pole mass of the Z boson.
The latest world average is $\alpha_{S}(M_{Z})$ = 0.1184 $\pm$ 0.0007 \cite{PhysRevD.86.010001}.
This average and its remarkable precision are driven by results obtained at relatively low
energies, namely from hadronic decays of $\tau$ leptons and from lattice QCD.
Jet and multijet production rates are directly sensitive to $\alpha_S$,
in principle even allow for measurements of the ``running'' at high energies,
and they provide constraints on PDFs, especially at medium and high parton momentum fractions, $x$.
For both PDF and $\alpha_S$ analyses, the measured cross sections have to be compared to predictions,
which for most jet observables are currently available to next-to-leading order (NLO) in perturbative QCD.
In this article, the latest jet measurements at HERA, the Tevatron and the LHC
with sensitivity to PDFs and $\alpha_S$ are summarized.
\section{Measurements at HERA}
\label{sec:HERA}
\subsection{Deep-inelastic scattering}
\label{sec:DIS}
Based on an earlier publication with double-differential cross sections for
inclusive jet, dijet and trijet production in deep-inelastic scattering (DIS) of electrons/positrons on protons
at a center-of-mass energy, $\sqrt{s}$, of 318~GeV in $Q^2$ bins from 150 to 15000 GeV${}^2$ \cite{H1prelim-11-032},
which confirmed a reasonable description of the data by the NLO prediction and sensitivity to differences in the
proton PDFs,
the H1 Collaboration has recently released updated results based on a regularized unfolding that is performed for all
bins and multijet categories simultaneously \cite{H1prelim-12-031}.
This allows for an optimized correction of detector effects and yields a complete correlation matrix,
valuable for the inclusion of these data into different QCD fits.
The H1 Collaboration used the covariance matrix to normalize the different
jet rates to the inclusive neutral-current DIS cross section. This normalization reduces both
the experimental and the theoretical uncertainties.
The results are shown in figure~\ref{fig:H1prelim-12-031_fig7}.
The cross-section predictions at next-to-leading and leading order were required to agree within 30\%,
which was fulfilled for 42 out of 65 bins of this measurement, before performing a combined fit of $\alpha_S (M_{Z})$
to the normalized inclusive jet, dijet and trijet data.
The result is
$\alpha_S (M_Z)$ = 0.1163 $\pm$ 0.0011 (exp.) $\pm$ 0.0014 (PDF) $\pm$ 0.0008 (hadr.) $\pm$ 0.0039 (th.).
\begin{figure}
\centering
\includegraphics[width=\columnwidth,viewport=15 25 515 505,clip]{H1prelim-12-031_fig7}
\caption{Normalized inclusive jet, dijet and trijet cross sections measured by the H1 Collaboration
as a function of the transverse momentum in different $Q^2$ bins and compared to the NLO prediction
\cite{H1prelim-12-031}.}
\label{fig:H1prelim-12-031_fig7}
\end{figure}
The H1 and ZEUS Collaborations explicitly studied $\alpha_S$-PDF correlations and the impact of HERA jet data on their
PDF fits \cite{H1prelim-11-034}.
When using inclusive DIS data only, taking $\alpha_S$ as a free parameter of the PDF fit increases the uncertainty on
the gluon density at low $x$.
Adding jet data, reduces the correlation between $\alpha_S$ and the gluon PDF and allows for an in-situ $\alpha_S$
determination.
The result obtained within the HERAPDF1.6 fit is $\alpha_S (M_Z)$ = 0.1202 $\pm$ 0.0013 (exp.) $\pm$ 0.0007 (model/param.)
$\pm$ 0.0012 (hadr.) ${}^{+0.0045}_{-0.0036}$ (scale).
\subsection{Photoproduction}
\label{sec:gammap}
Besides the DIS regime ($Q^2 >$ 1 GeV${}^2$), where a virtual boson is exchanged,
there is photoproduction ($Q^2 \approx 0$~GeV${}^2$), where a quasi-real photon is emitted from the incoming lepton.
In direct photoproduction, the photon directly takes part in the interaction, while
the photon acts as a source of partons in events with resolved photoproduction.
The latter are rather similar to hadron-hadron collisions since the cross section involves a photon PDF,
there will be a photon remnant and the event can be affected by multi-parton interactions.
As shown in figure~\ref{fig:DESY-12-045},
the ZEUS Collaboration published new double-differential jet cross sections in photoproduction events
with a center-of-mass energy of the photon-proton system between 142 and 293~GeV \cite{Abramowicz:2012jz}.
Compared to DIS measurements, this analysis has a relatively high reach of transverse jet energies, $E_T$, up to 80~GeV.
Overall, a reasonable agreement between data and the NLO prediction is observed.
Discrepancies at low $E_T$ and large pseudorapidities, $\eta$, could come from non-perturbative effects,
in particular related to multi-parton interaction, or the photon PDF.
The impact of the proton PDF uncertainty is in turn much smaller and relevant only at high $E_T$.
The ZEUS Collaboration employed these jet cross sections to measure $\alpha_S$.
The considered range in $d \sigma / d E_T$ was restricted to 21 $< E_T <$ 71~GeV in order to reduce
potential non-perturbative effects and the impact of the proton PDF uncertainty.
Based on the $k_T$ jet algorithm, an $\alpha_S (M_Z)$ of $0.1206\ ^{+0.0023}_{-0.0022}\ {\rm (exp.)}\ ^{+0.0042}_{-0.0035}\ {\rm (th.)}$
was obtained.
The dominant uncertainties are related to the photon PDF and terms beyond NLO.
Results based on the anti-$k_T$ and SIScone algorithms are almost identical.
\begin{figure}
\centering
\includegraphics[width=.96\columnwidth,clip]{DESY-12-045_8}
\caption{Inclusive jet cross section in photoproduction measured by the ZEUS Collaboration as a function of
the transverse jet energy in different rapidity regions and compared to the NLO prediction \cite{Abramowicz:2012jz}.}
\label{fig:DESY-12-045}
\end{figure}
\section{Measurements at the Tevatron}
\label{sec:Tevatron}
The inclusive jet cross-section in collisions of protons with anti-protons at $\sqrt{s}$ = 1.96~TeV
measured with high precision by the CDF and D0 Collaborations up to jet $p_T$ of 600~GeV
\cite{Aaltonen:2008eq,Abazov:2008ae}, serve as one of the main constraints on the gluon density for $x >$ 0.2-0.3
in global PDF fits already since several years.
Both Collaborations also measured differential dijet and trijet cross sections with significant sensitivity to
differences between the available PDF sets (see e.g. \cite{Abazov:2011ub}).
The inclusion of such multijet cross sections in PDF fits could again help to decorrelate the value of
$\alpha_S$ and the gluon density.
The D0 Collaboration performed an $\alpha_S$ measurement from the inclusive jet cross section \cite{Abazov:2009nc}.
Since the same data had already been used as input for the MSTW2008 PDF set, on which this $\alpha_S$ analysis was based,
the phase space with $x \gtrsim$ 0.25, where the impact on the gluon PDF had been significant, was neglected.
Although that restricted the analysis to 22 data points up to jet $p_T$ of 145~GeV,
the extracted $\alpha_S (M_Z)$ of 0.1161 ${}^{+0.0034}_{-0.0033}$ (exp.) ${}^{+0.0010}_{-0.0016}$ (non-pert.) ${}^{+0.0011}_{-0.0012}$ (PDF)
${}^{+0.0025}_{-0.0029}$ (scale) is still the most precise based on jet data from hadron-hadron collisions to date.
Recently, the DO Collaboration performed a new $\alpha_S$ determination using the observable
$R_{\Delta R} (p_T,\Delta R,p_{T \rm min}^{\rm nbr})
= \sum_{i=1}^{N_{\rm jet}(p_T)} N_{\rm nbr}^{(i)}(\Delta R,p_{T \rm min}^{\rm nbr}) / N_{\rm jet}(p_T)$ with $N_{\rm nbr}$ being the number
of neighboring jets within a distance of $\Delta R$ that have a transverse momentum above $p_{T \rm min}^{\rm nbr}$
\cite{:2012xib}.
This normalized triple-differential cross section, shown in figure~\ref{fig:1207-4957},
was found to be well described by the NLO prediction
for $p_{T \rm min}^{\rm nbr}$ = 50~GeV and higher.
$R_{\Delta R}$ allowed probing $\alpha_S (p_T)$ up to jet $p_T$ of 400~GeV
and mostly independent of assumptions on the running.
By combining all data points with $p_{T \rm min}^{\rm nbr}$ = 50, 70 and 90 GeV, an $\alpha_S (M_Z)$ of
0.1191 ${}^{+0.0008}_{-0.0009}$ (exp.) ${}^{+0.0002}_{-0.0001}$ (non-pert.) ${}^{+0.0010}_{-0.0024}$ (PDF)
${}^{+0.0046}_{-0.0066}$ (scale) was obtained.
This result has a remarkable experimental precision and reduced uncertainties related to non-perturbative
corrections but a stronger dependence on the choice of the renormalization and factorization scales,
which emphasizes the need for predictions beyond NLO for multijet topologies.
\begin{figure}
\centering
\includegraphics[width=\columnwidth,clip]{1207-4957_fig01}
\caption{Average number of neighboring jets measured by the D0 Collaboration, shown in comparison to the NLO prediction
and as a function of the inclusive jet transverse momentum, for three different intervals in the jet-jet distance
and four different requirements of minimum transverse momentum of the neighboring jet \cite{:2012xib}.}
\label{fig:1207-4957}
\end{figure}
\section{Measurements at the LHC}
\label{sec:LHC}
Both the ATLAS and the CMS Collaboration measured double-differential cross sections for
inclusive jet and dijet production at $\sqrt{s} =$~7~TeV using up to 37~pb${}^{-1}$ from
the 2010 dataset \cite{Aad:2011fc,CMS:2011ab},
where the lower event pile-up allowed to measure down to relatively low jet momenta and dijet masses,
and using the 5~fb${}^{-1}$ of the
2011 dataset \cite{ATLAS-CONF-2012-021,:2012bz}, where the jet trigger and selection thresholds had
to be raised but the huge amount of data collected yields small statistical uncertainties even at jet momenta up to
2~TeV and dijet masses up to 5~TeV.
Overall, a reasonable agreement is observed between data and the NLO prediction.
Experimental and theoretical uncertainties on these cross sections are of comparable size, where
the jet energy scale is the dominant experimental uncertainty
and PDF uncertainties give a significant contribution to the total uncertainty on the predicted cross sections.
There are some discrepancies between the central predictions obtained with different PDF sets.
However, these differences are well covered by the uncertainties.
Beginning of 2011, the LHC delivered proton-proton collisions at $\sqrt{s} =$~2.76~TeV,
which the ATLAS Collaboration used to measure the inclusive jet cross section up to jet $p_T$ of 430~GeV
\cite{ATLAS-CONF-2012-128}.
For given ranges in jet $p_T$ and rapidity, $y$,
these cross sections probe jet production at significantly different $x$ and $Q^2$
compared to $\sqrt{s} =$~7~TeV.
Despite the small size of the dataset (0.2~pb${}^{-1}$), the statistical uncertainty on these cross sections
at $\sqrt{s} =$~2.76~TeV is smaller than the total systematic uncertainty.
A crucial aspect here is that the jet reconstruction, energy scale and resolution are common to the analyses at 2.76
and 7~TeV, which means that the main systematic uncertainties can be assumed to be fully correlated.
This results in drastically reduced uncertainties on cross-sections ratios,
$\rho (p_T, y) = \sigma_{\text{jet}}^{\text{2.76 TeV}} (p_T, y) / \sigma_{\text{jet}}^{\text{7 TeV}} (p_T, y)$.
PDF uncertainties are found to dominate the uncertainty on the prediction for these ratios and to be generally larger than
the uncertainty on the measured ratio.
The ATLAS Collaboration studied the impact of the jet cross sections at $\sqrt{s} =$~2.76 and 7~TeV
with all their correlations using the HERAFitter framework.
Compared to parton distribution functions obtained based on HERA DIS data only,
the inclusion of the ATLAS jet data was found to result in a harder gluon distribution with a reduced uncertainty
at high $x$, which is illustrated in figure~\ref{fig:ATLAS-CONF-2012-128},
while slightly decreasing the high-$x$ sea-quark density.
\begin{figure}
\centering
\includegraphics[width=.9\columnwidth,clip]{ATLAS-CONF-2012-128_fig_18a}
\caption{Gluon density, shown together with its relative uncertainty and as a function of the fractional parton momentum,
resulting from a fit to HERA data only (filled area) and when adding ATLAS jet data (hatched area) \cite{ATLAS-CONF-2012-128}.}
\label{fig:ATLAS-CONF-2012-128}
\end{figure}
A first $\alpha_S$ determination based on LHC data was performed using the inclusive jet cross section
from the ATLAS experiment at $\sqrt{s} =$~7~TeV up to jet $p_T$ of 600~GeV \cite{Malaescu:2012ts}.
The result obtained by
simultaneously fitting different rapidity bins up to $|y| <$ 4.4
is $\alpha_S (M_Z)$ = 0.1151 $\pm$ 0.0047 (stat.) $\pm$ 0.0014 ($p_T$ range) $\pm$ 0.0060 (jet size)
${}^{+0.0044}_{-0.0011}$ (scale) ${}^{+0.0024}_{-0.0018}$ (PDF) ${}^{+0.0009}_{-0.0034}$ (non-pert. corr.), where
the largest uncertainty arises from a disagreement seen between the values obtained with jet radii $R = 0.4$ and 0.6.
The CMS Collaboration has recently performed a measurement of $R_{32}$, the ratio between the inclusive trijet and dijet
production rates \cite{CMS-PAS-QCD-11-003}.
Again, both the experimental and the theoretical uncertainties are drastically reduced by taking a cross-section ratio.
When selecting jets with $p_T >$ 150~GeV, the 2011 dataset allows probing the kinematic region
of 250 $< \langle p_{T1,2} \rangle <$ 1400 GeV, where $\langle p_{T1,2} \rangle$ is the average $p_T$ of the two
leading jets in the event.
Overall, the measured $R_{32}$ is well described by the NLO prediction.
Only when using the ABM11 PDF set, the prediction significantly underestimated the data up to $\langle p_{T1,2} \rangle$ of
600~GeV.
$R_{32}$ is proportional to $\alpha_S$, as illustrated in figure~\ref{fig:QCD-11-003},
and it has the advantage of a reduced dependence on assumption for the low-$Q^2$ evolution
made in the PDFs.
In order to reduce the impact of uncertainties related to choice and variation of the factorization and renormalization
scales, events with $\langle p_{T1,2} \rangle <$ 400~GeV are neglected when determining the best $\alpha_S$ in a
simultaneous $\chi^2$ fit to the different bins up to $\langle p_{T1,2} \rangle$ = 1.4 TeV,
which yields a logarithmic mean of 764~GeV for the considered range.
The result is then evolved downwards in energy to obtain
$\alpha_{S}(M_{Z})$ = 0.1143 $\pm$ 0.0064 (exp.) $\pm$ 0.0019 (PDF)$\, _{-0.0000}^{+0.0050}$(scale). This result
is based on the NNPDF2.1 distribution functions.
Compatible results are obtained with CT10 and MSTW2008 PDF sets,
while the low gluon density in ABM11 would yield an $\alpha_{S}(M_{Z})$ larger than 0.1200, which is the upper edge
of the $\alpha_{S}(M_{Z})$ values supported by that PDF group.
\begin{figure}
\centering
\includegraphics[width=.9\columnwidth,clip]{QCD-11-003}
\caption{Ratio of the inclusive 3-jet to 2-jet cross sections
measured by the CMS Collaboration as a function of the average transverse momentum of the two leading
jets and compared to the NLO prediction with different $\alpha_S (M_Z)$ values \cite{CMS-PAS-QCD-11-003}.}
\label{fig:QCD-11-003}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth,clip]{alphas_summary}
\caption{All jet-data based $\alpha_S (M_Z)$ results discussed in this article compared to the latest world average
for $\alpha_S (M_Z)$. \cite{Kogler}}
\label{fig:alphas_summary}
\end{figure}
\bigskip
\section{Conclusions}
\label{sec:conclusions}
Cross sections for jet and multijet production have been measured with high precision at HERA, the Tevatron and the
LHC, probing QCD predictions over impressive ranges in $x$ and $Q^2$.
They provide strong constraints on the gluon PDF at medium to high $x$,
allow measuring $\alpha_S$ up to the TeV scale already and disentangling gluon PDF and $\alpha_S$.
Figure~\ref{fig:alphas_summary} compares recent jet-based $\alpha_S (M_Z)$ results,
all of them perfectly compatible with the latest world average but affected by significant theory uncertainties.
Given the sizable uncertainties related to choice and variation of factorization and renormalization scales as well as from
non-perturbative corrections, all such jet measurements and subsequent extractions of QCD parameters are eagerly
awaiting cross-section predictions beyond NLO.
|
1,314,259,995,809 | arxiv | \section{Introduction}\label{sec:intro}
A \emph{residuated binar} is an algebra ${\mathbf A}=(A,\wedge,\vee,\cdot,\backslash,\slash)$, where $(A,\wedge,\vee)$ is a lattice, $\cdot$ is a binary operation on $A$, and for all $x,y,z\in A$,
$$x\cdot y\leq z \iff x\leq z\slash y \iff y\leq x\backslash z.$$
A \emph{residuated semigroup} is a residuated binar for which $\cdot$ is associative, and a residuated binar possessing an identity element $e$ for $\cdot$ is called \emph{unital}. An expansion of a unital residuated semigroup by a constant designating the identity is called a \emph{residuated lattice} \cite{GJKO}. All of the aforementioned algebras satisfy the distributive laws\footnote{Here and throughout, to reduce the need for parentheses we assume that $\cdot$ has priority over $\backslash,\slash$, which in turn have priority over $\wedge,\vee$. We also write $x\cdot y$ as $xy$.}
\begin{equation}\label{eq:fj}
x (y\vee z) = x y\vee x z \tag*{$(\cdot\vee)$}
\end{equation}
\begin{equation}\label{eq:jf}
(x\vee y) z = x z\vee y z \tag*{$(\vee\cdot)$}
\end{equation}
\begin{equation}\label{eq:lm}
x\backslash (y\wedge z) = x\backslash y\wedge x\backslash z \tag*{$(\backslash\wedge)$}
\end{equation}
\begin{equation}\label{eq:mr}
(x\wedge y)\slash z = x\slash z\wedge y\slash z \tag*{$(\wedge\slash)$}
\end{equation}
\begin{equation}\label{eq:rj}
x\slash (y\vee z) = x\slash y\wedge x\slash z \tag*{$(\slash\vee)$}
\end{equation}
\begin{equation}\label{eq:jl}
(x\vee y)\backslash z = x\backslash z\wedge y\backslash z \tag*{$(\vee\backslash)$}
\end{equation}
However, in general neither lattice distributivity nor any of the equations
\begin{equation}\label{eq:fm}
x (y\wedge z) = x y\wedge x z \tag*{$(\cdot\wedge)$}
\end{equation}
\begin{equation}\label{eq:mf}
(x\wedge y) z = x z\wedge y z \tag*{$(\wedge\cdot)$}
\end{equation}
\begin{equation}\label{eq:lj}
x\backslash (y\vee z) = x\backslash y\vee x\backslash z \tag*{$(\backslash\vee)$}
\end{equation}
\begin{equation}\label{eq:jr}
(x\vee y)\slash z = x\slash z\vee y\slash z \tag*{$(\vee\slash)$}
\end{equation}
\begin{equation}\label{eq:ml}
(x\wedge y)\backslash z = x\backslash z\vee y\backslash z \tag*{$(\wedge\backslash)$}
\end{equation}
\begin{equation}\label{eq:rm}
x\slash (y\wedge z) = x\slash y\vee x\slash z \tag*{$(\slash\wedge)$}
\end{equation}
hold in these algebras.
If $t$ is a term in the language of residuated binars (or residuated semigroups), then the \emph{opposite of $t$} is the term $t^{\op}$ defined recursively as follows. For $x$ a variable, set $x^{\op}=x$, and if $s$ and $t$ are terms then set \mbox{$(s\cdot t)^{\op}=t^{\op}\cdot s^{\op}$}, \mbox{$(s\slash t)^{\op}=t^{\op}\backslash s^{\op}$}, $(s\backslash t)^{\op}=t^{\op}\slash s^{\op}$, $(s\wedge t)^{\op}=t^{\op}\wedge s^{\op}$, and $(s\vee t)^{\op}=t^{\op}\vee s^{\op}$ (and $e^{\op}=e$ in the presence of a multiplicative identity $e$). The opposite of an equation $s=t$ is defined by $(s=t)^{\op}=(s^{\op} = t^{\op})$. \emph{Mirror duality} for residuated binars provides that an equation $\epsilon$ holds in the variety of all residuated binars if and only if $\epsilon^{\op}$ does as well. If $\Sigma\cup\{\epsilon\}$ is a set of equations in the language of residuated binars and $\Sigma^{\op} = \{\sigma^{\op} : \sigma\in\Sigma\}$, then $\Sigma\models\epsilon$ holds in the variety of residuated binars if and only if $\Sigma^{\op}\models \epsilon^{\op}$ holds. Observe that \ref{eq:fm}$^{\op}$, \ref{eq:lj}$^{\op}$, and \ref{eq:ml}$^{\op}$ are respectively \ref{eq:mf}, \ref{eq:jr}, and \ref{eq:rm}.
In the presence of a multiplicative identity $e$, left and right prelinearity
\begin{equation}\label{eq:lp}
e\leq x\backslash y\vee y\backslash x \tag*{$(lp)$}
\end{equation}
\begin{equation}\label{eq:rp}
e\leq x\slash y\vee y\slash x \tag*{$(rp)$},
\end{equation}
have a connection to the six nontrivial distributive laws given above. In particular, \cite[Proposition 6.10]{BT2003} shows that in residuated lattices satisfying $e$-distributivity
\begin{equation}\label{eq:ed}
(x\vee y)\wedge e = (x\wedge e)\vee (y\wedge e),\tag*{$(ed)$}
\end{equation}
the equations \ref{eq:lp}, \ref{eq:ml}, and \ref{eq:lj} are pairwise equivalent, as are the equations \ref{eq:rp}, \ref{eq:rm}, and \ref{eq:jr}. Because \ref{eq:lp} and \ref{eq:rp} axiomatize semilinear residuated lattices (i.e., those that are subdirect products of totally-ordered residuated lattices) under appropriate technical hypotheses (see \cite{BT2003}), this provides one explanation of the well-known fact that all six nontrivial distributive laws hold in semilinear residuated lattices. However, a residuated lattice may satisfy all six nontrivial distributive laws even though it is not semilinear (this is the case, e.g., in lattice-ordered groups).
The dependencies among the six nontrivial distributive laws are more complicated in the absence of a multiplicative identity. Sections \ref{sec:implications} and \ref{sec:countermodels} provide a complete description of the dependencies among the nontrivial distributive laws under the hypothesis of lattice distributivity, both for residuated binars and residuated semigroups. Section \ref{sec:additional properties} provides some additional implications among the distributive laws in unital residuated binars, and in the presence of lattice complements. We conclude in Section \ref{sec:open problems} by proposing some open problems.
\section{Implications among the nontrivial distributive laws}\label{sec:implications}
A residuated binar with a distributive lattice reduct may be associated with its \emph{frame}. The frame of a lattice-distributive residuated binar $\mathbf A$ may be obtained by taking the poset of prime filters of the lattice reduct of $\mathbf A$ and endowing it with a ternary relation $R$ defined by
$$R(F,G,H) \iff F\subseteq G\cdot H,$$
where $F\cdot G = \{xy : x\in G, y\in H\}$ is the complex product of $F$ and $G$. Observe that the ternary relation $R$ on the frame of a residuated binar is antitone in its first coordinate and isotone in its second and third coordinates.
Satisfaction of either of the identities \ref{eq:lj} and \ref{eq:jr} has significant consequences for the frame of a lattice-distributive residuated binar \cite{FP2018}, and the nontrivial distributive laws may be profitably analyzed from the point of view of frames. In fact, for lattice-distributive residuated binars, each of the distributive laws introduced in the previous section may be rendered in terms of an equivalent first-order condition on the corresponding frames by application of ALBA \cite{CP2012}. For instance, the identity \ref{eq:jr} is equivalent to the condition that for all $x,y,p,q,j$,
$$[R(x,j,p) \;\&\; R(y,j,q)]\implies \exists z [x,y\leq z \;\&\; (R(z,j,p) \text{ or } R(z,j,q))].$$
On the other hand, \ref{eq:ml} is equivalent to the condition that for all $x,y,p,q,j$,
$$[R(p,x,j) \;\&\; R(q,y,j)]\implies \exists z [z\leq x,y\;\&\; (R(p,z,j)\text{ or }R(q,z,j))],$$
whereas \ref{eq:lj} is equivalent to the condition that for all $x,y,p,q,j$,
$$[R(x,p,j) \;\&\; R(y,q,j)]\implies \exists z [x,y\leq z\;\&\; (R(z,p,j)\text{ or }R(z,q,j))].$$
\begin{proposition}\label{prop:jr and ml implies lj frame}
Let $\mathbf A$ be a residuated binar with a distributive lattice reduct. If $\mathbf A$ satisfies both \ref{eq:jr} and \ref{eq:ml}, then $\mathbf A$ also satisfies \ref{eq:lj}.
\end{proposition}
\begin{proof}
Suppose that both \ref{eq:jr} and \ref{eq:ml} hold. We use the equivalent frame conditions to verify \ref{eq:lj}, so suppose that $x,y,p,q,j$ are points in the frame of $\mathbf A$ such that $R(x,p,j)$ and $R(y,q,j)$. By the frame condition for \ref{eq:ml} there exists $z'$ with $z'\leq p,q$ and one of $R(x,z',j)$ or $R(y,z',j)$. Suppose first that $R(x,z',j)$ holds. Then $R(x,z',j)$ and $R(y,q,j)$, and by monotonicity and $z'\leq q$ we have $R(x,q,j)$ and $R(y,q,j)$. Using the frame condition for \ref{eq:jr} we obtain $z$ such that $z,y\leq z$ and $R(z,q,j)$. On the other hand, if $R(y,z',j)$ holds then $R(y,z',j)$ and $R(x,p,j)$. Monotonicity and $z'\leq p$ then gives $R(y,p,j)$ and $R(x,p,j)$, and by the frame condition for \ref{eq:jr} there exists $z$ with $x,y\leq z$ and $R(z,p,j)$. In either case, there exists $z$ with $x,y\leq z$ and either $R(z,p,j)$ or $R(z,q,j)$, which completes the proof.
\end{proof}
Other results of this kind may be discovered by appealing to equivalent conditions on frames. However, an entirely algebraic treatment is also possible. The next lemma is an important step in this.
\begin{lemma}\label{lem:four variables}
Each of the following gives a pair of identities that are equivalent in residuated binars.
\begin{enumerate}
\item \ref{eq:fm} and $xz\wedge yw\leq (x\vee y)(z\wedge w)$.
\item \ref{eq:mf} and $xz\wedge yw\leq (x\wedge y)(z\vee w)$.
\item \ref{eq:lj} and $(x\vee y)\backslash (z\vee w) \leq x\backslash z \vee y\backslash w$.
\item \ref{eq:jr} and $(z\vee w)\slash (x\vee y) \leq z\slash x\vee w\slash y$.
\item \ref{eq:ml} and $(x\wedge y)\backslash (z\wedge w)\leq x\backslash z\vee y\backslash w$.
\item \ref{eq:rm} and $(z\wedge w)\slash (x\wedge y)\leq z\slash x\vee w\slash y$.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove (1) and (3); (2) and (4) follow by a symmetric argument, and (5) and (6) follow by a proof similar to (3) and (4).
For (1), note that if $xz\wedge yw\leq (x\vee y)(z\wedge w)$ holds then by instantiating $y=x$ we obtain $xz\wedge xw\leq x(z\wedge w)$. The reverse inequality follows from the isotonicity of multiplication, so \ref{eq:fm} holds. Conversely, if \ref{eq:fm} holds then we have $xz\wedge yw\leq (x\vee y)z\wedge (x\vee y)w = (x\vee y)(z\wedge w)$.
For (3), taking $y=x$ in the inequality $(x\vee y)\backslash (z\vee w)\leq x\backslash z\vee y\backslash w$ gives \mbox{$x\backslash (z\vee w)\leq x\backslash z\vee x\backslash w$.} The reverse inequality holds because $\backslash$ is isotone in its numerator, whence \ref{eq:lj} holds. For the converse, note that \ref{eq:lj} implies $(x\vee y)\backslash (z\vee w)=(x\vee y)\backslash z\vee (x\vee y)\backslash w\leq x\backslash z\vee y\backslash w$, where the last step follows because $\backslash$ is antitone in its denominator.
\end{proof}
\begin{theorem}\label{thm:algebraic implications}
Let $\mathbf A$ be a residuated binar with a distributive lattice reduct. Then:
\begin{enumerate}
\item If $\mathbf A$ satisfies both \ref{eq:jr} and \ref{eq:ml}, then $\mathbf A$ also satisfies \ref{eq:lj}.
\item If $\mathbf A$ satisfies both \ref{eq:lj} and \ref{eq:rm}, then $\mathbf A$ also satisfies \ref{eq:jr}.
\item If $\mathbf A$ satisfies both \ref{eq:fm} and \ref{eq:jr}, then $\mathbf A$ also satisfies \ref{eq:rm}.
\item If $\mathbf A$ satisfies both \ref{eq:mf} and \ref{eq:lj}, then $\mathbf A$ also satisfies \ref{eq:ml}.
\item If $\mathbf A$ satisfies both \ref{eq:ml} and \ref{eq:fm}, then $\mathbf A$ also satisfies \ref{eq:mf}.
\item If $\mathbf A$ satisfies both \ref{eq:rm} and \ref{eq:mf}, then $\mathbf A$ also satisfies \ref{eq:fm}.
\end{enumerate}
\end{theorem}
\begin{proof}
We provide proofs for (1) and (5); (2) and (6) follow by mirror duality. The others follow similarly.
For (1), suppose that $u\leq (x\vee y)\backslash (z\vee w)$. Then by residuation we get $x,y\leq x\vee y\leq (z\vee w)\slash u$, and by \ref{eq:jr} we have $x\leq z\slash u\vee w\slash u$ and also $y\leq z\slash u\vee w\slash u$. Observe that $x= x\wedge (z\slash u \vee w\slash u)$ and $y= y\wedge (z\slash u \vee w\slash u)$, and by distributivity we obtain that $x=x_1\vee x_2$ and $y=y_1\vee y_2$, where
$$x_1=x\wedge (z\slash u),$$
$$x_2=x\wedge (w\slash u),$$
$$y_1=y\wedge (z\slash u),$$
$$y_2=y\wedge (w\slash u).$$
Note that
$$x_1\leq z\slash u\implies u\leq x_1\backslash z\leq (x_1\wedge y_2)\backslash z,$$
$$x_2\leq w\slash u\implies u\leq x_2\backslash w\leq (x_2\wedge y_1)\backslash w,$$
$$y_1\leq z\slash u\implies u\leq y_1\backslash z\leq (x_2\wedge y_1)\backslash z,$$
$$y_2\leq w\slash u\implies u\leq y_2\backslash w\leq (x_1\wedge y_2)\backslash w.$$
Hence we get that $u\leq (x_1\wedge y_2)\backslash (z\wedge w)\leq x_1\backslash z\vee y_2\backslash w$ and likewise \mbox{$u\leq (x_2\wedge y_1)\backslash (z\wedge w)\leq x_2\backslash z\vee y_1\backslash w$}. Also, $u\leq x_1\backslash z\leq x_1\backslash z\vee y_1\backslash w$ and $u\leq y_2\backslash w\leq x_2\backslash z\vee y_2\backslash w$. This implies that:
\begin{align*}
u &\leq (x_1\backslash z\vee y_2\backslash w)\wedge (x_2\backslash z\vee y_1\backslash w)\wedge (x_1\backslash z\vee y_1\backslash w)\wedge (x_2\backslash z\vee y_2\backslash w)\\
&= ((x_2\backslash z\wedge x_1\backslash z)\vee y_1\backslash w)\wedge ((x_1\backslash z\wedge x_2\backslash z)\vee y_2\backslash w)\\
&= (x_1\backslash z\wedge x_2\backslash z)\vee (y_1\backslash w\wedge y_2\backslash w)\\
&= (x_1\vee x_2)\backslash z \vee (y_1\vee y_2)\backslash w\\
&= x\backslash z\vee y\backslash w.
\end{align*}
This proves that $(x\vee y)\backslash (z\vee w)\leq x\backslash z\vee y\backslash w$, whence (1) follows by Lemma \ref{lem:four variables}(3).
To prove (5), suppose that $(x\wedge y)(z\vee w)\leq u$. By residuating and \ref{eq:ml}, we obtain $z,w\leq z\vee w\leq (x\wedge y)\backslash u = x\backslash u\vee y\backslash u$. Define
$$z_1=z\wedge (x\backslash u),$$
$$z_2=z\wedge (y\backslash u),$$
$$w_1=w\wedge (x\backslash u),$$
$$w_2=w\wedge (y\backslash u),$$
and note that by the distributivity of the lattice reduct we have $z=z_1\vee z_2$ an $w=w_1\vee w_2$. This provides
$$z_1\leq x\backslash u \implies xz_1\leq u,$$
$$z_2\leq y\backslash u \implies yz_2\leq u,$$
$$w_1\leq x\backslash u \implies xw_1\leq u,$$
$$w_2\leq y\backslash u\implies yw_2\leq u,$$
whence from the isotonicity of multiplication and the middle two items above, we obtain that $y(z_2\wedge w_1)\leq u$ and $x(z_2\wedge w_1)\leq u$. This provides that $(x\vee y)(z_2\wedge w_1)=x(z_2\wedge w_1)\vee y(z_2\wedge w_1)\leq u$, and from the assumption \ref{eq:fm} and Lemma \ref{lem:four variables}(1) we conclude that $xz_2\wedge yw_1\leq u$. Now note that
\begin{align*}
xz\wedge yw &= x(z_1\vee z_2)\wedge y(w_1\vee w_2)\\
&= (xz_1\vee xz_2)\wedge (yw_1\vee yw_2)\\
&= (xz_1\wedge yw_1)\vee (xz_1\wedge yw_2)\vee (xz_2\wedge yw_1)\vee (xz_2\wedge yw_2)\\
&\leq u,
\end{align*}
where the third equation above follow from lattice distributivity. It follows that $xz\wedge yw\leq (x\wedge y)(z\vee w)$, so \ref{eq:mf} follows by Lemma \ref{lem:four variables}(2). This gives (5).
\end{proof}
The implications articulated in Theorem \ref{thm:algebraic implications} are described by the directed graph in Figure \ref{fig:implications}. Each pair of identities given on the left-hand side (respectively, right-hand side) of the graph jointly imply their common successor on the right-hand side (respectively, left-hand side). Note that these consequences are hidden in the special case of $e$-distributive residuated lattices addressed in \cite{BT2003}, where taken individually \ref{eq:ml} and \ref{eq:lj} are equivalent, as are \ref{eq:jr} and \ref{eq:rm}.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.4]
\tikzset{vertex/.style = {shape=circle,draw,fill=white,inner sep=1.5pt, minimum size=2em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (a) at (0,8) {$\vee\slash$};
\node[vertex] (b) at (0,4) {$\wedge\backslash$};
\node[vertex] (c) at (0,0) {$\cdot\wedge$};
\node[vertex] (d) at (8,8) {$\backslash\vee$};
\node[vertex] (e) at (8,4) {$\slash\wedge$};
\node[vertex] (f) at (8,0) {$\wedge\cdot$};
\draw[edge] (4,7) to (d);
\draw[edge] (4,3) to (e);
\draw[edge] (4,-1) to (f);
\draw[edge] (4,9) to (a);
\draw[edge] (4,5) to (b);
\draw[edge] (4,1) to (c);
\draw (a) to (4,7);
\draw (b) to (4,7);
\draw (a) to (4,3);
\draw (c) to (4,3);
\draw (b) to (4,-1);
\draw (c) to (4,-1);
\draw (d) to (4,9);
\draw (e) to (4,9);
\draw (d) to (4,5);
\draw (f) to (4,5);
\draw (e) to (4,1);
\draw (f) to (4,1);
\end{tikzpicture}
\end{center}
\caption{Dependencies among the nontrivial distributive laws.}
\label{fig:implications}
\end{figure}
\section{The poset of subvarieties}\label{sec:countermodels}
The class of residuated binars with distributive lattice reducts forms a finitely-based variety $\sf RB$, and the implications announced in Theorem \ref{thm:algebraic implications} entail some inclusions among the subvarieties of $\sf RB$ determined by the nontrivial distributive laws. We will show that these are all of the inclusions among such subvarieties, completely describing the subposet of the subvariety lattice of $\sf RB$ whose elements are axiomatized (modulo the theory of $\sf RB$) by any collection of the nontrivial distributive laws. The same analysis holds for residuated semigroups as well.
\begin{figure}\label{fig:lattice reducts}
\begin{center}
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,fill=white,inner sep=1.5pt}}
\tikzset{edge/.style = {-,> = latex'}}
\node[vertex,label=left:$\top$] (b) at (0,-1) {};
\node[vertex,label=left:$a$] (c) at (-1,-2) {};
\node[vertex,label=right:$b$] (d) at (1,-2) {};
\node[vertex,label=left:$\bot$] (e) at (0,-3) {};
\draw[edge] (b) to (c);
\draw[edge] (b) to (d);
\draw[edge] (c) to (e);
\draw[edge] (d) to (e);
\end{tikzpicture}\hspace{0.25 in}
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,fill=white,inner sep=1.5pt}}
\tikzset{edge/.style = {-,> = latex'}}
\node[vertex,label=left:$\top$] (a) at (0,0) {};
\node[vertex,label=left:$c$] (b) at (0,-1) {};
\node[vertex,label=left:$a$] (c) at (-1,-2) {};
\node[vertex,label=right:$b$] (d) at (1,-2) {};
\node[vertex,label=left:$\bot$] (e) at (0,-3) {};
\draw[edge] (a) to (b);
\draw[edge] (b) to (c);
\draw[edge] (b) to (d);
\draw[edge] (c) to (e);
\draw[edge] (d) to (e);
\end{tikzpicture}\hspace{0.25 in}
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,fill=white,inner sep=1.5pt}}
\tikzset{edge/.style = {-,> = latex'}}
\node[vertex,label=left:$\top$] (b) at (0,-1) {};
\node[vertex,label=left:$a$] (c) at (-1,-2) {};
\node[vertex,label=right:$b$] (d) at (1,-2) {};
\node[vertex,label=left:$c$] (e) at (0,-3) {};
\node[vertex,label=left:$\bot$] (f) at (0,-4) {};
\draw[edge] (b) to (c);
\draw[edge] (b) to (d);
\draw[edge] (c) to (e);
\draw[edge] (d) to (e);
\draw[edge] (e) to (f);
\end{tikzpicture}
\end{center}
\caption{Labeled Hasse diagrams for the lattice reducts of $\mathbf A_1$, $\mathbf A_2$, $\mathbf A_3$ (left), $\mathbf A_4$, $\mathbf A_5$ (middle) and $\mathbf A_6$ (right).}
\end{figure}
\begin{proposition}\label{prop:no other implications}
Theorem \ref{thm:algebraic implications} gives the only implications among the six nontrivial distributive laws modulo the theory of residuated binars. The same holds for residuated semigroups.
\end{proposition}
\begin{proof}
For each $i\in\{1,2,3,4,5,6\}$ we define a residuated binar $\mathbf A_i$. The lattice reducts of each $\mathbf A_i$ is given in Figure \ref{fig:lattice reducts}. We provide operation tables for $\cdot$ in each $\mathbf A_i$ below; the operation tables for $\backslash$ and $\slash$ are uniquely determined by these in each case. For $\mathbf A_1$, $\mathbf A_2$, and $\mathbf A_3$:
$$\begin{array}{c|cccc}
\cdot&\bot&a&b&\top\\\hline
\bot&\bot&\bot&\bot&\bot\\
a&\bot&\bot&\bot&\bot\\
b&\bot&\bot&\top&\top\\
\top&\bot&\bot&\top&\top\\
\end{array}
\qquad
\begin{array}{c|cccc}
\cdot&\bot&a&b&\top\\\hline
\bot&\bot&\bot&\bot&\bot\\
a&\bot&\bot&\bot&\bot\\
b&\bot&a&b&\top\\
\top&\bot&a&b&\top\\
\end{array}
\qquad
\begin{array}{c|cccc}
\cdot&\bot&a&b&\top\\\hline
\bot&\bot&\bot&\bot&\bot\\
a&\bot&\bot&a&a\\
b&\bot&\bot&b&b\\
\top&\bot&\bot&\top&\top\\
\end{array}$$
For $\mathbf A_4$, $\mathbf A_5$, and $\mathbf A_6$:
$$\begin{array}{c|ccccc}
\cdot&\bot&a&b&c&\top\\\hline
\bot&\bot&\bot&\bot&\bot&\bot\\
a&\bot&\top&\bot&\top&\top\\
b&\bot&b&\bot&b&b\\
c&\bot&\top&\bot&\top&\top\\
\top&\bot&\top&\bot&\top&\top\\
\end{array}
\hspace{0.08 in}
\begin{array}{c|ccccc}
\cdot&\bot&a&b&c&\top\\\hline
\bot&\bot&\bot&\bot&\bot&\bot\\
a&\bot&\top&b&\top&\top\\
b&\bot&\bot&\bot&\bot&\bot\\
c&\bot&\top&b&\top&\top\\
\top&\bot&\top&b&\top&\top\\
\end{array}
\hspace{0.08 in}
\begin{array}{c|ccccc}
\cdot&\bot&a&b&c&\top\\\hline
\bot&\bot&\bot&\bot&\bot&\bot\\
a&\bot&a&\bot&\bot&a\\
b&\bot&\bot&b&\bot&b\\
c&\bot&\bot&\bot&\bot&\bot\\
\top&\bot&a&b&\bot&\top\\
\end{array}$$
Direct calculation verifies that:
\begin{itemize}
\item $\mathbf A_1\models$ \ref{eq:rm}, \ref{eq:ml}, \ref{eq:mf}, \ref{eq:fm} and $\mathbf A_1\not\models$ \ref{eq:lj}, \ref{eq:jr}.
\item $\mathbf A_2\models$ \ref{eq:lj}, \ref{eq:ml}, \ref{eq:mf}, \ref{eq:fm} and $\mathbf A_2\not\models$ \ref{eq:jr}, \ref{eq:rm}.
\item $\mathbf A_3\models$ \ref{eq:jr}, \ref{eq:rm}, \ref{eq:mf}, \ref{eq:fm} and $\mathbf A_3\not\models$ \ref{eq:lj}, \ref{eq:ml}.
\item $\mathbf A_4\models$ \ref{eq:jr}, \ref{eq:lj}, \ref{eq:rm}, \ref{eq:fm} and $\mathbf A_4\not\models$ \ref{eq:ml}, \ref{eq:mf}.
\item $\mathbf A_5\models$ \ref{eq:jr}, \ref{eq:lj}, \ref{eq:ml}, \ref{eq:mf} and $\mathbf A_5\not\models $ \ref{eq:rm}, \ref{eq:fm}.
\item $\mathbf A_6\models$ \ref{eq:jr}, \ref{eq:lj}, \ref{eq:rm}, \ref{eq:ml} and $\mathbf A_6\not\models $ \ref{eq:fm}, \ref{eq:mf}.
\end{itemize}
Let $\epsilon\in\{$\ref{eq:jr}, \ref{eq:lj}, \ref{eq:rm}, \ref{eq:ml}, \ref{eq:mf}, \ref{eq:fm}$\}$. Then there exists a unique implication listed in Theorem \ref{thm:algebraic implications} having $\epsilon$ as its consequent. Let $\epsilon_1,\epsilon_2$ be the identities in the antecedent of the aforementioned implication. Then the above countermodels show that if $\epsilon\notin\Sigma\subseteq\{$\ref{eq:jr}, \ref{eq:lj}, \ref{eq:rm}, \ref{eq:ml}, \ref{eq:mf}, \ref{eq:fm}$\}$ and $\epsilon_1\notin\Sigma$ or $\epsilon_2\notin\Sigma$, then $\epsilon$ is not entailed by $\Sigma$.
Note that each $\mathbf A_i$, $i\in\{1,2,3,4,5,6\}$, is an associative residuated binar. The result therefore holds for residuated semigroups as well.
\end{proof}
The left-hand side of Figure \ref{fig:subvariety poset} gives the Hasse diagram of the poset of subvarieties of $\sf RB$ determined by the six nontrivial distributive laws. The coatoms in this diagram are subvarieties axiomatized modulo $\sf RB$ by a single nontrivial distributive law, and the atoms are subvarieties axiomatized by one of the four-element subsets of $\{$\ref{eq:jr}, \ref{eq:lj}, \ref{eq:rm}, \ref{eq:ml}, \ref{eq:mf}, \ref{eq:fm}$\}$ satisfied in one of the models $\mathbf A_i$ given in the proof of Proposition \ref{prop:no other implications}.
The meets in this diagram correspond to intersection of subvarieties, but in general the joins do not correspond to joins in the lattice of subvarieties.
The same diagram describes the corresponding subvariety poset for residuated semigroups since the models ${\mathbf A}_i$, $i\in\{1,2,3,4,5,6\}$, are associative.
When $\cdot$ is commutative in a residuated binar $\mathbf A$, the two residuals satisfy $x\backslash y = y\slash x$ for all $x,y\in A$ and therefore $\backslash$ and $\slash$ coincide. In this event, \ref{eq:lj} is equivalent to \ref{eq:jr}, \ref{eq:ml} is equivalent to \ref{eq:rm}, and \ref{eq:fm} is equivalent to \ref{eq:mf}. The poset of subvarieties axiomatized by the three pairwise independent nontrivial distributive laws is pictured on the right-hand side of Figure \ref{fig:subvariety poset}. The correctness of this diagram can be verified by observing that the models ${\mathbf A}_1$ and ${\mathbf A}_6$ are commutative. Since they are also associative, the same diagram describes the subvariety poset for commutative residuated semigroups.
\begin{figure}\label{fig:subvariety poset}
\begin{center}
\begin{tikzpicture}[xscale=.5, yscale = .5,
every node/.style={circle, draw, fill=white, inner sep=1.5pt},
t/.style={rectangle,draw=white,fill=white,inner sep=0pt},
g/.style={draw,fill=white,inner sep=1.5pt}
]
\draw(6,0)node{}--(1,2)node{}--(0,4)node{}--(1,6)node{}--(2,8)
--(3,6)node{}--(4,4)node{}--(5,2)node{}--(6,4)node{}--(7,6)node{}
--(8,8)--(9,6)node{}--(10,4)node{}--(11,2)node{}--(6,0)node{}
--(3,2)node{}--(2,4)node{}--(1,6)node{}--(0,8)--(3,5)node{}
--(3,2)node{}--(4,4)node{}--(5,6)node{}--(6,8)--(7,6)node{}
--(8,4)node{}--(9,2)node{}--(10,4)node{}--(11,6)node{}--(10,8)
--(7,5)node{}--(7,2)node{}--(6,0)node{}--(5,2)node{}--(5,5)node{}
--(2,8)node{\scriptsize$\wedge\!\backslash$}--(5,10)node{}--(4,8)--(3,6)node{}--(2,4)node{}
--(1,2)node{}--(7,5)node[g]{}--(4,8)node{\scriptsize$\backslash\!\vee$}--(5,6)node{}--(6,4)node{}
--(7,2)node{}--(8,4)node{}--(9,6)node{}--(10,8)node{\scriptsize$\cdot\wedge$}--(5,10)[]node{}
--(8,8)node{\scriptsize$\slash\!\wedge$}--(5,5)node[g]{}--(11,2)node{}(3,5)node{}--(9,2)node{}
--(6,0)node{}(3,5)node[g]{}--(6,8)node{\scriptsize$\vee\!\slash$}--(5,10)node{}--(0,8)node{\scriptsize$\wedge\cdot$}
(11,2)--(20,0.5)node{}--(17,-2.5)node{}--(14,0.5)node{}--(14,3.5)node{}
--(17,6.5)node{}--(20,3.5)node{}--(20,0.5)node{}--(17,3.5)node{}--(14,0.5)node{}
(5,5)--(17,3.5)--(17,6.5)--(5,10)
(6,0)--(17,-2.5)
(5,2)--(14,0.5)
(11,6)--(20,3.5)
(5,6)--(14,3.5)
(0,4)--(11,5.9)
(0,7.8)--(11,6)
(0,4)--(11,2)
(5,10.7)node[t]{$\mathsf{RB}$}
(17,7.2)node[t]{$\mathsf{CRB}$};
\end{tikzpicture}
\end{center}
\caption{The poset of subvarieties determined by the nontrivial distributive laws in varieties of residuated binars ${\sf RB}$ and commutative residuated binars ${\sf CRB}$.}
\end{figure}
\section{Identity elements, complements, and prelinearity}\label{sec:additional properties}
We say that a residuated binar is \emph{complemented} if its lattice reduct is complemented, and \emph{Boolean} if its lattice reduct is a Boolean lattice. A unital residuated binar is called \emph{integral} if it satisfies the identity $x\leq e$, where $e$ is the multiplicative identity.\footnote{This usage of \emph{integral} is typical in the study of residuated lattices, and we caution that it conflicts with the common usage in the theory of relation algebras.}
Boolean (unital) residuated binars are called $(u)r$-algebras in \cite{JJR1995}.
Note that if $\cdot$ and $\wedge$ coincide in a residuated binar $\mathbf A$, then $\mathbf A$ is term-equivalent to a Brouwerian algebra (i.e., to the bottom-free reduct of a Heyting algebra). If additionally $\mathbf A$ is a Boolean residuated binar, then $\mathbf A$ is (term-equivalent to) a Boolean algebra.
The presence of complements and an identity element in a residuated binar can have a profound impact on whether it satisfies any of the six non-trivial distributive laws, a stark example of which is illustrated by the following lemma.
\begin{lemma}\label{lem:integral implies Boolean}
Let ${\bf A}$ be a unital complemented residuated binar. If ${\bf A}$ is integral, then $\wedge$ and $\cdot$ coincide.
\end{lemma}
\begin{proof}
Since $\mathbf A$ is integral, we have $x\cdot y \leq x\wedge y$ for all $x,y\in A$. This implies for any $x\in A$ we have that $x\cdot x'\leq x\wedge x' = \bot$, where $x'$ is a complement of $x$. On the other hand, since the identity element $e$ is the greatest element of ${\bf A}$ we have also that $x\vee x'=e$ for any $x\in A$. Multiplying by $x$ and using \ref{eq:fj}, we obtain $x=x\cdot e = x\cdot(x\vee x') = x^2 \vee x\cdot x'= x^2\vee\bot = x^2$. This gives that ${\bf A}$ is idempotent, whence for any $x,y\in A$, $x\wedge y =(x\wedge y)\cdot (x\wedge y)\leq x\cdot y\leq x\wedge y$, i.e., $x\cdot y = x\wedge y$.
\end{proof}
Thus the only complemented integral residuated binars are Boolean algebras, which satisfy all six nontrivial distributive laws as well as lattice distributivity. Satisfaction of nontrivial distributive laws also often forces integrality in this setting.
\begin{lemma}\label{lem:integral}
Let $\mathbf A$ be a unital residuated binar. If $e$ has a complement $e'$ and $\mathbf A$ satisfies any one of the distributive laws \ref{eq:fm}, \ref{eq:mf}, \ref{eq:ml}, \ref{eq:rm}, then $\mathbf A$ is integral.
\end{lemma}
\begin{proof}
We prove the result for \ref{eq:fm} and \ref{eq:ml}. The result follows for \ref{eq:mf} and \ref{eq:rm} by a symmetric argument.
First, suppose that $\mathbf A$ satisfies \ref{eq:fm}. Then:
\begin{align*}
e' &= e\cdot e'\\
&\leq \top\cdot e'\\
&= \top\cdot e' \wedge \top\\
&= \top\cdot (e'\wedge e)\\
&= \top\cdot\bot\\
&= \bot
\end{align*}
where the last equality uses the identity $x\cdot\bot=\bot$, which holds in all residuated binars. It follows that $e=e\vee\bot=e\vee e'=\top$, hence $e=\top$.
Second, suppose that $\mathbf A$ satisfies \ref{eq:ml}. Note that:
\begin{align*}
\top &= \bot\backslash\bot\\
&= (e\wedge e')\backslash \bot\\
&= (e\backslash\bot)\vee (e'\backslash\bot)\\
&= \bot\vee (e'\backslash\bot)\\
&= e'\backslash\bot,
\end{align*}
giving $\top\leq e'\backslash\bot$, and by residuation $e'\cdot\top\leq\bot$. As $e\leq\top$ and $\cdot$ is isotone, we get $e'\cdot e\leq e'\cdot\top\leq\bot$. Therefore $e'\leq\bot$, so $e'=\bot$. It follows that $e'=\bot$, yielding again $e=\top$ and completing the proof.
\end{proof}
Combining the previous two lemmas gives the following result.
\begin{corollary}\label{cor:complemented to Boolean}
Let $\mathbf A$ be a complemented unital residuated binar. If $\mathbf A$ satisfies any one of the distributive laws \ref{eq:fm}, \ref{eq:mf}, \ref{eq:ml}, \ref{eq:rm}, then $\mathbf A$ is a Boolean algebra.
\end{corollary}
\begin{proof}
Since $\mathbf A$ is complemented, $e$ has a complement. Lemma \ref{lem:integral} then gives that $\mathbf A$ is integral, and so by Lemma \ref{lem:integral implies Boolean} it follows that $\mathbf A$ is a Boolean algebra.
\end{proof}
\begin{lemma}
Let $\mathbf A$ be a unital Boolean residuated binar. If $\mathbf A$ satisfies any one of the distributive laws \ref{eq:fm}, \ref{eq:mf}, \ref{eq:lj}, \ref{eq:jr}, \ref{eq:ml}, or \ref{eq:rm}, then $\mathbf A$ is integral, and hence is a Boolean algebra.
\end{lemma}
\begin{proof}
Corollary \ref{cor:complemented to Boolean} settles the claim if $\mathbf A$ satisfies any of \ref{eq:fm}, \ref{eq:mf}, \ref{eq:ml}, or \ref{eq:rm}. We therefore prove the claim for $\mathbf A$ satisfying \ref{eq:lj}; it will follow if $\mathbf A$ satisfies \ref{eq:jr} by a symmetric argument. Suppose that $\mathbf A$ satisfies \ref{eq:lj}, and note that $e\leq\top$ implies $\top\backslash e'\leq e\backslash e'=e'$. By \ref{eq:lj} and the isotonicity of $\backslash$ in its numerator, we have:
\begin{align*}
\top &= \top\backslash\top\\
&= \top\backslash (e\vee e')\\
&= \top\backslash e\vee \top\backslash e'\\
&\leq \top\backslash e\vee e'.\\
\end{align*}
Hence $\top\backslash e\vee e' = \top$, so $(\top\backslash e)'\wedge e =\bot$. Because $\wedge$ has a residual $\to$ in any Boolean residuated binar, we get $e\leq(\top\backslash e)'\to \bot=(\top\backslash e)''=\top\backslash e$. By residuating with respect to $\cdot$, we obtain that $\top\leq e$, and hence $\top = e$.
\end{proof}
\begin{corollary}
In a unital Boolean residuated binar each of the identities \ref{eq:fm}, \ref{eq:mf}, \ref{eq:lj}, \ref{eq:jr}, \ref{eq:ml}, and \ref{eq:lm} is logically-equivalent to the other five.
\end{corollary}
The two prelinearity equations \ref{eq:lp} and \ref{eq:rp} are not expressible in the absence of a multiplicative identity $e$, but for unital residuated binars they enjoy a connection to the nontrivial distributive laws even in the absence of associativity. In particular, inspection of the proofs offered in \cite{BT2003} verifies that in a unital residuated binar satisfying
$$(x\vee y)\wedge e = (x\wedge e)\vee (y\wedge e),$$
each of \ref{eq:rm} and \ref{eq:jr} implies \ref{eq:lp}, and each of \ref{eq:ml} and \ref{eq:lj} implies \ref{eq:rp}. Without associativity, the converse implications fail. To see this, we may define a five-element residuated binar $\mathbf A_7$ whose lattice reduct is pictured in Figure \ref{fig:prelinearity}. The multiplication $\cdot$ on $\mathbf A_7$ is given in the following table:
$$\begin{array}{c|ccccc}
\cdot&\bot&a&b&e&\top\\\hline
\bot&\bot&\bot&\bot&\bot&\bot\\
a&\bot&a&\bot&a&e\\
b&\bot&\bot&b&b&\top\\
e&\bot&a&b&e&\top\\
\top&\bot&a&\top&\top&\top\\
\end{array}$$
The residuals $\backslash$ and $\slash$ are determined uniquely by the above table as well, and with these operations we have $\mathbf A_7\models$ \ref{eq:lp}, \ref{eq:rp}, but each of \ref{eq:rm}, \ref{eq:jr}, \ref{eq:ml}, and \ref{eq:lj} fail in $\mathbf A_7$. Note also that $\mathbf A_7\not\models$ \ref{eq:fm},\ref{eq:mf}, whence prelinearity does not entail either of the latter distributive laws.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,fill=white,inner sep=1.5pt}}
\tikzset{edge/.style = {-,> = latex'}}
\node[vertex,label=left:$\top$] (a) at (0,0) {};
\node[vertex,label=left:$e$] (b) at (0,-1) {};
\node[vertex,label=left:$a$] (c) at (-1,-2) {};
\node[vertex,label=right:$b$] (d) at (1,-2) {};
\node[vertex,label=left:$\bot$] (e) at (0,-3) {};
\draw[edge] (a) to (b);
\draw[edge] (b) to (c);
\draw[edge] (b) to (d);
\draw[edge] (c) to (e);
\draw[edge] (d) to (e);
\end{tikzpicture}
\end{center}
\caption{Hasse diagram for the lattice reduct of $\mathbf A_7$.}
\label{fig:prelinearity}
\end{figure}
\section{Open problems}\label{sec:open problems}
Lattice distributivity is a key ingredient in the known proofs of Theorem \ref{thm:algebraic implications}, whether purely algebraic or by equivalent frame conditions. We do not know whether any of the implications announced hold in all residuated binars (without assuming lattice distributivity), nor do we know whether any of these implications fail in this more general setting.
When present, a multiplicative identity element plays a decisive role in shaping the connection between the nontrivial distributive laws. Known characterizations of when a residuated binar may be embedded in a unital residuated binar crucially involve terms of the form $x\backslash x$ and $x\slash x$ (see \cite{B1999,JJR1995}), and we conjecture that conditions involving terms of this form may provide a more satisfying account of the role of a multiplicative identity in this context. In particular, it would be interesting to identify analogues of prelinearity in the non-unital setting and explicate their connection to the nontrivial distributive laws and semilinearity.
\bibliographystyle{plain}
|
1,314,259,995,810 | arxiv | \section{Introduction}
\label{Introduction}
\begin{wrapfigure}{r}{0.5\textwidth}
\centering
\begin{tikzpicture}[very thick,scale=0.63
\draw[black] (0,0)circle (2cm);
\draw[gray, thick] (-2.1,0) node[left,black]{$0$} -- (-1.9,0)
(2.1,0)node[right,black]{$\pi r_c$} -- (1.9,0);
\draw[<-,>=stealth,thick,gray] (0,2) -- (0,1.7);
\draw[<-,>=stealth,thick,gray] (0,-2) -- (0,-1.7);
\draw[<->,>=stealth,thick,gray] (1,1.7) -- (1,-1.7);
\draw[<->,>=stealth,thick,gray] (-1,1.7) -- (-1,-1.7);
\draw[thick, gray] (0,1)node[above]{$+y$} -- (0,-1)node[below]{$-y$};
\draw[<->, ultra thick] (3.5,0) --(4.5,0);
\draw[|-|,thick] (5,0)node[below=2mm]{UV} -- node[above]{$0\leq y\leq L$} (9,0)node[below=2mm]{IR};
\end{tikzpicture}
\caption{Cartoon of RS1 geometry.}
\label{fig_rs_geometry}
\end{wrapfigure}
The seminal work of Randall and Sundrum (RS) \cite{Randall:1999ee} provides an elegant solution to {\it the hierarchy problem}. Their proposal involves an extra-dimension with a non-trivial warp factor due to the assumed anti-de Sitter (AdS) geometry along the extra-dimension. In their model an AdS geometry on an $S_1/\mathbb{Z}_2$ orbifold is considered which is equivalent to a line-element $0\leq y\leq L$, where $y$ is the coordinate of the fifth-dimension and $L=\pi r_c$,
with $r_c$ being the radius of the circle in the fifth-dimension. Moreover, their model involves two D3-branes localized on the fixed points of the orbifold, a ``UV-brane'' at $y=0$ and an ``IR-brane'' at $y=L$ (our nomenclature will become clear below), see Fig. \ref{fig_rs_geometry}.
The solution for the RS geometry is \cite{Randall:1999ee,Randall:1999vf},
\beq
ds^2=e^{2A(y)}\eta_{\mu\nu}dx^\mu dx^\nu+dy^2, \lsp \text{with}\lsp A(y)=-k|y|, \label{metric}
\eeq
where $k$ is the inverse of the AdS radius. In the original RS1 model \cite{Randall:1999ee} it was assumed that the Standard Model (SM) is localized on the IR-brane, whereas gravity is localized on the UV-brane and propagates through the bulk to the IR-brane. They famously showed that if the 5D fundamental theory involves only one mass scale
$M_\ast$ -- the Planck mass in 5D -- then, due to the presence of non-trivial warping along the extra-dimension, the effective mass scale on the IR-brane is rescaled to $m_{KK}\sim ke^{-kL}\sim\co(\;\hbox{TeV})$ and hence ameliorates the hierarchy problem for mild values of $kL\sim\co(35)$.
Soon after the RS proposal, many important improvements to the model were considered. First, a stabilization mechanism for the RS1 setup was proposed by Goldberger and Wise \cite{Goldberger:1999uk}; it employs a real scalar field in the bulk of AdS geometry with localized potentials on both of the branes, see also \cite{DeWolfe:1999cp}. A second interesting observation, which could potentially solve the fermion mass hierarchy problem within the SM, was made by many groups \cite{Davoudiasl:1999tf,Grossman:1999ra,Chang:1999nh,Gherghetta:2000qt,Huber:2000ie}. The core idea of these works was to allow all the SM fields to propagate in the RS1 bulk, except the Higgs field which was kept localized on the IR-brane. In this way, the zero-modes of these bulk fields correspond to the SM fields and the overlap of $y$-dependent profiles of fermionic fields with the Higgs field could generate the required fermion mass hierarchy. To suppress the electroweak (EW) precision observables, the symmetry of the gauge group was enhanced by introducing custodial symmetry in Ref. \cite{Agashe:2003zs}. The common lore, in the RS1 model and its extensions, was to keep the Higgs field localized on the IR-brane in order to solve the hierarchy problem. The first attempt to consider the Higgs field in the bulk of RS1 was made by Luty and Okui \cite{Luty:2004ye}. They employed AdS/CFT duality~\footnote{For the phenomenological applications of AdS/CFT with RS1 geometry, see for example \cite{ArkaniHamed:2000ds,Rattazzi:2000hs}.} to argue that a {\it bulk Higgs} scenario can address the hierarchy problem by making the Higgs mass operator marginal in the dual CFT.
A study of electroweak symmetry breaking (EWSB) within the bulk Higgs scenario was first performed in the RS1 setup by Davoudiasl et al. \cite{Davoudiasl:2005uu}; they showed that the zero-mode of the bulk Higgs is tachyonic and hence could lead to a vacuum expectation value (vev) at the TeV scale. Recently there have been many studies where a bulk Higgs scenario is considered from different perspectives --- see for example: a study with custodial symmetry in the Higgs sector\cite{Cacciapaglia:2006mz}; models with a soft wall setup \cite{Falkowski:2008fz}; bulk Higgs mediated FCNC's \cite{Azatov:2009na}; suppression of EW precision observables by modifying the warped metric near the IR-brane \cite{Cabrer:2010si,Cabrer:2011fb,Cabrer:2011vu}; and, a bulk Higgs as the modulus stabilization field (Higgs--radion unification) \cite{Geller:2013cfa}. Different phenomenological aspects after the Higgs discovery were explored in \cite{Archer:2012qa,Frank:2013un,Malm:2013jia,Cox:2013rva,Archer:2014jca,Dillon:2014zea,Agashe:2014jca,Iyer:2015ywa}. These phenomenological studies show that the RS1 model with bulk SM fields and its descendants with modified geometry (RS-like warped geometries in general) are consistent with the current experimental bounds and EW precision data.
A separate category of generalization of the RS models is based on the assumption that the singular branes are replaced
with thick branes which are smooth field configurations of the bulk scalar field, see e.g. \cite{DeWolfe:1999cp} and
\cite{Ahmed:2012nh,Ahmed:2013mea}.
As we discussed above, RS-like warped geometries, being consistent with the experimental data, offer an attractive solution to many of the fundamental puzzles of the SM, mostly through geometric means. In the same spirit, one can ask if RS-like warped extra-dimensions can shed some light on another outstanding puzzle of SM, the lack of a candidate for dark matter (DM) which constitutes 83\% of the observed matter in the universe \cite{Planck:2015xua}. It appears that unlike (flat) universal extra-dimensions (UED), where the KK-modes of the bulk fields can be even and odd under KK-parity (implying that the lowest KK-odd particle (LKP) could be a natural dark matter candidate \cite{Servant:2002aq,Cheng:2002ej}), RS1-like models (involving two branes and warped bulk) are unable to offer an analogue of KK-parity. The reason lies in the fact that the RS1 geometry is just a single slice of AdS space and, since warped, cannot be symmetric around any point along the extra-dimension and hence does not allow a {\it KK-parity}. As a result it cannot accommodate a realistic dark matter candidate. To cure this problem in the warped geometries, usually extra discrete symmetries are introduced such that the SM fields are even while the DM is odd under such discrete symmetries in order to make it stable \cite{Agashe:2004ci,Panico:2008bx,Ponton:2008zv,Vecchi:2013xra}. Another way to mend this problem in warped geometries is to introduce an additional {\it hidden sector} with some local gauge symmetries such that only DM is charged under the hidden sector gauge symmetries and it couples to the SM very weakly \cite{Gherghetta:2010cq,vonHarling:2012sz}, (see also \cite{Frey:2009qb}).
An alternative to introducing additional symmetries, is to extend the RS1-like warped geometry in such a way that the whole geometric setup becomes symmetric around a fixed point in the bulk. Two $\mathbb{Z}_2$ symmetric warped configurations are possible. In the first, two identical AdS patches are symmetrically glued together at a UV fixed point, while in the second two identical AdS pathes are symmetrically glued together at an IR fixed point. The geometric configuration when the
two AdS copies are glued together at the UV fixed point will be referred as ``IR-UV-IR geometry'', whereas the
geometry corresponding to the setup when two AdS copies are glued at the IR fixed point is called ``UV-IR-UV geometry''. We will only consider the IR-UV-IR geometric setup --- it is straight forward to extend our analysis to the UV-IR-UV geometries. (A common pathology associated with this latter type of geometry is the appearance of ghosts.) We consider an interval $y\in[-L, L]$ in the extra-dimension, where on each end of the interval $y=\pm L$ there is a D3 brane with negative tension
(in Sec. \ref{IR-UV-IR background solution} it will be clear why we need negative tension branes) and at the center
of the interval, $y=0$, we place a positive tension brane where we assume that gravity is localized~\footnote{One can {\it smooth} the singular branes in our setup by appropriate scalar field configurations --- for smooth brane modeling see for example \cite{Ahmed:2012nh} and references therein.}. We call the boundary
branes ``IR-branes'' and the brane at $y=0$ we term the ``UV-brane''. The IR-UV-IR geometry and a pictorial description of such a geometric setup is shown in Fig. \ref{IR-UV-IR}. Since the brane tensions of the two IR-branes are the same, this geometry is $\mathbb{Z}_2$ symmetric. We are aware of only two earlier attempts to construct a similar setup. The first \cite{Agashe:2007jb} treated the lowest odd KK gauge mode as the DM candidate. The second employed a kink-like UV thick brane \cite{Medina:2010mu} and the corresponding dark-matter was the first odd KK-radion \cite{Medina:2011qc}.
\begin{figure}
\centerline{
\begin{tikzpicture}[very thick,rounded corners=0.5pt,line cap=round,scale=0.6]
\shadedraw[top color=blue!50,bottom color=blue!10,yslant=0.1]
(0,0) -- (2,2) -- node[above=-13pt,rotate=-90]{\small UV-brane} (2,7) -- (0,5) -- cycle;
\shadedraw[bottom color=orange!75,top color=orange!10,yslant=0.1](8,0.5) -- (9,1.5) -- node[above,rotate=-90]{\small IR-brane} (9,4) -- (8,3) -- cycle;
\shadedraw[bottom color=orange!75,top color=orange!10,yslant=0.1](-7,2.1) -- (-6,3.1) -- node[above=-1pt,rotate=-90]{\small IR-brane} (-6,5.6) -- (-7,4.6) -- cycle;
\draw[thick,black,yslant=0.1,opacity=0.5](0,5) to [out=-30,in=170]node[above right,blue,opacity=1]{\small $e^{-2k|y|}$} (8,3)
(2,7) to [out=-50,in=170] (9,4)
(2,2) to [out=8,in=180] (9,1.5)
(0,0) to [out=20,in=175] (8,0.5);
\draw[thick,black,yslant=0.1,opacity=0.5](0,5) to [out=200,in=0]node[above,blue,opacity=1]{\small $e^{-2k|y|}$} (-7,4.6)
(2,7) to [out=210,in=5] (-6,5.6)
(2,2) to [out=165,in=-5] (-6,3.1)
(0,0) to [out=145,in=-5] (-7,2.1);
\draw (0,0) node[below,blue]{\small $y=0$}
(8,0.5)node[above,blue]{\small $y=L$}
(-7,0.5)node[above,blue]{\small $y=-L$}
(5,2.5)node[above,black]{\small $\Lambda_B$}
(-3,2.5)node[above,black]{\small $\Lambda_B$};
\draw[->,>=stealth,thick,yslant=0.1] (1,1) -- (1,2.1) node[above]{\small $x^\mu$};
\draw[->,>=stealth,thick,yslant=0.1](1,1)-- (2.1,1) node[right]{\small $y$};
\end{tikzpicture}
}
\caption{The geometric configuration for IR-UV-IR setup, the parameters are defined in Sec. \ref{IR-UV-IR background solution}.}
\label{IR-UV-IR}
\end{figure}
In this work, we place all the SM fields, including the Higgs doublet, in the bulk of the IR-UV-IR geometry. We calculate the background solutions for our geometric setup without and with taking into account the backreaction of matter fields. Since only 5D Higgs doublet, present in the bulk as well on the branes, acquires $y$-dependent vev, therefore we solve the full 5D scalar-gravity coupled set of Einstein equations to get solutions which address the gauge hierarchy problem. Here 5D $SU(2)$ Higgs doublet plays the role of the Goldberger-Wise stabilization field and the values of Higgs vevs at the UV- and IR-brane fix the distance between the branes. We find that for a weak backreaction, the UV-brane Higgs vev has to be much smaller than that of the IR-brane. Moreover we show that in order to have 4D cosmological constant zero at the IR branes one needs precisely one fine-tuning, similar to RS1 \cite{Goldberger:1999uk,DeWolfe:1999cp}.
The geometric $\mathbb{Z}_2$ parity ($y\to-y$ symmetry) leads to ``warped KK-parity'', i.e. there are towers of even and
odd KK-modes corresponding to each bulk field. In the weak backreaction scenario we focus on EWSB induced by the bulk Higgs doublet and low energy aspects of the 4D effective theory for the even and odd zero-modes assuming the KK-mass scale is high enough $\sim\co(\text{few})\;\hbox{TeV}$. In the effective theory the even and odd Higgs doublets mimic a two-Higgs-doublet model (2HDM) scenario -- the truncated inert-doublet model -- with the odd doublet similar to the inert doublet but without corresponding pseudoscalar and charged scalars. All the parameters of this {\it truncated} 2HDM are determined by the fundamental 5D parameters of the theory and the choice of boundary conditions (b.c.) for the fields at $y=\pm L$. (Note that the boundary or ``jump'' conditions at $y=0$ follow from the bulk equations of motion in the case of even modes, whereas odd modes are required to be zero by symmetry.) There are many possible alternative choices for the b.c. at $\pm L$. We allow the $y$-derivative of a field to have an arbitrary value at $\pm L$ as opposed to requiring that the field value itself be zero, i.e. we employ Neumann or mixed b.c. rather than Dirichlet b.c. at $\pm L$. Only the former yields a non-trivial theory allowing spontaneous symmetry breaking (SSB), whereas the latter leads to an explicit symmetry breaking scenario in which there are no Goldstone modes and the gauge bosons do not acquire mass. With these choices, the symmetric setup yields an odd Higgs zero-mode that is a natural candidate for dark matter. We compute the one-loop quadratic (in cutoff) corrections to the two scalar zero modes within the effective theory and discuss their mass splitting. The dark matter candidate is a WIMP --- we calculate its relic abundance in the cold dark matter paradigm.
The paper is composed as follows. In Sec. \ref{KK-parity from warped geometry}, we setup the IR-UV-IR geometric configuration and provide background solutions without and with backreaction due to the presence of matter contents in the bulk and on the branes. We also discuss the manifestation of KK-parity due the $\mathbb{Z}_2$ geometric setup. Section \ref{SM EWSB due to bulk Higgs doublet} contains the main part of our work. There, we focus on EWSB for the SM gauge sector due to the bulk Higgs doublet in our ${\mathbb Z}_2$ symmetric geometry and obtain a low-energy 4D effective theory containing all the SM fields plus a real scalar -- a dark matter candidate -- which is odd under the discrete $\mathbb{Z}_2$ symmetry. In the subsequent two subsections of Sec. \ref{SM EWSB due to bulk Higgs doublet}, we consider the quantum corrections to the scalar masses below the KK-scale $\sim\co(\text{few})\;\hbox{TeV}$ and explore the possible implications of the dark-matter candidate by calculating its relic abundance. We summarize and give our conclusions in Sec. \ref{Summary}. We supplement the main text with an Abelian Higgs mechanism, with a complex scalar field and a gauge field, in our background geometry in Appendix \ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism}. In the Abelian case we lay down the foundation for SSB due to bulk Higgs, which is useful in the main text for the case of EWSB of the SM. Two apparently different approaches are considered to study SSB in the Abelian case: ({\it i}) SSB by vacuum expectation values of the KK modes; and, ({\it ii}) SSB via a vacuum expectation value of the 5D Higgs field. Low energy (zero-mode) 4D effective theories are obtained within the two approaches and we find that the effective theories are identical up to corrections of order $\co\big(m_0^2/m_{KK}^2\big)$, where $m_0$ and $m_{KK}$ are the zero-mode mass and KK-mass scale, respectively.
\section{A ${\mathbb Z}_2$ symmetric warped extra-dimension and KK-parity}
\label{KK-parity from warped geometry}
In this section we provide the background solution for the $\mathbb{Z}_2$ symmetric background (IR-UV-IR) geometry without and with backreaction due to the 5D matter fields and show how KK-parity is manifested within this warped geometric setup.
\subsection{The IR-UV-IR model: without backreaction}
\label{IR-UV-IR background solution}
We consider the IR-UV-IR warped geometry compactified on an interval,
$-L\leq y\leq L$, where a UV-brane with positive tension is located at $y=0$, and two negative tension IR-branes are located at $y=\pm L$. Note that the end points of the interval at $y=\pm L$ are not the fixed points of the $\mathbb{Z}_2$, the only fixed point
is at $y=0$, which is different from the $S_1/\mathbb{Z}_2$ orbifold (RS1 geometry) where $y=0$ and $y=L$ are both fixed points of the $\mathbb{Z}_2$.
The 5D gravity action for such a geometry, without taking into account any backreaction due to the presence of matter content, can be written as~\footnote{We use the metric signature $(-,+,+,+,+)$
and the unit system such that the 5D Planck mass $M_\ast=1$.},
\beq
S_G=\int d^5x\sqrt{-g}\left\{\frac R2-\Lambda_B-\lambda_{UV}\delta(y)-\lambda_{IR}\delta(y+L)-\lambda_{IR}\delta(y-L)\right\}+S_{GH},
\label{gravity_action}
\eeq
where $R$ is the Ricci scalar, $\Lambda_B$ is the bulk cosmological constant and $\lambda_{UV}(\lambda_{IR})$ are the brane
tensions at the UV(IR) fixed points. Above and henceforth the Dirac delta functions at $y=\pm L$ are defined in such a way that their integral is 1/2. Since our geometry is compact with boundaries, the action contains the Gibbons-Hawking boundary term, \footnote{The Gibbons--Hawking boundary term is needed in order to cancel variation of the Ricci scalar at the boundaries so that the RS metric \eqref{metric} is indeed a solution of the Einstein equations of motion.}
\beq
S_{GH}=-\int_{\partial{\cal M}}d^4x\sqrt{-\hat g} {\cal K}, \label{GH_term}
\eeq
where ${\cal K}$ is the intrinsic curvature of the surface of the boundary manifold $\partial{\cal M}$, given by
\beq
{\cal K}=-\hat g^{\mu\nu}\nabla_\mu n_\nu=\hat g^{\mu\nu}\Gamma^M_{\mu\nu}n_M,
\eeq
with $n_M$ being the unit normal vector to the surface of the boundary manifold $\partial{\cal M}$ and $\hat g_{\mu\nu}$ is the
induced boundary metric. For the 5D manifold with 4D Poincar\'e invariance ($n^5=1$ and $n^\mu=0$), the intrinsic curvature reduces to
\beq
{\cal K}=-\frac12\hat g^{\mu\nu}\partial_5\hat g_{\mu\nu}.
\eeq
The solution of the Einstein equations resulting from the above action is the RS metric \eqref{metric}, where the AdS curvature $k$ is related to $\Lambda_B$ by
\beq
\Lambda_B=-6k^2. \label{lambda_k}
\eeq
Since the above setup is compactified on an interval $y\in [-L,L]$, rather than on a circle as in RS1, one needs
to be careful and show that the solution \eqref{metric} is compatible with the boundaries and that the effective 4D
cosmological constant is zero, see also \cite{Lalak:2001fd}. We will see below that we need a {\it fine tuning} between the 5D cosmological constant
$\Lambda_B$ and the brane tensions $\lambda_{UV,IR}$ in order to get zero 4D cosmological constant. One can calculate
the effective 4D cosmological constant $\Lambda_{4}$ from the action \eqref{gravity_action} by integrating out
the extra-dimension,
\beq
\Lambda_4=-\int_{-L}^L dy\sqrt{-g}\left\{\frac R2-\Lambda_B-\lambda_{UV}\delta(y)-\lambda_{IR}\big[\delta(y+L)+\delta(y-L)\big]\right\}+\sqrt{-\hat g} {\cal K}\Big\vert_{-L}^{L}, \label{Lambda_4_a}
\eeq
where $R=-20 A^{\p2}-8A^{\p\p}$ and $\Lambda_B=-6A^{\p2}$ corresponding to the solution \eqref{metric}. Using $A(y)=-k|y|$ we find,
\beq
\Lambda_4=\left(\lambda_{UV}-6k\right)+\left(\lambda_{IR}+6k\right)e^{-4kL}, \label{Lambda_4_b}
\eeq
which can only be zero if
\beq
\lambda_{UV}=- \lambda_{IR}=6k.
\eeq
This result explicitly shows that one needs a positive tension brane at $y=0$ and two negative tension branes at $y=\pm L$ in order to obtain zero 4D cosmological constant. This is the usual fine tuning which appears in
brane world scenarios \cite{Randall:1999ee,Goldberger:1999uk,DeWolfe:1999cp}. Hence we have a 5D geometry with AdS
solution \eqref{metric} with negative bulk cosmological constant and a positive tension brane in the middle and two equal negative
tension branes at the end of the interval, see Fig. \ref{IR-UV-IR}.
We would like to mention here that we are considering a rigid IR-UV-IR geometry where the distance $L$ is tuned in order to solve the hierarchy problem. To stabilize the IR-UV-IR setup one may consider a mechanism like the one proposed by Goldberger and Wise (GW) \cite{Goldberger:1999uk,DeWolfe:1999cp} introducing a bulk scalar field with appropriate brane potentials such that the energy minimization would set the size of the 5D interval and yields a compactification scale that would solve the hierarchy problem. Since one of our aims is to analyze EWSB due to a 5D $SU(2)$ Higgs doublet in the IR-UV-IR model, therefore it is natural to consider the bulk $SU(2)$ Higgs doublet as the GW stabilizing field. We explore this option in the following subsection. A similar analysis for the case of RS1 has been considered in Ref.~\cite{Geller:2013cfa}. However, there the full scalar-gravity coupled equations are not solved,
the authors adopt a small backreaction anstaz. In this work, we are using so-called {\it superpotential method} to solve the full scalar-gravity coupled equations analytically.
\subsection{The IR-UV-IR model: with backreaction}
\label{The IR-UV-IR model: with backreaction}
In this subsection we employ an $SU(2)$ Higgs doublet in the bulk of IR-UV-IR model and obtain the background solutions for the 5D scalar-gravity coupled theory. Although our solution generating technique is adopted for a specific geometric configuration (IR-UV-IR model), this approach can be used to solve the Higgs-gravity backreaction in any warped extra-dimensional model (e.g. RS1) with a bulk Higgs. We use the following most general 4D Poincar\'e invariant metric ansatz:
\beq
ds^2=e^{2A(y)}\eta_{\mu\nu}dx^\mu dx^\nu+dy^2, \label{metric_sg}
\eeq
where $A(y)$ is a general $y$-dependent warp-function and $\eta_{\mu\nu}$ represents the 4D Minkowski metric. We consider the following scalar-gravity action for our model,
\begin{align}
S_{SG}=\int d^5x \sqrt{-g}&\Big\{\frac{R}{2}-\left\vert D_M H\right\vert^2 -V_{B}(H) \notag\\
&-V_{UV}(H)\delta(y)-V_{IR}(H)\Big[\delta(y+L)+\delta(y-L)\Big]\Big\}+S_{GH}, \label{action_sg}
\end{align}
where $R$ is the 5D Ricci scalar and $H$ is the $SU(2)$ Higgs doublet. Whereas, $V_{B}(H)$ and $V_{UV(IR)}(H)$ are the bulk and UV(IR) brane potentials, respectively.
Above, the $S_{GH}$ is the Gibbons-Hawking boundary action defined in \hbox{Eq.~}\eqref{GH_term}.
We can write the $y$-dependent vacuum expectation value (vev) of the $SU(2)$ Higgs doublet as:
\beq
\langle H\rangle=\frac{1}{\sqrt2}\bpm 0\\ \phi(y) \epm. \label{higgs_vev}
\eeq
In order to find the background solutions we need to solve the coupled scalar-gravity Einstein equations. The equations for the background fields $A(y)$ and $\phi(y)$ following from the above action \eqref{action_sg} and the metric ansatz \eqref{metric_sg} can be written as,
\begin{align}
6A^{\prime2}&=\frac{1}{2}\phi^{\prime2}-V_B(\phi),\label{eom01}\\
3A^{\prime\prime}+6A^{\prime2}&=-\frac{1}{2}\phi^{\prime2}-V_B(\phi)-V_{UV}(\phi)\delta(y)-V_{IR}(\phi)\Big[\delta(y+L)+\delta(y-L)\Big],\label{eom02}\\
\phi^{\prime\prime}+4A^{\prime}\phi^{\prime}&=\frac{\partial V_B(\phi)}{\partial \phi}+\frac{\partial V_{UV}(\phi)}{\partial \phi}\delta(y)+\frac{\partial V_{IR}(\phi)}{\partial \phi}\Big[\delta(y+L)+\delta(y-L)\Big]. \label{eom03}
\end{align}
\paragraph{Superpotential method:} In the following we will layout the so-called {\it superpotential method} for solving the above set of coupled scalar-gravity equations \cite{DeWolfe:1999cp}. Although the use of this method is motivated by supersymmetry, no supersymmetry is involved in
our setup. The method is elegant and
very efficient, in particular it applies to the system of second order differential equations
\eqref{eom01}-\eqref{eom03} and reduces them to a set of first order ordinary differential equations which are much easier to deal with. It is assumed that the scalar potential $V_B(\phi)$ could be expressed
in terms of the superpotential $W(\phi)$ as \cite{DeWolfe:1999cp,Csaki:2000zn},
\begin{equation}
V_B(\phi)=\frac{1}{8}\left( \frac{\partial W(\phi)}{\partial \phi}\right)^{2}-\frac{1}{6}W(\phi)^{2},
\label{potential}
\end{equation}
where the superpotential $W(\phi)$ satisfies,
\begin{align}
\phi^{\prime}&=\frac12\frac{\partial W(\phi)}{\partial \phi}, &A^{\prime}&=-\frac{1}{6} W(\phi),
\label{super_potential_eqs}
\end{align}
along with the following jump at $y=0$ and boundary conditions $y=\pm L$:
\begin{align}
\frac12\Big[W(\phi)\Big]_0&=V_{UV}(\phi)\Big\vert_{\phi=\phi(0)}, &\frac12\Big[\frac{\partial W(\phi)}{\partial \phi}\Big]_0&=\frac{\partial V_{UV}(\phi)}{\partial \phi}\Big\vert_{\phi=\phi(0)}. \label{jump_conditions_W}\\
W(\phi)\Big\vert_{\pm L}&=-V_{IR}(\phi)\Big\vert_{\phi=\phi(\pm L)}, &\frac{\partial W(\phi)}{\partial \phi}\Big\vert_{\pm L}&=-\frac{\partial V_{IR}(\phi)}{\partial \phi}\Big\vert_{\phi=\phi(\pm L)}. \label{boundary_conditions_W}
\end{align}
Above, the jump across the UV-brane of $j(y)=W(\phi) \text{ and }\partial W(\phi)/\partial \phi$, is defined as follows:
\beq
\big[j(y)\big]_{0}\equiv\lim_{\epsilon\to0}\big\{j(0+\epsilon)-j(0-\epsilon)\big\}. \label{jump}
\eeq
Let us consider the following form of the superpotential $W(\phi)$
\beq
W(\phi)=\left\{\begin{array}{ll} \enspace\;6k+(2+\beta)k\phi^2 &\lsp\text{for}\lsp 0<y<L\\
-6k-(2+\beta)k\phi^2 &\lsp\text{for}\hsp -L<y<0\end{array}\right., \label{superpotential}
\eeq
where $\beta\equiv\sqrt{4+\mu_B^2/k^2}$ parameterises the bulk mass $\mu_B$ of the Higgs field and $k$ is a constant of order $M_\ast$. The above form of $W(\phi)$ is $\mathbb{Z}_2$ odd under $y\to-y$, which ensures, by the virtue of \hbox{Eq.~}\eqref{super_potential_eqs}, that $A(y)$ and $\phi(y)$ are $\mathbb{Z}_2$ even. We get the scalar potential $V_B(\phi)$ from \hbox{Eq.~}\eqref{potential} as
\beq
V_B(\phi)=-6k^2+\frac12\mu_B^2\phi^2-\frac{k^2}2(2+\beta)^2\phi^4. \label{bulk_potenial}
\eeq
We employ the following forms of the brane-localized potentials,
\begin{align}
V_{UV}(\phi)&=W(\phi)+\frac{\lambda_{UV}}{4k^2}\big(\phi^2-\phi_{UV}^2\big)^2, \label{Vuv}\\
V_{IR}(\phi)&=-W(\phi)+\frac{\lambda_{IR}}{4k^2}\big(\phi^2-\phi_{IR}^2\big)^2, \label{Vir}
\end{align}
where $\phi_{UV(IR)}$ is a constant value of the background vev at $y=0 (\pm L)$ and $\lambda_{UV(IR)}$ is the quartic coupling at the UV(IR) brane.
The background vev $\phi(y)$ and the warp-function $A(y)$ can be obtained by integrating the first order equations \eqref{super_potential_eqs} as
\begin{align}
\phi(y)&=\phi_{IR}e^{(2+\beta)k(|y|-L)}, \label{scalar_phi}\\
A(y)&=-k|y| -\frac{1}{12}\phi_{IR}^2e^{-2(2+\beta)kL}\Big[e^{2(2+\beta)k|y|}-1\Big]. \label{warp_function_A}
\end{align}
Moreover, we impose the following normalization condition for the background vev:
\beq
\int_{-L}^{L}dy e^{2A(y)}\phi^2(y)=v_{SM}^2, \label{phi_vev_norm}
\eeq
where $v_{SM}$ is the SM vev.
The $\phi_{IR}$ resulting from the above normalization reads (see also \hbox{Eq.~}\eqref{vp_sol_toy}):
\beq
\phi_{IR}=v_{SM}\sqrt{k(1+\beta)}e^{kL}, \label{phi_ir}
\eeq
As we will see below in order to solve the gauge hierarchy problem, one needs $kL\simeq 37$ and for $\phi_{IR}\sim \co(M_{\text{Pl}}^{3/2})$, $k\approx \co(M_{\text{Pl}})$ and $\beta\approx\co(1)$, the above expression implies that then $v_{SM}\sim \co(\text{TeV})$.
In Figure \ref{phiAWV}, we have plotted the $y$-dependent background vev $\phi(y)$ and the warp factor $e^{A(y)}$ as function of $y$ in the left panel, while the right panel shows the superpotential $W(\phi)$ and the bulk scalar potential $V_B(\phi)$ as a function of $\phi$.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figphiA.pdf}\hsp
\includegraphics[width=0.45\textwidth]{figWV.pdf}
\end{center}
\caption{The left graph shows the $y$-dependent background vev $\phi(y)$ and the warp factor $e^{A(y)}$ as function of $y$, while the right graph illustrates the shapes of superpotential $W(\phi)$ and the bulk scalar potential $V_B(\phi)$ as a function of background vev $\phi$. The parameter choice adopted for the graphs is: $\beta=0$, $k=1$ and $\phi_{IR}=1$.}
\label{phiAWV}
\end{figure}
In order to gain some intuition, it is instructive to count number of parameters and constraints to see how many fine-tunings are necessary to obtain solutions with Minkowski D3-branes. There are three integration constants coming from the first order Eqs.~\eqref{potential}
and \eqref{super_potential_eqs}. One of these three, $A(0)$, is trivial additive constant hence we are left with two parameters to be fixed. Let us count the number of constraints: there are two constraints from the jump at $y=0$ from \hbox{Eq.~}\eqref{jump_conditions_W} and two constraints from the boundaries at $\pm L$ Eq.~\eqref{boundary_conditions_W} (note that the boundary conditions are same at $\pm L$). We can integrate \hbox{Eq.~}\eqref{potential} to fix the integration constant by requiring $W(\phi_{UV})=V_{UV}(\phi_{UV})$. This requirement together with \hbox{Eq.~}\eqref{potential} will trivially satisfy the second jump condition of \hbox{Eq.~}\eqref{jump_conditions_W}. We can then integrate the first equation of \hbox{Eq.~}\eqref{super_potential_eqs} to get $\phi(y)$ and the position of the IR-brane is fixed w.r.t. the UV-brane by requiring $\phi(L)=\phi_{IR}$. This fixes all the parameters of the theory but we are still left with two constraints \hbox{Eq.~}\eqref{boundary_conditions_W} which are not independent of each other as fixing $W(\phi_{IR})=-V_{IR}(\phi_{IR})$ will trivially satisfy second equation of \hbox{Eq.~}\eqref{boundary_conditions_W}. Hence we are left with precisely one fine-tuning which is required to ensure flat-brane solutions and is a common pathology of RS-like models, see also Ref.~\cite{DeWolfe:1999cp}.
Let us check that the 4D cosmological constant $\Lambda_4$ is indeed zero after having this fine-tuning is achieved.
The 4D cosmological constant for this scalar-gravity theory \eqref{action_sg} is given by the following integral calculated for the background solutions
\begin{align}
\Lambda_4&=-\int_{-L}^L dy\sqrt{-g}\left\{\frac R2-\frac12 \phi^{\p2}-V_B(\phi)-V_{UV}(\phi)\delta(y)-V_{IR}(\phi)\big[\delta(y+L)+\delta(y-L)\big]\right\}+\sqrt{- \hat g} {\cal K}\Big\vert_{-L}^{L} \notag\\
&=-\int_{-L}^L dye^{4A}\bigg\{6\left(A^\p+\frac16W\right)^2-\frac12\left(\phi^\p-\frac12\frac{\partial W}{\partial\phi}\right)^2-e^{-4A}\frac{d}{dy}\left[e^{4A}\left(4A^\p+\frac12W\right)\right] \notag\\
&\Lsp\hsp+\frac12\frac{\partial W}{\partial y}-V_{UV}(\phi)\delta(y)-V_{IR}(\phi)\big[\delta(y+L)+\delta(y-L)\big]\bigg\}-4e^{4A}A^\p\Big\vert_{-L}^{L} \notag\\
&=-\int_{-L}^L dye^{4A}\bigg\{6\left(A^\p+\frac16W\right)^2-\frac12\left(\phi^\p-\frac12\frac{\partial W}{\partial\phi}\right)^2\bigg\} \notag\\
&\hsp-\frac12\big[ W\big]_0+V_{UV}[\phi(0)]+\frac{e^{4A}}2W\Big\vert_{-L}^L+\frac{e^{4A(L)}}2V_{IR}[\phi(L)]+\frac{e^{4A(L)}}2V_{IR}[\phi(-L)]. \label{4d_cc}
\end{align}
Above, in the second line we have only used \hbox{Eq.~}\eqref{potential} and the value of the intrinsic curvature of the boundary manifolds ${\cal K}$. In the last line we have adopted the fact that the superpotential $W(\phi)$ \eqref{superpotential} is discontinuous at $y=0$, hence the jump at $y=0$ follows from the total derivative term. Now it is straightforward to see in the above equation that the full squares vanish due to \hbox{Eq.~}\eqref{super_potential_eqs}, whereas the last line vanishes due to the jump and boundary conditions \eqref{jump_conditions_W}-\eqref{boundary_conditions_W}. Hence we conclude that the 4D cosmological constant on the branes is indeed zero, as anticipated from our metric ansatz \eqref{metric_sg}.
The brane separation $L$ can be determined by \hbox{Eq.~}\eqref{scalar_phi} as:
\beq
kL=\frac{1}{2+\beta}\ln\left(\frac{\phi_{IR}}{\phi_{UV}}\right).
\eeq
In order to solve the gauge hierarchy problem, the number of $e$-foldings is required to be as follows
\beq
A(0)-A(L)\simeq \ln\left(\frac{M_{\text{Pl}}}{m_{EW}}\right)\approx 37. \label{e_folding}
\eeq
By using the \hbox{Eq.~}\eqref{warp_function_A}, we get:
\begin{align}
A(0)-A(L
&= \frac{1}{2+\beta}\ln\left(\frac{\phi_{IR}}{\phi_{UV}}\right)+\frac{1}{12}\big(\phi_{IR}^2-\phi_{UV}^2\big). \label{e_folding}
\end{align}
Note that for $\beta\geq0$, both terms in the above relation can contribute to give the desired number of $e$-foldings.
When the first term contributes mostly ({\it weak backreaction}), so $\phi_{IR}\gg\phi_{UV}$, then $kL\simeq37$.
Whereas, when the major contribution to the number of $e$-foldings comes from the second term ({\it strong backreaction}), i.e. $\phi_{UV}\sim\co(1)$ and $\phi_{IR}\sim\co(20)$
then $kL\sim \co(1)$.
In this paper we consider the weak backreaction scenario, i.e. $kL\simeq37$, which implies that the second term in the warp-function \eqref{warp_function_A} is negligible and hence the form of warp-function is $A(y)\simeq -k|y|$ which is the same as in the previous subsection with no backreaction. Note that the weak backreaction scenario is also required in order to insure the SM vev $v_{SM}\simeq246\;\hbox{GeV}$ below the KK-scale, see e.g. Ref.~\cite{Geller:2013cfa}. The case of strong backreaction is also interesting as the fine-tuning required is much less than that of the weak backreaction scenario but we will not consider it here.
\subsection{Warped KK-Parity}
\label{KK-Parity}
\noindent In this section we employ the background solution for the $\mathbb{Z}_2$ symmetric background (IR-UV-IR) geometry considered in Sec.~\ref{IR-UV-IR background solution} and show how KK-parity is manifested within this geometric setup. The IR-UV-IR geometry of Sec.~\ref{IR-UV-IR background solution} is $\mathbb{Z}_2$-symmetric and we will consider this symmetry to be exact for our 5D theory. If the 5D theory has this $\mathbb{Z}_2$-parity (symmetry) then the Schr\"odinger-like potential for all the fields is symmetric, resulting in even (symmetric) and odd (antisymmetric) eigenmodes. Thus, a general field $\Phi(x,y)$ can be KK decomposed
as follows
\beq
\Phi(x,y)=\sum_{n}\phi_n(x)f_n(y),
\eeq
where, due to the $\mathbb{Z}_2$ geometry, the wave functions $f_n(y)$ are either even or odd, so that
\beq
\Phi(x,y)\equiv\Phi^{(\pm)}(x,y),
\eeq
with
\begin{align}
\Phi^{(+)}(x,y)&=\sum_n\phi^{(+)}_n(x)f^{(+)}_n(y)\xrightarrow{y\to-y}+\Phi^{(+)}(x,y), \\
\Phi^{(-)}(x,y)&=\sum_n\phi^{(-)}_n(x)f^{(-)}_n(y)\xrightarrow{y\to-y}-\Phi^{(-)}(x,y).
\end{align}
Due to the geometric $\mathbb{Z}_2$ symmetry, a single odd KK-mode cannot couple to two even KK-modes in the 4D effective theory,
therefore the lowest odd KK-mode will be stable and may serve as a dark matter candidate.
Furthermore, as the geometry is $\mathbb{Z}_2$ symmetric in $y\in[-L,L]$, the continuity conditions for odd and even modes at $y=0$ strongly impact the physics scenario. Our choice will be that the odd (even) modes
satisfy Dirichlet (Neumann or mixed) boundary (jump) conditions (b.c.) at $y=0$, respectively. As for the odd modes, continuity implies that they must be zero at $y=0$, but we could also have demanded the Neumann conditions that their $y$ derivative be zero at $y=0$. We choose not to impose this additional b.c. in this work. As regards the even modes, one cannot choose Dirichlet b.c. at $y=0$ because of the presence of the UV-brane and associated ``jump" conditions following from the equations of motion.
\section{SM EWSB by bulk Higgs doublet -- the truncated-inert-doublet model}
\label{SM EWSB due to bulk Higgs doublet}
\noindent In this section we consider all the SM fields in the bulk and study phenomenological implications of our
symmetric geometry. Hereafter, we consider only the weak backreaction scenario discussed in Sec.~\ref{The IR-UV-IR model: with backreaction} and we employ the metric \eqref{metric} as a solution of the IR-UV-IR geometric background. Note also that vev of effective 4D Higgs field is of the electroweak scale which is much smaller than the gravity (Planck mass) scale (see \hbox{Eq.~}\eqref{phi_ir} and discussion below), therefore their back-reaction on the background geometry would be negligible, see sub-Sec.~\ref{The IR-UV-IR model: with backreaction} and Refs.~\cite{Geller:2013cfa,Cox:2013rva}. Moreover, we follow closely the Abelian case~\ref{EWSB by vacuum expectation value of KK modes} for EWSB of the 5D SM gauge group and as shown in Appendix~\ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism} this approach is equivalent, at the zero-mode level, to the canonical approach (Appendix~\ref{EWSB by vacuum expectation value of 5D Higgs field}), i.e. when the bulk Higgs field is expanded around the background vev.
The 5D action for the electroweak sector of the SM can be written as~\footnote{Note that the Higgs part of the Lagrangian is not exactly same as considered in Sec.~\ref{The IR-UV-IR model: with backreaction} for the analysis of the backreaction, here we neglect the quartic terms in the bulk and UV-brane potentials solely for phenomenological convenience since these terms are suppressed, see the main text below. Moreover, we introduce some convenient and familiar notations for brane potential parameters.}
\begin{align}
S=-\int d^5x \sqrt{-g}\bigg\{&\frac14 F^{a}_{MN}F^{aMN}+\frac14 B_{MN}B^{MN}+\left\vert D_M H\right\vert^2 +\mu_B^2|H|^2 \notag\\
&+V_{IR}(H)\delta(y+L)+V_{UV}(H)\delta(y)+V_{IR}(H)\delta(y-L)\bigg\}, \label{action_SM}
\end{align}
where $F^a_{MN}$ and $B_{MN}$ are the 5D field strength tensors for $SU(2)$ and $U(1)_Y$, respectively with $a$
being the number generators of $SU(2)$. Above, $H$ is the $SU(2)$ doublet and its brane potentials are
\beq
V_{UV}(H)=\frac{m^2_{UV}}{k}|H|^2, \Lsp V_{IR}(H)=-\frac{m^2_{IR}}{k}|H|^2+\frac{\lambda_{IR}}{k^2}|H|^4.
\label{boudary_potentials}
\eeq
In our approach, we do not put the Higgs quartic terms in the bulk nor on the UV-brane since we want EWSB to take place near the IR-brane. The covariant derivative $D_M$ is defined as follows:
\beq
D_M =\partial_M-i\frac{g_5}{2} \tau^aA^a_M -i\frac{g_5^\p}{2} B_M, \label{co_dir_M_SM}
\eeq
where $\tau^a$ are Pauli matrices and $g_5(g^\p_5)$ is the coupling constant for the $A_M^a(B_M)$ fields.
There is an important comment in order here concerning the above particular forms of bulk and brane-localized potentials. We have dropped quartic terms in the UV and the bulk potentials, even though there is no symmetry that would protect this choice. However there are phenomenological arguments that support such an option. As one can notice the UV Higgs quartic operator, i.e. $V_{UV}(H)\supset\lambda_{UV}/k^2 |H|^4$ is highly suppressed as $\lambda_{UV}/k^2\sim {\cal O}(M_{\text{Pl}}^{-2})$. Whereas for the IR Higgs quartic operator, suppression of $\lambda_{IR}/k^2$ is reduced to $\sim{\cal O}(m_{KK}^{-2})$ due to the non-trivial warp factor at the IR-brane, see also Refs.~\cite{Archer:2012qa,Archer:2014jca}. Similarly, the bulk quartic term would also be suppressed by some intermediate scale. For simplicity we ignore those terms.
We shell emphasize that the reasoning behind ignoring the bulk and the UV-brane localized quartic terms is purely phenomenological in its nature.
It is also worth recalling that in this work we are going to investigate the possibility of providing a dark matter candidate as a lowest mass odd scalar KK mode.
It is fair to expect that those results will not receive any substantial corrections in the presence of quartic UV and brane terms.
Besides that, this kind of phenomenological approximation of ignoring the bulk and UV-brane quartic terms is widely used in the literature on bulk Higgs scenarios in RS1-like models, see for example \cite{Cacciapaglia:2006mz,Falkowski:2008fz,Azatov:2009na,Cabrer:2010si,Cabrer:2011fb,Cabrer:2011vu,Geller:2013cfa,Archer:2012qa,Frank:2013un,Malm:2013jia,Cox:2013rva,Archer:2014jca,Agashe:2014jca}.
Note also that due to the $\mathbb{Z}_2$ geometric symmetry, physics of the full IR-UV-IR setup can be described completely by a single copy of RS1, i.e. UV-IR setup, however in that scenario each bulk field would be a subject of Neumann (or mixed) and Dirichlet boundary conditions at $y=0$ for even and odd fields, respectively.
It is instructive to make the usual redefinition of the gauge fields,
\begin{align}
W^{\pm}_M(x,y)&\equiv\frac{1}{\sqrt2}\Big(A^1_M\mp iA^2_M\Big), \\
Z_M(x,y)&\equiv\frac{1}{\sqrt{g^2_5+g^{\p2}_5}}\Big(g_5A^3_M-g^\p_5B_M\Big),\notag\\
A_M(x,y)&\equiv\frac{1}{\sqrt{g^2_5+g^{\p2}_5}}\Big(g^\p_5A^3_M+g_5B_M\Big).
\end{align}
Analogous to the 4D procedure, we define the 5D Weinberg angle $\theta$
as follows:
\beq
\cos\theta\equiv \frac{g_5}{\sqrt{g^2_5+g^{\p2}_5}}, \lsp
\sin\theta\equiv \frac{g^\p_5}{\sqrt{g^2_5+g^{\p2}_5}}. \label{sin_cos}
\eeq
The 5D gauge fields corresponding to the gauge group $SU(2)\times U(1)_Y$ are then
\beq
\mathbb{A}_M(x,y)\equiv \bpm \sin\theta A_M+\frac{\cos^2\theta-\sin^2\theta}{2\cos\theta}Z_M & \frac{1}{\sqrt2}W^+_M\\
\frac{1}{\sqrt2}W^-_M& -\frac{1}{2\cos\theta}Z_M \epm. \label{A_matrix}
\eeq
The gauge transformations for the Higgs doublet $H(x,y)$ and gauge matrix $\mathbb{A}_M$ under the gauge group
$SU(2)\times U(1)_Y$ can be written as
\begin{align}
H(x,y)\to H^\p(x,y)&=U(x,y)H(x,y), \label{H_gauge_trans}\\
\mathbb{A}_M(x,y)\to \mathbb{A}_M^\p(x,y)&=U(x,y)\mathbb{A}_M(x,y)U^{-1}(x,y)-\frac{i}{g_5}(\partial_M U(x,y)) U^{-1}(x,y),
\label{A_gauge_trans}
\end{align}
where $U(x,y)$ is the unitary matrix corresponding to the fundamental representation of $SU(2)\times U(1)_Y$
gauge transformations.
We will choose the 5D axial gauge analogous to the Abelian case~\ref{EWSB by vacuum expectation value of KK modes} by taking $\mathbb{A}_5(x,y)=0$. Note that we can always find $U(x,y)$
such that the axial gauge is manifest, i.e. $\mathbb{A}_5(x,y)=0$. We employ an axial gauge choice for the
non-Abelian case of the form
\begin{align}
U(x,y)=\widehat U(x){\cal P}e^{-ig_5\int_0^ydy^\p \mathbb{A}_5(x,y^\p)},
\label{axial_gauge_cond}
\end{align}
where $\widehat U(x)$ is the residual 4D gauge transformation and ${\cal P}$ denotes path-ordering
of the exponential. Another key point for the later discussion is that this 4D residual gauge
transformation $\widehat U(x)$ is independent of $y$ and thus even under the geometric parity.
As we have demonstrated in Appendix~\ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism}, due to the
symmetric geometry the background fields in the IR-UV-IR setup separate into even and odd bulk
wave functions. Hence, it is straightforward to
generalize the results obtained in Appendix~\ref{EWSB by vacuum expectation value of KK modes} for the
Abelian model to the electroweak sector of the SM. Let us start by decomposing the Higgs doublet and gauge
fields into components of definite parity as follows:
\beq
H(x,y)=H^{(+)}(x,y)+H^{(-)}(x,y), \lsp V_M(x,y)=V^{(+)}_{M}(x,y)+V^{(-)}_{M}(x,y), \label{even_odd_H_A_SM}
\eeq
where $V_M\equiv (A_M, W^{\pm}_M, Z_M)$.
We can write the action \eqref{action_SM} up to quadratic level in the $\mathbb{A}_5(x,y)=0$ gauge as
\begin{align}
S^{(2)}=&-\int d^5x \sqrt{-g}\bigg\{\frac12 {\cal W}^{+}_{(+)\mu\nu}{\cal W}_{(+)}^{-\mu\nu}+ \partial_5W^{+}_{(+)\mu}\partial^5W_{(+)}^{-\mu}+\frac14 {\cal Z}^{(+)}_{\mu\nu}{\cal Z}_{(+)}^{\mu\nu}+\frac12 \partial_5Z^{(+)}_{\mu}\partial^5Z_{(+)}^{\mu}\notag\\
&+\frac12 {\cal W}^{+}_{(-)\mu\nu}{\cal W}_{(-)}^{-\mu\nu}+\partial_5W^{+}_{(-)\mu}\partial^5W_{(-)}^{-\mu}+\frac14 {\cal Z}^{(-)}_{\mu\nu}{\cal Z}_{(-)}^{\mu\nu}+\frac12 \partial_5Z^{(-)}_{\mu}\partial^5Z_{(-)}^{\mu}\notag\\
&+\frac14 {\cal F}^{(+)}_{\mu\nu}{\cal F}_{(+)}^{\mu\nu}+\frac12 \partial_5A^{(+)}_{\mu}\partial_5A_{(+)}^{\mu} +\frac14 {\cal F}^{(-)}_{\mu\nu}{\cal F}_{(-)}^{\mu\nu}+\frac12 \partial_5A^{(-)}_{\mu}\partial^5A_{(-)}^{\mu} \notag\\
&+ \mathbb{D}_M H^{(+)\dag}\mathbb{D}^MH^{(+)} +\mu_B^2|H^{(+)}|^2 + \mathbb{D}_M H^{(-)\dag}\mathbb{D}^MH^{(-)} +\mu_B^2|H^{(-)}|^2 \notag\\
&+\frac{m^2_{UV}}{k}|H^{(+)}|^2\delta(y)-\frac{m^2_{IR}}{k}\big(|H^{(+)}|^2+|H^{(-)}|^2\big)\big[\delta(y+L)+\delta(y-L)\big]\bigg\}, \label{action_5d_sm}
\end{align}
where we have adopted the following definitions:
\begin{align}
{\cal \tilde V}^{(\pm)}_{\mu\nu}\equiv \partial_\mu \tilde V^{(\pm)}_{\nu}&-\partial_\nu \tilde V^{(\pm)}_{\mu}, \lsp {\cal F}^{(\pm)}_{\mu\nu}\equiv \partial_\mu A^{(\pm)}_{\nu}-\partial_\nu A^{(\pm)}_{\mu}, \label{photon_emt}\\
\mathbb{D}_\mu \bpm H^{(+)} \\ H^{(-)}\epm&\equiv\left[\partial_\mu -ig_5\bpm \mathbb{A}^{(+)}_\mu & \mathbb{A}^{(-)}_\mu \\
\mathbb{A}^{(-)}_\mu &\mathbb{A}^{(+)}_\mu \epm \right] \bpm H^{(+)} \\ H^{(-)}\epm, \label{c_dir_mu_pm}\\
\mathbb{D}_5 \bpm H^{(+)} \\ H^{(-)}\epm&\equiv \left[ \partial_5-ig_5\bpm \mathbb{A}^{(-)}_5 & \mathbb{A}^{(+)}_5 \\
\mathbb{A}^{(+)}_5 &\mathbb{A}^{(-)}_5 \epm\right] \bpm H^{(+)} \\ H^{(-)}\epm, \label{c_dir_5_pm}
\end{align}
where $\tilde V_\mu\equiv(W^\pm_\mu, Z_\mu)$ and $\mathbb{A}_M^{(\pm)}$ was defined in \eqref{A_matrix}.
It is convenient to write the Higgs doublets in the following form:
\beq
\bpm H^{(+)} \\ H^{(-)}\epm=e^{ig_5 (\Pi^{(+)}\mathds{1}+\Pi^{(-)}\tau_1)} \bpm {\cal H}^{(+)} \\ {\cal H}^{(-)}\epm, \label{higgs_redef}
\eeq
where ${\cal H}$ and $\Pi$ are defined as (the parity indices are suppressed)
\begin{align}
{\cal H}(x,y)&\equiv \frac{1}{\sqrt2}\bpm 0\\ h(x,y)\epm, \label{hat_H}\\
\Pi(x,y)&\equiv \bpm \frac{\cos^2\theta-\sin^2\theta}{2\cos\theta}\pi_Z & \frac{1}{\sqrt2}\pi^+_W\\
\frac{1}{\sqrt2}\pi^-_W& -\frac{1}{2\cos\theta}\pi_Z \epm. \label{pi_matrix}
\end{align}
We KK-decompose the Higgs doublets $H^{(\pm)}(x,y)$ and the gauge fields $V^{(\pm)}_\mu(x,y)$ as
\begin{align}
{\cal H}^{(\pm)}(x,y)&=\sum_n {\cal H}^{(\pm)}_n(x)f^{(\pm)}_n(y), \label{KK_H_sm}\\
\pi^{(\pm)}_{\tilde V}(x,y)&=\sum_n \pi_{{\tilde V}n}^{(\pm)}(x)a^{(\pm)}_{\tilde Vn}(y), \label{KK_pi_sm}\\
V^{(\pm)}_\mu(x,y)&=\sum_n V^{(\pm)}_{\mu n}(x)a^{(\pm)}_{V_n}(y), \label{KK_A_sm}
\end{align}
where the wave-functions $f^{(\pm)}_n(y)$ and $a^{(\pm)}_{V_n}(y)$ satisfy
\begin{align}
-\partial_5 (e^{4A(y)}\partial_5 f^{(\pm)}_n(y))+\mu_B^2e^{4A(y)}f^{(\pm)}_n(y)&=m_n^{2(\pm)}e^{2A(y)}f^{(\pm)}_n(y), \label{eom_fn_H}\\
-\partial_5 (e^{2A(y)}\partial_5 a^{(\pm)}_{V_n}(y))&=m_{V^{(\pm)}_n}^2a^{(\pm)}_{V_n}(y), \label{eom_fn_A}
\end{align}
and, with our background geometry $A(y)=-k|y|$.
The $y$-dependent wave functions $f^{(\pm)}_n(y)$ and $a^{(\pm)}_{V_n}(y)$ satisfy the following
orthonormality conditions:
\begin{align}
\int_{-L}^{+L}dy e^{2A(y)}f^{(\pm)}_m(y)f^{(\pm)}_n(y)=\delta_{mn}, \lsp \int_{-L}^{+L}dy a^{(\pm)}_{V_m}(y)a^{(\pm)}_{V_n}(y)=\delta_{mn}. \label{norm_condition_HA_sm}
\end{align}
The even modes are subject to jump conditions at $y=0$ while the odd modes are constrained by continuity at $y=0$, resulting in the following boundary conditions:
\begin{align}
\left(\partial_5 -\frac{m^2_{UV}}{k}\right)f^{(+)}_n(y)\Big\vert_{0}&=0, \Lsp f^{(-)}_n(y)\Big\vert_{0}=0, \label{H_bc0}\\
\partial_5a^{(+)}_{V_n}(y)\Big\vert_{0^+}&=0, \Lsp a^{(-)}_{V_n}(y)\Big\vert_{0}=0. \label{A_bc0}
\end{align}
The b.c. at $y=\pm L$ are:
\beq
\left(\pm\partial_5-\frac{m^2_{IR}}{k}\right)f^{(\pm)}_{n}(y)\Big\vert_{\pm L}=0, \lsp \partial_5a^{(\pm)}_{V_n}(y)\Big\vert_{\pm L}=0. \label{HA_bcL}
\eeq
As pointed out in the Abelian case~\ref{EWSB by vacuum expectation value of KK modes}, the choices of b.c. for $a^{(+)}_n(y)$ at $y=0,\pm L$ are motivated by the requirement that the even zero-mode profiles for gauge bosons be non-zero.
It is worth mentioning here that the choice of writing the Higgs doublets $H^{(\pm)}$ in the form of Eq. \eqref{higgs_redef} and using the KK decomposition
for the pseudoscalars $\pi^{(\pm)}_{\tilde V}$ as given in Eq.~\eqref{KK_pi_sm} are both motivated by model-building considerations discussed below.
The other possibility is to choose different KK bases and b.c. for the pseudoscalars $\pi^{(\pm)}_{\tilde V}$ such that after SSB these
pseudoscalars become Nambu-Goldstone bosons (NGB). The even zero-mode gauge bosons would then acquire masses by eating up the even-parity NGB,
whereas the odd-parity NGB would remain in the spectrum (the odd zero-mode gauge boson fields being zero, see below). An effective potential for the odd-parity NGB
would be generated through their interactions with gauge bosons, hence making them pseudo-NGB.
We don't follow this approach here but it is an interesting possibility in which the neutral odd pseudo-NGB would be a composite dark Higgs in the dual
CFT description.\footnote{At the final stages of the present
work, Ref.~\cite{Carmona:2015haa} appeared where the authors considered composite dark sectors. A similar construction can be naturally realized as a CFT dual to the model considered here.}
We assume that the KK-scale is high enough, i.e. $m_{KK}\sim\co(\text{few})\;\hbox{TeV}$, that we can consider
an effective theory where only the lowest modes (zero-modes with masses much below $m_{KK}$) are kept.
It is important to note that the odd zero-mode wave functions obey $a^{(-)}_{V_0}(y)=0$, as
can be easily seen from Eq. \eqref{eom_fn_A} along with the b.c. \eqref{A_bc0} and \eqref{HA_bcL}.
As a consequence of $a^{(-)}_{V_0}(y)=0$, the odd zero-mode gauge fields $V^{(-)}_{0\mu}(x)$
and the odd Goldstone modes $\pi^{(-)}_{\tilde V_0}(x)$ will not be present in the effective 4D theory. Moreover, the even zero-mode gauge
profile is constant, i.e. $a^{(+)}_{V_0}(y)=1/\sqrt{2L}$. Using the results from Appendix~\ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism}, we can determine the values of the couplings and mass parameters in the effective 4D
theory in terms of the parameters of the fundamental 5D theory. The result is that we can write down the
effective 4D action for the zero-modes as
\begin{align}
S^{(2)}_{eff}=&-\int d^4x \bigg\{\frac14 {\cal F}^{0(+)}_{\mu\nu}{\cal F}_{0(+)}^{\mu\nu}+\frac14 {\cal Z}^{0(+)}_{\mu\nu}{\cal Z}_{0(+)}^{\mu\nu}+\frac12 {\cal W}^{+0(+)}_{\mu\nu}{\cal W}_{0(+)}^{-\mu\nu}+ \partial_\mu {\cal H}^{(+)\dag}_0\partial^\mu {\cal H}^{(+)}_0 \notag\\
&+ \partial_\mu {\cal H}^{(-)\dag}_0\partial^\mu {\cal H}^{(-)}_0 +m_0^{2(+)}|{\cal H}^{(+)}_0|^2 +m_0^{2(-)}|{\cal H}^{(-)}_0|^2 -ig_{4}\partial_\mu{\cal H}^{(+)\dag}_0\mathbb{M}^\mu{\cal H}^{(+)}_0 \notag\\
&+ig_4{\cal H}^{(+)\dag}_0\mathbb{M}_\mu^\dag\partial^\mu{\cal H}^{(+)}_0+g^{2}_{4}{\cal H}^{(+)\dag}_0\mathbb{M}_\mu^\dag\mathbb{M}^\mu{\cal H}^{(+)}_0 +g^{2}_{4}{\cal H}^{(-)\dag}_0\mathbb{M}_\mu^\dag\mathbb{M}^\mu{\cal H}^{(-)}_0\bigg\}, \label{eff_action_zero_0}
\end{align}
where $\mathbb{M}_\mu$ is defined as
\beq
\mathbb{M}_\mu\equiv\mathbb{U}^\dag\mathbb{\hat A}^{(+)}_{0\mu}\mathbb{U}+\frac{i}{g_4}\mathbb{U}^\dag\partial_\mu\mathbb{U}, \label{M_def}
\eeq
with $\mathbb{U}\equiv e^{ig_4 \widehat \Pi_0^{(+)}}$ and $g_4\equiv g_5/\sqrt{2L}$. In the above action ${\cal H}^{(\pm)}_0$ are real doublets defined in Eq. \eqref{hat_H}, implying that ${\cal H}^{(\pm)\dag}_0={\cal H}^{(\pm)\intercal}_0$, whereas $\mathbb{\hat A}^{(+)}_{0\mu}$ and $\widehat \Pi_0^{(+)}$ are defined as (below we suppress the parity indices and zero-mode index):
\begin{align}
\mathbb{\hat A}_{\mu}(x)&\equiv \bpm \sin\theta A_{\mu}+\frac{\cos^2\theta-\sin^2\theta}{2\cos\theta}Z_{\mu} & \frac{1}{\sqrt2}W^+_{\mu}\\
\frac{1}{\sqrt2}W^-_{\mu}& -\frac{1}{2\cos\theta}Z_{\mu} \epm, \label{A_matrix_0}\\
\widehat \Pi(x)&\equiv \bpm \frac{\cos^2\theta-\sin^2\theta}{2\cos\theta}\pi_{Z} & \frac{1}{\sqrt2}\pi^+_{W}\\
\frac{1}{\sqrt2}\pi^-_{W}& -\frac{1}{2\cos\theta}\pi_{Z} \epm. \label{pi_matrix_0}
\end{align}
It is important to comment here that the above action is manifestly gauge invariant under the following
$SU(2)\times U(1)_Y$ gauge transformation,
\beq
\mathbb{\hat A}^{(+)}_{\mu}\to \widehat U\mathbb{\hat A}^{(+)}_{\mu}\widehat U^\dag-\frac{i}{g_4}(\partial_\mu \widehat U)\widehat U^\dag, \lsp \mathbb{U}\to \widehat U e^{ig_4 \widehat \Pi^{(+)}}, \label{gauge_trans_sm}
\eeq
whereas the ${\cal H}^{(\pm)}_0$ are gauge invariant under the 4D residual gauge transformation $\widehat U$.
Equation~\eqref{eff_action_zero_0} is a non-Abelian analog of the Abelian zero-mode
action given by \eqref{eff_action_Abelian_zm}.
We introduce a convenient notion for our effective theory by redefining $V_{0\mu}^{(+)}(x)\equiv V_\mu(x)$, $\pi^{(+)}_{\tilde V0}(x)\equiv \pi_{\tilde V}(x)$, $\widehat \Pi^{(+)}_{0}(x)\equiv \widehat \Pi(x)$ and
\beq
H_1(x)\equiv e^{ig_4 \widehat \Pi(x)}{\cal H}^{(+)}_0(x), \lsp H_2(x)\equiv e^{ig_4 \widehat \Pi(x)}{\cal H}^{(-)}_0(x). \label{H_12}
\eeq
Now the above effective action \eqref{eff_action_zero_0}, including the scalar interaction terms, can be written in a nice gauge covariant form as\footnote{Note that the action of Eq.~\eqref{eff_action_nab_gauge_cov} is a non-Abelian version of the
Abelian zero-mode action \eqref{eff_action_ab_gauge_cov}.
}
\begin{align}
S_{eff}=-\int d^4x &\bigg\{\frac14 {\cal F}_{\mu\nu}{\cal F}^{\mu\nu}+\frac14 {\cal Z}_{\mu\nu}{\cal Z}^{\mu\nu}+\frac12 {\cal W}^{+}_{\mu\nu}{\cal W}^{-\mu\nu} \notag\\
&+ \big({\cal D}_\mu H_1\big)^{\dag}{\cal D}^\mu H_1+ \big({\cal D}_\mu H_2\big)^{\dag}{\cal D}^\mu H_2 +V(H_1,H_2) \bigg\}, \label{eff_action_nab_gauge_cov}
\end{align}
where the scalar potential can be written as
\begin{align}
V(H_1,H_2) =&-\mu^{2}|H_1|^2 -\mu^{2}|H_2|^2 +\lambda|H_1|^4 +\lambda|H_2|^4+6\lambda|H_1|^2|H_2|^2. \label{potenial_sm}
\end{align}
The covariant derivative ${\cal D}_\mu$ is defined as
\begin{align}
{\cal D}_\mu&=\partial_\mu-ig_4\mathbb{\hat A}_{\mu}, \label{co_dir_4_sm}
\end{align}
where $\mathbb{\hat A}_{\mu}$ is defined in Eq.~\eqref{A_matrix_0}.
In the above scalar potential the mass parameter $\mu$ and quartic coupling $\lambda$ are defined as (see Appendix \ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism}),
\beq
\mu^2\equiv-m_{0}^{2(\pm)}=(1+\beta)m_{KK}^2\delta_{IR}, \Lsp \lambda\equiv\lambda_{IR}(1+\beta)^2, \label{mu_lambda}
\eeq
where $\delta_{IR}$, $m_{KK}$ and $\beta$ are defined in Eq.~\eqref{delta_IR_mKK}.
Concerning the symmetries of the above potential, one can notice that $V(H_1,H_2)$ is invariant under $[SU(2)\times U(1)_Y]^\prime \times [SU(2)\times U(1)_Y]$, where one of the blocks has been gauged while the other one survived as a global symmetry.
The zero-modes of the four odd vector bosons $(W_{0\mu}^{(-)\pm}, Z_{0\mu}^{(-)} \text{ and } A_{0\mu}^{(-)})$ and the three would-be-Goldstone
bosons $\Pi^{(-)}_0$ have been removed by appropriate b.c., implying that the
corresponding gauge symmetry has been broken explicitly. What remains is {\it the
truncated inert doublet model}, that contains ${H}_{1,2}$, and the corresponding residual $SU(2)\times U(1)_Y$ global symmetry of the action.
Symmetry under the above mentioned $U(1)^\prime \times U(1)$ implies in particular that
$V(H_1,H_2)$ is also invariant under various $\mathbb{Z}_2$'s, for example
$H_1\to -H_1$, $H_2\to -H_2$ and $H_1\to \pm H_2$.
As explained in the Abelian case~\ref{EWSB by vacuum expectation value of KK modes}, we choose the vacuum such that the even parity Higgs field $H_1$ acquires a vev,
whereas the odd parity Higgs field $H_2$ does not, i.e.
\beq
v^2_1\equiv v^2=\frac{\mu^2}{\lambda}, \Lsp v_2=0. \label{v1_v2}
\eeq
Let us now consider fluctuations around the vacuum of our choice
\beq
H_1(x)=\frac{1}{\sqrt2}e^{ig_4\widehat \Pi}\bpm 0\\v+h\epm, \Lsp H_2(x)=\frac{1}{\sqrt2}e^{ig_4\widehat \Pi}\bpm 0\\ \chi \epm, \label{H1_H2_def_sm}
\eeq
where $\widehat \Pi$ (defined in Eq. \eqref{pi_matrix_0}) contains the pseudoscalar Goldstone
bosons $\pi_{W^\pm,Z}$. We choose the unitary gauge in which $\pi_{W^\pm,Z}$ are gauged away, that is they are
eaten up by the massive gauge bosons $W^\pm_\mu$ and $Z_\mu$.
Hence in the unitary gauge our effective action up to the quadratic order
in fluctuations is
\begin{align}
S^{(2)}_{eff}=-\int d^4x &\bigg\{\frac12 {\cal W}^{+}_{\mu\nu}{\cal W}^{-\mu\nu}+ \frac14 {\cal Z}_{\mu\nu}{\cal Z}^{\mu\nu}+\frac14 {\cal F}_{\mu\nu}{\cal F}^{\mu\nu}+ m^2_W W^+_\mu W^{-\mu}+\frac12 m^2_Z Z_\mu Z^\mu\notag\\
&+\frac12\partial_\mu h\partial^\mu h+ \frac12m^2_h h^2 +\frac12\partial_\mu \chi\partial^\mu \chi+ \frac12m^2_\chi \chi^2\bigg\}, \label{eff_action_quadratic}
\end{align}
where the masses are,
\begin{align}
m^2_h&=m^2_\chi=2\mu^2, \lsp m^2_{W}=\frac{1}{4}g^2_4\frac{\mu^2}{\lambda}, \lsp m^2_{Z}=\frac14\Big(g^2_4+g^{\p2}_4\Big)\frac{\mu^2}{\lambda}=\frac{m^2_{W}}{\cos^2\theta_W}. \label{masses_higgs_WZ}
\end{align}
It is worth noticing here that the Higgs mass $m_h$ and the dark scalar mass $m_\chi$ are degenerate at the tree level. However, as we demonstrate below, this degeneracy is lifted by the quantum corrections predicted by the effective theory below the KK-mass scale.
The interaction terms for effective theory can be written as
\begin{align}
S_{int}&=-\int d^4x \bigg\{\lambda vh^3+\frac\lambda4 h^4+\frac\lambda4 \chi^4+3\lambda v h\chi^2+\frac32\lambda h^2\chi^2+\frac{g_4^2}{2}vW_\mu^+W^{-\mu}h \notag\\
&+\frac{g_4^2}{4}W_\mu^+W^{-\mu}(h^2+\chi^2)+\frac14(g_4^2+g_4^{\p2})vhZ_\mu Z^\mu +\frac18(g_4^2+g_4^{\p2})Z_\mu Z^\mu (h^2+\chi^2)\bigg\}, \label{eff_action_int}
\end{align}
where we have omitted terms involving gauge interactions alone as they are irrelevant to our discussion below.
\subsection{Quantum corrections to scalar masses}
\label{Quantum corrections to scalar masses}
\noindent In this subsection we will study the quantum corrections to the tree-level scalar masses of the Higgs boson
$h$ and the dark matter candidate $\chi$.
Before proceeding further, we want to point out here that in this work we have not studied fermions in our geometric
setup since our focus is on the bosonic sector of the SM and EWSB.
For the sake of self-consistency, we mention here three possibilities for fermion localization and their implications in our geometric setup:
\ben\itemsep0em
\item In this first scenario, one takes the heavy (top) quarks to be localized towards the IR-brane, while the light quarks and leptons are localized towards the UV-brane.
Through this geometric localization one can address the fermion mass hierarchy problem.
In this scenario the even and odd zero-modes corresponding to the heavy quarks
will be almost degenerate in our symmetric geometry, whereas the odd zero-modes corresponding to the light quarks could be
much heavier than their corresponding even zero-modes \cite{Agashe:2007jb,Medina:2010mu}.
\item In the second scenario, {\it all} the fermions have flat zero-mode profiles. This can be achieved by the choice of appropriate bulk mass parameters for the fermions. As a consequence of flat profiles the odd fermion zero-modes have to disappear
and the even zero-modes will correspond to the SM fermions (in this case the fermion mass hierarchy problem is reintroduced).
\item In the third scenario {\it all} the fermions are localized towards UV-brane. In this case the masses of {\it all} odd zero-modes of the fermions could be heavier than their corresponding even zero-modes.
\een
In this study we implicitly limit ourselves to the last two cases in order that the dark Higgs be the lightest odd particle and all the other odd zero-modes
are either not present in our low-energy effective theory or they are much heavier that the dark Higgs, which will therefore be the only relevant dark matter candidate. For either of the choices 2. or 3. above, the top Yukawa coupling $y_t$ in the low-energy effective
theory will be the same
as in the SM and the top-quark loop correction to the SM Higgs boson mass will be exactly as in the SM up to the KK cutoff. In case 2., the $n\neq 0$ fermion KK-modes are all much heavier than the KK cutoff, $m_{KK}$, and will not significantly influence the radiative corrections to the SM Higgs mass.
We leave the study of the complete fermionic sector associated with our geometric setup for future studies.
The quantum corrections to the Higgs boson ($h$) mass and the dark-Higgs ($\chi$) mass within our effective
theory below the KK-scale are quite essential for breaking the mass degeneracy of Eq.~\eqref{masses_higgs_WZ}. For instance, at the 1-loop level of the perturbative expansion, the main contributions (quadratically divergent) to the masses of the SM
Higgs and the dark-Higgs come from the exchanges of the top quark ($t$), massive gauge bosons
($W,~Z$), Higgs boson ($h$) and the dark-Higgs ($\chi$) in the loop~\footnote{Another scalar which could be potentially present in our effective theory is the {\it radion}, which is responsible for the stabilization of the setup. The stabilization mechanism is beyond the scope of the present work, as here we assume a rigid geometrical background with no fluctuations of the 5D metric. However, we want to comment here that if the radion were present in our effective theory, because of it bosonic nature it would likely reduce the fine-tuning much in the manner that the $\chi$ does.}, see Fig. \ref{loop_diagrams}.
\begin{figure}
\centering
\begin{tikzpicture}[node distance=1cm,very thick, rounded corners=0pt,line cap=round]
\begin{scope}
\coordinate[] (a1);
\coordinate[right=of a1] (v1);
\coordinate[right=1.3cm of v1] (v2);
\coordinate[right=of v2] (a2);
\draw[scalar] (a1) --node[below]{$h$} (v1);
\draw[scalar] (v2)--node[below]{$h$}(a2);
\draw[fermion](v1) arc (180:0:0.65);
\draw[fermion](v2) arc (0:-180:0.65)
(1.7,0.7) node[below]{$t$}
(1.7,-0.7) node[above]{$\bar t$};
\filldraw [orange] (v1) circle (2pt)
(v2) circle (2pt);
\end{scope}
\begin{scope}[xshift=4cm]
\coordinate[] (a1);
\coordinate[right=of a1] (v1);
\coordinate[right=of v1] (a2);
\draw[scalar] (a1) --node[below]{$h$}(v1);
\draw[scalar] (v1)--node[below]{$h$}(a2);
\draw[boson] (v1) .. controls (-1,2) and (3.3,2) .. (1,-0.2)
(1,1) node[]{$W,Z$};
\filldraw [orange] (v1) circle (2pt);\end{scope}
\begin{scope}[xshift=7cm]
\coordinate[] (a1);
\coordinate[right=of a1] (v1);
\coordinate[right=of v1] (a2);
\draw[scalar] (a1) --node[below]{$h$}(v1);
\draw[scalar] (v1)--node[below]{$h$}(a2);
\draw[scalar] (v1) .. controls (-1,2) and (3,2) .. (v1)
(1,1) node[]{$h,\chi$};
\filldraw [orange] (v1) circle (2pt);
\end{scope}
\begin{scope}[xshift=10cm]
\coordinate[] (a1);
\coordinate[right=of a1] (v1);
\coordinate[right=of v1] (a2);
\draw[scalar] (a1) --node[below]{$\chi$}(v1);
\draw[scalar] (v1)--node[below]{$\chi$}(a2);
\draw[boson] (v1) .. controls (-1,2) and (3.3,2) .. (1,-0.2)
(1,1) node[]{$W,Z$};
\filldraw [orange] (v1) circle (2pt);\end{scope}
\begin{scope}[xshift=13cm]
\coordinate[] (a1);
\coordinate[right=of a1] (v1);
\coordinate[right=of v1] (a2);
\draw[scalar] (a1) --node[below]{$\chi$}(v1);
\draw[scalar] (v1)--node[below]{$\chi$}(a2);
\draw[scalar] (v1) .. controls (-1,2) and (3,2) .. (v1)
(1,1) node[]{$h,\chi$};
\filldraw [orange] (v1) circle (2pt);
\end{scope}
\end{tikzpicture}
\caption{One-loop diagrams in the unitary gauge contributing to the Higgs boson mass and the DM scalar mass.}
\label{loop_diagrams}
\end{figure}
It is instructive to write the general 1-loop effective scalar potential $V_{eff}(H_1,H_2)$ for our effective theory, described in the previous section, as~\footnote{Note that in this section we are considering the Higgs doublets $H_{1,2}$ in the unitary gauge, such that $H_1(x)=\frac{1}{\sqrt2}\bpm 0\\v_1+h\epm$ and $H_2(x)=\frac{1}{\sqrt2}\bpm 0\\ v_2+\chi \epm$, where at tree level our choice was $v_1=v$ and $v_2=0$.}
\beq
V_{eff}(H_1,H_2)=V_0(H_1,H_2)+V_1(H_1,H_2), \label{effective_potential}
\eeq
where $V_0(H_1,H_2)$ is the tree level scalar potential given by Eq. \eqref{potenial_sm} and $V_1(H_1,H_2)$ is the 1-loop effective potential, given by (see for example Refs. \cite{Grzadkowski:2009mj,Kolda:2000wi,Casas:2004gh})
\begin{align}
V_1(H_1,H_2)&=\frac{\Lambda^2}{32\pi^2}\left[ 3\Big(g_4^2+\frac12(g_4^2+g^{\p2}_4)+8\lambda\Big)(|H_1|^2+|H_2|^2)-12y_t^2|H_1|^2\right] + \cdots, \label{potential_loop}
\end{align}
where $y_t$ is the top Yukawa coupling, related to top mass through $m_t^2=y_t^2v^2/2$.
We use the momentum cut-off regularization. Also it is important to comment here
that $H_2$ is odd under the geometric $\mathbb{Z}_2$ parity, implying that it does not couple
to the even zero-mode fermions. Moreover, we consider only the quadratically divergent part
of the effective scalar potential and the ellipses in the above equation
represent the terms which are not quadratically divergent.
The minimization of the effective potential $V_{eff}(H_1,H_2)$, i.e.
\begin{equation}
\frac{\partial V_{eff}}{\partial H_i}\Big\vert_{H_i=\langle H_i\rangle}=0, \hsp\text{where}\hsp \langle H_i\rangle=\frac{1}{\sqrt2}\bpm0\\v_{i}\epm \hsp i=1,2
\end{equation}
gives the following set of conditions for the global minimum,
\beq
\lambda v_1^2=\mu^2-\delta\mu^2-3\lambda v_2^2, \hsp \text{or}\hsp v_1=0, \label{v1_eff}\\
\eeq
and
\beq
\lambda v_2^2=\mu^2-\delta\mu^2 + \frac38\frac{\Lambda^2}{\pi^2}y_t^2 -
3\lambda v_1^2, \hsp \text{or}\hsp v_2=0, \label{v2_eff}
\eeq
where $\delta\mu^2$ is given by
\beq
\delta\mu^2=\frac{3\Lambda^2}{32\pi^2}\Big[g_4^2+\frac12(g_4^2+g^{\p2}_4)+8\lambda-4y_t^2\Big]. \label{delta_mu}
\eeq
Of the four possible global minima of Eqs.~\eqref{v1_eff} and \eqref{v2_eff}, we will choose the vacuum such that $H_1$ acquires the vev, whereas $H_2$ does not:
\[
v_1=v, \Lsp v_2=0,
\]
where $v\simeq246 \;\hbox{GeV}$ is the vacuum expectation value of the SM Higgs doublet.
With this choice of vacuum, the 1-loop corrected masses for the fluctuations around the vacuum are
\begin{align}
m^2_h=\frac{\partial^2V_{eff}(H_1,H_2)}{\partial H_1^2}\Big\vert_{H_1=v,H_2=0}&=\Big(-\mu^2+\delta\mu^2\Big)+3\lambda v^2=2\lambda v^2, \label{hat_mh}\\
m^2_\chi=\frac{\partial^2V_{eff}(H_1,H_2)}{\partial H_2^2}\Big\vert_{H_1=v,H_2=0}&=\Big(-\mu^2+\delta\mu^2\Big)+3\lambda v^2+\frac38\frac{\Lambda^2}{\pi^2}y_t^2, \notag\\
&=2\lambda v^2+\frac34\frac{\Lambda^2}{\pi^2v^2}m_t^2. \label{hat_mchi}
\end{align}
To get $m_h=125\;\hbox{GeV}$, equivalent to $v\simeq 246\;\hbox{GeV}$, we need to fine-tune the parameters of the theory. To quantify the level of fine-tuning, we employ the Barbieri--Giudice fine-tuning measure $\Delta_{m_h}$ \cite{Barbieri:1987fn,Kolda:2000wi,Casas:2004gh}:
\beq
\Delta_{m_h}\equiv \left\vert\frac{\delta\mu^2}{\mu^2}\right\vert=\left\vert\frac{\delta m_h^2}{m_h^2}\right\vert. \label{fine-tuning}
\eeq
We plot the fine-tuning measure $\Delta_{m_h}$ as a function of the effective cutoff scale $\Lambda$ in Fig. \ref{mchi_lambda}. If one allows fine-tuning of about 10\%, i.e. $\Delta_{m_h}=10$, then the effective cutoff scale is $\Lambda\simeq2\;\hbox{TeV}$. The most stringent bounds on the KK-scale $m_{KK}$ in RS1 geometry with a bulk Higgs come from electroweak precision tests (EWPT) by fitting the $S,~T$ and $U$ parameters \cite{Archer:2014jca}. The lower bound on the KK mass scale in our model (AdS geometry, i.e. $A(y)=-k|y|$) is $m_{KK}\gtrsim2.5\;\hbox{TeV}$ for $\beta=0$ and $m_{KK}\gtrsim4.3\;\hbox{TeV}$ for $\beta=10$ at $95\%$ C.L. \cite{Archer:2014jca}. This implies a tension between fine-tuning (naturalness) and the lower bound on the KK mass scale $m_{KK}$. The region within the gray lines in Fig. \ref{mchi_lambda} shows the current bounds on the KK mass scale for our geometry and the associated fine-tuning. It is important to comment here that a slight modification to the AdS geometry (for example, models with soft wall or thick branes) leads to a considerable relaxation of the above mentioned lower bound on KK mass scale \cite{Cabrer:2010si,Cabrer:2011fb,Cabrer:2011vu}. For instance, a mild modification to the AdS metric in the vicinity of the IR-brane can relax the KK mass scale to $m_{KK}\gtrsim 1\;\hbox{TeV}$ \cite{Cabrer:2010si,Cabrer:2011fb,Cabrer:2011vu,Iyer:2015ywa,Carmona:2011ib}. Needless to say, the generalization of our model to modified AdS geometries with soft walls or thick branes is possible.
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{figdmh.pdf}\hsp
\includegraphics[scale=0.45]{figmchi.pdf}
\end{center}
\caption{The left plot gives the value of the fine-tuning measure $\Delta_{m_h}$ for a Higgs mass of $125$ GeV as a function of the cutoff $\Lambda$. The right plot shows the dependence of $m_\chi$ on $\Lambda$ for $m_h=125\;\hbox{GeV}$. In our model $\Lambda=m_{KK}$. The vertical gray line indicates the current lower bound on the KK mass scale coming from EWPT as computed in our model for $\beta=0$, $m_{KK}\gtrsim2.5\;\hbox{TeV}$.}
\label{mchi_lambda}
\end{figure}
The 1-loop quantum corrected dark matter squared mass $m^2_\chi$ is:
\beq
m_\chi^2=m_h^2+ \frac34\frac{\Lambda^2}{\pi^2v^2}m_t^2\,.
\label{m-shift}
\eeq
Hence, $m_\chi$ is raised linearly with the cut-off scale $\Lambda$. This is illustrated in Fig.~\ref{mchi_lambda}. An interesting aspect of our model is that dark matter is predicted to be heavier than the SM Higgs boson.
A natural value of the cutoff coincides with the mass of the first KK excitations, which are experimentally
limited~\cite{Agashe:2014kda} to lie above a few TeV (depending on model details and KK mode sought). Requiring that the fine-tuning measure $\Delta_{m_h}$ be less than $100$ implies that $m_{KK}$ should be below about 6 TeV. Meanwhile, the strongest version of the EWPT bound requires $m_{KK}\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 2.5\;\hbox{TeV}$, corresponding to $m_\chi\sim 500\;\hbox{GeV}$, for which $\Delta_{m_h}$ is a very modest $\sim 18$. In short, our model is most consistent for $500\;\hbox{GeV}\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} m_\chi \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 1200\;\hbox{GeV}$.
\subsection{Dark matter relic abundance}
\label{Dark matter relic abundance}
\noindent In this subsection we calculate the dark matter relic abundance. The diagrams contributing to dark matter annihilation are shown in Fig.~\ref{DM_annihi_diagrams}. The squared amplitudes $|{\cal M}|^2$ corresponding to the contribution of each final state to dark matter annihilation are:
\begin{align}
\left| \mathcal{M} (\chi\chi \to \tilde V\tilde V)\right|^2 &= {4 m_{\tilde V}^4 \over S_{\tilde V} v^4} \left( 1+ {3m_h^2 \over s-m_h^2} \right)^2 \left[ 2+ \left( 1- {s \over 2m^2_{\tilde V}} \right)^2 \right], \label{M2VV}\\
\left| \mathcal{M} (\chi\chi \to f\bar{f}) \right| ^2&= 18 N_c {m_f^2 m_h^4 \over v^4} {s-4m^2_f \over (s-m_h^2)^2}, \label{M2ff}\\
\left| \mathcal{M} (\chi\chi \to hh) \right|^2 &= {9 m_h^4 \over 2 v^4} \left[ 1+ 3m_h^2 \left( {1 \over s-m_h^2} + {1 \over t-m_\chi^2} + {1 \over u-m_\chi^2} \right) \right]^2, \label{M2hh}
\end{align}
where $\tilde V={W,Z}$ and $S_W=1$ and $S_Z=2$ are the symmetry factors accounting for the identical particles in the final state; $N_c$ refers to the number of ``color'' degrees of freedom for the given fermion and $s,~t,~u$ are the Mandelstam variables. Here, we ignore the loop-induced $\gamma\gamma$ and $Z\gamma$ final states, which are strongly suppressed.
Note that the first term in the parenthesis in Eq.~(\ref{M2VV}) and the first term in the square bracket in Eq.~(\ref{M2hh}) arise from the $\chi\chi \tilde V\tilde V$ and the $\chi\chi hh$ contact interactions, respectively.
The former channel is present in our model since $\chi$ is a component of the (truncated) odd $SU(2)$ doublet.
\begin{figure}
\centering
\begin{tikzpicture}[node distance=1cm,very thick, rounded corners=0pt,line cap=round]
\begin{scope}[xshift=0.5cm]
\coordinate[] (v1);
\coordinate[above left=of v1] (a1);
\coordinate[below left=of v1] (a2);
\coordinate[above right=of v1] (b1);
\coordinate[below right=of v1] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v1);
\draw[boson](v1)--(b1)node[right]{$W,Z$};
\draw[boson](v1)--(b2)node[right]{$W,Z$};
\filldraw [orange] (v1) circle (2pt);
\end{scope}
\begin{scope}[xshift=4.5cm]
\coordinate[] (v1);
\coordinate[above left=of v1] (a1);
\coordinate[below left=of v1] (a2);
\coordinate[right=1.3cm of v1] (v2);
\coordinate[above right=of v2] (b1);
\coordinate[below right=of v2] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v1);
\draw[scalar] (v1)--node[above]{$h$}(v2);
\draw[boson](v2)--(b1)node[right]{$W,Z$};
\draw[boson](v2)--(b2)node[right]{$W,Z$};
\filldraw [orange] (v1) circle (2pt)
(v2) circle (2pt);
\end{scope}
\begin{scope}[xshift=9.5cm]
\coordinate[] (v1);
\coordinate[above left=of v1] (a1);
\coordinate[below left=of v1] (a2);
\coordinate[right=1.3cm of v1] (v2);
\coordinate[above right=of v2] (b1);
\coordinate[below right=of v2] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v1);
\draw[scalar] (v1)--node[above]{$h$}(v2);
\draw[fermion](v2)--(b1)node[right]{$f$};
\draw[fermion](b2)node[right]{$\bar f$} -- (v2);
\filldraw [orange] (v1) circle (2pt)
(v2) circle (2pt);
\end{scope}\newline
\begin{scope}[yshift=-2.5cm]
\coordinate[] (v1);
\coordinate[above left=of v1] (a1);
\coordinate[below left=of v1] (a2);
\coordinate[above right=of v1] (b1);
\coordinate[below right=of v1] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v1);
\draw[scalar](v1)--(b1)node[right]{$h$};
\draw[scalar](v1)--(b2)node[right]{$h$};
\filldraw [orange] (v1) circle (2pt);
\end{scope}
\begin{scope}[xshift=3cm,yshift=-2.5cm]
\coordinate[] (v1);
\coordinate[above left=of v1] (a1);
\coordinate[below left=of v1] (a2);
\coordinate[right=1.3cm of v1] (v2);
\coordinate[above right=of v2] (b1);
\coordinate[below right=of v2] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v1);
\draw[scalar] (v1)--node[above]{$h$}(v2);
\draw[scalar](v2)--(b1)node[right]{$h$};
\draw[scalar](v2) -- (b2)node[right]{$h$};
\filldraw [orange] (v1) circle (2pt)
(v2) circle (2pt);
\end{scope}
\begin{scope}[xshift=7.5cm,yshift=-2cm]
\coordinate[] (v1);
\coordinate[below= of v1] (v2);
\coordinate[position=150 degrees from v1] (a1);
\coordinate[position=-150 degrees from v2] (a2);
\coordinate[position=30 degrees from v1] (b1);
\coordinate[position=-30 degrees from v2] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v2);
\draw[scalar] (v1)--node[left]{$\chi$}(v2);
\draw[scalar](v1)--(b1)node[right]{$h$};
\draw[scalar](v2)--(b2)node[right]{$h$};
\filldraw [orange] (v1) circle (2pt)
(v2) circle (2pt);
\end{scope}
\begin{scope}[xshift=11cm,yshift=-2cm]
\coordinate[] (v1);
\coordinate[below= of v1] (v2);
\coordinate[position=150 degrees from v1] (a1);
\coordinate[position=-150 degrees from v2] (a2);
\coordinate[above right=1.8cm of v2] (b1);
\coordinate[below right=1.8cm of v1] (b2);
\draw[scalar] (a1)node[left]{$\chi$} -- (v1);
\draw[scalar] (a2)node[left]{$\chi$} -- (v2);
\draw[scalar] (v1)--node[left]{$\chi$}(v2);
\draw[scalar](v2)--(b1)node[right]{$h$};
\draw[scalar](v1)--(b2)node[right]{$h$};
\filldraw [orange] (v1) circle (2pt)
(v2) circle (2pt);
\end{scope}
\end{tikzpicture}
\caption{Dark matter annihilation diagrams.}
\label{DM_annihi_diagrams}
\end{figure}
In the left panel of Fig. \ref{sigma_omega} we plot the annihilation cross-section for
the contributing channels as a function of $m_\chi$. (Note that the parameter $\Lambda$ would only enter if we performed this calculation at the one-loop level.)
As seen from the plot, the total cross section is dominated by $WW$ and $ZZ$ final states.
The main contributions for these final states are those generated by the contact interactions $\chi\chi \tilde V\tilde V$.
In fact, in our model, the $\tilde V\tilde V$ final states are additionally enhanced by a constructive interference of
the contact $\chi\chi \tilde V\tilde V$ interaction with the s-channel Higgs-exchange diagram.
In addition, for low $m_\chi$, there is a comparable contribution from $\chi\chi$ annihilation into $hh$.
(The dip at $m_\chi\sim 210\;\hbox{GeV}$ is caused by cancellation between the contact
interaction and $s,~t,~u$-channel diagrams.) Fermionic final states are always irrelevant;
even $\chi\chi\to t \bar t$ production is very small in comparison to $\chi\chi \to \tilde V\tilde V$.
Then, adopting the standard s-wave cold dark matter approximation \cite{Kolb:1990vq}, we
find the present $\chi$ abundance $\Omega_\chi h^2$ shown in the right panel of Fig. \ref{sigma_omega}. We observe that $\Omega_\chi h^2\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 10^{-4} $ once the EWPT bound of $m_\chi\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 500\;\hbox{GeV}$ is imposed.
Clearly, some other dark matter component is needed within this model to satisfy the Planck
measurement, $\Omega_\chi h^2 \sim 0.1$ \cite{Planck:2015xua}.
\begin{figure}
\begin{center}
\includegraphics[width=0.47\textwidth]{figsigma0.pdf}\hsp
\includegraphics[width=0.45\textwidth]{figomegah2.pdf}
\end{center}
\caption{The above graphs show the annihilation cross-section $\sigma_0$ for different final states (left)
and the $\chi$ abundance $\Omega_\chi h^2\times 10^{4}$ (right) as a function of dark matter mass $m_\chi$.}
\label{sigma_omega}
\end{figure}
\section{Summary}
\label{Summary}
\noindent In this paper, we constructed a model with ${\mathbb Z}_2$ geometric symmetry such that two identical AdS patches are glued together at $y=0$, where $y$ is the coordinate of the fifth dimension. We considered three D3-branes, one at $y=0$ referred to as the UV-brane where gravity is assumed to be localized and two branes at $y=\pm L$ referred to as IR-branes -- the IR-UV-IR model. For this $\mathbb{Z}_2$ symmetric geometric setup we found that the RS metric \eqref{metric} is the background solution of pure gravity when the matter backreaction is neglected. To investigate possible backreaction of matter fields (bulk $SU(2)$ Higgs doublet) on the geometry, we solved the 5D coupled scalar-gravity equations of motion adopting the superpotential method. We found analytic solutions where the Higgs background vev is highly localized towards the IR brane. It was also verified that the backreaction is negligible for $kL\simeq 37$ which is required in order to address the gauge hierarchy problem. The technique employed to find solutions is very general and can be used to any bulk Higgs RS1-like constructions to stabilize geometry taking into account backreaction.
The motivation of this work is twofold: ({\it i}) to analyze the situation where EWSB is due to the bulk Higgs in this $\mathbb{Z}_2$ symmetric geometry; and ({\it ii}) to discuss the lowest odd KK-mode as a dark matter candidate. Concerning EWSB, we discussed in detail many important aspects of SSB due to a bulk Higgs. In the Appendix \ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism} we considered the Abelian gauge group where we discussed two alternative approaches to SSB due to a bulk Higgs field acquiring a vacuum expectation value. In one approach, the symmetry breaking is triggered by a vev of the KK zero-mode of the bulk Higgs field. The second approach is based on the expansion of the bulk Higgs field around an extra-dimensional vev with non-trivial $y$ profile. The comparison between the two Abelian scenarios is summarized in Tables \ref{comparison} and \ref{comp_para}. The (zero-mode) effective theories obtained from the two approaches are identical and the most intriguing feature of the Abelian Higgs mechanism is that the even and odd Higgs zero-modes have degenerate mass at the tree-level --- a feature that is also present in the SM case.
To achieve SSB, the choice of boundary conditions for the fields at $\pm L$ is critical. In both approaches to the Abelian case, we allowed the $y$-derivative of a field to have an arbitrary value at $\pm L$ as opposed to requiring that the field value itself be zero, i.e. we employed Neumann or mixed b.c. rather than Dirichlet b.c. at $\pm L$. The latter choice would have led to an explicit symmetry breaking scenario in which there are no Goldstone modes and the gauge bosons do not acquire mass. (Note that the boundary or ``jump'' conditions at $y=0$ follow from the bulk equations of motion in the case of even modes, whereas odd modes are required to be zero by symmetry.)
Following this introductory material in Appendix \ref{SSB in the IR-UV-IR model: the Abelian Higgs mechanism}, we considered EWSB assuming that the SM gauge group is present in the bulk of our $\mathbb{Z}_2$ symmetric 5D warped model. The zero-mode effective theory appropriate at scales below the KK scale, $m_{KK}$, was obtained. For appropriate Higgs field potentials in the bulk and localized at the UV and IR-branes, SM-like EWSB is obtained when only the IR-branes have a quartic potential term. In contrast, quadratic mass-squared terms are allowed both on the branes and in the bulk. Of course, to achieve spontaneous EWSB, we employed the same boundary conditions as delineated above for the Abelian model. The resulting model has the following features.
\bit \itemsep0em
\item Due to warped KK-parity all fields develop even and odd towers of KK-modes in the 4D effective theory.
\item Assuming that the KK-scale is high enough ($m_{KK}\sim\co(\text{few})\;\hbox{TeV}$), we derive the low energy effective theory which includes only zero-modes of the theory.
\item In the effective theory, the symmetry of the model is $[SU(2)\times U(1)_Y]^\prime \times [SU(2)\times U(1)_Y]$, where the unprimed symmetry group is gauged while the other stays as a global symmetry.
The zero-mode odd gauge fields and the corresponding Goldstone modes from the odd Higgs doublet are {\it eliminated} due to the b.c..
\item In the low energy effective theory, we are left with all the SM fields plus a {\it dark-Higgs} -- the odd zero-mode Higgs. This dark-Higgs and the SM Higgs (the even zero-mode) are degenerate at tree level.
\item In order to get the SM Higgs mass of $125\;\hbox{GeV}$, we need to fine-tune the 5D fundamental parameters of theory to about $1\% - 5\%$, where the upper bound is determined by the lower bound on the KK scale coming from EWPT requirements.
\item We computed the one-loop quantum corrections to the tree-level masses of the SM Higgs and the dark Higgs assuming that the cutoff scale of our effective theory is the KK-scale, $m_{KK}$. One finds that the dark-Higgs mass is necessarily larger than the SM Higgs mass, the difference being quadratically dependent on $m_{KK}$.
\item Requiring that the fine-tuning measure $\Delta_{m_h}$ be less than $100$ implies that $m_{KK}$ should be below about 6 TeV. Meanwhile, the strongest version of the EWPT bound requires $m_{KK}\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 2.5\;\hbox{TeV}$, corresponding to $m_\chi\sim 500\;\hbox{GeV}$, for which $\Delta_{m_h}$ is a very modest $\sim 18$. In short, our model is most consistent for $500\;\hbox{GeV}\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} m_\chi \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 1200\;\hbox{GeV}$.
\item We calculate the relic abundance of the dark-Higgs in the cold dark matter approximation. For $m_\chi$ in the above preferred range, $\Omega_\chi h^2 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 10^{-4}$ as compared to the current experimental value of $\sim 0.1$. To obtain a more consistent dark matter density, one needs to either assume another DM particle or perform a more rigorous analysis of our model by considering the even and odd higher KK-modes in the effective theory.
\eit
\section*{Acknowledgements}
\noindent JFG and YJ are supported in part by US DOE grant DE-SC-000999. AA and BG acknowledge partial support by the National Science Centre (Poland) research project, decision no DEC-2014/13/B/ST2/03969.
The work of AA was partially supported by the Foundation for Polish Science International PhD Projects Programme
co-financed by the EU European Regional Development Fund. AA and BG are grateful to the Mainz Institute for Theoretical Physics (MITP) and the University of California Davis for their hospitality and partial support during the completion of this work.
JFG and YJ acknowledge hospitality and partial support by Warsaw University during the course of the project.
AA would like to thank Adam Falkowski, Matthias Neubert and Mariano Quiros for useful discussions.
|
1,314,259,995,811 | arxiv | \section{Why a statistical method based on photoionization models}
\label{ojo}
When direct temperature measurements are missing, statistical methods
are used to infer abundances in giant \hii regions. There are two
families of statistical methods. One is based on calibrating samples of
objects for which the abundance could be derived from temperature-based
methods. The other is based on photoionization model grids. The latter
is free from observational biases, but the grid must cover all the
configurations that could be found in nature. \bond (Bayesian Oxygen and
Nitrogen abundance Determinations) belongs to the second family. \bond
infers oxygen and nitrogen abundances using carefully selected strong
and semistrong lines by comparing them to a grid of photoionization
models. The source code is open and freely available at
\url{http://bond.ufsc.br/}. Full details can be found at Vale Asari
\etal (2016). This manuscript is intended to be a quick-start guide to
highlight the most important aspects of the method.
\section{What sets \bond apart}
Common strong line methods based on simple calibrations (\oiii/\nii,
\nii/\Ha, $(\oii+\oiii)/\Hb$) assume that emission-line nebulae are a
one-parameter family, and that such parameter is the oxygen abundance.
The first to realise the importance of introducing a secondary parameter
to measure abundances was McGaugh (1991), who considered the effect of
the ionization parameter $U$. Nowadays there is a plethora of methods
to measure abundances based on the comparison of observed to theoretical
emission lines from a grid of photoionization models (McGaugh 1991;
Kewley \& Dopita 2002; Tremonti \etal 2004; Dopita \etal 2013;
P\'erez-Montero 2014; Blanc \etal 2015). We will discuss the difference
of our method with respect to others.
One novelty in our grid of photoionization models is that we do not
impose any a priori relation between N/O and O/H. The only other method
that also does not tie in the nitrogen and oxygen abundances is the one
by P\'erez-Montero (2014), with the difference that his method uses
auroral lines (so it is not a strong line method), and it selects models
around the empirical N/O versus O/H relation if auroral lines are not
available.
Another novelty in our method is that it uses Bayesian inference to
measure abundances, not quite unlike the one by Blanc \etal (2015). The
curious reader is referred to Section \ref{sec:bayes} for the key
points; we warn that, although Bayesian inference is part and parcel of
our method, it is not the most important aspect of \bond.
The killer features in \bond are: (a) N/O is free to vary, (b) it
distinguishes between the lower and upper metallicity branches, and (c)
it considers the effect of varying the hardness of the ionizing
radiation field. The next section shows how we have tackled (a), (b) and
(c), and explains the reasoning behind choosing which emission lines
need to be fitted.
\section{Input emission lines for \bond}
Our grid of photoionization models spans a wide range in O/H, N/O, and
$U$. The ionizing radiation field is provided by the instantaneous
starburst models of Moll\'a \etal (2009) for six different ages. Two
nebular geometries are considered (thin shell and filled sphere). We
thus have five parameters in our models. Even though we are interested
in measuring only two of them, O/H and N/O, we still ought to have good
constraints for the other three `uninteresting' parameters: $U$, the
hardness of the ionizing radiation field, and the density
structure. That is why we have chosen a set of emission lines carefully
tailored to constrain those five parameters all at once.
In the following we list these emission lines and explain the reason
behind each of them. Note that the (reddening-corrected) intensities of
\emph{all} of those lines with respect to \Hb\ are needed to run
\bond.
\begin{itemize}
\item The strong lines \Hb, \Oii, \Oiii, and \Nii. They were chosen
because, to first order, the strong line ratios $(\oiii+\oii)/\Hb$,
$\nii/\oii$, and $\oiii/\oii$ map into O/H, N/O and $U$,
respectively. We fit line intensities with respect to \Hb, and not the
latter strong line ratios directly, because we assume that the
intensities with respect to \Hb follow a Gaussian distribution when we
calculate likelihood probabilities.
\item The semistrong lines \Ariii and \Neiii (plus upper limits for
auroral lines of \Opp and \Np). The ratio $(\oiii+\oii)/\Hb$ is
bi-valued with respect to the oxygen abundance (see the inverted U
shape of the relation in Fig.~\ref{fig:bond-grid}, left). That is why
some methods impose a fixed relation between N/O versus O/H in their
photoionization models (e.g.\ Dopita \etal 2013; Blanc \etal
2015). Since we are interested in inferring both N/O and O/H, we did
not want to use nitrogen lines to break the bimodality in the oxygen
abundance. Restricting our search to emission lines easy to measure
in typical optical spectra (though not always reported in the
literature), we found an ideal candidate in the emission line ratio
\ariii/\neiii. \Arpp and \Nepp are formed in roughly the same zone,
but the excitation potentials of \ariii and \neiii are very different
(1.7 and 3.2 eV, respectively), so the ratio of these lines is
sensitive to the electronic temperature. Argon and neon are primary
elements and their global abundance ratio is expected to be
constant. Besides, they are both inert, thus do not suffer dust
depletion, so their abundance ratio in the ionized gas phase remains
constant. Fig.~\ref{fig:bond-grid} (centre) shows the line ratio
\ariii/\neiii in our grid as a function of \oiii/\oii (the latter
traces the ionization parameter $U$). The points are colour-coded as
falling in the lower or upper metallicity branch (blue and red,
respectively), showing that \ariii/\neiii can break the bimodality in
the oxygen abundance. An extra help on finding the right metallicity
branch can come from \emph{upper limits} in auroral lines.
\item The semistrong line \Hei. This is the crucial part of
\bond. Since we consider different ionization scenarios, e.g.\
different spectral energy distributions (SED) of the ionizing
radiation, we need to infer which of those scenarios is more
appropriate. We expect the SED to be different in different \hii
regions due to ageing (so the most massive stars have disappeared) or
due to stochastic effects in low luminosity \hii regions when the
upper part in the stellar initial mass function of the ionizing
cluster is not fully sampled. Fig.~\ref{fig:bond-grid} (left) shows
how the emissivity of $(\oiii+\oii)/\Hb$ depends not only on the
oxygen abundance (abscissa), but also on he hardness of the ionizing
radiation field (colour code). Note that at high metallicities a
single value of $(\oiii+\oii)/\Hb$ can span 1 dex in oxygen abundance
for different ionizing radiation fields. Fig.~\ref{fig:bond-grid}
(right) shows that the \Hei/\Hb line can be used as a proxy of the
hardness of the ionizing radiation field.
\end{itemize}
\section{Why go Bayesian}
\label{sec:bayes}
First, let us emphasise that the Bayesian inference is \emph{not} what
sets \bond apart. The heart and soul of \bond is the set of carefully
selected emission line ratios, tailor-made both to infer the oxygen and
nitrogen abundances in \hii regions, and to take into account important
secondary parameters. As reasoned in the previous section, this allows
us to (a) break the bimodality in oxygen abundance without any a priori
relation between N/O versus O/H, and (b) consider the (previously
neglected) role of the ionizing radiation field in abundance
determinations.
That said, we have opted for Bayesian inference for good reasons.
Before diving into the most important Bayesian aspects, let us examine
the method's acronym. \bond stands for Bayesian Oxygen and Nitrogen
abundance Determinations. Our method is not called Oxygen and Nitrogen
abundance, Ionization parameter, and Hardness of the ionizing radiation
field Determinations ({\scshape onihd}). The reason why the letters
{\scshape ih} are not in the method's name is the same reason why we
have decided to introduce the {\scshape b} for Bayesian in its name: the
Ionization parameter and Hardness of the ionizing radiation field are
\emph{nuisance} parameters, and the only way to get rid of them
respecting dimensional analysis is by going Bayesian.
Let us lay out the problem to see how its resolution points to Bayesian
inference. We start with a carefully designed photoionization grid,
finely spaced and spanning a wide range in O/H, N/O, and $U$. It also
considers a few values of hardness of the ionizing radiation field
mimicked by different SED ages. Even though the latter two parameters
($U$ and ionizing radiation field) are important and need to be well
modelled, they are of \emph{secondary} interest. This is why our acronym
shifts from {\scshape onihd} to {\scshape ond}.
How do we get rid of parameters for which we do not care (a.k.a.\
\emph{nuisance} parameters)? If we want to consider the probabilities of
all models in our grid at the same time, then we can simply
\emph{marginalise} over the nuisance parameters. Marginalising is
nothing more than integrating over a parameter. For instance, for a
fixed O/H and N/O, we marginalise over $U$ simply by adding up all the
probabilities of all models of a given O/H and N/O for all $U$. The
trick is that to integrate, say, in $dU$, the probability density
function (PDF) in the integrand must have physical units of
$U^{-1}$. Ordinary likelihood PDFs (e.g.\ $e^{-0.5 \chi^2}$ for Gaussian
distributions\footnote{
$\chi^2 = \sum_j {(c_{j} - o_j)^2}/{\sigma_j^2}$, where $o_j$ and
$\sigma_j$ are the observed line intensity and its uncertainty, and
$c_{j}$ is the computed line intensity.}) have units which are the
inverse of the observational data being fitted. So, from a
\emph{pragmatic} point of view, we are obliged to write out the
posterior PDFs, which have the correct physical units when we integrate
over a model parameter. For a thorough argument on the dimensional
analysis of PDFs, see Hogg (2012), especially the discussion around his
equation 3.
In other words, going Bayesian gives us licence to kill the nuisance
parameters, so we need to add {\scshape b} to {\scshape ond}.
To write the posterior PDFs, we need to spell out our priors. This has a
two-fold benefit.
First, we can plug in an informative prior: if we have
empirical evidence that some models are more probable in nature than
others, we can give them more weight by setting the prior probabilities
just right. In our code so far, we have taken the most conservative
approach we can and, following Blanc \etal (2015), we use an
uninformative prior (specifically, a Jeffrey's prior that is
logarithmic in O/H, N/O and $U$).
The second benefit of setting a prior is that we have an explicit
prescription for making a finer grid. The problem of comparing data to a
uniformly spaced grid of models by using a $\chi^2$ likelihood is that
most models will be very distant from the observed data as measured by
the uncertainties $\sigma_j$. If a grid is very very rough, the closest
model might even be a few $\sigma_j$ away from the model with the
highest likelihood. An ad hoc prescription to deal with this problem is
by setting up cooking factors to increase the observed uncertainties,
thus decreasing the distances between observed and computed emission
lines. A cooking factor has no real justification and needs to be
tailored to work for each new data point. Since we are using a Bayesian
prescription, we can do much better than relying on ad hoc
prescriptions. We simply interpolate our grid where we need---and
interpolation in $\log$ N/O, O/H and $U$ is reasonable once the grid is
fine enough. The interpolation is informed by posterior probability of
each element in the grid: if an element has a high probability, it is
worth creating more grid points inside its volume\footnote{The algorithm
to do importance sampling in \bond is the octree sampling, which is
computationally inexpensive. The reader might be more familiar with
MCMC samplers, which are more adequate when one has to compute models
on the fly and does not have a pre-defined grid. In our case, it is
much more sensible to compute many photoionization models a priori and
interpolate them on the fly than generating photoionization models on
the fly. }.
Just a final note on the Bayesian parlance. The outcome of a Bayesian
inference is the posterior PDF for a set of model parameters (say, O/H,
N/O, $U$, hardness of the ionizing radiation field). Since we are
interested only in O/H and N/O, we can integrate out all other
parameters and obtain the \emph{joint} O/H and N/O posterior PDF. The
joint PDF is simply a two-dimensional function that gives the
probability of each point in the O/H versus N/O plane, which is the
ultimate goal of \bond.
However, sometimes it is unpractical to work with the full joint PDF, so
we need a summarised description in the form
$12 + \log \mathrm{O/H} = 8.35 \pm 0.02$. There are many ways to
transform a two-dimensional function into a nominal value and a
dispersion. One way is to set the nominal value for N/O and O/H to be
the point where the joint PDF is the highest. We call this number the
maximum a posteriori (MAP), because it is calculated after (i.e.\ a
posteriori) the marginalisation of nuisance parameters. For the
dispersion, we can define ellipses of credible regions that encompass,
say, 5, 50, 68 or 95\% of the total joint PDF.
If we want to marginalise away either O/H or N/O (i.e.\ if we are
interested in one of those parameters alone), we can integrate over the
other parameter. Summarising the fully marginalised PDF also opens up a
menu of choices. The nominal value can be taken as the mean, median or
mode (that is, its peak) of the PDF. For the dispersion, the usual
choices are either the 50, 68, or 95 percent equal-tailed or highest
density intervals. Note that the equal-tailed intervals are related to
the median. The median is the point in the PDF curve where 50 percent of
the probability is to the left and 50 to the right. The equal-tailed
68\% interval is the region of a curve where 16 percent of the
probability is to the right and 16 percent to the left. The highest
density intervals, on the other hand, are related to the mode. The mode
is the point in the curve of highest probability. The 68\% highest
density interval is the region around the mode that adds up to 68
percent of the total probability. A nice visualisation tool for those
descriptions can be found at
\url{http://www.sumsar.net/blog/2014/10/probable-points-and-credible-intervals-part-one/}.
Another minor nuisance parameter we marginalise away are the
uncertainties. For \ariii/\Hb, \neiii/\Hb and \hei/\Hb, we consider an
extra noise source added in quadrature to the observational
uncertainties which we allow to vary from 2 to 100 percent of the
line intensity. The extra noise source is needed because, in nature, the
Ar/O and Ne/O ratios may differ somewhat from the ones assumed in our
model grid. Regarding \hei/\Hb, the problem is that our grid is only
coarsely meshed as regards the hardness of the ionizing radiation field.
We then calculate the marginalised likelihood PDF for those lines by
using all values of this extra noise and then marginalising it away. We
do so because we do not expect those lines to be completely correct in
our photoionization models, and they are used only to infer secondary
parameters.
\section{Conclusions}
\label{sec:conclusions}
We have highlighted the main characteristics of \bond, a method based on
a grid of photoionization models to measure oxygen and nitrogen
abundances in giant \hii regions using strong and semistrong lines. We
show why it is important to consider secondary parameters in abundance
determinations, especially the hardness of the ionizing radiation
field. The SEDs in \hii regions can vary from region to region due to
stellar ageing or to the stochastically sampling of the IMF. We also
show how one can break the metallicity bimodality without recourse
either to auroral lines or a fixed relation between N/O and O/H. We use
a selective set of emission lines to infer all those
parameters. Finally, we argue why using Bayesian inference is the
correct way (motivated by the dimensional analysis of probability
density functions) to treat the secondary parameters in abundance
determinations.
\section{Acknowledgements}
\label{sec:acknowledgements}
NVA is grateful to Oli Dors for having organised such a useful
workshop. NVA acknowledges the support from Programa de
P\'os-Gradua\c{c}\~ao em F\'{i}sica da UFSC and CAPES/PROAP to attend the
workshop. GS and NVA acknowledge the support from the CAPES CsF--PVE
project 88881.068116/2014-01. The grid of models has been run on
computers from the CONACyT/CB2010:153985, UNAM-PAPIIT-IN107215 and UNAM
Posgrado de Astrof\'{i}sica projects.
\begin{figure}
\centering
\includegraphics[width=0.33\columnwidth, trim=10 10 10 7]{ValeAsari-fig1a}
\includegraphics[width=0.27\columnwidth, trim= 0 10 10 9]{ValeAsari-fig1b}
\includegraphics[width=0.37\columnwidth, trim= 0 10 8 8]{ValeAsari-fig1c}
\caption{\textbf{Left:} $(\oiiis+\oii)/\Hb$ versus O/H coloured by
\qhe/\qh, which traces the hardness of the ionizing radiation
field. This represents the two secondary effects considered in
\bond. First, for a given \qhe/\qh, $(\oiiis+\oii)/\Hb$ maps into two
different O/H values. We find the correct metallicity branch by using
the \ariii/\neiii ratio (centre). Second, for high metallicities
$(\oiiis+\oii)/\Hb$ span almost a decade in O/H. We use \Hei/\Hb
(right) to find the correct hardness of the ionizing radiation field.
\textbf{Centre:} \ariii/\neiii versus \oiii/\oii, blue for models in
the lower metallicity and red for models in the upper metallicity
branch. \textbf{Right:} \Hei/\Hb versus O/H coloured by
\qhe/\qh. \Hei/\Hb can be used as a proxy for \qhe/\qh, except for the
highest values of \qhe/\qh. All figures are based on our grid of
photoionization models and taken from Vale Asari \etal (2016).}
\label{fig:bond-grid}
\end{figure}
|
1,314,259,995,812 | arxiv | \section{Introduction}
How a many-body quantum system thermalizes --or fails to do so-- under its own interaction is a fundamental yet elusive problem. Localization serves as a prototypical example for the absence of thermalization, first studied in the non-interacting single particle regime known as Anderson localization \cite{Anderson1958, Abrahams2010}, and then revived in the context of interacting systems (many-body localization, MBL) \cite{Abanin2018}. The existence of MBL as a phase of matter was demonstrated theoretically \cite{Basko2006, Imbrie2016a, Imbrie2016} and numerically \cite{Znidaric2008, Pal2010, Oganesyan2007, Berkelbach2010, Gornyi2005}. Recently, the MBL phase was observed in cold atoms \cite{Schreiber2015a, Choi2016, Bordia2016a, Kondov2015, Lukin2018, An2018}, trapped ions \cite{Smith2016, Roushan2017} and natural crystals using nuclear magnetic resonances \cite{Wei2018}.
Most characteristics of MBL, such as area law entanglement \cite{Serbyn2013, Bauer2013}, Poisson level statistics \cite{Pal2010, Oganesyan2007}, logarithmic growth of entanglement \cite{Bardarson2012, Znidaric2008, Abanin2013, Huse2014a, Vosk2013, Lukin2018, Kim2014a} and power law dephasing \cite{Serbyn2014a, Serbyn2014, DeTomasi2017, Chen2017, Serbyn2017}, can be understood via a phenomenological model that expresses the Hamiltonian in terms of a complete set of local integrals of motion (LIOMs) \cite{Huse2014a, Serbyn2013}.
However, the explicit computation of LIOMs and their interactions is a challenging task, complicated by the fact that the set of LIOMs is not unique.
LIOMs have been calculated by the infinite-time averaging of initially local operators \cite{Chandran2015, Geraedts2017}, however, the obtained LIOMs does not have binary eigenvalues and thus cannot form complete basis. Binary-eigenvalue LIOMs can be obtained using perturbative treatment of interactions \cite{Imbrie2016a, Ros2015, Rademaker2016, Rademaker2017, You2016}, Wegner-Wilson flow renormalization \cite{Pekker2017}, minimizing the commutator with the Hamiltonian \cite{OBrien2016}. The previous methods either requires strong disorder field strength, or assumes a cutoff of LIOMs in real space, so a complete numerical study of localization lengths is missing.
Here we design and implement a method to compute a complete set of binary LIOMs (i.e., with eigenvalues $\pm1$) in a non-perturbative way, by maximizing the overlap with physical spin operators.
This criterion enables a recursive determination, similar to quicksort, of the LIOMs matrix elements in the energy eigenbasis, without the need to exhaust all the eigenstate permutations, which would be prohibitive for system size $L>5$.
We verify that in the MBL phase the LIOMs are exponentially localized in real space, and their interaction strength decays exponentially as a function of interaction range.
LIOM localization lengths and interaction localization lengths can be extracted from the two exponential behavior respectively. Deep in the MBL phase, the two localization lengths are well characterized by the inequality derived in Ref.~\cite{Abanin2018}.
Near the transition point, that our construction enables exploring, the interaction localization length diverges, while the LIOM localization length remains finite: this should be expected given the constraints imposed by our construction, even if it contradicts the inequality in Ref.~\cite{Abanin2018}.
The explicit form of the LIOMs further enables exploring the system evolution, and we show that the LIOMs display a similar dynamics to the physical spin operators, and extract the dynamical localization length from the power law dephasing process~\cite{Serbyn2014a}.
Interestingly, we find that the dynamical localization length is much shorter than would be given by a conjectured relationship to the above two localization lengths~\cite{Serbyn2014a, Abanin2018}, suggesting that the dynamics does not only depend on the typical value of LIOMs and their interactions, but also on higher order correlations.
\section{Algorithm}
\begin{figure}[!htbp]\centering
\includegraphics[width=0.46\textwidth]{schematic2.pdf}
\caption{\label{fig:schematic}
The flow diagram shows an example of the construction of a complete set of LIOMs in a system with $L=3$. Grey block represents undetermined matrix element and orange (green) block represents +1 (-1) matrix element. First diagonalize the Hamiltonian, then find $j_M$ that maximizes $\langle \tilde{\tau}_z^j \sigma_z^j\rangle$ ($j_M=2$ here). Divide the 8 eigenstates into two sectors each containing 4 states according to $\langle n|\sigma_z^2|n\rangle$ and assign $\tau_z^2=\tilde{\tau}_z^2$. For each sector, find $j_M$ within the sector, divide into two sectors each containing 2 states and assign $\tau_z^{j_M}=\tilde{\tau}_z^{j_M}$. Repeat the step one more time and then all LIOMs are determined. }
\end{figure}
To understand the construction algorithm, we first review the properties of integrals of motions in the many-body localized phase. LIOMs $\{\tau_z^j\}$ are diagonal in the Hamiltonian eigenbasis $[H,\tau_z^j]=0$.
A complete set of LIOMs can be related to physical spin operators by a local unitary transformation $\tau_z^j=U\sigma_z^j U^\dagger$, which implies that (i) half of the eigenvalues of $\tau_z^j$ are +1 and the other half are -1; (ii) LIOMs are mutually independent (orthonormal) $\mathrm{Tr}(\tau_z^j \tau_z^k)/2^L=\delta_{jk}$; (iii) the weight of $\tau_z^j$ decays exponentially in real space for localized Hamiltonians.
In particular, property (ii) requires that, for any $j$, in either +1 or -1 sector of $\tau_z^j$, half of the diagonal elements of $\tau_z^k$ are +1 and the other half are -1 for all $k\neq j$. In another word, the +1 and -1 sectors of $\tau_z^j$ are effectively two manifolds that represent two instances of a new system with $L-1$ spins, containing all sites except $j$.
With only constraints (i-ii), there are $2^L!/L!$ different sets of IOMs among which we want to find the most local one.
However, enumerating the $2^L!/L!$ different sets, and quantifying the localization of the related $\tau_z^j$, is numerically prohibitive.
Therefore, instead of explicitly demanding the exponential localization, we maximize the overlap of LIOMs and physical spin operators $\mathrm{Tr}(\tau_z^j \sigma_z^j)$, which enables a systematic and efficient way to find a unique set of LIOMs, and then we verify that these LIOMs are indeed exponentially localized in the MBL phase.
Expanding the IOMs $\tau_z^j$ in the energy eigenbasis $\{\ket{n}\}$, $n=1,2,\cdots,2^L$, as $\tau_z^j=\sum_n a^j_n |n\rangle\langle n|$, our goal is to find $a_n^j\in\pm1$ under the constrains (i-iii). (We thus assume that we have diagonalized the Hamiltonian).
The algorithm
is reminiscent of quicksort (see Fig. \ref{fig:schematic}):
\begin{enumerate}
\item For all eigenstates $|n\rangle$ and spin $j$, evaluate $s_n^j=\langle n|\sigma_z^j|n\rangle$.
\item For each $j$, sort the eigenstates according to $s_n^j$, and define candidates $\tilde\tau_z^j=\sum_{n\in S^j_{max}}\ket{n}\bra{n}-\sum_{n\in S^j_{min}}\ket{n}\bra{n}$, where $S^j_{max(min)}$ is the set of eigenstates giving the $2^L/2$ largest (smallest) overlaps $s_n^j$.
\item For each $j$, compute the overlaps $\langle\tilde\tau_z^{j}\sigma_z^{j}\rangle=\sum_{n\in S^j_{max}}s_n^j-\sum_{n\in S^j_{min}}s_n^j$ and find the site $j_M$ that maximizes it. For this site, set $\tau_z^{j_M}\equiv\tilde\tau_z^{j_M}$.
\item Consider the two manifolds $\mathbb S^{j_M}_\pm$
corresponding to the $\pm1$ eigenstates of $\tau_z^{j_M}$. Each of these manifolds represents two instances of a new system with $L-1$ spins, containing all sites except $j_M$. In this new system, perform the same protocol 1-3 to set another LIOM. This results in 4 sectors, each containing $2^{L-2}$ states.
\item By repeating the previous steps $L-2$ times we finally reduce the dimension of each sector to just 1 and all $a_n^j$ are assigned.
\end{enumerate}
We note that our scheme does not necessarily find the most local set of $\tau_z^j$, since once the matrix elements of a LIOM are determined at a given step, the subsequent search for the rest of the LIOMs is restricted to its perpendicular complement to satisfy orthogonality (that is, we are not ensured to find a global optimum).
Therefore, we choose to divide sectors using the most local LIOM (largest $\langle \tilde\tau_z^j \sigma_z^j\rangle$), so that this division sets the least constrains to later divisions.
In Fig. \ref{fig:AveH_seq} of the Appendix/SM we show that this choice indeed gives the most local results among all alternate algorthms we tried. Because we only utilize the overlaps $s_n^j=\langle n|\sigma_z^j|n\rangle$ in the computation, the scheme is immune to accidental resonances in the spectrum.
\section{Results}
\begin{figure}[!htbp]\centering
\includegraphics[width=0.48\textwidth]{TauzH2}
\caption{\label{fig:TauzH}
(a-b) Median of the LIOM weights $|f_{n,k}^j|$ as a function of distance $n$ for two disorder strengths: (a) $W=20$, deep in the MBL phase, where the median decays exponentially; and (b) $W=1$, in the ergodic phase, where the median saturates. For each $j$, the median is taken over the index $k$ in $|f_{n,k}^j|$ as well as 20 different disorder realizations. Darker color represents LIOMs in the middle of the chain [as shown in the bottom of (b)], and left (right) half of the LIOMs are represented by dashed (blue) curves.
(c-d) Median of the interaction strength as a function of range $r$ for two disorder strengths. Dotted curves represent $l$-body interaction terms $ |V_{ij}|, |V_{ijk}|, \cdots$ ($l=2,\dots,9$), where the median is taken over all indices $i,j,\cdots$, as well as 100 disorder realizations. The solid curve represents median of all interaction terms for a given range $V(r)$, regardless of how many LIOMs are involved. $L=10$ in all subplots.
}
\end{figure}
\subsection{Localization of operators and interactions}
To test the proposed algorithm and characterize the LIOMs that it finds we consider a prototypical example of an MBL-supporting system, a Heisenberg spin-1/2 chain with random fields,
\begin{equation}
\label{eq:Hamiltonian}
H=\sum_{i=1}^L h_i \sigma_z^i +\sum_{i=1}^{L-1}\vec{\sigma}^i\cdot \vec{\sigma}^{i+1},
\end{equation}
where $h_i$ is uniformly distributed in $[-W,W]$. It is known~\cite{Pal2010} that in the thermodynamic limit there is a MBL phase transition at $W_c\approx 7\pm2$. Although this model conserves the total magnetization along $z$, the validity of the algorithm does not depend on this symmetry. To quantitatively check the locality of LIOMs, we decompose them into tensor products of Pauli operators
\begin{equation}
\tau_z^j=\sum_{n=0}^L \sum_k f_{n,k}^j \hat{O}^j_{n,k},
\end{equation}
where $\hat{O}^j_{n,k}$ is a tensor product of Pauli operators whose furthest non-identity Pauli matrix from $j$ is of \textit{distance} $n$, e.g. $\sigma_x^1 \otimes \sigma_x^2 \otimes \sigma_y^3 \otimes \mathbb{I}^4$ is of distance $n=2$ to $j=1$, because $\sigma_y^3$ is the furthest non-identity Pauli matrix. $k$ labels operators with the same $n$. $f_{n,k}^j=\mathrm{Tr}(\tau_z^j \hat{O}^j_{n,k})$ is the weight of $j$-th LIOM on $\hat{O}^j_{n,k}$.
Figures~\ref{fig:TauzH}(a) and (b) show the median of $|f_{n.k}^j|$ as a function of distance $n$. In the MBL phase, the median weight decays exponentially with distance $n$, while in the ergodic phase it saturates at large $n$.
Because the LIOMs form an orthonormal basis, the Hamiltonian can be decomposed into this basis unambiguously and efficiently:
\begin{equation}
\label{eq:Htauz}
H=\sum_i \xi_i\tau_z^i+\sum_{ij} V_{ij}\tau_z^i\tau_z^j+\sum_{ijk} V_{ijk}\tau_z^i\tau_z^j\tau_z^k+\cdots.
\end{equation}
For non-interacting models, only the $\xi_i$ coefficients are nonzero. We can define the range $r$ of each coupling term $V_{ij\cdots}$ as the largest difference among the indices. For example, the range for 2-body interaction $V_{ij}$ is simply $r=|i-j|$, while for 3-body interactions is $r=\mathrm{max}(|i-j|,|i-k|,|j-k|)$.
Figures~\ref{fig:TauzH}(c) and (d) show the median interaction strength as a function of interaction range. In the MBL phase, the interaction strength decays exponentially.
The behavior of two-body interactions $|V_{ij}|$ and three body interactions $|V_{ijk}|, \cdots$ show no significant difference~\cite{Rademaker2016, Pekker2017} and can be essentially captured by the median of all interaction terms for a given range $V(r)$.
We considered the median instead of the mean in order to exclude rare events, i.e., instances where the disorder strength is small in a local region.
\begin{figure}[t]\centering
\includegraphics[width=0.48\textwidth]{PDFtauV}
\caption{\label{fig:PDF}
(a) Probability distribution of LIOM weights $\log_{10}|f_{n,k}^j|$. For a given $n$, samples are taken from all possible $j$ and $k$ as well as 200 disorder realizations. The distribution shows one single Gaussian peak that shifts toward smaller weights with increasing distance $n$, signaling the localization of IOMs. (b) Probability distribution of the interaction strength $\log_{10}(|V|)$. For given range $r$, samples are taken from all terms in Eq. \ref{eq:Htauz} as well as 10000 disorder realizations. Two peaks can be observed: the left peak is due to the localized cases as it shifts to smaller interaction strengths for longer range; the right peak shows the delocalized cases (rare events) as it is independent of interaction range. $L=10$ and $W=20$ for both (a) and (b).
}
\end{figure}
\begin{figure}[!htbp]\centering
\includegraphics[width=0.49\textwidth]{ll14}
\caption{\label{fig:ll6}
(a) Dephasing of the physical spin operator $\sigma_x^L$ (dark green, dashed curve) and LIOM $\tau_x^L$ (green, solid curve). Initial state is a product state with each spin pointing randomly in xy plane, i.e. $|\psi(0)\rangle=\otimes_{j=1}^L (|+\rangle_j +e^{i\phi}|-\rangle_j)/\sqrt{2}$, with $\phi$ randomly sampled in $[0,2\pi]$, $\sigma_z^j|+\rangle_j=|+\rangle_j$ $\sigma_z^j|-\rangle_j=-|-\rangle_j$ for red curve and $\tau_z^j|+\rangle_j=|+\rangle_j$ $\tau_z^j|-\rangle_j=-|-\rangle_j$ for blue curve. $L=10$, $W=20$. Averaging is performed over 20 different initial state and 20 disorder realizations. Error bar represents the standard deviation of all configurations.
(b) and (c) Localization length as a function of disorder strength $W$ for $L=12$. The LIOM localization lengths are extracted from $\mathrm{Tr}(\tau_z^j\sigma_z^k)\sim \exp(-|k-j|/\xi)$ with $j=1$, interaction localization lengths from $V(r)\sim\exp(-r/\kappa)$ and dynamical localization lengths from $\langle\langle\tau_x^L\rangle^2\rangle\sim t^{-\xi'\ln2}$.
Error bar only shows the fitting error. $\xi'$ curve is extracted from the median of 50 disorder realizations and 50 initial states. $\xi$ and $\kappa$ are extracted from the median of 5000 disorder realizations. (b) is a zoom-in of (c) near the transition point.
}
\end{figure}
To gain more insight into the localization of IOMs and interactions and observe the occurrence of rare events, in Figure~\ref{fig:PDF} we further study the probability distribution of weight $f_{n,k}^j$ versus $n$, and the probability distribution of interaction strength $V(r)$ versus $r$ in the localized regime (strong disorder).
The distribution of $\log(|f_{n,k}^j|)$ can be described by a single Gaussian peak, centered at smaller values of $|f_{n,k}^j|$ when the distance $n$ increases,
confirming the localization of IOMs. Instead, two peaks can be observed in the distribution of $\log(|V|)$. The left peak shifts to smaller $|V|$ with increasing $r$, while the right peak (larger $|V|$) shows no significant shift. Moreover, the area of the right peak decreases for larger $W$ and smaller $L$. Therefore, we identify the left peak as describing localized cases, the right one as rare events.
The exponential localization of the LIOMs and their interactions are usually the two criteria that define the LIOM. In the rare region of low disorder, however, the two requirements cannot be satisfied simultaneously and there is no universal criteria on how to choose LIOMs in this case.
Here we require the IOM $\tau_z$ to be local by construction, so the presence of a rare region shows up only in the interaction strengths; choosing different criteria for the LIOM construction may lead to different results.
\subsection{Localization lengths}
From the explicit form of the LIOMs and their interactions, we can extract the \textit{LIOM localization length} $\xi$, via $|f_{n,k}^j|\sim \exp(-n/\xi)$, and \textit{interaction localization length} $\kappa$, via $|V(r)|\sim \exp(-r/\kappa)$~\cite{Abanin2018}. In Figure~\ref{fig:ll6} we show $\kappa$ and $\xi$ as a function of disorder strength $W$.
The LIOM localization length $\xi$ is extracted using the relation $\mathrm{Tr}(\tau_z^j\sigma_z^k)\sim \exp(-|k-j|/\xi)$ \cite{Chandran2015,Rademaker2016} because calculating $f_{n,k}^j$ is numerically demanding (see SM).
The interaction localization length $\kappa$ is extracted by fitting the distribution of $\log|V|$ (as in Fig.~\ref{fig:PDF}) to two Gaussian peaks and then fitting the localized peak center to a linear function of $r$. Because our method forces $\tau_z$ to be local, $\xi$ is always finite, while $\kappa$ diverges around $W=8.1$ [Fig. \ref{fig:ll6}(b)], which agrees with the critical point $W_c=7\pm 2$ reported in Ref.~\cite{Pal2010}.
It has been shown in \cite{Abanin2018} that the two localization lengths satisfy the inequality $\kappa^{-1}\ge (\xi^{-1}-\mathrm{ln}2)/2$. From the numerical results in Fig.~\ref{fig:ll6}(c), we find that this inequality is satisfied in the localized phase, except in the vicinity of the phase transition point.
\begin{figure}[t]\centering
\includegraphics[width=0.5\textwidth]{lbit_int.pdf}
\caption{\label{fig:lbit_int}
LIOM in non-interacting model. (a) (b) Median interaction strength in the basis of $\{ \tau_z^j\}$. $r=1$ denotes the single-particle Hamiltonian $\xi_j \tau_z^j$.
(c) (d) Median overlap between LIOMs and physical spins $\mathrm{Tr}(\hat{\mathcal{O}} \sigma_z^k)$, with $\hat{\mathcal{O}}=\Sigma_z^j$ (red) for single-particle LIOM and $\hat{\mathcal{O}}=\tau_z^j$ (blue) for LIOMs obtained using the scheme proposed in this paper. Different curves stand for different $j$.
In (a) and (c) $W=20$. $r>1$ interaction strength is below machine precision $\sim10^{-15}$. $\Sigma_z$ and $\tau_z$ show little difference.
In (b) and (d) $W=0.5$. $\tau_z^j$ is more localized at site $j$, but the interaction among LIOMs is not zero.
L=10 and 500 disorder realizations are used in all plots.
}
\end{figure}
\subsection{Non-interacting model: tradeoff of localization}
\label{sec:noninteracting}
We can better understand why the interaction localization length $\kappa$ diverges at the critical point while the LIOM localization length $\xi$ remains finite by applying our LIOM construction to a non-interacting model
$
H=\sum_{i=1}^L h_i \sigma_z^i +\sum_{i=1}^{L-1}\left(\sigma^i_x \sigma^{i+1}_x+\sigma^i_y \sigma^{i+1}_y\right)
$. Due to the lack of interactions, the system is effectively localized for arbitrarily small $W$. This Hamiltonian can be mapped to a free fermionic Hamiltonian via a Jordan-Wigner transformation \cite{Jordan1928}. The Hamiltonian can be diagonalized by single-particle IOMs $\{\Sigma_z^i\}$:
$H=\sum_i \tilde{\xi}_i \Sigma_z^i$, that is, the interaction localization length in the $\{\Sigma_z^i\}$ basis is zero. However, note that the single-particle IOMs $\{\Sigma_z^i\}$ can be highly non-local for small $W$.
We can instead apply our algorithm to find LIOMs $\{ \tau_z^j\}$ for this model as done for the interacting Hamiltonian and compare $\{\Sigma_z^j\}$ and $\{ \tau_z^j\}$ (see Fig. \ref{fig:lbit_int}). For large disorder strength, $W=20$, the Hamiltonian is practically interaction-free even in the $\tau_z^j$ basis, and indeed the LIOMs $\tau_z^j$ approach the IOMs, $\tau_z^j\approx\Sigma_z^j$.
The trade off between the two interaction strength $\kappa$ and $\xi$ becomes evident for small disorder, $W=0.5$, where $\tau_z^j\neq\Sigma_z^j$. In this regime, the single-particle IOMs $\Sigma_z^j$ are delocalized, $\xi\gg1$, but the Hamiltonian still has no interactions, $\kappa=0$. Instead, the LIOMs obtained by our construction, $\{ \tau_z^j\}$, are localized but they give rise to long-range interactions in the Hamiltonian, $\kappa\gg1$.
For interacting models, it is difficult to obtain IOMs that minimize the interactions in a non-perturbative way. Still, we expect that if one were indeed able to find such a set of IOMs, there would be a similar tradeoff between how local they are (small $\xi$) versus how local the interactions are (small $\kappa$) outside the well-localized phase. Our choice of criterion for constructing LIOMs not only allows a simple and efficient algorithm; by keeping the operators local even when crossing the localization transition, the $\tau_z^j$ are always well-defined and can be used to explore properties of the system, such as its dynamics, around the localization-delocalization transition point.
\subsection{Dephasing Dynamics}
Since physical spin and LIOM operators are related by a local unitary transformation, they are expected to exhibit a similar dynamics [Fig.~\ref{fig:ll6}(a)]. In particular, the higher order interaction terms in Eq.~(\ref{eq:Htauz}) induce dephasing of the transverse operators by creating an effective magnetic field $H_\mathrm{eff}$ at the location of spin $j$ due to all the other spins.
The dephasing of the expectation values $\langle \tau_x(t)\rangle$ and $\langle \sigma_x(t)\rangle $ is closely related to the logarithmic light cone in the MBL phase \cite{Serbyn2014a}. It was previously shown that $\langle \langle \sigma_x(t)\rangle ^2\rangle \approx \langle \langle \tau_x(t)\rangle ^2\rangle\propto t^{-\alpha}$, where we took the average of the expectation values over random initial states and disorder realizations.
For an initial state given by a product state with each individual spin pointing randomly in the xy plane, $\alpha=2\xi' \ln2$ for bulk spins and $\alpha=\xi' \ln2$ for boundary spins, where $\xi'$ is a localization length different from $\xi$ and $\kappa$ \cite{Serbyn2014a}. This length $\xi'$, that we name \textit{dynamical localization length}, describes the strength of the contribution to the effective magnetic field felt by spin $j$ due to spins at distance $l$: $H_\mathrm{eff}^l\sim\exp(-l/\xi')$ (see SM).
By assuming exponentially decaying interactions, $|V(r)|=\exp(-r/\kappa)$, it was conjectured that $\xi'^{-1}\ge\kappa^{-1}+(\ln2)/2$~\cite{Abanin2018}. We find instead a much larger dephasing rate [Fig.~\ref{fig:ll6}(c)]. To investigate whether this is due solely to our LIOMs construction which does not explicitly enforces an exponentially decaying interaction strength, we artificially generate an Hamiltonian satisfying $|V(r)|\propto\exp(-r/\kappa)$ (see SM). Still, although we indeed find a power law decay, this is even
faster than what we observe in Fig.~\ref{fig:ll6}(a). We conjecture that the dephasing process cannot be simply described by a mean interaction strength (the model used to justify the relationship to $\kappa$), and higher order correlations may play an important role.
\section{Conclusion and Outlook}
We provide a novel method to efficiently compute the LIOMs for MBL systems by maximizing the overlap between LIOMs and physical spin operators. The method is non-perturbative and thus immune to resonances in the spectrum, and can be applied at the phase transition point. The only quantity we use in computing the LIOMs and their interactions is the expectation value of physical spin operators on energy eigenstates $\langle n|\sigma_z^j|n\rangle$. Although we use exact diagonalization here, our scheme is compatible with renormalization group methods and matrix product state representations \cite{Khemani2016a,You2016}, which can potentially be applied to much larger system and beyond one dimension. We show the power of the constructed LIOMs by extracting the localization length of the LIOMs and the Hamiltonian interactions from their respective exponential decays. We also show that in the MBL phase, the LIOMs and physical spin operators exhibit similar dephasing dynamics, even if it cannot be simply explained by the typical weights of LIOMs and typical interaction strengths.
|
1,314,259,995,813 | arxiv | \section{Introduction}\label{intro}
While being relatively rare in the Universe, cooling-core clusters of galaxies hold a special place in discussions of active galactic nucleus (AGN) feedback. These are the systems in which the need for AGN feedback is most apparent; X-ray observations of the host intracluster medium (ICM) in such clusters find short radiative cooling times within the central 100\,kpc, yet there is an order of magnitude less cold gas and star formation in and around the central brightest cluster galaxy (BCG) than expected from such cooling \citep{peterson:06a,fabian:12a,liu:19a}. There must clearly be a source of heat for the ICM core, and AGN heating is strongly implicated via observations of strong ICM/AGN-jet interactions including jet-blown cavities \citep{fabian:00a,heinz:02a,birzan:04a}, sound waves and weak shocks \citep{fabian:05a,graham:08a,million:10a}, and AGN-driven turbulence \citep{zhuravleva:14a,hitomi:16a}. As such, cool core clusters are the ideal laboratory for understanding, at a detailed mechanistic level, at least one face of AGN feedback.
The process by which BCG AGN heat the ICM has been the subject of intense theoretical work and computational modelling \citep{churazov:01a,reynolds:02a,vernaleo:06a,li:14a,reynolds:15b,yang:16a,yang:16b,ruszkowski:17a,bambic:19a}. Questions still remain about the dominant mode of heating (with weak shocks/sound waves, turbulent dissipation, and cosmic ray streaming as the primary contenders) but the basic ingredients of the heating problem now seem clear. More mysterious is the other side of the feedback loop, the feeding of the black hole on sub-parsec scales in a manner that is self-regulated so as to produce the appropriate level of heating throughout the ICM core on the $>$10\,kpc scales. Broadly, the self-regulation of the feedback loop will be achieved if the AGN fuel supply results directly from the ICM cooling process, either by cooling-mediated Bondi-like accretion of the hot ICM \citep{allen:06a}, or the accretion of cold gas that condenses from the hot ICM due to thermal instability \citep{gaspari:13a}. Either way, we may expect an unusual mode of accretion to be operating for BCG AGN compared with typical AGN. Thus, it is particularly interesting to examine the circumnuclear environment of BCG AGN, search for evidence of inflows/outflows that can be traced back to the ICM, and seek signs of any localized processes that may act to stabilize (or destabilize) the overall thermal regulation of the cluster core.
Here, we use high-resolution X-ray spectroscopy to examine the circumnuclear environment of the AGN in NGC~1275, the BCG at the centre of the Perseus cluster of galaxies. Perseus is the closest \cite[$z=0.0176$;][]{hitomi:16a} massive \citep[$M=6.6\pm 0.4\times 10^{14}\hbox{$\rm\thinspace M_{\odot}$}$;][]{simionescu:11a} cool-core cluster and has become the Rosetta Stone of AGN-cluster feedback studies. This gives studies of NGC~1275 special significance. While our investigation uses X-ray observations, it is important and relevant to review the findings from longer wavelength studies. Near infrared integral field unit data have revealed significant amounts of molecular hydrogen within 100\,pc of the black hole \citep{wilman:05a}. Some of the observed molecular material forms a rotating circumnuclear disk with radius $r\sim 50{\rm\thinspace pc}$ \citep[][hereafter S13]{scharwachter:13a}, the dynamics of which imply a total enclosed mass of $M=8^{+7}_{-2}\times 10^8\hbox{$\rm\thinspace M_{\odot}$}$. Using data from the {\it Gemini} Near-infrared Integral Field Spectrograph (NIFS), S13 argue that the observed H$_2$ lines are shock excited, and estimate that the circumnuclear disk contains $\sim 4\times 10^8\hbox{$\rm\thinspace M_{\odot}$}$ of molecular gas orbiting the SMBH. They also find evidence for a streamer of molecular gas that appears to be inflowing towards this central disk in a retrograde sense, highlighting the kinematic complexity of this region. \cite{nagai:19a} present data from the Atacama Large Millimeter Array (ALMA) that traces the circumnuclear molecular gas with CO(2-1), HCN(3-2) and HCO$^+$(3-2) lines at high spatial resolution (20\,pc). They also find a complex set of molecular filaments and a cold rotating molecular disk extending 100\,pc from the SMBH, and highlight the similarity of this structure with predictions from the cold chaotic accretion model \citep{gaspari:17a}. The fact that the observed radio jet is oriented orthogonal to this molecular disk is suggestive that we are seeing the outer regions the SMBH accretion flow, and that this flow preserves its orientation down to close to the black hole. \cite{nagai:19a} discover HCN(3-2) and HCO$^+$(3-2) absorption against the radio continuum of the central pc-scale jet emission blueshifted by 300--600\hbox{$\km\s^{-1}\,$}, suggesting a fast molecular outflow from the AGN with an estimated H$_2$ column density of $N_{H_2}\approx 2.3\times 10^{22}\hbox{$\cm^{-2}\,$}$. The existence of a fast molecular outflow on $R\sim 100{\rm\thinspace pc}$ scales has been strengthened recently by new {\it Gemini} NIFS data that find a high velocity-dispersion component to the H$_2$ line which extends across the $3\arcsec\times 3\arcsec$ field of view of NIFS \citep{riffel:20a}.
In this paper, we study the AGN and its circumnuclear environment using a deep (490\,ks) observation with the High-Energy Transmission Gratings (HETG) on the {\it Chandra X-ray Observatory}. The 1--9\,keV band data is well described by a power-law\citep[modified by the expected Galactic absorption $N_H=1.32\times 10^{22}\hbox{$\cm^{-2}\,$}$][]{kalberla:05a}. We recently used this fact to set the tightest limits to date on the coupling of photons and light axion-like-particles (ALP) in the magnetic field of the ICM \citep[][hereafter R20]{reynolds:20a}. Here, we examine the implications of these data for the astrophysics of this AGN. We quantify the absence of emission/absorption lines in the soft X-ray spectrum and show that NGC~1275 does not possess the ``warm absorber'' outflows that are typical of many Seyfert-like AGN. We do, however. find evidence that part (15--20\%) of the X-ray emission is subject to cold absorption ($N_H\sim 8\times 10^{22}\hbox{$\cm^{-2}\,$}$) suggesting a composite X-ray source and sub-parsec scale structure in the circumnuclear molecular gas. We detect the iron-K$\alpha$ fluorescence line, previously seen by {\it XMM-Newton} and {\it Hitomi}, at high confidence. Motivated by hints of anomalous iron-K$\alpha$ broadening in the dispersed spectra, we discuss a methodology that can combined the strengths of the dispersive HETG data, non-dispersive microcalorimeter data from {\it Hitomi}, and high signal-to-noise but medium spectral resolution data from the {\it XMM-Newton}/EPIC in order to search for sub-arcsecond spatial extension of the iron-K$\alpha$ emission region.
The paper is organised as follows. Section~\ref{data} describes the {\it Chandra}/HETG observations that form our core dataset and sketches the reduction steps followed to produce science-ready products. Section~\ref{broadbandspec} examines the broad-band HETG spectrum, reporting evidence for partial covering of the X-ray source by a substantial cold column. Section~\ref{blind} reports a systematic ``blind'' search for emission and absorption lines in the 1.4--9\,keV spectrum, highlighting the importance of the look-elsewhere effect when assessing the significance of features. Section~\ref{outflows} presents a more global search for photoionized outflows using self-consistent photoionization models, that allow us to exclude the presence of absorbers with column densities and ionization parameters typically found in Seyfert nuclei. Section~\ref{reflection} examines the iron-K$\alpha$ fluorescence line and presents our methodology for constraining spatial broadening. Our conclusions and the implications of our results are discussed in Section~\ref{discussion}. Throughout this paper we adopt {\it Planck}-2018 cosmological parameters \citep[$H_0=67.4\hbox{$\kmps\Mpc^{-1}$}, \Omega_M=0.31, \Omega_\Lambda=0.69$; ][]{planck:18a}. With a cosmological redshift to NGC~1275 of $z=0.0173$, this gives a luminosity distance of 78.0\,Mpc and a scale of 365\,pc per arc-second. We note that, unless otherwise stated, all errors are quoted at the 90\% confidence level (CL).
3
\section{The {\it Chandra}/HETG data reduction}\label{data}
\begin{table}
\caption{Log of the Cycle-19 {\it Chandra}/HETG observations of NGC~1275.}
\begin{center}
\begin{tabular}{cccc}
ObsID & Start date & Good exposure & Roll angle \\\hline\hline
& & (ks) & (deg) \\\hline
20823 & 2017-10-24 & 53.3 & $-45$ \\
20450 & 2017-10-27 & 29.6 & $-45$ \\
20826 & 2017-10-28 & 29.6 & $-49$ \\
20451 & 2017-10-30 & 36.5 & $-73$ \\
20837 & 2017-10-31 & 35.5& $-60$ \\
20838 & 2017-11-01 & 24.7 & $-60$\\
20839 & 2017-11-03 & 21.7 & $-66$\\
20840 & 2017-11-04 & 20.7 & $-70$\\
20449 & 2017-11-06 & 45.3 & $-129$\\
20841 & 2017-11-09 & 54.2 & $-154$\\
20842 & 2017-11-10 & 18.3 & $-143$\\
20843 & 2017-11-11 & 22.1 & $+120$\\
20824 & 2017-12-02 & 49.3 & $+117$\\
20827 & 2017-12-04 & 17.0 & $+111$\\
20844 & 2017-12-05 & 34.5 & $+111$\\\hline
\end{tabular}
\end{center}
\label{tab:obs}
\end{table}
The {\it Chandra}/HETG data that form the core of the present study have previously been described by R20. Here, we use an updated but otherwise identical reduction of the data. In brief, {\it Chandra} observed NGC~1275 as part of a Cycle-19 Large Project, with the observing being split into 15 segments (ObsIDs) between 2017 October 24 and 2017 December 5 (Table~\ref{tab:obs}). The HETG was placed in the X-ray beam resulting in two independent dispersed spectra, one from the High Energy Grating (HEG) and one from the Medium Energy Grating (MEG). The HETG is a slit-less grating array resulting in dispersion of the extended ICM as well as the compact AGN in NGC~1275. Still, as explicitly illustrated in Figure~1 of R20, the dispersed point source emission can be isolated from much of the background and dispersed ICM emission resulting in a high-quality and high-resolution spectrum of the AGN.
The raw data were reprocessed with CIAO-4.12 and CALDBv4.9.1. The preparation of science-ready HEG and MEG spectra for the AGN then followed the standard recommended threads\footnote{http://cxc.cfa.harvard.edu/ciao/threads/spectra\_hetgacis/}, except for three modifications. Firstly, the width of the extraction region was reduced by a factor of two from the default (i.e. we use {\tt width\_factor\_hetg=18}) in order to permit the extension of the HEG spectrum up to 9\,keV by reducing overlap of the MEG and HEG extraction regions at the center of the dispersion pattern. Secondly, it was found that the automatic zeroth-order finder algorithm became confused by the extended ICM emission which, if uncorrected, would lead to a mis-identification of the center of the dispersed spectrum and hence an incorrect wavelength scale. This was overcome by hard-coding the initial guess for the location of the zeroth-order in the call to {\tt tgdetect} to be at the coordinates of the nucleus of NGC~1275 as determined by the spacecraft astrometry. We used sub-pixel imaging on a 0.05\arcsec grid to verify visually that, for each of the 15 ObsIDs, this leads to a correct placement of the center of the dispersed spectrum to within 0.2\arcsec or less. We then extracted the (positive and negative) first order HEG and MEG spectra and background spectra for each of the 15 ObsIDs and computed the associated response matrices and effective area files. The spectra were combined with the CIAO tool {\tt combine\_grating\_spectra} in order to produce a single first-order MEG and a single first-order HEG spectrum with a total of 490\,ks exposure time. Higher (second and third) order spectra were produced and examined but do not have the signal-to-noise to be useful and hence will not be discussed further.
Thirdly, as discussed in R20, care was needed with the background subtraction of the AGN spectra. The ``background'' for the AGN spectrum is principally due to dispersed (and some zeroth order) ICM light. The ICM emission is centrally peaked around the AGN itself, but the background spectra must be extracted from offset strips; hence it is expected that the pipeline processing will produce background spectra that underestimate the true dispersed-ICM emission captured in the AGN extraction region. Indeed, examining the default/pipeline products show that the background-subtracted AGN spectra still, actually, contain structure reminiscent of that in the background spectra. As shown in R20, MARX simulations find that this can be accounted for to a high accuracy with a simple renormalization of the background spectrum which we achieve by adjusting the AREASCAL keyword in the background spectra. We find that the background features are formally minimized according to a C-statistic measure by setting AREASCAL to be 0.474 (HEG) and 0.525 (MEG), implying that the background spectra are scaled up in normalization by approximately a factor of two.
\section{Broad-band HETG spectrum and the need for a composite X-ray source}\label{broadbandspec}
As shown in R20 (their Fig.~3), the HETG spectrum of NGC~1275 is well approximated by a power-law. As a first look at the spectrum, we use C-statistic minimization to fit a power-law model simultaneously to the 1--7\,keV MEG and 1.5--9\,keV HEG spectra. Cold/neutral Galactic absorption is included in the model fit with a column density of $N_H=1.32\times 10^{21}\hbox{$\cm^{-2}\,$}$ \citep{kalberla:05a}. We allow a multiplicative offset when modeling two spectra to allow for absolute flux calibration offsets. The best fitting model has a photon index $\Gamma=1.894\pm 0.009$, HEG power-law normalization at 1\,keV $A=(8.38\pm 0.08)\times 10^{-3}\,{\rm ph}\,{\rm s}^{-1}\,{\rm keV}^{-1}\,{\rm cm}^{-2}$, and a MEG offset factor of $1.026\pm 0.009$ (suggesting a slightly higher flux calibration for the MEG), with a goodness-of-fit of $C=4893$ for 4873 degrees of freedom. The corresponding 2--10\,keV flux is $(2.48\pm 0.02)\times 10^{-11}\hbox{$\erg\cm^{-2}\s^{-1}\,$}$, and the 2--10\,keV (rest-frame) luminosity is $(1.67\pm 0.02)\times 10^{43}\hbox{$\erg\s^{-1}\,$}$.
As discussed in the Introduction, ALMA has discovered HCN and HCO$^+$ absorption towards the parsec-scale jet \citep{nagai:19a} leading to an estimated H$_2$ column density of $N_{H_2}\approx 2.3\times 10^{22}\hbox{$\cm^{-2}\,$}$ (assuming an excitation temperature of 100K, an HCN-to-H$_2$ conversion factor of $10^{-9}$, and full covering of the mm-band continuum). X-ray variability \citep{fabian:15b,imazato:21a} requires that a significant fraction of the observed X-ray continuum must originate from parsec scales or less, so it is interesting to ask whether there are any signatures of this molecular gas column in the X-ray spectrum. If we assume that a cold (atomic$+$molecular) absorbing screen covers the entire X-ray source, the lack of soft X-ray spectral curvature leads to very tight limits on the column density --- adding a cold absorber at the redshift of NGC~1275 (xspec model {\tt zTBabs}) gives a formal limit of $N_H<3\times 10^{19}\hbox{$\cm^{-2}\,$}$ (90\% confidence level), three orders of magnitude smaller than the molecular column inferred from ALMA. This difference between the X-ray and mm-band absorbing column densities appears robust to the underlying assumptions. The difference is too great to be bridged by modifications of the assumed HCN excitation temperature or reasonable changes in the HCN-to-H$_2$ conversion factor. On the X-ray side, a line-of-sight H$_2$ column of $\sim 2\times 10^{22}\hbox{$\cm^{-2}\,$}$ covering the whole source would be impossible to reconcile with the HETG data given any reasonable intrinsic spectrum (imparting a factor of 100 deviation from the observed power-law at 1\,keV). The solution must be geometric, requiring the molecular gas to be clumpy or structured on sub-parsec scales.
\begin{figure}
\includegraphics[width=0.45\textwidth]{pcf_contours.pdf}
\caption{Confidence contours on the $(\log N_H, \log f_{\rm cov})$-plane from the fit of a partially-covered power-law model to the HETG data. Shown here the $68\%$ (white),$90\%$ (light grey), $99\%$ (dark grey) and $99.9\%$ (black) confidence levels, with the underlying C-stat in the background colour map. }\label{fig:pcf_contours}
\end{figure}
The observed X-ray emission in NGC~1275 may be a composite such that it possesses absorbed and unabsorbed subcomponents, with possible contributions to the X-ray emission including the inner accretion disk, a sub-parsec scale jet core, and the hot-spot of the current jet activity which is currently 1.2{\rm\thinspace pc}\ (projected) south of the nucleus \cite[radio component C3; ][]{nagai:10a,hodgson:21a}. It is then possible that some component of the X-ray {\it does} experience the absorption from the molecular gas. This motivates us to examine partial-covering solutions. The inclusion of a partial-covering absorber (xspec model {\tt TBpcf}) to our baseline power-law model leads to a significant improvement in the goodness of fit ($\Delta C=-24$ for two additional degrees of freedom) and flattens out some of the remaining ``wave-like'' residuals noted in R20. In this pure HETG-fit, we infer an intrinsic column of $N_H=7.8^{+3.1}_{-1.9}\times 10^{22}\hbox{$\cm^{-2}\,$}$ covering a fraction $f_{\rm cov}=0.13\pm 0.04$ of the X-ray emission (Figure~\ref{fig:pcf_contours}). The inclusion of this partial-covering X-ray absorber steepens the best-fit power-law slope to $\Gamma=1.97\pm 0.03$. This X-ray column density translates to a molecular column of $N_{H_2}=3.9^{+1.5}_{-1.0}\times 10^{22}\hbox{$\cm^{-2}\,$}$ which is approximately twice that inferred from {\it ALMA} assuming full covering of the mm-band continuum. This suggests that the mm-band absorption itself may also composite. We discuss this further in Section~\ref{discussion}
\section{A blind search for emission and absorption lines}\label{blind}
\begin{figure*}
\includegraphics[width=0.95\textwidth]{blindsearch2_nicefigure.pdf}
\caption{Results of the blind line search in the MEG/HEG joint spectrum. Shown are contours in the improvement in goodness of fit $\Delta C$ upon the inclusion of a Gaussian line with rest-frame energy $E_{\rm line}$ and normalization $N_{\rm line}$ relative to a baseline model consisting of a simple power-law continuum subject to Galactic absorption ($N_{}\rm H)=1.32\times 10^{21}\hbox{$\cm^{-2}\,$}$. Contour levels correspond to $\Delta C=-1.0$ (thin black line; 68\% single-trial CL), $\Delta C=-2.706$ red line; (90\% single-trial CL), $\Delta C=-6.65$ (green line; 99\% single-trial CL), $\Delta C=-13.8$ (blue line; 90\% CL for 500-trials). Upper panel shows the results of the soft band analysis, with MEG and HEG fitted in the range 1--5{\rm\thinspace keV} and 1.5--5\,keV respectively. Lower panel shows the hard band analysis, with MEG and HEG fitted in the range 5--7{\rm\thinspace keV} and 5--9\,keV respectively. }\label{fig:blindsearch}
\end{figure*}
These {\it Chandra}/HETG data permit the most sensitive search to date for X-ray emission and absorption lines from this AGN. Starting with the baseline power-law model discussed in Section~\ref{broadbandspec}, we examine the improvement in the C-statistic upon the inclusion of a Gaussian line with (rest frame) energy $E_{\rm line}$ and a normalization $N_{\rm line}$ that can be positive or negative. The line is taken to be sufficiently narrow as to be unresolved by the HETG ($\sigma=1{\rm\thinspace eV}$). We conduct the line search in two parts, treating the soft band (1.3--5{\rm\thinspace keV}) and hard band (5--9\,keV) separately so that the results are not biased by any subtle curvature in the underlying continuum. For this reason, this analysis is also insensitive to whether we use the simple power-law or the partial-covering model from Section~\ref{broadbandspec}; for simplicity, we use the simple power-law model. For each of these two bands, we scan through a dense grid on the $(E_{\rm line}, N_{\rm line})$-plane, refit the continuum parameters, and record the change in C-statistic ($\Delta C$).
\begin{table*}
\caption{Results of fitting a narrow ($\sigma=1{\rm\thinspace eV}$) Gaussian line to the most likely detections resulting from the blind line search of Section~\ref{blind}. }
\begin{center}
\begin{tabular}{cccclcl}\hline\hline
Energy & Norm & EW & $\Delta C$ & Possible ID & Velocity & Notes \\
(keV) & ($10^{-6}$ph/s/cm$^2$) & (eV) & & &($\hbox{$\km\s^{-1}\,$}$) &\\\hline
$1.469\pm 0.006$ & $2.5^{+2.0}_{-1.3}$ & $0.6\pm 0.4$ & $-8.4$ & MgXII Ly$\alpha$(1.472keV) & $+610\pm 220$ &\\
$1.593\pm 0.001$ & $-3.3\pm 0.1$ & $-0.9\pm 0.3$ & $-22.2$ & AlXII K$\alpha$(1.598keV) & $+940\pm 110$ &\\
$1.871\pm 0.001$ & $1.8\pm 1.0$ & $0.7\pm 0.4$ & $-8.9$ & SiXIII K$\alpha$ (1.855keV) & $-2590\pm 190$ & close to detector calibration feature\\
$2.011\pm 0.001$ & $-2.0\pm 1.0$ & $-0.8\pm 0.4$ & $-10.4$ & SiXIV Ly$\alpha$ (2.007keV) & $-600\pm 190$ &\\
$2.619\pm 0.002$ & $-2.2\pm 1.4$ & $-1.5\pm 1.1$ & $-5.3$ & SXVI Ly$\alpha$ (2.620keV) & $+110\pm 230$ &\\
$6.411\pm 0.015$ & $3.7^{+2.2}_{-2.0}$ & $14\pm 8$ & $-9.43$ & FeI K$\alpha$ (6.40keV) & $-520\pm 700$ & \\
$7.06^{-0.07}_{+0.12}$& $4.0^{+3.6}_{-2.1}$ & $18\pm 12$ & $-8.9$ & FeXXVI Ly$\alpha$ (6.97keV) & $-3870\pm 3000$ & \\\hline
\end{tabular}
\end{center}
{Notes : All errors quoted at the 90\% level for one interesting parameter. Line energy is given in the rest-frame of NGC~1275 assuming a redshift of $z=0.0173$. The improvement in the goodness of fit upon including the line ($\Delta C$) is with respect to the pure continuum model (powerlaw subject to Galactic absorption with $N_H=1.32\times 10^{21}\hbox{$\cm^{-2}\,$}$). Velocity is quoted with respect to the rest-frame of NGC~1275, with positive velocity implying recession.}
\label{tab:lines}
\vspace{0.5cm}
\end{table*}%
Figure~\ref{fig:blindsearch} shows the results of this exercise. In the soft-band (upper panel), approximately 50 features exceed the 90\% single-trial confidence level (red contours), and eight exceed the 99\% single-trial confidence level (green contours). Four of these are close to, but not exactly at, energies of important transitions that might be expected from a photoionized AGN wind; MgXII\,Ly$\alpha$ (emission), AlXII\,K$\alpha$ (absorption), SiXIII\,K$\alpha$ (emission), and SiXIV\,Ly$\alpha$ (absorption). We also note a less significant feature (present at the 90\% but not 99\% single-trial CL) that is consistent with SXVI\,Ly$\alpha$. Table~\ref{tab:lines} reports the detailed properties of these tentative line detections.
\begin{figure}
\includegraphics[width=0.49\textwidth]{al12_line.pdf}
\caption{Zoom-in of the 1.57--1.62{\rm\thinspace keV} (rest-frame) region of the MEG (red) and HEG (blue) spectra showing the putative AlXII\,K$\alpha$ absorption line. The solid red/blue lines show the corresponding best fitting power-law plus Gaussian model. The laboratory rest frame energy of the AlXII\,K$\alpha$ line is shown as the vertical dotted line.}\label{fig:al12_line}
\end{figure}
When interpreting the features seen in this blind line search, care is needed; there are a large number of independent spectral bins within this band and so it is important to account for the look-elsewhere effect when assessing statistical significance. We take the fiducial soft-band resolution of the HETG ($E/\Delta E\sim 500$) as an estimate for the number of independent trials. Only one feature, the putative AlXII\,K$\alpha$ feature at 1.593\,keV, exceeds the 90\% (500 trial) confidence level. Figure~\ref{fig:al12_line} shows the AlXII\,K$\alpha$ region of the HEG and MEG with the best fitting Gaussian model.
We note that, on the basis of the calculations by \cite{kockert:06a}, we do not expected to detect any absorption lines from the thermal ICM plasma. The strongest ICM absorption line in this soft band is predicted to be SXV\,K$\alpha$ (2.46\,keV) with an equivalent width (EW) of $\sim 1{\rm\thinspace eV}$. Such a weak line will fall below our sensitivity threshold and, indeed, we do not find such a feature. The predicted EW of the ICM SiXIV\,Ly$\alpha$ and SXVI\,Ly$\alpha$ are approximately $0.2{\rm\thinspace eV}$. Thus, it seems highly unlikely that ICM absorption is responsible for the putative line detections reported in Table~\ref{tab:lines}. An ICM origin for these lines is also inconsistent with the velocity shifts needed to bring the observed features in line with laboratory wavelengths.
Figure~\ref{fig:blindsearch} (bottom panel) shows the result of the blind line search in the hard-band. We clearly detect the 6.4\,keV fluorescent K$\alpha$-line of cold iron at the 99\% (single trial) CL; this line is discussed much more extensively in Section~\ref{reflection}. We also detect, again at the 99\% (single trial) CL, an emission features that lies close to the FeXXVI\,Ly$\alpha$ line. Making this identification suggests a significant flow of highly ionized plasma with a line of sight velocity of $V= -3900\pm 3000\hbox{$\km\s^{-1}\,$}$ (negative sign indicating flow towards us relative to the rest-frame of the AGN). There is a weaker and less significant feature ($E_{\rm line}=6.928\pm 0.018{\rm\thinspace keV}$, $\Delta C=-5.6$) that could be a redshifted/counterflowing partner of this line with line of sight velocity of $V= +1700\pm 700\hbox{$\km\s^{-1}\,$}$. This emission may result from a fast, recombining, highly-ionized wind driven by the central engine.
Caution must again be exercised in interpreting any other features in this band. Away from the physical energies for the astrophysically important lines, we must again take into account the look elsewhere effect. We take the spectral resolution in this band ($E/\Delta E\sim 200$) as an estimate for the number of independent energy bins. We find that none of the features in this band exceed a confidence level corresponding to 90\% for 200 trials and hence cannot be considered significant detections.
\section{Constraints on photoionized outflow signatures}\label{outflows}
The blind line search reveals tentative detections of several absorption lines of high ionization species (AlXII\,K$\alpha$, SiXIII\,K$\alpha$, SiXIV\,Ly$\alpha$ and SXVI\,Ly$\alpha$) with blueshifts indicating velocities up to $\sim 2500\hbox{$\km\s^{-1}\,$}$. These may hint at the existence of a photoionized outflow from the AGN, similar to that found in half of all Seyfert nuclei \citep{reynolds:97b,tombesi:13a,laha:21a}. We also find a significant number of other absorption features further away from the rest-frame energies of expected strong transitions. Especially interesting are the features in the 7--8\,keV band which are reminiscent of the highly ionized ultrafast outflows found in some Seyfert nuclei and broad-line radio galaxies. While these lines are not individually significant given the look-elsewhere effect, it is interesting to note that pairs of these lines have separations close to the FeXXV~K$\alpha$/FeXXVI~Ly$\alpha$ separation raising the possibility that multi-species modeling might yield greater significance.
In this Section, we examine the HETG spectrum using self-consistent photoionization models. We use the XSTAR photoionization code to compute the absorption spectrum from a column density $N_W$ of plasma photoionized by the AGN continuum. The ionization state of the plasma is described by an ionization parameter $\xi=4\pi F_{\rm ion}/n_e$ where $F_{\rm ion}$ is the energy flux of ionizing ($E>13.6{\rm\thinspace eV}$) radiation incident on the plasma and $n_e$ is the electron number density. For the purposes of the photoionization calculation, this ionizing continuum is modelled as a power-law with photon index $\Gamma=1.9$ between 0.1--100\,keV. We compute a grid of models that uniformly sample the $(\log\xi, \log N_W)$-plane with $20\times 20$ models spanning the range $\log\xi\in(0,6)$ and $\log N_W\in(20,24)$ (cgs units). The resulting multiplicative absorption models can then be applied to any continuum model, passed through the instrumental response, and compared with the HETG data.
Guided by the results of Section~\ref{blind}, we model the full usable range of the MEG (1--7\,keV) and HEG (1.5--9\,keV) spectra with a baseline model consisting of a power-law continuum (modified by Galactic absorption) and narrow Gaussian emission lines at 6.4\,keV (fixed redshift $z=0.0173$) and 6.97\,keV (redshift free parameter). Including the XSTAR photoionization model and minimizing CSTAT gives an improvement in the goodness of fit of $\Delta C=-11.8$, and suggests the presence of a highly ionized fast outflow with $\log\xi=3.92^{+0.37}_{-0.14}$, $\log N_W=22.3^{+0.17}_{-0.31}$ and an outflow velocity $v=(2.30^{+0.10}_{-0.05})\times 10^4\hbox{$\km\s^{-1}\,$}$. Here, the 90\% CL error bars are computed from the usual $\Delta C=2.706$ criterion. However, as in the blind line search, great care must be taken in assessing the significance of any such outflow due to the unknown velocity giving rise to a strong look-elsewhere effect.
\begin{table*}
\caption{Priors for our MCMC analysis of photoionized absorbers in NGC~1275. }
\begin{tabular}{llc}\hline\hline
Parameter & Prior & Range \\\hline
{\bf Photoionized absorber} & & \\
Column density $N_W$ & uniform in $\log N_W$ & $(10^{20},10^{24}\hbox{$\cm^{-2}\,$})$ \\
Ionization parameter $\xi$ & uniform in $\log\xi$ & $(10,10^5\,{\rm erg}\,{\rm s}^{-1}\,{\rm cm})$ \\
Redshift (wrt to AGN) $z_{\rm ph}$ & uniform in $z_{\rm ph}$ & $(-0.3,0.1)$ \\\hline
{\bf Powerlaw continuum} & & \\
Photon index $\Gamma$ & uniform in $\Gamma$ & $(1,3)$ \\
Normalization@1\,keV $N_{\rm pl}$ & uniform in $N_{\rm pl}$ & $(0,10^{-1}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2}\,{\rm keV}^{-1})$ \\\hline
{\bf Emission lines} & & \\
Normalization of 6.4\,keV line $N_{\rm 6.4}$ & uniform in $N_{\rm 6.4}$ & $(0,10^{-4}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2})$ \\
Normalization of 6.97\,keV line $N_{\rm 6.97}$ & uniform in $N_{\rm 6.97}$ & $(0,10^{-4}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2})$ \\
Redshift of 6.97\,keV line $z_{\rm 6.97}$ & uniform in $z_{\rm 6.97}$ & $(0,0.05)$ \\\hline
MEG/HEG scaling ratio, ${\cal R}$ & uniform in ${\cal R}$ & (0.8,1.2) \\\hline\hline
\end{tabular}
\label{tab:mcmc}
\end{table*}
It is common practice to assess the significance of such detections in the presence of the look elsewhere effect by generating many simulated spectra with noise characteristics similar to the real data. Instead, here we adopt a more Bayesian philosophy (i.e. one that avoids the invention of any data) and employ a Monte-Carlo Markov Chain (MCMC) analysis to map out the likelihood $P({\cal D}|{\cal M})$ where $\cal{D}$ is the spectral dataset and ${\cal M}=\{{\cal M}_i\}$ is the set of model parameters. If we specify the prior probability for the set of model parameters $P(\cal{M})$, Bayes' theorem then allows us to calculate the basic quantity of interest, the posterior probability $P({\cal M}|{\cal D})\propto P(\cal{M})P({\cal D}|{\cal M})$. The probability distribution of any single parameter (or sub-set of ``interesting'' parameters) is then obtained by marginalizing $P({\cal M}|{\cal D})$ over the all uninteresting parameters. Marginalization naturally accounts for any look-elsewhere effect that may be at play.
\begin{figure}
\includegraphics[width=0.48\textwidth]{xinh_plot2.pdf}
\vspace{-0.5cm}
\caption{Constraints on the column densiity $N_W$ and ionization parameter $\xi$ of any photoionized absorber along the line of sight to X-ray source in NGC~1275. Shown here are the 68\%, 95\%, 99\% and 99.9\% exclusion limits resulting from the MCMC analysis described in Section~\ref{outflows}. For comparison, red datapoints show detected warm absorbers from the sample of Tombesi et al. (2013), and cyan datapoints show the detected ultrafast outflows from the sample of Tombesi et al. (2011). These datapoints are shown with 1$\sigma$ errors. }
\label{fig:outflow}
\end{figure}
We use the Goodman-Weare MCMC algorithm incorporated within the XSPEC spectral fitting package to compute the likelihood for our problem. The full set of model parameters and their relevant priors are listed in Table~\ref{tab:mcmc}. We run eight independent MCMC chains each consisting of 100 walkers that undergo $1\times 10^4$ steps, giving chains with $1\times 10^6$ elements each. Each chain has an initial burn-in consisting of $1\times 10^3$ steps (i.e. 10\% of the length of the full chain). We find consistency between the chains in the distribution of all parameters, so we can combine them to give $8\times 10^6$ sample points for the likelihood and the corresponding posterior probability.
Figure~\ref{fig:outflow} shows the resulting joint probability distribution on the $(\log\xi, \log N_W)$ marginalizing over all other parameters. We fail to detect a photoionized absorber, and exclude at high confidence the presence of an absorber across a large swath of the $(\log\xi, \log N_W)$-plane. To give these constraints context, Fig.~\ref{fig:outflow} shows the warm absorbers from the Seyfert sample of \cite{tombesi:13a} as well as the ultrafast outflows from the sample of \cite{tombesi:11a}. We see that when photoionized absorbers are detected, it is usually within the part of parameter space that we can already rule out for NGC~1275. Thus the non-detection of a photoionized absorber in this object does indeed show our line of sight to be devoid of photoionized plasma columns that are commonly detected in bright AGN.
We note that if we repeat the MCMC analysis but freeze the outflow velocity of any absorber at $v=23,000\hbox{$\km\s^{-1}\,$}$ (i.e. the value determined by simple minimization of CSTAT), then we would infer the presence of a high-ionization absorber with 97\% confidence. This confirms that it is the look-elsewhere effect associated with the unknown velocity that compromises any such detection.
\section{X-ray reflection from circumnuclear gas}\label{reflection}
Prior studies have already established the presence of the 6.4\,keV fluorescent K$\alpha$ line of cold iron in the X-ray spectrum of NGC~1275. This line was clearly detected in a 2001 observation by the {\it XMM-Newton}/EPIC with a reported equivalent width $\sim 165{\rm\thinspace eV}$ \citep{churazov:2003hr}. A follow-up {\it XMM-Newton}/EPIC in 2006 found the line with a similar flux, but the overall increase in the continuum flux led to the equivalent width decreasing to 70--80\,keV \citep{yamazaki:13a}. More recently, non-dispersive high-resolution spectroscopy with the Soft X-ray Spectrometer (SXS) on {\it Hitomi} also clearly identifies and, for the first time, resolves the line \citep{hitomi:18a} finding a full-width half maximum (FWHM) 500--1600\hbox{$\km\s^{-1}\,$} (90\% confidence level). Given this relatively low velocity width, \cite{hitomi:18a} suggest that the line originates from a low-covering fraction atomic/molecular disk that extends anywhere from $\sim {\rm few}{\rm\thinspace pc}$ to hundreds of pc from the black hole. The flux of the line found by {\it Hitomi} is consistent with that seen by {\it XMM-Newton}, entirely expected given that the light-crossing time of the fluorescent structure will be at least $\sim 10$ years.
\subsection{Fluorescent iron emission in the HETG spectrum}
\begin{figure}
\includegraphics[width=0.48\textwidth]{ironbandlines.pdf}
\vspace{-0.5cm}
\caption{Iron K-band region of the HEG spectrum showing the neutral Fe-K$\alpha$ and the possible red/blue shifted FeXXVI-Ly$\alpha$ lines on top of the power-law continuum discussed in Section~\ref{blind}. The spectrum has been lightly binned by a factor of two, resulting in oversampling of the spectral resolution by a factor of two.}
\label{fig:ironbandlines}
\end{figure}
As reported in Section~\ref{blind} and Table~\ref{tab:lines}, we also find the fluorescent iron-K$\alpha$ line in our HETG spectrum. Figure~\ref{fig:ironbandlines} shows the best fitting single narrow Gaussian model overplotted on part of HEG spectrum, lightly binned for plotting purposes. To allow a more direct comparison with the {\it Hitomi} result, we proceed to refine our spectral model of the line to account for the fact that it is a doublet with energies 6.404\,keV (Fe-K$\alpha_1$) and 6.391\,keV (Fe-K$\alpha_2$) and a K$\alpha_1$/K$\alpha_2$ branching ratio of 2. Initially, we fix the redshift of the doublet to be strictly that of NGC~1275 ($z=0.0173$), and fix the velocity width of the emission line to be the best fitting value found by {\it Hitomi} ($\sigma=9{\rm\thinspace eV}$). Jointly fitting to the HEG/MEG spectrum (in the 5--9\,keV/5--7\,keV bands, respectively) we find the Fe-K$\alpha_1$ normalization to be $2.6^{+1.6}_{-1.5}\times 10^{-6}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2}$ (90\% CL). This is marginally in tension with the flux found by {\it Hitomi} ($4.5^{+1.5}_{-1.3}\times 10^{-6}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2}$ at 90\% CL) and in somewhat stronger tension with reported {\it XMM-Newton} \cite[$8.4^{+4.1}_{-3.9}\times 10^{-6}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2}$ at 90\% CL, dividing the total 2001 Fe-K$\alpha$ flux reported in][ by 1.5 to convert into a Fe-K$\alpha_1$ flux]{hitomi:18a}.
Given the physical size of the fluorescing region implied by the {\it Hitomi} measurements of the line width, we consider a genuine change in line flux to be unlikely. It is interesting to note that the HETG constraints on line flux can be brought into line with both {\it Hitomi} and {\it XMM-Newton} if we invoke additional line broadening. Permitting the width of the Gaussian line model to be a free parameter in the HETG fits results in a very modest improvement in the goodness of fit ($\Delta C=-0.7$ with $\sigma<86{\rm\thinspace eV}$ at 90\% CL) but this freedom increases the allowed range of Fe-K$\alpha_1$ line fluxes to $3.6^{+3.3}_{-2.2}\times 10^{-6}\,{\rm ph}\,{\rm s}^{-1}\,{\rm cm}^{-2}$ (90\% CL), entirely consistent with that found by {\it Hitomi} and {\it XMM-Newton}.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{ironband_heg.pdf}
\hspace{0.5cm}
\includegraphics[width=0.45\textwidth]{ironband_meg.pdf}
\vspace{-0.2cm}
\caption{Illustration of the consequences of spatial extension on the inferred iron K$\alpha$ line in the unbinned HEG (left panel) and MEG (right panel) spectra. The black line shows a model consisting of a point-like source of the iron-K$\alpha_1$/K$\alpha_2$ line with velocity dispersion $\sigma_v=420\hbox{$\km\s^{-1}\,$}$ (corresponding to 9\,eV). The green line shows the effect of extending the source of ron-K$\alpha$ photons by $\sigma_\theta=1\arcsec$. }
\label{fig:broadenedhetg}
\end{figure*}
The HETG is a slit-less dispersive spectrometer and so anomalous broadening of the Fe-K$\alpha$ line can result from spatial extension of the emitting gas \citep{marshall:17a}. Assuming a Gaussian distribution of velocity (with dispersion $\sigma_{\rm v}$ in velocity units) and spatial radial profile (with standard deviation $\sigma_\theta$ in angular units), the overall measured line width $\sigma$ (in energy units) will be given by
\begin{equation}\label{eq:grating1}
\sigma^2=\left(\frac{E_0}{c}\right)^2\sigma_{\rm v}^2+\left(\frac{fFPE_0^2}{2\pi\hbar cmD_{\rm Row}}\right)^2\sigma_{\theta}^2,
\end{equation}
where $E_0$ is the rest-frame energy of the line centroid, $F=10.0548{\rm\thinspace m}$ is the focal length of {\it Chandra}'s High Resolution Mirror Assembly, $P$ is the grating period ($P_{\rm MEG}=4001.95$\AA, $P_{\rm HEG}=2000.81$\AA), $m$ is the order of the dispersed spectrum, $D_{\rm Row}=8.63265{\rm\thinspace m}$ is the Rowland distance of the HETG from the ACIS-S array, and $c$ and $\hbar$ are the usual fundamental constants. We also include the correction factor of $f=0.73$ identified by Masterson \& Reynolds (in prep) on the basis of MARX simulations of extended iron-K$\alpha$ sources. Evaluating eqn.~\ref{eq:grating1} for the first order ($m=1$) observations of the 6.4\,keV fluorescent lines gives HEG and MEG line widths of
\begin{equation}\label{eq:gratingheg}
\sigma_{\rm HEG}^2=45.5\left(\frac{\sigma_{\rm v}}{1000\hbox{$\km\s^{-1}\,$}}\right)^2+7.37\times 10^2\left(\frac{\sigma_\theta}{1\,{\rm arcsec}}\right)^2\,{\rm\thinspace eV}^2
\end{equation}
and
\begin{equation}\label{eq:gratingmeg}
\sigma_{\rm MEG}^2=45.5\left(\frac{\sigma_{\rm v}}{1000\hbox{$\km\s^{-1}\,$}}\right)^2+2.95\times 10^3\left(\frac{\sigma_\theta}{1\,{\rm arcsec}}\right)^2\,{\rm\thinspace eV}^2
\end{equation}
As an illustration of this effect, Fig.~\ref{fig:broadenedhetg} shows the HEG and MEG spectra overlaid with two models for the iron line doublet, (i) a point-like source with velocity broadening corresponding to $\sigma_{\rm v}=9{\rm\thinspace eV}$ corresponding to a FWHM of 990\hbox{$\km\s^{-1}\,$} (black line), and (ii) a source with the same velocity broadening but spatially extended as a Gaussian radial profile with width $\sigma_\theta=1\arcsec$ (green line). In both cases the normalization is set to be consistent with {\it Hitomi} and {\it XMM-Newton} (using the procedure outlined below). It is apparent that the additional broadening due to spatial extension allows a higher flux line to be consistent with the HETG data. We can also infer that, if the source were extended by too much, it would no longer be possible to fit the line seen most clearly in HEG.
\subsection{Multi-observatory study of spatial extension}\label{reflectionmcmc}
Here, we present a methodology which allows rigorous constraints to be obtained on spatial broadening of the iron-K$\alpha$ fluorescing gas by demanding consistency in the properties of the emission line between {\it Chandra}/HETG, {\it Hitomi}/SXS and {\it XMM-Newton}/EPIC observations. While we ultimate conclude that extension of the iron-K$\alpha$ source is not required in NGC~1275 at the 90\% confidence level, we describe this analysis in detail as we believe that it may find application to future datasets once additional microcalorimeter observations of AGN are made by the {\it X-ray Imaging Spectroscopy Mission (XRISM)}.
We perform a joint analysis of the {\it Chandra}/HETG, {\it Hitomi}/SXS and {\it XMM-Newton}/EPIC data. For the {\it Chandra}/HETG, we consider the 1--7\,keV MEG and 1.5--9\,keV HEG spectra already described. For {\it Hitomi}, we use the non-dispersive high-resolution micro-calorimeter SXS spectrum of the core of the Perseus cluster spectrum previously described by \cite{hitomi:18a}. This spectrum will drive our constraints on the true velocity broadening of the fluorescent iron line independently of any spatial extent. To recap the {\it Hitomi} reduction, we use the {\it Hitomi} observations of the Perseus cluster obtained on 25--27 February 2016 and 4--6 March 2016 (OBSIDs 100040020--100040050) giving a total on-source exposure time of 240\,ksec. The cleaning of the events list, spectral calibration, and characterization of the non-cosmic background is discussed in detail in \cite{hitomi:18a}. From the cleaned events list, we extract a spectrum of the $3\times 3$ pixel$^2$ region centred on NGC~1275; this choice of region is estimated to capture at least 95\% of the AGN flux. We employ the spectral response matrix that includes the ``parabolic correction'', and the effective area curve appropriate for a point-like source.
We also include the {\it XMM-Newton} observations of NGC~1275 taken 30--31 January 2001; the {\it XMM-Newton} data provide the most stringent constraints on the flux of the fluorescent iron line. After obtaining the data from the {\it XMM-Newton} Science Archive (XSA) archives, we reprocessed the data using the {\it XMM-Newton} Science Analysis Software (SAS) version 18.0.0 and the Current Calibration Files (CCF) as of 28-June-2020. We then followed the standard data analysis threads\footnote{https://www.cosmos.esa.int/web/xmm-newton/sas-threads} in order to clean the events files of background flares and extract EPIC-MOS and EPIC-pn spectra of NGC~1275. A preliminary AGN spectrum was extracted from a circular region of radius 15\,arcsec centred on the core of NGC~1275, and a background spectrum (which is dominated by ICM emission) is extracted from an annulus centred on the AGN with inner--outer radii of 15--30\arcsec. An examination of the single-to-double event ratio suggests that the EPIC spectra are affected by modest photon pile-up. We mitigate the effects of pileup by excising counts from the central 3\arcsec (radius). An additional {\it XMM-Newton} observation taken on 29--31 January 2006 was examined, but the source was brighter thereby increasing the effects of pileup to the extent that we decided not to use these data.
We perform a joint fit of six datasets spanning three epochs; a single {\it Hitomi}/SXS spectrum (3--10\,keV band), the three {\it XMM-Newton}EPIC-MOS1/MOS2/pn spectra (2-10\,keV band), and the two {\it Chandra}/HETG spectra (using 1.5--9\,keV for HEG, 1--7\,keV for MEG). We use a spectral model consisting of an AGN power-law continuum subject to a partially-covering cold absorber, the fluorescent iron-K$\alpha_1$/K$\alpha_2$ doublet, and optical thin thermal emission from the plasma of the ICM core, all subject to neutral Galactic absorption with a column density $N_H=1.32\times 10^{21}\hbox{$\cm^{-2}\,$}$. The ICM emission is described with the APEC model that includes thermal and turbulent line broadening (XSPEC model {\tt bapec}); the explicit inclusion of ICM line broadening is crucial given the inclusion of the high resolution {\it Hitomi}/SXS data. Both the photon index and normalization of the AGN power-law continuum are allowed to be free parameters for each of the three epochs. The ICM component needs to be treated with more care in the joint fit of this rather heterogeneous dataset. For the {\it Hitomi} observation, the spectrum includes the AGN and a dominant contribution from the ICM; we freely fit for the plasma temperature $T$, metallicity $Z$ \citep[with respect to cosmic values of ][]{anders:89a}, ICM redshift $z_{\rm icm}$, velocity dispersion $\sigma_{\rm icm}$, and normalization. Due to the smaller extraction region and ability to define an appropriate background region, the {\it XMM-Newton} spectra have a sub-dominant thermal plasma component from the centralmost regions of the cluster. Thus, in these spectra, the ICM temperature and normalization for the block of {\it XMM-Newton} are free parameters, but the metallicity, redshift and velocity dispersion are tied to the {\it Hitomi} model. The {\it Chandra}/HETG spectra, given the background renormalization described in Section~\ref{data}, have no contribution from ICM emission and hence that component is given zero normalization when fitting the HEG and MEG spectra.
Given that the AGN emission is sub-dominant in the {\it Hitomi}/SXS spectrum, it is not possible for these data to provide meaningful constraints on the absorber that partially-covers the AGN X-ray continuum. We choose to tie the absorber parameters in this epoch to those of the HETG spectra which were taken approximately 20 months later. We do, however, allow the parameters of this absorption to be different at the time of the {\it XMM-Newton}/EPIC observation,15 years earlier than the {\it Hitomi}/SXS campaign.
For the fluorescent iron line, we demand that the flux and velocity width $\sigma_v$ of the iron-K$\alpha_1$/K$\alpha_2$ doublet (with rest-frame energy 6.404\,keV/6.391\,keV and branching ratio 2:1 respectively) is fixed across all datasets. Additional broadening of the line in the dispersed {\it Chandra}/HETG spectra due to spatial extent $\sigma_\theta$ is including according to eqns.~\ref{eq:gratingheg} and \ref{eq:gratingmeg}.
We again follow a Bayesian approach, using the Goodman-Weare MCMC algorithm to map out the likelihood through the 23-dimensional parameter space defining this model. Priors for the parameters are listed in Table~\ref{tab:jointfitting}. We use 640 walkers to span the parameter space, and perform an MCMC run with a burn-in length of $6.4\times 10^5$ (1000 steps for each of the walkers), and a main chain with length $1.28\times 10^6$ (2000 steps for each walker). Likelihood is assessed using the total CSTAT for the model fit to the six spectra. The {\it Hitomi}/SXS and {\it Chandra}/HETG spectra are unbinned apart from the elimination of energy channels with zero counts. For computational expediency, the {\it XMM-Newton}/EPIC spectra are binned to a minimum of 25 photons per bin.
\begin{table*}
\caption{Results of MCMC analysis of the iron-K$\alpha$ line from the joint {\it XMM-Newton}/{\it Hitomi}/{\it Chandra}. All error are quoted at the 90\% CL.}
\begin{tabular}{llcccc}\hline\hline
Parameter & Prior & {\it XMM-Newton} & {\it Hitomi}/SXS & {\it Chandra}/HETG \\
& &(Jan-2001) & (Feb/Mar-2016) & (Oct/Nov-2017) \\\hline
{\bf AGN power-law} & & & & \\
Photon index $\Gamma$ & uniform in $\Gamma\in(0,3)$ & $1.65^{+0.03}_{-0.04}$ & $2.12^{+0.10}_{-0.11}$ & $2.03^{+0.03}_{-0.04}$ \\
Normalization $A$ & uniform in $A\in (0,10^{-1})$ & $(2.4^{+0.6}_{-0.3})\times 10^{-3}$ & $(1.4^{+0.5}_{-0.4})\times 10^{-2}$ & $(1.1\pm 0.1)\times 10^{-2}$ \\
{\bf X-ray partial covering cold absorber} & & & & \\
Column Density $N_H$ ($\hbox{$\cm^{-2}\,$}$) & uniform in $N_H\in (0,10^{24})$ & $(5.7^{+4.5}_{-1.6})\times 10^{23}$ & ={\it Chandra}& $(8.2^{+2.5}_{-1.3})\times 10^{22}$\\
Covering fraction $f_{\rm cov}$ & uniform in $f_{\rm cov}\in (0,1)$ & $0.37^{+0.14}_{-0.09}$ & ={\it Chandra} & $0.19\pm 0.04$\\
\hline
{\bf Thermal ICM} & & & & \\
Temperature $T$ (keV) & uniform in $T\in (10^{-2},64)$ & $0.82^{+0.05}_{-0.06}$ & $3.53^{+0.09}_{+0.08}$ & ---\\
Metallicity $Z$ ($Z_\odot$)& uniform in $Z\in (0,5)$ & ={\it Hitomi} & $0.49^{+0.07}_{-0.05}$ & ---\\
Redshift $z_{\rm icm}$ & uniform in $z_{\rm icm}\in (0,0.1)$ & ={\it Hitomi} & $0.01777^{+0.00001}_{-0.00003}$ & ---\\
Velocity dispersion $\sigma_{\rm icm}$ (\hbox{$\km\s^{-1}\,$})& uniform in $\sigma_{\rm icm}\in (0,10^4)$ & ={\it Hitomi} & $180^{+7}_{-8}$ & ---\\
Normalization $A_{\rm icm}$ & uniform in $A_{\rm icm}\in (0,10^{-1})$ & $(3.0^{+0.6}_{-0.5})\times 10^{-4}$ & $(5.9^{+0.6}_{-0.7})\times 10^{-2}$ & --- \\\hline
{\bf Iron-K$\alpha$ emission} & & & & & \\
Redshift $z_{\rm Fe}$ & uniform in $z_{\rm Fe}\in (0,0.1)$ & ={\it Hitomi} & $0.0176^{+0.0008}_{-0.0007}$ & $0.011^{+0.009}_{-0.028}$\\
Velocity dispersion $\sigma_v$ (\hbox{$\km\s^{-1}\,$})& uniform in $\sigma_v\in (0,10^4)$ & \multicolumn{3}{c}{$424^{+539}_{-114}$ (tied across all datasets)}\\
Spatial extent $\log\sigma_\theta$ (arcsec)& uniform in $\log\sigma_\theta\in (-2,1)$ & \multicolumn{3}{c}{$0.1^{+0.8}_{-1.5}$ (tied across all datasets)}\\
K$\alpha_1$ normalization $A_{\rm Fe}$ & uniform in $A_{\rm Fe}\in (0,10^{-1})$ & \multicolumn{3}{c}{$(4.5^{+1.4}_{-0.9})\times 10^{-6}$ (tied across all datasets)}\\
K$\alpha_1$+K$\alpha_1$ equiv. width $W_{K\alpha}$ (eV) & (derived) & $77^{+24}_{-15}$ & $24^{+7}_{-5}$ & $27^{+9}_{-6}$ \\\hline
2--10\,keV AGN flux ($10^{-11}$ cgs) & (derived; absorbed flux) & 0.74 & 2.8 & 2.5 \\
2--10\,keV AGN luminosity ($10^{43}$ cgs) & (derived; de-absorbed) & 0.71 & 2.2 & 2.0\\\hline\hline
\end{tabular}\label{tab:jointfitting}
\end{table*}
\begin{figure}
\includegraphics[width=0.45\textwidth]{ironextent_pcf.pdf}
\caption{Constraints on the velocity dispersion $\sigma_v$ and spatial extent $\sigma_\theta$ of the fluorescing iron K$\alpha$ emitting gas from a joint analysis of the {\it Chandra}/HETG, {\it Hitomi}/SXS and {\it XMM-Newton}/EPIC spectra. Shown here are the 68\%, 90\%, 95\%, 99\%, and 99.9\% enclosed probability contours resulting from the MCMC analysis described in Section~\ref{reflectionmcmc}. Also shown is the velocity-radius relation relevant for gravitational motions, specifically circular motion (red-dashed line) and radial infall (red-dotted line), assuming a stellar velocity dispersion of $\sigma_*=259\hbox{$\km\s^{-1}\,$}$ \citep{riffel:20a} and central supermassive black hole of mass $M=1\times 10^9\hbox{$\rm\thinspace M_{\odot}$}$.}
\label{fig:ironextent}
\end{figure}
Table~\ref{tab:jointfitting} lists the best fitting parameters (minimizing CSTAT) and the single-parameter error bounds resulting from the MCMC chain. The ICM component as seen by {\it Hitomi} agrees well with the findings of \cite{hitomi:16a} and \cite{hitomi:18a}. Since the {\it XMM-Newton}/EPIC spectra are extracted from a smaller region than for {\it Hitomi} and are subject to background subtraction that includes ICM emission, the residual ICM component in the EPIC spectra has a significantly lower normalization and is cooler (reflecting the fact that the coolest plasma reside at the center of the system).
Marked changes are seen in the AGN component between the 2001 and 2016/2017 epochs. We find an AGN 2--10\,keV flux that increases from $\sim 7\times 10^{-12}\hbox{$\erg\cm^{-2}\s^{-1}\,$}$ in 2001 \citep[close to its historical minimum, ][]{fabian:15b} up to $\sim 2.5\times 10^{-11}\hbox{$\erg\cm^{-2}\s^{-1}\,$}$ in 2016 and 2017. Concurrent with the historical brightening, we find that the AGN continuum becomes softer. There is also a marked decrease in both the covering fraction and column density of the cold absorption from the 2001 to 2016/2017 epochs.
The main motivation of this particular modelling is to obtain constraints on the velocity broadening and spatial extent of the iron-K$\alpha$ fluorescent line emission. Figure~\ref{fig:ironextent} shows confidence contours on the velocity width and spatial extent of the fluorescent iron line emitting gas. The best-fitting model and 68\% enclosed probability contour suggest spatial broadening of the line emitting region at the 1\arcsec-scale. However, the 90\% enclosed probability contour reaches down to the smallest extents probed ($0.01\arcsec$) and so in this case we do not claim the significant detection of extension. Thus our result is not in strong tension with \cite{miller:17a} who used the novel technique of X-ray ``lucky imaging'' to constrain the extent of the iron-K$\alpha$ emission region to be $<0.3\arcsec$.
This joint fit constrains the velocity width of the iron-K$\alpha$ line to be 310--963\hbox{$\km\s^{-1}\,$} (90\% credible interval), translating into a full-width at half-maximum (FWMH) of 760--2300\hbox{$\km\s^{-1}\,$} (90\% credible interval). Interestingly, as evident from Fig.~\ref{fig:ironextent}, this analysis {\it disfavours} models with small spatial extent ($<50{\rm\thinspace pc}$) and large velocity width ($\sigma_v>600\hbox{$\km\s^{-1}\,$}$); this is a manifestation of the tension between the HETG and SXS line normalizations discussed above. If we suppose that the dynamics of the iron-K$\alpha$ emitting material is gravitational, we can relate the distance from the black hole and the expected line-of-sight velocity dispersion $\sigma_g$,
\begin{equation}
\sigma_{\rm g}^2=\frac{f_v GM_{\rm BH}}{R}+\sigma_*^2
\end{equation}
where $\sigma_*$ is the velocity dispersion of the galaxy potential (approximated as an isothermal sphere). Here, the factor $f_v=\sin i\approx 1/\sqrt{2}$ for circular motions (where $i$ is the viewing inclination that we take to be $i\sim 45^\circ$), and $f_v=2$ for radial infall. These two gravitational velocities are shown on Fig.~\ref{fig:ironextent} (dotted and dashed red line). We see that there is a tension between our measured line velocity and the expected gravitational motions if the fluorescing matter is closer than $R\sim 10{\rm\thinspace pc}$.
\section{Discussion and conclusions}\label{discussion}
The deep (490{\rm\thinspace ks}) {\it Chandra}/HETG exposure presented in this work provides the most detailed view of the X-ray spectrum of NGC~1275 to date and further adds to the picture of a complex circumnuclear environment around this BCG-AGN.
To recap our new findings:
\begin{enumerate}
\item The dominant component of the X-ray emission is well-approximated by an unabsorbed power-law. We can clearly reject the hypothesis that the entire X-ray source is absorbed by the molecular gas column of $N_{H_2}\approx 2\times 10^{22}\hbox{$\cm^{-2}\,$}$ seen by ALMA towards the parsec-scale jets as reported by \cite{nagai:19a}. However, we do find evidence that during our 2017 {\it Chandra} campaign approximately $15-20\%$ of the X-ray emission is absorbed by a cold (atomic and/or molecular) column density of $N_H\sim 8\times 10^{22}\hbox{$\cm^{-2}\,$}$, suggesting the possibility that some fraction of the X-ray emission does experience absorption by this molecular gas. Applying the partial covering model to archival {\it XMM-Newton} EPIC data, we infer that both the covering fraction and column density of the absorber were higher in the 2001 epoch, a time when the overall X-ray luminosity was only 35\% of its 2017 level.
\item A rigorous search that accounts for the look-elsewhere effect reveals no evidence for photoionized absorption in our HETG X-ray spectrum. While a blind analysis of the high-resolution X-ray spectrum finds tentative detections several high ionization absorption/emission lines, the lack of a common velocity shift precludes them arising from a single absorber and we conclude that they are likely statistical fluctuations. We rule out the presence of absorbers with ionization-parameters/column-densities commonly seen in Seyfert nuclei.
\item We detect the 6.4{\rm\thinspace keV} iron-K$\alpha$ fluorescent line from cold circumnuclear gas that has been previously studied most recently by {\it Hitomi}. Noting modest tensions between the iron-K$\alpha$ normalzations and/or widths between the {\it Chandra}/HETG, {\it Hitomi}/SXS, and {\it XMM-Newton}/EPIC spectra, we conduct a multi-satellite joint analysis that accounts for any anomalous broadening of the iron-K$\alpha$ line in the dispersed HETG that would result from spatial extension. While we ultimately conclude that spatial broadening is not required at the 90\% confidence level, we believe that this combination of microcalorimeter data (which measures the true velocity width of a line) with high-quality grating data (with spectra that are convolved with the source spatial structure) will become importance once the {\it X-ray Imaging and Spectroscopy Mission (XRISM)} deploys.
\end{enumerate}
To put these results into context, we must consider the parsec-scale structure of this AGN. This AGN is remarkable in that it has been observed to go through very significant changes in its parsec scale structure on human timescales, as seen by high-resolution radio imaging of the associated radio source 3C~84. From the earliest days of Very Long Baseline Interferometry (VLBI), it was known that 3C~84 exhibits complex structure on milli-arcsecond (mas) scales \citep{paulinytoth:76a}. At that epoch, the radio source was dominated by an inverted core and a second component located 1\,mas (0.4{\rm\thinspace pc}) away on a position angle (PA) of 210$^\circ$ \citep{unwin:82a,readhead:83a,Readhead:83b}. On slightly larger scales of 10--15\,mas (4--6{\rm\thinspace pc}), fainter VLBI knots traced out a probably jet channel directly south (at a PA of 170--180$^\circ$) from the inverted core which appears to connect onto the kpc-scale jet with a PA of 170$^\circ$ \citep{pedlar:83a}. VLBI monitoring revealed complex motions and flux changes of these jet subcomponents \citep{wright:88a,marr:89a,krichbaum:92a}. The northern counter-jet was first reported in 22\,GHz VLBI image by \cite{vermeulen:94a} with an inverted spectrum that strongly suggested the influence of free-free absorption by an ionized screen with temperature $T\sim 10^4{\rm\thinspace K}$ and column density $N_H\sim 10^{23}\hbox{$\cm^{-2}\,$}$ \citep{vermeulen:94a,walker:94a,walker:00a}. The fact that the free-free screen absorbs the counter-jet but not the jet suggests that it be identified with the outer regions of the accretion disk.
Both high-frequency (90\,GHz) radio monitoring and X-ray observations show that the AGN significantly faded from the time of these early VLBI studies and reached a minimum of activity around c2000 \citep{dutson:14a,fabian:15a}. Since that time, we have been witnessing the onset of a remarkable new period of activity. \cite{nagai:10a} analyzed 22\,GHz VLBI data to discover a new jet component (C3) propagating away from the core (C1) with PA of 180$^\circ$. They initially suggest that the component was launched in 2005, although a later analysis of higher resolution 43\,GHz data showed that the C3 could already be distinguished from the C1 as early as 2003-November \citep{suzuki:12a}. A picture emerged whereby C3 is the working surface of a new radio-lobe associated with restarted jet activity \citep{nagai:16a,kino:17a} and, c.2016, was approximately 3\,mas (1\,pc projected distance) directly south of the core C1. This picture is supported by monitoring with the Korean VLBI Network (KVN). This instrument shows weaker features in the jet that propagate from the core with apparent speeds ranging from 0.2--0.9c until they reach the vicinity of C3, after which they deflect and/or break apart, often coincident with a major $\gamma$-ray flare \citep{hodgson:21a}. \cite{kino:18a} finds that the location of C3 skips eastwards by 0.4\,mas in 2015 August--September before resuming its southernly propagation, suggesting that the jet's working surface is pushing into a clumpy and dense medium; the density required to provide the necessary ram pressure, $n\sim 4\times 10^3\hbox{$\cm^{-3}\,$}-2\times 10^5\hbox{$\cm^{-3}\,$}$, suggests a clumpy molecular medium.
Within this complex source, there are several possibilities for the origin of the nuclear X-rays including coronal emission from the inner accretion disk, the innermost regions of the jet (C1), the current working surface of the jet (C3), or the new jet-blown cocoon (seen in VLBI as the halo enveloping C1 and C3). Indeed, it would be natural for the observed X-rays to have multiple origins, the dominance of which can change over time. Monitoring by the {\it Fermi} Large Area Telescope (LAT) and the {\it Swift} X-ray Telescope (XRT) shows long term correlations between the nuclear 5--10\,keV X-ray flux and the 0.1--300\,GeV $\gamma$-ray flux which clearly has a jet/cocoon origin \citep{fukazawa:18a}. However, given the small physical size of this source, this may simply be a reflection of the overall level of activity in the disk/jet system and does not necessarily imply that the bulk of the X-rays originate from the jet. Short timescale $\gamma$-ray/X-ray correlations that would unambiguously signal a jet-origin for X-rays are more complex; some week-timescale $\gamma$-ray events show correlated X-ray variability whereas others do not \citep{fukazawa:18a,imazato:21a}.
In this context, our finding that the X-ray source is partially covered by a cold absorber fits naturally with the picture of a composite source embedded in a clumpy molecular medium. What is less clear is the precise identification of the absorbed and unabsorbed components. Comparing the spectral fits to the 2001 {\it XMM-Newton} and 2017 {\it Chandra} datasets, we find that the factor of three increase in X-ray luminosity must be dominated by the unabsorbed component. At the same time, the spectrum undergoes a significant softening, with the photon index increasing from $\Gamma\approx 1.65$ to $\Gamma\approx 2.0$. This may suggest a transition from a jet/cocoon (non-thermal inverse Compton) dominated to a disk-corona (thermal Comptonization) dominated X-ray spectrum.
Following this scenario, we infer that 80--85\% of the 2017-emission originates from the inner disk and is viewed along an unabsorbed line-of-sight, with the remaining 15--20\% coming from the parsec-scale jet structures including C3 absorbed by a column density $N_H\sim 10^{23}\hbox{$\cm^{-2}\,$}$ of molecular gas. While the spectral models presented in this paper have formally assumed equal photon indices for the absorbed and unabsorbed components, we have verified that the fit to the 2017-HETG data is relatively insensitive to a decoupled photon index for the absorbed component (with allowable values in the range $\Gamma=1.1-2.5$ at 90\% confidence, safely bracketing the value of $\Gamma=1.65$). This is not a unique conclusion, however --- fresh-injection of relativistic particles associated with jet shocks can create X-ray synchrotron components with $\Gamma\approx 2$ spectrum \citep{fukazawa:18a}. Significantly more comparison of both short-and-long term X-ray variability with radio- and $\gamma$-ray variability will be needed to fully disentangle this composite source.
These results may be more general applicability to BCG AGN. \cite{russell:13a} examine X-ray absorption in a sample of BCG-AGN, finding that 9/25 objects have significant columns ($N_H>10^{22}\hbox{$\cm^{-2}\,$}$) whereas 8/25 have no detectable absorption (with limits of $N_H<10^{21}\hbox{$\cm^{-2}\,$}$). Extrapolating our findings for NGC~1275, this diversity in the broader population may result from the X-ray source in BCG-AGN being disk/jet composites with parsec-scale jets that are frustrated by dense molecular gas. A natural consequence would be common mismatches between molecular line absorption and X-ray absorption. Assuming that the remarkable decadal time variability of NGC~1275 is not unusual (applying Copernican reasoning), we may also expect to see year-to-decade timescale variability in the measurable X-ray absorption of other BCG-AGN as the dominance of the (absorbed) jet component waxes and wanes.
\section*{Acknowledgements}
We thank the Chandra Science Center, and especially Hermann Marshall, for advice and guidance in the execution of these observations. C.S.R. thanks the STFC for support under the New Applicant grant ST/R000867/1 and Consolidated Grant ST/S000623/1, as well as the European Research Council (ERC) for support under the European Union's Horizon 2020 research and innovation programme (grant 834203).
R.S. and S.V. acknowledge support from NASA under the Chandra Guest Observer Program (grants G08-19088X and G09-20119X).
\section*{Data Availability}
All of the raw data used in this work are available via the public data archives, specifically the Chandra Data Archive ({\it Chandra}/HETG), the NASA-GSFC High Energy Astrophysics Science Archive ({\it Hitomi}) and the {\it XMM-Newton} Science Archive ({\it XMM-Newton}). Reduced data products and analysis scripts are available upon written request to the first author.
|
1,314,259,995,814 | arxiv | \section{Introduction}
The most popular public key cryptosystem in use today is
the RSA cryptosystem, introduced by Rivest, Shamir, and Adleman \cite{RSA}.
Its security is based on the intractability of
the integer factorization problem.
The modulus $n$ of a RSA cryptosystem is the product of two large
primes $p$ and $q$. The public exponent $e$ and the secret
exponent $d$ are related by
\begin{equation} \label{phi1}
ed\equiv 1 \pmod{\varphi(n)},
\end{equation}
where $\varphi(n)= (p-1)(q-1)$. In a typical RSA cryptosystem,
$p$ and $q$ have approximately the same number of bits, while $e<n$.
The encryption and decryption algorithms are given by $C= M^e
\bmod n$, $M=C^d \bmod n$.
To speed up the RSA decryption one may try to use small secret
decryption exponent $d$. The choice of a small $d$ is especially
interesting when there is a large difference in computing power
between two communicating devices, e.g. in communication between
a smart card and a larger computer.
In this situation, it would be desirable that the smart card has a small secret exponent,
while the larger computer has a small public exponent, to reduce the processing
required in the smart card.
In 1990, Wiener \cite{Wiener} described a polynomial time
algorithm for breaking a typical (i.e. $p$ and $q$ are of the
same size and $e<n$) RSA cryptosystem if the secret exponent $d$ has at most
one-quarter as many bits as the modulus $n$.
From (\ref{phi1}) it follows that there is an
integer $k$ such that $ed- k\varphi(n)=1$. Since $ \varphi(n)
\approx n$, we have that $\frac{k}{d} \approx \frac{e}{n}$.
Wiener's attack is usually described in the following form (see
\cite{B-notices,Smart}):
\emph{If $p<q<2p$, $e<n$ and $d<\frac{1}{3}\sqrt[4]{n}$, then $d$ is the
denominator of some convergent of the continued fraction expansion of
$\frac{e}{n}$.}
Indeed, under these assumptions it is easy to show that
$$ \left|\frac{e}{n} - \frac{k}{d} \right| < \frac{1}{2d^2}. $$
By the classical Legendre's theorem, $\frac{k}{d}$ is
some convergent $\frac{p_m}{q_m}$ of the continued fraction expansion of $\frac{e}{n}$, and
therefore $d$ can be computed efficiently from the public key
$(n,e)$. Namely, the total number of convergents is of order $O(\log n)$, and each convergent can be
tested in polynomial time.
In 1997, Verheul and van Tilborg \cite{V-vT} proposed an extension
of Wiener's attack that allows the RSA cryptosystem to be broken when $d$ is a
few bits longer than $n^{0.25}$. For $d>n^{0.25}$ their attack
needs to do an exhaustive search for about $2t+8$ bits (under
reasonable assumptions on involved partial convergents), where
$t=\log_2(d/n^{0.25})$.
\medskip
In \cite{DujeRSA}, we proposed a slight
modification of the Verheul and van Tilborg attack, based on
Worley's result on Diophantine approximations \cite{Worley},
which implies that all rationals $\frac{p}{q}$
satisfying the inequality
\begin{equation} \label{eq:zzz}
\left| \alpha - \frac{p}{q} \right| < \frac{c}{q^2},
\end{equation}
for a positive real number $c$,
have the form
\begin{equation} \label{eq:zzzz}
\frac{p}{q} = \frac{rp_{m+1} \pm sp_m}{ rq_{m+1} \pm sq_m }
\end{equation}
for some $m\geq -1$
and nonnegative integers $r$ and $s$ such that $rs < 2c$.
It has been shown recently in \cite{DuIbr} that Worley's result is sharp,
in the sense that the condition $rs<2c$ cannot be replaced by
$rs < (2-\varepsilon)c$ for any $\varepsilon$.
In both mentioned extensions of Wiener's attack, the candidates
for the secret exponent are of the form $d = r q_{m+1} + s q_m$.
Then we test all possibilities for $d$. The number of possibilities is
roughly the product of the number of possibilities for $r$ and the number of
possibilities for $s$, which is $O(D^2)$, where $d=D n^{0.25}$.
More precisely, the number of possible pairs $(r,s)$ in the Verheul and van Tilborg attack
is $O(D^2A^2)$, where $A=\max\{a_i : i=m\!+\!1,m\!+\!2,m\!+\!3\}$, while in our
variant the number of pairs is $O(D^2\log A)$ (and also $O(D^2\log D)$).
Another modification
of the Verheul and van Tilborg attack has been recently proposed by Sun, Wu an Chen \cite{SWC}.
It requires (heuristically) an exhaustive search for about $2t-10$ bits, so its complexity
is also $O(D^2)$.
We cannot expect drastic improvements here, since, by a result of
Steinfeld, Contini, Wang and Pieprzyk \cite{SCWP},
there does not exist an attack in this class with subexponential running time.
\medskip
Boneh and Durfee \cite{B-D1} and Bl\"omer and May \cite{B-M}
proposed attacks based on Coppersmith's lattice-based technique for
finding small roots of modular polynomials equations using LLL-algorithm. The attacks work
if $d< n^{0.292}$.
The conjecture is that the right bound below which a typical version of RSA
is insecure is $d< n^{0.5}$.
\medskip
In the present paper, we propose a new variant of Wiener's attack. It also uses
continued fractions and searches for candidates for the secret key in the form
$d=rq_{m+1}+sq_{m}$. However, the searching phase of this variant is significantly faster.
Its complexity is $O(D\log{D})$,
and it works efficiently for $d<10^{30}n^{0.25}$.
Although this bound is asymptotically weaker than the bounds in the above mentioned
attacks based on the LLL-algorithm (note however that these bounds are not strictly proved since
Coppersmith's theorem in the bivariate case is only a heuristic result -
see also \cite{Hinek,HLT}), for practical values of $n$ (e.g. for 1024-bits) these bounds
are of comparable size.
\section{The Verheul and van Tilborg attack}
In this section we briefly describe
the Verheul and van Tilborg attack \cite{V-vT} and its modification from \cite{DujeRSA}.
We assume that $p<q<2p$ and $e<n$.
Then it is easy to see that
\begin{equation} \label{eq:nej1}
\left| \frac{e}{n} - \frac{k}{d} \right| < \frac{2.122\, e}{n\sqrt{n}}.
\end{equation}
Let $m$ be the largest (odd) integer satisfying
$\frac{p_m}{q_m} - \frac{e}{n} > \frac{2.122\, e}{n\sqrt{n}}$.
Verheul and van Tilborg proposed to search for $\frac{k}{d}$ among the fractions of the form
$\frac{r p_{m+1}+s p_m}{rq_{m+1}+ s q_m}$. This leads to the system
\begin{eqnarray*}
r p_{m+1}+s p_m &=& k, \\
r q_{m+1}+ s q_m &=& d.
\end{eqnarray*}
The determinant of the system satisfies
$|p_{m+1}q_m - q_{m+1}p_m| = 1$, and therefore
the system has (positive) integer solutions:
\begin{eqnarray*}
r &=& dp_{m} -k q_m, \\
s &=& kq_{m+1}- d p_{m+1} .
\end{eqnarray*}
If $r$ and $s$ are small, then they can be found
by an exhaustive search. Let $[a_0;a_1,a_2,\ldots]$ be the continued
fraction expansion of $e/n$ and $D=d/n^{0.25}$.
In \cite{DujeRSA}, the following upper bounds for $r$ and $s$ were derived:
\begin{eqnarray*}
r &<& \max \{ \sqrt{2.122(a_{m+3}+2)}(a_{m+2}+1)D,\, \sqrt{2.122(a_{m+2}+2)}D \}, \\
s &<& \max \{ 2\sqrt{2.122(a_{m+3}+2)}D,\,\sqrt{2.122(a_{m+2}+2)}(a_{m+1}+1)D \} .
\end{eqnarray*}
The modified attack proposed in \cite{DujeRSA} searches
for $\frac{k}{d}$ among the fractions of the forms
$\frac{r p_{m+1}+s p_m}{rq_{m+1}+ s q_m}$,
$\frac{r p_{m+2}-s p_{m+1}}{rq_{m+2}- s q_{m+1}}$ and
$\frac{r p_{m+3}+s p_{m+2}}{rq_{m+3}+ s q_{m+2}}$.
It results with bounds for $r$ and $s$ which are (almost)
independent on the partial quotients $a_m$'s.
Hence, in both attacks bounds for $r$ and $s$ are of the form
$O(D)$, but in the case of \cite{DujeRSA} the implied constants
are much smaller (indeed, the table in Section \ref{impl} shows
that with high probability we have $r<4D$ and $s<4D$).
\section{Testing the candidates}
There are two principal methods for testing candidates for the secret exponent $d$.
{\bf Method I} (\cite{Wiener}): Compute $p$ and $q$, assuming $d$ is the correct guess,
using the following formulas:
$$ \varphi(n)=(de-1)/k, \quad p+q=n+1-\varphi(n), $$
$$ (q-p)^2 = (p+q)^2 - 4n, $$
$$ p = \frac{p+q}{2} - \frac{q-p}{2}, \quad q = \frac{p+q}{2} + \frac{q-p}{2}. $$
{\bf Method II} (\cite[Chapter 17]{Smart}): Test the congruence $(M^e)^d \equiv M \!\!\pmod{n}$,
for some random value of $M$, or simply for $M=2$.
\medskip
Both methods are very efficient. But in the situation where we have to test huge amount of
candidates for $d$ of the form $rq_{m+1}+sq_{m}$, there is a significant difference between them.
With the Method I it seems that we cannot avoid testing separately all possible pairs $(r,s)$.
On the other hand, here we present a new idea, which is to apply
``meet-in-the-middle'' to the Method II.
We want to test whether
\begin{equation} \label{eq:z}
2^{e(rq_{m+1}+sq_m)} \equiv 2 \pmod{n}.
\end{equation}
Note that $m$ is (almost) fixed. Indeed, let $m'$ be the largest odd integer such that
$$ \frac{p_{m'}}{q_{m'}} > \frac{e}{n} + \frac{2.122 e}{n\sqrt{n}}. $$
Then $m\in \{m', m'+1, m'+2 \}$ (see \cite{DujeRSA} for details).
Let $2^{e q_{m+1}}
\bmod{n} = a$, $(2^{e q_m})^{-1} \bmod{n} = b$. Then we test the
congruence
\begin{equation} \label{eq:zz}
a^r \equiv 2 b^s \!\!\pmod{n}.
\end{equation}
We can do it by
computing $a^r \bmod{n}$ for all $r$, sorting the list of results,
and then computing $2b^s \bmod{n}$ for each $s$ one at a time, and
checking if the result appears in the sorted list.
This decreases
the time complexity of the testings phase to $O(D\log{D})$ (with the
space complexity $O(D)$).
\section{Implementation issues and improvements} \label{impl}
The theoretic base for the extension of Wiener's attack is Worley's theorem
on Diophantine approximations of the form (\ref{eq:zzz}).
We have already mentioned a result from \cite{DuIbr} which shows
that Worley's result is in some sense the best possible. However, some improvements are
possible if we consider unsymmetrical variants of Worley's result
(with different bounds on $r$ and $s$). Roughly speaking, in
solutions of (\ref{eq:zzz}) in form (\ref{eq:zzzz}), if $r<s$ then we
may take $rs<c$ instead of $rs<2c$. Due to such unsymmetrical
results, a space-time tradeoff might be possible. The following
table shows the chance of success of our attack for various
(symmetrical and unsymmetrical) bounds on $r$ and $s$. We can see
that, with the same bound for $rs$, the better results are obtained for
smaller bounds on $r$ and larger bounds on $s$. In the implementations, this fact can be used
to decrease the memory requirements (up to factor $16$).
\begin{center}
\begin{tabular}{|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|}
\hline\rule{0pt}{24pt} bound for $r$ & bound for $s$ &
chance of success \\[2pt]
\hline\rule{0pt}{15pt}%
$4D$ & $4D$ & $98 \%$ \\
$2D$ & $2D$ & $89 \%$ \\
$D$ & $D$ & $65 \%$ \\
$D$ & $4D$ & $86 \%$ \\
$4D$ & $D$ & $74 \%$ \\
$D/2$ & $2D$ & $70 \%$ \\
$2D$ & $D/2$ & $47 \%$ \\
$D/4$ & $4D$ & $54 \%$ \\
$4D$ & $D/4$ & $28 \%$ \\[2pt]
\hline
\end{tabular}
\end{center}
In the implementation of the proposed attack, we can use hash functions instead of sorting.
Furthermore, it is not necessary to store all bits of $a^r \bmod{n}$ in the hash table.
Indeed, values of $a^r \bmod{n}$ are from the set $\{0,1,\ldots,n\}$, and the number of $r$'s
is typically much smaller than $n$. Therefore, around $2\log_2{D}$ stored bits will suffice
in order to avoid too many accidental collisions. Note that a reasonable number of collisions
is not big problem here, since each such collision can be efficiently tested by Method I.
Hash tables can be used to take into account the condition $\gcd(r,s)=1$.
This condition was easy to use in brute-force testing of all possible pairs $(r,s)$,
but the direct application of our ``meet-in-the-middle'' variant seemingly ignores it.
But if we create rows in the hash table according to divisibility properties of exponents $r$
modulo small primes,
we may take again an advantage of this condition and speed up the algorithm up to 39\%.
We have implemented several variants of the proposed attack in PARI and C++,
and they work efficiently for values of $D$ up to $2^{30}$, i.e. for $d<2^{30}n^{0.25}$.
For larger values of $D$ the memory requirements become too demanding for ordinary computers.
The following table compares this bound with the bound of $d$ in the best known
attacks on RSA with small secret exponent based on LLL-algorithm.
\begin{center}
\begin{tabular}{|@{\quad}r@{\quad}|@{\quad}c@{\quad}|@{\quad}c@{\quad}|}
\hline\rule{0pt}{24pt} $\log_2{n}$ & $\log_2(2^{30}n^{0.25})$ &
$\log_2(n^{0.292})$ \\[2pt]
\hline\rule{0pt}{15pt}
512 & 158 & 150 \\
768 & 222 & 224 \\
1024 & 286 & 299 \\
2048 & 542 & 598 \\[2pt]
\hline
\end{tabular}
\end{center}
The attack can be also slightly improved by using better approximations to $\frac{k}{d}$,
e.g. $\frac{e}{n+1-2\sqrt{n}}$ instead of $\frac{e}{n}$. Namely,
\begin{equation} \label{eq:nej2}
\left| \frac{e}{n+1-2\sqrt{n}} - \frac{k}{d} \right| < \frac{0.1221\, e}{n\sqrt{n}} \,.
\end{equation}
Comparing (\ref{eq:nej2}) with (\ref{eq:nej1}),
we see that by replacing $\frac{e}{n}$ by $\frac{e}{n+1-2\sqrt{n}}$
we can gain the factor $4$ in bounds for $r$ and $s$,
so decreasing both, time and memory requirements.
With these improvements, for
1024-bits RSA modulus $n$, the range in which our attack can be applied
becomes comparable and competitive with best known attacks based on the LLL-algorithm.
\medskip
{\bf Acknowledgements.} The author would like to thank Vinko Petri\v{c}evi\'{c} for his
help with C++ implementation of the various variants
of the attack described in this paper.
The author was supported by the Ministry of Science, Education and
Sports, Republic of Croatia, grant 037-0372781-2821.
{\small
|
1,314,259,995,815 | arxiv |
\section{Introduction}
\label{sec:introduction}
\begin{comment}
\end{comment}
After 20 years from when the human knowledge crossed the borders
of the solar system and found a planet orbiting another main
sequence star \citep{mayor:1995}, we can now count more than 1800
exoplanets in our Galaxy, and marvel how physically varied and
intriguing most of them are. The first class of \emph{unexpected}
planets with which we faced is composed by the so-called \emph{hot
Jupiters}, i.e. giant gaseous planets in close orbits
around their host stars, able to perform a complete orbit in a
relatively short time ($\sim 0.1-10$\,days). Even though they are
rarer than small-size rocky and Neptunian planets
\citep{fressin:2013,dressing:2013,petigura:2013}, there are
numerous reasons that make them very interesting to study,
especially those that transit their parent stars. Indeed, since
hot Jupiters are more massive and larger than rocky planets, it is
possible to measure their physical parameters with a much better
accuracy: \emph{in primis} mass and radius, but also their
spin-orbit alignment (from the Rossiter-McLaughlin effect), their
thermal flux and reflected light (from \emph{occultations} and
\emph{phase curve}), the chemical composition of their atmosphere
(from \emph{emission} and \emph{transmission spectra}), etc.
However, although all these parameters are accessible, even with
moderate-sized ground-based telescopes, there are various aspects
of the hot-Jupiter population that were not well understood. We
did not find, for example, any convincing way to group them in
classes based on some of their features (e.g.,
\citealp{hansen:2007,fortney:2008,schlaufman:2010,madhusudhan:2012}).
It is also very puzzling to determine what are the physical
mechanisms that regulate the formation, accretion, evolution and
cause the migration of giant planets from the snow line ($\sim
3$\,au) up to roughly $0.01$\,au from their host stars. In this
context, several scaling laws have been suggested between their
parameters (e.g.,
\citealp{southworth:2007,knutson:2010,hartman:2010}), but none
seems to be generally apply to all planets.
Answering the above questions is possible only by enlarging the
sample at our disposal, in particular the regions of the parameter
space which are currently deserted because of observational
biases. In the last three \emph{lustra}, ground-based transit
surveys have played a major role in exoplanet detection and thus
in the growth of our scientific knowledge about planetary systems.
In a fair competition with other teams (e.g., HATNet:
\citealp{bakos:2004}; WASP: \citealp{pollacco:2006}; KELT:
\citealp{pepper:2007}; MEARTH: \citealp{charbonneau:2009}; QES:
\citealp{alsubai:2013}; APACHE: \citealp{sozzetti:2013}; NGTS:
\citealp{wheatley:2013}), we are undertaking the HATSouth project,
which consists of the monitoring of millions of stars in the
southern sky to look for new exoplanet transit signals. Our survey
is carried out by a network of 6 telescope systems, employing 24
astrographs, distributed over three continents (South America,
Africa, and Australia), thus increasing the sensitive to
long-period ($>10$\,days) planets \citep{bakos:2013}.
Here we present two new transiting extrasolar planets: HATS-13b
and HATS-14b. The paper is organized as follows: in
Sect.\,\ref{sec:obs} we summarize the detection of the photometric
transit signal and the subsequent spectroscopic and photometric
observations of each star to confirm the planets. In
Sect.\,\ref{sec:analysis} we analyze the data to rule out false
positive scenarios, and to determine the stellar and planetary
parameters. Our findings are summarized and discussed in
Sect.\,\ref{sec:discussion}.
\section{Observations}
\label{sec:obs}
\subsection{Photometric detection}
\label{sec:detection}
The \emph{modus operandi} of the HATSouth survey is
comprehensively described in \citet{bakos:2013}. In brief,
HATSouth is a network of completely automated wide-field
telescopes, consisting in six homogeneous units located at three
different places in the southern hemisphere, i.e. Las Campanas
Observatory (LCO) in Chile, the H.E.S.S. site in Namibia, and
Siding Spring Observatory (SSO) in Australia. Each unit is
equipped with four 18\,cm $f/2.8$ Takahasi astrographs, each
working in pairs with Apogee U16M Alta 4k$\times$4k CCD cameras,
with a total mosaic field-of-view on the sky of $8^{\circ} \times
8^{\circ}$ at a scale of $3.7$\,arcsec\,pixel$^{-1}$. Observations
are performed through a Sloan-$r$ filter with an exposure time of
4\,minutes. Scientific images are automatically calibrated and
light curves are extracted by aperture photometry. They are then
treated with decorrelation and detrending
algorithms\footnote{External Parameter Decorrelation (EPD;
\citealp{bakos:2010}); Trend Filtering Algorithm (TFA;
\citealp{kovacs:2005}).} and finally run through with BLS
(Box-fitting Least Squares; \citealp{kovacs:2002}) to find
periodic signals by transiting exoplanets.
The stars HATS-13 (aka \hatcurCCtwomass{13}; $\alpha =
\hatcurCCra{13}$, $\delta = \hatcurCCdec{13}$; J2000) and HATS-14
(aka \hatcurCCtwomass{14}; $\alpha = \hatcurCCra{14}$, $\delta =
\hatcurCCdec{14}$; J2000) are two moderately bright
($V=13.89$\,mag and $V=13.79$\,mag, respectively) stars. They were
monitored between Nov 2009 and September 2010 by three of the
HATSouth units, which collected more than 10\,000 images for both
of them. Details of the observations are reported in
Table\,\ref{tab:photobs}. The corresponding light curves, folded
with a period of $P \sim 3.04$ and $2.77$\,days, respectively, are
plotted in Fig.\,\ref{fig:hatsouth}, both clearly showing
transiting-planet signals with depths of $\sim 2\%$ and $\sim
1\%$, respectively.
\begin{table*}
\caption{Summary of photometric observations} %
\label{tab:photobs} %
\centering %
\tiny %
\setlength{\tabcolsep}{8pt}
\begin{tabular}{llrrcr}
\hline\hline %
Instrument/Field$^{\mathrm{a}}$ & UT Date(s) & \# Images & Cadence$^{\mathrm{b}}$ & Filter & Precision$^{\mathrm{c}}$ \\
& & & (sec)~~~ & & (mmag)~~ \\
\hline \\%
\multicolumn{2}{l}{\textbf{HATS-13}} \\ [2pt]%
~~~~HS-2/G582 & 2009 Nov--2010 Sep & 2486 & 288~~~~ & $r$ & 12.6 ~~~~\\
~~~~HS-4/G582 & 2009 Sep--2010 Sep & 8565 & 288~~~~ & $r$ & 12.2 ~~~~\\
~~~~HS-6/G582 & 2010 Apr--2010 Sep & 356 & 265~~~~ & $r$ & 13.1 ~~~~\\
~~~~CTIO~0.9m$^{\mathrm{d}}$ & 2012 Aug 26 & 68 & 237~~~~ & $z$ & 2.9 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2012 Oct 17 & 82 & 87~~~~ & $g$ & 1.4 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2012 Oct 17 & 83 & 87~~~~ & $r$ & 1.2 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2012 Oct 17 & 82 & 87~~~~ & $i$ & 1.7 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2012 Oct 17 & 82 & 87~~~~ & $z$ & 1.6 ~~~~\\
~~~~PEST & 2013 May 3 & 99 & 130~~~~ & $R$ & 4.7 ~~~~\\
~~~~PEST & 2013 Jun 30 & 189 & 130~~~~ & $R$ & 4.7 ~~~~\\ [6pt] %
\multicolumn{2}{l}{\textbf{HATS-14}} \\ [2pt]%
~~~~HS-2/G582 & 2009 Nov--2010 Sep & 4866 & 284~~~~ & $r$ & 11.5 ~~~~\\
~~~~HS-4/G582 & 2009 Sep--2010 Sep & 8889 & 288~~~~ & $r$ & 12.6 ~~~~\\
~~~~HS-6/G582 & 2010 Aug--2010 Sep & 200 & 290~~~~ & $r$ & 11.6 ~~~~\\
~~~~PEST & 2013 Jun 06 & 131 & 131~~~~ & $R$ & 4.8 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2013 Jun 12 & 114 & 192~~~~ & $g$ & 1.6 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2013 Jun 12 & 114 & 192~~~~ & $r$ & 1.8 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2013 Jun 12 & 114 & 192~~~~ & $i$ & 2.6 ~~~~\\
~~~~GROND$^{\mathrm{d}}$ & 2013 Jun 12 & 114 & 192~~~~ & $z$ & 2.0 ~~~~\\
\hline
\end{tabular}
\tablefoot{\\
\tiny{
$^{\mathrm{a}}$
For HATSouth data we list the HATSouth unit and field name from
which the observations are taken. HS-1 and -2 are located at Las
Campanas Observatory in Chile, HS-3 and -4 are located at the
H.E.S.S. site in Namibia, and HS-5 and -6 are located at Siding
Spring Observatory in Australia. Each field corresponds to one of
838 fixed pointings used to cover the full 4$\pi$ celestial
sphere. All data from a given HATSouth field are reduced together,
while detrending through External Parameter Decorrelation (EPD) is
done independently for each unique field+unit combination.
\\ [2pt]%
$^{\mathrm{b}}$
The median time between consecutive images rounded to the nearest
second. Due to weather, the day--night cycle, guiding and focus
corrections, and other factors, the cadence is only approximately
uniform over short timescales.
\\ [2pt]%
$^{\mathrm{c}}$
The RMS of the residuals from the best-fit model.
\\ [2pt]%
$^{\mathrm{d}}$
The \emph{telescope-defocussing} technique \citep{southworth:2009} was used for this transit observation.}}
\end{table*}
\begin{figure*}%
\centering
{{\includegraphics[width=8cm]{\hatcurhtr{13}-hs.eps} }}%
\qquad
{{\includegraphics[width=8cm]{\hatcurhtr{14}-hs.eps} }}%
\caption{Phase-folded unbinned HATSouth light curves for HATS-13
(\emph{left}) and HATS-14 (\emph{right}). In each case we show two
panels. The top panel shows the full light curve, while the bottom
panel shows the light curve zoomed-in on the transit. The solid
lines show the model fits to the light curves. The dark filled
circles in the bottom panels show the light curves binned in phase
with a bin
size of 0.002.}%
\label{fig:hatsouth}%
\end{figure*}
\subsection{Spectroscopic Observations}
\label{sec:obsspec}
After being selected as \emph{HATSouth planet} candidates, HATS-13
and HATS-14 underwent spectral reconnaissance through low- and
medium-resolution observations with the Wide Field Spectrograph
(WiFeS; \citealp{dopita:2007}) mounted on the ANU 2.3\,m telescope
at SSO. This first step is very useful in the planet confirmation
process because it can immediately rule out possible false
positive cases, mainly caused by giant stars, F-M binary systems
and blending with faint eclipsing-binary systems.
Using WiFES, we identified both the targets as dwarf stars.
HATS-13 and HATS-14 were then accurately monitored with an array
of telescopes equipped with high-resolution spectrographs,
covering wide ranges of optical wavelengths, to look for possible
radial-velocity (RV) variations compatible with the presence of
planetary companions.
Four and five spectra were observed in May 2012 for HATS-13 and
HATS-14, respectively, with CYCLOPS mounted on the 3.9\,m
Anglo-Australian Telescope at SSO. A better RV accuracy was
achieved between May and November 2012 thanks to FEROS
\citep{kaufer:1998} on the MPG 2.2\,m telescope at the ESO
Observatory in La Silla and Coralie \citep{queloz:2001} on the
Euler 1.2\,m telescope, also located in La Silla. In total, with
these two instruments, we collected 32 and 31 spectra for HATS-13
and HATS-14, respectively, with an average precision of some tens
of meters per second. Information about these spectropic
observations are summarized in Table\,\ref{tab:specobs}, yet we
did not use all the spectra in the analysis, as some of them were
discarded due to high-sky contamination. Additional details about
the instruments and data-reduction processes are exhaustively
discussed in previous works of the HATS team and we refer the
reader to those (i.e.
\citealp{penev:2013,mohler:2013,bayliss:2013}). In particular,
Coralie and FEROS spectra were reduced using the new procedure
described in \citet{jordan:2014} and \citet{brahm:2015}.
To better characterize the periodic signal of the RV variation of
HATS-13, it was necessary to observe this target with higher RV
precision. On September 2012, we used the High Dispersion
Spectrograph (HDS; \citealp{noguchi:2002}) on the Subaru telescope
at the Observatory of Mauna Kea, Hawaii. Observations were spread
over four nights and performed in a way similar to those for
HATS-5 \citep{zhou:2014}, i.e. using a $0^{\prime\prime}.6 \times
2^{\prime\prime}.0$ slit and a setup which guaranteed a
wavelength-range coverage of $3500-6200$\,\AA, with a resolution
of $R=60\,000$. Ten spectra were taken using an I$_2$ cell and
another three without it (Table\,\ref{tab:specobs}). All of HDS
observations were reduced following \citet{sato:2002,sato:2012}.
All the RV measurements, extracted from the spectra here
discussed, are listed in Table\,\ref{tab:rvs1} and \ref{tab:rvs2}.
Phased RV and BS measurements are shown for each system in
Fig.\,\ref{fig:rvbis}.
\begin{table*}
\caption{Summary of spectroscopy observations} %
\label{tab:specobs} %
\centering %
\tiny %
\setlength{\tabcolsep}{8pt}
\begin{tabular}{llrrrrr}
\hline\hline %
Telescope/Instrument & UT Date(s) & \# Spec. & Res.~~~~ & S/N Range$^{\mathrm{a}}$ & $\gamma_{\rm RV}$$^{\mathrm{b}}~~$ & RV Precision$^{\mathrm{c}}$ \\ %
& & & $\Delta \lambda$/$\lambda$/1000 & & (\ensuremath{\rm km\,s^{-1}}) & (\ensuremath{\rm m\,s^{-1}})~~~~~ \\ %
\hline \\%
\multicolumn{2}{l}{\textbf{HATS-13}} \\ [2pt]%
ANU~2.3\,m/WiFeS & 2012 Apr 10 & 1 & 3 & 90 & $\cdots$ & $\cdots$ \\
ANU~2.3\,m/WiFeS & 2012 Apr 11-12 & 2 & 7 & 30--40 & 25.0 & 1000 \\
AAT~3.9\,m/CYCLOPS & 2012 May 5-11 & 4 & 70 & 10--20 & 25.8 & 190 \\
Euler~1.2\,m/Coralie & 2012 Jun--Nov & 9 & 60 & 12--17 & 25.8 & 100 \\
MPG~2.2\,m/FEROS & 2012 May--Oct & 23 & 48 & 29--72 & 25.8 & 68 \\
Subaru 8\,m/HDS & 2012 Sep 19 & 3 & 60 & 17--32 & $\cdots$ & $\cdots$ \\
Subaru 8\,m/HDS+I$_{2}$ & 2012 Sep 20--22 & 10 & 60 & 15--27 & $\cdots$ & 21 \\ [6pt] %
\multicolumn{2}{l}{\textbf{HATS-14}} \\ [2pt]%
ANU~2.3\,m/WiFeS & 2012 Apr 10 & 1 & 3 & 70 & $\cdots$ & $\cdots$ \\
ANU~2.3\,m/WiFeS & 2012 Apr 11-13 & 3 & 7 & 23--31 & 28.5 & 380 \\
AAT~3.9\,m/CYCLOPS & 2012 May 5-11 & 5 & 70 & 15--25 & 29.9 & 110 \\
Euler~1.2\,m/Coralie & 2012 Jun--Nov & 14 & 60 & 9--16 & 30.2 & 37 \\
MPG~2.2\,m/FEROS & 2012 May--2013 Jul & 17 & 48 & 25--68 & 30.2 & 12 \\
\hline %
\end{tabular}
\tablefoot{\\
\tiny{
$^{\mathrm{a}}$
S/N per resolution element near 5180\,\AA.
\\ [2pt]%
$^{\mathrm{b}}$
For Coralie, FEROS and CYCLOPS this is the systemic RV from fitting an orbit to the observations in \ref{sec:globmod}.
For WiFeS it is the mean of the observations, and for the du~Pont Echelle it is the measured RV of the single observation.
We do not provide this quantity for instruments for which only relative RVs are measured, or for the lower resolution WiFeS
observations which were only used to measure stellar atmospheric parameters.
\\ [2pt]%
$^{\mathrm{c}}$
For High-precision RV observations included in the orbit
determination this is the RV residuals from the best-fit orbit,
for other instruments used for reconnaissance spectroscopy this is
an estimate of the precision, or the measured standard
deviation.}}
\end{table*}
\begin{figure*}%
\centering
{{\includegraphics[width=8cm]{\hatcurhtr{13}-rv.eps} }}%
\qquad
{{\includegraphics[width=8cm]{\hatcurhtr{14}-rv.eps} }}%
\caption{Phased high-precision RV measurements for
\hbox{\hatcur{13}{}} (\emph{left}), and \hbox{\hatcur{14}{}}
(\emph{right}) from HDS (filled circles), FEROS (open triangles),
Coralie (filled triangles), and CYCLOPS (stars). In each case we
show three panels. The top panel shows the phased measurements
together with our best-fit model (see
Table\,\ref{tab:planetparam}) for each system. Zero-phase
corresponds to the time of mid-transit. The center-of-mass
velocity has been subtracted. The second panel shows the velocity
O--C residuals from the best fit. The error bars include the
jitter terms listed in Table\,\ref{tab:planetparam} added in
quadrature to the formal errors for each instrument. The third
panel shows the bisector spans (BS), with the mean value
subtracted. Note the different vertical scales of the panels.}
\label{fig:rvbis}%
\end{figure*}
\subsection{Photometric follow-up observations}
\label{sec:phot}
High-quality photometric follow-up observations of additional
transit events of the two targets were subsequently performed with
larger telescopes than the HATSouth units. This is also an
important step because it allowed us to have a precise light-curve
anatomy of the planetary transits (depth, duration and sharpness)
and -- by constraining the eccentricity via RV variations --
measure the mean density of the parent stars with high accuracy
and with no systematic errors \citep{seager:2003}. As we will see
in Sect.\,\ref{sec:stelparam} the knowledge of the stellar mean
density is a very useful constraint for the determination of the
other physical parameters of the two systems.
Concerning HATS-13, two complete and two incomplete transits were
observed using the MPG 2.2\,m, the CTIO 0.9\,m, and the PEST
0.3\,m telescopes. Two complete transit events were successfully
monitored for HATS-14 with the MPG 2.2\,m and PEST telescopes.
Relevant information about these observations (i.e. dates,
cadence, filter, precision) are reported in
Table\,\ref{tab:photobs}. In particular, the MPG 2.2\,m telescope
is equipped with GROND, a multi-imaging camera, able to observe a
field-of-view (FOV) of $5.4^{\prime} \times 5.4^{\prime}$ in four
different filters (similar to Sloan $g,r,i,z$) simultaneously
\citep{greiner:2008}. Details of the GROND camera and data
reduction are reported in \citet{penev:2013} and
\citet{mohler:2013}, while studies of the accuracy and S/N
expectations for this instrument were done by \citet{pierini:2012}
and \citet{mancini:2014}. The PEST telescope and data reduction
method are discussed in \citet{bayliss:2013}. The same information
for the CTIO 0.9\,m telescope have been reported by
\citet{hartman:2014}.
The light curves for HATS-13 and HATS-14 are shown in
Fig.\,\ref{fig:lc13} and Fig.\,\ref{fig:lc14}, respectively. The
corresponding data, including those from the HATS units, are given
in Table 3.
\begin{figure*}
\centering
\includegraphics[width=18cm]{\hatcurhtr{13}-lc.eps}
\caption{\emph{Left panel}: Unbinned transit light curves for
\hatcur{13}. The light curves have been corrected for quadratic
trends in time fitted simultaneously with the transit model. The
dates of the events, filters and instruments used are indicated.
Light curves following the first are displaced vertically for
clarity. Our best fit from the global modeling described in
Sect.\,\ref{sec:globmod} is shown by the solid lines. \emph{Right
panel}: residuals from the fits are displayed in the same order as
the left curves. The error bars represent the photon and
background shot noise, plus the readout noise.} %
\label{fig:lc13}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=18cm]{\hatcurhtr{14}-lc.eps}
\caption{Similar to Fig.\,\ref{fig:lc13}; here we show the follow-up
light curves for \hatcur{14}.} %
\label{fig:lc14}
\end{figure*}
\section{Analysis}
\label{sec:analysis}
Based on the data previously presented, this section is dedicated
to the derivation of the physical parameters of the HATS-13 and
HATS-14 planet hosts.
\subsection{Properties of the parent stars}
\label{sec:stelparam}
We used 17 and 14 high-resolution FEROS spectra to determine the
\emph{atmospheric} properties (metallicity, effective temperature
and surface gravity) of the stars HATS-13 and HATS-14,
respectively. This was accomplished by using the new routine ZASPE
(Zonal Atmospherical Stellar Parameter Estimator), which is fully
described in Brahm et al. (2015). The other principal stellar
parameters (like mass, radius, luminosity, age, etc.) and
corresponding uncertainties were estimated thanks to a Markov
chain Monte Carlo (MCMC) global analysis of our photometric and
spectroscopic data, following the methodology of
\citet{sozzetti:2007}. This is based on stellar effective
temperature $\ensuremath{T_{\rm eff\star}}$, which we determined with ZASPE, the
stellar mean density $\ensuremath{\rho_\star}$, estimated from the light-curve
fitting (see Sect.\,\ref{sec:globmod}), and from the Yonsei-Yale
(YY; \citealp{yi:2001}) evolutionary tracks.
Spanning a range of reliable values for the metallicity, we
calculated the YY isochrones for each of the two systems over a
wide a range of ages and compared the resulting $\ensuremath{T_{\rm eff\star}}$ and
$\ensuremath{\rho_\star}$ with those estimated from the data. The best agreement
returned the values of the other stellar parameters. In
particular, the better estimation of the stellar logarithmic
surface gravity (\ensuremath{\log{g_{\star}}}$=4.524 \pm 0.017$ for HATS-13 and
\ensuremath{\log{g_{\star}}}$=4.484 \pm 0.020$ for HATS-14), was used for a second
iteration of ZASPE, by fixing these values, to revise the other
atmospheric parameters.
The stellar properties that we derived are reported in
Table\,\ref{tab:stellar}, along with their 1$\sigma$
uncertainties. Model isochrones are shown in the panels of
Fig.\,\ref{fig:iso}, in which the positions of the two stars in
the $\ensuremath{T_{\rm eff\star}}-\ensuremath{\rho_\star}$ diagram are also marked.
We found that both the stars are slightly smaller and less massive
than the Sun, with parameters listed in Table\,\ref{tab:stellar}.
In particular, with \ensuremath{T_{\rm eff\star}}$=5523 \pm 69$\,K, $\ensuremath{M_\star}=0.962 \pm
0.029 \, M_{\sun}$, $\ensuremath{R_\star}=0.887 \pm 0.019\, R_{\sun}$, $B-V=0.80
\pm 0.03$, $V-H=1.84 \pm 0.04$, HATS-13 is a G5\,V star, whereas
HATS-14, characterized by \ensuremath{T_{\rm eff\star}}$=5346 \pm 60$\,K,
$\ensuremath{M_\star}=0.967 \pm 0.024\, M_{\sun}$,
$\ensuremath{R_\star}=0.933_{-0.015}^{+0.023}\, R_{\sun}$, $B-V=0.83 \pm 0.2$,
$V-H=1.87 \pm 0.3$, has a spectral class close to the K/G
transition \citep{pecaut:2013}. The preferred metallicities are
$\ensuremath{\rm [Fe/H]}=0.050 \pm 0.060$ and $\ensuremath{\rm [Fe/H]}=0.330 \pm 0.060$ for HATS-13 and
HATS-14, respectively.
Table\,\ref{tab:stellar} also shows the magnitudes of the two
stars in the optical bands (taken from APASS as listed in the UCAC
4 catalog; \citealp{zacharias:2012}) and in the NIR bands (from
2MASS). We compared these values with the predicted magnitudes in
each filter from the isochrones, determining the distance of the
two stars, that is $476 \pm 12$\,pc for HATS-13 and $513 \pm
14$\,pc for HATS-14. Here the extinction was estimated by assuming
an $R_{V}=3.1$ law from \citet{cardelli:1989}.
\begin{figure*}%
\centering
{{\includegraphics[width=8cm]{\hatcurhtr{13}-iso-rho.eps} }}%
\qquad
{{\includegraphics[width=8cm]{\hatcurhtr{14}-iso-rho.eps} }}%
\caption{Model isochrones from \citet{yi:2001} for the measured
metallicities of \hatcur{13} (\emph{left panel}) and \hatcur{14}
(\emph{right panel}). In each case we show models for ages of
0.2\,Gyr and 1.0 to 14.0\,Gyr in 1.0\,Gyr increments (ages
increasing from left to right). The adopted values of $\ensuremath{T_{\rm eff\star}}$
and \ensuremath{\rho_\star}\ are shown together with their 1$\sigma$ and
2$\sigma$ confidence ellipsoids. The initial values of \ensuremath{T_{\rm eff\star}}\
and \ensuremath{\rho_\star}\ from the first ZASPE and light curve\ analyses are
represented with a triangle.}
\label{fig:iso}%
\end{figure*}
\begin{table*}
\caption{Stellar parameters for \hatcur{13} and \hatcur{14}} %
\label{tab:stellar} %
\centering %
\tiny %
\setlength{\tabcolsep}{8pt}
\begin{tabular}{lccl}
\hline\hline %
& HATS-13b & HATS-14b \\ %
~~~~~~~~~~~~~~~Parameter~~~~~~~~~~~~~~~ & Value & Value & Source \\ %
\hline \\%
\multicolumn{2}{l}{Astrometric properties and cross-identifications} \\ [1pt] %
~~~~2MASS-ID\dotfill & \hatcurCCtwomass{13} & \hatcurCCtwomass{14} & \\
~~~~GSC-ID\dotfill & \hatcurCCgsc{13} & \hatcurCCgsc{14} & \\
~~~~R.A. (J2000)\dotfill & \hatcurCCra{13} & \hatcurCCra{14} & 2MASS\\
~~~~Dec. (J2000)\dotfill & \hatcurCCdec{13} & \hatcurCCdec{14} & 2MASS\\
~~~~$\mu_{\rm R.A.}$ (\ensuremath{\rm mas\,yr^{-1}}) & \hatcurCCpmra{13} & \hatcurCCpmra{14} & UCAC4\\
~~~~$\mu_{\rm Dec.}$ (\ensuremath{\rm mas\,yr^{-1}}) & \hatcurCCpmdec{13} & \hatcurCCpmdec{14} & UCAC4\\[3pt]
\multicolumn{2}{l}{Spectroscopic properties} \\ [1pt] %
~~~~$\ensuremath{T_{\rm eff\star}}$ (K)\dotfill & \hatcurSMEteff{13} & \hatcurSMEteff{14} & ZASPE $^{\mathrm{a}}$\\
~~~~$\ensuremath{\rm [Fe/H]}$\dotfill & \hatcurSMEzfeh{13} & \hatcurSMEzfeh{14} & ZASPE \\
~~~~$\ensuremath{v \sin{i}}$ (\ensuremath{\rm km\,s^{-1}})\dotfill & \hatcurSMEvsin{13} & \hatcurSMEvsin{14} & ZASPE \\
~~~~$\gamma_{\rm RV}$ (\ensuremath{\rm km\,s^{-1}})\dotfill & $25.804 \pm 0.014$ & $30.190\pm 0.008$ & Coralie, FEROS \\[3pt]
\multicolumn{2}{l}{Photometric properties} \\ [1pt] %
~~~~$B$ (mag)\dotfill & \hatcurCCtassmB{13} & \hatcurCCtassmB{14} & APASS $^{\mathrm{b}}$ \\
~~~~$V$ (mag)\dotfill & \hatcurCCtassmv{13} & \hatcurCCtassmv{14} & APASS $^{\mathrm{b}}$ \\
~~~~$g$ (mag)\dotfill & \hatcurCCtassmg{13} & $\cdots$ & APASS $^{\mathrm{b}}$ \\
~~~~$r$ (mag)\dotfill & \hatcurCCtassmr{13} & $\cdots$ & APASS $^{\mathrm{b}}$ \\
~~~~$i$ (mag)\dotfill & \hatcurCCtassmi{13} & $\cdots$ & APASS $^{\mathrm{b}}$ \\
~~~~$J$ (mag)\dotfill & \hatcurCCtwomassJmag{13} & \hatcurCCtwomassJmag{14} & 2MASS \\
~~~~$H$ (mag)\dotfill & \hatcurCCtwomassHmag{13} & \hatcurCCtwomassHmag{14} & 2MASS \\
~~~~$K_s$ (mag)\dotfill & \hatcurCCtwomassKmag{13} & \hatcurCCtwomassKmag{14} & 2MASS \\[3pt]
\multicolumn{2}{l}{Derived properties} \\ [1pt] %
~~~~$\ensuremath{M_\star}$ ($\ensuremath{M_\sun}$) \dotfill & \hatcurISOmlong{13} & \hatcurISOmlong{14} & YY+$\ensuremath{\rho_\star}$+ZASPE $^{\mathrm{c}}$ \\
~~~~$\ensuremath{R_\star}$ ($\ensuremath{R_\sun}$)\dotfill & \hatcurISOrlong{13} & \hatcurISOrlong{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~$\ensuremath{\log{g_{\star}}}$ (cgs)\dotfill & \hatcurISOlogg{13} & \hatcurISOlogg{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~$\ensuremath{\rho_\star}$ (\ensuremath{\rm g\,cm^{-3}})\dotfill & \hatcurISOrho{13} & \hatcurISOrho{14} & YY+$\ensuremath{\rho_\star}$+ZASPE $^{\mathrm{d}}$ \\
~~~~$\ensuremath{L_\star}$ ($\ensuremath{L_\sun}$)\dotfill & \hatcurISOlum{13} & \hatcurISOlum{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~$M_V$ (mag)\dotfill & \hatcurISOmv{13} & \hatcurISOmv{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~$M_K$ (mag,\hatcurjhkfilset{13})\dotfill & \hatcurISOMK{13} & \hatcurISOMK{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~Age (Gyr)\dotfill & \hatcurISOage{13} & \hatcurISOage{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~$A_{V}$ (mag)\dotfill & \hatcurXAv{13} & \hatcurXAv{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\
~~~~Distance (pc)\dotfill & \hatcurXdistred{13} & \hatcurXdistred{14} & YY+$\ensuremath{\rho_\star}$+ZASPE \\ [2pt] %
\hline %
\end{tabular}
\tablefoot{\\
\tiny{
%
$^{\mathrm{a}}$
ZASPE = Zonal Atmospherical Stellar Parameter Estimator routine
for the analysis of high-resolution spectra \citep{brahm:2015},
applied to the FEROS spectra of \hatcur{13} and
\hatcur{14}. These parameters rely primarily on ZASPE, but have a
small dependence also on the iterative analysis incorporating the
isochrone search and global modeling of the data.
\\ [2pt]%
$^{\mathrm{b}}$
From APASS DR6 for \hatcur{13}, \hatcur{14} as
listed in the UCAC 4 catalog \citep{zacharias:2012}.
\\ [2pt]%
$^{\mathrm{c}}$
\hatcurisoshort{13}+\ensuremath{\rho_\star}+ZASPE = Based on the \hatcurisoshort{13}
isochrones \citep{\hatcurisocite{13}}, \ensuremath{\rho_\star}\ as a luminosity
indicator, and the ZASPE results.
\\ [2pt]%
$^{\mathrm{d}}$
The parameter $\ensuremath{\rho_\star}$ is primarily determined
from the global fit to the light curves and RV data. The value
shown here also has a slight dependence on the stellar models and
ZASPE parameters due to restricting the posterior distribution to
combinations of $\left[ \ensuremath{\rho_\star}, \ensuremath{T_{\rm eff\star}}, \mathrm{Fe}/\mathrm{H} \right]$ that match to a
\hatcurisoshort{13} stellar model.
}}
\end{table*}
\subsection{Excluding blend scenarios}
\label{sec:blend}
To rule out the possibility that either \hatcur{13} or \hatcur{14}
is a blend between an eclipsing binary and a third star
(potentially in the foreground or background of the binary), we
carried out a blend analysis following \citet{hartman:2012}. We
find that for both objects the single star with a transiting
planet model fits the light curves and broad-band photometric
color data better than a blended eclipsing binary model. For
\hatcur{13} the best-fit transiting planet model is preferred with
$2\sigma$ confidence over the best-fit blend model, while for
\hatcur{14} the best-fit transiting planet model is preferred with
$4\sigma$ confidence. Moreover, we find that any blend model that
comes close to fitting the photometric data would have been easily
detected as a composite object based on the spectroscopic data
(there would be two clear peaks in the CCFs, the RVs from the
highest peak would vary by more than 1\,\ensuremath{\rm km\,s^{-1}}, as would the
bisector spans). We conclude that both \hatcur{13} and \hatcur{14}
are transiting planet systems. We cannot, however, rule out the
possibility that either object is a blend between a transiting
planet system and a third star that is fainter than the
planet-hosting star. For \hatcur{13} we find that including a
physical stellar companion with a mass greater than $0.84$\,\ensuremath{M_\sun}\
leads to a worse fit than not including the companion, however
even a companion up to the mass of the primary star cannot be
ruled out with greater than $5\sigma$ confidence. For \hatcur{14}
we can rule out companions with a mass greater than $0.92$\,\ensuremath{M_\sun}\
with greater than 5$\sigma$ confidence, while including a
companion with a mass greater than $0.5$\,\ensuremath{M_\sun}\ leads to a worse
fit of the data than a non-composite system. High-resolution
imaging and/or long-term RV monitoring are needed to determine if
either source has a stellar companion. For the remainder of the
paper we assume both objects are single stars with transiting
planets, however if either system has a stellar companion the true
radius and mass of the planet would be larger than what we infer
here.
\subsection{Global modelling of the data}
\label{sec:globmod}
We modeled the HATSouth photometry, the follow-up photometry, and
the high-precision RV measurements following \citet{pal:2008},
\citet{bakos:2010}, \citet{hartman:2012}. We fit
\citet{mandel:2002} transit models to the light curves, allowing
for a dilution of the HATSouth transit depth as a result of
blending from neighboring stars and over-correction by the
trend-filtering method. For the follow-up light curves we include
a quadratic trend in time in our model for each event to correct
for systematic errors in the photometry. We fit Keplerian orbits
to the RV curves allowing the zero-point for each instrument to
vary independently in the fit, and allowing for RV jitter which we
also vary as a free parameter for each instrument. We used a
Differential Evolution Markov Chain Monte Carlo procedure to
explore the fitness landscape and to determine the posterior
distribution of the parameters. One may see that for \hatcur{14}
the scatter in the Coralie and FEROS RV residuals is consistent
with the uncertainties (see Fig.\,\ref{fig:rvbis}), so our
modelling finds jitter values of $0$ for both instruments.
The resulting parameters for each system are listed in
Table\,\ref{tab:planetparam}. They were determined assuming circular
orbits. We have also explored non-zero eccentricities, by varying
$\sqrt{e}\cos{\omega}$ and $\sqrt{e}\sin{\omega}$ in the fitting
process, $e$ being the eccentricity and $\omega$ the argument of
the periastron. In this case, we got that $e<0.181\,(<0.142)$ at
$95\%$ confidence for HATS-3\,(HATS-4).
By inspecting Table\,\ref{tab:planetparam}, we can note that while
HATS-14b has mass ($M_{\mathrm{p}}=1.071 \pm 0.070\,\ensuremath{M_{\rm J}}$) and
size ($R_{\mathrm{p}}=1.039^{+0.032}_{-0.022}\,\ensuremath{R_{\rm J}}$) slightly
larger than those of Jupiter, HATS-13 is much less massive (only
$M_{\mathrm{p}}=0.543 \pm 0.072\,\ensuremath{M_{\rm J}}$), but bloated
($R_{\mathrm{p}}=1.212 \pm 0.035\,\ensuremath{R_{\rm J}}$). The above values lead
to mean densities extremely different, i.e.
$\rho_{\mathrm{p}}=0.377 \pm 0.058$\,g\,cm$^{-3}$ for HATS-13b and
$\rho_{\mathrm{p}}=1.191_{-0.140}^{+0.098}$\,g\,cm$^{-3}$ for
HATS-14b. Curiously, even though they have different physical
properties, their orbital periods (3.04 and 2.77 days) and
separation from the own host star (0.041 and 0.038 \,au) are
similar to each other.
\begin{table*}
\caption{Orbital and planetary parameters for \hatcurb{13} and
\hatcurb{14}} %
\label{tab:planetparam} %
\centering %
\tiny %
\setlength{\tabcolsep}{8pt}
\begin{tabular}{lcc}
\hline\hline %
& HATS-13b & HATS-14b \\ %
~~~~~~~~~~~~~~~Parameter~~~~~~~~~~~~~~~ & Value & Value \\ %
\hline \\%
\multicolumn{2}{l}{Light curve parameters} \\ [2pt] %
~~~$P$ (days) \dotfill & $\hatcurLCP{13}$ & $\hatcurLCP{14}$ \\
~~~$T_c$ (${\rm BJD}$) $^{\mathrm{a}}$ \dotfill & $\hatcurLCT{13}$ & $\hatcurLCT{14}$ \\
~~~$T_{14}$ (days) $^{\mathrm{a}}$ \dotfill & $\hatcurLCdur{13}$ & $\hatcurLCdur{14}$ \\
~~~$T_{12} = T_{34}$ (days) $^{\mathrm{a}}$ \dotfill & $\hatcurLCingdur{13}$ & $\hatcurLCingdur{14}$ \\
~~~$\ensuremath{a/\ensuremath{R_\star}}$ \dotfill & $\hatcurPPar{13}$ & $\hatcurPPar{14}$ \\
~~~$\ensuremath{\zeta/\ensuremath{R_\star}}$ $^{\mathrm{b}}$ \dotfill & $\hatcurLCzeta{13}$ & $\hatcurLCzeta{14}$ \\
~~~$\ensuremath{R_{p}}/\ensuremath{R_\star}$ \dotfill & $\hatcurLCrprstar{13}$ & $\hatcurLCrprstar{14}$ \\
~~~$b^2$ \dotfill & $\hatcurLCbsq{13}$ & $\hatcurLCbsq{14}$ \\ [2pt] %
~~~$b \equiv a \cos i/\ensuremath{R_\star}$ \dotfill & $\hatcurLCimp{13}$ & $\hatcurLCimp{14}$ \\
~~~$i$ (deg) \dotfill & $\hatcurPPi{13}$ & $\hatcurPPi{14}$ \\
\multicolumn{2}{l}{Limb-darkening coefficients $^{\mathrm{c}}$} \\ [2pt] %
~~~$c_1,g$ (linear term) \dotfill & $\hatcurLBig{13}$ & $\hatcurLBig{14}$ \\
~~~$c_2,g$ (quadratic term) \dotfill & $\hatcurLBiig{13}$ & $\hatcurLBiig{14}$ \\
~~~$c_1,r$ \dotfill & $\hatcurLBir{13}$ & $\hatcurLBir{14}$ \\
~~~$c_2,r$ \dotfill & $\hatcurLBiir{13}$ & $\hatcurLBiir{14}$ \\
~~~$c_1,i$ \dotfill & $\hatcurLBii{13}$ & $\hatcurLBii{14}$ \\
~~~$c_2,i$ \dotfill & $\hatcurLBiii{13}$ & $\hatcurLBiii{14}$ \\
~~~$c_1,z$ \dotfill & $\hatcurLBiz{13}$ & $\hatcurLBiz{14}$ \\
~~~$c_2,z$ \dotfill & $\hatcurLBiiz{13}$ & $\hatcurLBiiz{14}$ \\
~~~$c_1,R$ \dotfill & $\hatcurLBiR{13}$ & $\hatcurLBiR{14}$ \\
~~~$c_2,R$ \dotfill & $\hatcurLBiiR{13}$ & $\hatcurLBiiR{14}$ \\
\multicolumn{2}{l}{RV parameters} \\ [2pt] %
~~~$K$ (\ensuremath{\rm m\,s^{-1}}) \dotfill & $\hatcurRVK{13}$ & $\hatcurRVK{14}$ \\
~~~$e$ $^{\mathrm{d}}$ \dotfill & $\hatcurRVeccentwosiglimeccen{13}$ & $\hatcurRVeccentwosiglimeccen{14}$ \\
~~~RV jitter HDS (\ensuremath{\rm m\,s^{-1}}) $^{\mathrm{e}}$ \dotfill & \hatcurRVjitterA{13} & $\cdots$ \\
~~~RV jitter FEROS (\ensuremath{\rm m\,s^{-1}}) \dotfill & \hatcurRVjitterB{13} & \hatcurRVjitterB{14} \\
~~~RV jitter Coralie (\ensuremath{\rm m\,s^{-1}}) \dotfill & \hatcurRVjitterC{13} & \hatcurRVjitterA{14} \\
~~~RV jitter CYCLOPS (\ensuremath{\rm m\,s^{-1}}) \dotfill & \hatcurRVjitterD{13} & $\cdots$ \\
\multicolumn{2}{l}{Planetary parameters} \\ [2pt] %
~~~$\ensuremath{M_{p}}$ ($\ensuremath{M_{\rm J}}$) \dotfill & $\hatcurPPmlong{13}$ & $\hatcurPPmlong{14}$ \\
~~~$\ensuremath{R_{p}}$ ($\ensuremath{R_{\rm J}}$) \dotfill & $\hatcurPPrlong{13}$ & $\hatcurPPrlong{14}$ \\
~~~$C\,(\ensuremath{M_{p}},\ensuremath{R_{p}})$ $^{\mathrm{f}}$ \dotfill & $\hatcurPPmrcorr{13}$ & $\hatcurPPmrcorr{14}$ \\
~~~$\ensuremath{\rho_{p}}$ (\ensuremath{\rm g\,cm^{-3}}) \dotfill & $\hatcurPPrho{13}$ & $\hatcurPPrho{14}$ \\
~~~$\log g_p$ (cgs) \dotfill & $\hatcurPPlogg{13}$ & $\hatcurPPlogg{14}$ \\
~~~$a$ (AU) \dotfill & $\hatcurPParel{13}$ & $\hatcurPParel{14}$ \\
~~~$T_{\rm eq}$ (K) \dotfill & $\hatcurPPteff{13}$ & $\hatcurPPteff{14}$ \\
~~~$\Theta$ $^{\mathrm{g}}$ \dotfill & $\hatcurPPtheta{13}$ & $\hatcurPPtheta{14}$ \\
~~~$\log_{14}\langle F \rangle$ (cgs) $^{\mathrm{h}}$ \dotfill & $\hatcurPPfluxavglog{13}$ & $\hatcurPPfluxavglog{14}$ \\ [2pt] %
\hline %
\end{tabular}
\tablefoot{\\
\tiny{
$^{\mathrm{a}}$
Times are in Barycentric Julian Date calculated directly from UTC {\em without} correction for leap seconds.
\ensuremath{T_c}: Reference epoch of mid transit that minimizes the correlation with the orbital period.
\ensuremath{T_{14}}: total transit duration, time between first to last contact;
\ensuremath{T_{12}=T_{34}}: ingress/egress time, time between first and second, or third and fourth contact.
\\ [2pt]%
$^{\mathrm{b}}$
Reciprocal of the half duration of the transit used as a jump parameter in our MCMC analysis in place of $\ensuremath{a/\ensuremath{R_\star}}$. It is
related to $\ensuremath{a/\ensuremath{R_\star}}$ by the expression $\ensuremath{\zeta/\ensuremath{R_\star}} = \ensuremath{a/\ensuremath{R_\star}}(2\pi(1+e\sin\omega))/(P\sqrt{1-b^2}\sqrt{1-e^2})$
\citep{bakos:2010}.
\\ [2pt]%
$^{\mathrm{c}}$
Values for a quadratic law, adopted from the tabulations by \cite{claret:2004} according to the spectroscopic (ZASPE)
parameters listed in Table\,\ref{tab:stellar}.
\\ [2pt]%
$^{\mathrm{d}}$
As discussed in Sect.\,\ref{sec:globmod} the adopted parameters for all four systems are determined assuming circular orbits. We also list
the 95\% confidence upper limit on the eccentricity determined when $\sqrt{e}\cos\omega$ and $\sqrt{e}\sin\omega$ are allowed to
vary in the fit.
\\ [2pt]%
$^{\mathrm{e}}$
Term added in quadrature to the formal RV uncertainties for each instrument. This is treated as a free parameter in the fitting routine.
\\ [2pt]%
$^{\mathrm{f}}$
Correlation coefficient between the planetary mass $\ensuremath{M_{p}}$ and radius $\ensuremath{R_{p}}$ estimated from the posterior parameter distribution.
\\ [2pt]%
$^{\mathrm{g}}$
The Safronov number is given by $\Theta = \frac{1}{2}(V_{\rm esc}/V_{\rm orb})^2 = (a/\ensuremath{R_{p}})(\ensuremath{M_{p}} / \ensuremath{M_\star} )$
\citep[see][]{hansen:2007}.
\\ [2pt]%
$^{\mathrm{h}}$
Incoming flux per unit surface area, averaged over the orbit.
%
}}
\end{table*}
\section{Discussion and conclusions}
\label{sec:discussion}
After having monitored more than 3 million stars in its almost
first five years of life, the HATSouth survey is now entering in a
phase of continuous flow of exoplanet discoveries. In this work we have presented two new
hot-Jupiter transiting planets, HATS-13b and HATS-14b, both
orbiting around slightly metal rich, mild main-sequence stars with
a period of $\sim 3$\,days. Their detection is robustly based on
extensive photometric observations and numerous RV measurements,
as we described in the previous sections.
Orbiting around similar stars at similar distances, the stellar
radiation that the two planets receive are quite similar, i.e.
$\sim 5.4$ and $\sim 6.0 \times 10^{8}$\,erg\,s$^{-1}$\,cm$^{-2}$
for HATS-13b and HATS-14b, respectively, putting them in the pL
class, according to the terminology of \citet{fortney:2008}. Based
on their equilibrium temperature and surface gravity (see
Table\,\ref{tab:planetparam}), their scale heights are $\sim740$
and $\sim230$\,km, respectively. So, HATS-13b would be a suitable
target for transmission-spectroscopy follow-up observations, but,
since it is a pL planet, we do not expect that its atmosphere
hosts a large amount of absorbing molecules in the optical
wavelength range \citep{fortney:2010}. However, past observations
of transiting gas giants reveal a wide diversity (e.g.,
\citealp{wakeford:2015}) and a more sophisticated classification
scheme for hydrogen-dominated exoplanetary atmospheres would be
necessary (see \citealp{madhusudhan:2014} and references therein).
If we look to their Safranov number, HATS-13b and HATS-14b would
belong to separate classes of planets and should have had quite
different evolution, migration and evaporation processes
\citep{hansen:2007}. Actually, even though the parent stars have
similar masses, their inferred ages differ by a factor of $\sim2$
(see Table\,\ref{tab:stellar}). Fig.\,\ref{fig:diagrams} shows the
positions of the two new HATS planets in the current planet
mass-radius plot (left panel) and planet mass-density plot (right
panel). They are shown together with those of all the other known
transiting exoplanets (data taken from the TEPCat
catalogue\footnote{The Transiting Extrasolar Planet Catalogue
(TEPCat) is available at www.astro.keele.ac.uk/jkt/tepcat/
\citep{southworth:2011}.} on March 9, 2015). It can be noted
immediately that they occupy two quite different positions in both
the diagrams. In the left panel, HATS-14b appears to be a bit out
from the population of Jupiters with masses near
$1\,M_{\mathrm{J}}$, whereas HATS-13b is in the middle of a
cluster of planets with masses around $0.5\,M_{\mathrm{J}}$ and
inflated radii. In addition to the position of the planets, the
right panel also shows 3.2\,Gyr isochrones of giant planets at
0.045\,au orbital separation from a solar analogue
\citep{fortney:2007}. The plot suggests that HATS-13b should be a
core-free planet, while HATS-14 should have a massive core of
$\sim 50\,M_{\oplus}$ (we stress that, although we cannot rule out
the possibility that HATS-14 has a stellar companion which is
diluting the transit -- see discussion in Sect.\,\ref{sec:blend}
-- our 3$\sigma$ upper limit on the radius of the planet under
this scenario is $1.11\,\ensuremath{R_{\rm J}}$).
\begin{figure*}
\centering
\includegraphics[width=18.0cm]{diagrams.eps}
\caption{\emph{Left panel}: Masses and radii of the known
transiting extrasolar planets. The grey points denote values taken
from TEPCat. Their error bars have been suppressed for clarity.
HATS-13b and HATS-14b are shown with red points with error bars.
Dotted lines show where density is 2.5, 1.0, 0.5, 0.25 and 0.1
$\rho_{\mathrm{J}}$. \emph{Right panel}: the mass-density diagram
of the currently known transiting exoplanets (taken from TEPCat).
Again HATS-13b and HATS-14b are shown with red points with error
bars. Four planetary models with various core masses (10, 25, 50
and 100 Earth mass) and another without a
core \citep{fortney:2007} are plotted for comparison.} %
\label{fig:diagrams}
\end{figure*}
\begin{acknowledgements}
Development of the HATSouth project was funded by NSF MRI grant
NSF/AST-0723074, operations have been supported by NASA grants
NNX09AB29G and NNX12AH91H, and follow-up observations receive
partial support from grant NSF/AST-1108686. A.J. acknowledges
support from FONDECYT project 1130857, BASAL CATA PFB-06, and
project IC120009 ``Millennium Institute of Astrophysics (MAS)'' of
the Millenium Science Initiative, Chilean Ministry of Economy.
R.B. and N.E. are supported by CONICYT- PCHA/Doctorado Nacional.
R.B. and N.E. acknowledge additional support from project IC120009
``Millenium Institute of Astrophysics (MAS)'' of the Millennium
Science Initiative, Chilean Ministry of Economy. V.S. acknowledges
support form BASAL CATA PFB-06. K.P. acknowledges support from
NASA grant NNX13AQ62G. This work is based on observations made
with telescopes at the ESO Observatory of La Silla. This paper
also uses observations obtained with facilities of the Las Cumbres
Observatory Global Telescope. Work at the Australian National
University is supported by ARC Laureate Fellowship Grant
FL0992131. We acknowledge the use of the AAVSO Photometric All-Sky
Survey (APASS), funded by the Robert Martin Ayers Sciences Fund,
and the SIMBAD database, operated at CDS, Strasbourg, France.
Operations at the MPG 2.2m Telescope are jointly performed by the
Max Planck Gesellschaft and the European Southern Observatory. The
imaging system GROND has been built by the high-energy group of
MPE in collaboration with the LSW Tautenburg and ESO. We thank
R\'{e}gis Lachaume for his technical assistance during the
observations at the MPG 2.2m Telescope. We are grateful to P.
Sackett for her help in the early phase of the HATSouth project.
The reduced light curves presented in this work will be made
available at the CDS (http://cdsweb.u-strasbg.fr/). We acknowledge
the use of the following internet-based resources: the ESO
Digitized Sky Survey; the TEPCat catalog; the SIMBAD data base
operated at CDS, Strasbourg, France; and the arXiv scientific
paper preprint service operated by Cornell University.
\end{acknowledgements}
|
1,314,259,995,816 | arxiv |
\section{Introduction}
The focus of this article is the two-dimensional uniform spanning tree (UST), which is a random subgraph of $\mathbb{Z}^2$ that will henceforth be denoted by $\mathcal{U}$. Since the introduction of this object in \cite{Pem91}, considerable progress has been made in our understanding of the geometry of USTs (and, more generally, uniform spanning forests), see \cite{BLPS} for background. In this direction, a particularly useful viewpoint was provided by Wilson, who gave a construction of USTs via loop erased random walks (LERWs) \cite{Wil}. Indeed, the latter description was at the heart of Schramm's seminal work describing the subsequential scaling limits of two-dimensional LERW and $\mathcal{U}$ in terms of what is now called the Schramm-Loewner evolution (SLE) \cite{Schramm}, see also \cite{LSW}. In recent years, building on Lawler and Viklund's convergence result for the LERW in its natural parametrisation \cite{LV}, a more detailed picture of the scaling limit of $\mathcal{U}$ has been established \cite{BCK,HS}. And, closely related to this, properties of the simple random walk (SRW) on $\mathcal{U}$ have also been explored \cite{BCK, BM10, BM11}. The goal here is to provide further insight into the behaviour of the heat kernel (transition density) of the latter process.
Let us proceed to present some of the basic notation that will be used throughout the article. We will assume that the two-dimensional UST $\sU$ is built on a probability space $(\Omega, \sF, {\mathbf P}} \def\pE{{\mathbf E})$; we write $\pE$ for the associated expectation. Note that, $\mathbf{P}$-a.s., $\sU$ is a one-ended tree with vertex set $\mathbb{Z}^2$ \cite{Pem91}. We write $\gam(x,y)$ for the unique self-avoiding path between $x,y \in \bZ^2$, and $\gam(x,\infty)$ for the unique infinite self-avoiding path started at $x$. By Wilson's algorithm (see \cite{Wil}, and the recollection of this at the start of Section \ref{sec:LERW}), $\gam(x,y)$ is equal in law to the loop erasure of a SRW started at $x$ and run until it hits $y$. We will denote by ${d_\sU}$ the intrinsic metric on the graph $\sU$, so that ${d_\sU}(x,y)$ is the length of the geodesic $\gam(x,y)$. We write $\mu_\sU$ for the measure on $\mathbb{Z}^2$ such that $\mu_\sU(\{x\})$ is given by the number of edges of $\sU$ that contain $x$; this is the invariant measure of the simple random walk. We denote balls in the intrinsic metric ${d_\sU}$ by ${B_\sU}(x,r) =\{ y \in \bZ^2: {d_\sU}(x,y) \le r\}$. We use $d_\infty$ to denote the $\ell_\infty$ metric on $\bZ^2$, and $B_\infty(x,r)$ to denote balls in the $d_\infty$-metric; these balls are of course boxes.
Many of the exponents that describe the behaviour of $\sU$ and the associated random walk can be expressed in terms of the {\em growth exponent of the two-dimensional LERW}, which is given by $\kappa=5/4$. More precisely, let $L_n$ be the loop erasure of a SRW in $\bZ^2$ run until its first exit from $[-n,n]^2$, $M_n$ be the number of steps in $L_n$, and $G(n) = \bE(M_n)$. By \cite{Law14}, we have
\begin{equation} \label{e:growth}
G(n) = \bE M_n \asymp n^\kappa,
\end{equation}
where $\asymp$ means `bounded above and below by constant multiples of'. (This improves earlier estimates in \cite{Ken, Mas09}, which establish that $\lim_{n\rightarrow\infty} \log G(n)/\log n = \kappa$.) The papers \cite{BM10, BM11} gave estimates for the heat kernel of $\sU$ in terms of the function $G$; these can now be written more simply using \eqref{e:growth}. When we cite results from \cite{BM10, BM11} we will give the simplified versions without further comment.
Next, we introduce the simple random walk on $\sU$, which is the discrete-time Markov process $X^\sU=((X^\sU_n)_{n\geq0},({P}_x^\sU)_{x\in \mathbb{Z}^2})$ that at each time step jumps from its current location to a uniformly chosen neighbour in the graph $\sU$. For $x\in \mathbb{Z}^2$, the law ${P}_x^\sU$ is called the {\em quenched} law of the simple random walk on $\sU$ started at $x$, and we write
\[p_n^\sU(x,y)=\frac{P_x^\sU\left(X^\sU_n=y\right)}{\mu_\sU\left(\{y\}\right)},\qquad \forall x,y\in\mathbb{Z}^2,\]
for the corresponding quenched heat kernel.
To understand the properties of random walk on a space such as $\sU$, a by now well-established approach is to first study volume growth and resistance growth (see, for example, \cite{BJKS, Kum, KM}). Regarding the volume growth, one would expect from \eqref{e:growth} and Wilson's algorithm that ${B_\sU}(x,r^\kappa)$ should be approximately the same as $B_\infty(x,r)$, and hence that $|{B_\sU}(x,R)|$ should be of order $R^{2/\kappa}$. This expectation was confirmed by \cite[Theorem 1.2]{BM11}, which gives stretched exponential estimates for the upper and lower tails of $R^{-2/\kappa} |B_\sU(0,R)|$. We define the `fractal dimension' of $\sU$ by
\begin{equation}\label{dfdef}
d_f = \frac2\kappa=\frac85.
\end{equation}
Using the estimates in \cite[Theorem 1.2]{BM11} an easy Borel-Cantelli argument gives that there exist deterministic constants $c_1,c_2\in(0,\infty)$ such that, $\mathbf{P}$-a.s.,
\[c_1r^{d_f}(\log\log r)^{-9}\leq\mu_\sU\left(B_\mathcal{U}(0, r)\right)\leq c_2r^{d_f}(\log\log r)^{3}\]
for large $r$. The first main result of this paper is that volume fluctuations of log-logarithmic magnitude really do occur.
{\thm\label{mainthm1} $\mathbf{P}$-a.s.,
\begin{equation}\label{bigvolas1}
\limsup_{r \rightarrow \infty} \frac{ \mu_\sU\left(B_\mathcal{U}(0, r) \right) } {r^{d_f}(\log\log r)^{1/5}} =\infty,
\end{equation}
and also
\begin{equation}\label{smallvolas1}
\liminf_{r \rightarrow \infty}
\frac{ (\log\log r)^{3/5} \mu_\sU\left(B_\mathcal{U}(0, r) \right) }{r^{d_f}} =0.
\end{equation}}
\medskip
Similar fluctuations have also been observed for Galton-Watson trees \cite[Proposition 2.8]{BK} (see also \cite[Lemma 5.1]{CrKu}). The proof here is more complicated as the correlations between different parts of the space are harder to control. The key ingredient is the argument of Section \ref{sec:control} below, in which we provide a general technique for estimating from below the probability of seeing a particular path configuration in the initial stages of the construction of the UST via Wilson's algorithm. This enables us to control the probability of seeing especially short or long paths in some region of $\sU$.
The volume fluctuations of Theorem \ref{mainthm1} are associated with corresponding fluctuations in the on-diagonal part of the quenched heat kernel. From \cite[Theorem 4.5]{BM11}, we know there exist deterministic constants $c_1,c_2\in(0,\infty)$ and $\alpha_1,\alpha_2\in(0,\infty)$ such that, $\mathbf{P}$-a.s.,
\[c_1n^{-d_f/d_w}(\log\log n)^{-\alpha_1}\leq p^\mathcal{U}_{2n}(0,0)\leq c_2n^{-d_f/d_w}(\log\log n)^{\alpha_2}\]
for large $n$. Here
\begin{equation}\label{dwdef}
d_w = 1 + d_f = \frac{2+\kappa}{\kappa} =\frac{13}5
\end{equation}
is the so-called walk dimension; this represents the space-time scaling exponent with respect to the intrinsic metric. Applying Theorem \ref{mainthm1}, we are able to deduce that log-log fluctuations in the quenched heat kernel actually occur.
{\cor\label{cor1} There exists $\beta>0$ such that, $\mathbf{P}$-a.s.,
\begin{align*}
\liminf_{n\rightarrow \infty}(\log\log n)^{1/13}n^{d_f/d_w}p^\mathcal{U}_{2n}(0,0)&=0, \\
\limsup_{n\rightarrow \infty}(\log\log n)^{-\beta} n^{d_f/d_w}p^\mathcal{U}_{2n}(0,0)&=\infty.
\end{align*}}
\medskip
These volume and heat kernel fluctuations arise from unlikely configurations of $\sU$ inside $B_\infty(0,r_k)$ at a (random) sequence of scales $r_k \rightarrow \infty$. Another consequence of the occurrence of such exceptional configurations is the failure of the elliptic Harnack inequality in this setting. For a precise description of the particular form of the elliptic Harnack inequality that we consider, see Definition \ref{ehidef} below.
{\cor\label{ehicor} The large-scale elliptic Harnack inequality does not hold for the random walk on $\sU$.}
\medskip
We now consider the off-diagonal heat kernel. To avoid the issues of parity that arise from the fact $\sU$ is a bipartite graph, we introduce the following smoothed version of the heat kernel
\[ \tilde{p}_n^\sU(x,y):=\frac{p_n^\sU(x,y)+p_{n+1}^\sU(x,y)}{2}.\]
In \cite[Theorem 4.7]{BM11} it was shown that there exist deterministic constants $\alpha,C\in (0,\infty)$ such that, $\mathbf{P}$-a.s.:
\[\frac{n^{-\frac{d_f}{d_w}}}{A} \exp\left\{-A \left(\frac{d_\sU(0,x)^{{d_w}}}{n}\right)^{\frac{1}{{d_w}-1}} \right\}\leq\tilde{p}_n^\sU(0,x)\leq {A}{ n^{-\frac{d_f}{d_w}}}\exp\left\{-\frac{1}{A} \left(\frac{d_\sU(0,x)^{ {d_w}}}{n}\right)^{\frac{1}{{d_w}-1}} \right\}\]
holds whenever $n \ge d_\mathcal{U}(0,x)$ and $\max\{n^{d_w},|x|\}$ is suitably large, where
\begin{equation} \label{e:Adef}
A= A(n,x) :=C \left(\log\left(\max\{n^{d_w},|x|\} \right)\right)^\alpha.
\end{equation}
The logarithmic correction factor $A$ represents the possible influence of exceptional environments on the heat kernel.
We are unlikely to see an exceptional configuration at any particular scale, so it is not surprising that for the averaged heat kernel the fluctuations of Corollary \ref{cor1} disappear: by \cite[Theorem 4.4(c)]{BM11}, we have that
\begin{equation} \label{e:annondub}
c_1n^{-d_f/d_w} \leq \pE \, p^\mathcal{U}_{2n}(0,0)\leq c_2n^{-d_f/d_w},\qquad \forall n\geq 1.
\end{equation}
As for the off-diagonal part of the averaged heat kernel, one might hope that one could replace the random distance $d_\sU(0,x)$ with its typical order with respect to the Euclidean metric, that is, $|x|^\kappa$, and that, as with \eqref{e:annondub} one would be able to remove the errors associated with the term $A$ in \eqref{e:Adef}. We show that this is almost the case, however, in the annealed off-diagonal bounds the exponent $\frac{1}{d_w-1}$ needs to be replaced by a strictly smaller number.
\begin{thm}\label{mainthm3} There exist constants $c_1,c_2,c_3,c_4\in (0,\infty)$ and
$0< \theta_2 \le \theta_1 <1$ such that: for every $x=(x_1,x_2)\in\mathbb{Z}^2$ and $n\geq |x_1|+|x_2|$,
\[
n^{-\frac{d_f}{d_w}} \exp\left\{-c_2 \left(\frac{|x|^{\kappa {d_w}}}{n}\right)^{\frac{\theta_1}{{d_w}-1}} \right\}\leq
\pE \tilde{p}_n^\sU(0,x) \le c_3 n^{-\frac{d_f}{d_w}}\exp\left\{-c_4 \left(\frac{|x|^{\kappa {d_w}}}{n}\right)^{\frac{\theta_2}{{d_w}-1}} \right\}. \]
\end{thm}\medskip
Our argument indicates that we can take $\theta_1<1$ due to contribution to the averaged heat kernel from realisations of $\sU$ where the intrinsic distance from $0$ to $x$ is unusually short, and thus where the heat kernel $\tilde{p}^{\sU}_n(0,x)$ is unusually large. This phenomenon was not observed in the earlier study of random walk on a Galton-Watson tree of \cite{BK} (see Theorem 1.5 in particular), since the intrinsic metric of the trees was the only one involved there.
\begin{remark}
{\rm We have $\theta} \def\Th{\Theta_1 = \frac{ {d_w}-1 }{\kappa {d_w} -1}=\frac{32}{45}$, and we conjecture that this is also the correct value for $\theta} \def\Th{\Theta_2$. This would mean that the averaged heat kernel estimates of Theorem \ref{mainthm3} are of the usual sub-Gaussian form, but with respect to the extrinsic walk dimension $\kappa d_w$, rather than the intrinsic walk dimension that appears in the quenched bounds.
}\end{remark}
In the course of our proofs we obtain some new tail estimates on the length of the path $\gam(x,y)$ between points $x$ and $y$; by Wilson's algorithm this is also the length of a LERW run from $x$ to $y$.
\begin{thm} \label{T:dxy}
(a) There exist constants $c_i$ such that for $\lambda} \def\Lam {\Lambda \ge 1$, $x, y \in \bZ^2$,
\[ c_1 e^{ -c_2 \lambda} \def\Lam {\Lambda^4 } \le {\mathbf P}} \def\pE{{\mathbf E}\big( {d_\sU}(x,y) < \lambda} \def\Lam {\Lambda^{-1} d_\infty(x,y)^\kappa \big) \le c_3 e^{ -c_4 \lambda} \def\Lam {\Lambda^4 }.\]
(b) There exist constants $c,q$ such that for $\lambda} \def\Lam {\Lambda \ge 1$, $x, y \in \bZ^2$,
\[ {\mathbf P}} \def\pE{{\mathbf E}\left({d_\sU}(x,y) \ge \lambda} \def\Lam {\Lambda d_\infty(x,y)^\kappa\right) \le c ( \log \lambda} \def\Lam {\Lambda )^q\lambda} \def\Lam {\Lambda^{-(2-\kappa)/\kappa}. \]
\end{thm}\medskip
The upper bound in (a) is proved in Theorem \ref{T:LERW-lb}, (b) is proved at the end of Section \ref{sec:LERW}, and the lower bound in (a) is proved at the end of Section \ref{sec:control}.
\smallskip
We now consider the scaling limit of the UST and its heat kernel. Schramm's original work encoded $\mathcal{U}$ in terms of a path ensemble (consisting of the shortest paths in $\mathcal{U}$ between pairs of vertices), which enabled basic topological properties of any possible scaling limit to be deduced. In \cite{BCK}, building on the work of \cite{BM10, BM11}, this scaling picture was extended to incorporate the intrinsic (i.e.\ shortest path) metric on $\mathcal{U}$, as well as the uniform measure, with the result of \cite{BCK} being expressed in terms of the tightness under rescaling of $\mathcal{U}$ in a certain Gromov-Hausdorff-type topology for metric-measure spaces with an embedding into Euclidean space. The main obstacle to extending the work of \cite{BCK} to a full (i.e.\ non-subsequential) scaling limit was the need to prove the existence of the scaling limit of the two-dimensional LERW as a stochastic process, rather than simply as a compact subset of the plane. This was subsequently established in \cite{LV}, and Holden and Sun \cite{HS} then proved that $\sU$ has a full scaling limit as a metric-measure space.
Let us now describe the setting of \cite{BCK} more precisely. To retain information about $\sU$ in the Euclidean topology, $(\sU,{d_\sU})$ can be considered as a spatial tree (cf.\ \cite{DuL}) -- that is, as a real tree (see \cite[Definition 1.1]{rrt}, for example) obtained from the graph by including unit line segments along edges, embedded into $\mathbb{R}^2$ via a continuous map $\phi_\sU:\sU\rightarrow \mathbb{R}^2$, which is given by the identity on vertices, with linear interpolation along edges. In addition, suppose the space is rooted at the origin of $\mathbb{Z}^2$, giving a random `measured, rooted spatial tree' $(\sU,{d_\sU},\mu_\sU,\phi_\sU, 0)$. For this quintuplet, it follows from \cite[Theorem 1.1]{HS} (see also \cite[Theorem 1.1]{BCK}) that
\begin{equation}\label{scaling}
\left(\sU,\delta^{\kappa}{d_\sU},\delta^{2}\mu_\sU,\delta\phi_\sU,0\right)\buildrel{d}\over{\rightarrow}\left(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT\right)
\end{equation}
as $\delta\rightarrow0$ with respect to the Gromov-Hausdorff-type topology introduced in \cite[Section 3]{BCK}. The random limit space is such that, $\mathbf{P}$-a.s.: $(\mathcal{T},d_\mathcal{T})$ is a complete and locally compact real tree; $\mu_\mathcal{T}$ is a locally finite Borel measure on $(\mathcal{T},d_\mathcal{T})$; $\phi_\mathcal{T}$ is a continuous map from $(\mathcal{T},d_\mathcal{T})$ into $\mathbb{R}^2$; and $\rho_\mathcal{T}$ is a distinguished vertex in $\mathcal{T}$. In the original result of \cite{BCK}, the measure $\mu_\sU$ considered was the uniform measure on the vertices, but it is no problem to replace this with the measure we consider here since, after scaling the uniform measure by a factor of two, the Prohorov distance between the two measures is bounded above by two, and so the discrepancy disappears in the scaling limit. Moreover, it readily follows from \eqref{scaling} that the space $(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT)$ satisfies the following scale invariance property: for any $\lambda>0$,
\begin{equation}\label{scaleinvar}
\left(\sT,\lambda^{\kappa}d_\sT,\lambda^{2}\mu_\sT,\lambda\phi_\sT,\rho_{\mathcal{T}}\right)\buildrel{d}\over{=}\left(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT\right).
\end{equation}
Further from the SLE description of the limit in \cite{HS}, one also has rotational invariance, i.e.\ for any $\theta\in[0,2\pi)$,
\begin{equation}\label{rotinvar}
\left(\sT,d_\sT,\mu_\sT,R_\theta\phi_\sT,\rho_{\mathcal{T}}\right)\buildrel{d}\over{=}\left(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT\right),
\end{equation}
where $R_\theta$ is a rotation of Euclidean space about the origin by the angle $\theta$. In Proposition \ref{reroot} below, we further establish an invariance under a rerooting property for the limit space.
One of the motivations for proving \eqref{scaling} was to show that the random walks on $\sU$ converge to a limiting process. It was shown in \cite{BCK} that the random walks on $\sU$ started from 0 satisfy
\begin{equation}\label{rwconvh}
\left(\delta X^\mathcal{U}_{\delta^{-\kappa d_w}t}\right)_{t\geq 0}\rightarrow \left(\phi_{\sT}\left(X^\mathcal{T}_{t}\right)\right)_{t\geq 0}
\end{equation}
in distribution under the averaged or annealed law. (Cf.\ the more general statements concerning the convergence of random walks on trees of \cite{ALW,Cr}.) Here, for $\mathbf{P}$-a.e.\ realisation of $(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT)$, $X^\mathcal{T}=(X^\mathcal{T}_t)_{t\geq 0}$ is the canonical diffusion, or Brownian motion, on $(\sT,d_\sT,\mu_\sT)$ started from $\rho_\sT$, and $\phi_\sT(X^\mathcal{T})$ is the corresponding random element of $C(\mathbb{R}_+,\mathbb{R}^2)$. In this article, we connect the heat kernel of the discrete process $X^\sU$ to that of $X^\sT$, for which off-diagonal estimates were given in \cite{BCK}. As our first result in this direction, we show the convergence of the quenched and averaged on-diagonal part of the heat kernel. From \eqref{scaling} and \eqref{rwconvh}, the first claim of the following result, which concerns $(p^\sT_t(x,y))_{x,y\in\sT,t>0}$, the quenched heat kernel on the tree $\sT$ (as defined in \cite{BCK}), is essentially an application of the local limit theorem of \cite{CrHa}. To adapt this to yield the corresponding statement for the averaged heat kernels, we check the uniform integrability of the on-diagonal part of the discrete heat kernel by applying an argument similar to that applied to deduce averaged heat kernel estimates for Galton-Watson trees in \cite[Theorem 1.5]{BK}. We note that the exact form of the on-diagonal part of the limiting averaged heat kernel is a simple consequence of the scale invariance property \eqref{scaling}. Moreover, the result at \eqref{odhkscale} improves part of \cite[Theorem 4.4]{BM11}, where it was shown that $n^{{d_f}/{d_w}}\pE \tilde{p}_{n}^\sU(0,0)$ is bounded above and below by constants.
\begin{thm}\label{mainthm2} It holds that
\[\left(n^{{d_f}/{d_w}} \tilde{p}_{\lfloor tn\rfloor }^\sU(0,0)\right)_{t>0}\buildrel{d}\over{\rightarrow}\left( p_{t}^\sT(\rho_\sT,\rho_\sT)\right)_{t>0}\]
in distribution with respect to the topology of uniform convergence on compact subsets of $(0,\infty)$, and moreover,
\begin{equation}\label{odhkscale}
\left(n^{{d_f}/{d_w}}\pE \tilde{p}_{\lfloor tn\rfloor }^\sU(0,0)\right)_{t>0} \rightarrow\left(\pE p_{t}^\sT(\rho_\sT,\rho_\sT)\right)_{t>0}=\left( Ct^{-d_f/d_w}\right)_{t>0}
\end{equation}
in the same topology, where $C\in(0,\infty)$ is a constant.
\end{thm}
\medskip
We next turn our attention to the off-diagonal part of the heat kernel. Whilst it is natural to ask whether the scaling limit of \eqref{odhkscale} can be extended to include the off-diagonal part, we recall that $\phi_{\mathcal{T}}$ is not a bijection (see \cite[Theorem 1.3]{BCK}), and so one cannot \emph{a priori} assume that the limit of $n^{{d_f}/{d_w}}\pE \tilde{p}_{\lfloor tn\rfloor }^\sU(0,[xn^{1/\kappa d_w}])$ (where we write $[ xn^{1/\kappa d_w}]$ for the closest lattice point to $xn^{1/\kappa d_w}$) can be written as $\pE p_{t}^\sT(\rho_\sT,\phi_\sT^{-1}(x))$, or indeed that this latter expectation is well-defined.
This being the case, the following result is presented in terms of the density of the embedded process $\phi_\sT(X^\sT)$, where $X^\sT$ is the canonical Brownian motion on the limiting space; we note $\phi_\sT(X^\sT)$ is not Markov under the annealed law (or strong Markov under the quenched law, see Remark \ref{zqrem} below).
Nevertheless, as we will show, $\phi_\sT^{-1}$ is well defined except on a set of Lebesgue
measure zero, and so the averaged density of $\phi_\sT(X^\sT)$
is in fact given by the expression
$\pE p_{t}^\sT(\rho_\sT,\phi_\sT^{-1}(x))$.
The key additional input to the proof of this result is an equicontinuity property for the discrete heat kernel under scaling (see Proposition \ref{equicont}), which in turn depends on our estimate for the probability of seeing long paths in the uniform spanning tree (see Theorem \ref{T:dxy}).
\begin{thm}\label{denslimit}
For each $t\in(0,\infty)$, $\phi_\mathcal{T}(X^\mathcal{T}_t)$ admits a continuous probability density
${q}_t=({q}_t(x))_{x\in\mathbb{R}^2}$ under the annealed probability law ${\mathbf P}} \def\pE{{\mathbf E} \cdot P^\sU_0$, so that
\[\pE\left(P^\sU_0( \phi_\mathcal{T}(X^\mathcal{T}_t) \in B )\right) = \int_B q_t(x) dx\]
for all Borel $B \subseteq \bR^2$. The functions $({q}_t)_{t>0}$ satisfy the following.\\
(a) There exists a constant $C \in(0,\infty)$ such that
\[ |{q}_t(x)-{q}_t(y)| \leq C t^{-{d_f}/2{d_w}} |x-y|^{\kappa/2},\qquad \forall x,y\in \bR^2, t >0.\]
(b) For any $\lambda>0$ and $\theta\in[0,2\pi)$, it holds that
\[\left(\lambda^{d_f/d_w}q_{ t\lambda}\left(\lambda^{\frac{1}{\kappa d_w}}R_\theta x\right)\right)_{x\in\mathbb{R}^2}=\left(q_{ t}(x)\right)_{x\in\mathbb{R}^2}.\]
(c) For each $t\in(0,\infty)$, it holds that
\[\left(n^{{d_f}/{d_w}}\pE \tilde{p}_{\lfloor tn\rfloor }^\sU(0,[ x n^{\frac{1}{\kappa d_w}}])\right)_{x\in\mathbb{R}^2} \rightarrow
\left(q_t(x)\right)_{x\in\mathbb{R}^2}\]
uniformly on compact subsets of $\mathbb{R}^2$.\\
(d) For each $t\in(0,\infty)$ and $x\in\mathbb{R}^2$, it holds that
\[q_t(x)=\pE\left( p_{t}^\sT(\rho_\sT,\phi_\sT^{-1}(x))\right).\]
\end{thm}
\medskip
\begin{remark}
{\rm By (b) there exists a continuous function $f: \bR_+ \rightarrow \bR_+$ such that
\[ q_t(x)= t^{-{d_f}/{d_w}} f( |x| t^{-1/\kappa {d_w}} ). \]
}\end{remark}
\medskip
As an almost immediate corollary of Theorems \ref{mainthm3} and \ref{denslimit}, we obtain the following.
\begin{cor}\label{cor2} There exist constants $c_1,c_2,c_3,c_4\in (0,\infty)$ and $\theta_1,\theta_2\in(0,1)$ such that the averaged density of $\phi^\mathcal{T}(X^\mathcal{T}_t)$, as given by Theorem \ref{denslimit}, satisfies: for every $x\in\mathbb{R}^2$ and $t>0$,
\[ c_1 t^{-{d_f}/{d_w}} \exp\left\{-c_2 \left(\frac{|x|^{\kappa {d_w}}}{t}\right)^{\frac{\theta_1}{{d_w}-1}} \right\}\leq
{q}_t(x) \le c_3 t^{-{d_f}/{d_w}}\exp\left\{-c_4 \left(\frac{|x|^{\kappa {d_w}}}{t}\right)^{\frac{\theta_2}{{d_w}-1}} \right\}. \]
\end{cor}
\medskip
As a further consequence of Theorem \ref{mainthm3}, together with the
the estimate on the probability of seeing long paths of Theorem \ref{T:dxy}, we
obtain the following upper bounds on
the averaged behaviour of the distance travelled by $X^\sU$ up to a given time,
both in terms of the Euclidean and the intrinsic distances.
Bounds for $\pE \big( E^\sU_0 \big ( {d_\sU} (0,X^\sU_n )^p \big)$ were considered
in \cite[Theorem 4.6]{BM11}, but the upper bound there has an additional
term $(\log n)^{c p}$.
\begin{cor}\label{cor-dist} For every $p>0$, it holds that for $n \ge 1$,
\begin{align*}
& c'_p n^{p/{\kappa d_w}} \le \pE \Big( E^\sU_0 \big |X^\sU_n \big |^p \Big) \le c_p n^{p/{\kappa d_w}} , \\
& c'_p n^{p/{d_w}} \le \pE \Big( E^\sU_0 \big ( {d_\sU} (0,X^\sU_n ) ^p \Big) \le c_p n^{p/{ d_w}} .
\end{align*}
\end{cor}
It follows from the argument used to establish the random walk convergence result of \cite[Theorem 1.4]{BCK} that, under the averaged distribution, not only do we have \eqref{rwconvh}, but also $(n^{-1/{\kappa d_w}}|X^\sU_{\lfloor tn\rfloor}|)_{t\geq0}\buildrel{d}\over{\rightarrow}(|\phi_\sT(X^\sT_{t})|)_{t\geq0}$ and $(n^{-1/{d_w}}d_\mathcal{U}(0,X^\sU_{\lfloor tn\rfloor}))_{t\geq0}\buildrel{d}\over{\rightarrow}(d_\mathcal{T}(\rho_\sT,X^\sT_{t}))_{t\geq0}$.
Combining this with the integrability given by Corollary \ref{cor-dist} we obtain the following convergence
result.
\begin{cor}\label{cor3} (a) For every $p>0$, it holds that
\[\left(n^{-p/{\kappa d_w}} \pE \left( E^\sU_0 \left|X^\sU_{\lfloor tn\rfloor }\right|^p\right)\right)_{t\geq 0}\rightarrow
\left(\pE \left( E^\sT_{\rho_\sT} \left(\left|\phi_\sT(X^\sT_{t})\right|^p\right)\right)\right)_{t\geq 0}=\left(C_pt^{p/\kappa d_w}\right)_{t\geq 0}\]
with respect to the topology of uniform convergence on compact subsets of $[0,\infty)$, where $C_p\in(0,\infty)$ is a constant depending only upon $p$.\\
(b) For every $p>0$, it holds that
\[\left(n^{-p/{d_w}} \pE \left( E^\sU_0 \left(d_\mathcal{U}\left(0,X^\sU_{\lfloor tn\rfloor }\right)^p\right)\right)\right)_{t\geq 0}\rightarrow
\left(\pE \left( E^\sT_{\rho_\sT} \left(d_\mathcal{T}\left(\rho_\sT,X^\sT_{t}\right)^p\right)\right)\right)_{t\geq 0}=\left(C_pt^{p/d_w}\right)_{t\geq 0}\]
with respect to the topology of uniform convergence on compact subsets of $[0,\infty)$, where again $C_p\in(0,\infty)$ is a constant depending only upon $p$.
\end{cor}
\medskip
The remainder of the article is organised as follows. In Section \ref{sec:LERW} we review and refine some previous estimates for LERWs and the two-dimensional UST, proving the upper bound of Theorem \ref{T:dxy}(a) and Theorem \ref{T:dxy}(b) in particular. Section \ref{sec:control} provides an approach to showing that particular anomalous paths occur within the UST. This allows us to check the remaining part of Theorem \ref{T:dxy}, as well as the volume and heat kernel fluctuation results of Theorem \ref{mainthm1} and Corollary \ref{cor1} respectively, which will be done in Section \ref{sec:fluct}. Section \ref{sec:VRest} adapts results of \cite{BCK} concerning the structure of the UST to the case where we condition on a particular path being present, and these preliminary statements are then applied in Section \ref{sec:ann-b} to deduce the heat kernel bounds of Theorem \ref{mainthm3}. Then, in Section \ref{failEHI}, we confirm the failure of the elliptic Harnack inequality, as stated in Corollary \ref{ehicor}. And, in Section \ref{sec:scaling}, we apply the random walk scaling limit result of \cite{BCK} in conjunction with the estimates of this article to deduce Theorems \ref{mainthm2} and \ref{denslimit}, as well as Corollaries \ref{cor2} -- \ref{cor3}. Finally, we postpone to the appendix the proofs of some estimates from Section \ref{sec:LERW} that are relatively close variations on the proofs of the corresponding results in \cite{BM10}. NB.\ We will often use a continuous variable in places where a discrete one is required; in this case we implicitly mean that the floor of the relevant variable should be considered.
\section{Loop erased random walk and the UST} \label{sec:LERW}
This section contains some refinements of previous estimates on the geometry of the UST and the behaviour of the LERW. The key input we need for the averaged heat kernel upper bound (Proposition \ref{P:PF1n}) is a relatively straightforward adaptation of \cite[Proposition 2.10]{BCK}, adding resistance estimates to the volume estimates of the latter result. We also set out some new results, which include the upper bounds of Theorem \ref{T:dxy}.
We begin by introducing some notation for paths and operations on paths. A \emph{path} $\gam$ is a (finite or infinite) sequence of adjacent vertices in $\bZ^2$, i.e.\ $\gam=(\gam_0, \gam_1, \dots )$ with $\gam_{i-1}\sim \gam_i$, where for $x,y\in \mathbb{Z}^2$ we write $x\sim y$ if $|x-y|=1$. Given a set $A \subseteq \bZ^2$, we define $\tau_A = \min\{i \ge 0:\gam_i \notin A \}$, and set $\sE_A (\gam) = (\gam_0, \dots, \gam_{\tau_A})$. Given a finite path $\gam$, we write $\sL (\gam)$ for the chronological loop erasure of $\gam$, see \cite{La1, Law99}.
We now recall Wilson's algorithm, see \cite{Wil}. For $x \in \bZ^2$ let $S^x$ be a simple random walk (SRW) on $\bZ^2$ started at $x$; we take $(S^x)_{x\in\mathbb{Z}^2}$ to be independent. Write $\bZ^2$ as a sequence $\{ z_0,z_1, z_2 , \dots\}$, and define a sequence of trees as follows:
\begin{align}
\sU_0 &= \{ z_0\},\nonumber \\
\sU_i &= \sU_{i-1} \cup \mathcal{L}(\sE_{\sU_{i-1}^c}(S^{z_{i}})), \quad} \def\qq{\qquad i \ge 1, \label{wilsonalg}\\
\sU &= \cup_i \sU_i.\nonumber
\end{align}
By \cite{Wil}, the random tree $\sU$ has the law of the UST. It follows that the law of $\sU$ does not depend on the particular sequence $(z_i)$. In fact, more is true: the $z_i$ can be chosen adaptively as a function of $\sU_{i-1}$. However, if we use independent $(S^x)$ as above, then the final tree $\sU$ depends as a random variable on both the random walks and the sequence $(z_i)$. To circumvent this, for a finite graph Wilson \cite{Wil} defined
a family of random variables (called `stacks') that enable one to define (non-independent) random walks $(\widetilde S^x)_{x\in\mathbb{Z}^2}$, and in this setup the final tree $\sU$ does not depend on $(z_i)$ for $i \ge 1$.
(Note though that the random walk $\widetilde S^{(z_n)}$ does depend on the sequence
$(z_1, \dots z_{n-1})$.) It is straightforward to check that this also holds with probability 1 for a recurrent graph, and it will sometimes be useful for us to apply this construction.
We write $L(x,\infty)$ for the loop-erased random walk from $x$ to infinity; this is the weak limit as $m \rightarrow \infty$ of $\sL(\sE_{B_m(x)}(S^x))$. By Wilson's algorithm $L(x,\infty)$ has the same law as the $\gam(x, \infty)$, the unique injective path from $x$ to infinity in $\sU$. (See \cite[Proposition 14.1]{BLPS}, for example.) We moreover write $\gam_x = \gam(x,\infty)$, $\gam_x[i]$ for the $i$th point on $\gam_x$, define the segment of the path $\gam_x$ between its $i$th and $j$th points by $\gam_x[i,j]= (\gam_x[i], \gam_x[i+1], \dots, \gam_x[j])$, and define $\gam_x[i, \infty)$ in a similar fashion. Furthermore, we let $\tau_{y,r}(\gam_x) = \min\{i\geq 0: \gam_x[i] \not\in B_\infty(y,r)\}$; whenever we use notation such as $\gam_x[\tau_{y,r}]$, the exit time $\tau_{y,r}$ will always be for the path $\gam_x$. For $x,y\in \mathbb{Z}^2$, we introduce the `Schramm distance' on $\sU$ (after \cite{Schramm}) by setting
\[{d_\sU^\sS}(x,y):=\mathrm{diam}(\gam(x,y)), \]
where the right-hand side is the diameter of $\gam(x,y)$ (considered as a subset of $\mathbb{Z}^2$) with respect to $d_{\infty}$.
In the next sequence of results of this section we collect and refine some properties of loop-erased random walks from \cite{BCK, BM10, BM11, Mas09}. In the following result, we write $\partial B_{\infty}(0,r)$ for the outer boundary of $ B_{\infty}(0,r)$, i.e.\ those vertices of $\mathbb{Z}^2\backslash B_{\infty}(0,r)$ that have as a neighbour a vertex in $ B_{\infty}(0,r)$.
\begin{lemma} \label{L:abscty} Let $\theta} \def\Th{\Theta>1$, $n\ge 1$, and suppose that $D_1$, $D_2$ are subsets of $\bZ^2$ with ${B_\infty}(0,\theta} \def\Th{\Theta n) \subseteq D_1 \cap D_2$. There exists a constant $c_1=c_1(\theta)$ such that if $\gam$ is a self-avoiding path from $0$ to $\partial {B_\infty}(0,n)$ then
\[ {\mathbf P}} \def\pE{{\mathbf E}\left(\sE_{{B_\infty}(0,n)}( \sL( \sE_{D_1}(S^0))) = \gam\right) \le c_1 {\mathbf P}} \def\pE{{\mathbf E}\left( \sE_{{B_\infty}(0,n)} (\sL( \sE_{D_2}(S^0))) =\gam\right). \]
If for $i=1$ or $i=2$ one has $D_i=\bZ^d$, then $\sL( \sE_{D_i}(S^0))$ should be taken to be $L(0,\infty)$.
\end{lemma}
{ \sm {\em Proof. }} See \cite[Proposition 4.4]{Mas09} for the result when $\theta} \def\Th{\Theta\ge 4$. Checking the proof of the latter result, one finds that the result as stated above holds for any $\theta} \def\Th{\Theta>1$. (In \cite{Mas09} the emphasis was on the fact that one can take $C(\theta) = 1 + c (\log \theta} \def\Th{\Theta)^{-1}$ for large $\theta} \def\Th{\Theta$.)
{\hfill $\square$ \bigskip}
\begin{definition}\label{regdef}
{\rm Let $D\subseteq \bZ^2$. Let $\lambda} \def\Lam {\Lambda>1$, $1 \le r_1 \le r_2$.
We say that $D$ is {\em $(\lambda} \def\Lam {\Lambda, r_1, r_2)$-regular} if we have for each $x,y\in D$,
\begin{align*}
\lambda} \def\Lam {\Lambda^{-1} {d_\sU^\sS}(x,y)^\kappa\le {d_\sU}(x,y) \le\lambda} \def\Lam {\Lambda {d_\sU^\sS}(x,y)^\kappa,&\qquad\hbox{when }r_1 \le {d_\sU^\sS}(x,y) \le r_2,\\
{d_\sU}(x,y) \le \lambda} \def\Lam {\Lambda r_1^\kappa,&\qquad\hbox{when }{d_\sU^\sS}(x,y) \le r_1,\\
{d_\sU}(x,y) \ge \lambda} \def\Lam {\Lambda^{-1} r_2^\kappa,&\qquad\hbox{when }{d_\sU^\sS}(x,y) \ge r_2.
\end{align*}
}\end{definition}
It is straightforward to check the following.
\begin{lemma}
(a) If $D$ is $(\lambda} \def\Lam {\Lambda, r_1, r_2)$-regular and $r_2 \ge \lambda} \def\Lam {\Lambda^{2/\kappa} r_1$, then
\begin{align*}
\lambda} \def\Lam {\Lambda^{-1} {d_\sU}(x,y) \le {d_\sU^\sS}(x,y)^\kappa \le \lambda} \def\Lam {\Lambda {d_\sU}(x,y)^\kappa,
&\qquad\hbox{when } \lambda} \def\Lam {\Lambda r_1^\kappa\leq {d_\sU}(x,y) \leq \lambda} \def\Lam {\Lambda^{-1} r_2^\kappa,\\
{d_\sU^\sS}(x,y)^\kappa \le \lambda} \def\Lam {\Lambda r_1^\kappa, &\qquad\hbox{when } {d_\sU}(x,y) \le \lambda} \def\Lam {\Lambda r_1^\kappa, \\
{d_\sU^\sS}(x,y) \ge \lambda} \def\Lam {\Lambda^{-1} r_2^\kappa, &\qquad\hbox{when } {d_\sU}(x,y) \ge \lambda} \def\Lam {\Lambda^{-1} r_2^\kappa.
\end{align*}
(b) Let $\sT \subset \bZ^2$ be a tree, $w \not\in \sT$, $\gam(w,\sT)$ be a self-avoiding path from $w$ to $\sT$, and let $\sT'= \sT \cup \gam(w,\sT)$. If $\sT$ and $\gam(w,\sT)$ are $(\lambda} \def\Lam {\Lambda, r_1, r_2)$-regular, then $\sT'$ is $( 2^\kappa \lambda} \def\Lam {\Lambda, 2r_1, r_2)$-regular.
\end{lemma}
The next result is a consequence of \cite[Proposition 2.8]{BCK}.
\begin{lemma} \label{L:regbox}
Let $n\ge1$ and $\lambda} \def\Lam {\Lambda \ge \lambda} \def\Lam {\Lambda_0$. Then
\[ {\mathbf P}} \def\bE{{\mathbf E}\left( B_\infty(0,n) \hbox{ is $(\lambda} \def\Lam {\Lambda, e^{-c_1 \lambda} \def\Lam {\Lambda^{1/2} } n,n)$-regular}\right) \ge 1 - c_2 e^{-c_3 \lambda} \def\Lam {\Lambda^{1/2} }. \]
\end{lemma}
\begin{lemma} \label{L:regpath} Let $\theta} \def\Th{\Theta >1$,
$n\ge 1$, and suppose $D \subseteq \bZ^2$ is such that ${B_\infty}(0,\theta} \def\Th{\Theta n) \subseteq D$. It then holds that there exist constants $c_i=c_i(\theta)$ such that for $\lambda} \def\Lam {\Lambda>\lambda} \def\Lam {\Lambda_0$, where $\lambda} \def\Lam {\Lambda_0$ is some large, finite constant,
$$ {\mathbf P}} \def\pE{{\mathbf E}\big( \sE_{{B_\infty}(0,n)} ( \sL ( \sE_D (S^0))) \hbox{ is $(\lambda} \def\Lam {\Lambda, ne^{-c_1 \lambda} \def\Lam {\Lambda^{1/2}},n)$-regular} \big)
\ge 1 - c_2 e^{-c_3 \lambda} \def\Lam {\Lambda^{1/2} }. $$
\end{lemma}
{ \sm {\em Proof. }} By Lemma \ref{L:abscty} it is enough to prove this when $ \sL ( \sE_D (S^0))$ is replaced by $L(0,\infty)$. The bound then follows from Lemma \ref{L:regbox} and Wilson's algorithm. {\hfill $\square$ \bigskip}
\begin{lemma} \label{L:gam0y}
There exists $\lambda} \def\Lam {\Lambda_0 \ge 1$ and constant $c_1>0$ with the following properties. Let $n\ge 1$, $\lambda} \def\Lam {\Lambda \ge \lambda} \def\Lam {\Lambda_0$ and $x \in B_\infty(0, 3n/4)$, let $\pi$ be a shortest path in $\bZ^2$ between $0$ and $x$, and set $A =\{ y \in \bZ^2: {d_\infty}(y,\pi) \le n/8 \}$. Then
$$ {\mathbf P}} \def\pE{{\mathbf E}\big( \gam(0,x) \subseteq A \hbox{ and is $(\lambda} \def\Lam {\Lambda, ne^{-c_1 \lambda} \def\Lam {\Lambda^{1/2}},n)$-regular} \big) \ge c_1. $$
\end{lemma}
{ \sm {\em Proof. }} Let $G_1$ be the event that $B_\infty(0,n)$ is $(\lambda} \def\Lam {\Lambda, ne^{-c_1 \lambda} \def\Lam {\Lambda^{1/2}},n)$-regular. Choose $k \ge 16$ and let $y$ satisfy $d_\infty(0,y) = n/k$. Let $G_2 = \{ \gam(0,y) \subseteq {B_\infty}(0, n/8) \}$; by \cite[Lemma 2.6]{BM11} we have ${\mathbf P}} \def\pE{{\mathbf E}( G_2^c )\le c_2 k^{-1/3}$; choose $k$ so that $c_2 k^{-1/3} \le {\tfrac12}$. Let $S^x$ be a SRW started at $x$, and $G_4$ be the event that $S^x$ makes a closed loop around 0 which separates $0$ and $y$ before it leaves $A$; we have ${\mathbf P}} \def\bE{{\mathbf E}(G_4) \ge c_3 >0$. Then ${\mathbf P}} \def\bE{{\mathbf E}( G_2 \cap G_4) \ge {\tfrac12} c_3$. We now choose $\lambda} \def\Lam {\Lambda_0$ large enough so that ${\mathbf P}} \def\bE{{\mathbf E}( G_1^c) \le \frac14 c_3$ and hence writing $G= G_1 \cap G_2 \cap G_4$ we have ${\mathbf P}} \def\bE{{\mathbf E}( G) \ge \frac14 c_3$. On the event $G$ the SRW $S^x$ hits $\gam(0,y)$ before it exits $A$, so $\gam(0,x) \subseteq A$. Since $B_\infty(0,n)$ is regular, so is the path $\gam(0,x)$. {\hfill $\square$ \bigskip}
\begin{thm} \label{T:LERW-lb}
Let $n\ge 1$, and suppose $D \subseteq \bZ^2$ is such that ${B_\infty}(0, n) \subseteq D$. If $D \neq \bZ^2$, set $L_{n,D}= \sE_{{B_\infty}(0,n)} (\sL(\sE_{D}(S^0)))$, and set $L_{n,\bZ^2}= \sE_{{B_\infty}(0,n)} (L(0,\infty))$. It then holds that there exist constants $c_i$ such that, for $\lambda} \def\Lam {\Lambda\geq 1$,
\begin{equation} \label{e:LERW-lb}
{\mathbf P}} \def\pE{{\mathbf E}\left( | L_{n,D}| < \lambda} \def\Lam {\Lambda^{-1} n^\kappa \right) \le c_1 e^{ -c_2 \lambda} \def\Lam {\Lambda^{1/(\kappa-1)} }.
\end{equation}
In particular, for any $x,y\in\mathbb{Z}^2$,
\begin{equation} \label{e:duxy-lb}
{\mathbf P}} \def\pE{{\mathbf E}\left( {d_\sU}(x,y) < \lambda} \def\Lam {\Lambda^{-1} d_\infty(x,y)^\kappa \right) \le c_1 e^{- c_2 \lambda} \def\Lam {\Lambda^{1/(\kappa-1)} }.
\end{equation}
\end{thm}
{ \sm {\em Proof. }} A bound with exponent $\lambda} \def\Lam {\Lambda^{4/5 - \eps}$ is given in \cite[Theorem 6.7]{BM10}, and with some more care one can obtain \eqref{e:LERW-lb} by essentially the same arguments -- see the Appendix for details. Taking $x=0$, $D= \bZ^2\backslash\{y\}$ and $n =\lfloor d_\infty(0,y)\rfloor$, we have $ {d_\sU}(x,y) = | \sL(\sE_{D}(S^0))| \ge |L_{n,D}|$, which (with translation invariance) gives \eqref{e:duxy-lb}.
{\hfill $\square$ \bigskip}
To state our next result, Proposition \ref{P:PF1n}, we need to introduce some more notation and basic definitions. Specifically, we write $R_{\rm eff}$ for the effective resistance on $\sU$ considered as an electrical network with unit conductances along each edge. (See \cite{Bar17, LP} for background.) We recall from \eqref{dfdef} and \eqref{dwdef} the definition of ${d_f}$ and ${d_w}$.
\begin{definition}\label{def:defjlam}
{\rm We say a ball $B_\sU(x,r)$ is $\lambda} \def\Lam {\Lambda$-good if we have the following:\\
(1) $\lambda} \def\Lam {\Lambda^{-1} r^{d_f} \le |B_\sU(x,r) | \le \lambda} \def\Lam {\Lambda r^{d_f}$, \\
(2) $R_{\rm eff}(x, B_\sU(x,r)^c) \ge r/\lambda} \def\Lam {\Lambda$, \\
(3) $B_\sU(x,r) \subseteq {B_\infty}(x, \lambda} \def\Lam {\Lambda r^{1/\kappa})$.}\end{definition}
\noindent
We moreover define
\begin{equation}\label{f1def}
F_1(\lambda} \def\Lam {\Lambda,n) =\{ B_\sU(x,r) \hbox{ is $\lambda} \def\Lam {\Lambda$-good for all } x \in {B_\infty}(0,n), \, e^{-\lambda} \def\Lam {\Lambda^{1/40}} n^\kappa \le r \le n^\kappa\},
\end{equation}
and note that on the event $F_1(\lambda} \def\Lam {\Lambda,n)$ we have $B_\sU(x,n^\kappa) \subseteq {B_\infty}(x,\lambda} \def\Lam {\Lambda n)$ for all $x \in {B_\infty}(0,n)$.
\begin{propn}\label{P:PF1n}
There exist constants $c_1,c_2,\lambda_0$ such that
\[{\mathbf P}} \def\pE{{\mathbf E}\left(F_1(\lambda, n)^c\right) \le c_1\exp (-c_2\lambda^{1/16}),\qquad \forall n\ge 1,\:\lambda\ge \lambda} \def\Lam {\Lambda_0.\]
\end{propn}
{ \sm {\em Proof. }}
The proof below is a modification of that of \cite[Proposition 2.10]{BCK}. Let $r=ne^{-\lambda} \def\Lam {\Lambda^{1/32}}$, and assume first that $n$ is large enough so that $r\ge \lambda} \def\Lam {\Lambda$. Let $J(x,\lambda)$ be the set of those $r\in [1,\infty) $ such that the three conditions in Definition \ref{def:defjlam} hold.
Set $R_1=n$, $R_2= r e^{\lambda} \def\Lam {\Lambda^{1/16}}$, and let $D_2$ be as in \cite[Proposition 2.9]{BCK}, with $|D_2 | \leq c\lambda} \def\Lam {\Lambda^4e^{2\lambda} \def\Lam {\Lambda^{1/16}}$. Set $m_0:=\inf\{m: m \ge e^{\kappa\lambda} \def\Lam {\Lambda^{1/32}} \}$, and let $ E(r,\lambda} \def\Lam {\Lambda):=\cap_{x\in D_2}\cap_{m=1}^{m_0+1} \{ m r^{\kappa}\in J(x,\lambda} \def\Lam {\Lambda)\}$. A simple union bound allows us to deduce from \cite[Theorem 1.1(a) and Proposition 4.2(a)]{BM11} that
\[\mathbf{P}\left(E(r,\lambda} \def\Lam {\Lambda)^c\right) \le |D_2| e^{\kappa\lambda} \def\Lam {\Lambda^{1/32}} c e^{-c'\lambda} \def\Lam {\Lambda^{1/9}}\le C e^{-c'' \lambda} \def\Lam {\Lambda^{1/9}}.\]
Let $A_5(r,\lambda} \def\Lam {\Lambda)$ be the event given in the statement of \cite[Proposition 2.9]{BCK}; we have $\mathbf{P}( E(r,\lambda} \def\Lam {\Lambda)^c \cup A_5(r,\lambda} \def\Lam {\Lambda)^c) \le C \exp(-c \lambda} \def\Lam {\Lambda^{1/16} )$. Moreover, if $E(r,\lambda} \def\Lam {\Lambda)\cap A_5(r,\lambda} \def\Lam {\Lambda)$ holds, then, by \cite[(2.14) and the last display in the proof of Proposition 2.9]{BCK}, for each $x \in {B_\infty}(0,n)$, there exists $y=y_x \in D_2$ with $d_\mathcal{U}(x,y)\leq 4r^{\kappa}/\lambda} \def\Lam {\Lambda^{1/4}$ and $d_\infty(x,y)\le 2r/\lambda} \def\Lam {\Lambda$. Choosing $\lambda} \def\Lam {\Lambda_0$ large enough, we have $d_\mathcal{U}(x,y_x)\leq r^{\kappa}$ and $d_{\infty}(x,y_x) \le r$.
Now, suppose $E(r,\lambda} \def\Lam {\Lambda)\cap A_5(r,\lambda} \def\Lam {\Lambda)$ holds, and let $x\in {B_\infty}(0,n)$, and $s \in [ 4 \lambda} \def\Lam {\Lambda r^\kappa , n^\kappa]$. We will prove that $s \in J(x,2\lambda} \def\Lam {\Lambda)$ by verifying the conditions (1)--(3) in Definition \ref{def:defjlam}. Choose $m \in[4\lambda} \def\Lam {\Lambda,m_0+1]$ so that $(m-1) r^\kappa \le s \le mr^\kappa$. It then holds that
\[|B_\sU \left(x,s \right)| \le |B_\sU(y_x, (m+1)r^{\kappa})| \le \lambda} \def\Lam {\Lambda ((m+1)/(m-1))^{d_f} s^{d_f} \le 2\lambda} \def\Lam {\Lambda s^{d_f}. \]
Similarly, $|B_\sU(x,s^{\kappa})|\geq (2 \lambda} \def\Lam {\Lambda)^{-1} s^{d_f}$, so that the volume bound (1) holds. Next, applying the triangle inequality for resistances (and the fact that $R_{\rm eff}(x,y_x)=d_\sU(x,y_x)$),
\begin{align*}
R_{\rm{eff}} \left(x,B_\sU(x,s)^c \right)
&\geq R_{\rm{eff}}\left(y_x ,B_\sU(x,s)^c \right) - d_\sU(x,y_x) \\
&\geq R_{\rm{eff}} (y_x , B_\sU(y_x, s - r^\kappa)^c ) - r^\kappa\\
&\ge \lambda} \def\Lam {\Lambda^{-1} (m-2) r^\kappa - r^\kappa \geq (2 \lambda} \def\Lam {\Lambda)^{-1} s,
\end{align*}
which gives (2). Finally for (3) we have
\[B_\sU(x,s) \subseteq B_\sU\left(y_x, (m+1) r^{\kappa}\right) \subseteq {B_\infty}(y_x, {\lambda (m+1)^{1/\kappa} r} ) \subseteq {B_\infty}(x,{2\lambda} \def\Lam {\Lambda s}),\]
where the last inclusion holds since $r ( 1+ \lambda} \def\Lam {\Lambda (m+1)^{1/\kappa} ) \le 2 \lambda} \def\Lam {\Lambda (m-1)^{1/\kappa} r \le 2\lambda} \def\Lam {\Lambda s$. Thus, for $\lambda} \def\Lam {\Lambda\geq\lambda} \def\Lam {\Lambda_0$ with $\lambda} \def\Lam {\Lambda_0$ suitably large, $E(r,\lambda} \def\Lam {\Lambda)\cap A_5(r,\lambda} \def\Lam {\Lambda) \subseteq F_1(2 \lambda,n)$, and this completes the proof of the proposition in the case when $r \ge \lambda} \def\Lam {\Lambda$.
Finally, suppose $r =ne^{-\lambda} \def\Lam {\Lambda^{1/32}}<\lambda} \def\Lam {\Lambda$, i.e.\ $n\leq \lambda} \def\Lam {\Lambda e^{\lambda} \def\Lam {\Lambda^{1/32}}$. A union bound then gives
\[{\mathbf P}} \def\pE{{\mathbf E}\left(F_1(\lambda, n)^c\right)\leq \sum_{ x \in B_n(0)}\sum_{s=1}^{\lceil n^\kappa\rceil}{\mathbf P}} \def\pE{{\mathbf E}\left(s\notin J(x,\lambda)\right)\leq cn^{2+\kappa}e^{-c'\lambda^{1/9}}\leq Ce^{-c''\lambda^{1/9}},\]
where the last inequality is again an application of \cite[Theorem 1.1(a) and Proposition 4.2(a)]{BM11}. This is enough to complete the proof.
{\hfill $\square$ \bigskip}
A further observation that will be useful in the proof of the averaged heat kernel upper bound is the following.
\begin{lemma} \label{L:F1dist}
Suppose that $F_1(\lambda} \def\Lam {\Lambda,n)$ occurs, and let $x,y \in B_\infty(0,n)$. It then holds that
$d_{\infty}(x,y)\in[\lambda} \def\Lam {\Lambda e^{- \lambda} \def\Lam {\Lambda^{1/40}/\kappa} n,n]$ implies ${d_\sU}(x,y) \ge \lambda} \def\Lam {\Lambda^{-\kappa} d_{\infty}(x,y)^\kappa$.
\end{lemma}
{ \sm {\em Proof. }} Let $r < \lambda} \def\Lam {\Lambda^{-\kappa} d_{\infty}(x,y)^\kappa$, so that $y \not\in B_\infty(x,{\lambda} \def\Lam {\Lambda r^{1/\kappa}})$. The condition on $d_{\infty}(x,y)$ implies that we can choose $r$ so that $r \in [e^{-\lambda} \def\Lam {\Lambda^{1/40}} n^\kappa, n^\kappa]$, and thus property (3) in the definition of a good ball implies that $y \not\in B_\sU(x,r)$, and so ${d_\sU}(x,y) >r$. {\hfill $\square$ \bigskip}
The next few results will lead to the proof of Theorem \ref{T:dxy}(b), beginning with the case when $x$ and $y$ are neighbours in $\mathbb{Z}^2$.
\begin{prop} \label{nnp} There exist constants $c_i,q$ such that
\[c_1s^{-(2-\kappa)/\kappa}\leq {\mathbf P}} \def\pE{{\mathbf E}\left({d_\sU}(0,e_1) \ge s \right) \le c_2 ( \log s )^qs^{-(2-\kappa)/\kappa},\]
for all $s\geq 2$, where $e_1=(1,0)$.
\end{prop}
We start with a proof of the lower bound. For this, it will be convenient to introduce $\sU'$, the dual of $\sU$. This is the graph with vertex set $(\mathbb{Z}+\frac12)^2$ whose edges are precisely those nearest neighbour edges that do not cross an edge of $\sU$. It is known that $\sU'$ has the same law as $\sU$ (see \cite{BLPS}). We set $0'=(1/2,1/2)$ for the root of the dual graph.
\bigskip
\noindent
\emph{Proof of the lower bound of Proposition \ref{nnp}.} Applying \cite[Proposition 2.8]{BCK} and Proposition \ref{P:PF1n}, for $r\geq 1$, $\lambda} \def\Lam {\Lambda\geq \lambda} \def\Lam {\Lambda_0$, we can find an event $G_0(\lambda} \def\Lam {\Lambda,r)$ with ${\mathbf P}} \def\pE{{\mathbf E}( G_0(\lambda} \def\Lam {\Lambda,r)^c ) \le c_1 e^{-c_2 \lambda} \def\Lam {\Lambda^{1/40}}$ such that if this event holds then we have $B_{\infty}(0,R)$ is $(c_3\lambda^{1/20},r,R)$-regular and also $F_1(\lambda} \def\Lam {\Lambda,R)$ holds, where $R:=re^{\lambda} \def\Lam {\Lambda^{1/40}/\kappa}$. Let $G'_0(\lambda} \def\Lam {\Lambda,r)$ be the corresponding event for the dual graph, and define $G(\lambda} \def\Lam {\Lambda,r) = G_0(\lambda} \def\Lam {\Lambda,r) \cap G'_0(\lambda} \def\Lam {\Lambda,r)$.
On $G(\lambda} \def\Lam {\Lambda,r)$, if $\gam'$ is the unique injective path from $0'$ to infinity in $\sU'$, then we have that the section of $\gam'$ from its last exit from $B_{\infty}(0',r)$ to $B_{\infty}(0',2r)^c$ has length greater than $c\lambda^{-1/20}r^\kappa$. (We assume that $\lambda} \def\Lam {\Lambda$ is chosen large enough so that $e^{\lambda} \def\Lam {\Lambda^{1/40}} \ge 2$.) Denote by $\gam'_r$ this section of $\gam'$, and let $\{x_1', x_2'\}$ be an edge crossed by $\gam'_r$. If $\{x_1,x_2\}$ is the dual edge to $\{x_1',x_2'\}$, then it must be the case that ${d^S_\sU}(x_1,x_2) \ge r$, and thus ${d_\sU}(x_1,x_2) \ge c\lambda^{-1/20}r^\kappa$.
Finally, for $x_1\sim x_2$ (in $\bZ^2$), set $F(x_1,x_2) =\{ {d_\sU}(x_1,x_2) \ge c\lambda^{-1/20}r^\kappa\}$. The argument above gives that
$$ \sum_{x_1 \in B_{\infty}(0,2r)} \sum_{x_2 \sim x_1} \mathbf{1}_{ F(x_1,x_2) } \ge \mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r)} \sum_{x_1 \in B_{\infty}(0,2r)} \sum_{x_2 \sim x_1}\mathbf{1}_{ F(x_1,x_2) } \ge c\lambda^{-1/20}r^\kappa \mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r)}. $$
Hence taking expectations
\[\left(1-c_1 e^{-c_2 \lambda} \def\Lam {\Lambda^{1/40}}\right)c\lambda^{-1/20}r^\kappa \leq \sum_{x_1 \in B_{\infty}(0,2r)} \sum_{x_2 \sim x_1} {\mathbf P}} \def\pE{{\mathbf E}( F(x_1,x_2) ) \le c' r^2 {\mathbf P}} \def\pE{{\mathbf E}( F(0,e_1) ),\]
and the result follows by a simple reparameterisation.
{\hfill $\square$ \bigskip}
A similar idea gives an upper bound. We begin by looking at the size of the finite component rooted at a vertex. In particular, for $x \in \bZ^2$, this is defined to be the set $A_x =\{ y : x \in \gam(y,\infty) \}$. We also define the depth of $A_x$ by $ \mathrm{dep}(A_x) = \max\{ {d_\sU}(x,y) : y \in A_x \}$.
\begin{lemma}
For $\lambda} \def\Lam {\Lambda \in \bN$, we have that
\begin{align}
\label{e:Atail-lb1}
{\mathbf P}} \def\pE{{\mathbf E}( \mathrm{dep}(A_0) = \lambda} \def\Lam {\Lambda) &\le c_1 \lambda} \def\Lam {\Lambda^{-2/\kappa}, \\
\label{e:Atail-lb2}
{\mathbf P}} \def\pE{{\mathbf E}( \mathrm{dep}(A_0) \ge \lambda} \def\Lam {\Lambda ) &\le c_2 \lambda^{-(2-\kappa) /\kappa}.
\end{align}
\end{lemma}
{ \sm {\em Proof. }} Suppose that the event $G(\lambda} \def\Lam {\Lambda,r)$ defined in the proof of the lower bound holds, and again set $R:=re^{\lambda} \def\Lam {\Lambda^{1/40}/\kappa}$. NB. We suppose that $\lambda$ is large enough so that $(32\lambda)^{1/\kappa} r\leq R$. Let $x \in {B_\infty}(0,r)$, and suppose that $\mathrm{dep}(A_x) =s$, where $r^\kappa \le s\le (16\lambda} \def\Lam {\Lambda)^{-1} R^\kappa$. We then claim that $A_x \subseteq B_{\infty}(0,R/2)$. Indeed, suppose $y \in A_x \cap {B_\infty}(0,{R/2})^c$, and let $y'$ be the first point on $\gam(y,x)$ which is in $B_\infty(0,R)$. Since $r\leq \frac14 R \le d_\infty(x,y') \le {d^S_\sU}(x,y')$, we must then have $s=\mathrm{dep}(A_x) \ge {d_\sU}(x,y') \ge \min\{\lambda} \def\Lam {\Lambda^{-1} R^\kappa,\lambda} \def\Lam {\Lambda^{-1} {d^S_\sU}(x,y')^\kappa \}\ge \lambda} \def\Lam {\Lambda^{-1} 4^{-\kappa} R^\kappa$, which is a contradiction. Now, there exists $y_x \in A_x$ such that ${d_\sU}(x,y_x) =s$, and it must be the case that $y_x \in B_{\infty}(0,R/2)$. Thus the ball $B_\sU(y_x, s)$ is $\lambda} \def\Lam {\Lambda$-good, and we obtain
\begin{equation} \label{e:lbvolAx}
| A_x| \ge | B_\sU(y_x, s)| \ge \lambda} \def\Lam {\Lambda^{-1} \mathrm{dep}(A_x)^{2/\kappa}.
\end{equation}
Next, let $s \in \bN \cap [r^\kappa, 2r^\kappa]$, and set $\tilde H_s = \{ x \in B_{\infty}(0,r): \mathrm{dep}(A_x) =s \}$.
By \eqref{e:lbvolAx}, we then have that
\begin{equation}\label{sedf}
\mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r) } \sum_{x \in \tilde H_s} |A_x| \ge \lambda} \def\Lam {\Lambda^{-1} s^{2/\kappa} |\tilde H_s| \mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r) } .
\end{equation}
If $x \in \tilde H_s$ and $y \in A_x$, then $y \in B_{\infty}(0,R/2)$ and ${d_\sU}(x,y) \le s$. Hence $d_\infty(x,y)^\kappa \le {d^S_\sU}(x,y)^\kappa \le \max\{\lambda} \def\Lam {\Lambda r^\kappa,\lambda} \def\Lam {\Lambda {d_\sU}(x,y)\} \le 2\lambda} \def\Lam {\Lambda r^\kappa$, and so $ y \in B_{\infty}(0,(1+(2\lambda} \def\Lam {\Lambda)^{1/\kappa})r)$. Since the sets $(A_x)_{x \in \tilde H_s}$ are disjoint, it follows that $\mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r) } \sum_{x \in \tilde H_s} |A_x| \le c \lambda} \def\Lam {\Lambda^{2/\kappa} r^2 \mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r) }$, and combining this with the estimate at \eqref{sedf} yields
\begin{equation}\label{lamest}
|\tilde H_s|\mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r) } \le c\lambda} \def\Lam {\Lambda^{1+2/\kappa} \mathbf{1}_{G(\lambda} \def\Lam {\Lambda,r) }.
\end{equation}
Finally, let $\Lambda_*=\inf\{\lambda\geq \lambda_0:\:G(\lambda} \def\Lam {\Lambda,r)\mbox{ holds}\}$, and note that ${\mathbf P}} \def\pE{{\mathbf E}(\Lambda_*>\lambda} \def\Lam {\Lambda)\leq {\mathbf P}} \def\pE{{\mathbf E}(G(\lambda} \def\Lam {\Lambda,r)^c)\leq c_1e^{-c_2\lambda} \def\Lam {\Lambda^{1/40}}$. Thus $\Lambda_*$ is almost-surely finite,
and there exist finite constants $c_p$ such that $\bE (\Lam_*^p) \le c_p$; note that these constants
can be chosen not to depend on $r$.
Hence, from \eqref{lamest}, we deduce that, for $s\in \bN \cap [r^\kappa, 2r^\kappa]$,
\[{\mathbf P}} \def\pE{{\mathbf E}( \mathrm{dep}(A_0)=s)\leq r^{-2}|B_\infty(0,r)|{\mathbf P}} \def\pE{{\mathbf E}( \mathrm{dep}(A_0)=s)=r^{-2}\mathbf{E}(|\tilde H_s|)\leq cr^{-2}\mathbf{E}(\Lam_*^{1+2/\kappa})\leq cs^{-2/\kappa}.\]
This gives the bound \eqref{e:Atail-lb1}, and the bound \eqref{e:Atail-lb2} readily follows. {\hfill $\square$ \bigskip}
\noindent
\emph{Proof of the upper bound of Proposition \ref{nnp}.}
Again, suppose that the event $G(\lambda} \def\Lam {\Lambda,r)$ defined in the proof of the lower bound holds, and $\lambda$ is chosen large enough so that $2\leq e^{\lambda} \def\Lam {\Lambda^{1/40}/\kappa}$. Suppose that ${d_\sU}(0,e_1) >2^\kappa c_3 \lambda^{1/20} r^\kappa$ (where $c_3$ is as in the definition of the aforementioned event). Note that the dual vertices enclosed by the path $\gam(0,e_1)$ (combined with the edge from $0$ to $e_1$) are all elements of the finite component of $\sU'$ rooted at $0'$, which we denote $A'_{0'}$. On $G(\lambda} \def\Lam {\Lambda,r)$, we have ${d^S_\sU}(0,e_1) \ge 2r$, and so there exists a point $x_1 \in \gam(0,e_1)$ such that $d_\infty(0,x_1) \ge 2r$. Let $x_2$ be a point on $\gam(0,e_1)$ adjacent to $x_1$, and let $\{x'_1, x'_2\}$ be the edge dual to $\{x_1, x_2\}$. One of the points $x'_1$, $x_2'$ is in $A'_{0'}$; we call this point $x'$. Since $d_{\sU'}^{\mathcal S}} \def\sT {{\mathcal T}} \def\sU {{\mathcal U}(0',x') \ge r$ and $G(\lambda} \def\Lam {\Lambda,r)$ holds, we thus obtain that $\mathrm{dep}(A'_{0'} ) \ge c_3^{-1}\lambda^{-1/20}r^\kappa$. Hence $ G(\lambda} \def\Lam {\Lambda,r)\cap \{ {d_\sU}(0,e_1) \ge 2^\kappa c_3 \lambda^{1/20} r^\kappa \} \subseteq G(\lambda} \def\Lam {\Lambda,r) \cap \{ \mathrm{dep}(A'_{0'} ) \ge c_3^{-1}\lambda^{-1/20}r^\kappa \}$, and so, setting $\tilde{r}:=c_3^{-1}\lambda^{-1/20}r^\kappa$ and $\lambda=c'(\log\tilde{r})^{40}$,
\begin{eqnarray*}
{\mathbf P}} \def\pE{{\mathbf E}( {d_\sU}(0,e_1) \ge\tilde{r} ) &\le& {\mathbf P}} \def\pE{{\mathbf E}(G(\log\tilde{r},r)^c) + {\mathbf P}} \def\pE{{\mathbf E}( \mathrm{dep}(A'_{0'} ) \ge 2^{-\kappa}c_3^{-2}(c')^{-1/10}(\log\tilde{r})^{-4}\tilde{r}^\kappa )\\
& \le& c_1 e^{-c_2c'\log\tilde{r}} + c \tilde{r}^{-(2-\kappa)/\kappa}(\log\tilde{r})^{4(2-\kappa)/\kappa},
\end{eqnarray*}
which, taking $c'$ suitably large, yields the desired result.
{\hfill $\square$ \bigskip}
\noindent
\emph{Proof of Theorem \ref{T:dxy}(b).} Suppose that the event $G(\lambda} \def\Lam {\Lambda,r)$ defined in the proof of Proposition \ref{nnp} holds, and write $R:=re^{\lambda} \def\Lam {\Lambda^{1/40}/\kappa}$. We will assume that we also have a parameter $t$ that satisfies $4^\kappa c_3\lambda^{1/20}\leq t\leq c_3^{-1}\lambda^{-1/20}e^{\lambda^{1/40}}$. Let $d_\infty(0,x)=r$, and $L$ be a shortest path in $\bZ^2$ between $0$ and $x$. If ${d_\sU}(0,x) \ge tr^\kappa$, then $d_{\sU}^{{\mathcal S}} \def\sT {{\mathcal T}} \def\sU {{\mathcal U}}(0,x)\geq c_3^{-1/\kappa}\lambda^{-1/20\kappa}t^{1/\kappa}r$, and so there exists a point $y$ on $\gam(0,x)$ with $d_\infty(0,y) \ge c_3^{-1/\kappa}\lambda^{-1/20\kappa}t^{1/\kappa}r$. Let $y'$ be a dual point with $d_\infty(y,y')=\frac12$ which is separated from infinity by $\gam(0,x) \cup L$. The path in the dual tree $\gam'(y',\infty)$ must pass through $L$ at a point $z'$, and the length of the section of $\gam'(y',\infty)$ inside $B_\infty(z',r)$ will be of length at least $c_3^{-1}\lambda^{-1/20}r^\kappa$. Moreover, for each vertex $z'$ of $\gam'(y',\infty)$ inside $B_\infty(z',r)$, it must be the case that $A'_{z'}$ has $d_\infty$-diameter greater than $ d_\infty(0,y')-2r$, and so $\mathrm{dep}(A'_{z'})\geq 2^{-\kappa}c_3^{-1}\lambda^{-1/20}tr^\kappa$. Now, let $H=\{ {d_\sU}(0,x) \ge tr^\kappa\}$ and $F(z') =\{\mathrm{dep}(A'_{z'})\geq 2^{-\kappa}c_3^{-1}\lambda^{-1/20}tr^\kappa\}$. We then have that
$$\mathbf{1}_{H\cap G(\lambda} \def\Lam {\Lambda,r)} \sum_{z'\in B_{\infty}(0',2r)}\mathbf{1}_{ F(z') }
\ge \mathbf{1}_{H\cap G(\lambda} \def\Lam {\Lambda,r)} c_3^{-1}\lambda^{-1/20}r^\kappa. $$
So, by \eqref{e:Atail-lb2},
\begin{align*}
{\mathbf P}} \def\pE{{\mathbf E}( H \cap G(\lambda} \def\Lam {\Lambda,r))
&\le c_3 r^{-\kappa}\lambda^{1/20} \bE \left( \mathbf{1}_{H\cap G(\lambda} \def\Lam {\Lambda,r)} \sum_{z'\in B_{\infty}(0',2r)}\mathbf{1}_{ F(z') } \right) \\
&\le c_3 r^{-\kappa} \lambda^{1/20} \sum_{z'\in B_{\infty}(0',2r)} {\mathbf P}} \def\pE{{\mathbf E}(F(z') ) \\
&\le c r^{2-\kappa} \lambda^{1/20} (2^{-\kappa}c_3^{-1}\lambda^{-1/20}tr^\kappa)^{-(2-\kappa)/\kappa}\\
&= c \lambda^{1/10\kappa} t^{-(2-\kappa)/\kappa}.
\end{align*}
Hence, taking $\lambda=(\log{t})^{41}$, the result follows similarly to the end of the previous proof. {\hfill $\square$ \bigskip}
\section{Controlling paths} \label{sec:control}
In this section, we provide a general technique for estimating from below the probability of seeing a particular path configuration in the UST. This will enable us to estimate from below the probability of seeing especially short paths between given points, and so prove the lower bound in Theorem \ref{T:dxy}(a). These estimates will also be a key ingredient in establishing volume and heat kernel fluctuations for the UST, as we do in the subsequent section.
Let $x\in \mathbb{Z}^2$, and $m\geq 1$. A \emph{scale $m$ path} from 0 to $x$, $\pi$ say, is a sequence of distinct vertices $0=x_0,x_1,\dots,x_N=x$ such that $x_i\in (m\mathbb{Z})^2$ and the Euclidean (i.e.\ $\ell^2$) distance between $x_{i-1}$ and $x_i$ is equal to $m$ for each $i=1,\dots,N-1$, and also $x_N\in B_m(x_{N-1})$, where we define
\[B_r(x):=B_\infty(x,r/2).\]
We write $|\pi| = N$ for the length of the path. Now, fix a path $\pi$ of length $N$. The rest of this section is devoted to defining an event ${F}_m(x,\pi)$ with ${\mathbf P}} \def\pE{{\mathbf E}( {F}_m(x,\pi) ) \ge e^{-c N}$ such that on this event the path from 0 to $x$ in the UST is contained in $\cup_{i=0}^NB_m(x_i)$ and (up to constants) has length $Nm^{\kappa}$.
To complete the program described in the previous paragraph, we will again appeal to Wilson's algorithm. As at the start of Section \ref{sec:LERW}, let $(S^x)_{x\in\mathbb{Z}^2}$ be a collection of independent simple random walks on $\mathbb{Z}^2$, where $S^x$ is started from $x$. Slightly modifying the algorithm at \eqref{wilsonalg}, we use these to construct the part of the UST containing both $0$ and $x$ via an iterative procedure. In particular, let $k$ be some integer that will be fixed later. We begin our construction by taking $\mathcal{U}_0=\gamma_0$ to be the loop-erasure of $S^{(0,\lfloor m/k\rfloor)}$ run until it first hits the origin $x_0=0$. We then continue as at \eqref{wilsonalg}: for $i\geq 1$, let $\mathcal{U}_i=\mathcal{U}_{i-1}\cup \gamma _i$, where $\gamma_i$ is the loop-erasure of $S^{x_i}$ run until it first hits $\mathcal{U}_{i-1}$. We will later use the notation $x_i'$ to represent the unique point in $\mathcal{U}_{i-1}\cap \gamma _i$. From Wilson's algorithm, we obtain that the path from $0$ to $x$ in the graph tree $\mathcal{U}_N$ is distributed identically to the path from 0 to $x$ in $\mathcal{U}$. For convenience, we will henceforth assume that $\mathcal{U}$ has been constructed by continuing with Wilson's algorithm from $\mathcal{U}_N$, and so this equality is almost-sure.
We next define a sequence of `good' events $G_i$. Given $\lambda\geq 2$, which will also be chosen later, set $G_0:=\{\gamma_0\subseteq B_m(0)\}\cap\{|\gamma_0|\leq \lambda m^{\kappa}\}$, where $|\gamma_0|$ is the number of elements of the path $\gamma_0$. To define $G_i$ for $i=1,\dots,N-1$, first let $R_i$ be the $m\lambda^{-2}\times 2m\lambda^{-2}$ rectangle consisting of $B_{m\lambda^{-2}}(x_i)$ and the adjacent square of side-length ${m\lambda^{-2}}$ that is closest to $x_{i-1}$; this is the rectangle about $x_i$ with a solid border shown in Figure \ref{config}. Moreover, let $Q_i$ be the union of the $m\times \frac{m(1-\lambda^{-2})}{2}$ rectangle contained in $B_m(x_i)$ that is closest to $x_{i-1}$, the $m\times \frac{m(1-\lambda^{-2})}{2}$ rectangle contained in $B_m(x_{i-1})$ that is furthest from $x_{i-2}$ (if $i=1$, take this to be the rectangle closest to $x_i$), and $B_{{m\lambda^{-2}}}(x_{i-1})$; this is the dotted region shown in Figure \ref{config}. Note in particular $Q_i$ has essentially two forms, depending on whether $x_{i-2},x_{i-1},x_i$ are co-linear or not; these are the two configurations are shown in Figure \ref{config}. For $i=1,\dots N-1$, we then set $G_i=\cap_{j=1}^3G_i^j$, where:
\begin{itemize}
\item $G_i^1$ is the event that $S^{x_i}$ exits $R_i$ on the side closest to $x_{i-1}$ -- call this exit time $\tau_i$, and also $S^{x_i}_{\tau_i+\cdot}$ hits $\gamma_{i-1}$ before exiting $Q_i$;
\item $G_i^2$ is the event that $|\gamma_i|\in [\lambda^{-1}m^\kappa,\lambda m^\kappa]$;
\item $G_i^3$ is the event that $|\gamma_i\cap B_{3m\lambda^{-2}}(x_i)|\leq \lambda(3m\lambda^{-2})^{\kappa} = 3^{\kappa}\lambda^{-3/2}m^{\kappa}$.
\end{itemize}
Finally, we take
\begin{align}\nonumber
G_N &:=\left\{\gamma_N\subseteq B_m(x_{N-1})\cup B_m(x_{N})\right\}\cap\left\{|\gamma_N|
\leq \lambda m^{\kappa}\right\}, \\
\nonumber
F_m(x,\pi) &:= \cap_{i=0}^N G_i.
\end{align}
\begin{figure}[t]
\begin{center}
\input{Ai.tex}
\rput(-3,10.15){$x_i$}
\rput(-6,12){$\gamma_{i}$}
\rput(-7.95,8.5){$\gamma_{i-1}$}
\rput(-8.9,10.98){$x_i'$}
\rput(-8.4,10.15){$x_{i-1}$}
\rput(-8.5,6.9){$B_m(x_{i-2})$}
\rput(-3,2.55){$x_i$}
\rput(-6,4.3){$\gamma_{i}$}
\rput(-10.6,3.0){$\gamma_{i-1}$}
\rput(-8.6,3.4){$x_i'$}
\rput(-8.4,2.55){$x_{i-1}$}
\rput(-12.9,2.8){$B_m(x_{i-2})$}
\end{center}
\caption{Configuration within $B_m(x_{i-1})\cup B_m(x_i)$ on the event $G_0\cap\dots\cap G_i$.}\label{config}
\end{figure}
To highlight the relevance of ${F}_m(x,\pi)=\cap_{i=0}^NG_i$ to controlling path lengths, we note that on this event we have that
\begin{equation}\label{distupper}
d_\mathcal{U}(0,x)\leq \sum_{i=1}^Nd_\mathcal{U}(x_{i-1},x_i)\leq \sum_{i=0}^N|\gamma_i|\leq \lambda (N+1)m^\kappa\leq 2\lambda Nm^\kappa.
\end{equation}
Moreover, from the construction it is possible to deduce that, on ${F}_m(x,\pi)$,
\begin{eqnarray}
d_\mathcal{U}(0,x)&\geq&\sum_{i=1}^{N-1}d_\mathcal{U}(x_{i}',x_{i+1}')\nonumber\\
&\geq & \sum_{i=1}^{N-1}\left(d_\mathcal{U}(x_{i}',x_i)-d_\mathcal{U}(x_i,x_{i+1}')\right)\nonumber\\
&\geq &\sum_{i=1}^{N-1}\left(|\gamma_i|-|\gamma_i\cap B_{3m\lambda^{-2}}(x_i)|\right)\nonumber\\
&\geq & (N-1) \left(\lambda^{-1}-3^\kappa \lambda^{-3/2}\right)m^\kappa,
\label{distlower}
\end{eqnarray}
where the bound $d_\mathcal{U}(x_i,x_{i+1}')\leq |\gamma_i\cap B_{3m\lambda^{-2}}(x_i)|$ is a consequence of the fact that the random walk $S^{x_i}$ does not backtrack to $B_{m\lambda^{-2}}(x_i)$ once it exits $R_i\subseteq B_{3m\lambda^{-2}}(x_i)$, which means that neither does $\gamma_i$, and so the path in $\mathcal{U}$ from $x_i$ to $x_{i+1}'$ must be contained within $B_{3m\lambda^{-2}}(x_i)$. In particular, for $N\geq 2$ and $\lambda$ suitably large, this implies $d_\mathcal{U}(0,x)\geq \frac{1}{4\lambda}Nm^\kappa$, and so we have the desired control over the lengths of paths.
The following is the key estimate of this section.
{\propn \label{pathprop} If $k$ and $\lambda$ are chosen large enough, then there exists a constant $c\in(0,\infty)$ and $m_0\in\mathbb{N}$ such that
\begin{equation}\label{pathpropest}
\mathbf{P}\left({F}_m(x,\pi)\right)\geq e^{-c |\pi| }
\end{equation}
whenever $m\geq m_0$, $x\in\mathbb{Z}^2$, and $\pi$ is a scale $m$ path $\pi$ from $0$ to $x$.}
{ \sm {\em Proof. }} It is enough to show that if $k$ and $\lambda$ are chosen large enough, then there exists a constant $c>0$ such that
\begin{equation}\label{aoest}
\mathbf{P}(G_0)\geq c
\end{equation}
and also, for $i=1,\dots, N$,
\begin{equation}\label{aiest}
\mathbf{P}\left(G_i\:\vline\:G_0,\dots,G_{i-1}\right)\geq c
\end{equation}
uniformly over $m$, $x$ and $\pi$.
To establish \eqref{aoest}, we first note that \cite[Lemma 2.6 and Proposition 2.7]{BM11} imply
\[\mathbf{P}(G_0)\geq \mathbf{P}\left(\gamma_0\subseteq B_m(0)\right)-\mathbf{P}\left(|\gamma_0|>\lambda m^\kappa\right)\geq 1-ck^{-1/3}-c(\lambda k^\kappa)^{-1/5},\]
whenever $m\geq k\geq 1$. Taking any $\lambda\geq 1$ and $k$ large yields a bound of the desired form.
For \eqref{aiest} when $i=1,\dots,N-1$, we start by bounding $\mathbf{P}\left(G_i\:\vline\:G_0,\dots,G_{i-1}\right)$ below by
\[\mathbf{P}\left(G_i^1\:\vline\:G_0,\dots,G_{i-1}\right)
-\mathbf{P}\left(G_i^1\cap (G_i^2)^c\:\vline\:G_0,\dots,G_{i-1}\right)
-\mathbf{P}\left(G_i^1\cap (G_i^3)^c\:\vline\:G_0,\dots,G_{i-1}\right).\]
Now, elementary random walk estimates yield that the first term here is bounded below by $c \lambda^{-4}$ whenever $\lambda^{2}\geq k$ (this latter inequality is required for the case $i=1$). Next, let $\tilde{\gamma}_i$ be the loop-erased random walk from $x_i$ to the boundary of $(B_m(x_i)\cup Q_i)\backslash\gamma_{i-1}$. On $G_i^1$, we have that $\gamma_i=\tilde{\gamma}_i$. Hence, by \cite[Theorem 5.8 and 6.7]{BM10},
\[\mathbf{P}\left(G_i^1\cap (G_i^2)^c\:\vline\:G_0,\dots,G_{i-1}\right)
\leq \mathbf{P}\left(|\tilde{\gamma}_{i}|\not\in [\lambda^{-1}m^k,\lambda m^k]\:\vline\:G_0,\dots,G_{i-1}\right)\leq c_1e^{-c_2\lambda^{3/5}}.\]
Similarly, let $\tilde{\tilde{\gamma}}_i$ be the loop-erased random walk from $x_i$ to the boundary of $(R_i\cup Q_i)\backslash\gamma_{i-1}$. On $G_i^1$, we have that $\gamma_i=\tilde{\tilde{\gamma}}_i$. So, by applying \cite[Theorem 5.8]{BM10} again,
\begin{eqnarray*}
\mathbf{P}\left(G_i^1\cap (G_i^3)^c\:\vline\:G_0,\dots,G_{i-1}\right)
&\leq& \mathbf{P}\left(|\tilde{\tilde{\gamma}}_{i}\cap B_{3m\lambda^{-2}}(x_i)|>\lambda (3m\lambda^{-2})^\kappa\:\vline\:G_0,\dots,G_{i-1}\right)\\
&\leq& c_1e^{-c_2\lambda}.
\end{eqnarray*}
Combining these estimates we obtain
\[\mathbf{P}\left(G_i\:\vline\:G_0,\dots,G_{i-1}\right)\geq c \lambda^{-4}-c_1e^{-c_2\lambda^{3/5}},\]
which is greater than $\frac{c}2\lambda^{-4}>0$ for large $\lambda$.
The estimate (\ref{aiest}) for $i=N$ is obtained similarly.
{\hfill $\square$ \bigskip}
We can now conclude the proof of Theorem \ref{T:dxy} -- recall that (b) was proved at the end of Section \ref{sec:LERW}, and the upper bound in (a) follows from Theorem \ref{T:LERW-lb}.
\bigskip
\smallskip\noindent{\em Proof of the lower bound in Theorem \ref{T:dxy}(a)}.
Without loss of generality, we may assume $y=0$. Moreover, in view of Proposition \ref{pathprop}, we can choose constants $c_1,c_2,c_3,c_4,c_5,c_6$ such that if $k=c_1$, $\lambda=c_2$, then the estimate (\ref{pathpropest}) holds with $c=c_3$ and $m_0=c_4$, and also on ${F}_m(x,\pi)$ we have that
\[c_5Nm^\kappa \leq d_\mathcal{U}(0,x)\leq c_6Nm^\kappa.\]
Now, for any $m\geq1$ and $x\in\mathbb{Z}^2$ with $d_{\infty}(0,x)\geq m$, one can choose a scale $m$ path $\pi$ from $0$ to $x$ such that $N\leq c_7 d_{\infty}(0,x) /m$. On ${F}_m(x,\pi)$, we therefore have that
\[d_\mathcal{U}(0,x)\leq c_6c_7d_{\infty}(0,x)m^{\kappa-1}.\]
It readily follows that if $d_{\infty}(0,x)\geq c_4(c_6c_7 \lambda)^4$ and we choose $m={d_{\infty}(0,x)}/{(c_6c_7 \lambda)^4}$, which implies $m\geq c_4$ and $d_{\infty}(0,x)\geq m$, then on ${F}_m(x,\pi)$ it holds that $d_\mathcal{U}(0,x)\leq \lambda^{-1}d_{\infty}(0,x)^\kappa$. So we conclude that
\[\mathbf{P}\left(d_\mathcal{U}(0,x)\leq \lambda^{-1} d_{\infty}(0,x)^\kappa\right)\geq\mathbf{P}\left({F}_m(x,\pi)\right)\geq e^{-c_3 N}\geq e^{-c_3c_6^4c_7^5\lambda^4}.\]
{\hfill $\square$ \bigskip}
{\remark \rm Whilst it would be straightforward to apply our approach to construct a corresponding exponential estimate from below for the probability of seeing exceptionally long paths in the UST, a stronger polynomial bound for such an event is already known. Indeed, by considering that with polynomially large probability the loop-erased random walk from $x$ to $y$ exits $B_\infty(x,\lambda d_{\infty}(x,y))$, it was established in \cite[Proposition 2.7]{BM11} that
\[\mathbf{P}\left(d_\mathcal{U}(x,y)\geq \lambda d_{\infty}(x,y)^\kappa\right)\geq c\lambda^{-4/5-\varepsilon}\]
for $x,y\in\mathbb{Z}^2$, $\lambda\geq \lambda_0$ (cf. Proposition \ref{nnp}). The point of our approach is that it also gives control of the macroscopic shape of the long path.}
\section{Volume fluctuations} \label{sec:fluct}
In this section, we prove Theorem \ref{mainthm1}. The main ingredient in the proofs of these results is the following lemma, which provides tail bounds for the volume of balls in the UST.
{\lem
There exist constants $c_1, c_2$ such that, for all $r,\lambda\geq 1$,
\begin{equation}\label{bigvol}
\mathbf{P}\left( |B_\mathcal{U}(0, r) | \geq \lambda r^{2/\kappa} \right) \geq c_1 e^{-c_2\lambda^{\kappa/(\kappa-1)}}=c_1 e^{-c_2\lambda^{5}},
\end{equation}
and also
\begin{equation}\label{smallvol}
\mathbf{P}\left( |B_\mathcal{U}(0, r) | \leq \lambda^{-1} r^{2/\kappa} \right) \geq c_1 e^{-c_2\lambda^{\kappa/(2-\kappa)}}
=c_1 e^{-c_2\lambda^{5/3}}.
\end{equation}}
{\remark \rm See \cite[Theorem 1.2]{BM11} for upper bounds of $\exp( -c \lambda} \def\Lam {\Lambda^{1/3})$ for the probability in \eqref{bigvol} and of $\exp( -c \lambda} \def\Lam {\Lambda^{1/9})$ for the probability in \eqref{smallvol}.}
{ \sm {\em Proof. }} Consider a square of $N\times N$ boxes, each of size $m\times m$, with the bottom left box centred on the origin. Let $\pi$ be the scale $m$ horizontal path from 0 to the point $x=((N-1)m,0)$, and suppose that the part of the UST containing $0$ and $x$ is constructed as in the event ${F}_m(x,\pi)$ of the previous section. Then, for each string of vertical boxes, assume that one has a similar construction, where at the bottom level we assume that the algorithm attaches to the horizontal part of the construction. If both such stages of this construction occur, we say that the event ${F}(N,m)$ holds. (See Figure \ref{grid}.) Similarly to the proof of Proposition \ref{pathprop}, we have that
\begin{equation}\label{fbound}
\mathbf{P}\left({F}(N,m)\right)\geq e^{-cN^2}
\end{equation}
for all $N\geq 1$, $m\geq m_0$. Moreover, similarly to \eqref{distupper}, we deduce that, on ${F}(N,m)$,
\begin{equation}\label{b1}
d_\mathcal{U}(0,x)\leq c Nm^\kappa
\end{equation}
for every $x\in \mathcal{U}_{N,m}$, where $\mathcal{U}_{N,m}$ is the subset of $\mathcal{U}$ built in defining the event ${F}(N,m)$. Now, on ${F}(N,m)$, we have that every vertex in the $Nm\times Nm$ region of boxes is within a $d_\infty$-distance of $m$ from a vertex in $\mathcal{U}_{N,m}$. Thus, conditioning on $\mathcal{U}_{N,m}$ and continuing to construct the remainder of $\mathcal{U}$ from this tree as the root, by a minor adaptation of the `filling-in' argument of \cite[Proposition 3.2]{BM11}, it is possible deduce that on an event of (conditional) probability greater than $1-c_1e^{-c_2 N^{{1/3}}}$ every vertex $x$ contained in the bottom $Nm \times \frac{Nm}{2}$ squares that is inside the outer paths (i.e.\ the shaded region of Figure \ref{grid}) satisfies
\begin{equation}\label{b2}
d_{\mathcal{U}}(x,\mathcal{U}_{N,m})\leq (N^{1/2}m)^\kappa\leq Nm^\kappa.
\end{equation}
In particular, putting the bounds at \eqref{b1} and \eqref{b2} together, we deduce that
\[\mathbf{P}\left(|B_\mathcal{U}(0,cNm^\kappa)|\geq c(Nm)^2\right)\geq
\left(1-c_1e^{-c_2 N^{{1/3}}}\right) e^{-cN^2}\geq c_3e^{-c_4N^2}.\]
Setting $r=cNm^\kappa$ and $\lambda = cN^{2(\kappa-1)/\kappa}/c^{2/\kappa}$ yields the result at \eqref{bigvol}.
\begin{figure}[t]
\begin{center}
\scalebox{0.3}{\includegraphics{grid.eps}}
\rput(-5.55,.4){$0$}
\rput(0.3,.4){$((N-1)m,0)$}
\end{center}
\caption{The tree $\mathcal{U}_{N,m}$ on the event ${F}(N,m)$.}\label{grid}
\end{figure}
For the result at (\ref{smallvol}), we argue similarly, though with a different initial tree configuration. We again consider a square of $N\times N$ boxes, each of size $m\times m$, centred on the origin. Let $\pi$ be the scale $m$ path that starts at 0 and spirals outwards around the boxes. Denoting the centre of the final box by $x$, we write ${G}(N,m)={F}_m(x,\pi)$. (See Figure \ref{spiral}.) From Proposition \ref{pathprop}, we have that
\[\mathbf{P}\left({G}(N,m)\right)\geq e^{-cN^2}.\]
Furthermore, let $y$ and $y'$ be centres of two adjacent boxes at a Euclidean distance approximately $Nm/3$ from the origin, but with $y$ one circuit closer to the origin than $y'$. (See Figure \ref{spiral}.) By arguing as at \eqref{distlower}, we have on ${G}(N,m)$ that $d_\mathcal{U}(0,y)\geq cN^2m^\kappa$.
\begin{figure}[t]
\begin{center}
\scalebox{0.3}{\includegraphics{spiral.eps}}
\rput(-3.15,2.8){$0$}
\rput(-1.9,2.8){$y$}
\rput(-1.9,1.6){$y'$}
\rput(-0.7,0.3){$x$}
\end{center}
\caption{The tree $\tilde{\mathcal{U}}_{N,m}$ on the event ${G}(N,m)$.}\label{spiral}
\end{figure}
Next, denote by $\tilde{\mathcal{U}}_{N,m}$ the tree constructed in the definition on ${G}(N,m)$, and note that every vertex in $B_{Nm}(0)$ is within $d_\infty$-distance $m$ of this set. Thus, similarly to the first part of the proof, we can again apply the `filling-in' argument of \cite[Proposition 3.2]{BM11} to deduce that on an event of (conditional on $\tilde{\mathcal{U}}_{N,m}$) probability greater than $1-c_1e^{-c_2 N^{{1/3}}}$ every vertex $x$ contained in $B_{Nm/2}(0)$ is within a $d_{\mathcal{U}}$-distance of $Nm^\kappa$. If this is the case and ${G}(N,m)$ occurs, it further holds that every point $z$ on the straight line between $y$ and $y'$ satisfies
\[d_\mathcal{U}(0,z)\geq d_\mathcal{U}(0,y)-d_\mathcal{U}(y,z)\geq cN^2m^\kappa -Nm^\kappa\geq cN^2m^\kappa,\]
where we have applied the lower bound on $d_\mathcal{U}(0,y)$ from the previous paragraph. In particular, since by construction any path in $\mathcal{U}$ from $B_{Nm}(0)^c$ to $0$ must pass through the line between $y$ and $y'$, it follows that $B_{Nm}(0)^c\subset B_{\mathcal{U}}(0, cN^2m^\kappa)^c$, which implies in turn that
\[\mathbf{P}\left(|B_\mathcal{U}(0,cN^2m^\kappa)|\leq (Nm)^2\right)\geq
\left(1-c_1e^{-c_2 N^{{1/3}}}\right) e^{-cN^2}\geq c_3e^{-c_4N^2}.\]
Setting $r=cN^2m^\kappa$ and $\lambda = cN^{\frac{2(2-\kappa)}{\kappa}} $ yields \eqref{smallvol}.
{\hfill $\square$ \bigskip}
\noindent
{\em Proof of Theorem \ref{mainthm1}.} We start by showing large volumes occur almost-surely, i.e.\ \eqref{bigvolas1}. To this end, we define a sequence of scales:
\[D_i=e^{i^2},\qquad m_i =e^{i^2}/\varepsilon(\log i)^{1/2}.\]
We now run Wilson's algorithm, using the family of independent SRW $(S^x, x \in \bZ^2)$. At stage $i$ we use all the vertices in $B_{2 D_i}(0)$ which have not already been explored, in an order described in more detail below; write $\sU_i$ for the tree obtained. Let $\sF_i$ be the $\sigma$-field generated by the construction at the end of stage $i$.
By \cite[Theorem 1.1]{BM11}, we have that
\begin{equation}\label{inclusion}
B_{2D_i}(0)\subseteq B_\mathcal{U}(0,\lambda^4 D_i^\kappa)\subseteq B_{\lambda^5 D_i}(0)
\end{equation}
with probability greater than $1-c\lambda^{-17/16}$.
Hence, if we run Wilson's algorithm from the vertices contained inside $B_{2D_i}(0)$ (in any order), taking $0$ as the root, then the probability of seeing the part of the tree we generate, $\mathcal{U}_i$ say, leaving $B_{\lambda^5 D_i}(0)$ is less than $c\lambda^{-17/16}$. By applying a Borel-Cantelli argument we thus obtain that
\begin{equation}\label{inclusion2}
\mathcal{U}_i\subseteq B_{i^5D_i}(0)\subseteq B_{m_{i+1}}(0)
\end{equation}
for large $i$, almost-surely. Moreover, from \eqref{inclusion}, we see that we may also assume that the $d_\mathcal{U}$-diameter of $\mathcal{U}_i$ is bounded above by $i^4D_i^\kappa\leq m_{i+1}^\kappa$ for large $i$, almost-surely. Define the event ${F}(i)$ to be the event that \eqref{inclusion2} and the diameter estimate for $\mathcal{U}_i$ both hold.
Next, we define an event ${G}(i+1)$ as follows. In particular, we first suppose that it incorporates the event ${F}(i)$ holding. We then mimic the definition of the event
\[{F}(D_{i+1}/m_{i+1}, m_{i+1})={F}(c(\log {(i+1)})^{1/2}, m_{i+1})\]
from the proof of (\ref{bigvol}). However, we run the first random walk in the box $B_{m_{i+1}}(0)$ until it hits the root $\mathcal{U}_i\subseteq B_{m_{i+1}}(0)$, rather than the root $0$. Let $\mathcal{U}'(i+1)$ be the part of the UST that is thus constructed. Next, extend $\mathcal{U}'(i+1)$ to a tree $\mathcal{U}(i+1)$ by running loop-erased random walks from each of the vertices contained in the bottom $\frac{D_{i+1}}{m_{i+1}}\times \frac{D_{i+1}}{2m_{i+1}}$ squares that is inside the outer paths until they hit the part of the tree already constructed as in Wilson's algorithm (again, we refer to the shaded region of Figure \ref{grid}). We then complete the definition of ${G}(i+1)$ by supposing on this event that the $d_\mathcal{U}$-diameter of $\mathcal{U}(i+1)$ is bounded above by $c D_{i+1}m_{i+1}^{\kappa-1}$. (Since on ${F}(i)$ we also have an estimate for the $d_\mathcal{U}$-diameter of $\mathcal{U}(i)$ of $m_{i+1}^\kappa$, we can control the lengths of paths in the appropriate way.) Similarly to \eqref{fbound}, this construction yields that, for large $i$,
\[\mathbf{P}\left({G}(i+1)\:\vline\:\mathcal{F}_i\right)=\mathbf{P}\left({G}(i+1)\:\vline\:\mathcal{F}_i\right)\mathbf{1}_{{F}(i)}\geq e^{-c (D_{i+1}/m_{i+1})^2}\geq i^{-c}\]
for some $c<1$. Since it is clear that ${G}(i)$ is $\mathcal{F}_i$-measurable, then it follows from the conditional Borel-Cantelli lemma that ${G}(i)$ occurs infinitely often, almost-surely. Finally, we note that on ${G}(i)$ we have that $|B_\mathcal{U}(0,c D_{i+1}m_{i+1}^{\kappa-1})|\geq cD_{i+1}^2$. From this, the reparameterisation $r_i=cD_im_i^{\kappa-1}$ yields the result.
To prove (\ref{smallvolas1}), we proceed in essentially in the same way. In particular, define an event ${H}(i+1)$ similarly to ${G}(i+1)$, but based on the event ${G}(D_{i+1}/m_{i+1},m_{i+1})$ from the proof of (\ref{smallvol}) (i.e.\ using the spiral path of Figure \ref{spiral}, rather than the finger-like structure of Figure \ref{grid}), and then `filling-in' from all vertices in $B_{D_{i+1}/2}(0)$. Arguing as in the proof of (\ref{smallvol}), we deduce that $\mathbf{P}({H}(i+1)\:\vline\:\mathcal{F}_i)\geq i^{-c}$ for some $c<1$, ${H}(i)$ is $\mathcal{F}_i$-measurable, and moreover, on ${H}(i)$ we have that $|B_\mathcal{U}(0,cD_im_i^{\kappa-2})|\leq D_{i}^2$.
{\hfill $\square$ \bigskip}
\section{Volume and resistance estimates on the UST} \label{sec:VRest}
The aim of this section is to derive estimates for `good events', on which we have control on the volume and resistance of the two-dimensional UST. These will be applied in the subsequent sections to deduce the heat kernel estimates and other results stated in the introduction. Much of what we do here will build on previous work from \cite{BCK, BM11}. As already noted, the main input for the averaged heat kernel upper bound was the adaptation of \cite[Proposition 2.10]{BCK} that was established in Proposition \ref{P:PF1n}. A key difference in deriving the averaged heat kernel lower bound is that we will be need to understand the structure of the UST conditional on the presence of a given path, and deriving the relevant estimates requires substantial effort; our main result is Theorem \ref{thm:thm5-8}.
Our first two lemmas relate to the following situation. Suppose we have begun the construction of the UST using Wilson's algorithm, and have constructed a tree $\sU_0$. Write ${\mathbf P}} \def\pE{{\mathbf E}_\sT$ for the law of $\sU$ conditional on the event $\{ \sU_0 = \sT\}$. We wish to adapt the unconditioned results of \cite{BCK, BM11} to the law ${\mathbf P}} \def\pE{{\mathbf E}_\sT$. We begin with the following `filling-in' lemma, based on \cite[Lemma 2.3]{BCK} and \cite[Proposition 3.2]{BM11}. If $\sT$ is a tree contained in $\sU$, then for each $x \in \bZ^2$ there exists a unique self-avoiding path in $\sU$ connecting $x$ and $\sT$; we denote this by $\gam(x,\sT)$. We write ${d_\sU}(x,\sT) = \min\{ {d_\sU}(x,y), y \in \sT\}$ for the length of this path.
\begin{lem} \label{L:fillin} Let $r\ge 1$, and $\sT$ be a finite connected tree. There exist constants $c_i\in(0,\infty)$ not depending on $r$ and $\sT$ such that for each $\delta \le \frac14 $ the following holds. Let $A_1 \subseteq A_2$ be subsets of $\bZ^2$, with the property that any path in $\bZ^2 \setminus \sT$ between $A_1$ and $A_2^c$ is of length greater than $r$. Suppose that ${d_\infty}(x,\sT) \le \delta r$ for all $x \in A_2$.
Then there exists an event $G$ such that $${\mathbf P}} \def\pE{{\mathbf E}_\sT(G^c) \le c_1 r^{-2} |A_2| \exp(-c_2 \delta^{-1/2}),$$ and on $G$ we have that, for all $x \in A_1$:
\[{d_\sU}(x, \sT) \le ( \delta^{1/2} r)^\kappa;\qquad{d_\sU^\sS}(x, \sT) \le \delta^{1/2} r; \qquad \gam(x,\sT) \subseteq A_2. \]
\end{lem}
{ \sm {\em Proof. }} If $A_1$ is a Euclidean ball of radius $r/2$, and $A_2$ is a Euclidean ball of radius $r$ centred on the same point, then this is immediate from the proof of \cite[Lemma 2.3]{BCK}. (Checking the proof in \cite{BM11} one sees that one can take the power of $\delta$ to be $\delta^{-1/2}$ rather than $\delta^{-1/3}$.) The proof for more general sets $A_i$ is similar.
{\hfill $\square$ \bigskip}
{\lem[{Cf.\ \cite[Lemma 2.5]{BCK}}] \label{L:basicLP}
Let $x\in\mathbb{Z}^2$, $r\ge 1$, $k\geq 2$, and $D_0 \subseteq \bZ^2$ satisfy $B_{9r/8}(x) \subseteq \cup_{y \in D_0} B_{r/18k}(y)$. Let $\sT$ be a finite connected tree such that $\mathcal{T}\subseteq B_{2r}(x)^c$, and write $\gam = \gam(x,\sT)$. There exists an event $F_1=F_1(x,r,k)$ which satisfies
\[ {\mathbf P}} \def\pE{{\mathbf E}_\sT( F_1^c ) \le e^{-c_1 k^{1/8}},\]
and on $F_1(x,r,k)$ there exists $T \le \tau_{x,r}(\gam)$ such that, writing $W_{x}=\gam[T]$:\\
(a) $k^{-1/4} r^\kappa \le T \le k^{1/4} r^\kappa$;\\
(b) $ a^{-2}r \le d_{\infty}(x,W_{x}) \le r$;\\
(c) there exists $Y_{x} \in D_0$ such that $d_{\infty}( Y_{x}, W_{x})\le {r}/{3k}$, ${d_\sU^\sS}(Y_{x}, W_{x} ) \le {2r}/{3k}$ and also ${d_\sU}(Y_{x},W_{x})\le c_1 (r/k)^\kappa$.}
\smallskip\noindent { \sm {\em Proof. }} This follows as in \cite{BCK}. The most delicate part of the argument is to verify that \cite[Lemma 2.4]{BCK} holds in this context. For this, we need to show that if $\gam = \gam(x, \sT)$, then $\gam$ does not make too many close returns to the segment $\gam[x, \tau_{x,r}]$ after time $\tau_{x,(1 + k^{-1/8})r}$. The argument in \cite[Lemma 2.4]{BCK} is for $\gam(x,\infty)$, and the proof for $\gam(x, \sT)$ is very similar.
{\hfill $\square$ \bigskip}
\begin{figure}[t]
\begin{center}
\scalebox{0.4}{\includegraphics{section4edit.eps}}
\end{center}
\caption{The part of $\sU$ constructed on the event $H_{N_1}$ (with
$N_1=3$).}\label{sec4fig}
\end{figure}
Towards stating Theorem \ref{thm:thm5-8}, we now introduce an event similar to those of the kind considered in Sections \ref{sec:control} and \ref{sec:fluct}, but incorporating the regularity of Definition \ref{regdef}. We now choose $N \in \bN$ to be suitably large; in particular $N \ge 128$. Let $(\tilde{x}_i)_{i=0}^{N_1}$ be the path with an `S shape' given by
\begin{align*}
&((-N,-1),(-N+1,-1),\dots,(N,-1),\\
&\qquad(N,0),(N-1,0),\dots,(-N,0),\\
&\qquad\qquad(-N,1),(-N+1,1),\dots,(N,1));
\end{align*}
note that $N_1=3(2N+1)-1$. Let $m\in\mathbb{N}$ with $m\geq 256$, we then let $({x}_i)_{i=0}^{N_1}$ be the corresponding scale $m$ path given by setting $x_i=m\tilde{x}_i$. (Ultimately we will only be interested in the situation when both $N$ and $m$ are very large.) Let $\Gamma} \def\gam{\gamma(0)$ be the path in $\bR^2$ which is the union of the line segments $[x_{i-1},x_i]$ for $i=1, \dots, N_1$. For $t >0$ we write
\[\Gamma} \def\gam{\gamma(t) = \{ x\in \bR^2: {d_\infty}(x, \Gamma} \def\gam{\gamma(0)) \le mt \}.\]
We now use Wilson's algorithm to construct $\sU$, and begin the construction using the points $x_i$, $0\le i \le N_1$. We wish the tree constructed to be inside $\Gamma} \def\gam{\gamma(1/8)$ -- see Figure \ref{sec4fig}, and also to have some additional regularity properties given below. As above, let $(S_n^x)_{n\geq 0}$, $x\in\mathbb{Z}^2$, be independent SRW in $\bZ^2$, where $S^x$ is started from $x$. For $i=1, \dots, N_1$ let $R'_i$ be the rectangle, with sides $m/4$ and $5m/4$ which contains both ${B_\infty}(x_{i-1}, m/8)$ and ${B_\infty}(x_i, m/8)$. Let $R_i \subset R'_i$ be the rectangle with sides $m/4$ and $m/2$ which contains ${B_\infty}(x_i, m/8)$, and let $Q_i = R'_i \setminus {B_\infty}(x_i, m/8)$. Let $\sU_0= \gam_0= \{0\}$, and for $i \ge 1$ let
$$ \gam_i = \sL( \sE_{\sU_{i-1}} (S^{x_i})), \qquad \sU_i = \sU_{i-1} \cup \gam_i,\qquad \hbox{for } i=1, \dots, N_1. $$
Define events $G^j_i$, $i=1, \dots, N_1$, $j=1,2$, as follows.
\begin{itemize}
\item $G_i^1$ is the event that $S^{x_i}$ first exits $R_i$ on the side closest to $x_{i-1}$, within a $d_\infty$-distance $m/16$ of the line segment between $x_{i-1}$ and $x_i$ -- call this exit time $\tau_i$, and also $S^{x_i}_{\tau_i+\cdot}$ hits $\gamma_{i-1}$ before exiting $Q_i$;
\item $G_i^2$ is the event that $\gam_i$ is $(\lambda} \def\Lam {\Lambda, (3/2)m e^{-c_1 \lambda} \def\Lam {\Lambda^{1/2}}, 3m/2)$-regular.
\end{itemize}
Let $G_i = G^1_i \cap G^2_i$, and
\[H_i = \bigcap_{j=1}^i G_j, \qquad i=1, \dots, N_1.\]
Similarly to Proposition \ref{pathprop}, we then have the following.
\begin{proposition} \label{P:HN1}
There exist constants $c_1 \in(0,\infty)$ and $\lambda} \def\Lam {\Lambda_0,m_0\ge 2$ such that if $\lambda} \def\Lam {\Lambda\ge \lambda} \def\Lam {\Lambda_0$ and $m\geq m_0$, then
\begin{equation} \label{e:PGNlb}
{\mathbf P}} \def\pE{{\mathbf E}( H_{N_1} ) \ge e^{- c_1 N_1}.
\end{equation}
\end{proposition}
{ \sm {\em Proof. }} Set $H_0 = \Omega$. Arguing as in the proof of Lemma \ref{L:gam0y}, we have that ${\mathbf P}} \def\pE{{\mathbf E}(H_1)= {\mathbf P}} \def\pE{{\mathbf E}( G_1) \ge c_2>0$ for $m$ sufficiently large. From this, the result at \eqref{e:PGNlb} will follow if we can prove that for some $c_3$ that
\begin{equation} \label{e:indh}
{\mathbf P}} \def\pE{{\mathbf E}( H_i | H_{i-1} ) \ge e^{-c_3}, \qquad i=2, \dots, N_1.
\end{equation}
We use induction. Suppose that \eqref{e:indh} holds for $i-1$ for some $i\geq 2$. Since $\sU_{i-1}$ contains a path from $x_{i-1}$ to the boundary of ${B_\infty}(x_{i-1},m/8)$, standard properties of the simple random walk $S^{x_i}$ give us that
\begin{equation}\label{aaa}
{\mathbf P}} \def\pE{{\mathbf E}( G_{i}^1 |H_{i-1}) \ge 2e^{-c_4}.
\end{equation}
Now, let $\widetilde \gam_i$ be the path in $\sU_i$ from $x_i$ to $\sU_{i-2}$. Note that, if we condition on $\sU_{i-2}$, then $\widetilde \gam_i$ is equal in law to $\sL( \sE_{\sU^c_{i-2}} (S^{x_i}) )$. Thus, if we set
$$ \widetilde G_{i}^2 = \{ \sE_{B_{3m}(x_i)} (\widetilde \gam_i) \hbox{ is $(\lambda} \def\Lam {\Lambda, (3/2)me^{-c_1 \lambda} \def\Lam {\Lambda^{1/2}}, 3m/2)$-regular}\},$$
then Lemma \ref{L:regpath} yields that
\begin{equation}\label{bbb}
\mathbf{P}\left( (\widetilde G_{i}^2)^c | H_{i-2}\right) \le c_5e^{-c_6 \lambda^{1/2}}.
\end{equation}
It is also straightforward to verify that $G_i^{1} \cap \widetilde G_{i}^2 \cap H_{i-1} \subseteq G_{i}^1 \cap G_{i}^2\cap H_{i-1}$, and it therefore follows that
\[ {\mathbf P}} \def\pE{{\mathbf E}( H_i | H_{i-1} )={\mathbf P}} \def\pE{{\mathbf E}( G_i^1\cap{G}_i^2 | H_{i-1} )\geq {\mathbf P}} \def\pE{{\mathbf E}( G_i^1\cap \tilde{G}_i^2 | H_{i-1} )\geq {\mathbf P}} \def\pE{{\mathbf E}( G_{i}^1 |H_{i-1})-\mathbf{P}\left(\left(\widetilde G_{i}^2\right)^c | H_{i-1}\right).\]
Using \eqref{bbb} and the inductive hypothesis we have that
\begin{equation} \label{e:pf3c}
\mathbf{P}\left(\left(\widetilde G_{i}^2\right)^c | H_{i-1}\right)=\frac{\mathbf{P}\left(\left(\widetilde G_{i}^2\right)^c\cap H_{i-1} | H_{i-2}\right)}{\mathbf{P}\left(H_{i-1} | H_{i-2}\right)}
\le e^{c_3} \times c_5 e^{-c_6\lambda^{1/2}} \leq e^{-c_4},
\end{equation}
where the final inequality holds by taking $\lambda_0$ suitably large. Combining \eqref{aaa} and \eqref{e:pf3c}, we thus obtain that \eqref{e:indh} holds for $i$.
{\hfill $\square$ \bigskip}
We now fix $\lambda \ge \lambda} \def\Lam {\Lambda_0$, where $\lambda_0\ge 2$ is as in the previous proposition, and consider the uniform spanning tree obtained by conditioning on the event $H_{N_1}$. Let $\sT$ be a fixed tree such that ${\mathbf P}} \def\pE{{\mathbf E}( \sU_{N_1} = \sT \:|\: H_{N_1})>0$, and, as above, write ${\mathbf P}} \def\pE{{\mathbf E}_\sT( \cdot) = {\mathbf P}} \def\pE{{\mathbf E}( \cdot | \sU_{N_1} = \sT )$. We will derive volume and resistance bounds for balls ${B_\sU}(x,r)$ where $x$ is close to the middle section of $\sT$ and $r$ is of order $m^\kappa$ that will hold with ${\mathbf P}} \def\pE{{\mathbf E}_\sT$ probability close to 1. To this end, we introduce some more notation. The tree $\sT$ contains a path from $\tilde x_0$ to $\tilde x_{N_1}$; denote this by $\sT_{trunk}$. For $a>0$, let
\begin{align*}
\sT_{0} &:= \sT \cap \left( [- (N-1) m, (N-1) m]\times [-m/8,m/8]\right), \\
\sT_{0,+} &:= \sT \cap \left( [-(N-1) m, (N-1) m]\times [7m/8,9m/8]\right).
\end{align*}
The following lemma, relating to distances on $\sT$, follows easily from the definition of the event $H_{N_1}$.
\begin{lemma} \label{L:dUonT}
Let $z_1, z_2 \in \sT_0$ with $d_\infty(z_1,z_2 ) \ge 3m$. Then
\[ c_1 \lambda} \def\Lam {\Lambda^{-1} m^{\kappa -1} d_\infty(z_1,z_2 ) \le {d_\sU}(z_1,z_2) \le c_2 \lambda} \def\Lam {\Lambda m^{\kappa -1} d_\infty(z_1,z_2 ).\]
\end{lemma}
Now, for $a\in(0,1)$, let
\begin{equation}\label{ddef}
D(a):= [-aNm,aNm] \times [-9m/8,9m/8].
\end{equation}
We wish to define the region in $D(a)$ which lies `between' $\sT_0\cap \sT_{trunk}$ and $\sT_{0,+}\cap \sT_{trunk}$. To do this precisely, write $\sT_{trunk}^\bR$ for the continuous piecewise linear self-avoiding path in $\bR^2$ obtained by connecting neighbouring points in $\sT_{trunk}$ by a line segment. Let $D^+_\bR(a)$ be the closure of the connected component of $D(a) \setminus \sT_{trunk}^\bR$ which contains the point $(0, m/2)$, and define $D^+(a) = D^+_\bR(a) \cap \bZ^2$. To simplify our notation we will concentrate on the regions $D^+(a)$; exactly the same arguments apply to the corresponding region $D^-(a)$ lying `below' $\sT_0\cap \sT_{trunk}$.
Let $k \in \bN$ satisfy $k \ge \lambda} \def\Lam {\Lambda^4$ and $k \le m$. We now choose a grid $\Lam_1 \subset D^+(7/8)$ of points with separation of order $\frac14 k^{-1/4} m$ such that
\begin{align*}
&D^+(3/4) \subseteq \bigcup_{ z \in \Lam_1 } {B_\infty}\left(z, \fract14 k^{-1/4} m\right).
\end{align*}
Since $| D(3/4)| \le c N m^2$, we can choose this set so that $ |\Lam_1| \le c N k^2$. Let $z \in \Lam_1$ and set
\begin{align*}
G_{11}(z) &:= \left\{ S^z \hbox{ hits $\sT$ before it leaves ${B_\infty}(z, k^{1/7} m )$} \right\}, \\
G_{12}(z) &:= \left\{ | \gam(z,\sT)| \le k^{1/7} m^\kappa \right\}, \\
G_{13}(z) &:= \left\{ | \gam(z,\sT)| \ge k^{-1/7} d_\infty(z, \sT)^\kappa \right\}.
\end{align*}
\begin{lemma} \label{L:connect2}
If $\lambda^4\leq k\leq m \wedge (N/8)^7$ and $z \in \Lam_1$ then
\[ {\mathbf P}} \def\pE{{\mathbf E}_\sT (G_{1j}(z)^c ) \le c e^{-c k^{1/10} } \hbox{ for } j=1,2,3.\]
\end{lemma}
{ \sm {\em Proof. }} Note that $S^z$ can only leave ${B_\infty}(z, k^{1/7} m )$ without hitting $\sT$ if it leaves horizontally at a distance of order $k^{1/7}m$ from $z$. Since every point in $D^+(3/4)$ is within a ${d_\infty}$ distance $5m/8$ of $\sT$ we obtain the bound on ${\mathbf P}} \def\pE{{\mathbf E}(G_{11}(z)^c)$. The bound for $G_{12}$ follows by part 1 of \cite[Theorem 2.2]{BM11} (with $D=D'=D(1)$ and $n=m$), and the bound for $G_{13}$ follows by part 2 of the same theorem.
{\hfill $\square$ \bigskip}
Now let
$$ F_2(k) = \bigcap_{ z \in \Lam_1} \left(G_{11}(z) \cap G_{12}(z) \cap G_{13}(z)\right); $$
by Lemma \ref{L:connect2} we have
\begin{equation} \label{e:F2prob}
{\mathbf P}} \def\pE{{\mathbf E}( F_2(k)^c ) \le c N k^{2} e^{-c k^{1/10} } \le c N e^{- c' k^{1/10} }.
\end{equation}
\begin{proposition}\label{P:volub}
There exists $\delta_1>0$ such that the following holds.
Suppose that $\lambda} \def\Lam {\Lambda^7 \leq k \le m \wedge (\delta_1 N)^7$.
There exists an event $F_3=F_3(k)$ with
\begin{equation} \label{e:PF5c}
{\mathbf P}} \def\pE{{\mathbf E}_\sT( F_3^c) \le c N e^{-c_3 k^{1/10}}
\end{equation}
such that on $F_3(k)$ the following properties hold. \\
(a) If $y \in D^+(5/8)$, then there exists $x \in \sT$ with
$$ {d^S_\sU}(y,x) \le 5 k^{1/7}m, \quad} \def\qq{\qquad {d_\sU}(y,x) \le 2 k^{1/7}m^\kappa. $$
(b) If $y \in D^+(5/8)$ and $1 \le s \le \delta_1 \lambda} \def\Lam {\Lambda^{-1} N $, then
\begin{equation} \label{e:ballcontain}
{B_\sU}(y, s m^\kappa) \subseteq B_\infty(y, c (\lambda} \def\Lam {\Lambda s + k^{1/7}) m ) \cap D(3/4),
\end{equation}
and thus
\begin{equation} \label{e:volub}
| {B_\sU}(y, s m^\kappa) | \le c' (\lambda} \def\Lam {\Lambda s + k^{1/7})^2 m^2.
\end{equation}
\end{proposition}
{ \sm {\em Proof. }}
We continue the construction of the UST from $\sU_{N_1}$ by adding in the points in the grid $\Lam_1 \cap D^+(3/4)$; write $\sU_1^*$ for the tree thus obtained. We then complete the uniform spanning tree inside $D^+(5/8)$. We use the filling-in of Lemma \ref{L:fillin} with $\delta = k^{-1/2}$, $r=m/8$, $A_1 = D^+(5/8)$, $A_2=D^+(3/4)$, and write $\tilde F_{3}(k)$ for the `good event' given by Lemma \ref{L:fillin}. Then
$$ {\mathbf P}} \def\pE{{\mathbf E}( \tilde F_{3}(k)^c ) \le c N e^{- k^{1/4} }. $$
Now let $ F_3(k) = F_{2}(k) \cap \tilde F_{3}(k)$; the bound \eqref{e:PF5c} follows from \eqref{e:F2prob} and the bound on ${\mathbf P}} \def\pE{{\mathbf E}(\tilde F_{3}^c(k))$ given above.
In the remaining part of the proof, we assume $F_3(k)$ holds. Let $y_1 \in D^+(5/8)$. Then the event $\tilde F_{3}$ implies that there exists $w_1 \in \sU_1^*$ with ${d_\sU^\sS}(y_1,w_1)\le k^{-1/4} m$ and ${d_\sU}(y_1,w_1) \le ( k^{-1/4} m )^\kappa$. By the construction of $\sU_1^*$ there exists a point $z_1 \in \Lam_1$ such that $w_1 \in \gam(z_1, \sT)$, and $\gam(z_1, \sT) \subset B_\infty(z_1, k^{1/7}m)$. Let $x_1$ be the point where $\gam(z_1, \sT)$ meets $\sT$. The events $G_{1i}(z_1)$ imply that ${d^S_\sU}(w_1, x_1) \le 4 k^{1/7} m$, and ${d_\sU}(x_1, w_1) \le k^{1/7} m^\kappa$, and the bounds in (a) follow immediately.
For part (b), let $y_1, y_2 \in D^+(5/8)$ with ${d_\sU}(y_1, y_2) \le s m^\kappa$. Let $x_2$ be the point where $\gam(y_2,\sT)$ meets $\sT$ -- we may have $x_1=x_2$. As $\sU$ is a tree, we have ${d_\sU}(x_1,x_2) \le {d_\sU}(y_1,y_2) \le s m^\kappa$. Using Lemma \ref{L:dUonT}, and taking $\lambda} \def\Lam {\Lambda_0$ large enough so that $c_1^{-1} \lambda} \def\Lam {\Lambda s \ge 3$, we obtain
$$ d_\infty(x_1 , x_2 ) \le c_1^{-1} \lambda} \def\Lam {\Lambda m s . $$
Since ${d_\infty}(y_j , x_j) \le 5 k^{1/7} m$, it follows that ${d_\infty}(y_1,y_2) \le c \lambda} \def\Lam {\Lambda m s + 10 m k^{1/7}$. This proves \eqref{e:ballcontain} and the volume upper bound \eqref{e:volub} is then immediate. {\hfill $\square$ \bigskip}
We now consider resistance bounds.
\begin{proposition}
There exist $\delta_2>0$ and $c_1$ such that the following holds. Suppose that $\lambda} \def\Lam {\Lambda^7 \le k \le m \wedge (\delta_2 N)^7$, and $c_1 k^{1/7} \le s \le \delta_2 N$. Let $F_3(k)$ be as in the previous proposition. On the event $F_3(k)$, we have
\begin{equation} \label{e:rebound}
s m^\kappa \ge R_{\rm eff}(x, {B_\sU}(x, s m^\kappa)^c ) \ge \fract14 s m^\kappa
\hbox{ for } y \in D^+(9/16).
\end{equation}
\end{proposition}
{ \sm {\em Proof. }}
The upper bound is immediate. For the lower bound let $y \in D^+(5/8)$, and write $B_1= {B_\sU}(y, {\tfrac12} s m^\kappa)$, $B_2 = {B_\sU}(y, s m^\kappa)$. It is sufficient to prove that there are exactly two points in $\partial B_1$ which are connected to $\partial B_2$ by a path outside $B_1$; a cut set argument then gives the bound \eqref{e:rebound}.
Note first that by the construction of $\sT$ there exists $c$
such that each component $\sT'$ of $\sT \setminus \sT_{trunk}$ satisfies
$$ {d_\sU}(z,z') \le c \lambda} \def\Lam {\Lambda m^\kappa, \quad} \def\qq{\qquad {d^S_\sU}(z,z') \le c m, \hbox{ for } z,z' \in \sT'. $$
By Proposition \ref{P:volub} and the observation above there is a point $x \in \sT_{trunk}$ with ${d_\sU}(x,y) \le c_1 k^{1/7} m^\kappa$ and ${d^S_\sU}(x,y) \le c_1 k^{1/7} m$; we used here the fact that $k \ge \lambda} \def\Lam {\Lambda^7$. Note that $x \in B_1$.
Let $w_1, w_2$ be the two points in $\sT_{trunk}\cap \partial B_1$; it is clear that each of these is connected to $\partial B_2$ by a path outside $B_1$. Now let $z \in \partial B_2$ and suppose there exists $w \in \partial B_1$ with $w \neq w_1, w_2$ such that $\gam(w,z)$ is disjoint from $B_1$. By Proposition \ref{P:volub} we have ${d_\infty}(y,z) \le c (\lambda} \def\Lam {\Lambda s + k^{1/7}) m \le c' s m$, so choosing $\delta_2$ small enough we have that $z \in D^+(5/8)$. Let $z'$ be the closest point in $\sT$ to $z$. By Proposition \ref{P:volub} we have ${d_\sU}(z,z') \le 2 k^{1/7} m^\kappa$, and it follows that the path $\gam(z,y)$ must intersect $\sT$. Let $z''$ be the closest point in $\sT_{trunk}$ to $z'$. The definition of $z$ implies that $z''$ must lie on $\sT_{trunk}$ between $w_1$ and $w_2$, and hence
${d_\sU}(y,z'') \le {\tfrac12} s m^\kappa$. Thus
$$ {d_\sU}(y,z) \le {d_\sU}(y,z'') + {d_\sU}(z'',z') + {d_\sU}(z',z) \le {\tfrac12} s m^\kappa +c \lambda} \def\Lam {\Lambda m^\kappa
+ 2 k^{1/7} m^\kappa, $$
which contradicts the fact that $z\in \partial B_2$ if $c_1$ is large enough.
{\hfill $\square$ \bigskip}
\begin{proposition}
There exist $\delta_3>0$ and $c_1$ such that the following holds. Suppose that $\lambda} \def\Lam {\Lambda \le k^{1/7} \le \delta_3 N$ and $k \le m^{1/2}$. There is an event $F_4(k)$ with ${\mathbf P}} \def\pE{{\mathbf E}_\sT( F_4^c) \le c N e^{-c' k^{1/24} }$ such that on $F_4(k)$, if $c_1 k^{1/4} \le s \le \delta_3 N$, then
\[ | {B_\sU}(x,s) | \ge c \lambda^{-1} k^{-5/2} sm^2 \quad} \def\qq{\qquad \hbox { for all $x \in D^+({\tfrac12})$}. \]
\end{proposition}
{ \sm {\em Proof. }}
We follow the general lines of \cite[Theorem 3.4]{BM11}, but note that the event $H_{N_1}$ means that the path $\sT$ cannot loop back on itself too much. This means that the hardest part of the proof in \cite{BM11}, which uses \cite[Lemma 3.7]{BM11} is not needed.
We choose points $z_i \in \sT$, $1\le i \le N_2$, such that ${B_\infty}(z_i, m/2)$ are disjoint and $\sT \subseteq \cup_i {B_\infty}(z_i, m)$. We have $c N \le N_2 \le c' N$. Write $m_1 = k^{-1/4} m$. For each $i$ choose points $w_{ij} \in \sT \cap B_\infty(z_i, m/2)$ with $1\le j\le N_3$ such that ${B_\infty}(w_{ij}, m_1)$ are disjoint and $N_3 \ge c k^{1/4}$.
The event $H_{N_1}$ implies that if $y_1, y_2 \in \mathcal{T}\cap{B_\infty}(z_i, m_1)$, then ${d_\sU}(y_1,y_2) \le c \lambda} \def\Lam {\Lambda m_1^\kappa$. Choose also $a, b >0$. Let $Q_1(w_{ij}) = {B_\infty}(w_{ij}, k^{-a} m_1)$ and $Q_2(w_{ij}) = {B_\infty}(w_{ij}, 2 k^{-a} m_1)$. We cover $Q_2(w_{ij})$ by a grid $\Lam_3$ of points with separation $k^{-b} k^{-a} m_1$, so that $|\Lam_3| =4 k^{2b}$. We run Wilson's algorithm for the points in $\Lam_3$, and declare this stage of the construction a success for $w_{ij}$ if for all $y \in \Lam_3$, the random walk $S^y$ hits $\sT$ before it leaves ${B_\infty}(w_{ij}, m_1)$. By the discrete Beurling estimate, \cite{LL}, the probability of failure $p_1$ satisfies
\[ p_1 \le |\Lam_3| c k^{-a/4} \le c k^{ 2b - a/4}. \]
We choose $a=1,b=1/12$ and $k$ large enough so that $p_1 < {\tfrac12}$.
Using the `stacks' construction in Wilson's algorithm, we can successively explore the UST in each box ${B_\infty}(w_{ij}, m_1)$, for $j=1, \dots, N_3$, and continue until we get a success. (See the argument in Theorem 5.4 of \cite{BJ} for more details.) Conditional on previous failures, the probability of a failure at stage $j$ is still bounded by $p_1$, and so since we have $N_3$ tries, the overall probability of failure is less than $c e^{- c k^{1/4}}$.
If this stage is a success for some $j$, we write $j_i =j$ and $w_i'= w_{i {j_i}}$. We then fill in the UST inside $Q_2(w'_i)$. By the filling-in result of Lemma \ref{L:fillin} (with $A_j = Q_j(w'_i)$, $j=1,2$ and $r = k^{-1} m_1$, $\delta = k^{-1/12}$) the `good' event $G=G(w'_i)$ given there satisfies $ {\mathbf P}} \def\pE{{\mathbf E}_\sT (G^c) \le c e^{-c k^{1/24}}$. Moreover, if $G$ holds, then every path from a point in ${B_\infty}(w'_i, k^{-1} m_1)$ to $\sT$ is contained in ${B_\infty}(w'_i, m_1)$. Write $G'_i$ for the event that both stages of the construction are successful.
Now set
$$ \tilde F_{4} = \bigcap_{i=1}^{N_2} G'_i, $$
so that
$$ {\mathbf P}} \def\pE{{\mathbf E}_\sT ( \tilde F_{4}^c ) \le c N e^{ -c k^{1/24} } e^{- c k^{1/4}}\leq c N e^{ -c' k^{1/24} } , $$
and define $F_4 := \tilde F_{4} \cap F_3$, where $F_3$ is the event defined in Proposition \ref{P:volub}.
Suppose now that $F_4$ holds. Let $y \in {B_\infty}(w'_i, k^{-1} m_1)$. The event $F_3$ implies that ${d_\sU}(y, \sT) \le c k^{1/4} m^\kappa$, and the event $G'_i$ implies that $\gam(y, \sT) \subseteq {B_\infty}(w'_i,m_1)$. It follows that ${d_\sU}(y, z_i) \le c_2 ( k^{1/4} + \lambda} \def\Lam {\Lambda) m^\kappa$, and thus we obtain that
$$ | {B_\sU}(z_i, c_2 ( k^{1/4} + \lambda} \def\Lam {\Lambda) m^\kappa )| \ge | {B_\infty}(w'_i, k^{-1} m_1) | \ge m^2 k^{-5/2}. $$
Next let $x \in D^+(1/2)$, and set $J = \{ z_i: {d_\sU}(z_i, x) \le {\tfrac12} s m^\kappa \}$. By Lemma \ref{L:dUonT} we have $|J| \ge c \lambda} \def\Lam {\Lambda^{-1} s$. Taking $c_1$ large enough we have ${B_\sU}(z_i, c_2 ( k^{1/4} + \lambda} \def\Lam {\Lambda) m^\kappa ) \subset {B_\sU}(x,sm^\kappa)$, and so
$$ {B_\sU}(x,sm^\kappa) \ge |J| m^2 k^{-5/2} = c \lambda} \def\Lam {\Lambda^{-1} s m^2 k^{-5/2}. $$
{\hfill $\square$ \bigskip}
We summarize the estimates of this section in the following theorem.
\begin{thm}\label{thm:thm5-8}
There exist constants $c_i$ such that the following holds. Suppose that $\lambda} \def\Lam {\Lambda^7 \le k \le m^{1/2} \wedge c_1 N^4$. There exists an event $F_*=F_*(k)$ with ${\mathbf P}} \def\pE{{\mathbf E}_\sT( F_*^c) \le c_2 N e^{-c_3 k^{1/24}}$, such that on $F_*(k)$ if $x\in D^+(1/2)$, and $c_4 k^{1/4} \le s \le c_5 N$ then
\begin{equation}\label{vest}
c\lambda^{-1} k^{-5/2} m^2 s \le | {B_\sU}(x, s m^\kappa)| \le c\lambda} \def\Lam {\Lambda m^2 s,
\end{equation}
\begin{equation}\label{rest}
\fract14 s m^\kappa \le R_{\rm eff}(x, {B_\sU}(x, s m^\kappa )^c ) \le s m^\kappa .
\end{equation}
\end{thm}
Finally, we give a local version of the previous result. For $x\in \mathcal{T}$, and $s \in [c_4 k^{1/4}, c_5 N]$ define $F_*(x,k,s)$ to be the event that the estimates at \eqref{vest} and \eqref{rest} hold. By only considering boxes of size $m$ within a distance $c sm$ of $x$, rather than the order $N$ boxes considered in the previous result, one readily obtains the following.
\begin{cor}\label{corest}
Let $\lambda} \def\Lam {\Lambda^7 \le k \le m^{1/2} \wedge c_1 N^4$ and $c_4 k^{1/4} \le s \le c_5 N$.
Then if $x \in \sT_0 \cap D^+(1/2)$,
\[{\mathbf P}} \def\pE{{\mathbf E}_\sT( F_*(x,k,s)^c) \le c\lambda s e^{-c' k^{1/24}}.\]
\end{cor}
\section{Heat kernel bounds} \label{sec:ann-b}
In this section, we will obtain heat kernel bounds using the estimates given in the previous section, starting with the quenched fluctuations.
\bigskip
\noindent
{\em Proof of Corollary \ref{cor1}.} By \cite[Theorem 4.1]{BK}, for any realisation of $\mathcal{U}$ we have that
\begin{equation}\label{eq:th41bk}
p^\mathcal{U}_{2r|B_{\mathcal{U}}(0,r)|}(0,0)\leq \frac{2}{|B_{\mathcal{U}}(0,r)|}.\end{equation}
Let $a_n=|B_{\mathcal{U}}(0,n)|n^{-d_f}$, and $t_n=n|B_{\mathcal{U}}(0,n)|$.
Plugging these into \eqref{eq:th41bk}, we have
\[
p^\mathcal{U}_{2t_n}(0,0)\leq \frac{2}{|B_{\mathcal{U}}(0,n)|}=2t_n^{-d_f/d_w}a_n^{-(1+2/\kappa)}
=2t_n^{-d_w/d_f}a_n^{-5/13}.\]
By \eqref{bigvolas1}, $a_n>(\log \log n)^{1/5}$ infinitely often, almost-surely, giving the
liminf statement.
We next prove the limsup statement. We use the construction of $\sU$ given after the proof of Lemma \ref{L:basicLP} with a $3(2N+1)$ array of boxes of side $m$. As in Proposition \ref{P:HN1}, this has probability of success of at least $e^{-c_1 N}$.
Given $m$, we define
$R_*=m^\kappa$,
$N = \frac 1{2c_1} \log \log R_*$, $R={\tfrac12} N m^\kappa$, and define
\begin{align*}
v(t) &= \begin{cases}t^{d_f}, &~~\hbox{ if }~ t \le R_*,\\
t R_*^{{d_f}-1}, &~~\hbox{ if }~ t \ge R_*.\end{cases}
\end{align*}
Let $\sT$ be the tree given right after Proposition \ref{P:HN1}. Then by Theorem \ref{thm:thm5-8}, there is an event $F_*(k)$ with ${\mathbf P}} \def\pE{{\mathbf E}_\sT( F_*(k)^c) \le c N e^{-c' k^{1/24}}$ such that the following hold on $F_*(k)$:
\begin{align*}
c_{1,k}v(R_0)\le | {B_\sU}(0, R_0) | \le c_{2,k}v(R_0),\\
c_{3,k}R_0\le R_{\rm eff}(0, {B_\sU}(0, R_0)^c ) \le R_0,~~~~~~
\end{align*}
where $R_0:=ck^\theta m^\kappa$ and $c_{i,k}:=k^{q_i}$, where $q_i\in\mathbb{R}$. We now follow \cite{KM}. Let $r(t)=t$ and let $\sI(t)$ be the inverse function of $r(t)\cdot v(t)$. After some calculations we get, noting ${d_w}=1+{d_f}$,
\begin{align*}
\sI(t) =
\begin{cases}
t^{1/{d_w}},& ~\hbox{ if }~~ t \le R_*^{d_w}, \\
t^{1/2} R_*^{-({d_f}-1)/2},& ~\hbox{ if }~~ t \ge R_*^{{d_w}},
\end{cases}
\end{align*}
and the function $\tilde k(t) = v(\sI(t))$ is
\begin{align*}
\tilde k(t) = \begin{cases}
t^{{d_f}/(1+{d_f})}=t^{{d_f}/{d_w}}, &\hbox{ if }~~ t \le R_*^{{d_w}}, \\
t^{1/2} R_*^{({d_f}-1)/2} , &\hbox{ if }~~ t \ge R_*^{{d_w}}.
\end{cases}
\end{align*}
We can rewrite the final line as
$$ \tilde k(t) = t^{{d_f}/{d_w}} \left( \frac{R_*^{{d_w}}}{t} \right)^\alpha, ~~~\mbox{where}~~~
\alpha = \frac{ {d_f}-1}{2{d_w}} >0.$$
One then finds from \cite[Proposition 3.3]{KM} that
$$ p^\sU_{2n}(0,0) \ge \frac{ c_{1,k}} { \tilde k(n) } \, \hbox { for } \, \frac {c_{2,k}}2 R^{{d_w}} \le n \le c_{2,k}R^{{d_w}},$$
for some $c_{i,k}=c_ik^{-q_i}$, $i=1,2$, where $c_i,q_i>0$. (Note that \cite[Proposition 3.3]{KM} holds for $R\ge R_*$ if the assumption of the proposition holds for $R\ge R_*$.) So taking $T = c_{2,\lambda} R^{{d_w}}/2= \frac{c_{2,\lambda}}{2(4c_1)^{{d_w}}}(R_* \log \log R_*)^{{d_w}}$, it holds that, given $\sT$, with probability greater than $1-c N e^{-c' k^{1/24}}$, we have
$$ T^{{d_f}/{d_w}} \tilde{p}^\sU_T(0,0) \ge c_{3,k}(\log \log R_*)^{\alpha {d_w}} \ge c_{3,k}' (\log \log T)^{\alpha {d_w}}. $$
In order to have $1-c N e^{-c' k^{1/24}}\ge 1/2$, it is enough to take $k\asymp (\log N)^{24}$ which is comparable to $(\log\log\log R_*)^{24}$. (Note that this choice of $k$ enjoys $k\leq \sqrt{m}\wedge (N/8c\lambda)^{1/\theta}$ that is required in Theorem \ref{thm:thm5-8}.) Hence we have
$$ T^{{d_f}/{d_w}} \tilde{p}^\sU_T(0,0) \ge c_6 (\log \log T)^{\alpha {d_w}-\varepsilon}, $$
for some $c_6>0$ and $\varepsilon>0$ which is small.
The rest of the argument goes through similarly to the proof of Theorem \ref{mainthm1} in Section \ref{sec:fluct}.
We choose $m(i) = e^{i^2/\kappa}$, so $R_*(i) = e^{i^2}$,
$N(i) = (2c_1)^{-1} \log\log R_*(i)$, and
$\sum_i e^{-c_1 N(i)} = \sum_i i^{-1}=\infty$.
Similarly to \eqref{inclusion2}, we have good separation of scales. Using the conditional Borel-Cantelli lemma, we obtain the desired lower bound.
{\hfill $\square$ \bigskip}
\subsection{Averaged heat kernel upper bound}
To establish the upper bound of Theorem \ref{mainthm3}, we start by deducing upper estimates for the transition density that hold on the event $F_1(\lambda} \def\Lam {\Lambda,n)$, which was defined at \eqref{f1def}. In this subsection, we fix $\eps_0= 1/40$. Moreover, throughout this and the next subsection, we will write
\[\Phi(t,r) = \Big( \frac{r^{d_w}}{t} \Big)^{1/({d_w}-1)},\]
where $d_w=13/5$ was introduced at \eqref{dwdef}. We also set
\begin{equation}\label{sigxr}
\sigma_{x,r} = \inf\left\{n \ge 0: {d_\sU}(x,X^\sU_n)=r\right\}, \qquad T_x= \inf\left\{ n\ge 0: X^\sU_n =x \right\}.
\end{equation}
\begin{lemma}[{\cite[Proposition 3.3]{KM}}]\label{L:KM34} There exist constants $c_i$ and $q_i$ such that if $B_\sU(x,r)$ and $B_\sU(x,c_1\lambda^{-q_1}r)$ are $\lambda} \def\Lam {\Lambda$-good, then
\begin{align*}
&p^\sU_t(x,x) \le c_2 \lambda} \def\Lam {\Lambda^{q_2} t^{-{d_f}/{d_w}},\qquad\hbox{if }{\tfrac12} r^{d_w} \le t\le r^{d_w}~\hbox{and } t\in \bN,\\
&P_x^\sU\left( \sigma_{x,r} > c_3 \lambda} \def\Lam {\Lambda^{-q_3} r^{d_w}\right) \ge c_4 \lambda} \def\Lam {\Lambda^{-q_4}.
\end{align*}
\end{lemma}
\begin{lemma} \label{L:F1easyub} There exists $\lambda} \def\Lam {\Lambda_0$ such that if $\lambda} \def\Lam {\Lambda\ge \lambda} \def\Lam {\Lambda_0$ then
on the event $F_1(\lambda} \def\Lam {\Lambda,n)$ it holds that
\begin{align}
&p^\sU_{t}(x,x) \le c_2\lambda} \def\Lam {\Lambda^{q_2} t^{-{d_f}/{d_w}}, \quad} \def\qq{\qquad \hbox{ for all $x \in B_\infty(0,n)$, $n^{{d_w} \kappa} e^{- \lambda} \def\Lam {\Lambda^{\eps_0}/2} \le t \le n^{{d_w} \kappa}$
and $t\in \bN$}, \nonumber\\
\label{e:smallexit}
&P^\sU_x( \sigma_{x,r} > c_3 \lambda} \def\Lam {\Lambda^{-q_3} r^{d_w}) \ge c_4 \lambda} \def\Lam {\Lambda^{-q_4}, \quad} \def\qq{\qquad \hbox{ for all $x \in B_\infty(0,n)$
and $n^{\kappa} e^{- \lambda} \def\Lam {\Lambda^{\eps_0}/2 } \le r \le n^{\kappa}$. }
\end{align}
\end{lemma}
{ \sm {\em Proof. }} This follows immediately from Lemma \ref{L:KM34} and the definition of $F_1(\lambda} \def\Lam {\Lambda,n)$.
{\hfill $\square$ \bigskip}
\begin{lemma} \label{L:tailhit}
There exist $\lambda} \def\Lam {\Lambda_0$, $q>0$ such that the follows holds. Suppose $F_1(\lambda} \def\Lam {\Lambda,n)$ holds with $\lambda} \def\Lam {\Lambda \ge \lambda} \def\Lam {\Lambda_0$. If $n/8 \le d_\infty(0,x) \le 7n/8$, then
\begin{equation} \label{e:med-hit}
P^\sU_0( T_x \le t ) \le 2\exp( - c \lambda} \def\Lam {\Lambda^{-q} \Phi(t,n^{\kappa}) )
\quad} \def\qq{\qquad \hbox { for } \; t \ge n^{\kappa {d_w}} e^{-\lambda} \def\Lam {\Lambda^{\eps_0}/2}.
\end{equation}
\end{lemma}
{ \sm {\em Proof. }} Let $y$ be the first point on the path $\gam(0,x)$ with $d_\infty(0,y)\ge n/9$. By Lemma \ref{L:F1dist} we have ${d_\sU}(0,y) \ge c \lambda} \def\Lam {\Lambda^{-\kappa} n^{\kappa}$, and it is clear that $T_y \le T_x$. Let $m \ge 1$, and set
$$ R = c \lambda} \def\Lam {\Lambda^{-\kappa} n^{\kappa}, \quad} \def\qq{\qquad r = \frac{R}{m}, \quad} \def\qq{\qquad s = \frac{t}{m}. $$
Moreover, let $x_i$ be points on $\gam(0,x)$ such that $x_0=0$ and $d_\sU(x_{i-1},x_i)=r$, and $\xi_i$ be the duration from $T_{x_i}$ until $X^\sU$ leaves $B_\sU(x_i, r)$. Using this notation, we have that
$$ T_y \ge \sum_{i=0}^{m-1} \xi_i \ge s \sum_{i=0}^{m-1} \mathbf{1}_{\{\xi_i>s\}}. $$
We next choose $m \in \bN$ so that $m \in [m_1,m_1+1]$ where
$$ m_1^{{d_w}-1} = \lambda} \def\Lam {\Lambda^{-b} \frac{ n^{\kappa {d_w}}}{t}, $$
and $b >0$ will be chosen later. We set $q = ({d_w} q_4 + b)/({d_w}-1)$. If $t \ge \lambda} \def\Lam {\Lambda^{-b} n^{\kappa {d_w}}$, then $\Phi(t,n^\kappa)\leq \lambda^{b/(d_w-1)}$, and so our choice of $q$ ensures that the probability bound in \eqref{e:med-hit} holds. Thus we will assume that $m_1\ge 1$, so that $m_1 \le m \le 2m_1$ and $r = {R}/{m} \ge {R}/{2m_1}$. Now, the condition on $r$ in \eqref{e:smallexit} holds if ${R}/{2m_1} = {c \lambda} \def\Lam {\Lambda^{-\kappa} n^\kappa }/{2m_1} \ge e^{-\lambda} \def\Lam {\Lambda^{\eps_0}/2} n^\kappa$. This is equivalent to
$$ 2^{{d_w}-1} \lambda} \def\Lam {\Lambda^{-b } \frac{ n^{\kappa {d_w}}}{t} \le (c\lambda} \def\Lam {\Lambda^{-\kappa} e^{\lambda} \def\Lam {\Lambda^{\eps_0}/2})^{d_w-1}, $$
i.e.\ $ t \ge c \lambda} \def\Lam {\Lambda^{\kappa(d_w-1)-b} e^{-\lambda} \def\Lam {\Lambda^{\eps_0}(d_w-1)/2 } n^{\kappa {d_w}}$, and we observe that this holds if $t \ge n^{\kappa {d_w}} e^{-3\lambda} \def\Lam {\Lambda^{\eps_0}/5}$ and $\lambda$ is large enough.
To apply \eqref{e:smallexit}, we will also need that $s = t/m$ satisfies $s \le c_3\lambda} \def\Lam {\Lambda^{-q_3} r^{d_w}$. After some algebra we find this requires $\lambda} \def\Lam {\Lambda^{-b} \le c_3 \lambda} \def\Lam {\Lambda^{-q_3-\kappa d_w}$. So, taking $b > q_3+ \kappa d_w$ and $\lambda} \def\Lam {\Lambda_0$ large enough, this condition is also satisfied.
With the choice of $m$ in the previous paragraph, we can apply the bounds in \eqref{e:smallexit} to deduce that $\sum_{i=0}^{m-1} \mathbf{1}_{\{\xi_i>s\}}$ stochastically dominates a binomial random variable with parameters $m$ and $p= c_4 \lambda} \def\Lam {\Lambda^{-q_4}$. Applying the following general bound for a binomial random variable $\eta$,
$$ \mathbf{P}\left( |\eta- \mathbf{E}\eta | > t \right) \le 2 \exp( -t^2/(2 \mathbf{E}\eta + 2t/3) ),$$
we thus deduce that
\begin{align*}
P^\sU_0( T_y < {\tfrac12} c_4\lambda^{-q_4}t )=P^\sU_0( T_y < {\tfrac12} s mp ) \le 2 e^{ - c m p} = 2 \exp( -c \lambda} \def\Lam {\Lambda^{-q_4} m ),
\end{align*}
and the result follows by a reparameterisation of $t$.
{\hfill $\square$ \bigskip}
The following result improves upon the corresponding bound in \cite[Proposition 4.15]{BM11} by obtaining an upper bound for $p^\sU_T(x,y)$ on a set for which the probability has a uniform (in $n$) lower bound. Whilst the estimate holds for a more limited range of times, it is enough for our purposes. We take $\eps_1 < \eps_0=1/40$.
\begin{proposition} \label{P:G-ub}
Suppose $F_1(\lambda} \def\Lam {\Lambda,n)$ holds with $\lambda} \def\Lam {\Lambda \ge \lambda} \def\Lam {\Lambda_0$, then
\[ p^\sU_t(0,x) \le \lambda} \def\Lam {\Lambda^{q_5} t^{-{d_f}/{d_w}} \exp\left( -\lambda} \def\Lam {\Lambda^{-q_5} \Phi(t,n^{\kappa}) \right)\]
whenever $x \in B_{3n/4}(0)\setminus B_{n/2}(0)$, $ e^{-\lambda} \def\Lam {\Lambda^{\eps_1}} n^{\kappa {d_w}}\leq t\leq n^{\kappa d_w}$ and $t\in \bN$.
\end{proposition}
{ \sm {\em Proof. }} Let $z_1$ be the first point on the path $\gam(0,x)$ with $d_\infty(0,z_1)\geq n/8$, and $z_2$ be the first point on the path $\gam(x,0)$ with $d_\infty(x,z_2)\geq n/8$. Let $A_0$ be the set of points $y$ in $\bZ^2$ such that the path $\gam(0,y)$ does not contain $z_1$, and $A_x= \bZ^2 \setminus A_0$. Then, as in the proof of \cite[Theorem 4.9]{BK},
\begin{align}
P^\sU_0( X^\sU_t =x) &= P^\sU_0( X^\sU_t =x, X^\sU_{[t/2]} \in A_0) + P^\sU_0( X^\sU_t =x, X^\sU_{[t/2]} \in A_x) \nonumber\\
&\le 4 P^\sU_x( X^\sU_t =0, X^\sU_{[t/2]} \in A_0)+ P^\sU_0( X^\sU_t =x, X^\sU_{[t/2]} \in A_x) \nonumber\\
&\le 4 P^\sU_x( X^\sU_t =0, T_{z_2} < t/2 ) + P^\sU_0( X^\sU_t =x, T_{z_1} < t/2).\label{eeee}
\end{align}
For the second term above,
\begin{align*}
P^\sU_0( X^\sU_t =x, T_{z_1} < t/2)
&= E^\sU_0 \left( \mathbf{1}_{\{T_{z_1} < t/2\}} P^\sU_{z_1}( X^\sU_{t-T_{z_1}}=x) \right)\\
&\le P^\sU_0(T_{z_1} < t/2 ) \sup_{t/2\le s \le t, s\in \bN} P^\sU_{z_1}( X^\sU_s=x) \\
&\le 4P^\sU_0(T_{z_1} < t/2 ) \sup_{t/2\le s \le t, s\in \bN} \sqrt{p^\sU_s(z_1,z_1)p^\sU_s(x,x)} \\
&\le c\lambda} \def\Lam {\Lambda^{q } \exp\left( -c'\lambda} \def\Lam {\Lambda^{-q} \Phi(t,n^{\kappa}) \right) \lambda} \def\Lam {\Lambda^q t^{-{d_f}/{d_w}}.
\end{align*}
Here we used the Cauchy-Schwarz for the penultimate bound, and Lemmas \ref{L:F1easyub} and \ref{L:tailhit} to obtain the final one. The first term of \eqref{eeee} is bounded in the same way. {\hfill $\square$ \bigskip}
We now have all the pieces in place, and the one remaining lemma we give provides the means to put these together.
\begin{lemma} \label{L:Epsum} Let $G_k$, $k \ge 1$, be a
sequence of sets with ${\mathbf P}} \def\pE{{\mathbf E}(G_k) \rightarrow 1$
and let $T\in \bN$.
If we have $p^\sU_T(0,x) \le a_k$ on $G_k$ for each $k$, then
\begin{equation} \label{e:aksum}
\pE p^\sU_T(0,x) \le a_1 + \sum_{k=2}^\infty a_{k} {\mathbf P}} \def\pE{{\mathbf E}( G_{k-1}^c).
\end{equation}
\end{lemma}
{ \sm {\em Proof. }} Set $A_1=G_1$ and $A_k = G_k\backslash G_{k-1}$ for $k\ge 2$. Since
${\mathbf P}} \def\pE{{\mathbf E}(\cup_k G_k)=1$, we have ${\mathbf P}} \def\pE{{\mathbf E}(\cup_k A_k)=1$, and thus
$$ \pE p^\sU_T(0,x) = \sum_{k=1}^\infty \pE( p^\sU_T(0,x)\mathbf{1}_{A_k})
\le \sum_{k=1}^\infty a_k {\mathbf P}} \def\pE{{\mathbf E}(A_k)
\le a_1 {\mathbf P}} \def\pE{{\mathbf E}(A_1) + \sum_{k=2}^\infty a_k {\mathbf P}} \def\pE{{\mathbf E}( G_{k-1}^c). $$
{\hfill $\square$ \bigskip}
\noindent
\emph{Proof of the upper bound of Theorem \ref{mainthm3}.} By \cite[Theorem 4.4]{BM11}, we have that $ \pE p^\sU_{2T} (0,0) \le c T^{-{d_f}/{d_w}}$. Hence, applying the Cauchy-Schwarz as in the proof of Proposition \ref{P:G-ub}, we further have that, for all $x \in \bZ^2$,
\begin{equation}\label{e:aub-near}
\pE p^\sU_{2T} (0,x)\le \pE[p^\sU_{2n}(0,0)^{1/2} p^\sU_{2T}(x,x)^{1/2}]\le \pE(p^\sU_{2T}(0,0))^{1/2} \pE(p^\sU_{2T}(x,x))^{1/2} \le c T^{-{d_f}/{d_w}}.
\end{equation}
Hence if $d_\infty(0,x)\leq 16$, then the result follows.
Now let $d_\infty(0,x) \ge 16$.
Choose $n$ such that $x \in B_{3n/4}(0)\backslash B_{n/2}(0)$, and set $\Phi=\Phi(T,n^\kappa)$.
Set for $k \ge 1$,
$$ \lambda} \def\Lam {\Lambda_k = k^{1/\eps_0} \Phi^{1/(\eps_0+q_5)}.$$
Choose $c_1$ so that $\Phi \ge c_1$ implies that
$\lambda_k^{-(d_w-1)(\varepsilon_0+q_5)}\ge e^{-\lambda_k^{\varepsilon_1}}$.
If $\Phi \le c_1$, then the estimate again follows from \eqref{e:aub-near}, so we
assume that $\Phi > c_1$.
We now use Lemma \ref{L:Epsum} with $G_k = F_1(\lambda} \def\Lam {\Lambda_k,n)$.
The definition of $\lambda} \def\Lam {\Lambda_k$ gives that
$T \geq n^{\kappa d_w}\lambda_k^{-(d_w-1)(\varepsilon_0+q_5)}\geq n^{\kappa d_w}e^{-\lambda_k^{\varepsilon_1}}$,
so Proposition \ref{P:G-ub} allows us to take
$$ a_k = T^{-{d_f}/{d_w}} \lambda} \def\Lam {\Lambda_k^{q_5} \exp\big( -\lambda} \def\Lam {\Lambda_k^{-q_5} \Phi \big). $$
Thus the first term in the sum \eqref{e:aksum} is given by
$$ a_1= T^{-{d_f}/{d_w}} \Phi^{q_5/(q_5+\eps_0)} \exp( - \Phi^{\eps_0/(q_5+\eps_0)} ). $$
Appealing to Proposition \ref{P:PF1n}, and for convenience replacing $\exp(- c\lambda} \def\Lam {\Lambda^{1/16})$ with the weaker
bound $\exp(-c \lambda} \def\Lam {\Lambda^{\eps_0})$,
we see the $k$th term for $k \ge 2$ is bounded above by
$$ T^{-{d_f}/{d_w}} k^{q_5/\eps_0} \Phi^{q_5/(q_5+\eps_0)}
\exp\big( - \Phi^{\eps_0/(q_5 + \eps_0)} ( k^{-q_5/\eps_0} + (k-1) ) \big). $$
Summing this series, the bound follows with $\theta_2 = \eps_0/(q_5 + \eps_0)$. {\hfill $\square$ \bigskip}
\subsection{Averaged heat kernel lower bound}
In this subsection, we will use Theorem \ref{thm:thm5-8} to establish the averaged heat kernel lower bound from Theorem \ref{mainthm3}. The ideas of the following arguments are from \cite[Section 4]{BK}. We first obtain deterministic diagonal and near-diagonal lower bounds that hold on realisations of $\sU$ that occur with suitably high probability. We recall the notation $D^+(a)$ from $4$ lines
below \eqref{ddef}, define $D^-(a)$
analogously for the corresponding part of the UST below $\mathcal{T}_{0}\cap\mathcal{T}_{trunk}$, and set $D^\pm(a):=D^+(a) \cup D^-(a)$.
\begin{lemma}\label{on-dialbqq}
Let $\lambda} \def\Lam {\Lambda\geq \lambda} \def\Lam {\Lambda_0$, $m\geq m_0$ and $\lambda} \def\Lam {\Lambda^7\leq k\leq m^{1/2}\wedge c_1N^4$. Moreover, let $\tilde{F}_*(k)$ be an event with the properties described in the statement of Theorem \ref{thm:thm5-8} for both ${D}^+(1/2)$ and ${D}^-(1/2)$, and in particular satisfies ${\mathbf P}} \def\pE{{\mathbf E}_\sT( \tilde{F}_*(k)^c) \le c N e^{-c' k^{1/24}}$. Then there exist constants $c_i,q_i$ such that on $\tilde{F}_*(k)$, if $x\in D^\pm(3/8)$ and $c_2k^{1/4}\leq s\leq c_3 N$, then
\begin{equation}\label{eq:niebi2-4}
\tilde{p}^{\mathcal{U}}_{n}(x,x) \ge c_4 \lambda^{-3}k^{-5/2}s^{-1}m^{-2}\geq c_5 \lambda^{-q_1}k^{-q_2}n^{-d_f/d_w}
\end{equation}
for $c_6 \lambda^{-1}k^{-5/2}s^2 m^{\kappa d_w}\le n \le c_7\lambda^{-1}k^{-5/2}s^2 m^{\kappa d_w}$.
\end{lemma}
{ \sm {\em Proof. }} This can be obtained by modifying standard arguments. By a line-by-line modification of the proof of \cite[Proposition 4.4.1, 4.4.3]{Kum}, for example, we have on $\tilde{F}_*(k)$ that
\[ c'\lambda^{-1}k^{-5/2}s^2m^{\kappa d_w}\leq E^\sU_x\left(\sigma_{x,sm^\kappa}\right)\le c\lambda s^2m^{\kappa d_w}\]
for all $x\in D^\pm(3/8)$ and $s$ in the given range. The above estimates and the Markov property (see
\cite[Proposition 4.4.3]{Kum}) imply that the following holds on $\tilde{F}_*(k)$,
\begin{eqnarray*}
P^\sU_x\left(\sigma_{x,s_k}>n\right)&\ge& \frac{c'\lambda^{-1}k^{-5/2}s^2m^{\kappa d_w}-n}
{c\lambda s^2m^{\kappa d_w}},
\end{eqnarray*}
for all $x\in D^\pm(3/8)$ and $n\geq 0$. Given this and the upper volume estimate that holds on $\tilde{F}_*(k)$, \eqref{eq:niebi2-4} can be proved as in \cite[Proposition 4.4.4]{Kum}. {\hfill $\square$ \bigskip}
\begin{lemma}\label{near-diag-lower}
(a) Let $\lambda} \def\Lam {\Lambda\geq \lambda} \def\Lam {\Lambda_0$, $m\geq m_0$, $\lambda} \def\Lam {\Lambda^7\leq k\leq m^{1/2}\wedge c_1N^4$ and $\alpha\in(0,1)$.
Moreover, let $\tilde{F}_*(k)$ be an event as in Lemma \ref{on-dialbqq} that satisfies
${\mathbf P}} \def\pE{{\mathbf E}_\sT( \tilde{F}_*(k)^c) \le c N e^{-c' k^{1/24}}$.
Then there exist constants $c'_i,q_i$ such that on $\tilde{F}_*(k)$, we have for $x\in D^\pm(3/8)$ and $y\in\sU$ satisfying $d_{\sU}(x,y)\le 2s^{\alpha}m^\kappa$ for some $c'_1 \lambda^{4/(1-\alpha)} k^{5/(1-\alpha)}\leq s\leq c'_2 N$,
\begin{equation} \label{e:ndlb67}
\tilde{p}^\sU_n(x,y) \ge c'_3 \lambda^{-q_1}k^{-q_2}n^{-d_f/d_w}
\hbox{ for $c'_4 \lambda^{-1}k^{-5/2}s^2 m^{\kappa d_w}\le n \le c'_5\lambda^{-1}k^{-5/2}s^2 m^{\kappa d_w}$}.
\end{equation}
(b) If $x_0\in D(3/8)\cap\mathcal{T}_0$ and $x,y\in B_\sU(x_0,s^{1/2}m^\kappa)$, then the same lower bound holds on an event $\tilde{F}_*(x_0,k,s)$ that satisfies ${\mathbf P}} \def\pE{{\mathbf E}_\sT( \tilde{F}(x_0,k,s)_*^c) \le c \lambda s e^{-c' k^{1/24}}$.
\end{lemma}
{ \sm {\em Proof. }} By the discrete-time adaptation of \cite[Lemma 4.3]{BK} (which can be obtained by applying estimates in \cite[Section 4]{llt}) and Lemma \ref{on-dialbqq}, we have
\begin{eqnarray*}
\left|\frac{\tilde{p}^\sU_n(x,y)}{\tilde{p}^\sU_n(x,x)}-1\right|^2\le \frac{c d_\sU(x,y)}{n\tilde{p}^\sU_n(x,x)}
\le \frac{c'\lambda^4 k^5 }{ s^{1-\alpha}}\leq \frac{1}{4}.
\end{eqnarray*}
Hence $|\tilde{p}^\sU_n(x,y)-\tilde{p}^\sU_n(x,x)|\le \tilde{p}^\sU_n(x,x)/2$, so we obtain
\begin{eqnarray*}
\tilde{p}^\sU_n(x,y)\ge \tilde{p}^\sU_n(x,x)-|\tilde{p}^\sU_n(x,y)-\tilde{p}^\sU_n(x,x)|\ge \tilde{p}^\sU_n(x,x)/2
\ge c \lambda^{-q_1}k^{-q_2}n^{-d_f/d_w},
\end{eqnarray*}
where we used Lemma \ref{on-dialbqq} in the last inequality. This establishes part (a), and part (b) is obtained in the same way, but using Corollary \ref{corest} in place of Theorem \ref{thm:thm5-8}.
{\hfill $\square$ \bigskip}
\begin{defn}
{\rm
Let $M,N\in {\mathbb N}$, $\alpha=\frac12$, $\lambda\geq \lambda_0$, $m\geq m_0$, $\mathcal{T}$ be a fixed tree as described after Proposition \ref{P:HN1}, and $x\in \sT_{0}$ with $x\ne 0$. Set $r=d_\sU(0,x)/N$ and let $z_0=0,z_1,\cdots,z_N=x$ be points on the path between $0$ and $x$ with $|d_{\sU}(z_{i-1},z_i)-r|\le 1$ that are chosen in some fixed way. For $i=1,\dots,N$, let $\xi_i$ be the smallest integer $k$ such that $\tilde{F}_*(z_i,k,k^{12})$, $\{|B_\sU(z_{i-1},\frac{1}{4}k^{6}m^\kappa)|\geq m^2\}$ and $\{|B_\sU(z_i,\frac14k^{6}m^\kappa)|\geq m^2\}$ hold. (Set $\xi_i=\infty$ if the requirements are not satisfied.) We then say that $G(q,x,N,M)$ holds if $\sum_{i=1}^N\xi_i^q\le MN$.
}\end{defn}
\begin{propn} It holds that
\[{\mathbf P}} \def\pE{{\mathbf E}_\sT(G(q,x,N,M))\ge 1-\frac {c_q}M- cN\lambda \left(m^{6}\wedge N\right)e^{-c' m^{1/48}\wedge (cN^{1/288})},\]
where $c_q$ is a constant that depends on $q$.
\end{propn}
{ \sm {\em Proof. }} By Corollary \ref{corest} and a simple union bound, it holds that
\[\mathbf{P}_{\mathcal{T}}\left(\xi_i> m^{1/2}\wedge (cN^{1/12})\mbox{ for some }i\right)\leq N\times c\lambda \left(m^{6}\wedge N\right)e^{-c' m^{1/48}\wedge (cN^{1/288})}.\]
Moreover, Corollary \ref{corest} and the Markov inequality yield that
\[\mathbf{P}_{\mathcal{T}}\left(\sum_{i=1}^N\xi_i^q\mathbf{1}_{\{\xi_i\leq m^{1/2}\wedge (cN^{1/12})\}}> MN\right)\leq \frac{c_q}{M}.\]
Putting these estimates together completes the proof.
{\hfill $\square$ \bigskip}
\medskip
\noindent
\emph{Proof of the lower bound of Theorem \ref{mainthm3}.}
For simplicity, we only consider the case when $x=(R,0)$, where $R\in\mathbb{Z}_+$; see Remark \ref{R:genx}
for the modifications necessary for the general case. Let $T \geq R$, and set
$y = R^{\kappa {d_w}}/T$. We need to consider several cases. These will depend on constants $b,b'\geq 8$, which will be chosen below.
\smallskip\noindent {\em Case 1: $R \le T \le b R$.} Let $F$ be the event that the UST $\sU$ contains the
straight path along the $x_1$-axis between $0$ and $x$. By considering the construction of $\sU$
which starts by running $S^x$ until it hits $0$, we have ${\mathbf P}} \def\bE{{\mathbf E}(F) \ge 4^{-R}$.
Let $z$ be the point adjacent to $x$ on the path $\gam(0,x)$.
If the event $F$ holds, then $P^\sU_0( X_T \in \{x,z\}) \ge 4^{-T}$, and it follows immediately
that $\bE \tilde p^\sU_T(0,x) \ge 4^{-R+T} \ge e^{-c T}$, which yields the desired lower bound for
these values of $R$ and $T$.
\smallskip\noindent {\em Case 2: $T\geq b'R^{\kappa d_w}$}. To begin with, suppose that $R\geq 1$.
We use Lemma \ref{near-diag-lower}, and take $\lambda} \def\Lam {\Lambda = \lambda} \def\Lam {\Lambda_0$.
Given $k$ we set $c_1 N^4= k$, and choose $k=k_0\geq\lambda} \def\Lam {\Lambda_0$ large enough so that
${\mathbf P}} \def\bE{{\mathbf E}_\sT( \tilde F_*(k_0)^c) \le {\tfrac12}$.
We take $s = c'_1 \lambda} \def\Lam {\Lambda_0^8 k_0^{10}$. As the constants
$\lambda} \def\Lam {\Lambda,k,s$ do not depend on $R$ or $T$, we can absorb them into constants $c_i$.
The bound \eqref{e:ndlb67} holds for $T \in [c_1 m^{\kappa {d_w}}, c_2 m^{\kappa {d_w}}]$, so we
choose $m$ so that $T$ is in this range; this gives that $m \ge c R$. (NB. By increasing the value of $b'$ if necessary, we can further ensure that $m\geq m_0$ and $m^{1/2}\geq k_0$.) The construction of $ \tilde F_*(k_0)$ in Section \ref{sec:VRest} implies that
on this event ${d_\sU}(0,x) \le c s^{1/2} m^\kappa$, and thus we have the lower bound
$$ \tilde p^\sU_T(0,x) \ge c T^{-{d_f}/{d_w}}. $$
Since ${\mathbf P}} \def\bE{{\mathbf E}( H_{N_1}) \ge \exp(-c N) \ge \exp( -c' k_0)$ and
${\mathbf P}} \def\bE{{\mathbf E}_\sT( \tilde F_*(k_0)) \ge {\tfrac12}$, the averaged lower bound $ \bE \tilde p^\sU_T(0,x) \ge c T^{-{d_f}/{d_w}}$
follows. If $R=0$, then one can use the same event as for $R=1$ to deduce the result, since one also has that $ \tilde p^\sU_T(0,0) \ge c T^{-{d_f}/{d_w}}$ on that event.
\smallskip\noindent {\em Case 3: $bR < T < b' R^{\kappa {d_w}}$}.
Choose $N,m \in\mathbb{N}$ to satisfy
\[N\le (b')^{\frac{\kappa d_w}{\kappa d_w-1}}\left(\frac{|x|^{\kappa d_w}}T\right)^{1/(\kappa d_w-1)}< N+1, \quad} \def\qq{\qquad m \le R/N< m+1. \]
Note that $N+1 \ge (b')^{\frac{\kappa d_w}{\kappa d_w-1}}y^{1/(\kappa {d_w}-1)} \ge 8$, and $R/N \ge b^{1/(\kappa {d_w}-1)}$.
Hence choosing $b$ large enough we can ensure that $m \ge m_0$ and also that
if $\sT$ is a tree selected in the way described after Proposition \ref{P:HN1}, then
${\mathbf P}} \def\bE{{\mathbf E}_\sT(G(43/2,x, N,M))\geq \frac{1}{2}$.
The reason we take $q=43/2$ is that this is the power of $\xi_i$ that arises in the time range
for the estimate \eqref{e:ndlb67}. More precisely
on $G(q,x, N,M)$, for each $i=1,\dots,N$, it holds that
\begin{equation} \label{e:chain-lb}
\tilde{p}^\sU_{n_i}(z,y) \ge c_1 \lambda^{-q_1} \xi_i^{-q_2}n_i^{-{d_f}/{d_w}}
\end{equation}
for $c_2\lambda^{-1} \xi_i^{-5/2}(\xi_i^{12})^2m^{\kappa {d_w}}\le n_i \le c_3\lambda^{-1} \xi_i^{-5/2}(\xi_i^{12})^2 m^{\kappa {d_w}}$ and $z,y\in B_\sU(z_i,\xi_i^{6}m^\kappa)$.
Since $24-5/2=43/2$, in
the argument below, we will need to sum over the quantities $\xi^{43/2}_i$; restricting to the event $G(43/2,x, N,M)$ ensures that we can control this sum.
Now, since it holds that $d_\sU(z_{i-1},z_{i})\leq c \lambda m^\kappa$, the estimate
\eqref{e:chain-lb} includes the case when $z\in B_\sU(z_{i-1},\frac14\xi_i^{6}m^\kappa)$ and $y\in B_\sU(z_{i},\frac14\xi_i^{6}m^\kappa)$. Setting $\tilde{T}:=\sum_{i=1}^Nn_i$, where $n_i$, $i=1,\dots, N$,
satisfy the previous constraints, we then have that
\begin{equation}\label{eq:8-11do20}
c_2 \lambda^{-1}\sum_i\xi_i^{43/2} m^{\kappa {d_w}}\le
\tilde{T} \le c_3\lambda^{-1} \sum_i\xi_i^{43/2} m^{\kappa {d_w}}.
\end{equation}
Moreover, writing $B_i:=B_\sU(z_i,\frac14(\xi_{i}^{6}\wedge \xi_{i+1}^{6})m^\kappa)$, we have that
\begin{eqnarray*}
\lefteqn{\tilde{p}^\sU_{\tilde{T}}(0,x)}\\
&\ge &\frac12\sum_{y_1\in B_1}\cdots \sum_{y_{N-1}\in B_{N-1}}\tilde{p}^\sU_{n_1}(0,y_1)
\cdots \tilde{p}^\sU_{n_N}(y_{N-1},x)\mathbf{1}_{\{n_i-d_\sU(y_{i-1},y_i)\mbox{ is even for each }i\}}\\
&\ge & \prod_{j=1}^{N-1}\Big(|B_j|\, c_1 \lambda^{-q_1} \xi_j^{-q_2}n_j^{-{d_f}/{d_w}}\Big)
c_1 \lambda^{-q_1} \xi_N^{-q_2}n_N^{-{d_f}/{d_w}}\\
&\ge &c_\lambda\tilde{T}^{-d_f/d_w}\exp\left(-c'_\lambda N-(q_2+(43/2)d_f/d_w)\sum_{i=1}^N \log \xi_i-\frac{d_f}{d_w}\log\left(\sum_{i=1}^N \xi_i^{43/2}\right)\right)\\
&\ge& c_\lambda\tilde{T}^{-d_f/d_w}\exp(-c_\lambda'N),
\end{eqnarray*}
where in the last inequality we used $\sum_{i=1}^N \log \xi_j\le \sum_{i=1}^N\xi^{43/2} \le MN$.
Note that in \eqref{eq:8-11do20}, we may take $c_2>0$ as small as we like. (This is because $\tilde{p}^\sU_n(x,x)$ is monotone decreasing, which means we can take $c_6$ in Lemma \ref{on-dialbqq} as small as desired. Moreover, we can take $c_4'$ in Lemma \ref{near-diag-lower} to match this). In particular, taking $c_2\leq 8^{-\kappa d_w}M^{-1}$, we obtain
\[c_2 \lambda^{-1}\sum_i\xi_i^{43/2} m^{\kappa {d_w}}\leq 8^{-\kappa d_w}Nm^{\kappa {d_w}}\le T.\]
Hence we may take $\tilde{T}\le T$. If $T\le c_3 \lambda^{-1} \sum_i\xi_i^{43/2} m^{\kappa {d_w}}$, then we can choose $n_j$ so that $T=\tilde{T}$. If not, let $T'=T-\tilde{T}\le Nr^{d_w}$. Let $j_0$ be such that $\xi_{j_0}$ is minimal, and add $N'$ extra steps between $B_{j_0-1}$ and $B_{j_0}$ in the chaining argument above, each with time length satisfying the constraint of $n_{j_0}$ and the total time of the additional steps is equal to $T'$. The latter constraints readily imply that $N'\leq cN$, and we further observe that each extra step contributes a factor of $c_\lambda \xi_{j_0}^{-(q_2+(43/2)d_f/d_w)}$ to the lower bound. Thus the total contribution is no less than $e^{-c_\lambda N}$. Taking the average over $G(43/2,x,N,M)$ and $\mathcal{T}$, we obtain the result.
{\hfill $\square$ \bigskip}
\begin{remark} \label{R:genx}
{\rm For a general $z = (z_1, z_2) \in \bZ^2$ we need to replace the tree $\sT$ defined in Section \ref{sec:VRest}
by a tree which connects $0$ and $z$. We replace the `S-shaped' path $(\tilde x_i)_{i=0}^{N_1}$
defined just after Lemma \ref{L:basicLP} with a path for the which the central section has `L' shape
which connects $0$ and $(z_1/m, z_2/m)$, and the rest of the path shields the central section from
the remainder of $\bZ^2$.
The estimates of Section \ref{sec:VRest} and \ref{sec:ann-b} all work for this path, and the proof of the
lower bound on $\bE \tilde p^\sU_n(0,z)$ then follows.
} \end{remark}
\section{Failure of the elliptic Harnack inequalities}\label{failEHI}
The aim of this section is to make precise and prove Corollary \ref{ehicor}. We start by giving the definition of the elliptic Harnack inequality that we consider, as well as a related metric doubling property.
\begin{defn}\label{ehidef}
{\rm Let $(X_\omega,d_\omega, \mu_\omega)$ be a weighted random graph.\\
(i) We say that the \emph{large scale elliptic Harnack inequalities (LS-EHI)} hold (for the random walk associated with $(X_\omega,d_\omega, \mu_\omega)$) if there exists a deterministic constant $C>1$ and, for each $x_0\in X_\omega$, there exists an $R_{1,x_0}(\omega)>0$ such that the following inequality is satisfied
\begin{equation*}
\sup_{B_{d_\omega} (x_{0},R) }{u}\leq C\inf_{B_{d_\omega} (x_{0},R) }{u}.
\end{equation*}
for any $x_0\in X_\omega$, $R\ge R_{1,x_0}(\omega)$ and any non-negative bounded harmonic function $u$ on $B_{d_\omega}( x_{0},2R)$.\\
(ii) We say that the \emph{large scale metric doubling property (LS-MD)} holds if there exists a deterministic constant $M\in {\mathbb N}$ and, for each $x_0\in X_\omega$, there exists $R_{2,x_0}(\omega)>0$ such that, for any $x_0\in X_\omega$ and $R\ge R_{2,x_0}(\omega)$, $B_{d_\omega}(x_0,R)$ can be covered by $M$ balls of radius $R/2$.}
\end{defn}
The main result in this section is the following.
\begin{thm}\label{thm:MD-LSthm}
(LS-EHI) does not hold for the random walk on $\sU$.
\end{thm}
For the proof, we use the following proposition.
\begin{prop}\label{thm:MD-LSpr}
(LS-EHI) implies (LS-MD).
\end{prop}
{ \sm {\em Proof. }} The proof is a line-by-line modification of
\cite[Theorem 3.11]{BM}. Hence we omit it.
{\hfill $\square$ \bigskip}
The following lemma will be used to check that (LS-MD) is violated for $\sU$.
\begin{lemma} \label{L:noMD}
There exists a constant $\delta>0$ such that, $\mathbf{P}$-a.s., one can find a divergent sequence $(R_n)_{n\geq 1}$ for which there exist at least $n$ disjoint $d_\sU$-balls of radius $\delta R_n$ contained in $B_\sU(0,R_n)$.
\end{lemma}
{ \sm {\em Proof. }} Let $(G(i))_{i\geq 1}$ be the events described in the proof of Theorem \ref{mainthm1},
where it was shown that $G(i)$ holds infinitely often, $\mathbf{P}$-a.s. Now, let
$(z_j)_{j=1}^{\varepsilon(\log i)^{1/2}}$ be the vertices at the centres of the top
row of boxes in the configuration shown in Figure \ref{grid} for $N=D_i/m_i$ and $m=m_i$.
On $G(i)$, we have that
\[d_\sU(0,z_j)\leq C\left(\frac{D_i}{m_i}\right)m_i^{\kappa},\qquad \forall j=1,2,\dots,\varepsilon(\log i)^{1/2},\]
and also
\[d_\sU(z_j,L_i)\geq c\left(\frac{D_i}{m_i}-2\right)m_i^{\kappa},\qquad \forall j=1,2,\dots,\varepsilon(\log i)^{1/2},\]
where $L_i$ is the bottom row of boxes in the configuration shown in Figure \ref{grid}.
It readily follows that there exist at least $\varepsilon(\log i)^{1/2}$ disjoint $d_\sU$-balls of
radius $\frac{c}{2}\left(\frac{D_i}{m_i}-2\right)m_i^{\kappa}$ contained in
$B_\sU(0,C\left(\frac{D_i}{m_i}\right)m_i^{\kappa})$. Hence taking
\[R_n=C\left(\frac{D_i}{m_i}\right)m_i^{\kappa},\qquad \delta=\frac{c}{4C},\]
where $i=e^{(n/\varepsilon)^2}$ yields the result.
{\hfill $\square$ \bigskip}
\noindent
\emph{Proof of Theorem \ref{thm:MD-LSthm}.} Let $\delta>0$, and suppose that LS-MD holds. Then there exists a constant $M' = M'(M,\delta)$ such that: for each $x \in X_\omega$, there exists $R'_{x,\omega} < \infty$, such that if $R \ge R'_{x,\omega}$, then
the ball $B_{\sU}(x,R)$
can be covered by $M'$ balls of radius $\delta R$. However, Lemma \ref{L:noMD} shows that this fails for $\sU$. Hence Proposition \ref{thm:MD-LSpr} yields the result.
{\hfill $\square$ \bigskip}
\section{Scaling limits}\label{sec:scaling}
In this section, we prove the results stated in the introduction concerning scaling limits of the random walk, namely Theorems \ref{mainthm2} and \ref{denslimit}, and Corollaries \ref{cor2} and \ref{cor3}.
\bigskip
\noindent
{\em Proof of Theorem \ref{mainthm2}.} By the separability of the Gromov-Hausdorff-vague topology (see, for example, \cite[Proposition 5.12]{ALWGap}), it is possible to suppose that we have a sequence $(\sU_n)_{n\geq 1}$ of copies of $\sU$, all built on the same probability space, so that
\[\left(\sU_n,n^{-\kappa}d_{\sU_n},n^{-2}\mu_{\sU_n},0\right){\rightarrow}\left(\sT,d_\sT,\mu_\sT,\rho_\sT\right)\]
holds $\mathbf{P}$-a.s. (Note that for this part of the article, we do not need the spatial embeddings into $\mathbb{R}^2$.) It follows from \cite[Proposition 5.9]{ALWGap} that, $\mathbf{P}$-a.s., there exists a metric space $(M,d_M)$ so that the spaces $(\sU_n,n^{-\kappa}d_{\sU_n})$, $n\geq 1$, and $(\sT,d_\sT)$ can be isometrically embedded into $(M,d_M)$ in such a way that: $0$ and $\rho_{\sT}$ are mapped to a common point, $0_M$ say; the embedded measures $n^{-2}\mu_{\sU_n}$ converge vaguely to the embedded version of $\mu_\sT$; and, for all but countably many $r$, the sets $\sU_n\cap \bar{B}_M(0_M,r)$, where $\bar{B}_M(0_M,r)$ is the closed ball in $M$ of radius $r$ centred at $0_M$, converge to $\sT\cap \bar{B}_M(0_M,r)$ with respect to the Hausdorff distance between compact subsets of $(M,d_M)$. As a consequence (see, \cite[Theorem 7.1]{Cr}), we moreover have that the laws of the random walks $(X^{\sU_n}_{tn^{\kappa+2}})_{t\geq0}$ converge weakly to the law of $(X^\sT_t)_{t\geq0}$, when these are considered as measures on $D(\mathbb{R}_+,M)$. Consequently, we have that the Assumptions 1 and 5 of \cite{CrHa} are satisfied (actually Assumption 1 requires the convergence of measures of balls under the various laws, but this condition is readily relaxed to the requirement that the balls in question are continuity sets for the limiting measure), and hence we can apply \cite[Theorem 1 and Proposition 14]{CrHa} to deduce that the associated transition densities satisfy, $\mathbf{P}$-a.s.,
\begin{equation}\label{hkllt}
\left(n^{2} \tilde{p}_{\lfloor tn^{2+\kappa}\rfloor }^{\sU_n}(0,0)\right)_{t>0}{\rightarrow}\left( p_{t}^\sT(\rho_\sT,\rho_\sT)\right)_{t>0}.
\end{equation}
Reparameterising this, the first part of the theorem follows.
In view of the distributional limit we have just proved, to prove the scaling limit at \eqref{odhkscale} it will suffice to check the following integrability condition: for any $p\geq 1$, there exists a constant $C\in(0,\infty)$ such that
\begin{equation}\label{ofmoments}
\sup_{n\geq 1} n^{d_f/d_w}\left\|\tilde{p}_{n}^\sU\left(0,0 \right)\right\|_{p}\leq C,
\end{equation}
where $\|\cdot\|_p$ is the $L_p$ norm with respect to $\mathbf{P}$. Now, by Lemma \ref{L:F1easyub}, on the event $F_1(\lambda, n^{1/d_w\kappa})$, it holds that $\tilde{p}_{n}^\sU(0,0 )\leq c\lambda^q n^{-d_f/d_w}$. Hence, if $\Lambda_n :=\inf\{\lambda\geq 1:\:F_1(\lambda, n^{1/d_w\kappa})\mbox{ holds}\}$, then
\begin{equation}\label{lpbound}
n^{d_f/d_w}\left\|\tilde{p}_{n}^\sU\left(0,0 \right)\right\|_{p}\leq c\left\|\Lambda_n^q\right\|_{p}.
\end{equation}
Since Proposition \ref{P:PF1n} yields that the right-hand side above is uniformly bounded in $n$, this completes the proof. {\hfill $\square$ \bigskip}
In preparation for the proof of Theorem \ref{denslimit}, we verify the equicontinuity of the averaged heat kernel under scaling.
\begin{propn}\label{equicont} There exists a constant $C\in(0,\infty)$ such that
\[\sup_{n\geq 1}n^{{d_f}/{d_w}}\left|\pE \tilde{p}_{\lfloor tn\rfloor }^\sU\left(0,[x n^{\frac{1}{\kappa d_w}}]\right)
-\pE \tilde{p}_{\lfloor tn\rfloor }^\sU\left(0,[y n^{\frac{1}{\kappa d_w}}]\right)\right|\leq Ct^{-d_f/2d_w}|x-y|^{\kappa/2},\]
for all $x,y\in\mathbb{R}^2$, $t>0$.
\end{propn}
{ \sm {\em Proof. }} From \cite[Lemmas 9 and 10]{CrHa}, we have for every $x,y\in\mathbb{Z}^2$ and $n\geq 1$ that
\begin{equation}\label{ffffff}
\left( \tilde{p}_{n}^\sU\left(0,x \right)
- \tilde{p}_{n}^\sU\left(0,y \right)\right)^2\leq \frac{2d_\mathcal{U}(x,y)\tilde{p}_{2\lceil n/2\rceil}^\sU\left(0,0 \right)}{n}.
\end{equation}
Hence Jensen's and H\"{o}lder's inequalities yield that, for any $\varepsilon>0$,
\[\left| \pE\tilde{p}_{n}^\sU\left(0,x \right)
-\pE \tilde{p}_{n}^\sU\left(0,y \right)\right|\leq \sqrt{\frac{2}{n}}\left\|d_\mathcal{U}(x,y)^{1/2}\right\|_{1+\varepsilon}
\left\|\tilde{p}_{2\lceil n/2\rceil}^\sU\left(0,0 \right)^{1/2}\right\|_{\frac{1+\varepsilon}{\varepsilon}},\]
where we again write
$\|\cdot\|_p$ for the $L_p$ norm with respect to $\mathbf{P}$.
Now, by Theorem \ref{T:dxy},
it holds that, for suitably small $\varepsilon$,
\begin{equation}\label{dest1}
\left\|d_\mathcal{U}(x,y)^{1/2}\right\|_{1+\varepsilon}\leq C|x-y|^{\kappa/2}.
\end{equation}
Moreover, from \eqref{lpbound} (and Proposition \ref{P:PF1n}), we have that
\[\left\|\tilde{p}_{2\lceil n/2\rceil}^\sU\left(0,0 \right)^{1/2}\right\|_{\frac{1+\varepsilon}{\varepsilon}}\leq C n^{-d_f/2d_w}.\]
Since $n^{{d_f}/{d_w}}\times n^{-1/2} \times (n^{\frac{1}{\kappa d_w}})^{\kappa/2}\times n^{-d_f/2d_w}=1$, combining these estimates readily yields the result.
{\hfill $\square$ \bigskip}
We moreover note the following rerooting invariance property of the limiting tree.
\begin{propn}\label{reroot} (a) For any $x\in\mathbb{R}^2$,
\[\mathbf{P}\left(\left|\phi_{\sT}^{-1}(x)\right|>1\right)=0.\]
(b) For any $x\in\mathbb{R}^2$,
\begin{equation}\label{gggggg}
\left(\sT,d_\sT,\mu_\sT,\phi_\sT-x,\phi_{\sT}^{-1}(x)\right)\buildrel{d}\over{=}\left(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT\right).
\end{equation}
\end{propn}
{ \sm {\em Proof. }} We first prove the result of part (a) for $x\in\mathbb{R}^2\backslash\{0\}$. In particular, by the scale and rotational invariance properties of \eqref{scaleinvar} and \eqref{rotinvar}, respectively, we have that $\mathbf{P}(|\phi_{\sT}^{-1}(x)|>1)$ is a constant for $x\in\mathbb{R}^2\backslash\{0\}$. Moreover, as was noted in the proof of \cite[Theorem 1.3]{BCK}, we know that the Lebesgue measure of $\{x:\:|\phi_{\sT}^{-1}(x)|>1\}$ is zero, $\mathbf{P}$-a.s. Hence, it follows from Fubini's theorem that $\mathbf{P}(|\phi_{\sT}^{-1}(x)|>1)=0$ for all $x\in\mathbb{R}^2\backslash\{0\}$.
We next prove part (b) for $x\in\mathbb{R}^2\backslash\{0\}$. To begin with, we note from part (a) that the left-hand side of \eqref{gggggg} is a well-defined measured, rooted spatial tree, $\mathbf{P}$-a.s. Moreover, by the separability of the Gromov-Hausdorff-type topology that we are considering (see \cite[Proposition 3.4]{BCK}), it is possible to suppose that we have realisations of the relevant random objects built on a common probability space so that
\[\left(\sU,\delta_n^{\kappa}{d_\sU},\delta^{2}_n\mu_\sU,\delta_n\phi_\sU,0\right){\rightarrow}\left(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT\right),\]
almost-surely as $n\rightarrow\infty$ (cf.\ the proof of Theorem \ref{mainthm2}). It follows that it is almost-surely possible to choose a (random) $x_n^R\in\delta_n\mathbb{Z}^2$ such that
\[\left(\sU,\delta_n^{\kappa}{d_\sU},\delta^{2}_n\mu_\sU,\delta_n\phi_\sU-x_n^R,x_n^R\right){\rightarrow}\left(\sT,d_\sT,\mu_\sT,\phi_\sT-x,\phi_\sT^{-1}(x)\right).\]
In particular, this implies that $|x_n^R-x|\rightarrow 0$, almost-surely. Moreover, let $x_n\in\delta_n\mathbb{Z}^2$ be a deterministic sequence such that $|x_n-x|\rightarrow 0$. One can then deduce from \cite[Theorem 1.1]{BM11} (and the Borel-Cantelli lemma) that there exists a deterministic subsequence $n_i$ along which $\delta_{n_i}^\kappa d_\sU(x_{n_i},x_{n_i}^R)\rightarrow 0$, almost-surely. Hence we find that, almost-surely,
\[\left(\sU,\delta_{n_i}^{\kappa}{d_\sU},\delta^{2}_{n_i}\mu_\sU,\delta_{n_i}\phi_\sU-x_{n_i},x_{n_i}\right){\rightarrow}\left(\sT,d_\sT,\mu_\sT,\phi_\sT-x,\phi_\sT^{-1}(x)\right).\]
By the translation invariance of $\sU$ (see \cite[Theorem 2.3]{Pem91}), the left-hand side here has the same distribution as $(\sU,\delta_{n_i}^{\kappa}{d_\sU},\delta^{2}_{n_i}\mu_\sU,\delta_{n_i}\phi_\sU,0)$, which we know converges in distribution to $(\sT,d_\sT,\mu_\sT,\phi_\sT,\rho_\sT)$, and so the result follows.
Finally, let $x\in\mathbb{R}^2\backslash\{0\}$. Then, from part (b) (for such $x$), we know that $|\phi^{-1}_{\sT}(0)|$ is equal in distribution to $|\phi^{-1}_{\sT}(x)|$. And, from part (a) (again, for such $x$), we know the latter is $\mathbf{P}$-a.s.\ equal to 1. In particular, we find that $\phi^{-1}_{\sT}(0)=\{\rho_\sT\}$, $\mathbf{P}$-a.s. Hence both part (a) and part (b) are readily extended to include the point $x=0$.
{\hfill $\square$ \bigskip}
\noindent
{\em Proof of Theorem \ref{denslimit}.} From \cite[Theorem 1.4]{BCK}, we know that, under the averaged law $\int P^\sU_0(\cdot)\mathbf{P}(d\sU)$,
\begin{equation}\label{rwconv}
\left(n^{-1/d_w\kappa}X^{\sU}_{tn}\right)_{t\geq 0}\buildrel{d}\over{\rightarrow} \left(\phi_{\sT}\left(X^{\sT}_t\right)\right)_{t\geq 0}.
\end{equation}
Applying this in conjunction with Theorem \ref{mainthm2} (specifically \eqref{odhkscale}) and Proposition \ref{equicont}, elementary analysis arguments yield that, for each fixed $t\in(0,\infty)$, $\phi_{\sT}(X^{\sT}_t)$ admits a density $q_t(x)\in C(\mathbb{R}^2,\mathbb{R})$ satisfying the convergence result of part (c). From this, part (a) of the theorem is a simple consequence of Proposition \ref{equicont}. Moreover, given the continuity of the density $q_t$ in the spatial variable, part (b) follows from the scale and rotational invariance properties at \eqref{scaleinvar} \eqref{rotinvar}.
For part (d), we again recall from the proof of \cite[Theorem 1.3]{BCK} that the Lebesgue measure of $\{x:\:|\phi_{\sT}^{-1}(x)|>1\}$ is zero, and also from the latter result that $\mu_{\sT}=\mathcal{L}\circ\phi_\sT$, where $\mathcal{L}$ is two-dimensional Lebesgue measure. Putting these observations together yields
\begin{eqnarray*}
\int_B q_t(x) dx&=&\pE\left(P^\sU_0( X^\mathcal{T}_t \in\phi_\mathcal{T}^{-1}( B ))\right)\\
& =&\pE\left( \int_{\phi_\sT^{-1}(B)}p_{t}^\sT(\rho_\sT,x)\mu_\sT(dx)\right)\\
& =&\pE\left( \int_{B}p_{t}^\sT(\rho_\sT,\phi^{-1}_\sT(x))\mathbf{1}_{\{|\phi^{-1}_\sT(x)|=1\}}dx\right)\\
&=&\int_{B}\pE\left( p_{t}^\sT(\rho_\sT,\phi^{-1}_\sT(x))\right)dx,
\end{eqnarray*}
for all Borel $B \subseteq \bR^2$, where we have applied Fubini's theorem and Proposition 8.2(a) to obtain the final equality. It follows that the desired equality holds for Lebesgue almost-every $x$, and so to complete the proof, it will suffice to show that, for each fixed $t$, $p_t(x):=\pE( p_{t}^\sT(\rho_\sT,\phi^{-1}_\sT(x)))$ is continuous in $x$. Now, from the rotational invariance of \eqref{rotinvar}, we have that $p_t(x)$ is constant on circles centred at the origin. And thus, to check continuity at $x\neq 0$, it will suffice to show that $p_t(\lambda x)\rightarrow p_t(x)$ as $\lambda\rightarrow 1$. Moreover, by the scale invariance property \eqref{scaleinvar}, this is equivalent to checking that $p_{\lambda t}(x)\rightarrow p_t(x)$ as $\lambda\rightarrow1$, and doing this is our next aim. Arguing as in the proof of \cite[Theorem 10.4]{Kigq}, for example, and applying the monotonicity of the on-diagonal part of the heat kernel, one can deduce that, for $s,t>r$,
\[\left|p_{s}^\sT(\rho_\sT,\phi^{-1}_\sT(x))-p_{t}^\sT(\rho_\sT,\phi^{-1}_\sT(x))\right|\leq 2r^{-1}|t-s|\sqrt{p_{r/2}^\sT(\rho_\sT,\rho_\sT)p_{r/2}^\sT(\phi^{-1}_\sT(x),\phi^{-1}_\sT(x))}.\]
From this, the Cauchy-Schwarz inequality, the rerooting invariance of Proposition \ref{reroot}(b), and \eqref{odhkscale}, we see that
\[\left|p_s(x)-p_t(x)\right|\leq 2r^{-1}|t-s|p_{r/2}(0)=Cr^{-1-d_f/d_w}|t-s|,\]
which implies that $p_{\lambda t}(x)\rightarrow p_t(x)$ as $\lambda\rightarrow1$, as desired. To deal with the case $x=0$, we again argue as in the proof of \cite[Theorem 10.4]{Kigq}, for example (cf.\ \eqref{ffffff}), to deduce that
\[\left|p_{t}^\sT(\rho_\sT,\rho_\sT)-p_{t}^\sT(\rho_\sT,\phi^{-1}_\sT(x))\right|\leq t^{-1}\sqrt{d_\mathcal{T}(\rho_\sT,\phi_{\sT}^{-1}(x))p_{t}^\sT(\rho_\sT,\rho_\sT)}.\]
This implies
\[\left|p_{t}(0)-p_{t}(x)\right|\leq t^{-1}\left\|d_\mathcal{T}(\rho_\sT,\phi_{\sT}^{-1}(x))^{1/2}\right\|_{1+\varepsilon}\left\|p_{t}^\sT(\rho_\sT,\rho_\sT)^{1/2}\right\|_{\frac{1+\varepsilon}{\varepsilon}}.\]
From \eqref{hkllt} and \eqref{ofmoments}, we have that the term $\|p_{t}^\sT(\rho_\sT,\rho_\sT)^{1/2}\|_{\frac{1+\varepsilon}{\varepsilon}}$ is finite for any $\varepsilon>0$. Moreover, arguing as in the proof of Proposition \ref{reroot}, for each $x$, one has that there exists a sequence $(x_{n})$ such that $|x_{n}-x|\rightarrow 0$ and, along a subsequence $(n_i)$,
\[n_i^{-\kappa}d_{\sU}(0,x_{n_i})\buildrel{d}\over{\rightarrow}d_\mathcal{T}(\rho_\sT,\phi^{-1}_{\sT}(x)).\]
Hence from \eqref{dest1} we obtain that $\|d_\mathcal{T}(\rho_\sT,\phi^{-1}(x))^{1/2}\|_{1+\varepsilon}\leq C|x|^{\kappa/2}$, where the constant does not depend on $x$. In particular, these estimates imply that $p_t(x)\rightarrow p_t(0)$ as $|x|\rightarrow 0$, and so the proof is complete.
{\hfill $\square$ \bigskip}
\noindent
{\em Proof of Corollary \ref{cor2}.} This is an easy application of Theorems \ref{mainthm3} and \ref{denslimit}(c).
{\hfill $\square$ \bigskip}
\noindent
{\em Proof of Corollary \ref{cor-dist}.}
We begin with the bounds for $|X^\sU_n|$.
Integrating the upper bound of Theorem \ref{mainthm3}, we find that
\[ {n^{-p/d_w\kappa}} \bE ( E_0^\sU |X^{\sU}_{n} |^p )
\le n^{-p/d_w\kappa}\sum_{x\in\mathbb{Z}^2}c_1n^{-{d_f}/{d_w}}|x|^p\exp\left\{-c_2 \left(\frac{|x|^{\kappa {d_w}}}{n}\right)^{\frac{\theta_2}{{d_w}-1}}\right\}
\leq c_3,\]
as required. The lower bound follows in a similar fashion.
For the upper bound on ${d_\sU}(0, X^\sU_n)$
set $R_k = \lceil e^k n ^{1/d_w}\rceil$, $B_k = B_\sU(0, R_k)$ for $k \ge 0$, $B_{-1} =\emptyset$,
and $D_k = B_k \setminus B_{k-1}$ for $k \ge 0$.
Let $k_0 = ((d_w-1)/d_w) \log n$. Note that if $k > k_0$ then $R_k>n$, so that $P_0^\sU ( X^\sU_n \in D_k) =0$. (Recall we are looking at the discrete time walk.) Write $\sigma_k = \sigma_{0,R_k+1}$, where $\sigma_{x,r}$ was defined at \eqref{sigxr}. We then have
\begin{align}
\bE E_0^\sU d_\sU(0, X^\sU_n)^p
&\le \bE \sum_{k=0}^{k_0} 2^pe^{ pk} n^{p/d_w} P_0^\sU ( X^\sU _n \in D_k) \nonumber\\
&\le 2^pn^{p/d_w} \sum_{k=0}^{k_0} e^{pk} \bE P_0^\sU ( \sigma_k \le n).\label{sumtobound}
\end{align}
Now, by an almost identical argument to Lemma \ref{L:tailhit}, it is possible to check that on the event $F_1(\lambda_k,\lambda} \def\Lam {\Lambda_k R_k^{1/\kappa})$ with $\lambda_k:=(4k)^{40}$ we have
\[ P_0^\sU ( \sigma_k \le n) \le C \exp( - c \lambda} \def\Lam {\Lambda_k^{-q_4} m_k)=C\exp(-c\lambda} \def\Lam {\Lambda_k^{-q}e^{kd_w/(d_w-1)});\]
here, $m_k:=(c_3\lambda_k^{-q_3})^{1/(d_w-1)}\Phi(R_k^{d_w}/n)$ represents the number of steps into which the stopping time is decomposed, where $c_3,q_3,q_4$ are as in \eqref{e:smallexit}. Hence, by Proposition \ref{P:PF1n},
\[ \bE P_0^\sU( \sigma_k \le n)\le {\mathbf P}} \def\pE{{\mathbf E}\left( F_1(\lambda_k,\lambda} \def\Lam {\Lambda_k R_k^{1/\kappa})^c\right) + C\exp(-c\lambda} \def\Lam {\Lambda_k^{-q}e^{kd_w/(d_w-1)}) \leq Ce^{-ck^{40/16}},\]
which implies that the sum in \eqref{sumtobound} is finite, and so establishes
the upper bound. The lower bound is proved by the same argument as is used in \cite[Theorem 4.4]{BM11}.
{\hfill $\square$ \bigskip}
\noindent
{\em Proof of Corollary \ref{cor3}.}
From \eqref{rwconv} we have under the averaged law that
\[\left(n^{-1/d_w\kappa}\left|X^{\sU}_{tn}\right|\right)_{t\geq 0}\buildrel{d}\over{\rightarrow} \left(\left|\phi_{\sT}\left(X^{\sT}_t\right)\right|\right)_{t\geq 0}.\]
Part (a) now follows using the uniform integrability given by Corollary \ref{cor-dist}.
For part (b), we start by noting that the convergence at \eqref{scaling} implies that the same result holds if $\delta\phi_\sU$ is replaced by the map $x\mapsto \delta^{\kappa}d_{\sU}(0,x)$, and $\phi_\sT$ is replaced by the map $x\mapsto d_{\sT}(\rho_\sT,x)$. As a consequence, in place of the random walk convergence result of \eqref{rwconv}, one obtains that
\[\left(n^{-1/d_w}d_{\sU}(0,X^{\sU}_{tn})\right)_{t\geq 0}\buildrel{d}\over{\rightarrow} \left(d_{\sT}(\rho_\sT,X^{\sT}_t)\right)_{t\geq 0}.\]
(Concretely, apply \cite[Theorem 7.2]{Cr}.)
Part (b) then also follows from Corollary \ref{cor-dist}.
{\hfill $\square$ \bigskip}
\begin{remark}\label{zqrem}
{\rm
Let $\mathcal{R}:=\{x:\:|\phi^{-1}_\sT(\{x\})|=1\}$.
With $\mathbf{P}$-probability one, we have that $\mathcal{L}(\mathcal{R}^c)=0$, where we again use $\mathcal{L}$ to denote Lebesgue measure on $\mathbb{R}^2$, and moreover $0\in\mathcal{R}$ (see Proposition \ref{reroot} and its proof). Since $\mu_\sT(\phi^{-1}_\sT(\mathcal{R}^c))=\mathcal{L}(\mathcal{R}^c)$ (by \cite[Theorem 1.3]{BCK}), it follows that, $\mathbf{P}$-a.s., for any $x\in\mathcal{R}$ and $t\geq 0$,
\[P^\mathcal{T}_{\phi^{-1}_\sT(x)}\left(\phi_\sT(X^\mathcal{T}_t)\in\mathcal{R}\right)=\int_{\phi_\sT^{-1}(\mathcal{R})}p_t^\mathcal{T}(\phi_\sT^{-1}(x),y)\mu_\sT(dy)=1,\]
where $P^\mathcal{T}_{\phi^{-1}_\sT(x)}$ is the quenched law of $X^\mathcal{T}$ started from $\phi_\sT^{-1}(x)$. It readily follows that, when started from $x\in\mathcal{R}$ (including from $x=0$), $\phi_\sT(X^\mathcal{T})$ is a Markov process, and moreover has transition density that is determined by $(p^\mathcal{T}_t(\phi^{-1}_\sT(y),\phi^{-1}_\sT(z)))_{y,z\in\mathcal{R}}$ (and which is defined arbitrarily elsewhere). On the other hand, if $\tau$ is a stopping time for $\phi_\sT(X^\mathcal{T})$ such that $P^\mathcal{T}_{\phi^{-1}_\sT(x)}(\phi_\sT(X^\mathcal{T}_\tau)\in\mathcal{R}^c)>0$, then it is clear that the quenched law of $(\phi_\sT(X^\mathcal{T}_{\tau+t}))_{t\geq0}$ does not only depend on $\phi_\sT(X^\mathcal{T}_{\tau})$, and so $\phi_\sT(X^\mathcal{T})$ is not strong Markov. Indeed, the situation is somewhat similar to that of reflecting Brownian motion in a planar domain with a slit removed (cf.\ comments in \cite[Section 3]{BZ}), though the slit is replaced in our case by the dense set $\mathcal{R}^c\subseteq\mathbb{R}^2$, which we note coincides with the `dual trunk' studied in \cite[Section 10]{Schramm}.
} \end{remark}
|
1,314,259,995,817 | arxiv | \section{Introduction}
\label{sec:intro}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{images/overview.pdf}
\vspace{-0.8cm}
\caption{Training procedure of \textsc{FedAUX}. \textbf{Preparation phase:} P1) The unlabeled auxiliary data is used to pre-train a feature extractor (e.g. using contrastive representation learning). P2) The feature-extractor is sent to the clients, where it is used to initialize the client models. Based on extracted features, a logistic scoring head is trained to distinguish local client data from a subset of the auxiliary data. P3) The trained scoring head is sanitized using a $(\varepsilon, \delta)$-differentially private mechanism and then used to compute certainty scores on the distillation data.
\textbf{Training Phase:} T1) In each communication round, a subset of the client population is selected for training. Each selected client downloads a model initialization from the server, and then updates the full model $f_i$ (feature extractor \& scoring head) using their private local data. T2) The locally trained classifier and scoring models $f_i$ and $s_i$ are sent to the server, where they are combined into a weighted ensemble. T3) Using the unlabeled auxiliary data and the weighted ensemble as a teacher, the server distills a student model which is used as the initialization point for the next round of Federated training. \textbf{*}Note that in practice we perform computation of soft-labels and scores at the server to save client resources.}
\label{fig:overview}
\end{figure*}
Federated Learning (FL) allows distributed entities ("clients") to jointly train (deep) machine learning models on their combined data, without having to transfer this data to a centralized location \macrocite{mcmahan2017communication}. The Federated training process is orchestrated by a central server. The distributed nature of FL improves privacy \macrocite{li2019federated}, ownership rights \macrocite{sheller2020federated} and security \macrocite{mothukuri2020survey} for the participants. As the number of mobile and IoT devices and their capacities to collect large amounts of high-quality and privacy-sensitive data steadily grows, Federated training procedures become increasingly relevant.
While the client data in Federated Learning is typically assumed to be private, in most real-world applications the server additionally has access to unlabeled \emph{auxiliary} data, which roughly matches the distribution of the client data. For instance, for many Federated computer vision and natural language processing problems, such auxiliary data can be given in the form of public data bases such as ImageNet \macrocite{deng2009imagenet} or WikiText \macrocite{merity2016pointer}. These data bases contain millions to billions of data samples but are typically lacking the necessary label information to be useful for training task-specific models.
Recently, Federated Distillation (FD), a novel algorithmic paradigm for Federated Learning problems where such auxiliary data is available, was proposed. In contrast to classic parameter averaging based FL algorithms \macrocite{mcmahan2017communication, mohri2019agnostic, reddi2020adaptive, li2019fedprox, sattler2019robust}, which require all client's models to have the same size and structure, FD allows the clients to train heterogeneous model architectures, by distilling the client predictions on the auxiliary set of data into a student model. This can be particularly beneficial in situations where clients are running on heterogeneous hardware. Studies show that FD based training has favorable communication properties \macrocite{itahara2020distill, sattler2020communication}, and can outperform parameter averaging based algorithms \macrocite{lin2020ensemble}.
However, just like for their parameter-averaging-based counterparts, the performance of FD based learning algorithms falls short of centralized training and deteriorates quickly if the training data is distributed in a heterogeneous ("non-iid") way among the clients. In this work we aim to further close this performance gap, by exploring the core assumption of FD based training and deriving maximum utility from the available unlabeled auxiliary data. Our main contributions are as follows:
\begin{itemize}
\item We show that a wide range of (out-of-distribution) auxiliary data sets are suitable for self-supervised pre-training and can drastically improve FL performance across all baselines.
\item We propose a novel certainty-weighted FD technique, that improves performance of FD on non-iid data substantially, addressing a long-standing problem in FL research.
\item We propose an $(\varepsilon, \delta)$-differentially private mechanism to constrain the privacy loss associated with transmitting certainty scores.
\end{itemize}
These performance improvements are possible a) under the same assumptions made in the FD literature, b) with only negligible additional computational overhead for the resource-constrained clients and c) with small quantifiable excess privacy loss.
\section{Related Work}
\label{sec:related}
\textbf{Federated Distillation:}
Distillation \macrocite{bucila2006compression, hinton2015distill} is a common technique to transfer the knowledge of one or multiple \macrocite{you2017learning, anil2018large} machine learning classifiers to a different model, and is typically used in centralized settings before deployment in order to reduce the model complexity, while preserving predictive power. To this end, the predictions of the teacher model(s) on a distillation data set are used to guide the training process of the potentially less complex student model. Federated Distillation (FD) algorithms, which leverage these distillation techniques to aggregate the client knowledge, are recently gaining popularity, because they outperform conventional parameter averaging based FL methods \macrocite{lin2020ensemble, chen2020feddistill} like \textsc{FedAVG} or FedPROX \macrocite{mcmahan2017communication, li2019fedprox} and allow clients to train heterogeneous model architectures \macrocite{li2019fedmd, chang2019cronus, li2021fedh2l}. FD methods can furthermore reduce communication overhead \macrocite{jeong2018distill, itahara2020distill, seo2020fd, sattler2020communication}, by exploiting the fact that distillation requires only the communication of model predictions instead of full models. In contrast to centralized distillation, where training and distillation data usually coincide, FD makes no restrictions on the auxiliary distillation data\footnote{Recent work even suggests that useful distillation data can be generated from the teacher models themselves \macrocite{nayak2019zero}.},
making it widely applicable.
Our work, is in line with \macrocite{lin2020ensemble, chen2020feddistill} in that it aims to improve overall training performance in FL. Both \textsc{FedDF} \macrocite{lin2020ensemble} and \textsc{FedBE} \macrocite{chen2020feddistill} combine parameter averaging as done in FedAVG \macrocite{mcmahan2017communication} with ensemble distillation to improve FL performance. While \textsc{FedDF} combines client predictions by means of an (equally weighted) model ensemble, \textsc{FedBE} forms a Bayesian ensemble from the client models for better robustness to heterogeneous data. Taking \textsc{FedDF} as a starting point, we additionally leverage the auxiliary distillation data set for unsupervised pre-training and weight the client predictions in the distillation step according to their prediction certainty to better cope with settings where the client’s data generating distributions are statistically heterogeneous.
\textbf{Weighted Ensembles:}
Weighted ensemble methods were studied already in classical work \macrocite{hashem1993ensemble, perrone1993ensemble, opitz1999popular}, with certainty weighted ensembles of neural networks in particular being proposed for classification e.g. in \macrocite{jimenez1998dynamically}.
Mixture of experts and boosting methods \macrocite{yuksel2012twenty, masoudnia2014mixture, schapire1999brief} where multiple simple classifiers are combined by weighted averaging are frequently used in centralized settings.
A more detailed discussion of related work can be found in Appendix \ref{supp:related}.
\section{Federated Learning with Auxiliary Data}
\label{sec:method}
\begin{figure*}[t!]
\centering
\includegraphics[width=1.0\textwidth]{images/toy2_acc.pdf}
\vspace{-0.8cm}
\caption{\textbf{Weighted Ensemble Distillation} illustrated in a toy example on the Iris data set (data points are projected to their two principal components). Three Federated Learning clients hold disjoint non-iid subsets of the training data. Panels 1-3: Predictions made by linear classifiers trained on the data of each client. Labels and predictions are color-coded, client certainty (measured via Gaussian KDE) is visualized via the alpha-channel. The mean of client predictions (panel 4) only poorly captures the distribution of training data. In contrast, the certainty-weighted mean of client predictions (panel 5) achieves much higher accuracy.}
\label{fig:toy}
\end{figure*}
In this section, we describe our method for efficient Federated Learning in the presence of unlabeled auxiliary data (\textsc{FedAUX}). An illustration of our proposed approach is given in Figure \ref{fig:overview}. We describe \textsc{FedAUX} for the homogeneous setting were all clients hold the same model prototype. The detailed algorithm for the more general model-heterogeneous setting can be found in Appendix \ref{supp:algorithm}. An exhaustive \emph{qualitative} comparison between \textsc{FedAUX} and baseline methods is given in Appendix \ref{supp:qualitative}.
\subsection{Problem Setting} We assume the conventional FL setting where a population of $n$ clients is holding potentially non-iid subsets of private labeled data $D_1,..,D_n$, from a training data distribution $(\bigcup_{i\leq n}D_i)\sim \varphi(\mathcal{X}, \mathcal{Y})$. We further make the assumption that the server and the clients both have access to a public collection of unlabeled auxiliary data from a deviating distribution $D_{aux}\sim \psi(\mathcal{X})$. The latter assumption is common to all studies on FD.
One round of federated training is then performed as follows: A subset $\mathcal{S}_t$ of the client population is selected by the server and downloads a model initialization. Starting from this model initialization, each client then proceeds to train a model $f_i$ on it's local private data $D_i$ by taking multiple steps of stochastic gradient descent. We assume that these local models can be decomposed into a feature extractor $h_i$ and a classification head $g_i$ according to $f_i=g_i \circ h_i
. Finally, the updated models $f_i$, $i\in\mathcal{S}_t$ are sent back to the server, where they are aggregated to form a new server model $f$, which is used as the initialization point for the next round of FL. The goal of FL is to obtain a server model $f$, which optimally generalizes to new samples from the training data distribution $\varphi$, within a minimum number of communication rounds $t\leq T$.
\subsection{Federated Ensemble Distillation}
Federated Ensemble Distillation is a novel method for aggregating the knowledge of FL clients. Instead of aggregating the parameters of the client models (e.g. via an averaging operation), a student model is trained on the combined predictions of the clients on some public auxiliary data. Let $x\in D_{aux}$ be a batch of data from the auxiliary distillation data set. Then one iteration of student distillation is performed as
\begin{align}
\label{eq:distill_update}
\theta^{t, j+1} \leftarrow \theta^{t, j} - \eta\frac{\partial D_{KL}(\mathcal{A}(\{f_i(x)|i\in \mathcal{S}_t\}), \sigma(f(x, \theta^{t, j})))}{\partial \theta^{t, j}}.
\end{align}
Hereby, $D_{KL}$ denotes the Kullback-Leibler divergence, $\eta>0$ is the learning rate, $\sigma$ is the softmax-function and $\mathcal{A}$ is a mechanism to aggregate the soft-labels. Existing work \macrocite{lin2020ensemble} aggregates the client predictions by taking the mean according to
\begin{align}
\label{eq:agg_mean}
\mathcal{A}_{mean}(\{f_i(x)|i\in \mathcal{S}_t\}) = \sigma\left(\frac{\sum_{i\in \mathcal{S}_t} f_i(x)}{|\mathcal{S}_t|}\right).
\end{align}
Federated Ensemble Distillation is shown to outperform parameter averaging based techniques \macrocite{lin2020ensemble}.
\subsection{Self-supervised Pre-training}
\label{sec:pretrain}
Self-supervised representation learning can leverage large records of unlabeled data to create models which extract meaningful features. For the two types of data considered in this study - image and sequence data - strong self-supervised training algorithms are known in the form of contrastive representation learning \macrocite{chen2020simple, wang2020understanding} and next-token prediction \macrocite{devlin2018bert, radford2019language}.
As part of the \textsc{FedAUX} preparation phase (cf. Fig. \ref{fig:overview}, P1) we propose to perform self-supervised training on the auxiliary data $D_{aux}$ at the server. We emphasize that this step makes no assumptions on the similarity between the local training data and the auxiliary data. This results in a parametrization for the feature extractor $h_0$. Since the training is performed at the server, using publicly available data, this step inflicts neither computational overhead nor privacy loss on the resource-constrained clients.
\subsection{Weighted Ensemble Distillation}
\label{sec:weighted_distill}
Different studies have shown that both the training speed, stability and maximum achievable accuracy in existing FL algorithms deteriorate if the training data is distributed in a heterogeneous "non-iid" way among the clients \macrocite{zhao2018federated, sattler2019robust, li2020covergence}. Federated Ensemble Distillation makes no exception to this rule \macrocite{lin2020ensemble}.
The underlying problem of combining hypotheses derived from different source domains has been explored in multiple-source domain adaptation theory \macrocite{mansour2008domain, hoffman2018domain}, which shows that standard convex combinations of the hypotheses of the clients as done in \macrocite{lin2020ensemble} may perform poorly on the target domain. Instead, a distribution-weighted combination of the local hypotheses is shown to be robust \macrocite{mansour2008domain, hoffman2018domain}. A simple toy example, displayed in Figure~\ref{fig:toy}, further illustrates this point.
Inspired by these results, we propose to modify the aggregation rule of FD \eqref{eq:agg_mean} to a certainty-weighted average:
\begin{align}
\label{eq:weighted}
\mathcal{A}_{s}(\{(f_i(x), s_i(x))|i\in \mathcal{S}_t\}) = \sigma\left(\frac{\sum_{i\in \mathcal{S}_t} s_i(x)f_i(x)}{\sum_{i\in \mathcal{S}_t} s_i(x)}\right)
\end{align}
The question remains, how to calculate the certainty scores $s_i(x)$ in a privacy preserving way and for arbitrary high-dimensional data, where simple methods, such as Gaussian KDE used in our toy example, fall victim to the curse of dimensionality. To this end, we propose the following methodology:
We split the available auxiliary data randomly into two disjoint subsets,
$
D^-~\cup~ D_{distill}= D_{aux},
$
the "negative" data and the "distillation" data. Using the pre-trained model $h_0$ ($\rightarrow$ sec. \ref{sec:pretrain}) as a feature extractor, on each client, we then train a logistic regression classifier to separate the local data $D_i$ from the negatives $D^-$, by optimizing the following regularized empirical risk minimization problem
\begin{align}
\label{eq:ERM}
w_i^* = \arg\min_{w} J(w, h_0, D_i, D^-)
\end{align}
with
\begin{align}
\begin{split}
J(w, h_0, D_i, D^-) = &a\sum_{x\in D_i \cup D^-}l(t_x\langle w, \tilde{h}_0(x)\rangle)+\lambda R(w).
\end{split}
\end{align}
Hereby $t_x=2(\mathbb{1}_{x\in D_i})-1\in[-1,1]$ defines the binary labels of the separation task, $a=(|D_i|+|D^-|)^{-1}$ is a normalizing factor and $\tilde{h}_0(x)=h_0(x)(\max_{x\in D_i\cup D^-}\|h_0(x)\|)^{-1}$ are the normalized features. We choose $l(z) = \log(1+\exp(z))$ to be the logistic loss and $R(w) = \frac{1}{2}\|w\|^2_2$ to be the $\ell_2$-regularizer. Since $J$ is $\lambda$-strongly convex in $w$, problem \eqref{eq:ERM} is uniquely solvable. This step is performed only once on every client, during the preparation phase (cf. Fig.~\ref{fig:overview}, P2) and the computational overhead for the clients of solving \eqref{eq:ERM} is negligible in comparison to the cost of multiple rounds of training the (deep) model $f_i$.
Given the solution of the regularized ERM $w_i^*$, the certainty scores on the distillation data $D_{distill}$ can be obtained via
\begin{align}
s_i(x)=(1+\exp(-\langle w_i^*, \tilde{h}_0(x)\rangle))^{-1}+\xi.
\end{align}
A small additive $\xi>0$ ensures numerical stability when taking the weighted mean in \eqref{eq:weighted} (we set $\xi=1e-8$). In Appendix \ref{supp:domain_adaptation}, we provide further empirical results, suggesting that our certainty-weighted averaging method \eqref{eq:weighted} approximates a robust aggregation rule proposed in \macrocite{mansour2008domain}
\subsection{Privacy Analysis}
\label{sec:privacy}
Sharing the certainty scores $\{s_i(x)|x\in D_{distill}\}$ with the central server intuitively causes privacy loss for the clients. After all, a high score $s_i(x)$ indicates, that the public data point $x\in D_{distill}$ is similar to the private data $D_i$ of client $i$ (in the sense of \eqref{eq:ERM}). To protect the privacy of the clients, quantify and limit the privacy loss, we propose to use data-level differential privacy (cf. Fig. \ref{fig:overview}, P3). Following the classic definition of \macrocite{dwork2014algorithmic}, a randomized mechanism is called differentially private, if it's output on any input data base $d$ is indistinguishable from output on any neighboring database $d'$ which differs from $d$ in one element.
\begin{definition}
A randomized mechanism $\mathcal{M} : \mathcal{D} \rightarrow{\mathcal{R}}$ satisfies $(\varepsilon,\delta)$-differential privacy if for any two adjacent inputs $d$ and $d'$ that differ in only one element and for any subset of outputs $S\subseteq \mathcal{R}$, it holds that
\begin{align}
P[\mathcal{M}(d)\in S]\leq \exp(\varepsilon)P[\mathcal{M}(d')\in S]+\delta.
\end{align}
\end{definition}
Differential privacy of a mechanism $\mathcal{M}$ can be achieved, by limiting it's sensitivity
\begin{align}
\Delta(\mathcal{M}) = \max_{d_1,d_2\in\mathcal{D}}\|\mathcal{M}(d_1)-\mathcal{M}(d_2)\|
\end{align}
and then applying a randomized noise mechanism. We adapt a Theorem from \macrocite{chaudhuri2011differentially} to establish the sensitivity of \eqref{eq:ERM}:
\begin{theorem}
\label{theo:1}
If $R(\cdot)$ is differentiable and 1-strongly convex and $l$ is differentiable with $|l'(z)|\leq 1$ $\forall z$, then the $\ell^2$-sensitivity $\Delta_2(\mathcal{M})$ of the mechanism
\begin{align}
\mathcal{M} : D_i \mapsto \arg\min_{w} J(f, h_0, D_i, D^-)
\end{align}
is at most $2(\lambda(|D_i|+|D^-|))^{-1}$.
\end{theorem}
The proof can be found in Appendix \ref{supp:proof}. As we can see the sensitivity scales inversely with the size of the total data $|D_i|+|D^-|$. From Theorem \ref{theo:1} and application of the Gaussian mechanism \macrocite{dwork2014algorithmic} it follows that the randomized mechanism
\begin{align}
\label{eq:san}
\mathcal{M}_{san} : D_i \mapsto \arg\min_{f} J(f, h_0, D_i, D^-)+N
\end{align}
with $N\sim\mathcal{N}(\mathbf{0}, I\sigma^2)$ and $\sigma^2=\frac{8\ln(1.25\delta^{-1})}{\varepsilon^2\lambda^2(|D_i|+|D_{aux}|)^2}$ is $(\varepsilon, \delta)$-differentially private.
The post-processing property of DP ensures that the release of any number of scores computed using the output of mechanism $\mathcal{M}_{san}$ is still $(\varepsilon,\delta)$-private. Note, that in this work we restrict ourselves to the privacy analysis of the scoring mechanism. The differentially private training of deep classifiers $f_i$ is a challenge in it's own right and has been addressed e.g. in \macrocite{abadi2016deep}. Following the basic composition theorem \macrocite{dwork2014algorithmic}, the total privacy cost of running \textsc{FedAUX} is the sum of the privacy loss of the scoring mechanism $\mathcal{M}_{san}$ and the privacy loss of communicating the updated models $f_i$ (the latter is the same for all FL algorithms).
\section{Experiments}
\subsection{Setup}
\textbf{Datasets and Models:} We evaluate \textsc{FedAUX} and SOTA FL methods on both Federated image and text classification problems with large scale convolutional and transformer models respectively. For our image classification problems we train ResNet- \macrocite{he2016deep}, MobileNet- \macrocite{sandler2018mobilenetv2} and ShuffleNet- \macrocite{zhang2018shufflenet} type models on CIFAR-10 and CIFAR-100 and use STL-10, CIFAR-100 and SVHN as well as different subsets of ImageNet (Mammals, Birds, Dogs, Devices, Invertebrates, Structures)\footnote{The methodology for generating these subsets is described in Appendix \ref{supp:iamgenet_subsets}} as auxiliary data. In our experiments, we always use 80\% of the auxiliary data as distillation data $D_{distill}$ and 20\% as negative data $D^-$. For our text classification problems we train Tiny-Bert \macrocite{jiao2020tinybert} on the AG-NEWS \macrocite{Zhang2015CharacterlevelCN} and Multilingual Amazon Reviews Corpus \macrocite{marc_reviews} and use BookCorpus \macrocite{Zhu_2015_ICCV} as auxiliary data.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{images/summary_distillation_methods.pdf}
\vspace{-0.8cm}
\caption{Evaluation on \textbf{different neural networks} and client population sizes $n$. Accuracy achieved after $T=100$ communication rounds by different Federated Distillation methods at different levels of data heterogeneity $\alpha$. STL-10 is used as auxiliary data set. In the "Mixed" setting one third of the client population each trains on ResNet8, MobileNetv2 and Shufflenet respectively. Black dashed line indicates centralized training performance.}
\label{fig:summary_distillation}
\includegraphics[width=0.8\textwidth]{images/results_transformer.pdf}
\vspace{-0.5cm}
\caption{Evaluating \textsc{FedAUX} on \textbf{NLP Benchmarks}. Performance of \textsc{FedAUX} for different combinations of local datasets and heterogenity levels $\alpha$. 10 clients training TinyBERT at $\alpha=0.01$ and $C=100\%$. Bookcorpus is used as auxiliary data set. Black dashed line indicates centralized training performance. }
\label{fig:transformer}
\includegraphics[width=0.8\textwidth]{images/dp_analysis2.pdf}
\vspace{-0.5cm}
\caption{\textbf{Privacy Analysis}. Performance of \textsc{FedAUX} for different combinations of the privacy parameters $\varepsilon$, $\delta$ and $\lambda$. 40 clients training Resnet-8 for $T=10$ rounds on CIFAR-10 at $\alpha=0.01$ and $C=40\%$. STL-10 is used as auxiliary data set.}
\label{fig:dp_analysis}
\end{figure*}
\begin{table*}[t!]
\centering
\caption{Maximum accuracy achieved by \textsc{FedAUX} and other baseline FL methods after $T=100$ communication rounds, at \textbf{different participation rates $C$} and levels of data heterogeneity $\alpha$. 20 Clients training ResNet-8 on CIFAR-10. Auxiliary data used is STL10. $^*$Methods assume availability of auxiliary data. $^\dagger$Improved Baselines.}
\label{tab:participation_rate}
\begin{tabular}{llllllll}
\toprule
& \multicolumn{3}{c}{$\alpha=0.01$} & & \multicolumn{3}{c}{$\alpha=100.0$} \\
\cline{2-4}\cline{6-8}
Method & $C=0.2$ & $C=0.4$ & $C=0.8$ & & $C=0.2$ & $C=0.4$ & $C=0.8$ \\
\midrule
\textsc{FedAVG} \macrocite{mcmahan2017communication} & 19.9$\pm$0.7 & 23.6$\pm$2.0 & 28.9$\pm$2.0 & & 81.3$\pm$0.1 & 82.2$\pm$0.0 & 82.3$\pm$0.1 \\
\textsc{FedPROX} \macrocite{li2019fedprox} & 28.4$\pm$2.5 & 34.0$\pm$1.9 & 42.0$\pm$1.0 & & 81.4$\pm$0.1 & 82.3$\pm$0.2 & 82.0$\pm$0.3 \\
\textsc{FedDF}$^*$ \macrocite{lin2020ensemble} & 25.0$\pm$0.8 & 27.8$\pm$0.8 & 30.6$\pm$0.3 & & 80.8$\pm$0.1 & 81.4$\pm$0.3 & 81.5$\pm$0.3 \\
\textsc{FedBE}$^*$ \macrocite{chen2020feddistill} & 20.9$\pm$0.6 & 25.7$\pm$1.4 & 29.1$\pm$0.1 & & 81.4$\pm$0.7 & 82.0$\pm$0.1 & 82.2$\pm$0.2 \\
\textsc{FedAVG+P}$^*$$^\dagger$ & 30.4$\pm$7.9 & 32.1$\pm$2.0 & 38.4$\pm$0.5 & & 89.0$\pm$0.1 & \textbf{89.5$\pm$0.1} & \textbf{89.6$\pm$0.1} \\
\textsc{FedPROX+P}$^*$$^\dagger$ & 42.8$\pm$2.7 & 43.1$\pm$0.2 & 49.0$\pm$0.7 & & 88.9$\pm$0.0 & 89.1$\pm$0.1 & 89.4$\pm$0.0 \\
\textsc{FedDF+P}$^*$$^\dagger$ & 28.8$\pm$3.0 & 39.3$\pm$3.6 & 48.1$\pm$1.1 & & 88.8$\pm$0.0 & 88.9$\pm$0.1 & 88.9$\pm$0.1 \\
\textsc{FedBE+P}$^*$$^\dagger$ & 30.2$\pm$2.2 & 29.8$\pm$0.8 & 37.7$\pm$0.0 & & \textbf{89.1$\pm$0.1} & 89.5$\pm$0.2 & 89.5$\pm$0.0 \\
\textsc{FedAUX}$^*$ & \textbf{54.2$\pm$0.3} & \textbf{71.2$\pm$2.1} & \textbf{78.5$\pm$0.0} & & 88.9$\pm$0.0 & 89.0$\pm$0.0 & 89.0$\pm$0.1 \\
\bottomrule
\end{tabular}
\caption{Maximum accuracy achieved by \textsc{FedAUX} and other baseline FL methods after 100 communication rounds, when \textbf{different sets of unlabeled auxiliary data} are used for pre-training and/ or distillation. 40 Clients training ResNet-8 on CIFAR-10 at $C=40\%$.}
\label{tab:aux_data}
\begin{tabular}{llrrrrrrrr}
\toprule
& & \multicolumn{8}{c}{Auxiliary Data}\\
\cline{3-10}
$\alpha$ & Method & STL-10 & CIFAR-100 & SVHN & Invertebr. & Birds & Devices & Dogs & Structures \\
\midrule
0.01 & \textsc{FedDF} & 27.9$\pm$3.2 & 29.5$\pm$6.2 & 28.1$\pm$3.9 & 28.5$\pm$3.6 & 30.1$\pm$2.0 & 26.3$\pm$0.2 & 28.9$\pm$5.1 & 30.2$\pm$7.0 \\
& \textsc{FedDF+P} & 43.0$\pm$5.2 & 41.6$\pm$1.1 & 29.6$\pm$3.4 & 38.8$\pm$6.5 & 41.4$\pm$5.9 & 35.9$\pm$4.9 & 41.1$\pm$7.3 & 36.7$\pm$7.1 \\
& \textsc{FedAUX} & \textbf{76.8$\pm$0.9} & \textbf{71.5$\pm$2.5} & \textbf{43.7$\pm$1.5} & \textbf{68.2$\pm$0.7} & \textbf{65.7$\pm$3.1} & \textbf{71.5$\pm$0.1} & \textbf{71.8$\pm$3.8} & \textbf{64.1$\pm$3.3} \\
\midrule
100.00 & \textsc{FedDF} & 79.3$\pm$0.7 & 79.9$\pm$0.1 & 80.9$\pm$0.1 & 80.2$\pm$0.1 & 80.2$\pm$0.4 & 79.4$\pm$0.3 & 79.7$\pm$0.4 & 80.1$\pm$0.2 \\
& \textsc{FedDF+P} & 88.3$\pm$0.0 & 86.7$\pm$0.0 & \textbf{81.7$\pm$0.2} & 87.4$\pm$0.1 & 87.6$\pm$0.0 & 87.7$\pm$0.1 & 88.4$\pm$0.0 & \textbf{87.4$\pm$0.1} \\
& \textsc{FedAUX} & \textbf{88.5$\pm$0.0} & \textbf{86.7$\pm$0.1} & 81.6$\pm$0.0 & \textbf{87.8$\pm$0.1} & \textbf{87.8$\pm$0.1} & \textbf{87.8$\pm$0.0} & \textbf{88.6$\pm$0.0} & 87.3$\pm$0.1 \\
\bottomrule
\end{tabular}
\caption{\textbf{One-shot performance} of different FL methods. Maximum accuracy achieved after $T=1$ communication rounds at participation-rate $C=100\%$. Each client trains for $E=40$ local epochs.}
\label{tab:one-shot}
\begin{tabular}{lllllcllll}
\toprule
& \multicolumn{4}{c}{MobileNetv2, $n=100$} & & \multicolumn{4}{c}{Shufflenet, $n=100$} \\
\cline{2-5}
\cline{7-9}
Method & $\alpha=0.01$ & $\alpha=0.04$ & $\alpha=0.16$ & $\alpha=10.24$ & & $\alpha=0.01$ & $\alpha=0.04$ & $\alpha=0.16$ & $\alpha=10.24$ \\
\midrule
\textsc{FedAVG} & 10.3$\pm$0.0 & 13.6$\pm$2.3 & 23.6$\pm$0.0 & 30.5$\pm$0.9 & & 12.1$\pm$0.8 & 17.4$\pm$0.4 & 28.2$\pm$0.8 & 37.8$\pm$0.7 \\
\textsc{FedPROX} & 11.6$\pm$0.8 & 14.3$\pm$1.4 & 23.7$\pm$0.3 & 30.5$\pm$0.5 & & 12.9$\pm$1.7 & 18.9$\pm$0.2 & 29.4$\pm$0.3 & 38.9$\pm$0.5 \\
\textsc{FedDF} & 16.8$\pm$4.2 & 29.5$\pm$3.8 & 37.7$\pm$1.1 & 40.4$\pm$0.5 & & 16.0$\pm$5.1 & 27.3$\pm$0.1 & 38.7$\pm$0.2 & 45.5$\pm$0.5 \\
\textsc{FedAVG+P} & 24.3$\pm$1.1 & 44.0$\pm$4.4 & 57.6$\pm$3.7 & 69.9$\pm$0.0 & & 25.5$\pm$1.4 & 44.2$\pm$0.1 & 62.9$\pm$1.6 & 71.9$\pm$0.1 \\
\textsc{FedPROX+P} & 27.2$\pm$2.2 & 43.4$\pm$3.6 & 56.9$\pm$3.9 & 70.0$\pm$0.1 & & 28.4$\pm$0.2 & 47.1$\pm$1.5 & 63.3$\pm$1.2 & 71.9$\pm$0.1 \\
\textsc{FedDF+P} & 46.7$\pm$5.6 & 61.1$\pm$1.3 & 67.6$\pm$0.5 & 71.2$\pm$0.1 & & 40.4$\pm$2.7 & 59.4$\pm$0.8 & 68.8$\pm$0.2 & 72.7$\pm$0.0 \\
\textsc{FedAUX} & \textbf{64.8$\pm$0.0} & \textbf{65.5$\pm$1.0} & \textbf{68.2$\pm$0.2} & \textbf{71.3$\pm$0.1} & & \textbf{66.9$\pm$0.6} & \textbf{68.6$\pm$0.4} & \textbf{70.8$\pm$0.3} & \textbf{72.9$\pm$0.1} \\
\bottomrule
\end{tabular}
\end{table*}
\textbf{Federated Learning environment and Data Partitioning}: We consider Federated Learning problems with up to $n=100$ participating clients. In all experiments, we split the training data evenly among the clients according to a dirichlet distribution following the procedure outlined in \macrocite{hsu2019measuring} and illustrated in Fig. \ref{fig:alpha}. This allows us to smoothly adapt the level of non-iid-ness in the client data using the dirichlet parameter $\alpha$. We experiment with values for $\alpha$ varying between 100.0 and 0.01. A value of $\alpha=100.0$ results in an almost identical label distribution, while setting $\alpha=0.01$ results in a split, where the vast majority of data on every client stems from one single class. See Appendix \ref{supp:data_splitting} for a more detailed description of our data splitting procedure. We vary the client participation rate $C$ in every round between 20\% and 100\%.
\textbf{Pre-training strategy:} For our image classification problems, we use contrastive representation learning as described in \macrocite{chen2020simple} for pre-training. We use the default set of data augmentations proposed in the paper and train with the Adam optimizer, learning rate set to $10^{-3}$ and a batch-size of 512. For our text classification problems, we pre-train using self-supervised next-word prediction.
\textbf{Training the Scoring model and Privacy Setting:} We set the default privacy parameters to $\lambda=0.1$, $\varepsilon=0.1$ and $\delta=1e-5$ respectively and solve \eqref{eq:ERM} by running L-BFGS \macrocite{liu1989limited} until convergence ($\leq 1000$ steps).
\textbf{Baselines:} We compare the performance of \textsc{FedAUX} to state-of-the-art FL methods: \textsc{FedAVG} \macrocite{mcmahan2017communication}, \textsc{FedProx} \macrocite{li2019fedprox}, Federated Ensemble Distillation (\textsc{FedDF}) \macrocite{lin2020ensemble} and \textsc{FedBE} \macrocite{chen2020feddistill}. To clearly discern the performance benefits of the two components of \textsc{FedAUX} (unsupervised pre-training and weighted ensemble distillation), we also report performance metrics on versions of these methods where the auxiliary data was used to pre-train the feature extractor $h$ ("\textsc{FedAVG+P}", "\textsc{FedProx+P}", "\textsc{FedDF+P}" resp. "\textsc{FedBE+P}"). For \textsc{FedBE} we set the sample size to 10 as suggested in the paper. For \textsc{FedProx} we always tune the proximal parameter $\mu$.
\textbf{Optimization:} On all image classification task, we use the very popular Adam optimizer \macrocite{kingma2014adam}, with a fixed learning rate of $\eta=10^{-3}$ and a batch-size of 32 for local training. Distillation is performed for one epoch for all methods using Adam at a batch-size of 128 and fixed learning rate of $5e-5$. More detailed hyperparameter analysis in Appendix \ref{supp:hyperparameter} shows that this choice of optimization parameters is approximately optimal for all of the methods. If not stated otherwise, the number of local epochs $E$ is set to 1.
\subsection{Evaluating \textsc{FedAUX} on common Federated Learning Benchmarks}
We start out by evaluating the performance of \textsc{FedAUX} on classic benchmarks for Federated image classification. Figure \ref{fig:summary_distillation} shows the maximum accuracy achieved by different Federated Distillation methods after $T=100$ communication rounds at different levels of data heterogeneity. As we can see, \textsc{FedAUX} distinctively outperforms \textsc{FedDF} on the entire range of data heterogeneity levels $\alpha$ on all benchmarks. For instance, when training ResNet8 with $n=80$ clients at $\alpha=0.01$, \textsc{FedAUX} \emph{raises the maximum achieved accuracy from 18.2\% to 78.1\%} (under the same set of assumptions). The two components of \textsc{FedAUX}, unsupervised pre-training and weighted ensemble distillation, both contribute independently to the performance improvement, as can be seen when comparing with \textsc{FedDF+P}, which only uses unsupervised pre-training. Weighted ensemble distillation as done in \textsc{FedAUX} leads to greater or equal performance than equally weighted distillation (\textsc{FedDF+P}) across all levels of data heterogeneity. The same overall picture can be observed in the "Mixed" setting where clients train different model architectures. Detailed training curves are given in the Appendix \ref{supp:training_curves}.
Table \ref{tab:participation_rate} compares the performance of \textsc{FedAUX} and baseline methods at different client participation rates $C$. We can see that \textsc{FedAUX} benefits from higher participation rates. In all scenarios, methods which are initialized using the pre-trained feature-extractor $h_0$ distinctively outperform their randomly initialized counterparts. In the iid setting at $\alpha=100.0$ \textsc{FedAUX} is mostly en par with the (improved) parameter averaging based methods \textsc{FedAVG+P} and \textsc{FedPROX+P}, with a maximum performance gap of 0.8\%. At $\alpha=0.01$ on the other hand \textsc{FedAUX} outperforms all other methods with a margin of up to 29\%.
\subsection{Evaluating \textsc{FedAUX} on NLP Benchmarks}
Figure \ref{fig:transformer} shows learning curves for Federated training of TinyBERT on the Amazon and AG-News datasets at two different levels of data heterogeneity $\alpha$. We observe, that \textsc{FedAUX} significantly outperforms \textsc{FedDF+P} as well as \textsc{FedAVG+P} in the heterogeneous setting ($\alpha=0.01$) and reaches 95\% of its final accuracy after one communication round on both datasets, indicating suitability for one-shot learning. On more homogeneous data ($\alpha=1.0$) \textsc{FedAUX} performs mostly en par with pre-trained versions of \textsc{FedAVG} and \textsc{FedDF}, with a maximal performance gap of 1.1 \% accuracy on the test set.
We note, that effects of data heterogeneity are less severe as in this setting as both the AG News and the Amazon data set only have four and five labels respectively and an $\alpha$ of $1.0$ already leads to a distribution where each clients owns a subset of the private data set containing all possible labels. Further details on our implementation can be found the Appendix \ref{supp:transformer_implementation}.
\subsection{Privacy Analysis of \textsc{FedAUX}}
\label{sec:ex_privacy}
Figure \ref{fig:dp_analysis} examines the dependence of \textsc{FedAUX}' training performance of the privacy parameters $\varepsilon$, $\delta$ and the regularization parameter $\lambda$. As we can see, performance comparable to non-private scoring is achievable at conservative privacy parameters $\varepsilon$, $\delta$. For instance, at $\lambda=0.01$ setting $\varepsilon=0.04$ and $\delta=10^{-6}$ reduces the accuracy from 74.6\% to 70.8\%. At higher values of $\lambda$, better privacy guarantees have an even less harmful effect, at the cost however of an overall degradation in performance. Throughout this empirical study, we have set the default privacy parameters to $\lambda=0.1$, $\varepsilon=0.1$ and $\delta=1e-5$.
We also perform an empirical privacy analysis in the Appendix \ref{supp:epirical_privacy}, which provides additional intuitive understanding and confidence in the privacy properties of our method.
\subsection{Evaluating the dependence on Auxiliary Data}
\label{sec:auxiliary_data}
Next, we investigate the influence of the auxiliary data set $D_{aux}$ on unsupervised pretraining, distillation and weighted distillation respectively. We use CIFAR-10 as training data set and consider 8 different auxiliary data sets, which differ w.r.t their similarity to this client training data - from more similar (STL-10, CIFAR-100) to less similar (Devices, SVHN)\footnote{The CIFAR-10 data set contains images from the classes airplane, automobile, bird, cat, deer, dog, frog, horse, ship and truc.}.
Table \ref{tab:aux_data} shows the maximum achieved accuracy after $T=100$ rounds when each of these data sets is used as auxiliary data. As we can see, performance \emph{always} improves when auxiliary data is used for unsupervised pre-training. Even for the highly dissimilar SVHN data set (which contains images of house numbers) performance of \textsc{FedDF+P} improves by 1\% over \textsc{FedDF} in both the iid and non-iid regime. For other data sets like Dogs, Birds or Invertebrates performance improves by up to 14\%, although they overlap with only one single class of the CIFAR-10 data set. The outperformance of \textsc{FedAUX} on such a wide variety of highly dissimilar data sets suggest that beneficial auxiliary data should be available in the majority of practical FL problems and also has positive implications from the perspective of privacy. Interestingly, performance of \textsc{FedDF} seems to only weakly correlate with the performance of \textsc{FedDF+P} and \textsc{FedAUX} as a function of the auxiliary data set. This suggests, that the properties, which make a data set useful for distillation are not the same ones that make it useful for pre-training and weighted distillation. Investigating this relationship further is an interesting direction of future research.
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{images/alpha_small.pdf}
\vspace{-0.8cm}
\caption{Illustration of the \textbf{Dirichlet data splitting strategy} we use throughout the paper, exemplary for a Federated Learning setting with 20 Clients and 10 different classes. Marker size indicates the number of samples held by one client for each particular class. Lower values of $\alpha$ lead to more heterogeneous distributions of client data. Figure adapted from \macrocite{lin2020ensemble}.}
\label{fig:alpha}
\includegraphics[width=0.5\textwidth]{images/linear_resnet2.pdf}
\vspace{-0.8cm}
\caption{\textbf{Linear evaluation}. Training curves for different Federated Learning methods at different levels of data heterogeneity $\alpha$ when only the classification head $g$ is updated in the training phase. A total of $n=80$ clients training ResNet8 on CIFAR-10 at $C=40\%$, using STL-10 as auxiliary data set.}
\label{fig:linear}
\end{figure}
\subsection{\textsc{FedAUX} in hardware-constrained settings}
\textbf{Linear Evaluation:}
In settings where the FL clients are hardware-constrained mobile or IoT devices, local training of entire deep neural networks like ResNet8 might be infeasible. We therefore also consider the evaluation of different FL methods, when only the linear classification head $g$ is updated during the training phase. Figure \ref{fig:linear} shows training curves in this setting when clients hold data from the CIFAR-10 data set. We see that in this setting performance of \textsc{FedAUX} is high, independent of the data heterogeneity levels $\alpha$, suggesting that in the absence of non-convex training dynamics our proposed scoring method actually yields robust weighted ensembles in the sense of \macrocite{mansour2008domain}. We note, that \textsc{FedAUX} also trains much more smoothly, than all other baseline methods.
\textbf{One-Shot Evaluation:}
In many FL applications, the number of times a client can participate in the Federated training is restricted by communication, energy and/ or privacy constraints \macrocite{guha2019one-shot, papernot2018scalable}. To study these types of settings, we investigate the performance of \textsc{FedAUX} and other FL methods in Federated one-shot learning where we set $T=1$ and $C=100\%$. Table \ref{tab:one-shot} compares performance in this setting for $n=100$ clients training MobileNetv2 resp. ShuffleNet. \textsc{FedAUX} outperforms the baseline methods in this setting at all levels of data heterogeneity $\alpha$.
\section{Conclusion}
In this work, we explored Federated Learning in the presence of unlabeled auxiliary data, an assumption made in the quickly growing area of Federated Distillation. By leveraging auxiliary data for unsupervised pre-training and weighted ensemble distillation we were able to demonstrate that this assumption is rather strong and can lead to drastically improved performance of FL algorithms. These results reveal the limited merit in comparing FD based methods with parameter averaging based methods (which do not make this assumption) and thus have implications for the future evaluation of FD methods in general.
\clearpage
\setcounter{section}{0}
\renewcommand{\thesection}{\Alph{section}}
\twocolumn[
\icmltitle{\textsc{FedAUX}: Leveraging Unlabeled Auxiliary Data in Federated Learning\\- \textsc{Supplementary Materials} -}
\vskip 0.3in
]
\section{Extended Related Work Discussion}
\label{supp:related}
\textbf{Ensemble Distillation in Federated Learning:}
A new family of Federated Learning methods leverages model distillation \suppcite{hinton2015distill} to aggregate the client knowledge \suppcite{jeong2018distill, lin2020ensemble, itahara2020distill, chen2020feddistill}. These Federated Distillation (FD) techniques have at least three distinct advantages over prior, parameter averaging based methods and related work can be organized according to which of these aspects it primarily focuses on.
First, Federated Distillation enables aggregation of client knowledge independent of the model architecture and thus allows clients to train models of different structure, which gives additional flexibility, especially in hardware-constrained settings. \textsc{FedMD} \suppcite{li2019fedmd}, Cronus \suppcite{chang2019cronus} and \textsc{FedH2L} \suppcite{li2021fedh2l} address this aspect. FedMD additionally requires to locally pre-train on the \emph{labeled} public data which makes it difficult to perform a fair numerical comparison. FedH2L requires communication of soft-label information after every gradient descent step and is thus not suitable for most practical FL applications where communication channels are intermittent. Cronus addresses aspects of robustness to adversaries but is shown to perform consistently worse than \textsc{FedAVG} in conventional FL. While we do not focus on this aspect, our proposed approach is flexible enough to handle heterogeneous client models (c.f. Appendix \ref{supp:algorithm}).
Second, Federated Distillation has advantageous communication properties. As models are aggregated by means of distillation instead of parameter averaging it is no longer necessary to communicate the raw parameters. Instead it is sufficient for the clients to only send their soft-label predictions on the distillation data. Consequently, the communication in FD scales with the size of the distillation data set and not with the size of the jointly trained model as in the classical parameter averaging based FL. This leads to communication savings, especially if the local models are large and the distillation data set is small.
Jeong~et.~al~ and subsequent work \suppcite{jeong2018distill, itahara2020distill, seo2020fd, sattler2020communication} focus on this aspect. These methods however are computationally more expensive for the resource constrained clients, as distillation needs to be performed locally and perform worse than parameter averaging based training after the same number of communication rounds. Our proposed approach relies on communication of full models and thus requires communication at the order of conventional parameter averaging based methods.
Third, when combined with parameter averaging, Federated Distillation methods achieve better performance than purely parameter averaging based techniques. Both the authors in \suppcite{lin2020ensemble} and \suppcite{chen2020feddistill} propose FL protocols, which are based on classical \textsc{FedAVG} and perform ensemble distillation after averaging the received client updates at the server to improve performance. \textsc{FedBE}, proposed by \suppcite{chen2020feddistill}, additionally combines client predictions by means of a Bayesian model ensemble to further improve robustness of the aggregation. Our work primarily focuses on this latter aspect. Building upon the work of \suppcite{lin2020ensemble}, we additionally leverage the auxiliary distillation data for unsupervised pre-training and weigh the client predictions in the distillation step according to their certainty scores to better cope with settings where the client’s data generating distributions are statistically heterogeneous.
We also mention the related work by Guha~et~al.~\suppcite{guha2019one-shot}, which proposes a one-shot distillation method for convex models, where the server distills the locally optimized client models in a single round as well as the work of \suppcite{sun2020federated} which addresses privacy issues in Federated Distillation. Federated one-shot distillation is also addressed in \suppcite{zhou2020distilled}. Federated Distillation for edge-learning was proposed in \suppcite{ahn2019wireless}.
\textbf{Weighted Ensembles:}
The study of weighted ensembles started around the '90s with the work by \suppcite{hashem1993ensemble, perrone1993ensemble, sollich1995learning}.
A weighted ensemble of models combines the output of the individual models by means of a weighted average in order to improve the overall generalization performance. The weights allow to indicate the percentage of trust or expected performance for each individual model.
See \suppcite{sharkey1996combining, opitz1999popular} for an overview of ensemble methods.
Instead of giving each client a static weight in the aggregation step of distillation, we weight the clients on an instance base as in \suppcite{jimenez1998dynamically}, i.e., each clients prediction is weighted using a data-dependent certainty score. Weighted combinations of weak classifiers are also commonly leveraged in centralized settings in the context of of mixture of experts and boosting methods \suppcite{yuksel2012twenty, masoudnia2014mixture, schapire1999brief}.
\textbf{Data Heterogeneity in Federated Learning:}
As the training data is generated independently on the participation devices, Federated Learning problems are typically characterised by statistically heterogeneous client data \suppcite{mcmahan2017communication}. It is well known, that conventional FL algorithms like \textsc{FedAVG} \suppcite{mcmahan2017communication} perform best on statistically homogeneous data and suffer severely in this (“non-iid”) setting \suppcite{zhao2018federated, li2020covergence}. A number of different studies \suppcite{li2019fedprox, zhao2018federated, sattler2019robust, chen2020feddistill} have tried to address this issue, but relevant performance improvements so far have only been possible under strong assumptions. For instance \suppcite{zhao2018federated} assume that the server has access to \emph{labeled} public data from the \emph{same} distribution as the clients. In contrast, we only assume that the server has access to \emph{unlabeled} public data from a potentially \emph{deviating} distribution. Other approaches \suppcite{sattler2019robust} require high-frequent communication, with up to thousands of communication rounds, between server and clients, which might be prohibitive in a majority of FL applications where communication channels are intermittent and slow. In contrast, our proposed approach can drastically improve FL performance on non-iid data even after just one single communication round.
For completeness, we note that there exists also a different line of research, which aims to address data heterogeneity in FL via meta- and multi-task learning. Here, separate models are trained for each client \suppcite{smith2017fedMTL, wu2020dpFedMTL} or clients are grouped into different clusters with similar distributions \suppcite{ghosh2019noniid, sattler2020clustered}.
\textbf{Unlabeled Data in Federated Learning:}
To the best of our knowledge, there do not exist any prior studies on the use of unlabeled auxiliary data in FL outside of Federated Distillation methods.
Federated semi-supervised learning techniques \suppcite{zhang2020benchmarking, jeong2020federated} assume that clients hold both labeled and unlabeled private data from the local training distribution. In contrast, we assume that the server has access to public unlabeled data that may differ in distribution from the local client data. Federated self-supervised representation learning \suppcite{zhang2020federated} aims to train a feature extractor on private unlabeled client data. In contrast, we leverage self-supervised representation learning at the server to find a suitable model initialization.
\textbf{Personalization and Federated Transfer Learning:}
The aim of Transfer Learning is to transfer learned knowledge from a specific domain or task to related domains or tasks.
Transfer learning methods are of particular interest in FL settings where the client's local data generating distributions are statistically heterogeneous.
To address the statistical heterogeneity, methods for personalizing the server model to the client's local distributions, e.g. by using distillation \suppcite{li2019fedmd}, parameter fine-tuning \suppcite{wang2019finetuning, mansour2020personalization} or regularization \suppcite{li2019fedprox}, have been proposed. Transferring knowledge from one domain to another domain raises the question of the generalization capabilities and domain adaptation theory gives answers in the form of generalization bounds. Particularly, multiple-source domain adaptation theory \suppcite{mansour2008domain, bendavid2010domain, hoffman2018domain}, which considers the capabilities of transferring knowledge from multiple source domains to some target domain, is relevant for FL.
One interesting question when having knowledge in multiple source domains is how to weight each individual source domain in the process of transferring knowledge to the target domain.
In the \textsc{FedDF} algorithm \suppcite{lin2020ensemble}, the client's local hypotheses are uniformly averaged to obtain a global hypothesis and it is remarked that domain adaptation theory \suppcite{mansour2008domain, hoffman2018domain} has shown such standard convex combinations of source hypotheses not to be robust for the target domain.
A distribution-weighted combination of the local hypotheses, as suggested by domain adaptation theory \suppcite{mansour2008domain} \suppcite{hoffman2018domain}, based on a privacy-preserving local distribution estimation is posed as an open problem for FL in \suppcite{lin2020ensemble}. We address exactly this open question.
\begin{algorithm}[t!]
\caption{\textsc{FedAUX} Preparation Phase (with different model prototypes $\mathcal{P}$)}
\label{alg:FedAUXprep}
\begin{algorithmic}
\STATE \textbf{init:} Split $D^- \cup D_{distill} \leftarrow D_{aux}$
\STATE \textbf{init:} HashMap $\mathcal{R}$ that maps client $i$ to model prototype $P$
\STATE \underline{Server does:}
\FOR{each model prototype $P\in\mathcal{P}$}
\STATE $h^P_0\leftarrow \text{train\_self\_supervised}(h^P, D_{aux})$
\ENDFOR
\FOR{each client $i \in \{1,..,n\}$ \textbf{in parallel}}
\STATE \underline{Client $i$ does:}
\STATE $P\leftarrow \mathcal{R}[i]$
\STATE $\sigma^2\leftarrow \frac{8\ln(1.25\delta^{-1})}{\varepsilon^2\lambda^2(|D_i|+|D^-|)^2}$
\STATE $w_i^* \leftarrow \arg\min_{w} J(w, h^P_0, D_i, D^-) + \mathcal{N}(\mathbf{0}, I\sigma^2)$
\STATE $\gamma_i\leftarrow\max_{x\in D_i\cup D^-}\|h^P_0(x)\|$
\ENDFOR
\STATE \underline{Server does:}
\FOR{$i=1,..,n$}
\STATE create HashMap
\STATE $s_i \leftarrow \{x\mapsto(1+\exp(-\langle w_i^*, \gamma_i^{-1}h^{P}_0(x)\rangle))^{-1}+\xi$ for $x\in D_{distill}\}$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{Data Splitting Methodology}
\label{supp:data_splitting}
We split the training data among the clients using the common Dirichlet splitting strategy proposed in \suppcite{hsu2019measuring} and later used in \suppcite{lin2020ensemble} and \suppcite{chen2020feddistill}. This approach allows us to smoothly adapt the level of heterogeneity in the client data via the concentration parameter $\alpha$. To generate the data split, we sample $c$ vectors
\begin{align}
p_1,..,p_c\sim\text{Dir}(\alpha),
\end{align}
where c is the number of classes, from the symmetric $n$-categorical Dirichlet distribution. For all $p_i\in\mathbb{R}_{\geq 0}^n$ it then holds $\|p_i\|_1=1$. The vectors are then stacked To address the statistical heterogeneity, methods for personalizing the server model to the client's local distributions, e.g. by using distillation \suppcite{li2019fedmd}, parameter fine-tuning \suppcite{wang2019finetuning, mansour2020personalization} or regularization \suppcite{li2019fedprox}, have been proposed. Transferring knowledge from one domain to another domain raises the question of the general
into a matrix
\begin{align}
P = [p_1, .., p_c]\in\mathbb{R}^{n,c}
\end{align}
which is standardized, by repeatedly normalizing the columns and rows. This process converges quickly and is stopped after 1000 iterations. Let $M_j$ be the amount of data points belonging to class $j$ in the training data set. Each client $i$ is then assigned $P_{i,j}M_j$ (non-overlapping) data points from all classes $j=1,..,c$. Figure \ref{fig:splitting_method} illustrates the splitting procedure and displays random splits of data for $n=20$ and $c=10$. In all our experiments, the data splitting process is controlled by a random seed, to ensure that the different baseline methods are all trained on the same split of data.
\begin{figure*}[t!]
\centering
\includegraphics[width=1.0\textwidth]{images/alpha.pdf}
\caption{Illustration of the Dirichlet data splitting strategy used throughout the paper. Dot size represents number of data points each client holds from any particular class. Lower values of $\alpha$ lead to more heterogeneous splits of data.}
\label{fig:splitting_method}
\end{figure*}
\begin{algorithm}[t!]
\caption{\textsc{FedAUX} Training Phase (with different model prototypes $\mathcal{P}$). Training requires feature extractors $h_0^P$ and scores $s_i$ from Alg. \ref{alg:FedAUXprep}. The same $D^- \cup D_{distill} \leftarrow D_{aux}$ as in Alg. \ref{alg:FedAUXprep} is used. Choose learning rate $\eta$ and set $\xi=10^{-8}$.}
\label{alg:FedAUXtrain}
\begin{algorithmic}
\STATE \textbf{init:} HashMap $\mathcal{R}$ that maps client $i$ to model prototype $P$
\STATE \textbf{init:} Inverse HashMap $\tilde{\mathcal{R}}$ that maps model prototype $P$ to set of clients (s.t. $i\in\tilde{\mathcal{R}}[\mathcal{R}[i]]~\forall i$)
\STATE \textbf{init:} Initialize model prototype weights $\theta^P$ with feature extractor weights $h^P$ from Alg. \ref{alg:FedAUXprep}
\FOR{communication round $t=1,..,T$}
\STATE select subset of clients $\mathcal{S}_t\subseteq \{1,..,n\}$
\FOR{selected clients $i \in \mathcal{S}_t$ \textbf{in parallel}}
\STATE \underline{Client $i$ does:}
\STATE $\theta_i\leftarrow \text{train}(\theta_0\leftarrow\theta^{\mathcal{R}[i]}, D_i)$\hfill \# Local Training
\ENDFOR
\STATE \underline{Server does:}
\FOR{each model prototype $P\in\mathcal{P}$}
\STATE $\theta^P\leftarrow \sum_{i\in \mathcal{S}_t\cap \tilde{\mathcal{R}}[P]}\frac{|D_i|}{\sum_{l\in \mathcal{S}_t\cap \tilde{\mathcal{R}}[P]}|D_l|} \theta_i$ \hfill \# Parameter
\STATE \hfill \# Averaging
\FOR{mini-batch $x \in D_{distill}$}
\STATE $\tilde{y} \leftarrow \sigma\left(\frac{\sum_{i\in \mathcal{S}_t} s_i[x]f_i(x, \theta_i)}{\sum_{i\in \mathcal{S}_t} s_i[x]}\right)$\hfill \# Can be arbitrary
\STATE $\theta^P \leftarrow \theta^P - \eta\frac{\partial D_{KL}(\tilde{y}, \sigma(f(x, \theta^P)))}{\partial \theta^P}$ \hfill \# Optimizer
\ENDFOR
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\textwidth]{images_supplement/detailed_training_curves_ResNet8.pdf}
\caption{Detailed training curves for ResNet-8 trained on CIFAR-10, $n=80$ Clients, $C=40\%$.}
\label{fig:training_curves_resnet}
\includegraphics[width=0.95\textwidth]{images_supplement/detailed_training_curves_MobileNetv2.pdf}
\caption{Detailed training curves for MobileNetv2 trained on CIFAR-10, $n=100$ Clients, $C=40\%$.}
\label{fig:training_curves_mobilenet}
\includegraphics[width=0.95\textwidth]{images_supplement/detailed_training_curves_ShuffleNet.pdf}
\caption{ Shufflenet trained on CIFAR-10, $n=100$ Clients, $C=40\%$.}
\label{fig:training_curves_shufflenet}
\includegraphics[width=0.95\textwidth]{images_supplement/detailed_training_curves_Mixed.pdf}
\caption{Detailed training curves for mixed models trained on CIFAR-10. 20 each train ResNet8, MobileNetv2 and Shufflenet respectively.}
\label{fig:training_curves_mixed}
\end{figure*}
\section{Detailed Algorithm}
\label{supp:algorithm}
The training procedure of \textsc{FedAUX} can be divided into a preparation phase, which is given in Alg. \ref{alg:FedAUXprep} and a training phase, which is given in Alg. \ref{alg:FedAUXtrain}. We describe the general setting where clients may hold different model prototypes $P$ from a set of prototypes $\mathcal{P}$. This general setting simplifies to the setting described in Sec.~\ref{sec:method} if $|\mathcal{P}|=1$.
\textbf{Preparation Phase:} In the preparation phase, the server uses the unlabeled auxiliary data $D_{aux}$, to pre-train the feature extractor $h^P$ for each model prototype $P$ using self-supervised training. Suitable methods for self-supervised pre-training are contrastive representation learning \suppcite{chen2020simple}, or self-supervised language modeling/ next-token prediction \suppcite{devlin2018bert}. The pre-trained feature extractors $h^P_0$ are then communicated to the clients and used to initialize part of the local classifier $f=g\circ h$. The server also communicates the negative data $D^-$ to the clients (in practice we can instead communicate the extracted features $\{|h_0^P(x)|x\in D^-\}$ of the raw data $D^-$ to save communication). Each client then optimizes the logistic similarity objective $J$ \eqref{eq:ERM} and sanitizes the output by adding properly scaled Gaussian noise. Finally, the sanitized scoring model $w_i^*$ is communicated to the server, where it is used to compute certainty scores $s_i$ on the distillation data (the certainty scores can also be computed on the clients, however this results in additional communication of distillation data and scores).
\textbf{Training Phase:} The training phase is carried out in $T$ communication rounds. In every round $t\leq T$, the server randomly selects a subset $\mathcal{S}_t$ of the overall client population and transmits to them the latest server models $\theta^\mathcal{R}[i]$, which match their model prototype $P$ (in round $t=1$ only the pre-trained feature extractor $h^P_0$ is transmitted). Each selected client updates it's local model by performing multiple steps of stochastic gradient descent (or it's variants) on it's local training data. This results in an updated parameterization $\theta_i$ on every client, which is communicated to the server. After all clients have finished their local training, the server gathers the updated parameters $\theta_i$. For each model prototype $P$ the corresponding parameters are then aggregated by weighted averaging. Using the model averages as a starting point, for each prototype the server then distills a new model, based on the client's certainty-weighted predictions.
\section{Qualitative Comparison with Baseline Methods}
\label{supp:qualitative}
Table \ref{tab:compare_qualitative} gives a qualitative comparison between \textsc{FedAUX} and the baseline methods \textsc{FedAVG} and \textsc{FedDF}.
\begin{itemize}
\item Compared with \textsc{FedAVG} and \textsc{FedDF}, \textsc{FedAUX} additionally requires the clients to once solve the $\lambda$-strongly convex ERM \eqref{eq:ERM}. For this problem linearly convergent algorithms are known \suppcite{liu1989limited} and thus the computational overhead is negligible compared with the complexity of multiple rounds of locally training deep neural networks.
\item \textsc{FedAUX} also adds computational load to the server for self-supervised pre-training and computation of the certainty scores $s_i$. As the server is typically assumed to have massively stronger computational resources than the clients, this can be neglected.
\item Once, in the preparation phase of \textsc{FedAUX}, the scoring models $w_i^*$ need to be communicated from the clients to the server. The overhead of communicating these $H$-dimensional vectors, where $H$ is the feature dimension, is negligible compared to the communication of the full models $f_i$.
\item \textsc{FedAUX} also requires the communication of the negative data $D^-$ and the feature extractor $h_0$ from the server to the clients. The overhead of sending $h_0$ is lower than sending the full model $f$, and thus the total downstream communication is increased by less than a factor of $(T+1)/T$. The overhead of sending $D^-$ is small (in our experiments $|D^-|=0.2|D_{aux}|$) and can be further reduced by sending extracted features $\{|h_0^P(x)|x\in D^-\}$ instead of the full data. For instance, in our experiments with ResNet-8 and CIFAR-100 we have $|D^-|=12000$ and $h_0^P(x)\in\mathbb{R}^{512}$, resulting in a total communication overhead of $12000\times512\times4B=24.58$MB for $D^-$. For comparison the total communication overhead of once sending the parameters of ResNet-8 (needs to be done $T$ times) is $19.79$MB.
\item Communicating the scoring models $w_i^*$ incurs additional privacy loss for the clients. Using our proposed sanitation mechanism this process is made $(\varepsilon, \delta)$-differentially private. Our experiments in section \ref{sec:ex_privacy} demonstrate that \textsc{FedAUX} can achieve drastic performance improvements, even under conservative privacy constraints. All empirical results reported are obtained with $(\varepsilon,\delta)$ differential privacy at $\varepsilon=0.1$ and $\delta=10^{-5}$.
\item Finally, \textsc{FedAUX} makes the additional assumption that unlabeled auxiliary data is available to the server. This assumption is made by all Federated Distillation methods including \textsc{FedDF}.
\end{itemize}
\definecolor{Gray}{gray}{0.9}
\begin{table*}[t!]
\centering
\caption{\textbf{Qualitative Comparison:} Complexity, communication overhead, privacy loss after $T$ communication rounds as well as implicit assumptions made by different Federated Learning methods.}
\label{tab:compare_qualitative}
{\renewcommand{\arraystretch}{1.5}
\begin{tabular}{p{2.5cm}|p{2.5cm}p{2.5cm}p{3.9cm}p{3.6cm}}
\toprule
& \textsc{FedAVG} & \textsc{FedDF} & \textsc{FedAUX} (preparation phase) & \textsc{FedAUX} (training phase)\\
\midrule
\rowcolor{Gray}
Operations (Clients) & Local Training ($\times T$)& Local Training ($\times T$)& Solve $\lambda$-strongly convex ERM \eqref{eq:ERM} & Local Training ($\times T$) \\
Operations (Server) & Model Averaging ($\times T$)& Model Averaging, Distillation ($\times T$) & Self-Supervised Pre-training of $h_0$, Computation of certainty scores $s_i$ & Model Averaging, Distillation ($\times T$) \\
\rowcolor{Gray}
Communication Clients $\rightarrow$ Server & Model Parameters $f_i$ ($\times T$) & Model Parameters $f_i$ ($\times T$)& Scoring Models $w_i^*$ & Model Parameters $f_i$ ($\times T$) \\
Communication Server $\rightarrow$ Clients & Model Parameters $f$ ($\times T$) & Model Parameters $f$ ($\times T$) & Negative Data $D^-$, Feature Extractor $h_0$ & Model Parameters $f$ ($\times T$) \\
\rowcolor{Gray}
Privacy Loss & Privacy loss of communicating $f_i$ ($\times T$) & Privacy loss of communicating $f_i$ ($\times T$)& $(\varepsilon, \delta)$-DP & Privacy loss of communicating $f_i$ ($\times T$)\\
Assumptions & No Assumptions & Auxiliary Data & Auxiliary Data & Auxiliary Data
\end{tabular}
}
\end{table*}
\section{Additional Results and Detailed Training Curves}
\label{supp:training_curves}
In this sections we give detailed training curves for the results shown in Figure \ref{fig:summary_distillation}. As can be seen, in the highly non-iid setting at $\alpha\in\{0.01,0.04\}$, all methods exhibit convergence issues. This behavior is well known in FL and is described for instance in \suppcite{zhao2018federated, sattler2019robust}. Notably, the performance of \textsc{FedAUX} after one single communication round exceeds the maximum achieved performance of all other methods over the entire course of training. At higher values of $\alpha\geq 0.16$ all methods train smoothly and validation performance asymptotically increases over the curse of training. \textsc{FedAUX} dominates all baseline methods at all communication rounds in the heterogeneous settings. In the mostly iid-setting at $\alpha=10.24$ \textsc{FedAUX} is en par with the pre-trained version of \textsc{FedDF}.
Table \ref{tab:cifar100} compares performance of \textsc{FedAUX} to baseline methods on the CIFAR-100 data set. Again \textsc{FedAUX} outperforms \textsc{FedAVG} and \textsc{FedDF} across all level of data heterogeneity $\alpha$ and shows superior performance to the improved \textsc{FedDF+P} when data is highly heterogeneous at $\alpha=\{0.01, 0.04\}$. Interestingly in this setting \textsc{FedDF+P} manages to slightly outperform \textsc{FedAUX} at medium data heterogeneity levels $\alpha=\{0.16, 0.64\}$. This indicates that our proposed differentially private certainty scoring method may insufficiently approximate the true client certainty in this setting. We leave potential improvements of this mechanism for future work.
\begin{table}[t!]
\centering
\caption{Results on data sets with \textbf{higher number of classes.} Training ResNet-8 on \textbf{CIFAR-100}. Accuracy achieved after $T=100$ communication rounds by different Federated Distillation methods at different levels of data heterogeneity $\alpha$. STL-10 is used as auxiliary data set.}
\label{tab:cifar100}
\begin{tabular}{lrrrrrr}
\toprule
& \multicolumn{6}{c}{$\alpha$}\\
\cline{2-7}
{} & $0.01$ & $0.04$ & $0.16$ & $0.64$ & $2.56$ & $10.24$ \\
\midrule
FedAVG & 24.1 & 36.3 & 47.2 & 50.7 & 52.2 & 52.2 \\
FedDF & 11.4 & 24.4 & 45.0 & 49.5 & 52.5 & 51.2 \\
FedDF+P & 18.2 & 42.0 & \textbf{58.0} & \textbf{60.8} & 61.6 & 62.0 \\
FedAUX & \textbf{34.1} & \textbf{47.4} & 56.4 & 60.7 & \textbf{62.5} & \textbf{62.5} \\
\bottomrule
\end{tabular}
\end{table}
\section{Details on generating Imagenet subsets}
\label{supp:iamgenet_subsets}
To simulate the effects of a wide variety of auxiliary data sets on the training performance of \textsc{FedAUX}, we generate different structured subsets of the ImageNet data base (resized to $32\times 32\times 3$). Each subset is defined via a top-level Wordnet ID which is shown in Table \ref{tab:wordnetids}. To obtain the images from the subset, we select all leaf-node IDs of the respective top-level IDs via the Imagenet API
\begin{center}
\url{http://www.image-net.org/api/text/wordnet.structure.hyponym?wnid=<top-level ID>&full=1}
\end{center}
and then take only those classes from the full Imagenet data set, which match these leaf-node IDs. Table \ref{tab:wordnetids} also shows the number of samples contained in every subset that was generated this way.
\begin{table}[t!]
\centering
\caption{\textbf{Auxiliary data sets} used in this study and their defining Wordnet IDs and data sets sizes.}
\label{tab:wordnetids}
\begin{tabular}{lll}
\toprule
Data set & Wordnet ID & Dataset Size\\
\midrule
Imagenet Devices & n03183080 & 165747\\
Imagenet Birds & n01503061 & 76541\\
Imagenet Animals & n00015388 & 510530\\
Imagenet Dogs & n02084071 & 147873\\
Imagenet Invertebrates & n01905661 & 79300\\
Imagenet Structures & n04341686 & 74400\\
\bottomrule
\end{tabular}
\end{table}
\section{Details on the Implementation and Results of the NLP Benchmarks}
\label{supp:transformer_implementation}
As mentioned in section 4.3 \textit{Evaluating} \textsc{FedAUX} \textit{on NLP Benchmarks} we used TinyBERT as a model for our NLP experiments. TinyBERT was pre-trained on Bookcorpus\footnote{\url{https://huggingface.co/datasets/bookcorpus}} which led us to select the same dataset as a public dataset in order to follow the methodology outlined in section \ref{sec:pretrain}. As private datasets we chose the AG News dataset\footnote{\url{https://huggingface.co/datasets/ag_news}} \suppcite{Zhang2015CharacterlevelCN}, a topic classification dataset, and the english texts from the Multilingual Amazon Reviews Corpus\footnote{\url{https://huggingface.co/datasets/amazon_reviews_multi}} \suppcite{marc_reviews}, which we use for predicting how many stars a review gets. The pre-trained weights and the tokenizer for TinyBERT are available at the corresponding repository\footnote{\url{https://huggingface.co/huawei-noah/TinyBERT_General_4L_312D}}. All experiments were conducted using $\epsilon = 0.1$ and $\delta=10^{-5}$ as differential privacy parameters, 1 epoch for local training and distillation, ten clients and 100\% participation rate as well as 160000 disjoint data points, which were sampled from BookCorpus, for the public and distillation datasets respectively. Furthermore the ADAM optimizer with a learning rate of $10^{-5}$ was used for both local training and distillation. The regularization strength of the logistic regression classifier was set to $0.01$. The batch size for $D_i, D^{-}$ and $D_{distill}$ was 32. Detailed results for figure \ref{fig:transformer} are depicted in table \ref{tab:transformer_results}.
\begin{table*}[t!]
\centering
\caption{\textbf{NLP Benchmarks} of different FL methods. Maximum accuracy achieved after $T=20$ communication rounds at participation-rate $C=100\%$.}
\label{tab:transformer_results}
\begin{tabular}{lllcll}
\toprule
& \multicolumn{2}{c}{AG News} & & \multicolumn{2}{c}{Amazon} \\
\cline{2-3}
\cline{5-6}
Method &
$\alpha=0.01$ &
$\alpha=1.0$ &
&
$\alpha=0.01$ &
$\alpha=1.0$ \\
\midrule
\textsc{FedAVG+P}&
78.80$\pm$4.40 &
\textbf{92.17$\pm$1.98} &
&
41.70$\pm$0.58 &
\textbf{55.17$\pm$0.40} \\
\textsc{FedDF+P} &
78.05$\pm$7.64 &
90.83$\pm$0.25 &
&
38.04$\pm$0.84 &
54.63$\pm$0.66 \\
\textsc{FedAUX}&
\textbf{85.04$\pm$1.21} &
91.00$\pm$0.30 &
&
\textbf{49.11$\pm$0.22} &
54.86$\pm$0.61 \\
\bottomrule
\end{tabular}
\end{table*}
\section{Hyperparameter Evaluation}
\label{supp:hyperparameter}
In this section we provide a detailed hyperparameter analysis for our proposed method and the baseline methods used in this study. For all methods we use the very popular Adam optimizer for both local training and distillation. We vary the learning rate in $\{1e-2, 1e-3, 1e-4, 1e-5\}$ for local training an distillation. For FedPROX, we vary the parameter $\lambda_{prox}$, controlling the proximal term in the training objective in $\{1e-2, 1e-3, 1e-4, 1e-5\}$. Figure \ref{fig:HPO} compares the maximum achieved accuracy after 50 communication rounds for the different methods and hyperparameter settings, for a FL setting with 20 clients training ResNet-8 on CIFAR-10 at a participation-rate of 40\%. The auxiliary data set we use is STL-10.
For each method and each level of data heterogeneity, table \ref{tab:HPO} shows the accuracy of the best performing combination of hyperparameters. As we can see \textsc{FedAUX} matches the performance of the best performing methods in the iid setting with $\alpha=100.0$ and outperforms all other methods distinctively in the non-iid setting with $\alpha=0.01$.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{images_supplement/hyperparameter_optimization.pdf}
\caption{Results of our \textbf{hyperparameter optimization} for ResNet8. 20 Clients are trained for 50 communication rounds, at a participation rate of $C=40\%$. Both local training and distillation is performed for 1 epoch.}
\label{fig:HPO}
\end{figure*}
\begin{table*}[t!]
\centering
\caption{\textbf{Best performing hyperparameter combinations} for each method when training ResNet8 with $n=20$ clients for 50 communication rounds at a participation rate of $C=40\%$. Both local training and distillation is performed for 1 epoch. Methods sorted by top accuracy.}
\label{tab:HPO}
\begin{tabular}{lrrllr}
\toprule
Method & Alpha & Local LR & Distill LR & $\lambda$ FedProx & Accuracy \\
\midrule
FedPROX+P & 100 & 0.001 & - & 0.0001 & 0.8946 \\
FedAUX & & 0.001 & 1e-05 & - & 0.8941 \\
FedDF+P & & 0.001 & 1e-05 & - & 0.8936 \\
FedAVG+P & & 0.001 & - & - & 0.8924 \\
FedBE & & 0.001 & 1e-05 & - & 0.8246 \\
FedPROX & & 0.001 & - & 0.001 & 0.8232 \\
FedAVG & & 0.001 & - & - & 0.8228 \\
FedDF & & 0.001 & 1e-05 & - & 0.8210 \\
\midrule
FedAUX & 0.01 & 0.001 & 0.0001 & - & 0.7501 \\
FedPROX+P & & 0.01 & - & 0.01 & 0.6122 \\
FedDF+P & & 0.001 & 0.001 & - & 0.4786 \\
FedPROX & & 0.001 & - & 0.01 & 0.4145 \\
FedAVG+P & & 0.001 & - & - & 0.3929 \\
FedDF & & 0.001 & 0.001 & - & 0.3481 \\
FedBE & & 0.001 & 0.001 & - & 0.3196 \\
FedAVG & & 0.0001 & - & - & 0.2770 \\
\bottomrule
\end{tabular}
\end{table*}
\section{Domain-Adaptation-Theoretic Motivation for weighted ensemble distillation}
\label{supp:domain_adaptation}
Domain adaptation theory \suppcite{mansour2008domain, bendavid2010domain, hoffman2018domain}, and in particular with multiple sources, can be used in order to obtain generalization bounds for non-iid FL settings as it has been done in \suppcite{lin2020ensemble} for uniformly averaging of the client hypotheses to obtain a global hypothesis. From multiple-source adaptation theory we know that a distribution-weighted combination of the client hypotheses is robust w.r.t. generalization for any target domain that is a convex combination of the source domains. However, exact information about the local distributions is rarely present in practical applications of FL and if it is, then directly sharing this information with the server in order to get a better global hypothesis is often not feasible in FL settings due to privacy restrictions. Nonetheless, settings with exact or approximate information about the local distributions (e.g. obtained by KDE) show us, what is possible if the server had access to this information and thus leads to benchmarks with a solid theoretic foundation to which we can compare our approach. Consequently, we aim at a weighting of the client’s local hypotheses based on a privacy-preserving local distribution estimation that respects both the theoretical generalization capabilities and the privacy restrictions in FL.
With the help of a toy example in Fig.~\ref{fig:toy_domain_adaptation} we illustrates that the certainty scores $s_i(\cdot), i \in \{1,\ldots,n\}$, obtained via privacy-preserving logistic regression give a good approximation to the distribution-weights suggested by domain adaptation theory \suppcite{mansour2008domain}, i.e. we show that $s_i(x) / \sum_{j} s_j(x) \approx D_i(x) / \sum_j D_j(x)$ for $x \in \mathcal{X}$.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{images/toy_domain_adaptation_all.pdf}
\caption{Left: Toy example with 3 clients holding data sampled from multivariate Gaussian distributions $D_1$, $D_2$ and $D_3$. All clients solve optimization problem $J$ by contrasting their local data with the public negative data, to obtain scoring models $s_1$, $s_2$, $s_3$ respectively. As can be seen in the plots to the right, our proposed scoring method approximates the robust weights proposed in \suppcite{mansour2008domain} as it holds $s_i(x)/\sum_j s_j(x)\approx D_i(x)/\sum_j D_j(x)$ on the support of the data distributions.}
\label{fig:toy_domain_adaptation}
\end{figure*}
\section{Proof of Theorem \ref{theo:1}}
\label{supp:proof}
\begin{theorem}
If $R(\cdot)$ is differentiable and 1-strongly convex and $l$ is differentiable with $|l'(z)|\leq 1$ $\forall z$, then the $\ell^2$-sensitivity $\Delta_2(\mathcal{M})$ of the mechanism
\begin{align}
\mathcal{M} : D_i \mapsto \arg\min_{w} J(w, h_0, D_i, D^-)
\end{align}
is at most $2(\lambda(|D_i|+|D^-|))^{-1}$.
\end{theorem}
\begin{proof}
The proof is an adaptation of the result shown in \suppcite{chaudhuri2011differentially}.
We have
\begin{align}
\begin{split}
J(w, h_0, D_i, D^-) = &a\sum_{x\in D_i \cup D^-}l(t_x\langle w, \tilde{h}_0(x)\rangle)+\lambda R(w)
\end{split}
\end{align}
with $t_x=2(\mathbb{1}_{x\in D_i})-1\in[-1,1]$, $a=(|D_i|+|D^-|)^{-1}$ and $\tilde{h}_0(x)=h_0(x)(\max_{x\in D^-\cup D_i} \|h_0(x)\|)^{-1}$.
Let $D_i=\{x_1,..,x_N\}$ and $D_i'=\{x_1,..,x'_N\}$ be two local data sets that differ in only one element. For arbitrary $D^-$ and $h_0$ define
\begin{align}
w^* = \arg\min_{w} J(w, h_0, D_i, D^-),
\end{align}
\begin{align}
v^* = \arg\min_{w} J(w, h_0, D_i', D^-),
\end{align}
\begin{align}
n(w)=J(w, h_0, D_i, D^-)
\end{align}
and
\begin{align}
m(w) &= J(w, h_0, D_i, D^-)-J(w, h_0, D'_i, D^-)
\end{align}
Since
\begin{align}
m(w) = a (l(t_x\langle w, h_0(x_N)\rangle)-l(t_x\langle w, h_0(x'_N)\rangle))
\end{align}
we have
\begin{align}
\nabla m(w) = a (t_xl'(t_x\langle w, h_0(x_N)\rangle)h_0(x_N)^T-\\t_xl'(t_x\langle w, h_0(x'_N)\rangle)h_0(x'_N)^T)
\end{align}
which can be bounded in norm
\begin{align}
\|\nabla m(w)\| &= a (\|h_0(x_N)-h_0(x_N')\|)\\&\leq a (\|h_0(x_N)\|+\|h_0(x_N')\|)\\&\leq 2a
\end{align}
as $t_x\in[-1,1]$, $|l'(x)|\leq 1$ and
\begin{align}
\|\tilde{h}_0(x)\|=\|h_0(x)(\max_{x\in D_i\cup D^-}h_0(x))^{-1}\|\leq1.
\end{align}
Furthermore, since $n(w)$ is $\lambda$-strongly convex it follows by Shalev-Schwartz inequality
\begin{align}
(\nabla n(w^*) - \nabla n(v^*))^T(w^*-v^*)\geq\lambda\|w^*-v^*\|^2 .
\end{align}
Combining this result with Cauchy-Schwartz inequality and $\nabla m(v^*) = \nabla n(v^*) - \nabla n(w^*)$ yields
\begin{align}
\|w^*-v^*\|\|\nabla m(v^*)\|&\geq (w^*-v^*)^T\nabla m(v^*)\\&=(w^*-v^*)^T(\nabla n(v^*)-\nabla n(w^*))\\&\geq\lambda \|w^*-v^*\|^2
\end{align}
Thus
\begin{align}
\|w^*-v^*\|\leq\frac{\|\nabla m(v^*)\|}{\lambda}\leq\frac{2a}{\lambda}
\end{align}
which concludes the proof.
\end{proof}
\section{Empirical Privacy Evaluation}
\label{supp:epirical_privacy}
Our proposed method is provably differentially private and achieves state-of-the-art performance, even at very conservative privacy levels. If not explicitly stated otherwise, all results presented in this study were achieved with $(\varepsilon, \delta)$-differentially private certainty scores at conservative privacy parameters $\delta=10^{-5}$ and $\varepsilon=0.1$. In this section, we additionally evaluate the privacy properties of the certainty scores empirically. Figure \ref{fig:similar_images} shows, for four different clients, the 5 images $x$ from the distillation data set $D_{distill}$, which were assigned the highest certainty score $s_i(x)$ by the client's scoring model $w_i^*$ (left column). Displayed next to the images are their 4 nearest neighbors $x'$ in feature space which maximize the cosine-similarity
\begin{align}
\text{sim}(x,x') = \frac{\langle h_0(x), h_0(x')\rangle}{\|h_0(x)\|\|h_0(x')\|}.
\end{align}
In this example the clients hold non-iid subsets of CIFAR-10 ($\alpha=0.01$) and the "Imagenet Dogs" (c.f. Appendix \ref{supp:iamgenet_subsets}) data set is used as auxiliary data. Using weighted ensemble distillation in this setting improves training performance from 48.46\% to 75.59\%.
As we can see, while certainty scores are able to inform the distillation process and allow \textsc{FedAUX} to outperform baseline methods on heterogeneous data, they reveal only fuzzy, indirect information about the local training data. For instance, client 1, which in this example is mainly holding data from the airplane class, assigns the highest scores to pictures in the auxiliary data set that show dogs in cars or in front of blue skies. From this it could be concluded that a majority of the clients training data contains man-made objects in front of blue backgrounds, but direct exposure of single data points is improbable.
Note that there exist also many FL scenarios in which the server is assumed to be trustworthy, and only the final trained model which is released to the public needs to be privately sanitized. In these settings, direct inspection of certainty scores by outside adversaries is not possible and thus privacy loss through certainty scores is even less critical. Future work could also explore the use encryption-based techniques for secure weighted aggregation of client predictions.
\begin{figure*}[t!]
\centering
\subfigure[Client 1: Images $x$ from the distill data set with the highest scores $s_i(x)$ and their nearest neighbors in feature space in the local data set $D_i$.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_4.pdf}}\hfill
\subfigure[Client 2: Images $x$ from the distill data set with the highest scores $s_i(x)$ and their nearest neighbors in feature space in the local data set $D_i$.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_1.pdf}}
~
\subfigure[Client 3: Images $x$ from the distill data set with the highest scores $s_i(x)$ and their nearest neighbors in feature space in the local data set $D_i$.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_9.pdf}}\hfill
\subfigure[Client 4: Images $x$ from the distill data set with the highest scores $s_i(x)$ and their nearest neighbors in feature space in the local data set $D_i$.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_3.pdf}}
\caption{Data points $x$ from the auxiliary data set which were assigned the highest scores $s_i(x)$ and their nearest neighbors in the data of 4 randomly selected clients $D_i$. Clients hold non-iid subsets from the CIFAR-10 data set ($\alpha=0.01$). Auxiliary data used is ImageNet Dogs (cf. Appendix \ref{supp:iamgenet_subsets}). No differential privacy is used.}
\label{fig:similar_images}
\end{figure*}
\begin{figure*}[t!]
\centering
\subfigure[Images from the distill data set with the higher scores and their nearest neighbors in feature space in the local data set of client 1.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_dp_4.pdf}}\hfill
\subfigure[Images from the distill data set with the higher scores and their nearest neighbors in feature space in the local data set of client 2.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_dp_1.pdf}}
~
\subfigure[Images from the distill data set with the higher scores and their nearest neighbors in feature space in the local data set of client 3.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_dp_9.pdf}}\hfill
\subfigure[Images from the distill data set with the higher scores and their nearest neighbors in feature space in the local data set of client 4.]{\includegraphics[width=0.48\textwidth]{images_supplement/privacy_neirest_neighbors_imagenet_dogs_dp_3.pdf}}
\caption{Data points $x$ from the auxiliary data set which were assigned the highest scores $s_i(x)$ and their nearest neighbors in the data of 4 randomly selected clients $D_i$. Clients hold non-iid subsets from the CIFAR-10 data set ($\alpha=0.01$). Auxiliary data used is ImageNet Dogs (cf. Appendix \ref{supp:iamgenet_subsets}). Scores obtained with differential privacy at $\varepsilon=0.1$, $\delta=10^{-5}$.}
\label{fig:similar_images_with_dp}
\end{figure*}
\newpage
\bibliographystyle{icml2021}\bibliography{references}
\end{document}
|
1,314,259,995,818 | arxiv | \section{Motivation}
Up to now, the options for constructing exact inhomogeneous cosmologies were: (i) a small range of non-homogeneous metrics such as the \LT\ (LT) metric \cite{Lem33,Tol34}, the \Sz (S) metric \cite{Szek75a,Szek75b}
\footnote{See \cite{Kra97} for a survey of inhomogeneous cosmologies, \cite{BoKrHeCe09,BoCeKr11} for an overview and review of recent developments, and \cite{Hel09} for a well illustrated quick introduction and discussion of selected recent results. Also see the CQG issue \cite{CQGIC11} on inhomogeneous cosmologies.}
their generalisations to non-zero pressure, the {\L} and Szafron metrics \cite{Lem33,Szaf77}, and a number of others with a less believable equation of state (EoS); (ii) the swiss cheese (SC) constructio
\footnote{For a good list of references see Grenon \& Lake \cite{GreLak10}, footnotes 1 to 4.}
that inserts spherical structures into a Friedmann-{\L}-Robertson-Walker (FLRW) `background', including multi-level swiss cheese structures; etc
\footnote{The interesting method of Lindquist \& Wheeler \cite{LinWhe57}, has surface layers between the Schwarzschild cells, the nature of which is not clear. A recent non-vaccum generalisation \cite{CliFer10,Clif10} is an approximate treatment. In the best current exact multi-black hole metric, there are multiple Reissner-Nordstrom bodies, which have gravitational and electromagnetic `forces' exactly balanced \cite{Pap45,Maj47}, with a $\Lambda$ acceleration added \cite{KasTra93}.}
We here present a new way to construct exact inhomogeneous cosmologies that are arbitrarily inhomogenous out to all distances. They don't have an all-enveloping `background' metric, or even an asymptotic one, and they could be inhomogeneous on any scale, but they can also be made to have the same `average' everywhere. The scale and strength of the inhomogeneities can vary across the spacetime, or be kept statistically similar in all regions, and repeating patterns are also possible. It should be feasible to generalise this construction to other metrics than those considered here.
In several recent works, Wiltshire \cite{Wilt07a,Wilt07b,Wilt08,Wilt09,SmaWil10,Wilt11} has discussed a range of unresolved questions about the assumptions underlying the standard approach to cosmological model building. In attempting to address them, he has propounded some alternative approaches, including the `Cosmological Equivalence Principle', and some deep questioning of what we mean by `averaging', and whether there is a well-defined relationship between an `average' model and real observations. A key concept, that we attempt to model here, is the idea that void regions expand faster than cluster regions, and that, as time goes by, they occupy an increasing fraction of space, so the `average' expansion rate becomes more and more dominated by the void expansion rate, while observers inhabit regions with little or no expansion, thus generating an apparent acceleration \cite{Buch08,Buch11,Rasa11,KolMarMat09}. The model below is relatively simple, being a first attempt of its kind, and it does not capture as many aspects of this particular proposal as we hope will be possible with subsequent generalisations.
\subsection{Inhomogeneity in \LT, \Sz\ and \SC\ Models}
The {\LT} (LT) metric is spherically symmetric, but radially inhomogeneous, describing a ball of dust particles in comoving coordinates, for which the density and the dynamics depend on both time and radius. It contains the dust FLRW and Schwarzschild-Kruskal-Szekeres metrics as special cases, and is well suited to describing a black hole in a cosmological background. It is an excellent first approximation for non-linear gravitational collapse. As a cosmological model, it is certainly good for putting exact inhomogeneities, with a variety of scales, in an asymptotically uniform spacetime, but strong spherical inhomogeneities at very large radii are not observed. If one thinks of an LT model as an angular average of an inhomogeneous cosmology that is homogeneous on a sufficiently large scale, then the inhomogeneity should die off with radial distance. Nevertheless, it has seen extensive use in studying the non-linear evolution of cosmic structures, and in offering an explanation of the dimming of the supernovae. An asymptotically inhomogeneous model has been proposed to solve the horizon problem.
The {\Sz} (Sz) metric is even more interesting, since it has no Killing vectors, and thus doesn't suffer from the drawbacks of a high degree of symmetry noted above in the LT models. It has been used to model voids next to clusters, and even triple structures, and the range of possible structures it can describe is still not known. Nearly all studies have looked at the quasi-spherical class, which may be thought of as an LT model in which each spherical shell has been displaced relative to the others. This produces a dipole effect in the shell separation and in the density distribution around each shell. Thus it may be that this `radial-inhomogeneity-plus-varying-dipole' eventually becomes rather unrealistic at large enough radii. The little studied quasi-pseudo-spherical class can be thought of as an irregular stacking of hyperboloids (pseudo-spherical shells) that vary in density and evolution, each of which contains an underdensity, or overdensity, due to the `pseudo-dipole'. The principle inhomogeneity runs from one side of the universe to the other, with an extra variation snaking through the middle. It does not allow arbitrary inhomogneity in arbirary directions. Therefore this latter class appears to offer the possibility of very interesting structures and cosmologies, but it hasn't really been explored.
In swiss cheese models, one starts with an FLRW spacetime, and then cuts spherical holes in it, which may be filled with some other spacetime metric. For the construction to be valid, the Darmois junction conditions must be satisfied on the boundary between the two spacetimes; each boundary being a timelike 3-surface --- that is, the history of a spherical 2-surface. Thus the possible interiors are restricted to metrics such as Schwarzschild, {\LT}, Vaidya, {\Sz}. {\L} and Szafron interiors, that have non-zero pressure, are also possible but have not to our knowledge been used. Very often, the matching also restricts the FLRW EoS to be that of dust. Thus there is an obvious `average' FLRW model, and the behaviour of the average is known from the start.
The new construction has features significantly different from each of these. Further, it may be combined with all of the above, thereby much expanding the range of possible exact inhomogeneous cosmological models.
\section{Assembling Multiple `Voids' \& `Clusters' in a Single Exact Spacetime}
\noindent
${}$ \hfill
\pb{14cm}{
${}$ \hfill
\includegraphics{MultiKasnerDiag.eps}
\hfill ${}$ \\
{\small {\bf Figure 1.} A sample from a multi-component inhomogeneous universe at one moment in time, showing several `repeats' of a `block' of component cuboids. In this illustration, each block ($4 \times 4 \times 4$ of them shown) has 8 components; 1 of type I, 3 of type II, 3 of type III, and 1 of type IV, and their sizes in each block are actually different, so they are never exactly repeated. These components are marked on the block at the near corner --- the 3rd type II component (IIy) is of course hidden. In the main example of the paper, each component is a distinct Kasner-type region. The different K components have different expansion behaviour, so the fraction of a block they occupy changes with time. One could construct blocks with $3 \times 3 \times 3$ components, or $5 \times 2 \times 7$, or even component sequences in each of the $x$, $y$ \& $z$ directions that never repeat. What matters is that the two expansion rates in the surfaces where two components join are the same in both component regions. Note that even when the set of expansion rates in a block repeats, the block sizes do not have to. This is the case illustrated above.}
}
\hfill ${}$
\\[2mm]
In this section we construct a truly inhomogeneous universe, that does not have an obvious ``background" metric. We use the Darmois junction conditions in General Relativity (GR) to join many regions of different matter content and evolution type. If the Darmois conditions are obeyed, the result is an exact solution of the Einstein Field Equations (EFEs).
The essential idea is to do a 3-d tesselation, to fill space with an array of spacetime regions that are properly matched together at their boundaries. It is the need to ensure a proper 3-surface matching at each surface where component regions join, that renders this otherwise simple idea distinctly non-trivial. We begin by considering a kind of irregular cubic lattice, in which the basic `unit' or `building block' is composed of 8 `pieces', or `component regions', each one a different Kasner-type (K) metric \cite{Kasn25} --- see Fig.\ 1. The hope is that this construction may be generalisable to a variety of other metrics. In fact, an inhomogenous cosmology consisting of multiple slabs of FLRW and Kasner has been constructed in the ``cheese slice" models of \cite{DyeLanSha93,DyeOli01}, but the model presented here is much more realistic in that it is inhomogeneous in all 3 spatial dimensions, instead of just 1
\footnote{Some properties were investigated in \cite{GiaDye09}. Although \cite{CenMat79} adjoined different Bianchi I cosmologies on an initial time slice, and mentioned the case of FLRW next to Kasner, it is evident that a full Darmois-type matching was not achieved, since ``delta-function discontinuities in the Riemann tensor" are found in section VI - i.e ``surface layers" or ``shock fronts" developed. A matching of planar dust (Ellis) metrics to vacuum was considered in \cite{Lak92}.}
Therefore, at any given moment, the 3-d patchwork is like a cubic lattice, but each of the 8 components of a block has different expansion behaviour, so the relative sizes of each component will evolve greatly. The example illustrated in Fig.\ 1, needs 4 types of region, types I - IV. We could try to use type I regions to model `voids', type II `walls', type III `filaments`, and type IV `superclusters' (`clusters' for short). Each type I region adjoins 6 type II regions across a timelike 3-surface; each type IV region adjoins 6 type III regions; each type II region adjoins 4 type III regions and 2 type I; and each type III region adjoins 4 type II regions and 2 type IV. To make sure the result is a regular solution of the EFEs, we will have to apply the Darmois junction conditions to the loci where 2 or more of these regions meet; see \S \ref{JCs} for the details.
In the simplest case, there is only one kind of type II \& type III region, though oriented differently, and the basic block is cuboidal and repeats indefinitely. However, as subsequent sections will show, this much regularity is not required by the construction. In the more general case, the 3 type II \& 3 type III regions may all be different metrics, and the expansion rates and sizes of the blocks and their components need not repeat, but rather have a random distribution of parameters. See section \ref{MKU}.
\section{The Kasner-type Metric}
The Kasner metric\cite{Kasn25}, and its non-vacuum generalisation, is a spatially flat, anisotropic Bianchi model of type I. It has a different `expansion law' in each of 3 perpendicular directions. The metric is
\begin{align}
ds^2 & = - dt^2 + t^{2 \alpha} \, dx^2 + t^{2 \beta} \, dy^2 + t^{2 \gamma} \, dz^2 \showlabel{Kds2} ~.
\end{align}
The Einstein field equations (EFEs), and the expansion are given by
\begin{align}
\kappa \rho & = \frac{\alpha \beta + \beta \gamma + \gamma \alpha}{t^2} ~, \\
\kappa p_x & = \frac{\beta + \gamma - (\beta^2 + \beta \gamma + \gamma^2)}{t^2} ~, \\
\kappa p_y & = \frac{\gamma + \alpha - (\gamma^2 + \gamma \alpha + \alpha^2)}{t^2} ~, \\
\kappa p_z & = \frac{\alpha + \beta - (\alpha^2 + \alpha \beta + \beta^2)}{t^2} ~, \\
\Theta & = \frac{\alpha + \beta + \gamma}{t} ~.
\end{align}
The pressures are all proportional to the density, but different in the 3 perpendicular directions.
The unit vector $u^a = \delta^a_t$ is geodesic and comoving with the matter.
\paragraph{The Minkowski Case}
To make the Riemann tensor zero requires e.g.
\begin{align}
\alpha = 0 = \beta = \gamma ~;~~~~~~\mbox{or}~~~~~~ \alpha = 0 = \beta ~,~ \gamma = 1 ~.
\end{align}
\paragraph{The Vacuum Case}
The requirement $\rho = 0 = p_x = p_y = p_z$ leads to $(\alpha, \beta, \gamma)$ being an equally spaced triplet round the Kasner circle:
\begin{align}
\alpha & = \frac{1 + 2 \cos\theta}{3} ~, \\
\beta & = \frac{1 + 2 \cos(\theta + 2 \pi/3)}{3} ~, \\
\gamma & = \frac{1 + 2 \cos(\theta + 4 \pi/3)}{3} ~.
\end{align}
It is not possible for more than two of $\alpha$, $\beta$ \& $\gamma$ to be the same, so vacuum Kasner cannot be isotropic, unless it is Minkowski. The only vacuum case with two of them the same is $\{\alpha, \beta, \gamma\} = \{2/3, 2/3, -1/3\}$.
\paragraph{The Isotropic Case}
Putting $\gamma = \beta = \alpha$ gives the spatially flat FLRW models,
\begin{align}
S = t^\alpha ~,~~~~~~
\kappa \rho = \frac{3 \alpha^2}{t^2} ~,~~~~~~
\kappa p = \frac{2 \alpha - 3 \alpha^2}{t^2} = \left( \frac{2}{3 \alpha} - 1 \right) \rho ~,
\end{align}
where $S$ is the FLRW scale factor.
\paragraph{The Zero Pressure Case}
The requirement $0 = p_x = p_y = p_z$ is satisfied by the Minkowski case, the set of vacuum cases, and the $\alpha = 2/3$ FLRW case.
\subsection{Physicality Conditions}
\showlabel{PhysCond}
Since we will be assembling many different K regions to construct an inhomogeneous universe, it is useful to check their physical behaviour. The condition for non-negative density is easy to satisfy,
\begin{align}
\rho \geq 0 & ~~~~\to~~~~ \alpha \beta + \beta \gamma + \gamma \alpha \geq 0 ~,
\end{align}
and that for non-negative pressure,
\begin{align}
p_x \geq 0 & ~~~~\to~~~~ \beta + \gamma - (\beta^2 + \beta \gamma + \gamma^2) \geq 0 ~, \nn \\
& ~~~~\to~~~~ \frac{1}{2} \left( (1 - \beta) - \sqrt{(1 - \beta)(1 + 3\beta)}\; \right) \leq \gamma \leq \frac{1}{2} \left( (1 - \beta) + \sqrt{(1 - \beta)(1 + 3\beta)}\; \right) ~, \nn \\
& ~~~~~~~~~~~ -1/3 \leq \beta \leq 1 ~~~~~~ \big[ -1/3 \leq \gamma \leq 1 \big] ~, \\
p_y \geq 0 & ~~~~\to~~~~ \gamma + \alpha - (\gamma^2 + \gamma \alpha + \alpha^2) \geq 0 ~, \\
p_z \geq 0 & ~~~~\to~~~~ \alpha + \beta - (\alpha^2 + \alpha \beta + \beta^2) \geq 0 ~,
\end{align}
which is a fattened region around the Kasner circle, with a 3-lobed shape, a bit like a cardamom pod, is also not hard to satisfy. For a non-relativistic gas, we expect,
\begin{align}
p_x < \frac{\rho}{3} & ~~~~\to~~~~ 3(\beta + \gamma) - 3(\beta^2 + \beta \gamma + \gamma^2) < \alpha \beta + \beta \gamma + \gamma \alpha \nn \\
& ~~~~\to~~~~ \gamma < \frac{1}{6} \left( 3 - 4 \alpha - \beta + \sqrt{9 + 12 \alpha - 6 \beta - 20 \alpha^2 - 4 \alpha \beta + \beta^2}\; \right) \nn \\
& ~~~~~~\mbox{\&}~~~~~ \gamma > \frac{1}{6} \left( 3 - 4 \alpha - \beta - \sqrt{9 + 12 \alpha - 6 \beta - 20 \alpha^2 - 4 \alpha \beta + \beta^2}\; \right) ~, \\
p_y < \frac{\rho}{3} & ~~~~\to~~~~ 3(\gamma + \alpha) - 3(\gamma^2 + \gamma \alpha + \alpha^2) < \alpha \beta + \beta \gamma + \gamma \alpha \nn \\
& ~~~~\to~~~~ \gamma < \frac{1}{6} \left( 3 - 4 \beta - \alpha + \sqrt{9 + 12 \beta - 6 \alpha - 20 \beta^2 - 4 \beta \alpha + \alpha^2}\; \right) \nn \\
& ~~~~~~\mbox{\&}~~~~~ \gamma > \frac{1}{6} \left( 3 - 4 \beta - \alpha - \sqrt{9 + 12 \beta - 6 \alpha - 20 \beta^2 - 4 \beta \alpha + \alpha^2}\; \right) ~, \\
p_z < \frac{\rho}{3} & ~~~~\to~~~~ 3(\alpha + \beta) - 3(\alpha^2 + \alpha \beta + \beta^2) < \alpha \beta + \beta \gamma + \gamma \alpha \nn \\
& ~~~~\to~~~~ \gamma < 3 - 3 \alpha - \beta - \frac{2 \beta^2}{\alpha + \beta} ~.
\end{align}
Similarly, to ensure the sound speed less is than light speed, $p < \rho$, we obtain a similar set of conditions.
These are not always so easy to obey, because, in the transition from dust FLRW, $(\alpha,\beta,\gamma) = (2/3, 2/3, 2/3)$ to Minkowski vacuum, $(\alpha,\beta,\gamma) = (0, 0, 0)$, the matter becomes increasingly stiff. Thus in the general case we have an exotic fluid. Nevertheless, as a first simple example of this type of construction, the above behaviour is not too bad. In any case, these component regions may themselves be thought of as averages over a more complicated matter distribution, so the above is only an effective bulk EoS (equation of state). The fact that the pressures are different in different directions is only to be expected in regions with ``pancake" or ``cigar" expansion. More general metrics for the component spacetimes that are stitched together in such a patchwork, will no doubt allow us to improve this aspect.
\subsection{Cosmological Units}
\showlabel{CGU}
We choose geometric units such that $G = 1 = c$, and for the remaining freedom, we specify that $1$ time unit = $10$~gigayears. We call these cosmological geometric units, and the values of the cosmological time, length and distance units are:
\[
\begin{tabular}{llll}
1 ctu & = & 10 Gy \\
1 clu & = & 3.066 Gpc \\
1 cmu & = & 6.409 $\times 10^{22}$ M$_\odot$
\end{tabular}
\]
In these units, $100$~km/s/Mpc is very close to $1$/ctu.
\section{Junction Conditions}
\showlabel{JCs}
We now implement the Darmois \cite{Darm27} junction conditions. If they are satisfied, the combined spacetime metric is $C^1$ and piecewise $C^3$, and it may be shown \cite{HelDra94
\footnote{Although this paper is about conservation failing at a signature change surface, the non-signature-changing case was done first.}
that, due to the Israel identities \cite{Isra66}, the conservation laws $\nabla_\nu G^{\mu\nu} = 0$ are satisfied even through the $C^1$ junctions.
For the 3-surface $x = X =$~const, we define the surface $\Sigma$ and the surface coordinates $\xi^i$ to be
\begin{align}
x_\Sigma^\mu & = (t, X, y, z) ~, \\
\xi^i & = (t, y, z) ~.
\end{align}
Here, greek indices range $0$ to $3$, and latin indices range $1$ to $3$, but note that, in the context of junction conditions, index $0$ indicates the direction orthogonal to $\Sigma$, and is not necessarily time. We then calculate, in order, the surface basis vectors, their derivatives, the surface normal, the intrinsic metric and the extrinsic curvature:
\begin{align}
(e_i)^\mu & = \pd{x^\mu}{\xi^i} =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix} ~, \\
\pdt{x^\mu}{\xi^i}{\xi^j} & = 0 ~, \\
n_\mu & = (0, t^\alpha, 0, 0) ~, \\
{}^3\!\!g_{ij} & = g_{\mu\nu} \, (e_i)^\mu \, (e_j)^\nu = \diag(-1, t^{2\beta}, t^{2\gamma}) ~, \\
K_{ij} & = - n_\sigma \left( \pdt{x^\sigma}{\xi^i}{\xi^j} + \Gamma^\sigma{}_{\mu\nu} \, (e_i)^\mu \, (e_j)^\nu \right) = 0 ~.
\end{align}
The two manifolds we wish to join are labelled ``$+$" and ``$-$", and are both Kasner-type (K) spacetimes:
\begin{align}
V^+ :~~~~ x_+^\mu = (t_+, x_+, y_+, z_+) ~,~~~~ g_{\mu\nu} = \diag(-1, t_+^{\alpha_+}, t_+^{\beta_+}, t_+^{\gamma_+}) ~, \\
V^- :~~~~ x_-^\mu = (t_-, x_-, y_-, z_-) ~,~~~~ g_{\mu\nu} = \diag(-1, t_-^{\alpha_-}, t_-^{\beta_-}, t_-^{\gamma_-}) ~.
\end{align}
If the two surfaces to be identified, $\Sigma_+$ \& $\Sigma_-$, are located at $x = X_+$ \& $x = X_-$, and the mapping between them is
\begin{align}
t_+ & = t_- = \xi^1 ~, \\
y_+ & = y_- = \xi^2 ~, \\
z_+ & = z_- = \xi^3 ~,
\end{align}
then the Darmois conditions, which use square brackets to denote the jump in a quantity across the junction, consequently require
\begin{align}
\big[ {}^3\!\!g_{ij} \big] = {}^3\!\!g_{ij} |_\Sigma^+ & - {}^3\!\!g_{ij} |_\Sigma^- = 0 ~, \\
\big[ K_{ij} \big] = K_{ij} |_\Sigma^+ & - K_{ij} |_\Sigma^- = 0 ~.
\end{align}
For our matching, these lead to
\begin{align}
\beta_+ = \beta_- ~,~~~~~~ \gamma_+ = \gamma_- ~.
\end{align}
This is consistent with the Israel requirement \cite{Isra66}, which means that, for an observer moving with a timelike 3-surface, the pressures must match, but the density does not have to
\footnote{At the level of bulk fluid parameters, the Darmois matching is complete. If however one introduces a kinetic theory description of the matter --- far too complex for present purposes --- then there would be further conditions needed to match all the modes.}
Therefore, two K spaces may be joined across any pair of constant $x$ surfaces if the two expansion indices in the plane of the surfaces are the same on either side of the junction. The expansion index perpendicular to $\Sigma$ may differ in $V_+$ \& $V_-$. The same goes for matching pairs of constant $y$ or $z$ surfaces. Thin boundaries where the tangential pressures are different on either side are of course unrealistic; at the atomic level, one would expect streaming of particles and photons to blur the boundaries. However, junction condition methods are understood to be useful approximations to transitions that happen within a relatively thin region, for which the exact smooth equations are unsolvable.
Now the Darmois conditions allow the regular junction of two manifolds at a pair of identified 3-surfaces, the 3-surfaces being unbounded. In our case, each matching surface is bounded by other matching surfaces, so the size of the two identified surfaces must also be the same. Where 2 cuboidal regions meet, in our array of different K regions, they must have 2 of the coordinate dimensions, $\Delta x$, $\Delta y$ \& $\Delta z$, in common --- see below.
Furthermore, where 4 cuboids touch at a 2-surface (a 1-d curve existing through time), the corner conditions \cite{Tayl04} should be checked
\footnote{See \cite{GiaDye09} for an application.}
However, we expect no difficulty, as the spatial coordinates in the K metric are Cartesian. Similarly, at the worldlines of the vertices where 8 cuboids meet, an appropriate modification of those corner conditions should be applied.
\section{Multi-Kasner Universes}
\showlabel{MKU}
We are now ready to patch together Kasner-type (K) cuboids. Using the matching rules just obtained, a general, repeating $2 \times 2 \times 2$ model would be as in Table 1:
\\[2mm]
\noindent
${}$ \hfill
\pb{14cm}{
${}$ \hfill
\begin{tabular}{l|l|l|l}
Region & Region type & $(\alpha, \beta, \gamma)$ & $(\Delta x, \Delta y, \Delta z)$ \\
\hline
I & `void' & $(a, b, c)$ & $(A, B, C)$ \\
IIx & $x$ `wall' & $(d, b, c)$ & $(D, B, C)$ \\
IIy & $y$ `wall' & $(a, e, c)$ & $(A, E, C)$ \\
IIz & $z$ `wall' & $(a, b, f)$ & $(A, B, F)$ \\
IIIx & $y$-$z$ `filament' & $(a, e, f)$ & $(A, E, F)$ \\
IIIy & $z$-$x$ `filament' & $(d, b, f)$ & $(D, B, F)$ \\
IIIz & $x$-$y$ `filament' & $(d, e, c)$ & $(D, E, C)$ \\
IV & `cluster' & $(d, e, f)$ & $(D, E, F)$
\end{tabular}
\hfill ${}$ \\[1mm]
{\small {\bf Table 1.} A repeating $2 \times 2 \times 2$ multi-Kasner model. The region numbers correspond to those marked in Fig. 1. The triplet $(\alpha, \beta, \gamma)$ is the set of exponents in the metric \er{Kds2}, and the particular values $a$, $b$, $c$, $d$, $e$, $f$ may be chosen freely, subject to appropriate physicality conditions, as in section \ref{PhysCond}. Similarly the triplet $(\Delta x, \Delta y, \Delta z)$ gives the three coordinate dimensions of a region, as at the end of section \ref{JCs}, and $A$, $B$, $C$, $D$, $E$, $F$ may be chosen freely.}
}
\hfill ${}$
\\[2mm]
For a simple model, one may set
\begin{align}
c = b = a ~,~~~~ f = e = d ~,~~~~
C = B = A ~~~~\mbox{and}~~~~ F = E = D ~.
\showlabel{SimpEx}
\end{align}
Thus, although this is not necessary, one of the 8 components could easily be an isotropic (i.e. FLRW) region, and we could even have two distinct FLRW regions per block.
In fact, the building blocks do not have to repeat exactly. Referring again to Fig. 1, as we follow a line of component regions in the $x$ direction, say, the particular pairs $(\beta, \gamma)$ and $(B, C)$ must be the same for every region along that line, but need not be the same as adjacent lines. Similarly, following a line in the $y$ direction, the particular pairs $(\alpha, \gamma)$ and $(A, C)$ must be the same all down that row; etc. for following a $z$ line. Thus, in each of the $x$, $y$ \& $z$ directions respectively, we may have non-repeating sequences of expansion rates and widths:
$(\alpha_1,\;A_1),\;(\alpha_2,\;A_2),\;(\alpha_3,\;A_3),\; \cdots$;
$(\beta_1,\;B_1),\;(\beta_2,\;B_2),\;(\beta_3,\;B_3),\; \cdots$;
$(\gamma_1,\;C_1),\;(\gamma_2,\;C_2),\;(\gamma_3,\;C_3),\; \cdots$.
Each $(\alpha_i,\;A_i)$ stays constant in a constant $(y, z)$ surface, and so on. Thereby we may create an arbitrarily inhomogeneous universe. If desired, these parameter values may be chosen from some statistical distribution.
Furthermore, if there are FLRW regions, each one of them may be filled with a variety of swiss-cheese inhomogeneities, or even multi-level swiss-cheese inhomogeneites, and exact inhomogeneous models, such as the {\LT} and {\Sz} metrics, can be used in the SC constructions.
\section{Averaging}
We now consider how to choose an FLRW model that best approximates this universe in some kind of `average' sense, and we look at the evolution of the `average' Hubble, deceleration and EoS parameters.
The simplest approach is to do a volume-weighted average. When the 3-spaces are flat, this is quite simple. For simplicity, we consider a repeating $2 \times 2 \times 2$ block, so that there is a uniform large-scale average. The volume occupied by each of the 8 component regions is just
\begin{align}
V = \Delta x \, \Delta y \, \Delta z \, t^{\alpha+\beta+\gamma} ~,
\end{align}
and its derivatives are
\begin{align}
\dot{V} & = \Delta x \, \Delta y \, \Delta z \, (\alpha + \beta + \gamma) \, t^{\alpha+\beta+\gamma-1} ~, \\
\ddot{V} & = \Delta x \, \Delta y \, \Delta z \, (\alpha + \beta + \gamma)(\alpha + \beta + \gamma - 1) \, t^{\alpha+\beta+\gamma-2} ~.
\end{align}
Thus the volume fraction of any given region $i$ is
\begin{align}
f_i = \frac{V_i}{\sum_{j=1}^8 V_j} ~,
\end{align}
the volume-averaged expansion rate is
\begin{align}
\ol{\Theta} = \frac{\sum_{i=1}^8 \dot{V}_i}{\sum_{j=1}^8 V_j} = 3 \ol{H} ~,
\end{align}
and the volume-averaged deceleration parameter is
\begin{align}
\ol{q} = 2 - \frac{3 \big( \sum_{j=1}^8 V_j \big) \big( \sum_{k=1}^8 \ddot{V}_k \big) }{ \big( \sum_{i=1}^8 \dot{V}_i \big)^2 } ~.
\end{align}
If we try to fit an FLRW model
\begin{align}
\kappa \rho = \frac{3 (\dot{S}^2 + k)}{S^2} = 3 H^2 (1 + \Omega_k) ~,~~~~~~ \kappa p = \frac{( 2 S \ddot{S} + \dot{S}^2 + k)}{S^2} = H^2 (1 - 2 q + \Omega_k) ~,
\end{align}
to this inhomogeneous cosmology, then we would adjust the EoS, $p = w \rho$, so as to reproduce the actual expansion, i.e.
\begin{align}
w & = \frac{1}{3} \left( 1 - \frac{2 q}{1 + \Omega_k} \right) ~,
\end{align}
and setting $k = 0$ (since the K component regions are all flat), we have the effective EoS parameter
\begin{align}
\ol{w} & = \frac{1 - 2 \ol{q}}{3} ~.
\end{align}
For the simple example \er{SimpEx} above, these would be,
\begin{align}
f_i & = \frac{(t^{3a},~ t^{2a+d},~ t^{a+2d},~ t^{3d})}{(t^a + t^d)^3} ~, \\
\ol{\Theta} & = \frac{3 (a t^a + d t^d)}{t (t^a + t^d)} ~, \\
\ol{q} & = - \frac{(t^a + t^d) \big( a (a - 1) t^a + d (d - 1) t^d \big) }{(a t^a + d t^d)^2} ~;
\end{align}
and for a single K component they are
\begin{align}
f = 1 ~,~~~~~~ \Theta = \frac{\alpha + \beta + \gamma}{t} ~,~~~~~~ q = - 1 + \frac{3}{\alpha + \beta + \gamma} ~,~~~~~~ w = 1 - \frac{2}{\alpha + \beta + \gamma} ~. \showlabel{fThqw}
\end{align}
\section{Model Details and Results}
We now present 3 variations of a specific case, a repeating $2 \times 2 \times 2$ block. The details of models (A) to (C) are given in tables 2 to 4. In each, the type I \& type IV regions are specified, and the remaining regions are fixed by the junction conditions. The repeating pattern allows an average to be calculated that is globally uniform, and thus the evolution of the component parts can be compared with the average. If the comoving size of a component region is $A \times B \times C$ in the $x$, $y$ \& $z$ directions, then the $x$ size of the region today ($t \sim 1$~ctu) is $\sim A$~clu; so $A = 0.02$ corresponds to $\sim 60$Mpc --- the scale of voids, walls, filaments, etc. $B$ \& $C$ would have a similar order of magnitude.
\\[2mm]
\noindent
${}$ \hfill
\pb{14cm}{
${}$ \hfill
\begin{tabular}{l|l|l|l|l|l|l}
Region & Region type & Expansion rate triplet $\alpha$-$\beta$-$\gamma$ & $\rho$ & $(p_x, p_y, p_z)$ & $\Theta$ & $q$ \\
\hline
&&&&&\\[-3.5mm]
I & `void' & $(1/2, 1/2, 1/2)$ & $\frac{3}{4 t^2}$ & $(\frac{1}{4 t^2}, \frac{1}{4 t^2}, \frac{1}{4 t^2})$ & $\frac{3}{2 t}$ & $1$ \\[2mm]
IIx & $y$-$z$ `wall' & $(2/3, 1/2, 1/2)$ & $\frac{11}{12 t^2}$ & $(\frac{1}{4 t^2}, \frac{5}{36 t^2}, \frac{5}{36 t^2})$ & $\frac{5}{3 t}$ & $\frac{4}{5}$ \\[2mm]
IIy & $z$-$x$ `wall' & $(1/2, 2/3, 1/2)$ & $\frac{11}{12 t^2}$ & $(\frac{5}{36 t^2}, \frac{1}{4 t^2}, \frac{5}{36 t^2})$ & $\frac{5}{3 t}$ & $\frac{4}{5}$ \\[2mm]
IIz & $x$-$y$ `wall' & $(1/2, 1/2, 2/3)$ & $\frac{11}{12 t^2}$ & $(\frac{5}{36 t^2}, \frac{5}{36 t^2}, \frac{1}{4 t^2})$ & $\frac{5}{3 t}$ & $\frac{4}{5}$ \\[2mm]
IIIx & $x$ `filament' & $(1/2, 2/3, 2/3)$ & $\frac{10}{9 t^2}$ & $(0, \frac{5}{36 t^2}, \frac{5}{36 t^2})$ & $\frac{11}{6 t}$ & $\frac{7}{11}$ \\[2mm]
IIIy & $y$ `filament' & $(2/3, 1/2, 2/3)$ & $\frac{10}{9 t^2}$ & $(\frac{5}{36 t^2}, 0, \frac{5}{36 t^2})$ & $\frac{11}{6 t}$ & $\frac{7}{11}$ \\[2mm]
IIiz & $z$ `filament' & $(2/3, 2/3, 1/2)$ & $\frac{10}{9 t^2}$ & $(\frac{5}{36 t^2}, \frac{5}{36 t^2}, 0)$ & $\frac{11}{6 t}$ & $\frac{7}{11}$ \\[2mm]
IV & `cluster' & $(2/3, 2/3, 2/3)$ & $\frac{4}{3 t^2}$ & $(0, 0, 0)$ & $\frac{2}{t}$ & $\frac{1}{2}$
\end{tabular}
\hfill ${}$ \\[1mm]
{\small {\bf Table 2.} Model (A). Type I regions (voids) are radiation FLRW, type IV regions (clusters) are dust FLRW.}
}
\hfill ${}$
\\[5mm]
\noindent
${}$ \hfill
\pb{14cm}{
${}$ \hfill
\begin{tabular}{l|l|l|l|l|l|l}
Region & Region type & Expansion rate triplet $\alpha$-$\beta$-$\gamma$ & $\rho$ & $(p_x, p_y, p_z)$ & $\Theta$ & $q$ \\
\hline
&&&&&\\[-3.5mm]
I & `void' & $(0, 0, 0)$ & $0$ & $(0, 0, 0)$ & $0$ & - \\[2mm]
IIx & $y$-$z$ `wall' & $(2/3, 0, 0)$ & $0$ & $(0, \frac{2}{9 t^2}, \frac{2}{9 t^2})$ & $\frac{2}{3 t}$ & $\frac{7}{2}$ \\[2mm]
IIy & $z$-$x$ `wall' & $(0, 2/3, 0)$ & $0$ & $(\frac{2}{9 t^2}, 0, \frac{2}{9 t^2})$ & $\frac{2}{3 t}$ & $\frac{7}{2}$ \\[2mm]
IIz & $x$-$y$ `wall' & $(0, 0, 2/3)$ & $0$ & $(\frac{2}{9 t^2}, \frac{2}{9 t^2}, 0)$ & $\frac{2}{3 t}$ & $\frac{7}{2}$ \\[2mm]
IIIx & $x$ `filament' & $(0, 2/3, 2/3)$ & $\frac{4}{9 t^2}$ & $(0, \frac{2}{9 t^2}, \frac{2}{9 t^2})$ & $\frac{4}{3 t}$ & $\frac{5}{4}$ \\[2mm]
IIIy & $y$ `filament' & $(2/3, 0, 2/3)$ & $\frac{4}{9 t^2}$ & $(\frac{2}{9 t^2}, 0, \frac{2}{9 t^2})$ & $\frac{4}{3 t}$ & $\frac{5}{4}$ \\[2mm]
IIIz & $z$ `filament' & $(2/3, 2/3, 0)$ & $\frac{4}{9 t^2}$ & $(\frac{2}{9 t^2}, \frac{2}{9 t^2}, 0)$ & $\frac{4}{3 t}$ & $\frac{5}{4}$ \\[2mm]
IV & `cluster' & $(2/3, 2/3, 2/3)$ & $\frac{4}{3 t^2}$ & $(0, 0, 0)$ & $\frac{2}{t}$ & $\frac{1}{2}$
\end{tabular}
\hfill ${}$ \\[1mm]
{\small {\bf Table 3.} Model (B). Type I regions (voids) are $(0, 0, 0)$ Minkowski vacuum, type IV regions (clusters) are dust FLRW.}
}
\hfill ${}$
\\[5mm]
\noindent
${}$ \hfill
\pb{14cm}{
${}$ \hfill
\begin{tabular}{l|l|l|l|l|l|l}
Region & Region type & Expansion rate triplet $\alpha$-$\beta$-$\gamma$ & $\rho$ & $(p_x, p_y, p_z)$ & $\Theta$ & $q$ \\
\hline
&&&&&\\[-3.5mm]
I & `void' & $(1/10, 1/10, 1/10)$ & $\frac{3}{100 t^2}$ & $(\frac{17}{100 t^2}, \frac{17}{100 t^2}, \frac{17}{100 t^2})$ & $\frac{3}{10 t}$ & $9$ \\[2mm]
IIx & $y$-$z$ `wall' & $(2/3, 1/10, 1/10)$ & $\frac{43}{300 t^2}$ & $(\frac{221}{900 t^2}, \frac{221}{900 t^2}, \frac{17}{100 t^2})$ & $\frac{13}{15 t}$ & $\frac{32}{13}$ \\[2mm]
IIy & $z$-$x$ `wall' & $(1/10, 2/3, 1/10)$ & $\frac{43}{300 t^2}$ & $(\frac{221}{900 t^2}, \frac{17}{100 t^2}, \frac{221}{900 t^2})$ & $\frac{13}{15 t}$ & $\frac{32}{13}$ \\[2mm]
IIz & $x$-$y$ `wall' & $(1/10, 1/10, 2/3)$ & $\frac{43}{300 t^2}$ & $(\frac{17}{100 t^2}, \frac{221}{900 t^2}, \frac{221}{900 t^2})$ & $\frac{13}{15 t}$ & $\frac{32}{13}$ \\[2mm]
IIIx & $x$ `filament' & $(1/10, 2/3, 2/3)$ & $\frac{26}{45 t^2}$ & $(0, \frac{221}{900 t^2}, \frac{221}{900 t^2})$ & $\frac{43}{30 t}$ & $\frac{47}{43}$ \\[2mm]
IIIy & $y$ `filament' & $(2/3, 1/10, 2/3)$ & $\frac{26}{45 t^2}$ & $(\frac{221}{900 t^2}, 0, \frac{221}{900 t^2})$ & $\frac{43}{30 t}$ & $\frac{47}{43}$ \\[2mm]
IIIz & $z$ `filament' & $(2/3, 2/3, 1/10)$ & $\frac{26}{45 t^2}$ & $(\frac{221}{900 t^2}, \frac{221}{900 t^2}, 0)$ & $\frac{43}{30 t}$ & $\frac{47}{43}$ \\[2mm]
IV & `cluster' & $(2/3, 2/3, 2/3)$ & $\frac{4}{3 t^2}$ & $(0, 0, 0)$ & $\frac{2}{t}$ & $\frac{1}{2}$
\end{tabular}
\hfill ${}$ \\[1mm]
{\small {\bf Table 4.} Model (C). Type I regions (voids) are $(1/10, 1/10, 1/10)$ FLRW near-vacuum, type IV regions (clusters) are dust FLRW.}
}
\hfill ${}$
\\[2mm]
The behaviour of (B) \& (C) should be quite similar, and we particularly focus on (C), even though the EoS $p = (17/3) \rho$ is unrealistic, as it allows the evolution of both FLRW regions, as well as the average, to appear together on the graphs. The evolution of model (C) is plotted in Fig.\ 2.
\noindent
\pb{\textwidth}{
${}$ \hfill
\includegraphics[scale=0.4]{MultiKasner_log_f_vs_log_t.eps}
\hfill
\includegraphics[scale=0.4]{MultiKasner_log_Theta_vs_log_t.eps}
\hfill ${}$ \\
${}$ \hfill
\includegraphics[scale=0.4]{MultiKasner_Theta_Comparison_vs_log_t.eps}
\hfill
\includegraphics[scale=0.4]{MultiKasner_q_vs_log_t.eps}
\hfill ${}$ \\
${}$ \hfill
\includegraphics[scale=0.4]{MultiKasner_w_vs_log_t.eps}
\hfill ${}$ \\
${}$ \hfill
\pb{14cm}{
{\small
{\bf Figure 2.}
Evolution of the Dust-Near-Vacuum Model, with $0.02^3 = A B C = D E F$. Top left: evolution of the volume fraction of the 4 component types [type I (FLRW dust) - dashed, type II - dotted, type III - solid, type IV (FLRW near-vacuum) - dot-dashed]; for types II \& III there are 3 regions each with the fraction shown. Top right: evolution of the expansion [dashed - FLRW dust, dot-dashed - FLRW near-vacuum, solid - average]. Middle left: comparison of expansion rates [dashed - $\Theta_{Av}/\Theta_I$, dot-dashed - $\Theta_{Av}/\Theta_{IV}$]. Middle right: evolution of the deceleration [dashed - FLRW dust, dot-dashed - FLRW near-vacuum, solid - average]. Bottom: evolution of the effective equation of state parameter [dashed - FLRW dust, dot-dashed - FLRW near-vacuum, solid - average]. The units are cosmological geometric units, as given in \S\ref{CGU}, for which $1$~ctu = 10~Gy, so the age of the universe is around $0.14$ on the $\log_{10} t$ axis. The ordinate variables are dimensionless except for $\Theta$, which has units of fractional volume increase per ctu, so $\log_{10} \Theta = 1$ indicates a ten-fold increase in volume in $10$~Gy, and a Hubble rate of $65$~km/s/Mpc corresponds to $\log_{10} \Theta = 0.3$.
}
}
\hfill ${}$
}
Figure 2 shows the time evolution of the volume fraction of each component region, the expansion rates of the average compared with regions I \& IV, the average deceleration, and the effective EoS parameter. This model clearly demonstrates how inhomogeneity can cause average parameters to evolve differently from what might be expected in a totally smoothed out universe.
Although the vacuum or near-vacuum regions could model voids, the spatial flatness means they don't (or hardly) expand, whereas low density regions that develop in an initially expanding universe would retain their initial expansion --- or rather the galaxies at their borders would. For greater realism, we would need to give the empty (type I) regions negative curvature (Milne model), and the high-density (type IV) regions positive curvature. The type II \& III regions would need to be able to join them up across suitable surfaces and satisfy Darmois. Thus it is important to generalise this construction method by finding matchings for other types of component region.
\section{Discussion and Conclusions}
A new method for constructing exact inhomogeneous cosmological models is presented. The resulting cosmologies may have arbitrarily strong inhomogeneities on arbitrary scales, and yet they may be made very random, or exactly homogeneous on average. They are not based on a `background' or `enveloping' metric that is effectively the large-scale average. The method can easily be combined with the swiss-cheese approach and exact inhomogeneous metrics, and thereby it provides for a much wider range of interesting comological models, with more degrees of inhomogeneity, than hitherto. Though the class of example models explored here is relatively simple, it does illustrate some significant features, and points the way to more complex possibilities.
The above calculations have clearly demonstrated how the deceleration and equation-of-state parameters of the best-fit FLRW average model evolve with time, while those in each component of the construction are constant; and this even though the construction is exactly spatially uniform above a certain scale. This behaviour is due to the component regions having different expansion behaviour. Our finding therefore lends strong support to the contention that inhomogeneity means different regions evolve at different rates, and the evolution of the `average' model is not the just the average of those rates, but strongly depends on which regions dominate the volume. Since we live in a region with little or no expansion, the average Riemann and Ricci fields can be very different from those felt locally.
With the model presented here, the range of possibilities is not as large as one might like, since the spatial curvature of all the component regions is zero. More realistically, one would expect regions of both positive and negative spatial curvature, and one would expect some regions to expand and recollapse while others are ever-expanding. In this case, we expect the vacuum regions would expand fastest, and effective acceleration would emerge with time. However, it is a distinctly more interesting challenge to see how a collection of such regions could be patched together, using the technique presented here. While more general Bianchi models \cite{Bian1896,EllMcC69,RyaShe75} seem to be the obvious extension, application of the Darmois conditions between positive and negative curvature regions is likely to result in much trickier constraints. Similarly, the limited range of constructions for this zero curvature case means that it is not possible to find completely reasonable equations of state for all the regions. We expect generalisation of this initial model will allow for some very interesting and realistic possibilities. Thus one should regard this current model as more illustrative than physical.
\acknowledgments
CH thanks David L Wiltshire \& family for excellent hosting and warm hospitality during a 7 week visit to the University of Canterbury, where long discussions were the genesis of this paper. David should really be a co-author, but the earthquake severly curtailed his time, and he preferred to withdraw. Thanks also to Ishwaree Neupane \& family of UC for kind hospitality. Many thanks to the University of Canterbury for an Erskine fellowship; and the University of Cape Town for rated researcher financial support.
|
1,314,259,995,819 | arxiv | \section{Introduction}\label{sec:intro}
After the discovery of a Higgs-like resonance, with a mass of 125 GeV, at the Large Hadron Collider (LHC) in 2012, various properties of this new particle have been studied. The spin and parity measurements have established it as a $0^{+}$ state at 99.9\% CL against alternative scenarios~\cite{Aad:2015mxa}. Couplings of this new particle with the fermions and gauge bosons predicted in the standard model are getting constrained as more and more data
are being analyzed by the LHC experiments~\cite{Khachatryan:2016vau,Sirunyan:2018koj,ATLAS:2019slw}. To this end, the vector-boson fusion production of the Higgs boson, associated production of $VH (V=Z,W)$, and Higgs boson's decay into vector bosons set limits on the $HVV$ couplings~\cite{Anderson:2013afp,CMS:2020dkv}. The gluon-gluon ($gg$) channel production of the Higgs boson helps in constraining the $H t\bar{t}$ coupling~\cite{CMS:2020dkv}. In addition, the evidence for the associated production of Higgs boson with a top-quark pair~\cite{Aaboud:2017jvq,Sirunyan:2018hoz} will provide the direct measurement of $Ht\bar{t}$ coupling. We still need to measure the trilinear and quartic Higgs self-couplings in order to know the form of the Higgs potential which will in turn reveal the exact symmetry breaking mechanism.
The Higgs self-couplings can be probed directly in multi-Higgs production processes~\cite{TheATLAScollaboration:2014scd,Chen:2015gva,Fuks:2017zkg}. Recently, indirect methods of probing them at hadron
and lepton colliders have also been proposed~\cite{Rossi:2020xzx,McCullough:2013rea,Borowka:2018pxx}.
Similarly, the quartic couplings involving Higgs and vector bosons $HHVV$ are also not constrained independently. This coupling can be probed in the vector-boson fusion production of a Higgs boson pair~\cite{Bishara:2016kjn,Aad:2020kub}. In order to find the signals of new physics, it is important that we improve our theoretical predictions for the processes
involving Higgs boson at current and future colliders.
\\
Loop-induced decay and scattering processes can play an important role in searching for new physics. In the presence of new physics (new particles and/or interactions), the rates for such processes can differ significantly from their standard model predictions. In this regard, many $gg$ channel scattering processes in $2 \to 2$ and $2 \to 3$ category have been studied~\cite{deFlorian:1999tp, Melia:2012zg, Agrawal:2012df, Campanario:2011cs, Agrawal:2012as,
Campanario:2012bh, Shivaji:2013cca, Campbell:2014gua, Agrawal:2014tqa, Mao:2009jp, Hespel:2015zea, Gabrielli:2016mdd, Caola:2015psa, Caola:2015rqy, Campbell:2016yrh, Caola:2016trd, Granata:2017iod, Shivaji:2016lnu, Plehn:2005nk, Binoth:2006ym, Fuks:2017zkg, Maltoni:2014eza, Papaefstathiou:2015paa, Kilian:2017nio, Hirschi:2015iia,Agrawal:2017cbs}. In the present work, we are interested in loop-induced $gg$ channel contribution to $VVH$ ($\gamma \gamma H,~ \gamma ZH,~ ZZH$, and $W^{+}W^{-}H$) production. In QCD perturbation
theory, the leading order $gg$ channel contribution to $pp \to VVH$ is an NNLO contribution at the cross section level. Due to many
electroweak couplings involved and loop-induced nature of $gg \to VVH$ processes, their cross sections are expected to be small.
However, they can be important at high energy hadron colliders like 100 TeV $pp$ collider such as proposed hadronic
Future Circular Collider (FCC-hh) facility at
CERN~\cite{Mangano:2270978} and Super Proton-Proton Collider (SPPC) facility in China~\cite{CEPC-SPPCStudyGroup:2015csa}.
At such energy scale, the gluon flux inside the proton becomes very large.
In fact, for $\gamma\gamma H$, the $gg$ channel gives the dominant contribution.
\\
Unlike the quark-quark contributions, which are mainly sensitive to $HVV$
couplings, the gluon-gluon contribution allows access to $Ht{\bar t}, HHH$, and $HHVV$ couplings as well.
Note that the processes under consideration are background to $pp \to HH$ when one of the Higgs bosons decays into $\gamma\gamma/ \gamma Z/ ZZ^*$
or $WW^*$ final states. The process $pp \to ZZH$ is also a background to $pp \to HHH$ when two of the three Higgs bosons decay into
$b{\bar b}$ final states.
In this work, we present a detailed study of $gg \to \gamma\gamma H$ and $\gamma ZH$ for the first time in the SM.
The $gg$ channel contribution to $ZZH$ and $WWH$ in the SM have been studied in the past~\cite{Mao:2009jp,Baglio:2015eon,Baglio:2016ofi}.
We have presented the $ZZH$ and $WWH$ calculations in detail and have proposed
methods to enhance the relative contribution of gluon-gluon
channel over quark-quark channel.
Since loop-induced processes are sensitive to new physics,
we also study the effect of new physics in all $VVH$ processes using a common BSM framework --- the $\kappa$-framework. Going beyond the $\kappa$-framework, we have treated the $HHVV$ coupling independently and emphasized
its effect in $ZZH$ and $WWH$ processes. BSM study in a more sophisticated
framework is desirable but it is beyond the scope of the present work.
\\
Experimentally, $W$ and $Z$-boson polarizations have been measured at hadronic
colliders \cite{Chatrchyan:2011ig,Aad:2012ky,Aaboud:2019gxl}. We also compute the cross sections for the processes when
these bosons are polarized. For each process, the different production channels contribute predominantly to specific polarization configurations. This can help in enhancing
the contribution of the $gg$ channel, as compared to the $qq$ channel. The
$gg$ channel have sometimes stronger dependence on the kappa parameters,
in particular on $\kappa_V$. Therefore, an event sample with larger $gg$ channel
contribution can be helpful.
\\
The paper is organized as follows. In Sec.~\ref{sec:gg-fuse-VVH}, we discuss the Feynman diagrams which contribute to $gg \to VVH$
amplitudes. The model independent framework to study new physics effects is outlined in Sec.~\ref{sec:bsm-para}. In Sec.~\ref{sec:calc-check}, we provide details on the calculation techniques and various checks that we have performed in order to ensure the correctness of our calculation. In Sec.~\ref{sec:numrRes}, we present numerical results in SM and BSM scenarios for all the $VVH$ processes. Finally, we summarize our results and conclude in Sec.~\ref{sec:concl}.
\section{Gluon fusion Contribution to $VVH$}
\label{sec:gg-fuse-VVH}
The $gg$ channel contribution to $pp \to VVH$ is due to a loop-induced scattering process mediated by a quark-loop. The classes of diagrams contributing to $gg \to VVH$ processes are shown in Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}\footnote{Feynman diagrams have been made using Jaxodraw~\cite{Binosi:2008ig}.}.
For convenience, the diagrams contributing to $gg \to WWH$ process are shown separately in Fig.~\ref{fig:feyn-WWH-pen-bx-tr}. The $gg \to \gamma\gamma H$ process receives contribution only from the pentagon diagrams, while, $\gamma ZH$ receives contribution from both
pentagon and box class of diagrams. In case of $gg \to ZZH,~WWH$ processes, triangle class of diagrams also contribute.
We have taken all quarks but the top-quark as massless. Therefore, the top-quark contribution is relevant in
diagrams where Higgs boson is directly attached to the quark loop.
In the diagrams where Higgs boson does not directly couple to the quark loop, light quarks can also contribute.
The complete set of diagrams for each process can be obtained by permuting external legs.
These permutations imply that there are 24 diagrams in pentagon topology, 6 diagrams in
each box topology and 2 diagrams in each triangle topology.
The diagrams in which only one type of quark flavor runs
in the loop, are not independent.
Due to Furry's theorem only half of them
are independent~\cite{10.1143/ptp/6.4.614}. This observation leads to a significant simplification in the overall calculation.
This simplification, however, is not applicable to the $WWH$ case, where flavor changing interaction is
involved in the quark loop. For example, see (a) and (b) in Fig.~\ref{fig:feyn-WWH-pen-bx-tr}.
\\
\begin{figure}[h]
\includegraphics[angle=0,width=1\linewidth]{ggxxh_topo.pdf}\\
\caption{Different classes of diagrams for $gg \to VVH,~{V=\gamma, Z}$. In diagram (b), $q$ represents all quark flavors.
Process $gg \to \gamma\gamma H$ receives contribution only from (a) type diagrams, while $gg \to \gamma ZH$ gets contribution from both (a) and (b) type diagrams. In the case of $ZZH$, all the diagrams contribute; the diagrams (b) and (f) cover the situation in which $H$ is attached to a $Z$ boson. }
\label{fig:feyn-A-Z-H-pen-bx-tr}
\end{figure}
Thus, there are 12 independent pentagon diagrams (Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(a)) due to top-quark loop
contributing to $gg \to \gamma\gamma H$ process.
Similarly, the $gg \to\gamma ZH$ process receives contribution from 12 independent pentagon diagrams (Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(a)) due to top-quark loop and 3 independent box
diagrams (Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(b)) for each quark flavor. In principle, 5 light quarks ($u, d, c, s, b$) and 1 heavy quark ($t$)
contribute. The box class of diagrams arise due to $ZZH$ coupling and has effective box topology of $gg \to \gamma Z^*$
amplitude. Furry's theorem, in this case, implies that the axial vector coupling of $Z$ boson with quark does not
contribute to $gg \to \gamma ZH$ amplitude.
\\
Like the $gg \to \gamma ZH$ process, the $gg \to ZZH$ amplitude receives contribution from 12 independent pentagon diagrams with top-quark in the loop
(Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(a)). However, there are 6 independent box diagrams with effective box topology of $gg \to ZZ^*$ amplitude for each quark flavor
which covers the possibilities of $H$ coupling with any of the two external $Z$ bosons (Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(b)).
Further, a new box type contribution arises which has effective box topology of $gg \to HH^*$ amplitude
(Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(c)). Once again there are 3 such independent diagrams with only top-quark in the loop. In addition to that,
there are 4 independent triangle diagrams with top-quark in the loop and having effective triangle topology
of $gg \to H^*$ amplitude (Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr} (d), (e), (f)). In $gg \to ZZH$ amplitude, the Furry's theorem implies that the vector and axial vector coupling of $Z$ boson with quarks can contribute at quadratic level only.
\\
\begin{figure}[H]
\includegraphics[angle=0,width=1\linewidth]{ggwwh_topo_v2.pdf}\\
\caption{Different classes of diagrams contributing to $gg \to WWH$ process. With respect to $ZZH$, new
classes of box and triangle diagrams appear due to $ZWW$
coupling. In (a) and (b), due to the flavor changing interaction of $W$ with quarks, both the quark
flavors of a given generation enter in the loop. The diagrams (b), (g) and (i) cover the case when
$H$ is attached to a $W$ boson. }
\label{fig:feyn-WWH-pen-bx-tr}
\end{figure}
Among all $VVH$ amplitudes, the structure of $gg \to WWH$ amplitude is the most complex. Due to the involvement
of flavor changing interactions in Fig.~\ref{fig:feyn-WWH-pen-bx-tr} (a) and (b), the Furry's theorem is not applicable to these diagrams.
Therefore, 24 independent pentagon diagrams contribute to $gg \to WWH$ process for each generation of quarks. However,
since we neglect Higgs coupling with light quarks including the $b$ quark, there are only 12 non-zero independent
pentagon diagrams. In Fig.~\ref{fig:feyn-WWH-pen-bx-tr} (b), all the three quark generations contribute. Taking into account the possibility
of Higgs boson coupling with any of the two external $W$ bosons, there are total 12 independent box diagrams of type (b) for each generation.
{In diagrams (a) and (b), the axial vector coupling of $W$ with quarks contributes at quadratic as well as at linear level.}
Like in the $gg \to ZZH$ process, there are 3 independent box diagrams of type (c). Due to $ZWW$ coupling, a new box contribution of type (d) having
effective box topology of $gg \to HZ^*$ amplitude appears. Furry's theorem for diagram (d) implies that the vector coupling
of $Z$ with quarks does not contribute to the amplitude. The same explains the absence of similar box diagram due to
$\gamma WW$ coupling. Further, there are 4 independent triangle diagrams with top-quark loop (Fig.~\ref{fig:feyn-WWH-pen-bx-tr} (e), (f) (g)) as in case
of the $gg \to ZZH$ process. A new type of 3 independent triangle diagrams for each quark flavor with effective triangle topology of
$gg \to Z^*$ amplitude appears, once again due to $ZWW$ coupling
(Fig.~\ref{fig:feyn-WWH-pen-bx-tr} (h), (i)). These triangle diagrams are anomalous and they can receive contribution only from the third generation
quarks as the bottom and top-quarks have very different masses. This is indeed the case for (h) type diagrams.
However, we find that (i) type diagrams do not contribute. This is explained in the appendix A.
\section{BSM Parametrization}
\label{sec:bsm-para}
Measuring the couplings of the Higgs boson with fermions, gauge bosons and with itself is an important
aspect of finding the signatures of new physics at colliders. With the help of the data collected
so far at the LHC, we now know couplings of the Higgs boson with the top quark with an accuracy of 10-20\% and with vector bosons with an accuracy of 10\% at 1$\sigma$ ~\cite{Tanabashi:2018oca}. The Higgs self-couplings, on the other hand,
are practically unconstrained~\cite{Aad:2019uzh}.
\\
To study the new physics effects in $VVH$ processes, we take the simplest approach of modifying the SM
like couplings only, also known as the kappa framework for the parametrization of new physics~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn,Ghezzi:2015vva}.
In this framework, no new Lorentz structures and no new interaction vertices appear. The LHC
experiments have interpreted the data using this framework so far.
The couplings of our interest are $Ht{\bar t}$, $HVV$, $HHH$ and $HHVV$. Out of these couplings, $gg \to \gamma\gamma H$
is sensitive to only $Ht{\bar t}$ coupling. The $HVV$ coupling affects all other processes. The couplings $HHH$ and $HHVV$ affect only $gg \to VVH,~ {V=Z,W}$ processes.
\\
The modification in these couplings due
to new physics is implemented through scale factor $\kappa_{i }$ for various couplings of the Higgs boson in the SM.
In kappa framework, there
are three such scale factors namely $\kappa_t$ for Higgs coupling with top-quark, $\kappa_V$ for Higgs coupling with
vector bosons ($\kappa_{HZZ} = \kappa_{HWW} = \kappa_V$)~\footnote{Note that in the SM, the tree level interaction vertices
$H\gamma\gamma$ and $H\gamma Z$ do not exist.} and $\kappa_\lambda$ for Higgs coupling with itself.
Since in the SM both $HVV$ and $HHVV$ couplings are related, the scaling of $HHVV$ coupling is also parametrized by $\kappa_V$. In a more generic BSM framework, the $HHVV$ coupling, in principle, can be independent of $HVV$ coupling.
\\
In the presence of BSM effects, the amplitudes for the $gg$ channel processes depend on $\kappa_t$, $\kappa_V$, and $\kappa_\lambda$
as follows.
\begin{eqnarray}
{\cal M}^{\rm BSM}(gg \to \gamma\gamma H) &=& \kappa_t {\cal M}^{\rm SM}_{\rm PEN} \label{eq:aaH}\\
%
{\cal M}^{\rm BSM}(gg \to \gamma Z H) &=& \kappa_t {\cal M}^{\rm SM}_{\rm PEN} + \kappa_V {\cal M}^{\rm SM}_{\rm BX_1} \label{eq:aZH}\\
%
{\cal M}^{\rm BSM}(gg \to Z Z H) &=& \kappa_t {\cal M}^{\rm SM}_{\rm PEN} + \kappa_V {\cal M}^{\rm SM}_{\rm BX_1} +
\kappa_t^2\kappa_V {\cal M}^{\rm SM}_{\rm BX_2} + \nonumber \\
&& \kappa_t\kappa_V\kappa_\lambda {\cal M}^{\rm SM}_{\rm TR_1} +
\kappa_t\kappa_V {\cal M}^{\rm SM}_{\rm TR_2} + \kappa_t\kappa_V^2 {\cal M}^{\rm SM}_{\rm TR_3} \label{eq:ZZH}\\
%
{\cal M}^{\rm BSM}(gg \to WW H) &=& \kappa_t {\cal M}^{\rm SM}_{\rm PEN} + \kappa_V {\cal M}^{\rm SM}_{\rm BX_1} +
\kappa_t^2\kappa_V {\cal M}^{\rm SM}_{\rm BX_2} + \nonumber \\
&& \kappa_t {\cal M}^{\rm SM}_{\rm BX_3} + \kappa_t\kappa_V\kappa_\lambda {\cal M}^{\rm SM}_{\rm TR_1} +
\kappa_t\kappa_V {\cal M}^{\rm SM}_{\rm TR_2} + \nonumber \\
&& \kappa_t\kappa_V^2 {\cal M}^{\rm SM}_{\rm TR_3} + \kappa_V {\cal M}^{\rm SM}_{\rm TR_4} \label{eq:WWH}
\end{eqnarray}
In the above, the amplitude ${\cal M}^{\rm SM}_i$ is related to one of the diagram classes displayed in Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr} (Fig.~\ref{fig:feyn-WWH-pen-bx-tr} for $WWH$).
This can be easily identified by looking at $\kappa$-factors in front of the amplitude. Note that in $WWH$ amplitude,
${\cal M}^{\rm SM}_{\rm TR_4}$ includes both (h) and (i) type diagrams of Fig.~\ref{fig:feyn-WWH-pen-bx-tr}.
This parametrization does not affect the gauge invariance of the amplitudes with respect to the gluons as
it will become clear in the next section. The standard model prediction can be obtained by setting
$\kappa_t=\kappa_V=\kappa_\lambda = 1$. Thus, except in $gg \to \gamma\gamma H$, we can expect nontrivial interference effects on total and differential cross sections for $gg \to VVH$ processes due to new physics
in $\kappa$-framework.
\section{Calculation and Checks}
\label{sec:calc-check}
The calculation of quark-loop diagrams is carried out using a semi-automated in-house package {\tt OVReduce}~\cite{Agrawal:1998ch} which allows
the calculation of any one-loop amplitude with maximum five propagators in the loop. The main steps involved in our calculation are: quark-loop trace evaluation, one-loop tensor reduction to master integrals and evaluation of master integrals. Trace calculation and
simplification of the amplitude are done using symbolic manipulation software {\tt FORM}~\cite{Vermaseren:2000nd}. Tensor reduction
of one-loop amplitudes into one-loop master integrals is done numerically following the method of Oldenborgh-Vermaseren~\cite{vanOldenborgh:1989wn}.
Further, the one-loop master integrals are also calculated numerically using the {\tt OneLOop} package~\cite{vanHameren:2010cp}.
More details on this can be found in ~\cite{Shivaji:2013cca}.
We perform the calculation in $4-2\epsilon$ space-time dimensions to regulate ultraviolet (UV) and infrared (IR)
singularities of one-loop
master integrals. Since the couplings of $Z$ and $W$ bosons with quarks involve $\gamma_5$, the trace calculation
needs special care. We have used 4-dimensional properties of $\gamma_5$ in the calculation. This works because the SM is anomaly free. We have chosen Unitary gauge for the calculation
of the amplitudes.
\\
The amplitude calculation for each process can be efficiently organized using prototype
amplitudes for each class of diagrams. For example, amplitudes for all the 12 independent pentagon diagrams in $gg \to \gamma\gamma H$ process can be obtained using only one prototype pentagon amplitude.
Similarly, prototype amplitudes can be identified for each topology contributing to each process.
The full amplitude for each process is a function of external momenta and polarization vectors/helicities.
Due to huge expressions of the amplitudes, we calculate helicity amplitudes and the squaring of the
amplitude for each process is done numerically. The number of helicity amplitudes for
$gg \to \gamma\gamma H,~\gamma ZH,~ ZZH,~ WWH$ processes are 16, 24, 36, and 36, respectively.
\\
There are a number of checks that we have performed in order to ensure the correctness of the amplitudes. We have checked
that the amplitudes are separately UV and IR finite. In $4-2\epsilon$ dimensions,
these divergences appear as poles in $1/\epsilon$ (for UV and IR) and $1/\epsilon^2$ (for IR only). Each pentagon
diagram is UV finite. This we
expect from the naive power counting. The individual box diagram is not UV finite, however, the full box
amplitude, in each class, is UV finite. The UV finiteness of triangle amplitudes holds for each diagram.
One-loop diagrams with all massive internal lines are IR finite, as expected. Thus, IR finiteness check is relevant
to the diagrams with massless quarks in the loop. This includes box class of diagrams of Fig.~\ref{fig:feyn-A-Z-H-pen-bx-tr}(b) in $gg \to \gamma ZH$
and $ZZH$. In $gg \to WWH$ case, potentially IR divergent diagrams include Fig.~\ref{fig:feyn-WWH-pen-bx-tr}(a), (b), (h) and (i).
Unlike UV, the IR finiteness holds for each diagram~\cite{Shivaji:2013cca}.
\\
We have also checked the gauge invariance of the amplitudes
with respect to the external gluons. For that we numerically replace the
gluon polarization vector $\epsilon^\mu(p)$ by its four momentum $p^\mu$ and
expect a gauge invariant amplitude to vanish.
We find that the gauge invariance check holds for each class of diagrams.
This is expected because different box and triangle topologies for each process arise due to
the existence of various electroweak couplings.
This is a very strong check on our calculation which is organized using
only prototype amplitudes. However, this check
cannot verify relative signs between different classes of diagrams. In order to verify such
relative signs, one needs to perform gauge invariance check in electroweak theory
which is a non-trivial task\footnote{A wrong relative sign between different class of diagrams {\it may} lead to violation of unitarity in certain
processes~\cite{Maltoni:2001hu}.}. We rather rely on cross-checking the calculation
using different methods and tools.
We have compared our matrix element for each process with those calculated using
{\tt MadLoop}~\cite{Alwall:2014hca} and have found an excellent agreement.
{Being process specific, our code is efficient and provides greater flexibility when producing phenomenological result.}
\\
Numerical predictions for cross section and kinematic distributions are obtained
using Monte Carlo techniques for phase space integration. We use {\tt AMCI}~\cite{Veseli:1997hr}
package for Monte Carlo phase space integration which is based on {\tt VEGAS}~\cite{Lepage:123074}
algorithm and allows parallelization of phase space point generation and matrix-element
computation using {\tt PVM} software~\cite{Geist:1995:PPV:207505}.
\section{Numerical Results}\label{sec:numrRes}
The cross section and kinematic distributions for $pp \to VVH$ processes in SM and in BSM constitute
the main results of this section.
The numerical results are produced using following basic selection cuts unless stated otherwise,
\begin{eqnarray}
p_T^{\gamma} > 50~{\rm GeV},~ |\eta^\gamma| < 2.5,~ \Delta R_{\gamma\gamma} > 0.4,~ |y^{H,Z,W}| < 5.
\label{eq:cuts}
\end{eqnarray}
The results for the $gg$ channel processes are calculated using {\tt CT14LO}~\cite{Dulat:2015mca} parton distribution function (PDF)
and partonic center-of-mass energy $(\sqrt{\hat s})$ is chosen as common scale for renormalization ($\mu_R$) and
factorization ($\mu_F$). The results are obtained for three different choices of collider energies: $\sqrt{s} = 14, 27,$
and 100 TeV. From phenomenological point of view we will focus on
$p_T(H)$ and $M(VV)$ distributions.
\\
We compare the $gg$ channel contribution to $pp \to VVH$ with contribution arising from the $qq$ channels. The
$qq$ channel contribution at LO and NLO (QCD) is calculated using {\tt MadGraph5\_aMC@NLO}~\cite{Alwall:2014hca} in five flavor scheme for all but WWH production. The $qq$ channel contribution to WWH production is instead calculated in four flavor scheme\footnote{For WWH production, currently {\tt MadGraph5\_aMC@NLO} cannot produce NLO correction to the $bb$ channel.}.
The LO $qq$ channel contributions are pure electroweak processes and they do not depend on $\alpha_s$.
For LO and NLO (QCD) results, we use {\tt CTEQ14LO} and {\tt CT14NLO} PDFs, respectively~\cite{Dulat:2015mca}. The scale choice is same as in the $gg$ channel calculation.
In both $gg$ and $qq$ channel calculations, the
scale uncertainties are estimated by varying $\mu_R$ and $\mu_F$ independently by a factor of two. We
quote only minimum and maximum uncertainties thus obtained.
\\
To quantify the relative importance of the $gg$ channel contribution
in processes dominated by the $qq$ channel, we
define following ratio,
\begin{eqnarray}
R_1 = \frac{\sigma^{\rm VVH, LO}_{gg}}{\sigma^{\rm VVH, NLO}_{qq}-\sigma^{\rm VVH, LO}_{qq}}.
\label{eq:R1}
\end{eqnarray}
This ratio compares the leading order $gg$ channel contribution with NLO QCD correction in the $qq$ channel. Recall that technically $gg$ channels contribute at NNLO.
Similarly, at differential level we define another ratio,
\begin{eqnarray}
R_2 = \frac{\frac{d\sigma}{dX}\bigr\rvert^{\rm VVH, LO}_{gg}}{
\frac{d\sigma}{dX}\bigr\rvert^{\rm VVH, NLO}_{qq}},
\label{eq:R2}
\end{eqnarray}
where, $X$ denotes a kinematic variable.
\\
As mentioned in section~\ref{sec:bsm-para},
the BSM effects are parametrized in terms of scale factors
$\kappa_t$, $\kappa_V$ and $\kappa_\lambda$.
In order to compare their relative importance, we vary
them independently by 10\% about their SM values.
Further, we comment on the effect of $\kappa_\lambda$
and $\kappa_{HHVV}$ (the scale factor for the $HHVV$ coupling\footnote{Note this is different from $k_V$, which scales both $HVV$ and $HHVV$ couplings at the same time. })
which are least constrained at present,
in $ZZH$ and $WWH$ processes.
\subsection{Predictions for the $pp \to \gamma \gamma H$ process}\label{sec:results-AAH}
\begin{table}[H]
\begin{center}
\resizebox{0.7\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$\sqrt{\rm s}\;(\rm TeV)$
& {$\sigma^{\gamma\gamma\tin{\rm H,\,LO}}_{\tin{gg}}\;~[\rm ab]$}
& {$\sigma^{\gamma\gamma\tin{\rm H,\,LO}}_{\tin{qq}}\;~[\rm ab]$}
& {$\sigma^{\gamma\gamma\tin{\rm H,\,NLO}}_{\tin{qq}}\;[\rm ab]$}
\\
\hline
& {} & {} & {} \\
14
& {$5.36^{+28\%}_{-20\%}$}
& {$0.033^{+13\%}_{-14\%}$}
& {$0.046^{+5\%}_{-6\%}$}
\\
& {} & {} & {} \\
27
& {$22.0^{+22\%}_{-19\%}$}
& {$0.153^{+15\%}_{-17\%}$}
& {$0.234^{+5\%}_{-7\%}$}
\\
& {} & {} & {} \\
100
& {$220.1^{+27\%}_{-21\%}$}
& {$1.4^{+20\%}_{-20\%}$}
& {$2.25^{+5\%}_{-8\%}$}
\\
& & & \\
\hline \end{tabular}}
\end{center}
\caption{A comparison of different perturbative orders in QCD coupling contributing to $pp \to \gamma \gamma H$
hadronic cross section at $\sqrt{s}=$ 14, 27, and 100 TeV. } \label{table:xs-aaH}
\end{table}
The cross section for this process is dominated by the $gg$ channel. In the $qq$ channel, only bottom-quark initiated
subprocess contribute to $\gamma\gamma H$ production.
However, this cross section is quite small, owing to small
bottom Yukawa coupling. In Tab.~\ref{table:xs-aaH}, we compare the $gg$ and $qq$ channel
contributions to the hadronic cross section at 14, 27 and 100 TeV colliders. The results are with minimum 50 GeV transverse momentum of photons. We find that the $gg$ channel contribution increases 40 times as the collider center-of-mass energy goes
from 14 TeV to 100 TeV. Due to a small cross-section, this process cannot be observed at the HL-LHC; FCC-pp will be more suitable.
The $gg$ channel contribution becomes important at higher center-of-mass energy collider, as in this case smaller partonic momentum fractions ($x$) are accessible, where gluon flux is significantly large.
The scale uncertainties on the
cross sections for the $gg$ channel are in the range of 20-30\%. It is clear
from the table that the $qq$ channel contribution is negligible compared to
the $gg$ channel contribution. It is merely 1\% of the $gg$ channel contribution even after
including the NLO-QCD corrections.
\\
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggaaH-pt-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggaaH-M_aa-100TeV_v8.pdf}
\end{center}
\caption{ Kinematic distributions for $gg \to \gamma \gamma H$ process in the SM at 100 TeV. In the $p_T$ distribution plot, $\gamma_1$ and $\gamma_2$ refer to the hardest and second hardest photons in $p_T$, respectively. In the right plot, we show $M(\gamma \gamma)$ distribution for $gg \to \gamma\gamma H$. In addition, the total cross section for the $gg \to H H \to \gamma\gamma H$ process has been shown at 125 GeV. ``$\times 200$" implies that the height of purple vertical line should be multiplied by a factor of 200 in order to get the correct cross section for the $gg \to H H \to \gamma\gamma H$ process. }
\label{fig:dists-AAH-sm}
\end{figure}
In Fig.~\ref{fig:dists-AAH-sm}, we have plotted $p_T$ distributions for hardest photon, next-to-hardest photon, and Higgs in the left figure, and
diphoton invariant mass distribution ($M(\gamma\gamma)$) in the right figure for the 100 TeV collider (FCC-hh). The $ p_T$ distributions for them peak around 150 GeV, 90 GeV, and 70 GeV, respectively.
We find that the tail of $p_T(H)$ is softer than that of photons. The $M(\gamma\gamma)$ distribution shows an interesting feature -- it has two peaks. The right peak occurs at around 350 GeV, exhibiting the $t\bar{t}$ threshold effect in the distribution. To verify that the second peak is indeed due to $t\bar{t}$ threshold effect, we changed in our code the value of $m_t$ from 173 GeV to 200 GeV and the second peak was found to get shifted to 400 GeV.
\\
As mentioned before, this process is a background to double Higgs production process when one of the Higgs bosons decays into a photon pair. To manage the background one usually looks at `$\gamma \gamma b \bar{b}$' final state, instead of `$ b \bar{b} b \bar{b}$', as the signature of the double Higgs boson production. At a 100 TeV collider, while the cross section for the $gg \to \gamma \gamma H$ production, with the cuts in Eq. \ref{eq:cuts}, is about 220 ab, the cross section for $gg \to H H \to \gamma\gamma H$, with the same set of cuts, is about 2600 ab.
From the right panel of Fig.~\ref{fig:dists-AAH-sm}, it can be seen that the cross section for $\gamma\gamma H$ production in the bin from 120 GeV to 140 GeV is about 3 ab. On the other hand, all the cross section for $gg \to H H \to \gamma\gamma H$ is concentrated in a very narrow width around the mass of Higgs, 125 GeV \footnote{In the right panel of Fig.~\ref{fig:dists-AAH-sm}, at 125 GeV, rather than showing a very narrow Breit-Wigner distribution, we have shown the total cross section for $gg \to H H \to \gamma\gamma H$ by a single vertical line.}. As a result, $gg \to \gamma\gamma H$ is an insignificant background to the process $gg \to H H \to \gamma\gamma H$.
\\
Regarding anomalous coupling contributions, we note that as only pentagon diagrams contribute to the process $gg \to \gamma\gamma H$, its cross section scales as $\kappa_t^2$. So a 10\% change in $\kappa_t$ will change the cross section and
distributions by about 20\%. For the $qq$ channel process, the cross section is too small. It depends on $\kappa_b$, which we do not change from the standard model value.
\subsection{Predictions for the $pp \to \gamma ZH$ process}\label{sec:results-AZH}
Unlike $\gamma\gamma H$ case, the $\gamma ZH$ production receives dominant contribution from the $qq$ channel.
With $ p_T^\gamma > 50$ GeV, the $gg$ channel contributions to $\gamma ZH$ production at 14, 27, and 100 TeV colliders are 4 ab, 16 ab, and 168 ab, respectively.
The corresponding values for the LO $qq$ channel contribution are 689 ab, 1733 ab, and 7498 ab, respectively.
From Tab.~\ref{table:xs-aZH}, it can be seen that $R_1$, which is the ratio of the $gg$ channel contribution to NLO correction in the $qq$ channel, is as small as 0.06 for 100 TeV collider, and even smaller for HE-LHC (27 TeV) and LHC (14 TeV).
The scale uncertainties for the $gg$ channel are around 20\% while those for the $qq$ channel at NLO are in the range of $2-3\%$. A larger
scale dependence in the $gg$ channel contribution can be attributed to the presence of higher power of $\alpha_s$ factor in the $gg$ amplitudes.
\begin{table}[H]
\begin{center}
\resizebox{0.8\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|}
\hline
$\sqrt{\rm s}\;(\rm TeV)$ & \multicolumn{1}{c|}{$\sigma^{\gamma\tin{\rm ZH,\,LO}}_{\tin{gg}}\;~[\rm ab]$} & \multicolumn{1}{c|}{$\sigma^{\gamma\tin{\rm ZH,\,LO}}_{\tin{qq}}\;~[\rm ab]$} & \multicolumn{1}{c|}{$\sigma^{\gamma\tin{\rm ZH,\,NLO}}_{\tin{qq}}\;[\rm ab]$} & \multicolumn{1}{c|}{$R_1$}\\
\cline{2-5}
\hline
& \multicolumn{1}{c|}{} &\multicolumn{1}{c|}{} &\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} \\
14 & \multicolumn{1}{c|}{$4.0^{+26\%}_{-20\%}$} & \multicolumn{1}{c|}{$689^{\,+\, 0\%}_{-0.2\%}$} & \multicolumn{1}{c|}{$909^{+1.7\%}_{-1.3\%}$} & \multicolumn{1}{c|}{0.02} \\
& \multicolumn{1}{c|}{} &\multicolumn{1}{c|}{} &\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} \\
27 & \multicolumn{1}{c|}{$16^{+22\%}_{-17\%}$} & \multicolumn{1}{c|}{$1773^{+3.0\%}_{-3.6\%}$} & \multicolumn{1}{c|}{$2349^{+1.7\%}_{-2.1\%}$} & \multicolumn{1}{c|}{0.03} \\
& \multicolumn{1}{c|}{} &\multicolumn{1}{c|}{} &\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} \\
100 & \multicolumn{1}{c|}{$168^{+21\%}_{-19\%}$} & \multicolumn{1}{c|}{$7498^{+8.8\%}_{-9.4\%}$} & \multicolumn{1}{c|}{$10430^{+2.2\%}_{-3.8\%}$} & \multicolumn{1}{c|}{0.06}\\
& & & & \\
\hline \end{tabular}}
\end{center}
\caption{ A comparison of different perturbative orders in QCD coupling contributing to $pp \to \gamma ZH$
hadronic cross section at $\sqrt{s}=$ 14, 27, and 100 TeV. $R_1$ compares
the $gg$ channel contribution with correction at NLO and it is defined in Eq~\ref{eq:M1}.} \label{table:xs-aZH}
\end{table}
In Tab.~\ref{table:xs-aZH-cuts}, the effect of various $p_T^\gamma$ cuts in $gg$ and $qq$ channels has been shown. As the cut on $p_T^\gamma$ increases, the $qq$ channel cross section decreases faster than the $gg$ channel. In going from 50 GeV to 200 GeV cut, the cross section of the $gg$ channel decreases roughly by a factor of 6, while that of the $qq$ channel decreases by a factor of 9. Thus relative
contribution from the $gg$ channel can be enhanced with the help of harder
$p_T^\gamma$ cut. We find that the $p_T(H)$ cuts have opposite effect \emph{i.e.}
the $gg$ channel is favored at low $p_T(H)$.
\begin{table}[H]
\begin{center}
\resizebox{0.95\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|}
\hline
$p_{T, min}^\gamma$ (GeV) & ${gg \rightarrow \gamma \rm Z H }$ [ab] & ${qq \rightarrow \gamma \rm Z H (LO)}$ [ab] & ${qq \rightarrow \gamma \rm Z H (NLO)}$ [ab] \\
\hline
50 & 168 & 7498 & 10430 \\
\hline
100 & 95 & 2812 & 4072 \\
\hline
150 & 47 & 1366 & 2069 \\
\hline
200 & 28 & 765 & 1190 \\
\hline
\end{tabular}
}
\end{center}
\caption{Effect of $p_T^\gamma$ cut on the cross section of $pp \rightarrow \gamma Z H$ production at the 100 TeV collider (FCC-hh).}
\label{table:xs-aZH-cuts}
\end{table}
In Fig.~\ref{fig:dists-AZH-sm}, we have displayed $p_T$ distributions for the final state particles on the left, and $\gamma Z$ pair invariant mass distribution on the right for the 100 TeV collider. The $p_T$ distributions peak around 100 GeV while the $M(\gamma Z)$ distribution peaks around 200 GeV. Like the case of $gg \to \gamma \gamma H$ process as a background to $gg \to H H \to \gamma \gamma H$, the $gg \to \gamma Z H$ process is also an insignificant background to $gg \to H H \to \gamma Z H$. This is because at a 100 TeV collider, with the cuts in Eq.~\ref{eq:cuts}, the cross section for $gg \to H H \to \gamma Z H$ is about 2000 ab, while
the cross section for $gg \to \gamma Z H$ process is about 170 ab. Moreover, all the cross section for the $gg \to H H \to \gamma Z H$ process congregates around the mass of the decaying Higgs boson, 125 GeV
\footnote{However, instead of showing a very narrow Breit-Wigner distribution for Higgs' decay, we have depicted the total cross section at 125 GeV by a single vertical line.}, while, as can be seen from the right panel of the Fig.~\ref{fig:dists-AZH-sm}, the cross section for the $gg \to \gamma Z H$ process in the bin from 120 GeV to 140 GeV is about 3 ab. However, the $qq$ channel for $\gamma Z H$ production may act as an important background for the $gg \to H H \to \gamma Z H$ process.
\\
In Fig.~\ref{fig:dist-aZH-pen-bx-tr}, we show $ p_T(H)$ distributions for different classes of diagrams -- pentagon, box, and sum of their individual contributions, their interference, and total at the 100 TeV collider.
The contribution of the box diagrams is more than the pentagon diagrams mainly because of the light quark contributions.
The interference effect between the pentagon and box diagrams has kinematic dependence.
We find that in the region of our kinematic interest, it is always destructive and, near the peak, its effect is close to -30\%.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggaZH-pt-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggaZH-M_aZ-100TeV_v8.pdf}
\end{center}
\caption{ Kinematic distributions for $gg \to \gamma ZH$ in the SM at 100 TeV. The purple vertical line in the right plot at 125 GeV shows the total cross section for the process $gg \to H H \to \gamma Z H$. ``$\times 200$" means that the height of the purple vertical line needs to be scaled by a factor of 200 to get the correct cross section for the $gg \to H H \to \gamma Z H$ process. }
\label{fig:dists-AZH-sm}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggaZH-pth-interference-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggaZH-multi-pth-With-And-Without-TOP-100TeV_v8.pdf}
\end{center}
\caption{Left: The contribution of pentagon (blue) and box (green) diagrams, as well as their squared sum, interference, and total contribution to $p_T(H)$ distributions for the $gg \to \gamma ZH$ process at the 100 TeV FCC-hh collider. Right: The effect of excluding top-quark contribution from the diagrams in Fig. 1(b) to the full amplitude.} \label{fig:dist-aZH-pen-bx-tr}
\end{figure}
Since the $gg\gamma Z^*$ type box amplitude does
not depend on the axial-vector coupling of the off-shell longitudinal $Z$-boson with the quarks, the top-quark contribution is not very significant at the level of total cross section. This is shown in the right panel of the Fig.~\ref{fig:dist-aZH-pen-bx-tr}. We can see that in the
tail where top quark is effectively light, the cross section increases by about 20\%.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{100TeV_ggazh-histogram_v4.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{100TeV_qqazh-histogram_v8.pdf}
\end{center}
\caption{ LO cross section for $\gamma ZH$ production in different helicity configurations in the $gg$ (left) and $qq$ (right) channels. Legends correspond to different helicities of initial states.}\label{fig:xs-hist-helicity-aZH}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{multi_pTH_100TeV_azh-hel_both_1_or_m1_v8.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{multi_pTa_100TeV_azh-hel_both_1_or_m1_v8.pdf}
\end{center}
\caption{The kinematic distributions for $gg$ and $qq$ channels when both $\gamma$ and $Z$-boson have same helicity. The ratio of the distributions from the two channels is shown in the lower panel of each figure.}\label{fig:dist-helicity-aZH}
\end{figure}
We have noted that the relative importance of gluon fusion
channel can be enhanced by applying higher $p_T(\gamma)$ cuts.
To distinguish the $gg$ channel contribution from the dominant $qq$ channel, one can use the polarized cross sections and distributions. In Fig. \ref{fig:xs-hist-helicity-aZH}, we have displayed the LO cross sections for various helicity states of the final state
particles, $\gamma$ and $Z$ boson. The figure also shows the contribution of
various polarization states of the initial state particles. We cannot measure the polarization of the initial
state particles that are in a bound state, proton. However, experimentally, one can
measure the $Z$-boson polarization~\cite{Chatrchyan:2011ig,Aad:2012ky,Aaboud:2019gxl}. The polarization of photon has been measured by the LHCb collaboration in $b$-baryon's decay\cite{Aaij:2016ofv,Legger:2006cq, Hiller:2007ur, Orlovsky:2007hv, Shchutska:2008dba, Oliver:2010im, Martin:2019xcc}. At a 100 TeV collider,
the contribution of the $gg$ channel process to the production of
$\gamma Z H$ is only $2.2\%$. However, if we look at those final states where
photon and $Z$-boson have the same transverse polarization, then this ratio increases to $10-11\%$. (The $qq$ channel makes largest contribution when the $Z$ boson is longitudinally polarized.)
This is a non-trivial contribution, and can be measured, if enough integrated luminosity
is available. In Fig.\ref{fig:dist-helicity-aZH}, we have plotted the Higgs boson and $Z$-boson $p_T$ distributions. By making appropriate cuts on the small and large $p_T$
of these particles, we can further enhance the $gg$ channel contribution.
\\
Turning to the effect of anomalous couplings, we find that the $gg$ channel shows very small dependence on the $\kappa_t$, as it is present only in pentagon diagrams whose contribution is small (see Fig.~\ref{fig:dist-aZH-pen-bx-tr}). However, it strongly depends on $\kappa_V$, as the box-diagrams contribution is much more than the pentagon-diagram contribution.
We find that the change in cross section for $\kappa_t= 1.1 (0.9)$ is
5.4\% (-1.2\%). On the other hand, for $\kappa_V= 1.1 (0.9)$ the
cross section changes by 18\% (15\%).
{ We do not show the effect of anomalous couplings on the distribution. It can be understood qualitatively from Eq.~\ref{eq:aZH} and Fig.~\ref{fig:dist-aZH-pen-bx-tr} in the $gg$ channel}.
The $qq$ channel is sensitive to $\kappa_V$ only. The amplitude has
overall linear dependence on $\kappa_V$ due to which the effect of anomalous coupling
$k_V$ is flat for both total and differential cross sections.
\subsection{Predictions for $pp \to ZZH$}
The cross sections for $ZZH$ production via various channels have been tabulated in Table~\ref{table:xs-ZZH} along with the corresponding scale uncertainties. The total cross section for $gg \to ZZH$ is significantly larger than that of $gg \to \gamma Z H$. This increase is mainly due to the
contribution from axial-vector coupling of $Z$ with quarks.
The $gg$ channel contributions to $ZZH$ production at 14, 27, and 100 TeV colliders are 124 ab, 579 ab, and 7408 ab, respectively. The corresponding values of the LO $qq$ channel contributions are 2184, 5997, and 36830 ab, respectively. The ratio, $R_1$, is found to be 0.25, 0.4, and 1.05, respectively.
Thus at 100 TeV, the $gg$ channel contribution is as important as the QCD NLO correction. As has already been discussed, this increase in ratio $R_1$ with collider energy is due to the large gluon flux.
\\
\begin{table}[h]
\begin{center}
\resizebox{0.8\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c||c||}
\hline
$\sqrt{\rm s}\;(\rm TeV)$ & $\sigma^{\tin{ \rm ZZH,\,LO}}_{\tin{gg}}\;~[\rm ab]$ & $\sigma^{\tin{\rm ZZH,\,LO}}_{\tin{qq}}\;~[\rm ab]$ & $\sigma^{\tin{\rm ZZH,\,NLO}}_{\tin{qq}}\;[\rm ab]$ & ${R}_{1} $\\
\hline
& & & & \\
14 & $124^{+28.2\%}_{-21.0\%}$ & $2184^{+0.2\%}_{-0.6\%}$ & $2710^{+1.4\%}_{-1.0\%}$ & 0.24 \\
& & & & \\
27 & $579^{+23.3\%}_{-18.5\%}$ & $5997^{+2.4\%}_{-3.0\%}$ & $7396^{+1.3\%}_{-1.6\%}$ & 0.41 \\
& & & & \\
100 & $7408^{+22\%}_{-18\%}$ & $36830^{+8.0\%}_{-8.7\%}$ & $43940^{+1.2\%}_{-2.6\%}$ & 1.04 \\
& & & & \\
\hline
\end{tabular}}
\end{center}
\caption{ A comparison of different perturbative orders in QCD coupling contributing to $pp \to ZZH$ cross section at $\sqrt{s}=$ 14, 27, and 100 TeV. The ratio $R_1$, defined in Eq.~\ref{eq:R1}, quantifies the $gg$ channel contribution with respect to the NLO correction in $qq$ channel process. } \label{table:xs-ZZH}
\end{table}
In the $gg$ channel, the scale uncertainties of the total cross sections are in the range of 20-30\% which is similar to the scale uncertainties observed for $\gamma\gamma H$ and $\gamma ZH$. We find that the uncertainty due to the renormalization scale variation is more than that due to the factorization scale variation. While the change in the renormalization scale mainly changes $\alpha_s$, the change in the factorization scale changes the parton distribution function.
The uncertainty for the renormalization scale variation is nearly same at all the collider energies. This happens as the contribution to the total cross section comes from nearly same region of partonic center of mass energy of the process and in every bin of this region, $\alpha_s$ changes by nearly same factor for the change in the renormalization scale.
However, uncertainty for the factorization scale variation is different for different colliders.
This happens as for different collider energies, different $x$ regions contribute to the process and for different $x$ regions change in parton distribution function with the factorization scale is different, where $x$ is partonic momentum fraction. We have also observed that with an increase in the factorization scale, for 14 and 27 TeV colliders, the cross-section decreases; however for 100 TeV collider the cross-section increases.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggZZH-pt-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggZZH-M_ZZ-100TeV_v3.pdf}
\end{center}
\caption{ Kinematic distributions for $gg \to ZZH$ in the SM at the 100 TeV collider. $Z_1\ {\rm and}\ Z_2$ refer to the hardest, and second hardest in $p_T$, respectively. }
\label{fig:dists-ZZH-sm}
\end{figure}
In the tree level $qq$ channel, there is no QCD vertex. So here change in the renormalization scale does not affect the cross section. But, the change in the factorization scale can affect the cross section, and uncertainty increases with collider energy. However, when NLO QCD correction is considered, change in either of renormalization and factorization scales changes the cross section. The uncertainty in the cross section due to the renormalization scale variation is small as NLO QCD correction is much smaller than the LO results.
The overall uncertainty in this case is smaller than the LO case, which is expected for higher order calculation.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggZZH-pth-interference-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggZZH-multi-pth-With-And-Without-TOP-100TeV_v3.pdf}
\end{center}
\caption{Left : SM contribution of pentagon (blue), box (green), triangle (gray) diagrams, as well as their squared sum (black), interference (orange) and total (red) contribution to $p_T(H)$ distributions in $gg \to ZZH$ at 100 TeV collider (FCC-hh).
Right: The effect of excluding top-quark contribution from Fig. 1(b) to full amplitude. } \label{fig:dist-ZZH-pen-bx-tr}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth,height=0.35\linewidth]{ppZZH-pth-comparison_qq_gg_100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth,height=0.35\linewidth]{ppZZH-multi-pth-comparison_qq_gg_100TeV_v3.pdf}
\end{center}
\caption{The left figure shows the normalized distribution for $p_T(H)$ in $gg$ and $qq$ channel processes. In the top panel of the right figure, we show the distribution of $qq$ (NLO) + $gg$ (LO) and $qq$ (NLO) production with $p_T(H)$. The lower panel shows the ratio of them.
}
\label{fig:dist-ZZH-QQ-GG}
\end{figure}
In Fig.~\ref{fig:dists-ZZH-sm}, we have plotted $p_T$ distributions for leading $ p_T(Z_1)$, next-to-leading $ p_T(Z_2)$, and Higgs boson in the left figure, and $Z$-pair invariant mass distribution in the right figure for the 100 TeV collider. The $p_T$ distributions peak around 100 GeV, 60 GeV, and 80 GeV, respectively. The $M(ZZ)$ distribution peaks around
$Z$ pair threshold.
\\
Interference of various diagrams plays a major role in $gg \to ZZH$ production. In Fig.~\ref{fig:dist-ZZH-pen-bx-tr},
we have shown the $p_T(H)$ distributions for penta, box, triangle, sum of their individual contributions,
interference, and total at the 100 TeV collider (FCC-hh). As can be seen, the box diagrams give the largest contribution,
then comes the triangle contribution and the penta contributes the least.
Like in $\gamma ZH$ case, the large box contribution is due to
the light quarks in the loop.
Further, because of large {\it destructive}
interference, the
total contribution is smaller by about a factor of five than the box contribution.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{100TeV_ggzzh-histogram_v4.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{100TeV_qqzzh-histogram_v8.pdf}
\end{center}
\caption{LO cross section for $ZZH$ production in different helicity configurations in $gg$ (left) and $qq$ (right) channels. Legends correspond to different helicities of initial states.
}\label{fig:xs-hist-helicity-ZZH}
\end{figure}
We have found that the top-quark contribution in $ggZZ^*$-type box diagram is quite significant despite the propagators suppression. This is due to the
coupling of off-shell longitudinal $Z$ boson (effectively the Goldstone boson) with top-quark and it is proportional to $m_t$~\footnote{The results for $ZZH$ process presented in the
conference proceeding~\cite{Shivaji:2016lnu} did not include top-quark contribution. We also fixed a bug in the code, numerical impact of which has
been found to be small.}. We show the effect of excluding the top-quark contribution in $ggZZ^*$-type box diagram (Fig.1(b))
on $p_T(H)$ distribution in the right panel of Fig.~\ref{fig:dist-ZZH-pen-bx-tr}. As we expect, excluding top-quark contribution in $ggZZ^*$-type box diagram leads to non-unitary behavior in the full amplitude.
\\
In the left figure of Fig.~\ref{fig:dist-ZZH-QQ-GG}, we see that the shape of $p_T$ distribution for Higgs boson in the $gg$ and $qq$ channel processes is nearly same at 100 TeV collider (FCC-hh). The relative importance of the $gg$ channel over the $qq$ channel is visible in the tail.
In the right plot, we give $p_T(H)$ distribution
combining $gg$ and $qq$ (NLO) contributions as the best prediction from our
calculations.
In the bottom panel of the plot, $R_2$ signifies the ratio of differential cross section from the $gg$ channel to that from NLO $qq$ channel process. The dashed line shows the ratio of corresponding total cross sections, which is 0.17. At the tail of the distribution, we see the $gg$ channel contribution becomes further important, but there differential cross section itself is quite small.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{multi_pTH_100TeV_zzh-hel_both_0_v8.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{multi_pTZ1_100TeV_zzh-hel_both_0_v8.pdf}
\end{center}
\caption{The kinematic distributions from $gg$ and $qq$ channels when the final state $Z$-bosons are longitudinal. The ratio of the distributions from the two channels have been shown in the lower panel of each figure. In the right figure, $Z_1$ denotes the harder of two $Z$-bosons in $p_T$.}\label{fig:dist-helicity-ZZH}
\end{figure}
Once again we find that if we categorize events based on the helicity states
of the two $Z$ bosons, the relative importance of the $gg$ channel contribution over
the $qq$ channel contribution can be increased.
From Fig.~\ref{fig:xs-hist-helicity-ZZH}, we see that in the $gg$ channel
the longitudinal $Z$ bosons contribute the most,
while in the $qq$ channel their transverse helicity states give dominant contribution.
The relative cross section of the $gg$ channel with respect to the $qq$ channel is
about $20\%$. However, if we restrict ourselves to the case when both $Z$-bosons are
longitudinally polarized, then this ratio almost doubles. Since the cross section for these polarized states for the $gg$ channel is about 2000 ab, there will be enough events
to observe this process at a $100$ TeV machine.
At the distribution level, from the Fig. \ref{fig:dist-helicity-ZZH}, we observe that if we
restrict ourselves to the contributions from
the longitudinal $Z$ bosons with $p_T(H)$ beyond 150 GeV, the relative contribution of the $gg$ channel increases significantly. Experimentally, one may look at the signature $l^{+}l^{-}l^{+}l^{-}b\bar{b}$. This signature is obtained when $Z \rightarrow l^{+}l^{-} (l=e/\mu)$ and for $H \rightarrow \bar{b}b$. Taking into account the branching ratios, and
$b$-tagging efficiency, one may expect about 75 events at the FCC-hh collider (with
$\rm{30\ ab}^{-1}$ integrated luminosity) from $gg$ channel and about
$210$ events from $qq$ channel. This is when both $Z$ bosons are longitudinally polarized. This number will go down when detection and kinematic-cut efficiency factors are included. However if in future,
one could use hadronic decay modes of a $Z$ boson to measure its
polarization, then the number of events would increase.
\\
As can be seen from Eq.~\ref{eq:ZZH}, the $gg$ channel depends on $\kappa_t,\ \kappa_V,\ \rm{and}\ \kappa_\lambda$. We vary these $\kappa$'s by 10\% from their SM values. The $gg$ channel strongly depends on both $\kappa_t\ \rm{and}\ \kappa_V$. In the $gg$ channel, $\pm$10\% change in $\kappa_t$ causes 68\% and -18\% variations in the cross section, respectively. And $\pm$10\% change in $\kappa_V$ causes 45\% and -28\% changes in the cross section, respectively. Similar variation in $\kappa_\lambda$ does not lead to much variation in the total cross section. Since this coupling is not yet well constrained, we will discuss it in detail in subsection~\ref{Remarks:HHH_and_HHVV}.
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth,height=0.35\linewidth]{ggZZH-multi-pth-comparison_gg_kt_100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth,height=0.35\linewidth]{ggZZH-multi-pth-comparison_gg_kV_100TeV_v3.pdf}
\end{center}
\caption{Effect of anomalous values of $\kappa_{t}$ and $\kappa_{V}$ on
$p_T(H)$ distribution for $ZZH$ production via the $gg$ channel. The
lower panels display ratio of BSM and SM distributions.}
\label{fig:dist-ano-gg-ZZH}
\end{figure}
In Fig.~\ref{fig:dist-ano-gg-ZZH}, we display the effect of $\kappa_t$ and $\kappa_V$ on $p_T(H)$ distribution.
We show the absolute distribution in the top panel, while in the bottom panel we show the ratio of distribution with anomalous coupling to that with the SM coupling. We can see that in the presence of anomalous $\kappa_t$ and $\kappa_V$,
the shape of the distribution remains more or less same.
However, due to non-trivial interference effects, the modifications in
presence of anomalous couplings are not
same in all the bins.
We see that for $\kappa_t=1.1$ the cross section in the bins near
tail of the distribution increases by a factor of 2. On the other
hand for $\kappa_V=1.1$, the maximum change in the cross section
is around 1.5. Thus tail of the distributions are
more sensitive to modifications in couplings due to high scale new physics.
The $qq$ channel depends mainly on $\kappa_V$. However, as we have considered bottom quark contribution also, the $qq$ channel depends on $ \kappa_\lambda$ as well. In the $qq$ channel, $\kappa_V$ comes as an overall factor both for LO and NLO amplitude, and so the effect of 10\% change in $\kappa_V$ causes around 20\% change in the cross section,
both at total and differential levels. We find a very mild dependence on $\kappa_\lambda$.
\subsection{Predictions for $pp \to WWH$}\label{sec:results-WWH}
The cross section for this process is the largest among all the $VVH$
processes considered in this paper. In Tab.~\ref{table:xs-WWH}, we report
the cross section predictions for $WWH$ process at different collider
center-of-mass energies.
The $gg$ channel contributions to $WWH$ production at 14, 27, and 100 TeV colliders are 290 ab, 1344 ab, and 17403 ab, respectively.
These numbers are roughly 2.3 times higher than $ZZH$ cross sections.
As regards scale uncertainties, the $gg \to WWH$ cross sections follow the same pattern as observed in $gg \to ZZH$.
The corresponding values of the LO $qq$ channel cross sections are 8658, 23040, and 128000 ab, respectively\footnote{
Due to technical reasons in the NLO calculation using {\tt MG5AMC\@NLO},
the $qq$ results are provided in 4 flavor scheme.}.
The ratio, $R_1$, is
found to be 0.15, 0.19, and 0.43, respectively. Unlike $ZZH$ production, the contribution of the $gg$ channel is relatively smaller.
\begin{table}[H]
\begin{center}
\resizebox{0.8\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c||c||}
\hline
$\sqrt{\rm s}\;(\rm TeV)$ & $\sigma^{\tin{ \rm WWH,\,LO}}_{\tin{gg}}\;~[\rm ab]$ & $\sigma^{\tin{\rm WWH,\,LO}}_{\tin{qq}}\;~[\rm ab]$ & $\sigma^{\tin{\rm WWH,\,NLO}}_{\tin{qq}}\;[\rm ab]$ & ${R}_{1} $\\
\hline
& & & & \\
14 & $290^{+27.6\%}_{-21.0\%}$ & $8658^{+0.3\%}_{-0.7\%}$ & $11220^{+1.5\%}_{-1.1\%}$ & 0.11 \\
& & & & \\
27 & $1344^{+22.5\%}_{-18.8\%}$ & $23040^{+2.1\%}_{-2.7\%}$ & $30090^{+1.7\%}_{-1.8\%}$ & 0.19 \\
& & & & \\
100 & $17403^{+20.6\%}_{-17.8\%}$ & $128000^{+7.5\%}_{-8.1\%}$ & $167300^{+2.0\%}_{-3.3\%}$ & 0.44 \\
& & & & \\
\hline
\end{tabular}}
\end{center}
\caption{A comparison of different perturbative orders in QCD coupling contributing to $pp \to WWH$ hadronic cross section at $\sqrt{s}=$ 14, 27, and 100 TeV. The ratio $R_1$ defined in Eq.~\ref{eq:R1} quantifies the $gg$ channel contribution with respect to the $qq({\rm NLO})$ correction. The $qq$ results are reported in four flavor scheme.} \label{table:xs-WWH}
\end{table}
In the left figure of Fig.~\ref{fig:dists-WWH-sm}, we can see that the $ p_T$ distribution of $W^{+}$ and $W^{-}$ overlap with each other, which is expected in the case of the $gg$ channel. The $p_T(H)$ distribution peaks around 100 GeV, and its fall in the tail is slower than that of $p_T(W^\pm)$ distributions. { In the right of Fig.~\ref{fig:dists-WWH-sm}, the distribution for invariant mass of $W^+$ and $W^-$ has been shown, which peaks around 200 GeV.}
\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggWpWmH-pt-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggWpWmH-M_WpWm-100TeV_v3.pdf}
\end{center}
\caption{ $p_T$ and $M(WW)$ distributions for $gg \to WWH$ in the SM at the 100 TeV collider (FCC-hh).}
\label{fig:dists-WWH-sm}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{ggWWH-pth-interference-100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{ggWWH-multi-pth-With-And-Without-TOP-100TeV_v3.pdf}
\end{center}
\caption{Left : SM contribution of pentagon (blue), box(green), triangle (gray) diagrams, as well as their square sum, interference, and total contribution to $p_T(H)$ distributions in $gg \to WWH$ at 100 TeV FCC-hh collider.
Right: The effect of excluding third generation quark contribution from Fig. 2(b) to full amplitude.} \label{fig:dist-WWH-pen-bx-tr}
\end{figure}
Like $gg \to ZZH$ production case, in $gg \to WWH$ production also, interference of various diagrams plays a major role. On the left of Fig.~\ref{fig:dist-WWH-pen-bx-tr}, we have shown $ p_T(H)$ distributions
for individual topologies as well as for their interference at a 100 TeV collider. The box contribution is the largest in all the bins while the
pentagon contribution is the lowest beyond $p_T > 100$ GeV. The total contribution is much smaller than the box contribution because of strong destructive interference effect which is shown by orange line in the figure.
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth,height=0.35\linewidth]{ppWpWmH-pth-comparison_qq_gg_100TeV_v3.pdf}
\includegraphics[angle=0,width=0.48\linewidth,height=0.35\linewidth]{ppWpWmH-multi-pth-comparison_qq_gg_100TeV_v3.pdf}
\end{center}
\caption{The left figure shows the normalized distribution for $p_T(H)$ in the $gg$ and $qq$ channel process. In the top panel of the right figure, we show the distribution due to $qq$ (NLO)+ $gg$ (LO) and $qq$ (NLO) production with $p_T(H)$. The lower panel shows their ratio. Results do not include contribution of the $bb$ channel process.}
\label{fig:dist-WWH-QQ-GG}
\end{figure}
Due to the presence of top quark propagators in
$ggWW^*$ type box diagram, one may naively
think of a suppressed contribution from the
third generation quarks at low $p_T(H)$.
In Fig.~\ref{fig:dist-WWH-pen-bx-tr}, we show the effect of excluding the third
generation quark contribution from the $ggWW^*$ type box diagram, on the $p_T(H)$ distribution.
Like in $gg \to ZZH$, the third generation quark contribution in $ggWW^*$ type box diagram is necessary for the unitarization of the full amplitude.
\\
In the left plot of Fig.~\ref{fig:dist-WWH-QQ-GG}, the normalized $p_T$ distributions for Higgs boson in the $gg$ and $qq$ channel processes have been shown for 100 TeV collider (FCC-hh). The $p_T(H)$ distribution in
the $gg$ channel peaks slightly on the harder side
making the channel more relevant in higher $p_T(H)$ bins.
To quantify it better we also plot the the ratio of distributions due to $qq$ (NLO) + $gg$ (LO) and $qq$ (NLO). At differential level
the ratio varies between 1.05 and 1.18 compared to its value (1.1) for the total cross section. Once again, we find that the $gg$ channel contribution is more relevant at higher $p_T$ where its contribution reaches 18\%.
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{100TeV_ggwwh-histogram_v4.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{100TeV_qqwwh-histogram_v8.pdf}
\caption{LO cross section for $WWH$ production in different helicity configurations in $gg$ (left) and $qq$ (right) channels. Legends correspond to different helicities of initial states.}\label{fig:xs-hist-helicity-WWH}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.48\linewidth]{multi_pTH_100TeV_wpwmh-hel_both_0_v8.pdf}
\includegraphics[angle=0,width=0.48\linewidth]{multi_pTW+_100TeV_wpwmh-hel_both_0_v8.pdf}
\caption{Comparing the $gg$ and $qq$ channel contributions to $W^+W^-H$ for events with longitudinal $W$ bosons.}\label{fig:dist-helicity-WWH}
\end{center}
\end{figure}
Similar to the case of $ZZH$, for this process also, the cross section in the $gg$
channel is dominated by longitudinally polarized $W$-bosons (Fig.~\ref{fig:xs-hist-helicity-WWH}). The relative contribution of this channel is about $13\%$, with respect to the $qq$ channel. However, when both $W$-bosons are longitudinally polarized, then this ratio increases to $32\%$. There will
also be enough events at a 100 TeV collider to observe the $gg$ channel contribution.
The relative
contribution of the $gg$ channel over the $qq$ channel
can be further increased by requiring the $p_T(W)$ to be beyond a certain value between 50 and 100 GeV, see Fig. \ref{fig:dist-helicity-WWH}. Here also one may consider leptonic decay channel for $W$ bosons, as that will help in the measurement of its polarization. We consider $l^{+}\nu_{l}l^{-}{\bar{\nu}_l}b\bar{b}$ final state as the signature. Here, as
before $l = e/\mu$. In the literature, various techniques, including Neural Network methods have been discussed to measure the $W$ boson momentum \cite{Grossi:2020orx}. Taking into account the branching ratios and the $b$-tagging efficiency, one may expect about 1750 events from $gg$ channel and $5900$ events from the $qq$ channel at the FCC-hh collider with
$\rm{30\ ab}^{-1}$ integrated luminosity. The number of these events would change depending
on the detector and kinematic-cut efficiency factors.
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width=0.45\linewidth,height=0.35\linewidth]{ggWpWmH-multi-pth-comparison_gg_kt_100TeV_v3.pdf}
\includegraphics[angle=0,width=0.45\linewidth,height=0.35\linewidth]{ggWpWmH-multi-pth-comparison_gg_kV_100TeV_v3.pdf}
\end{center}
\caption{Effect of anomalous values of $\kappa_{t}$ and $\kappa_{V}$ on $WWH$ production via the $gg$ channel. The upper panel shows absolute distribution, and the lower panel shows the ratio of the BSM and SM distributions.}
\label{fig:dist-ano-gg-WWH}
\end{figure}
Next, we focus on the effect of anomalous couplings on the total and differential cross sections. The $gg$ channel depends on $\kappa_t,\ \kappa_\lambda,\ \rm{and}\ \kappa_V$ (see Eq.~\ref{eq:WWH}).
We find that the channel is mostly sensitive to $\kappa_V$
and $\kappa_t$. For $\kappa_V=1.1(0.9)$ the cross section
changes by about 38\%(-26\%). While, for $\kappa_t=1.1(0.9)$ the
cross section changes by about 54\% (-3\%). The dependence on
$\kappa_\lambda$ is found to be relatively small.
In Fig.~\ref{fig:dist-ano-gg-WWH}, we show the effect of $\kappa_t$ and $\kappa_V$ on the $p_T(H)$ distribution for the $gg$ channel. We do not show the distribution for anomalous $\kappa_\lambda$ as its effect on cross section is very small for 10\% variation. We see that the shape remains more or less same in presence of anomalous couplings.
We see that in the bins around 400 GeV, this ratio is around 1.5 for $\kappa_t=1.1$ and $\kappa_V=1.1$. For $\kappa_t=0.9$, the ratio remains close to 1 throughout all the bins and for $\kappa_V=0.9$, it is in the range 0.7--0.8. Similar to the case for $qq \to ZZH$, the $qq \to WWH$ cross section
is also proportional to $\kappa_V^2$ at LO and NLO(QCD). So here as well, a 10\% change in $\kappa_V$ gives around 20\% obvious change in cross section, both at the total and differential levels.
\subsection{Remarks on anomalous $HHH$ and $HHVV$ couplings} \label{Remarks:HHH_and_HHVV}
{
We have seen that the gluon fusion $ZZH$ and $WWH$ processes
are most relevant for BSM physics due to their large cross sections.
We found that their cross sections do not
change much for a 10\% variation in $\kappa_\lambda$. However, we know that this coupling is presently unconstrained
by the experimental data. According to the future projections for
HL-LHC, only values $\kappa_\lambda \lesssim -2$ and $\kappa_\lambda \gtrsim 8$ can
be ruled out~\cite{TheATLAScollaboration:2014scd}. In this range the cross section for $ZZH$ and
$WWH$ processes in the $gg$ channel varies significantly.
In fact, it can change maximum by a factor of 3.
This is shown in the left panel of Fig.~\ref{fig:dep-lam3-hhvv}. Notice that the $WWH$ process is more
affected by anomalous $HHH$ coupling than $ZZH$ process.
\\
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.6]{zzh-wpwmh-lam3_plot_14TeV_v3.pdf}
\includegraphics[scale=0.6]{zzh-wpwmh-hhVV_plot_14TeV_v3.pdf}
\caption{Dependence of $gg \to ZZH, WWH$ cross sections on $HHH$ (left) and
$HHVV$ (right) couplings at 14 TeV. The vertical lines in the left plot represent projected sensitivity on $\kappa_\lambda$ at HL-LHC and those on the right represent current sensitivity on $\kappa_{HHVV}$ at the LHC.}
\label{fig:dep-lam3-hhvv}
\end{center}
\end{figure}
\begin{table}[h]
\begin{center}
\resizebox{0.7\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c||c|c|}
\hline
Collider & $gg$ process & $c^{\kappa_\lambda}_1$ & $c^{\kappa_\lambda}_2$ & $c^{\kappa_{HHVV}}_1$ & $c^{\kappa_{HHVV}}_2$ \\
\hline
\hline
\multirow{2}{*}{14 TeV} & $ZZH$ & -0.275 & 0.053 & -0.458 & 0.335 \\
\cline{2-6}
& $WWH$ & -0.318 & 0.071 & -0.440 & 0.301\\
\hline
\hline
\multirow{2}{*}{100 TeV} & $ZZH$ & -0.256 & 0.046 & -0.563 & 0.772 \\
\cline{2-6}
& $WWH$ & -0.281 & 0.057 & -0.524 & 0.672\\
\hline
\end{tabular}}
\caption{$c^i_1$ and $c^i_2$ that appear in the definition of
signal strengths for $gg \to ZZH, WWH$ processes at 14 TeV LHC and 100 TeV collider.}\label{table:c1-c2}
\end{center}
\end{table}
Although in SM model, the $HHVV (V = Z, W)$ coupling is correlated to the $HVV$ coupling, in presence of new physics this
correlation may not exist. Keeping this possibility in mind, we have varied the $HHVV$ coupling independently\footnote{It should be noted that independent variation of $HVV$ and $HHVV$ couplings
can be done systematically in an EFT framework which is beyond the scope of
the present work.} and we find that the cross section changes
very strongly.
This is shown in the right panel of Fig.~\ref{fig:dep-lam3-hhvv}. We can see that the effect of the $HHVV$ coupling
is relatively larger on $gg \to ZZH$ than on $gg \to WWH$. Close to SM values, the difference is negligible. According to a recent search
for Higgs boson pair production via vector boson fusion carried out
by the ATLAS collaboration using 126 $fb^{-1}$ data collected at
13 TeV LHC, the allowed values of $\kappa_{HHVV}$ lie in the range
(-0.56, 2.89) at 95\% confidence level~\cite{Aad:2020kub}.
\\
The quantity plotted in Fig.~\ref{fig:dep-lam3-hhvv} is known as signal strength ($\mu$) which has been
utilized by experimentalists as observable for data analyses.
The signal strength for each process can be parametrized as
\begin{eqnarray}
\mu = \frac{\sigma^{\rm BSM}}{\sigma^{\rm SM}} = 1 + c^i_1 (\kappa_i-1) + c^i_2(\kappa_i-1)^2,
\end{eqnarray}
where $\kappa_i = \kappa_\lambda, \kappa_{HHVV}$. In table~\ref{table:c1-c2}, we have
provided the values of $c_1^i$ and $c_2^i$ for $ZZH$ and $WWH$ processes for the 14 TeV LHC and a 100 TeV pp collider.
We note that $c_2^{k_\lambda}$ is smaller by an order of magnitude than $c_1^{k_\lambda}$, suggesting a strong interference effect mentioned before. Therefore, $c_2^{\kappa_\lambda}$ is relevant mostly for large values of $\kappa_i$. On the other hand, $c_2^{\kappa_{HHVV}}$ is of the same order as $c_1^{\kappa_{HHVV}}$. Since $c_1^i$ is negative, the cross section increase observed in the figures for $\kappa_i < 1$ is quite significant, which causes the (negative) lower bound on $\kappa$ to be tighter than the (positive) upper one. At a 100 TeV pp collider, while the other $c_i$s remain more or less same as that in 14 TeV collider, $c_2^{\kappa_{HHVV}}$ increases by around a factor of two, implying the possibility of a far more stringent bound on the $HHVV$ couplings. }
\\
Since the $gg$ fusion channel contribution to $ZZH$ and $WWH$ processes cannot be fully separated from the corresponding contributions from the $qq$ channel, the above result should be interpreted carefully. A realistic estimate of the BSM effects discussed above must include the contributions from $qq$ channel. Since $qq$ channel contributions are insensitive to $\kappa_\lambda$ and $\kappa_{HHVV}$, they can be seen as one of the major backgrounds to the gluon fusion processes. As we have pointed out, the measurement of the
polarization of the $W/Z$ boson can help in reducing this background. A systematic signal-background analysis is beyond the scope of the present work. For the benefit of the reader, in Fig~\ref{fig:pp-dep-lam3-hhvv}, we present the ratio $\sigma/\sigma_{\rm SM}$ for $pp \to ZZH, WWH$ which includes both $qq$ and $gg$ channel contributions as functions of $\kappa_\lambda$ and $\kappa_{HHVV}$. In obtaining these results only standard cuts mentioned in the previous sections have been applied. We can see that at the 14 TeV, the ratio of BSM and SM cross sections due to $qq+gg$ channels is significantly smaller than that due to $gg$ channel alone. Moreover, the $ZZH$ process turns out to be more affected by $\kappa_\lambda$ and $\kappa_{HHVV}$ than the $WWH$.
\\
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.6]{pp-zzh-wpwmh-lam3_plot_14TeV_v4.pdf}
\includegraphics[scale=0.6]{pp-zzh-wpwmh-hhVV_plot_14TeV_v4.pdf}
\caption{Dependence of $pp \to ZZH, WWH$ cross sections on $HHH$ (left) and
$HHVV$ (right) couplings at 14 TeV. The vertical lines in the left plot represent projected sensitivity on $\kappa_\lambda$ at HL-LHC and those on the right represent current sensitivity on $\kappa_{HHVV}$ at the LHC.}
\label{fig:pp-dep-lam3-hhvv}
\end{center}
\end{figure}
To be more precise, we find that by changing $\kappa_\lambda$ in the range $(-2,8)$, the cross section for $ZZH$ process changes in the range $7-4\%$ at the 14 TeV. The corresponding change at the 100 TeV falls in the range of $20-8\%$. On the other hand, when changing $\kappa_{HHVV}$ in the range $(-0.56,2.89)$, the maximum cross section change in $ZZH$ process is found to be $\sim 8\%$ and $\sim~46\%$ at the 14 TeV and the 100 TeV, respectively. Again, we may mention that the polarization measurements would increase the
fraction of $gg$ channel events, thus increasing the dependence on $\kappa_{HHVV}$.
\section{Conclusions}\label{sec:concl}
In this paper, we have considered production of $VV^\prime H$ ($\gamma\gamma H$, $\gamma ZH$, $ZZH$, and $WWH$) at
proton-proton colliders.
We investigated the sensitivity of these processes to various couplings of the Higgs boson, in particular to $HHH$ and $HHVV$ couplings
which are practically unconstrained.
Our main focus was the $gg$ channel contribution, which occurs at NNLO in $\alpha_s$. The scale uncertainties on the total cross sections are found to be of the order of 20\%.
A number of checks like UV and IR finiteness and gauge invariance of the amplitudes with respect to the gluons
have been performed to ensure the correctness of the calculation.
At a 100 TeV collider, the cross sections for these processes via the $gg$ channel range from 0.2 fb to 17 fb, $gg \to WWH$ being the dominant channel among all. We have seen the $gg \to \gamma \gamma H$ and $gg \to \gamma Z H$ processes are insignificant background to $gg \to HH \to \gamma \gamma H$ and $gg \to HH \to \gamma Z H$, respectively.
\\
We have also compared the $gg$ channel contribution with the fixed order NLO QCD correction to $pp \to VV^\prime H$ in order to
emphasize their relative importance.
For $\gamma \gamma H$ production, the $gg$ channel can be said to be the only production channel, as the $bb$ channel process contribution is negligibly small.
At a 100 TeV collider, the $gg \to \gamma ZH$ channel contribution is around 6\% of the NLO QCD correction in the $qq$ channel. The $\gamma Z H$ production shows one interesting feature: with an increase in the $p_T$ cut on photon, the $qq$ channel contribution decreases faster than the $gg$ channel contribution.
At this collider, the contribution of the $gg$ channel to $ZZH$ production is as important as the fixed order QCD NLO correction to the $qq$ channel. On the other hand, the $gg \to WWH$ channel cross section is around half the fixed order NLO QCD correction to the $qq$ channel. We have observed strong destructive interference effects among various classes of diagrams in $gg \to \gamma ZH, ZZH, WWH$.
Besides total cross sections at the LHC, HE-LHC, and FCC-hh, we have obtained relevant kinematic distributions at FCC-hh
in the $gg$ channel. We find that the $p_T(H)$ spectrum from the $gg$ channel is harder than that from the $qq$ channel for ZZH and WWH productions.
We have also shown that by selecting events based on the polarization of final state
vector bosons, the relative contribution of the $gg$ channel over the $qq$ channel can be enhanced.
\\
In addition to the SM results, effect of anomalous couplings ($\kappa_t$, $\kappa_V$, and $\kappa_\lambda$) for
$Ht\bar{t}$, $HVV$, $HHVV$, and $HHH$ vertices have been studied in the kappa framework.
We find that the new physics effects are quite important in $gg \to ZZH, WWH$ processes due to non-trivial interference effects in these processes.
A 10\% change in $\kappa_t$ on the higher side can enhance the $gg \to ZZH$ and $WWH$ cross sections by 68\% and 54\%, respectively. Similar
change in $\kappa_V$ enhances these cross sections by about 40\%.
Unlike in $qq$ channels, the kinematic distributions in $gg$
channels display non-trivial changes in presence of new physics. The dependence of the $g g$ channel on the $\kappa_V$ is stronger
than that of the $qq$ channel. By considering events with longitudinally polarized
vector bosons for the processes $ p p \to ZZH, WWH$, we can enhance the
fraction of the $gg$ channel contribution. This event sample will have even
stronger dependence on $\kappa_V$.
Since the $HHH$ and $HHVV$ couplings are not well constrained, we have
also considered larger independent variations in $\kappa_\lambda$ and $\kappa_{HHVV}$.
We find that the effect of $\kappa_{HHVV}$ on the cross section
is much stronger than that of $\kappa_\lambda$. Therefore the process
$ p p \to ZZH, WWH$ with longitudinally polarized $Z$ and $W$ bosons
can help in determining the $HHVV$ coupling.
\clearpage
\section*{Acknowledgements}
DS would like to acknowledge the use of HPC cluster facility, SAMKHYA, in Institute of Physics, Bhubaneswar. AS would
like to acknowledge fruitful discussions with Xiaoran Zhao.
|
1,314,259,995,820 | arxiv | \section{Introduction}
Since the discovery of quantum Hall effect \cite{PRL45-494,PRL48-1559,PRL49-405,RMP58-519}, the topological phase of matter has attracted extensive attention. Although the topological phase of matter is first found in electronic systems, it is later extended to the field of quasi-particles, such as phonons \cite{PRL105-225901}, photons \cite{PRL100-013904,NM12-233} and even magnons \cite{Sci329-297}, {\it etc}. Katsura {\it et al.} \cite{PRL104-066403} theoretically predicted the thermal Hall effect of magnons. Subsequently, Onose {\it et al.} \cite{Sci329-297} experimentally confirms the existence of magnon Hall effect in Lu$_{2}$V$_{2}$O$_{7}$ with pyrochlore structure, which sets off an upsurge of studying the topological properties of magnons. So far, various non-trivial topological phases of magnons for different crystal structures and magnetic interactions had been presented. In two-dimensional (2D) system, it is found that the anisotropic magnetic interaction, such as the magnetic dipole-dipole interaction \cite{PRB87-174402,PRB90-104417,PRB98-224409}, Dzyaloshinskii-Moriya interaction (DMI) \cite{PRB97-174413,PRB99-214424,PRB99-174412,JPCM29-185801} and pseudo-dipolar interaction (PDI) \cite{PRB95-014435,PRApp9-024029} rather than the isotropic magnetic interaction (Heisenberg-type exchange interaction), usually produce the non-trivial topological phase of magnon in different types of lattices, {\it e.g.} honeycomb lattice, Kagome lattice and triangular lattice, {\it etc}. These anisotropic interactions often open an energy gap and lead to topological magnon insulator, and can be seen as effective spin-orbit couplings in electronic topological insulator. Due to the inherent topological protection, the topological magnons generally have a wider application prospect than the traditional ones.
Due to the abundance of magnetic materials and magnetic structures \cite{EPL103-47010}, as well as the manipulation of various magnetic interactions, magnons have become an important platform for topological physics and quantum technology \cite{PR915-1,ARCMP13-171}.
As a consequence, it has great potential application for designing novel quantum devices in magnonics \cite{JPD43-264001,NP11-453}.
As we know, there also are abundant topological phases in one-dimensional (1D) electronic systems, {\it e.g.} Su-Schrieffer-Heeger (SSH) model and Kitaev chain, {\it etc}. Indeed unlike electrons, the magnon topological phases are easier to be realized by regulating magnetic interactions. Although the 2D magnons are extensively studied in previous years and numerous topological magnon states ({\it e.g.} magnonic topological insulator, magnonic topological semimetal with Dirac or Weyl magnon, {\it etc}.) are discovered, the topological properties of magnons in 1D system remain unclear. We can expect more abundant topological phases in 1D magnon systems than in electronic systems due to the rich regulation of magnetic interactions. We notice that recently Pirmoradian {\it et al.} \cite{PRB98-224409} constructed a 1D magnetosphere chain \cite{PRL42-1698}, in which there is only dipole-dipole interaction rather than Heisenberg-type interaction between macrospins on the magnetospheres, and they discovered topological magnons by manipulating the external magnetic field. As a matter of fact, the magnetosphere chain they studied is a classical magnetic macrosystem. However, the mechanism to induce intrinsic topological magnons without external field is still lacking, especially the topological properties in the lattice model of the realistic magnetic materials is deserved to explore.
In this paper, here we investigate the possible magnonic topological phases and the related topological phase transitions in a 1D ferromagnetic (FM) SSH model by manipulating both the isotropic Heisenberg exchange interaction and the anisotropic exchange interactions, including DMI and PDI. To characterize the topological phase, we adopt the Zak phase for the infinite system and the real-space topological number for the finite system, respectively. We first investigate the pure Heisenberg-type interaction, and find that the topological phase can be realized by adjusting the ratio of intracellular interaction $J_{1}$ and intercellular interaction $J_{2}$, {\it i.e.} $J_{2}$$/$$J_{1}$. This result is very different from that in the 2D honeycomb ferromagnet, in which there is no topological phase transition for only isotropic Heisenberg-type interaction. Moreover, the introduction of anisotropic interactions, {\it {\it i.e.}} DMI (intracellular interaction $D_{1}$ and intercellular interaction $D_{2}$) and PDI (intracellular interaction $F_{1}$ and intercellular interaction $F_{2}$), also plays an important role in the realization of the topological phases. We demonstrate that the intercellular interactions ($J_{2}$, $D_{2}$, and $F_{2}$) favor topological non-trivial phase, while the intracellular interactions ($J_{1}$, $D_{1}$, and $F_{1}$) hinder it, in the 1D FM SSH model. This paper is organized as follows: the theoretical model and method are introduced in Sec. II; the numerical results and discussions are presented in Sec. III; and the last section is the remarks and conclusions.
\section{Model and Method}
Analogous to the electronic SSH model \cite{PRL42-1698}, as shown in Fig.~\ref{Fig1}(a), we construct a 1D FM SSH model including the intracellular and intercellular interactions, {\it i.e.} $J_{1}$ and $J_{2}$, which presents a pure Heisenberg-type model. In general, there are two types of FM ground states, that is, the spin orientation is perpendicular to or along the direction of the 1D chain, as shown in Fig. ~\ref{Fig1}(c), and thus these two quantization axes are considered, respectively.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=1.0 \columnwidth]{Fig1.eps}
\caption{(Color online) (a) 1D electronic SSH model, (b) 1D FM SSH model. Each unit cell contains two lattice sites A and B. The solid and dotted lines indicate the strong and weak exchange interactions, respectively. $J_{1}$ and $J_{2}$ denote the intracellular and intercellular interactions, respectively. (c) Two types of FM ground states: the spin quantization axes are $z$-axis (top panel) and $x$-axis (bottom panel), respectively.}
\label{Fig1}
\end{figure}
The Hamiltonian of the 1D spin chain with anisotropic exchange interactions can be written as
\begin{eqnarray}
H&=&H_{0}+H_{AE}+H_{DM}+H_{PD}.
\label{Eq1}
\end{eqnarray}
$H_{0}$ is the Heisenberg exchange interaction, reading
\begin{eqnarray}
H_{0}&=&-J_{1}\sum_{i=1}^{N}{\mathbf{S}_{i,A}\cdot \mathbf{S}_{i,B}}-J_{2}\sum_{i=1}^{N-1}{\mathbf{S}_{i,B}\cdot \mathbf{S}_{i+1,A}},
\label{Eq2}
\end{eqnarray}
where $J_{1}$ and $J_{2}$ represent the exchange interaction in the unit cell (intracellular) and between the two unit cells (intercellular), respectively. And $H_{AE}=-\frac{1}{2}\sum_{i}{K_{i}{{(S}_{i}^{z})}^{2}}$
is the easy-axial anisotropy term for the quantization $z$-axis, where $K_{i}$ is the axial anisotropy energy ($K_{i}$=10$J_{1}$ is used for all the calculations). $H_{DM}$ denotes the DMI term \cite{JPCS4-241,PR120-91}, originating from the spin-orbit coupling, while $H_{PD}$ represents the PDI \cite{PR52-1178,PRL102-017205}, mainly contributed from both the strong spin-orbit coupling and the orbital degree of freedom.
The DMI and PDI are covered by the following forms:
\begin{eqnarray}
H_{DM}&=&\mathbf{D}_{1}\cdot\sum_{i}\left(\mathbf{S}_{i,A}\times \mathbf{S}_{i,B}\right)\\ \nonumber
&&+\mathbf{D}_{2}\cdot\sum_{i,i+1}\left( \mathbf{S}_{i,B}\times\mathbf{S}_{i+1,A}\right),
\label{Eq3}
\end{eqnarray}
\begin{eqnarray}
H_{PD}&=&-F_{1}\sum_{i=1}^{N}{\left(\mathbf{S}_{i,A}\cdot \mathbf{e}_{i,AB}\right)\left(\mathbf{S}_{i,B}\cdot\mathbf{e}_{i,AB} \right)}\\ \nonumber
&&-F_{2}\sum_{i=1}^{N-1}{\left(\mathbf{S}_{i,B}\cdot \mathbf{e}_{i,i+1,BA}\right)\left(\mathbf{S}_{i+1,A}\cdot \mathbf{e}_{i,i+1,BA}\right)},
\label{Eq4}
\end{eqnarray}
where $\mathbf{D}_{1}$ ($F_{1}$) and $\mathbf{D}_{2}$ ($F_{2}$) represent intracellular and intercellular nearest neighbor DMIs (PDIs), respectively. $\mathbf{e}_{i,AB}$ is the unit vector between lattice sites A and B in the $i$-th unit cell, and $\mathbf{e}_{i,i+1,BA}$ is the unit vector between lattice site B of the $i$-th unit cell and lattice site A of the $(i+1)$-th unit cell. $\mathbf{e}_{i,AB}$ and $\mathbf{e}_{i,i+1,BA}$ point in the direction of the chain.
We firstly analyze the magnetic ground state before calculating the magnon spectrum. The Heisenberg term with $-J<0$ favors the FM state, and the easy-axial anisotropy term with $K_{z}$ tends to align the spins parallel along the $z$ axis. However, in the presence of anisotropic magnetic exchange interactions, the nearest-neighboring DMI and PDI, the magnetic ground state may change. In the classical spin limit, the derivation process with respect to the spin canting angles \cite{PRB97-094412} is performed to obtain the minimal energy of the total Hamiltonian $H$ (Eq.(\ref{Eq1})). Two possible ground states are found to be the FM states along $z$-axis and $x$-axis with the total energies per unit cell $-(J_{1}+J_{2})-K$ and $-(J_{1}+J_{2})-(F_{1}+F_{2})$, respectively. Hence the phase boundary is $K=F_{1}+F_{2}$, that is, when $K>F_{1}+F_{2}$, it is FM state with spins oriented along the $z$ axis ($z$-FM phase), otherwise, it is FM state with the spins oriented along the $x$ axis ($x$-FM phase). Through the analysis of the total energy, it is found that the component of the DMI $D_{z}$, will not affect the $z$-FM phase because it does not contribute to the total energy. While the PDI $F>0$ prefers each pair of spins to be parallel to the bond between them ($x$ axis), so it can change the magnetic ground state. Note that all the parameter values used ($K=10J_{1}$, $F=J_{1}$, $0<F_{2}/F_{1}\leq 5$, for the convenience of theoretical analysis) fall into the $z$-FM phase, even for the large values of $D_{z}$ or $F$.
By the linear Holstein-Primakoff (HP) transformation \cite{PR58-1098} with bosonic generation (annihilation) operator (for A sublattice, $a^{\dag}$ ($a$); for B sublattice, $b^{\dag}$ ($b$)), the Hamiltonian of magnon in momentum $k$-space can be expressed as $H=\frac{1}{2}\sum_{k}{\Psi_{k}^{\dag}H_{k}\Psi_{k}}$, where $\Psi_{k}=\left(a_{k},a_{-k}^{\dag},b_{k},b_{-k}^{\dag}\right)^{T}$. Using the linear HP transformation, the spin operators for the A (or B) sublattice are given by
\begin{equation}
\begin{cases}
S_{A}^{\dag}=\sqrt{2S}a \\ \nonumber
S_{A}^{-}=\sqrt{2S}a^{\dag} \\ \nonumber
S_{A}^{z}=S-a^{\dag}a. \\
\end{cases}
\end{equation}
By the Bogoliubov transformation \cite{INC7-794}, we can diagonalize the magnon Hamiltonian and obtain the energy spectrum of the magnon. The magnon Hamiltonian $H_{k}$ satisfies the generalized eigenvalue problem \cite{PR139-A450},
\begin{eqnarray}
\eta H_{k}\Gamma_{k}&=&\Gamma_{k}\eta E_{k},
\label{Eq5}
\end{eqnarray}
where $\Gamma_{k}$ is the transformation matrix that diagonalizes the
magnon Hamiltonian, $\eta$ is a metric matrix,
$\eta=I_{2 \times 2} \otimes \sigma_{z}$, $I$ is the identity
matrix, and $\sigma_{z}=\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}$ is the Pauli matrix. In the generalized eigenvalue
problem of boson, $E(k)=-E(-k)$ is artificially introduced. As a
result, we only need to consider the non$-$negative eigenvalues of the
energy spectrum. On the other hand, for a finite 1D system with $N$
unit cells, the generalized eigenvalue problem in real space is
$\eta H\Gamma=\Gamma\eta E$, where $H$ is the $2N \times 2N$
real space matrix, and $\eta=I_{N \times N} \otimes \sigma_{z}$.
As is known that, the electronic 1D SSH model possesses the chiral ($\Gamma$), spatial ($P$) and temporal ($T$) inversion symmetries \cite{PE119-113973} defined as $\Gamma H(k)\Gamma^{-1}=-H(k)$, $PH(k)P^{-1}=H(-k)$, $TH(k)T^{-1}=H(-k)$, respectively, in which
$\Gamma=I_{2 \times 2} \otimes \sigma_{z}$, $P=I_{2 \times 2} \otimes \sigma_{x}$, $T=I_{2 \times 2} \otimes \begin{pmatrix}
0 & i \\
i & 0
\end{pmatrix}$. The chiral symmetry requires on-site energy is 0, while
the spatial and temporal inversion symmetry only require the identical
on-site energies \cite{PRL127-147401}. In consequence, for the 1D FM SSH model with pure Heisenberg-type interaction, the chiral symmetry is broken, while the spatial and temporal inversion symmetries are preserved. When the anisotropic DMI and PDI exist in the system, the spatial inversion symmetry is broken while only the temporal inversion symmetry is preserved. Consequentially, the 1D FM SSH model of magnons has no direct correspondence to the electronic systems because of the different symmetries; thus, the topological magnons of the magnetic SSH model are not superficial extensions of the electronic SSH system. The most important difference is that the topological magnons involving anisotropic interaction features do not have direct correspondence to any electronic systems, though the isotropic Heisenberg term ($J_{1}$ and $J_{2}$) corresponds to the electronic interactions ($t_{1}$ and $t_{2}$). In addition, it should be pointed out that the change of quantization axis does not affect the symmetry of the system.
For a 1D infinite system, Zak phase \cite{PRL62-2747} is often used to describe its topological properties within the momentum $k$-space,
\begin{eqnarray}
\varphi_{Zak}&=&-i\int_{BZ}^{\ }{\Psi_{k}^{\dag}\frac{d}{dk}\Psi_{k}{dk}}.
\label{Eq6}
\end{eqnarray}
When $\varphi_{Zak}$ is not 0, the topological phase is non-trivial,
and the system has topological edge state; otherwise, the topological
number is 0, and the topological phase is trivial, which means no
topological edge state in the system.
For a 1D finite system, we extend the real-space topological number of
the electronic (fermionic) version \cite{AP356-383} to the magnonic (bosonic) case. The
topological number of real space $N_{R}$ is described as
\begin{eqnarray}
N_{R}&=&\frac{1}{2}Sig((X+iH)\eta),
\label{Eq7}
\end{eqnarray}
where $X$ is a matrix composed of rescaled coordinates consistent with
the size of $H$. For the $i$-th unit cell, its reduced coordinates are
$X_{i,j}=\frac{i\delta_{ij}}{N}$. $Sig(Q)$ is equal to the number
of positive eigenvalues of $Q=(X+iH)\eta$.
\section{Results and Discussions}
\subsection{Heisenberg-type interaction}
We first investigate the 1D FM SSH model with only pure Heisenberg-type isotropic exchange interaction. Through the linear HP transformation mentioned above, the Heisenberg interaction term $H_{0}$ can be expressed as
\begin{eqnarray}
H_{0}&=&\left(J_{1}+J_{2}+K \right)S\sum_{i} \left(a_{i}^{\dag}a_{i}+ b_{i}^{\dag}b_{i} \right)\\ \nonumber
&&-J_{1}S\sum_{i} \left(a_{i}b_{i}^{\dag}+ a_{i}^{\dag}b_{i} \right)
-J_{2}S\sum_{i} \left(b_{i}a_{i+1}^{\dag}+ b_{i}^{\dag}a_{i+1} \right).
\label{Eq8}
\end{eqnarray}
The matrix of $H_{0}$ can be seen in the Appendix.
The band structures dependent on the ratio $J_{1}/J_{2}$ are presented in Fig.~\ref{Fig2}(a)-(c).
It is obviously found that a topological phase transition occurs at a critical ratio $J_{1}/J_{2}$=1, accompanied by an energy gap close. This is very similar to the case of the electronic 1D SSH model \cite{TI2017}, that is the phase transition between topologically trivial and non-trivial phases can be realized by regulating the ratio of the intracellular and intercellular interactions, distinctly different from that in the 2D honeycomb ferromagnet in which the topological phase transition is mainly induced by the anisotropic interaction rather than the isotropic interaction \cite{JPCM29-185801}.
Notice that for the critical ratio $J_{1}/J_{2}$=1, the system exhibits a linear Dirac-like dispersion, but it is trivial owning to the band accidental degeneracy. Therefore, for $J_{1}/J_{2}$$\neq$1, although the energy band opens a gap, it is not necessarily topologically non-trivial.
It is demonstrated that the intercellular (intracellular) interaction $J_{2}$ ($J_{1}$) favors (unfavors) the topological non-trivial phase. Indeed, for $J_{1}<J_{2}$ (in comparison with $\delta t=t_{1}-t_{2}>0$ for the electronic SSH model), the magnon state is a 1D topological magnon insulator, as shown in Fig.~\ref{Fig2}(c). In addition, we note that the position of the gap close at $k=\pi$ or $-\pi$ is different from that of the electronic SSH model at $k=0$.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig2.eps}
\caption{(Color online) Band structures dependent on the ratio $J_{1}/J_{2}$ for 1D FM SSH model and the real-space energy spectrum of finite 1000 lattice sites of 1D chain, respectively: $J_{1}/J_{2}$=1.5 (a) and (d), 1.0 (b) and (e), 0.5 (c) and (f).}
\label{Fig2}
\end{figure}
Owning to the finite energy gap of 1D topological magnon insulator, a topologically protected edge state can be expected according to the bulk-edge correspondence \cite{PRB95-035421}. In order to investigate the edge state of this topological magnon insulator, the energy spectrums of finite lattice sites in real space are also given, as displayed in Fig.~\ref{Fig2}(d)-(f). Accordingly, two degenerate edge states are found located within the band gap for the topological magnon insulating phase, which can be seen in Fig.~\ref{Fig2}(f).
Through the analysis of the real-space-resolved wave functions for the two degenerate edge states, it is found that the two edge states are separately localized at the two ends of the 1D chain, as seen in Fig.~\ref{Fig3}. In fact, this degenerate edge states can be simply lifted by introducing an inequivalence between A and B sites in the SSH model.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig3.eps}
\caption{(Color online) Wave-function amplitude dependent on the lattice index. The two degenerate edge states in Fig. 2(f) are separately localized at both ends of the 1D chain.}
\label{Fig3}
\end{figure}
Considering the topological characteristic of this in-gap edge states, it can be regarded as zero-energy mode of magnon at finite frequency in 1D quantum magnets, which is very like the Majorana zero mode in 1D chain (nanowire) of electronic system \cite{Science336-1003}. It should be emphasized that the in-gap edge state as a novel type of bound state is distinctly different from the two-magnon bound state in 1D Heisenberg spin chain \cite{Nature502-76}.
The corresponding topological invariants in both the momentum $k$-space (Zak phase $\varphi_{Zak}$) and the real space ($N_{R}$) are shown in Fig.~\ref{Fig4}. It can be found that the topological number changes from 0 to 1 at the critical point $J_{1}$=$J_{2}$ after the energy gap is closed and reopened, indicating the phase transition process between the topological trivial and non-trivial phases. Notice that it can be seen that the topological number of $k$-space and real space are consistent with each other in describing the topological phase transition, implying our methods in calculating the topological invariants are effective.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.6 \columnwidth]{Fig4.eps}
\caption{(Color online) Momentum $k$-space Zak phase $\varphi_{Zak}$ and real-space topological number $N_{R}$ in the 1D FM SSH model with the Heisenberg-type interaction.}
\label{Fig4}
\end{figure}
For the quantization $x$-axis, as displayed in Fig.~\ref{Fig1}(c), the Heisenberg interaction term $H_{0}^{'}$ can be rewritten as
\begin{eqnarray}
H_{0}^{'}&=&\left(J_{1}+J_{2} \right)S\sum_{i} \left(a_{i}^{\dag}a_{i}+ b_{i}^{\dag}b_{i} \right)\\ \nonumber
&&-J_{1}S\sum_{i} \left(a_{i}^{\dag}b_{i}+h.c. \right)
-J_{2}S\sum_{i} \left(b_{i}^{\dag}a_{i+1}+h.c. \right)
\label{Eq9}
\end{eqnarray}
Obviously, it is consistent with the form of the quantization $z$-axis which indicates that the change of the quantization axis does not affect our results for the pure Heisenberg-type interaction.
\subsection{In the presence of DMI}
Although the isotropic Heisenberg interaction can induce the topological phase in 1D spin chain, the anisotropic interactions play an essential role in topological phase transition in 2D lattices, such as honeycomb \cite{PRL117-227201,PRB95-014435,PRB97-174413,PRB99-214424} and Kagome \cite{PRB87-144101,PRB90-024412} lattices, {\it etc}. Therefore the influence of the anisotropic exchange interactions deserves investigation. Here we first focus on the DMI originating from the spin-orbit coupling. Using the linear HP transformation, the DMI term $H_{DM}$ (more details are shown in Appendix) can be written as
\begin{eqnarray}
H_{DM}&=&-iD_{1z}S\sum_{i}\left(a_{i}^{\dag}b_{i}-a_{i}b_{i}^{\dag} \right)\\ \nonumber
&&-iD_{2z}S\sum_{i,i+1}\left(b_{i}^{\dag}a_{i+1}-b_{i}a_{i+1}^{\dag} \right),
\label{Eq10}
\end{eqnarray}
where $D_{1z}$ and $D_{2z}$ are the components of intracellular
$\mathbf{D}_{1}$ and intercellular $\mathbf{D}_{2}$ on the
quantization $z$$-$axis of spin, respectively. For simplicity, we fix
$D_{1z}=J_{2}=1$ for the $J_{1}/J_{2}$ case and $D_{1z}=J_{1}=1$ for the $J_{2}/J_{1}$ case in all the calculations. Note that due to the chiral characteristic of DMI, the values of both $D_{1z}$ and $D_{2z}$ can be positive or negative for 1D chain,
and all cases of four possible signs are considered. It is found that the change of sign does not change the topological phase diagram, but only the position of the closing gap in the energy spectrum during the phase transition. For $D_{2z}/D_{1z}>0$, the energy gap closes at $k=\pi/2$ (Note that for the pure DMI alone, the gap closes at $k=0$, not shown here, different from the case of the pure Heisenberg interaction.); otherwise, the position where energy gap closes is associated with $D_{2z}/D_{1z}$. Therefore, we only consider two cases with the relative sign of the $D_{2z}/D_{1z}$ being positive or negative.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig5.eps}
\caption{(Color online) The $J$-$D$ phase diagram of
$J_{1}/J_{2}$ (a) and $J_{2}/J_{1}$ (b) for the 1D FM SSH model with
the anisotropic DMI. The white dotted lines represent the phase transition
boundaries between the trivial and non-trivial phases given by the relations $\left|D_{2z}/D_{1z}\right|=J_{1}/J_{2}$ ($D_{1z}=J_{2}=1$ is fixed) and $\left(J_{2}/J_{1}\right)^{2}+\left(D_{2z}/D_{1z}\right)^{2}=2$ ($D_{1z}=J_{1}=1$ is fixed), respectively. (c) and (d) are $J_{1}-J_{2}$ phase diagrams at $D_{1z}=1.0>D_{2z}=0.5$ and $D_{1z}=1.0<D_{2z}=1.5$, respectively. The black dashed lines denote the original phase transition boundaries of the pure Heisenberg interactions.}
\label{Fig5}
\end{figure}
Compared with the case of pure Heisenberg interaction, the phase
transition condition (more details are shown in Appendix) in the presence of the anisotropic DMI is
$J_{1}^{2}+{D}_{1z}^{2}-J_{2}^{2}-D_{2z}^{2}=0$. Thus, the phase transition occurs at
$\left|D_{2z}/D_{1z}\right|=J_{1}/J_{2}$ for the
fixed $D_{1z}=J_{2}=1$ case, and
$\left(J_{2}/J_{1}\right)^{2}+\left(D_{2z}/D_{1z}\right)^{2}=2$
for the fixed $D_{1z}=J_{1}=1$ case. Obviously, the intercellular
$D_{2}$ (intracellular $D_{1}$) DMI boosts
(weakens) and even induces (eliminates) the topologically non-trivial
phase, though the Heisenberg interaction alone can lead to the
topological magnon phase, which obviously can be seen in Figs.~\ref{Fig5}-\ref{Fig7}.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig6.eps}
\caption{(Color online) Band structures dependent on the ratio $D_{2z}/D_{1z}$ at
$J_{1}/J_{2}$=1.5: (a) $D_{2z}/D_{1z}$=1.0 (b) $D_{2z}/D_{1z}$=1.5 (c) $D_{2z}/D_{1z}$=2.0. The vertical black dotted line in (b) indicates the position where the energy gap is closed.}
\label{Fig6}
\end{figure}
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.6 \columnwidth]{Fig7.eps}
\caption{(Color online) Momentum $k$-space Zak phase $\varphi_{zak}$ and real-space topological number $N_{R}$ in the 1D FM SSH model with the
Heisenberg and DMI interactions for $J_{1}/J_{2}$=1.5.}
\label{Fig7}
\end{figure}
For the quantization $x$-axis, as displayed in Fig.~\ref{Fig1}(c), the DMI term $H_{DM}^{'}$ can be rewritten as
\begin{eqnarray}
H_{DM}^{'}&=&-iD_{1x}S\sum_{i}\left(a_{i}^{\dag}b_{i}-a_{i}b_{i}^{\dag} \right)\\ \nonumber
&&-iD_{2x}S\sum_{i,i+1}\left(b_{i}^{\dag}a_{i+1}-b_{i}a_{i+1}^{\dag} \right),
\label{Eq11}
\end{eqnarray}
where $D_{1x}$ and $D_{2x}$ are the components of
$\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ on the quantization $x$-axis of
spin, respectively. From the analysis of the DMI, we can find that only
the components in the direction of quantization axis will affect the
topological properties of the system. Therefore, the conclusions drawn above are still valid.
\subsection{In the presence of PDI}
Apart from the DMI, the PDI is another essential anisotropic exchange interaction, which is generally from the combination of the strong spin-orbit coupling and the orbital degree of freedom \cite{PRL102-017205}. Consequently, the PDI has the different influence on magnons due to its origin distinctly different from that of the DMI. For the quantization $z$-axis, using the linear HP transformation, the PDI term $H_{PD}$ (more details are shown in Appendix) are described as
\begin{eqnarray}
H_{PD}&=&-\frac{F_{1}}{2}S\sum_{i}\left(a_{i}b_{i}+a_{i}b_{i}^{\dag}+ a_{i}^{\dag}b_{i}+a_{i}^{\dag}b_{i}^{\dag} \right) \\ \nonumber
&&-\frac{F_{2}}{2}S\sum_{i,i+1}\left(b_{i}a_{i+1}+b_{i}a_{i+1}^{\dag}+ b_{i}^{\dag}a_{i+1}+b_{i}^{\dag}a_{i+1}^{\dag} \right).
\label{Eq12}
\end{eqnarray}
In this 1D chain case, the direction of the PDI is along the $x$-axis and
perpendicular to the quantization $z$-axis. For simplicity,
$F_{1}=J_{2}=1$ for the $J_{1}/J_{2}$ case and $F_{1}=J_{1}=1$ for the $J_{2}/J_{1}$ case is fixed in all the
calculations. The phase transition condition (more details are shown in Appendix) in the case of the Heisenberg interaction and PDI is $F_{1}+2J_{1}-F_{2}-2J_{2}=0$. Thus the boundary between the trivial and non-trivial phases is determined by the relation $2J_{1}/J_{2}-1=F_{2}/F_{1}$ for $F_{1}/J_{2}=1$, and $-2J_{2}/J_{1}+3=F_{2}/F_{1}$ for $F_{1}/J_{1}=1$. Similar to the Heisenberg interaction and DMI cases, the intercellular $F_{2}$ (intracellular $F_{1}$) PDI favors (unfavors) the topologically non-trivial phase, as demonstrated in Figs.~\ref{Fig8}-\ref{Fig10}.
Compared with the cases of the Heisenberg and DMI interactions, the position of gap close for the pure PDI case is at $k=\pi$, same as the Heisenberg one. This is a result of the different symmetries of these three magnetic interactions.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig8.eps}
\caption{(Color online) Band structures dependent on the ratio
$F_{2}/F_{1}$ for $J_{1}/J_{2}$=1.5: $F_{2}/F_{1}$=1.5 (a), 2.0 (b), and 2.5 (c). The vertical black dotted lines in (b) denote the positions of the gap closed for the PDI case, different from that of the DMI case.}
\label{Fig8}
\end{figure}
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig9.eps}
\caption{(Color online) The $J$-$F$ phase diagram of $J_{1}/J_{2}$ (a) and
$J_{2}/J_{1}$ (b) for the 1D FM SSH model with the anisotropic PDI. The white dotted lines represent the phase transition boundaries between the trivial and non-trivial phases given by the relations $2J_{1}/J_{2}-1=F_{2}/F_{1}$ ($F_{1}=J_{2}=1$ is fixed) and $-2J_{2}/J_{1}+3=F_{2}/F_{1}$ ($F_{1}=J_{1}=1$ is fixed), respectively. (c) and (d) are $J_{1}-J_{2}$ phase diagrams at $F_{1}=1.0>F_{2}=0.5$ and $F_{1}=1.0<F_{2}=1.5$, respectively. The black dashed lines denote the original phase transition boundaries of the pure Heisenberg interactions.}
\label{Fig9}
\end{figure}
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.6 \columnwidth]{Fig10.eps}
\caption{(Color online) Momentum $k$-space Zak phase $\varphi_{zak}$ and real-space topological number $N_{R}$ in the 1D FM SSH model with the
Heisenberg and PDI interactions for $J_{2}/J_{1}=1.0$.}
\label{Fig10}
\end{figure}
For the quantization $x$-axis, as displayed in Fig.~\ref{Fig1}(c), the PDI term $H_{PD}^{'}$ can be rewritten as
\begin{eqnarray}
H_{PD}^{'}&=&-\left(F_{1}+F_{2} \right)S\sum_{i}\left( a_{i}^{\dag}a_{i}+b_{i}^{\dag}b_{i} \right),
\label{Eq13}
\end{eqnarray}
and then the PDI becomes a term similar to the easy-axial anisotropy $K$ term, equivalent to the on-site energy of magnon. Therefore, the PDI has no influence on the topological properties in this case.
\subsection{In the presence of both DMI and PDI}
Finally, we turn to the case of coexistence of both the anisotropic DMI and PDI. In fact, the Heisenberg interaction, DMI and PDI may possibly coexist in the same realistic magnetic materials due to the the existence of multiple degrees of freedom, especially in 3$d$, 4$d$ and even 5$d$ correlated materials, such as magnetic materials in iridates \cite{PRL105-027204,PRB84-054409}, {\it etc}. However, the phase transition condition is no longer linear. For simplicity, both $D_{1z}=J_{1}$, $F_{1}=J_{1}$ and $J_{1}=1$ are fixed in all the calculations. Then the phase transition boundary can be determined numerically.
As shown in Figs.~\ref{Fig11}, comparing the $J$-$D$, $J$-$F$ and $D$-$F$ phase diagrams, it is found that the coexistence of the isotropic Heisenberg interaction, anisotropic DMI and PDI show different topological phase diagrams from these in Fig.~\ref{Fig5}(a) and Fig.~\ref{Fig9}(a) as a result of the interplay among them.
\begin{figure}[htbp]
\hspace*{-2mm}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, angle=0, width=0.8 \columnwidth]{Fig11.eps}
\caption{(Color online) The $J$-$D$, $J$-$F$ and $D$-$F$ phase diagrams for the 1D FM SSH model with both the anisotropic DMI and PDI.}
\label{Fig11}
\end{figure}
Evidently, these three interactions have synergistic effects on the generation of topologically non-trivial phase. In essence, it is the intercellular interactions ($J_{2}$, $D_{2}$ and $F_{2}$) rather than the intracellular interactions ($J_{1}$, $D_{1}$ and $F_{1}$) that contribute to the emergence of the topological phases. As a consequence, these different magnetic exchange interactions introduce a rich topological phase diagram in the 1D magnon system, suggesting a realization of multiple manipulation of topological magnonics. As matter of fact, the relative strength between the isotropic and anisotropic exchange interactions, can be changed by the strain and pressure regulations in the realistic materials, subsequently the topological phase transition of magnons can be achieved.
\section{Remarks and Conclusions}
We notice that different from our local spins, topological magnons in 1D itinerant flatband ferromagnet are also proposed \cite{PRB97-245111}. Indeed, the 1D topological magnons can be explored in realistic one-dimensional (1D) magnets \cite{PRB104-085429}, quasi-1D magnetic materials \cite{NJP15-093043}, or 1D organic magnetic systems \cite{NM17-308,NRM5-87}. As can be expected, more 1D magnetic models rather than the SSH model can be extended to topological magnons. Another way to implement the 1D ferromagnetic SSH model is to strip an edge from the magnonic 2D SSH model in honeycomb ferromagnets \cite{PRB103-014407}.
On the other hand, our results also imply that the topological magnons in the 1D systems are distinctly different from these in the 2D ones, introducing a new platform for the topological magnons.
In summary, we construct a one-dimensional (1D) ferromagnetic (FM) SSH model with the anisotropic interactions, and find a topological magnon non-trivial phase described by the Zak phase, induced not only by the pure strong intercellular isotropic Heisenberg interaction $J_{1}/J_{2}<1$, but also by the strong intercellular anisotropic DMI and PDI. The intercellular anisotropic DMI and PDI in combination with the intercellular isotropic Heisenberg-type interaction manifest a synergistic effect on the topological phase transition. In addition, the quantization axis of spin also substantially affect the topological magnon phase diagram owing to the anisotropic interactions. Of particular interest, the existence of topologically protected in-gap edge states (magnon bound states) in 1D magnets with FM SSH model provides a possible route to Majorana-like particle realization. Due to the rich manipulated magnetic interactions and the 1D structural characteristics, abundant topological magnon states and magnonic crystals can be designed from bottom to top, suggesting potential applications in the field of topological magnonics.
\acknowledgements
This work was supported by the National Sciences Foundation of China under Grant Nos. 11974354, 11774350 and 11574315. The calculations were performed in Center for Computational Science of CASHIPS, the ScGrid of Supercomputing Center and Computer Network Information Center of Chinese Academy of Science.
\section{Appendix}
For the quantization $z$-axis, the explicit matrix form of $H_{0}$,
$H_{DM}$ and $H_{PD}$ are
\begin{eqnarray}
H_{0}=\begin{pmatrix}
\varepsilon_{0} & 0 & h(k) & 0 \\
0 & \varepsilon_{0} & 0 & h^{*}(k) \\
h^{*}(k) & 0 & \varepsilon_{0} & 0 \\
0 & h(k) & 0 & \varepsilon_{0}
\end{pmatrix}
\end{eqnarray}
\begin{eqnarray}
H_{DM}=\begin{pmatrix}
0 & 0 & f(k) & 0 \\
0 & 0 & 0 & f^{*}(k) \\
f^{*}(k) & 0 & 0 & 0 \\
0 & f(k) & 0 & 0
\end{pmatrix}
\end{eqnarray}
\begin{eqnarray}
H_{PD}=\begin{pmatrix}
0 & 0 & g(k) & g(k) \\
0 & 0 & g^{*}(k) & g^{*}(k) \\
g^{*}(k) & g(k) & 0 & 0 \\
g^{*}(k) & g(k) & 0 & 0
\end{pmatrix}
\end{eqnarray}
where $\varepsilon_{0}=\left(J_{1}+J_{2}+K \right)S$, $h(k)=-J_{1}S- J_{2}Se^{-ika}$, $f(k)=-iD_{1}S+iD_{2}Se^{-ika}$, $g(k)=\left(-F_{1}S-F_{2}Se^{-ika}\right)/2$.
For $H_{1}=H_{0}+H_{DM}$, we assume $J_{1}/J_{2}=\alpha>0$,
$D_{1z}/J_{2}=\beta_{1}>0$, $D_{2z}/D_{1z}=\gamma_{1}$,
$D_{2z}/J_{2}=\gamma_{1}\beta_{1}$. When the topological phase transition occurs at the closed energy gap, and the parameters are satisfied by
\begin{eqnarray}
\begin{cases}
-\alpha-\cos(ka)+\gamma_{1}\beta_{1}\sin(ka)=0 \\
\sin(ka)-\beta_{1}+\gamma_{1}\beta_{1}\cos(ka)= 0 \\
\end{cases}
\end{eqnarray}
Cancelling $sin(ka)$ and $cos(ka)$, we find $\beta_{1}^{2}+\alpha^{2}=1+\gamma_{1}^{2}\beta_{1}^{2}$. Here
we take $\beta_{1}=1$ and $a=1$. As a result, the condition of
the phase transition ({\it i.e.} the gap closed) is $\alpha^{2}=\gamma_{1}^{2}$. When $\gamma_{1}=\alpha$, $\sin(ka)=1$, that is, $k=\pi/2a$; when $\gamma_{1}=-\alpha$, $\sin(ka)=(1-\alpha^{2})/(1+\alpha^{2})$, $\cos(ka)=-2\alpha/(1+\alpha^{2})$.
For $H_{2}=H_{0}+H_{PD}$, we assume $F_{1}/J_{2}=\beta_{2}>0$,
$F_{2}/F_{1}=\gamma_{2}>0$, $F_{2}/J_{2}=\gamma_{2}\beta_{2}$. The parameters are satisfied by
\begin{eqnarray}
(2\alpha+\beta_{2})^{2}+(2+\gamma_{2}\beta_{2} )^{2}+2(2\alpha+\beta_{2})(2+\gamma_{2}\beta_{2} )\cos(ka)=0
\end{eqnarray}
Obviously, $-1\leq \cos(ka)<0$. Assuming $n=\cos(ka)-(-1)$, it can be rewritten as
\begin{eqnarray}
((2\alpha+\beta_{2})-(2+\gamma_{2}\beta_{2} ))^{2}+2n(2\alpha+\beta_{2})(2+\gamma_{2}\beta_{2} )=0
\end{eqnarray}
We can see that it is valid only for $n=0$, {\it i.e.} $\cos(ka)=-1$.
We take $\beta_{2}=1$, and then $\gamma_{2}=2\alpha-1$. Then the condition of topological phase transition can be determined analytically.
|
1,314,259,995,821 | arxiv | \section{\sf Introduction}\label{introduction}
Kolmogorov's phenomenological statistical theory of turbulence, based on a set of axioms, displays certain well-known
characteristics, such as a $k^{-5/3}$ spectrum in an inertial range with a wavenumber cut-off at $L^{-1}Re^{3/4}$,
together with a dissipation range beyond this [2-5].
In contrast, from the perspective of Navier-Stokes analysis, much remains open in the three-dimensional case [6-19].
A proof of the existence and uniqueness of solutions is missing so the existence of a global attractor remains an open question [10-13].
Moreover, characteristics of an energy spectrum, such as its steepness and wavenumber cut-off, are hard to extract from a
time-evolving PDE. An interesting question is whether numerical experiments on the Navier-Stokes equations can inform
the analysis by suggesting a new and different way of looking at Navier-Stokes turbulence? In the early days of Navier-Stokes
simulations [20-25]
less resolution was available but, in recent years, several very large simulations (up to a maximum of $4096^{3}$) have
been performed [26-32].
The data from two of these, together with additional computations, are used in an attempt to understand the behaviour of
the solutions from a range of initial conditions.
\par\smallski
The variables that will be used in this paper are defined in terms of the Navier-Stokes vorticity field $\mbox{\boldmath$\omega$} = \mbox{curl}\,
\mbox{\boldmath$u$}$ in the following manner [1,\,33-36]\,:
\beq{s1}
\Omega_{m}(t) = \left(L^{-3}\int_{\mathcal{V}} |\mbox{\boldmath$\omega$}|^{2m}dV\right)^{1/2m}\,;
\qquad D_{m} = \left(\varpi_{0}^{-1}\Omega_{m}\right)^{\alpha_{m}}\,;
\qquad\alpha_{m} = \frac{2m}{4m-3}\,,
\eeq
where $\varpi_{0} = \nu L^{-2}$ is the frequency on the periodic box $[0,\,L]^{3}$. Note that $D_{1} = \left(\varpi_{0}^{-1}
\Omega_{1}\right)^{2}$ is proportional to the $H_{1}$-norm of the velocity field.
A recent set of numerical experiments, using a variety of initial conditions, each with periodic boundary conditions \cite{DGGKPV13},
has suggested that the $D_{m}$ are ordered on a descending scale such that $D_{m+1} < D_{m}$ for $m\geq 1$. In itself this is not
surprising\,: while H\"older's inequality necessarily enforces the $\Omega_{m}$ to be ordered on an ascending scale such that
$\Omega_{m} \leq \Omega_{m+1}$, the decreasing nature of the $\alpha_{m}$ means that if the $\Omega_{m}$ are bunched
sufficiently close, the ordering of the $D_{m}$ could easily be the reverse of the $\Omega_{m}$, as indeed is observed numerically.
What is more surprising is the observed strong separation on a logarithmic scale in the descending sequence of the $D_{m}$, in
particular from $D_{1}$. This separation is observed to be of the form\footnote{\sf The exponent has been changed to $A_{m}$
from $a_{m}$ in \cite{DGGKPV13} to avoid confusion with $\alpha_{m}$.} (see \S\ref{threereg})
\begin{equation}\label{dep1a}
D_{1}^{\alpha_{m}/2} \leq D_{m} \leq D_{1}^{A_{m}}\,,\qquad\qquad m\geq 2\,,
\end{equation}
where $\shalf\alpha_{m} < A_{m}(t) < \shalf$\,: the lower bound arises from $\Omega_{1} \leq \Omega_{m}$ expressed in
the $D_{m}$-notation.
\par\smallski
The main intention of this paper is to investigate how the numerically observed depletion in (\ref{dep1a}) severely reduces the
strength of the vortex stretching, thereby opening a window through which we can examine its effect on the regularity problem.
To illustrate how this comes about, let us summarize the results which standard methods (H\"older and Sobolev inequalities)
yield when attempting to estimate the rate of enstrophy production $\dot{D}_{1}$. The result in the unforced case is
\begin{equation}\label{D1est1a}
\shalf\dot{D}_{1} \leq \varpi_{0}\left(- D_{1}^{2}/4E + c\,D_{1}^{3}\right)\,,
\end{equation}
where the dimensionless, bounded energy is $E = \nu^{-2}L^{-1}\int_{\mathcal{V}}|\mbox{\boldmath$u$}|^{2}dV$. This result has been known for a generation
\cite{CF88,IDDS, DG95,FMRT01} and is derived for the reader in \S\ref{threereg}. As it stands, (\ref{D1est1a}) allows no control
over $D_{1}$ beyond short times for arbitrarily large initial data or for long times from very small initial data. Moreover, dimensional
scaling arguments suggest that no improvement on the $D_{1}^{3}$-term can be obtained when standard methods are used. However,
\S\ref{threereg} shows that a re-working of this term by the insertion of the nonlinear depletion
\begin{equation}\label{dep1b}
D_{m} \leq D_{1}^{A_{m,\lambda}}\,,\qquad\mbox{where}\qquad A_{m,\lambda} = \max_{t} A_{m}(t)
\end{equation}
results in the $D_{1}^{3}$-term being replaced by one proportional to $D_{1}^{\xi_{m,\lambda}}$ (see (\ref{b9})) where\,:
\begin{equation}\label{ximdef}
\xi_{m,\lambda} = \frac{\chi_{m,\lambda}+2m-3}{2(m-1)}\,,\qquad\mbox{with}\qquad\chi_{m,\lambda} = A_{m,\lambda}(4m-3)\,.
\end{equation}
The parameter $\lambda$, lying in the range $1\leq \lambda \leq 4$, appears through a scaling argument in \S\ref{amchoice} which
suggests that $A_{m,\lambda}$ and $\chi_{m,\lambda}$ take the form
\begin{equation}\label{amlamdef}
A_{m,\lambda} = \frac{m\lambda + 1-\lambda}{4m-3}\quad\mbox{and}\quad\chi_{m,\lambda} = m\lambda + 1-\lambda\,.
\end{equation}
Note that when $\lambda = 4$, then $A_{m,4} = 1$. The value of $\lambda$ chosen in the above range depends on the initial
conditions of a given numerical simulation. Equations (\ref{ximdef}) and (\ref{amlamdef}) yield
\begin{equation}\label{ximdep}
\xi_{m,\lambda} = 1+\shalf\lambda
\end{equation}
which is explicitly independent of $m$.
To gain control over $D_{1}$, for long times and large initial data, it is thus necessary to restrict $\xi_{m,\lambda}$ to $\xi_{m,\lambda} < 2$
and $\lambda$ to the range\footnote{\sf The lower bound $\lambda \geq 1$ derives from the lower bound on $D_{m}$ in (\ref{dep1a}).}
$1 \leq \lambda < 2$\,: see Fig. 1. It appears that the numerical data in \cite{DGGKPV13} can be fitted to (\ref{amlamdef}) with $\lambda$
sitting well within this range\,: $\lambda_{\rm min}$ is chosen as the minimum value of $\lambda$ for any given numerical fit. In \S\ref{num} we suggest
that the range $1.15 \leq \lambda_{\rm min} \leq 1.5$ is appropriate for a range of initial conditions.
\begin{figure}
\centering
\setlength{\unitlength}{5mm}
\begin{picture}(11,11)
\thicklines
\put(0,0){\vector(0,1){10}}
\put(0,0){\vector(1,0){15}}
\thinlines
\put(0,10){\makebox(0,0)[b]{$D_{m}$}}
\thinlines
\put(15.7,-0.25){\makebox(0,0)[b]{$D_{1}$}}
\put(0,0){\line(1,1){10}}
\put(12,9.8){\makebox(0,0)[b]{\scriptsize $D_{m} = C_{m}D_{1}$}}
\qbezier(0,0)(6,6)(11,7)
\put(4,8){\makebox(0,0)[b]\textbf{\sf\scriptsize regime III}}
\put(8.3,7.8){\makebox(0,0)[b]\textbf{\sf\scriptsize regime II (weak)}}
\put(8.2,5.6){\makebox(0,0)[b]\textbf{\sf\scriptsize regime I (regular)}}
\put(6,2){\makebox(0,0)[b]\textbf{\sf\scriptsize not allowed}}
\put(12,6.9){\makebox(0,0)[b]{\scriptsize $\lambda = 2$}}
\qbezier(0,0)(6,4)(11,4)
\put(12,3.8){\makebox(0,0)[b]{\scriptsize $\lambda = 1$}}
\linethickness{.4mm}
{\qbezier[25](0,.2)(5,4.5)(9.7,5)}
{\qbezier[25](0,.2)(5,4.5)(11,4.7)}
\put(13.5,4.6){\makebox(0,0)[b]{\scriptsize $D_{m} = D_{1}^{A_{m,\lambda}}$}}
\end{picture}
\caption{\sf\scriptsize A cartoon of the three regimes in the $D_{1}-D_{m}$ plane represented by the inequalities in
(\ref{reg2}) at some value of $m > 1$, with $A_{m,\lambda}$ defined in (\ref{dep1b}). Two solid concave curves bound
regime I\,: the lower curve derives from H\"older's inequality and corresponds to $\lambda=1$, whereas the upper curve
is the upper limit of the regular regime (see \S\ref{1streg}) which corresponds to $\lambda=2$. The dotted curves
approximately denote the region where the computations of \S\ref{num} lie for various values of $\lambda$ in the range
$1.15 \leq \lambda \leq 1.5$. The line $D_{m} = C_{m}D_{1}$ separates regimes II and III.}\label{phase}
\end{figure}
\par\smallski
While (\ref{dep1b}) is designated as regime I it is nevertheless theoretically possible that there exist other regimes beyond
this (see Fig 1). The following regimes are defined and analyzed in \S\ref{1streg} and \S\ref{2ndreg}\,:
\beq{reg2}
D_{1}^{\alpha_{m}/2} &\leq& D_{m} \leq D_{1}^{A_{m,\lambda}}\,,\qquad\qquad~~\mbox{(regime~I)}\,;\nonumber\\
D_{1}^{A_{m,\lambda}} &<& D_{m} \leq C_{m}D_{1}\,,~~\qquad\qquad\mbox{(regime~II)}\,;\\
C_{m}D_{1} &<& D_{m}\,,\qquad\qquad\qquad\qquad~~\mbox{(regime~III)}\,,\nonumber
\eeq
with regime I corresponding to the range $1\leq\lambda \leq 2$.
The constant $C_{m}$ is determined in \S\ref{2ndreg}, where it is shown that regime II leads to no improvement in the $D_{1}^{3}$
estimate. Solutions are actually regular in regime III, but it is an open question whether this regime is physical. Fig. 1 is a cartoon of the
regimes in (\ref{reg2}).
\par\smallski
Remarkably, the two respective values of the exponents $\xi_{m,\lambda} = 1+ \shalf\lambda$ and $\xi_{m,4}=3~(\lambda = 4)$ in
regimes I and II, are close to those found in a paper by Lu and Doering \cite{LD08}, who used a numerical calculus of variations argument
to find the value(s) of the exponent $\xi_{m,\lambda}$ when the rate of enstrophy production is maximized subject to the constraint
$\mbox{div}\,\mbox{\boldmath$u$} = 0$. They found that two branches existed, the lower being $D_{1}^{1.78}$ and the uppermost $D_{1}^{2.997}$.
Later, Schumacher, Eckhardt and Doering \cite{SED10} suggested that 7/4 and 3 were the likely values of these two exponents\,; the
exponent $\xi_{m,\lambda} = 7/4$ corresponds to $\lambda = 1.5$ which lies at the upper end of our observed range $1.15 \leq \lambda
\leq 1.5$.
\par\smallski
Boundedness from above of $D_{1}$ establishes existence and uniqueness and is the missing ingredient in the search for the existence of a
global attractor $\mathcal{A}$ [10-13], albeit limited to regime I. In \S\ref{att} it is shown that estimates for the Lyapunov dimension of
$\mathcal{A}$ are found to be (Proposition \ref{attdim})
\begin{equation}\label{introattdim}
d_{L}(\mathcal{A}) \leq c_{m}Re^{\frac{3(6-\lambda)}{5(2-\lambda)}}~~~\mbox{or}~~~
c_{m}Gr^{\frac{3(4-\lambda)}{5(2-\lambda)}}\,,
\end{equation}
where $Re$ and $Gr$ are respectively the Reynolds and Grashof numbers defined in \S\ref{threereg}. \S\ref{spectra} shows that there
is a corresponding energy spectrum in an inertial range for which $\mathcal{E}(k)\sim k^{-q_{m,\lambda}}$ where $q_{m,\lambda}=
3-4/3\lambda$, with a cut-off at $L^{-1}Re^{3\lambda/4}$. The lower concave curve in Fig. 1 corresponds to $\lambda =1$ for which
$q_{m,1} = 5/3$ with a cut-off at $L^{-1}Re^{3/4}$. Regime I corresponds to $5/3 \leq q_{m,\lambda} < 7/3$.
\par\smallski
If these properties of regime I turn out to be typical of Navier-Stokes flows in periodic domains, then the existence and uniqueness results
derived here are consistent with the observation that both numerical solutions [20-32] and experimental data \cite{Sreeni91,MS91}, while
providing evidence of strong intermittency, have shown none of the violent super-exponential or singular growth observed in the 3D Euler
equations \cite{HouLi06,RMK2013b}, nor have they shown any positive evidence of a lack of uniqueness. A related question is why a regime
with such heavy depletion is favoured? Moreover, what vortical structures would correspond to it? Formally, using
Sobolev and H\"older inequalities in $d$ dimensions ($1 \leq d \leq 3$) to estimate the vortex-stretching term, as in (\ref{D1est1a}),
results in $\xi_{m,\lambda} = (6-d)/(4-d)$, which takes the expected value $\xi_{m,4}=3$ when $d=3$. The value of $d$ corresponding to
$\xi_{m,\lambda} = 1+\shalf\lambda$ is $d_{\lambda} = 4(\lambda - 1)/\lambda$ which takes values from $d_{1.15}\approx 0.52$ to
$d_{1.5} = 4/3$. This suggests that the dominant structures which give rise to the depletion observed in regime I could be the pasta-mix of
tubes on which both vorticity and strain have long been numerically observed to accumulate \cite{Sreeni91,VM94} but also suggests that
some vortical structures may lie closer to scattered points.
\par\smallski
In contrast, \S\ref{2ndreg} shows that in regime II (labelled in Fig 1) these methods fail to find a proof of the existence of an attractor.
Only Leray's weak solutions are known to exist and $q_{m,\lambda}$ lies at its outer limit with a value of $8/3$ for the sustenance of an
energy cascade \cite{DG02,SF75}. In regime III, vorticity norms are under control, although it is possible that this regime represents an
extreme state. While numerical evidence suggests that the Navier-Stokes equations operate in regime I only, it is still possible that solutions
could jump between regimes, corresponding to some unusual initial conditions or higher Reynolds numbers. These possibilities are discussed
in \S\ref{con}.
\section{\sf\large\textbf{Three Navier-Stokes regimes}}\label{threereg}
Consider the forced $3D$ Navier-Stokes equations on the periodic domain $[0,\,L]^{3}$\,:
\begin{equation}\label{NSE1}
\partial_{t}\mbox{\boldmath$u$} + \mbox{\boldmath$u$} \cdot\nabla\mbox{\boldmath$u$} = \nu\Delta\mbox{\boldmath$u$} - \nabla p + \mathbi{f}(\mathbi{x})\,,
\end{equation}
with $\mbox{div}\,\mbox{\boldmath$u$} = 0$. The forcing function $\mathbi{f}(\mathbi{x})$ and its derivatives are considered
to be $L^{2}$-bounded \cite{DF02}. Estimates will be made in terms of the Grashof number $Gr$ and the Reynolds
number whose definitions are \cite{DF02}
\beq{GRdef}
Gr &=& \frac{L^{3}f_{rms}}{\nu^2}\,,\qquad\qquad f_{rms}^2 = L^{-3}\|\mathbi{f}\|_{2}^{2}\,,\\
Re &=& \frac{LU_{0}}{\nu}\,,\qquad\qquad\quad U_{0}^{2} = L^{-3}\left<\|\mbox{\boldmath$u$}\|_{2}^{2}\right>_{T}\,,
\eeq
and
where the time average to time $T$ is given by
\begin{equation}\label{timeav}
\left<g(\cdot)\right>_{T} =
\frac{1}{T}\int_{0}^{T}g(\tau)\,d\tau\,.
\end{equation}
Doering and Foias \cite{DF02} have introduced a simplified form of forcing with the mild restriction that involves it
peaking around a length scale $\ell$, which, for simplicity, is taken here to be the box length $L$. Then they have
shown that Navier-Stokes solutions obey $Gr \leq c\,Re^{2}$ and that the global enstrophy satisfies
\begin{equation}\label{DF1}
\left<D_{1}\right>_{T} \leq GrRe + O\left(T^{-1}\right)\leq c\,Re^{3} + O\left(T^{-1}\right)\,.
\end{equation}
In fact, all the $\left<D_{m}\right>_{T}$ for $1\leq m\leq \infty$ are bounded \cite{JDGCMS11}.
\subsection{\sf\textbf{A summary of numerical work}}\label{num}
The results from several numerical experiments, some of which were reported in \cite{DGGKPV13}), are summarized in Figs. 2, 3
and 4 which show plots of
\begin{equation}\label{amdef}
A_{m}(t) = \frac{\ln D_{m}(t)}{\ln D_{1}(t)}
\end{equation}
versus time $t$, with the exception of Fig. 3c, in which the horizontal axis is $Re_{\lambda}$\,: this the conventional notation for
the Taylor micro-scale Reynolds number so the subscript $\lambda$ should not be confused with the parameter $\lambda$ in (\ref{dep1b}).
The $D_{m}$ are replaced by the time averages $\left<D_{m}\right>_{T}$ and the $A_{m}$ by $\overline{A}_{m}$\,:
\begin{enumerate}\itemsep -1mm
\item Figs. 2a-d come from a pseudo-spectral $512^{3}$ simulation of the forced Navier-Stokes equations on a $(2\pi)^{3}$
domain with random initial conditions\,: in all cases $\max_{t}A_{m} \leq 0.46$, with $m=2$ at the upper limit, but with values
dropping close to about $0.37$ as $m\to9$. Fig. 2a is the result of Kolmogorov forcing $f(x,y,z) = f_{0} \sin(k_{1}x)$ with $f_{0}
= 0.005$ and $k_{1}=1$, which keeps the Grashof number $Gr$ constant ($Gr = 8.8\times 10^{7}$). Figs. 2b-d are the result of
white-noise forcing restricted to those modes for which $|\mbox{\boldmath$k$}| = 1$, i.e.,
\begin{equation}\label{wn1}
f(x,y,z) = f_{0}(t)\cos \{k_{1} x + k_{2}y + k_{3}z\}\,.
\end{equation}
The amplitude $f_0 (t)$ is a zero-mean ($Gr = L^{3} f_{A}\nu^{-2}$), Gaussian white noise with variance
$\left< f_{0}(t)\,f_{0}(t')\right> = f_{A}\,\delta (t - t')$. The values of $Re_{\lambda}$ for the simulations shown in Figs.
2b-d are $Re_{\lambda_{1,2,3}}= 97, 117$ and $192$ respectively.
\item Fig. 3a is a decaying simulation of fully developed Navier-Stokes turbulence performed by Kerr \cite{RMK2012a,DGGKPV13}
who used an anisotropic $1024\times2048\times 512$ mesh in a $2\pi (2\times8\times1)$ domain, with symmetries applied to the
$y$ and $z$ directions. As summarized in \cite{DGGKPV13,RMK2012a}, the simulation has long anti-parallel vortices as initial
conditions from which develop three sets of reconnections at $t=16,\,96$ and $256$. The figure is a plot of $A_{m}$ for $m=2$
descending to $m=9$ where $\max_{t}A_{m}$ takes its maximum at $m=2$ (0.46), and decreases to about $3/8$ as $m\to9$.
\item Fig. 3b shows a plot from a decaying version of the simulation in Figs. 2a-d. $\max_{t}A_{m} \leq 0.43$, but decreases close to
$0.37$ as $m\to9$.
\item Fig. 3c derives from a DNS data-base using a massively parallel pseudo-spectral code run on $10^{5}$ processors, which
includes simulations with resolutions up to $4096^{3}$ and Taylor-Reynolds number up to $Re_{\lambda} \sim 1000$
\cite{DYS2008,DY2010,YDS2012}. In order to maintain a stationary state, turbulence is forced numerically at the large scales.
Results are shown using the stochastic forcing of Eswaran \& Pope \cite{EP88} (denoted as EP), as well as a deterministic
scheme described in \cite{DY2010} (denoted as FEK). The figure shows the $m=2$ case descending to $m=6$\,: open and
closed symbols in the figure correspond to EP and FEK forcing, respectively. These schemes are summarized in more detail in
\cite{DGGKPV13}. Here $\overline{A}_{m}$ is defined by $\overline{A}_{m} = \ln \left<D_{m}\right>_{T}/\ln \left<D_{1}
\right>_{T}$ while the horizontal axis denotes values of $Re_{\lambda}$ which goes up to $10^{3}$, while $\max\overline{A}_{m}
\leq 0.42$.
\item The simulations above have been performed in the range $2 \leq m \geq 9$. In Fig. 4 we give one example of a simulation in
the range $1 \leq m \leq 2$. Three values of $\left(m,\,A_{m,\lambda}\right)$ are given in table 2. There it can be seen that the
range of $\lambda$ is $1.19 \leq \lambda \leq 1.5$.
\end{enumerate}
\begin{figure}[ht]\label{fig2}
\centering
\includegraphics[scale=.30]{Am_kolmo.pdf}
\includegraphics[scale=.30]{Am_Gr1-2.pdf}\qquad\\
\includegraphics[scale=.30]{Am_Gr2-2.pdf}
\includegraphics[scale=.30]{Am_Gr3-2.pdf}
\caption{\sf\scriptsize Plots of $A_{m}$ versus time for four forced simulations\,: in all of these $\max_{t} A_{m} < 0.5$. Figs 2a-d
are the result of pseudo-spectral $512^{3}$ simulations on a cubical $[0,\,2\pi]^{3}$-domain with random initial conditions.
Fig 2a is the result of Kolmogorov forcing with $|\mathbi{f}_{0}|=0.005$ and $k_{1}=1$, while figs 2b-d are plots for three different
values of $Re_{\lambda}$, namely $Re_{\lambda_{1,2,3}} = 97, 117$ and $192$ respectively, under the influence of white-noise
forcing\,: see the text for a more detailed explanation.}
\end{figure}
\begin{figure}[ht]\label{fig3}
\centering
\includegraphics[scale=.25]{am_kerr.pdf}
\includegraphics[scale=.24]{Am_unforced.pdf}\quad
\includegraphics[scale=.25]{Am_DD.pdf}
\caption{\sf\scriptsize Plots 3a,\,b are of $A_{m}$ defined in (\ref{amdef}) versus time for two decaying simulations for which
$\max_{t}A_{m} < 0.5$.
Fig. 3a is a $1024\times2048\times512$ pseudo-spectral simulation on a long $4\pi\times16\pi\times2\pi$ domain with anti-parallel initial
conditions \cite{RMK2012a,DGGKPV13}. Fig. 3b is a decaying version of the $512^{3}$ simulation as in Figs. 2a-d. Fig. 3c is a plot of
$\overline{A}_{m}$ defined by $\overline{A}_{m} = \ln \left<D_{m}\right>_{T}/\ln \left<D_{1}\right>_{T}$ arising from the TAMU
database with the horizontal axis denoting values of $Re_{\lambda}$. Open and closed symbols denote results from two types of
forcing (EP and FEK) for statistically steady flows\,: see \cite{DGGKPV13}.}
\end{figure}
\par\noindent
\begin{figure}[ht]\label{fig4}
\centering
\includegraphics[scale=.33]{Am_fract_m.pdf}
\hspace{5mm}
\includegraphics[scale=.33]{Dm_fract_m.pdf}
\caption{\sf\scriptsize Example of a simulation with white-noise forcing, as in Fig. 2d ($Re_{\lambda}=192$) performed
in the range $1 \leq m \leq 2$. In the right-hand figure $D_{1}$ is included whereas it is not in Fig. 2d.}
\end{figure}
\subsection{\sf\textbf{How to choose $\max_{t} A_{m}$}}\label{amchoice}
\begin{table}[h]\label{amtab}
\begin{center}
\begin{tabular}{||c||c|c|c|c|c|c|c|c|c||}
\hline
$\lambda$ & 1.1 & 1.15 & 1.2 & 1.25 & 1.3 & 1.35 & 1.4 & 1.45 & 1.5\\\hline
$A_{2,\lambda}$ & 0.42 & 0.43 & 0.44 & 0.45& 0.46 & 0.47 & 0.48 & 0.49 & 0.5\\\hline
$A_{6,\lambda}$ & 0.31 & 0.32 & 0.33 & 0.35& 0.36 & 0.36 & 0.38 & 0.39 & 0.40\\\hline
$A_{9,\lambda}$ & 0.30 & 0.31 & 0.32 & 0.33& 0.35 & 0.36 & 0.37 & 0.38 & 0.39\\\hline
\end{tabular}
\caption{\sf\scriptsize Table of values of $\lambda$ and $A_{2,\lambda},~A_{6,\lambda}$ and $A_{9,\lambda}$
corresponding to Figs. 2 and 3.}
\end{center}
\end{table}
The numerical experiments reported above show that $A_{m}$ has values lying in a wide range. Is there a way of choosing $\max_{t}A_{m}$
as a function of $m$ in a simple manner consistent with the results of these simulations? The following is a consistency argument based on
the inequalities the $D_{m}$ must obey.
\par\smallski
Firstly it is easy to prove that
\begin{equation}\label{ap1}
\Omega_{m}^{m(p+q)} \leq \Omega_{m-p}^{q(m-p)}\Omega_{m+q}^{p(m+q)}\,.
\end{equation}
Let $p=m-1$ and $q=1$ to give $\Omega_{m}^{m^2} \leq \Omega_{1}\Omega_{m+1}^{m^{2}-1}$. In terms of $D_{m}$, this
translates to
\begin{equation}\label{ap3}
D_{m}^{m^2} \leq D_{1}^{\alpha_{m}/2}D_{m+1}^{\alpha_{m}(m^{2}-1)/\alpha_{m+1}}\,.
\end{equation}
Suppressing the $\lambda$-label on $A_{m+1}$ in the depletion $D_{m+1} \leq D_{1}^{A_{m+1}}$, we obtain
\begin{equation}\label{ap4}
D_{m}\leq D_{1}^{\frac{1+A_{m+1}(m-1)(4m+1)}{m(4m-3)}}\,.
\end{equation}
For the exponent on the right-hand side of (\ref{ap3}) to be consistent with $D_{m} \leq D_{1}^{A_{m}}$, we require
\begin{equation}\label{ap5}
\frac{1+A_{m+1}(m-1)(4m+1)}{m(4m-3)} = A_{m}\,.
\end{equation}
By using the definition $\chi_{m} = A_{m}(4m-3)$, this reduces to
\begin{equation}\label{ap6}
1+ (m-1)\chi_{m+1} = m\chi_{m}\,,
\end{equation}
which is solved to give\,:
\begin{proposition}\label{deltaprop}
The solution of (\ref{ap6}) is given by
\begin{equation}\label{ap7}
\chi_{m,\lambda} = m\lambda + 1-\lambda\,,\qquad\mbox{or}\qquad
A_{m,\lambda} = \frac{m\lambda + 1-\lambda}{4m-3}\,,
\end{equation}
where the constant $\lambda$ lies in the range $1 \leq \lambda \leq 4$.
\end{proposition}
The fit of (\ref{ap7}) to the figures in \S\ref{num} is not perfect in the sense that numerical trajectories do not
follow \textit{exactly} the concave curves of Fig. 1, so the appropriate value of $\lambda$ from an initial condition
needs to be estimated. To achieve this, we label as $\lambda_{m}$ those values computed from $\max_{t}A_{m}$
in a given figure. These increase slightly with $m$\,; for example, in Fig 3a, $\lambda_{2} =1.4$ at $m=2$, whereas
$\lambda_{9}=1.45$ at $m=9$ (see table 1). $\lambda_{\rm min}$, defined in (\ref{lamindef}), is taken as the minimum
of a set of values of $\lambda$ computed over a range of $m$ from a given initial condition. This can then be used in
the estimates for the attractor dimension or energy spectrum in the following sections.
\begin{equation}\label{lamindef}
\lambda_{\rm min} = \min \left\{\lambda_{m}\,: 2 \leq m \leq N\right\}\,.
\end{equation}
\begin{table}[h]\label{mlesstab}
\begin{center}
\begin{tabular}{||c||c|c|c||}
\hline
$m$ & 1.1 & 1.5 & 1.9 \\\hline
$A_{m,\lambda}$ & 0.85 & 0.55 & 0.45\\\hline
$\lambda$ & 1.5 & 1.3 & 1.19\\\hline
\end{tabular}
\caption{\sf\scriptsize Table of values of $\lambda$ corresponding to $A_{1.1,\lambda},~A_{1.5,\lambda}$ and
$A_{1.9,\lambda}$ corresponding to Fig. 4.}
\end{center}
\end{table}
\subsection{\sf\textbf{A division into three regimes}}
The scaling \textit{ansatz} for $A_{m,\lambda}$ in (\ref{ap7}) derived in Proposition 1 suggests that the $D_{1}-D_{m}$ plane
can be divided into different regimes for the range\footnote{\sf The two regimes I and II could be merged by using the range $1\leq
\lambda\leq 4$ but this leaves a gap between $D_{1}$ and $C_{m}D_{1}$ which causes technical difficulties.} $1\leq\lambda\leq 2$\,:
\par\vspace{2mm}\noindent
\textbf{Regime I\,:} $D_{1}^{\frac{m}{4m-3}} \leq D_{m} \leq D_{1}^{A_{m,\lambda}}$\,: the lower bound is $\Omega_{1}
\leq \Omega_{m}$ expressed in the $D_{m}$-notation.
\par\vspace{2mm}\noindent
\textbf{Regime II\,:} $D_{1}^{A_{m,\lambda}} < D_{m} \leq C_{m} D_{1}$, where\footnote{\sf The constants $c_{i,m}$ in $\varpi_{1,m}
= \varpi_{0}c_{1,m}^{-1}$ and $\varpi_{2,m} = \varpi_{0}c_{2,m}$ have the following properties \cite{JDGPRS10,JDGJMP12}\,:
$c_{1,m}$ is a Sobolev constant multiplied by $m^{2}/(m-1)$ whereas $c_{2,m}$ derives from the constant in $\|\nabla\mbox{\boldmath$u$}\|_{p}
\leq c_{p}\|\mbox{\boldmath$\omega$}\|_{p}$ for $1 < p < \infty$.} $C_{m}^{\eta_{m} } = \varpi_{2,m}/\varpi_{1,m} = c_{1,m}c_{2,m}$, and
$\eta_{m} = \frac{2m}{3(m-1)}$.
\par\vspace{2mm}\noindent
\textbf{Regime III\,:} $ C_{m} D_{1} < D_{m}$.
\par\vspace{2mm}\noindent
Regime III appears in the following way. Using the standard contradiction method\footnote{\sf This method assumes the existence of a maximal
interval of existence and uniqueness on an interval $[0,\,T^{*})$, which means that $D_{1}$ must be infinite at $T^*$\,: then, in any subsequent
calculation, one considers the behaviour of $D_{1}$ as $t\to T^*$. If this limit is finite then a contradiction has occurred, thus invalidating the original
assumption of a \textit{maximal} interval. This cannot be zero so it must be infinite. The value of the method is that it allows the differentiation
of the $D_{m}$ on $[0,\,T^{*})$.},
for $1 < m < \infty$ ($\varpi_{i,m}$ are constants), the $D_{m}$ obey the differential inequality \cite{JDGJMP12,JDGiutam13}
\beq{s3}
\dot{D}_{m} &\leq& D_{m}^{3}\left\{-\varpi_{1,m}\left(\frac{D_{m+1}}{D_{m}}\right)^{\rho_{m}}
+ \varpi_{2,m}\right\} + \varpi_{3,m}Gr D_{m}^{\frac{\alpha_{m}-1}{\alpha_{m}}}\,,\\
\rho_{m} &=& \twothirds m(4m+1)\label{rhomdef}\,.
\eeq
Moreover, for $m > 1$ it is easily proved that
\begin{equation}\label{s4}
\frac{D_{m}}{D_{1}}\leq \left(\frac{D_{m+1}}{D_{m}}\right)^{(m-1)(4m+1)}\,,
\end{equation}
which changes (\ref{s3}) and (\ref{rhomdef}) into
\beq{s5}
\dot{D}_{m} &\leq& D_{m}^{3}\left\{-\varpi_{1,m}\left(\frac{D_{m}}{D_{1}}\right)^{\eta_{m}} +
\varpi_{2,m}\right\} + \varpi_{3,m}Gr D_{m}^{\frac{\alpha_{m}-1}{\alpha_{m}}}\,,\\
\eta_{m} &=& \frac{2m}{3(m-1)}\label{delmdef}\,.
\eeq
Clearly, in regime III the combination of terms within the braces is negative and can be neglected. In this regime
the dissipation is sufficiently strong to control solutions rather than depletion reducing the nonlinearity. In the
unforced case, the $D_{m}$ always decay\,; at most, they grow only algebraically in time in the forced case (see
\S\ref{2ndreg}). Moreover, $\Omega_{1} \leq \Omega_{m}$ universally implies that
\begin{equation}\label{A1}
\left(\frac{D_{m}}{D_{1}}\right)^{\eta_{m}}D_{m}^{2} \geq 1\,,
\end{equation}
which means that $D_{m} \geq 1$ in regime I, while in regime II there is a lower bound $D_{m}^{2} \geq
C_{m}^{-\eta_{m}}$.
\section{\sf\large\textbf{Regime I}}\label{1streg}
\subsection{\sf\large\textbf{Depletion resulting in an absorbing ball for $D_{1}$}}\label{ball}
It has long been understood that the $H_{1}$-norm of the velocity field ($D_{1}$ in the notation of this paper) controls
all the regularity
properties of $3D$ Navier-Stokes equations \cite{CF88,IDDS,DG95,FMRT01}. It is also the essential missing ingredient in
the search for the proof of the existence of a $3D$ Navier-Stokes global attractor. What is required is an ``absorbing ball'' for
this norm, which consists of a ball of finite radius into which all solutions are drawn for large times. In what follows, estimates
are made for the forced case in terms of the Grashof number $Gr$ or Reynolds number $Re$. In the unforced case the conclusions
regarding the finiteness of $D_{1}$ still stand except that the radius of the ball decays and the attractor is just the origin.
\par\smallski
In this context it is difficult to handle a wide variety of forcing functions analytically. For simplicity we shall remain with the
properties of the forcing as in Doering and Foias \cite{DF02} who took forcing at a single scale $\ell$, taken here to be the
box-scale $L$, to make estimates in terms of the Grashof number $Gr$ or Reynolds number $Re$ defined in (\ref{GRdef}).
\par\smallski
The first task is to illustrate why the standard estimate for $D_{1}$ produces an apparently unsurmountable problem. Note
that from the definition of the $D_{m}$ in (\ref{s1}) $D_{1} = L\nu^{-2}\|\mbox{\boldmath$\omega$}\|_{2}^{2}$
so, using the standard contradiction method (see footnote 10), a formal differential inequality for $D_1$ is
\begin{equation}\label{b2}
\shalf \dot{D}_{1} \leq L\nu^{-2}\left\{-\nu \int_{\mathcal{V}} |\nabla\mbox{\boldmath$\omega$}|^{2}\,dV + \int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}\,dV +
L^{-1}\left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2}\,dV\right)^{1/2}\|\mathbi{f}\|_{2}\right\}\,.
\end{equation}
Dealing with the negative term first, an integration by parts gives
\begin{equation}\label{b3}
\int_{\mathcal{V}} |\mbox{\boldmath$\omega$}|^{2}\,dV \leq \left(\int_{\mathcal{V}}|\nabla\mbox{\boldmath$\omega$}|^{2}dV\right)^{1/2}\left(\int_{\mathcal{V}}|\mbox{\boldmath$u$}|^{2}dV\right)^{1/2}\,,
\end{equation}
where the dimensionless energy $E$ is defined as
\begin{equation}\label{b4}
E = \nu^{-2}L^{-1}\int_{\mathcal{V}}|\mbox{\boldmath$u$}|^{2}\,dV\,.
\end{equation}
which is always bounded such that [10-13]
\begin{equation}\label{b7}
\overline{\lim}_{t\to\infty}E \leq c\,Gr^{2}\,.
\end{equation}
Then the nonlinear term in (\ref{b2}) can be estimated in two ways\,:
\begin{enumerate}\itemsep -1mm
\item By using a Sobolev inequality in the standard way [10-13]\,;
\item By invoking the nonlinear depletion of regime I.
\end{enumerate}
(1) The standard method simply involves a Schwarz inequality to estimate the nonlinear term as
\begin{equation}\label{b5a}
\int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}\,dV \leq \|\mbox{\boldmath$\omega$}\|_{2}\|\mbox{\boldmath$\omega$}\|_{4}^{2}\,.
\end{equation}
After the application of the Sobolev inequality $\|\mbox{\boldmath$\omega$}\|_{4} \leq c\,\|\nabla\mbox{\boldmath$\omega$}\|_{2}^{3/4}\|\mbox{\boldmath$\omega$}\|_{2}^{1/4}$,
this becomes
\beq{b5b}
\int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}\,dV &\leq& c\,\|\nabla\mbox{\boldmath$\omega$}\|_{2}^{3/2}\|\mbox{\boldmath$\omega$}\|_{2}^{3/2}\nonumber\\
&\leq&
\frac{3\nu}{4}\|\nabla\mbox{\boldmath$\omega$}\|_{2}^{2} + \frac{c}{4\nu^{3}}\|\mbox{\boldmath$\omega$}\|_{2}^{6}\,.
\eeq
(\ref{b2}) then becomes
\begin{equation}\label{b6}
\shalf \dot{D}_{1} \leq \varpi_{0}\left(-\frac{1}{4}\frac{D_{1}^{2}}{E} + c\,D_{1}^{3} + Gr D_{1}^{1/2}\right)\,.
\end{equation}
Clearly the cubic nonlinearity is too strong for the quadratic negative term\,: all we can deduce is that $D_{1}$ is bounded from
above only for short times or for small initial data. The difficulty caused by this term has been known for many decades\,: see
\cite{CF88,IDDS,DG95,FMRT01} and also Lu and Doering \cite{LD08}.
\par\medskip\noindent
(2) Now we turn to using the nonlinear depletion of regime I. How might the insertion of $D_{m} \leq D_{1}^{A_{m,\lambda}}$ mollify
the cubic exponent in (\ref{b6})? We return to (\ref{b2}) and estimate the nonlinear term as
\beq{b8a}
\int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}dV &=&\int_{\mathcal{V}} |\mbox{\boldmath$\omega$}|^{\frac{2m-3}{m-1}}|\mbox{\boldmath$\omega$}|^{\frac{1}{m-1}}|\nabla\mbox{\boldmath$u$}|dV\nonumber\\
&\leq & \left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2}dV\right)^{\frac{2m-3}{2(m-1)}}
\left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2m}dV\right)^{\frac{1}{2m(m-1)}}\left(\int_{\mathcal{V}}|\nabla\mbox{\boldmath$u$}|^{2m}dV\right)^{\frac{1}{2m}}\nonumber\\
&\leq & c_{m}\left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2}dV\right)^{\frac{2m-3}{2(m-1)}}\left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2m}dV\right)^{\frac{1}{2(m-1)}}\nonumber\\
&=& c_{m}L^{3}\varpi_{0}^{3} D_{1}^{\frac{2m-3}{2m-2}}D_{m}^{\frac{4m-3}{2m-2}}\,,\qquad\qquad 1 < m < \infty\,.
\eeq
based on $\|\nabla\mbox{\boldmath$u$}\|_{p} \leq c_{p} \|\mbox{\boldmath$\omega$}\|_{p}$, for $1 < p < \infty$. Inserting the depletion $D_{m} \leq D_{1}^{A_{m,\lambda}}$,
\begin{equation}\label{b8b}
L\nu^{-2}\int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}\,dV \leq c_{m}\varpi_{0}D_{1}^{\xi_{m,\lambda}}\,,
\end{equation}
where $\xi_{m,\lambda}$ is defined as in (\ref{ximdef}) but repeated here
\begin{equation}\label{ximdefA}
\xi_{m,\lambda} = \frac{\chi_{m,\lambda}+2m - 3}{2(m-1)}\,,\qquad\qquad \chi_{m,\lambda} =
A_{m,\lambda}(4m-3) = m\lambda +1 -\lambda\,.
\end{equation}
Thus we have
\begin{equation}\label{ximdeldef}
\xi_{m,\lambda} = 1+ \shalf \lambda\,,
\end{equation}
\textbf{which is explicitly $m$-independent.} Thus the equivalent of (\ref{b6}) is
\begin{equation}\label{b9}\boxed{
\shalf \dot{D}_{1} \leq \varpi_{0}\left(-\frac{D_{1}^{2}}{E} + c_{m} D_{1}^{1+ \shalf\lambda} + Gr D_{1}^{1/2}\right)\,.}
\end{equation}
Given that $E$ is bounded above, $D_{1}$ is always under control provided $\lambda$ is restricted to the range $1 \leq \lambda< 2$.
This is expressed in the following\,:
\begin{proposition}\label{prop2}
If the solution always remains in regime I ($1 \leq \lambda < 2$), there exists an absorbing ball for $D_{1}$ of radius
\begin{equation}\label{D1ball}
\overline{\lim}_{t\to\infty} D_{1}\leq c_{m}Gr^{\frac{4}{2 - \lambda}} + O\left(Gr^{4/3}\right)\,.
\end{equation}
\end{proposition}
\textbf{Remark 1\,:} The range of control over $D_{1}$ in $1 \leq \lambda < 2$ can be extended to $\lambda = 2$ as (\ref{b9})
shows that there is an exponentially growing bound on $D_{1}$ at this value.
\par\smallskip\noindent
\textbf{Remark 2\,:} Note that the values of $\lambda = \lambda_{\rm min}$ corresponding to the numerical experiments in \S\ref{threereg}
lie well within the range ($1 \leq \lambda < 2$) of validity, as illustrated by Fig. 1.
\par\smallskip\noindent
\textbf{Remark 3\,:} From (\ref{D1ball}) and the standard properties of the Navier-Stokes equations \cite{CF88,IDDS,DG95,FMRT01},
we conclude that a global attractor $\mathcal{A}$ exists in this regime, which is a compact $L^{2}$-bounded ball for the
velocity field $\mbox{\boldmath$u$}$. $c_{m}$ is a generic constant dependent only on $m$.
\subsection{\sf\large\textbf{An estimate for the attractor dimension}}\label{att}
It is now possible to estimate the Lyapunov dimension of the global attractor $\mathcal{A}$, which has been
shown to exist as a result of Proposition \ref{prop2}, subject to the depletion in regime I. A connection
between the system dynamics and the
attractor dimension is provided by the notion of the Lyapunov exponents through the Kaplan-Yorke formula.
For ODEs the Lyapunov exponents control the exponential growth or contraction of volume elements in phase
space\,: the Kaplan-Yorke formula expresses the balance between volume growth and contraction realized
on the attractor. It has been rigorously applied to global attractors in PDEs by Constantin and Foias
\cite{CF85,CF88}\,: see also \cite{IDDS,DG95,GT97}. The formula is the following\,: for Lyapunov exponents
labelled in descending order and designated by $\mu_{n}$, the Lyapunov dimension $d_{L}$ is defined in
terms of these by
\begin{equation}\label{dldef}
d_{L} = N -1 + \frac{\mu_{1}+\ldots + \mu_{N-1}}{-\mu_{N}},
\end{equation}
where the number $N$ of $\mu_{n}$ is chosen to satisfy
\begin{equation}\label{KY2}
\sum_{n=1}^{N-1}\mu_{n}\geq 0\hspace{1cm}\mbox{but}\hspace{1cm}\sum_{n=1}^{N}\mu_{n} < 0\,.
\end{equation}
Note that according to the definition of $N$, the ratio of exponents in (\ref{dldef}) satisfies
\begin{equation}\label{KY1}
0 \leq \frac{\mu_{1}+\ldots + \mu_{N-1}}{-\mu_{N}} < 1\,,
\end{equation}
so the formula generally yields a non-integer dimension such that
\begin{equation}\label{KY3}
N - 1 \leq d_{L} < N\,.
\end{equation}
The value of $N$ that turns the sign of the sum of the Lyapunov exponents, as in (\ref{KY2}), is that value
of $N$ that bounds above $d_{L}$ and hence the Hausdorff and fractal dimensions $d_{H}$ and $d_{F}$.
For a discussion of generalized dimensions see the paper by Hentschel and Procaccia \cite{HP83}. To use the
method for PDEs as developed in \cite{CF85,CF88} the phase space is replaced by $\mbox{\boldmath$u$}\in L^{2}\cap\mbox{div}
\,\mbox{\boldmath$u$}=0$, which is infinite dimensional. The solution $\mbox{\boldmath$u$}(t)$ forms an orbit in this space, with different sets of
initial conditions $\mbox{\boldmath$u$}(0)+\delta\mbox{\boldmath$u$}_{i}(0)$, which evolve into $\mbox{\boldmath$u$}(t)+\delta\mbox{\boldmath$u$}_{i}(t)$ for $i =1,\ldots,N$.
The linearized form of the Navier-Stokes equations in terms of $\delta\mbox{\boldmath$u$}$ of $\mbox{\boldmath$u$}$ is
\begin{equation}\label{NS2}
\partial_{t}(\delta\mbox{\boldmath$u$}) + \mbox{\boldmath$u$}\cdot\nabla\delta\mbox{\boldmath$u$} + \delta\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$} = \nu \Delta\delta\mbox{\boldmath$u$} -\nabla\delta p\,,
\end{equation}
which can also be written in the form
\begin{equation}\label{trace1}
\partial_{t}(\delta\mbox{\boldmath$u$}) = \mathcal{M}\delta\mbox{\boldmath$u$}\,.
\end{equation}
If they are chosen to be linearly independent, initially these $\delta\mbox{\boldmath$u$}_{i}$ form an $N$-volume or parallelpiped of volume
\begin{equation}\label{vol1}
V_{N}(t) = \left|\delta\mbox{\boldmath$u$}_{1}\wedge\delta\mbox{\boldmath$u$}_{2}\ldots\wedge\delta\mbox{\boldmath$u$}_{N}\right|\,.
\end{equation}
It is now necessary to find the time evolution of $V_{N}$. This is given by
\begin{equation}\label{vol2}
\dot{V}_{N} = V_{N}Tr \left[\mathbf{P}_{N}\mathcal{M}\mathbf{P}_{N}\right]\,,
\end{equation}
which is easily solved to give
\begin{equation}\label{trace2}
V_{N}(t) = V_{N}(0) \exp \int_{0}^{t}Tr \left[\mathbf{P}_{N}\mathcal{M}\mathbf{P}_{N}\right](\tau )\,d\tau\,.
\end{equation}
$\mathbf{P}_{N}(t)$ is an $L^{2}$-orthogonal projection, using the orthonormal set of functions $\{\mbox{\boldmath$\phi$}_{i}\}$, onto the
finite dimensional subspace $\mathbf{P}_{N}L^{2}$, which spans the set of vectors $\delta\mbox{\boldmath$u$}_{i}$ for $i=1,\,...,\,N$.
In terms of the time average $\left<\cdot\right>_{t}$ up to time $t$, the sum of the first $N$ global Lyapunov
exponents is taken to be \cite{CF85,CF88}
\begin{equation}\label{global1}
\sum_{n=1}^{N}\mu_{n} = \left< Tr \left[\mathbf{P}_{N}\mathcal{M}\mathbf{P}_{N}\right]\right>_{t} \,.
\end{equation}
As in (\ref{KY2}), we want to find the value of $N$ that turns the sign of
$\left<Tr\left[\mathbf{P}_{N}\mathcal{M}\mathbf{P}_{N}\right]\right>_{t}$
and for which volume elements contract to zero. This value of $N$ bounds above
$d_{L}$ as in (\ref{dldef}). To estimate this we write
\begin{equation}\label{tr1}
Tr \left[\mathbf{P}_{N}{\cal M}\mathbf{P}_{N}\right] = \sum_{n=1}^{N}
\int_{\mathcal{V}}\mbox{\boldmath$\phi$}_{n}\cdot\left\{\nu\Delta\mbox{\boldmath$\phi$}_{n}
- \mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$\phi$}_{n} - \mbox{\boldmath$\phi$}_{n}\cdot\nabla\mbox{\boldmath$u$}
- \nabla\tilde{p}\left(\mbox{\boldmath$\phi$}_{n}\right)\right\}\,dV.
\end{equation}
Since $\mbox{div}\,\delta_{m}\mbox{\boldmath$u$}_{n} = 0$ for all $n$, then
$\mbox{div}\,\mbox{\boldmath$\phi$}_{n}=0$ also and so the pressure term
integrates away, as does the second term
\begin{equation}\label{tr2}
Tr \left[\mathbf{P}_{N}{\cal M}\mathbf{P}_{N}\right]
\leq - \nu \sum_{n=1}^{N} \int_{\mathcal{V}} |\nabla\mbox{\boldmath$\phi$}_{n}|^{2}\,dV
+ \sum_{n=1}^{N} \int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}|\,|\mbox{\boldmath$\phi$}_{n}|^{2}\,dV .
\end{equation}
Because the $\mbox{\boldmath$\phi$}_{n}$ are orthonormal they obey the relations
\begin{equation}\label{tr3}
\sum_{n=1}^{N} \int_{\mathcal{V}}|\mbox{\boldmath$\phi$}_{n}|^{2}\,dV = N,
\hspace{1cm}\mbox{and}\hspace{1cm}
Tr \left[\mathbf{P}_{N}(-\Delta)\mathbf{P}_{N}\right]
= \sum_{n=1}^{N} \int_{\mathcal{V}}|\nabla\mbox{\boldmath$\phi$}_{n}|^{2}\,dV.
\end{equation}
In $3D$ the $\mbox{\boldmath$\phi$}_{n}$ satisfy the Lieb-Thirring inequalities \cite{CF85,CF88,FMRT01,IDDS}
for orthonormal functions
\begin{equation}\label{LT1}
\int_{\mathcal{V}}\left(\sum_{n=1}^{N}
|\mbox{\boldmath$\phi$}_{n}|^{2}\right)^{5/3}\!\!dV
\leq c\,\sum_{n=1}^{N} \int_{\mathcal{V}} |\nabla\mbox{\boldmath$\phi$}_{n}|^{2}\,dV,
\end{equation}
where $c$ is independent of $N$. Moreover, it is known that the first $N$ eigenvalues of the
Stokes operator in three-dimensions satisfy
\begin{equation}\label{lower1}
Tr\left[\mathbf{P}_{N}(-\Delta )\mathbf{P}_{N}\right] \geq c\,N^{5/3}L^{-2}\,.
\end{equation}
To exploit the Lieb-Thirring inequality (\ref{LT1}) to estimate the last term in (\ref{tr2}) we write it as
\begin{equation}\label{nltA}
\sum_{n=1}^{N} \int_{\mathcal{V}} |\nabla \mbox{\boldmath$u$}|\,|\mbox{\boldmath$\phi$}_{n}|^{2}\,dV
\leq \left[\int_{\mathcal{V}}|\nabla \mbox{\boldmath$u$} |^{5/2}\,dV\right]^{2/5}\,\left[\int_{\mathcal{V}}
\left(\sum_{n=1}^{N}|\mbox{\boldmath$\phi$}_{n}|^{2}\right)^{5/3}\!dV\right]^{3/5}.
\end{equation}
Hence, using (\ref{LT1}) and time averaging $\left<\cdot\right>_{t}$, we find
\begin{eqnarray}\label{nlt3}
\left<\sum_{n=1}^{N} \int_{\mathcal{V}}|\nabla \mbox{\boldmath$u$} |\,|\mbox{\boldmath$\phi$}_{n}|^{2}\,dV\right>_{t}
& \leq & c\,\left<\left(Tr \left[\mathbf{P}_{N}(-\Delta)\mathbf{P}_{N}
\right]\right)^{3/5}\left(\int_{\mathcal{V}}|\nabla\mbox{\boldmath$u$}|^{5/2}\,dV
\right)^{2/5}\right>_{t}\nonumber\\
& \leq & \frac{3\nu}{5}\left<Tr \left[\mathbf{P}_{N}(-\Delta)\mathbf{P}_{N} \right]\right>_{t}
+ \frac{2c}{5\nu^{3/2}}\left<\int_{\Omega}|\nabla\mbox{\boldmath$u$}|^{5/2}\,dV\right>_{t}
\end{eqnarray}
and so (\ref{tr2}) can be written as
\begin{equation}\label{tr4}
\left<Tr \left[\mathbf{P}_{N}{\cal M}\mathbf{P}_{N}\right]\right>_{t}
\leq - \frac{2}{5}\nu\left<Tr \left[\mathbf{P}_{N}(-\Delta)\mathbf{P}_{N}\right]\right>_{t} + \frac{2}{5}
c\,\nu^{-3/2}\left<\int_{\mathcal{V}} |\nabla \mbox{\boldmath$u$}|^{5/2}\,dV\right>_{t}\,.
\end{equation}
To estimate the nonlinear term we use H\"older's inequality to obtain ($m > 1$)
\beq{ad2a}
\int_{\mathcal{V}}\left|\nabla\mbox{\boldmath$u$}\right|^{5/2}\,dV &\leq& c\, \int_{\mathcal{V}} |\mbox{\boldmath$\omega$}|^{5/2}\,dV\nonumber\\
&\leq& c\,\left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2}\,dV\right)^{\frac{4m-5}{4(m-1)}}\left(\int_{\mathcal{V}}|\mbox{\boldmath$\omega$}|^{2m}\,dV\right)^{\frac{1}{4(m-1)}}\nonumber\\
&\leq& c\, \varpi_{0}^{5/2}L^{3}D_{1}^{\frac{4m-5}{4(m-1)}}D_{m}^{\frac{4m-3}{4(m-1)}}\,.
\eeq
Therefore, using this and (\ref{lower1}), we find
\begin{equation}\label{ad3a}
\varpi_{0}^{-1}\left<Tr \left[\mathbf{P}_{N}{\cal M}\mathbf{P}_{N}\right]\right>_{t} \leq
- c_{1}N^{5/3} + c_{2}\left<D_{1}^{\frac{4m-5}{4(m-1)}}D_{m}^{\frac{4m-3}{4(m-1)}}\right>_{t}\,.
\end{equation}
It is at this point where the depletion of nonlinearity $D_{m} \leq D_{1}^{A_{m,\lambda}}$ is used, thereby giving
\begin{equation}\label{b15}
\left<D_{1}^{\frac{4m-5}{4(m-1)}}D_{m}^{\frac{4m-3}{4(m-1)}}\right>_{t} \leq \left<D_{1}\right>_{t}
\left(\overline{\lim}_{t\to\infty}D_{1}\right)^{\frac{\chi_{m,\lambda}-1}{4(m-1)}}\,,
\end{equation}
where $\chi_{m,\lambda} = A_{m,\lambda}(4m-3)$ as defined in (\ref{ximdef}).
Proposition \ref{prop2} and the estimate $\left<D_{1}\right>_{t} \leq c\, GrRe$ from \cite{DF02} then allow us to write
\begin{equation}\label{ad4a}
\left<D_{1}\right>_{t} \left(\overline{\lim}_{t\to\infty}D_{1}\right)^{\frac{\chi_{m,\lambda}-1}{4(m-1)}}
\leq c\,(Gr Re) Gr^{\frac{\chi_{m,\lambda}-1}{2m-1 - \chi_{m,\lambda}}}
\leq c\, Re^{\frac{6m - 5 - \chi_{m,\lambda}}{2m-1-\chi_{m,\lambda}}}\,,
\end{equation}
and so (\ref{ad3a}) can be written as
\begin{equation}\label{ad4b}
\left<Tr \left[\mathbf{P}_{N}{\cal M}\mathbf{P}_{N}\right] \right>_{t} \leq \varpi_{0}\left(- c_{1}N^{5/3} +
c_{2}Re^{\frac{6m - 5 - \chi_{m,\lambda}}{2m-1-\chi_{m,\lambda}}}\right)\,.
\end{equation}
To find an estimate solely in terms of $Gr$ the $(Gr Re)$-term of (\ref{ad4a}) is replaced by $Gr^{2}$. Choosing
$\chi_{m,\lambda}$ as in (\ref{ap7}), we have proved\,:
\begin{proposition}\label{attdim}
If the solution always remains in regime I the Lyapunov dimension of the global attractor $\mathcal{A}$ is estimated as
\begin{equation}\label{ad5a}
d_{L}(\mathcal{A}) \leq c_{1,m}Re^{\frac{3}{5}\left(\frac{6 - \lambda}{2-\lambda}\right)}\,,
\end{equation}
or, alternatively, as
\begin{equation}\label{ad5b}
d_{L}(\mathcal{A}) \leq c_{2,m}Gr^{\frac{3}{5}\left(\frac{4-\lambda}{2-\lambda}\right)}\,.
\end{equation}
\end{proposition}
\section{\sf\large\textbf{Regimes II and III}}\label{2ndreg}
In \S\ref{1streg}, regime I has been defined to lie in the region $D_{m} \leq D_{1}^{A_{m,\lambda}}$ for $1 \leq \lambda < 2$, with
regime II defined as the region where this inequality has been reversed up to $C_{m}D_{1}$. One could fuse regimes I and II together
by taking $\lambda$ in the wider range $1 \leq \lambda \leq 4$ but we have no control over $D_{1}$ for $2 < \lambda \leq 4$. In this
section we choose to remain with the definition of regime II as in (\ref{reg2}).
\par\smallski
To test whether there is any depletion in regime II let us repeat inequality (\ref{b8a}) for the nonlinear term and use $D_{m}
\leq C_{m} D_{1}$
\beq{sec1}
\int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}\,dV &\leq& c\,L^{3}\varpi_{0}^{3} D_{1}^{\frac{2m-3}{2m-2}}D_{m}^{\frac{4m-3}{2m-2}}\nonumber\\
&\leq& c\,L^{3}\varpi_{0}^{3}C_{m}^{\frac{4m-3}{2m-2}} D_{1}^{3}\,.
\eeq
Thus $\xi_{m,4}=3$ and there is no depletion of nonlinearity in the upper bound $D_{m} \leq C_{m} D_{1}$. Moreover, when the scaling
argument in \S\ref{amchoice} is repeated, this too shows no depletion. To test whether the dissipation term in (\ref{s5}) is changed by
the use of the lower bound $D_{1}^{A_{m,\lambda}} < D_{m}$ we consider first (\ref{A1})
\begin{equation}\label{sec3a}
D_{m}^{2}\left(\frac{D_{m}}{D_{1}}\right)^{\eta_{m}} > D_{1}^{A_{m,\lambda}(2+\eta_{m}) - \eta_{m}} \equiv
D_{1}^{\Delta_{m,\lambda}}\,,
\end{equation}
which improves the lower bound of unity in (\ref{A1}) and thereby increases the dissipation. In fact
\begin{equation}\label{Deltalam}
\Delta_{m,\lambda} = \frac{2}{3}(\lambda - 1)\,.
\end{equation}
Let us assume that initial data is placed in regime II at a time $t_{0}$\,: then dividing (\ref{s5}) by
$D_{m}^{3}$ we find
\begin{equation}\label{sec4}
\shalf \frac{d~}{dt}D_{m}^{-2} \geq D_{m}^{-2}\left\{\varpi_{1,m}D_{1}^{\Delta_{m,\lambda}}\right\} - \varpi_{2,m}\,,
\end{equation}
where, for convenience, we have taken the unforced case \cite{JDGJMP12}. An integration over $[t_{0},\,t]$ gives
\begin{equation}\label{sec5}
e^{-(t-t_{0})g(t)}D_{m}^{-2}(t) \geq D_{m}^{-2}(t_{0}) -
2\varpi_{2,m}\int_{t_{0}}^{t}e^{-(\tau-t_{0})g(\tau)}\,d\tau
\end{equation}
where
\begin{equation}\label{gdef1}
g(t) = \frac{ 2 \varpi_{1,m}}{t-t_{0}}\int_{t_{0}}^{t}D_{1}^{\Delta_{m,\lambda}}\,d\tau\,.
\end{equation}
The main question here is whether there exists a sufficiently large \textit{lower} bound on the time average $g(t)$ to prove that the right
hand side of (\ref{sec5}) never develops a zero for some wide range of initial data? The problem is that the size of $\int_{t_{0}}^{t}
D_{1}^{\Delta_{m,\lambda}}d\tau$ over \textit{very short intervals} $[t_{0},\,t]$ is indeterminate. This lower bound would have to be
large enough \textit{on arbitrarily small intervals} for the negative integral of the exponential in (\ref{sec5}) to be always smaller than
$D_{m}^{-2}(t_{0})$ to prevent a zero forming on the right-hand side.
\par\smallski
Finally, regime III is easily dealt with because the condition $C_{m} D_{1} < D_{m}$ allows us to drop two of the three terms in
the (\ref{s5}) leaving us with $\dot{D}_{m} \leq 0$ in the unforced case, thus implying decay from initial data. In the forced case
$\dot{D}_{m} \leq \varpi_{3,m}Gr D_{m}^{1-1/\alpha_{m}}$ and so it follows that any $D_{m}$ that satisfies this is bounded for
all time as in
\begin{equation}\label{sec7}
D_{m} \leq \left[D_{m}^{\alpha_{m}^{-1}}(t_{0}) + \alpha_{m}^{-1}\varpi_{3,m}Gr\left(t - t_{0}\right)\right]^{\alpha_{m}}
\end{equation}
\section{\sf\large\textbf{Energy spectra and typical length scales in regimes I \& II}}\label{spectra}
Some ideas are explained in this section on how information might be extracted from the analysis on the properties of an
energy spectrum $\mathcal{E}(k)$ corresponding to regimes I and II. Doering and Gibbon \cite{DG02} have shown how
to associate bounds of time averages with the moments of this spectrum by following some ideas in \cite{Frisch95,SF75}.
It is these arguments we shall summarize first.
\par\smallski
In the standard manner, we define
\begin{equation}\label{Hndef}
H_{n}(t) = \int_{\mathcal{V}} |\nabla^{n}\mbox{\boldmath$u$}|^{2}\,dV\qquad\mbox{with}\qquad H_{0} = \int_{\mathcal{V}} |\mbox{\boldmath$u$}|^{2}\,dV\,,
\end{equation}
where the label $n$ refers to derivatives. Then it was shown in \cite{DG02} that to take proper account of the forcing
these require an additive adjustment such that
\begin{equation}\label{Fndef}
F_{n} = H_{n} + \tau^{2}\|\nabla^{n}\mathbi{f}\|_{2}^{2}\,,
\end{equation}
where $\tau^{-1} \sim \varpi_{0}Gr^{\shalf+ \varepsilon}$ for any $\varepsilon > 0$. This formalism now allows us to
define the set of `wave-numbers' $\kappa_{n,0}$ and $\kappa_{n,1}$ such that
\begin{equation}\label{kappadef}
\kappa_{n,0}^{2n} = F_{n}/F_{0}\,,\qquad\qquad \kappa_{n,1}^{2(n-1)} = F_{n}/F_{1}\,.
\end{equation}
Using the fact that
\begin{equation}\label{H1}
\shalf\dot{H}_{1} \leq -\nu H_{2}+ \int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}dV + \mbox{forcing}\,,
\end{equation}
which is just another way of expressing (\ref{b2}), we can re-visit the inequality in (\ref{b8a}) to estimate the integral in (\ref{H1})
with the application of the depletion of regime I
\begin{equation}\label{H2a}
\int_{\mathcal{V}} |\nabla\mbox{\boldmath$u$}||\mbox{\boldmath$\omega$}|^{2}dV \leq \varpi_{0}\left(L^{3}\varpi_{0}^{2}\right)^{1-\xi_{m,\lambda}}H_{1}^{\xi_{m,\lambda}}
= \varpi_{0}H_{1}D_{1}^{\xi_{m,\lambda}-1}\,,
\end{equation}
which, again, is just another expression of (\ref{b8b}). The bounds $1 \leq\lambda < 2$ mean that
\begin{equation}\label{ximbd}
3/2 \leq \xi_{m,\lambda} < 2\,,
\end{equation}
and so
\begin{equation}\label{H2b}
\shalf\dot{H}_{1} \leq \varpi_{0}\left\{-L^{2} H_{2} + H_{1}D_{1}^{\xi_{m,\lambda}-1}\right\} + \mbox{forcing}\,,
\end{equation}
which, when the $H_{n}$ are adjusted to the $F_{n}$ defined in (\ref{Fndef}) as in \cite{DG02}, becomes
\begin{equation}\label{H3}
\shalf\dot{F}_{1} \leq \varpi_{0}\left\{-L^{2} F_{2} + F_{1}D_{1}^{\xi_{m,\lambda}-1}\right\} + c_{n}\varpi_{0}Gr\,F_{1}\,.
\end{equation}
Dividing (\ref{H3}) by $F_{1}$ and time averaging, we get \cite{DG02}
\beq{kapest1}
L^{2}\left<\kappa_{2,1}^{2}\right>_{T} \leq \left<D_{1}\right>_{T}^{\xi_{m,\lambda}-1}
\leq c\,Re^{3\left(\xi_{m,\lambda}-1\right)}\,.
\eeq
Moreover, we can also write
\begin{equation}\label{kapest2}
\left<\kappa_{2,0}\right>_{T} \leq \left<\kappa_{2,1}\kappa_{1,0}\right>_{T}^{1/2}
\leq \left<\kappa_{2,1}^{2}\right>_{T}^{1/4}\left<\kappa_{1,0}^{2}\right>_{T}^{1/4}\,.
\end{equation}
In \cite{DG02} it was shown that Leray's energy inequality leads to an estimate for $L^{2}\left<\kappa_{1,0}^{2}\right>_{T}
\leq Re^{1+\varepsilon}$, although from now on we ignore the infinitesimal $\varepsilon >0$. We combine this with (\ref{kapest2})
to show that\footnote{\sf To find a good estimate for $\left<\kappa_{n,0}\right>_{T}$ for $n > 2$ using the depletion is a difficult task.
The estimate for this, found in \cite{DG02} and quoted in (\ref{app8}), is valid in regime II where no depletion result has been used.}
\begin{equation}\label{kapest3a}
\left<\kappa_{2,0}\right>_{T} \leq c\, Re^{\sigma_{m,\lambda}} + O\left(Gr^{1/4}\right)\,,
\end{equation}
where
\begin{equation}\label{kapest3b}
\sigma_{m,\lambda} = \frac{3(\chi_{m,\lambda}-1) + 2(m-1) }{8(m-1)} = (3\lambda + 2)/8\,.
\end{equation}
\par\smallski
To interpret this estimate physically in terms of statistical
turbulence theory (restricting attention to forcing at the longest wavelength $\ell = L$), suppose that $Gr$ is high enough and the resulting
flow is turbulent, ergodic and isotropic enough in the limit $T\to\infty$ that the wave-numbers $\left<\kappa_{n,0}\right>_{T}$
may be identified with the moments of the energy spectrum $\mathcal{E}(k)$ according to
\begin{equation}\label{app1}
\left<\kappa_{n,0}\right>_{T} := \left(\frac{\int_{L^{-1}}^{\infty}k^{2n}\mathcal{E}(k)\,dk}
{\int_{L^{-1}}^{\infty}\mathcal{E}(k)\,dk}\right)^{1/2n}\,.
\end{equation}
The {\em a priori} constraints on $\mathcal{E}(k)$ are that the velocity $U$ and energy dissipation rate $\epsilon$ obey
\begin{equation}\label{app2}
U^{2} = \int_{L^{-1}}^{\infty}\mathcal{E}(k)\,dk
\hspace{2cm}
\epsilon = \int_{L^{-1}}^{\infty}\nu k^{2}\mathcal{E}(k)\,dk\,.
\end{equation}
Suppose also that $\mathcal{E}(k)$ displays an ``inertial range'' in the sense that it scales with a power of $k$ up to an effective
cut-off wavenumber $k_{c}$. For simplicity, let us write
\begin{equation}\label{app3}
\mathcal{E}(k) = \left\{
\begin{array}{cr}
A\,k^{-q}, & \ \ \ L^{-1}\leq k \leq k_{c}\,,\\
0, & k > k_{c}\,,
\end{array}
\right.
\end{equation}
We also assume that $k_{c}$ diverges as $\nu\to 0$, while $U^{2}$ and $\epsilon$ remain finite, and that $A$
depends only upon the energy flux $\epsilon$ and the outer length scale $\ell = L$. Then we have the asymptotic relations
\begin{equation}\label{app4}
\epsilon \sim \frac{U^3}{L}\hspace{1cm}\mbox{and}\hspace{1cm}
L k_{c} \sim \left(\frac{\epsilon}{\nu^3}\right)^{\frac{1}{9-3q}}
L^{\frac{4}{9-3q}} \sim Re^{\frac{1}{3-q}}\,.
\end{equation}
Then the moments of the spectrum $\left<\kappa_{n,0}\right>_{T}$ satisfy
\begin{equation}\label{app5}
L\left<\kappa_{n,0}\right>_{T}\sim (L\,k_{c})^{1-\frac{q-1}{2n}}
\sim Re^{\frac{1}{3-q} - \frac{1}{2n}\left(\frac{q-1}{3-q}\right)}\,.
\end{equation}
Now let us compare this scaling result with the estimate in (\ref{kapest3a}) for $n=2$ with $q= q_{m,\lambda}$\,:
this correspondence tells us that
\begin{equation}\label{app6}
q_{m,\lambda} = \frac{12\sigma_{m,\lambda} -5}{4\sigma_{m,\lambda}-1}\,,\qquad\mbox{with}\qquad
q_{m,\lambda} = 3 - \frac{4}{3\lambda}\,.
\end{equation}
In fact, for regime I, $q_{m,\lambda}$ lies between
\begin{equation}\label{qran1}
5/3 \leq q_{m,\lambda} < 7/3\,.
\end{equation}
The 5/3 at the lower end is the conventional Kolmogorov result which rises to just under 7/3. The cut-off of the
inertial range as (\ref{app3}) is given by
\begin{equation}\label{qran2}
Lk_{c} \sim Re^{1/(3-q)}\qquad\mbox{so}\qquad Lk_{c,\lambda} \sim Re^{3\lambda/4}\,.
\end{equation}
A resolution length is inbuilt into this formalism\,: the estimate for $L\left<\kappa_{2,0}\right>_{T}$, with an exponent of
$\sigma_{m}$, can be interpreted as an average length scale. Thus, the first $L\left<\kappa_{1,0}\right>_{T}$ is followed by
an estimate for $L\left<\kappa_{2,0}\right>_{T}$ at $\chi_{m,\lambda} = m\lambda +1-\lambda$\,:
\begin{equation}\label{app7}
L\left<\kappa_{1,0}\right>_{T} \leq Re^{1/2}\,,\qquad\qquad
L\left<\kappa_{2,0,\lambda}\right>_{T} \leq Re^{\sigma_{m,\lambda}}\,,
\end{equation}
where $\sigma_{m,\lambda}$ is defined as in (\ref{kapest3b}). This is roughly consistent with scaling arguments
found in other parts of the literature \cite{YS2005,DYS2008}.
\par\smallski
In regime II we are forced to revert to the weak solution results in \cite{DG02} where it was shown that for $n \geq 2$,
\begin{equation}\label{app8}
\left<\kappa_{n,0}\right>_{T} \leq c\, Re^{3 - \frac{5}{2n}}\,.
\end{equation}
For $n=2$ this means $\sigma_{m,\lambda} = 7/4$ and thus $q_{m,\lambda}=8/3$. Table 3 summarizes the results for
both regimes I and II.
\begin{table}
\bc
\begin{tabular}{||c||c|c|c||}\hline
$\lambda$ & $1$ &$2$ & $4 $\\\hline
$\sigma_{m,\lambda}=(3\lambda +2)/8$ & $5/8$ & $1$ & 7/4\\\hline
$q_{m,\lambda}=3-4/3\lambda$ & $5/3$ & 7/3 & 8/3\\\hline
$Lk_{c} \leq Re^{3\lambda/4}$ & $Re^{3/4}$ & $Re^{3/2}$ & $Re^{3}$\\\hline
\end{tabular}
\ec
\vspace{-5mm}
\caption{\sf\scriptsize The entries in the second and third columns are the lower and upper bounds of $\sigma_{m,\lambda},~q_{m,\lambda}$
and $Lk_{c}$ corresponding to the two concave curves in Fig. 1. The fourth column lists values of these at $\lambda =4$, which is near the
extreme end of regime II.}
\end{table}
\par\smallski
Interestingly, Sulem and Frisch \cite{SF75} showed that a $k^{-8/3}$ energy spectrum is the borderline steepness
capable of sustaining an energy cascade. This spectrum corresponds to the extreme limit, where the energy dissipation
is concentrated on sets of dimension zero (points) in space \cite{Man75,FSN78}. It provides some physical setting
in which to interpret the result of Caffarelli, Kohn \& Nirenberg that the space-time dimension of the Navier-Stokes
singular set is unity \cite{CKN82}.
\section{\sf\large\textbf{Conclusion}}\label{con}
Three regimes have been identified based on the size of the $D_{m}$ for $m\geq 2$ relative to that of $D_{1}$. Regime I has
been shown to have a sufficiently depleted nonlinearity that an absorbing ball exists for $D_{1}$. The consequence of this is
that a global attractor exists, provided solutions remain in regime I. A diagrammatic description of the relation between the
three regimes is given below\,:
\beq{pic1}\nonumber
\underbrace{........~\mbox{Regime~I}~.......}_{D_{m}\leq D_{1}^{A_{m,\lambda}} - \mbox{\small regular}}~D_{1}^{A_{m,\lambda}}~
\underbrace{...............~\mbox{Regime~II}~.................}_{D_{1}^{A_{m,\lambda}} \leq D_{m} \leq C_{m} D_{1} - \mbox{\small weak solutions}}~
C_{m}D_{1}~\underbrace{...........~\mbox{Regime~III}~..........}_{C_{m}D_{1} < D_{m} - \mbox{\small regular}}
\eeq
Fig. 1 in \S\ref{introduction} depicts these regions in the $D_{m}-D_{1}$ plane. Specifically, the region between $\lambda=1$ and
$\lambda = 2$ is the region where solutions are regular. The dotted curves within this show the approximate region (not the exact
trajectories) where the computations of \S\ref{num} lie.
\par\smallski
These results also prompt the following set of questions.
\par\smallski
The first question is why should the Navier-Stokes equations choose to operate in regime I, as observed? While the numerical experiments
in \cite{DGGKPV13} have shown no evidence of a transition from regime I to II, nevertheless, such a transition cannot be discounted for
different sets of initial conditions or higher Reynolds numbers. This raises the question whether solutions with initial conditions lying in
regime I remain there for all time?
If a transition does occur, how might it come about? Are regimes II and III physical in the sense that while mathematically allowable,
do they represent recognizable turbulent states? Regimes I and II appear to be consistent with the two branches discovered by Lu and Doering
\cite{LD08} in their maximization of the rate of enstrophy production. The $\xi_{m,\lambda}=1+\shalf\lambda$ result in regime I takes the
value of 1.75 at $\lambda =1.5$\,: the value at the lower branch in \cite{LD08} is $\xi_{m,\lambda} = 1.78$. The value $\xi_{m,4} =3$ for
regime 2 is also consistent with $\xi_{m,\lambda}=2.997$ on the upper branch in \cite{LD08}. In a further paper, Schumacher, Eckhardt and
Doering \cite{SED10} found numerically that $\xi_{m,\lambda}=3/2$. This, however, was derived from an analysis of local concentrations of
vorticity, not the full volume calculations in this paper. Nevertheless, it is worth pointing out that bounds on $\xi_{m,\lambda}$ are
\begin{equation}\label{ximlam}
3/2 \leq \xi_{m,\lambda} < 2\,,\qquad\qquad 1\leq \lambda < 2\,,
\end{equation}
so the result in \cite{SED10} lies exactly at the extreme lower bound where the energy spectrum is $q_{m,1}=5/3$.
\par\smallski
Secondly, what of initial conditions that are the reverse of the observed ordering\,: that is, initial conditions that are in an ascending scale
and thus satisfy $D_{m} < D_{m+1}$? A recent numerical experiment by Kerr \cite{RMK2013b} on the $3D$ Euler equations found that,
in the late stage, the $D_{m}$ did indeed reverse in order to this ascending scale $D_{m} < D_{m+1}$. Then in a further experiment Kerr
\cite{Kerr2014priv} took this reversed state as initial conditions for the Navier-Stokes equations to discover that the ordering immediately
switched back again to the descending scale $D_{m+1} < D_{m}$.
\par\smallski
Thirdly, the magnitude of the vortex stretching term is locally dependent on the angle between $\mbox{\boldmath$\omega$}$ and eigenvectors of the strain
matrix. Overall, this is averaged within the norms buried within the $D_{m}$. Is it possible that a more direct connection could be made in
the analysis between these results and the work of Constantin and Fefferman and others on the direction of vorticity [55-61,41]?
\par\smallski
Fourthly, there is a growing body of work on so-called Navier-Stokes-$\alpha$ models, which includes the Leray-$\alpha$, LANS--$\alpha$,
Clark-$\alpha$ and Bardina models [62-67],
plus the Navier-Stokes-Voight model \cite{Voight08}. All of these models have better regularity properties,
in differing degrees, than the original Navier-Stokes equations themselves. A comparison between these and the results of regime I might
be a useful exercise.
\par\smallski
Finally, the depletion of nonlinearity in regime I is sufficiently strong to suggest that vorticity may be accumulating on low-dimensional sets.
A generation of graphics has suggested that this is indeed the case\,: vortex sheets rolling up into tubes is typically the situation as a turbulent
Navier-Stokes flow matures beyond intermediate times. An analytical proof of this poses formidable technical problems as no proof exists for
the Divergence theorem nor the Sobolev inequalities on a fractal domain
with evolving fractal boundary conditions. Given these hurdles all that can be done at present is to re-estimate \textit{formally}
(\ref{D1est1a}) in $d$-dimensions using dimensional analysis. This suggests that the formal equivalent of (\ref{D1est1a}) is
\begin{equation}\label{D1est2}
\shalf \varpi_{0}^{-1}\dot{D}_{1} \leq - \left(\frac{4-d}{4}\right)\frac{D_{1}^{2}}{E} + c_{d}D_{1}^{\frac{6-d}{4-d}}\,.
\end{equation}
Note that when $d=3$ this reduces to (\ref{D1est1a}). With $\xi_{m,\lambda} = \frac{6-d}{4-d}$ it is easy to calculate the value of $d$
corresponding to $\xi_{m,\lambda}=1+ \shalf\lambda$ which is
\begin{equation}\label{dlam}
d_{\lambda} =4\left(1 - \lambda^{-1}\right)\,.
\end{equation}
For instance, this takes the value of $d_{1.15} \approx 0.52$ when $\lambda = 1.15$ to $d_{4/3} = 1$ when $\lambda = 4/3$.
This suggests that the low-dimensional set corresponding to a nonlinearity of $D_{1}^{\xi_{m,\lambda}}$ is one which may
run from being a set of points to tube-like vortical structures [26-32,43].
\par\vspace{3mm}\noindent
\textbf{Acknowledgments\,:} DAD acknowledges the computing resources provided by the NSF-supported XSEDE and DOE INCITE
programs under whose auspices some of these calculations were performed. DV acknowledges the support of the F\'ed\'eration
Wolfgang Doeblin and, together, RP and DV acknowledge the support of the ``Indo-French Center for Applied Mathematics'', UMI
IFCAM -- Bangalore\,; AG and RP thank DST, CSIR and UGC (India) and the SERC (IISC) for computational resources. AG also thanks
the European Research Council for support under the EU's 7th Framework Programme (FP7/2007-2013)/ERC grant agreement number
297004.
\vspace{-3mm}
|
1,314,259,995,822 | arxiv | \section{Approach}
\label{sec:approach}
We begin by describing the $\mathtt{SegNet}$\xspace module in our approach, and
then explain how it is used to score semantic segmentation proposals.
\subsection{$\mathtt{SegNet}$\xspace: Predicting Coarse Segmentations}
\begin{comment}
\textbf{Using Segmentation Structure}
As previously mentioned, segmentation can be reduced
to patch classification where the model is trained to minimize pixel-wise error
averaged over all pixels. The main difference between this approach,
classification, and detection is how the CNN classifier is trained.
The task is more fine grained, so dense ground truth labels
are available, meaning that each label has a different
meaning than it would for other tasks.
But this description leaves something out.
In the case of image classification and object detection there isn't
nearly as much structure in the output as there is for segmentation.
Classification has a single output for one image, so there is nothing to relate.
Detection might have a couple outputs which might relate to each other, but
might not. However, pixels which are close together in segmentations
are highly related due to the nature of \emph{things} and \emph{stuff}.
To take this difference into account we can think of segmentation in CNNs in
a slightly different way. Instead of taking patches as inputs and producing
labels as outputs, CNNs can take (RGB) images as input and produce (label)
images as outputs. This is \textit{exactly} how patch classification methods
work \cite{ciresan2012deep, farabet2013pami, pinheiro2014rcnn}, so it's only
a different way of thinking at first, but the thought reveals that the
objective function can take into account output structure
across patches.
\end{comment}
\textbf{Architecture.}
As shown in \figref{fig:teaser}, our architecture contains 8 convolutional
layers and each is fed through rectified linear non-linearities except the last,
which is fed through a pixel-wise $C$-way softmax to label an image with
$C$ classes.
There are no fully connected layers.
\textbf{Comparison with a classification net.}
The first 5 layers in $\mathtt{SegNet}$\xspace are convolutional layers ($\mathtt{conv}$\xspace),
with 96, 256, 384, 384, then 256 filters,
Max pooling ($\mathtt{pool}$\xspace) and local response normalization follow the first
two layers, similar to AlexNet.
\footnote{More specifically, CaffeNet \cite{jia13caffe}, which is AlexNet,
except Local Response Normalization and Max
Pooling layers are swaped. Differences are summarized in
\url{https://github.com/BVLC/caffe/issues/296}.}
The 5th $\mathtt{conv}$\xspace layer produces 256 feature maps of size $13 \times 13$, but
we do not pool after this layer,
and
this is where the architectures diverge. In standard classification CNN,
$\mathtt{conv}$\xspace layers are typically followed by fully-connected ($\mathtt{fc}$\xspace)
layers.
We do not have $\mathtt{fc}$\xspace layers.
Instead, we add two more $\mathtt{conv}$\xspace layers with 128 feature maps
each and a third $\mathtt{conv}$\xspace layer with as many feature maps as the number of classes
(including the `background' class).
These $C$ final feature maps can be thought of as `semantic feature maps' since
they give pixel-wise probabilities for each class and those are interpretable.
Crucially, we initialize the first 5 $\mathtt{conv}$\xspace layers with weights from CaffeNet
trained on ImageNet, thus utilizing the large classification corpus.
During training, we keep these $\mathtt{conv}$\xspace-layer weights fixed
and only learn the weights of the newly added layers.
We apply dropout \cite{hinton2012dropout} before each of the added
feature maps during training. By feeding each pixel at the output
through a softmax activation function (normalized over classes)
we can output `semantic feature maps' which collectively give
a distribution over classes at each output pixel in the 13 $\times$ 13 grid.
An example which shows 21 feature maps for the 21 PASCAL classes is shown in
\figref{fig:coarse}.
Since $\mathtt{SegNet}$\xspace contains no fully-connected layers,
the only weights are the filters.
This is greatly beneficial since a majority of the parameters in standard classification nets
lie in the fully-connected layers.
Indeed, $\mathtt{SegNet}$\xspace contains less than 10\% of CaffeNet parameters.
Interestingly, in previous work, Zeiler and Fergus \cite{zeiler2013convnet} have observed that
weights in convolutional filters provide more information per weight because
removing fully connected layers (containing most parameters)
does not lead to a proportional decline in classification performance.
Since $\mathtt{SegNet}$\xspace predictions are coarse (low-resolution),
we need to down-sample high-resolution segmentation ground truths to derive
the annotation for training $\mathtt{SegNet}$\xspace parameters. Each pixel in a down-sampled
segmentation corresponds to a patch in the high-res version, so
we compute distributions over classes in this patch, yielding a soft segmentation
ground-truth, similar to our predictions.
Our baseline loss for training $\mathtt{SegNet}$\xspace is the standard cross-entropy computed
between a pixel's predicted class distribution and the ground truth's distribution.
Notice that this loss function is ``decomposable'' over pixels -- it
treats segmentation as independent classification problems at each pixel.
\subsection{Optimizing a Segmentation-Specific Loss}
Recall that the standard evaluation criteria used in segmentation tasks is
Intersection-over-Union(IOU) averaged across classes.
Although imperfect (in the sense that it does not reward boundary alignment),
it does captures some notions of a good segmentation better than decomposable metrics such as
Hamming.
Unfortunately, this metric does not decompose over pixels or even images.
In fact, it is a \emph{corpus-level} metric, and can only be computed for
an entire dataset, not individual images.
Fortunately, we only need a loss's gradient to train a CNN, so we can directly
optimize such a metric. The supplementary material shows our our derivation
of IOU's gradient.
Before going further, it's worth taking a detailed look at how
this loss behaves when optimized via gradient descent.
Consider high-resolution predictions and ground truth.
Let $TP_k$ denote the number of true positive for class $k$ across the dataset,
\ie the number of pixels across all images that are annotated and predicted as class $k$.
Analogously, let $FP_k$ denote the false positives,
$FN_k$ the false negatives, and $GT_k$ the sum of ground truth
pixels for class $k$.
Then, the Jaccard Index for class $k$ can be defined as:
\begin{align}
IOU_k = \frac{TP_k}{TP_k+FP_k+FN_k},
\end{align}
which is averaged across categories to yield the final metric:
\begin{align}
IOU &= \frac{1}{K}\sum_{k=1}^K IOU_k
\end{align}
It might seem intuitive to aim to maximize IOU, but in our experiments we
have found the optimization to be easier when we minimized the Union-over-Intersection (UOI) instead. In our preliminary experiments with random initialisation, the IOU objective function always led to an all-background prediction whereas UOI minimization leads to a much better solution.
We explain why this might be the case.
\input{figs/fig4}
\textbf{Optimizing IOU.}
First, let us use the fact that $TP_k + FN_k = GT_k$ to
rewrite the gain function as:
\begin{align}
IOU_k &= \frac{GT_k-FN_k}{GT_k+FP_k}
\end{align}
For the sake of building an understanding, consider the gradients of the
$IOU_k$ gain function
with respect to the two kinds of mistakes ($FP_k$ and $FN_k$):
\begin{align}
\frac{\partial (IOU_k)}{\partial (FP_k)} &= \frac{-(GT_k-FN_k)}{(GT_k+FP_k)^2}\\
\frac{\partial (IOU_k)}{\partial (FN_k)} &= \frac{-1}{GT_k+FP_k}
\end{align}
Notice that there are two things non-ideal about these gradients.
First, as the number of mistakes ($FP_k$ or $FN_k$) increase the gradients \emph{diminish}.
Second, as the number of mistakes reduce, the gradients \emph{increase}.
Such a behavior hampers convergence of first-order methods.
Note that we only analyzed the effect of $FP_k, FN_k$ on $IOU_k$, and but not on
$IOU_{k'}$ for other categories $k'$.
Each pixel can be assigned only one
category, and thus the mistakes $\{FP_k, FN_k\}_1^K$ are not independent of each other.
Thus, we need to also analyze the other terms $\del(IOU_{k'})/\del(FP_k)$.
\figref{fig:sim} shows a simulation where we computed the
behavior of IOU as a function of increasing $FP_k$ and $FN_k$.
Our illustration of the gradients of $IOU_k$ provides an intuition for the behavior of the IOU function.
\textbf{Optimizing UOI.}
Now we show that Union-over-Intersection (UOI) is a smoother optimization
function based on the behavior of its gradient and that it shares a
natural relation with IOU. In a manner similar to IOU, the $UOI$ function can be written as:
\begin{align}
UOI &= \frac{1}{K}\sum_{k=1}^{K} UOI_k\\
&= \frac{1}{K}\sum_{k=1}^{K} \frac{TP_k + FP_k + FN_k}{TP_k}\\
&= \frac{1}{K}\sum_{k=1}^{K} \frac{GT_k+FP_k}{GT_k-FN_k}
\end{align}
Consider the gradient of $UOI_k$ \wrt the number of mistakes $FP_k$ and $FN_k$:
\begin{align}
\frac{\partial UOI_k}{\partial FP_k} &= \frac{1}{GT_k - FN_k}\\
\frac{\partial UOI_k}{\partial FN_k} &= \frac{GT_k+FP_k}{(GT_k - FN_k)^2}
\end{align}
We can see that $UOI_k$ has more desirable properties compared to $IOU_k$,
as illustrated in \figref{fig:sim}.
When the number of mistakes are large, the gradient is large.
As the number of mistakes decrease, the gradient decreases as well.
\textbf{Does UOI optimise IOU?}
Since we have now established that the UOI function has more desirable traits,
we should understand whether the two objectives are related. Does minimization
of UOI lead to the maximization of IOU?
We show that IOU can be \emph{lower-bounded} by a decreasing function
of UOI:
\begin{equation}
\sum_k IOU_k \ge f\left(\sum_k UOI_k\right).
\end{equation}
Since $f(x)$ is a decreasing function in $x$,
we can see that decreasing UOI leads to
increasing the lower-bound on IOU.
Importantly, we can show that the bound is \emph{tight} --
that the maximum possible value of $\sum_k IOU_k$ (=K) is achievable,
although it requires that $TP_k \ne 0$ for all classes $k$.
In our coarse segmentation setting this can always be achieved by not allowing any
soft outputs to be 0.
\begin{proof}
Assume $0 < IOU_i$.
Now consider the decreasing function $f(x) = \frac{1}{x}$.
We need to show that:
\begin{enumerate}
\item $\sum_k IOU_k \ge \frac{1}{\sum_k UOI_k}$
\item There exists a value of UOI for which IOU=1
\end{enumerate}
Let $x_i = IOU_i$. Notice that $0 < x_i \le 1$. This means that:
\begin{align}
\sum_{k=1}^K x_k \ge x_i
\quad
\Rightarrow
\quad
\frac{1}{x_i} \ge \frac{1}{\sum_k x_k}
\end{align}
Now, if we notice
\begin{equation}
\frac{1}{x_i} + \sum_{k \in \{1, \ldots, K\} \setminus \{i\}} \frac{1}{x_k} = \sum_{k \in \{1, \ldots, K\}} \frac{1}{x_k}
\end{equation}
then we can see
\begin{align}
\sum_k\frac{1}{x_k} \ge \frac{1}{\sum_k x_k}
\quad
\Rightarrow
\quad
\sum_k{x_k} \ge \frac{1}{\sum_k\frac{1}{x_k}}\\
\Rightarrow \sum_k IOU_k \ge \frac{1}{\sum_k UOI_k}
\end{align}
Hence, the lower-bound of IOU has been shown as a decreasing function of UOI.
If $UOI_i=1 \,\,\, \forall i$, clearly $\sum_{k=1}^K IOU_k=K$ and $IOU=1$. This implies that UOI acts as
a good surrogate objective for optimizing IOU.
\end{proof}
\subsection{Semantic Segmentation Proposals}
Our second module is a pipeline which uses a graphical model to generate semantic segmentation proposals.
We directly use the O$_{2}$P+DivMBest\xspace approach of \cite{yadollahpour2013rerank}.
They use the O$_{2}$P model~\cite{carreira_eccv12} which
generates approximately 150 CPMC segments \cite{carreira_cvpr10} for each image,
then scores them using Support Vector Regressors trained over
second-order pooled features \cite{carreira_eccv12}. These segmentations are
greedily pasted to form a semantic segmentation. Finally, DivMBest\xspace is used to generated
multiple diverse semantic segmentation proposals.
\cite{yadollahpour2013rerank} showed that one of these proposals tends to be significantly
more accurate that the 1-best semantic segmentation. Let the \texttt{oracle}\xspace segmentation be the
most accurate segmentation in the set.
The \texttt{oracle}\xspace accuracy at just 10 proposals is $15$\%-points\xspace higher than the 1-best segmentation.
\section{Post-Processing: Coarse to Full}
To evaluate segmentations from $\mathtt{SegNet}$\xspace we need to post-process these coarse
segmentations to produce ``full-sized'' semantic segmentation.
We'll refer to the up-sampled segmentation as $\hat{P}_{jk}$ where each index $j$ corresponds to a
pixel in the original image and $k$ indexes classes.
Let $\hat{p}_{ik}$ denote the probability of class $k$ predicted at coarse pixel $i$ (which corresponds to
a patch in the full resolution image)
by the last layer of a $\mathtt{SegNet}$\xspace.
We propose four post-processing strategies that are arranged by increasing sophistication;
the first two methods are simple heuristics while the last two methods try to pick good proposals.
\textbf{(Naive)}
A simple way to do this, which we'll call
naive upsampling just copies the argmax of a coarse pixel into all
pixels in the patch it came from:
\begin{equation} \label{eq:dump_ups}
\hat{P}_{jk} = \argmax_{k} \hat{p}_{ik}
\end{equation}
\textbf{(Superpixel)}
The next step is to try and respect object boundaries using small superpixels (using
SLIC \cite{achanta2012slic}), which are labeled by coarse segmentations.
For each superpixel, we aggregate distributions over categories from
patches overlapping with this superpixel. Each distribution is weighted by the percentage of
pixels in the superpixel that are also in the patch.
This gives a distribution for superpixels. We take the $\argmax$ for each patch distribution.
Neither this smart upsampling, nor the previous \naive upsampling
are competitive.
\textbf{($\mathtt{SegNet}$\xspace)}
Next, we use $\mathtt{SegNet}$\xspace outputs to pick proposals.
We down-sample each DivMBest\xspace segmentation to
$13 \times 13$ soft segmentations, similar to how ground truth was
downsampled for training.
Call $\hat{q}_{ik}^m$ the probability of the $m^{th}$ DivMBest downsampled
segmentation at patch $i$.
We score the consistency of $\hat{p}$ and $\hat{q}^m$ with the
symmetric-KL augmented by a background penalty term:
\begin{equation}
\small
S(m) = \sum_{i} \left[ D_{KL}(\hat{p}_i || \hat{q}_i^m) + D_{KL}(\hat{q}_i^m || \hat{p}_i) + 0.02 \hat{q}_{i,0}^m \right].
\end{equation}
where $D_{KL}$ is the Kullback-Leibler divergence and $0.02\hat{q}_{i, 0}^m$ is a
regularizer that penalizes background prediction.
The background penalty comes from observing that background (class 0) is frequently
overpredicted;
adding this term consistently improved validation performance.
\textbf{($\mathtt{SegNet}$\xspace+SVM)}
The is our final approach, which works best, and uses $\mathtt{SegNet}$\xspace segmentations
as a feature, and training a re-ranker\xspace to pick the best proposal from DivMBest\xspace.
Specifically, similar to \cite{yadollahpour2013rerank}, we train a
ranking Support Vector Machine to choose the best proposal
according to a variety of features which describe
proposals. We use both sophisticated hand-engineered features taken from \cite{yadollahpour2013rerank},
and simple features based on $\mathtt{SegNet}$\xspace outputs.
This is similar to R-CNN~\cite{girshick2013convnet} and SDS~\cite{hariharan2014sds}, which each use an SVM
trained on CNN features to evaluate proposals.
\textbf{Segmentation features:}
\begin{enumerate}
\item[]
\textbf{($\mathtt{SegNet}$\xspace)}
As in the previous section, we calculate KL
divergence between proposals and CNN segmentations,
and consider each direction the divergence can be computed separately
(\textbf{2} dimensions).
We also considered expected intersection, expected union,
expected intersection over expected union, and expected union over expected intersection.
Each of these statistics is computed for all PASCAL classes plus background
(\textbf{84} dimensions).
\item[]
\textbf{(CNN Classification)}
We use class-wise scores from an SVM
trained on DeCAF features \cite{donahue2013decaf}
for PASCAL classification (\textbf{20} dimensions).
In addition, we extend image classification to predict not just the existene of
objects, but also whether a category in an object is greather than a certain size or not.
To train an SVM for class $C$ and threshold $t \in [0, 1]$
we set the ground truth label for class $C$ to 0 if the percent of $C$
pixels in an image is below $t$.
The thresholds are chosen per-class by sorting images by percentage of $C$ pixels
then using the $C$-percent of the image at the 20th, 40th, 60th, and 80th percentile.
(\textbf{80} dimensions).
\item[]
\textbf{(DivMBest+ReRank)}
Finally, we use use all features used by \cite{yadollahpour2013rerank} (\textbf{1966} dimensions).
\end{enumerate}
\section{Conclusion and Future Work}
To summarize, we present a two-module approach for semantic segmentation.
Module 1 uses a graphical model to produce
multiple semantic segmentation proposals.
Module 2 uses \textbf{$\mathtt{SegNet}$\xspace}, a novel CNN which
is specifically trained for semantic segmentation task-loss, and is
used to to score then re-rank\xspace these proposals, resulting in a final segmentation.
Our approach achieves $52.5\%$ on the PASCAL 2012 segmentation challenge.
Our experiments with and without proposals
reach findings that are consistent with those observed in previous work --
that methods which use proposals consistently outperform those which don't, even
among methods which use CNNs.
In object detection non-proposals methods (sliding window) still
are efficient enough to be viable for CNNs \cite{sermanet2013overfeat}.
However, for segmentation, proposals are vital.
Perhaps unsurprisingly, it is clear that more work needs to be done with CNNs.
Our results suggest that some information is not captured by our
CNN.
Achieving peak performance in our pipeline requires
hand crafted features from \cite{yadollahpour2013rerank}, though our $\mathtt{SegNet}$\xspace-based
features are significant contributors.
Another path of future experiments involves Microsoft's Common Objects in
Context dataset \cite{coco}, which contains an order of magnitude more
segmentation ground truth than has existed before.
Following AlexNet, it seems reasonable to expect better performance from a larger
net trained on a bigger dataset. Since our net predicts entire segmentations
instead of single labels it could have more to learn from this dataset than
classification nets.
\begin{comment}
TODO: say something about the size of data
even though there weren't many annotations
we were able to leverage CNNs
CNNs are not the be-all end-all of vision
* because hand selected features still help
* this does not mean that the other ranker features should not be themselves learned
* it does mean that they can't be learned with the CNN model + data we currently have
* CNNs are still the best single method we know of
* CNNs do know where things are
* I'm still not sure whether we should use CNNs that know where things are or not
* CNNs aren't that great at localization... perhaps by wiggling the
receptive field for one pixel around they'll work a lot better, but
proposals (MCG or DivMBest) seem to trump this version of their localization
* you need a holistic environment to get the best performance
-> research should focus on figuring out what different methods reason
about so we can ignore some and keep the ones with complementary information
* this could be more than using CNN based features
-> research should focus on how to combine the complementary information
that comes from the left over methods
* this doesn't just mean coming up with a better ranking function
* can we create a framework in which to learn the rest of the features from raw images?
-> this is a great direction to pursue
* we will train the net with COCO, this could do better than SDS because we get to reason about more at once (rather than just boxes)
* note that we're still doing pixel wise classification, so the model isn't that more complex
* they did have many more examples than us, so we could have more potential growth
* We don't think that the difference between the SSVM re-ranker and
the purely CNN based re-ranker can be closed simply by improving
the CNN. Though the difference added by the remaining features is small,
their variety is enough that we suspect the CNN may not be able to learn
about some of the information they provide. A better CNN should improve
performance, but this gap will probably remain.
\end{comment}
\section{Introduction}
\label{sec:intro}
Training deep Convolutional Neural Networks (CNNs) with large amounts of labeled
data has produced impressive results for classification and detection of
objects and attributes~\cite{krizhevsky2012imagenet,
girshick2013convnet, sermanet2013overfeat, zhang2013panda}.
The natural next question to ask is -- can these deep models be generalized
beyond simple prediction spaces (as in multi-way classification)
to complex, structured prediction spaces as in
semantic segmentation, keypoint/pose estimation, and coarse 3D estimation?
There are two main challenges in this generalization:
\begin{compactitem}
\item \textbf{Does vision $=$ ``lots of classification''?}
Most recent applications of CNNs to new tasks
such as detection and segmentation have framed these tasks as
``lots of classification'',
either of scanning window patches~\cite{farabet2013pami, ciresan2012deep, pinheiro2014rcnn,
grangier2009deep, schulz2012cnn} or region
proposals \cite{girshick2013convnet,hariharan2014sds,he2014spatial}.
While these results are encouraging,
such formulations ignore the rich structure in the output space.
In semantic segmentation, the goal is to label each pixel
with an object class. Labels of nearby pixels tend to be correlated,
and independent per-pixel predictions loose this valuable signal.
These intuitions are also reflected in the choice of the evaluation metrics
used by community -- for instance, mean Jaccard Index (or Intersection-over-Union (IOU))
used by PASCAL segmentation, as opposed to the \naive Hamming distance.
\item \textbf{Limited training data.}
Unlike classification, which requires image-level labels, and detection,
which requires bounding boxes, higher-level scene understanding tasks such as
semantic segmentation, or coarse 3D estimation often require
\emph{dense pixel-level} annotations that are time consuming and
expensive to collect. Thus, such datasets are significantly smaller
in scale than classification, despite ongoing valiant efforts~\cite{coco}.
\end{compactitem}
\input{figs/teaser/teaser.tex}
\textbf{Goal.}
At a high level, the goal of this paper is to address the above two
challenges -- to leverage improvements in CNN-based classification for
higher-level vision tasks in a manner that uses the large training corpus
available for classification without ``shoe-horning'' the task at hand into
repeated classification.
\textbf{Overview.}
We present a novel CNN-based approach for semantic segmentation, the task of
labeling each pixel in an image with an object class.
\figref{fig:teaser} illustrates our two-module approach.
Module 1 uses a graphical model to produce
multiple semantic segmentation proposals.
Module 2 uses a novel CNN called \textbf{$\mathtt{SegNet}$\xspace}, which
is used to score and re-rank\xspace these proposals, resulting in the final prediction.
\textbf{Contributions.}
Our primary technical contribution is $\mathtt{SegNet}$\xspace, a novel CNN that
directly outputs a (coarse) semantic segmentation. Importantly, $\mathtt{SegNet}$\xspace is
task-aware, and \emph{specifically trained to optimize the corpus-level PASCAL
IOU loss function}.
To the best of our knowledge, this is the first CNN
specifically designed for semantic segmentation.
While our experiments focus on this one specific application (semantic segmentation),
at a high-level, our approach presents a general recipe for combining
the strengths of graphical models (modeling dependencies) and deep learning (learning rich features)
for a range of applications. The recipe is simple --
use graphical models to generate a small set of proposals and CNNs to score them.
Formulating the problem this way has a number of advantages:
\begin{compactitem}
\item \textbf{Wider receptive field without loss in resolution:}
As CNNs get deeper, each output pixel gets to see a larger patch of input and reason
about more context.
Unfortunately, the output also gets coarser due to the pooling layers.
Thus, practitioners are left with a dilemma -- either build shallow networks that
have limited performance or deeper richer networks that loose localization information.
Our 2-module approach does not face this problem;
$\mathtt{SegNet}$\xspace gets to look at not just a patch or a segment, but the
entire image to make its predictions. The loss in resolution is acceptable
because the $\mathtt{SegNet}$\xspace prediction simply needs to re-rank\xspace holistic proposals, which are full resolution.
\item
\textbf{Leveraging classification corpus while learning output structure:}
The first few layers of $\mathtt{SegNet}$\xspace are warm-started with
with Krizhevsky~\etal's classification network (AlexNet) trained on ImageNet \cite{krizhevsky2012imagenet}.
These weights have learned the expected Gabor-like filters, and are good low-level features
for natural images.
We make the last few layers task-aware by optimizing corpus-level structured loss on PASCAL.
\item \textbf{Graphical models encode knowledge about output structure:}
Re-ranking\xspace proposals produced by graphical models allows us to
reason about segmentation structure in a \emph{second} way --
through the large body of work tying graphical models and structured prediction.
\end{compactitem}
\section{Acknowledgements}
\nocite{jia13caffe}
{
\bibliographystyle{ieee}
\subsection{Does UOI optimise IOU?}
Since we have now established that the UOI function has more desirable traits, we have to understand whether there exists a correlation in between the two objectives defined in [eq(1) and eq(6)]. In other words, we have to show that minimisation of UOI leads to the maximisation of IOU.
We argue that there exists a lower bound on the IOU for a fixed UOI. In other words, if we can show that there exists a decreasing function `$f$' such that,
\begin{equation}
\sum IOU_k \ge f(\sum UOI_k)
\end{equation}
then we can say that decreasing value of UOI leads to increasing value of the lower-bound on IOU. Further, we also have to show that the maximum possible value of IOU(=K) is achievable.
Suppose the decreasing function `$f$' is defined as:
$$f(x)=\frac{1}{x}$$
We have to now show that:
\begin{enumerate}
\item $\sum IOU_k \ge \frac{1}{\sum UOI_k}$
\item There exists a value of UOI for which IOU=K
\end{enumerate}
Assume that $x_k=IOU_k$, we know that
\begin{gather}
0\le x_k \le 1\\
\Rightarrow \frac{1}{x_k} \ge \frac{1}{\sum_k x_k}\\
\Rightarrow \sum_k\frac{1}{x_k} \ge \frac{1}{\sum_k x_k}\\
\Rightarrow \sum_k{x_k} \ge \frac{1}{\sum_k\frac{1}{x_k}}\\
\Rightarrow \sum_k IOU_k \ge \frac{1}{\sum_k UOI_k}
\end{gather}
Hence, the lower-bound of IOU has been shown as a decreasing function of UOI. In the case where each of $UOI_k=1$, clearly, $\sum_{k=1}^K IOU_k=K$. This implies that UOI acts as a good objective function for optimising IOU.
\paragraph{Losses}
We tried two loss functions, the first being cross entropy averaged over each
pixels of of the soft $13 \times 13$ segmentations and the second
being Intersection Over Union. The first is standard, but we'll describe
IOU in more detail.
Let $\hat{p}_{ik}$ denote the probability of class $k$ predicted at pixel $i$
by the last layer of a $\mathtt{SegNet}$\xspace and $p_{ik}$ denote the same thing for ground truth.
Expected intersection is given by
\begin{equation} \label{eq:int}
E[I_k] = \sum_i \hat{p}_{i, k} p_{i, k}^m
\end{equation}
and the expected union is given by
\begin{equation} \label{eq:union}
E[U_k] = \sum_i \hat{p}_{i, k} + p_{i, k}^m - \hat{p}_{i, k} p_{i, k}^m
\end{equation}
Now we can define IOU loss:
\begin{equation} \label{eq:iou}
\mathcal{L}(\bm{\hat{p}}, \bm{p}) = \sum_{k} \frac{E[I_{k}(\bm{\hat{p}}, \bm{p})]}{E[U_{k}(\bm{\hat{p}}, \bm{p})]}
\end{equation}
Theoretically, the idea of IOU aligns with semantic segmentation better than
cross-entropy. Notice that
\begin{enumerate}
\item
Classes are weighted equally instead contributing to the loss
in proportion to their pixel count.
\item
The loss does not break down over pixels.
\item
If equations \ref{eq:int} and \ref{eq:union} are summed over pixels in a
batch instead of pixels in an image then the loss doesn't even break down
over examples. Since the PASCAL metric computes
this corpous wide IOU, it is a closer approximation than optimizing
an IOU which is computed per-image.
When the batch size is the size of the dataset the approximation is exact.
\end{enumerate}
\section{Related Work}
As \figref{fig:teaser} suggests,
our work relates to two themes -- deep learning and proposal-based vision pipelines.
\textbf{Convolutional Neural Networks.}
Image classification and object detection are formulated as patch
classification problems, where the patch is either the whole image or comes from
a set of boxes sampled across scale and aspect ratio.
Segmentation is a natural extension of this view, and a number of recent approaches have
classified uniform patches sampled in a grid
\cite{farabet2013pami, ciresan2012deep, pinheiro2014rcnn, grangier2009deep,
schulz2012cnn}.
The size of the patch in consideration (or ``receptive field'') determines the
amount of context available --
\cite{farabet2013pami} and \cite{schulz2012cnn} use
multi-scale CNNs to increase the receptive field while limiting increase in
model complexity; \cite{schulz2012cnn} simply makes their convolution filters
larger. Most notably, \cite{pinheiro2014rcnn} uses a
recurrent CNN
to gain depth and a larger receptive field,
while limiting the parameters that need be learned.
Our network is deeper, so our receptive field is naturally large.
Interestingly, a number of these approaches find that the structure
in natural images isn't well respected by their CNN predictions.
Thus, the CNN predictions are post-processed using graphical models
\cite{farabet2013pami} to insert structural knowledge back into the pipeline.
To contrast, our proposal re-ranking\xspace step can be thought of as a sophisticated form of
post-processing.
\begin{comment}
Other deviations from this basic approach have to do with postprocessing of
the resulting segmentations, most extensively explored in three stages by
Farabet et al.\ \cite{farabet2013pami}.
From simple to complex, they (i) votes on superpixel labels, (ii) try to enforce
smoothness with a CRF
over superpixels, and (iii) solve an optimal cover
problem which can merge superpixels to produce a better overall labeling.
In \cite{ciresan2012deep} they explore some simpler post-processing methods
to achieve smoothness (e.g. median filter) and in \cite{schulz2012cnn}
they sometimes add a convolutional layer with large filters on top of the output
\todo{how large?}
to reason about cross-pixel dependencies over long ranges.
The proposals described next can be thought of as a more
sophisticated form of post-processing.
\end{comment}
\nocite{mussi2007object}
\nocite{gattaunrolling}
\nocite{sermanet2009multirange}
\textbf{Graphical Models and Proposals.}
Modern approaches for object detection and semantic segmentation increasingly rely
on category independent bounding-box and segment proposals
\cite{carreira_cvpr10, uijlings2013selective, arbelaez2014mcg}.
In both cases, the search space (\#boxes, \#segments) is overwhelmingly large,
and the goal is to reduce the search space to enable expensive processing,
without throwing out good solutions.
\cite{girshick2013convnet} and \cite{hariharan2014sds} achieved state of the art performance on
detection and segmentation respectively
by classifying bounding-box and region proposals using a CNN.
Most proposal methods need to produce on the order of
200-5000 proposals to get sufficiently high recall.
Our approach may be viewed as an instantiation of the same philosophy,
only operating a step ``downstream''.
Specifically, we produce entire image labelings, not category-independent box/segment proposals.
Interestingly, this allows us to use significantly fewer proposals --
on the order of 10-30 per image.
We are motivated by the
observation made in recent work~\cite{yadollahpour2013rerank}
-- \emph{a set of just 10 image labelings has the potential to improve PASCAL
segmentation by $15$\%-points\xspace ($33\%$ relative gain)}.
Using fewer proposals allows even more complex scoring of those proposals
by sophisticated secondary modules, as we do in this work.
In a manner similar to us,
the most successful detection and segmentation methods,
do \emph{not} use their CNNs for localization; rather the CNNs are used to score
proposals~\cite{girshick2013convnet, hariharan2014sds}.
On the other hand, CNNs generate dense features in
~\cite{farabet2013pami,sermanet2013overfeat}, but are outperformed by proposal-based
methods.
\begin{comment}
Barring SDS \cite{hariharan2014sds}, none of these approaches have taken
advantage of the AlexNet architecture \cite{krizhevsky2012imagenet} or a
dataset at the scale of ImageNet dataset \cite{ilsvrc13} to do segmentation.
Proposal based
methods haven't used it to better reason about semantic content and CNN
based segmentation methods haven't adapted their architecture and training set.
\end{comment}
\begin{comment}
\cite{pinheiro2014rcnn} points out that
"The main limitation of scene labeling ap-
proaches based on graphical models is the computational
cost at test time, which limits the model to simple contex-
tual features."
\end{comment}
\section{Experiments and Results}
\input{figs/fig5/fig5}
\input{figs/fig6/fig6}
\input{figs/fig8/fig8}
\input{figs/table}
\input{figs/table_val}
\textbf{Setup.}
We report our results on the PASCAL VOC 2012 segmentation dataset.
We used the $\mathtt{trainval}$\xspace data provided by the challenge, and the additional annotations
collected by Berkeley~\cite{bharathICCV2011}.
We trained $\mathtt{SegNet}$\xspace in a cross-val manner -- we split the entire dataset into 10 folds,
trained $\mathtt{SegNet}$\xspace on 9 folds and computed $\mathtt{SegNet}$\xspace outputs in the 10th fold.
Finally, we trained the SSVM re-ranker\xspace on all training data other than $\mathtt{val}$\xspace.
All results and analyses reported in this paper are on $\mathtt{val}$\xspace. We picked our best performing
approach and uploaded to the PASCAL evaluation server to report results on $\mathtt{test}$\xspace.
Table \ref{tab:valresults} shows the results of all approaches on PASCAL 2012 $\mathtt{val}$\xspace set.
\Naive upsampling performs worst at $31.3\%$.
Superpixel upsampling gives a
small improvement at $31.9\%$.
Neither of these are competitive, which we suspect
is due of the coarseness of the segmentations.
It's possible that more sophisticated up-sampling strategies from
\cite{ciresan2012deep, farabet2013pami, pinheiro2014rcnn}
would result in more competitive segmentations directly from the CNN.
On the other hand, even our simple re-ranking\xspace of DivMBest proposals is competitive,
at $48.6\%$.
Our final method of $\mathtt{SegNet}$\xspace features with the SVM re-ranker\xspace yielded best results at
$53.1\%$. We uploaded our best performing method on the PASCAL evaluation server
and table \ref{tab:results} shows the results
Contrary to most recent CNN results, our setting allows $\mathtt{SegNet}$\xspace to
perform well with relatively little data.
The PASCAL segmentation dataset \cite{pascal-voc-2012}
augmented with extra annotations from \cite{bharathICCV2011}
only has about 12000 images.
We think a variety of decisions combined to allow competitive performance
with such little data.
Foremost is our ability to initialize weights for the first few layers from AlexNet,
which was trained with a larger dataset (ImageNet).
By keeping those weights fixed, we constrained learning to the small set of
parameters contained in deeper $\mathtt{conv}$\xspace layers.
The lack of fully connected layers also helps keep the parameter count low.
This gives $\mathtt{SegNet}$\xspace much less opportunity to overfit to our smaller dataset.
Furthermore, forcing the final segmentation to be a choice from proposals
(1) constrained the model even more (pick 1 of 30) and (2) allowed us to
incorporate a variety of information from other methods to compensate for things
$\mathtt{SegNet}$\xspace could not learn.
In essence, we constructed a deep model without learning a deep model.
In figure \figref{fig:uoivsce} we show performance of $\mathtt{SegNet}$\xspace re-ranking\xspace
to compare losses. After training a net for 4000 iterations with cross-entropy
we continue training from that net using 3 losses. Optimizing UOI clearly
outperforms cross-entropy. Because the losses might be complementary, we
also optimize a linear comination of the two ($0.7UOI + 0.3CE$, found with grid search).
This further improves performance by a bit.
\input{figs/fig7}
\paragraph{Ablation Studies.}
We tried to tease apart the influence of different components in our pipeline.
First, if we train an SVM re-ranker\xspace with $\mathtt{SegNet}$\xspace features alone, it
performs about the same as simple
KL divergence based ranking ($47.4\%$).
Adding (\textbf{CNN classification}) features and (\textbf{DivMBest+ReRank})
features from \cite{yadollahpour2013rerank} increases this performance
by about $3.5\%$ and $4.0\%$ respectively.
Using both yields an extra percent of performance.
To get a better idea of which features are considered important by SVM re-ranker,
we considered various subsets of the features
Recall that the three types of features are (1) $\mathtt{SegNet}$\xspace features,
(2) Classification features, and
(3) DivMBest+ReRank features from \cite{yadollahpour2013rerank}.
For reference, using all three resulted in $53\%$ on $\mathtt{val}$\xspace.
If we only use (1) then we get $49.0\%$, only (2) gives $47.8\%$, and
only (3) gives
$48.1\%$.
Thus we see that the learned $\mathtt{SegNet}$\xspace features
outperform non-segmentation CNN features (classification)
and 1000s of dimensions of hand crafted features
from \cite{yadollahpour2013rerank}.
However, doing both works best.
Using all features but $\mathtt{SegNet}$\xspace features, \ie, (2)+(3)
gives $50.5\%$, which shows that $\mathtt{SegNet}$\xspace features are
important, even in the presence of other CNN-based features.
Using just DivMBest+ReRank features and $\mathtt{SegNet}$\xspace features, \ie (1)+(3)
performs at $52.0\%$, so $\mathtt{SegNets}$\xspace again appear to be more important than simple CNN classification
features.
Some qualitative results were also interesting. In \figref{fig:bird}
we note that (probabilistic) softness of our segmentations helps alleviate
some problems with coarseness, but we point out how such problems still
manifest in \figref{fig:coarse} and \figref{fig:tradeoff}.
\section{Introduction and Notation}
This document is meant to accompany the paper ``Combining the Best of Graphical
Models and ConvNets for Semantic Segmentation''. Herein we compute the derivatives
of Intersection-over-Union (IOU) and Union-over-Intersection (UOI) with respect
to an input feature map $\mathbf{z}$. Two forms of the IOU derivative can be
seen in \eqref{eq:IOU_expect} and \eqref{eq:IOU_prob}. Corresponding forms
of the UOI derivative are in \eqref{eq:UOI_expect} and \eqref{eq:UOI_prob}.
We'd like to use a supervised learning algorithm to train a model for labeling
image pixels using $K$ classes. Let $\bm{\hat{Y}}$ be the predicted
segmentation and $\bm{Y}$ the ground truth so that $\hat{Y}_i$ and $Y_i$ are
random variables indicating the class of pixel $i$; each takes values in
$\{1, \ldots, K\}$ and pixels labeled with indices $\{1, \ldots, N\}$. The set of pixels
can either be all pixels in an image or all pixels in a minibatch, though we take
it to be the set of all pixels in a minibatch. Thus,
$P(\hat{Y}_i = k)$ is the probability of predicting pixel $i$ is class $k$.
However, we'll introduce the following short hand to condense the notation:
\begin{equation}
\hat{p}_{i, k} = P(\hat{Y}_i = k)
\end{equation}
and
\begin{equation}
p_{i, k} = P(Y_i = k)
\end{equation}
Here, the model produces a score $z_{i, k}$ for each pixel $i$ and class $k$
then predicts $\bm{\hat{Y}}$ using a softmax. It assigns
\begin{equation}
\hat{p}_{i, k}
:= \frac{\exp(z_{i, k})}{\sum_{\tilde{k}} \exp(z_{i, \tilde{k}})}
\end{equation}
First we'll build up some machinery useful for both IOU and UOI, then we'll
compute their derivatives.
\section{Intersection and Union}
The intersection function captures the notion of agreement between ground
truth and prediction. It's defined for class $k \in \{1, \ldots, K\}$ as
\begin{equation}
I_k(\bm{\hat{Y}}, \bm{Y}) = \sum_i^N \ind{\hat{Y}_i = k \land Y_i = k}
\end{equation}
where $\ind{\cdot}$ is the indicator function.
We're interested in the expected intersection,
\begin{align}
E\left[ I_k(\bm{\hat{Y}}, \bm{Y}) \right]
&= E\left[ \sum_i^N \ind{\hat{Y}_i = k \land Y_i = k} \right] \\
&= \sum_i^N E\left[ \ind{\hat{Y}_i = k \land Y_i = k} \right]
\end{align}
which can be reduced to the following, because the expectation of an indicator
function is the probability of the event inside:
\begin{align}
E\left[ I_k(\bm{\hat{Y}}, \bm{Y}) \right]
&= \sum_i^N \hat{p}_{i, k} p_{i, k}
\end{align}
The union function is about the total "footprint" in pixels of ground truth and
prediction; it is defined as
\begin{equation}
U_k(\bm{\hat{Y}}, \bm{Y}) = \sum_i^N \ind{\hat{Y}_i = k \lor Y_i = k}
\end{equation}
Expected union is analogous to expected intersection:
\begin{align}
E\left[ U_k(\bm{\hat{Y}}, \bm{Y}) \right]
&= E\left[ \sum_i^N \ind{\hat{Y}_i = k \lor Y_i = k} \right] \\
&= \sum_i^N E\left[ \ind{\hat{Y}_i = k \lor Y_i = k} \right] \\
&= \sum_i^N \left( \hat{p}_{i, k} + p_{i, k} - \hat{p}_{i, k} p_{i, k} \right)
\end{align}
In fact, it can be expressed using Intersection (inclusion-exclusion principle)
\begin{align}
E\left[ U_k(\bm{\hat{Y}}, \bm{Y}) \right]
&= \sum_i^N \left( \hat{p}_{i, k} + p_{i, k} \right) - E\left[ I_k(\bm{\hat{Y}}, \bm{Y}) \right]
\end{align}
\section{Softmax Gradient}
A softmax function takes a bunch of scores and outputs probabilities.
In the next section we'll need the softmax's gradient at any output with
respect to any input. Normally we care about the gradient of an output with
respect its corresponding input, but this case is a bit more general.
Here $k' \in \{1, \ldots, K\}$.
\begin{align}
\frac{\del \hat{p}_{i, k'}}{\del z_{i, k}}
&= \frac{\del}{\del z_{i, k}} \frac{\exp(z_{i, k'})}{\sum_{\tilde{k}} \exp(z_{i, \tilde{k}})} \\
&= \frac{\frac{\del}{\del z_{i, k}} \left( \exp(z_{i, k'}) \right) \sum_{\tilde{k}} \exp(z_{i, \tilde{k}}) -
\exp(z_{i, k'}) \frac{\del}{\del z_{i, k}} \left( \sum_{\tilde{k}} \exp(z_{i, \tilde{k}}) \right)}
{\left( \sum_{\tilde{k}} \exp(z_{i, \tilde{k}}) \right)^2} \\
\end{align}
If $k = k'$ then
\begin{align}
\frac{\del \hat{p}_{i, k'}}{\del z_{i, k}}
&= \frac{\exp(z_{i, k}) \sum_{\tilde{k}} \exp(z_{i, \tilde{k}}) -
\exp(z_{i, k}) \exp(z_{i, k})}
{\left( \sum_{\tilde{k}} \exp(z_{i, \tilde{k}}) \right)^2} \\
&= \hat{p}_{i, k} - \hat{p}_{i, k}^2
\end{align}
When $k \ne k'$,
\begin{align}
\frac{\del \hat{p}_{i, k'}}{\del z_{i, k}}
&= \frac{- \exp(z_{i, k'}) \exp(z_{i, k})}
{\left( \sum_{\tilde{k}} \exp(z_{i, \tilde{k}}) \right)^2} \\
&= - \hat{p}_{i, k'} \hat{p}_{i, k}
\end{align}
Now the whole derivative can be written as one expression and simplified a bit.
\begin{align}
\frac{\del \hat{p}_{i, k'}}{\del z_{i, k}}
&= \ind{k = k'} (\hat{p}_{i, k} - \hat{p}_{i, k}^2) -
\ind{k \ne k'} \hat{p}_{i, k'} \hat{p}_{i, k} \\
\end{align}
In the first case, substitute $k'$ for $k$ to get
\begin{align}
\ind{k = k'} (\hat{p}_{i, k} - \hat{p}_{i, k}^2) -
\ind{k \ne k'} \hat{p}_{i, k'} \hat{p}_{i, k}
&= \ind{k = k'} \hat{p}_{i, k'} (1 - \hat{p}_{i, k}) -
\ind{k \ne k'} \hat{p}_{i, k'} \hat{p}_{i, k} \\
&= \hat{p}_{i, k'} \left( \ind{k = k'} (1 - \hat{p}_{i, k}) +
\ind{k \ne k'} (-\hat{p}_{i, k}) \right) \\
&= \hat{p}_{i, k'} \left( \ind{k = k'} (1 - \hat{p}_{i, k}) +
\ind{k \ne k'} (0 - \hat{p}_{i, k}) \right) \\
&= \hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k})
\end{align}
\section{Gradient of Expected Intersection and Expected Union}
Next, compute the derivatives of expected intersection and expected union.
For intersection,
\begin{align}
\frac{\del}{\del \hat{p}_{i, k}} E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]
&= \frac{\del}{\del \hat{p}_{i, k}} \sum_j^N \hat{p}_{j, k'} p_{j, k'} \\
&= p_{i, k'} \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}} \label{eq:I_grad}
\end{align}
In the case of union,
\begin{align}
\frac{\del}{\del \hat{p}_{i, k}} E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]
&= \frac{\del}{\del \hat{p}_{i, k}} \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'} - \hat{p}_{j, k'} p_{j, k'} \right) \\
&= (1 - p_{i, k'}) \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}} \label{eq:U_grad}
\end{align}
\section{IOU Loss and its Gradient}
Now, write out the IOU loss function
\begin{equation} \label{eq:exp_iou}
\mathcal{L}_{IOU}(\bm{\hat{Y}}, \bm{Y}) = \sum_{k'}^K \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}
\end{equation}
and compute the gradient of IOU
\begin{align}
\frac{\del \mathcal{L}_{IOU}(\bm{\hat{Y}}, \bm{Y})}{\del z_{i, k}}
&= \left( \sum_{k'}^K \frac{\del}{\del z_{i, k}} \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]} \right) \\
&= \sum_{k'}^K \left( \frac{\del}{\del \hat{p}_{i, k}} \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}
\frac{\del \hat{p}_{i, k}}{\del z_{i, k}} \right) \label{eq:IOU_chain}
\end{align}
First we'll focus on
\begin{align}
\frac{\del}{\del \hat{p}_{i, k}} \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}
\end{align}
By substituting these into the following we get
\begin{align}
\frac{\del}{\del \hat{p}_{i, k}} \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}
&= \frac{
E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] \frac{\del}{\del \hat{p}_{i, k}} E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]
- E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] \frac{\del}{\del \hat{p}_{i, k}} E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]
}
{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \\
&= \frac{
E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'} \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}} - E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'}) \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}}
}
{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \\
&= \frac{
E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'} - E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})
}
{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}}
\label{eq:IU_version}
\end{align}
Finally, substituting \eqref{eq:IU_version} into \eqref{eq:IOU_chain} gives
\begin{align}
\frac{\del \mathcal{L}_{IOU}(\bm{\hat{Y}}, \bm{Y})}{\del z_{i, k}}
&= \sum_{k'}^K \left( \frac{\del}{\del \hat{p}_{i, k}} \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}
\frac{\del \hat{p}_{i, k}}{\del z_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'}
- E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})}
{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}}
\frac{\del \hat{p}_{i, k}}{\del z_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'}
- E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})}
{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \frac{\del \hat{p}_{i, k'}}{\del \hat{z}_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'}
- E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})}
{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]^2}
\hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k}) \right) \label{eq:IOU_expect} \\
&= \sum_{k'}^K \left( \frac{p_{i, k'} \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'} - \hat{p}_{j, k'} p_{j, k'} \right)
- (1 - p_{i, k'}) \sum_j^N \left( \hat{p}_{j, k'} p_{j, k'} \right) }
{\left( \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'} - \hat{p}_{j, k'} p_{j, k'} \right) \right)^2}
\hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k}) \right) \\
&= \sum_{k'}^K \left( \frac{p_{i, k'} \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'}\right)
- \sum_j^N \left( \hat{p}_{j, k'} p_{j, k'} \right) }
{\left( \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'} - \hat{p}_{j, k'} p_{j, k'} \right) \right)^2}
\hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k}) \right) \label{eq:IOU_prob}
\end{align}
\section{Union over Intersection}
The gradient of Union over Intersection can be computed in a similar fashion.
UOI loss is defined as
\begin{equation} \label{eq:exp_uoi}
\mathcal{L}_{UOI}(\bm{\hat{Y}}, \bm{Y}) = \sum_{k'}^K \frac{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}
\end{equation}
Given \eqref{eq:I_grad} and \eqref{eq:U_grad}, we can compute the derivative of UOI:
\begin{align}
\frac{\del \mathcal{L}_{UOI}(\bm{\hat{Y}}, \bm{Y})}{\del z_{i, k}}
&= \sum_{k'}^K \left( \frac{\del}{\del \hat{p}_{i, k}} \frac{E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]}{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}
\frac{\del \hat{p}_{i, k}}{\del z_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] \frac{\del}{\del \hat{p}_{i, k}} E[U_{k'}(\bm{\hat{Y}}, \bm{Y})]
- E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] \frac{\del}{\del \hat{p}_{i, k}} E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]}
{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]^2}
\frac{\del \hat{p}_{i, k}}{\del z_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})
- E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'}}
{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \frac{\del \hat{p}_{i, k'}}{\del \hat{p}_{i, k}}
\frac{\del \hat{p}_{i, k}}{\del z_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})
- E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'}}
{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]^2} \frac{\del \hat{p}_{i, k'}}{\del \hat{z}_{i, k}} \right) \\
&= \sum_{k'}^K \left( \frac{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})] (1 - p_{i, k'})
- E[U_{k'}(\bm{\hat{Y}}, \bm{Y})] p_{i, k'}}
{E[I_{k'}(\bm{\hat{Y}}, \bm{Y})]^2}
\hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k}) \right) \label{eq:UOI_expect} \\
&= \sum_{k'}^K \left( \frac{(1 - p_{i, k'}) \sum_j^N \left( \hat{p}_{j, k'} p_{j, k'} \right)
- p_{i, k'} \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'} - \hat{p}_{j, k'} p_{j, k'} \right)}
{\left( \sum_j^N \left( \hat{p}_{j, k'} p_{j, k'} \right) \right)^2}
\hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k}) \right) \\
&= \sum_{k'}^K \left( \frac{\sum_j^N \left( \hat{p}_{j, k'} p_{j, k'} \right)
- p_{i, k'} \sum_j^N \left( \hat{p}_{j, k'} + p_{j, k'}\right)}
{\left( \sum_j^N \left( \hat{p}_{j, k'} p_{j, k'} \right) \right)^2}
\hat{p}_{i, k'} (\ind{k = k'} - \hat{p}_{i, k}) \right) \label{eq:UOI_prob}
\end{align}
|
1,314,259,995,823 | arxiv | \section{Introduction}
Flag kernels on homogeneous groups have been introduced by Nagel-Ricci-Stein \cite{nagel} in their study of quadratic $CR$-manifolds. They can be regarded as a generalization of Calder\'on-Zygmund singular kernels with singularities extending over the whole of the hyperspace $x_1=0$, where $x_1$ is the top level variable. The definition is complex (see below), as it involves cancellation conditions for each variable separately. However, the descritption of flag kernels in terms of their Fourier transforms is much simpler and bears a striking resemblance to that of the symbols of convolution operators considered independently by the author (in, e.g. \cite{arkiv2007}).
In Nagel-Ricci-Stein \cite{nagel} we find an $L^p$-boundedness theorem for the very special flag kernels where the associated gradation consists of commuting subalgebras of the underlying Lie algebra of the homogeneous group. The natural question of what happens if the gradation is the natural gradation of the homogeneous Lie algebra is left open. The aim of this paper is to answer the question in the affirmative. We prove that such flag kernels give rise to bounded operators.
The smooth symbolic calculus mentioned above has been adapted to an extended class of flag kernels of small (positive and negative) orders and combined with a variant of the Littlewood-Paley theory built on a stable semigroup of measures with smooth densities very similar to the Poisson kernel on the Euclidean space. The strong maximal function of Christ \cite{christ} is also instrumental. The approach has been inspired by the well-known paper by Duoandicoetxea and Rubio de Francia \cite{duoandi}. The dependence of the present paper on Duoandicoetxea and Rubio de Francia \cite{duoandi} is evident throughout.
The class of flag kernels dealt with here is in fact an algebra. For this the reader is referred to \cite{colloquium2010} where also the $L^2$-boundedness of flag kernels is proved solely by means of the symbolic calculus.
After this paper had been completed, a preprint of Nagel-Ricci-Stein-Wainger \textit{Singular integrals with flag kernels on homogeneous groups I,} has been made available, where the $L^p$-boundedness theorem for flag kernels is proved. This comprehensive treatment of flag kernels on homogeneous groups has been announced for some time. Professor Stein has lectured a couple of times on the subject, see, e.g. \cite{stein}. The authors also use a version of Littlewood-Paley theory but otherwise the approach differs from the one presented here in many respects, the most important being our use of the symbolic calculus and partitions of unity related to a stable semigroup of measures. That is why we believe that what is presented here has an independent value and may count as a contribution to the theory.
\section{Preliminaries}
Let $\Ge$ be a nilpotent Lie algebra with a fixed Euclidean structure and $\Ges$ its dual. Let $\delta_tx=tx$, $t>0$ be a family of dilations on $\Ge$ and let
\[
\Ge_j=\{x\in\Ge: \delta_tx=t^{p_j}\cdot x\},
\hspace{2em}
1\le j\le d,
\]
where $1=p_1< p_2<\dots <p_d$. Denote by
\[
Q_j=p_j\cdot\dim\Ge_j
\]
the homogenous dimension of $\Ge_j$. The homogeneous dimension of $\Ge$ is
\[
Q=\sum_{j=1}^dQ_j.
\]
We have
\begin{equation}\label{grad}
\Ge=\bigoplus_{j=1}^d\Ge_j,
\qquad
\Ges=\bigoplus_{j=1}^d\Ges_j
\end{equation}
and
\[
[\Ge_i,\Ge_j]\subset \left\{
\begin{array}{ll}
\Ge_k, & {\rm if \ } p_i+p_j=p_k,\cr \{0\}, & {\rm if \ } p_i+p_j\notin{\mathcal{P},}
\end{array}
\right.
\]
where $\mathcal{P}=\{p_j:1\le j\le d\}$.
Let
\[
x\to|x|\approx\sum_{j=1}^d\|x_j\|^{1/p_j}
\]
be a homogeneous norm on $\Ge$ smooth away from the origin. Let also
\[
|x|_j=|(x_1,x_2,\dots,x_j,0,\dots,0)|,
\qquad
1\le j\le d.
\]
In particular, $|x|_1=|x_1|$, and $|x|_d=|x|$. Another notation will be applied to $\Ges$. For $\xi\in\Ges$,
\[
|\xi|_j=|(0,\dots,0,\xi_j,\xi_{j+1},\dots,\xi_d)|,
\qquad
1\le j\le d.
\]
In particular, $|\xi|_1=|\xi|$, and $|\xi|_d=|\xi_d|$.
We shall also regard $\Ge$ as a Lie group with the
Campbell-Hausdorff multiplication
\[
xy=x+y+r(x,y),
\]
where $r(x,y)$ is the (finite) sum of terms of order at least $2$ in the Campbell-Hausdorff
series for $\Ge$. Under this identification the homogeneous ideals
\[
\Ge^{(k)}=\bigoplus_{j=k}^d\Ge_j
\]
are normal subgroups.
In expressions like $D^{\alpha}$ or $x^{\alpha}$ we shall use multiindices
\[
\alpha=(\alpha_1,\alpha_2,\dots,\alpha_d),
\]
where
\[
\alpha_k=(\alpha_{k1},\alpha_{k1},\dots,\alpha_{kn_k}),
\qquad
n_k=\dim\Ge_k=\dim\Ges_k,
\]
are themselves multiindices with positive integer entries corresponding to the spaces $\Ge_k$ or $\Ges_k$. The homogeneous length of $\alpha$ is defined by
\[
|\alpha|=\sum_{k=1}^d|\alpha_k|,
\qquad|\alpha_k|=p_k(\alpha_{k1}+\alpha_{k2}+\dots+\alpha_{kn_k}).
\]
The Schwartz space of smooth functiions which vanish rapidly at infinity along with their derivatives will be denoted by $\Schw(\Ge)$. For a tempered distribution $K$, that is a continuous linear functional on $\Schw(\Ge)$, we shall write
\[
\langle K,f\rangle=\int_{\Ge}f(x)K(x)\,dx,
\qquad
f\in\Schw(\Ge),
\]
without implying thereby that $K$ is a locally integrable function.
Even though the flag kernels are our prime concern here we need a broader class of kernels to properly deal with them. In \cite{studia2010}, we proposed a natural generalization of the flag kernels of Nagel-Ricci-Stein. Let
\[
\|f\|_{(k)}=\max_{|\alpha|\le Q_k+1}\sup_{x\in\Ge_k}(1+|x|)^{Q_k+1}|D^{\alpha}f(x)|
\]
be a fixed norm in the Schwartz space $\Schw(\Ge_k)$. Let
\[
\mathcal{N}=\{\nu=(\nu_1,\nu_2,\dots,\nu_d): |\nu_k|<Q_k, \ 1\le k\le d\}.
\]
Let $\nu\in\mathcal{N}$. We define the class $\Fe(\nu)$ by induction on the homogeneous step $d$. When $d=0$ the elements of $\Fe(\emptyset)$ are simply constants. If $d\ge1$, we say that a distribution $K\in\Schw^{\star}(\Ge)$ is in $\Fe(\nu)$ if it is smooth away from the hyperspace $x_1=0$ and satisfies the following conditions:
i) For every multiindex $\alpha$,
\begin{equation}\label{size}
|D^{\alpha}K(x)|\le C_{\alpha}|x|_1^{-\nu_1-Q_1-|\alpha_1|}|x|_2^{-\nu_2-Q_2-|\alpha_2|}\dots
|x|_d^{-\nu_d-Q_d-|\alpha_d|}
\end{equation}
for $x_1\neq0$;
ii) For any $1\le k\le d$,
\begin{equation}\label{cancellation}
<K_{R,\phi},f>=R^{-\nu_k}\int_{\Ge}\phi(Rx_k)f(x_1,\dots,x_{k-1},x_{k+1},\dots,x_d)K(x)\,dx
\end{equation}
is in $\Fe(\nu_{(k)})$ on $\oplus_{j\neq k}\Ge_j$, where $\nu_{(k)}=(\nu_1,\dots,\nu_{k-1},\nu_{k+1},\dots,\nu_d)$, and this is uniform in $\phi\in\Schw(\Ge_1)$ with $|\phi\|_{(k)}\le1$ and $R>0$. (Note that the meaning of \textit{uniform boundedness} of a family of members of $\Fe(\nu)$ is obvious in the case $d=0$ and, for $d\ge1$, can be defined by induction.)
For every $N$, we define a norm $\|\cdot\|_{\nu,N}$ in $\Fe(\nu)$ as the maximum of all the bounds occurring in the definition. First, we let
\[
s_N^{\nu}(P)=\max_{|\alpha|\le N}\sup_{x_1\neq0}\prod_{k=1}^d|x|_k^{Q_k+\nu_k+|\alpha_k|}|D^{\alpha}K(x)|.
\]
and, if $d=1$,
\[
\|K\|_{\nu_1,N}=s_N^{\nu_1}(K)+\sup_{|\phi\|_{(1)}\le1}\sup_{R>0}R^{-\nu_1}|<K,\phi\circ\delta_R>|.
\]
If $d>1$, we let
\[
\|K\|_{\nu,N}=s_N^{\nu}(K)+\max_{1\le k\le d}\sup_{\|\phi\|_{(k)}\le1}\sup_{R>0}\|K_{R,\phi}\|_{\nu_{(k)},N}.
\]
Thus, $\Fe(\nu)$ can be regarded as a locally convex topological vector space. Let us remark that $\Fe(0)=\Fe(0,0,\dots,0)$ is exactly the class of flag kernels of Nagel-Ricci-Stein \cite{nagel} (see Corollary 3.7 of \cite{studia2010}).
For a $K\in\Schw^{\star}(\Ge)$, let
\[
<\widetilde{K},f>=\int_{\Ge}f(x^{-1})K(dx),
\qquad
f\in\Schw(\Ge).
\]
The following three propositions have been proved in \cite{colloquium2010} and \cite{studia2010}.
\begin{proposition}[(Theorem 2.5 of \cite{colloquium2010})]\label{l2bound_flag}
Let $K\in\Fe(0)$ be a flag kernel on $\Ge$. The convolution operator
$f\to f\star\widetilde{K}$
defined initially on $\Schw(\Ge)$ extends uniquely to a bounded operator on $L^2(\Ge)$. \end{proposition}
\begin{proposition}[(Proposition 1.5 of \cite{studia2010})]\label{gen_flag}
Let $\nu\in\mathcal{N}$. A distribution $K$ is in $\Fe(\nu)$ if and only if its Fourier transform is locally integrable, smooth for $\xi_d\neq0$, and satisfies
\begin{equation}\label{multiplier}
|D^{\alpha}\widehat{K}(\xi)|\le C_{\alpha}|\xi|_1^{\nu_1-|\alpha_1|}\dots|\xi|_d^{\nu_d-|\alpha_d|},
\qquad
\xi_d\neq0.
\end{equation}
\end{proposition}
\noindent
Cf. also the original Theorem 2.3.9 of Nagel-Ricci-Stein \cite{nagel} for kernels $K\in\Fe(0)$.
\begin{proposition}[Theorem 4.8 of \cite{studia2010}]\label{composition_flag}
Let $\nu,\mu,\nu+\mu\in\mathcal{N}$. Let $K\in\Fe(\nu)$, $L\in\Fe(\mu)$. Let $\phi=\otimes_{k=1}^d\phi_k\in C_c^{\infty}(\Ge)$ be equal to $1$ in a neighbourhood of $0$. There exists a $P=P_{K,L}\in\Fe(\nu+\mu)$ such that
\[
P=\lim_{\e\to0}K_{\e}\star L
\]
in the sense of distributions, where
\[
<K_{\e},f>=\int_{\Ge}\phi(\e x)f(x)K(dx),
\qquad
f\in\Schw(\Ge).
\]
Moreover, the mapping $(K,L)\to P_{K,L}$ is continuous.
\end{proposition}
\section{Semigroups of measures}
Following Folland-Stein \cite{folland}, we say that a function $\phi$ belongs to the class $\mathcal{R}(a)$, where $a>0$, if it is smooth and
\begin{equation}\label{rclass}
|D^{\alpha}\phi(x)|\le C_{\alpha}(1+|x|)^{-Q-a-|\alpha|},
\qquad
{\rm all \ } \alpha.
\end{equation}
\begin{proposition}\label{fourier}
Let $\phi\in\mathcal{R}(a)$ for some $0<a<1$ and let $\int\phi=0$. Then $\phi\in\Fe(a)$.
\end{proposition}
\begin{proof}
The size condition (\ref{size}) follows by (\ref{rclass}). To verify the cancellation condition (\ref{cancellation}) let $f\in\Schw(\Ge)$ and $R>0$. Then
\begin{align*}
\int_{\Ge}&f(Rx)\phi(x)\,dx=\int_{\Ge}\big(f(Rx)-f(0)\big)\phi(x)\,dx \\
&\le\int_{|x|\le R^{-1}}\big(f(Rx)-f(0)\big)\phi(x)\,dx
+\int_{|x|\ge R^{-1}}\big(f(Rx)-f(0)\big)\phi(x)\,dx \\
&\le\|f\|\left(R\int_{|x|\le R^{-1}}|x|^{-Q-a+1}\,dx+2\int_{|x|\ge R^{-1}}|x|^{-Q-a}\,dx\right)\\
&\le CR^a\|f\|,
\end{align*}
where $\|\cdot\|$ is a Schwartz class norm.
\end{proof}
Let
\[
\langle P,f\rangle
=\lim_{\e\to}\int_{|x|\ge\e}\Big(f(0)-f(x)\Big)\frac{dx}{|x|^{Q+1}},
\qquad
f\in\Schw(\Ge).
\]
The distribution $P$ is an infinitesimal generator of a continuous semigroup of probability measures with smooth densities
\[
h_t(x)=t^{-Q}h(t^{-1}x),
\]
where $h\in\eR(1)$ and $P^Nh\in\eR(N)$ for $N=1,2,\dots$. In other words,
\[
h_t\star h_s=h_{t+s},
\qquad
t,s>0.
\]
and
\[
\frac{d}{dt}\big|_{t=0}<h_t,f>=-<P,f>,
\qquad
f\in\Schw(\Ge),
\]
The operator $Pf=f\star P$ is essentially selfadjoint with $\Schw(\Ge)$ for its core domain. The reader is referred to \cite{inventiones1986} for proofs and details.
For $0<{a}<1$
\begin{equation}\label{m}
\langle P^{a},f\rangle=\frac{1}{\Gamma(-{a})}\int_0^{\infty}t^{-1-{a}}\langle
\delta_0-h_t, f\rangle\,dt=\frac{1}{\Gamma(1-{a})}\int_0^{\infty}t^{-{a}}\langle
Ph_t, f\rangle\,dt
\end{equation}
defines a homogeneous distribution smooth away from the origin (cf., e.g. Yosida \cite{yosida}).
\begin{proposition}\label{pm}
For every $0<a<1$,
\[
P^{a}h\in\eR(a)
\qquad
{\rm and}
\qquad
\int_{\Ge}P^{a}h(x)\, dx=0.
\]
\end{proposition}
\begin{proof}
By (\ref{m}),
\[
P^{a}h(x)=\frac{1}{\Gamma(1-{a})}\int_0^{\infty}t^{-{a}}Ph_{t+1}(x)\,dt,
\]
whence
\begin{align*}
|D^{\alpha}P^{a}h(x)|&\le\frac{C_{\alpha}}{\Gamma(1-{a})}\int_0^{\infty}\frac{t^{-{a}}\,dt}{(t+1+|x|)^{Q+1+|\alpha|}}\\
&\le C_{\alpha}'\int_0^{\infty}\frac{t^{-{a}}\,dt}{(\frac{t}{1+|x|}+1)^{Q+1+|\alpha|}}\cdot(1+|x|)^{-Q-1-|\alpha|}\\
&\le C_{\alpha}''\int_0^{\infty}\frac{t^{-{a}}\,dt}{(t+1)^{Q+1+|\alpha|}}\cdot(1+|x|)^{-Q-{a}-|\alpha|},
\end{align*}
as required.
Now, for every $t>0$,
\[
\int h_t\,dx=1.
\]
Therefore,
\[
\int Ph_t\,dx=-\frac{d}{dt}\int h_t\,dx=0,
\qquad
t>0.
\]
which combined with (\ref{m}) gives the second part of the assertion.
\end{proof}
\section{Littlewood-Paley theory}
From now on we fix the function $\phi=P^{1/2}h_{1/2}$.
\begin{remark}\label{fixfi}
By the results of the previous section, $\phi$ is a smooth function satisfying the estimates
\begin{equation}\label{fiestim}
|D^{\alpha}\phi(x)|\le C_{\alpha}(1+|x|)^{-Q-1/2-|\alpha|}.
\end{equation}
Moreover, $\phi\in\Fe(1/2)$.
\end{remark}
\begin{lemma}\label{fi}
We have
\[
f=\int_0^{\infty}f\star\phi_t\star\phi_t\,\frac{dt}{t},
\qquad
f\in\Schw(\Ge).
\]
\end{lemma}
\begin{proof}
By the semigroup properties,
\[
-\frac{d}{dt}f\star h_t=f\star Ph_t=\frac{1}{t}f\star(\phi_t\star\phi_t),
\]
whence
\[
\int_{\e}^Mf\star\phi_t\star\phi_t\frac{dt}{t}=f\star h_{\e}- f\star h_M.
\]
Now, if $\e\to0$ and $M\to\infty$, the expression on the right hand side tends to $f$ in the sense of distributions.
\end{proof}
Let $T=(t_1,\dots,t_d)\in\R_+^d$. We shall regard $\R_+^d$ as a product of copies of the multiplicative group $\R^+$. We shall write
\[
T^{a}=(t_1^{a},\dots, t_d^{a}),
\qquad
TS=(t_1s_1,\dots,t_ds_d),
\qquad
\frac{dT}{T}=\frac{dt_1\dots dt_d}{t_1\dots t_d},
\qquad
a\in\R.
\]
Let $\phi_k$ be the counterpart of $\phi$ for $\Ge$ replaced by $\Ge^{(k)}$, $1\le k\le d$. Let
\[
\Phi_k=\delta_k\otimes\phi_k,
\]
where $\delta_k$ stands for the Dirac delta at $0\in\oplus_{j=1}^{k-1}\Ge_j$. Let
\[
\Phi=\Phi_1\star\Phi_2\star\dots\star\Phi_d,
\]
and
\[
\Phi_T=(\Phi_1)_{t_1}\star\dots\star(\Phi_d)_{t_{d}},
\qquad
T\in\R_+^d.
\]
\begin{corollary}\label{unity}
We have
\[
\Phi\in|\Fe|(1/2):=\bigcap_{\e\in\{-1,1\}}\Fe(\e_1/2,\dots,\e_d/2).
\]
Furthermore,
\[
f=\int_{\R_+^d}f\star\Phi_T\star\widetilde{\Phi_T}\,\frac{dT}{T},
\qquad
f\in\Schw(\Ge).
\]
\end{corollary}
\begin{proof}
By Remark \ref{fixfi},
\[
\Phi_k\in\Fe(0,\dots0,1/2,0,\dots,0)\cap\Fe(0,\dots0,-1/2,0,\dots,0),
\]
where the only nonzero term stands on the $k$-th position. Therefore the first part of our assertion
follows by Proposition \ref{composition_flag}. The second one is a consequence of Lemma\nobreak \ \ref{fi}.
\end{proof}
\begin{proposition}\label{paley}
The Paley-Littlewood square function
\[
G_{\Phi}(f)(x)=\left(\int_{\R_+^d}|f\star\Phi_T(x)|^2\,\frac{dT}{T}\right)^{1/2},
\]
is bounded as an operator on $L^p(\Ge)$. In other words, for every $1<p<\infty$, there is a constant $C_{\phi,p}>0$ such that
\[
\|G_{\Phi}(f)\|_p\le C_{\phi,p}\|f\|_p,
\qquad
f\in\Schw(\Ge).
\]
\end{proposition}
\begin{proof}
The proof is implicitly contained in Folland-Stein \cite{folland} (see Theorem 6.20.b and Theorem 7.7) so we dispense ourselves with presenting all details.
We start with defining some Hilbert spaces and operators. Let $X_0=\C$ and
\[
X_k=L^2(\R^k_+,\frac{dT}{T}),
\qquad
1\le k\le d.
\]
For a given $x\in\Ge$, let $F_k(x):X_{k-1}\to X_k$ be given by
\[
F_k(x)m(t_1,\dots,t_{k-1},t_k)=(\phi_k)_{t_k}(x_k,\dots,x_d)m(t_1,\dots,t_{k-1}).
\qquad
m\in X_{k-1}.
\]
Finally, let $W_k:C_c(\Ge,X_{k-1})\to C_0(\Ge,X_k)$ be the operator
\[
W_kf(x)(T,t_k)=(f\star F_k)(x)(T,t_k)=\int_{\Ge^{(k)}}(\phi_k)_{t_k}(y)f(xy)(T)\,dy,
\]
where $T=(t_1,\dots,t_{k-1})$. Note that $W_k$ acts only on $(x_k,\dots,x_d)$-variable.
We claim that
$$
W_k:L^2(\Ge,X_{k-1})\to L^2(\Ge,X_k)
$$
is an isometry. In fact, by definition of $\Phi_k$,
\begin{align*}
\|W_kf\|_{L^2(\Ge,X_k}^2&=\int_{\Ge}\|W_kf(x)\|_{X_k}^2\,dx\\
&=\int_{\Ge}dx\int_0^{\infty}\frac{dt}{t}
\int_{\R^{k-1}_+}\frac{dT}{T}\int_{\Ge^{(k)}}|(\phi_k)_t(y)f(xy)(T)|^2\,dy\\
&=\int_{\R^{k-1}_+}\frac{dT}{T}\int_0^{\infty}\frac{dt}{t}
\int_{\Ge}\int_{\Ge^{(k)}}|(\phi_k)_t(y)f(xy)(T)|^2\,dydx,\\
&=\int_{\R^{k-1}_+}\frac{dT}{T}\int_0^{\infty}\frac{dt}{t}<f_T\star(\Phi_k)_t,f_T\star(\Phi_k)_t>
=\|f\|_{L^2(\Ge,X_{k-1})}^2,
\end{align*}
where $f_T(x)=f(x)(T)$.
Another property of $W_k$ that is needed is the following. For every $\alpha$
\begin{equation}\label{Kestimate}
\|D^{\alpha}F_k(x)\|_{(X_{k-1},X_k)}\le C_{\alpha}|x|_k^{-Q-|\alpha|}.
\end{equation}
This follows readily from (\ref{fiestim}) specialized to $\phi_k$:
\[
|D^{\alpha}\phi_k(x)|\le C_{\alpha}(1+|x|_k)^{-Q-1/2-|\alpha|}.
\]
As a bounded operator from $L^2(\Ge,X_{k-1})$ to $L^2(\Ge,X_k)$ satisfying (\ref{Kestimate}) is $W_k$ a vector-valued kernel of type $0$, and, by Theorem 6.20.b of Folland-Stein \cite{folland}, maps $L^p(\Ge,X_{k-1})$ into $L^p(\Ge,X_k)$ boundedly for every $1<p<\infty$.
This implies our assertion. In fact,
\[
G_{\Phi}(f)(x)=\|f\star F_1\star\dots\star F_d(x)\|_{X_d},
\]
and therefore
\[
\|G_{\Phi}(f)\|_{L^p(\Ge)}=\|T_dT_{d-1}\dots T_1f\|_{L^p(\Ge,X_d)}
\le C\|f\|_{L^p(\Ge,X_0)}=C\|f\|_p.
\]
\end{proof}
A word of comment on the symbol $\Phi_T$ would be appropriate here. The notation may suggest that the functions $\Phi_T$ are dilates of a single function. They are not, but they have estimates of this form, which is our justification. The same applies to the symbol $K_T$ below. In the next section we are going to use the same notation for the ``real'' dilates of a function. We hope the reader will not get confused.
\section{The strong maximal function}
For a function $F$ on $\Ge$ and a $T\in\R^d_+$, let
\[
F_T(x)=F_{(t_1,t_2,\dots t_d)}(x)=t_1^{-Q_1}t_2^{-Q_2}\dots t_d^{-Q_d}F(t_1x_1,t_2x_2,\dots,t_dx_d).
\]
\textit{The strong maximal function} on $\Ge$ is defined by
\[
{\bf M}f(x)=\sup_{T\in\R_+^d}\int_{|y|\le1}|f(x(Ty)^{-1})\,dy=\sup_{T}|f\star(\chi_B)_T(x)|,
\]
where $\chi_B$ stands for the characteristic function of the unit ball $B=\{x\in\Ge:|x|\le1\}$, and $Ty=(t_1y_1,\dots,t_dy_d)$. A theorem of Michael Christ asserts that for every $1<p<\infty$ there exists a constant $C>0$ such that
\[
\|{\bf M}f\|_p\le C\|f\|_p,
\qquad
f\in L^p(\Ge),
\]
that is, ${\bf M}$ is of $(p,p)$ type (see Christ \cite{christ}).
We shall need the following corollary to the Christ theorem. Let
\[
\gamma(t)=\min\{t,t^{-1}\},
\qquad
t>0.
\]
\begin{corollary}\label{mchrist}
Let
\[
F(x)=\Pi_{j=1}^d\gamma(|x_j|)^a|x_j|^{-Q_j},
\qquad
x\neq0,
\]
for some $a>0$.Then the maximal fuction
\[
M_Ff(x)=\sup_{T\in\R_+^d}|f\star F_T(x)|
\]
is of $(p,p)$ type for $1<p<\infty$.
\end{corollary}
\begin{proof}
Let $B_j$ be the unit ball in $\Ge_j$ and let $|B_j|$ be the Lebesgue measure of $B_j$. Let $D=B_1\times\dots\times B_d$. Then for every simple positive function $h\le F$ of the form
\[
h(x)=\sum_{R}c_R\chi_{D}(R^{-1}x),
\qquad
R=(r_1,r_2,\dots,r_d)\in\R_+^d,
\]
we have
\[
h_T(x)=\sum_{R}c_Rr_1^{Q_1}r_2^{Q_2}\dots r_d^{Q_d}(\chi_D)_{RT}(x)
=\frac{C\|h\|_1}{|D|}(\chi_D)_{RT}(x),
\]
and therefore
\[
M_Ff(x)\le\frac{C\|F\|_1}{|D|}{\bf M}f(x),
\]
which completes the proof.
\end{proof}
\section{Flag kernels}
We keep the notation established in previous sections.
\begin{lemma}
Let
\[
K_{T,S}=\widetilde{\Phi_{TS}}\star K\star{\Phi}_{T},
\qquad
T,S\in\R^d_+.
\]
Then $K_{T,S}\in\Fe(0)$ uniformly, and satisfy the estimates
\begin{equation}\label{F0}
|D^{\alpha}\widehat{K}_{T,S}(\xi)|\le C_{\alpha}\gamma(S)^{1/2}|\xi|_1^{-|\alpha_1|}\dots|\xi|_d^{-|\alpha_d|},
\end{equation}
where
\[
\gamma(S)=\gamma(s_1)\gamma(s_2)\cdots\gamma(s_d).
\]
\end{lemma}
\begin{proof}
By the first part of Corollary \ref{unity}, $\Phi_T\in|\Fe|(1/2)$ with bounds uniformly proportional to $\gamma(T)^{1/2}$. Note that
\[
\gamma(TS)\le\gamma(T)\cdot\gamma(S).
\]
Thus, our assertion follows by Proposition \ref{composition_flag}.
\end{proof}
We let
$$
K_T=K\star\Phi_T,
\qquad
T\in\R_+^d.
$$
\begin{lemma}\label{maxi}
For every $T$, $K_{T}$ is an integrable function, and the maximal operator
\begin{equation}\label{max}
K_{\Phi}^{\star}f(x)=\sup_{T}|f\star|\widetilde{K}_{T}|(x)|
\end{equation}
is of type $(p,p)$ for all $1<p<\infty$.
\end{lemma}
\begin{proof}
Observe that by Proposition \ref{l2bound_flag}, $K_{T}\in L^2(\Ge)$ so it is a function. Moreover,
by Corollary \ref{unity} and Proposition \ref{composition_flag}, it is a smooth away from $x_1=0$, and satisfies
\[
|K_{T}(x)|\le C\gamma(T)^{1/2}
\gamma(|x|_1)^{1/2}|x|_1^{-Q_1}\dots\gamma(|x|_d)^{1/2}|x|_d^{-Q_d}
\]
uniformly in $T$ so that $K_{T}\le CF_T$, where $F_T$ is a dilate of
\[
F(x)=\gamma(|x|_1)^{1/2}|x|_1^{-Q_1}\dots\gamma(|x|_d)^{1/2}|x|_d^{-Q_d}.
\]
This shows that $K_{T}$ is integrable. The second part of our claim follows by Corollary \ref{mchrist} and the above.
\end{proof}
We turn to the main result of this paper. The reader may wish to compare the proof we give with that of Theorem B and the preceding lemma of Duoandicoetxea-Rubio de Francia \cite{duoandi}.
\begin{theorem}\label{main}
Let $K$ be a flag kernel on $\Ge$. Then the singular integral operator
\[
f\to f\star\widetilde{K},
\qquad
f\in\Schw(\Ge),
\]
extends uniquely to a bounded operator on $L^p(\Ge)$ for all $1<p<\infty$.
\end{theorem}
\begin{proof}
Let $f,h\in\Schw(\Ge)$. We have
\begin{align*}\label{decomposition}
<f\star\widetilde{K},h>
&=\int_{\R_+^d}\frac{dS}{S}\int_{\R_+^d}\frac{dT}{T}
<f\star\Phi_{T},h\star\Phi_{TS}\star\widetilde{\Phi_{TS}}\star K\star\Phi_T>\\
&=\int_{\R_+^d}\frac{dS}{S}\int_{\R_+^d}\frac{dT}{T}
<f_T,h_{TS}\star K_{T,S}>,
\end{align*}
where
\[
f_{T}=f\star{\Phi}_{T},
\qquad
h_{TS}=h\star\Phi_{TS},
\qquad
K_{T,S}=\widetilde{\Phi_{TS}}\star K\star\Phi_T.
\]
We are going to estimate
\[
<L_Sf,h>=\int_{\R_+^d}\frac{dT}{T}<f_{T},h_{TS}\star K_{T,S}>
\]
for a given $S$. Let us start with $L^2$-estimates. We have
\[
|<L_Sf,h>|
\le\left(\int_{\Ge}\int_{\R_+^d}|f_{T}(x)|^2\,\frac{dT}{T}dx\right)^{1/2}
\cdot\left(\int_{\Ge}\int_{\R_+^d}|h_{TS}\star K_{T,S}(x)|^2\,\frac{dT}{T}dx\right)^{1/2}.
\]
By (\ref{F0}) and Proposition \ref{l2bound_flag}, the operators $f\to f\star K_{T,S}$ are bounded with norm estimates uniformly proportional to $\gamma(S)^{1/2}$ so that, by Proposition \ref{paley},
\begin{align*}
|<L_Sf,h>|&\le C\gamma(S)^{1/2}\,\|G_{\Phi}(f)\|_2\,\|G_{\Phi}(h)\|_2\\
&\le C_1\gamma(S)^{1/2}\|f\|_2\|h\|_2,
\end{align*}
that is,
\begin{equation}\label{l2}
\|L_Sf\|_2\le C_1\gamma(S)^{1/2}\|f\|_2,
\qquad
f\in\Schw(\Ge).
\end{equation}
For $1<p<2$ and $f,h\in\Schw(\Ge)$,
\begin{align*}
|<L_Sf,h>|&
\le\int_{\Ge}\left(\int_{\R_+^d}|f_{T}(x)|^2\frac{dT}{T}\right)^{1/2}
\left(\int_{\R_+^d}|h_{TS}\star K_{T,S}(x)|^2\,\frac{dT}{T}\right)^{1/2}\,dx\\
&\le C_1\|G_{\Phi}(f)\|_p\left(\int_{\Ge}\left(\int_{\R_+^d}|h_{TS}\star K_{T,S}(x)|^2\,\frac{dT}{T}\right)^{q/2}dx\right)^{1/q}\\
&=C_2\|f\|_p\cdot\left\|\int_{\R_+^d}|h_{TS}\star K_{T,S}(\cdot)|^2\,\frac{dT}{T}\right\|_{q/2}^{1/2},
\end{align*}
where $1/p+1/q=1$. Note that $q>2$. Thus, there exists a nonnegative function $u$ with $\|u\|_r=1$, where $2/q+1/r=1$, such that
\[
\left\|\int_{\R_+^d}|h_{TS}\star K_{T,S}(\cdot)|^2\,\frac{dT}{T}\right\|_{q/2}
=\int_{\Ge}\int_{\R_+^d}|h_{TS}\star K_{T,S}(x)|^2\,\frac{dT}{T}\cdot u(x)\, dx.
\]
Now,
\begin{align*}
h_{TS}\star K_{T,S}&=(h\star\Phi_{TS})\star(\widetilde{\Phi_{TS}}\star K\star\Phi_T)\\
&=(h\star\Phi_{TS}\star\widetilde{\Phi_{TS}})\star(K\star\Phi_T)=h'_{TS}\star K_T.
\end{align*}
Recall also that, by Lemma \ref{maxi}, $K_T$ are integrable functions. Therefore, by Lemma\nobreak\ \ref{maxi} again,
\begin{align}\label{nomax}
\left\|\int_{\R_+^d}|h'_{TS}\star K_{T}(\cdot)|^2\,\frac{dT}{T}\right\|_{q/2}
&\le C_1\int_{\R_+^d}\int_{\Ge}|h'_{TS}|^2\star|K_{T}|(x)\cdot u(x)\, dx\frac{dT}{T}\notag\\
&\le C_2\int_{\Ge}\int_{\R^d}|h'_{TS}(x)|^2\,\frac{dT}{T}\cdot K_{\Phi}^{\star}u(x)\,dx\notag\\
&\le C_3\|G_{\Phi}(h)\|_q^2\cdot\|K^{\star}_{\Phi}u\|_r\le C_4\|h\|_q^2,
\end{align}
where we have used the estimate
\begin{align*}
|h'_{TS}\star K_{T}(x)|^2&\le\left(\int_{\Ge}|h'_{TS}(xy^{-1})|\cdot|K_{T}(y)|^{1/2}\cdot|K_{T}(y)|^{1/2}\,dy\right)^2\\
&\le\int_{\Ge}|h'_{TS}|^2(xy^{-1})\cdot|K_{T}|(y)\,dy\cdot\int_{\Ge}|K_{T}(y)|\,dy\\
&\le C|h'_{TS}|^2\star|K_{T}|(x),
\end{align*}
the integrals
\[
\int_{\Ge}|K_{T}(x)|\,dx\le C
\]
being uniformly bounded, as can be seen from the proof of Lemma \ref{maxi}. Therefore,
\begin{equation}\label{lp}
\|L_Sf\|_p\le C_1\|f\|_p.
\end{equation}
\medskip
Now, by interpolating between (\ref{l2}) and (\ref{lp}), we get
\[
\|L_Sf\|_p\le C_2\gamma(S)^{\e_p}\|f\|_p,
\]
where $\e_p>0$ depends only on $p$, and, finally,
\[
\|f\star\widetilde{K}\|_p\le C_3\left(\int_{\R_+^d}\gamma(S)^{\e_p}\,\frac{dS}{S}\right)\cdot\|f\|_p
=C_4\|f\|_p,
\]
which proves our case for $1<p\le2$. The result for $2<p<\infty$ follows by duality.
\end{proof}
\section*{Acknowledgements}
I wish to extend thanks to Alexander Nagel, Fulvio Ricci, and Elias M. Stein for their interest in my work and an inspiring conversation. I am also indebted to Fran{\c c}ois Piquard for pointing out an editorial omission in the initial version of the manuscript.
\newpage
|
1,314,259,995,824 | arxiv |
\section{Hierarchical matrix approximation of spectral projectors}
\label{hmatrices}
Introduced in the context of integral and partial differential equations,
hierarchical matrices allow for the data-sparse representation of a certain class of dense matrices. In the following, we briefly recall the concept of hierarchical matrices and
some operations; see, e.g.,~\cite{Bebendorf2008,Hackbusch2015} for more details.
\subsection{Matrices with hierarchical low-rank structures}
\subsubsection{HODLR matrices}
We first discuss \textit{hierarchically off-diagonal low-rank} (HODLR) matrices. For convenience,
we assume that $n = 2^p$ for $p \in \mathbb{N}$. Given a prescribed maximal off-diagonal rank $k\in \mathbb{N}$, we suppose that a matrix $M\in {\mathbb R}^{n\times n}$
admits the representation
\begin{equation}
\label{eq:HODLR}
M = \begin{bmatrix}
M^{(1)}_{1} &U_1^{(1)}V_1^{(1)^{*}}\\
U_2^{(1)}V_2^{(1)^{*}} & M^{(1)}_{2}\\
\end{bmatrix},
\end{equation}
where $M^{(1)}_i \in {\mathbb R}^{\frac{n}{2} \times \frac{n}{2}}, U^{(1)}_i, V^{(1)}_i \in {\mathbb R}^{\frac{n}{2}\times k}$, for $i = 1,2$, and $k\ll n$. A HODLR matrix
is obtained by applying~\eqref{eq:HODLR} recursively to the diagonal blocks $M^{(l-1)}_i$, where $i = 1,\ldots, 2^{l-1}$ for the $l$th level of recursion, $2 \leq l \leq p$.
The recursion terminates when the diagonal blocks are sufficiently small, that is, $\frac{n}{2^l} \leq n_{\min}$ for a minimal block size $n_{\min} \in \mathbb{N}$; see Figure~\ref{fig:h_hodlr_matrices} below
for an illustration. Formally, we define the set of HODLR matrices with block-wise rank $k$ as
\begin{equation*}
\mathcal{H}(k) := \left\lbrace M \in \mathbb{R}^{n \times n}: \rank M\vert_{\off} \leq k \text{ $\forall$off-diagonal block $M\vert_{\off}$ in recursive subdivision} \right\rbrace\text{.}
\end{equation*}
Any matrix $M \in \mathcal{H}(k)$ admits a data-sparse representation. By storing the off-diagonal blocks in
terms of their low-rank factors and the diagonal blocks as dense matrices, the memory required for representing $M$ is $\mathcal{O}(kn\log n)$, assuming that $k$ is constant with respect to $n$.
Given a general matrix $A \in \mathbb R^{n\times n}$, an approximation $M \in \mathcal{H}(k)$ to $A$ is obtained by computing
truncated singular value decompositions of the off-diagonal blocks of $A$. The quality of such an approximation is governed by the truncated singular values.
For simplifying the presentation, we have assumed that the ranks in the off-diagonal blocks are all bounded by the same integer $k$. In practice, we choose these ranks adaptively based on an absolute truncation tolerance $\epsilon$ and they may be different for each block.
As explained in~\cite{Bebendorf2008,Hackbusch2015}, several matrix operations can be performed approximately and efficiently within the HODLR format.
The use of formatted arithmetics leads to linear-polylogarithmic complexity for these operations. Table~\ref{table:complexity_HODLR} summarizes the complexity of operations needed by the QDWH algorithm
for $M_1,M_2,R\in \mathcal{H}(k)$, where $T$ is triangular, and $v\in \mathbb{R}^n$.
\begin{table}[ht!]
\caption{Complexity of some arithmetic operations in the HODLR format.}
\label{table:complexity_HODLR}
\centering \begin{tabular}{rcc}
\hline
Operation & & Computational complexity \\ \hline
Matrix-vector mult. $M_1*_{\mathcal{H}} v$ & & $\mathcal{O}(kn\log n)$ \\
Matrix addition $M_1 +_{\mathcal{H}} M_2 \in \mathcal{H}(k)$ & & $\mathcal{O}(k^2n\log n)$ \\
Matrix multiplication $M_1 *_{\mathcal{H}} M_2 \in \mathcal{H}(k)$ & & $\mathcal{O}(k^2n\log^2 n)$ \\
Cholesky decomposition $\h\operatorname{-Cholesky}(M_1) \in \mathcal{H}(k)$ & & $\mathcal{O}(k^2n\log^2 n)$ \\
Solving triangular system $M_1 *_{\mathcal{H}} T = M_2 \in \mathcal{H}(k)$ & & $\mathcal{O}(k^2n\log^2 n)$ \\ \hline
\end{tabular}
\end{table}
\begin{remark}
\label{remark:qr}
The QR-based iteration~\eqref{eq:qdwh_qr_reccur} of QDWH requires the computation of the QR decomposition~\eqref{eq:qdwh_qr_decomposition}.
Unlike for $\h$-Cholesky, there is no straightforward way of performing QR decompositions in hierarchical matrix arithmetics. To our knowledge, three different algorithms~\cite{Bebendorf2008,BennerMach2010,Lintner2004} have been proposed for this purpose.
However, each of them seems to have some drawbacks, e.g., failing to achieve a highly accurate decomposition or leading to loss of orthogonality in the orthogonal factor. Hence, instead of using any of the existing algorithms, we develop a
novel method in Section~\ref{qr-based-iteration} to compute the QR decomposition~\eqref{eq:qdwh_qr_reccur} that exploits the particular structure of the matrix in the first iteration of the QDWH algorithm.
\end{remark}
\subsubsection{Hierarchical matrices} \label{sec:hierarchicalmatrices}
Let $I = \lbrace 1, 2,\ldots, n\rbrace$ denote the row and column index sets of a matrix $M\in {\mathbb R}^{n\times n}$. To consider more general hierarchical matrices, we define a partition $P$ of $I \times I$ as follows. On level $l = 0$, the index set $I^0 := I$
is partitioned into $I^0 = I_1^1 \cup I_2^1$, with $I_1^1 = \lbrace 1,\ldots,\frac{n}{2} \rbrace$ and $I^1_2 = \lbrace \frac{n}{2}+1,\ldots,n \rbrace$.
At this point, the partition $P$ contains five blocks: $I\times I$ and $I^1_i\times I^1_j$ for $i,j = 1,2$. The subdivision continues as follows: on each level $l = 1,\ldots, p-1$ the index sets $I^l_i$ are
partitioned into sets $I^{l+1}_{2i-1}$ and $I^{l+1}_{2i}$ of equal size, contributing the blocks $I^{l+1}_i\times I^{l+1}_j$ for $i,j = 1,\ldots, 2^l$ to the partition $P$. The recursion terminates when a
block $I^l_i\times I^l_j$ satisfies a certain admissibility condition or when $\min\lbrace \vert I^l_i\vert, \vert I^l_j\vert\rbrace \leq n_{\min}$ holds.
\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.55]
\fill[gray] (0.5,7)--(1,7)--(1,7.5)--(0.5,7.5);
\fill[gray] (0,8)--(0,7.5)--(0.5,7.5)--(0.5,8);
\fill[gray] (1.5,6)--(2,6)--(2,6.5)--(1.5,6.5);
\fill[gray] (1,7)--(1,6.5)--(1.5,6.5)--(1.5,7);
\fill[gray] (2.5,5)--(3,5)--(3,5.5)--(2.5,5.5);
\fill[gray] (2,6)--(2,5.5)--(2.5,5.5)--(2.5,6);
\fill[gray] (3.5,4)--(4,4)--(4,4.5)--(3.5,4.5);
\fill[gray] (3,5)--(3,4.5)--(3.5,4.5)--(3.5,5);
\fill[gray] (4.5,3)--(5,3)--(5,3.5)--(4.5,3.5);
\fill[gray] (4,4)--(4,3.5)--(4.5,3.5)--(4.5,4);
\fill[gray] (5.5,2)--(6,2)--(6,2.5)--(5.5,2.5);
\fill[gray] (5,3)--(5,2.5)--(5.5,2.5)--(5.5,3);
\fill[gray] (6.5,1)--(7,1)--(7,1.5)--(6.5,1.5);
\fill[gray] (6,2)--(6,1.5)--(6.5,1.5)--(6.5,2);
\fill[gray] (7.5,0)--(8,0)--(8,0.5)--(7.5,0.5);
\fill[gray] (7,1)--(7,0.5)--(7.5,0.5)--(7.5,1);
\draw (0,0) rectangle (8,8);
\draw (0,0) rectangle (4, 4);
\draw (4,0) rectangle (8, 4);
\draw (0,4) rectangle (4, 8);
\draw (0,6)--(4,6);
\draw (2,4)--(2,8);
\draw (4,2)--(8,2);
\draw (6,0)--(6,4);
\draw (0,7)--(2,7);
\draw (1,6)--(1,8);
\draw (0,7.5)--(1,7.5);
\draw (0.5,7)--(0.5,8);
\draw (1,6.5)--(2,6.5);
\draw (1.5,6)--(1.5,7);
\draw (3,4)--(3,6);
\draw (2,5)--(4,5);
\draw (2,5.5)--(3,5.5);
\draw(2.5,5)--(2.5,6);
\draw (3,4.5)--(4,4.5);
\draw (3.5,4)--(3.5,5);
\draw (4,3)--(6,3);
\draw (5,2)--(5,4);
\draw (4,3.5)--(5,3.5);
\draw (4.5,3)--(4.5,4);
\draw[] (5,2.5)--(6,2.5);
\draw (5.5,2)--(5.5,3);
\draw (6,1)--(8,1);
\draw (7,0)--(7,2);
\draw (6,1.5)--(7,1.5);
\draw (6.5,1)--(6.5,2);
\draw (7,0.5)--(8,0.5);
\draw (7.5,0)--(7.5,1);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=0.55]
\fill[gray] (0,8)--(0,7)--(1,7)--(1,8);
\fill[gray] (1,7)--(1.5,7)--(1.5,7.5)--(1,7.5);
\fill[gray] (1,7)--(1,6.5)--(0.5,6.5)--(0.5,7);
\fill[gray] (1,7)--(1,6)--(2,6)--(2,7);
\fill[gray] (2,6)--(2.5,6)--(2.5,6.5)--(2,6.5);
\fill[gray] (2,6)--(2,5.5)--(1.5,5.5)--(1.5,6);
\fill[gray] (2,6)--(2,5)--(3,5)--(3,6);
\fill[gray] (3,5)--(3.5,5)--(3.5,5.5)--(3,5.5);
\fill[gray] (3,5)--(3,4.5)--(2.5,4.5)--(2.5,5);
\fill[gray] (3,5)--(3,4)--(4,4)--(4,5);
\fill[gray] (4,4)--(4.5,4)--(4.5,4.5)--(4,4.5);
\fill[gray] (4,4)--(4,3.5)--(3.5,3.5)--(3.5,4);
\fill[gray] (4,4)--(4,3)--(5,3)--(5,4);
\fill[gray] (5,3)--(5.5,3)--(5.5,3.5)--(5,3.5);
\fill[gray] (5,3)--(5,2.5)--(4.5,2.5)--(4.5,3);
\fill[gray] (5,3)--(5,2)--(6,2)--(6,3);
\fill[gray] (6,2)--(6.5,2)--(6.5,2.5)--(6,2.5);
\fill[gray] (6,2)--(6,1.5)--(5.5,1.5)--(5.5,2);
\fill[gray] (6,2)--(6,1)--(7,1)--(7,2);
\fill[gray] (7,1)--(7.5,1)--(7.5,1.5)--(7,1.5);
\fill[gray] (7,1)--(7,0.5)--(6.5,0.5)--(6.5,1);
\fill[gray] (7,1)--(7,0)--(8,0)--(8,1);
\draw (0,0) rectangle (8,8);
\draw (0,0) rectangle (4, 4);
\draw (4,0) rectangle (8, 4);
\draw (0,4) rectangle (4, 8);
\draw (0,2)--(8,2);
\draw (0,6)--(8,6);
\draw (2,0)--(2, 8);
\draw (6,0)--(6,8);
\draw (2,3)--(8,3);
\draw (4,1)--(8,1);
\draw (5,0)--(5,6);
\draw (7,0)--(7,4);
\draw (0,7)--(4,7);
\draw (0,5) -- (6,5);
\draw (3,2)--(3,8);
\draw (1,4)--(1,8);
\draw (0,7.5)--(2,7.5);
\draw (0.5, 6)--(0.5,8);
\draw (1.5,5)--(1.5,8);
\draw (0,6.5)--(3,6.5);
\draw (1,5.5)--(1,8);
\draw (2.5, 4)--(2.5,7);
\draw (1,5.5)--(4,5.5);
\draw (2,4.5)--(5,4.5);
\draw (3.5,3)--(3.5,6);
\draw (4.5,2)--(4.5,5);
\draw (3,3.5)--(6,3.5);
\draw(4,2.5)--(7,2.5);
\draw (5.5,1)--(5.5,4);
\draw (5,1.5)--(8,1.5);
\draw (6.5,0)--(6.5,3);
\draw (6,0.5)--(8,0.5);
\draw (7.5,0)--(7.5, 2);
\end{tikzpicture}
\caption{Left: HODLR matrix. Right: $\h$--matrix with admissibility condition~\eqref{eq:stand_adm}. Blocks colored grey are stored as dense matrices.}
\label{fig:h_hodlr_matrices}
\end{figure}%
Inspired by discretizations for 1D integral equations~\cite{Hackbusch1999}, we make use of the following admissibility condition:
\begin{equation}
\label{eq:stand_adm}
\text{block } \tau = t \times s\text{ is admissible } \Longleftrightarrow\ \min\{\diam(t), \diam(s) \}\leq \dist(t,s)\text{,}
\end{equation}
with $$\diam(t):= \underset{i,j\in t} \max \hskip 3pt \vert i-j\vert, \quad \dist(t,s):= \underset{i\in t, j\in s} \min\vert i-j\vert.$$
See Figure~\ref{fig:h_hodlr_matrices} for an illustration of the resulting partition $P$. Given $P$, the set of $\h$--matrices with block-wise rank $k$ is defined as
\begin{equation*}
\mathcal{H}(P,k) := \left\lbrace M \in \mathbb{R}^{n \times n}: \rank M\vert_{\tau} \leq k \text{ for all admissible blocks $\tau \in P$} \right\rbrace\text{.}
\end{equation*}%
The complexity of arithmetic operations displayed in Table~\ref{table:complexity_HODLR} extends to $\h(P,k)$.
\begin{example} \label{ex:hodlrvshmatrix} \rm
We investigate the potential of the HODLR and $\h$--matrix formats to efficiently store spectral projectors of banded matrices. For this purpose, we have generated, as explained in Section~\ref{sec:construct_test_matrices}, a symmetric $b$-banded matrix $A\in \mathbb{R}^{16000\times 16000}$ with
eigenvalues in $[-1,\hskip 3pt -\gap] \cup [\gap, \hskip 3pt 1]$. The memory
needed to store the full spectral projector $\Pi_{<0}(A)$ in double precision is $2048$ MB. We choose $n_{\min} = 250$, a truncation tolerance $\epsilon = 10^{-10}$, and $\gap \in \{10^{-1},10^{-4}\}$. Table~\ref{table:storage_spec} reveals that the HODLR format often requires less memory to approximately store $\Pi_{<0}(A)$, unless both $\gap$ and the bandwidth are large. In terms of computational time, the outcome is even clearer.
For bandwidth $b = 8$ and $\gap = 10^{-1}$, a situation that favors the $\h$--matrix format in terms of memory, we have run the algorithm described in Section~\ref{overall_algorithm} in both formats. It turned out that the use of the HODLR format
led to an overall time of $608$ seconds, while the $\h$--matrix format required $792$ seconds.
\end{example}
\begin{table}[h!]
\caption{Memory required to approximately store spectral projectors for the banded matrices from Example~\ref{ex:hodlrvshmatrix} in HODLR and $\h$--matrix format.}
\label{table:storage_spec}
\centering
\begin{tabular}[t]{c||c||c}
$\gap = 10^{-1}$ &HODLR &\text{$\h$--matrix} \\
\hline
\hline
b = 1 &$55.72$ MB & $95.16$ MB\\
b = 2 &$79.38$ MB &$96.42$ MB\\
b = 4 &$127.04$ MB &$106.54$ MB\\
b = 8 &$219.92$ MB &$151.06$ MB \\
b = 16 &$395.91$ MB &$291.85$ MB \\
\hline
\end{tabular}
\quad
\begin{tabular}[t]{c||c||c}
$\gap = 10^{-4}$ &HODLR &\text{$\h$--matrix} \\
\hline
\hline
b = 1 &$86.03$ MB & $128.58$ MB\\
b = 2 &$129.71$ MB &$160.56$ MB\\
b = 4 &$206.32$ MB &$225.72$ MB\\
b = 8 &$340.88 $ MB &$352.54$ MB \\
b = 16 &$567.69$ MB &$583.93$ MB \\
\hline
\end{tabular}
\end{table}
Based on the evidence provided by Example~\ref{ex:hodlrvshmatrix}, we have concluded that more general $\h$--matrix formats bring little advantage and thus focus on the HODLR format for the rest of this paper.
\subsection{A priori bounds on singular values and memory requirements}
To study the approximation of $\Pi_{<0}(A)$ in the HODLR format, we first derive bounds for the singular values of the off-diagonal ranks based on rational approximations to the $\sign$ function. In the
following, we say that a rational function $r$ is of type $(k,s)$ and write $r\in \mathcal{R}_{k,s}$ if $r = p/q$ holds for polynomials $p$ and $q$ of degree at most $k$ and $s$, respectively.
\subsubsection{Rational approximation of sign function}
\label{sec:rational_approx}
Given $R>0$, the min-max problem
\begin{equation}
\label{eq:min_max_rational}
\underset{r\in \mathcal{R}_{2m-1, 2m}}\min \underset{x \in [-R,-1]\cup [1,R]}\max \vert \sign(x) - r(x)\vert
\end{equation}
has a unique solution $s_{m}$. Called a Zolotarev function of type $(2m-1, 2m)$ corresponding to $R$ (see e.g.~\cite[Chapter 9]{Akhiezer1990}), this function takes the form
\begin{equation*}
s_{m}(x) := Cx \frac{\prod_{i = 1}^{m-1} (x^2 +c_{2i})}{ \prod_{i = 1}^{m} (x^2 +c_{2i-1})}.
\end{equation*}
The coefficients $c_i, i = 1,\ldots,2m$ are given in terms of the Jacobi elliptic function $\operatorname{sn}(\cdot;\kappa)$:
\begin{equation}
\label{eq:zolo_coeff}
c_i = \frac{\sn^2(\frac{iK(\kappa)}{2m};\kappa)}{1 - \sn^2(\frac{iK(\kappa)}{2m};\kappa)},
\end{equation}
where $\kappa = \sqrt{1 - 1/R^2}$ and $K(\kappa)$ is defined as the complete elliptic integral of the first kind
$$K(\kappa) = \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1 - \kappa^{2}\sin^2 \theta}} =\int_{0}^{1} \frac{dt}{\sqrt{(1-t^2)(1 - \kappa^2t^2)}}.$$
The constant $C$ is uniquely determined by the condition
\begin{equation*}
\underset{x \in [-R,-1]}\min 1 + s_{m}(x) = \underset{x \in [1,R]}\max 1 - s_{m}(x).
\end{equation*}
As shown in~\cite{GuetPoliTang2015}, the approximation error $E_{m} := \underset{x \in [-R,-1]\cup [1,R]}\max \vert \sign(x) - s_{m}(x)\vert$ is bounded as
\begin{equation}
\label{eq:error_sign_rational}
\frac{4\rho^{m}}{\rho^{m} +1} \leq E_{m} \leq 4\rho^{m},
\end{equation}
where $\rho = \rho(\mu) = \exp\big(-\frac{\pi K(\mu')}{2 K(\mu)}\big)$ with $\mu = \big(\frac{\sqrt{R}-1}{\sqrt{R}+1}\big)^{2}$ and $\mu' = \sqrt{1 - \mu^2}$.
The following lemma derives a bound from~\eqref{eq:error_sign_rational} that reveals the influence of the gap on the error.
\begin{lemma} \label{lemma:approxerror}
With the notation introduced above and $\gap = 1/R$, it holds that
\begin{equation}
\label{eq:upper_bound_simplified}
E_{m} \leq 4 \exp\bigg(-\frac{\pi^2m}{ 4\log \big( 4/ \sqrt[4]{\gap} + 2\big)}\bigg).
\end{equation}
\end{lemma}
\begin{proof}
Following Braess and Hackbusch~\cite{BraessHack2005}, we have
$$K(\mu') \geq \pi/2, \hskip 10pt K(\mu) \leq \log (4/ \mu' + 2)\text{.}$$
Thus, the upper bound in~\eqref{eq:error_sign_rational} implies
\begin{equation*}
E_{m} \leq 4\exp\bigg(-\frac{\pi^2m}{ 4\log ( 4/ \mu' + 2 )}\bigg).
\end{equation*}
From
\[
\mu' = \sqrt{1 - \left(\frac{1 -\sqrt{\gap}}{1+ \sqrt{\gap}}\right)^{4}} = \frac{\sqrt{8\sqrt{\gap}(1+\gap)}}{(1+\sqrt{\gap})^2} \geq \sqrt[4]{\gap}
\]
it follows that $\log ( 4/\mu' + 2 ) \leq \log ( 4/ \sqrt[4]{\gap} + 2)$, which completes the proof.
\end{proof}
It is simple to bound the ranks of the off-diagonal blocks for a rational function applied to a banded matrix.
\begin{lemma}
\label{lemma:rational_ranks}
Consider a $b$-banded matrix $A \in \mathbb{R}^{n\times n}$ and a rational function $r_{m}$ of type $(2m-1, 2m)$, with poles disjoint from the spectrum of $A$. Then the off-diagonal blocks of
$r_{m}(A)$ have rank at most $2mb$.
\end{lemma}
\begin{proof}
Assuming that $r$ has simple poles, let $r_{m}(x) = \sum_{i=1}^{2m} \omega_i (x - \mu_i)^{-1}$ be a partial fraction expansion of $r_{m}$, with $\omega_i, \mu_i\in\mathbb{C}, i = 1,\ldots, 2m$.
Thus, $r_{m}(A)$ is a sum of $2m$ shifted inverses of $A$. By a well known result (see, e.g.,~\cite{VandeVanBarelMastro2008}),
the off-diagonal blocks of each summand $B = A-\mu_i I$ satisfy $\rank B^{-1}|_{\off} = \rank B |_{\off}$. Noting that $\rank B |_{\off}=b$, because $B$ has bandwidth $b$, this completes the proof for simple poles. The result extends to non-simple poles by the semi-continuity of the rank function.
\end{proof}
\subsubsection{Singular value decay of off-diagonal blocks}
The results of Lemma~\ref{lemma:approxerror} and Lemma~\ref{lemma:rational_ranks} allow us to establish exponential decay
for the singular values of the off-diagonal blocks in $\Pi_{<0}(A)$ or, equivalently, in $\sign(A)$ for any symmetric banded matrix $A$. By rescaling $A$, we may assume without loss of generality that its
spectrum is contained in $[-R, \hskip 3pt -1] \cup [1, \hskip 3pt R]$. We let $\sigma_i(\cdot)$ denote the $i$th largest singular value of a matrix.
\begin{theorem}
\label{sign_sing_value_decay}
Consider a symmetric $b$-banded matrix $A\in\mathbb{R}^{n\times n}$ with the eigenvalues contained in $[-R, \hskip 3pt -1] \cup [1, \hskip 3pt R]$,
and $m \in \mathbb{N}$. Letting $\gap = 1/R$, the singular values of any off-diagonal block $\Pi_{<0}(A)\vert_{\off}$ satisfy
\begin{equation*}
\sigma_{2mb+1}(\Pi_{<0}(A)\vert_{\off}) \leq 2\exp\bigg(-\frac{\pi^2m}{ 4\log (4/\sqrt[4]{\gap} + 2)}\bigg).
\end{equation*}
\end{theorem}
\begin{proof}
Let $s_{m}$ denote the solution of the min-max problem~\eqref{eq:min_max_rational}.
Because
$s_{m}(A)\vert_{\off}$ has rank at most $2mb$ by Lemma~\ref{lemma:rational_ranks}, and the best rank-$i$ approximation error is governed by the $(i+1)$th largest singular value, it follows
from~\eqref{eq:upper_bound_simplified} that
\begin{align*}
\sigma_{2mb+1}(\sign(A)\vert_{\off}) &\le \Vert \sign(A) - s_{m}(A) \Vert_2 \leq \underset{x \in [-R,-1]\cup [1,R]}\max \vert \sign(x) - s_{m}(x)\vert \notag \\
&\leq 4\exp\bigg(-\frac{\pi^2m}{ 4\log (4/\sqrt[4]{\gap} + 2)}\bigg)\text{.}
\end{align*}
The statement thus follows from the relation $\Pi_{<0}(A)\vert_{\off} = -\frac{1}{2}\sign(A)\vert_{\off}$.
\end{proof}
\subsubsection{Memory requirements with respect to gap}
Theorem~\ref{sign_sing_value_decay} allows us to study the memory required to approximate $\Pi_{<0}(A)$ in the HODLR format to a prescribed accuracy. For this purpose,
let $\Pi^{\h}$ denote the best approximation in the Frobenius norm of $\Pi_{<0}(A)$ in the HODLR format with all off-diagonal ranks bounded by $2mb$.
Necessarily, the diagonal blocks of $\Pi^{\h}$ and $\Pi_{<0}(A)$ are the same. For an off-diagonal block of size $k$, Theorem~\ref{sign_sing_value_decay} implies
\begin{align*}
\Vert \Pi_{<0}(A)|_{\off} - \Pi^{\h}|_{\off} \Vert_F^2 & = \sum_{i = 2mb+1}^k \sigma_{i}(\Pi_{<0}(A)\vert_{\off})^2 \le
\sum_{j = m}^{\lceil k/2b \rceil -m } 2 b\, \sigma_{2j b+1}(\Pi_{<0}(A)\vert_{\off})^2 \\
&\le 8b \sum_{j = m}^{\lceil k/2b \rceil -m } \tau^{2j} \le \frac{8b}{1-\tau^2} \tau^{2m},
\end{align*}
with $\tau = \exp\Big(-\frac{\pi^2}{ 4\log (4/\sqrt[4]{\gap} + 2)}\Big)$. Taking into account the total number of off-diagonal blocks, we arrive at
\[
\Vert \Pi_{<0}(A) - \Pi^{\h} \Vert_F^2 \le \frac{16 b}{1-\tau^2} (n/n_{\min} - 1) \tau^{2m}
\]
Thus, the value of $m$ needed to attain $\Vert \Pi_{<0}(A) - \Pi^{\h} \Vert_F\le \delta$ for a
desired accuracy $\delta > 0$ satisfies $m = \mathcal{O}\big( \vert\log \gap \vert \cdot \log\big(bn \delta^{-1} \vert\log \gap \vert\big) \big)$.
The corresponding approximation $\Pi^{\h}$ requires
\begin{equation}
\label{eq:memory_sproj_rational}
\mathcal{O}\Big(\vert\log \gap \vert \cdot \log\big(bn \delta^{-1} \vert\log \gap \vert\big) bn \log n \Big)
\end{equation}
memory. Up to a double logarithmic factor, this shows that the memory depends logarithmically on the spectral gap.
\subsubsection{Comparison to approximate sparsity}
We now compare~\eqref{eq:memory_sproj_rational} with known results for approximate sparsity.
Assuming we are in the setting of Theorem~\ref{sign_sing_value_decay}, it is shown in~\cite{BenBoiRaz2013} that the off-diagonal entries of $\Pi_{<0}(A)$ satisfy
$$\vert (\Pi_{<0}(A))_{ij} \vert \leq C \e^{-\alpha \vert i-j \vert}, \qquad \alpha = \frac{1}{2b} \log\left(\frac{1+\gap}{1-\gap}\right),$$
for some constant $C>0$ depending only on $R$.
Let $\Pi^{(m)}$ denote the best approximation in the Frobenius norm to $\Pi_{<0}(A)$ by a matrix of bandwidth $m$. Following~\cite[Theorem 7.7]{BenBoiRaz2013}, we obtain
\[
\Vert \Pi_{<0}(A) - \Pi^{(m)} \Vert_{F} \leq \frac{C}{\sqrt{\alpha}}\sqrt{n} \e^{-\alpha m}\text{.}
\]
Choosing a value of $m$ that satisfies $m = \mathcal{O}\big( b \gap^{-1} \log\big(C bn \delta^{-1} \gap^{-1}\big) \big)$ thus ensures
an accuracy of $\delta > 0$, where we used $\alpha \approx \gap / b$.
Since the storage of $\Pi^{(m)}$ requires $\mathcal{O} (mn)$ memory, we arrive at
\begin{equation}
\label{eq:memory_sproj_poly}
\mathcal{O}\left(\frac{1}{\gap} \log\big(C bn \delta^{-1} \gap^{-1}\big) bn\right)
\end{equation}
memory. In contrast to the logarithmic dependence in~\eqref{eq:memory_sproj_rational}, the spectral gap now enters the asymptotic complexity inversely proportional.
On the other hand,~\eqref{eq:memory_sproj_rational} features a factor $\log n$ that is not present in~\eqref{eq:memory_sproj_poly}.
For most situations of practical interest, we expect that the much milder dependence on the gap far outweighs this additional factor. In summary,
the comparison between~\eqref{eq:memory_sproj_rational} and~\eqref{eq:memory_sproj_poly} provides strong theoretical justification for favoring the HODLR format over approximate sparsity.
\section{Conclusion}
In this paper we have developed a fast algorithm for computing spectral projectors of large-scale symmetric banded matrices. For this purpose, we have tailored the
ingredients of the QDWH algorithm, such that the overall algorithm has linear-polylogarithmic complexity. This allows us to compute highly accurate approximations to the spectral projector for very large sizes (up to $n = 1\,000\,000$ on a desktop computer) even when the involved spectral gap becomes small.
The choice of hierarchical low-rank matrix format is critical to the performance of our algorithm. Somewhat surprisingly,
we have observed that the relatively simple HODLR format outperforms a more general $\h$-matrix format. We have not investigated the choice of a format with nested low-rank factors,
such as HSS matrices. While such a nested format likely lowers asymptotic complexity, it presumably only pays off for larger values of $n$.
\section{Numerical experiments}
\label{experiments}
In this section, we demonstrate the performance of our preliminary {\sc Matlab}{} implementation of the hQDWH algorithm. All
computations were performed in {\sc Matlab}{} version 2014a on an Intel Xeon CPU with 3.07GHz, $4096$ KByte of level $2$ cache
and $192$ GByte of RAM. To be able to draw a fair comparison, all experiments were performed on a single core.
To measure the accuracy of the QDWH algorithm, we use the functions $e^{Q}_{\id}$, $e^{Q}_{\trace}$, $e^{Q}_{\SP}$ defined in~\eqref{eq:error_measures}. The error measures $e^{\h}_{\id}$, $e^{\h}_{\trace}$, $e^{\h}_{\SP}$ for the hQDWH algorithm are defined analogously.
In all experiments, we used the tolerance $\delta = 10^{-15}$ for stopping the QDWH/hQDWH algorithms. Unless stated otherwise, the truncation tolerance for recompression in the HODLR format is set to
$\epsilon = 10^{-10}$; the minimal block-size is set to
$n_{\min} = 250$ for tridiagonal matrices and $n_{\min} = 500$ for banded matrices.
The performance of the algorithm is tested on various types of matrices, including synthetic examples as well as examples from widely used sparse matrix collections.
\subsection{Construction of synthetic test matrices}
\label{sec:construct_test_matrices}
Given a prescribed set of eigenvalues $\lambda_1,\ldots, \lambda_n$ and a bandwidth $b$, we construct a symmetric $b$--banded matrix
by an orthogonal similarity transformation of $A = \diag(\lambda_1,\ldots, \lambda_n)$.
For this purpose, we perform the following operation for $i = n,n-1,\ldots,2$:
First, a Givens rotations $G(i-1,i,\alpha_i)$ is created by annihilating the second component of the vector $\big[ {a_{ii} \atop 1 } \big]$. The update $A \gets G(i-1,i,\alpha_i)^* A G(i-1,i,\alpha_i)$ introduces nonzero off-diagonal elements in $A$. For
$i=n,\ldots,n-b+1$, this fill-in stays within the $b$ bands. For $i \le n-b$, two undesired nonzero elements are created in row $i-1$ and column $i-1$ outside the $b$ bands. These nonzero elements are immediately chased off to the bottom right corner by applying $n-b-i+1$ Givens rotations, akin to Schwarz band reduction~\cite{Schwarz1968}.
When the procedure is completed, the $b$ bands of $A$ are fully populated.
In all examples below, we choose the eigenvalues to be uniformly distributed in $[-1,-\gap] \cup [\gap,1]$. Our results indicate that the
performance of our algorithm is robust with respect to the choice of eigenvalue distribution. In particular, the timings stay
almost the same when choosing a distribution geometrically graded towards the spectral gap.
\subsection{Results for tridiagonal matrices}
\begin{example}[\bfseries{Accuracy versus $\gap$}]
\label{ex:error_vs_gap_tridiag}
\rm First we investigate the behavior of the errors for hQDWH and QDWH with respect to the spectral gap. Using the construction from Section~\ref{sec:construct_test_matrices}, we consider $10000 \times 10000$ tridiagonal matrices with
eigenvalues in $[-1,\hskip 3pt -\gap] \cup [\gap, \hskip 3pt 1]$, where
$\gap$ varies from $10^{-15}$ to $10^{-1}$. From Figure~\ref{fig:error_vs_gap_tol_tridiag} (left), it can be seen that
tiny spectral gaps do not have a significant influence on the distance from identity and the trace error for both algorithms.
On the other hand, both $e^{\h}_{\SP}$ and $e^{Q}_{\SP}$ are sensitive to a decreasing gap, which reflects the ill-conditioning of the spectral projector for small gaps.
\end{example}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_tridiag/error_vs_gap_tridiag.eps}
\includegraphics[width=0.48\textwidth]{figures_tridiag/error_vs_tolerance_tridiag.eps}
\end{center}
\caption{Left (Example~\ref{ex:error_vs_gap_tridiag}): Comparison of accuracy for hQDWH and QDWH applied to tridiagonal matrices. Right (Example~\ref{ex:error_vs_tolerance_tridiag}):
Accuracy of hQDWH for different truncation tolerances.}
\label{fig:error_vs_gap_tol_tridiag}
\end{figure}
\begin{example}[\bfseries{Accuracy versus $\epsilon$}]
\label{ex:error_vs_tolerance_tridiag}
{\rm We again consider a tridiagonal matrix $A\in\mathbb{R}^{10000\times 10000}$, with the eigenvalues
in $[-1, \hskip 3pt -10^{-4}] \cup [10^{-4}, \hskip 3pt 1]$. The truncation tolerance $\epsilon$ for recompression in the HODLR format is varied in the interval
$[10^{-15}, 10^{-5}]$. Figure~\ref{fig:error_vs_gap_tol_tridiag} shows the resulting errors in the hQDWH algorithm. As
expected, the errors $e^{\h}_{\id}$, $e^{\h}_{\trace}$, $e^{\h}_{\SP}$ increase as $\epsilon$ increases. Both $e^{\h}_{\id}$ and $e^{\h}_{\SP}$ grow linearly with
respect to $\epsilon$, while $e^{\h}_{\trace}$ appears to be a little more robust. }
\end{example}
\begin{example}[\bfseries{Accuracy for examples from matrix collections}]
\label{ex:accuracy_applications}
\textnormal{We tested the accuracy of the hQDWH algorithm for the matrices from applications also considered in~\cite{MarqVomeDemmParl2009}:}
\begin{itemize}
\item \textnormal{Matrices from the BCSSTRUC1 set in the Harwell-Boeing Collection~\cite{Davis2007}. In these examples, a finite element discretization leads to a generalized eigenvalue problem $Kx = \lambda Mx$ with $K,M$ symmetric positive definite.
We consider the equivalent standard eigenvalue problem $L^{-1}KL^{-T}$, where $L$ denotes the Cholesky factor of $M$, and shift the matrix such that
approximately half of its spectrum is negative. Finally, the matrix is reduced to a tridiagonal matrix using the Matlab{} function \texttt{hess}.}
\item \textnormal{Matrices from UF Sparse Matrix Collection~\cite{Davis2007}. We consider the symmetric Alemdar and Cannizzo matrices, as well as
a matrix from the NASA set. Again, the matrices are shifted such that roughly half of their spectrum is negative and then reduced to tridiagonal form.}
\end{itemize}%
\begin{table}[h!]
\centering
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c||c|c|c|c|c|c}
&matrix &$n$ &$e^{\h}_{\id}$ &$e^{\h}_{\trace}$ &$e^{\h}_{\SP}$ &$\Vert \cdot \Vert_2$ &$\gap$\\
\hline
\hline
\multirow{3}{*}{\begin{sideways}\tiny{BCSSTRUC1}\end{sideways}} &\text{bcsst08} &$1074$ &$10^{-11}$ &$10^{-13}$ &$10^{-9}$ &$1.68\cdot10^{7}$ &$1.2615$\\
\cline{2-8}
&\text{bcsst09} &$1083$ &$10^{-10}$ &$10^{-11}$ &$10^{-8}$ &$4.29\cdot 10^{14}$ &$2.13\cdot 10^{10}$\\
\cline{2-8}
&\text{bcsst11} &$1474$ &$10^{-10}$ &$10^{-12}$ &$10^{-7}$ &$4.75\cdot 10^{9}$ &$1.58\cdot 10^{5}$\\
\hline
\hline
&\text{Cannizzo matrix} &$4098$ &$10^{-10}$ &$10^{-10}$ &$10^{-7}$ &$3.07\cdot 10^{8}$ &$0.728$\\
\cline{2-8}
&\text{nasa4704} &$4704$ &$10^{-10}$ &$10^{-12}$ &$10^{-9}$ &$2.07\cdot 10^{8}$ &$20.182$\\
\cline{2-8}
&\text{Alemdar matrix} &$6245$ &$10^{-10}$ &$10^{-11}$ &$10^{-7}$ &$69.5$ &$0.0079$\\
\cline{2-8}
\end{tabular}
}
\caption{Accuracy of hQDWH for tridiagonal matrices from Example~\ref{ex:accuracy_applications}.}
\label{table:error_test_matrices}
\end{table}%
{\rm Table~\ref{table:error_test_matrices} reveals that the hQDWH algorithm yields accurate approximations also
for applications' matrices. More specifically, in all examples, $e^{\h}_{\id}$ and $e^{\h}_{\trace}$ obtain values of order of $\epsilon$,
while $e^{\h}_{\SP}$ shows dependence on the relative spectral gap. }
\end{example}
\begin{example}[\textbf{Breakeven point relative to {\tt eig}}]
\textnormal{To compare the computational times of the hQDWH algorithm with \texttt{eig}, we
consider tridiagonal matrices with eigenvalues contained in $[-1, \hskip 3pt -\gap] \cup [\gap, \hskip 3pt 1]$ for various gaps.
Table~\ref{table:break_even_point_tridiag} shows the resulting breakeven points, that is, the value of $n$ such that hQDWH is faster than {\tt eig} for matrices of size at least $n$. Not surprisingly,
this breakeven point depends on the gap, as ranks are expected to increase as the gap decreases.
However, even for $\gap = 10^{-4}$, the hQDWH
algorithm becomes faster than \texttt{eig} for matrices of moderate size ($n \ge 3250$) and the ranks of the off-diagonal blocks in the HODLR representation of the spectral projector remain reasonably small.}
\begin{table}[ht!]
\centering
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c||c}
gap &breakeven point &max off-diagonal rank \\
\hline
\hline
$10^{-1}$ &$n=2250$ &$18$\\
\hline
$10^{-2}$ &$n=2500$ &$28$\\
\hline
$10^{-3}$ &$n=2750$ &$35$ \\
\hline
$10^{-4}$ &$n=3250$ &$37$\\
\hline
\end{tabular}
}
\caption{Breakeven point of hQDWH relative to \texttt{eig} for tridiagonal matrices. The last column shows the maximal off-diagonal rank in the output of hQDWH.}
\label{table:break_even_point_tridiag}
\end{table}
\end{example}
\begin{example}[\bfseries{Performance versus $n$}]
\label{ex:scaling_tridiag}
{\rm In this example, we investigate the asymptotic behavior of the hQDWH algorithm, in terms of computational time and memory, for tridiagonal matrices with eigenvalues in $[-1, \hskip 3pt -10^{-6}] \cup
[10^{-6}, \hskip 3pt 1]$. Figure~\ref{fig:gap_6_tridiag_res} indicates that the expected $\mathcal{O}(n\log^2n)$ computational time and $\mathcal{O}(n\log n)$ are nicely matched. The faster increase for smaller $n$ is due to fact that the off-diagonal ranks first grow from $30$ to $64$ until they settle around $64$ for sufficiently large $n$. }
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_tridiag/gap_6_tridiag_timing_new.eps}
\includegraphics[width=0.48\textwidth]{figures_tridiag/gap_6_tridiag_storage_new.eps}
\end{center}
\caption{Example~\ref{ex:scaling_tridiag}. Performance of hQDWH and \texttt{eig} applied to tridiagonal matrices with respect to $n$. Left: Computational time. Right: Memory requirements.}
\label{fig:gap_6_tridiag_res}
\end{figure}
\end{example}
\begin{example}[\bfseries{Performance for 1D Laplace}]
\label{ex:scaling_tridiag_gallery}
{\rm It is interesting to test the performance of the hQDWH algorithm for matrices for which the spectral gap decreases as $n$ increases. The archetypical example is the (scaled) tridiagonal matrix from the central difference discretization of the $1$D Laplace operator, with eigenvalues $\lambda_k = 2 - 2\cos\frac{k\pi}{n+1}$ for $k = 1,\ldots,n$. The matrix is shifted by $2$, such that half of its spectrum is negative and the eigenvalues become equal to $\lambda_k = - 2\cos\frac{k\pi}{n+1}$. The spectral gap is given by $\gap = 2\sin\frac{\pi}{n+1} = {\mathcal O}(1/n^2)$. According to Theorem~\ref{sign_sing_value_decay}, the numerical ranks of the off-diagonal blocks depend logarithmically on the spectral gap. Thus, we expect that the hQDWH algorithm
requires $\mathcal{O}(n\log^4n)$ computational time and $\mathcal{O}(n\log^2 n)$ memory for this matrix. Figure~\ref{fig:gallery_tridiag} nicely confirms this expectation. }
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_tridiag/gallery_tridiag_time.eps}
\includegraphics[width=0.48\textwidth]{figures_tridiag/gallery_tridiag_storage.eps}
\end{center}
\caption{Example~\ref{ex:scaling_tridiag_gallery}. hQDWH and \texttt{eig} for discretized 1D Laplace. Left: Computational time with respect to $n$. Right: Memory requirements with respect to $n$.}
\label{fig:gallery_tridiag}
\end{figure}
\end{example}
\begin{example}[\bfseries{Performance versus $n_{\min}$}]
\label{ex:time_vs_nb_tridiag}
{\rm The choice of the minimal block size $n_{\min}$ in the HODLR format influences the performance of hQDWH.
We have investigated this dependence for $50\,000\times 50\,000$ tridiagonal matrices with eigenvalues contained
in $[-1, \hskip 3pt -\gap] \cup [\gap, \hskip 3pt 1]$,
and $\gap \in \lbrace 10^{-1}, 10^{-4}, 10^{-6}\rbrace$. Figure~\ref{fig:time_vs_nb_tridiag} indicates that the optimal
value of $n_{\min}$ increases for smaller gaps. However, the execution time is not overly
sensitive to this choice; a value of $n_{\min}$ between $200$ and $500$ leads to
good performance.}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_tridiag/cpu_vs_nb_tridiag.eps}
\end{center}
\caption{Example~\ref{ex:time_vs_nb_tridiag}. Computational time of hQDWH versus $n_{\min}$.}
\label{fig:time_vs_nb_tridiag}
\end{figure}
\end{example}
\subsection{Results for banded matrices}
\begin{example}[\bfseries{Accuracy versus $\gap$}]
\label{ex:band_error_vs_gap}
{\rm Similarly to Example~\ref{ex:error_vs_gap_tridiag}, we study the impact of the spectral gap on the accuracy of hQDWH and QDWH for banded matrices.
Using once again the construction from Section~\ref{sec:construct_test_matrices}, we consider $10000 \times 10000$ banded matrices with bandwidth $8$
and eigenvalues in $[-1,\hskip 3pt -\gap] \cup [\gap, \hskip 3pt 1]$, where
$\gap$ varies from $10^{-15}$ to $10^{-1}$. The left plot of Figure~\ref{fig:error_vs_gap_tol_band} reconfirms the observations from Example~\ref{ex:error_vs_gap_tridiag}.}
\end{example}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_banded/error_vs_gap_band_8.eps}
\includegraphics[width=0.48\textwidth]{figures_banded/error_vs_tol_band_8.eps}
\end{center}
\caption{Left (Example~\ref{ex:band_error_vs_gap}): Comparison of accuracy for hQDWH and QDWH applied to banded matrices with bandwidth $8$. Right (Example~\ref{ex:band_error_vs_tol}):
Accuracy of hQDWH for different truncation tolerances.}
\label{fig:error_vs_gap_tol_band}
\end{figure}
\begin{example}[\bfseries{Accuracy versus $\epsilon$}]
\label{ex:band_error_vs_tol}
\textnormal{We investigate the influence of the truncation tolerance $\epsilon$ on accuracy for an $10\,000\times 10\,000$ banded matrix with bandwidth $b = 8$ and
the eigenvalues contained in $[-1, \hskip 3pt -10^{-4}] \cup [10^{-4}, \hskip 3pt 1]$. The right plot of Figure~\ref{fig:error_vs_gap_tol_band} reconfirms the observations from Example~\ref{ex:error_vs_tolerance_tridiag}. }
\end{example}
\begin{example}[\bfseries{Breakeven point relative to {\tt eig}}]
{\rm Table~\ref{table:break_even_point_banded} shows when hQDWH becomes faster than \texttt{eig} for $n\times n$ banded matrices with eigenvalues
contained in $[-1, \hskip 3pt -\gap] \cup [\gap, \hskip 3pt 1]$ for $\gap = 10^{-1}$ and $10^{-4}$. Compared to Table~\ref{table:break_even_point_tridiag}, the breakeven point is lower for bandwidths $b = 2$ and $b= 4$ than for bandwidth $1$. This is because {\tt eig} needs to perform tridiagonal reduction when $b\ge 2$. }
\begin{table}[ht!]
\centering
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c|c|c|c}
\backslashbox[5mm]{gap}{b} &$2$ &$4$ &$8$ &$16$\\
\hline
\hline
$10^{-1}$ &$n=1250$ &$n=1750$ &$n=2500$ &$n=5250$\\
\hline
$10^{-4}$ &$n=1750$ &$n=2500$ &$n=5000$ &$n=9500$\\
\end{tabular}
}
\caption{Breakeven point of hQDWH relative to \texttt{eig} applied for banded matrices with various bandwidths and spectral gaps.}
\label{table:break_even_point_banded}
\end{table}
\end{example}
\begin{example}[\bfseries{Performance versus $n$}]
\label{ex:scaling_banded_wrt_n}
\textnormal{We consider banded matrices with bandwidth $4$ and with eigenvalues contained in $[-1, \hskip 3pt -10^{-1}] \cup [10^{-1}, \hskip 3pt 1]$.
As in Example~\ref{ex:scaling_tridiag}, Figure~\ref{fig:gap_1_band_4_res} confirms that the computational time of hQDWH scales like $\mathcal{O}(n\log^2n)$ while memory scales like $\mathcal{O}(n\log n)$. Note that the maximal rank in the off-diagonal blocks
is $66$ for $n = 1\,000$, and $71$ for $n = 500\,000$.}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_banded/gap_1_band_4_timing_new.eps}
\includegraphics[width=0.48\textwidth]{figures_banded/gap_1_band_4_storage_new.eps}
\end{center}
\caption{Example~\ref{ex:scaling_banded_wrt_n}. Performance with respect to $n$ of hQDWH and \texttt{eig} applied to banded matrices with bandwidth $4$. Left: Computational time. Right: Memory requirements.}
\label{fig:gap_1_band_4_res}
\end{figure}
\end{example}
\begin{example}[\bfseries{Performance versus $b$}]
\label{ex:scaling_banded_wrt_b}
\textnormal{To verify the influence of the matrix bandwidth on the performance of our algorithm, we consider $100\,000\times 100\,000$ banded matrices with
eigenvalues contained in $[-1, \hskip 3pt -10^{-6}] \cup [10^{-6}, \hskip 3pt 1]$. Figure~\ref{fig:gap_6_band_res} clearly demonstrates that
computational time grows quadratically while memory grows linearly with respect to the bandwidth $b$.}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{figures_banded/gap_6_time_vs_band_new.eps}
\includegraphics[width=0.48\textwidth]{figures_banded/gap_6_store_vs_band_new.eps}
\end{center}
\caption{Example~\ref{ex:scaling_banded_wrt_b}.
Performance with respect to bandwidth $b$ of hQDWH applied to $100\,000\times 100\,000$ banded matrices. Left: Computational time. Right: Memory requirements.}
\label{fig:gap_6_band_res}
\end{figure}
\end{example}
\section{Computation of spectral projectors via QDWH}
\label{qdwh_algorithm}
In the following, we assume $\mu = 0$ without loss or generality, and thus consider the computation of the spectral projector $\Pi_{< 0}(A)$ associated with the negative eigenvalues of a
symmetric nonsingular matrix $A \in {\mathbb R}^{n\times n}$. Following~\cite{NakaHigh2013}, our approach is based on a well-known connection to the polar decomposition.
The polar decomposition~\cite[Chapter~9]{GolVanL2013} of $A$ takes the form $A = UH$ for an orthogonal matrix $U$ and a symmetric positive definite matrix $H$. Let
$A = V\Lambda V^{*}$ be a spectral decomposition of $A$ such that $\Lambda = \operatorname{diag}(\Lambda_{-}, \Lambda_{+})$, where $\Lambda_{-}$ and $\Lambda_{+}$
are diagonal matrices containing the $\nu$ negative and the $n-\nu$ positive eigenvalues of $A$, respectively. Then
\begin{align}
A &= V \operatorname{diag}(\Lambda_{-}, \Lambda_{+}) V^{*} \notag\\
&= \underbrace{V \operatorname{diag}(-I_{\nu}, I_{n-\nu}) V^{*}}_{=:U} \cdot \underbrace{V\operatorname{diag}(\vert \Lambda_{-}\vert, \vert \Lambda_{+}\vert) V^{*}}_{=:H} \notag
\end{align}
gives the polar decomposition of $A$.
In particular, this shows that the matrix sign function $\sign(A)$ coincides with the orthogonal factor $U$ from the polar decomposition.
More importantly, $\Pi_{< 0}(A) = \frac{1}{2}(I - U)$.
\subsection{QDWH algorithm}
\label{qdwh_algorithm1}
The QDWH algorithm~\cite{NakaBaiGygi2010} computes the polar factor $U$ of $A$ as the limit of the sequence $X_k$ defined by
\begin{align}
\label{eq:reccur}
X_0 &= A/\alpha\text{,}\notag \\
X_{k+1} &= X_k(a_k I + b_kX_k^{*}X_k)(I + c_kX_k^{*}X_k)^{-1}\text{.}
\end{align}
The parameter $\alpha > 0$ is an estimate of $\Vert A \Vert_2$. The parameters $a_k, b_k, c_k$ are computed via the relations
\begin{equation}
\label{eq:qdwh_parameters_abc}
a_k = h(l_k),\quad b_k = (a_k-1)^2/4, \quad c_k = a_k + b_k - 1.
\end{equation}
Representing a lower bound for the smallest singular value of $X_k$, the parameter $l_k$ is determined by the recurrence
\[
l_k = l_{k-1}(a_{k-1} + b_{k-1}l^{2}_{k-1})/(1 + c_{k-1}l^{2}_{k-1}), \quad k \geq 1,
\]
where $l_{0}$ is a lower bound for $\sigma_{\min}(X_0)$.
The function $h$ is given by
\begin{equation*}
h(l) = \sqrt{1+ \gamma} + \frac{1}{2}\sqrt{8 - 4\gamma + \frac{8(2 - l^2)}{l^2\sqrt{1 + \gamma}}},\quad \gamma = \sqrt[3]{\frac{4(1 - l^2)}{l^4}}.
\end{equation*}
The efficient estimation of $\alpha$ and $l_{0}$, required to start the recurrence, will be discussed in Section~\ref{overall_algorithm}.
The QDWH algorithm is cubically convergent and it has been shown in~\cite{NakaBaiGygi2010} that at most $k = 6$ iterations are needed to obtain convergence within tolerance $10^{-16}$, i.e.
$\Vert X_6 - U \Vert_2 < 10^{-16}$ for every matrix $A$ with $\kappa(A) \leq 10^{16}$.
The recurrence~\eqref{eq:reccur} has the equivalent form
\begin{subequations}
\label{eq:qdwh_qr_reccur}
\begin{align}
X_0 &= A/\alpha \label{eq:qdwh_it_0},\\
X_{k+1} &= \frac{b_k}{c_k}X_k + \frac{1}{\sqrt{c_k}}\left(a_k - \frac{b_k}{c_k}\right)Q_1Q_2^{*}\label{eq:qdwh_qr} \text{,}
\end{align}
\end{subequations}
with the QR decomposition
\begin{equation}
\label{eq:qdwh_qr_decomposition}
\begin{bmatrix}
\sqrt{c_k}X_k \\
I
\end{bmatrix} =
\begin{bmatrix}
Q_1 \\
Q_2
\end{bmatrix}R\text{.}
\end{equation}
Throughout the paper, we refer to~\eqref{eq:qdwh_qr_reccur} as a \emph{QR-based iteration}. On the other hand, as observed in~\cite{NakaHigh2013}, the recurrence~\eqref{eq:reccur} can also be rewritten in terms of the \emph{Cholesky-based iteration}
\begin{subequations}
\label{eq:qdwh_chol_reccur}
\begin{align}
Z_k &= I + c_k X_k^{*}X_k, \hskip 5pt W_k = \chol(Z_k), \label{eq:qdwh_chol_1}\\
X_{k+1} &= \frac{b_k}{c_k}X_k + \left( a_k - \frac{b_k}{c_k}\right)(X_kW_k^{-1})W_k^{-*}\label{eq:qdwh_chol2},
\end{align}
\end{subequations}
where $\chol(Z_k)$ denotes the Cholesky factor of $Z_k$.
Following~\cite{NakaBaiGygi2010}, either variant of the QDWH
algorithm is terminated when $l_k$ is sufficiently close to $1$, that is, $\vert 1 - l_k\vert \leq \delta$ for some stopping tolerance $\delta$, say $\delta = 10^{-15}$.
We mention that a higher--order variant of QDWH, called Zolo-pd, has recently been proposed by Freund and Nakatsukasa~\cite{NakaFreund2015}. This method approximates the polar
decomposition in at most two iterations but requires more arithmetic per iteration.
\subsection{Switching between QR-based and Cholesky-based iterations}
Due to its lower operation count, it can be expected that one Cholesky-based iteration~\eqref{eq:qdwh_chol_reccur} is faster than one QR-based iteration~\eqref{eq:qdwh_qr_reccur}. However, when $Z_k$ is ill-conditioned,
which is signaled by a large value of $c_k$,
the numerical stability of~\eqref{eq:qdwh_chol_reccur} can be jeopardized. To avoid this, it is proposed in~\cite{NakaHigh2013} to switch from~\eqref{eq:qdwh_qr_reccur} to~\eqref{eq:qdwh_chol_reccur} as soon as $c_k \leq 100$.
Since $c_k$ converges monotonically from above to $3$, this implies that this hybrid approach will first perform a few QR-based iterations and then switch for good to Cholesky-based iterations.
In fact, numerical experiments presented in~\cite{NakaHigh2013} indicate that at most two QR-based iterations are performed.
For reasons explained in Remark~\ref{remark:qr} below, we prefer to perform only one QR-based iteration and then switch to Cholesky-based iterations. To explore the impact of this choice on numerical accuracy,
we perform a comparison of the QDWH algorithm proposed in~\cite{NakaHigh2013} with a variant of QDWH that performs only one QR-based iteration. We consider the following error measures:
\begin{align} \label{eq:error_measures}
\begin{split}
e^{Q}_{\id} &:= \Vert U^2 - I\Vert_2,\\
e^{Q}_{\trace}&:= \vert \trace(U) - \trace(\sign(A))\vert,\\
e^{Q}_{\SP} &:= \Big\Vert \frac{1}{2}(I-U) - \Pi_{<0}(A)\Big\Vert_2,
\end{split}
\end{align}
where $U$ denotes the output of the QDWH algorithm, and $\Pi_{<0}(A)$ the spectral projector returned by the {\sc Matlab}{} function \texttt{eig}.
\begin{example} \rm
Let $A\in \mathbb{R}^{2000\times 2000}$ be a symmetric tridiagonal matrix constructed as described in Section~\ref{sec:construct_test_matrices}, such that
half of the spectrum of $A$ is contained in $[-1, \hskip 3pt -\gap]$ and the other half in $[\gap, \hskip 3pt 1]$, for
$\gap \in \left\lbrace 10^{-1}, 10^{-5}, 10^{-10}, 10^{-15}\right\rbrace$.
\begin{table}[h!]
\centering
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c|c|c|c|c|c}
Algorithm~\cite{NakaHigh2013} &gap &$10^{-1}$ &$10^{-5}$ &$10^{-10}$ &$10^{-15}$\\
\hline
\hline
\multirow{3}{6.5em}{one QR-based iteration~\eqref{eq:qdwh_qr_reccur}}&$e^{Q}_{\trace}$ &$5.55\cdot 10^{-17}$ &$7.22\cdot 10^{-16}$ &$2.22\cdot 10^{-16}$ &$1.11\cdot 10^{-16}$\\
\cline{2-6}
&$e^{Q}_{\id}$ &$1.15\cdot 10^{-15}$ &$2.41\cdot 10^{-15}$ &$1.84\cdot 10^{-15}$ &$1.82\cdot 10^{-15}$ \\
\cline{2-6}
&$e^{Q}_{\SP}$ &$1.87\cdot 10^{-14}$ &$4.35\cdot 10^{-12}$ &$1.88\cdot 10^{-6}$ & $1.91\cdot 10^{-2}$\\
\hline
\hline
\multirow{4}{6.5em}{several QR-based iterations~\eqref{eq:qdwh_qr_reccur}}&$e^{Q}_{\trace}$ &$5.55\cdot 10^{-17}$ &$1.22\cdot 10^{-15}$ &$1.53\cdot 10^{-16}$ &$6.25\cdot 10^{-16}$\\
\cline{2-6}
&$e^{Q}_{\id}$ &$1.15\cdot 10^{-15}$ &$2.58\cdot 10^{-15}$ &$1.81\cdot 10^{-15}$ &$2.04\cdot 10^{-15}$ \\
\cline{2-6}
&$e^{Q}_{\SP}$ &$1.87\cdot 10^{-14}$ &$2.12\cdot 10^{-12}$ &$2.82\cdot 10^{-6}$ & $3.06\cdot 10^{-2}$\\
\cline{2-6}
&\text{$\#$ of \eqref{eq:qdwh_qr_reccur}} &$1$ &$2$ &$2$ &$3$\\
\hline
\end{tabular}
}
\caption{Comparison of errors in the QDWH algorithm with one or several QR-based iterations.}
\label{table:iterations_vs_error}
\end{table}
As can be seen in Table~\ref{table:iterations_vs_error}, the errors obtained by both variants of the QDWH algorithm exhibit a similar behavior. Even for tiny spectral gaps,
no significant loss of accuracy is observed if only one QR-based iteration is performed.
\end{example}
\section{QR-based first iteration of QDWH}
\label{qr-based-iteration}
The first QR-based iteration of the QDWH algorithm requires computing the QR decomposition
\begin{equation} \label{eq:qrdecomp}
\begin{bmatrix}
c A \\
I
\end{bmatrix} =
\begin{bmatrix}
Q_1 \\
Q_2
\end{bmatrix}R
\end{equation}
for some scalar $c>0$. Without loss of generality, we suppose that $c = 1$. In this section, we develop an algorithm that requires $\mathcal O(b^2 n)$ operations for
performing this decomposition when $A$ is a $b$--banded matrix. In particular, our algorithm directly computes $Q_1$ and $Q_2$ in the HODLR format.
Since it is significantly simpler, we first discuss the case of a tridiagonal matrix $A$ before treating the case of general $b$.
It is interesting to note that the need for computing a QR decomposition of the form~\eqref{eq:qrdecomp} also arises in the solution of ill-posed inverse problems with
Tikhonov regularization; see, e.g.,~\cite{Bjorck1996}. However, when solving ill-posed problems, usually only the
computation of the upper-triangular factor $R$ is required, while the QDWH algorithm requires the computation of the orthogonal factor.
\subsection[QR decomposition for tridiagonal A]{QR decomposition of $\big[ {A \atop I} \big]$ for tridiagonal $A$}
\label{qr_tridiag}
For the case of a bidiagonal matrix $A$, Eld{\'e}n~\cite{Elden1977} proposed a fast algorithm for reducing a matrix $\big[ {A \atop I} \big]$ to upper triangular form. In the following, we
propose a modification of Eld{\'e}n's algorithm suitable for tridiagonal $A$.
\begin{figure}[ht!]
\begin{align*}
&\begin{array}{c@{\hspace{1mm}}}
\Rc\\[-0.4cm]
\\
\clr \\
\clr \\
\clr \\
\rc \\
\\
\\
\\
\end{array}
\underset{G(1,5,\beta_1)^{*}}{
\begin{lbmatrix}{4}
\circledr{$\times$} &\times \\
\times &\times &\times\\
&\times &\times &\times \\
& &\times &\times\\ \hline
\circled{1} &\textcolor{red}{\times} & &\\
&1 & &\\
& &1 &\\
& & &1\\
\end{lbmatrix}}\rightarrow
\begin{array}{c@{\hspace{1mm}}}
\Rc \\
\rc \\[-0.4cm]
\\
\\
\\
\\
\\
\\
\\
\end{array}
\underset{G(1,2,\gamma_1)^{*}}{\begin{lbmatrix}{4}
\circledr{$\times$} &\times &\textcolor{red}{\times} \\
\circledt{$\times$} &\times &\times\\
&\times &\times &\times \\
& &\times &\times\\ \hline
0 &\times & &\\
&1 & &\\
& &1 &\\
& & &1\\
\end{lbmatrix}}\rightarrow
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\Rc \\[-0.01cm]
\rc \\[-0.4cm]
\\
\\
\\
\end{array}
\underset{G(5,6,\alpha_2)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times\\
&\times &\times &\times \\
& &\times &\times\\ \hline
0 &\circledr{$\times$} & &\\
&\circled{1} & &\\
& &1 &\\
& & &1\\
\end{lbmatrix}} \rightarrow
\begin{array}{c@{\hspace{1mm}}}
\\
\Rc \\[0.05cm]
\clr \\ [0.05cm]
\clr \\
\rc \\[-0.05cm]
\\
\\
\\
\end{array}
\underset{G(2,5,\beta_2)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\circledr{$\times$} &\times\\
&\times &\times &\times \\
& &\times &\times\\ \hline
0 &\circledt{$\times$} &\textcolor{red}{\times} &\\
&0 & &\\
& &1 &\\
& & &1\\
\end{lbmatrix} }\rightarrow \\%\xrightarrow{G(2,n+1,\beta)^{*}}\\
&\begin{array}{c@{\hspace{1mm}}}
\\
\Rc \\[0.01cm]
\rc \\[-0.02cm]
\\
\\
\\
\\
\\
\end{array}
\underset{G(2,3,\gamma_2)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\circledr{$\times$} &\times &\textcolor{red}{\times}\\
&\circledt{$\times$} &\times &\times \\
& &\times &\times\\ \hline
0 &0 &\times &\\
&0 & &\\
& &1 &\\
& & &1\\
\end{lbmatrix}} \rightarrow
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\Rc \\[-0.01cm]
\clr \\%[-0.4cm]
\rc \\[-0.4cm]
\\
\\
\end{array}
\underset{G(5,7,\alpha_3)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times &\times\\
&0 &\times &\times \\
& &\times &\times\\ \hline
0 &0 &\circledr{$\times$}&\\
&0 & &\\
& &\circled{1} &\\
& & &1\\
\end{lbmatrix}}\rightarrow
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\Rc \\ [-0.4cm]
\\
\clr \\
\rc \\[-0.05cm]
\\
\\
\\
\end{array}
\underset{G(3,5,\beta_3)^{*}}{
\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times &\times\\
&0 &\circledr{$\times$} &\times \\
& &\times &\times\\ \hline
0 &0 &\circledt{$\times$} &\textcolor{red}{\times}\\
&0 & &\\
& &0 &\\
& & &1\\
\end{lbmatrix}}\rightarrow
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\Rc \\ [-0.4cm]
\\
\rc \\
\\
\\
\\
\\
\end{array}
\underset{G(3,4,\gamma_3)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times &\times\\
&0 &\circledr{$\times$} &\times \\
& &\circledt{$\times$} &\times\\ \hline
0 &0 &0 &\times\\
&0 & &\\
& &0 &\\
& & &1\\
\end{lbmatrix}}\rightarrow\\
&\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\Rc \\[-0.1cm]
\clr \\[0.1cm]
\clr \\ [0.05cm]
\rc \\[-0.5cm]
\\
\end{array}
\underset{G(5,8,\alpha_4)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times &\times\\
&0 &\times &\times \\
& &0 &\times\\ \hline
0 &0 &0 &\circledr{$\times$}\\
&0 & &\\
& &0 &\\
& & &\circled{1}\\
\end{lbmatrix}}\rightarrow
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\Rc \\ [-0.45cm]
\\
\rc \\[-0.01cm]
\\
\\
\\
\end{array}
\underset{G(4,5,\beta_4)^{*}}{\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times &\times\\
&0 &\times &\times \\
& &0 &\circledr{$\times$}\\ \hline
0 &0 &0 &\circledt{$\times$}\\
&0 & &\\
& &0 &\\
& & &0\\
\end{lbmatrix}}\rightarrow
\begin{lbmatrix}{4}
\times &\times &\times\\
0 &\times &\times &\times\\
&0 &\times &\times \\
& &0 &\times\\ \hline
0 &0 &0 &0\\
&0 & &\\
& &0 &\\
& & &0\\
\end{lbmatrix} \text{.}
\end{align*}
\caption{Fast QR decomposition of $\big[ {A \atop I} \big]$ for tridiagonal $A$ and $n = 4$. In each step, a Givens rotation is applied to the rows denoted by the arrows. Crosses denote generically nonzero elements, boxed/circled crosses are used to define Givens rotations, while
red crosses denote the fill-in during the current operation.}
\label{fig:qr_example}
\end{figure}
Our proposed algorithm is probably best understood from the illustration in Figure~\ref{fig:qr_example} for $n = 4$. In the $i$th step of the algorithm, all subdiagonal elements in
the $i$th column of $\big[ {A \atop I} \big]$ are annihilated by performing Givens rotations either with the diagonal element, or with the element $(n+1,i)$. By carefully choosing the order of annihilation,
only one new nonzero subdiagonal element is created in column $i+1$. The detailed pseudocode of this procedure is provided in Algorithm~\ref{alg:alg1}. We use $G(i,j,\alpha)$ to denote a Givens rotation of angle $\alpha$ that is applied to rows/columns $i$ and $j$.
\begin{algorithm}[ht!]
\caption{\text{Fast QR decomposition~\eqref{eq:qrdecomp} for tridiagonal $A$}}
\label{alg:alg1}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{algorithmic}[1]
\REQUIRE Tridiagonal matrix $A$.
\ENSURE Factors $Q,R$ of a QR decomposition of $\big[ {A \atop I} \big]$.
\STATE $Q \gets I_{2n}, R \gets \big[ {A \atop I} \big]$.
\STATE Construct $G(1,n+1,\beta_1)$ to annihilate $R(n+1,1)$.
\STATE Update $R \gets G(1,n+1,\beta_1)^{*}R$ and $Q \gets QG(1,n+1,\beta_1)$
\STATE Construct $G(1,2,\gamma_1)$ to annihilate $R(2,1)$.
\STATE Update $R \gets G(1,2,\gamma_1)^{*}R$ and $Q \gets QG(1,2,\gamma_1)$.
\vskip 1pt
\FOR{$i = 2,\ldots,n$}
\STATE Construct $G(n+1,n+i,\alpha_i)$ to annihilate $R(n+i,i)$.
\STATE Update $R \gets G(n+1,n+i,\alpha_i)^{*}R$ and $Q \gets QG(n+1,n+i,\alpha_i)$. \label{alpha}
\STATE Construct $G(i,n+1,\beta_i)$ to annihilate $R(n+1,i)$.
\STATE Update $R \gets G(i,n+1,\beta_i)^{*}R$ and $Q \gets QG(i,n+1,\beta_i)$.
\IF{$i<n$}
\STATE Construct $G(i,i+1,\gamma_i)$ to annihilate $R(i+1,i)$.
\STATE Update $R \gets G(i,i+1,\gamma_i)^{*}R$ and $Q \gets QG(i,i+1,\gamma_i)$.
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{alg:alg1} performs $3n-2$ Givens rotations in total. By exploiting its sparsity in a straightforward manner, only
$\mathcal{O}(n)$ operations and memory are required to compute the upper triangular factor $R$.
The situation is more complicated for the orthogonal factor. Since $Q$ is dense, it would require $\mathcal{O}(n^2)$ operations and memory to form $Q$ using Algorithm~\ref{alg:alg1}.
In the following section, we explain how the low-rank structure of $Q$ can be exploited to reduce this cost to $\mathcal{O}(n)$ as well.
\subsubsection{Ranks of off-diagonal blocks and fast computation of orthogonal factor}
For our purposes, it suffices to compute the first $n$ columns of the $2n\times 2n$ matrix $Q$, that is, the $n\times n$ matrices $Q_1 = Q(1:n,1:n)$ and $Q_2 = Q(n+1:2n,1:n)$.
The order of Givens rotations in Algorithm~\ref{alg:alg1} implies that $Q_1$ is an upper Hessenberg matrix while $Q_2$ is an upper triangular matrix.
The following theorem shows that all off-diagonal blocks of $Q_1,Q_2$ have rank at most two.
\begin{theorem}
\label{thm:ranks_tridiag}
For the orthogonal factor $Q$ returned by Algorithm~\ref{alg:alg1}, it holds that the matrices $Q(1:k,k+1:n)$ and $Q(n+1:n+k,k+1:n)$ have rank at most two for all $1 \leq k < n$.
\end{theorem}
\begin{proof}
We only prove the result for $Q(1:k,k+1:n)$; the proof for $Q(n+1:n+k,k+1:n)$ is analogous.
During steps $1,\ldots,k-1$ of Algorithm~\ref{alg:alg1}, $Q(1:k,k+1:n)$ is not modified and remains zero. In step $k$ of Algorithm~\ref{alg:alg1}, column $k+1$ of $Q$ is modified, while $Q(1:k,k+2:n)$ remains zero. After step $k$ has been completed,
let us set
\begin{equation}
\label{eq:span_tridiag}
\mathcal{U} := \operatorname{span} \lbrace Q(1:k,k+1), Q(1:k,n+1)\rbrace \subset {\mathbb R}^k.
\end{equation}
By construction, $\myspan Q(1:k,k+1:n) \subset {\mathcal U}$. In the following, we show by induction that this relation holds for all subsequent steps of Algorithm~\ref{alg:alg1}.
Suppose that $\myspan Q(1:k,k+1:n) \subset {\mathcal U}$ holds after $i$ steps for some $i$ with $k\le i \le n-1$. In step $i+1$, the following operations are performed:
\begin{enumerate}
\item $G(n+1,n+i+1,\alpha_{i+1})$ is applied to columns $n+1$ and $n+i+1$ of $Q$. Because $Q(1:k, n+i+1)$ is zero before applying the rotation, this simply effects a rescaling of column $n+1$ and thus $Q(1:k, n+1)\in \mathcal{U}$ remains true.
\item $G(i+1,n+1,\beta_{i+1})$ is applied to columns $i+1$ and $n+1$ of $Q$, which preserves $\myspan Q(1:k,k+1:n) \subset {\mathcal U}$.
\item If $i<n$, $G(i+1,i+2,\gamma_{i+1})$ is applied to columns $i+1$ and $i+2$ of $Q$, which again preserves $\myspan Q(1:k,k+1:n) \subset {\mathcal U}$.
\end{enumerate}
After completion of the algorithm, the column span of $Q(1:k,k+1:n)$ is thus contained in a subspace of dimension at most two. This proves the statement of the theorem.
\end{proof}
\begin{remark}
\label{construct_off}
The proof of Theorem~\ref{thm:ranks_tridiag} can be turned into a procedure for directly computing low-rank representations for the off-diagonal blocks of $Q_1, Q_2$ in the HODLR format. Due to the structure of $Q_1$ and $Q_2$,
all lower off-diagonal blocks have ranks $1$ and $0$ respectively, and the computation of their low-rank representations is straightforward. In the following, we therefore only discuss the computation of a low-rank representation for an upper off-diagonal $p \times s$ block
$Q_1|_{\off} = U_1V_1^{*} $ with $U_1\in\mathbb{R}^{p\times 2}$, $V_1\in\mathbb{R}^{s\times 2}$.
Let $r+1$ and $k+1$ denote the row and column in $Q_1$ that correspond to the first row and column of $Q_1|_{\off}$, respectively. The construction of
$Q_1|_{\off}$ begins in step $k$ of Algorithm~\ref{alg:alg1}, because $Q_1|_{\off}$ is zero before step $k$. During step $k$ only the first column of $Q_1|_{\off}$ is affected
by $G(k,k+1,\gamma_k)$; it becomes a scalar multiple of $Q_1(r+1:r+p,k)$.
After step $k$ of Algorithm~\ref{alg:alg1} is completed, we set $U_1 = [Q_1|_{\off}(:,1), Q(r+1:r+p,n+1)]$, as in~\eqref{eq:span_tridiag}. The matrix $V_1$ stores the coefficients in the basis $U_1$ of the columns in $Q_1|_{\off}$. Initially, $V_1 = [e_1, \mathbf{0}]$ with the first unit vector $e_1$. As we also need
to update the basis coefficients of $Q(r+1:r+p,n+1)$, we actually consider the augmented matrix
$V^{*} = \big[V_1^{*} \quad {0 \atop 1}\big]$. In all subsequent steps of Algorithm~\ref{alg:alg1},
we only apply Givens rotations to the corresponding columns of $V^{*}$. Note that the last column of $V^{*}$ is only rescaled, as it is always combined with a zero column.
After completing step $k+s$ of Algorithm~\ref{alg:alg1},
$Q_1|_{\off}$ remains unchanged and we extract the factor $V_1$ from the first $s$ columns of $V$.
Using the described procedure, the overall complexity to compute a low rank representation of $Q_1|_{\off}$ is
$\mathcal{O}(\max\lbrace p,s \rbrace)$.
The off-diagonal blocks of $Q_2$ are treated analogously.
\end{remark}
The QDWH algorithm makes use of the matrix product $Q_1 Q_2^*$, see~\eqref{eq:qdwh_qr}. Theorem~\ref{thm:ranks_tridiag}, together with the
upper Hessenberg/triangular structure, directly implies that the ranks of the lower and upper off-diagonal blocks of $Q_1 Q_2^*$ are bounded by three and two, respectively. In fact,
the following theorem shows a slightly stronger result.
\begin{theorem}
\label{thm:rank_tridiag_product}
Every off-diagonal block of $Q_1 Q_2^{*}$ for the orthogonal factor returned by Algorithm~\ref{alg:alg1} has rank at most $2$.
\end{theorem}
\begin{proof}
By Algorithm~\ref{alg:alg1} and Theorem~\ref{thm:ranks_tridiag}, the matrices $Q_1$ and $Q_2$ admit for any $1 \leq k < n$ a partitioning of the from
\begin{equation*}
Q_1 = \left[ \begin{array}{@{\,} c|c @{\,}}
X_1 & U_1V_1^{*}\\
\hline
\sigma e_1e_k^{*} & X_2 \\
\end{array} \right], \quad
Q_2 = \left[ \begin{array}{@{\,} c|c @{\,}}
Y_1 & U_2V_2^{*}\\
\hline
\phantom{e_1}\mathbf{0}\phantom{e} & Y_2 \\
\end{array} \right],
\end{equation*}
where $X_1\in {\mathbb R}^{k\times k}, X_2\in{\mathbb R}^{n-k\times n-k}$ are upper Hessenberg, $Y_1 \in {\mathbb R}^{k\times k}, Y_2\in{\mathbb R}^{n-k\times n-k}$ are upper triangular, $U_1, U_2 \in {\mathbb R}^{k\times 2}$, $V_1, V_2\in{\mathbb R}^{n-k\times 2}$, $\sigma \in{\mathbb R}$, and
$e_1, e_k$ denote unit vectors of appropriate lengths. The upper off-diagonal block of $Q_1Q_2^{*}$ equals to a product of rank-$2$ matrices
\begin{align*}
(Q_1 Q_2^{*})(1:k, k+1:n) &= X_1 \cdot \mathbf{0} + U_1 \underbrace{V_1^{*} Y_2^{*}}_{\widetilde{V}_1^{*}} = U_1\widetilde{V}_1^{*}\text{.}
\end{align*}
Moreover, the lower off-diagonal block amounts to a sum of a rank-$1$ and a rank-$2$ matrix
\begin{align*}
(Q_1 Q_2^{*})(k+1:n, 1:k) &= \sigma e_1 e_k^{*} Y_1^{*} + \underbrace{X_2 V_2}_{ \widetilde{V}_2 } U_2^{*}\\
&= \sigma e_1 Y_1(:,k)^{*} + \widetilde{V}_2 U_2^{*}\text{.}
\end{align*}
If $\sigma = 0$, the statement holds. Otherwise, we first show that the vectors $Y_1(:, k)$ and $U_2(:,1)$ are collinear. Let us recall that the vectors $Y_1(:,k)$, $U_2(:,1)$ coincide with the vectors
$Q(n+1:n+k+1,k)$, $Q(n+1:n+k+1,k+1)$ computed during step $k$ of Algorithm~\ref{alg:alg1}. As $Q(n+1:n+k+1,k)$ and $Q(n+1:n+k+1,k+1)$ are collinear
after performing step $k$, the same holds for $Y_1(:,k)$, $U_2(:,1)$, and $Y_1(:,k) = \eta U_2(:,1)$ for some $\eta \in \mathbb{R}$. Hence, we obtain
\begin{align*}
(Q_1 Q_2^{*})(k+1:n, 1:k) &= \sigma \eta e_1 U_2(:,1)^{*} + \widetilde{V}_2 U_2^{*} = \widehat{V}_2 U_2^{*}\text{,}
\end{align*}
which completes the proof.
\end{proof}
From Theorem~\ref{thm:rank_tridiag_product} and the recurrence~\eqref{eq:qdwh_qr_reccur} it follows that
the first iterate of the QDWH algorithm can be exactly represented in the HODLR format with off-diagonal ranks at most $3$.
\subsection[QR decomposition for banded A]{QR decomposition of $\big[ {A \atop I} \big]$ for banded $A$}
In this section, we discuss the QR decomposition of $\big[ {A \atop I} \big]$ for a banded symmetric matrix $A$ with bandwidth $b> 1$. Let us first note that
Eld{\'e}n~\cite{Elden1984} proposed a fast algorithm for reducing a matrix $\big[ {A \atop L} \big]$ with an upper triangular banded matrix $L$.
Eld{\'e}n's algorithm does not cover the fast computation of the orthogonal factor and requires the application of $(2b+1)n + nb - \frac{5}{2}b^2 - \frac{3}{2}b$ Givens rotations. In the following, we propose a different algorithm that only requires $(2b+1)n - b^2 - b$ Givens rotations.
Figure~\ref{fig:gap_6_band_res} illustrates the idea of our algorithm for $n = 6$ and $b = 3$. In the $i$th step of the algorithm, the subdiagonal elements in the $i$th column of $\big[ {A \atop I} \big]$ are annihilated as follows. A first group of Givens rotations ($\alpha_{i,j}$) annihilates all elements in row $n+i$, which consists of the diagonal element of $I$ and fill-in from the previous step. Then a Givens rotation ($\beta_{i}$) annihilates the element $(n+1,i)$. Finally, a second group of Givens rotations ($\gamma_{i,j}$) annihilates all subdiagonal elements of $A$.
The detailed procedure is given in Algorithm~\ref{alg:alg2}.
\begin{figure}[!ht]
\begin{align*}
&\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\\
\Rc \\[-0.01cm]
\rc \\[-0.4cm]
\\
\\
\\
\\
\end{array}
\underset{G(2,1,\alpha_{2,1})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\times &\times &\times &\times & \\
0 &\times &\times &\times &\times &\times \\
0 &\times &\times &\times &\times &\times\\
&\times &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &\circledr{$\times$} &\times &\times\\
&\circled{1} &\textcolor{red}{\times} &\textcolor{red}{\times}\\
& &1 \\
& & &1 \\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \hskip -7pt
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\\
\\
\Rc \\[-0.01cm]
\rc \\[-0.4cm]
\\
\\
\\
\end{array}
\underset{G(2,3,\alpha_{2,3})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\times &\times &\times &\times & \\
0 &\times &\times &\times &\times &\times \\
0 &\times &\times &\times &\times &\times\\
&\times &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &\times &\times &\times\\
&0 &\circledt{$\times$} &\times\\
& &\circledr{1} &\textcolor{red}{\times}\\
& & &1 \\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \hskip -7pt
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\\
\\
\Rc \\[-0.01cm]
\clr \\
\rc \\[-0.4cm]
\\
\\
\end{array}
\underset{G(2,4,\alpha_{2,4})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\times &\times &\times &\times & \\
0 &\times &\times &\times &\times &\times \\
0 &\times &\times &\times &\times &\times\\
&\times &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &\times &\times &\times\\
&0 &0 &\circledt{$\times$}\\
& &\times &\times\\
& & &\circledr{1}\\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \\
&\begin{array}{c@{\hspace{1mm}}}
%
\Rc \\[-0.35cm]
\\
\clr \\
\clr \\
\clr \\
\clr \\
\rc \\
\\
\\
\\
\\
\end{array}
\underset{G(2,n+1,\beta_2)^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\circledr{$\times$} &\times &\times &\times & \\
0 &\times &\times &\times &\times &\times \\
0 &\times &\times &\times &\times &\times\\
&\times &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &\circledt{$\times$} &\times &\times &\textcolor{red}{\times}\\
&0 &0 &0\\
& &\times &\times\\
& & &\times\\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \hskip -7pt
\begin{array}{c@{\hspace{1mm}}}
%
\Rc \\[0.0cm]
\rc \\[-0.35cm]
\\
\\
\\
\\
\\
\\
\\
\\
\\
\end{array}
\underset{G(2,3,\gamma_{2,3})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\circledr{$\times$} &\times &\times &\times &\textcolor{red}{\times} \\
0 &\circledt{$\times$} &\times &\times &\times &\times \\
0 &\times &\times &\times &\times &\times\\
&\times &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &0 &\times &\times &\times\\
&0 &0 &0\\
& &\times &\times\\
& & &\times\\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \hskip -7pt
\begin{array}{c@{\hspace{1mm}}}
%
\Rc \\[0.0cm]
\clr \\
\rc \\[-0.45cm]
\\
\\
\\
\\
\\
\\
\\
\\
\end{array}
\underset{G(2,4,\gamma_{2,4})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\circledr{$\times$} &\times &\times &\times &\times\\
0 &0 &\times &\times &\times &\times \\
0 &\circledt{$\times$} &\times &\times &\times &\times\\
&\times &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &0 &\times &\times &\times\\
&0 &0 &0\\
& &\times &\times\\
& & &\times\\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \hskip -7pt\\
&\begin{array}{c@{\hspace{1mm}}}
%
\Rc \\[0.0cm]
\clr \\
\clr \\
\rc \\[-0.45cm]
\\
\\
\\
\\
\\
\\
\\
\end{array}
\underset{G(2,5,\gamma_{2,5})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\circledr{$\times$} &\times &\times &\times &\times\\
0 &0 &\times &\times &\times &\times \\
0 &0 &\times &\times &\times &\times\\
&\circledt{$\times$} &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &0 &\times &\times &\times\\
&0 &0 &0\\
& &\times &\times\\
& & &\times\\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \hskip -7pt
\begin{array}{c@{\hspace{1mm}}}
\\
\\
\\
\\
\\
\Rc \\[-0.01cm]
\clr \\
\rc \\[-0.4cm]
\\
\\
\\
\end{array}
\underset{G(3,1,\alpha_{3,1})^{*}}{\begin{lbmatrix}{6}
\times &\times &\times &\times &\times &\times\\
0 &\times &\times &\times &\times &\times\\
0 &0 &\times &\times &\times &\times \\
0 &0 &\times &\times &\times &\times\\
&0 &\times &\times &\times &\times \\
& &\times &\times &\times &\times \\ \hline
0 &0 &\circledr{$\times$} &\times &\times\\
&0 &0 &0\\
& &\circledt{$\times$} &\times &\textcolor{red}{\times}\\
& & &\times\\
& & & &1 \\
& & & & &1 \\
\end{lbmatrix}} \rightarrow \cdots
\end{align*}
\caption{Second step of fast QR decomposition (Algorithm~\ref{alg:alg2}) of $\big[ {A \atop I} \big]$ for banded $A$ with $n = 6$ and $b = 3$.
In each step, a Givens rotation is applied to the rows denoted by the arrows. Crosses denote generically nonzero elements, boxed/circled crosses are used to define Givens rotations, while
red crosses denote the fill-in during the current operation.}
\label{fig:qr_example_banded}
\end{figure}
\begin{algorithm}[ht!]
\caption{\text{Fast QR decomposition~\eqref{eq:qrdecomp} for banded $A$}}
\label{alg:alg2}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{algorithmic}[1]
\REQUIRE Banded matrix $A$ with bandwidth $b$.
\ENSURE Factors $Q,R$ of a QR decomposition of $\big[ {A \atop I} \big]$.
\STATE $Q \gets I_{2n}$, $R \gets \big[ {A \atop I} \big]$.
\STATE Construct $G(1,n+1,\beta_1)$ to annihilate $R(n+1,1)$.
\STATE Update $R \gets G(1,n+1,\beta_1)^{*}R$ and $Q \gets QG(1,n+1,\beta_1)$
\FOR{$j = 2,\ldots,b+1$}
\STATE Construct $G(1,j,\gamma_{1,j})$ to annihilate $R(j,1)$.
\STATE Update $R \gets G(1,j,\gamma_{1,j})^{*}R$ and $Q \gets QG(1,j,\gamma_{1,j})$.
\ENDFOR
\vskip 1pt
\FOR{$i = 2,\ldots,n$}
\STATE Construct $G(n+1,n+i,\alpha_{i,i})$ to annihilate $R(n+i,i)$.
\STATE Update $R \gets G(n+1,n+i,\alpha_{i,i})^{*}R$ and $Q \gets QG(n+1,n+i,\alpha_{i,i})$.
\vskip 1pt
\FOR{$j = i+1,\ldots,\min \lbrace n,b+i-1\rbrace$}
\STATE Construct $G(n+i,n+j,\alpha_{i,j})$ to annihilate $R(n+i,j)$.
\STATE Update $R \gets G(n+i,n+j,\alpha_{i,j})^{*}R$ and $Q \gets QG(n+i,n+j,\alpha_{i,j})$
\ENDFOR
\vskip 1pt
\STATE Construct $G(i,n+1,\beta_i)$ to annihilate $R(n+1,i)$.
\STATE Update $R \gets G(i,n+1,\beta_i)^{*}R$ and $Q \gets QG(i,n+1,\beta_i)$.
\vskip 1pt
\IF{$i<n$}
\FOR{$j = i+1,\ldots,\min\lbrace n,b+i\rbrace$}
\STATE Construct $G(i,j,\gamma_{i,j})$ to annihilate $R(j,i)$.
\STATE Update $R \gets G(i,j,\gamma_{i,j})^{*}R$ and $Q \gets QG(i,j,\gamma_{i,j})$
\ENDFOR
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsubsection{Ranks of off-diagonal blocks and fast computation of orthogonal factor}
Due to the order of annihilation in Algorithm~\ref{alg:alg2}, it follows that $Q_1 = Q(1:n,1:n)$ is a $b$-Hessenberg matrix (that is, the matrix is zero below the $b$th subdiagonal) while
$Q_2 = Q(n+1:2n,1:n)$ is an upper triangular matrix.
The following result and its proof yield an ${\mathcal O}(b^2 n)$ algorithm for computing $Q_1$ and $Q_2$, analogous to Theorem~\ref{thm:ranks_tridiag} and Remark~\ref{construct_off}.
\begin{theorem}
\label{thm:ranks_banded}
For the orthogonal factor $Q$ returned by Algorithm~\ref{alg:alg2}, it holds that the matrices $Q(1:k,k+1:n)$ and $Q(n+1:n+k,k+1:n)$ have rank at most $2b$ for all $1 \leq k < n$.
\end{theorem}
\begin{proof}
Again, we prove the result for $Q(1:k,k+1:n)$ only.
After $k$ steps of Algorithm~\ref{alg:alg2} have been performed, we define the subspace
\begin{align*}
\mathcal{U} := \myspan \lbrace & Q(1:k,k+1),\ldots,Q(1:k,k+b),Q(1:k,n+1),\\
& Q(1:k,n+k+1),\ldots, Q(1:k,n+k+b-1) \rbrace,
\end{align*}
which is of dimension not larger than $2b$. At this point, the columns $Q(1:k,j)$ are zero for $j = k+b+1,\ldots,n$ and $j = n+k+b,\ldots,2n$. Thus,
\begin{equation} \label{eq:relationbanded}
\myspan Q(1:k,k+1:n+1) \subset {\mathcal U}, \qquad \myspan Q(1:k, n+k+1:2n) \subset {\mathcal U}
\end{equation}
hold after $k$ steps of Algorithm~\ref{alg:alg2}. We now show by induction that this relation holds for all subsequent steps.
Suppose that~\eqref{eq:relationbanded} holds after $i$ steps with $k\le i\le n-1$. In step $i+1$, the following
operations are performed by Algorithm~\ref{alg:alg2}:
\begin{enumerate}
\item $G(n+1,n+i+1,\alpha_{i+1,i+1})$ is applied to columns $n+1$ and $n+i+1$ of $Q$, which affects and preserves both inclusions in~\eqref{eq:relationbanded}. Then
$G(n+i+1,n+j,\alpha_{i+1,j})$ is applied to columns $n+i+1$ and $n+j$ of $Q$, for $j=i+2:\min \lbrace n,i+b\rbrace$, hence
$\myspan Q(1:k, n+k+1:2n) \subset {\mathcal U}$ remains true.
\item $G(i+1,n+1,\beta_{i+1})$ is applied to columns $i+1$ and $n+1$ of $Q$, preserving $\myspan Q(1:k,k+1:n+1) \subset {\mathcal U}$.
\item If $i+1< n$, $G(i+1,j,\gamma_{i+1,j})$ is applied to columns $i+1$ and $j$ of $Q$, for $j = i+2:\min \lbrace n, i+b+1\rbrace$, which
retains $\myspan Q(1:k,k+1:n+1) \subset {\mathcal U}$.
\end{enumerate}
Therefore~\eqref{eq:relationbanded} holds after Algorithm~\ref{alg:alg2} has been completed, which completes the proof of the theorem.
\end{proof}
The following result is an extension of Theorem~\ref{thm:rank_band_product} from the tridiagonal to the banded case. Its proof is very similar and therefore omitted.
\begin{theorem}
\label{thm:rank_band_product}
Every off-diagonal block of $Q_1Q_2^{*}$ for the orthogonal factor returned by Algorithm~\ref{alg:alg2} has rank at most $2b$.
\end{theorem}
\section{hQDWH algorithm}
\label{overall_algorithm}
Algorithm~\ref{alg:alg3}, summarizes the hQDWH algorithm proposed in this paper.
\begin{algorithm}[h!]
\caption{\text{hQDWH algorithm}}
\label{alg:alg3}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{algorithmic}[1]
\REQUIRE Symmetric banded matrix $A$ with bandwidth $b\ge 1$, minimal block-size $n_{\min} \ge 2$, truncation tolerance $\epsilon>0$, stopping tolerance $\delta>0$.
\ENSURE Approximation $P$ in HODLR format to spectral projector $\Pi_{<0}(A)$.
\STATE Choose initial parameters $\alpha, l_0$ of QDWH according to~\eqref{eqref:param_estimation}.\label{alg_alpha_l}
\STATE $X_0 = A/\alpha$.
\STATE $k = 0$.
\WHILE{$\vert 1 - l_k\vert > \delta$} \label{alg_stopping}
\STATE Compute $a_k$, $b_k$, $c_k$ according to the recurrence~\eqref{eq:qdwh_parameters_abc}.
\IF{$k = 0$}
\STATE Apply $\begin{displaystyle}\begin{cases}
\text{Algorithm~\ref{alg:alg1},} &\text{for } b = 1\\
\text{Algorithm~\ref{alg:alg2},} &\text{for } b > 1\\
\end{cases}\end{displaystyle}$
to $\begin{bmatrix}\sqrt{c_0}X_0\\ I\end{bmatrix}$ and store resulting array $G$ of Givens rotations.\label{alg_compute_givens}
\STATE Compute $Q_1$ and $Q_2$ from $G$ in HODLR format; see Remark~\ref{construct_off}. \label{alg_compute_Q}
\STATE $X_{1} = \frac{b_0}{c_0}*_{\h}X_0 +_{\h}\frac{1}{\sqrt{c_0}}\left(a_0 - \frac{b_0}{c_0}\right)*_{\h}Q_1*_{\h}Q_2^{*}$. \label{alg_QR_it}
\ELSE
\STATE $W_k = \h\operatorname{-Cholesky}(I +_{\h} c_k*_{\h}X_k^{*}*_{\h}X_k)$. \label{alg_chol1}
\STATE Solve upper-triangular system $Y_kW_k = X_k$ in HODLR format.
\STATE Solve lower-triangular system $V_kW_k^{*} = Y_k$ in HODLR format.
\STATE $X_{k+1} = \frac{b_k}{c_k}*_{\h}X_k +_{\h} \left(a_k - \frac{b_k}{c_k}\right)*_{\h}V_k$. \label{alg_chol}
\ENDIF
\STATE $k = k+1$.
\STATE $l_k = l_{k-1}(a_{k-1} + b_{k-1}l^{2}_{k-1})/(1 + c_{k-1}l^{2}_{k-1})$.
\ENDWHILE
\STATE $U = X_k$. \label{alg_sign}
\STATE $P =\frac{1}{2}*_{\h}(I -_{\h}U)$. \label{alg_sp}
\end{algorithmic}
\end{algorithm}
\noindent In the following, we comment on various implementation details of Algorithm~\ref{alg:alg3}.
\begin{description}
\item[line~\ref{alg_alpha_l}] As proposed in~\cite{NakaHigh2013}, the parameters $\alpha \gtrsim \Vert A \Vert_2$ and $l_0 \lesssim \sigma_{\min}(X_0)$ needed to start the QDWH algorithm are estimated as
\begin{equation}
\label{eqref:param_estimation}
\alpha = \texttt{normest}(A), \quad l_0 = \Vert A/\alpha \Vert_1 / (\sqrt{n}\cdot \texttt{condest}(A/\alpha))\text{,}
\end{equation}
where \texttt{normest} and \texttt{condest} denote the {\sc Matlab}{} functions for estimating the matrix $2$--norm using the power method and the $1$--norm condition number using~\cite{HighTiss2000}, respectively. Both functions exploit that $A$ is sparse and require ${\mathcal O}(bn)$ and ${\mathcal O}(b^2 n)$ operations, respectively.
\item[lines~\ref{alg_compute_givens}--~\ref{alg_QR_it}] This part of the algorithm deals with the implementation of the first QR-based iterate~\eqref{eq:qdwh_qr_reccur}. The generation of Givens rotations by Algorithms~\ref{alg:alg1} and~\ref{alg:alg2} for reducing $\big[ {\sqrt{c_0}X_0 \atop I} \big]$ to triangular form has been implemented in a \textsc{C} function, making use of the LAPACK routine {\tt DLARTG}. The function is called via a MEX interface and returns an array $G$ containing the cosines and sines of all rotations. This array is then used
in~\ref{alg_compute_Q} to generate $Q_1$ and $Q_2$ in the HODLR format, whose precise form is defined by the input parameter $n_{\min}$.
\item[lines~\ref{alg_chol1}--~\ref{alg_chol}] The computation of the $k$th iterate $X_k$, $k>1$, involves the Cholesky decomposition, addition, and the solution of triangular linear systems in the HODLR format. Existing techniques for HODLR matrices have been used for this purpose, see Section~\ref{hmatrices}, and repeated recompression with the absolute truncation tolerance $\epsilon$ is applied.
\end{description}
\begin{remark}
Algorithm~\ref{alg:alg3} extends in a straightforward way to the more general hierarchical matrix format from Section~\ref{sec:hierarchicalmatrices}.
The only major difference is the need for converting the matrices after line~\ref{alg_QR_it} from the HODLR to the hierarchical matrix format.
This extension of Algorithm~\ref{alg:alg3} was used in Example~\ref{ex:hodlrvshmatrix}.
\end{remark}
Assuming that all ranks in the off-diagonal blocks are bounded by $k \ge b$, Algorithm~\ref{alg:alg3} requires ${\mathcal O} (kn \log n)$ memory and $\mathcal{O}(k^2n\log^2 n)$ operations.
\section{Introduction}
Given a symmetric banded matrix $A\in \mathbb{R}^{n\times n}$ with eigenvalues
\begin{equation*}
\lambda_{1} \leq \cdots \leq \lambda_{\nu} < \mu < \lambda_{\nu+1} \leq \cdots \leq \lambda_{n},
\end{equation*}
we consider the computation of the spectral projector $\Pi_{< \mu}(A)$ associated with the eigenvalues $\lambda_{1}, \ldots ,\lambda_{\nu}$. We specifically target the
situation where both $n$ and $\nu$ are large, say $n = 100\,000$ and $\nu = 50\,000$, which makes approaches based on computing eigenvectors computationally expensive. For a tridiagonal matrix,
the MRRR algorithm requires $\mathcal{O}(\nu n)$ operations and memory~\cite{DhillParlVoemel2006}
to compute the $\nu$ eigenvectors needed to define $\Pi_{< \mu}(A)$.
There are a number of applications giving rise to the problem under consideration. First and foremost,
this task is at the heart of linear scaling methods for the calculation of the electronic structure of molecules with a large number of atoms. For insulators at zero
temperature, the density matrix is the spectral projector associated with the eigenvalues of the Hamiltonian below the so called
HOMO-LUMO gap; see~\cite{Goedecker1999} for an overview. The Hamiltonian is usually symmetric and, depending on the discretization and the structure of the molecule,
it can be (approximately) banded. A number of existing linear scaling methods use that this sometimes implies that the spectral projector may also admit a good approximation by a
banded matrix; see~\cite{BenBoiRaz2013} for a recent survey and a mathematical justification. For this approach to work well, the HOMO-LUMO gap should not become too small. For metallic systems,
this gap actually converges to zero, which makes it impossible to apply an approach based on approximate bandedness or, more generally, sparsity.
Another potential important application for banded matrices arises in dense symmetric eigenvalue solvers. The eigenvalues and eigenvectors of a symmetric dense
matrix $A$ are usually computed by first reducing $A$ to tridiagonal form and then applying either divide-and-conquer method or MRRR; see, e.g.~\cite{AuckBungHuckLang2011,DemmMarqParlVoem2008}
for recent examples. It is by no means trivial to implement the reduction to tridiagonal form efficiently so that it performs well on a modern computing architecture with a memory hierarchy. Most
existing approaches~\cite{AuckBlumBungHuck2011,BienIgualKressPet2011,HaidLtaiDong2011,HaidSolcGates2013,SoloBallDemmHoef016}, with the notable exception of~\cite{PetsPeisBien2013},
are based on successive band reduction~\cite{BiscLangSun2000}. In this context, it would be preferable to design an eigenvalue solver
that works directly with banded matrices, bypassing the need for tridiagonal reduction. While we are not aware of any such extension of MRRR, this possibility has been explored
several times for the divide-and-conquer method, e.g., in~\cite{Arbenz1992,HaidLtaiDong2012}. The variants proposed so far seem to suffer either from numerical instabilities
or from a complexity that grows significantly with the bandwidth. The method proposed in this paper can be used to directly compute the spectral
projector of a banded matrix, which in turn could potentially be used as a basis for a fast spectral divide and conquer algorithm in the spirit of Nakatsukasa and Higham~\cite{NakaHigh2013}.
To deal with small spectral gaps, one needs to go beyond sparsity. It turns out that
hierarchical matrices~\cite{Hackbusch1999}, also called $\h$--matrices, are much better suited in such a setting.
Intuitively, this can be well explained by considering the approximation of the Heaviside function $\Pi_{< \mu}(x)$ on the eigenvalues of $A$.
While a polynomial approximation of $\Pi_{< \mu}$ corresponds to a sparse approximation of $\Pi_{< \mu}(A)$~\cite{BenBoiRaz2013}, a rational approximation corresponds to
an approximation of $\Pi_{< \mu}(A)$ that features hierarchical low-rank structure. It is well known, see, e.g.,~\cite{PetrPop1987}, that a rational
approximation is more powerful in dealing with nearby singularities, such as $x = \mu$ for $\Pi_{< \mu}(x)$.
There are a number of existing approaches to use hierarchical low-rank structures for the fast computation of matrix functions,
including spectral projectors. Beylkin, Coult, and Mohlen\-kamp~\cite{BeylCoultMohl1999} proposed a combination of the Newton--Schulz iteration
with the HODLR format, a subset of $\h$--matrices, to compute spectral projectors for banded matrices. However,
the algorithm does not fully exploit the potential of low-rank formats; it converts a full matrix to the HODLR format in each iteration. In the context of Riccati and Lyapunov matrix equations,
the computation of the closely related sign function of an $\h$--matrix has been discussed in~\cite{GrasHackKhor2003,BaurBenner2006}.
The work in~\cite{GavrHackKhor2002,GavrHackKhor2004,GrasHackKhor2003} involves the $\h$--matrix approximation of resolvents, which is then used to compute the matrix exponential and related matrix functions.
Other hierarchical matrix techniques for eigenvalue problems include slicing-the-spectrum, which uses LDL decompositions to
compute eigenvalues in a specified interval for symmetric HODLR and HSS matrices~\cite{BennerMach2012} as well as $\h^{2}$--matrices~\cite{BennBoermMach2013}. Approximate
$\h$--matrix inverses can be used as preconditioners in iterative eigenvalue solvers; see~\cite{Lintner2002,Mach2012} for examples. Recently, Vogel et al.~\cite{VogelXiaCauBal2016}
have developed a fast divide-and-conquer method for computing all eigenvalues and eigenvectors in the HSS format. However, as the matrix of eigenvectors is represented in a factored form, it would be a
nontrivial and possibly expensive detour to compute spectral projectors via this approach.
In this paper we propose a new method based on a variant~\cite{NakaHigh2013} of the QR-based dynamically weighted Halley algorithm (QDWH)
for computing a polar decomposition~\cite{NakaBaiGygi2010}. Our method exploits the fact that the iterates of QDWH applied to a banded matrix
can be well approximated in the HODLR format. In fact, we show that the memory needed for storing the approximate spectral projector depends only logarithmically
on the spectral gap, a major improvement over approximate sparsity. The implementation of QDWH requires some care, in particular,
concerning the representation of the first iterate. One major contribution of this work is to show how this can be done efficiently.
The remainder of the paper is organized as follows. In Section~\ref{qdwh_algorithm}, we review the QDWH algorithm for computing a spectral projector $\Pi_{< \mu}(A)$. Section~\ref{hmatrices} recalls well-known
facts about the HODLR format and the corresponding formatted arithmetics. Based on the best rational approximation to the sign function, we derive new a priori bounds on the singular values
for off-diagonal blocks of $\Pi_{< \mu}(A)$, from which we deduce bounds on the memory required to store $\Pi_{< \mu}(A)$ approximately in the HODLR format. Section~\ref{qr-based-iteration} discusses
the efficient realization of the QR decomposition required in the first iterate of the QDWH algorithm. Section~\ref{overall_algorithm} summarizes our newly proposed QDWH algorithm in the HODLR format and
provides implementation details. Finally, numerical experiments both for tridiagonal and banded matrices are shown in Section~\ref{experiments}.
|
1,314,259,995,825 | arxiv | \section{Introduction}
Conformal field theories (CFT) are interesting for a variety of
reasons. One of the most important reason is that a theory critical at
a continuous phase transition is expected to acquire conformal
invariance which imposes strong constraints on the correlation
functions\cite{Polyakov1970}. This has motivated the idea of
bootstrap\cite{Polyakov1974}. Particularly in two dimensions these
ideas have been very fruitful \cite{Polyakov1984}. Reviews of later
developments and references are given in
\cite{DiFrancesco1997,Rychkov2016}.
The advent of the AdS/CFT correpondence
\cite{Maldacena,Polyakov,Witten1,Witten2} or ``holography'' between a
boundary CFT and a bulk gravity theory opened up another approach to
solving CFT's. \footnote{It also opens up the amazing possibility of
rewriting quantum gravity as a quantum field theory in flat space.}
There is a large amount of literature on this. See, for example,
\cite{Penedones2016} for a review.
In the AdS/CFT correspondence the radial direction can be interpreted
as the scale of the boundary field theory. Thus, a radial evolution
can be thought of as an RG evolution and has been dubbed ``holgraphic
RG''
\cite{Akhmedov,Akhmedov1,Akhmedov2,Alvarez,Kraus,Warner,Verlinde,Boer,Faulkner,Klebanov:1999tb,Heemskerk,Morris,Bzowski:2015pba,deHaro:2000vlm}.
The precise connection between the boundary RG and holographic RG is,
however, still an open question.
Recently a connection has been proposed between the Exact
Renormalization Group (ERG) equation
\cite{Wilson,Wegner,Wilson2,Polchinski} and the Holographic
Renormalization Group (Holographic RG) equation. It was shown in
\cite{Sathiapalan:2017frk} that the RG evolution operator for a Wilson
action of a D-dimensional field theory obeying the Polchinski ERG
equation can be formulated as a $D+1$-dimensional functional
integral. The extra dimension, corresponding to the moving scale $\Lambda$
of the ERG, makes it a ``holographic'' formulation. Furthermore, a
change of field variables or field redefinition maps the $D+1$
dimensional action for the functional integral to the action of a free
massive scalar field in $AdS_{D+1}$. It was then shown that the
calculation of the two point function reduces to the familiar
calculation using the AdS/CFT correspondence.
This proposal is quite general, and detailed calculations were done
for the Gaussian theory \cite{Sathiapalan:2017frk}. The scalar field
theory action has a free parameter, i.e., the mass of the scalar
field, which is related to the anomalous dimension of the boundary
operator in the AdS/CFT context . This parameter appears to come out
of nowhere. To understand the origin of the anomalous dimension
parameter, an ERG equation with anomalous dimension was analysed in
\cite{Sathiapalan:2019zex}. The same change of variables mapped this
to a scalar field theory in the AdS space-time, and this time it was
easy to see that the mass parameter is naturally related to the
anomalous dimension parameter in the ERG. Normally, interactions are
required for a field to have anomalous dimension. Since the exact RG
for interacting theories is difficult, a Gaussian theory with an
anomalous dimension introduced by hand was studied in
\cite{Sathiapalan:2019zex}.
In order to improve our understanding of the connection between ERG
and the AdS/CFT correspondence, it is necessary to have an interacting
example --- one needs a non-trivial boundary CFT and a fixed-point
Wilson action for this CFT \footnote{Note that the ``Wilson action''
always has a finite UV cutoff --- this is a point of departure from
the usual CFT actions written in the continuum.}. Then the RG
evolution of small perturbations to this theory can be studied by ERG.
Using the ideas of \cite{Sathiapalan:2017frk,Sathiapalan:2019zex} this
can be mapped to a scalar field theory in $D+1$-dimensional AdS
space. This would make a contact with more detailed AdS/CFT
calculations of higher point correlators. A well studied field theory
is the $\lambda \phi^4$ scalar field theory in $4-\epsilon$ dimensions that
has the famous Wilson-Fisher fixed point. When there are $N$ scalar
fields, this is often referred to as the $O(N)$ model. In this paper,
as a first step, we construct a fixed-point Wilson action for this
theory to order $\epsilon^2$. It is at this order that the anomalous
dimension first shows up. The action is obtained by solving the
fixed-point ERG equation perturbatively. The fixed-point equation
imposes the constraint of scale invariance.
In fact the theory is also
conformally invariant. This follows from the properties of the energy
momentum tensor --- if it is traceless the theory is conformally
invariant. Indeed the tracelessness of the energy-momentum tensor
defines what we mean by a CFT \cite{Callan:1970ze, Coleman2,Brown,
Polchinski2}. It is thus important to study the energy-momentum tensor and we construct it in this paper.
The energy-momentum tensor is also important in the context of
AdS/CFT: one of the really interesting aspects of the AdS/CFT
correspondence is that the $D+1$-dimensional bulk theory has dynamical
gravity. In addition to the scalar field, there is the gravitational
field that couples to the energy momentum tensor of the boundary CFT. Thus to extend the ideas
of \cite{Sathiapalan:2017frk,Sathiapalan:2019zex} to understand bulk gravity in AdS/CFT correspondence, from ERG
one has to construct the energy momentum operator.
The energy-momentum tensor for $\phi^4$ field theory
has been worked out in the dimensional regularization scheme
\cite{Brown}. The construction of the energy-momentum tensor from the
ERG point of view has been studied in general in
\cite{Sonoda-emt,Rosten2}. The main idea is to solve the Ward Identity associated with
coordinate transformations. This can be done in perturbation theory. We construct the leading terms that corresponds to the zero momentum
energy momentum tensor. One can also check that the trace
of the energy momentum tensor is proportional to the number operator. We apply this prescription here and
construct the zero momentum energy momentum tensor to $O(\lambda^2)$.
This paper is organized as follows: In Section 2 we give a review of
ERG and the fixed-point equation. We also give some background
material on the energy-momentum tensor. In Section 3 we construct the
solution to the fixed-point equation and obtain the fixed-point
action. In Section 4 we give a different approach to obtaining the fixed point equation and also calculate some correlation functions. In Section 5 the construction of the energy-momentum tensor is
given. We conclude the paper in Section 6.
\section{Background}
\label{secA}
\subsection{Exact Renormalization Group and Fixed Point equation}
\label{secA1}
We review the necessary background in this section. It depends mostly
on \cite{Igarashi, Sonoda-equiv}.
\subsubsection{ Exact Renormalization group}
\vspace{0.1 in} Renormalization means essentially going from one scale
$\Lambda_0$ to a lower scale $\Lambda$, where the initial scale
$\Lambda_0$ is typically called a bare scale. One will want to see how
the physics changes with scale. What do we mean by physics at
$\Lambda_0$? It means our theory will not be sensitive to momentum
$p >\Lambda_0$. The partition function of the full theory is given by
\begin{align*}
Z= \int\mathcal{D}\phi \, e^{-S[\phi]}
\end{align*}
where
\begin{align*}
S= \int_{p}\frac{1}{2} p^2 \phi^2 +S_{I}[\phi]
\end{align*}
To make it a partition function at scale $\Lambda_0$ we will try to
suppress the kinetic energy term for
$\infty < p < \Lambda_0 $. To execute this we will put a smooth cutoff
in the kinetic energy term to obtain the bare action
\begin{align}
S_B [\phi] \equiv \frac{1}{2}\int_p
\phi\frac{p^2}{K(p^2/\Lambda_0^2)} \phi + S_{I,B} [\phi]
\end{align}
and the bare partition function
\begin{equation}
Z_B \equiv \int \mathcal{D}\phi\, e^{- S_B [\phi]}
\end{equation}
We will choose the cutoff function will follow the condition \hspace{0.05 in}$K(0)=1$ and $K(\infty)=0$. In general cutoff functions satisfy stronger properties , but that will not affect the fixed point values of the couplings \cite{Igarashi-gamma}.
\vspace{0.1 in}
\vspace{0.02 in} Now we want to go to a lower scale $\Lambda$. For
that, observe the following identity
\begin{align*}
&\int \mathcal{D}\phi \exp \left[ -\frac{1}{2} \int_p
\phi(-p)\frac{1}{A(p)+B(p)}\phi(p)-S_{I,B}[\phi]\right]\\
=& \int \mathcal{D}\phi_1 \mathcal{D} \phi_2 \exp
\left[-\frac{1}{2}\int_p
\frac{1}{A(p)}\phi_1(-p)\phi_1(p)-\frac{1}{2}\int_p
\frac{1}{B(p)}\phi_2(-p)\phi_2(p)- S_{I,B}[\phi_1+\phi_2]\right]
\end{align*}
Using this we can write
\begin{align}
\nonumber Z_B= &\int \mathcal{D}\phi_l \mathcal{D} \phi_h \exp \bigg
\lbrace-\frac{1}{2}\int_p
\frac{p^2}{K(p^2/\Lambda^2)}\phi_l(-p)\phi_l(p)\\
\nonumber
-&\frac{1}{2}\int_p
\frac{p^2}{K(p^2/\Lambda_0^2)-K(p^2/\Lambda^2)}\phi_h(-p)\phi_h(p)-
S_{I,B}[\phi_l+\phi_h]\bigg \rbrace
\end{align}
We can effectively call $\phi_l(\phi_h)$ as low(high) energy field as
it is propagated by low(high) momentum propagator $\Delta_l(\Delta_h)$ defined below
\begin{align}\label{prop}
\Delta_l= \frac{K(p^2/\Lambda^2)}{p^2},\quad
\Delta_h=\frac{K(p^2/\Lambda^2)-K(p^2/\Lambda_0^2)}{p^2}
\end{align}
So we can write
\begin{align*}
Z_B= &\int \mathcal{D}\phi_l \exp \left[ -\frac{1}{2}\int_p \phi_l
\Delta_l^{-1} \phi_l\right] \int \mathcal{D}\phi_h \exp
\left[-\frac{1}{2} \int_p \phi_h \Delta_h^{-1} \phi_h- S_{I,B}[\phi_l+\phi_h]\right]\\
=& \int \mathcal{D} \phi_l \exp \left[ -\frac{1}{2}\int_p \phi_l
\Delta_l^{-1} \phi_l\right] \exp \lbrace -S_{I,\Lambda}
[\phi_l]\rbrace
\end{align*}
where
\begin{align}\label{SILambda}
\exp \lbrace -S_{I,\Lambda}[\phi_l] \rbrace \equiv \int
\mathcal{D}\phi_h \exp\bigg \lbrace
-\frac{1}{2}\int_p \phi_h\Delta_h^{-1}\phi_h-S_{I,B}[\phi_l+\phi_h]\bigg
\rbrace
\end{align}
$S_{I, \Lambda}$ is the interaction part of an effective low energy
field theory with a UV cutoff $\Lambda$.
\newline
Let
\begin{equation}\label{SLambda}
S_\Lambda[\phi] \equiv \frac{1}{2} \int_p \phi_l \Delta_l^{-1}
\phi_l+ S_{I,\Lambda} [\phi_l]
\end{equation}
be the whole action so that
\begin{equation}
Z_B = \int \mathcal{D} \phi_l\, e^{- S_\Lambda [\phi_l]}
\end{equation}
Using (\ref{SILambda}), we obtain
\begin{align}
e^{- S_\Lambda [\phi]}
&= \int \mathcal{D}\varphi\, \exp \left[ - S_B [\varphi] + \frac{1}{2}
\int_p \frac{p^2}{K(p/\Lambda_0)} \varphi (p) \varphi (-p)
- \frac{1}{2} \int_p \frac{p^2}{K(p/\Lambda)} \phi (p)
\phi (-p) \right.\notag\\
&\left.\quad - \frac{1}{2} \int_p \frac{p^2}{K(p/\Lambda_0) -
K(p/\Lambda)} \left( \varphi (p) - \phi (p)\right)\left(\varphi
(-p)-\phi (-p)\right) \right]\label{SB-SLambda}
\end{align}
where we have written $\phi_l$ as $\phi$ and $\phi_h$ as $\varphi - \phi$.
This will be useful later.
\vspace{0.1 in}
It is to be noted that one can go back to the bare partition function
anytime . For this reason this scheme is called \textbf{``exact''},
i.e. we lose no physical information by varying the scale. It is
easy to see this explicitly. Using (\ref{SB-SLambda}), we can
calculate the generating functional of $S_B$ using $S_\Lambda$ as
\begin{align}
&\int \mathcal{D} \phi\, \exp \left( - S_B [\phi] - \int_p J(-p) \phi
(p) \right)\notag\\
&= \exp \left[ \frac{1}{2} \int_p J(p) J(-p) \frac{1}{p^2} \left\lbrace
K(p/\Lambda_0)\left(1 - K(p/\Lambda_0)\right) -
\left(\frac{K(p/\Lambda_0)}{K(p/\Lambda)}\right)^2
K(p/\Lambda)\left(1 - K(p/\Lambda)\right) \right\rbrace
\right]\notag\\
&\qquad \times \int \mathcal{D} \phi\, \exp \left( - S_\Lambda [\phi] - \int_p
J(-p) \frac{K(p/\Lambda_0)}{K(p/\Lambda)}\, \phi (p)\right)
\label{ZBJ-SL}
\end{align}
We observe that the correlation functions of $S_B$ are the same as
those of $S_\Lambda$ up to the trivial (short-distance) contribution
to the two-point function and up to the momentum-dependent rescaling
of the field by $\frac{K (p/\Lambda_0)}{K (p/\Lambda)}$
\cite{Sonoda-equiv}. If we ignore the small corrections to the
two-point functions, we can write
\begin{equation}
\prod_{i=1}^n \frac{1}{K(p_i/\Lambda)}\, \vev{\phi (p_1) \cdots \phi
(p_n)}_{S_\Lambda}
= \prod_{i=1}^n \frac{1}{K(p_i/\Lambda')}\, \vev{\phi (p_1) \cdots \phi
(p_n)}_{S_{\Lambda'}}\label{SL-correlations}
\end{equation}
\subsubsection{Polchinski's ERG equation}
We have given an integral formula (\ref{SILambda}) for $S_{I,\Lambda}$ and
(\ref{SB-SLambda}) for $S_\Lambda$. It is easy to derive differential equations
from these. From (\ref{SILambda}), we obtain Polchinski's ERG
equation
\begin{equation}\label{polpsi}
- \Lambda \frac{\partial S_{I,\Lambda} [\phi]}{\partial \Lambda}
= \int_p (-) \frac{d K(p/\Lambda)}{dp^2} \left(-
\frac{\delta S_{I,\Lambda}[\phi]}{\delta \phi (p)}\frac{\delta
S_{I,\Lambda} [\phi]}{\delta \phi (-p)} + \frac{\delta^2
S_{I,\Lambda} [\phi]}{\delta \phi (p) \delta \phi (-p)} \right)
\end{equation}
for $S_{I,\Lambda}$. From (\ref{SB-SLambda}) we obtain
\begin{equation}
- \Lambda \frac{\partial S_\Lambda [\phi]}{\partial \Lambda}
= \int_p \left[ - 2 p^2 \frac{d \ln K(p/\Lambda)}{dp^2} \, \phi (p)
\frac{\delta S_\Lambda}{\delta \phi (p)} + \frac{d K(p/\Lambda)}{dp^2} \left(
- \frac{\delta S_\Lambda}{\delta \phi (p)} \frac{\delta
S_\Lambda}{\delta \phi (-p)} + \frac{\delta^2 S_\Lambda}{\delta
\phi (p) \delta \phi (-p)} \right)\right]\label{ERG-SL}
\end{equation}
for the entire Wilson action.
\subsubsection{The limit $\Lambda \to 0+$}
In the limit $\Lambda \to 0+$ we expect $S_\Lambda [\phi]$ approaches something related
to the partition function. If we substitute
\begin{equation}
\lim_{\Lambda \to 0+} K(p/\Lambda) = 0
\end{equation}
into (\ref{SB-SLambda}), we get
\begin{align}
&\lim_{\Lambda \to 0+} e^{- S_\Lambda [\phi] + \frac{1}{2} \int_p
\frac{p^2}{K(p/\Lambda)} \phi (p) \phi (-p)} = \lim_{\Lambda \to 0+}
e^{- S_{I,\Lambda} [\phi]} \notag\\
&= e^{- \frac{1}{2} \int_p \frac{p^2}{K (p/\Lambda_0)} \phi (p) \phi
(-p)} \int \mathcal{D} \varphi\, \exp \left[ - S_B [\varphi] +
\int_p \frac{p^2}{K(p/\Lambda_0)} \varphi (p) \phi (-p) \right]
\end{align}
Hence, rewriting $\phi (p)$ by $\frac{K(p/\Lambda_0)}{p^2} J (p)$, we
obtain the generating functional of the bare theory as the $\Lambda
\to 0+$ limit of $S_{I,\Lambda}$:
\begin{align}
Z_B [J] &\equiv \int \mathcal{D} \varphi\, \exp \left[ -S_B [\varphi] - \int_p \varphi
(p) J(-p)\right]\notag\\
&= e^{- \frac{1}{2} \int_pJ(p) J(-p) \frac{K(p/\Lambda_0)}{p^2}}
\lim_{\Lambda \to 0+} \exp \left( - S_{I,\Lambda} \left[
\frac{K(p/\Lambda_0)}{p^2} J(p) \right] \right)\label{ZBJ-limit}
\end{align}
\subsubsection{IR limit of a critical theory}
For the bare theory at criticality, we expect that the correlation
functions
\begin{equation}
\vev{\varphi (p_1) \cdots \varphi (p_n)}_B
\equiv \int \mathcal{D} \varphi\, \varphi (p_1) \cdots \varphi
(p_n)\,e^{-S_B [\varphi]}
\end{equation}
to become scale invariant in the IR limit, i.e., for small momenta.
To be more precise, we can define the limit
\begin{equation}
\mathcal{C} (p_1,\cdots,p_n) \equiv \lim_{t \to \infty} e^{\frac{n}{2}
\left( -(D+2) + \eta \right) t}\vev{\varphi (p_1 e^{-t}) \cdots
\varphi (p_n
e^{-t})}_B \label{IRlimit}
\end{equation}
where $\frac{\eta}{2}$ is the anomalous dimension.
What does this mean for $S_\Lambda$ in the limit $\Lambda \to 0+$? As
we have seen above, the interaction part $S_{I,\Lambda}$ becomes the
generating functional of the bare theory in this limit. Since only
the IR limit of the correlation functions are scale invariant, only
the low momentum part of $\lim_{\Lambda \to 0+} S_{I,\Lambda}$
corresponds to the scale invariant theory defined by the IR limit
(\ref{IRlimit}).
To understand the IR limit better, we follow Wilson \cite{Wilson} and
reformulate the ERG trasnformation in two steps:
\begin{enumerate}
\item introduction of an anomalous dimension (section
\ref{subsub:anomalous}) --- the anomalous dimension is an important
ingredient of the IR limit. We need to introduce an anomalous
dimension of the field within ERG.
\item introduction of a dimensionless framework (section \ref{subsub:dimensionless}) ---
each time we lower the cutoff $\Lambda$ we have to rescale space to
restore the same momentum cutoff. This is necessary to realize
scale invariance within ERG.
\end{enumerate}
\subsubsection{Anomalous dimension in ERG\label{subsub:anomalous}}
The cutoff dependent Wilson action $S_\Lambda [\phi]$ has two parts:
\begin{equation}
S_\Lambda [\phi] = \frac{1}{2} \int_p \frac{p^2}{K(p/\Lambda)} \phi
(p) \phi (-p) + S_{I,\Lambda} [\phi]
\end{equation}
The first term is a kinetic term, but this is not the only kinetic
term; part of the interaction quadratic in $\phi$'s also contains the
kinetic term. The normalization of $\phi$ has no physical
meaning, and it is natural to normalize the field so that
$S_{I,\Lambda}$ contains no kinetic term.
To do this, we modify the ERG differential equation (\ref{ERG-SL}) by
adding a number operator \cite{Igarashi, Igarashi-gamma}:
\begin{align}
- \Lambda \partial_\Lambda S_\Lambda [\phi]
&= \int_p \left( - 2 p^2 \frac{d}{d p^2} \ln K(p/\Lambda)\, \phi (p)
\frac{\delta S_\Lambda}{\delta \phi (p)} - \frac{d}{dp^2}
K(p/\Lambda) \left\lbrace \frac{\delta^2 S_\Lambda}{\delta \phi (p) \delta
\phi (-p)} - \frac{\delta S_\Lambda}{\delta \phi (p)} \frac{\delta
S_\Lambda}{\delta \phi (-p)} \right\rbrace \right)\notag\\
&\quad - \frac{\eta_\Lambda}{2} \mathcal{N}_\Lambda [\phi]\label{ERG-SL-eta}
\end{align}
where the number operator $\mathcal{N}_\Lambda [\phi]$ is defined by
\begin{equation}
\mathcal{N}_\Lambda [\phi] \equiv \int_p \left[ \phi (p) \frac{\delta
S_\Lambda}{\delta \phi (p)} + \frac{K(p/\Lambda) \left(1 -
K(p/\Lambda)\right)}{p^2} \left\lbrace \frac{\delta^2 S_\Lambda}{\delta \phi (p) \delta
\phi (-p)} - \frac{\delta S_\Lambda}{\delta \phi (p)} \frac{\delta
S_\Lambda}{\delta \phi (-p)} \right\rbrace \right]\label{NL}
\end{equation}
This counts the number of fields:
\begin{equation}
\vev{\mathcal{N}_\Lambda [\phi]\,\phi (p_1) \cdots \phi (p_n)}_{S_\Lambda} = n \vev{\phi
(p_1) \cdots \phi (p_n)}_{S_\Lambda}
\end{equation}
(Again we are ignoring small corrections to the two-point functions.)
Under (\ref{ERG-SL-eta}) the correlation functions change as
\begin{equation}
\prod_{i=1}^n \frac{1}{K(p_i/\Lambda)} \, \vev{\phi (p_1) \cdots \phi
(p_n)}_{S_\Lambda} =
\left(\frac{Z_\Lambda}{Z_{\Lambda'}}\right)^{\frac{n}{2}}
\prod_{i=1}^n \frac{1}{K(p_i/\Lambda')}\, \vev{\phi (p_1) \cdots \phi
(p_n)}_{S_{\Lambda'}}\label{SL-correlators-eta}
\end{equation}
where $Z_\Lambda$ is the solution of
\begin{equation}
- \Lambda \frac{\partial}{\partial \Lambda} Z_\Lambda = \eta_\Lambda\,
Z_\Lambda
\end{equation}
satisfying the initial condition
\begin{equation}
Z_{\Lambda_0} = 1
\end{equation}
We can choose $\eta_\Lambda$ so that $S_\Lambda$ has the same kinetic
term independent of $\Lambda$. For (\ref{ERG-SL-eta}), the integral
formula (\ref{SB-SLambda}) must be changed to \cite{Sonoda-equiv}
\begin{align}
e^{S_\Lambda [\phi]}
&= \int \mathcal{D} \varphi\, e^{S_0 [\varphi]}\notag\\
&\quad \times \exp \left[ - \frac{1}{2} \int_p
\frac{p^2}{\frac{1-K(p/\Lambda)}{Z_\Lambda K(p/\Lambda)} -
\frac{1-K(p/\Lambda_0)}{K(p/\Lambda_0)}} \left( \frac{\varphi (p)}{K(p/\Lambda_0)} -
\frac{\phi (p)}{\sqrt{Z_\Lambda}\,K(p/\Lambda)}\right)
\left( \frac{\varphi (-p)}{K(p/\Lambda_0)} -
\frac{\phi (-p)}{\sqrt{Z_\Lambda}\,K(p/\Lambda)}\right) \right]\label{SB-SLambda-eta}
\end{align}
This reduces to (\ref{SB-SLambda}) for $Z_\Lambda = 1$.
\subsubsection{Dimensionless framework\label{subsub:dimensionless}}
To reach the IR limit (\ref{IRlimit}) we must look at smaller and
smaller momenta as we lower the cutoff $\Lambda$. We can do this by
measuring the momenta in units of the cutoff $\Lambda$. At the same
time we render all the dimensionful quantities such as $\phi (p)$
dimensionless by using appropriate powers of $\Lambda$.
We introduce a dimensionless parameter $t$ by
\begin{equation}
\Lambda = \mu \, e^{-t}
\end{equation}
where $\mu$ is an arbitary fixed momentum scale. We then define the
dimensionless field with dimensionless momentum by
\begin{equation}
\bar{\phi} (p) \equiv \Lambda^{\frac{D+2}{2}} \phi (p \Lambda)
\end{equation}
and define a Wilson action parametrized by $t$:
\begin{equation}
\bar{S}_t [\bar{\phi}] \equiv S_\Lambda [\phi]
\end{equation}
We can now rewrite (\ref{ERG-SL-eta}) for $\bar{S}_t$:
\begin{align}
\partial_t \bar{S}_t [\bar{\phi}]
&= \int_p \left( - 2 p^2 \frac{d}{dp^2} \ln K(p) + p \cdot \partial_p
+ \frac{D+2}{2} \right) \bar{\phi} (p) \cdot \frac{\delta \bar{S}_t
[\bar{\phi}]}{\delta \bar{\phi} (p)}\notag\\
&\quad + \int_p (-) \frac{d}{dp^2} K(p) \, \left\lbrace \frac{\delta^2
\bar{S}_t}{\delta \bar{\phi} (p) \delta \bar{\phi} (-p)} -
\frac{\delta \bar{S}_t}{\delta \bar{\phi} (p)} \frac{\delta
\bar{S}_t}{\delta \bar{\phi} (-p)} \right\rbrace
- \frac{\eta_t}{2} \mathcal{N}_t [\bar{\phi}]
\label{ERG-St-eta}
\end{align}
where we have replaced $\eta_\Lambda$ by $\eta_t$, and
\begin{equation}
\mathcal{N}_t [\bar{\phi}]
\equiv \int_p \bar{\phi} (p) \frac{\delta \bar{S}_t
[\bar{\phi}]}{\delta \bar{\phi} (p)} + \int_p
\frac{K(p)\left(1-K(p)\right)}{p^2} \left(
\frac{\delta^2 \bar{S}_t}{\delta \bar{\phi} (p) \delta \bar{\phi}
(-p)} - \frac{\delta \bar{S}_t}{\delta \bar{\phi} (p)}
\frac{\delta \bar{S}_t}{\delta \bar{\phi} (-p)} \right)\label{Noperator}
\end{equation}
is the number operator for $\bar{S}_t$.
Rewriting (\ref{SL-correlators-eta}) in terms of dimensionless fields, we
obtain
\begin{align}
&\prod_{i=1}^n \frac{1}{K(p_i)} \,\vev{\bar{\phi} (p_1 ) \cdots \bar{\phi} (p_n)}_{\bar{S}_{t}}\notag\\
&= \left(\frac{Z_t}{Z_{t'}}\right)^{\frac{n}{2}} e^{- \frac{n}{2} \left(D-2\right) (t-t')} \prod_{i=1}^n
\frac{1}{K(p_i e^{-(t-t')})}\, \vev{\bar{\phi} (p_1 e^{-(t-t')}) \cdots \bar{\phi}
(p_n e^{-(t-t')})}_{\bar{S}_{t'}} \label{St-correlators-eta}
\end{align}
where $Z_t$ satisfies
\begin{equation}
\partial_t Z_t = \eta_t \, Z_t
\end{equation}
(The corrections to the two-point functions are ignored.) Comparing
(\ref{St-correlators-eta}) with (\ref{IRlimit}), the existence of the
IR limit implies that
\begin{equation}
\lim_{t \to \infty} \eta_t = \eta
\end{equation}
and
\begin{equation}
\lim_{t \to \infty} \prod_{i=1}^n \frac{1}{K(p_i)}\, \vev{\bar{\phi}
(p_1) \cdots \bar{\phi}}_{\bar{S}_t} = \mathcal{C} (p_1, \cdots,
p_n)
\end{equation}
In other words $\bar{S}_t$ approaches a limit as $t \to +\infty$:
\begin{equation}
\lim_{t \to +\infty} \bar{S}_t = \bar{S}_\infty
\end{equation}
We call $\bar{S}_\infty$ a fixed point because the right-hand side of
(\ref{ERG-St-eta}) vanishes for it:
\begin{align}
0 &= \int_p \left( - 2 p^2 \frac{d}{dp^2} \ln K(p) + p \cdot \partial_p
+ \frac{D+2}{2} \right) \bar{\phi} (p) \cdot \frac{\delta \bar{S}_\infty
[\bar{\phi}]}{\delta \bar{\phi} (p)}\notag\\
&\quad + \int_p (-) \frac{d}{dp^2} K(p) \, \left\lbrace \frac{\delta^2
\bar{S}_\infty}{\delta \bar{\phi} (p) \delta \bar{\phi} (-p)} -
\frac{\delta \bar{S}_\infty}{\delta \bar{\phi} (p)} \frac{\delta
\bar{S}_\infty}{\delta \bar{\phi} (-p)} \right\rbrace - \frac{\eta}{2}
\mathcal{N}_\infty [\bar{\phi}]
\end{align}
\subsubsection{Fixed-point equation}
Instead of choosing $\eta$ dependent on $t$, we may choose $\eta$ as a
constant so that there is a non-trivial fixed-point solution
$\bar{S}_\infty$ for which the right-hand side of (\ref{ERG-St-eta})
vanishes. With a constant anomalous dimension, the dimensionless ERG
equation is given by
\begin{align}
\partial_t \bar{S}_t [\bar{\phi}]
&= \int_p \left( - 2 p^2 \frac{d}{dp^2} \ln K(p) + \frac{D+2}{2} -
\frac{\eta}{2} + p \cdot \partial_p \right) \bar{\phi} (p) \cdot
\frac{\delta \bar{S}_t [\bar{\phi}]}{\delta \bar{\phi} (p)}\notag\\
&\quad + \int_p \left( - 2 \frac{d}{dp^2} K(p) - \eta \frac{K(p)
\left(1 - K(p)\right)}{p^2} \right) \frac{1}{2} \left( \frac{\delta^2
\bar{S}_t [\bar{\phi}]}{\delta \bar{\phi} (p) \delta \bar{\phi}
(-p)} - \frac{\delta \bar{S}_t [\bar{\phi}]}{\delta \bar{\phi} (p)}
\frac{\delta \bar{S}_t [\bar{\phi}]}{\delta \bar{\phi} (-p)} \right)
\end{align}
For the O($N$) model with $N$ fields $\phi^i\,(i=1,\cdots,N)$, the ERG
equation becomes
\begin{align}\label{ERG-St-eta-ON}
\partial_t \bar{S}_t [\bar{\phi}]
&= \int_p \left( - 2 p^2 \frac{d}{dp^2} \ln K(p) + \frac{D+2}{2} -
\frac{\eta}{2} + p \cdot \partial_p \right) \bar{\phi}^i (p) \cdot
\frac{\delta \bar{S}_t [\bar{\phi}]}{\delta \bar{\phi}^i (p)}\notag\\
&\quad + \int_p \left( - 2 \frac{d}{dp^2} K(p) - \eta \frac{K(p)
\left(1 - K(p)\right)}{p^2} \right) \frac{1}{2} \left( \frac{\delta^2
\bar{S}_t [\bar{\phi}]}{\delta \bar{\phi}^i (p) \delta \bar{\phi}^i
(-p)} - \frac{\delta \bar{S}_t [\bar{\phi}]}{\delta \bar{\phi}^i (p)}
\frac{\delta \bar{S}_t [\bar{\phi}]}{\delta \bar{\phi}^i (-p)} \right)
\end{align}
where the repeated indices $i$ are summed over.
\subsection{Energy Momentum Tensor: Scale Invariance and Conformal Invariance}
\label{secA2}
\subsubsection{ Energy Momentum Tensor in the Classical Theory}
In this paper we will focus on the following Euclidean action whenever
a concrete action is required for a calculation
\[
S_E=\int d^Dx \sqrt g [{1\over 2} g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi +{1\over 2} m^2\phi^2 +\frac{\lambda}{4!}\phi^4]
\]
Using
\[
\delta g =g g^{\mu\nu}\delta g_{\mu\nu}, \quad\delta \sqrt g = {1\over 2} \sqrt g
g^{\mu\nu}\delta g_{\mu\nu},\quad \delta g_{\mu\nu}= -g_{\mu\rho}\delta
g^{\rho \sigma}g_{\sigma \nu}
\]
we get
\begin{equation}
\delta S_E =-\int d^Dx~{1\over 2} \delta g_{\mu\nu}\sqrt g [ \partial^\mu \phi \partial^\nu
\phi - g^{\mu\nu}{\cal L}]\equiv -\int d^Dx~{1\over 2}\delta g_{\mu\nu}\sqrt g
T^{\mu\nu}
\end{equation}
where
\begin{equation}
T^{\mu\nu}\equiv -\frac{2}{\sqrt g}\frac{\delta S}{\delta g_{\mu\nu}}
= \partial^\mu \phi \partial^\nu \phi - g^{\mu\nu}{\cal L}
\end{equation}
One can check that
\begin{equation}\label{cons}
\partial^\nu T_{\mu\nu}= -\partial_\mu \phi \left[\frac{\partial {\cal L}}{\partial \phi}
-\partial^\rho \left(\frac{\partial {\cal L}}{\partial ^\rho \phi}\right)\right]=-\partial_\mu\phi
\frac{\delta S_E}{\delta \phi}
\end{equation}
Thus, classically the energy momentum tensor is conserved on-shell.
\newline
Now we rewrite $T_{\mu\nu}$ in a form that will be useful later. Define the traceless tensor
\begin{equation}
t_{\mu\nu}= D \partial_\mu\partial_\nu - g_{\mu\nu}\Box~~
\end{equation}
and the transverse tensor
\begin{equation}
\sigma_{\mu\nu}= (g_{\mu\nu}\Box - \partial_\mu \partial_\nu )\phi^2
\end{equation}
Using the identity
\[
\partial_\mu \phi \partial_\nu \phi = \partial_\mu \partial_\nu {1\over 2} \phi^2 -\phi \partial_\mu\partial_\nu \phi
\]
one can rewrite
\begin{align}
\label{classicalem}
T_{\mu\nu} &= \frac{1}{4(D-1)} t_{\mu\nu}\phi^2 + \frac{D-2}{4(D-1)}
(\partial_\mu\partial_\nu -g_{\mu\nu}\partial^2)\phi^2 -\frac{1}{D} \phi t_{\mu\nu} \phi \notag\\
&\quad - \frac{1}{D} g_{\mu\nu} \left[m^2\phi^2 + (4-D)\frac{\lambda}{4!}\phi^4 +\frac{D-2}{2}E\right]
\end{align}
The trace which is proportional to $g_{\mu\nu}\frac{\delta S}{\delta
g_{\mu\nu}}$ can be written as $\frac{\partial S}{\partial t}$ when
$g_{\mu\nu}=e^{2t}\delta_{\mu\nu}$ and is the response to scale
transformations.
\begin{equation} \label{Tmn}
T^\mu_\mu= \frac{(2-D)}{4}\Box \phi^2 -
\left[m^2\phi^2+(4-D)\frac{\lambda}{4!}\phi^4 +\frac{D-2}{2}E\right]
\end{equation}
with
\[
E =\phi \frac{\delta S_E}{\delta \phi}
\]
proportional to the equation of motion. The terms proportional to
$m^2$ and $\lambda$ are genuine violations of scale invariance. But the
first term can be gotten rid of by defining the improved energy
momentum tensor
\begin{equation} \label{improvement} \Theta_{\mu\nu}=T_{\mu\nu}+
\frac{D-2}{4(D-1)}\sigma_{\mu\nu}\phi^2 \end{equation}
which is still conserved. So in a genuinely classically scale
invariant theory with $m^2=0$ and $\lambda=0 ~\mathrm{or}~ D=4$ one expects
\[
\Theta^\mu_\mu= \frac{2-D}{2}E
\]
\subsubsection{Trace of the Energy Momentum Tensor in the Quantum Theory: Perturbative}
When quantum corrections \footnote{We are working in Euclidean
space. So ``quantum" fluctuations are actually statistical
fluctuations} are included the condition for scale invariance is
modified. The trace will be defined as before proportional to
$\frac{\partial S}{\partial t}$. Before we turn to the exact RG let
us see what happens in the usual lowest order perturbation theory.
Let us start at $\Lambda_0$ and evolve to $\Lambda$ with $\Lambda$ close to $\Lambda_0$.
\begin{equation} S_{\Lambda_0} = \int _x \left[{1\over 2} \partial_\mu\phi \partial^\mu \phi + {1\over 2} m_0^2
\phi^2 +\lambda _0 \frac{\phi^4}{4!}\right]
\end{equation}
and
\[
S_\Lambda = \int _x \left[(1-\delta Z(t)){1\over 2} \partial_\mu\phi \partial^\mu \phi + {1\over 2} (m_0^2+
\delta m_0(t)^2)\phi^2 + (\lambda_0+\delta \lambda_0(t)) \frac{\phi^4}{4!} +
O(1/\Lambda)\right]
\]
Here $\delta Z$ is the correction to the kinetic term coming from the two
loop diagram at $\mathcal{O}(\lambda^2)$, $\delta m_0^2\approx O(\lambda)$ and $\delta \lambda_0
\approx \mathcal{O}(\lambda^2)$ are the corrections starting at one loop.
\newline
We rewrite $S_\Lambda$ in a suggestive way by adding and subtracting some terms proportional to $\delta Z$:
\begin{align}
S_\Lambda &=
\int _x \Big[{1\over 2} \partial_\mu\phi \partial^\mu \phi + {1\over 2} \underbrace{(m_0^2+
\delta m_0(t)^2 + \delta Z m_0^2)}_{m^2(t)=m^2_R}\phi^2 +
\underbrace{(\lambda_0+\delta \lambda_0(t)+2\delta Z \lambda_0)}_{\lambda(t)=\lambda_R}
\frac{\phi'^4}{4!} + \mathcal{O}(1/\Lambda)\Big] \notag\\
&\quad -\delta Z \underbrace{\left[{1\over 2} \partial_\mu\phi \partial^\mu \phi + {1\over 2}
m_0^2\phi^2+2 \lambda_0 \frac{\phi^4}{4!}\right]}_{\phi \frac{\partial {\cal L}}{\partial
\phi}}
\end{align}
If we think of $S_{\Lambda_0}$ as the bare action $S_B$ and $S_\Lambda$ as the
renormalized action $S_R$ so that $S_B = S_R + S_{counter-term}$, then
$\lambda_0=\lambda_B$ and $\lambda (t)=\lambda_R$. The relation between renormalized
and bare quantities is
\[
\lambda_B = \frac{\lambda_R + \delta \lambda_R}{Z^2}
\]
Here $\delta \lambda_R$ is the {\em counterterm} and is chosen to {\em
cancel} the correction $\delta \lambda_0$ so $\delta \lambda_R =- \delta \lambda_0$. Let
us write everything in terms of $\lambda_B$:
\begin{align*}
\lambda_B
&= \lambda_R +\delta \lambda_R - 2\delta Z \lambda_R \approx \lambda_R +\delta \lambda_R - 2\delta Z \lambda_0\\
\lambda_B + 2 \delta Z \lambda _0 - \delta \lambda _R
&= \lambda_0 + 2 \delta Z \lambda_0 + \delta \lambda_0=\lambda_R=\lambda(t)
\end{align*}
Thus for small $t$:
\[
\lambda(t)= \lambda_0+\beta(\lambda_0)t~~;~~~m^2(t)=m^2(0)(1+\gamma_m t)~~~;~~\delta Z =-2\gamma t
\]
Furthermore define
\[
x = \bar x \Lambda^{-1} = \bar x \Lambda_0 e^t
\]
The trace of the energy momentum tensor is given by the dependence on $t$
\begin{align} \label{dtS}
-T^\mu_{~\mu}
&=\frac{\partial S_{\Lambda_0}}{\partial t}\notag\\
&=\Lambda^{-D}\left\lbrace
\int _{\bar x} \left[ {1\over 2} m_0^2\gamma_m (\lambda_0)\phi^2 + \beta(\lambda_0)
\frac{\phi^4}{4!}\right]+2\gamma \int_x {1\over 2} \phi \frac{\delta
S_{\Lambda_0}}{\delta \phi(x)} \right.\notag\\
&\qquad\left.+D \int _{\bar x} [ {1\over 2} m_0^2\phi^2+ \lambda_0
\frac{\phi^4}{4!}]+(D-2)\int_{\bar x}{1\over 2} \partial_{\bar \mu}\phi \partial^{\bar
\mu} \phi
+ O(1/\Lambda_0)]\right\rbrace
\end{align}
Define dimensionless variables as
\[
m_0^2 = \bar m^2 \Lambda_0^2 = \bar m^2 e^{2t} \Lambda^2
\]
and
\[
\lambda_0 = (\Lambda_0)^{4-D}\bar \lambda_0 = \bar \lambda_0e^{(4-D)t}(\Lambda)^{4-D}
\]
and fields
\[
\phi=(\Lambda)^{\frac{D-2}{2}}\bar \phi= e^{-\frac{D-2}{2}t}\Lambda_0^{\frac{D-2}{2}}\bar \phi
\]
Now add and subtract
\[
(\frac{D-2}{2}) \int_{\bar x} \bar \phi \frac{\delta S_{\Lambda_0}}{\delta \bar \phi(x)}
\]
to get
\begin{equation} \label{dtSdimless}
-T^\mu_{~\mu}= \int _{\bar x} \underbrace{\Big[ {1\over 2}
\bar m^2(2+\gamma_m (\lambda_0))\bar \phi^2 +( \beta(\lambda_0)-(D-4)\lambda_0)
\frac{\bar \phi^4}{4!}\Big]}_{\textrm{``$\beta$-function''}}+\left(\frac{D-2}{2}+\gamma\right)
\int_{\bar x} \bar \phi \frac{\delta S_{\Lambda_0}}{\delta \bar \phi(x)} + O(1/\Lambda_0)
\end{equation}
LHS can be identified with the trace of the energy momentum tensor in
the quantum theory and can be compared with the corresponding
classical expression in \eqref{Tmn}.
The above gives an idea of how the quantum corrections modify $T_{\mu\nu}$. A detailed calculation of the energy momentum tensor in the renormalized theory in terms of composite operators and using dimensional regularization is given in \cite{Brown}.
A systematic and precise treatment is provided by ERG and is given in \cite{Sonoda-emt,Rosten2} and is summarized below.
\subsubsection{Energy Momentum Tensor in Exact RG}
\label{emerg}
We summarize the properties of the energy momentum tensor in ERG, given in \cite{Sonoda-emt}.
\newline
The Ward Identity almost \footnote{up to transverse terms of the form $\partial_\mu\partial_\nu-\Box \delta_{\mu\nu}$ that do not contribute} defines the energy momentum.
Because of general coordinate invariance
\[
\delta x^\mu = -\epsilon^\mu~~;~~~~\phi'(x) = \phi(x) + \epsilon^\mu \partial_\mu \phi(x)
\]
is equivalent to (Assume that $g_{\mu\nu}=\eta_{\mu\nu}$)
\[
\delta g_{\mu\nu}= \epsilon_{(\mu,\nu)}
\]
and
\[
\int {\cal D}\phi'=\int {\cal D}\phi_{g+\delta g}~~~;~~~S[\phi,g+\delta g]=S[\phi',g]
\]
Thus the following identity must hold
\[
Z[J]=\int {\cal D}\phi' e^{-S[\phi'(x)] + \int _{x} J(x)\phi'(x)}=\int {\cal D}\phi_{g+\delta g}e^{-S[\phi(x),g+\delta g] + \int _xJ(x)(\phi(x) +\epsilon^\mu \partial_\mu \phi(x))}
\]
Then using the definition of the energy momentum tensor, i.e.
\begin{equation} \label{defn}
Z[J=0,g+\delta g]=\int {\cal D}\phi_{g+\delta g} e^{-S[\phi,g+\delta g]}\equiv \int {\cal D}\phi _g e^{-S[\phi,g]+{1\over 2}\int \sqrt g \delta g_{\mu\nu}T^{\mu\nu}}
\end{equation}
we get the Ward identity
\begin{equation} \label{wi}
-\partial_\mu \langle T^\mu_{~\nu}(x) \phi(x_1)...\phi(x_n)\rangle + \sum_{i=1}^n \delta(x-x_i)\langle \phi(x_1)....\partial_\nu\phi(x_i)...\phi(x_n)\rangle=0
\end{equation}
This is a statement of the conservation of $T_{\mu\nu}$ corresponding to the classical statement \eqref{cons}.
\newline
In ERG this can be written as a Ward identity for the composite operator $[T_{\mu\nu}]$
\begin{equation} \label{wierg}
q^\mu [T_{\mu\nu}(q)]=\int_p e^{S[\phi]}K(p) (p+q)_\nu\frac{\delta}{\delta \phi(p)}( [\phi(p+q)]e^{-S[\phi]})
\end{equation}
The equation corresponding to \eqref{dtSdimless} and \eqref{Tmn}
is
\begin{equation} \label{Tmnerg}
T^\mu_\mu(0)= -\frac{\partial S}{\partial t} - (\frac{D-2}{2} +\gamma) {\cal N}
\end{equation}
where $-\frac{\partial S}{\partial t}$ gives the ERG evolution, with anomalous dimension, in terms of dimensioness variables - the ``$\beta$-function". It vanishes at the fixed point. ${\cal N}$ is the number operator. Note that this equation is obtained for zero momentum or as an integral over space-time in position space. The classical analog of this is \eqref{Tmn}, which was obtained for arbitrary momentum.
Note that in equations \eqref{wierg} and \eqref{Tmnerg}, both LHS and RHS are composite operators. So one strategy will be to evaluate $T_{\mu\nu}$ using these equations in the bare theory at some scale $\Lambda_0$ which will be taken to be infinity. The bare theory is very simple so the calculations can be done exactly. Then one can evolve $T_{\mu\nu}$ down to a scale $\Lambda<<\Lambda_0$ order by order using the ERG evolution operator. If we choose $\lambda$ and $m$ to be on the critical surface we are guaranteed that at $\Lambda$ the theory flows to the fixed point action. Thus we will have evaluated the energy momentum tensor at the fixed point.
Another approach is to work directly with the known fixed point action and solve the Ward identity order by order. In this paper we follow the second approach.
\section{Wilson-Fisher Fixed Point for the $O(N)$ Model}
\label{sec 3}
We will find the fixed-point Wilson action by putting $\frac{\partial \bar S_t}{\partial t}=0$ in (\ref{ERG-St-eta-ON}). As we will work mostly with dimensionless variables we will remove the bar sign from the dimensionless variables unless otherwise mentioned. Also t dependence of actions and fields being readily implied, the subscript t will be omitted too. We give the fixed point action $S$ in the following form:
\[
S= S_2 + S_4 +S_6
\]
where $S_2$ and $S_4$ are given by
\begin{align}
S_2 &= \int \frac{d^D p}{(2\pi)^D} U_2(p) {1\over 2} \phi^I(p) \phi^I(-p) \\
S_4 &= \frac{1}{2}\prod_{i=1}^3\int \frac{d^Dp_i}{(2\pi)^D} U_4 (p _1,
p _2; p_3,p_4)
{1\over 2} \phi^I(p_1)\phi^I(p_2){1\over 2} \phi^J(p_3)\phi^J(p_4)
\end{align}
where $p_1+p_2+p_3+p_4=0$ is implied. Instead of putting an explicit delta
function and integrating over $p_4$ we will simply impose momentum
conservation at every stage. Accordingly $S_6$ is given by
\begin{equation}
S_6= \frac{1}{3!}\prod_{i=1}^5\int \frac{d^Dp_i}{(2\pi)^D} U_6 (p _1,
p _2; p_3,p_4;p_5,p_6)
{1\over 2}\phi^I(p_1)\phi^I(p_2)
{1\over 2}\phi^J(p_3)\phi^J(p_4){1\over 2}\phi^K(p_5)\phi^K(p_6)
\end{equation}
\subsection{Equations for the vertices}
We get the following equations for $U_2$,$U_4$ and $U_6$:
\paragraph{Equation for $U_2$}
\begin{align}\label{V2}
\nonumber 0=\int &\frac{d^D p}{(2\pi)^D} \Bigg\{\bigg(\frac{-\eta}{2} \frac{K(1-K)}{p^2}-K'(p^2)\bigg)\frac{1}{8} \bigg[4N U_4(p_1,-p_1;p,-p)+8 U_4(p_1,p;-p_1,-p)\bigg]\\
-&\nonumber \frac{1}{2!}2U_2(p)U_2(p)\delta^D(p - p_1)\Bigg\}+\bigg ( \frac{-\eta}{2}+1-2\frac{p_1^2}{K(p_1^2)} K'(p_1^2)\bigg ) U_2(p_1)
-\frac{1}{2!}p_1\frac{dU_2(p_1)} {dp_1} \\
\end{align}
\paragraph{Equation for $U_4$}
\begin{align}\label{V4}
0=& \nonumber \int \frac{d^D p}{(2\pi)^D} \bigg(\frac{-\eta}{2} \frac{K(1-K)}{p^2}-K'(p^2)\bigg)
\frac{1}{48} \\
\nonumber \times \bigg \lbrace & 6 N U_6 (p _1, p _2;p_3,p_4; p,-p)+ 12 U_6 (p _1,p ;p_2,-p; p_3,p_4)+12 U_6(p_1,p_2;p_3, p;p_4,- p) \bigg \rbrace\\
- \nonumber & \sum_{j=1}^4\bigg(\frac{-\eta}{2} \frac{K(1-K)}{p_j^2}-K'(p_j^2)\bigg)U_2(p_j)~\frac{2}{8} U_4 (p _1, p _2; p_3,p_4)+\sum_{j=1}^4\bigg(\frac{-\eta}{2}-2\frac{p^2}{K(p_j^2)} K'(p_j^2)\bigg)~\frac{1}{8} U_4 (p _1, p _2; p_3,p_4)\\
+& \bigg[4-D - \sum_{i=1}^4 p_i \frac{d}{d p_i} \bigg]\frac{1}{8} U_4 (p _1, p _2; p_3,p_4)
\end{align}
Here $p = p_a+p_b+p_n=-(p_i+p_j+p_m)$.
\paragraph{Equation for $U_6$}
\begin{align}\label{V6}
0=&\nonumber \frac{2}{48} \sum _{6~perm~of~(m,n)}\bigg(\frac{-\eta}{2} \frac{K(1-K)}{(p_i+p_j+p_m)^2)}-K'((p_i+p_j+p_m)^2)\bigg) U_4 (p _i, p _j;p_m,p) U_4 (p _a, p _b;p_n,-p)\\
+ & \nonumber \sum_{j=1}^6\bigg(K'(p_j^2)-\frac{-\eta}{2} \frac{K(1-K)}{p_j^2}\bigg)U_2(p_j)\frac{2}{48} U_6 (p _1, p _2; p_3,p_4; p_5,p_6) \\
+& \sum_{j=1}^6\bigg(\frac{-\eta}{2}-2\frac{p^2}{K(p_j^2)} K'(p_j^2)\bigg)\frac{1}{48} U_6 (p _1, p _2; p_3,p_4;p_5,p_6)+\bigg[6 - 2D - \sum_{i=1}^6 p_i \frac{d}{d p_i} \bigg]\frac{1}{48} U_6 (p _1, p _2; p_3,p_4; p_5,p_6)
\end{align}
\subsection{Solving the Equations}
We know that $U_4 \approx \mathcal{O}(\epsilon)$ and $U_6 \approx \mathcal{O}(\epsilon ^2)$ and $\eta \approx \mathcal{O}(\epsilon^2)$, where $\epsilon = 4-D$.
\subsubsection{$\mathcal{O}(1)$: Retrieving Gaussian theory}
We start with \eqref{V2} for $U_2$. Neglecting $U_4$ and $\eta$ and collecting coefficients of $\phi^2$ we get
\begin{equation}
0= K'(p^2) U_2(p)U_2(p)
+
\bigg(1-2\frac{p^2}{K(p^2)} K'(p^2)\bigg)U_2(p)
-p^2\frac{dU_2(p)} {dp^2}
\end{equation}
$U_2(p)= \frac{p^2}{K(p^2)}$ solves this equation. This is expected since the Gaussian theory is expected to be a fixed point - and this ERG was obtained from Polchinski's ERG by adding on the kinetic term
$ {1\over 2} \int \frac{d^D p}{(2\pi)^D} \phi(p) \frac{p^2}{K(p^2)}\phi(-p)$.
\vspace{0.1 in}
Thus our solution can be written as
\begin{equation} \label{U2general} \boldmath
U_2(p)= \frac{p^2}{K(p^2)} +\underbrace{U_2^{(1)}(p)}_{\mathcal{O}(\epsilon)}+ \mathcal{O}(\epsilon^2)
\end{equation}
\subsubsection{$\mathcal{O}(\epsilon)$: Fixed Point value of $m^2$}
We go back to \eqref{V2} and keep $U_4$ which is $\mathcal{O}(\epsilon)$ but drop $\eta$ which is $\mathcal{O}(\epsilon^2)$.
\begin{align}
\nonumber 0=\int & \frac{d^D p}{(2\pi)^D} \bigg(\frac{-\eta}{2} \frac{K(1-K)}{p^2}-K'(p^2)\bigg)\times\\
\nonumber \bigg \lbrace & \frac{1}{8}\Big[4N U_4(p_1,p_2;p,-p)+8 U_4(p_1,p;-p,-p_1)\Big] -\frac{1}{2!}2U_2(p)U_2(p)\delta^D(p - p_1)\bigg \rbrace\\
+\nonumber&\bigg(\frac{-\eta}{2}+1-2\frac{p_1^2}{K(p_1^2)} K'(p_1^2)\bigg)U_2(p_1)
-\frac{1}{2!}p_1\frac{dU_2(p_1)} {dp_1} \\
\end{align}
We use \eqref{U2general} in the above equation and look at the terms of order $\epsilon$. To leading order we set $U_4=\lambda $, which is $\mathcal{O}(\epsilon)$. The equation for $U_2^{(1)}$ is given by
\begin{align*}
0=-\lambda\frac{4N+8}{8} \int \frac{d^D p}{(2\pi)^D} K'(p^2) +2 \frac{p_1^2}{K(p_1^2)} U_2^{(1)}\bigg(p_1)K'(p_1^2)+(1-2\frac{p_1^2}{K(p_1^2)} K'(p_1^2)\bigg)U_2^{(1)}(p_1)
-p_1^2\frac{dU_2^{(1)}(p_1)} {dp_1^2}
\end{align*}
To leading order this equation is solved by a constant $U_2^{(1)}$, i.e.
\begin{equation}
0= -\lambda\frac{4N+8}{8} \int \frac{d^D p}{(2\pi)^D} K'(p^2) + U_2^{(1)}
\end{equation}
Thus
\begin{equation} \label{U2epsilon}
\boldsymbol{ U_2^{(1)}=\lambda\frac{N+2}{2} \int \frac{d^D p}{(2\pi)^D} K'(p^2)}
\end{equation}
\vspace{0.1 in}
Here
\begin{equation*}
\int \frac{d^D p}{(2\pi)^D} = \frac{1}{2^D\pi^{D/2}\Gamma(D/2)}\int (p^2)^{\frac{D-2}{2}}dp^2
\end{equation*}
To get leading results we can set $D=4$:
\begin{equation} \label{U2main}
U_2^{(1)}= \lambda\frac{4N+8}{8}\frac{1}{(4\pi)^2}\int _0^\infty dp^2 p^2 K'(p^2)=-\lambda\frac{4N+8}{8}\frac{1}{(4\pi)^2}\int _0^\infty dp^2 K(p^2)
\end{equation}
We have used $K(0)=1, K(\infty)=0$. This gives the fixed point value of the dimensionless mass parameter:
\begin{equation} \label{U2}
U_2^{(1)}=m_\star^2= -\lambda\frac{N+2}{2}\frac{1}{(4\pi)^2}\int _0^\infty dp^2 K(p^2)
\end{equation}
To evaluate the integral explicitly we need a specific form for $K$. We use $K(p^2)=e^{-p^2}$. Then the integral is equal to 1.
\subsubsection {$\mathcal{O}(\epsilon^2)$: Expression for the six-point vertex }
Let us turn to \eqref{V6} reproduced below:
\begin{align}
0=\nonumber -&\frac{2}{48 }\sum _{6~perm~of~(i,j,m)}\bigg(\frac{-\eta}{2} \frac{K(1-K)}{(p_i+p_j+p_m)^2}-K'((p_i+p_j+p_m)^2)\bigg) U_4 (p _i, p _j;p_m,p) U_4 (p _a, p_b;p_n,-p)\\
+& \nonumber \sum_{j=1}^6 \bigg \lbrace\bigg(K'(p_j^2)-\frac{-\eta}{2} \frac{K(1-K)}{p_j^2}\bigg)2 U_2(p_j)+\bigg(\frac{-\eta}{2}-2\frac{p^2}{K(p_j^2)} K'(p_j^2)\bigg) \bigg \rbrace \frac{1}{48} U_6 (p _1, p _2; p_3,p_4;p_5,p_6)\\
+& \bigg[6 - 2D - \sum_{i=1}^6 p_i \frac{d}{d p_i} \bigg]\frac{1}{48} U_6 (p _1, p _2; p_3,p_4; p_5,p_6)
\end{align}
where $p = p_a+p_b+p_n=-(p_i+p_j+p_m)$.
\vspace{0.05 in}
In this equation we keep terms of $\mathcal{O}(\epsilon^2)$. Since $\eta$ is $\mathcal{O}(\epsilon^2)$, and multiplies terms of $\mathcal{O}(\epsilon^2)$, it contributes only at $\mathcal{O}(\epsilon^4)$ in this equation, so it can be dropped here. Furthermore then, if we use the leading order solution for $U_2= \frac{p^2}{K(p^2)}$, the second and third terms cancel each other. So we are left with
\begin{align}
0=-&\nonumber\frac{2}{48}\sum _{6~perm~(i,j,m)}K'\big((p_i+p_j+p_m)^2\big) U_4 (p _i, p _j;p_n,p) U_4 (p _a, p _b;p_n,-p)\\
+&\bigg[(6 - 2D - \sum_{i=1}^6 p_i \frac{d}{d p_i}) \bigg]\frac{1}{48} U_6 (p _1, p _2; p_3,p_4;p_5,p_6)
\end{align}
Since $U_4=\lambda$ to this order, we obtain
\begin{equation}
0=\lambda^2\frac{2}{48}\sum _{6~perm~(i,j,m)}K'((p_i+p_j+p_m)^2)+\bigg[6 - 2D - \sum_{i=1}^6 p_i \frac{d}{d p_i} \bigg]\frac{1}{48} U_6 (p _1, p _2; p_3,p_4;p_5,p_6)
\end{equation}
The solution for one permutation is
\[
U_6(p _1,p _2;p_3,p_4;p_5,p _6)=\lambda^2 \frac{K((p _1+p _2+p _3)^2)-K(0)}{(p_1+p _2+p _3)^2}
\]
The full solution is given by
\begin{align}
\nonumber U_6(p _1,p _2;p_3,p_4;p_5,p _6)= -\lambda^2 & \big \lbrace h(p_1+p_2+p_3)+h(p_1+p_2+p_4)+ h(p_1+p_2+p_5)\\
+&h(p_1+p_2+p_6)+h(p_1+p_3+p_4)+h(p_2+p_3+p_4)\big \rbrace
\end{align}
where $h(x)=\frac{K(0)-K(x)}{x^2}$.
\subsubsection{Fixed Point value of $\lambda$: Solution for $ U_4$ at $\mathcal{O}(\epsilon)$}
The $U_4$ equation is given by \eqref{V4}. In this equation $\eta$ can be neglected as $-\eta \approx \mathcal{O}(\epsilon^2)$ . Also we put the value of $U_2$ upto order of $\epsilon$ found above. There is a cancellation between the second and third terms on the R.H.S and we obtain
\begin{align} \label{952}
\nonumber &\bigg[\bigg(4-D - \sum_{i=1}^4 p_i \frac{d}{d p_i}\bigg) -\sum_{j=1}^4 2 K'(p_j^2)\frac{\lambda}{16\pi^2}\frac{N+2}{2}\bigg]\frac{1}{8} U_4 (p _1, p _2; p_3,p_4)\\ \nonumber
=&\int \frac{d^D p}{(2\pi)^D} K'(p^2)\frac{1}{48}\bigg \lbrace 6N U_6(p _1, p _2;p_3,p_4; p,-p)+12 U_6 (p _1, p ;p_2,-p; p_3,p_4)+12 U_6(p_1,p_2;p_3,p;p_4,-p)\bigg \rbrace\\
\end{align}
The solution is given in the Appendix \eqref{evaluatev4}. The fixed point value $\lambda^*$ given below solves the above equation:
\begin{equation} \label{lambda}
\boldmath \lambda^* = (4-D) \frac{16\pi^2}{N+8}
\end{equation}
\subsection{Determining Anomalous Dimension}
{\bf $U_2$ equation at $\mathcal{O}(\epsilon^2)$}
\begin{align*}
0 = \int & \bigg \lbrace \frac{d^D p}{(2\pi)^D} \bigg(\frac{-\eta}{2} \frac{K(1-K)}{p^2}-K'(p^2)\bigg)\left[\frac{\delta^2 S_4}{\delta \phi^I (p) \delta \bar \phi^I(-p)} - \frac{\delta S_2}{\delta \phi^I (p)}\frac{\delta S_2}{\delta \phi^I (-p)}\right ] \bigg \rbrace \\
+\bigg \lbrace &-\frac{\eta}{2}-2\frac{p^2}{K(p^2)}K^\prime(p^2)\bigg \rbrace \phi(p).\frac{\delta S}{\delta \phi(p)} +\mathcal{G}_{dil}^c S_2
\end{align*}
where we plug in:
\begin{eqnarray}
U_4(p_1,p_2;p_3,p_4)&=& \lambda + \underbrace{\tilde U_4(p_1,p_2;p_3,p_4)}_{O(\epsilon^2)}\nonumber \\
U_2(p)&=& \frac{p^2}{K} -\lambda\frac{N+2}{2} \int \frac{d^D p}{(2\pi)^D} K'(p^2) + \underbrace{\tilde U_2(p)}_{O(\epsilon^2)}
\end{eqnarray}
and keep only $\mathcal{O}(\epsilon^2)$ terms in the above equation to get
\begin{align}
\nonumber 0=\int &\frac{d^D p}{(2\pi)^D} \bigg(\frac{-\eta}{2} \frac{K(1-K)}{p^2}-K'(p^2)\bigg)\times\\
\nonumber \bigg \lbrace & \frac{1}{8}\bigg[4N\tilde{U}_4(p_1,-p_1;p,-p)+8\tilde{U}_4(p_1,p;-p,-p_1)\bigg] -\frac{1}{2!}2U_2(p)U_2(p)\delta^D(p - p_1)]\bigg \rbrace\\
+& \bigg(\frac{-\eta}{2}+1-2\frac{p_1^2}{K(p_1^2)} K'(p_1^2)\bigg)U_2(p_1)-p_1^2\frac{dU_2(p_1)} {dp_1^2}
\end{align}
\vspace{0.2 in}
On simplification it gives
\[
-\frac{-\eta}{2} \frac{(1-K)}{K} p_1^2 - \int \frac{d^D p}{(2\pi)^D} K'(p^2)
\frac{1}{8}\bigg[4N\tilde{U}_4(p_1,-p_1;p,-p)+8\tilde{U}_4(p_1,p_1;-p,-p_1)\bigg]+ K^\prime(p_1^2)U_2(p_1)U_2(p_1)\]
\begin{equation} \label{U2.1}
+ \frac{-\eta}{2} \frac{p_1^2}{K} + \tilde U_2(p_1) -p_1^2\frac{d\tilde U_2(p_1)} {d p_1^2}=0
\end{equation}
In the L.H.S the third term will cancel with part of the second term (shown in \ref{U2cal}). Also the raison d'etre for introducing $\eta$ is to ensure that $U_2 = p^2 +\mathcal{O}(p^4)$. So we let $\tilde U_2 =\mathcal{O}(p^4)$. So The anomalous dimension is given by
\begin{equation} \label{eta}
\boldmath
\frac{\eta}{2} = -\frac{d}{dp_1^2} \int \frac{d^D p}{(2\pi)^D} K'(p^2)\frac{1}{8}\bigg[4N\tilde{U}_4^{II}(p_1,-p_1;p,-p)+8\tilde{U}_4^{II}(p_1,p;-p_1,-p)\bigg]~\Bigg|_{p_1^2=0}
\end{equation}
Here the superscript $II$ is explained in Appendix A and refers to a class of Feynman diagrams.
\vspace{0.05 in}
$\tilde U_4$ is determined by solving \eqref{952}. So using \eqref{eta} and \eqref{U4.1} one can determine $\eta$. This is done in the Appendix \eqref{ETA}. The result is of course well known \cite{Wilson}:
\begin{equation}
\frac{\eta}{2} =\lambda^2\frac{N+2}{4} \frac{1}{(16\pi^2)^2}=\frac{N+2}{(N+8)^2}\frac{\epsilon^2}{4}
\end{equation}
\vspace{0.1 in}
Collecting results we have (we have put D=4 for $\mathcal{O}(\epsilon^2)$ terms),
\begin{align}
\boldsymbol{U_2(p)= \frac{p^2}{K(p^2)} -\lambda\frac{N+2}{2} \int \frac{d^D p}{(2\pi)^D} K'(p^2) +\tilde U_2(p)}
\end{align}
The expression for $\tilde U_2(p)$ is given in \eqref{tildeU2full} (also in the next section a neater expression is presented).
\begin{align}
\nonumber \boldsymbol {U_4(p_1,p_2;p_3,p_4) = }& \boldsymbol{(4-D) \frac{16\pi^2}{N+8} +\frac{(N+2)}{2} \frac{\lambda^2}{16\pi^2}\sum_{j=1}^4 h(p_j)}\\
\boldsymbol{-}& \boldsymbol{\lambda^2 \bigg[(N+4)F(p_1+p_2)+2 F(p_1+p_3)+2 F(p_1+p_4)\bigg]}
\end{align}
where
\begin{align*}
F(p)= \frac{1}{2}\int \frac{d^D p}{(2\pi)^D} h(q)\bigg[h(p+q)-h(q)\bigg]
\end{align*}
and
\begin{align*}
h(p)= \frac{K(0)-K(p^2)}{p^2}
\end{align*}
\begin{boldmath}
\begin{align}
\nonumber U_6(p _1,p _2;p_3,p_4;p_5,p _6)=-\lambda^2\bigg \lbrace & h(p_1+p_2+p_3)+h(p_1+p_2+p_4)+h(p_1+p_2+p_5)\\
+& h(p_1+p_2+p_6)+h(p_1+p_3+p_4)+h(p_2+p_3+p_4)\bigg \rbrace
\end{align}
\end{boldmath}
and the anomalous dimension is given by
\begin{align}
\boldsymbol {\frac{\eta}{2} =\lambda^2\frac{N+2}{4} \frac{1}{(16\pi^2)^2}=\frac{N+2}{(N+8)^2}\frac{\epsilon^2}{4}}
\end{align}
To evaluate the integrals we have put $D=4$ and used specific form of $K(p^2)= e^{-p^2}$.
\vspace{0.1 in}
This completes the solution of the fixed point ERG equation and determination of the eigenvalue $\eta$ corresponding to anomalous dimension up to $O(\epsilon ^2)$. In the next section we give a slightly different approach to obtaining the fixed point action and evaluate correlation functions.
\setcounter{subsection}{3}
\section{Correlation functions}
\subsection{A more general equation}
In the previous section we set $\frac{\partial S}{\partial t}=0 $ and solved the fixed point equation for the action order by order. One can also solve a more general equation where the LHS is not set to zero but to
$\frac{\partial S}{\partial t}= \beta _J \frac{\partial S}{\partial \lambda _J}$. The parameters can be chosen so that the beta functions are zero. This has the effect that the equations are modifed at each order by terms of higher order. The advantage is that the solutions are easier to write down.
\vspace{0.2 in}
We want to obtain the fixed-point Wilson action to order
$\lambda^2$ in the following form:
\begin{align}\label{action}
S [\phi^I]
&= \int_p \frac{1}{2} \phi^I (p) \phi^I (-p)\, \left( \frac{p^2}{K(p)} + U_2 (p)
\right)\notag\\
&\quad + \frac{1}{2} \int_{p_1,p_2,p_3,p_4} \frac{1}{2} \phi^I (p_1)
\phi^I (p_2) \frac{1}{2} \phi^J (p_3) \phi^J (p_4)\,\delta
\left(\sum_{i=1}^4 p_i\right)\, \bigg( \lambda + V_4 (p_1,p_2;
p_3,p_4)\bigg)\notag\\
&\quad + \frac{1}{3!} \int_{p_1,\cdots,p_6} \frac{1}{2} \phi^I (p_1)
\phi^I (p_2) \frac{1}{2} \phi^J (p_3) \phi^J (p_4) \frac{1}{2}
\phi^K (p_5) \phi^K (p_6) \,\delta
\left(\sum_{i=1}^6 p_i\right)\\
&\qquad\qquad \times V_6 (p_1,p_2; p_3,p_4; p_5,p_6)\notag
\end{align}
As we will all vertex function in powers of
$\lambda$ we have to put the general expression for $\frac{\partial \lambda}{\partial t}$ i.e
\begin{align*}
\frac{\partial \lambda}{\partial t}= (\epsilon \lambda+ \beta_{N}^{(1)}\lambda^2)
\end{align*}
Where $\beta_N^{(1)}$ , the leading term in the beta function, is given by
\begin{align*}
\beta_N^{(1)}=2(N+8) \int \frac{d^D p}{(2\pi)^D} K^\prime(p) \frac{K((0)-K(p)}{p^2}\equiv -(N+8) \int_p f(p)h(p)
\end{align*}
where $f(p)=-2K^\prime(p^2)$.
If we assume $ V_2(p)= \lambda v_2^{(1)}(p)+ \lambda^2 v_2^{(2)}(p)= \lambda v_2^{(1)}(p)+ \Big( V_2^I(p)+ V_2^{II}(p)\Big)$, where $V_2^{I(II)}$ is analog of $\tilde U_2^{I(II)}$ in \ref{U2cal}, then
\begin{align*}
\frac{\partial V_2(p)}{\partial t}= \Big(\epsilon \lambda+ \beta_{N}^{(1)}\lambda^2\Big)v_2^{(1)}(p)+ 2\lambda^2 \epsilon v_2^{(2)}(p)+ 2\lambda^3\beta_N^{(1)}v_2^{(2)}(p)
\end{align*}
Similarly if $ V_4(p_1,p_2;p_3,p_4)= V_4^I(p_1,p_2;p_3,p_4)+ V_4^{II}(p_1,p_2;p_3,p_4)$, where $ V_4^{I(II)}(p_1,p_2;p_3,p_4)$ is equivalent to $\tilde U_4^{I(II)}(p_1,p_2;p_3,p_4)$ in \ref{U4cal}.
\begin{align*}
\frac{\partial}{\partial t} \Big[\lambda+V_4(p_1,p_2;p_3,p_4)\Big]= \Big(\epsilon \lambda+ \beta_{N}^{(1)}\lambda^2\Big)+ 2V_4(p_1,p_2;p_3,p_4)\Big(\epsilon+ \beta_{N}^{(1)}\lambda\Big)
\end{align*}
\vspace{0.2 in}
\textbf{A}. \eqref{U2epsilon} is modified to
\begin{align*}
\frac{1}{2}\epsilon~ v_2^{(1)}(p)=- \frac{4N+8}{8}\int \frac{d^D p}{(2\pi)^D} K^\prime(p^2)+v_2^{(1)}(p).
\end{align*}
gives
\begin{align}
v_2^{(1)}(p)= &- \frac{N+2}{2-\epsilon}\frac{1}{2} \int \frac{d^D p}{(2\pi)^D} f(p)\\
\equiv \nonumber & -(N+2)v_2
\end{align}
where $v_2= \frac{1}{2-\epsilon}\frac{1}{2} \int \frac{d^D p}{(2\pi)^D} f(p)$
\vspace{0.2 in}
\textbf{B}. \eqref{U4tilde} turns into
\begin{align}
& \nonumber \bigg[ \epsilon + \sum_{j=1}^4p_j\frac{d}{dp_j} \bigg ] V_4^{II}(p_1,p_2;p_3,p_4))\\
=& -2\lambda^2\int _{\bar p} K'(p^2)\bigg[(N+4)h(p_1+p_2+p)+2 h(p+p_1+p_3)+2 h(p+p_1+p_4)-(N+8)h(p)\bigg]
\end{align}
If we write $V_4^{II}(p_1,p_2;p_3,p_4))=-\lambda^2 \Big\lbrace (N+4)F(p_1+p_2)+2F(p_1+p_3)+2F(p_1+p_4)\Big\rbrace$ the equation for $F(p)$ can be written as,
\begin{align}
\big(p.\partial p+\epsilon \big)F(p)=\int \frac{d^D p}{(2\pi)^D} f(q)h(q+p)+\frac{1}{3}\beta^{(1)}
\end{align}
where
\begin{align*}
\frac{1}{3}\beta^{(1)}= -\int \frac{d^D p}{(2\pi)^D} f(p)h(p)
\end{align*}
The solution , analytic at $p=0$ is,
\begin{align}
F(q)= \frac{1}{2} \int \frac{d^D p}{(2\pi)^D} h(p)\Big(h(q+p)-h(p)\Big)
\end{align}
\vspace{0.2 in}
\textbf{C}. Similarly \eqref{U4I} gets modified to,
\begin{align}
\bigg[\epsilon+ \sum_{j=1}^4 p_j\frac{d}{d p_j}\bigg] \frac{1}{8} V_4^I(p_1,p_2;p_3,p_4)= \lambda^2 (N+2) \int \frac{d^D p}{(2\pi)^D} K^\prime(p^2) \bigg \lbrace -\frac{1}{8}\sum_{j=1}^4 h(p_j)- \frac{1}{4(2-\epsilon)} K^\prime(p_j^2)\bigg \rbrace
\end{align}
whose solution is,
\begin{align}
V_4^I(p_1,p_2;p_3,p_4)&= \lambda^2\frac{(N+2)}{2-\epsilon}\int \frac{d^D p}{(2\pi)^D}(- K^\prime(p^2)) \sum_{j=1}^4 h(p_j)
\end{align}
Also
\begin{align*}
\frac{1}{8} \Big \lbrace 4N V_4^I(p_1,-p_1;p,-p)+8 V_4^I(p,p_1;-p,-p_1) \Big\rbrace &= \frac{(N+2)^2}{2-\epsilon} \lambda^2 \int \frac{d^D q}{(2\pi)^D}(-K^\prime( q^2)) \Big[ h(p_1)+h(p) \Big]
\end{align*}
\vspace{0.2 in}
\textbf{D}. \eqref{U2.1} turns into,
\begin{align}
(2-2\epsilon)V_2^I-2p_1^2\frac{d V_2^I(p_1)}{dp_1^2}=~&-\frac{2\lambda^2}{2-\epsilon} (N+2)^2 \bigg \lbrace\int \frac{d^D p}{(2\pi)^D} (-K^\prime(p^2)) \bigg \rbrace^2 h(p_1)-2\big(v_2^{(1)}\big)^2 K^\prime(p_1^2)
\end{align}
The solution is
\begin{align}
\nonumber V_2^I(p_1)= -(N+2)^2\lambda^2 \frac{1}{(2-\epsilon)^2} \frac{1}{4}\bigg \lbrace \int \frac{d^D p}{(2\pi)^D} f(p) \bigg \rbrace^2 h(p_1)
\end{align}
\vspace{0.2 in}
\textbf{E}. \eqref{tildeU2.2} changes to
\begin{align*}
& \nonumber \Big(-2+2\epsilon\Big)V_2^{II}(p_1)+\beta_{N}^{(1)}\lambda^2 v_2^{(1)}(p)+2p_1^2\frac{dV_2^{II}(p_1)} {dp_1^2}\\
=& -3\lambda^2(N+2)\int_{ r, p}(- K'( p^2))h( r)
\Big[h(p_1 + p+ r)- h( r)\Big]+\frac{2}{2-\epsilon}\Big[(N+2)^2\lambda^2 \int (-K^\prime( q^2))\Big]\int_{p}(-K^\prime(p^2))h(p)-\eta ~p_1^2
\end{align*}
If we assume
\begin{align*}
V_2^{II}(p)= -3\lambda^2 (N+2) G(p)
\end{align*}
Then $G(p)$ satisfies the following equation,
\begin{equation}
(p.\partial p-2+2\epsilon)G(p)= \int f(q)F(p+q)+\frac{2v_2}{3}\int_p f(p) h(p) +\eta^{(2)} p^2
\end{equation}
From \eqref{eta} we get $\eta= 3(N+2) \lambda^2 \eta^{(2)}$ where,
\begin{align*}
\eta^{(2)}= -\frac{d}{dp^2}\int f(q)F(q+p)\Big |_{p=0}
\end{align*}
The solution , analytic at $p=0$ is
\begin{align}
G(p)= \frac{1}{3} \int h(q)(F(p+q)-F(q))+\frac{1}{\epsilon} \frac{\eta^{(2)}}{2}p^2-\frac{1}{2-2\epsilon}\Bigg(\int f(q)F(q)+\frac{2v_2}{3}\int_p f(p) h(p) \Bigg)
\end{align}
$V_2^I(p)+V_2^{II}(p)$ when calculated in the limit $\epsilon \rightarrow 0$ gives the expression of $\tilde{U}_2(p)$ mentioned in the previous section.
\vspace{0.2 in}
The solutions are given by,
\begin{subequations}
\begin{align}
\boldsymbol { V_2 (p)}
& \boldsymbol {= - \lambda (N+2) v_2 - \lambda^2 \left( 3 (N+2) G (p) +
(N+2)^2 \left(v_2\right)^2 h (p) \right)}\\
\boldsymbol {V_4 (p_1,p_2; p_3,p_4)}
& \boldsymbol {= - \lambda^2 \Big( (N+4) F(p_1+p_2) + 2 F(p_1+p_3) + 2
F(p_1+p_4)} \notag\\
&\qquad\qquad \boldsymbol { - (N+2) v_2 \sum_{i=1}^4 h(p_i)} \Big)\\
\boldsymbol {V_6 (p_1,p_2; p_3,p_4; p_5,p_6)}
& \boldsymbol { = - \lambda^2} \left( \boldsymbol {h(p_1+p_2+p_3) + h(p_1+p_2+p_4) +
h(p_1+p_2+p_5)} \right.\notag\\
&\qquad\left. \boldsymbol { + h(p_1+p_2+p_6) + h(p_3+p_4+p_1) +
h(p_3+p_4+p_2)} \right)
\end{align}
\end{subequations}
where
\begin{equation*}
f(p)=-2K^\prime(p^2);~h(p)= \frac{K(0)-K(p^2)}{p^2}
\end{equation*}
and
\begin{equation}
\boldsymbol { v_2 = \frac{1}{2-\epsilon} \frac{1}{2} \int \frac{d^D p}{(2\pi)^D} f(p) }
\label{summary-v2}
\end{equation}
If we take the limit $\epsilon \rightarrow 0$ and $K(p^2)=e^{-p^2}$ we get
\begin{align*}
v_2= \frac{1}{2}\int \frac{d^4 p}{(2\pi)^4} e^{-p^2} = \frac{1}{2}\frac{1}{16\pi^2}
\end{align*}
\begin{equation*}
F(p)= \frac{1}{2}\int \frac{d^D p}{(2\pi)^D} h(q)\Big[h(p+q)-h(q)\Big]
\end{equation*}
The coupling constant $\lambda$ is given, to order $\epsilon = 4-D$, as
\begin{equation}
\boldsymbol {\lambda = \frac{\epsilon}{- \beta_N^{(1)}} = \frac{(4
\pi)^2}{N+8}}\,\epsilon \label{summary-lambda}
\end{equation}
The anomalous dimension is given, to order $\epsilon^2$, as
\begin{equation}
\boldsymbol {\eta = \frac{N+2}{2 (N+8)^2}}\,
\boldsymbol {\epsilon^2} \label{summary-eta}
\end{equation}
\subsection{Calculation of correlation functions}
In this section we will calculate two-, four-, and six-point correlation functions. Recall that our Wilson action has a fixed momentum cutoff
of order $1$. If we consider the momenta much larger than the cutoff,
the vertices of the Wilson action gives the correlation functions \cite{Sonoda-correlation}. We first rescale the
field
\begin{equation}
J^I (p) \equiv \frac{1}{h(p)} \phi^I (p)
\end{equation}
and define
\begin{equation}
W [J^I] \equiv - S [\phi^I] + \frac{1}{2} \int_p J^I (p) J^I (-p)
\frac{h(p)}{K(p)}
\end{equation}
For our Wilson action, this is given by
\begin{align}
W[J^I]
&= \int_p \frac{1}{2} J^I (p) J^I (-p) \, h(p)^2 \left( \frac{1}{h(p)} - V_2 (p)
\right)\notag\\
&\quad + \frac{1}{2} \int_{p_1,p_2,p_3,p_4} \frac{1}{2} J^I (p_1) J^I
(p_2) \frac{1}{2} J^J (p_3) J^J (p_4)\, \delta \left(\sum_{i=1}^4
p_i\right)\notag\\
&\qquad\qquad \times \prod_{i=1}^4 h(p_i)\, \cdot \left( - \lambda -
V_4 (p_1,p_2; p_3,p_4) \right)\notag\\
&\quad + \frac{1}{3!} \int_{p_1,\cdots,p_6} \frac{1}{2} J^I (p_1) J^I
(p_2) \frac{1}{2} J^J (p_3) J^J (p_4) \frac{1}{2} J^K (p_5) J^K
(p_6)\, \delta \left(\sum_{i=1}^6 p_i\right)\notag\\
&\qquad\qquad \times \prod_{i=1}^6 h(p_i)\, \cdot (-) V_6 (p_1, p_2;
p_3,p_4; p_5,p_6)
\end{align}
In the high momentum limit we obtain the generating functional of the
connected correlation functions
\begin{equation}
\mathcal{W} [J^I] = \lim_{t \to +\infty} W [J_t^I]
\end{equation}
where
\begin{equation}
J_t^I (p) \equiv \exp \left( - t \frac{D-2+\eta}{2} \right) J^I (p e^{-t})
\end{equation}
In our case we obtain
\begin{align}
W [J_t^I]
&=\int_p \frac{1}{2} J^I (p)
J^I (-p) \, \exp \left( t (2-\eta) \right) h(p e^t)^2 \left(
\frac{1}{h (p e^{t})} - V_2 (p e^t)
\right)\notag\\
&\quad + \frac{1}{2}
\int_{p_1,p_2,p_3,p_4} \frac{1}{2} J^I (p_1) J^I
(p_2) \frac{1}{2} J^J (p_3) J^J (p_4)\, \delta \left(\sum_{i=1}^4
p_i\right)\notag\\
&\qquad\quad \times \exp \left( t (D+4-2\eta \right) \prod_{i=1}^4
h(p_i e^t)\, \cdot \left( -
\lambda - V_4 (p_1 e^t,p_2 e^t; p_3 e^t,p_4 e^t) \right)\notag\\
&\quad + \frac{1}{3!}
\int_{p_1,\cdots,p_6} \frac{1}{2} J^I (p_1) J^I
(p_2) \frac{1}{2} J^J (p_3) J^J (p_4) \frac{1}{2} J^K (p_5) J^K
(p_6)\, \delta \left(\sum_{i=1}^6 p_i\right)\notag\\
&\qquad\quad \times \exp \left( t (2D + 6 - 3 \eta\right)
\prod_{i=1}^6 h(p_i e^t)\, \cdot (-) V_6 (p_1 e^t, p_2 e^t; p_3
e^t,p_4 e^t; p_5 e^t,p_6 e^t)
\end{align}
In the limit $t \to +\infty$ we obtain
\begin{align}
\mathcal{W} [J^I]
&= \int_p \frac{1}{2} J^I (p) J^I (-p)\, C_2 (p)\notag\\
&\quad + \frac{1}{2} \int_{p_1,p_2,p_3,p_4} \frac{1}{2} J^I (p_1) J^I
(p_2) \frac{1}{2} J^J (p_3) J^J (p_4)\, \delta \left( \sum_{i=1}^4
p_i \right) \, C_4 (p_1, p_2; p_3, p_4)\notag\\
&\quad + \frac{1}{3!} \int_{p_1,\cdots, p_6} \frac{1}{2} J^I (p_1) J^I
(p_2) \frac{1}{2} J^J (p_3) J^J (p_4) \frac{1}{2} J^K (p_5) J^K
(p_6) \, \delta \left( \sum_{i=1}^6 p_i \right)\notag\\
&\qquad\qquad \times C_6 (p_1, p_2; p_3, p_4; p_5, p_6)
\end{align}
\subsubsection{Two-point function}
\begin{align}
C_2 (p)
&= \lim_{t \to +\infty} \exp \left( t (2-\eta)\right) h(p e^t)^2
\left( \frac{1}{h(p e^{t})} - V_2 (p e^t) \right)\notag\\
&= \lim_{t \to +\infty} \frac{1}{(p^2)^2}\left[p^2 (1 - \eta \,t) +
\lambda^2 3 (N+2) e^{-2t} G (p e^t) \right]
\end{align}
Using
\begin{equation}
G (p e^t) \overset{t \to \infty}{\longrightarrow} p^2 e^{2t}
\frac{1}{12 (4\pi)^4} \ln \left( p^2 e^{2t}\right)
\end{equation}
we obtain
\begin{equation}
\boxed{
C_2 (p) = \frac{1}{p^2} \left( 1 + \frac{\eta}{2} \ln p^2 \right) =
\frac{1}{p^{2 - \eta}}}
\end{equation}
\subsubsection{Four-point function}
\begin{align}
& C_4 (p_1, p_2; p_3,p_4)\notag\\
&= \lim_{t \to +\infty} \exp \left( t (D+4-2\eta) \right)
\prod_{i=1}^4 h(p_i e^t) \, \cdot \left( - \lambda - V_4 (p_1 e^t,
p_2 e^t; p_3 e^t, p_4 e^t ) \right)\notag\\
&= \prod_{i=1}^4 \frac{1}{p_i^2} \lim_{t \to +\infty} \left(1 - \epsilon\,
t \right) \Big[ - \lambda \notag\\
&\qquad + \lambda^2 \left( (N+4) F \left(
(p_1+p_2) e^t \right) + 2 F \left( (p_1+p_3) e^t \right) + 2 F
\left( (p_2+p_3) e^t \right) \right)\Big]
\end{align}
Using
\begin{equation}
F (p e^t) \overset{t \to +\infty}{\longrightarrow}- \frac{1}{(4 \pi)^2}
\ln \left( p e^t \right)
\end{equation}
we obtain
\begin{equation}
\prod_{i=1}^4 p_i^2 \cdot C_4 (p_1,p_2; p_3,p_4)
= - \lambda \left( 1 + \epsilon \frac{1}{N+8} \ln
\left\lbrace (p_1+p_2)^{N+4} (p_1+p_3)^2 (p_2+p_3)^2 \right\rbrace \right)
\end{equation}
\subsubsection{Six-point function}
Since $V_6$ is already of order $\lambda^2$, we can take $D=4$ and
$\eta=0$ to obtain
\begin{align}
C_6 (p_1,p_2;p_3,p_4;p_5,p_6)
&= \lim_{t\to +\infty} e^{t \left(2 D + 6 - 3 \eta\right)}
\prod_{i=1}^6 h(p_i e^t)\, (-) V_6 (p_1 e^t, p_2 e^t; p_3 e^t, p_4
e^t; p_5 e^t, p_6 e^t)\notag\\
& = \lim_{t \to +\infty} e^{14 t}
\prod_{i=1}^6 \frac{1}{p_i^2 e^{2t}}
\cdot \lambda^2 \left( h \left( (p_1+p_2+p_3) e^t \right) + \cdots
\right)\notag\\
&= \lambda^2 \prod_{i=1}^6 \frac{1}{p_i^2} \left(
\frac{1}{(p_1+p_2+p_3)^2} + \cdots + \frac{1}{(p_3+p_4+p_2)^2}
\right)
\end{align}
\section{Construction of the energy-momentum tensor at the fixed
point}
Given a fixed-point Wilson action, we wish to construct the
energy-momentum tensor $\Theta_{\mu\nu} (p)$. It is a
\textbf{symmetric} tensor implicitly determined by the Ward identity
\begin{equation}
p_\mu \Theta_{\mu\nu} (p) = e^{S} \int_q K(q) (q+p)_\nu
\frac{\delta}{\delta \phi^I (q)} \left( \op{\phi^I (q+p)}\,e^{-S}
\right) \label{Ward-EM}
\end{equation}
where
\begin{equation}
\op{\phi^I (p)} \equiv \frac{1}{K(p)} \left(\phi^I (p) -
\frac{K(p)\left(1-K(p)\right)}{p^2} \frac{\delta S}{\delta \phi^I
(-p)} \right)
\end{equation}
is the composite operator corresponding to $\phi^I (p)$. The Ward
identity leaves an additive ambiguity of the form
\[
\left(p^2 \delta_{\mu\nu} - p_\mu p_\nu \right) \mathcal{O} (p)
\]
where $\mathcal{O} (p)$ is a scalar composite operator. Since
$\Theta_{\mu\nu}$ must have zero scale dimension, $\mathcal{O}$ must have
scale dimension $-2$. There is no such $\mathcal{O}$, since the squared mass
operator $\frac{1}{2} \phi^2$ acquires a positive anomalous dimension
at the fixed point. Hence, the Ward identity determines
$\Theta_{\mu\nu}$ unambiguously. In fact we are going to calculate
$\Theta_{\mu\nu} (p)$ only at $p=0$; we need not worry about this
ambiguity anyway.
It is convenient to expand $\Theta_{\mu\nu} (p)$ in powers of $\op{\phi^I}$:
\begin{align}
\Theta_{\mu\nu} (p)
&= \sum_{n=0}^\infty \frac{1}{n!}
\int_{p_1,\cdots,p_{2n}} \prod_{i=1}^n \frac{1}{2} \op{\phi^{I_i}
(p_{2i-1})} \op{\phi^{I_i} (p_{2i})}\, \delta \left(\sum_{i=1}^{2n} p_i -
p\right)\notag\\
&\quad \times c_{\mu\nu, 2n} (p_1,p_2; \cdots; p_{2n-1}, p_{2n})
\end{align}
To order $\lambda^2$, we only have three coefficients
$c_{\mu\nu,0}, c_{\mu\nu, 2}, c_{\mu\nu,4}$. Since the
field-independent term ($n=0$) is proportional to $\delta (p)$, we
cannot determine $c_{\mu\nu,0}$ from the Ward identity. So, we will
determine only $c_{\mu\nu,2}$ and $c_{\mu\nu,4}$.
From (\ref{action}), we obtain
\begin{align}
&\op{\phi^I (p)}
= \phi^I (p) - h(p) \Big\lbrace V_2 (p) \phi^I (p) \notag\\
&\quad + \int_{p_1,p_2,p_3} \frac{1}{2} \phi^J (p_1) \phi^J (p_2)
\phi^I (p_3) \, \delta \left(\sum_{i=1}^3 p_i - p \right)\, \left(
\lambda + V_4(p_1, p_2; p_3, -p) \right)\notag\\
&\quad + \frac{1}{2} \int_{p_1,\cdots,p_5} \frac{1}{2} \phi^J (p_1)
\phi^J (p_2) \frac{1}{2} \phi^K (p_3) \phi^K (p_4) \phi^I
(p_5)\,\delta \left(\sum_{i=1}^5 p_i - p \right)\notag\\
&\qquad\qquad\qquad \times V_6 (p_1,p_2; p_3,p_4; p_5,-p) \Big\rbrace
\end{align}
Inverting this we obtain, to order $\lambda^2$,
\begin{align}
&\phi^I (p)
= \op{\phi^I (p)} + h (p) \Big\lbrace V_2^{1PI} (p) \op{\phi^I (p)} \notag\\
&\quad + \int_{p_1,p_2,p_3} \frac{1}{2} \op{\phi^J (p_1)}\op{\phi^J (p_2)}
\op{\phi^I (p_3)}\,\delta \left(\sum_{i=1}^3 p_i - p \right) \,
\left( \lambda + V_4^{1PI}
(p_1, p_2; p_3, -p)\right) \Big\rbrace
\end{align}
where we have defined the 1PI vertices as
\begin{subequations}
\begin{align}
V_2^{1PI} (p)
&= - \lambda (N+2) v_2 - \lambda^2 3 (N+2) G(p)\\
V_4^{1PI} (p_1, p_2; p_3,p_4)
&= - \lambda^2 \left( (N+4) F(p_1+p_2) + 2 F(p_1+p_3) + 2
F(p_1+p_4)\right)
\end{align}
\end{subequations}
Note that $\phi^I$ has no sixth order term expanded in $\op{\phi}$'s
to order $\lambda^2$.
The rhs of (\ref{Ward-EM}) gives
\begin{align}
& e^S \int_q K(q) (q+p)_\nu \frac{\delta}{\delta \phi^I (q)}
\left( \op{\phi^I (q+p)} e^{-S} \right)\notag\\
&= \int_q K(q) (q+p)_\nu \left( - \op{\phi^I (q+p)} \frac{\delta
S}{\delta \phi^I (q)} + \frac{\delta}{\delta \phi^I (q)}
\op{\phi^I (q+p)} \right)\label{rhs-Ward-EM}
\end{align}
Expanding this in powers of $\op{\phi}$'s, we obtain from
(\ref{Ward-EM}) the following equations that determine the
coefficients $c_{\mu\nu, 2}$ and $c_{\mu\nu, 4}$.
\begin{subequations}
\begin{align}
& p_\mu c_{\mu\nu,2} (p_1, p_2)
= - p_{1\nu} p_2^2 - p_{2\nu} p_1^2 \notag\\
&\quad + \lambda (N+2) \left(v_2 p_\nu -
\int_q (q+p)_\nu R(q) h(q) h(q+p)\right)\notag\\
&\quad + \lambda^2 (N+2) \Big[ 3 \left( p_{1\nu} G(p_2) + p_{2\nu}
G(p_1) \right) \notag\\
&\qquad - (N+2) v_2 \int_q (q+p)_\nu R(q) h(q) h(q+p)
\left(h(q)+h(q+p)\right) \notag\\
&\qquad + \frac{1}{2} \int_q \left\lbrace (q+p)_\nu R(q) - q_\nu R(q+p) \right\rbrace
h(q) h(q+p) \notag\\
&\qquad\qquad\quad\times \left\lbrace (N+2) F (p) + 3 F(q+p_1) + 3 F(q+p_2) \right\rbrace \Big]
\label{pc2}
\end{align}
and
\begin{align}
&p_\mu c_{\mu\nu,4} (p_1, p_2; p_3,p_4)
= - \lambda p_\nu \notag\\
&\quad + \lambda^2 \Big\lbrace (N+4) \left( F(p_1+p_2) (p_3+p_4)_\nu
+ F(p_3+p_4) (p_1+p_2)_\nu \right)\notag\\
&\qquad+ 2 p_{1\nu} \left(F(p_2+p_3) + F(p_2+p_4)\right) + 2 p_{2\nu}
\left( F(p_2+p_3) + F(p_2+p_4) \right)\notag\\
&\qquad+ 2 p_{3\nu} \left(F(p_4+p_1) + F(p_4+p_2) \right) + 2
p_{4\nu} \left( F(p_3+p_1) + F(p_3+p_2) \right) \Big\rbrace\notag\\
&\quad + \lambda^2 \frac{1}{2} \int_q \left\lbrace (q+p)_\nu R(q) - q_\nu R(q+p)
\right\rbrace h(q) h(q+p)\notag\\
&\,\times \left\lbrace (N+4) \left( h(q+p_1+p_2) + h(q+p_3+p_4) \right) +
4 \left( h(q+p_1+p_3) + h(q+p_1+p_4) \right)\right\rbrace
\label{pc4}
\end{align}
\end{subequations}
To determine $c_{\mu\nu,2} (p_1, p_2)$ at $p=0$, we substitute
$p_2 = p-p_1$ into the rhs of (\ref{pc2}), and expand the result to
first order in $p$. This gives
\begin{align}
c_{\mu\nu,2} (p_1, -p_1)
&= - p_1^2 \delta_{\mu\nu} + 2 p_{1\mu} p_{1\nu}\notag\\
&\quad + \lambda (N+2) \delta_{\mu\nu} \left\lbrace v_2 - \int_q R(q) \left(
h(q)^2 + \frac{1}{D} h(q) q \cdot \partial_q h(q) \right)\right\rbrace\notag\\
&\quad + \lambda^2 (N+2) \Big\lbrace 3 \left( \delta_{\mu\nu} G(p_1) - 2
p_{1\mu} p_{1\nu} G' (p_1) \right)\notag\\
&\qquad + \int_q \left( \delta_{\mu\nu} R(q) - 2 q_\mu q_\nu
R' (q) \right) h(q)^2 \left( - (N+2) v_2 h(q) + 3 F(q+p_1)\right)
\Big\rbrace \label{cmunu2}
\end{align}
Similarly, substituting $p_4 = p - (p_1+p_2+p_3)$ into the rhs of
(\ref{pc4}) and expanding the result to first order in $p$, we obtain
\begin{align}
&c_{\mu\nu,4} (p_1,p_2;p_3, - (p_1+p_2+p_3)) = - \lambda
\delta_{\mu\nu}\notag\\
&\quad + \lambda^2 \Big\lbrace
(N+4) \left( \delta_{\mu\nu} F(p_1+p_2) - 2 (p_1+p_2)_\mu
(p_1+p_2)_\nu F' (p_1+p_2) \right)\notag\\
&\qquad + 2 \left( \delta_{\mu\nu} F(p_1+p_3) - 2 (p_1+p_3)_\mu
(p_1+p_3)_\nu F' (p_1+p_3) \right)\notag\\
&\qquad + 2 \left( \delta_{\mu\nu} F(p_2+p_3) - 2 (p_2+p_3)_\mu
(p_2+p_3)_\nu F' (p_2+p_3) \right)\notag\\
&\qquad + \int_q \left( \delta_{\mu\nu} R(q) - 2 q_\mu q_\nu R' (q)
\right) h(q)^2 \notag\\
&\qquad \quad \times \left( (N+4) h(q+p_1+p_2) + 2 h(q+p_1+p_3) + 2
h(q+p_2+p_3)\right) \Big\rbrace\label{cmunu4}
\end{align}
\subsection*{Check of the trace anomaly}
Using the energy-momentum tensor obtained above, we can verify the
trace anomaly
\begin{equation}
\Theta (0) = - \left( \frac{D-2}{2} + \frac{1}{2} \eta \right) \mathcal{N} (0)
\end{equation}
where the anomalous dimension is given by (\ref{summary-eta}) to order
$\epsilon^2$.
The trace is easily obtained from (\ref{cmunu2}, \ref{cmunu4}) as
\begin{align}
\Theta (0)
&= \int_p \frac{1}{2} \op{\phi^I (p)} \op{\phi^I (-p)} \Bigg[ - (D-2)
p^2 \notag\\
&\quad + \lambda (N+2) D \left\lbrace v_2 - \int_q R(q) \left( h(q)^2 +
\frac{1}{D} h(q) q\cdot\partial_q h(q) \right)\right\rbrace\notag\\
&\quad + \lambda^2 (N+2) \Big\lbrace
3 (D - p \cdot \partial_p ) G (p)\notag\\
&\qquad + \int_q \left(D - q \cdot \partial_q\right) R(q)
\cdot h(q)^2 \left( - (N+2) v_2 + 3 F(q+p) \right)
\Big\rbrace\Bigg]\notag\\
&\quad + \frac{1}{2} \int_{p_1,\cdots,p_4} \frac{1}{2} \op{\phi^I
(p_1)} \op{\phi^I (p_2)} \frac{1}{2} \op{\phi^J (p_3)} \op{\phi^J
(p_4)}\, \delta \left(\sum_{i=1}^4 p_i\right)\notag\\
&\qquad \times \Bigg[ - \lambda D\notag\\
&\qquad\quad + \lambda^2 \Big\lbrace
(N+4) \left(D - p \cdot \partial_p \right)
F(p)\Big|_{p=p_1+p_2}\notag\\
&\qquad\qquad\quad + 2 \left(D - p \cdot \partial_p \right)
F(p)\Big|_{p=p_1+p_3} + 2 \left(D - p \cdot \partial_p \right)
F(p)\Big|_{p=p_2+p_3} \notag\\
&\qquad\qquad + \int_q (D-q \cdot \partial_q) R(q) \cdot h(q)^2\notag\\
&\qquad\qquad\quad \times \left( (N+4) h(q+p_1+p_2) + 2 h(q+p_1+p_3)
+ 2 h (q+p_2+p_3) \right) \Big\rbrace \Bigg]
\end{align}
On the other hand the number operator, defined by
\begin{equation}
\mathcal{N} (0) \equiv - e^{S} \int_q K(q) \frac{\delta}{\delta \phi^I (q)}
\left( \op{\phi^I (q)} e^{-S}\right)\,,
\end{equation}
is calculated as
\begin{align}
\mathcal{N} (0)
&= \int_p \frac{1}{2} \op{\phi^I (p)} \op{\phi^I (-p)} \Big[ 2 p^2 +
(N+2) \lambda \left( - 2 v_2 + \int_q R(q) h(q)^2 \right)\notag\\
&\quad + \lambda^2 (N+2) \left\lbrace - 6 G(p) + 2 (N+2) v_2 \int_q R(q) h(q)^3 - 6
\int_q R(q) h(q)^2 F(q+p) \right\rbrace \Big]\notag\\
&\quad + \frac{1}{2} \int_{p_1,\cdots,p_4} \frac{1}{2} \op{\phi^I
(p_1)} \op{\phi^I (p_2)} \frac{1}{2} \op{\phi^J (p_3)} \op{\phi^J
(p_4)}\, \delta \left(\sum_{i=1}^4 p_i\right)\notag\\
&\qquad \times \Big[ 4 \lambda - 4 \lambda^2 \left\lbrace
(N+4) F(p_1+p_2) + 2 F(p_1+p_3) + 2 F(p_2+p_3) \right\rbrace \notag\\
&\qquad\quad - 2 \lambda^2 \int_q R(q) h(q)^2 \big\lbrace (N+4)
h(p+p_1+p_2) \notag\\
&\qquad\qquad\qquad\qquad\qquad + 2 h(p+p_1+p_3) + 2 h(p+p_2+p_3)
\big\rbrace \Big]
\end{align}
Using
\begin{equation}
f(q) = \left(q \cdot \partial_q + 2 \right) h(q) = (2 - q
\cdot \partial_q) R(q) \cdot h(q)^2
\end{equation}
and the equations satisfied by $F$ and $G$
\begin{subequations}
\begin{align}
\left( p \cdot \partial_p + \epsilon \right) F(p)
&= \int_q f(q) \cdot \left(h(q+p)
- h(q) \right)\\
\left( p \cdot \partial_p - 2 + 2 \epsilon \right) G (p)
&= \frac{2}{3} v_2 \int_q f(q)
\cdot h(q) + \eta \,p^2 + \int_q f(q) \cdot F(q+p)
\end{align}
\end{subequations}
we obtain
\begin{align}
& \Theta (0) + \left(\frac{D-2}{2} + \gamma_N^{(2)} \lambda^2 \right)
\mathcal{N} (0)\notag\\
&= \left(\epsilon \lambda + \beta_N^{(1)} \lambda^2 \right) \Bigg[
\int_p \frac{1}{2} \op{\phi^I (p)} \op{\phi^I (-p)} \, (N+2) v_2\notag\\
&\quad - \frac{1}{2} \int_{p_1,p_2,p_3, p_4} \frac{1}{2} \op{\phi^I
(p_1)} \op{\phi^I (p_2)} \frac{1}{2} \op{\phi^J (p_3)} \op{\phi^J
(p_4)} \,\delta \left(\sum_{i=1}^4 p_i \right)\Bigg]
\end{align}
where we have dropped $\epsilon \lambda^2 G(p)$ and $\epsilon \lambda^2 F(p)$,
which are terms of order $\epsilon^3$. This vanishes at the fixed point,
where
\[
\epsilon \lambda + \beta_N^{(1)} \lambda^2 = 0,
\]
to order $\epsilon^2$.
\subsection*{Correlation functions}
In the previous section we saw how the fixed-point Wilson action gives
the correlation functions. Similarly, the coefficient functions
$c_{\mu\nu, 2} (p_1, p_2)$ and $c_{\mu\nu, 4} (p_1,p_2; p_3,p_4)$ give
the 1PI correlation functions of the energy-momentum tensor at $p=0$:
\begin{align}
\vev{\Theta_{\mu\nu} (0) \phi^I (p) \phi^J (q)}^{1PI}
&= p^{2-\eta} q^{2 -\eta} \vev{\Theta_{\mu\nu} (0) \phi^I (p)
\phi^J (q)}\notag\\
&= \delta (p+q) \delta^{IJ} \lim_{t \to \infty} e^{(-2 + \eta) t}
c_{\mu\nu,2} (p e^t, - p e^t)
\end{align}
and
\begin{align}
&\vev{\Theta_{\mu\nu} (0) \phi^I (p_1) \phi^J (p_2) \phi^K (p_3) \phi^L
(p_4)}^{1PI} \notag\\
&\qquad= \prod_{i=1}^4 p_i^{2-\eta} \cdot
\vev{\Theta_{\mu\nu} (0) \phi^I (p_1) \phi^J (p_2)
\phi^K (p_3) \phi^L (p_4)} \notag\\
&\qquad= \delta \left( \sum_{i=1}^4 p_i \right) \lim_{t \to \infty} e^{(-\epsilon+4\eta)t}
\left[ \delta^{IJ} \delta^{KL} c_{\mu\nu,4}
(p_1 e^t,p_2 e^t; p_3 e^t,p_4 e^t) \right.\notag\\
&\qquad\qquad \left.+ \delta^{IK} \delta^{JL} c_{\mu\nu,4} (p_1
e^t,p_3 e^t; p_2 e^t,p_4 e^t) + \delta^{IL} \delta^{JK}
c_{\mu\nu,4} (p_1 e^t,p_4 e^t; p_2 e^t,p_3 e^t) \right]
\end{align}
We obtain the two-point function as
\begin{align}
\lim_{t\to\infty} e^{(-2 + \eta) t} c_{\mu\nu,2} (p e^t, -p e^t)
&= \lim_{t\to\infty} \left\lbrace\left(1 + \eta \, t\right) \left( - p^2
\delta_{\mu\nu} + 2 p_\mu p_\nu \right) \right.\notag\\
&\left.\quad + \lambda^2 (N+2) 3 e^{- 2
t} \left(
\delta_{\mu\nu} G( p e^t) - 2 p_\mu p_\nu e^{2t} G' (p e^t) \right)
\right\rbrace\notag\\
&= p^{-\eta} \left( - p^2 \delta_{\mu\nu} + 2 p_\mu p_\nu\right)
\end{align}
where we have used the asymptotic form
\begin{equation}
G (p) - 2 p_\mu p_\nu G' (p) \overset{p \to \infty}{\longrightarrow}
\frac{1}{12 (4 \pi)^4} \left( p^2 \delta_{\mu\nu} - 2 p_\mu p_\nu
\right) \ln p^2
\end{equation}
We obtain the four-point function as
\begin{align}
& \lim_{t \to \infty} e^{(- \epsilon + 4 \eta) t} c_{\mu\nu,4} (p_1 e^t, p_2 e^t;
p_3 e^t, p_4 e^t)\notag\\
&= \lambda \lim_{t \to \infty} (1 - \epsilon \, t) \Bigg[
\delta_{\mu\nu} \big\lbrace - 1 \notag\\
&\qquad\qquad + \lambda \left( (N+4) F\left((p_1+p_2)e^t\right) + 2
F\left((p_1+p_3)e^t\right) + F\left((p_2+p_3)e^t\right) \right)
\big\rbrace \notag\\
&\,\,- \lambda \left\lbrace (N+4) \frac{(p_1+p_2)_\mu
(p_1+p_2)_\nu}{(p_1+p_2)^2} + \frac{(p_1+p_3)_\mu
(p_1+p_3)_\nu}{(p_1+p_3)^2} + \frac{(p_2+p_3)_\mu
(p_2+p_3)_\nu}{(p_2+p_3)^2} \right\rbrace \Bigg]\notag\\
&= - \lambda \delta_{\mu\nu}\left[ 1+ \frac{\epsilon}{N+8} \ln \left\lbrace
(p_1+p_2)^{N+4}(p_1+p_3)^2(p_2+p_3)^2\right\rbrace\right]
\end{align}
where we have kept only the logarithms of momenta at order $\epsilon^2$.
\section{Summary and Conclusions:}
In this paper we have studied some aspects of the $O(N)$ model using the Exact RG formalism. We have done two things:
1) We have constructed the Wilson action for the $O(N)$ model at the Wilson Fisher fixed point in $4-\epsilon$ dimensions up to order $\epsilon^2$. This is done by solving the fixed point equation, order by order in $\epsilon$. Some correlation functions have also been calculated.
2) We have constructed the energy momentum tensor for this theory. This is done by solving the Ward Identity for diffeomorphism invariance.
The traceless-ness of the energy momentum tensor implies that the Wilson action is scale and conformal invariant. It is important to note that all this is in the presence of a {\em finite cutoff} $\Lambda$.
As mentioned in the introduction, one of the motivations for this construction is the use the ideas in \cite{Sathiapalan:2017frk,Sathiapalan:2019zex} and construct the AdS action corresponding to this CFT. A related problem is to construct the AdS action for sources for composite operators such as $\phi^i \phi^i$. Even more interesting would be to study the massless spin 2 field that would be the source for the energy momentum tensor. This would give dynamical gravity in the bulk as a consequence of Exact RG in the boundary by a direct change of variables similar to what was done for the scalar field in \cite{Sathiapalan:2017frk,Sathiapalan:2019zex}.
\begin{appendices}
\section{Fixed Point Action}
\subsection{Evaluation of $U_4$}\label{evaluatev4}
We need to solve
\begin{align} \label{952.1}
\nonumber &\bigg[\bigg(4-D - \sum_{i=1}^4 p_i \frac{d}{d p_i}\bigg) +\sum_{j=1}^4 2 K'(p_j^2) U_2^{(1)}(p_j)\bigg]\frac{1}{8} U_4 (p _1, p _2; p_3,p_4)\\
=\nonumber & \int \frac{d^D p}{(2\pi)^D} K'(p^2)) \frac{1}{48}\bigg \lbrace 6N U_6 (p _1, p _2;p_3,p_4; p,-p)+ 12 U_6 (p _1, p ;p_2,-p;p_3,p_4)+12 U_6(p_1,p_2;p_3,p;p_4,-p)\bigg \rbrace\\
= &\nonumber \int \frac{d^D p}{(2\pi)^D} K'(p^2))\bigg \lbrace -\frac{(N+2)}{8} \bigg( h(p_1) +h(p_2)+ h(p_3)+h(p_4)\bigg) \\
&-\frac{(N+4)}{4}\bigg(h(p+p_1+p_2) + 2 h(p+p_1+p_3) +2 h(p+p_1+p_4)\bigg)\bigg \rbrace
\end{align}
where
\begin{equation}\label{type1}
\int \frac{d^D p}{(2\pi)^D} K'(p^2)) \bigg \lbrace -\frac{(N+2)}{8} \bigg( h(p_1) +h(p_2)+ h(p_3)+h(p_4)\bigg ) \bigg \rbrace
\end{equation}
corresponds to the kind of diagrams shown in \ref{type1diagram}. Here the external loop does not involve momenta $p_i+p_j$. We will call it Type I diagrams. Considering only leading order terms in $p_j^2$ the contribution from type I diagram in \eqref{952.1} is
\begin{align} \label{type1value}
= -\frac{N+2}{8} \frac{\lambda^2}{16\pi^2} ~ 4 K^\prime(p_j^2)\bigg\vert_{p_j=0}
\end{align}
Now consider the second term in L.H.S of \eqref{952.1}. In the limit of small external momenta after putting the value of $U_2^{(1)}(p)=-\frac{N+2}{2}\frac{\lambda}{16\pi^2}$( as we are considering terms of $\mathcal{O}(\epsilon^2)$ we have put D=4 to find $U_2^{(1)}$) we get
\begin{align}
\nonumber -&\sum_{j=1}^4 2 K^\prime(p_j^2)\bigg \vert_{p_j\rightarrow 0}\frac{\lambda}{16\pi^2}\frac{N+2}{2}\frac{1}{8} V_4 (p _1, p _2; p_3,p_4)\\
=& -4 K^\prime(p_j^2)\bigg\vert_{p_j\rightarrow 0}\frac{\lambda^2}{16\pi^2}\frac{N+2}{8}
\end{align}
This cancels exactly with \eqref{type1value}.
\begin{figure}[b]
\centering
\begin{tikzpicture}
\begin{feynman}
\vertex (e) at (1,0);
\vertex (m) at ( 0, 0);
\vertex (n) at ( 3,0);
\vertex (x) at (-2,0){\(\phi^I\)};
\vertex (y) at (5,0);
\vertex (a) at (-1,-2){\(\phi^J\)};
\vertex (b) at ( 5,0){\(\phi^J\)};
\vertex (c) at (-1, 2){\(\phi^I\)};
\diagram* {
(m) -- [ edge label= $p$, near end](n),
(m)-- [edge label =$p_2$](x),
(a) -- [edge label'=$p_3$](m),
(c) -- [edge label = $p_1$](m),
(b) -- [near start,edge label = $p_4$, edge label'=$-p$, near end] (n),
(n) -- [out=120, in=60, min distance= 3cm] n,
};
\end{feynman}
\end{tikzpicture}
\caption{Type I diagram }
\label{type1diagram}
\end{figure}
Similarly in \eqref{952.1} the term
\begin{align}
\int \frac{d^D p}{(2\pi)^D} K'(p^2)\bigg \lbrace-\frac{(N+4)}{4}\bigg(h(p+p_1+p_2) + 2 h(p+p_1+p_3) +2 h(p+p_1+p_4)\bigg)\bigg \rbrace
\end{align}
corresponds to the kind of diagram shown in \ref{type2diagram}. We will call it Type II diagram. In the limit $p_i \rightarrow 0$ the above term becomes
\begin{figure}
\centering
\begin{tikzpicture}
\begin{feynman}
\vertex (r) at (0,0);
\vertex (m) at (-2, 0);
\vertex (n) at ( 2,0);
\vertex (x) at (-4,0);
\vertex (y) at (4,0);
\vertex (a) at (-3,-2);
\vertex (c) at (-3, 2);
\vertex (d) at (3,2);
\vertex (e) at (3,-2);
\node at (-3.5,2) {$\phi^I$};
\node at (3.5,-2) {$\phi^J$};
\node at (3.5,-2) {$\phi^J$};
\node at (3.5,2) {$\phi^I$};
\node at (-3.5,-2){$\phi^K$};
\node at (4.5,0){$\phi^J$};
\node at (-4.5,0){$\phi^K$};
\diagram* {
(c) -- [edge label= $p$](m),
(d) -- [edge label' = $-p$](n),
(x)-- [edge label =$p_1$](m)--(r)-- (n) -- [edge label=$p_3$](y),
(m) -- [edge label =$p_2$](a),
(n) -- [edge label'= $p_4$](e),
};
\end{feynman}
\draw [dashed] (c) .. controls (0,3).. (d);
\end{tikzpicture}
\caption{Type II diagram }
\label{type2diagram}
\end{figure}
\begin{align*}
& \lambda^2~\frac{(N+8)}{4}\frac{1}{16\pi^2}\int_0^\infty dp^2 K^\prime(p^2) \Big(K(p^2)-K(0)\Big)\\
=&\lambda^2 \frac{(N+8)}{4}\frac{1}{16\pi^2}\int_0^\infty dp^2 \bigg \lbrace \frac{1}{2}\frac{d(K^2)}{dp^2}-K(0)K^\prime(p^2) \bigg \rbrace
\end{align*}
Using $K(\infty)=0$ and $K(0)=1$, this integral gives $\frac{1}{2}$. Equating this contribution with $\epsilon \frac{\lambda}{4!}$ from L.H.S of \eqref{952.1} we obtain
\[
\frac{1}{8} (4-D) \lambda = \frac{N+8}{8}\frac{\lambda^2}{(4\pi)^2}
\]
Thus in addition to the trivial fixed point $\lambda=0$, we have a non trivial fixed point:
\begin{equation}
\boldmath \lambda = (4-D) \frac{16\pi^2}{N+8}
\end{equation}
\subsection{Solving for $\tilde U_4$}\label{u4}
\label{U4cal}
$\tilde{U}_4$ will have contribution from both type I and II diagram explained above. We write
\begin{align*}
\tilde{U}_4=\tilde{U}_4^{I}+\tilde{U}_4^{II}
\end{align*}
according to contributions from type I(II) diagrams.
\newline
(We shall set $D=4$ while evaluating integrations in those terms that are already of $\mathcal{O}(\epsilon^2)$.)
\paragraph{Type I diagram}
In \eqref{952.1} the first term on the LHS and the first terms on the RHS (Type I) cancel only in leading order. In general their difference is
\[
\lambda^2\frac{N+2}{8}\times \frac{1}{(4\pi)^2}\int_0^\infty dp^2 K^\prime(p^2)\Bigg[ \sum _j \frac{K(p_j^2)-K(0)}{p_j^2} - K'(p_j^2)\Bigg]
\]
Taylor expanding we find
\[
\lambda^2\frac{N+2}{8}\times \frac{1}{(4\pi)^2}\int dp^2 K^\prime(p^2)K''(0){1\over 2} \sum _j p_j^2 \equiv c \sum_j p_j^2
\]
This is a contribution to $\tilde U_4(p_1,p_2;p_3,p_4)$ that we can call $\Delta U_4^I(p_1,p_2;p_3,p_4)$.
Consider a type I graph where the line at one end has $p_1$ and lines with momenta $p_2,p_3,p_4$ are at the other end. This corresponds to the term
\[
\lambda^2\frac{N+2}{8}\times \frac{1}{(4\pi)^2}\int dp^2 K^\prime(p^2)K''(0){1\over 2} p_1^2 \equiv c p_1^2
\]
when contracted in a loop in order to contribute to $\tilde U_2$, so that say $p_3=-p_4$, we have $p_2=-p_1$. It contributes to $\tilde U_2(p_1^2)$ an amount
\begin{align*}
\int dp^2 K'(p^2){1\over 2} \Delta U_4^I(p_1,-p_1,p,-p)= \int dp^2 K'(p^2){1\over 2} c(p_1^2)
=\Big[c \int dp^2 K'(p^2)\Big] p_1^2 \equiv A p_1^2
\end{align*}
This is just a simple wave function renormalization that does not depend on $p_1$. There is no contribution to the mass.
The same argument applies to all the other permutations of the type I terms.
A simple wave function renormalization $\phi '^2=(1+A) \phi^2$ can ensure the normalization of the kinetic term..
They do not affect the physics or contribute to $\eta$. However, type-I term contributes to sub-leading order term of $m^2$ or $U_2$.
$\tilde{U}_4^I$ satisfies the following equation:
\begin{align}
-\sum_{i=1}^4p_i\frac{d}{d p_i} \frac{1}{8}\tilde{U}_4^I(p_1,p_2;p_3,p_4)=\lambda^2\frac{N+2}{8}\times \frac{1}{(4\pi)^2}\int_0^\infty dp^2 K^\prime(p^2)\Bigg[ \sum _j \frac{K(p_j^2)-K(0)}{p_j^2} - K'(p_j^2)\Bigg]
\end{align}
The solution is
\begin{subequations}
\begin{align}\label{U4I}
\tilde{U}_4^I(p_1,p_2;p_3,p_4)=&-\lambda^2 \frac{(N+2)}{2} \frac{1}{16\pi^2}\sum_{j=1}^4 \frac{K(p_j^2)-K(0)}{p_j^2}\\
=& \lambda^2 \frac{(N+2)}{2} \frac{1}{16\pi^2}\sum_{j=1}^4 h(p_j)
\end{align}
\end{subequations}
where $K(p)=e^{-p^2}$ is assumed.
\paragraph{Type II Diagram}
In \eqref{952.1} if we keep terms upto $\mathcal{O}(\epsilon^2)$,
\begin{align} \label{U4tilde}
\nonumber & \frac{1}{8}\bigg[ \sum_{j=1}^4p_j\frac{d}{dp_j}\bigg]\tilde U_4^{II}(p_1,p_2;p_3,p_4)\\
=&\frac{\lambda^2}{4}\int \frac{d^D p}{(2\pi)^D} K'(p^2)\bigg \lbrace (N+4)h(p+p_1+p_2)+2 h(p+p_1+p_3)+2 h(p+p_1+p_4)-(N+8) h(p)\bigg \rbrace
\end{align}
where $h(p)=\frac{K(0)-K(p)}{p^2}$.
It is to be noted in the momentum independent part $-\epsilon \frac{\lambda}{4!}$ we have written $\epsilon$ in terms of $\lambda$ using the fixed point value of $\lambda$.
\newline
The solution at $\mathcal{O}(\epsilon^2)$, analytic at zero external momenta, is given by
\begin{subequations}
\begin{align}\label{U4II}
& \nonumber \tilde U_4^{II}(p_1,p_2;p_3,p_4)\\
=&-\frac{\lambda^2}{2} \int \frac{d^D p}{(2\pi)^D} h(p)\Big[(N+4)h(p_1+p_2+p)+2 h(p+p_1+p_3)+2 h(p+p_1+p_4)-(N+8)h(p)\Big]\\
=&-\lambda^2 \Big[(N+4)F(p_1+p_2)+2 F(p_1+p_3)+2 F(p_1+p_4)\Big]
\end{align}
\end{subequations}
where $F(q)=\frac{1}{2}\int \frac{d^D p}{(2\pi)^D} h(p)\Big(h(p+q)-h(p)\Big)$.
\subsection{Equation for $\tilde U_2$}
\label{U2cal}
From \eqref{U2.1} we get
\begin{align}\label{U2.2}
\nonumber 0=&\int \frac{d^D p}{(2\pi)^D} \Big(-K'(p^2)\Big) \times\\
\nonumber \bigg \lbrace &\frac{1}{8}\Big[4N\tilde{U}_4^I(p_1,-p_1;p,-p)+4N \tilde{U}_4^{II}(p_1,-p_1;p,-p)+ 8\tilde{U}_4^{I}(p_1,p;-p_1,-p)+8\tilde{U}_4^{II}(p_1,p;-p_1,-p)\Big]\\
- &v_2^{(1)}(p)v_2^{(1)}(p)\delta^D(p - p_1)\bigg \rbrace- \frac{\eta}{2}p_1^2+ \tilde{U}_2(p_1)-p_1^2\frac{d\tilde{U}_2(p_1)} {dp_1^2}
\end{align}
From\eqref{U4I}
\begin{align}
\nonumber &\frac{1}{8}\bigg \lbrace 4N\tilde{U}_4^I(p_1,-p_1;p,-p)+8\tilde{U}_4^I(p_1,p;-p,-p_1)\bigg \rbrace\\
= & \frac{1}{2}(N+2)^2\frac{\lambda^2}{16\pi^2}\bigg \lbrace h(p)+ h(p_1) \bigg \rbrace
\end{align}
and from \eqref{U4II}
\begin{align}
\nonumber &\frac{1}{8}\bigg \lbrace 4N\tilde{U}_4^{II}(p_1,-p_1;p,-p)+8\tilde{U}_4^{II}(p_1,p;-p,-p_1)\bigg \rbrace\\
= & -\frac{3\lambda^2}{2}(N+2) \int_{ r} \bigg \lbrace h( r)\Big[h( r+ p_1+ p)-h( r)\Big] \bigg \rbrace
\end{align}
If we decompose $\tilde{U}_2$ in two parts namely $\tilde{U}_2^I$ and $\tilde{U}_2^{II}$ respectively, in the following way,
1.
\begin{align}\label{tildeU2.1}
\tilde{U}_2^I(p_1)-p_1^2\frac{d\tilde{U}_2^I(p_1)} {dp_1^2}= \int \frac{d^D p}{(2\pi)^D} K^\prime(p^2) \frac{1}{2}(N+2)^2\frac{\lambda^2}{16\pi^2} h(p_1) - \big(U_2^{(1)}\big)^2 K^\prime(p_1^2)
\end{align}
which gives
\begin{align}
\tilde{U}_2^I(p_1)= -\frac{\lambda^2}{(16\pi^2)^2}\frac{(N+2)^2}{4} h(p_1)
\end{align}
2.
\begin{align}\label{tildeU2.2}
& \nonumber -2\tilde{U}_2^{II}(p_1)+2p_1^2\frac{d\tilde{U}_2^{II}(p_1)} {dp_1^2}\\
=& -6\lambda^2(N+2)\int \frac{d^D p}{(2\pi)^D} \Big(- K'( p^2)\Big)F(p_1+p)+ (N+2)^2\frac{\lambda^2}{16\pi^2}\int \frac{d^D p}{(2\pi)^D}\Big(-K^\prime(p^2)\Big)h(p)-\eta ~p_1^2
\end{align}
which gives
\begin{align}
\tilde{U}_2^{II}(p_1)=~& p_1^2\int_{p^2 = 0}^{p_1^2} dp^2 \frac{\int \frac{d^D q}{(2\pi)^D}\Big\lbrace-6\lambda^2(N+2)(- K'( q^2))F(p+q)\Big\rbrace-\eta~p^2}{2p^4}- \frac{(N+2)^2}{4}\frac{\lambda^2}{(16\pi^2)^2}
\end{align}
The second term in the expression of $\tilde{U}_4^{II}$ is evaluated using $K(p)=e^{-p^2}$.
\vspace{0.05 in}
Hence The full expression of $\tilde{U}_2(p_1)$ is given by
\begin{boldmath}
\begin{align}\label{tildeU2full}
\nonumber \tilde{U}_2(p_1)=-&\frac{\lambda^2}{(16\pi^2)^2}\frac{(N+2)^2}{4} h(p_1)\\
+& p_1^2\int_{p^2 = 0}^{p_1^2} dp^2 \frac{\int \frac{d^D q}{(2\pi)^D}\Big\lbrace-6\lambda^2(N+2)(- K'( q^2))F(p+q)\Big\rbrace-\eta~p^2}{2p^4}- \frac{(N+2)^2}{4}\frac{\lambda^2}{(16\pi^2)^2}
\end{align}
\end{boldmath}
\subsection { Expression for $\eta$}\label{ETA}
Only Type II diagrams contribute to $\eta$.
Because we need the external momentum to flow through the loop - to get a momentum dependence in $U_2$. This can happen
only in Type II terms and that too for \textbf{certain contractions}.
\vspace{0.05 in}
(Calculation of this section requires us to go back to bar denoted variable as dimensionless variable. So $p$'s from last section are replaced with $\bar p$. )
\vspace{0.05 in}
From \eqref{eta} we have
\begin{equation} \label{eta1.1}
\frac{\eta}{2} = -\frac{1}{8}\frac{d}{d\bar r^2} \int _{\bar q} K'(\bar q^2)\bigg \lbrace 4N\tilde{U}_4^{II}(\bar q,-\bar q;\bar r,-\bar r)+ 8\tilde{U}_4^{II}(\bar q,\bar r;-\bar r,-\bar q) \bigg \rbrace ~\Bigg|_{\bar r^2=0}
\end{equation}
We can convert differentiation w.r.t $p_j$ into that w.r.t $\Lambda$ , i.e.
\begin{align*}
-\sum_{j=1}^4\bar p_j\frac{d}{d\bar p_j}=\Lambda\frac{d}{d\Lambda}
\end{align*}
So \eqref{U4tilde} gives following expression for $\tilde{U}_4^{II}$:
\begin{align} \label{U4.1}
\nonumber \frac{1}{8}&\tilde {U}_4^{II}(\frac{p_1}{\Lambda},\frac{p_2}{\Lambda};\frac{p_3}{\Lambda},\frac{p_4}{\Lambda})\\
= &\frac{\lambda^2}{4}\int _0^{\ln \Lambda}d \ln \Lambda'~\int_{\bar p} K'(\bar p^2)
\bigg[(N+4)h(\bar p +\frac{p_1}{\Lambda'}+\frac{p_2}{\Lambda'})+ 2 h(\bar p +\frac{p_1}{\Lambda'}+\frac{p_3}{\Lambda'})+ 2 h(\bar p+\frac{p_1}{\Lambda'}+\frac{p_4}{\Lambda'})-(N+8)h(\bar p)\bigg]
\end{align}
Hence
\begin{align}\label{eta1.2}
\nonumber &\frac{1}{8}\bigg \lbrace 4N\tilde{U}_4^{II}(\bar q,-\bar q;\bar r,-\bar r)+ 8\tilde{U}_4^{II}(\bar q,\bar r;-\bar r,-\bar q)\bigg \rbrace \\
=& \frac{\lambda^2}{4}\int _0^{\ln \Lambda}d \ln \Lambda'~\int_{\bar p,\bar r} K'(\bar p^2)\bigg \lbrace (12N+48)h(\bar p +\frac{q}{\Lambda'}+\frac{r}{\Lambda'})+(12N+48)h(\bar p +\frac{q}{\Lambda'}-\frac{r}{\Lambda'})-24(N+2) h(\bar p)\bigg \rbrace
\end{align}
So we need to find the coefficient of $\bar r^2$ in $\Big[ h(\bar p +\frac{q}{\Lambda'}+\frac{r'}{\Lambda'})+h(\bar p+\frac{q}{\Lambda'}-\frac{r'}{\Lambda'})\Big] $ which is calculated as
\begin{align}\label{eta1.3}
& \nonumber{1\over 2} \frac {r^\mu r^\nu}{\Lambda'^2}\frac{d^2}{dr'^\mu dr'^\nu}\Big[ h(\bar p +\frac{q}{\Lambda'}+\frac{r'}{\Lambda'})+h(\bar p+\frac{q}{\Lambda'}-\frac{r'}{\Lambda'})\Big]\Bigg|_{r'=0}\\
=\nonumber &\frac {r^\mu r^\nu}{\Lambda'^2}\bigg(\frac{d^2}{d\bar r'^\mu d\bar r'^\nu} h(\bar p +\frac{q}{\Lambda'}+\bar r'\bigg)\Bigg|_{\bar r'=0}\\
=& \nonumber \frac{\bar r^2}{4}\bigg( \frac{d^2}{d\bar r^\mu d\bar r_\mu} h(\bar p + \frac{ q}{\Lambda^\prime} +\bar r\bigg)\Bigg|_{r=0}\\
=& \nonumber -\frac{\bar r^2}{4}\frac{d^2}{d \bar r^\mu d \bar r_\mu}\frac {K(\bar r^2)-1}{\bar r^2}\Bigg|_{\bar r= \bar p+ \frac{q}{\Lambda^\prime}}\\
=& \bar r^2 K''((\bar p+\frac{q}{\Lambda'})^2)
\end{align}
where we have used the facts: in 4 dimensions $(\frac{d}{dp_\mu}\frac{1}{p^2})=\delta^4(p)$ and $K(0)=1$.
From \eqref{eta1.1},\eqref{eta1.2} and \eqref{eta1.3} we get
\begin{equation}
\frac{\eta}{2}=3\lambda^2 (N+2)\int _{\bar q} K'(\bar q^2)\int _0^{\ln \Lambda}d \ln \Lambda'~(\frac{\Lambda}{\Lambda'})^2\int_{\bar p} K'(\bar p^2)K''((\bar p + \frac{ q}{\Lambda'})^2)
\end{equation}
{\bf Evaluation of integral:}
Let us use $\bar q'= \frac{q}{\Lambda'}$ and $\Lambda'$ as variables of integration, rather than $\bar q=\frac{q}{\Lambda}$ and $\Lambda'$. So change variables:
\[
\bar q= \bar q' \frac{\Lambda'}{\Lambda} ~;~~~\bar q^2 = \bar q'^2\Big(\frac{\Lambda'}{\Lambda}\Big)^2 ~;~~~\int d^4\bar q = \int d^4\bar q \Big(\frac{\Lambda'}{\Lambda}\Big)^4
\]
to get
\[
\frac{\eta}{2}=-3\lambda^2(N+2)\int _0^{\ln \Lambda}d \ln \Lambda'~\int _{\bar q'} \Big(\frac{\Lambda'}{\Lambda}\Big)^{-2}K'(\bar q'^2)\Big(\frac{\Lambda'}{\Lambda}\Big)^2 \int_{\bar p} K'(\bar p^2)K''((\bar p + \frac{ q}{\Lambda'})^2)
\]
Using $ K'(\bar q'^2)= \frac{dK}{d\Lambda'}\frac{d\Lambda'}{d\bar q'^2}= -\frac{\Lambda'}{2\bar q'^2}\frac{dK}{d\Lambda'}$ we get
\[
\frac{\eta}{2}=-3\lambda^2(N+2)\int _0^{ \Lambda}d \Lambda'~\frac{dK}{d\Lambda'}\int _{\bar q'}\frac{1}{2\bar q'^2}\int_{\bar p} K'(\bar p^2)K''((\bar p + \bar q')^2)
\]
Since $\bar q'$ is an independent variable we can write this as
\[
\frac{\eta}{2}=-3\lambda^2(N+2)\int _{\bar q'}\int _0^{ \Lambda}d \Lambda'~\frac{dK}{d\Lambda'}\frac{1}{2\bar q'^2}\int_{\bar p} K'(\bar p^2)K''((\bar p + \bar q')^2)
\]
The integral over $\bar p$ is a function of $\bar q'$ and not $\Lambda'$. So we can do the $\Lambda'$ integral easily. Using $K(\infty)=0$ we get
\[
\frac{\eta}{2}=-\frac{3\lambda^2}{2}(N+2)\underbrace{\int _{\bar q'}~K(\bar q'^2)\frac{1}{\bar q'^2}\int_{\bar p} K'(\bar p^2)K''((\bar p + \bar q')^2)}_{-\frac{\pi^4}{6(2\pi)^8}}= \frac{1}{4}\lambda^2(N+2)\frac{1}{(16\pi^2)^2}
\]
The integral underbraced above is calculated to give $-\frac{\pi^4}{6(2\pi)^8}$ for $K(x)=e^{-x}$. But it can be shown to give identical result for any smooth $K(x)$ \cite{Liu-integral}.
Using $\lambda = \frac{16\pi^2}{N+8}\epsilon$ we can write the anomalous dimension as:
\begin{equation} \boldmath
\frac{\eta}{2} =\frac{1}{4}\lambda^2(N+2) \frac{1}{(16\pi^2)^2}=\frac{N+2}{(N+8)^2}\frac{\epsilon^2}{4}
\end{equation}
\section{ Asymptotic behaviors of $F (p)$ and $G(p)$}
\label{asymptotic}
The function $F(p)$ is defined by
\begin{equation}
\left( p \cdot \partial_p + \epsilon \right) F(p) = \int_q f(q) \Big(
h(q+p) - h(q) \Big)
\end{equation}
For large $p$, we obtain an equation satisfied by the
asymptotic form $F_{\mathrm{asymp}} (p)$:
\begin{equation}
\left( p \cdot \partial_p + \epsilon \right) F_{\mathrm{asymp}} (p) = -
\int_q f(q) h(q) = - \frac{1}{(4 \pi)^2} + \mathrm{O} (\epsilon)
\end{equation}
This implies
\begin{equation}
F_{\mathrm{asymp}} (p) = - \frac{1}{\epsilon} \int_q f(q) h(q) + C_F (\epsilon)
p^{-\epsilon}
\end{equation}
where $C_F(\epsilon)$ is independent of $p$. Since $F (p)$ is finite in
the limit $\epsilon \to 0+$, we must find
\begin{equation}
C_F (\epsilon) = \frac{1}{\epsilon} \frac{1}{(4\pi)^2} + \cdots
\end{equation}
Hence, expanding in $\epsilon$, we obtain
\begin{equation}
F_{\mathrm{asymp}} (p) = - \frac{1}{(4 \pi)^2} \ln p + \mathrm{const}
+ \mathrm{O} (\epsilon)
\end{equation}
We next consider $G(p)$ satisfying
\begin{equation}
\left( p \cdot \partial_p - 2 + 2 \epsilon \right) G(p) = \int_q f(q)
F(q+p) + 2 v_2 \int_q f(q) h(q) + \eta^{(2)} p^2
\end{equation}
where
\begin{equation}
\eta^{(2)} = - \frac{d}{dp^2} \int_q f(q) F(q+p)\Big|_{p=0} =
\frac{1}{6 (4\pi)^4} + \mathrm{O} (\epsilon)
\end{equation}
The asymptotic form $G_{\mathrm{asymp}} (p)$ satisfies
\begin{equation}
\left( p \cdot \partial_p - 2 + 2 \epsilon\right) G_{\mathrm{asymp}} (p) =
\eta^{(2)} p^2
\end{equation}
This gives
\begin{equation}
G_{\mathrm{asymp}} (p) = \frac{1}{2 \epsilon} \eta^{(2)} p^2 + C_G (\epsilon)
p^{2 - 2 \epsilon}
\end{equation}
Since $G (p)$ is finite as $\epsilon \to 0+$, we obtain
\begin{equation}
C_G (\epsilon) = - \frac{1}{\epsilon} \frac{1}{12 (4\pi)^4} + \cdots
\end{equation}
Hence,
\begin{equation}
G_{\mathrm{asymp}} (p) = p^2 \left( \frac{1}{6 (4 \pi)^4} \ln p +
\mathrm{const} \right) + \mathrm{O} (\epsilon)
\end{equation}
\end{appendices}
|
1,314,259,995,826 | arxiv | \section{Introduction}
Let $M$ be a Lie algebra over the field $\text{$\mathbf{F}$}$. Suppose $M$ is
nilpotent of nilpotency class $c$, so that $c$ is the smallest
number such that $M^{c+1}=0$. If $M$ has finite dimension
$n\ge 2$, it is well-known that $c\le n-1$. When $c=n-1$,
$M$ is said to be a Lie algebra of maximal class.
Consider the Lie powers $M^i$. Then $M$ is of maximal
class when the codimension of $M^i$ is exactly $i$,
for $i\le c+1$. It is natural to extend the definition
to an infinite-dimensional Lie algebra $M$ by saying that
$M$ is of maximal class when the codimension of $M^i$ is
$i$ for all $i$ (see \cite{Sha}).
One can grade $M$ with respect to the filtration of the
$M^i$: let
\begin{equation*}
L_{i} = M^{i} / M^{i+1} ,
\end{equation*}
and consider
\begin{equation}\label{eq:L}
L = \bigoplus_{i = 1}^{\infty} L_{i}.
\end{equation}
There is a natural way of defining a Lie product on $L$, and the
graded Lie algebra $L$ has the following properties: $\dim (L_1)=2$,
$\dim (L_i)\le 1$ for $i\ge 2$, and $L$ is generated by $L_1$. Note
that here too we allow all $L_i$ to be non-zero, thereby
including
infinite-dimensional algebras. A graded Lie algebra $L$ satisfying
these conditions is called a graded Lie algebra of maximal class
in~\cite{CMN, CN, J}. However, this definition does not capture all
possibilities. One of the other possibilities for a graded Lie algebra
$L=\bigoplus_{i=1}^\infty L_i$ to be of maximal class is to have $\dim
(L_i)\le 1$ for all $i\ge 1$, with $L$ generated by $L_1$ and $L_2$.
We call
a graded Lie algebra of this form an algebra of type
$2$, whereas we refer to
a graded Lie algebra of maximal class in the
sense of~\cite{CMN, CN, J} as an algebra of type $1$.
In studying algebras of type 2, we will mainly deal with
the infinite dimensional ones (as in~\cite{Sha, CMN, CN, J}).
However, our arguments also provide fairly complete
information about finite dimensional algebras.
If the characteristic of the underlying field $\text{$\mathbf{F}$}$ is zero,
it is well-known that there is only one infinite dimensional
algebra of type 1. This is the algebra
\begin{equation}\label{eq:a}
a = \Span{x, y : [y \U{x}{i} y] = 0, \ \text{for all $i \ge 1$}},
\end{equation}
where $x$ and $y$ have weight 1. The ideal generated by $y$ is
an abelian maximal ideal here. However, if $\text{$\mathbf{F}$}$ has prime
characteristic $p$ there are uncountably many algebras of
type 1 \cite{Sha, CMN}; these algebras were classified
in \cite{CN, J}.
Over a field $\text{$\mathbf{F}$}$ of characteristic zero there are three
infinite-dimensional algebras of type 2 \cite{SZ, F},
called $m$, $m_2$ and $W$, and these are defined over the
integers. The first one is a close analogue to $a$. It is
given as
\begin{equation}\label{eq:m}
m = \Span{e_{1}, e_{2} :
[e_{2} \U{e_{1}}{i} e_{2}] = 0, \ \text{for all $i \ge 1$}},
\end{equation}
where $e_1$ has weight 1 and $e_2$ has weight 2. The ideal
generated by $e_2$ is an abelian maximal ideal here.
The second one is defined as
\begin{equation}\label{eq:m_2}
\begin{aligned}
m_{2}
=
\big< \,
e_{i}, i \ge 1:\
&[e_{i} e_{1}] = e_{i+1}, \ \text{for all $i \ge 2$},
\\&[e_{i} e_{2}] = e_{i+2}, \ \text{for all $i \ge 3$}
\\&[e_{i} e_{j}] = 0, \ \text{for all $i, j \ge 3$}
\, \big>,
\end{aligned}
\end{equation}
where $e_i$ has weight $i$. Here
$m_2^2=\Span{ e_{i} : i \ge 3}$ is a maximal abelian ideal.
The third algebra is the positive part of the Witt algebra:
\begin{equation*}
W = \Span{ e_{i}, i \ge 1: [e_{i} e_{j}] = (i - j) e_{i+j} },
\end{equation*}
and is not soluble.
When one considers these algebras over a field $\text{$\mathbf{F}$}$ of prime
characteristic $p > 2$, ${m}$ and ${m_{2}}$ give algebras of
type 2, but $W$ does not.
We will show in the next section that there is a natural way
to obtain an algebra of type 2 from an uncovered algebra of
type 1. (See the next section for the relevant definition.)
In particular, $m$ arises from $a$ in this way. We will show
that for prime characteristic $p > 2$ the algebras of type 2
consist of
\begin{itemize}
\item algebras arising in this natural way from algebras of type 1,
\item ${m_{2}}$,
\item one further family of soluble algebras,
\item in the case $p=3$, one additional family of soluble algebras.
\end{itemize}
This yields a classification of algebras of type 2 over
fields of characteristic $p > 2$. We believe the case of
characteristic two to be considerably more complicated.
\section{Preliminaries}
\label{sec:preliminaries}
Let $L$ be an \emph{infinite-dimensional} Lie algebra over a field
$\text{$\mathbf{F}$}$ that is graded over the positive integers:
\begin{equation}\label{eq:L_again}
L = \bigoplus_{i = 1}^{\infty} L_{i}.
\end{equation}
If $\dim(L_{1}) = 2$, $\dim(L_{i}) = 1$ for $i > 1$,
and $L$ is generated by $L_{1}$, we say that $L$ is \emph{an
algebra of type $1$.} These are the algebras that are called
algebras of maximal class in \cite{CMN, CN, J}.
In these papers these algebras are classified over fields
of prime characteristic $p$.
As mentioned in the Introduction, over a field of
characteristic zero there is only one isomorphism class
of algebras of type 1. This is the algebra
$a$ of~\eqref{eq:a} generated by two elements $x$ and $y$ of
weight 1, subject to the relations $[y \U{x}{i} y] = 0$,
for all $i \ge 1$. This algebra is metabelian, and the
graded maximal ideal containing $y$ is abelian. Here we use
the notation
\begin{equation*}
[y \U{x}{i} y] = [y \OldU{x}{i} y].
\end{equation*}
If in the algebra~\eqref{eq:L_again} we have $\dim(L_{i}) = 1$
for all $i \ge 1$, and if $L$ is generated by $L_{1}$ and
$L_{2}$, we say that $L$ is \emph{an algebra of type $2$.}
Choose non-zero elements $e_{1} \in L_{1}$ and $e_{2}
\in L_{2}$. Since $L$ is of maximal class, for each $i \ge 2$
we have $[L_{i} L_{1}] = L_{i+1}$. Therefore we can recursively
define $e_{i+1} = [e_{i} e_{1}]$, for $i \ge 2$, and we have
$L_{i} = \Span{e_{i}}$ for all $i$. We keep this notation
fixed for the rest of the paper, allowing ourselves to
rescale $e_{2}$ when needed.
In \cite{CMN, CN}, to which we refer the reader for all details,
a theory of \emph{constituents} has been developed for
algebras of type $1$
over fields of positive characteristic $p$.
If $L$ is such an algebra, define its
\emph{$i$-th two-step centralizer} as
\begin{equation*}
C_{i} = C_{L_{1}}(L_{i}) =
\Set{ v \in L_{1} : \text{$[u v] = 0$ for $u \in L_{i}$}},
\end{equation*}
for $i > 1$. Each $C_{i}$ is a one-dimensional subspace
of $L_{1}$. A special role is played by the
\emph{first two-step centralizer $C_{2}$.}
In fact, the sequence of the two-step
centralizers consists of patterns, called constituents, of
the following type
\begin{equation*}
C_{i} \ne C_{2},
\quad
C_{i+1} = C_{i+2} = \dots = C_{i+l} = C_{2},
\quad
C_{i+l+1} \ne C_{2}.
\end{equation*}
Here $l$ is called the \emph{length of the constituent}.
(We are following the definition of \cite{CN}, which differs
from that of \cite{CMN}.) The first constituent requires a
special treatment: its length is defined as the smallest $f$
such
that $C_{f} \ne C_{2}$, and turns out to be of the form
$f = 2 q$, where $q = p^{h}$,
for some $h$.
It is proved in \cite{CMN} that
if the first constituent has length $2 q$, then
the constituents of $L$ can have lengths of the form
\begin{equation*}
2 q,
\qquad\text{or}\qquad
2 q - p^{t},
\quad\text{for $0 \le t \le h$.}
\end{equation*}
An algebra of type 1 is said to be \emph{uncovered} if the union of
the $C_{i}$ does not exhaust all of $M_{1}$.
It is proved
in~\cite{CMN} that over any field of
positive characteristic there are uncontably many uncovered
algebras of type 1. (On the other hand, if the field is
at most countable, there are algebras of type $1$ that are not
uncovered.) If $M= \bigoplus_{i = 1}^{\infty} M_{i}$ is uncovered,
there is an element $z \in M_{1}$ such that
\begin{equation}\label{eq:z}
\text{$[M_{i} z] = M_{i+1}$ for all $i \ge 1$.}
\end{equation}
We consider the maximal graded subalgebra
\begin{equation*}
L = \Span{z} \oplus \bigoplus_{i \ge 2} M_{i}
\end{equation*}
of $M$. Because of~\eqref{eq:z}, $L$ is an algebra of
type 2. In addition, the algebra $L$ inherits some kind
of constituent pattern from $M$,
as we will see in the following.
From now on we will
assume $p > 2$.
If we apply this procedure to the unique algebra $M = a$
of~\eqref{eq:a} of type 1 in
characteristic zero, which is clearly uncovered, we get the algebra
$L$ of type 2 generated by an element $e_{1}$ of weight one and an
element $e_{2}$ of weight two subject to the relations $[e_{2}
\U{e_{1}}{i} e_{2}] = 0$, for all $i \ge 1$. This is the algebra $m$
of~\eqref{eq:m}.
In positive characteristic, note first of all that in $L$
we may take $e_{1} = z$, $e_{2} = [y z]$ where
$0 \ne y \in C_{2}$, and take $e_{k} = [e_{k-1} e_1]$
for $k > 2$. Suppose that in $M$ we have a segment of the sequence
of two-step centralizers of the form
\begin{equation*}
C_{2} = C_{n-2} = C_{n-1}, C_{n} = \Span{y + \lambda z}
\ne C_{2}, C_{n+1} = C_{n+2} = C_{2},
\end{equation*}
so that $\lambda \ne 0$.
Note that the first constituent has length $2q \ge 6$ so
that, in particular,
\begin{equation*}
[e_{3} e_{2}]
=
[ [y z z] [y z] ]
=
[y z z y z] - [y z z z y]
= 0.
\end{equation*}
We have
\begin{align*}
[e_{n-1} e_{2}]
&=
[e_{n-1} [y z]]
\\&=
[e_{n-1} y z] - [e_{n-1} z y]
\\&=
- [e_{n-1} z y] && \text{as $C_{n-1} =C_{2} = \Span{y}$}
\\&=
- [e_{n-1} e_{1} y]
\\&=
- [e_{n} y]
\\&=
[e_{n}, \lambda z] - [e_{n}, y + \lambda z]
\\&=
\lambda e_{n+1}.
\intertext{Similarly}
[e_{n} e_{2}]
&=
[e_{n} [y z]]
\\&=
[e_{n} y z] - [e_{n} z y]
\\&=
[e_{n} y z] && \text{as $C_{n+1} =C_{2} = \Span{y}$}
\\&=
[e_{n} y e_{1}]
\\&=
[e_{n}, - \lambda z, e_{1}] + [e_{n}, y + \lambda z, e_{1}]
\\&=
- \lambda e_{n+2}.
\end{align*}
Finally
\begin{equation*}
[e_{n+1} e_{2}] = 0
\end{equation*}
as $C_{n+1} = C_{n+2} = C_{2}$.
In view of this, we introduce a definition of constituents for
algebras of type~2 that is compatible with the definition for algebras
of type 1.
Let $L$ be an arbitrary algebra of type 2.
If $[e_{3} e_{2}] = [e_{2} e_{1} e_{2}] \ne 0$, we have
no theory of constituents for $L$. Algebras of this
type are dealt with in Section 3 and Section 7. If
$[e_{2} e_{1} e_{2}] = 0$, and for some $n$ we have
$[e_{n-1} e_{2}] = 0$, but
$[e_{n} e_{2}] = \lambda e_{n+2} \ne 0$, for some
$\lambda \ne 0$, then
\begin{align*}
0
&=
[e_{n-1} [e_{2} e_{1} e_{2}]]
\\&=
- [e_{n-1} e_{1} e_{2} e_{2}]
+ 2 [e_{n-1} e_{2} e_{1} e_{2}]
- [e_{n-1} e_{2} e_{2} e_{1}]
\\&=
- [e_{n-1} e_{1} e_{2} e_{2}]
\\&=
- [e_{n} e_{2} e_{2}]
\\&=
- \lambda [e_{n+2} e_{2}],
\end{align*}
so that $[e_{n+2} e_{2}] = 0$. We are therefore led
to the following definition. Let $L$ be an algebra of
type 1 in which $[e_{2} e_{1} e_{2}] = 0$. Suppose there
are integers $m, n$ such that
\begin{align*}
&[e_{m-1} e_{2}] = 0,
\\&
[e_{m} e_{2}] = \eta e_{m+2}, && \text{with $\eta \ne 0$,}
\\&
[e_{m+1} e_{2}] = \theta e_{m+2},
\\&
[e_{m+2} e_{2}] = \dots = [e_{n-1} e_{2}] = 0,
\\&
[e_{n} e_{2}] = \lambda e_{n+2}, && \text{with $\lambda \ne 0$,}
\\&
[e_{n+1} e_{2}] = \mu e_{n+3}.
\end{align*}
We call this pattern a \emph{constituent of length $l = n - m$
and type $(\lambda,\mu)$.} Note that $\theta$ and $\mu$ might well be
zero.
Here, too, the first constituent requires an ad hoc
treatment. If in the algebra $L$ one has
$[e_{2} e_{1} e_{2}] = 0$, and $n$ is the smallest
integer greater than 1 such that $[e_{n} e_{2}] \ne 0$, we
say that the first constituent has length $n+1$. If there
is no such $n$, then $L$ is isomorphic to the algebra $m$
above.
We will see in Section~\ref{sec:first_constituent}
that the first constituent of an algebra of type 2 can
have length $q + 1$ or $2 q$, where $q$ is a power of the
characteristic of the underlying field. If the first
constituent has length $2 q$, we will see in
Section~\ref{sec:classical} that $L$ comes from an
algebra of type $1$ via the procedure described above.
If the first constituent has length $q + 1$, we will see in
Sections~\ref{sec:extra}--\ref{sec:construction_3} that we
obtain one soluble algebra of type 2 for $q > 3$, and a family of
soluble algebras for $q = 3$.
We have just seen that an algebra of type 2 that comes
from an algebra of type 1 has constituents of type
$(\lambda,-\lambda)$. We now prove that the converse
also holds.
Suppose all constituents of the algebra $L$ of type 2
are of type $(\lambda,-\lambda)$. Consider the following
partial linear map
\begin{equation*}
\begin{cases}
e_{1} \mapsto - e_{2}\\
e_{2} \mapsto 0.
\end{cases}
\end{equation*}
We show that we can extend this to a unique derivation $D$ of weight 1
on the whole of $L$. In the extension $M$ of $L$ by $D$, we have $[D
e_{1}] = - e_{1} D = e_{2}$. Thus $M$ is generated by the elements
$e_{1}$ and $D$ of weight 1, and it is an uncovered algebra of type 1.
We begin with
$e_{3}D=[e_{2} e_{1}]D=[e_{2}D, e_{1}]+[e_{2}, e_{1}D] = 0$.
Suppose now we come to the end of a constituent in $L$,
so that we have
\begin{equation*}
[e_{i-2} e_{2}] = 0,
[e_{i-1} e_{2} ] = \lambda e_{i+1},
[e_{i} e_{2} ] = - \lambda e_{i+2}.
\end{equation*}
We have so far, proceeding by induction, $e_{i-2} D = 0$.
Now
\begin{equation*}
e_{i-1} D = [e_{i-2} e_{1}] D =
[e_{i-2} D, e_{1}] + [e_{i-2}, e_{1} D]
= - [e_{i-2} e_{2}] = 0.
\end{equation*}
Then
\begin{equation*}
e_{i} D = [e_{i-1} e_{1}] D =
[e_{i-1} D, e_{1}] + [e_{i-1}, e_{1} D]
= - [e_{i-1} e_{2}] = - \lambda e_{i+1},
\end{equation*}
\begin{equation*}
e_{i+1} D = [e_{i} e_{1}] D =
[e_{i} D, e_{1}] + [e_{i}, e_{1} D]
= - \lambda [e_{i+1} e_{1}] - [e_{i} e_{2}] = 0,
\end{equation*}
and
\begin{equation*}
e_{i+2} D = [e_{i+1} e_{1}] D =
[e_{i+1} D, e_{1}] + [e_{i+1}, e_{1} D]
= - [e_{i+1} e_{2}] = 0,
\end{equation*}
so that we can continue by induction.
This definition of $D$ is compatible with
the relations $[e_{i-2}, e_{2}] = 0$,
$[e_{i-1}, e_{2}] = - \lambda e_{i+1}$,
$[e_{i}, e_{2}] = \lambda e_{i+2}$,
$[e_{i+1}, e_{2}] = 0$.
This is clear for all but the third one. For this we have
\begin{align*}
[e_{i} D, e_{2}] + [e_{i}, e_{2} D] = - \lambda [e_{i+1}, e_{2}]
= 0 = e_{i+2} D.
\end{align*}
In~\cite{CMN} a device for studying algebras of
type 1 called \emph{deflation} has been introduced.
We now show that this can be applied also to algebras
of type 2, and the result will be an algebra of
type 1. This is useful in simplifying some proofs later on.
Let $L$ be an algebra of type 2 as in~\eqref{eq:L_again}.
Consider its subalgebra
\begin{equation*}
S = \bigoplus_{i = 1}^{\infty} L_{i p}.
\end{equation*}
Grade $S$ by assigning weight $i$ to $L_{i p}$.
Now $S$ admits the derivation $D = \ad(e_{1})^{p}$ which,
in the new grading, has weight 1. We have
\begin{equation*}
L_{i p} \ad(e_{1})^{p} = [L_{i p} \U{e_{1}}{p}] = L_{(i+1) p}.
\end{equation*}
It follows that the extension of $S$ by $D$ is a graded
Lie algebra of maximal class, and it is generated by the
two elements $e_{p}$ and $D$ of weight 1. Therefore it is
an algebra of type 1.
\bigskip
In this section we have used several times the Jacobi identity $[z [y
x]] = [z y x] - [z x y]$, and its consequence
\begin{equation*}
[z [y \U{x}{n}]] = \sum_{i = 0}^{n}
(-1)^{i} \binom{n}{i} [z \U{x}{i} y \U{x}{n-i}].
\end{equation*}
In such a formula, to evaluate binomial coefficients modulo a prime we
will make use of Lucas' theorem, in the following form. Suppose $a, b$
are non-negative integers, and $q > 1$ is a power of a prime $p$.
Write $a = a_{0} + a_{1} q$, and $b = b_{0} + b_{1} q$, where the
$a_{i}$ and $b_{i}$ are non-negative integers, and $a_{0}, b_{0} < q$.
Then
\begin{equation*}
\binom{a}{b} \equiv \binom{a_{0}}{b_{0}} \cdot \binom{a_{1}}{b_{1}}
\pmod{p}.
\end{equation*}
\section{Characterizing $m_{2}$}
In this section we start dealing with algebras
of type 2 that do not admit a theory of constituents,
that is, in which $[e_3,e_2] \ne 0$ We may
thus assume without loss of generality $[e_3,e_2]=e_5$.
We obtain
\begin{align*}
0
&=
[e_3 e_3]
\\&=
[e_3 [e_2 e_1]]
\\&=
[e_3 e_2 e_1] - [e_3 e_1 e_2]
\\&=
e_6 - [e_4 e_2].
\end{align*}
Suppose that
\[
\lbrack e_i,e_1]=e_{i+1}\text{ for }i>1,
\]
\[
\lbrack e_3,e_2]=e_5,\;[e_4,e_2]=e_6,\;[e_5,e_2]=ae_7,%
\;[e_6,e_2]=be_8,
\]
\[
\lbrack
e_7,e_2]=ce_9,\;[e_8,e_2]=de_{10},\;[e_9,e_2]=fe_{11},%
\;[e_{10},e_2]=ge_{12}.
\]
Here $a,b,c,d,f,g,$ are parameters.
$[[e_2,e_1,e_1],[e_2,e_1,e_1]]=0$ gives $1-2a+b=0$,
so $b=-1+2a$.
$[[e_2,e_1,e_1,e_1],[e_2,e_1,e_1,e_1]]=0$ gives
$a-3b+3c-d=0$.
Now $a-3b+3c-d=-5a+3+3c-d$, and so $d=3-5a+3c$.
$[[e_2,e_1,e_1,e_1,e_1],[e_2,e_1,e_1,e_1,e_1]]=0$
gives $b-4c+6d-4f+g=0$.
\[
b-4c+6d-4f+g=17-28a+14c-4f+g,
\]
so $g=-17+28a-14c+4f$.
Note that $[e_2,e_1,e_1,e_1]=-[e_1,e_2,e_2]$.
$[[e_2,e_1,e_1,e_1],[e_1,e_2,e_2]]=0$ gives $bd-2ad+ac=0$.
\[
\begin{array}{l}
bd-2ad+ac \\
=\left(-1+2a\right) \left(3-5a+3c\right)-2a\left(3-5a+3c\right)+ac \\
=-3+5a-3c+ac,
\end{array}
\]
so either $a=3$ (which gives $12=0$), or
$\displaystyle c=\frac{3-5a}{a-3}$.
\[
\lbrack
[e_2,e_1,e_1,e_1,e_1],[e_2,e_1,e_1,e_1]]+[[e_2,e_1,e_1,e_1,e_1],[e_1,e_2,e_2]]=0
\]
gives
\[
b-3c+3d-f+cf-2bf+bd=0.
\]
\begin{align*}
\displaystyle
b-3c+3d-f+cf&-2bf+bd =
\\&=
\displaystyle
8-13a+6\frac{3-5a}{a-3}-f+\frac{3-5a}{a-3}f
\\&\phantom{=\ }
-2\left( -1+2a\right) f
+
\allowbreak \left( -1+2a\right) \left( 3-5a+3\frac{3-5a}{a-3}\right)
\\&=
\displaystyle
-2\frac{-7a+3+a^2-4fa+2fa^2+5a^3}{a-3}.
\end{align*}
So provided the characteristic is not 2, and provided $a\neq 3$,
\[
-7a+3+a^2+5a^3+(2a^2-4a)f=0.
\]
\[
\lbrack
[e_2,e_1,e_1,e_1,e_1,e_1],[e_2,e_1,e_1,e_1]]+[[e_2,e_1,e_1,e_1,e_1,e_1],[e_1,e_2,e_2]]=0
\]
gives
\[
c-3d+3f-g+dg-2cg+cf=0.
\]
\[
\begin{array}{l}
\displaystyle
c-3d+3f-g+dg-2cg+cf \\
\displaystyle
=-2 \frac{18-27f-123a+280a^2+78fa-53fa^2-253a^3+10fa^3+70a^4}{\left(
a-3\right) ^2}.
\end{array}
\]
So provided the characteristic is not 2, and provided $a\neq 3$,
\[
18-123a+280a^2-253a^3+70a^4+(-27+78a-53a^2+10a^3)f=0.
\]
Combining these two equations we obtain
\begin{multline*}
(-27+78a-53a^2+10a^3)(-7a+3+a^2+5a^3)
-\\-
(2a^2-4a)(18-123a+280a^2-253a^3+70a^4)=0
\end{multline*}
Expanding, we obtain
\begin{align*}
0
&=
495a-81-1260a^2+1710a^3-1305a^4+531a^5-\allowbreak 90a^6
\\&=
-9\times \left(10a-9\right) \left( a-1\right) ^5.
\end{align*}
So if the characteristic is not 2 or 3 or 5 then
$a=1$ or $a=\frac 9{10}$. If the characteristic is 5
then $a=1$. The cases when the characteristic is
2 or 3 have to be dealt with separately. We deal
with the latter in Section~\ref{sec:extra_3}.
When $a=\frac 9{10}$, it is proved in~\cite{Marina}
that the algebras one obtains are quotients of a
certain central extension of the positive part of
the infinite-dimensional Witt algebra. In any case,
there are no infinite-dimensional algebras of maximal
class here.
The choice $a = 1$ uniquely determines the following
metabelian Lie algebra \cite{F, SZ}:
\begin{align*}
m_{2} =
\big\langle
e_{i}, i \ge 1 :\
&\text{$[e_{i} e_{1}] = e_{i+1}$, for $i \ge 2$,}
\\&\text{$[e_{i} e_{2}] = e_{i+2}$, for $i \ge 3$,}
\\&\text{$[e_{i} e_{j}] = 0$, for $i, j \ge 3$ }
\big\rangle.
\end{align*}
Note that $\ad(e_{2})$ is the square of $\ad(e_{1})$
on $L^{2}$.
In fact, we have to show that $m_{2}$ has the
following presentation:
\begin{equation*}
\Span{e_{1}, e_{2} : [e_{2} e_{1} e_{2}] = [e_{2} e_{1}^{3}],
[e_{2} e_{1}^{3} e_{2}] = [e_{2} e_{1}^{5}]}.
\end{equation*}
We use the notation $[e_{i} e_{1}] = e_{i+1}$,
so that the two defining relations can be rewritten
as $[e_{3} e_{2}] = e_{5}$ and $[e_{5} e_{2}] = e_{7}$.
We have already seen that the first one implies
$[e_{4} e_{2}] = e_{6}$. Suppose now we have proved
\begin{equation*}
[e_{3} e_{2}] = e_{5},
[e_{4} e_{2}] = e_{6},
[e_{5} e_{2}] = e_{7},
\dots,
[e_{n-1} e_{2}] = e_{n+1},
\end{equation*}
for some $n > 5$, and want to prove
$[e_{n} e_{2}] = e_{n+2}$. We work out the expansion
\begin{align*}
0
&=
[ e_{n-3}, [e_{3} e_{2}] - e_{5}]
=
[ e_{n-3}, [e_{2} e_{1} e_{2}] - [e_{2} e_{1}^{3}]]
\\&=
[ e_{n-3} [e_{2} e_{1}] e_{2}]
\\&\phantom{=\ }
- [ e_{n-3} e_{2} [e_{2} e_{1}]]
\\&\phantom{=\ }
- [ e_{n-3} [e_{2} e_{1}^{3}]
\\&=
(1 - 1) [e_{n} e_{2}]
\\&\phantom{=\ }
- e_{n+2} + [e_{n} e_{2}]
\\&\phantom{=\ }
- (1 - 3 + 3) e_{n+2} + [e_{n} e_{2}]
\\&=
2 [e_{n} e_{2}] - 2 e_{n+2}.
\end{align*}
Note that this does not work for $n = 5$. From this it is
straightforward to see that the algebra is metabelian,
and thus is isomorphic to $m_{2}$. In fact we have for
$i, j \ge 3$
\begin{align*}
[ e_{i} e_{j} ]
&=
[ e_{i} [ e_{2} \U{e_{1}}{j-2} ] ]
\\&=
\sum_{k=0}^{j-2} (-1)^{k} \binom{j-2}{k}
[ e_{i} \U{e_{1}}{k} e_{2} \U{e_{1}}{j-2-k} ]
\\&=
\left(
\sum_{k=0}^{j-2} (-1)^{k} \binom{j-2}{k}
\right)
\cdot
e_{i+j}
\\&=
0.
\end{align*}
\section{The length of the first constituent}
\label{sec:first_constituent}
Suppose now $L$ is an algebra of type 2 over a field of positive
characteristic $p$. Suppose $L$ admits a theory of
constituents. Therefore $[e_{3} e_{2}] = [e_{2} e_{1} e_{2}] = 0$. If
$[e_{i} e_{2}] = 0$ for all $i \ge 3$, then $L$ is isomorphic to
$m$ of~\eqref{eq:m}. Suppose thus there is an $n > 3$ such that
\begin{math}
[e_{3} e_{2}] = [e_{4} e_{2}] = \dots = [e_{n-2} e_{2}] = 0,
\end{math}
but
\begin{math}
[e_{n-1} e_{2}] \ne 0.
\end{math}
We intend to show that $n$, the length of the first constituent, can
only assume the values
\begin{equation*}
\begin{cases}
2 q, & \text{for some power $q$ of $p$, or}\\
q + 1, & \text{for some power $q > 3$ of $p$.}
\end{cases}
\end{equation*}
We may assume, rescaling $e_{2}$, that $[e_{n-1} e_{2}] = e_{n+1}$. We
first prove that $n$ is even, with a simple argument similar to one
of~\cite{CMN}. In fact, if $n = 2 k - 1$ is odd, we have
\begin{align*}
0
&=
[ e_{k} e_{k} ] = [e_{k} [e_{2} \U{e_{1}}{k-2}]]
\\&=
\sum_{i = 0}^{k-2}
(-1)^{i} \binom{k-2}{i}
[ e_{k} \U{e_{1}}{i} e_{2} \U{e_{1}}{k-2-i} ]
\\&=
\sum_{i = 0}^{k-2}
(-1)^{i} \binom{k-2}{i}
[ e_{k+i} e_{2} \U{e_{1}}{k-2-i} ]
\approx
[e_{n-1} e_{2}]
=
e_{n+1},
\end{align*}
a contradiction.
Here and in the following we write $a \approx b$ to mean that $a$
is either $b$ or $- b$. Write $n = 2 k$. We aim at proving that the
only possible values for $k$ are $q$ and $(q+1)/2$.
We first compute
\begin{align*}
0
&=[e_{k+1} e_{k+1} ] =
[e_{k+1} [e_{2} \U{e_{1}}{k-1}] ]
\\&\approx
\binom{k - 1}{k - 2} [e_{2 k - 1} e_{2} e_{1}]
- \binom{k - 1}{k -1} [e_{2 k} e_{2}]
\\&=
(k - 1) e_{2 k + 2} - [e_{2 k} e_{2}],
\end{align*}
to show
\begin{equation*}
[e_{n} e_{2}]
=
(k - 1) e_{n + 2}.
\end{equation*}
We now have
\begin{equation}\label{eq:extend_by_one}
\begin{aligned}
0
&=
[e_{n - 2} [e_{2} e_{1} e_{2}]]
\\&\approx
[e_{n - 2} e_{1} e_{2} e_{2}]
\\&=
[e_{n + 1} e_{2}].
\end{aligned}
\end{equation}
Further,
\begin{equation}\label{eq:previous_calc}
\begin{aligned}
0
&=
[e_{n - 1} [e_{2} e_{1} e_{2}]]
\\&\approx
[e_{n - 1} e_{1} e_{2} e_{2}]
- 2 [e_{n - 1} e_{2} e_{1} e_{2}]
+ [e_{n - 1} e_{2} e_{2} e_{1}]
\\&=
(k - 1 - 2) [e_{n+2} e_{2}].
\end{aligned}
\end{equation}
This shows that $[e_{n+2} e_{2}] = 0$, except when $k \equiv 3
\pmod{p}$.
Suppose first we have $n = 6$, or $k = 3$. We have here
\begin{equation*}
[e_{5} e_{2}] = e_{7},\quad
[e_{6} e_{2}] = 2 e_{8}, \quad
[e_{7} e_{2}] = 0.
\end{equation*}
We want to show that $p = 3$ or $5$ here, so that this fits into the
$n = 2q$ or $q+1$ pattern above. Suppose $p > 5$. We compute
\begin{align*}
0
&=
[e_{5} [e_{2} e_{1}^{3}] ]
\\&=
[e_{5} e_{2} e_{1}^{3} ] - 3 [e_{6} e_{2} e_{1}^{2} ]
+ 3 [e_{7} e_{2} e_{1} ] - [e_{8} e_{2} ]
\\&=
-5 e_{10} - [e_{8} e_{2}],
\end{align*}
so that $[e_{8} e_{2}] = -5 e_{10}$.
\begin{align*}
0
&=
[e_{6} [e_{1} e_{2} e_{2}] ]
\\&=
[e_{7} e_{2} e_{2} ] - 2 [e_{6} e_{2} e_{1} e_{2} ]
+ [e_{6} e_{2} e_{2} e_{1} ]
\\&=
- 4 [e_{9} e_{2}] + 2 (-5) e_{11},
\end{align*}
so that
\begin{equation*}
[e_{9} e_{2}] = - \frac{5}{2} e_{11}.
\end{equation*}
Finally
\begin{align*}
0
&=
[e_{6} [e_{2} e_{1}^{4}] ]
\\&=
[e_{6} e_{2} e_{1}^{4} ]
- 4 [e_{7} e_{2} e_{1}^{3} ] + 6 [e_{8} e_{2} e_{1}^{2} ]
- 4 [ e_{9} e_{2} e_{1}] + [e_{10} e_{2} ]
\\&=
18 e_{12} + [e_{10} e_{2}]
\end{align*}
and
\begin{align*}
0
&=
[e_{7} [e_{1} e_{2} e_{2}] ]
\\&=
[e_{8} e_{2} e_{2} ] - 2 [e_{7} e_{2} e_{1} e_{2} ]
+ [e_{7} e_{2} e_{2} e_{1} ]
\\&=
-5 [e_{10} e_{2}]
\end{align*}
yield $e_{12} = 0$, a contradiction.
Suppose then $k > 3$, that is, $n > 6$. We have thus $[e_{5} e_{2}] =
0$, so that
\begin{align*}
0
&=
[e_{n - 3} [e_{5} e_{2}]]
=
[e_{n - 3} [e_{2} e_{1} e_{1} e_{1} e_{2}] ]
\\&=
[e_{n - 3} [e_{2} e_{1} e_{1} e_{1}] e_{2}]]
\\&=
3 [e_{n - 3} e_{1} e_{1} e_{2} e_{1} e_{2} ]
- [e_{n - 3} e_{1} e_{1} e_{1} e_{2} e_{2}]
\\&=
(3 - (k-1)) [e_{n+2} e_{2}].
\end{align*}
This shows that $[e_{n+2} e_{2}] = 0$, except when $k \equiv 4
\pmod{p}$, which was covered by~\eqref{eq:previous_calc}.
To find out what the possible values of $k$ are, we compute
\begin{align*}
0
&=
[e_{k+2} e_{k+2}] = [e_{k+2} [e_{2} \U{e_{1}}{k}] ]
\\&\approx
\binom{k}{k - 3} [e_{2 k - 1} e_{2} e_{1} e_{1} e_{1}]
- \binom{k}{k - 2} [e_{2 k} e_{2} e_{1} e_{1}]
\end{align*}
which yields
\begin{equation*}
0
=
\left( \binom{k}{3} - \binom{k}{2} (k-1) \right) e_{2k+4}
= \frac{k (k - 1) (-2k + 1)}{6} \, e_{2k+4}.
\end{equation*}
This shows that the only possibilities for $k$ are
\begin{equation*}
k \in \left\{ 0, 1,\frac{1}{2} \right\} \pmod{p},
\end{equation*}
for $p > 3$, whereas for $p = 3$ one has
\begin{equation*}
k \in \left\{ 0, 1, \frac{1}{2} \right\} \pmod{9}.
\end{equation*}
When $k \equiv 0 \pmod{p}$, we show that $k = q$, a power of $p$. (The
case when $p = 3$ is not special here, as we have already dealt with
$k = 3$ for $p = 3$ above.) This we do by exploiting the deflation
procedure, as described in Section~\ref{sec:preliminaries}. Suppose
in fact $k = q m$, with $q$ a power of $p$, and $m \not\equiv 0
\pmod{p}$. Thus $n = 2 q m$ here. We have $[e_{n - 1} e_{2}] = e_{n + 1}$
and $[e_{n} e_{2}] = - e_{n + 2}$. We have also proved
in~\eqref{eq:extend_by_one} that $[e_{n+1} e_{2}] = 0$.
We first extend this to
\begin{equation*}
[e_{n + 1} e_{2}]
=
[e_{n + 2} e_{2}]
=
\dots
=
[e_{n + p - 2} e_{2}]
=
0.
\end{equation*}
We proceed by induction on $l$, for $1 < l \le p - 2$:
\begin{equation}\label{eq:implicit_lower_bound}
\begin{aligned}
0
&=
[e_{n - 1} [e_{2} e_{1}^{l - 1} e_{2}]]
\\&=
[e_{n - 1} [e_{2} e_{1}^{l - 1}] e_{2}]
-
[e_{n - 1} e_{2} [e_{2} e_{1}^{l - 1}]]
\\&=
[e_{n - 1} e_{2} e_{1}^{l - 1} e_{2}]
- (l - 1)
[e_{n - 1} e_{1} e_{2} e_{1}^{l-2} e_{2}]
- (-1)^{l-1}
[e_{n - 1} e_{2} e_{1}^{l - 1} e_{2}]
\\&=
(1 + l - 1 - (-1)^{l - 1})
\cdot
[e_{n + l} e_{2}].
\end{aligned}
\end{equation}
Now
\begin{equation*}
1 + l - 1 - (-1)^{l - 1} = l + (-1)^{l} =
\begin{cases}
l - 1 & \text{when $l$ is odd,}\\
l + 1 & \text{when $l$ is even.}
\end{cases}
\end{equation*}
In any case the coefficient of $ [e_{n + l} e_{2}]$ is less than $p$
for $l < p - 1$, so that it is non-zero.
In the deflated algebra, we thus have
\begin{equation*}
[e_{2 q m - p} e_{p}] =
[e_{2 q m - p} [e_{2} e_{1}^{p-2}]] =
0
\end{equation*}
and
\begin{equation*}
[e_{2 q m} e_{p}] =
[e_{2 q m} [e_{2} e_{1}^{p-2}]] =
- e_{2 q m + p}.
\end{equation*}
In the deflated algebra the first constituent has thus length $2 q m$.
It follows from the theory of algebras of type~1 that $m = 1$. We will
show in Section \ref{sec:classical} that algebras of type 2 with $k =
q$ come from algebras of type 1.
When $k \equiv \displaystyle \frac{1}{2} \pmod{p}$, write
\begin{equation*}
k = \frac{q m + 1}{2},
\end{equation*}
where $p$ does not divide $m$. Thus $n = q m + 1$. We want to show
that $m = 1$. Suppose otherwise. We have
\begin{equation*}
[e_{n - 1} e_{2}] = e_{n + 1}
\qquad\text{and}\qquad
[e_{n} e_{2}] = - \frac{1}{2} e_{n + 2}.
\end{equation*}
We
begin with proving
\begin{equation}\label{eq:plenty_of_zeroes}
[e_{n + 1} e_{2}]
=
[e_{n + 2} e_{2}]
= \dots =
[e_{n + q - 1} e_{2}]
=
0.
\end{equation}
The identity
\begin{equation}\label{eq:an_equation}
[e_{2} e_{1}^{k} e_{2}] = 0
\end{equation}
holds for $k \le n - 4$. Note that $n - 4 = q m - 3 \ge 2 q - 3$,
as $m > 1$.
Let $l < q - 1$. Write $l + 1 = \beta p^{t}$, where $\beta \not\equiv 0
\pmod{p}$. Note that $p^{t} < q$, so that
\begin{equation*}
l + p^{t} \le q - 2 + p^{t} \le 2 q - 3,
\end{equation*}
and $[e_{2} e_{1}^{l + p^{t}} e_{2}] = 0$, by~\eqref{eq:an_equation}.
Suppose first $t > 0$.
We compute
\begin{align*}
0
&=
[ e_{n - 1 - p^{t}} [e_{2} e_{1}^{l + p^{t}} e_{2}] ]
\\&=
[ e_{n - 1 - p^{t}} [e_{2} e_{1}^{l + p^{t}}] e_{2}]
\\&\approx
\left(
\binom{l + p^{t}}{p^{t}}
+
\frac{1}{2}
\binom{l + p^{t}}{p^{t} + 1}
\right)
[e_{n + l + 1} e_{2}]
\\&=
\left(
\binom{\beta p^{t} + p^{t} - 1}{p^{t}}
+
\frac{1}{2}
\binom{\beta p^{t} + p^{t} - 1}{p^{t} + 1}
\right)
[e_{n + l + 1} e_{2}]
\\&=
\left( \beta + \frac{1}{2} (\beta \cdot (-1)) \right)
[e_{n + l + 1} e_{2}]
\\&=
\frac{\beta}{2}
\cdot
[e_{n + l + 1} e_{2}],
\end{align*}
so that $[e_{n + l + 1} e_{2}] = 0$.
Now consider the case when $p^{t} = 1$, so that $l + 1 \not\equiv 0
\pmod{p}$. An analogous calculation yields
\begin{equation*}
0
=
\frac{(l + 1) \cdot (l + 4)}{4}
\cdot
[e_{n + l + 1} e_{2}].
\end{equation*}
We obtain $[e_{n + l + 1} e_{2}] = 0$, except when $l + 4$ is divisible
by $p$. Note that we may assume $p > 3$ here, since we have already
dealt with the case when $l + 1 \equiv 0 \pmod{p}$. We compute
\begin{align*}
0
&=
[e_{n-3} [e_{2} e_{1}^{l + 2} e_{2}]]
\\&\approx
\left(
\binom{l + 2}{2} + \frac{1}{2} \cdot \binom{l + 2}{3}
\right)
[e_{n + l + 1} e_{2}]
\\&=
[e_{n + l + 1} e_{2}],
\end{align*}
as $p > 3$.
We now reach a contradiction by proving $e_{n + q +1} = e_{q m + q +
2} = 0$. Since
$n = q m + 1$ is even, $m$ is odd, and $q m + q + 2$ is even. Consider
the integer
\begin{equation*}
\frac{q m + q + 2}{2}
=
q \cdot \frac{m+1}{2} + 1.
\end{equation*}
Note that
\begin{equation*}
q \cdot \frac{m+1}{2} + 1 - 2 =
q \cdot \frac{m-1}{2} + q - 1
\end{equation*}
We obtain, using~\eqref{eq:plenty_of_zeroes},
\begin{equation}\label{eq:the_big_kill}
\begin{aligned}
0
&=
[
e_{q \cdot \frac{m+1}{2} + 1}
e_{q \cdot \frac{m+1}{2} + 1}
]
=
[
e_{q \cdot \frac{m+1}{2} + 1}
[e_{2}e_{1}^{q \frac{m-1}{2} + q - 1}]
]
\\&=
\begin{aligned}[t]
\Big(
&(-1)^{q \frac{m-1}{2} - 1}
\binom{q \frac{m-1}{2} + q - 1}{q \frac{m-1}{2} - 1}
\\&+
(-1)^{q \frac{m-1}{2}}
\binom{q \frac{m-1}{2} + q - 1}{q \frac{m-1}{2}}
\cdot \left( - \frac{1}{2}\right)
\Big)
\cdot
e_{q m + q + 2}.
\end{aligned}
\end{aligned}
\end{equation}
Now we have
\begin{equation*}
\binom{q \frac{m-1}{2} + q - 1}{q \frac{m-1}{2} - 1}
\equiv
\binom{q \frac{m-1}{2} + q - 1}{q \frac{m-3}{2} + q - 1}
\equiv
\frac{m-1}{2}
\pmod{p},
\end{equation*}
while
\begin{equation*}
\binom{q \frac{m-1}{2} + q - 1}{q \frac{m-1}{2}}
=
1.
\end{equation*}
Therefore, up to a sign, the overall coefficient of $e_{q m + q + 2}$
in~\eqref{eq:the_big_kill} is
\begin{equation*}
\frac{m-1}{2} + \frac{1}{2} = \frac{m}{2} \not\equiv 0.
\end{equation*}
This disposes of the case $m > 1$, so we obtain
\begin{equation*}
k = \frac{q + 1}{2},
\quad
n = q + 1.
\end{equation*}
We will deal with this case
in Sections \ref{sec:extra} and \ref{sec:Extra_Construction}.
Remember that when $p = 3$ we are taking $q \ge 9$ here. In fact when
$q =3$ we get $k = 2$, so that $[e_{3} e_{2}] \ne 0$, and the algebra
does not admit a theory of constituents.
We now deal with the case $k \equiv 1 \pmod{p}$, so $k = 1 + q m$,
where $q$ is a power of $p$, and $m \not\equiv 0 \pmod{p}$. Thus $n =
2 q m + 2$. We have thus $[e_{n - 1} e_{2}] = e_{n + 1}$ and $[e_{n}
e_{2}] = 0$. We want to show that this case does not occur.
Let $1 \le l < q$. Assume by induction
\begin{equation*}
[e_{n} e_{2}] = [e_{n+1} e_{2}] = \dots = [e_{n+l-1} e_{2}] = 0.
\end{equation*}
We compute
\begin{equation}\label{eq:l}
0
=
[e_{n-2} [e_{2} e_{1}^{l} e_{2}]]
=
[e_{n-2} [e_{2} e_{1}^{l}] e_{2}]
=
- l [e_{n + l} e_{2}].
\end{equation}
We obtain $[e_{n + l} e_{2}] = 0$ for $l < p$. We can use this and
deflation to show that $m = 1$.
Because of
\begin{equation*}
[e_{n - 2} e_{p}]
=
[e_{n - 2} [e_{2} e_{1}^{p-2}]]
=
2 e_{n + p - 2},
\end{equation*}
the length of the first constituent in the deflated algebra (which is
of type~1) is $2 q m / p$. If $m > 1$, this is not twice a power of
$p$. It follows that $m = 1$, and $n = 2 q + 2$.
We now show that $[e_{n + l} e_{2}] = 0$ holds in fact for all $l <
q$. Because of the argument of~\eqref{eq:l}, we have to deal with the
case $l \equiv 0 \pmod{p}$. If $p^{t}$ is the highest power of $p$
that divides $l$, and $l = \beta p^{t}$, with $\beta \not\equiv 0
\pmod{p}$, we compute
\begin{align*}
0
&=
[e_{n-p^{t}-1} [e_{2} e_{1}^{l+p^{t}-1} e_{2}]]
\\&=
[e_{n-p^{t}-1} [e_{2} e_{1}^{l+p^{t}-1}] e_{2}]
\\&=
\pm \binom{l + p^{t} - 1}{p^{t}} [e_{n + l} e_{2}].
\end{align*}
Here
\begin{equation*}
\binom{l + p^{t} - 1}{p^{t}}
=
\binom{\beta p^{t} + p^{t} - 1}{p^{t}} \equiv \beta
\not\equiv 0 \pmod{p}.
\end{equation*}
We can perform this calculation when $l + r - 1 < 2 q - 1$. Note that
this holds for $l < q$. We have thus proved
\begin{equation}\label{eq:these_vanish}
[e_{n} e_{2}] = [e_{n + 1} e_{2}] =
\dots = [e_{n + q - 1} e_{2}] = 0.
\end{equation}
Now we use the relation $[e_{n - 1} e_{2}] - e_{n + 1} = 0$ to prove
$e_{3 q + 3} = e_{n + q + 1} = 0$, a contradiction. We evaluate
\begin{align*}
0
&=
[e_{q}, [e_{n-1} e_{2}] - e_{n+1}]
\\&=
[e_{q} [e_{2} e_{1}^{2 q - 1} e_{2}]] - [e_{q} [e_{2} e_{1}^{2 q +
1}]]
\end{align*}
Note first that $2 q + 1$ is the only value $i$ in the range $2 \le i
\le 3 q + 1$ for which $[e_{i} e_{2}] \ne 0$. Now $[e_{q} [e_{2}
e_{1}^{2 q - 1} e_{2}]]$ expands as a combination of commutators of the
form $[e_{i} e_{2} e_{1}^{2 q + 1 - i}]$, for some $q + 2 \le i \le 3
q + 1$, so that it vanishes.
We obtain
\begin{align*}
0
&=
[e_{q} [e_{2} e_{1}^{2 q + 1}]]
\\&=
(-1)^{q+1} \binom{2 q + 1}{q+1}
e_{3 q + 3}
\\&=
2 e_{3 q + 3}.
\end{align*}
\section{First constituent of length $2 q$}
\label{sec:classical}
This is the case $k = q$ of the previous section. Suppose we have
\begin{gather*}
[e_{i} e_{2}] = 0, \quad \text{for $i < 2 q - 1$}\\
[e_{2 q - 1} e_{2}] = e_{2 q + 1},
\qquad
[e_{2 q} e_{2}] = - e_{2 q + 2}.
\end{gather*}
We want to show that the algebra comes from an algebra of type $1$ via
the procedure described in Section~\ref{sec:preliminaries}, by proving
that all constituents have type $(\lambda, - \lambda)$.
Proceeding by
induction, assume we have already proved this up to a certain
constituent, that ends as
\begin{equation}\label{eq:end_of_constituent}
[e_{m} e_{2}] = \lambda e_{m+2},
\qquad
[e_{m+1} e_{2}] = - \lambda e_{m+3},
\end{equation}
for some $\lambda \ne 0$. We first show, also by induction, that $2
q$ is an upper bound for the length of the next constituent, and $q$
is a lower bound.
Suppose the next constituent has length greater than $2 q$, so that
\begin{equation*}
[e_{m+k} e_{2}] = 0
\end{equation*}
for $2 \le k \le 2 q$. We obtain immediately
\begin{equation*}
[e_{m} [e_{2 q} e_{2}]]
=
[e_{m} [e_{2} e_{1}^{2 q - 2} e_{2}]]
=
0,
\end{equation*}
as this is a multiple of $[e_{m + 2 q} e_{2}] = 0$.
This yields
\begin{align*}
0
&=
[ e_{m}, [e_{2 q} e_{2}] + e_{2 q + 2}]
\\&=
[ e_{m} e_{2 q + 2}]
=
[ e_{m} [e_{2} e_{1}^{2 q}]]
=
[ e_{m} e_{2} e_{1}^{2 q}]
\\&=
\lambda e_{m + 2 + 2 q},
\end{align*}
a contradiction.
We now prove that the next constituent has length at least $q$, that
is,
\begin{equation*}
[e_{m + 2} e_{2}] = [e_{m + 3} e_{2}] = \dots
= [e_{m + q - 1} e_{2}] = 0.
\end{equation*}
This we do more generally for the case when the current constituent is
of the general form
\begin{equation}\label{eq:general_end_of_constituent}
[e_{m} e_{2}] = \mu e_{m+2},
\qquad
[e_{m+1} e_{2}] = \nu e_{m+3},
\end{equation}
as this will be useful later in this section. Recall that $\mu \ne 0$
here, but $\nu$ might be zero.
If $\nu = 0$ in ~\eqref{eq:general_end_of_constituent}, we compute,
proceeding by induction on $l$, for $0 < l < q - 1$,
\begin{align*}
0
&=
[e_{m-1} [e_{2} e_{1}^{l} e_{2}]]
\\&=
[e_{m-1} [e_{2} e_{1}^{l}] e_{2}]
\\&=
- l \mu [e_{m + l + 1} e_{2}].
\end{align*}
The coefficient vanishes when $l \equiv 0 \pmod{p}$. In this case,
write $l = \beta p^{t}$, with $\beta \not\equiv 0 \pmod{p}$. Note that
$p^{t} < q$ here, so that $l + p^{t} - 1 < q - 2 + q - 1 < 2 q - 3$
and $[e_{2} e_{1}^{l+p^{t}-1} e_{2}] = 0$. Also, $[e_{m-p^{t}} e_{2}]
= \dots = [e_{m-1} e_{2}] = 0$, since we are assuming by induction
that constituents have length at least $q$. We compute
\begin{align*}
0
&=
[e_{m-p^{t}} [e_{2} e_{1}^{l+p^{t}-1} e_{2}]]
\\&
- \binom{l + p^{t} - 1}{p^{t}} \mu [e_{m + l + 1} e_{2}].
\end{align*}
Here
\begin{equation*}
\binom{l + p^{t} - 1}{p^{t}}
=
\binom{\beta p^{t} + p^{t} - 1}{p^{t}}
\equiv
\beta
\not\equiv
0
\pmod{p}.
\end{equation*}
Suppose now $\nu \ne 0$. We have first
\begin{align*}
0
=
[e_{m-1} [e_{2} e_{1} e_{2}]]
=
- \mu [e_{m+2} e_{2}],
\end{align*}
so that $[e_{m+2} e_{2}] = 0$. We proceed now by induction on $l$,
for $0 < l < q - 2$.
\begin{align}
0
&=
[e_{m} [e_{2} e_{1}^{l} e_{2}]]
\notag
\\&=
[e_{m} [e_{2} e_{1}^{l}] e_{2}]
-
[e_{m} e_{2} [e_{2} e_{1}^{l-1}]]
\notag
\\&=
(\mu - l \nu - \mu (-1)^{l}) [e_{m + l + 2} e_{2}].
\label{eq:coeff}
\end{align}
For $l$ even, the coefficient is $- l \nu \ne 0$, so we get
$[e_{m + l + 2} e_{2}] = 0$, unless $l \equiv 0 \pmod{p}$. In this case,
we compute
\begin{align*}
0
&=
[e_{m+1} [e_{2} e_{1}^{l-1} e_{2}]]
\\&=
[e_{m+1} [e_{2} e_{1}^{l-1}] e_{2}]
-
[e_{m+1} e_{2} [e_{2} e_{1}^{l-1}]]
\\&=
(\nu - (-1)^{l-1} \nu)
[e_{m+l+2} e_{2}].
\end{align*}
As $l$ is even here, the coefficient is $2 \nu \ne 0$.
For $l$ odd, the coefficient in~\eqref{eq:coeff} is $2 \mu - l \nu$.
Suppose this vanishes. As $1 \le l < q - 2$, we have $q > 3$ here, so
that $[e_{m-2} e_{2}] = 0$. We compute
\begin{align*}
[e_{m-2} [e_{2} e_{1}^{l+2} e_{2}]]
&=
\left(
\binom{l + 2}{2} \mu - \binom{l + 2}{3} \nu
\right)
\cdot
[e_{m + l + 2} e_{2}]
\\&=
\frac{(l + 2) (l + 1)}{6} \mu
[e_{m + l + 2} e_{2}],
\end{align*}
where we have used the fact that $l \nu = 2 \mu$ here. The
coefficient vanishes when $l + 2 \equiv 0 \pmod{p}$, or $l + 1 \equiv
0 \pmod{p}$. (Except possibly when $p = 3$, and $l + 1$ or $l + 2$
are divisible by $3$ but not by $9$ -- in this case the rest of
the discussion is superfluous. Note that $l \not\equiv 0 \pmod{p}$
here, otherwise $\mu = \frac{1}{2} l \nu = 0$. Therefore $l \equiv -1,
-2 \pmod{3}$ when $p = 3$, so that $(l+2)(l+1)/6$ is an integer.)
When $l + 2 \equiv 0 \pmod{p}$, we have $0 = 2 \mu - l \nu = 2 (\mu +
\nu)$, so that we are in the case of~\eqref{eq:end_of_constituent},
with $\mu = \lambda$ and $\nu = -\lambda$ for some $\lambda \ne 0$.
Write $l + 2 = \beta p^{t}$, with $\beta \not\equiv 0$. It is easy to
see, with an argument we have employed before, that $l + p^{t} < 2 q -
3$, so that $[e_{2} e_{1}^{l+p^{t}} e_{2}] = 0$.
We have then
\begin{align*}
0
&=
[e_{m-p^{t}} [e_{2} e_{1}^{l+p^{t}} e_{2}]]
\\&=
[e_{m-p^{t}} [e_{2} e_{1}^{l+p^{t}}] e_{2}]
\\&=
\left(
- \binom{l + p^{t}}{p^{t}} \lambda
+ \binom{l + p^{t}}{p^{t} + 1} (- \lambda)
\right).
\cdot
[e_{m+l+2} e_{2}]
\\&=
- \lambda \cdot \binom{l + p^{t} + 1}{p^{t} + 1}
\cdot
[e_{m+l+2} e_{2}].
\end{align*}
As
\begin{equation*}
- \lambda \cdot \binom{l + p^{t} + 1}{p^{t} + 1}
=
- \lambda \cdot \binom{\beta p^{t} + p^{t} - 1}{p^{t} + 1}
\equiv
\lambda \beta
\not\equiv
0
\pmod{p},
\end{equation*}
we get $[e_{m+l+2} e_{2}] = 0$.
When $l + 1 \equiv 0 \pmod{p}$, write $l + 1 = \beta p^{t}$,
with $\beta \not\equiv 0 \pmod{p}$. Compute
\begin{align*}
0
&=
[e_{m-p^{t}} [e_{2} e_{1}^{l+p^{t}} e_{2}]]
\\&=
\left(
- \binom{l+p^{t}}{p^{t}} \mu + \binom{l+p^{t}}{p^{t} + 1} \nu
\right)
\cdot
[e_{m + l + 1} e_{2}].
\end{align*}
The coefficient here is, up to a sign,
\begin{math}
\beta (\mu + \nu).
\end{math}
This cannot vanish, otherwise the two relations $\mu + \nu = 0$ and $0
= 2 \mu - l \nu = 2 \mu + \nu$ would yield $\mu = \nu = 0$, a
contradiction.
We now provide the induction step for our assumption that all
constituents are of the form $(\lambda, - \lambda)$.
Suppose first the following constituent is of length $q$. Let
\begin{equation*}
[e_{m+q} e_{2}] = \mu e_{m+q+2},
\qquad\text{and}\qquad
[e_{m+q+1} e_{2}] = \nu e_{m+q+3},
\end{equation*}
We have
\begin{align*}
0
&=
[e_{m-1}, [e_{2 q - 1} e_{2}] - e_{2 q + 1}]
\\&=
[e_{m-1}, [e_{2} e_{1}^{2 q - 3} e_{2}]] - [e_{m-1}, e_{2 q + 1}]
\\&=
[e_{m-1} [e_{2} e_{1}^{2 q - 3}] e_{2}]
- [e_{m-1} e_{2} [e_{2} e_{1}^{2 q - 3}]]
- [e_{m-1} [e_{2} e_{1}^{2 q - 1}]]
\end{align*}
The second term vanishes because $[e_{m-1} e_{2}] = 0$. The first term
is a multiple of $[e_{m + 2 q - 2} e_{2}] = [e_{(m + q) + q - 2}
e_{2}]$. If this is non-zero, it exhibits a constituent of length $q
- 2$ or $q - 1$, whereas we have shown $q$ to be a lower bound for the
length of a constituent. Therefore the first term also vanishes.
We are left with
\begin{align*}
0
&=
[e_{m-1} [e_{2} e_{1}^{2 q - 1}]]
\\&\approx
(-1)^{1} \binom{2 q - 1}{1} [e_{m} e_{2} e_{1}^{2 q - 2}]
+
(-1)^{2} \binom{2 q - 1}{2} [e_{m+1} e_{2} e_{1}^{2 q - 3}]
\\&\phantom{\approx\ }+
(-1)^{1 + q} \binom{2 q - 1}{1 + q}
[e_{m+q} e_{2} e_{1}^{q - 2}]
+
(-1)^{1 + q + 1} \binom{2 q - 1}{1 + q + 1}
[e_{m+q+1} e_{2} e_{1}^{q - 3}]
\end{align*}
Now the first two binomial coefficients readily evaluate to $1$, while
for the last two we have, for $l \ge q$,
\begin{align*}
(-1)^{1 + l} \binom{2 q - 1}{1 + l}
&=
(-1)^{l+1}
\begin{pmatrix}
q & + & q - 1\\
q & + & l - q + 1
\end{pmatrix}
\\&\equiv
- (-1)^{l - q + 1} \binom{q - 1}{l - q + 1}
\\&=
-1.
\end{align*}
We obtain
\begin{equation*}
0
=
\left(
\lambda - \lambda - \mu - \nu
\right)
\cdot
e_{m + 2 q},
\end{equation*}
so that $\nu = - \mu$, as requested.
Suppose now the next constituent has length $l > q$, so that in
particular
\begin{equation*}
[e_{m + 2} e_{2}] = [e_{m + 3} e_{2}] = \dots = [e_{m + q} e_{2}] = 0.
\end{equation*}
We first extend this to show $[e_{m + q + 1} e_{2}] = 0$, so that $l > q +
1$. This follows from
\begin{align*}
0
&=
[e_{m} [e_{2} e_{1}^{q-1} e_{2}]]
\\&=
[e_{m} [e_{2} e_{1}^{q-1}] e_{2}]
-
[e_{m} e_{2} [e_{2} e_{1}^{q-1}]]
\\&=
\left( \lambda - \lambda - \lambda \right)
\cdot
[e_{m + q + 1} e_{2}].
\end{align*}
Suppose now
$[e_{m + l} e_{2}] = \mu e_{m + l + 2}$ and $[e_{m + l + 1} e_{2}] = \nu
e_{m + l + 3}$. We compute
\begin{equation*}
0
=
[e_{m + l - q}, [e_{2 q - 1} e_{2}] - e_{2 q + 1}]
=
[e_{m + l - q} [e_{2 q - 1} e_{2}] ]
-
[e_{m + l - q} e_{2 q + 1}].
\end{equation*}
Keeping in mind that $m + l - q \ge m + 2$, the first term is
immediately seen to vanish.
We are left with
\begin{align*}
0
&=
[e_{m + l - q} e_{2 q + 1}]
\\&=
[e_{m + l - q} [ e_{2} e_{1}^{2 q - 1}]]
\\&=
(-1)^{q} \binom{2 q - 1}{q}
[e_{m + l - q} e_{2} e_{1}^{q - 1}]
+
(-1)^{q + 1} \binom{2 q - 1}{q+ 1}
[e_{m + l - q + 1} e_{2} e_{1}^{q - 2}]
\\&=
( - \mu - \nu ) \cdot e_{m + l + 1}.
\end{align*}
In this case, too, we obtain $\nu = - \mu$. This completes the
induction step.
\section{First constituent of length $q$}
\label{sec:extra}
Let $q$ be a power of $p$ ($q>3$), and suppose that $[e_i,e_2]=0$ for $%
i=3,4,\ldots ,q-1$, and that $[e_q,e_2]\ne 0$. By scaling $e_2$ we may
suppose that $[e_q,e_2]=e_{q+2}$. We show that there is a unique infinite
dimensional Lie algebra $L$ of type 2 satisfying this condition. The Lie
algebra $L$ is defined by the following:
\begin{itemize}
\item $[e_i,e_2]=0$ for $i=3,4,\ldots ,q-1$,
\item $[e_q,e_2]=e_{q+2}$, $[e_{q+1},e_2]=-\frac 12e_{q+3}$,
\item $[e_{kq},e_2]=\frac 12e_{kq+2}$, $[e_{kq+1},e_2]=-\frac 12e_{kq+3}$
for $k=2,3,\ldots $,
\item $[e_k,e_2]=0$ for $k>q+1$ unless $k=0 \pmod{q}$ or $k=1 \pmod{q}$.
\end{itemize}
Note that in this Lie algebra, if $m>q$ and $n\ge 1$ then
\[
\lbrack [e_m,e_n,e_1^q]=[e_m,e_1^q,e_n]
\]
so that
\[
\lbrack [e_m,e_{n+q}]=[e_m,[e_n,e_1^q]]=0
\]
It follows that if $m,n>q$ then $[e_m,e_n]=0$, so that the Lie algebra is
soluble. We give a construction of $L$ in Section 8, and we make use of the
existence of $L$ in the following way. In $L$ we have $[e_n,e_2]=\mu
_ne_{n+2}$ for $n>2$, where $\mu _n=0,1,\frac 12,$ or $-\frac 12$ as
described above. Suppose that we have a Lie algebra $M$ of type 2, where $M$
is spanned by $\{e_i\,|\,i\geq 1\}$, with $[e_i,e_1]=e_{i+1}$ for $i>1$ and $%
[e_n,e_2]=\mu _ne_{n+2}$ for $2<n<2m-2$. Then the relation $[e_m,e_m]=0$ gives
\begin{eqnarray*}
0 &=&[e_m,[e_2,e_1^{m-2}]] \\
&=&\sum_{k=0}^{m-2}(-1)^k\binom{m-2}k[e_m,e_1^k,e_2,e_1^{m-2-k}] \\
&=&\sum_{k=0}^{m-3}(-1)^k\binom{m-2}k\mu
_{m+k}e_{2m}+(-1)^{m-2}[e_{2m-2},e_2].
\end{eqnarray*}
So $[e_{2m-2},e_2]=\mu e_{2m}$ for some $\mu $ which is uniquely determined
by $\{\mu _k\,|\,m\leq k<2m-2\}$. The existence of $L$ implies that $\mu
=\mu _{2m-2}$.
So we assume that $[e_i,e_2]=0$ for $i=3,4,\ldots ,q-1$, and that $%
[e_q,e_2]=e_{q+2}$. The argument just given implies that
\[
\lbrack e_{q+1},e_2]=\mu _{q+1}e_{q+3}=-\frac 12e_{q+3}.
\]
Since $q>3$, $[e_1,e_2,e_2]=0$, and so
\[
0=[e_{q-1},[e_1,e_2,e_2]]=[e_q,e_2,e_2]=[e_{q+2},e_2].
\]
It follows that $[e_{q+3},e_2]=\mu _{q+3}e_{q+5}=0.$
We now show by induction that $[e_k,e_2]=0$ for $k=q+2,q+3,\ldots ,2q-1$. We
have established the cases $k=q+2$ and $q+3$. So suppose that $q+3<m<2q$,
and suppose that $[e_k,e_2]=0$ for $k=q+2,q+3,\ldots ,m-1$.
Using the argument above, it is only necessary to consider the case when $m$
is odd. Then
\begin{eqnarray*}
0 &=&[e_{q+1},[e_2,e_1^{m-q-3},e_2]] \\
&=&[e_{q+1},[e_2,e_1^{m-q-3}],e_2]-[e_{q+1},e_2,[e_2,e_1^{m-q-3}]] \\
&=&[e_{q+1},e_2,e_1^{m-q-3},e_2]-(-1)^{m-q-3}[e_{q+1},e_2,e_1^{m-q-3},e_2] \\
&=&-[e_m,e_2].
\end{eqnarray*}
So $[e_k,e_2]=0$ for $k=q+2,q+3,\ldots ,2q-1$, as claimed. Also
\[
\lbrack e_{2q},e_2]=\mu _{2q}e_{2q+2}=\frac 12e_{2q+2}.
\]
The equations obtained so far leave $[e_{2q+1},e_2]$ undetermined, and so we
suppose that
\[
\lbrack e_{2q+1},e_2]=\lambda e_{2q+3},
\]
for some $\lambda $. We will show below that $\lambda $ must equal $-\frac 12
$ or $-\frac 14$, but first we show that $[e_k,e_2]=0$ for $2q+1<k<3q$. It
is convenient to subdivide the proof of this into the case when $\lambda =0$
and the case when $\lambda \neq 0$.
First consider the case when $\lambda =0$.
The equation $[[e_2,e_1^q],[e_2,e_1^q]]=0$ gives
\[
\lbrack e_2,e_1^q,e_2,e_1^q]=[e_2,e_1^{2q},e_2],
\]
which implies that $[e_{2q+2},e_2]=0$. And
\[
\lbrack e_{2q},[e_1,e_2,e_2]]=0
\]
gives
\[
-[e_{2q+3},e_2]=0,
\]
So we assume that $2q+3<m<3q$, and that $[e_k,e_2]=0$ for $2q<k<m$. If $m$
is odd then
\[
0=[e_{2q},[e_2,e_1^{m-2q-2},e_2]]=[e_m,e_2].
\]
If $m$ is even and $m<3q-1$ then $[e_{2q-1},[e_2,e_1^{m-2q-1},e_2]]=0$ gives
\begin{equation}
(m-2q-1)[e_m,e_2]=0. \label{eq1}
\end{equation}
Also if $2q+3<m<3q$ then $[e_{m-q},e_2]=0$, and so the equation
\[
\lbrack e_{m-q},[e_2,e_1^{q-2},e_2]]=[e_{m-q},[e_2,e_1^q]]
\]
gives
\begin{equation}
((3q-m+1)\frac 12+1)[e_m,e_2]=0. \label{eq3}
\end{equation}
\From (\ref{eq1}) we see that if $m$ is even and $2q+3<m<3q-1$ then $%
[e_m,e_2]=0$ unless $m=1 \pmod{p}$. But (\ref{eq3}) shows that $[e_m,e_2]=0$
in the case when $m=1 \pmod{p}$, as well as in the case when $m=3q-1$. So $%
[e_k,e_2]=0$ for $2q+1<k<3q$ in the case when $\lambda =0$.
So suppose that $\lambda \neq 0$. As above, we want to show that $%
[e_k,e_2]=0 $ for $2q+1<k<3q$. Since we need the following argument several
times, it is convenient to put it in the form of a lemma.
\begin{lemma}
\label{mrvl1}Let $t\geq 1$ and let $q=p^s>3$. Suppose that $[e_k,e_2]=0$ for
$1<k<2tq$ unless $k=0 \pmod{q}$ or $k=1 \pmod{q}$, that $[e_{2tq},e_2]=\alpha
e_{2tq+2}$ for some $\alpha \neq 0$, and that $[e_{2tq+1},e_2]=\lambda
e_{2tq+3}$ for some $\lambda \neq 0$. Then $[e_{2tq+k},e_2]=0$ for $1<k<q$.
\end{lemma}
\begin{proof}
The case $k=2$ follows from
\[
0=[e_{2tq-1},[e_1,e_2,e_2]]=\alpha [e_{2tq+2},e_2].
\]
Now suppose by induction that $m$ is odd, that $3\leq m\leq q-2$, and that $%
[e_{2tq+k},e_2]=0$ for all $k$ such that $1<k<m$. We show that $%
[e_{2tq+m},e_2]=[e_{2tq+m+1},e_2]=0$, and this establishes the lemma by
induction on (odd) $m$.
First we have
\begin{equation}
0=[e_{2tq+1},[e_2,e_1^{m-2},e_2]]=2\lambda [e_{2tq+m+1},e_2]-\lambda
(m-2)[e_{2tq+m},e_2,e_1]. \label{eq21}
\end{equation}
If we let $d=\frac{m+3}2$, we also have
\begin{eqnarray*}
0 &=&[e_{tq+d},e_{tq+d}] \\
&=&[e_{tq+d},[e_d,e_1^{tq}]] \\
&=&\sum_{r=0}^t(-1)^r\binom tr[e_{(t+r)q+d},e_d,e_1^{(t-r)q}].
\end{eqnarray*}
Now our hypotheses imply that $[e_{(t+r)q+d},e_d]=0$ if $r<t$. So this
equation implies that $[e_{2tq+d},e_d]=0$ also. Since $e_d=[e_2,e_1^{d-2}]$
this gives
\[
\sum_{r=0}^{d-2}(-1)^r\binom{d-2}r[e_{2tq+d+r},e_2,e_1^{d-2-r}]=0.
\]
But our inductive hypothesis implies that $[e_{2tq+d+r},e_2]=0$ for $r<d-3$.
So we obtain
\begin{equation}
\lbrack e_{2tq+m+1},e_2]=(d-2)[e_{2tq+m},e_2,e_1]. \label{eq22}
\end{equation}
Since $d-2=\frac{m-1}2$ (\ref{eq21}) and (\ref{eq22}) imply that $%
[e_{2tq+m+1},e_2]=[e_{2tq+m},e_2,e_1]=0$, which also implies that $%
[e_{2tq+m},e_2]=0$. This completes the proof of the lemma.
\end{proof}
So $[e_k,e_2]=0$ for $2q+1<k<3q$, whatever the value of $\lambda $.
Now consider the equation
\[
\lbrack e_{2q},[e_2,e_1^{q-2},e_2]]=[e_{2q},[e_2,e_1^q]].
\]
This gives
\begin{eqnarray*}
0
&=&[e_{2q},[e_2,e_1^{q-2}],e_2]-[e_{2q},e_2,[e_2,e_1^{q-2}]]-[e_{2q},e_2,e_1^q]+[e_{2q},e_1^q,e_2]
\\
&=&\frac 12[e_{3q},e_2]+2\lambda [e_{3q},e_2]+\frac 12[e_{3q},e_2]-\frac
12e_{3q+2}+[e_{3q},e_2] \\
&=&(2+2\lambda )[e_{3q},e_2]-\frac 12e_{3q+2},
\end{eqnarray*}
which implies that
\begin{equation}
e_{3q+3}=(4+4\lambda )[e_{3q},e_2,e_1] \label{eq5}
\end{equation}
And
\[
\lbrack e_{2q+1},[e_2,e_1^{q-2},e_2]]=[e_{2q+1},[e_2,e_1^q]]
\]
gives
\begin{equation}
2\lambda [e_{3q+1},e_2]-\lambda (q-2)[e_{3q},e_2,e_1]=\lambda
e_{3q+3}-[e_{3q+1},e_2]. \label{eq6}
\end{equation}
In addition, the equation
\[
\lbrack [e_2,e_1^{\frac{3q-1}2}],[e_2,e_1^{\frac{3q-1}2}]]=0
\]
gives
\begin{eqnarray}
0 &=&\left( (-1)^{\frac{q-3}2}\binom{\frac{3q-1}2}{\frac{q-3}2}\frac
12+(-1)^{\frac{q-1}2}\binom{\frac{3q-1}2}{\frac{q-1}2}\lambda \right)
e_{3q+3} \label{eq7} \\
&&+(-1)^{\frac{3q-3}2}\frac{3q-1}2[e_{3q},e_2,e_1]+(-1)^{\frac{3q-1}%
2}[e_{3q+1},e_2] \nonumber
\end{eqnarray}
Now
\[
\binom{\frac{3q-1}2}{\frac{q-3}2}=\binom{q+\frac{q-1}2}{\frac{q-1}2-1}=\frac{%
q-1}2\pmod{p},
\]
and
\[
\binom{\frac{3q-1}2}{\frac{q-1}2}=\binom{q+\frac{q-1}2}{\frac{q-1}2}=1\text{
mod }p,
\]
So (\ref{eq7}) gives
\[
(-1)^{\frac{q-1}2}\left( \frac 14+\lambda \right) e_{3q+3}-(-1)^{\frac{3q-3}%
2}\frac 12[e_{3q},e_2,e_1]+(-1)^{\frac{3q-1}2}[e_{3q+1},e_2]=0,
\]
which implies that
\begin{equation}
\left( \frac 14+\lambda \right) e_{3q+3}-\frac
12[e_{3q},e_2,e_1]-[e_{3q+1},e_2]=0. \label{eq8}
\end{equation}
\From (\ref{eq5}) and (\ref{eq8}) we obtain
\[
\lbrack e_{3q+1},e_2]=((4+4\lambda )(\frac 14+\lambda )-\frac
12)[e_{3q},e_2,e_1].
\]
So (\ref{eq6}) gives
\[
(2\lambda +1)((4+4\lambda )(\frac 14+\lambda )-\frac 12)+2\lambda =\lambda
(4+4\lambda ),
\]
which implies that $\lambda =-\frac 12$ or $-\frac 14$.
If $\lambda =-\frac 12$ then $[e_{3q},e_2]=\frac 12e_{3q+2}$, and $%
[e_{3q+1},e_2]=-\frac 12e_{3q+3}$. If $\lambda =-\frac 14$ and $p=3$, then (%
\ref{eq5}) gives $e_{3q+3}=0$, so $\lambda =-\frac 12$ is the only
possibility when $p=3$. If $\lambda =-\frac 14$ and $p\neq 3$, then we have $%
[e_{3q},e_2]=\frac 13e_{3q+2}$ and $[e_{3q+1},e_2]=-\frac 16e_{3q+3}$.
Thus we have established that if $q$ is a power of $p$ ($q>3$), and $%
[e_i,e_2]=0$ for $i=3,4,\ldots ,q-1$, and $[e_q,e_2]=e_{q+2}$, then
\[
\lbrack e_{q+1},e_2]=-\frac 12e_{q+3},
\]
\[
\lbrack e_k,e_2]=0\text{ for }q+1<k<2q,
\]
\[
\lbrack e_{2q},e_2]=\frac 12e_{2q+2},
\]
\[
\lbrack e_{2q+1},e_2]=\lambda e_{2q+3}\text{ where }\lambda =-\frac 12\text{
or }\lambda =-\frac 14,
\]
\[
\lbrack e_k,e_2]=0\text{ for }2q+1<k<3q,
\]
\[
\lbrack e_{3q},e_2]=\left\{
\begin{array}{l}
\frac 12e_{3q+2}\text{ when }\lambda =-\frac 12 \\
\frac 13e_{3q+2}\text{ when }\lambda =-\frac 14
\end{array}
\right. ,
\]
\[
\lbrack e_{3q+1},e_2]=\left\{
\begin{array}{l}
-\frac 12e_{3q+3}\text{ when }\lambda =-\frac 12 \\
-\frac 16e_{3q+3}\text{ when }\lambda =-\frac 14
\end{array}
\right. .
\]
Furthermore, the case $\lambda =-\frac 14$ can only arise when $p\neq 3$.
\subsection{Generic step for $\lambda =-\frac 12$.}
We assume that $q$ is a power of $p$ ($q>3$) and we assume that
\begin{itemize}
\item $[e_i,e_2]=0$ for $i=3,4,\ldots ,q-1$,
\item $[e_q,e_2]=e_{q+2}$, $[e_{q+1},e_2]=-\frac 12e_{q+3}$,
\item $[e_{kq},e_2]=\frac 12e_{kq+2}$, $[e_{kq+1},e_2]=-\frac 12e_{kq+3}$
for $k=2,3,\ldots ,2n-1$ ($n\geq 2$),
\item $[e_k,e_2]=0$ for $q+1<k<(2n-1)q$ unless $k=0 \pmod{q}$ or $k=1
\pmod{q}$.
\end{itemize}
We show that
\begin{itemize}
\item $[e_{(2n-1)q+k},e_2]=0$ for $1<k<q$,
\item $[e_{2nq},e_2]=\frac 12e_{2nq+2}$, $[e_{2nq+1},e_2]=\lambda e_{2nq+3}$
where $\lambda =-\frac 12$ or $-\frac 14$,
\item $[e_{2nq+k},e_2]=0$ for $1<k<q$,
\item if $\lambda =-\frac 12$ then $[e_{(2n+1)q},e_2]=\frac 12e_{(2n+1)q+2}$
and $[e_{(2n+1)q+1},e_2]=-\frac 12e_{(2n+1)q+3}$,
\item if $\lambda =-\frac 14$ then $[e_{(2n+1)q},e_2]=\frac 13e_{(2n+1)q+2}$
and $[e_{(2n+1)q+1},e_2]=-\frac 16e_{(2n+1)q+3}$.
\end{itemize}
First we show that $[e_{(2n-1)q+k},e_2]=0$ for $1<k<q$. Since $%
[e_{(2n-2)q+k},e_2]=0$, the equation
\[
\lbrack e_{(2n-2)q+k},[e_2,e_1^{q-2},e_2]]=[e_{(2n-2)q+k},[e_2,e_1^q]]
\]
gives
\[
(-1)^{q-k}\left( \binom{q-2}{q-k}+\binom{q-2}{q-k+1}\right) \frac
12[e_{(2n-1)q+k},e_2]=-[e_{(2n-1)q+k},e_2].
\]
This implies that $[e_{(2n-1)q+k},e_2]=0$, since
\begin{multline*}
1+(-1)^{q-k}\binom{q-2}{q-k}\frac 12+(-1)^{q-k}\binom{q-2}{q-k+1}\frac 12
=\\=
1+(q-k+1)\frac 12-(q-k+2)\frac 12\pmod{p}
=\\=
\frac 12\pmod{p}.
\end{multline*}
Next consider the equation
\[
\lbrack e_{(2n-1)q},[e_2,e_1^{q-2},e_2]]=[e_{(2n-1)q},[e_2,e_1^q]]
\]
This gives
\begin{eqnarray*}
&&2[[e_{(2n-1)q},e_2,e_1^{q-2},e_2]+2[[e_{(2n-1)q+1},e_2,e_1^{q-3},e_2] \\
&=&[e_{(2n-1)q},e_2,e_1^q]-[e_{(2n-1)q},e_1^q,e_2]
\end{eqnarray*}
which implies that
\[
\lbrack e_{2nq},e_2]=\frac 12e_{2nq+2}.
\]
The equations obtained so far leave $[e_{2nq+1},e_2]$ undetermined, and so
we suppose that
\[
\lbrack e_{2nq+1},e_2]=\lambda e_{2nq+3},
\]
for some $\lambda $. We will show below that $\lambda $ must equal $-\frac
12 $ or $-\frac 14$.
First note that the lemma implies that if $\lambda \neq 0$ then $%
[e_{2nq+k},e_2]=0$ for $1<k<q$. We show that $[e_{2nq+k},e_2]=0$ for $1<k<q$
in the case $\lambda =0$ also. So suppose that $\lambda =0$.
\[
0=[e_{2nq-1},[e_1,e_2,e_2]]=\frac 12[e_{2nq+2},e_2].
\]
Also
\begin{eqnarray*}
0 &=&[e_{2nq},[e_1,e_2,e_2]] \\
&=&[e_{2nq},e_1,e_2,e_2]-2[e_{2nq},e_2,e_1,e_2]]+[e_{2nq},e_2,e_2,e_1] \\
&=&-[e_{2nq+3},e_2],
\end{eqnarray*}
so $[e_{2nq+3},e_2]=0$.
We assume that $3<m<q$, and that $[e_{2nq+k},e_2]=0$ for $1<k<m$. If $m$ is
odd then
\[
0=[e_{2nq},[e_2,e_1^{m-2},e_2]]=[e_{2nq+m},e_2].
\]
If $m$ is even and $m<q-1$ then $[e_{2nq-1},[e_2,e_1^{m-1},e_2]]=0$ gives
\begin{equation}
(m-1)[e_{2nq+m},e_2]=0. \label{eq10}
\end{equation}
Also if $3<m<q$ then $[e_{(2n-1)q+m},e_2]=0$, and so the equation
\[
\lbrack e_{(2n-1)q+m},[e_2,e_1^{q-2},e_2]]=[e_{(2n-1)q+m},[e_2,e_1^q]]
\]
gives
\begin{equation}
((q-m+1)\frac 12+1)[e_{2nq+m},e_2]=0. \label{eq11}
\end{equation}
\From (\ref{eq10}) we see that if $m$ is even and $3<m<q-1$ then $%
[e_{2nq+m},e_2]=0$ unless $m=1 \pmod{p}$. But (\ref{eq11}) shows that $%
[e_{2nq+m},e_2]=0$ in the case when $m=1 \pmod{p}$, as well as in the case
when $m=q-1$. So $[e_{2nq+k},e_2]=0$ for $1<k<q$ in the case when $\lambda
=0 $, as well as in the case $\lambda \neq 0$.
Now consider the equation
\[
\lbrack e_{2nq},[e_2,e_1^{q-2},e_2]]=[e_{2nq},[e_2,e_1^q]].
\]
This gives
\begin{equation}
e_{(2n+1)q+3}=(4+4\lambda )[e_{(2n+1)q},e_2,e_1] \label{eq15}
\end{equation}
in exactly the same way as (\ref{eq5}) was obtained from $%
[e_{2q},[e_2,e_1^{q-2},e_2]]=[e_{2q},[e_2,e_1^q]]$. And
\[
\lbrack e_{2nq+1},[e_2,e_1^{q-2},e_2]]=[e_{2nq+1},[e_2,e_1^q]]
\]
gives
\begin{equation}
2\lambda [e_{(2n+1)q+1},e_2]-\lambda (q-2)[e_{(2n+1)q},e_2,e_1]=\lambda
e_{(2n+1)q+3}-[e_{(2n+1)q+1},e_2]. \label{eq16}
\end{equation}
Now consider the equation
\begin{equation}
\lbrack [e_{nq+2},e_1^{\frac{q-1}2}],[e_{nq+2},e_1^{\frac{q-1}2}]]=0
\label{eq17}
\end{equation}
If we expand $[[e_{nq+2},e_1^{\frac{q-1}2}],[e_{nq+2},e_1^{\frac{q-1}2}]]$
we obtain a sum of the form
\[
\sum_{r=\frac{q-1}2}^{q-1}\alpha _r[e_{nq+2+r},e_{nq+2},e_1^{q-1-r}].
\]
Now
\begin{eqnarray*}
&&[e_{nq+2+r},e_{nq+2}] \\
&=&[e_{nq+2+r},[e_2,e_1^{nq}]] \\
&=&\sum_{s=0}^n(-1)^s\binom ns[e_{nq+2+r},e_1^{sq},e_2,e_1^{(n-s)q}].
\end{eqnarray*}
If $r<q-2$, then $[e_{nq+2+r},e_1^{sq},e_2,e_1^{(n-s)q}]=0$ for all $s$. If $%
r=q-2$ then
\[
\lbrack e_{nq+2+r},e_1^{sq},e_2,e_1^{(n-s)q}]=\frac 12e_{(2n+1)q+2}
\]
for $s<n$, and
\[
\lbrack e_{nq+2+r},e_1^{sq},e_2,e_1^{(n-s)q}]=[e_{(2n+1)q},e_2]
\]
for $s=n$. It follows that if $r=q-2$ then
\[
\sum_{s=0}^n(-1)^s\binom
ns[e_{nq+2+r},e_1^{sq},e_2,e_1^{(n-s)q}]=(-1)^n(\frac
12e_{(2n+1)q+2}-[e_{(2n+1)q},e_2]).
\]
Similarly, if $r=q-1$ then
\[
\sum_{s=0}^n(-1)^s\binom
ns[e_{nq+2+r},e_1^{sq},e_2,e_1^{(n-s)q}]=(-1)^n(\lambda
e_{(2n+1)q+3}-[e_{(2n+1)q+1},e_2]).
\]
So (\ref{eq17}) gives
\[
\frac 12(\frac 12e_{(2n+1)q+3}-[e_{(2n+1)q},e_2,e_1])+\lambda
e_{(2n+1)q+3}-[e_{(2n+1)q+1},e_2]=0,
\]
which implies that
\begin{equation}
\left( \frac 14+\lambda \right) e_{(2n+1)q+3}-\frac
12[e_{(2n+1)q},e_2,e_1]-[e_{(2n+1)q+1},e_2]=0. \label{eq18}
\end{equation}
Equations (\ref{eq15}), (\ref{eq16}) and (\ref{eq18}) imply that $\lambda
=-\frac 12$ or $-\frac 14$ in exactly the same way as equations (\ref{eq5}),
(\ref{eq6}) and (\ref{eq8}) do. They similarly imply that if $\lambda
=-\frac 12$ then $[e_{(2n+1)q},e_2]=\frac 12e_{(2n+1)q+2}$, and $%
[e_{(2n+1)q+1},e_2]=-\frac 12e_{(2n+1)q+3}$. If $\lambda =-\frac 14$ and $%
p=3 $, then (\ref{eq15}) gives $e_{(2n+1)q+3}=0$, so $\lambda =-\frac 12$ is
the only possibility when $p=3$. If $\lambda =-\frac 14$ and $p\neq 3$, then
we have $[e_{(2n+1)q},e_2]=\frac 13e_{(2n+1)q+2}$ and $[e_{(2n+1)q+1},e_2]=-%
\frac 16e_{(2n+1)q+3}$.
This establishes the generic step for $\lambda =-\frac 12$.
\subsection{Generic step for $\lambda =-\frac 14$.}
We assume that $q$ is a power of $p$ ($p>3$) and we assume that
\begin{itemize}
\item $[e_i,e_2]=0$ for $i=3,4,\ldots ,q-1$,
\item $[e_q,e_2]=e_{q+2}$, $[e_{q+1},e_2]=-\frac 12e_{q+3}$,
\item $[e_{kq},e_2]=\frac 12e_{kq+2}$, $[e_{kq+1},e_2]=-\frac 12e_{kq+3}$
for $k=2,3,\ldots ,2n-1$ ($n\geq 2$),
\item $[e_{2nq},e_2]=\frac 12e_{2nq+2}$, $[e_{2nq+1},e_2]=-\frac
14e_{2nq+3} $,
\item There exists $s$ with $1\leq s<p-2$ such that $[e_k,e_2]=0$ for $%
q+1<k<(2n+s)q$ unless $k=0 \pmod{q}$ or $k=1 \pmod{q}$,
\item $[e_{(2n+k)q},e_2]=\frac 1{k+2}e_{(2n+k)q+2}$ and $%
[e_{(2n+k)q+1},e_2]=-\frac 1{2(k+2)}e_{(2n+k)q+3}$ for $k=1,2,\ldots ,s$.
\end{itemize}
Note that this situation arises from the case $\lambda =-\frac 14$ of the
last section, with $s=1$.
We show that $[e_{(2n+s)q+k},e_2]=0$ for $1<k<q$. In addition we show that
if $s<p-3$ then $[e_{(2n+s+1)q},e_2]=\frac 1{s+3}e_{(2n+s+1)q+2}$, $%
[e_{(2n+s+1)q+1},e_2]=-\frac 1{2(s+3)}e_{(2n+s+1)q+3}$, and we show that if $%
s=p-3$ then $e_{(2n+s+1)q+2}=0$. This contradiction shows that the case $%
\lambda =-\frac 14$ cannot arise in an infinite dimensional Lie algebra of
type 2.
For the moment we suppose that $s<p-3$.
First we show that $[e_{(2n+s)q+k},e_2]=0$ for $1<k<q$. The case $k=2$
follows from
\[
0=[e_{(2n+s)q-1},[e_1,e_2,e_2]]=\frac 1{s+2}[e_{(2n+s)q+2},e_2].
\]
For $k=3$ we have
\[
\lbrack e_{(2n+s-1)q+3},[e_2,e_1^{q-2},e_2]]=[e_{(2n+s-1)q+3},[e_2,e_1^q]]
\]
which gives
\[
((q-2)\frac 1{s+2}-(q-1)\frac
1{2(s+2)})[e_{(2n+s)q+3},e_2]=-[e_{(2n+s)q+3},e_2],
\]
and so implies that $[e_{(2n+s)q+3},e_2]=0$ unless $2s+1=0 \pmod{p}$. We also
have
\begin{eqnarray*}
0 &=&[e_{(2n+s)q},[e_1,e_2,e_2]] \\
&=&(-\frac 1{2(s+2)}-\frac 2{s+2})[e_{(2n+s)q+3},e_2],
\end{eqnarray*}
which implies that $[e_{(2n+s)q+3},e_2]=0$ unless $p=5$. Now if $p=5$ then
our assumption that $1\leq s<p-3$ implies that $s=1$ so that $2s+1\neq 0$
mod 5. So $[e_{(2n+s)q+3},e_2]=0$ in every case.
Now suppose that $3<m<q$ and that $[e_{(2n+s)q+k},e_2]=0$ for all $k$ such
that $1<k<m$. If $m$ is even then
\[
0=[e_{(2n+s)q+1},[e_2,e_1^{m-3},e_2]]=-\frac 1{s+2}[e_{(2n+s)q+m},e_2].
\]
So we may assume that $m$ is odd. In this case we have
\begin{eqnarray*}
0 &=&[e_{(2n+s)q},[e_2,e_1^{m-2},e_2]] \\
&=&(\frac 2{s+2}-(m-2)\frac{-1}{2(s+2)})[e_{(2n+s)q+m},e_2],
\end{eqnarray*}
so $[e_{(2n+s)q+m},e_2]=0$ unless $m=-2 \pmod{p}$. We also have
\[
\lbrack e_{(2n+s-1)q+m},[e_2,e_1^{q-2},e_2]]=[e_{(2n+s-1)q+m},[e_2,e_1^q]].
\]
This implies that
\[
((q-m+1)\frac 1{s+2}-(q-m+2)\frac 1{2(s+2)}+1)[e_{(2n+1)q+m},e_2]=0,
\]
which implies that $[e_{(2n+s)q+m},e_2]=0$ unless $m=2(s+2) \pmod{p}$. Since $%
s<p-3$, $[e_{(2n+s)q+m},e_2]=0$ in every case.
Next consider the equation
\[
\lbrack e_{(2n+s)q},[e_2,e_1^{q-2},e_2]]=[e_{(2n+s)q},[e_2,e_1^q]].
\]
This implies that
\[
(\frac 2{s+2}-\frac 2{2(s+2)}+1)[e_{(2n+s+1)q},e_2]=\frac
1{s+2}e_{(2n+s+1)q+2},
\]
and hence that
\[
\lbrack e_{(2n+s+1)q},e_2]=\frac 1{s+3}e_{(2n+s+1)q+2}.
\]
And the equation
\[
\lbrack e_{(2n+s)q+1},[e_2,e_1^{q-2},e_2]]=[e_{(2n+s)q+1},[e_2,e_1^q]]
\]
implies that
\begin{eqnarray*}
&&(-\frac 2{2(s+2)}+1)[e_{(2n+s+1)q+1},e_2]+\frac
1{2(s+2)}(q-2)[e_{(2n+s+1)q},e_2,e_1] \\
&=&-\frac 1{2(s+2)}e_{(2n+s+1)q+3}.
\end{eqnarray*}
So
\[
\lbrack e_{(2n+s+1)q+1},e_2]=-\frac 1{2(s+3)}e_{(2n+s+1)q+3}.
\]
Finally we consider the case when $s=p-3$. Let $2t=2n+s=2n+p-3$. Then $%
[e_{2tq},e_2]=-e_{2tq+2}$ and $[e_{2tq+1},e_2]=\frac 12e_{2tq+3}$.
The lemma implies that $[e_{2tq+k},e_2]=0$ for $1<k<q$.
Consider the equation
\[
\lbrack e_{2tq},[e_2,e_1^{q-2},e_2]]=[e_{2tq},[e_2,e_1^q]].
\]
This implies that
\[
(-2-\frac{q-2}2+1)[e_{(2t+1)q},e_2]=-e_{(2t+1)q+2},
\]
and hence that
\[
e_{(2t+1)q+2}=0.
\]
\section{The case $[e_{3} e_{2}] \ne 0$ and $p =3$}
\label{sec:extra_3}
Let $L$ be an $\mathbb{N}$-graded Lie algebra of maximal class over a
field $\text{$\mathbf{F}$}$
of characteristic 3, where $L$ has basis $\{e_i\,|\,i=1,2,\ldots \}$, with $%
[e_i,e_1]=e_{i+1}$ for $i>1$. We consider the case when $[e_3,e_2]\neq 0$.
By rescaling $e_2$ we may assume that $[e_3,e_2]=e_5$, which implies that $%
[e_4,e_2]=e_6$ but leaves $[e_5,e_2]$ undetermined. We show that for every $%
\lambda \in \text{$\mathbf{F}$}$ there is a unique infinite dimensional soluble algebra $%
L(\lambda )$ of type 2 satisfying these relations, together with the
relation $[e_5,e_2]=\lambda e_7$. The algebra $L(\lambda )$ has basis $%
\{e_i\,|\,i=1,2,\ldots \}$, and satisfies the following relations:
\begin{equation}\label{eq:identities_for_Llambda}
\begin{cases}{}
[e_i,e_1]=e_{i+1},& \text{for $i>1$,}\\{}
[e_3,e_2]=e_5,\\{}
\begin{aligned}[b]
&[e_{3k+1},e_2]=e_{3k+3}, [e_{3k+2},e_2]=\lambda e_{3k+4},\\
&\qquad[e_{3k+3},e_2]=(-1-\lambda )e_{3k+5},
\end{aligned}& \text{for $k\geq 1$,}\\{}
[e_k,e_3]=(1-\lambda )e_{k+3},& \text{for $k\geq 4$,}\\{}
[e_k,e_m]=0,& \text{for $k,m\geq 4$.}
\end{cases}
\end{equation}
We give a construction of $L(\lambda )$ in
Section~\ref{sec:construction_3}, but in fact it is easy
to show directly that these relations (together with the relations $%
[e_i,e_i]=0$, $[e_i,e_j]+[e_j,e_i]=0$) imply the Jacobi relations
\[
\lbrack e_i,e_j,e_k]+[e_j,e_k,e_i]+[e_k,e_i,e_j]=0.
\]
Note that $L(1)\cong m_2$ and that $L(-1)$ is the analogue for $q=3$ of the
algebra constructed in Section 6.
So we suppose that $L$ has basis $\{e_i\,|\,i=1,2,\ldots \}$, with $%
[e_i,e_1]=e_{i+1}$ for $i>1$, $[e_3,e_2]=e_5$, $[e_4,e_2]=e_6$, $%
[e_5,e_2]=\lambda e_7$. We show that if $n\geq 4$ then $[e_n,e_2]=\mu
_ne_{n+2}$, where
\[
\mu _n=\left\{
\begin{array}{l}
1\text{ if }n=1 \pmod{3} \\
\lambda \text{ if }n=2 \pmod{3} \\
-1-\lambda \text{ if }n=0 \pmod{3}.
\end{array}
\right.
\]
The fact that $[e_k,e_3]=(1-\lambda )e_{k+3}$ for $k\geq 4$, and that $%
[e_k,e_m]=0$ for $k,m\geq 4$, follows easily from this.
We will make use of the following argument. Suppose that we have shown that $%
[e_n,e_2]=\mu _ne_{n+2}$ for all $n$ with $4\leq n<2m$. Then the relation $%
[e_{m+1},e_{m+1}]=0$ implies that
\begin{eqnarray*}
0 &=&[e_{m+1},[e_2,e_1^{m-1}]] \\
&=&\sum_{k=0}^{m-1}(-1)^k\binom{m-1}k[e_{m+1},e_1^k,e_2,e_1^{m-1-k}],
\end{eqnarray*}
and so $[e_{2m},e_2]$ is determined by the values of $[e_n,e_2]$ for $%
m+1\leq n<2m$. So $[e_{2m},e_2]=\mu e_{2m+2}$, for some $\mu $ which is
uniquely determined by $\{\mu _n\,|\,m+1\leq n<2m\}$. But $L(\lambda )$ is a
Lie algebra which satisfies $[e_n,e_2]=\mu _ne_{n+2}$ for all $n\geq 4$. So $%
\mu =\mu _{2m}$. In particular, this argument implies that $%
[e_6,e_2]=(-1-\lambda )e_8$.
Now suppose that $[e_n,e_2]=\mu _ne_{n+2}$ for $4\leq n<m$ for some $m\geq 7$%
. We show that this implies that $[e_m,e_2]=\mu _me_{m+2}$. By the argument
above, we only need to consider the case when $m$ is odd. We use the fact
that $[e_2,e_1^3]+[e_1,e_2,e_2]=0$. So
\begin{align*}
0
&=
[e_{m-3},[e_2,e_1^3]]+[e_{m-3},[e_1,e_2,e_2]]
\\&=
[e_{m-3},e_2,e_1^3]-[e_{m-3},e_1^3,e_2]+
\\&\phantom{=\ }+
[e_{m-3},e_1,e_2,e_2]+[e_{m-3},e_2,e_1,e_2]+[e_{m-3},e_2,e_2,e_1]
\\&=
\mu _{m-3}(1+\mu _{m-1})e_{m+2}-(1-\mu _{m-2}-\mu _{m-3})[e_m,e_2]
\\&=
\mu _{m-3}(1+\mu _{m-1})e_{m+2}-(1+\mu _{m-1})[e_m,e_2].
\end{align*}
Provided $1+\mu _{m-1}\neq 0$, this gives $[e_m,e_2]=\mu _{m-3}e_{m+2}=\mu
_me_{m+2}$, as required. Note that $1+\mu _{m-1}=0$ can only occur when $%
\lambda =0$ and $m=1 \pmod{3}$, or when $\lambda =-1$ and $m=0 \pmod{3}$. So the
uniqueness of $L(\lambda )$ is established except in the cases when $\lambda
=0$ and $\lambda =-1$. We deal with these two cases separately.
\subsection{The case $\lambda =0$.}
Let $L$ be an $\mathbb{N}$-graded Lie algebra spanned by $\{e_i\,|\,i=1,2,%
\ldots \}$, with $[e_i,e_1]=e_{i+1}$ for $i>1$. Let $[e_3,e_2]=e_5$, $%
[e_4,e_2]=e_6$, $[e_5,e_2]=0$. As above, we suppose that for some $n\geq 1$
we have $[e_{3k+1},e_2]=e_{3k+3}$, $[e_{3k+2},e_2]=0$, $%
[e_{3k+3},e_2]=-e_{3k+5}$ for $1\leq k\leq n$, and we suppose that $%
[e_{3n+4},e_2]=\mu e_{3n+6}$ for some $\mu \neq 1$. As above, we may assume
that $n$ is odd. We prove that $L(0)$ is the unique infinite dimensional
algebra over $\text{$\mathbf{F}$}$ of type 2 satisfying $[e_3,e_2]=e_5$, $[e_5,e_2]=0$ by
showing that this implies that $L$ is nilpotent.
First note that
\begin{align*}
0
&=
[e_{3n+2},[e_2,e_1^3]]+[e_{3n+2},[e_1,e_2,e_2]]
\\&=
[e_{3n+2},e_2,e_1^3]-[e_{3n+2},e_1^3,e_2]+
\\&\phantom{=\ }+
[e_{3n+2},e_1,e_2,e_2]+[e_{3n+2},e_2,e_1,e_2]+[e_{3n+2},e_2,e_2,e_1]
\\&=
[e_{3n+5},e_2]
\end{align*}
We also have $[e_2,e_1^3,e_2]=0$ which implies that $%
[e_{3n+1},[e_2,e_1^3,e_2]]=0$. This gives $(1+\mu )[e_{3n+6},e_2]=e_{3n+8}$.
If $\mu =-1$ then we have $e_{3n+8}=0$ and $L$ is nilpotent (as claimed). So
we assume that $\mu \neq -1$ and that $[e_{3n+6},e_2]=\frac 1{1+\mu
}e_{3n+8} $. Next,
\[
\lbrack e_{3n+4},[e_2,e_1,e_2]]=[e_{3n+4},[e_2,e_1^3]]
\]
implies that $[e_{3n+7},e_2]=\frac \mu {1+\mu }e_{3n+9}$. And since $%
[e_4,e_2]=e_6=[e_3,e_1^3]$ we have
\[
\lbrack e_{3n+3},[e_4,e_2]]=[e_{3n+3},[e_3,e_1^3]],
\]
which gives
\[
\mu +\frac 1{1+\mu }=-1-\mu -\frac 1{1+\mu }+\frac \mu {1+\mu }.
\]
But this implies that $\mu =0$, and so $[e_{3n+4},e_2]=[e_{3n+5},e_2]=0$, $%
[e_{3n+6},e_2]=e_{3n+8}$, and $[e_{3n+7},e_2]=0$.
Now let $m=\frac{n+1}2$. Then
\begin{eqnarray*}
0 &=&[[e_3,e_1^{3m}],[e_3,e_1^{3m}]] \\
&=&\sum_{r=0}^m(-1)^r\binom mr[e_{3m+3},e_1^{3r},e_3,e_1^{3(m-r)}] \\
&=&\sum_{r=0}^m(-1)^r\binom mr[e_{3m+3+3r},e_3,e_1^{3(m-r)}].
\end{eqnarray*}
Our inductive hypothesis implies that $[e_{3k},e_3]=e_{3k+3}$ for $m+1\leq
k\leq n$. And $[e_{3n+3},e_3]=-e_{3n+6}$, $[e_{3n+6},e_3]=e_{3n+9}$. So this
equation gives $me_{3n+9}=0$. It follows that $m=0 \pmod{3}$, and hence that $%
n=-1 \pmod{3}$.
Since $n=-1 \pmod{3}$, $n>1$, and so $[e_7,e_2]=e_9$. Hence
\[
\lbrack e_{3n+1},[e_7,e_2]]=[e_{3n+1},e_9].
\]
Since $e_9=[e_3,e_1^3,e_1^3]$ we see that
\begin{eqnarray*}
[e_{3n+1},e_9]
&=&[e_{3n+1},e_3,e_1^3,e_1^3]+[e_{3n+1},e_1^3,e_3,e_1^3]+[e_{3n+1},e_1^3,e_1^3,e_3]
\\
&=&e_{3n+10}-[e_{3n+8},e_2].
\end{eqnarray*}
And since $[e_7,e_2]=[e_4,e_1^3,e_2]$ we have
\begin{align*}
[e_{3n+1},[e_7,e_2]]
&=
[e_{3n+1},e_4,e_1^3,e_2]-[e_{3n+1},e_1^3,e_4,e_2]
\\&\phantom{=\ }-[e_{3n+1},e_2,e_4,e_1^3]+[e_{3n+1},e_2,e_1^3,e_4]
\\&=
-e_{3n+10}.
\end{align*}
So $[e_{3n+8},e_2]=-e_{3n+10}$.
To summarize, we may assume that $n$ is odd and $n=-1 \pmod{3}$, and that
\begin{itemize}
\item $[e_{3k},e_2]=-e_{3k+2}$, $[e_{3k+1},e_2]=e_{3k+3}$, $[e_{3k+2},e_2]=0
$ for $2\leq k\leq n$,
\item $[e_{3n+3},e_2]=-e_{3n+5}$, $[e_{3n+4},e_2]=[e_{3n+5},e_2]=0$,
\item $[e_{3n+6},e_2]=e_{3n+8}$, $[e_{3n+7},e_2]=0$, $%
[e_{3n+8},e_2]=-e_{3n+10}$.
\end{itemize}
We let $n=2cq-1$, where $q$ is a power of 3 and where $c$ is coprime to 3.
Then we make a further inductive assumption that for some $t$ with $n+2\leq
t\leq n+q $ we have
\begin{itemize}
\item $[e_{3k},e_2]=e_{3k+2}$, $[e_{3k+1},e_2]=0$, $[e_{3k+2},e_2]=-e_{3k+4}
$ for $n+2\leq k\leq t$.
\end{itemize}
We show that this implies that $[e_{3t+3},e_2]=e_{3t+5}$, $[e_{3t+4},e_2]=0$%
, $[e_{3t+5},e_2]=-e_{3t+7}$. We have to divide the proof that $%
[e_{3t+3},e_2]=e_{3t+5}$ into two cases depending on whether $t$ is odd or
even.
If $t$ is odd let $m=\frac{t-1}2$. Then, since $e_{3m+4}=[e_4,e_1^{3m}]$,
we see that the equation $[e_{3m+4},e_{3m+4}]=0$ gives
\[
\sum_{r=0}^m(-1)^r\binom mr[e_{3m+4},e_1^{3r},e_4,e_1^{3(m-r)}]=0.
\]
Now $[e_{3k+1},e_4]=0$ for $m<k\leq n$ and for $n+1<k<t$, $%
[e_{3n+4},e_4]=e_{3n+8}$, $[e_{3t+1},e_4]=-e_{3t+5}+[e_{3t+3},e_2]$. So we
obtain
\[
(-1)^{n-m}\binom m{n-m}e_{3t+5}-(-1)^me_{3t+5}+(-1)^m[e_{3t+3},e_2]=0,
\]
which implies that
\[
\lbrack e_{3t+3},e_2]=(1+\binom m{n-m})e_{3t+5}.
\]
Now we can write $t=n+2s$ for some $s$ with $2\leq 2s<q$. So
\[
\binom m{n-m}=\binom m{2m-n}=\binom{cq+s-1}{2s-1}=0\pmod{3},
\]
and $[e_{3t+3},e_2]=e_{3t+5}$.
Now consider the case when $t$ is even. We have $%
e_{3(t-n)}=[e_{3(t-n)-2},e_2]$, and so
\[
\lbrack e_{3n+5},[e_{3(t-n)-2},e_2]]=[e_{3n+5},e_{3(t-n)}].
\]
Now
\begin{eqnarray*}
[e_{3n+5},e_{3(t-n)}]
&=&[e_{3n+5},[e_3,e_1^{3(t-n-1)}]] \\
&=&\sum_{r=0}^{t-n-1}(-1)^r\binom{t-n-1}%
r[e_{3n+5},e_1^{3r},e_3,e_1^{3(t-n-1-r)}].
\end{eqnarray*}
Since $[e_{3n+5},e_3]=-e_{3n+8}$, $[e_{3k+5},e_3]=e_{3k+8}$ for $n<k<t-1$, $%
[e_{3t+2},e_3]=-e_{3t+5}-[e_{3t+3},e_2]$, and since $t-n-1$ is even, this
implies that
\[
\lbrack e_{3n+5},e_{3(t-n)}]=-e_{3t+5}-[e_{3t+3},e_2].
\]
Also
\begin{eqnarray*}
[e_{3n+5},[e_{3(t-n)-2},e_2]]
&=&[e_{3n+5},e_{3(t-n)-2},e_2] \\
&=&[e_{3n+5},[e_4,e_1^{3(t-n-2)}],e_2].
\end{eqnarray*}
Since $[e_{3n+5},e_4]=e_{3n+9}$ and $[e_{3k+5},e_4]=0$ for $n<k\leq t-2$,
this implies that
\[
\lbrack e_{3n+5},[e_{3(t-n)-2},e_2]]=[e_{3t+3},e_2].
\]
So the equation
\[
\lbrack e_{3n+5},[e_{3(t-n)-2},e_2]]=[e_{3n+5},e_{3(t-n)}]
\]
implies that $[e_{3t+3},e_2]=e_{3t+5}$.
So $[e_{3t+3},e_2]=e_{3t+5}$ whether $t$ is odd or even.
Next note that the equations
\begin{eqnarray*}
\lbrack e_{3t+1},[e_2,e_1,e_2]] &=&[e_{3t+1},[e_2,e_1^3]], \\
\lbrack e_{3t+2},[e_2,e_1,e_2]] &=&[e_{3t+2},[e_2,e_1^3]]
\end{eqnarray*}
give $[e_{3t+4},e_2]=0$, $[e_{3t+5},e_2]=-e_{3t+7}$.
So, by induction, we may assume that
\begin{itemize}
\item $[e_{3k},e_2]=e_{3k+2}$, $[e_{3k+1},e_2]=0$, $[e_{3k+2},e_2]=-e_{3k+4}
$ for $n+2\leq k\leq n+q+1$.
\end{itemize}
Finally, let $m=\frac{n+q}2$. We have
\begin{eqnarray*}
0 &=&[[e_3,e_1^{3m}],[e_3,e_1^{3m}]] \\
&=&\sum_{r=0}^m(-1)^r\binom mr[e_{3m+3},e_1^{3r},e_3,e_1^{3(m-r)}].
\end{eqnarray*}
We have $[e_{3k},e_3]=e_{3k+3}$ for $m+1\leq k\leq n$ and for $n+1<k\leq
n+q+1$, $[e_{3n+3},e_3]=-e_{3n+6}$. Since $\sum_{r=0}^m(-1)^r\binom mr=0$,
we obtain $\binom mqe_{3n+3q+6}=0$. Since $m=cq+\frac{q-1}2$, $\binom
mq=c\neq 0 \pmod{3}$, and so $e_{3n+3q+6}=0$.
Thus the assumption that $[e_{3n+4},e_2]\neq e_{3n+6}$ implies that $L$ is
nilpotent in every case. This completes our analysis of the case when $%
\lambda =0$.
\subsection{The case $\lambda =-1$.}
Let $L$ be an $\mathbb{N}$-graded Lie algebra spanned by $\{e_i\,|\,i=1,2,%
\ldots \}$, with $[e_i,e_1]=e_{i+1}$ for $i>1$. Let $[e_3,e_2]=e_5$, $%
[e_4,e_2]=e_6$, $[e_5,e_2]=-e_7$. Repeating the argument above, we have $%
[e_6,e_2]=\mu _6e_8=0$ (since 6 is even), $[e_7,e_2]=\mu _7e_9=e_9$ (since $%
7\neq 0 \pmod{3}$), and $[e_8,e_2]=\mu _8e_{10}=-e_{10}$ (since 8 is even). And
we may suppose that for some even $n\geq 2$ we have $[e_{3k},e_2]=0$, $%
[e_{3k+1},e_2]=e_{3k+3}$, $[e_{3k+2},e_2]=-e_{3k+4}$ for $2\leq k\leq n$,
and that $[e_{3n+3},e_2]=\mu e_{3n+6}$ for some $\mu \neq 0$. We prove that $%
L(-1)$ is the unique infinite dimensional algebra over $\text{$\mathbf{F}$}$ of type 2
satisfying $[e_3,e_2]=e_5$, $[e_5,e_2]=-e_7$ by showing that this implies
that $L$ is nilpotent.
The relation
\[
\lbrack e_{3n+1},[e_2,e_1,e_2]]=[e_{3n+1},[e_2,e_1^3]]
\]
gives $[e_{3n+4},e_2]=(1+\mu )e_{3n+6}$. And the relation
\[
\lbrack e_{3n+2},[e_2,e_1,e_2]]=[e_{3n+2},[e_2,e_1^3]]
\]
gives
\[
-(1+\mu )[e_{3n+5},e_2]=(1-\mu )e_{3n+7}.
\]
If $\mu =-1$ then this gives $e_{3n+7}=0$, and $L$ is nilpotent. So we may
suppose that $\mu \neq -1$, and that $[e_{3n+5},e_2]=\frac{\mu -1}{\mu +1}%
e_{3n+7}$.
Since $[e_4,e_2]=e_6=[e_3,e_1^3]$ we obtain
\[
\lbrack e_{3n+2},[e_4,e_2]]=[e_{3n+2},[e_3,e_1^3]].
\]
This gives
\[
(1-\mu )[e_{3n+6},e_2]+(\frac{\mu -1}{\mu +1}+\mu
+1)e_{3n+8}=[e_{3n+6},e_2]-(\frac{\mu -1}{\mu +1}+\mu +1)e_{3n+8}.
\]
Since $\mu \neq 0$ we have $[e_{3n+6},e_2]=-\frac \mu {\mu +1}e_{3n+8}$.
Let $n=2m$. Then, since $e_{3m+4}=[e_4,e_1^{3m}]$, the equation $%
[e_{3m+4},e_{3m+4}]=0$ gives
\[
\sum_{r=1}^m(-1)^m\binom mr[e_{3m+4},e_1^{3r},e_4,e_1^{3(m-r)}]=0.
\]
Now $[e_{3k+1},e_4]=0$ for $1\leq k<n$, and $[e_{3n+1},e_4]=\mu e_{3n+5}$, $%
[e_{3n+4},e_4]=\frac{\mu (\mu -1)}{\mu +1}$. So we obtain
\[
(m\mu -\frac{\mu (\mu -1)}{\mu +1})e_{3n+8}=0.
\]
So either $e_{3n+8}=0$ (and $L$ is nilpotent), or $m\mu (\mu +1)=\mu (\mu
-1) $. But since $\mu \neq 0$, the only solution of $m\mu (\mu +1)=\mu (\mu
-1)$ is $\mu =1$ and $m=0 \pmod{3}$.
So we may assume that $n=2cq$ where $q$ is a power of 3 and where $c$ is
coprime to 3, and we may assume that
\begin{itemize}
\item $[e_3,e_2]=e_5$, $[e_4,e_2]=e_6$, $[e_5,e_2]=-e_7$,
\item $[e_{3k},e_2]=0$, $[e_{3k+1},e_2]=e_{3k+3}$, $[e_{3k+2},e_2]=-e_{3k+4}
$ for $2\leq k\leq n$,
\item $[e_{3n+3},e_2]=e_{3n+5}$, $[e_{3n+4},e_2]=-e_{3n+6}$, $%
[e_{3n+5},e_2]=0$, $[e_{3n+6},e_2]=e_{3n+8}$.
\end{itemize}
We make the further inductive hypothesis that for some $t$ with $n+1\leq
t<n+q$ we have
\begin{itemize}
\item $[e_{3k+1},e_2]=-e_{3k+3}$, $[e_{3k+2},e_2]=0$, $%
[e_{3k+3},e_2]=e_{3k+5}$ for $n+1\leq k\leq t$.
\end{itemize}
We show that this implies that $[e_{3t+4},e_2]=-e_{3t+6}$, $[e_{3t+5},e_2]=0$%
, $[e_{3t+6},e_2]=e_{3t+8}$.
The equation
\[
\lbrack e_{3t+1},[e_2,e_1,e_2]]=[e_{3t+1},[e_2,e_1^3]]
\]
gives $[e_{3t+4},e_2]=-e_{3t+6}$.
We have to divide the proof that $[e_{3t+5},e_2]=0$ into two cases depending
on whether $t$ is odd or even. First suppose that $t$ is odd and let $m=%
\frac{t+1}2$. Then $e_{3m+2}=[e_2,e_1^{3m}]$ and so the equation $%
[e_{3m+2},e_{3m+2}]=0$ gives
\[
\sum_{r=0}^m(-1)^r\binom mr[e_{3m+2},e_1^{3r},e_2,e_1^{3(m-r)}]=0.
\]
Now $[e_{3k+2},e_2]=-e_{3k+4}$ for $m\leq k\leq n$, $[e_{3k+2},e_2]=0$ for $%
n<k\leq t$, and so we obtain
\[
-\sum_{r=0}^{n-m}(-1)^r\binom mre_{3t+7}+(-1)^m[e_{3t+5},e_2]=0.
\]
We can write $t=n+2s-1=2cq+2s-1$ where $1\leq s<\frac{q+1}2$ so that $m=cq+s$
and $n-m=cq-s$. So
\[
\sum_{r=0}^{n-m}(-1)^r\binom mr=\sum_{r=0}^{cq-s}(-1)^r\binom{cq+s}r=\pm
\sum_{r=0}^{2s-1}(-1)^r\binom{cq+s}r.
\]
But $2s-1<q$ and so $\binom{cq+s}r=0 \pmod{3}$ for $s<r\leq 2s-1$. So, working
modulo 3,
\[
\sum_{r=0}^{n-m}(-1)^r\binom mr=\pm \sum_{r=0}^s(-1)^r\binom{cq+s}r=\pm
\sum_{r=0}^s(-1)^r\binom sr=0,
\]
and hence $[e_{3t+5},e_2]=0$.
Next suppose that $t$ is even. The equation
\[
\lbrack e_{3n+2},[e_{3(t-n+1)},e_2]]=0
\]
gives
\[
\lbrack e_{3n+2},e_{3(t-n+1)},e_2]+[e_{3n+4},e_{3(t-n+1)}]=0.
\]
Since $e_{3(t-n+1)}=[e_3,e_1^{3(t-n)}]$, this implies that
\[
\sum_{r=0}^{t-n}(-1)^r\binom{t-n}r\left(
[e_{3n+2},e_1^{3r},e_3,e_1^{3(t-n-r)},e_2]+[e_{3n+4},e_1^{3r},e_3,e_1^{3(t-n-r)}]\right) =0.
\]
Now $[e_{3n+2},e_3]=e_{3n+5}$, $[e_{3k+2},e_3]=-e_{3k+5}$ for $n<k\leq t$, $%
[e_{3k+4},e_3]=-e_{3k+7}$ for $n\leq k<t$, and $%
[e_{3t+4},e_3]=-e_{3t+7}-[e_{3t+5},e_2]$. Since $t-n$ is even this equation
implies that $[e_{3t+5},e_2]=0$.
So $[e_{3t+5},e_2]=0$ whether $t$ is odd or even.
Finally
\[
\lbrack e_{3t+3},[e_2,e_1,e_2]]=[e_{3t+3},[e_2,e_1^3]]
\]
gives $[e_{3t+6},e_2]=e_{3t+8}$. So we may assume by induction that
\begin{itemize}
\item $[e_{3k+1},e_2]=-e_{3k+3}$, $[e_{3k+2},e_2]=0$, $%
[e_{3k+3},e_2]=e_{3k+5}$ for $n+1\leq k\leq n+q$.
\end{itemize}
To complete our analysis of case 3 we let $t=\frac{n+q-1}2$, and we consider
the equation
\begin{eqnarray*}
0 &=&[[e_4,e_1^{3t}],[e_4,e_1^{3t}]] \\
&=&\sum_{r=0}^m(-1)^r\binom tr[e_{3t+4},e_1^{3r},e_4,e_1^{3(t-r)}].
\end{eqnarray*}
Since $[e_{3k+4},e_4]=0$ for $t\leq k<n-1$ and for $n\leq k\leq 2t$, and $%
[e_{3n+1},e_4]=e_{3n+5}$ this implies $\binom tqe_{3n+3q+5}=0$. Since $t=cq+%
\frac{q-1}2$, $\binom tq=c\neq 0 \pmod{3}$, and hence $e_{3n+3q+5}=0$.
Thus the assumption that $[e_{3n+3},e_2]\neq 0$ implies that $L$ is
nilpotent in every case. This completes our analysis of the case when $%
\lambda =-1$.
\section{Constructing the algebra\\
with first constituent of length $q$}
\label{sec:Extra_Construction}
In this section we construct the algebra $L$ with first constituent of
length $q$ which is described in Section 6. If $q=3$, this construction
gives the algebra $L(-1)$ of Section 7.
Let $p$ be an odd prime, and let $q$ be a power of $p$. Let $V$ be a vector
space of dimension $q$ over the field $\text{$\mathbf{F}$}(t)$ of rational functions over the
field $\text{$\mathbf{F}$}$ with $p$ elements. We grade $V$ over the cyclic group of order $q$%
,
\[
V=\Span{v_{0}}\oplus \Span{v_{1}}\oplus \dots \oplus \Span{v_{q-1}}.
\]
Consider the following endomorphisms $D$ and $E$, of $V$.
\begin{align*}
E& =\begin{cases} v_{i} \mapsto v_{i+1} & \text{if $i \ne q - 1$}\\ v_{q-1}
\mapsto t v_{0}. \end{cases} \\
D& =\begin{cases} v_{0} \mapsto v_{2}\\ v_{q-1} \mapsto - t v_{1}\\ v_{i}
\mapsto 0 & \text{otherwise}. \end{cases}
\end{align*}
Thus $E$ has weight $1$, and $D$ has weight $2$.
We construct the Lie algebra $A$ spanned by $E$ and $D$ in the endomorphism
algebra of $V$.
Consider $[DE^{q-2}]$, which has weight $q\equiv 0\pmod{q}$. For $0\leq j<q$
we have
\[
v_j[DE^{q-2}]=\sum_{i=0}^{q-2}(-1)^i\binom{q-2}i v_jE^iDE^{q-2-i}.
\]
If $j>0$ then $v_jE^iD=0$ unless $i=q-j-1$ or $q-j$. For $i=q-j$ we have $%
v_jE^i=tv_0$, and thus
\[
(-1)^i\binom{q-2}i v_jE^iDE^{q-2-i}=(-1)^{q-j}\binom{q-2}{q-j}tv_j,
\]
while for $i=q-1-j$ we have $v_jE^i=v_{q-1}$, and thus
\begin{align*}
(-1)^i\binom{q-2}i v_jE^iDE^{q-2-i}& =-(-1)^{q-1-j}\binom{q-2}{q-1-j}tv_j \\
& =(-1)^{q-j}\binom{q-2}{q-1-j}tv_j.
\end{align*}
It follows that
\[
v_j[DE^{q-2}]=v_j(-1)^{q-j}\binom{q-1}{q-j}tv_j=tv_j.
\]
Similarly (for $0\leq i\leq q-2$) we have $v_0E^iD=0$ unless $i=0$, and so $%
v_0[DE^{q-2}]=tv_0$. So $[DE^{q-2}]=t\cdot 1$ is scalar multiplication by $t$%
. It follows that all the $[DE^i]$, for $0\le i\le q-2$, are non-zero, and
thus linearly independent over $\text{$\mathbf{F}$}$, as they have distinct weights $2,\dots
,q$. We claim that $[DE^iD]=0$ for $0\le i<q-2$. To see this, consider the
associative expansion of $[DE^iD]$, which is a linear combination of
monomials of the form $E^\alpha DE^\beta D$, $DE^\beta DE^\alpha $, with $%
\alpha +\beta =i$. Note that if $E^\alpha DE^\beta D$ or $DE^\beta DE^\alpha
$ is a monomial which occurs in any of these expansions then $\beta <q-3$.
This is trivially true, except in the expansion of $[DE^{q-3}D]$. However in
the expansion of $[DE^{q-3}D]$, $DE^{q-3}D$ appears twice, \emph{but with
opposite signs}. So it is sufficient to show that if $\beta <q-3$ then $%
v_jDE^\beta D=0$ for all $j$. But $v_jD=0$ unless $j=0$ or $q-1$, and
\[
v_0DE^\beta D=v_{\beta +2}D=0,
\]
\[
v_{q-1}DE^\beta D=-tv_{\beta +1}D=0,
\]
since $0<\beta +1,\beta +2<q-2$.
Therefore
\[
A = \Span{E, [D E^{i}] : 0 \le i \le q - 2}
\]
is $q$-dimensional.
Let us now consider the semidirect product $V+\End(V)$, and in it the Lie
algebra $L$ over $\text{$\mathbf{F}$}$ generated by
\[
e_1=E,\qquad e_2=-\frac 1{2t}\cdot v_1-\frac 12D.
\]
Recursively define $e_{i+1}=[e_ie_1]$, for $i\ge 2$. Note that for $2\le
i\le q$ we have by induction
\[
e_i=-\frac 1{2t}v_{i-1}-\frac 12[DE^{i-2}].
\]
In particular for $i=q$ we have
\[
e_q=-\frac 1{2t}\cdot v_{q-1}-\frac 12[DE^{q-2}]=-\frac 1{2t}v_{q-1}-\frac
t2\cdot 1.
\]
Therefore
\[
e_{q+1}=-\frac 12v_0,\qquad e_{q+2}=-\frac 12v_1,
\]
and we are in $V$ from now on, and further commutation with $e_1$ and $e_2$
is straightforward. In particular, if $0\leq r<q$ and $k\geq 1$, then $%
e_{kq+r+1}=[e_{q+1}e_1^{(k-1)q+r}]=-\frac 12t^{k-1}v_r$, and
\[
\lbrack e_{kq+r+1},e_2]=\frac 14t^{k-1}v_rD=\left\{
\begin{array}{l}
0\text{ unless }r=0\text{ or }q-1 \\
-\frac 12e_{kq+3}\text{ if }r=0 \\
\frac 12e_{(k+1)q+2}\text{ if }r=q-1.
\end{array}
\right.
\]
For $2\le i\le q-1$ we have
\[
\lbrack e_ie_2]=\frac 14\left( \frac 1tv_{i-1}D-\frac
1tv_1[DE^{i-2}]+[DE^{i-2}D]\right) =0,
\]
because of the above, and the easy fact that $v_1[DE^{i-2}]=0$. And
\[
\lbrack e_qe_2]=\frac 14\left( \frac 1tv_{q-1}D-\frac
1tv_1[DE^{q-2}]+[DE^{q-2}D]\right) =-\frac 12v_1=e_{q+2}.
\]
So $L$ is of maximal class, graded as we want it to be. We have seen that
the first constituent has length $q$, and that $[e_qe_2]=e_{q+2}$, $%
[e_{q+1},e_2]=-\frac 12e_{q+3}$. For $n>q$ we have $[e_n,e_2]=0$
unless $n$ is congruent to $0$
or $1$ modulo $q$, and for $k\geq 2$ we have $[e_{kq},e_2]=\frac
12e_{kq+2}$,
$%
[e_{kq+1},e_2]=-\frac 12e_{kq+3}$.
\section{Constructing the extra algebras for $q = 3$}
\label{sec:construction_3}
In this section we construct the algebras $L(\lambda)$ of
Section~\ref{sec:extra_3}. These are defined over a field $\text{$\mathbf{F}$}$ of
characteristic $3$, for $\lambda \in \text{$\mathbf{F}$}$.
The construction is similar to the one of the previous section. We
rephrase it here in terms of matrices.
Let $t$ be an indeterminate over $K$. Let $v_{1}, v_{2}, v_{3}$ be the
standard basis of the space of row vectors $K(t)^{3}$.
Consider the $3 \times 3$ matrices over $K(t)$
\begin{equation*}
E =
\begin{bmatrix}
& 1 & \\
& & 1 \\
t & & \\
\end{bmatrix},
\qquad
D =
\begin{bmatrix}
& & 1 \\
\lambda t & & \\
& - (1 + \lambda ) t & \\
\end{bmatrix}.
\end{equation*}
where as usual zero entries are omitted. We have
\begin{equation*}
[D E] =
\begin{bmatrix}
(1 - \lambda) t & & \\
& (1 - \lambda) t & \\
& & (1 - \lambda) t \\
\end{bmatrix},
\end{equation*}
a scalar matrix, so that the Lie algebra spanned by $D$ and $E$ has
dimension $3$. Now consider the block $4 \times 4$ matrices
\begin{equation*}
e_{1} =
\begin{bmatrix}
E & 0\\
0 & 0\\
\end{bmatrix},
\qquad
e_{2} =
\begin{bmatrix}
D & 0\\
v & 0\\
\end{bmatrix}.
\end{equation*}
Here
\begin{equation*}
v = \frac{1}{t} \, v_{2} = [0, \frac{1}{t}, 0] \in K(t)^{3}.
\end{equation*}
Consider the Lie algebra $S$ spanned by $e_{1}$ and $e_{2}$.
We compute
\begin{equation*}
e_{3} = [e_{2} e_{1}] =
\begin{bmatrix}
[D E] & 0\\
v E & 0\\
\end{bmatrix},
\qquad
e_{4} = [e_{3} e_{1}] =
\begin{bmatrix}
0 & 0\\
v E^{2} & 0\\
\end{bmatrix}.
\end{equation*}
Here $v E = 1/t \cdot v_{3}$, and $v E^{2} = [1, 0, 0] = v_{1}$. If we
define $e_{i+1} = [e_{i} e_{1}]$, for $i \ge 2$, we find thus that for
$i \ge 4$ we have
\begin{equation*}
e_{i} =
\begin{bmatrix}
0 & 0\\
t^{j} v_{k} & 0\\
\end{bmatrix},
\end{equation*}
where $1 \le k \le 3$, and $i = 3 (j+1) + k$. It follows that the
algebra $S$ is infinite-dimensional over $K$, with basis $e_{i}$, for
$i \ge 1$.
We have
\begin{equation*}
[e_{3} e_{2}] =
\begin{bmatrix}
0 & 0\\
v_{2} & 0\\
\end{bmatrix}
= e_{5}
\end{equation*}
and
\begin{equation*}
[e_{5} e_{2}] =
\begin{bmatrix}
0 & 0\\
\lambda t v_{1} & 0\\
\end{bmatrix}
= \lambda e_{7}.
\end{equation*}
It is now straightforward to see that all
identities~\eqref{eq:identities_for_Llambda} are satisfied in $S$, so
that $S$ is isomorphic to the algebra $L(\lambda)$ of
Section~\ref{sec:extra_3}.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
1,314,259,995,827 | arxiv | \section{Introduction}
Choosing a pricing model, especially in risk management and for Valuation Adjustments (xVA), is a balancing act between accuracy and pricing speed. Often, the calculation speed is critical in determining the model of choice, especially when handling large portfolios. The modern approach for modelling interest rates is based on the Heath, Jarrow, and Morton (HJM)~\cite{HJM:1992} framework with an arbitrary term-structure of volatility and covariance of {\it forward rates} across maturities. The framework is generic, and essentially any sensible term structure dynamics can be modelled by the HJM technique. Unfortunately, the number of models giving rise to analytical formulae for option pricing or even zero-coupon bonds is minimal. Therefore, models that can generate realistic implied volatilities while allowing for efficient derivative pricing are in high demand, especially when the portfolios need to be evaluated multiple times and for different market scenarios due to regulatory pressure.
A family of the HJM models that has gained particular acceptance from practitioners and academia is the family of affine models~\cite{Duffie:2000}. Over many years, various affine models have been studied, leading to significant contributions regarding efficient pricing, simulation and calibration (for the overview, see~\cite{BrigoMercurio:2007}). In particular, the classic affine short-rate models, like the Hull-White~\cite{HullWhite:1990} model, are popularised due to the closed formula for zero-coupon bonds and semi-analytic swaption pricing. On the downside, they lack sufficient flexibility to calibrate to market-implied volatilities. Additional model flexibility without compromising numerical efficiency is highly desired. A straightforward approach to incorporate and control the implied volatility smile and skew into the short-rate models is to define a stochastic or a local volatility process, for example, in~\cite{casassus2005unspanned,fong1991fixed,gatarek2017nonparametric}. Although such models give us a closed-form expression for the zero-coupon bond, pricing more advanced derivatives like options often involves advanced numerical techniques like Fourier inversion or numerical integration, which is undesired for the pricing of high-volume derivatives. A survey of the most popular stochastic volatility models in the interest rate world can be found in~\cite{andreasen2010stochastic}.
This article focuses on extending existing interest rate models with model parameters to be random via the so-called randomization method (RAnD)~\cite{grzelakRAnD}. Randomization by making the model parameters stochastic is a technique for more flexibility in the stochastic model-thus, improving the calibration quality while preserving the analytic properties of the models considered. The concept of randomization represents the uncertainty of potentially hidden states that are not sufficiently captured by deterministic parameters. Conventionally, under the affine framework, an extension by a stochastic parameter would require the model to meet linearity conditions - this does not have to be the case under RAnD. The method requires an affinity to hold, given a specific realization of the stochastic parameter. In other words, it builds an outer layer over the affine models and allows a stochastic parameter setting in that layer. The quadrature rule handles the numerical complexity associated with an infinite number of parameter values and reduces it to only a few {\it critical} parameter realizations. The selection of these points is based on the moments of the stochastic parameter.
The RAnD method is generic and can be applied to any pricing model. For the sake of simplicity, however, we focus on one of the most popular interest rate models, namely the Hull-White (HW) model~\cite{HullWhite:1990}. In particular, we will show that applying the RAnD method to the HW model results in a local-volatility type of dynamics. The local volatility dynamics may seem like a ``deal-breaker''; typically, the local volatility models, especially in interest rates, are notoriously tricky to operate, i.e., closed-form option pricing is hardly possible. However, although the randomized Hull-White (rHW) model is of the local volatility type, the option pricing is as complex as the standard, {\it unrandomized} model. Furthermore, the presented model offers fast and accurate calibration as the local volatility dynamics provide more flexibility in generating volatility smiles and skews for various market conditions.
Our method also addresses a challenging problem stated by Brigo and Mercurio in~\cite{brigo2002lognormal}, where the mixture of lognormals was studied. In their paper, it was shown that great flexibility to standard models like Black-Scholes was introduced by considering a convex combination of the associated probability density functions: $\omega_1f_{1}(x;\sigma_1)+\dots+\omega_Nf_{N}(x;\sigma_N)$ with $\omega_1+\dots+\omega_N=1$ and some $\sigma_i>0$, $i=1,\dots,N.$ However, the problem encountered was the large number of the model parameters that are difficult to interpret and relate to the corresponding implied volatilities. As stated in their work: {``\it the absence of bounds on the parameter $N$ implies that a virtually unlimited number of parameters can be introduced in the dynamics so as to be used for a better calibration to market data. (...), one has to find the correct tradeoff between model flexibility and number of parameters so as to avoid both poor calibration and over-parametrization issues''}. However, this problem is resolved with the RAnD method. The method transitions from a continuous randomizer $\vartheta$ to a unique sequence of weights $\omega_i$ and associated values $\sigma_i$. In this setting, the model calibration is performed by varying the parameters of the randomizer, $\vartheta$; therefore, the number of free parameters is manageable. Moreover, as presented in this work, the varying parameters of $\vartheta$ enable a transparent control of the implied volatilities.
This article comprises five sections. First, the details of the HJM framework and pricing equations using the RAnD method are provided in Section~\ref{sec:HJM}, and a discussion on the arbitrage-free conditions is covered in Section~\ref{sec:arbitrageHJM}. In Section~\ref{sec:randomizedHW}, we consider the rHW model and derive the dynamics of the randomized stochastic processes. Section~\ref{sec:dynamics_rHW} discusses different variants of the rHW model parameter choices, namely the randomization using univariate or bivariate distributions. Section~\ref{sec:pricingUnder_rHW} focuses on derivative pricing; in particular, the pricing equations for options on zero-coupon bonds and swaptions are provided. Numerical results are presented in Section~\ref{sec:numericalExperiments}, where in Section~\ref{sec:evolutionIR}, the swaption implied volatility surfaces for HW and rHW models are compared. Section~\ref{sec:impact_on_IV} analyzes the randomizers' impact on controlling implied volatilities for swaptions. Next, the calibration results are presented in Section~\ref{sec:calibration_MarketData}, and in Section~\ref{sec:pricingBivariate}, the extension to the bivariate randomization case is established. Finally, the convergence results are illustrated in Section~\ref{sec:convergence} and Section~\ref{sec:conclusion} concludes.
\section{The Heath-Jarrow-Morton models, affinity and randomization}
\label{sec:HJM}
Here, we will give an informal presentation of the Heath-Jarrow-Morton (HJM) framework and its relation to arbitrage-free short-rate models. We define the probability space $\left(\Omega,\F(t),\Q\right)$ on which the HJM arbitrage-free condition requires that the instantaneous forward rates, $f(t,T)$, are modelled by the following arbitrage-free SDE:
\begin{equation}
\label{eqn:HJMDynamics}
\d f(t,T)=\alpha(t,T){\rm d}t+\gamma(t,T)\dW(t),\;\;\;\alpha(t,T)= \gamma(t,T)\int_t^T \gamma(t,z){\rm d}z.
\end{equation}
The instantaneous forward rate, $f(t,T)$, is determined by the volatility driver $\gamma(t,T)$, which can be defined as a deterministic function, stochastic process or by a random variable. The case where $\gamma(t,T)$ is defined as a random variable, thus time-invariant, will be associated with parameter randomization.
We focus here on the short-rate dynamics under the HJM framework. Suppose that $f(0,t)$, $\alpha(t,T)$ and $\gamma(t,T)$ are differentiable in their second argument, with $\int_0^T\left|\frac{\partial}{\partial t}f(0,t)\right|<\infty,$ then the short-rate process under the HJM framework is given by,
\begin{eqnarray}
\d r(t) = \zeta(t){\rm d}t + \gamma(t,t) \d W(t),
\label{eqn:short_rate_dynamics}
\end{eqnarray}
with $$\zeta(t)=\alpha(t,t)+\frac{\partial }{\partial t}f(0,t)+\int_0^t\frac{\partial}{\partial t}\alpha(z,t)\d z+\int_0^t\frac{\partial}{\partial t}\gamma(z,t)\d W(z).$$
A particular class of popular models are the models belonging to the Affine-Diffusion interest rate models. It is shown~\citep{Duffie:2000} that in this class, for process ${{r}}(t)$, the discounted characteristic function is of the following form: \begin{equation}
\label{eqn:ChFCond}
\phi_{{r}}({{u}};t,T)=\E_t\left[\exp\left(-\int_t^T r(s){\rm d}s+i{u}{{r}}(T)\right)\right]=\exp\left(A({ u};\tau)+{{B}}({ u};\tau){{r}}(t)\right),
\end{equation}
with the expectation under risk-neutral measure $\Q$ for $\tau=T-t$, where $A({u};\tau)$ and ${{ B}}({ u};\tau)$ form the fundamental solution to the corresponding pricing PDE and satisfying the complex-valued ODEs, see the work by Duffie-Pan-Singleton~\citep{Duffie:2000}. The affinity conditions are rather strict as they impose a linear structure on the model, i.e., the model dynamics and its covariance structure needs to be linear in the state variables. The representation in~(\ref{eqn:ChFCond}) is of particular importance as it gives an explicit closed-form expression for the integrated short-rate process and therefore determines the price of a zero-coupon bond (ZCB). Models not leading to an analytic ZCB price are rarely used for pricing purposes. Unfortunately, because of the affinity constraints, the model extensions for some model parameters to be stochastic are minimal. In this article, we focus on relaxing this constraint and letting the model parameters be random while benefiting from the closed-form solutions for the ZCBs and pricing, as presented in~(\ref{eqn:ChFCond}).
We consider a vector $\Theta=[{\vartheta}_1,\dots,{\vartheta}_d]^{\rm T}$, with $d\in\mathbb{N}$ representing the number of randomized parameters, where each ${\vartheta}_i$ is a possibly correlated, time-invariant, random variable. A realization of ${\vartheta_i}$ is indicated by $\theta_{i}$, $\vartheta_i(\omega)=\theta_i$, and consequently the realization for ${\Theta}$ is indicated by $\theta=[\theta_{1},\dots,\theta_d]^{\rm T}$. Under the RAnD framework, parameters are not driven by a stochastic process but only by a random variable; however, an extension to piece-wise stochastic parameters is also possible (see~\cite{grzelakRAnD} for details).
In a nutshell, the RAnD method relies on quadrature integration and conditional expectation. Therefore, the technique can be applied wherever we deal with an expectation. In principle, randomization can be applied to Equation~(\ref{eqn:ChFCond}), which would provide us with a {\it randomized} ZCB; on the other hand, the randomization can be applied at the valuation level, where the expectation of the expected payoff is computed. We prefer the latter approach to construct the randomized variant of any interest rate model by simply evaluating the model on a particular realization of the model parameters. This will be particularly important when pricing non-standard derivatives.
In the randomized HJM framework, the volatility in~(\ref{eqn:HJMDynamics}) may be dependent on random parameters $\vartheta_1,\dots,\vartheta_d$, i.e., $\gamma(t,T;\{\vartheta_1,\dots,\vartheta_d\}).$
Let us now consider a stochastic model under the HJM framework for derivative pricing with the price denoted as $V(t,r(t;\theta))$ with $\theta$ indicating a vector-valued realization of the model parameter, $\Theta$. $V(t,r(t;\theta))$ may correspond to a model of {Va\v si\v cek}, Hull-White, Black-Karasi{\'n}ski or any other model that belongs to the class of HJM~\cite{BrigoMercurio:2007}. Then, the pricing under randomized parameter space will be given by the following relation:
\begin{eqnarray}
\label{eqn:genericPricing}
V(t,r(t;\Theta)):=\E_t[V(t,r(t;\Theta=\theta))]=\int_{\R^d} V(t,r(t;\Theta=\theta))f_{\Theta}(\theta)\d\theta,
\end{eqnarray}
where the integration takes place over the parameter space $\Theta\in\R^d$ and $f_\Theta(\theta)$ corresponds to the associated $d-$dimensional probability density function.
Equation~(\ref{eqn:genericPricing}) shows that a random model parameter can be considered averaging over possible parameter realizations. From a numerical perspective, the integration in~(\ref{eqn:genericPricing}) is expensive.
The randomization enables for randomizing $d-$dimensional space of parameters; however, for simplicity, we will mainly focus on single or bivariate parameter randomization, $\vartheta_1(\omega)=\theta_1$ and $\vartheta_2(\omega)=\theta_2$, thus also letting for the correlation between stochastic parameters.
Our aim is twofold: we aim to provide numerically efficient methods for the computation of the pricing Equation in~(\ref{eqn:genericPricing}) and to determine the stochastic differential equation for the randomized short-rate processes, $r(t;\vartheta)$ where $\vartheta$ is the chosen random parameter. The insight into the dynamics of the randomized stochastic process will help us categorise the randomized processes and the structure of their drift and volatility coefficients.
\begin{theorem}[Pricing formula for the randomized model]
\label{thm:rHW_generic}
Consider a random variable $\vartheta$, defined on some finite domain $D_\vartheta:=[a,b]$, with its PDF, $f_\vartheta(x)$, CDF, $F_\vartheta(x)$ and a realization $\theta$, $\vartheta(\omega)=\theta$ such that for some $N\in\mathbb{N}$ the moments are finite, $\E[\vartheta^{2N}]<\infty$.
Let $V(t,r(t;\vartheta))$ be the price of a financial derivative depending on the short-rate $r(t;\theta)$ with parameter $\theta$, then the randomized price, $V(t,r(t;\vartheta))$ is given by:
\begin{eqnarray}
\label{eqn:rHW_Generic}
V(t,r(t;\vartheta))=\int_{[a,b]} V(t,r(t;\theta))\d F_{\vartheta}(\theta)=\sum_{n=1}^N\omega_n V(t,r(t;\theta_n))+\epsilon_{N},
\end{eqnarray}
where the pairs $\{\omega_n,\theta_n\}_{n=1}^N$ are the Gauss-quadrature weights and the nodes based on the parameter cumulative distribution function, $F_{\vartheta}(\cdot),$ determined by $\zeta(\vartheta):\R\rightarrow \{\omega_n,\theta_n\}_{n=1}^N$ defined in~\ref{res:zeta} and where the error $\epsilon_N$ is defined as:
\begin{equation}
\label{eqn:RAnDError}
\epsilon_{N}=\frac{1}{(2N)!}\frac{\partial^{2N}}{\partial\xi^{2N}}V(t;r(t,\vartheta=\xi)),\;\;\;a<\xi<b.
\end{equation}
\begin{proof}
Define a payoff depending on the short-rate $r(t)$, $0<t<T$, by $\chi(r,T)$, then, for a stochastic parameter $\vartheta$ with the help of conditional expectation, we find:
\begin{eqnarray}
\label{eqn:pricingEqnProof}
V(t,r(t;\vartheta))=\E_{t}\left[\E_{t}\left[\frac{M(t)}{M(T)}\chi(r,T)\Big|\vartheta=\theta\right]\right]=:\E_{t}\left[V(t,r(t;\theta))\right],
\end{eqnarray}
with $M(t)$ being the money-savings account. By application of the quadrature rule, Equation~(\ref{eqn:pricingEqnProof}) becomes:
\begin{eqnarray*}
V(t,r(t;\vartheta))=\int_{[a,b]} V(t,r(t;\theta))\d F_{\vartheta}(\theta)=\sum_{n=1}^N\omega_n V(t,r(t;\theta_n))+\epsilon_{N}.
\end{eqnarray*}
Here, the error is given by $\epsilon_{N}$, for $a<\xi<b$, and the pair $\{\omega_n,\theta_n\}_{n=1}^N$ forms the Gauss-quadrature weights and the nodes based on the parameter distribution, as proven in~\citep{bulirsch2002introduction} (p.180, Theorem 3.6.24).
\end{proof}
\end{theorem}
Theorem~\ref{cor:RAnDChF} gives us an explicit relation between a continuous randomizer $\vartheta$ and the {\it discretization} in terms of pairs $\{w_n,\theta_n\}_{n=1}^N$, leading to a simplification of the integral in~(\ref{eqn:genericPricing}). In this paper, these pairs are based on moments of the randomizer, $\vartheta,$ and, for completeness, the detailed computation procedure is given in~\ref{res:zeta}.
The associated error $\epsilon_N$ in~(\ref{eqn:RAnDError}) can be interpreted as a {\it cost} when moving from the continuous to a discretized random parameter, $\vartheta\rightarrow \{\omega_n,\theta_n\}_{n=1}^N$. To some extent, the pricing with a discrete set of parameter realizations, $\theta_n$, and the associated probabilities, $w_n$, resembles the so-called regime-switch method where a finite set of states is defined. However, from a practical perspective, it is challenging to deal with $N$ pairs of the possible variable states and the associated probabilities, especially in the calibration procedure.
The procedure described in~\ref{res:zeta}, however, simplifies this problem as we can control the quadrature pairs employing the parameters of the randomizer $\vartheta$. To illustrate this process, let us consider a stochastic parameter with a randomizer $\vartheta\sim\mathcal{N}(\mu,\sigma^2)$; thus, the stochastic parameter, $\vartheta$, is driven by two parameters $\mu$ and $\sigma^2$. By application of~\ref{res:zeta}, for any set of parameters $\mu$ and $\sigma^2$, we compute the corresponding pairs $\{\omega_n,\theta_n\}_{n=1}^N$; therefore, in the calibration procedure we will only vary $\mu$, and $\sigma^2$. This procedure drastically reduces the number of associated model parameters, thus facilitating the calibration. We also stress that the set of optimal pairs is solely obtained based on the moments of the randomizing random variable, $\vartheta$, implying that every random variable with, preferably, closed-form moments may be used for randomization. In Table~\ref{Tab:Moments}, a few selected random variables and their moments are tabulated. In Section~\ref{sec:calibration_MarketData}, we will also classify the randomizers, $\vartheta$, for which the computations can be greatly reduced or even tabulated.
To emphasize the flexibility of the RAnD method, we focus on the application in the Hull-White model, where we will apply randomization to each model parameter and consider an extension to a bivariate case. In the following sections, we consider a finite number of realizations of $\vartheta$. We will denote them as $\theta_1,\dots,\theta_N$, for some $N\in\mathbb{N}$. These ``specific'' realizations we will interchangeably call either ``collocation''~\citep{scmc2019} or ``quadrature'' points. Finally, by $\vartheta(\hat a,\hat b)$, we denote that $\vartheta$ is a random variable driven by parameters: $\hat a$, $\hat b$.
\subsection{Arbitrage-free conditions under the RAnD method}
\label{sec:arbitrageHJM}
Before we analyze the specifics of the randomized Hull-White (rHW) model, let us review the implications of randomization on the pricing, under the affine-diffusion framework, of ZCB and discuss arbitrage-free aspects of the RAnD method. By setting $u=0$ in~(\ref{eqn:ChFCond}), the risk-free pricing formula for a randomized ZCB, $P(t, T;\vartheta)$ is given by:
\begin{eqnarray}
\label{eqn:ZCB}
P(t,T;\vartheta)=\int_{[a,b]} P(t,T;\theta)\d F_{\vartheta}(\theta)=\int_{[a,b]} \e^{A(\tau;\theta)+{{B}}(\tau;\theta){{r}}(t;\theta)}\d F_{\vartheta}(\theta),\;\;a<b,
\end{eqnarray}
with $A(\tau;\theta):=A(0,\tau;\theta)$ and $B(\tau;\theta):=A(0,\tau;\theta)$ in~(\ref{eqn:ChFCond}) for $\tau=T-t,$ and $r(t;\theta)$ indicates a short-rate model with constant parameter realizations $\theta.$ By application of Theorem~\ref{thm:rHW_generic}, the randomized ZCB is known explicitly, and it is presented in Corollary~\ref{cor:RAnDChF}.
\begin{cor}[ZCBs under Randomized Affine Jump Diffusion Processes]
\label{cor:RAnDChF}
Let ${r}(t;\theta)$ represent an affine short-rate process with some constant parameter $\theta$. Assuming that the corresponding ChF, $\phi_{r(T)|\vartheta=\theta}(\cdot)$, is well defined and $2N$ times differentiable w.r.t. $\theta$, the randomized ZCB is given by,
\begin{eqnarray}
P(t,T;\vartheta)&=&\sum_{n=1}^{N}\omega_nP(t,T;\theta_n)+\epsilon_N=\sum_{n=1}^{N}\omega_n\e^{A(\tau;\theta_n)+{{ B}}(\tau;\theta_n)r(t;\theta_n)}+\epsilon_N\nonumber\\
\label{eqn:randChF_ZCB}
&=:&P(t,T;\{\theta_n\}_{n=1}^N)+\epsilon_N,
\end{eqnarray}
with ${A}(\tau;\theta_n)$, ${{ B}}(\tau;\theta_n)$ being the real-valued functions obtained from Riccati-type of ODEs available for affine models~\cite{OosterleeGrzelakBook}.
\begin{proof}
The proof is a consequence of Theorem~\ref{thm:rHW_generic}.
\end{proof}
\end{cor}
Representation~(\ref{eqn:randChF_ZCB}) illustrates that the ZCB, $P(t,T;\vartheta)$, randomized with stochastic parameter $\vartheta$ can be expressed as a weighted sum of the ZCBs evaluated at a given realization of $\vartheta$ and where $\omega_1+\dots+\omega_N=1$. Note that under the HJM framework, each ZCB, $P(t,T;\theta_n)$ in~(\ref{eqn:randChF_ZCB}) is arbitrage-free; therefore, the convex combination is too. This can be shown by checking whether $P(S,T;\{\theta_n\}_{n=1}^N)/M(S)$ is indeed a martingale for some $t<S<T$:
\begin{eqnarray*}
\E_t\left[\frac{P(S,T;\{\theta_n\}_{n=1}^N)}{M(S)}\right]=\E_t\left[\frac{\sum_{n=1}^{N}\omega_nP(S,T;\theta_n)}{M(S)}\right]=\sum_{n=1}^{N}\omega_n\E_t\left[\frac{P(S,T;\theta_n)}{M(S)}\right],
\end{eqnarray*}
since every discounted ZCB, $P(S,T;\theta_n)/M(S),$ is a martingale under the $\Q$ measure we have,
\begin{eqnarray*}
\E_t\left[\frac{P(S,T;\{\theta_n\}_{n=1}^N)}{M(S)}\right]=\sum_{n=1}^{N}\omega_n\frac{P(t,T;\theta_n)}{M(t)}=\frac{P(t,T;\{\theta_n\}_{n=1}^N)}{M(t)}.
\end{eqnarray*}
This result shows that the application of the RAnD method enables the transition from a continuous random variable to a discrete one; the cost of this transition is expressed by $\epsilon_N$. Moreover, as shown above, the error $\epsilon_N$ in~(\ref{eqn:randChF_ZCB}) does not impact the arbitrage conditions-the pricing under the RAnD method is simply a {\it weighted average} of arbitrage-free prices. However, from a practical perspective, keeping this error as small as possible is still essential. This is particularly important when calibrating the randomized model to market data, i.e., the calibration procedure is associated with varying parameters of the variable $\vartheta$. It is, therefore, important that the discretized version resembles the continuous version as closely as possible. In the numerical section, Section~\ref{sec:convergence}, we will investigate the convergence aspects of different randomizers and an optimal number of expansion terms, $N$.
\subsection{The randomized Hull-White (rHW) short-rate model}
\label{sec:randomizedHW}
This section provides a specification leading us to the randomized version of the famous HW model~\citep{HullWhite:1990}-a single-factor, no-arbitrage yield curve model, in which an extended Ornstein-Uhlenbeck drives the short-term interest rate mean-reverting process. Under the HJM framework and the arbitrage-free condition for the drift in~(\ref{eqn:HJMDynamics}), the rHW model is specified by:
\begin{equation}
\label{eqn:HJMHW}
\gamma(t,T)=\eta\cdot \e^{-\lambda(T-t)},\;\;\;t<T,
\end{equation}
where we consider three different randomization cases: the randomization of the volatility parameter, $\eta$, the mean-reversion, $\lambda$, or the randomization of both parameters using bivariate distribution:
\begin{equation}
\label{eqn:HW_RAnD_Parms}
\eta\stackrel{\d}{=}\vartheta_1,\;\;\text{or}\;\;\; \lambda\stackrel{\d}{=}\vartheta_2,\;\;\;\text{or}\;\;\; \lambda|\eta\stackrel{\d}{=}\vartheta_2|\vartheta_1.
\end{equation}
To simplify the notation, we consider one parameter $\theta\in\{\eta,\lambda\}$ with the corresponding random variable $\vartheta$. Particular choices of the parameters will be given explicitly.
Given the HJM volatility in~(\ref{eqn:HJMHW}), under the risk-free measure, the dynamics of the HW model read:
\begin{equation}
\label{eqn:HW_SDE}
\d {r}(t)=\lambda(\psi(t)-{r}(t)){\rm d}t + \eta\dW(t),\;\;\;r_0\equiv f(0,0),
\end{equation}
with \begin{equation}
\label{eqn:Psi_f_0_t}
\psi(t)=f(0,t)+\frac{1}{\lambda}f(0,t)+\frac{\eta^2}{2\lambda^2}\left(1-\e^{-2\lambda t}\right),\;\;\;f(0,t)=-\frac{\partial \log P(0,t)}{\partial t},
\end{equation}
where $\psi(t)$ is a time-dependent drift term, which ensures the fit of the calibration to a yield curve observed in the market, $W(t)$ is the Brownian motion under measure $\Q$, and $f(0,t)$ indicates the instantaneous forward rate computed from the yield curve, defined in terms of ZCBs, $P(0,t)$. Parameter $\eta$ determines the overall level of the volatility, and $\lambda$ is the reversion rate parameter. A large value of $\lambda$ causes short-term rate movements to dampen out rapidly, reducing the long-term volatility. Because of this interdependence between the model parameters, often, in practical applications, $\lambda$ is fixed and set to a constant. In contrast, $\eta$ is often set to be piece-wise constant and calibrated such that the ATM volatilities are calibrated. We will show that under the RAnD method, such a strategy is inadequate.
The short-rate $r(t)$ in~(\ref{eqn:HW_SDE}) is thus {\it normally distributed} with $r(t)\sim\mathcal{N}(\mu_{r(t)},\sigma^2_{r(t)}),$ with
\begin{eqnarray}
\label{eqn:r_t_distribution}
\mu_{r(t)}:=r_0\e^{-\lambda
t}+\lambda\int_0^t{\psi(z)\e^{-\lambda(t-z)} {\rm d}z},\;\;\;\sigma^2_{r(t)}:=\frac{\eta^2}{2\lambda}\left(
1-\e^{-2\lambda t}\right).
\end{eqnarray}
Moreover, for
$\psi(t)$ constant, i.e., $\psi(t)\equiv\psi$ (in this case we deal with the \index{Va\v si\v cek model}{\em Va\v si\v cek model}~\citep{Vasicek:1977}), we have $\lim_{t\rightarrow\infty}\E_{t_0}\left[r(t)\right]=\psi.$ This
means that the first moment of the process converges
to the mean-reverting level $\psi$, for large values of $t$.
As a first step, we check the impact of randomization on the probability density function, PDF, of the interest rate process $r(t)$ in~(\ref{eqn:HW_SDE}). Since the HW model belongs to the affine class of processes, we can benefit from the available ChF. Given a stochastic parameter ${\bf \vartheta}$, the ChF is given by:
\begin{eqnarray*}
\phi_{{ r}}({ u};t,T):=\E_t\Big[\e^{-\int_t^T r(s){\rm d}s+i{ u}{r}(T)}\Big]=\E_t\left[\E_t\Big[\e^{-\int_t^T r(s){\rm d}s+i{ u}{ r}(T)}\big|{\vartheta}={\theta}\Big]\right].
\end{eqnarray*}
By application of the quadrature rule in Theorem~\ref{thm:rHW_generic} and~\ref{res:zeta}, we find:
\begin{eqnarray}
\label{eqn:ChFIntegral}
\phi_{{ r}}({ u};t,T)=\int_{[a,b]} \phi_{{r}|{\vartheta}=\theta}({ u};t,T)\d F_{\vartheta}(\theta)=\sum_{n=1}^N\omega_n\phi_{{ {r}(T)}|\vartheta=\theta_n}({ u};t,T) + \epsilon_N^F,
\end{eqnarray}
and by utilizing the Fourier transform, the PDF of the rHW model reads,
\begin{eqnarray}
\nonumber
f_{{{r}(T)}}(x)=\frac{1}{2\pi}\int_{\R}\e^{-iux}\sum_{n=1}^{N}\omega_n\phi_{{ {r}}|\vartheta=\theta_n}({ u};t,T)\d u+\epsilon_N^F=\sum_{n=1}^{N}\omega_nf_{{ r(T;\theta_n)}}(x)+\epsilon_N^F,
\label{eqn:HW_ranDensity}
\end{eqnarray}
where $r(t)|\vartheta=\theta_n$ indicates the HW model with a particular realization, $\theta_n$, of the randomized variable. The representation above presents the relation between the densities of the rHW process, $r(t)$, as a convex combination of the HW processes, $r(t)|\theta_n$ with a particular parameter $\theta_n$. We also highlight that the error $\epsilon_N^F$ in~(\ref{eqn:ChFIntegral}) may differ from the quadrature error in~(\ref{eqn:RAnDError}).
Because of different randomization choices, we distinguish the following randomization types and their associated error:
\begin{equation}
\label{eqn:PDF_rand}
f_{{{r}(T)}}(y)=\sum_{n=1}^{N}\omega_nf_{{ r(T;\theta_n)}}(y)+\epsilon_N^F=:\left\{\begin{array}{ccc}
f_{\overline{r}(T)}(y)+\overline\epsilon_N,\;\;\;\text{for}\;\;\;\eta\stackrel{\d}{=}\vartheta,\\
f_{\widetilde{r}(T)}(y)+\widetilde\epsilon_N,\;\;\;\text{for}\;\;\;\lambda\stackrel{\d}{=}\vartheta,\\
f_{\widehat{r}(T)}(y)+\widehat\epsilon_N,\;\;\;\text{for}\;\;\;\eta\stackrel{\d}{=}\vartheta_1\;\;\&\;\;\lambda|\eta\stackrel{\d}{=}\vartheta_2.
\end{array}\right.
\end{equation}
Under the RAnD method, the PDF, $f_{{{r}(T)}}(y)$ in~(\ref{eqn:PDF_rand}) is given as a linear combination of normal densities that depend on different parameter realizations $\theta_n$, resembling a similar problem as presented in~\cite{brigo2002lognormal}.
In the next section, we associate the PDF given in~(\ref{eqn:PDF_rand}) and find the corresponding SDE for a 1D stochastic representation.
\subsection{Dynamics of the rHW model}
\label{sec:dynamics_rHW}
The combination of Equations~(\ref{eqn:HW_RAnD_Parms}) and~(\ref{eqn:HW_SDE}) shows us that the simulation of the randomized short-rate models is explicit using the Monte Carlo technique. However, by an explicit form of the corresponding SDE, we will get more insight into the randomization and its impact on the dynamics of the driving process.
As a start, under the rHW model, consider the randomization of the volatility parameter, $\theta:=\eta$, with $\vartheta(\omega)=\theta$, and where the corresponding quadrature pairs are given by $\{\omega_n,\eta_n\}_{n=1}^N$. For such a setting, under each realization $\eta_n$, we have the associated HW model, $\overline{r}_n(t):=\overline{r}_n(t;\eta_n)$, with the dynamics given by:
\begin{eqnarray}
\label{eqn:HW_eta_n}
\d \overline{r}_n(t)=\lambda(\overline\psi_n(t)-\overline{r}_n(t)){\rm d}t + \eta_n\dW(t),\;\;\;\overline{r}_n(t_0)=f(0,0),\;\;\;n=1,\dots,N,
\end{eqnarray}
with common Brownian motion, $W(t)$, and initial value $\overline{r}_n(t_0)=f(0,0)$ in~(\ref{eqn:Psi_f_0_t}), where $\lambda$ is constant and equal for all the $\overline{r}_n(t)$, and where $\overline\psi_n(t)$ is defined in~(\ref{eqn:Psi_f_0_t}) and is given as a function of $\lambda$ and $\eta_n$.
Given the sequence of HW model processes, $\overline{r}_1(t),\dots,\overline{r}_N(t)$, in~(\ref{eqn:HW_eta_n}) and the probability density relation in~(\ref{eqn:HW_ranDensity}), we consider the problem of finding the corresponding SDE for the rHW process, $\overline{r}(t)$. Formally, we seek an SDE, with the solution given in~(\ref{eqn:PDF_rand}) and where each of the constituent processes, $\overline{r}_n(t)$, is driven by~(\ref{eqn:HW_eta_n}).
Thus, we consider the following process,
\begin{eqnarray}
\label{eqn:HW_local}
\d \overline{r}(t)=\overline\lambda(t,\overline{r}(t)){\rm d}t + \overline{\eta}(t,\overline{r}(t))\dW(t),\;\;\;\overline{r}(t_0)=f(0,0),
\end{eqnarray}
with some state-dependent drift, $\overline\lambda(t,\overline{r}(t))$, and volatility, $\overline{\eta}(t,\overline{r}(t))$, and where Brownian motion $W(t)$ is as in~(\ref{eqn:HW_eta_n}).
As indicated in~\cite{brigo2008general}, the problem of finding an SDE when marginal distributions and the corresponding weights are given is the reverse of finding the marginal density function of the solution of an SDE when the coefficients are known. Moreover, the problem of a mixture of normal or lognormal processes is known~\cite{brigo2008general}. Here, in Proposition~\ref{prop:rand_eta}, we adapt those techniques and provide an explicit form for the SDE in~(\ref{eqn:HW_local}).
\begin{prop}[Local volatility process for the HW model with randomized volatility parameter, $\eta$]
\label{prop:rand_eta}
Let us assume a sequence of positive constants $\eta_n$, $n=1,\dots,N.$ Then, the SDE
\begin{equation}
\label{eqn:HW_localVol}
\d \overline{r}(t)=\overline\lambda(t,\overline{r}(t)){\rm d}t + \overline{\eta}(t,\overline{r}(t))\dW(t),\;\;\;\overline{r}(t_0)=f(0,0),
\end{equation}
with
\begin{eqnarray}
\overline\lambda(t,y)=\sum_{n=1}^{N}\overline{\Lambda}_n(t,y)\lambda(\overline\psi_n(t)-y),\;\;\;\overline{\eta}^2(t,y)=\sum_{n=1}^{N}\eta_n^2\overline\Lambda_n(t,y),
\end{eqnarray}
where:
\[\overline\Lambda_n(t,y)=\frac{\omega_nf_{\overline{r}(t;\eta_n)}(y)}{\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)},\]
has a strong solution whose marginal density is given by the mixture of normal probability density functions:
\begin{eqnarray}
\label{eqn:HW_rand_eta_density}
f_{{\overline{r}(t)}}(y)=\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y),
\end{eqnarray}
where $\sum_{n=1}^N\omega_n=1$ for $\omega_n\geq0$, $n=1,\dots,N$ with $f_{\overline{r}(t;\eta_n)}(x)$ the PDF of the HW model with dynamics, given by:
\begin{equation}
\label{eqn:HW_individual_asset}
\d \overline{r}_n(t)=\lambda(\overline\psi_n(t)-\overline{r}_n(t)){\rm d}t + \eta_n\dW(t),\;\;\;\overline{r}_n(t_0)=f(0,0),
\end{equation}
where $\overline{r}_n(t):=\overline{r}_n(t;\eta_n)$ with $\overline\psi_n(t)=f(0,t)+\frac{1}{\lambda}f(0,t)+\frac{\eta_n^2}{2\lambda^2}\left(1-\e^{-2\lambda t}\right).$
\begin{proof}
The problem we address is the derivation of the drift, $\overline\lambda(t,y)$, and the volatility function, $\overline\eta(t,\overline{r}(t))$, in the SDE:
\begin{equation}
\label{eqn:HW_eta}
\d \overline{r}(t)=\overline\lambda(t,\overline{r}(t)){\rm d}t + \overline\eta(t,\overline{r}(t))\dW(t),
\end{equation}
such that \begin{equation}
\label{eqn:density_r_T}
f_{{\overline{r}(t)}}(y)=\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y),
\end{equation}
where $f_{\overline{r}(t;\eta)}(y)$ is the PDF of the Hull-White process in~(\ref{eqn:HW_eta_n}), with parameter $\eta$. Under the HW model dynamics, the so-called linear-growth condition holds, i.e., $\eta_n^2\leq C_n(1+y^2)$ for $i=1,\dots,N.$ Assuming that the same non-explosion condition holds for~(\ref{eqn:HW_eta}), i.e., $\overline\eta^2(t,y)\leq C(1+y^2),$ uniformly in $t,$ we start by deriving the
Fokker-Planck equation for $\overline{r}(t)$:
\begin{eqnarray}
\label{eqn:FK_r_T}
\frac{\partial }{\partial t}f_{\overline{r}(t)}(y)=\frac12\frac{\partial^2 }{\partial y^2}\left(\overline\eta^2(t,y))f_{\overline{r}(t)}(y)\right)-\frac{\partial }{\partial y}\overline\lambda(t,y)f_{\overline{r}(t)}(y),
\end{eqnarray}
while for each individual interest rate process, $r_n(t)$, in~(\ref{eqn:HW_individual_asset}) we have:
\begin{eqnarray}
\label{eqn:SDE_individual_r_t}
\frac{\partial }{\partial t}f_{\overline{r}(t;\eta_n)}(y)=\frac12\frac{\partial^2 }{\partial y^2}\left(\eta_n^2f_{\overline{r}(t;\eta_n)}(y)\right)-\frac{\partial }{\partial y}\left(\lambda(\overline\psi_n(t)-y)\right)f_{\overline{r}(t;\eta_n)}(y).
\end{eqnarray}
After substituting~(\ref{eqn:density_r_T}) into~(\ref{eqn:FK_r_T}):
\begin{eqnarray*}
\frac{\partial }{\partial t}\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)=\frac12\frac{\partial^2 }{\partial y^2}\left[\overline\eta^2(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)\right]-\frac{\partial }{\partial y}\overline{\lambda}(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y).
\end{eqnarray*}
Due to the linearity of the derivative operator, we find,
\begin{eqnarray*}
\sum_{n=1}^{N}\omega_n\frac{\partial }{\partial t}f_{\overline{r}(t;\eta_n)}(y)=\frac12\frac{\partial^2 }{\partial y^2}\left[\overline\eta^2(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)\right]-\frac{\partial }{\partial y}\overline\lambda(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y).
\end{eqnarray*}
By substitution of~(\ref{eqn:SDE_individual_r_t}) and operator linearity,
\begin{eqnarray*}
\small
\frac{\partial^2 }{\partial y^2}\sum_{n=1}^{N}\omega_n\frac12\eta_n^2f_{\overline{r}(t;\eta_n)}(y)-\sum_{n=1}^{N}\omega_n\frac{\partial }{\partial y}\lambda(\overline\psi_n(t)-y)f_{\overline{r}(t;\eta_n)}(y)=\\\frac12\frac{\partial^2 }{\partial y^2}\left[\overline\eta^2(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)\right]-\frac{\partial }{\partial y}\overline\lambda(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y).
\end{eqnarray*}
Finally, by matching the appropriate terms and integration, we find:
\begin{eqnarray*}
\sum_{n=1}^{N}\omega_n\eta_n^2f_{\overline{r}(t;\eta_n)}(y)&=&\overline\eta^2(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)+C_1(t)y+C_2(t),\\
\sum_{n=1}^{N}\omega_n\lambda(\overline\psi_n(t)-y)f_{\overline{r}(t;\eta_n)}(y)&=&\overline\lambda(t,y)\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)+C_3(t),
\end{eqnarray*}
for some time-dependent functions $C_1(t)$, $C_2(t)$ and $C_3(t).$
Since the LHS and RHS of the equations above need to satisfy uniform convergence requirements implying that for $y\rightarrow\infty$, they converge to $0$, the following needs to hold: $C_1(t)=C_2(t)=C_3(t)=0$, $\forall t.$ Therefore, the expressions for $\overline\lambda(t,y)$ and $\overline\eta(t,y)$ read:
\begin{eqnarray}
\overline\eta^2(t,y)=\frac{\sum_{n=1}^{N}\omega_n\eta_n^2f_{\overline{r}(t;\eta_n)}(y)}{\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)},\;\;\;\;\;\overline\lambda(t,y)=\frac{\sum_{n=1}^{N}\omega_n\overline\lambda(\overline\psi_n(t)-y)f_{\overline{r}(t;\eta_n)}(y)}{\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)}.
\end{eqnarray}
By setting
\begin{equation}
\overline\Lambda_n(t,y)=\frac{\omega_nf_{\overline{r}(t;\eta_n)}(y)}{\sum_{n=1}^{N}\omega_nf_{\overline{r}(t;\eta_n)}(y)}\;\;\;\text{for}\;\;\;n=1,\dots,N,
\end{equation}
we can write
\begin{eqnarray}
\overline\lambda(t,y)=\sum_{n=1}^{N}\overline{\Lambda}_n(t,y)\lambda(\overline\psi_n(t)-y),\;\;\;\text{and}\;\;\;\overline{\eta}^2(t,y)=\sum_{n=1}^{N}\eta_n^2\overline\Lambda_n(t,y).
\end{eqnarray}
Finally, by taking $\eta_{*}:=\max_{i=1,\dots,N}\eta_n$ and since $\sum_{n=1}^N\overline{\Lambda}_n(t,y)=1,$ $\forall y,$ we have:
\begin{eqnarray*}
\overline\eta^2(t,y)=\sum_{n=1}^{N}\eta_n^2\overline\Lambda_n(t,y)\leq \sum_{n=1}^{N}\eta_{*}^2\overline\Lambda_n(t,y)=\eta_*^2=C.
\end{eqnarray*}
Since the volatility parameter $\overline\eta^2(t,y)$ is bounded by a constant, the uniform convergence criterion is satisfied. The uniqueness of the strong solution follows from Theorem 12.1 in~\cite{rogers_williams_2000} while in~\cite{brigo2008general} (Theorem 2.1), a proof for a generic case for a normal mixture is provided.
\end{proof}
\end{prop}
Proposition~\ref{prop:rand_eta} illustrates that under the randomized volatility parameter for the HW model, the normal mixture dynamics resemble a one-dimensional local-volatility-type diffusion process. The local volatility function is expressed as a weighted volatility squared, $\eta_n^2$, of the constituent processes, $\overline{r}_n(t)$, and where the weights are functions of the quadrature coefficients, $\omega_n$, and the corresponding PDFs, $f_{\overline{r}(t;\eta_n)}(y)$.
Following the same strategy, we derive the dynamics of a process of the rHW model with the randomized mean-reversion parameter, $\lambda$. Proposition~\ref{prop:rand_lambda} provides the details.
\begin{prop}[Dynamics of the HW model with randomized mean-reversion parameter, $\lambda$] \label{prop:rand_lambda}
Let us assume a sequence of positive constants $\lambda_m$, $m=1,\dots,M$, then the SDE
\begin{equation}
\label{eqn:HW_localVol_lambda}
\d \widetilde{r}(t)=\widetilde\lambda(t,\widetilde{r}(t)){\rm d}t + \widetilde{\eta}(t,\widetilde{r}(t))\dW(t),\;\;\;\widetilde{r}(t_0)=f(0,0),
\end{equation}
with $\widetilde\eta(t,y)=\eta$, and
\begin{eqnarray}
\widetilde\lambda(t,y)=\sum_{m=1}^M\widetilde\Lambda_m(t,y)\lambda_m(\psi_m(t)-y),\;\;\;\text{where}\;\;\;\widetilde\Lambda_m(t,y)=\frac{\varpi_mf_{\widetilde{r}(t;\lambda_m)}(y)}{\sum_{m=1}^{M}\varpi_mf_{\widetilde{r}(t;\lambda_m)}(y)},
\end{eqnarray}
has a strong solution whose marginal density is given by the mixture of normal probability density functions:
\begin{eqnarray}
\label{eqn:HW_rand_lambda_density}
f_{{\widetilde{r}(t)}}(x)=\sum_{m=1}^{M}\varpi_mf_{\widetilde{r}(t;\lambda_m)}(x),
\end{eqnarray}
where $\sum_{m=1}^M\varpi_m=1$ for $\varpi_m\geq0$, $m=1,\dots,M$, with $f_{\widetilde{r}(t;\lambda_m)}(x)$ is the PDF the HW model whose dynamics is given by:
\[\d \widetilde{r}_m(t)=\lambda_m(\psi_m(t)-\widetilde{r}_m(t)){\rm d}t + \eta\dW(t),\;\;\;\widetilde{r}_m(t_0)=f(0,0),\]
where $\widetilde{r}_n(t):=\widetilde{r}_n(t;\lambda_n)$ with $\psi_m(t)=f(0,t)+\frac{1}{\lambda_m}f(0,t)+\frac{\eta^2}{2\lambda_m^2}\left(1-\e^{-2\lambda_m t}\right).$
\begin{proof}
The proof is analogous to Proposition~\ref{prop:rand_eta}.
\end{proof}
\end{prop}
With the rHW model, the randomization of either model parameters enables us to determine the corresponding 1D SDE. For the random volatility parameter, $\eta$, we derived a local-volatility type short-rate process; however, in the case of random $\lambda$, the corresponding 1D process has a different structure, i.e., the volatility coefficient, $\widetilde\eta(t,y)=\eta$, in~(\ref{eqn:HW_localVol_lambda}), stays constant. When analyzing the dynamics of the corresponding ZCB (see Corollary~\ref{cor:RAnDChF}), which is a tradable asset, the dynamics will resemble a local-volatility type process. To derive the corresponding SDE, one needs to follow the same strategy as in Propositions~\ref{prop:rand_eta} and~\ref{prop:rand_lambda} with the process for the ZCB, instead of the short-rate process.
In the final part of this section, the case of joint randomization of both HW model parameters will be discussed.
\subsubsection{Dynamics of the rHW model with bivariate distribution for $\lambda$ and $\eta$.}
\label{sec:rHW_bivariateCase}
An extension of the RAnD method is to consider both HW model parameters random and follow a bivariate distribution. Such an extension may benefit from the interconnection between model parameters and possibly the correlation coefficient; however, it would require that the conditional moments are known explicitly. This is troublesome because only for a few random distributions the moment functions are known in the closed form. However, if we stay, for example, within the Gaussian world, such an extension to the 2D case is possible.
\begin{cor}[Random parameters with bivariate distribution]
\label{cor:bivariate}
Under a bivariate distribution $\Theta=[\vartheta_1,\vartheta_2]$ with $\zeta(\vartheta_1)=\{\omega_{1,n},\theta_{1,n}\}_{n=1}^N$ and conditioned on $\zeta(\vartheta_2|\vartheta_1)=\{\omega_{2,m},\theta_{2,m}\}_{m=1}^M$, the randomized ChF is given by:
\begin{eqnarray}
\phi_{{ X}}({ u};t,T)=\sum_{n=1}^{N}\omega_{n}\sum_{m=1}^{M}\omega_{m}\phi_{{ X}|\vartheta_1=\theta_{n},\vartheta_2=\theta_{m}}({ u};t,T)+\epsilon_{N,M}^b,
\end{eqnarray}
where $N$ and $M$ indicate the number of expansion terms for $\vartheta_1$ and $\vartheta_2|\vartheta_1$ respectively, $\vartheta_2|\vartheta_1$ indicates a conditional random variable, $\epsilon_{N,M}^b$ is the corresponding aggregated error, and the remaining specification follows Theorem~\ref{cor:RAnDChF}.
\end{cor}
The PDF computation for the bivariate case requires the sequential computation of the associated weights and the corresponding points, which can be established utilizing Propositions~\ref{prop:rand_eta} and~\ref{prop:rand_lambda}. In the first iteration step, we compute the grid points associated with the volatility parameter, $\eta$, and for each realization $\eta_n$, we establish the corresponding conditional PDF. By summing over all possible pairs, the unconditional PDF for the rHW model can be computed,
\begin{eqnarray}
\label{eqn:PDF_bivariate}
f_{r(T;\eta_n)}(x)=\sum_{m=1}^{M}\varpi_{m,n}f_{{ r(T;\eta_n,\lambda_m)}}(x),\;\;\;f_{{r(T)}}(x)=\sum_{n=1}^{N}\sum_{m=1}^{M}\omega_n\varpi_{m,n}f_{{ r(T;\eta_n,\lambda_m)}}(x),
\end{eqnarray}
where $\varpi_{m,n}$ is defined in~(\ref{eqn:HW_rand_lambda_density}), $\omega_n$ is given in~(\ref{eqn:HW_rand_eta_density}) and $f_{{ r(T;\eta_n,\lambda_m)}}(x)$ indicates the PDF of the HW model with parameters $\eta_n$ and $\lambda_m$, and is defined in~(\ref{eqn:r_t_distribution}).
Proposition~\ref{prop:rand_eta_and_lambda} provides the dynamics of the associated rHW model.
\begin{prop}[Local volatility process for the HW model with randomized parameters]
\label{prop:rand_eta_and_lambda}
Let us assume a sequence of positive constants $\eta_n$, $n=1,\dots,N$, and $\lambda_{m,n}$, $m=1,\dots,M$, then the SDE:
\begin{equation}
\label{eqn:HW_localVol}
\d \widehat{r}(t)=\widehat\lambda(t,\widehat{r}(t)){\rm d}t + \widehat{\eta}(t,\widehat{r}(t))\dW(t),\;\;\;\widehat{r}(t_0)=f(0,0),
\end{equation}
with
\begin{eqnarray*}
\widehat\eta^2(t,\widehat{r}(t))&=&\sum_{n=1}^N\sum_{m=1}^M\widehat\Lambda_{n,m}(t,y)\eta_n^2,\;\;\;\widehat\lambda(t,\widehat{r}(t))=\sum_{n=1}^{N}\sum_{m=1}^{M}\widehat\Lambda_{n,m}(t,y)\lambda_{m,n}(\psi_{m,n}(t)-y),
\end{eqnarray*}
where \[\widehat\Lambda_{n,m}(t,y)=\frac{\omega_n\varpi_{m,n}f_{{ r(t;\eta_n,\lambda_m)}}(y)}{\sum_{n=1}^{N}\sum_{m=1}^{M}\omega_n\varpi_{m,n}f_{r(t;\eta_n,\lambda_{m,n})}(y)},\]
has a strong solution whose marginal density is given by the following mixture of normal probability density functions:
\begin{eqnarray*}
f_{{\widehat{r}(t)}}(y)=\sum_{n=1}^{N}\sum_{m=1}^{M}\omega_n\varpi_{m,n}f_{{ r(t;\eta_n,\lambda_{m,n})}}(y),
\end{eqnarray*}
where $\sum_{n=1}^N\omega_n=1$, $\sum_{m=1}^M\varpi_{m,n}=1$ for $\omega_n,\varpi_{m,n}\geq0$, $n=1,\dots,N$, $m=1,\dots,M$ with $f_{r(t;\eta_n,\lambda_{m,n})}(y)$ being the PDF of the HW model whose dynamics are given by:
\[\d r_{m,n}(t)=\lambda_{m,n}(\psi_{m,n}(t)-r_{m,n}(t)){\rm d}t + \eta_{n}\dW(t),\;\;\;r_{m,n}(t_0)=f(0,0),\]
and $r_{n,m}(t):=r(t;\eta_n,\lambda_m)$ with $\psi_{m,n}(t)=f(0,t)+\frac{1}{\lambda_{m,n}}f(0,t)+\frac{\eta_{n}^2}{2\lambda_{m,n}^2}\left(1-\e^{-2\lambda_{m,n} t}\right).$
\begin{proof}
The proof is analogous to Proposition~\ref{prop:rand_eta}.
\end{proof}
\end{prop}
The concept of multivariate model parameters can be extended further. In the case of the HW model with piece-wise constant parameters, it is possible to randomize each piece-wise element, to mimic the parameter controlled by the stochastic process. Unfortunately, as the number of ranges increases, the number of terms in the summation in~(\ref{eqn:PDF_bivariate}) will grow exponentially.
The impact of bivariate model parameters on implied volatilities will be analyzed further in Section~\ref{sec:numericalExperiments}.
\section{Pricing under the randomized Hull-White (rHW) model}
\label{sec:pricingUnder_rHW}
This section focuses on the pricing under the rHW model. The presented pricing equations utilize Theorem~\ref{thm:rHW_generic}. Throughout the section, we denote by $V_{{\rm HW}}(t,r(t;\theta))$ with $\theta\in\{\eta,\lambda\}$, the Hull-White price of a derivative depending on the short-rate $r(t;\theta)$ with parameters $\lambda$ and $\eta$, by $V_{{\rm rHW}}(t,r(t;\vartheta))$ we denote the pricing under the rHW model.
\subsection{Swaptions under rHW model}
While in Equation~(\ref{eqn:rHW_Generic}), there is a relationship between HW and rHW prices, here explicit pricing formulas will be given. The valuation of swaption contracts in the rHW model can be carried out at the same complexity level as in the standard HW model.
The critical ingredient for pricing swaptions under the rHW model is an analytical expression for the options of the {\it pure} ZCB. Lemma~\ref{lem:optionsZCB} provides the pricing equations under the rHW model.
\begin{lem}[Pricing of options on a ZCB under the rHW model]
\label{lem:optionsZCB}
Under the randomized HW model parameters, $\theta:=\eta$ or $\theta:=\lambda$, with $\vartheta(\omega)=\theta$, the price of a European-style option on a ZCB, $P(T,S)$, with option's expiry, $T$, strike, $K$, and maturity of the underlying bond $S$, is given by:
\begin{eqnarray}
\label{eqn:optionZCB_rHW}
V^{\text{Z}}_{\chi}(t,T,S,K;\vartheta)=\sum_{n=1}^N\omega_nV^{\text{Z}}_{\chi}(t,T,S,K;\theta_n),
\end{eqnarray}
where,
\begin{eqnarray}
\nonumber
V^{\text{Z}}_{\chi}(t,T,S,K;\theta_n)&=&\E^\Q_{t}\left[\frac{M(t)}{M(T)}\max(\chi(P(T,S;\theta_n)-K),0)\right]\\
&=&\chi P(t,S)F_{\mathcal{N}(0,1)}(\chi d_n)-\chi KP(t,T)F_{\mathcal{N}(0,1)}(\chi(d_n-\bar\sigma_n)),
\label{eqn:ZCBOption}
\end{eqnarray}
with $P(t,S)$ and $P(t,T)$ the ZCB computed from the associated yield curve; $P(T,S;\theta_n)$ is the ZCB obtained from the model with the randomized parameter, $\vartheta$; $\chi=1$ and $\chi=-1$ corresponds to call and put options, respectively. Depending on the randomized parameter, we have:
\begin{eqnarray}
&&\text{for}\;\;\eta:\;\bar\sigma^2_n=\frac{\eta_n^2}{2\lambda}\left(1-\e^{-2\lambda(T-t)}\right)B^2(T,S;\lambda),\\
&&\text{for}\;\;\lambda:\;\bar\sigma^2_n=\frac{\eta^2}{2\lambda_n}\left(1-\e^{-2\lambda_n(T-t)}\right)B^2(T,S;\lambda_n),
\end{eqnarray}
with
\[d_n=\frac{1}{\bar{\sigma}_n}\log\frac{P(t,S)}{P(t,T)K}+\frac{\bar\sigma_n}{2},\;\;\;B(T,S;\lambda)=\frac{1}{\lambda}\left(1-\e^{-\lambda(S-T)}\right),\]
where the HW model is defined in~(\ref{eqn:HW_SDE}) and $F_{\mathcal{N}(0,1)}(\cdot)$ corresponds the standard normal CDF.
\begin{proof}
The proof is a direct consequence of combining Theorem~\ref{thm:rHW_generic} with the pricing of options on a ZCB under the HW model (as given in~\cite{BrigoMercurio:2007,OosterleeGrzelakBook}).
\end{proof}
\end{lem}
Now, we consider an interest rate swap with a fixed rate, $K$, and payment times $\mathcal{T} =\{T_i,T_{i+1},\dots, T_m\}$ and the corresponding reset rates $\{T_{i-1},T_{i+1},\dots, T_{m-1}\}$ with the payoff given by:
\[H^{Swap}_{\text{P/R}}(\mathcal{T},K)=\bar\alpha\sum_{k=i}^m\tau_k\left(\ell(T_{k-1};T_{k-1},T_{k})-K\right),\]
for $\tau_k=T_{k}-T_{k-1},$ with $P$ indicating a swap payer for $\bar\alpha=1$ and a swap receiver for $\bar\alpha=-1$ and where $\ell(t;T_{k-1},T_k)$ stands for the libor rate over the period $[T_{k-1},T_k]$ observed at time $t.$
To determine today's value of the swap, we evaluate the corresponding expectation of the \index{discounted cash flow} discounted future cash flows, i.e., each payment which takes place at the time points, $T_{i},\dots,T_m$, needs to be
discounted to today,
\begin{eqnarray}
\label{eqn:swap}
V_{\text{P/R}}^{\text{Swap}}(t,K,\mathcal{T})=\bar\alpha \sum_{k=i}^m\tau_k\E^\Q_t\left[\frac{M(t)}{M(T_k)}\big(\ell_k(T_{k-1})-K\big)\right]=\bar\alpha\sum_{k=i}^m\tau_kP(t,T_k)\big(\ell_k(t)-K\big),
\end{eqnarray}
with $\ell_k(t):=\ell(t;T_{k-1},T_k).$
We derive the valuation formula for a swaption contract with strike $K$ and option expiry $T=T_{i-1}$. Using~(\ref{eqn:swap}), the pricing equation reads:
\begin{eqnarray*}
V^{\text{Swpt}}_{\text{P/R}}(t,T,K,\mathcal{T})=\E^{\Q}_t\left[\frac{M(t)}{M(T)}\max\left(V_{\text{P/R}}^{\text{Swap}}(T,K,\mathcal{T}),0\right)\right]=P(t,T)\E^{T}_t\left[\max\left(V_{\text{P/R}}^{\text{Swap}}(T,K,\mathcal{T}),0\right)\right],
\end{eqnarray*}
where the expectation, $\E^T_t[\cdot]$, is taken under the $T-$forward measure and where the price of a swap price at $T=T_{i-1}$ is given by:
\begin{eqnarray*}
V_{\text{P/R}}^{\text{Swap}}(T,K,\mathcal{T})=\bar\alpha\left[1-P(T,T_m)-K\sum_{k=i}^m\tau_k
P(T_i,T_k)\right]=\bar\alpha-\bar\alpha\sum_{k=i}^mc_kP(T_i,T_k),\label{eqn:c_k_def}
\end{eqnarray*}
with $c_{k}=K\tau_k$ for $k=i,\dots,m-1$, $c_m=1+K\tau_m$, and $\tau_k=T_k-T_{k-1}$.
So, the pricing equation becomes:
\begin{eqnarray*}
V^{\text{Swpt}}_{\text{P/R}}(t,T,K,\mathcal{T})=\bar\alpha P(t,T)\E^{T}_t\left[\max\left(1-\sum_{k=i}^mc_kP(T_i,T_k),0\right)\right].
\end{eqnarray*}
Up to this point, the pricing equations do not depend on specific model choices. Now, however, we consider pricing under the rHW model, which, via the conditional expectation approach, as presented in Lemma~\ref{thm:rHW_generic}, yields:
\begin{eqnarray}
\label{eqn:swaptionInner}
V^{\text{Swpt}}_{\text{P/R}}(t,T,K,\mathcal{T};\vartheta)&=&\bar\alpha P(t,T)\E_t^T\left[\E^{T}_t\left[\max\Big(1-\sum_{k=i}^mc_kP(T_i,T_k),0\Big)\Big|\vartheta=\theta\right]\right]\\
&=&\bar\alpha P(t,T)\sum_{n=1}^N\omega_n\E^{T}_t\left[\max\Big(1-\sum_{k=i}^mc_kP(T_i,T_k;\theta_n),0\Big)\right]+\epsilon_N.\nonumber
\end{eqnarray}
Here, $\epsilon_N$ is the associated quadrature error defined in~(\ref{eqn:RAnDError}). Since the inner expectation in~(\ref{eqn:swaptionInner}) resembles the expression for a swaption under the classical HW, we follow the standard procedure and apply the so-called Jamshidian trick~\cite{Jamshidian:1989:ExactBond} to exchange the maximum operator and the expectation. The resulting pricing equations for swaptions under the rHW model are given in Lemma~\ref{lem:swaption_rHW}.
\begin{lem}[Pricing of Swaptions under randomized Hull-White model]
\label{lem:swaption_rHW}
Consider the rHW model, with parameters $\{\lambda,\eta\}$ and the randomizing random variable $\vartheta$, which randomizes either of the model parameters. For a unit notional, a constant strike, $K$, option expiry $T=T_{i-1}$ and a strip of swap payments $\mathcal{T} =\{T_i,\dots,T_m\}$, with $T_i>T_{i-1}$ and accruals $\tau_i=T_i-T_{i-1}$, the prices of swaption payer and receiver, $\text{P/R}:=\text{Payer/Receiver}$, are given by:
\begin{eqnarray}
\label{eqn:swaptionPricing_rHW}
V_{\text{P/R}}^{\text{Swpt}}(t_0,T,\mathcal{T},K;\vartheta)=\sum_{n=1}^N\omega_n\sum_{k=i}^mc_kV^{\text{Z}}_{\chi}(t_0,T,T_k,\hat{K}_k(\theta_n);\theta_n),
\end{eqnarray}
with a swaption payer, $\text{P}$, for $\chi=-1$, swaption receiver, $\text{R}$, with $\chi=1$, where $V^{\text{Z}}_{\chi}(\cdot)$ is defined in~(\ref{eqn:ZCBOption}) and where the strike price $\hat{K}_k(\theta_n)=\exp\left(A(T,T_k;\theta_n)+ B(T,T_k;\theta_n)r^*_n\right)$. Here, $r^*_n$ is determined by
solving, for each parameter realization $\theta_n$, the following equation:
\begin{equation}\label{eqn:Jamsh_eqn}
1-\sum_{k=i}^mc_k\exp\Big(A(T,T_k;\theta_n)- B(T,T_k;\theta_n)r^*_n\Big)=0,\;\;\;n=1,\dots,N,
\end{equation}
where
\begin{eqnarray*}
A(T,T_k;\{\lambda,\eta\})&=&\log\frac{P(0,T_k)}{P(0,T)}+ B(T,T_k;\{\lambda,\eta\})f(0,T)-\frac{\eta^2}{4\lambda}\left(1-\e^{-2\lambda T}B^2(T,T_k;\{\lambda,\eta\})\right),\\
B(T,T_k;\{\lambda,\eta\})&=&\frac{1}{\lambda}\left(1-\e^{-\lambda(T_k-T)}\right),
\end{eqnarray*}
with $c_{k}=K\tau_k$ for $k=i,\dots,m-1$, $c_m=1+K\tau_m$ and where the pairs $\{\omega_n,\theta_n\}$, $n=1,\dots,N$, are based on the randomizer $\vartheta$ and computed using~\ref{res:zeta}.
\end{lem}
Under the rHW model, the pricing Equation~(\ref{eqn:swaptionPricing_rHW}) for swaptions is a direct application of Theorem~\ref{thm:rHW_generic}. The pricing under randomized parameters is simply an {\it average} of non-randomized prices, with varying model parameters, accompanied by weights based on the randomizer.
From the computational perspective, the swaption pricing in~(\ref{eqn:swaptionPricing_rHW}) requires $N$ swaption prices for different realizations $\theta_n$, $n=1,\dots,N$; therefore, $N$ optimization problems in~(\ref{eqn:Jamsh_eqn}) need to be solved. This computational complexity can be greatly reduced by employing the multi-d Newton-Raphson algorithm to determine optimal $r^*_n$ for $n=1,\dots,N.$
\begin{rem}[Computation of sensitivities]
The computation of sensitivities under the RAnD method is straightforward, i.e., because the pricing is expressed as a convex combination, the sensitivities are expressed as a weighted sum of individual derivatives. For example, in the case of the swaption pricing, the sensitivity to a particular market quote $q$ is expressed as a sum of sensitivities of individual options on ZCBs:
\begin{eqnarray}
\frac{\partial}{\partial q}V_{\text{P/R}}^{\text{Swpt}}(t_0,T,\mathcal{T},K;\vartheta)=\sum_{n=1}^N\omega_n\sum_{k=i}^Mc_k\frac{\partial}{\partial q}V^{\text{Z}}_{\chi}(t_0,T,T_k,\hat{K}_k(\theta_n);\theta_n),
\end{eqnarray}
with the specification as given in Lemma~\ref{lem:swaption_rHW}.
On the other hand, the sensitivity to the quadrature pairs, $\{\omega_n,\theta_n\}$, $\partial \theta_n/\partial \hat a$ and $\partial \omega_n/\partial \hat a$ may be, for some specific cases, computed analytically (see Section~\ref{sec:calibration_MarketData}). Still, in a generic setting, it is recommended to compute these derivatives numerically, with, for example, finite differences:
\[\frac{\zeta(\vartheta(\hat a+\delta_{\hat a}))-\zeta(\vartheta(\hat a-\delta_{\hat a}))}{2\delta_{\hat a}}\approx \left\{\frac{\partial \omega_n}{\partial \hat a},\frac{\partial \theta_n}{\partial \hat a}\right\},\]
where $\vartheta(\hat a)$ indicates the dependence of the random variable and parameter $\hat a$, $\delta_{\hat a}$ is the {\it shock} size and $\zeta(\vartheta):\R\rightarrow \{\omega_n,\theta_n\}_{n=1}^N$ is defined in~\ref{res:zeta}. Due to the applied finite difference shocks to $\hat a$, an additional bias will be introduced. We expect, however, this error to be of acceptable magnitude, as is commonly observed in current financial practice.
\end{rem}
\section{Numerical experiments}
\label{sec:numericalExperiments}
In this section, several pricing experiments will be performed. First, we present a detailed analysis of the rHW model in realistic pricing scenarios. In the first experiment, we explore the implied volatility smile evolution in time, comparing the implied volatility surface from the HW and rHW models. As the next step, the study of the parameter randomization on shapes of implied volatilities will be illustrated. Finally, the calibration results with market data will be presented in the conclusive experiment. This section will end with numerical experiments involving bivariate distribution for the model parameters.
\subsection{Evolution of implied volatilities}
\label{sec:evolutionIR}
We analyze the swaption implied volatilities observed in the market and compare them to implied volatilities generated using the HW and rHW models. We focus here on the flexibility in generating realistic implied volatility shapes. Throughout the section, we will consider the {\it shifted} implied volatilities computed by inverting Black's formula, which, for $\bar\alpha\in\{1,-1\}$ for call and put options, respectively, reads:
\begin{eqnarray}
\label{eqn:BS}
V_{\text{B}}(T,K,F_0,\sigma_s,\bar\alpha,s)&=&\bar\alpha\cdot (F_0+s) \cdot F_{\mathcal{N}(0,1)}(\bar\alpha d_1)-\bar\alpha (K+s)F_{\mathcal{N}(0,1)}(\bar \alpha d_2),\\\nonumber
d_1&=&\frac{1}{\sigma\sqrt{T}}\left[\log (F_0+s)/(K+s)+1/2\sigma_s^2T\right],\\\nonumber
d_2&=&d_1-\sigma_s\sqrt{T},
\end{eqnarray}
with shift parameter $s$, $F_0$ being the forward rate, $\sigma_s$ is the corresponding volatility coefficient, $K$ being the strike, and $T$ corresponds to the time to option expiry.
Then, the shifted Black's formula for swaptions is given by:
\begin{eqnarray}
\label{eqn:swaptionBlack}
V^{\text{B,Swpt}}_{P/R}=V_{\text{B}}(T,K,S(t_0),\sigma_s,\bar\alpha,s)\sum_{i=1}^m\tau_iP(t_0,T_i),\;\;\;S(t_0)=\frac{P(t_0,T_{i-1})-P(t_0,T_m)}{\sum_{i=1}^m\tau_iP(t_0,T_i)},
\end{eqnarray}
with swaption payer, $P$, for $\bar\alpha=1$ and swaption receiver, $R$, for $\bar\alpha=-1$ and where $S(t_0)$ is the corresponding swap rate. The shift parameter, $s$, is typically perceived as the {\it lower bound} for the interest rates by the market participants and varies depending on the currency. It is important to note that when inverting the Black's formula in~(\ref{eqn:swaptionBlack}), the corresponding implied volatility $\sigma_s$ is a function of the shift parameter, i.e., for different choices of $s$, the implied volatilities are different. In principle, this is not a problem as long as the implied volatilities from the market and model volatilities are computed with the same shift coefficient.
In the first experiment, we compare the implied volatility surfaces of the HW model and the rHW model. In the rHW model, the speed of mean-reversion is randomized by a uniform distribution, $\lambda\sim\mathcal{U}([\hat{a},\hat{b}])$, on an interval $[\hat{a},\hat{b}].$ The numerical results are illustrated in Figure~\ref{fig:3D}. The results demonstrate that the HW model can only generate implied volatility skew. At the same time, the randomization of the mean-reversion parameter, $\lambda$, shows implied volatility skew and smile. Although the randomization is not {\it time-dependent}, i.e., the parameters are stochastic but stationary, we observe a time evolution of the implied volatilities. The same phenomenon has been observed in~\cite{brigo2002lognormal,grzelakRAnD}, where the randomized Black-Scholes model was considered.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/3d_HW.pdf}
\includegraphics[width=0.45\textwidth]{Figures/3d_rHW.pdf}
\caption{Swaption volatility evolution for the HW and rHW models implied by the shifted Black's model. The simulation was performed for varying swaption option expiry, $T$, and a fixed tenor of $1y$. The parameters specified in the experiment are: for the HW model: $\eta=0.005$, $\lambda=0.001$ and for the rHW model: $\eta=0.005$ and $\lambda~\sim\mathcal{U}([-0.15,0.6]).$ In the experiment, the implied volatilities are computed with zero shift parameter, $s=0.$ }
\label{fig:3D}
\end{figure}
The reported numerical results are promising. By a stochastic mean-reversion parameter, the HW model has an additional degree of freedom that improves the model's flexibility. In the following sections, we will analyze the randomization effect of all model parameters and the different choices of randomizing distributions.
\subsection{Randomization and impact on implied volatilities}
\label{sec:impact_on_IV}
Here, we study the impact of different randomization choices on the implied volatility shape. Understanding how parameters affect the generated volatilities is crucial in model calibration. A clear relationship between the model parameters and the associated skew, curvature and volatility level is fundamental, enabling traders to react swiftly to market movements. It is also the basis for setting up a calibration routine when a particular market movement can be encapsulated in a parameter change.
In the first experiment, we analyze the randomization of volatility parameter $\eta$. We consider two randomization cases where $\eta$ either follows a uniform or a normal distribution. The details regarding the computation of the associated moments and the corresponding quadrature points can be found in~\ref{sec:appendix}. The experiment is set up so that both approaches' mean values, $\E[\eta]$, are equal.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/impact_eta.pdf}
\includegraphics[width=0.45\textwidth]{Figures/impact_eta_normal.pdf}
\caption{Impact of randomized volatility parameter $\eta$ on implied volatilities with a fixed speed of mean-reversion parameter $\lambda=0.009.$ Left: $\eta\sim\mathcal{U}([\hat{a},\hat{b}])$. Right: $\eta\sim\mathcal{N}(\hat{a},\hat{b}).$}
\label{fig:impact_1}
\end{figure}
The results are presented in Figure~\ref{fig:impact_1}, where it is shown that the randomization of the volatility parameter $\eta$ has a pronounced effect on the level of implied volatility with minimal effect on the smile. The curvature is visible for uniform randomization. It is essential to note that a higher variance for the normal distribution may give rise to a higher curvature but may also cause issues related to negative volatilities. Our experiments have shown that even for distributions defined in the positive domain, the impact on the smile is limited, even for fat-tailed random variables.
A much richer spectrum of implied volatility shapes is obtained when the randomization technique is applied to the mean-reversion parameter $\lambda$. Figure~\ref{fig:impact_2} presents the randomization with either uniform or normal random variables. A substantial amount of curvature can be generated by taking the mean-reversion random. We also report that the curvature change affects the overall volatility level, i.e., it is impossible to keep the level fixed and only adjust the smile. However, the implied volatility level can be fixed by adjusting the volatility parameter $\eta$. This strategy will be discussed further in the context of model calibration.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/impact_lambda.pdf}
\includegraphics[width=0.45\textwidth]{Figures/impact_lambda_normal.pdf}
\caption{Impact of randomized volatility parameter $\lambda$ on implied volatilities with a fixed volatility parameter $\eta=0.0050.$ Left: $\lambda\sim\mathcal{U}([\hat{a},\hat{b}])$. Right: $\lambda\sim\mathcal{N}(\hat{a},\hat{b}).$}
\label{fig:impact_2}
\end{figure}
In the final experiment of this section, we consider the randomization of $\lambda\sim\mathcal{U}([\hat{a},\hat{b}])$ using uniform distribution and check how the parameters $\hat{a}$ and $\hat{b}$ affect the implied volatilities. In Figure~\ref{fig:impact_3}, the results show an interesting pattern: the curvature level is mainly driven by the distance $|\hat{b}-\hat{a}|$, i.e., the larger the distance, the more implied volatility smile is generated. Changes of either of the parameters affects the volatility level; therefore, some of the volatility effect, $\eta$, can be offset by the interval $[\hat{a},\hat{b}].$
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/impact_lambda_fixed_a.pdf}
\includegraphics[width=0.45\textwidth]{Figures/impact_lambda_fixed_b.pdf}
\caption{Impact of randomized volatility parameter $\lambda\sim\mathcal{U}([\hat{a},\hat{b}])$ on implied volatilities with a fixed volatility parameter $\eta=0.0050$. Left: varying parameter $\hat{a}$. Right: varying parameter $\hat{b}.$}
\label{fig:impact_3}
\end{figure}
Given the numerical results presented above, we conclude that the most considerable impact on implied volatilities comes from the mean-reversion, $\lambda$, and not from the volatility parameter, $\eta.$ This is particularly interesting when we confront these results with Proposition~\ref{prop:rand_eta} and Proposition~\ref{prop:rand_lambda}, where the dynamics of the corresponding short-rate processes were derived and where it was shown that randomization of the mean-reversion did not lead to the local volatility type of dynamics. In contrast, the randomization of $\eta$ does, which can be explained by the nature of interest rate derivatives, where derivative prices are driven by the dynamics of the ZCBs, but not directly by the short-rate process. However, since under the HW model, the volatility of the ZCBs is given by both parameters (see~\cite{OosterleeGrzelakBook}), the randomization of either of them will imply a local volatility type of dynamics for the ZCBs.
\subsection{Calibration of the randomized Hull-White model}
\label{sec:calibration_MarketData}
Every model that may be considered successful must also show its ability to calibrate to the market quotes. Moreover, the calibration and pricing process needs to be computationally efficient, especially when considering the pricing of large portfolios, for example, in the context of xVA. Therefore, this section focuses on calibrating the rHW model to swaption implied volatilities.
Calibration of model parameters requires multiple iterations over the parameter space until specific optimization criteria are met. This implies that the pricing needs to be repeated at every choice of parameter candidate. By randomization of the model parameters, the number of free parameters will increase. The additional degrees of freedom will depend on the type of randomizer chosen. Let us consider, for example, parameter randomization with $\vartheta(\hat a,\hat{b})$, which is driven by two model parameters $\hat a$ and $\hat b$. Two parameters of the randomizer $\vartheta$ correspond to one additional degree of freedom that can be used for model calibration.
From the computational perspective, using the RAnD method requires that for every iteration step in the calibration procedure, the mapping between the randomizing variable $\vartheta$ and the corresponding pairs $\{\omega_n,\theta_n\}$, $n=1,\dots,N$, needs to be employed. Although the algorithm given in~\ref{res:zeta} is straightforward (it mainly depends on the computation of eigenvalues), it can be further simplified. When randomizing variables can be expressed as a linear combination of some {\it base} random variable, for example, normal or uniform, the calculations of the weights and the corresponding nodes can be significantly simplified. In the case of a normal randomizer, $\vartheta\sim\mathcal{N}(\hat a,\hat b^2)$, we can benefit from the linearity of the normal distribution. For the standard normal and its associated points, we have $\theta_n=\hat a + \hat b \cdot \theta_{\mathcal{N},n}$, where $\theta_{\mathcal{N},n}$ are the nodes corresponding to standard normal, $\mathcal{N}(0,1).$ This implies that we can simply {\it tabulate} the results for the standard normal and scale the points accordingly. In the case of the weights $\omega_n$, they stay invariant to a linear transformation~\cite{scmc2019}. A similar property holds for $\vartheta\sim\mathcal{U}([\hat a,\hat b])$, where the nodes can be computed for $\vartheta\sim\mathcal{U}([0,1])$ and scaled appropriately.
In the calibration experiment, we consider the market data for the USD market as of the 18th of August, 2022. In all calibration exercises, we consider a fixed tenor of $1y$ and analyze the accuracy for varying expires and strikes. The calibrated model parameters for both HW and rHW models are tabulated in Table~\ref{Tab:Calibration}. The results are intriguing, i.e., in the randomization for $\lambda$, we were able to calibrate all swaptions while keeping the mean of the randomizer fixed at $0.1$. This shows that having more degrees of freedom is not necessary to improve the calibration results. As in the standard HW model, we have only used two parameters.
\begin{table}[htb!]
\centering\footnotesize
\caption{\footnotesize Calibration of the HW and rHW model: parameters determined in swaption calibration. }
\begin{tabular}{c|c|c||c|c}
\multicolumn{1}{c}{}&\multicolumn{2}{c||}{$\text{Hull-White}$}&\multicolumn{2}{c}{$\text{RAnD Hull-White}$}\\\hline
$T$, expiry &$\eta$&$\lambda$&$\eta$&$\lambda$\\\hline\hline
$1y$&0.0094&0.0090&0.0091&$\lambda\sim\mathcal{N}(0.1,0.45^2)$\\
$2y$&0.0082 &0.0035 & 0.0080&$\lambda\sim\mathcal{N}(0.1,0.33^2)$\\
$5y$&0.0069&0.0020&0.0079&$\lambda\sim\mathcal{N}(0.1,0.16^2)$\\
$8y$&0.0067&0.0095&0.0080&$\lambda\sim\mathcal{N}(0.1,0.12^2)$\\
$10y$&0.0067&0.0090&0.0082&$\lambda\sim\mathcal{N}(0.1,0.11^2)$\\
$15y$&0.0064&0.0080&0.0085&$\lambda\sim\mathcal{N}(0.1,0.09^2)$\\
$20y$&0.0060&0.0080&0.0086&$\lambda\sim\mathcal{N}(0.1,0.08^2)$
\end{tabular}
\label{Tab:Calibration}
\end{table}
The calibration fit is presented in Figures~\ref{fig:calib_1},~\ref{fig:calib_2} and~\ref{fig:calib_3}. We report an excellent calibration fit for all considered option expiries, varying from 1y to 20y. We have used two parameters in all the calibration cases, just as in the HW model. The results confirm that the RAnD method has great potential for improving existing pricing methods, even with the same number of degrees of freedom.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/calib_1y_2.pdf}
\includegraphics[width=0.45\textwidth]{Figures/calib_2y_2.pdf}
\caption{Calibration results of the HW and the rHW models. The market implied volatilities for swaptions were obtained on 18/08/2022 for the USD market. Option expiry: $T=1y$ and $T=2y$ and the implied volatility shift: $s=1\%$. Calibrated parameters are presented in Table~\ref{Tab:Calibration}.}
\label{fig:calib_1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/calib_5y_2.pdf}
\includegraphics[width=0.45\textwidth]{Figures/calib_10y_2.pdf}
\caption{Calibration results of the HW and the rHW models. The market implied volatilities for swaptions were obtained on 18/08/2022 for the USD market. Option expiry: $T=5y$ and $T=10y$ and the implied volatility shift: $s=1\%$. Calibrated parameters are presented in Table~\ref{Tab:Calibration}. }
\label{fig:calib_2}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/calib_15y_2.pdf}
\includegraphics[width=0.45\textwidth]{Figures/calib_20y_2.pdf}
\caption{Calibration results of the HW and the rHW models. The market implied volatilities for swaptions were obtained on 18/08/2022 for the USD market. Option expiry: $T=15y$ and $T=20y$ and the implied volatility shift: $s=1\%$. Calibrated parameters are presented in Table~\ref{Tab:Calibration}. }
\label{fig:calib_3}
\end{figure}
\subsection{Pricing under bivariate distributions for the model parameters}
\label{sec:pricingBivariate}
This section extends the rHW model and considers a bivariate distribution for both model parameters. Under a bivariate distribution $\Theta=[\vartheta_1,\vartheta_2]$ with $\zeta(\vartheta_1)=\{\omega_{1,n},\theta_{1,n}\}_{n=1}^N$ and conditioned on $\zeta(\vartheta_2|\vartheta_1)=\{\omega_{2,m},\theta_{2,m}\}_{m=1}^M$, the randomized prices are:
\begin{eqnarray}
V_{{\rm rHW}}(t,r(t);\vartheta_1,\vartheta_2)=\sum_{n=1}^{N}\omega_{n}\sum_{m=1}^{M}\omega_{m}V_{{\rm HW}}(t,r(t);\eta_n,\lambda_m)+\epsilon_{N,M}^b,
\end{eqnarray}
where $N$ and $M$ indicate the number of expansion terms for $\vartheta_1$ and $\vartheta_2|\vartheta_1$, respectively, $\vartheta_2|\vartheta_1$ indicates a conditional random variable, $\epsilon_{N,M}^b$ is the corresponding aggregated error. The remaining specification follows Theorem~\ref{cor:RAnDChF}.
As an example, let us consider a bivariate normal distribution for the pair $(\vartheta_1,\vartheta_2)$ with the corresponding realizations $(\eta,\lambda)$ for which we have:
\begin{eqnarray}
\vartheta_2|\vartheta_1=\eta_{n}\sim\mathcal{N}\left(\mu_{\lambda}+\frac{\sigma_\lambda}{\sigma_{\eta}}\rho(\eta_n-\mu_{\eta}),(1-\rho^2)\sigma_\lambda^2\right),
\end{eqnarray}
where $\rho$ is the correlation coefficient between $\vartheta_1$ and $\vartheta_2$.
In Figure~\ref{fig:impact_correlation}, we illustrate the impact of the correlation coefficient $\rho$ on the swaption implied volatilities. We report that similar to, e.g., the Heston model, the correlation controls the implied volatility skew. Furthermore, we observe that a higher positive correlation generates more skew, while a negative correlation generates more volatility curvature (smile).
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/impact_rho_1.pdf}
\includegraphics[width=0.45\textwidth]{Figures/impact_rho_2.pdf}
\caption{Impact of randomized volatility parameters $\eta$ and $\lambda$ on implied volatilities driven by bivariate normal distribution. Results are presented for varying correlation, $\rho$. Left: $\eta\sim\mathcal{N}(0.008,0.002^2)$ and $\lambda\sim\mathcal{N}(0.5,0.05^2)$. Right: $\eta\sim\mathcal{N}(0.01,0.002^2)$ and $\lambda\sim\mathcal{N}(0.5,0.2^2)$. }
\label{fig:impact_correlation}
\end{figure}
\begin{rem}[Volatility term structure and feasible strategy for model calibration]
Commonly, the volatility-parameter $\eta$ is piece-wise constant, so the ATM volatilities are well calibrated. This strategy will also work with the randomized mean-reversion parameter, i.e., the mean-reversion parameter can be used for smile/skew calibration, while the volatility parameter, $\eta$, will ensure a proper fit of the ATM level.
\end{rem}
\subsection{Convergence results}
\label{sec:convergence}
As presented in Theorem~\ref{thm:rHW_generic}, applying the RAnD method produces a quadrature error. In this section, we analyze the convergence of the error depending on the number of expansion terms, $N$.
The convergence speed depends on the distribution and its parameters, as presented in Figure~\ref{fig:convergence_N}. We report an excellent convergence factor; however, as expected, it depends on the variance of the randomizing random variable. Both cases will achieve satisfactory results for already $N=5$. Randomization using different random variables showed equivalent patterns of convergence.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Convergence_eta.pdf}
\includegraphics[width=0.45\textwidth]{Figures/convergence_lambda.pdf}
\caption{Convergence results for randomization of volatility parameter, and mean reversion parameter $\eta$ and $\lambda$ under the HW model respectively. The base parameters in the experiment were $\eta=0.00625$ and $\lambda= 0.002$.}
\label{fig:convergence_N}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have applied the randomization technique to enhance the flexibility of short-rate models for interest rates. We have shown that the model parameters driven by a random variable, instead of being deterministic, facilitate a practical extension of standard, well-popularized models. In addition, this article points out how the normal mixture (a sum of the Hull-White PDFs) can be expressed as a one-dimensional diffusion process with a local volatility function.
We have illustrated that, for the randomized Hull-White model (rHW), one can utilize the available closed-form pricing equations and benefit from flexibility in controlling implied volatilities. In particular, we have shown that the rHW model results in almost perfect swaption calibration.
Finally, the RAnD method is generic and is not limited to any particular modelling choice; therefore, it opens many possibilities for improving existing pricing frameworks.
\bibliographystyle{abbrv}
|
1,314,259,995,828 | arxiv | \section{Proof of the First Zonklar Equation}
\end{document}
\section{Acknowledgement}
The work is supported by Secure \& Trustworthy Intelligence Laboratory (STiL) established by Prof. Kai Zhou.
\section{Conclusion}
\label{sec-conclude}
In this paper, we initiate the study to investigate the vulnerability of graph-based anomaly detection under structural poisoning attacks. Specially, we target on $\mathsf{OddBall}$ as feature extraction based anomaly detector and $\mathsf{LGCN}$ as the surrogate model of the GCN-based anomaly detector, we further mathematically formulate it as a discrete one-level optimization problem by incorporating specially designed regression techniques (OLS estimation for $\mathsf{OddBall}$ and RWLS estimation for $\mathsf{LGCN}$). We then propose a novel training procedure $\mathsf{BinarizedAttack}$ which effectively damages the target GAD systems, significantly outperforming existing methods. Moreover, we explore the black-box attacks of the typical six GCN-based GAD systems (GCN-reweight, GAT-reweight, FdGars, GEM, Player2vec and GraphSMOTE) and prove that by universally attacking $\mathsf{LGCN}$ can effectively transfer attack other GCN-based GAD systems in black-box manner. In the future work, it is interesting to explore the possible countermeasure against such poisoning attacks on graph data to enhance the robustness of the existed graph anomaly detectors.
\section{Possible Countermeasures}
\label{sec-defense}
Defense approaches against data poisoning attacks have been extensive studied mainly in the computer vision domain. Typical methods include sanitizing the dataset by identifying and removing the maliciously injected data points, training robust models that could still perform well on poisoned data, and so on. More recently, there are also works that explore defense approaches in the graph learning domain.
For example, based on the observation that $\mathsf{netattack}$ \cite{Z_gner_2018} tends to increase the rank of the adjacency matrix, \cite{10.1145/3336191.3371789} proposed the low-rank approximation of the adjacency matrix by SVD for defense. Graph variational autoencoder is used in \cite{DBLP:journals/corr/abs-2006-08900} to generate a smooth adjacency matrix to defend two popular attacks $\mathsf{metattack}$\cite{DBLP:journals/corr/abs-1902-08412} and $\mathsf{netattack}$. A robust GCN~\cite{kipf2017semisupervised} model is proposed in \cite{zhu2019robust} by formulating the Gaussian distribution based on the hidden features of GCN and penalizing the high variance to enhance the robustness. \cite{DBLP:journals/corr/abs-1908-07558} and \cite{DBLP:journals/corr/abs-2006-08149} introduced the attention mechanism to decrease the effect of adversarial links in the attacked graph.
In this paper, we further explore possible countermeasures to defend against $\mathsf{BinarizedAttack}$. Our key observation is that, since OLS \cite{Zdaniuk2014} estimation is sensitive to poisoned data in the training process, we can use robust estimators instead to estimate the regression parameters. Huber \cite{10.1214/aoms/1177703732} proposed the Huber loss function to replace the mean square loss:
\begin{equation}
\rho_{Huber}(t)=\left\{
\begin{aligned}
&\frac{1}{2}t^{2},\quad \text{if} \ |t|\leq k, \\
&k|t|-\frac{1}{2}k^{2}, \quad \text{if} \ |t| > k.
\end{aligned}
\right.
\end{equation}
where $k$ is the hyper-parameter to penalize the outliers linearly instead of quadratically.
Random Sample Consensus (RANSAC \cite{ransac}) uses Huber loss with $k=1$ for robust estimation.
It was known that RANSAC is able to estimate the model parameters that achieve high accuracy even when the dataset contains a significant number of outliers. In light of this, we adopt the robust estimation method from RANSAC to estimate the parameters of $\mathsf{OddBall}$ (thus a robust version of $\mathsf{OddBall}$) from the poisoned graph generated by $\mathsf{BinarizedAttack}$. Our experiments show that this robust estimation approach can slightly mitigate the attack effect. The defense of structural poisoning attack remains as an important future work.
\section{Experiments}
\label{sec-exp}
\subsection{Datasets and Experiment Settings}
\subsubsection{Datasets} \textbf{BA} (Barabasi-Albert)~\cite{BA_graph} is a graph generative model which incorporate the idea of preferential attachment on the links probability. we set the number of edges attaching from a new node to existing nodes is $5$. \textbf{Blogcatalog}~\cite{zafarani2014users} is a social network indicating follower/followee relationships among bloggers in a blog sharing website. The network has 88,800 nodes and 2.1M edges. \textbf{Wikivote}~\cite{leskovec2010predicting} contains all the Wikipedia voting data from the inception of Wikipedia till January 2008. Nodes in the network represent Wikipedia users and the edge represents that user $i$ voted on user $j$. The dataset contains around 7000 nodes and 0.1M edges. \textbf{Bitcoin-Alpha}~\cite{kumar2018rev2} is a who-trusts-whom social network of traders who trade Bitcoin on the \textbf{Bitcoin-Alpha} platform. It is a weighted signed network with weights ranging from $+10$ to $-10$. It has more than 3000 nodes and 24186 edges. We pre-process the data by removing all the negative edges and erasing the weights of all the remaining edges, resulting in an unsigned unweighted version of \textbf{Bitcoin-Alpha}. For \textbf{Wikivote} and \textbf{Bitcoin-Alpha} we randomly sample the connected sub-graph with around 1000 nodes from the whole graph. \textbf{Citeseer} \cite{citeseer}, \textbf{Cora} \cite{cora} \textbf{Cora-ML} \cite{cora_ml} are the citation networks which well capture the citation relations and co-authors relations. \textbf{ca-GrQc} \cite{cagrqc} is a collaboration network from the e-print arXiv and covers scientific collaborations between authors papers submitted to General Relativity and Quantum Cosmology category. For all the real-world graph data we pick out the largest connnected component to prevent the present of the isolated nodes. The statistics of the real datasets are summarized in Tab.~\ref{table-datasets}. Here \textbf{Citeseer-Wo} means we only consider the structural information of \textbf{Citeseer} dataset.
\begin{table}[h]
\centering
\caption{Statistics of datasets.}
\label{table-datasets}
\resizebox{0.8\columnwidth}{!}{%
{\begin{tabular}{|c|c|c|c|c|}
\hline
Dataset & \# Nodes & \# Edges & \# Attributes & \# Anomalies\\
\hline
\hline
\textbf{BA} & $1000$ & $4975$ & NA & NA \\
\textbf{Wikivote} & $1012$ & $4860$ & NA & NA\\
\textbf{Bitcoin-Alpha} & $1025$ & $2311$ & NA & NA\\
\textbf{Cora-ML} & $2810$ & $7981$ & NA & NA\\
\textbf{Citeseer-Wo} & $3327$ & $4732$ & NA & NA\\
\textbf{ca-GrQc} & $5242$ & $14496$& NA & NA\\
\hline
\textbf{Cora} & $2708$ & $5803$ & $1433$ & $150$\\
\textbf{Citeseer} & $3327$ & $4732$ & $3703$ & $150$\\
\textbf{Blogcatalog} & $5196$ & $172652$ & $8189$ & $300$\\
\hline
\end{tabular}}
}
\end{table}
\subsubsection{Setting}
For datasets in Tab.~\ref{table-datasets}, we analyze the structural poisoning attacks against $\mathsf{OddBall}$ with the first six datasets. We determine our target set by sampling 10 target nodes from the top-$50$ nodes based on $\mathsf{AScore}$ (From now on we denote anomaly score as $\mathsf{AScore}$ for simplicity) rankings. For each experiment, we sample target nodes 5 times individually and report the mean values to measure the efficacy of the structural attack on $\mathsf{OddBall}$. For structural attack (add/delete edges) on node anomaly detection, we aim to minimize the $\mathsf{AScore}$ sums for target set $V$ under different budget constraints. We determine the max budget $B$ for each case by the convergence of $\mathsf{AScore}$ for the three structural attacks. We stop attacking until the changes of $\mathsf{AScore}$ saturated.
We explore the structural poisoning attacks against the $\mathsf{LGCN}$ with the last three datasets in Tab.~\ref{table-datasets}. We synthetically inject structural anomalies into these datasets and flag the perturbed nodes as anomalous. Referring to \cite{skillicorn2007detecting}, we randomly choose some nodes in the clean graph and inject cliques to make them fully connected. We split the dataset into training and testing part, the ratio of training to testing is $9:1$. We split the dataset $5$ times and report the mean AUC scores for model comparison.
\label{sec-case-study}
\subsection{Attack Performance}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/BA_5.pdf}
\caption{\textbf{BA}}
\label{fig:blogcatalog}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/target_bitcoin_alpha_rand10.pdf}
\caption{\textbf{Bitcoin-Alpha}}
\label{fig:bitcoin_alpha}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/target_wikivote_rand10.pdf}
\caption{\textbf{Wikivote}}
\label{fig:wikivote}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/target_citeseer_rand10.pdf}
\caption{\textbf{Citeseer-Wo}}
\label{fig:citeseer}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/target_cora_ml_rand10.pdf}
\caption{\textbf{Cora-ML}}
\label{fig:cora_ml}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/target_ca_grqc_rand10.pdf}
\caption{\textbf{ca-GrQc}}
\label{fig:ca_grqc}
\end{subfigure}
\caption{Mean changes in $\mathsf{AScores}$ for \textbf{BA}, \textbf{Bitcoin-Alpha}, \textbf{Wikivote} and \textbf{Citeseer-Wo}, \textbf{Cora-ML} and \textbf{ca-GrQc}.}
\label{fig-main-exp}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/add_edges.pdf}
\caption{Case 1: Add edges}
\label{fig:y equals x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/delete_edges.pdf}
\caption{Case 2: Delete edges}
\label{fig:three sin x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/mixed_edges.pdf}
\caption{Case 3: Add/Delete edges}
\label{fig:five over x}
\end{subfigure}
\caption{$\mathsf{BinarizedAttack}$ can add, delete, and both add and delete edges in the graph.}
\label{fig-visual}
\end{figure*}
\subsubsection{Effectiveness of attack $\mathsf{OddBall}$} Our main focus is to investigate
whether the proposed attack methods can effectively decrease $\mathsf{AScores}$ of the target nodes. We use the Decreasing
Percentage of $\mathsf{AScore}$ denoted as $\tau_{as}$ as the evaluation metric. Specifically, let $S_{\mathcal{T}}^0$ and $S_{\mathcal{T}}^B$ be the sum of $\mathsf{AScores}$ of the target nodes before attack and after an
attack with a budget $B$, respectively. Then, $\tau_{as}$ is defined as
$\tau_{as} = (S_{\mathcal{T}}^0 - S_{\mathcal{T}}^B) / S_{\mathcal{T}}^0$.
Fig.~\ref{fig-main-exp} presents the $\tau_{as}$ of the three attack methods,
$\mathsf{BinarizedAttack}$, $\mathsf{GradMaxSearch}$, and $\mathsf{ContinuousA}$, with varying attack power. In particular, the attack power is measured as a percentage $\frac{B}{|E|}$, where $|E|$ is the number of edges in the clean graph. We note that in all the cases, the attacker
has modified very limited edges in the graph: less than $2\%$
when $|V | = 10$.
We can observe from Fig.~\ref{fig-main-exp} that both $\mathsf{BinarizedAttack}$
and $\mathsf{GradMaxSearch}$ can significantly (up to $90\%$) decrease $\mathsf{AScores}$ with very limited power while
$\mathsf{ContinuousA}$ is not effective in several cases. This demonstrated the unpredictable feature of $\mathsf{ContinuousA}$ when converting continuous solutions to discrete ones. Our main method $\mathsf{BinarizedAttack}$ consistently achieves the best performance in
all cases. We note that the margin between $\mathsf{BinarizedAttack}$ and $\mathsf{GradMaxSearch}$ is significant (with a $> 10\%$ performance
improvement) when the attack power is high (although the two lines look close in the figures). For example, in Fig.~\ref{fig:cora_ml},
when the attack power is $0.5\%$, $\mathsf{BinarizedAttack}$ outperforms
$\mathsf{GradMaxSearch}$ by $41.01\%$. One intriguing observation is
that the gap between $\mathsf{BinarizedAttack}$ and $\mathsf{GradMaxSearch}$
becomes larger when we increase the attacker’s budget. The
reason is that $\mathsf{GradMaxSearch}$ is a greedy strategy in nature
and should performs well when modifying a few edges;
meanwhile, it is also myopic when the budget is large.
The computational bottleneck of $\mathsf{BinarizedAttack}$ lies in computing the egonet edge features $E_{i}$ for all nodes, which involves calculating $\mathbf{A}^3$. We use two techniques to allow $\mathsf{BinarizedAttack}$ to scale to large graphs. First, we use the sparse matrix representation of the adjacent matrix $\mathbf{A}$ and implement all the the computations based on sparse matrix techniques. Empirically, we observe that, on the largest dataset \textbf{ca-GrQc}, using sparse matrix can introduce an around $\times 10$ speed increase. Second, we observe through our experiments that $\mathsf{BinarizedAttack}$ will almost only modify the edges directly connected to the targeted nodes. For example, the percentage of direct edges are 94.7\% for \textbf{Bitcoin-Alpha}. As a result, we can design a variant of $\mathsf{BinarizedAttack}$, termed $\mathsf{BinarizedAttack}$-$\mathsf{Direct}$, where the attacker is retreated to altering the \textit{direct} connections to the target nodes. This will decrease the number of parameters involved in optimization; however, the cost is that the performance of attack might be sacrificed. We implemented $\mathsf{BinarizedAttack}$-$\mathsf{Direct}$ on the largest dataset \textbf{ca-GrQc}. The results shows that $\mathsf{BinarizedAttack}$-$\mathsf{Direct}$ will increase the speed by around $2$ times with a reasonable performance sacrifice (Fig.~\ref{fig:ca_grqc}).
We further show in Fig.~\ref{fig-visual} how $\mathsf{BinarizedAttack}$ will actually modify the structure of a real-world graph (\textbf{Wikivote} as the example). We
demonstrated three cases, where the attacker deletes edges
only, adds edges only, and deletes and adds edges simultaneously. It shows that $\mathsf{BinarizedAttack}$ will indeed break the
anomalous structural patterns (i.e., near-star and near-clique)
in the graph.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/compare-cora.pdf}
\caption{\textbf{Cora}}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/compare-citeseer.pdf}
\caption{\textbf{Citeseer}}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth,height=3.2cm]{figures/compare-BlogCatalog.pdf}
\caption{\textbf{Blogcatalog}}
\end{subfigure}
\caption{Mean testing AUC scores for \textbf{Cora}, \textbf{Citeseer} and \textbf{Blogcatalog} attribute graphs.}
\label{fig-LGCN-exp}
\end{figure*}
\subsubsection{Effectiveness of attack $\mathsf{LGCN}$}
For attacking supervised graph anomaly detection, we focus on analyzing the attack efficacy of three attack methods by measuring the changes in the overall AUC scores after attack. Fig.~\ref{fig-LGCN-exp} presents the mean testing AUC scores of the proposed three attack methods with different attacking powers. Since it is a global attack, we may increase the degree attack power to up to around $30\%$ to highlight the attacking performance on the supervised graph anomaly detection.
We observe that similar to attacking $\mathsf{OddBall}$, our main method $\mathsf{BinarizedAttack}$ consistently outperforms the other two baseline methods in all the three real-world datasets. Especially, when the attacking power increase to up to $30\%$ for \textbf{Citeseer}, $\mathsf{BinarizedAttack}$ can effectively misguide the $\mathsf{LGCN}$ for anomaly node detection ($\text{AUC}=0.5$ means the classification is totally a random guess). At the same time, the AUC margin between $\mathsf{BinarizedAttack}$ and $\mathsf{GradMaxSearch}$ increase to up to around $0.15$ for \textbf{Citeseer} and around $0.08$ for \textbf{Cora}.
\subsubsection{Attacks against GCN-Based GAD Systems}
\begin{table*}[h]
\centering
\caption{Black-box attacks to GCN-based GAD systems.}
\label{tab-black-box-attack}
\resizebox{1.8\columnwidth}{!}{%
\begin{tabular}{|c|cc|cc|cc|cc|cc|cc|}
\hline
\multirow{2.1}*{BR (\%)}&\multicolumn{2}{c|}{GCN-reweight}&\multicolumn{2}{c|}{GAT-reweight}&\multicolumn{2}{c|}{FdGars}&\multicolumn{2}{c|}{GEM}&\multicolumn{2}{c|}{Player2vec}&\multicolumn{2}{c|}{GraphSMOTE}\\
&\textbf{Cora}&\textbf{Citeseer}&\textbf{Cora}&\textbf{Citeseer}&\textbf{Cora}&\textbf{Citeseer}&\textbf{Cora}&\textbf{Citeseer}&\textbf{Cora}&\textbf{Citeseer}&\textbf{Cora}&\textbf{Citeseer}\\
\hline
\hline
$0$&$0.72$&$0.79$&$0.71$&$0.79$&$0.65$&$0.64$&$0.72$&$0.76$&$0.67$&$0.64$&$0.73$&$0.78$\\
$(0,1]$&$0.71$&$0.76$&$0.70$&$0.74$&$0.60$&$0.58$&$0.68$&$0.75$&$0.60$&$0.60$&$0.73$&$0.77$\\
$(1,2]$&$0.70$&$0.77$&$0.72$&$0.74$&$0.60$&$0.60$&$0.70$&$0.74$&$0.61$&$0.59$&$0.73$&$0.78$\\
$(2,5]$&$0.68$&$0.76$&$0.70$&$0.72$&$0.60$&$0.56$&$0.70$&$0.72$&$0.59$&$0.55$&$0.73$&$0.76$\\
$(5,7]$&$0.65$&$0.75$&$0.66$&$0.74$&$0.59$&$0.60$&$0.70$&$0.72$&$0.60$&$0.58$&$0.72$&$0.78$\\
$(7,10]$&$0.68$&$0.75$&$0.68$&$0.71$&$0.56$&$0.56$&$0.70$&$0.70$&$0.58$&$0.57$&$0.71$&$0.77$\\
$(10,15]$&$0.60$&$0.71$&$0.60$&$0.69$&$0.54$&$0.53$&$0.57$&$0.68$&$0.53$&$0.52$&$0.67$&$0.73$\\
$(15,20]$&$0.60$&$0.64$&$0.60$&$0.66$&$0.52$&$0.53$&$0.56$&$0.66$&$0.51$&$0.52$&$0.66$&$0.70$\\
$(20,25]$&$0.54$&$0.59$&$0.56$&$0.63$&$0.48$&$0.52$&$0.49$&$0.67$&$0.51$&$0.54$&$0.63$&$0.66$\\
\hline
\end{tabular}
}
\end{table*}
In this section, we explore the black box attacks of the $\mathsf{BinarizedAttack}$ against $\mathsf{LGCN}$ on other GCN-based GAD models: GCN-reweight, GAT-reweight, FdGars, GEM, Player2vec and GraphSMOTE on \textbf{Cora} and \textbf{Citeseer} dataset as exemplars. We measure the attack efficacy of black-box attacks with the mean AUC scores on the testing data. The results are shown in Tab.~\ref{tab-black-box-attack}. $BR$ is the modified edges percentage range. We can observe that $\mathsf{BinarizedAttack}$ against $\mathsf{LGCN}$ has a significant effect on the global accuracy of those six GCN-based GAD systems especially for FdGars. That is, when the attacking power increases to more than $20\%$, the mean AUC scores for \textbf{Cora} and \textbf{Citeseer} decrease to $0.48$ and $0.52$, resulting in an approximately useless anomaly detector. On the other hand, the biggest gap for mean AUC scores before and after an attack is the black box attack against GEM on \textbf{Cora} dataset, which is up to around $32\%$ (decreased from $0.72$ to $0.49$). In general, the results in Tab.~\ref{tab-black-box-attack} shows that $\mathsf{BinarizedAttack}$ against $\mathsf{LGCN}$ as a surrogate model can effectively influence the $\mathsf{GCN}$ backbones of the other GAD systems, resulting in respectable attack efficacy in black-box attack manner.
\subsection{Side Effects of Attack OddBall}
$\mathsf{BinarizedAttack}$ is a structural targeted attack that aims to mislead the predictions of a \textit{small set of} target nodes in poisoning manner. A desirable character is that the attack would not significantly change the data distribution to the extent that the poisoned data would look abnormal for a smart defender. In other words, an attacker wants to make the attack \textit{unnoticeable}. In this paper we assume the detector is specially caring about the ego-features distribution.
In our context, $\mathsf{BinarizedAttack}$ modifies the features $(\mathbf{N}_i, \mathbf{E}_i)$ of the ego-net centered at node $i$. By design, $\mathsf{BinarizedAttack}$ will significantly modify the features of targeted nodes; however, an inevitable \textit{side effect} for the structural attack against $\mathsf{OddBall}$ is that $\mathsf{BinarizedAttack}$ would also change the features of the rest of the nodes. In the worst case, the shift of the feature distributions caused by an attack is so large that the attacked graph would appear abnormal and be easily detected by the defender. Thus, ideally, we would expect the side effect of $\mathsf{BinarizedAttack}$ is reasonably small to achieve an \textit{unnoticeable} attack.
We thus experiment to investigate the shift of distribution of the ego-features $(\mathbf{N}, \mathbf{E})$ before and after $\mathsf{BinarizedAttack}$.
Specifically, we deploy the permutation test \cite{permutationtest}, which is a general non-parametric test to check whether two different independent data groups follow the same distribution. However, it will be NP-hard for the permutation test if we consider all kinds of perturbations in the concatenation of two groups data. We thus refer to the Monte Carlo way to approximate the \textit{p}-value with sufficient experiment times, which is computed as:
\begin{align}
\label{eqn-pvalue}
p(t\geq t_{0})=\frac{1}{M}\sum_{j=1}^{M}I(t_{j}\geq t_{0}),
\end{align}
where $t_{0}$ is the observed value of test statistic and $t$ is the statistic calculated from the re-samples, i.e., $t(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime},y_{1}^{\prime},y_{2}^{\prime},...,y_{n}^{\prime})=|\bar{x^{\prime}}-\bar{y^{\prime}}|,$ $M$ is the Monte Carlo sampling times (in the experiment we set $M=100000$). $x$ and $y$ can be either $\mathbf{N}^{clean}$ and $\mathbf{N}^{poisoned}$ or $\mathbf{E}^{clean}$ and $\mathbf{E}^{poisoned}$. $x^{\prime}$ and $y^{\prime}$ are re-samples of $\mathsf{Concat}[\mathbf{N}^{clean}||\mathbf{N}^{poisoned}]$ or $\mathsf{Concat}[\mathbf{E}^{clean}||\mathbf{E}^{poisoned}]$.
We report the $p$-value of ego-features $\mathbf{N}$ and $\mathbf{E}$ on two real datasets: \textbf{Bitcoin-Alpha} and \textbf{Wikivote}. The results are shown in Tab.~\ref{tab-unnotice}, where we consider ego-features of the poisoned graph with maximum perturbations and $|V|=30$.
\begin{table}[h]
\centering
\caption{$p$-values for ego-features.}
\label{tab-unnotice}
\resizebox{0.8\columnwidth}{!}{%
\begin{tabular}{|c|cc|cc|}
\hline
\multirow{2}*{\diagbox{$t$}{$p$}{features}}&\multicolumn{2}{c|}{\textbf{Bitcoin-Alpha}}&\multicolumn{2}{c|}{\textbf{Wikivote}}\\
&$\mathbf{N}$&$\mathbf{E}$&$\mathbf{N}$&$\mathbf{E}$\\
\hline
\hline
$1$&$0.72$&$0.04$&$0.56$&$0.02$\\
\hline
$2$&$0.66$&$0.03$&$0.58$&$0.02$\\
\hline
$3$&$0.75$&$0.04$&$0.56$&$0.01$\\
\hline
$4$&$0.72$&$0.04$&$0.57$&$0.02$\\
\hline
$5$&$0.71$&$0.03$&$0.56$&$0.005$\\
\hline
\end{tabular}
}
\end{table}
From Tab.~\ref{tab-unnotice} we notice that under 99\% significant level we cannot reject the null hypothesis that $\mathbf{N}^{clean}$ and $\mathbf{N}^{poisoned}$ follows the same distribution, that is, we can draw the conclusion that $\mathsf{BinarizedAttack}$ does not manipulate the distribution of $\mathbf{N}$. For ego-feature $\mathbf{E}$, in most cases, we cannot reject the null hypothesis. However, in experiment 5 for \textbf{Wikivote} the $p$-value is less than 1\%, so we reject the null hypothesis, that is, in \textbf{Wikivote} experiment 5 $\mathsf{BinarizedAttack}$ manipulate the distribution of $\textbf{E}$ under the significant level 99\%. As an example, the probability density function of $\textbf{N}^{clean}$ and $\textbf{N}^{poisoned}$ and $\textbf{E}^{clean}$ and $\textbf{E}^{poisoned}$ on \textbf{Wikivote} are presented in Fig.~\ref{fig-kde} (similar results are observed on other datasets).
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth,height=3cm]{figures/wikivote_N_kde.pdf}
\caption{\textbf{Wikivote N}}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth,height=3cm]{figures/wikivote_E_kde.pdf}
\caption{\textbf{Wikivote E}}
\end{subfigure}
\caption{The probability density function of ego-features $\textbf{N}$ and $\textbf{E}$ for clean and poisoned graphs on \textbf{Wikivote}.}
\label{fig-kde}
\end{figure}
\subsection{Ablation Study on WLS}
In this section, we start to discuss the contribution of the weighted least square estimation (WLS) in the attack model. If we set the diagonal matrix $\mathbf{D}=\text{Diag}([1,...,1]_{n\times 1})$, the point estimate of RWLS in Eqn.~\eqref{eqn-RWLS-estimation} changes to Ridge regression \cite{ridge}:
\begin{subequations}
\label{eqn-Ridge-estimation}
\begin{align}
&\tilde{\mathbf{A}}=\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}(\mathbf{A+I})\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}},\\
&\mathbf{W}^{*}=((\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\tilde{\mathbf{A}}^{2}\mathbf{X}+\xi \mathbf{I}_{p\times p})^{-1}(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{Y}.
\end{align}
\end{subequations}
We compare the decreasing percentage of mean AUC scores for RWLS and Ridge regression on \textbf{Cora} dataset as exemplar. Fig.~\ref{fig-ablation} illustrates that by using RWLS as the point estimate for $\mathsf{LGCN}$'s weights, the attacking performance is significantly higher than Ridge regression (the decreasing percentage of mean AUC scores for RWLS is much larger than Ridge). This phenomenon is attributed to the fact that WLS incorporate the imbalanced sample numbers and pay more attention to the minority class during the training phase, leading to more suitable close form solutions of $W^{*}$. Hence the introduction of RWLS makes a great contribution to the attack model.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth,height=3cm]{figures/Ridge-vs-RWLS-cora.pdf}
\caption{Ridge-vs-RWLS}
\label{fig-ablation}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth,height=3cm]{figures/Tuning-xi-cora.pdf}
\caption{Tuning $\xi$}
\label{fig-sensitivity}
\end{subfigure}
\caption{(a) RWLS vs Ridge; (b) is the sensitivity analysis on $\xi$.}
\label{fig-ablation-sensitivity}
\end{figure}
\subsection{Sensitivity Analysis on $\xi$}
Another important issue is the choice of the penalty parameter $\xi$ for Ridge regression. According to Tab.~\ref{table-datasets}, the dimension of the attributes for real-world graphs are usually more than one thousand, thus leading to a high-dimensional problem in the regression. The introduction of $L2$ penalty here can not only prevent the inexistence of the inverse of singularity matrix in the point estimate, but also highlight the important part of the nodal attributes to prevent over-fitting.
Fig.~\ref{fig-sensitivity} depicts the effect of choosing $\xi$ with varying order of magnitudes for attack efficacy. The results show that too high and too low value of $\xi$ will damage the attacking performance of the attack model. Specially, if $\xi$ is ranged between $[0.001,0.01]$, the attacking performance will be unstable. In this context, we regard that $\xi=0.1$ is a suitable choice for attacking $\mathsf{LGCN}$. Without loss of generality, we choose $\xi=0.1$ for attacking $\mathsf{LGCN}$ in Fig.~\ref{fig-LGCN-exp} for sake of fair comparison.
\section{Formulation of Attacks}
\label{sec-formulation}
\subsection{Threat Model}
We consider a system consisting of three parties: a defender, an attacker, and the outside environment, the interplay among which is illustrated in Fig.~\ref{fig-interaction}. Specifically, a defender deploys a GAD system to detect anomalous nodes in an existing network. In practice, the network is not readily available; instead, the defender will need to construct the network via data collection. We model data collections as a querying process where the defender sends queries consisting of pairs of nodes $(u,v)$ to the environment, which responds with the existence of the relation between $u$ and $v$ (i.e., whether there is an edge between $u$ and $v$). Then, based on the query results the defender can construct an observed network. This querying process models real-world scenarios such as taking surveys on friendships, conducting field experiments to measure the communication channels, etc.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth,height=4cm]{figures/interaction.pdf}
\caption{Interplay among defender, attacker and environment.
}
\label{fig-interaction}
\end{figure}
An attacker, sitting between the defender and the outside environment, can tamper with the above data collection procedure by modifying the query results sent from the environment. For example, as shown in Fig.~\ref{fig-interaction}, the defender makes a query ``Is there a relation between $C$ and $D$?" to the environment; an attacker can change the query result ``$\{C,D\} = 0$" (non-existence) to ``$\{C,D\} = 1$" (existence), which is obtained by the defender. Consequently, the attacker is inserting an edge in the network constructed by the defender. That is, by modifying the query results, the attacker is equivalently manipulating the network topology, resulting in structural attacks. We state the attacker's goal, knowledge, and capability as follows.
\begin{itemize}
\item[$\bullet$] \textbf{Attacker's Goal.} For attacking unsupervised learning approach like $\mathsf{OddBall}$, we assume that the attacker has \textit{a set of target nodes} within the network, which are \textit{risky} nodes that have relatively high anomaly scores initially. The attacker's goal is thus to enable these target nodes to evade the detection. For attacking supervised learning methods such as GCN, the attacker's goal is to minimize the overall classification accuracy of the GAD systems.
\item[$\bullet$] \textbf{Attacker's Knowledge} We assume a worst-case scenario where the attacker has full knowledge of the network structure. That is, the attacker knows all the queries as well as the results to all those queries. Regarding the deployed GAD systems, we consider different cases where the attacker may or may not know the exact model in deployment.
\item[$\bullet$] \textbf{Attacker's Capability.} Again, we consider a worst-case scenario where the attacker can modify the query results at her choice. That is the attacker has control of the global structure of the graph and is able to add/delete edges from the graph. To further constrain the attacker's ability, we assume that the attacker can add/delete up to $B$ edges.
\end{itemize}
\subsection{Problem Formulation}
In this section, we first universally formulate the structural poisoning attacks against various GAD systems as bi-level discrete optimization problems. Then, we instantiate the attack problems against $\mathsf{OddBall}$ and GCN-based methods, respectively, based on their unique characteristics.
We use $\mathcal{G}_{0}=\{\mathbf{V}_{0},\mathbf{E}_{0},\mathbf{X}_{0},\mathbf{Y}_{0}\}$ to represent the ground-truth graph in the environment, where $\mathbf{Y}_0$ are the binary (anomalous or not) labels of the nodes.
In particular, $\mathbf{X}_0 \in \mathbb{R}^{n\times p}$ denotes the attribute matrix for the nodes, where each node has an attribute vector of dimension $p$. When the node attributes are not available, we set $\mathbf{X}_{0}=\emptyset$.
In structural attacks, the attacker will manipulate the graph structure in the data collection phase, resulting in a manipulated graph $\mathcal{G}=\{\mathbf{V}_{0},\mathbf{E},\mathbf{X}_{0},\mathbf{Y}_{0}\}$ \textit{observed} by the defender. We note that only the graph structure $\mathbf{E}_0$ will be modified to $\mathbf{E}$. We use $\mathbf{A}_0$ and $\mathbf{A}$ to denote the adjacency matrices of $\mathcal{G}_{0}$ and $\mathcal{G}$, respectively.
Then the structural poisoning attacks against the GAD systems can be formulated as:
\begin{subequations}
\label{eqn-opt-universal}
\begin{align}
\mathbf{A}^* = &\argmin_{\mathbf{A}} \mathcal{L}_{atk}(\mathbf{A},\mathbf{X},\theta^{*},\mathbf{Y}), \label{eq:goal-universal}\\
\text{s.t.} \quad &\theta^{*} = \mathsf{Regression} (\mathbf{A},\mathbf{X},\theta,\mathbf{Y}), \label{eq:const1-universal}\\
&\frac{1}{2}||\mathbf{A}_0 - \mathbf{A}||_1 \leq B, \quad \mathbf{A} \in \{0,1\}^{n \times n}. \label{eq:const2-universal}
\end{align}
\end{subequations}
Here, in~\eqref{eq:goal-universal} the loss function $\mathcal{L}_{atk}(\cdot)$ quantifies the attacker's malicious goal of evading detection.
In~\eqref{eq:const1-universal}, we use $\mathsf{Regression}(\cdot)$ to denote the training process of a GAD system, where the output is the optimal model parameter $\theta^*$ by specially designed regression methods. \eqref{eq:const2-universal} specifies the budget constraint on the attacker's power. We emphasize that we solve the complex bi-level optimization problem where whenever one tries to optimize graph structure $\mathbf{A}$, the optimal model parameter $\theta^*$ would change according to a complex training process to the one-level case by finding an appropriate point estimation $\theta^{*}$ for different scenarios.
\subsubsection{Attacking FeXtra-Based GAD Systems}
We now formulate the attacks against FeXtra-Based GAD systems with $\mathsf{OddBall}$ as an instantiation.
We assume that the attacker
has a set of target nodes $\mathcal{T} \subset V_0$, and her goal is to reduce the probabilities that the nodes in $\mathcal{T}$ are detected as anomalous. To this end, the attacker is allowed to add or delete \textit{at most} $B$ edges in the graph $\mathcal{G}_{0}$
with the goal of minimizing the anomaly scores of the target nodes in $\mathcal{T}$. In our formulation, we consider the weighted sum of scores, i.e., $S_{\mathcal{T}}(\mathbf{A}) = \sum_{i:v_i\in \mathcal{T}}\kappa_{i} S_i(\mathbf{A}) = \sum_{i:v_i\in \mathcal{T}} \kappa_{i} \frac{\max(E_{i},e^{\beta_{0}^*}N_{i}^{\beta_{1}^*})}{\min(E_{i},e^{\beta_{0}^*}N_{i}^{\beta_{1}^*})}\ln(|E_{i}-e^{\beta_{0}^*}N_{i}^{\beta_{1}^*}|+1)$, where $S_i(\mathbf{A})$ denotes the anomaly score of node $v_i$ and $(\beta_0^*, \beta_1^*) = \theta^*$ are the optimal parameters of the regression model.
For simplicity, in this paper, we consider the equal weight case, i.e., $\forall i, \kappa_{i}=1$; however, our methods can be easily extended to the case with unequal weights.
Then the attack can be formulated as the following optimization problem:
\begin{subequations}
\label{eqn-opt-original}
\begin{align}
\mathbf{A}^* = &\argmin_{\mathbf{A}} S_{\mathcal{T}}(\mathbf{A}), \label{eq:goal}\\
\text{s.t.} \quad &\beta_0^*, \beta_1^* = \mathsf{Train} (\mathbf{A}), \label{eq:const1}\\
&\frac{1}{2}||\mathbf{A}_0 - \mathbf{A}||_1 \leq B, \quad \mathbf{A} \in \{0,1\}^{n \times n}. \label{eq:const2}
\end{align}
\end{subequations}
To solve the optimization problem \eqref{eqn-opt-original}, we re-formulate it from several aspects.
First, we note that the anomaly score $S_i(\mathbf{A})$ for node $v_i$ is a normalized distance to the regression line. To reduce the non-linearity and ease optimization, we omit the normalization term and use a proxy $\tilde{S}_i(\mathbf{A}) = \ln(|E_{i}-e^{\beta_{0}^*}N_{i}^{\beta_{1}^*}|+1)$ to approximate $S_i(\mathbf{A})$. Consequently, we will use an objective function $\tilde{S}_{\mathcal{T}}(\mathbf{A}) = \sum_{i:v_i\in \mathcal{T}} (E_{i}-e^{\beta_{0}^*}N_{i}^{\beta_{1}^*})^{2}$ acting as the surrogate of $S_{\mathcal{T}}(\mathbf{A})$ in the optimization process. We emphasize that $\tilde{S}_{\mathcal{T}}(\mathbf{A})$ is only used in the optimization process and we use the true anomaly scores $S_{\mathcal{T}}(\mathbf{A})$ for evaluation. Second, $\mathsf{OddBall}$ uses OLS point estimation to learn the model parameters $\beta_0^*$ and $\beta_1^*$, which have closed-form solutions (Eqn.~\eqref{eqn-parameter}). This allows us to directly substitute $\beta_0^*$ and $\beta_1^*$ with functions of $\mathbf{A}$.
At last, we can explicitly write the features $N_i$ and $E_i$ as
$N_{i}=\sum_{j=1}^{n}\mathbf{A}_{ij}$, $E_{i}=N_{i}+\frac{1}{2}\mathbf{A}_{ii}^{3}.$ Finally, we can re-formulate the attack problem as:
\begin{subequations}
\label{eqn-model}
\begin{align}
&\mathbf{A}^* = \argmin_{\mathbf{A}}\ \tilde{S}_{\mathcal{T}}(\mathbf{A}) \label{eq:goal2}\\
&= \argmin_{\mathbf{A}} \sum_{i:i \in \mathcal{T}}(E_{i}-e^{(1,\ln N_{i})^{T}([\mathbf{1},\ln \mathbf{N}]^{T}[\mathbf{1},\ln \mathbf{N}])^{-1}[\textbf{1},\ln \mathbf{N}]^{T}\ln \mathbf{E}})^{2}, \nonumber \\
&\text{s.t.}\quad N_{i}=\sum_{j=1}^{n}\mathbf{A}_{ij}, \quad E_{i}=N_{i}+\frac{1}{2}\mathbf{A}_{ii}^{3}; \label{eq:const21} \\
&\quad \quad \frac{1}{2}||\mathbf{A}_0 - \mathbf{A}||_1 \leq B, \quad \mathbf{A} \in \{0,1\}^{n \times n} \label{eq:const22}.
\end{align}
\end{subequations}
Solving the above one-level discrete optimization problem leads to the optimal structural poisoning attacks.
\subsubsection{Attacking GCN-based GAD Systems}
We now formulate the structural poisoning attacks against GCN-based GAD Systems. One challenge is that these GAD systems have their unique characteristics, meaning that designing specific attacks for each system would be troublesome. In addition, GCN-based systems involve sophisticated training processes, imposing extra challenges to solving the attack optimization problem. To address these challenges, we propose to construct a simplified GCN model termed $\mathsf{LGCN}$ as a surrogate model, which represents the common graph convolutional process shared by all those GCN-based GAD systems. Thus, by attacking this surrogate $\mathsf{LGCN}$ model, we can generalize the attacks to all GCN-based GAD systems. In addition, a nice feature of $\mathsf{LGCN}$ is that we can use close-form (Ridge weighted least square estimation ~\cite{ridge, WLS}) functions to approximate its training process, which will significantly reduce the complexity of solving the attack optimization problem by transforming the complex bi-level optimization problem to one-level version.
Referring to \cite{Z_gner_2018}, we construct the $\mathsf{LGCN}$ model by replacing the activation function (Like $\mathsf{ReLU}$~\cite{relu}) of the convolutional layer \eqref{eqn-convolutional-layer} with the linear activation function.
In this paper, we consider a two-layered $\mathsf{LGCN}$:
\begin{subequations}
\begin{align}
&\tilde{\mathbf{A}}=\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}(\mathbf{A+I})\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}},\\
&\mathbf{Z}=\text{sigmoid}(\tilde{\mathbf{A}}\tilde{\mathbf{A}}\mathbf{X}\mathbf{W}^{1}\mathbf{W}^{2})=\text{sigmoid}(\tilde{\mathbf{A}}^{2}\mathbf{X}\mathbf{W}),
\end{align}
\end{subequations}
here $\mathbf{W}^{1}$ and $\mathbf{W}^{2}$ represent the weights for the first and second $\mathsf{GCN}$ layer. We use $\text{sigmoid}$ instead of $\text{softmax}$ for sake of transferring the nonconvex deep learning model to a convex logistic regression. The corresponding reweighting binary cross-entropy loss is defined as:
\begin{equation}
\mathcal{L}_{\text{R-BCE}}=-\sum_{i=1}^{n}(\omega y^{i}\log(\mathbf{Z}_{i})+(1-y^{i})\log(1-\mathbf{Z}_{i})),
\end{equation}
where $\omega$ is the ratio of positive samples number to negative samples number. However, the optimal $\mathbf{W}^{*}$ can only be approximated by vanilla gradient descent under this scenario. Crucially, we instead replace the cross-entropy loss with mean square loss in order to reach $\mathbf{W}^{*}$ without tedious training. As a result, we have new loss as:
\begin{equation}
\begin{split}
\label{eqn-WLS}
\mathcal{L}_{\text{R-MSE}}&=\sum_{i=1,y^{i}=1}^{n}(\omega(y^{i}-(\tilde{\mathbf{A}}^{2}\mathbf{X})_{i}))^{2}+\sum_{i=1,y^{i}=0}^{n}(y^{i}-(\tilde{\mathbf{A}}^{2}\mathbf{X})_{i})^{2}\\
& = (\mathbf{Y}-\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}(\mathbf{Y}-\tilde{\mathbf{A}}^{2}\mathbf{X}),
\end{split}
\end{equation}
where \[
\mathbf{D}=\begin{pmatrix}
\omega^{y^{1}} & 0 & \dots & 0 \\
0 & \omega^{y^{2}} & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & \omega^{y^{n}}
\end{pmatrix}
\]
is a diagonal matrix. Eqn.~\eqref{eqn-WLS} is the loss function of weighted least square (WLS \cite{WLS}) problem. The point estimate of WLS is
\begin{equation}
\mathbf{W}^{*}=((\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\tilde{\mathbf{A}}^{2}\mathbf{X})^{-1}(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\mathbf{Y}.
\end{equation}
In consideration of the sigularity matrix $(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\tilde{\mathbf{A}}^{2}\mathbf{X}$ caused by $p>n$ (singular matrix does not have its inverse), we add Ridge penalty \cite{ridge} on $\mathcal{L}_{\text{R-MSE}}$ to ameliorate the high dimensional problem and obtain:
\begin{equation}
\mathcal{L}_{\text{RR-MSE}}=(\mathbf{Y}-\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}(\mathbf{Y}-\tilde{\mathbf{A}}^{2}\mathbf{X})+\xi\mathbf{W}^{T}\mathbf{W},
\end{equation}
the corresponding point estimate of $\mathbf{W}^{*}$ is:
\begin{equation}
\label{eqn-RWLS-estimation}
\mathbf{W}^{*}=((\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\tilde{\mathbf{A}}^{2}\mathbf{X}+\xi \mathbf{I}_{p\times p})^{-1}(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\mathbf{Y}.
\end{equation}
During the testing phase, we feed $\mathbf{W}^{*}$ to $\mathsf{LGCN}$ and get the prediction score $\mathbf{Z}^{*}=\text{sigmoid}(\tilde{\mathbf{A}}^{2}\mathbf{X}\mathbf{W}^{*})$ for prediction.
Since the $\mathsf{GCN}$-based GAD system is a supervised structural poisoning attacks, the attacker's goal is to decrease the classification accuracy of the surrogate model in the previous context under the limited budgets. The reformulation of the objective is:
\begin{equation}
\begin{split}
\mathbf{A}^{*}=&\argmin_{\mathbf{A}} -\mathcal{L}_{\text{R-BCE}}(\mathbf{A},\mathbf{X},\mathbf{W}^{*},\mathbf{Y});\\
\text{s.t.} \quad &\mathbf{W}^{*}=\argmin_{\mathbf{W}} \mathcal{L}_{\text{RR-MSE}}(\mathbf{A},\mathbf{X},\mathbf{W},\mathbf{Y}),\\
&\frac{1}{2}||\mathbf{A}_0 - \mathbf{A}||_1 \leq B, \quad \mathbf{A} \in \{0,1\}^{n \times n}.
\end{split}
\end{equation}
Here we emphasize that the attack loss $\mathcal{L}_{\text{R-BCE}}$ can be partitioned into two parts to incorporate the information of the training and testing data:
\begin{equation}
\label{eqn-RBCE-split}
\mathcal{L}_{\text{R-BCE}}(\mathbf{Y})=h\mathcal{L}_{\text{R-BCE}}(\mathbf{Y}^{train})+(1-h)\mathcal{L}_{\text{R-BCE}}(\hat{\mathbf{Y}}^{test}),
\end{equation}
where $\mathbf{Y}^{train}$ is the training node labels, $\hat{\mathbf{Y}}^{test}$ is the predictions for the testing node labels based on the pre-trained $\mathsf{LGCN}$. Referring to \cite{DBLP:journals/corr/abs-1902-08412}, we choose $h=0.5$ during training phase.
With the help of the close form of $\mathbf{W}^{*}$, we can transform the bi-level optimization problem to the one-level case:
\begin{equation}
\label{eqn-attack-LGCN}
\begin{split}
\mathbf{A}^{*}=&-\argmin_{\mathbf{A}} \mathcal{L}_{\text{R-BCE}}(\mathbf{A},\mathbf{X},\mathbf{W}^{*},\mathbf{Y});\\
\text{s.t.} \quad &\tilde{\mathbf{A}}=\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}(\mathbf{A+I})\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}},\\ &\mathbf{W}^{*}=((\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\tilde{\mathbf{A}}^{2}\mathbf{X}+\xi \mathbf{I}_{p\times p})^{-1}(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\mathbf{Y}^{train},\\
&\frac{1}{2}||\mathbf{A}_0 - \mathbf{A}||_1 \leq B, \quad \mathbf{A} \in \{0,1\}^{n \times n}.
\end{split}
\end{equation}
Solving $\mathbf{A}^{*}$ lead to a optimal solution to the above discrete optimization problem.
\section{Introduction}}
Anomaly detection is a long-standing task in the field of data science and engineering with the goal to spot unusual patterns from the massive amount of data. Recently, due to the powerful representation abilities of graphs as well as the advances in graph mining and learning techniques, Graph-based Anomaly Detection (GAD) is becoming prevalent across a wide spectrum of domains. Various GAD systems are proposed and deployed as an indispensable security component in detecting, for example, fake accounts in social networks \cite{8621913}, fraudulent transactions in the financial industry \cite{fraudpayments}, and Botnets in communication networks \cite{180611}.
Those GAD systems, however, expose a new attacking surface, ironically due to their unique advantage of being able to exploit the connections among data (i.e., edges in graphs). As a motivative example, consider the problem of \textit{misinformation diffusion} in social networks, where an attacker aims to identify a set of nodes as the sources (called \textit{seeds}) to diffuse misinformation (e.g., fake news or hate speeches) through social media platforms. To maximize the diffusion range, the attacker can employ some \textit{influence maximization}~\cite{IM} algorithms to identify those most promising seeds, which however are prone to be labeled as anomalous by GAD systems. Meanwhile, the attacker can proactively alter the social ties (e.g., friend/unfriend with other users) of those seeds so as to prevent them from being detected. That is, attackers now can evade GAD systems via manipulating the \textit{structure} of the graph; we term such attacks as \textit{structural attacks}. In Fig.~\ref{fig-case-study}, we demonstrate how our proposed structural attacks can help those seeds to evade a popular GAD system.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.234\textwidth}
\centering
\includegraphics[width=\textwidth,height=3cm]{figures/overlaps.pdf}
\caption{Detection of seeds}
\label{fig-detect}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.234\textwidth}
\centering
\includegraphics[width=\textwidth,height=3cm]{figures/celf_AScore.pdf}
\caption{Evade detection}
\label{fig-evade}
\end{subfigure}
\caption{(a) The attacker uses four strategies (CELF, Greedy, Degree and Betweenness centralities) to select seeds. However, these seeds can all be detected using a GAD
algorithm $\mathsf{OddBall}$ among the top-k (in terms of anomaly score AScore) nodes. (b) The attacker can use our proposed algorithm $\mathsf{BinarizedAttack}$ to significantly decrease the AScore and ranking of those target seeds, thus evading the detection of $\mathsf{OddBall}$.}
\label{fig-case-study}
\end{figure}
The previous example shows that there is an urgent need to investigate the \textit{adversarial robustness} of current GAD systems -- \textit{how robust could those GAD systems be under carefully designed attacks?}
To this end, we initiate the study of structural attacks against GAD systems, answering to what extend an attacker can evade the detection of prevalent GAD systems by solely manipulating graph structures.
Studying such structural attacks has significant practical implications. Previous research on evading detection tools (e.g., PDF/Android malware detection \cite{7791883}) primarily operate in a feature space, where attackers manipulate the \textit{extracted} feature vectors.
Consequently, it is known that the manipulated feature vectors are hard to be mapped back to real entities (e.g., PDF/Android software), affecting the effectiveness of the actual attacks.
In contrast, altering the structure of a graph always corresponds to manipulating actual connections among entities, making structural attacks highly realizable in practice.
Besides, there are scenarios where an attacker could have control of the whole graph structure, allowing the global optimization of the attacks. A representative example is the Command \& Control center in Botnets \cite{180611}, which can coordinate the communication among Bots, i.e., globally optimizing the structure of the communication graph, to evade Botnets detection tools. These practical threats imposed by structural attacks motivate us to systematically study the attacker's ability to evade GAD.
In this paper, we instantiate our study by attacking two families of widely used GAD systems. The first family is termed as FeXtra-Based GAD systems. Specifically, these systems (e.g., $\mathsf{OddBall}$~\cite{oddball}) extract \textit{structural features} of each node in the graph, then build regression models over those features, and finally compute the node anomaly scores. We focus on FeXtra-Based GAD systems, particularly $\mathsf{OddBall}$, mainly due to the following reasons. First, $\mathsf{OddBall}$ stands for a class of simple while effective approaches that require much less information from data, and thus are widely adopted in practice. Second, the hand-crafted features in $\mathsf{OddBall}$ are based on domain-specific knowledge, which well captures the endogenous structural anomalous in the graph.
In comparison, the second family relies on the ability of Graph Convolutional Networks (GCNs~\cite{kipf2017semisupervised}) as the essential component to automatically learn the anomalous patterns, thus termed as GCN-based GAD systems. Since GCNs are demonstrated to achieve state-of-the-art performances across various graph-related tasks, we also choose them as our attack targets.
However, attacking GAD systems faces several challenges. First, GAD systems naturally feature an unsupervised/semi-supervised setting, where all nodes reside in a single graph. Attacking GAD thus is poisoning the data from which the detection model is built, which is in strong contrast to those attacks that are manipulating the inputs to a \textit{fixed trained} model. These \textit{poisoning attacks} are mathematically modeled as bi-level optimization problems, which are notoriously hard to solve in the literature. Second, structural attacks operate in a discrete domain, resulting in a strategy space exponential in the graph size. Furthermore, the discrete nature makes it hard to adopt gradient-descent-based attack methods \cite{Z_gner_2018,DBLP:journals/corr/abs-1902-08412}, as the attacker now needs to make binary decisions on deleting or keeping the connections in the graph. Indeed as we will show later, trivially utilizing the gradient information may result in non-effective attacks.
To tackle these challenges, we propose $\mathsf{BinarizedAttack}$, which is inspired by \textit{Binarized Neural Networks} (BNN)~\cite{courbariaux2016binarized} designed for model compression. Specifically, BNN transforms the real-valued model weights to discrete values $+1$/$-1$ to reduce the model size. To find the discrete optimal weights, BNN uses a continuous version of the weights when computing the gradients in the backward propagation. In light of this, $\mathsf{BinarizedAttack}$ associates a \textit{discrete} as well as a \textit{continuous} decision variable for each edge/non-edge in the graph and uses a novel gradient-descent-based approach to solve the resulting discrete optimization problem. Specifically, in the forward pass, the discrete decision variables are used to evaluate the objective function. In the backward pass, the continuous decision variables are firstly updated based on the \textit{fractional} gradients, and the discrete ones are then updated accordingly. In essence, $\mathsf{BinarizedAttack}$ could better use the gradient information to guide the search for the optimal graph structure. Comprehensive experiments demonstrate that $\mathsf{BinarizedAttack}$ can effectively solve the discrete optimization problem and outperforms other gradient-descent-based approaches.
In summary, we identify and initiate the study of a new vulnerability of graph-based anomaly detection systems. Our results show that by slightly modifying the graph structure, attackers can successfully evade two families of widely used GAD systems. In particular, we propose a novel gradient-descent-based approach $\mathsf{BinarizedAttack}$ that can effectively solve discrete optimization problems, outperforming existing methods. The research opens the door to studying this new type of attacks against graph-based learning systems.
\section{Related Works}
\label{sec-related}
\subsection{Graph-based anomaly detection} The essential task of Graph-based Anomaly Detection (GAD) is to identify anomalous patterns from graph data. It becomes prevalent due to the powerful representation ability of graphs as well as the advances in graph mining and learning techniques. Our focus is anomaly detection on static graphs~\cite{akoglu2015graph}, where the graph structure and node features are fixed overtime. Representative GAD can be roughly classified into four classes. Specifically, \textit{feature-based} methods~\cite{oddball} extract hand-crafted features from the graph and use typical machine learning approaches to analyze those features. \textit{Proximity-based} methods~\cite{chen2013ascos} exploit the graph structure to measure the distances among nodes and anomalous nodes are assumed to have larger distances to other nodes. \textit{Community-based} methods~\cite{kuang2012symmetric} employ community detection approaches to cluster normal nodes. \textit{Relational-learning-based} methods~\cite{kang2011mining} design graphical models to capture the relations among nodes and cast anomaly detection as a classification problem. Importantly, all these methods rely on the graph structure as an essential input. More recently, there are methods based on the powerful graph neural network to capture anomalous features in the relational data~\cite{FdGars, GEM, Player2vec, GraphSMOTE}, the propagation of the convolutional layer naturally can distinguish the anomalous relation information of the anomaly nodes among the benign nodes.
\subsection{Adversarial graph analysis} Our work belongs to a long line of research that studies the adversarial robustness of various graph analysis tasks, such as node classification~\cite{Z_gner_2018,DBLP:journals/corr/abs-1902-08412,zhou2020robust}, link prediction~\cite{zhou2019attacking,waniek2019hide}, community detection~\cite{waniek2018hiding}, etc. All related works on attacking graph analytic tools (\cite{Z_gner_2018}, \cite{DBLP:journals/corr/abs-1902-08412}, \cite{zhou2020robust}, \cite{zhou2019attacking}, \cite{waniek2019hide}, \cite{waniek2018hiding}) formulate the structural attacks as discrete optimization problems. Those approaches for solving the optimization problems diverge into two categories: task-specific and general. The task-specific approaches heavily rely on the concrete scenarios of the optimization problem, which of course depends on the target model. For example, \cite{zhou2019attacking} studies the sub-modularity of the objective functions and accordingly proposes an approximation algorithm with nice theoretical guarantees.
In contrast, the general approach is to use the gradient-descent method to optimize the objective iteratively. The key to this approach lies in properly using the gradient information to guide the optimization process in the discrete process.
The most notable representative is a greedy approach wherein each iteration the edge corresponding to the largest gradient is selected (thus \textit{greedy}) to alter. For example, \cite{Z_gner_2018} generates the adversarial perturbations on both the node features and graph structure by maximizing the classification margin of the target nodes and make the decision by greedy search; \cite{DBLP:journals/corr/abs-1902-08412} utilizes the meta-gradient on the bi-level optimization problem of the structural poisoning attack on node classification.
Our proposed $\mathsf{BinarizedAttack}$ differs in the way of utilizing the gradient information by using two sets of discrete and continuous variables in optimization, which will lead to more precise updates in the discrete space. This idea of using two sets of variables is loosely connected to binary neural networks \cite{courbariaux2016binarized}, which feature binary network parameters with a rather different design goal of making the size of the neural network compact.
In fact, we adapted the greedy approach to our case, resulting in a strong baseline attack method termed $\mathsf{GradMaxSearch}$. We analyzed the advantages of $\mathsf{BinarizedAttack}$ compared to $\mathsf{GradMaxSearch}$ in Section~\ref{sec-binary} and our comprehensive experiments also demonstrated that $\mathsf{BinarizedAttack}$ outperforms $\mathsf{GradMaxSearch}$ on all datasets.
\section{Attack Methods}
\label{sec-methods}
Now, the task of designing structural poisoning attacks amounts to solving the two optimization problems~\eqref{eqn-model} and~\eqref{eqn-attack-LGCN}.
In this section, we introduce three methods, $\mathsf{GradMaxSearch}$, $\mathsf{ContinuousA}$, and $\mathsf{BinarizedAttack}$. Specifically, the first two proposed methods $\mathsf{GradMaxSearch}$ and $\mathsf{ContinuousA}$ are adapted from typical approaches in the literature for solving similar problems. We further propose $\mathsf{BinarizedAttack}$ to address their limitations.
\subsection{Conventional Methods}
Solving the optimization problem \eqref{eqn-model} to obtain the optimal graph structure is hard in general, mainly due to the integer variables involved. Thus a common approach is to relax the integral constraints, transforming the optimization problem to its continuous counterpart, for which gradient-descent-based optimization techniques could be employed. Ideally, a \textit{continuous} optimal solution $\tilde{\mathbf{A}}^*$ is obtained, which is then transformed to a \textit{discrete} solution $\mathbf{A}^*$.
This approach faces two central challenges: i) how to use the gradient information for the guidance of searching for $\tilde{\mathbf{A}}^*$, and ii) how to discretize $\tilde{\mathbf{A}}^*$ to obtain $\mathbf{A}^*$. Based on the previous works in the literature, we propose two methods.
\subsubsection{$\mathsf{GradMaxSearch}$}
Most of the previous works on structural attacks utilize a greedy strategy to solve the optimization problem in an iterative way. Specifically, the integer constraints on $\mathbf{A}$ are relaxed and in each iteration, the gradients of the objective with respect to each entry of $\tilde{\mathbf{A}}$ are calculated. Then, the entry with the largest gradient is picked for alternation (either deleting or adding an edge).
Intuitively, a larger gradient indicates a bigger impact on the objective value. The iteration continues until budget constraint $B$ is reached.
Attacking $\mathsf{LGCN}$ with $\mathsf{GradMaxSearch}$ is similar to attacking $\mathsf{OddBall}$.
Regarding the implementation of $\mathsf{GradMaxSearch}$, we note that when picking the entry associated with the largest gradient, one should pay attention to the \textit{signs of the gradients}. For example, when $\mathbf{A}_{ij}=0$ we need to ensure that the corresponding gradient $\frac{\partial AS(v_{a})}{\partial \mathbf{A}_{ij}}<0$ (and vice versa) to make the operation (add or delete) valid. In addition, to avoid repeatedly adding and deleting the same edge, we maintain a pool to record the edges that have not been modified. Meanwhile, we also avoid the operations that would result in singleton nodes.
\subsubsection{$\mathsf{ContinuousA}$}
Softening the discrete objective is another concerns. An apparent bottleneck of $\mathsf{GradMaxSearch}$ is that the one-level discrete objective function is only optimized through $B$ steps, where $B$ is the attacker's budget. An alternative way is thus to totally treat $\mathbf{A}$ as continuous variables $\tilde{\mathbf{A}} \in [0,1]^{n\times n}$ and optimize the objective function until it converges.
In detail, by solving the optimization problem in the continuous domain using gradient-descent, we obtain the sub-optimal continuous solution $\tilde{\mathbf{A}}^*$. We then calculate the differences in absolute values between the original $\mathbf{A}$ and $\tilde{\mathbf{A}}^*$, and pick those edges associated with the top-$B$ absolute differences to modify.
\subsection{BinarizedAttack}
\label{sec-binary}
Both $\mathsf{GradMaxSearch}$ and $\mathsf{ContinuousA}$ have their own limitations. For
$\mathsf{GradMaxSearch}$, one major limitation is that the gradient only indicates a relatively
\textit{small fractional} update on the corresponding entry in $\mathbf{A}$; while a discrete value ($\pm 1$) is
actually updated. This would not necessarily optimize the objective. Moreover, due
to budget constraint $B$, the objective is only optimized through $B$ steps. $\mathsf{ContinuousA}$ treating $\mathbf{A}$ as continuous in the whole process of optimization, totally ignoring the effect of discrete updates on the objective function. Furthermore, without careful design, converting from the fractional optimal solution $\tilde{\mathbf{A}}^*$ to $\mathbf{A}^*$ may lead to arbitrary bad performances.
We propose $\mathsf{BinarizedAttack}$ to mitigate these limitations. At a high level, $\mathsf{BinarizedAttack}$ is a gradient-descent-based approach that optimizes the objective in iterations. Each iteration consists of a forward pass, where the objective function is evaluated on some decision variables, and a backward pass, where the decision variables are updated based on calculated gradients. The core idea of $\mathsf{BinarizedAttack}$ lies in the design of two sets of decision variables as well as the way of utilizing gradient information. Specifically, we associate each entry $\mathbf{A}_{ij}$ with a \textbf{discrete dummy} (the use of \textit{dummy} will be explained later) decision variable $\mathbf{Z}_{ij}\in \{-1,+1\}$, where $\mathbf{Z}_{ij}=-1$ indicates that the corresponding entry $\mathbf{A}_{ij}$ will be modified and vice versa. For example, if $\mathbf{A}_{ij} = 1$ and $\mathbf{Z}_{ij}=-1$, we will change $\mathbf{A}_{ij}$ to $0$. Let $\mathbf{A}_0$ and $\mathbf{A}$ be the original and modified adjacency matrix, respectively, which are connected through the decision variables $\mathbf{Z}$ by
\begin{equation}
\mathbf{A}=(\mathbf{A}_0 - 0.5 \cdot \mathbf{1}^{n\times n})\odot \mathbf{Z}+0.5,
\end{equation}
where $\odot$ denotes element-wise multiplication between two matrices.
We further associate each entry $\mathbf{A}_{ij}$ with a \textbf{continuous soft} decision variable $\dot{\mathbf{Z}} \in [0,1]$ to facilitate gradient computation. The two sets of decision variables $\dot{\mathbf{Z}}$ and $\mathbf{Z}$ are related by
\begin{align}
\label{eqn-z-z}
\mathbf{Z} =- \mathsf{binarized}(2\cdot \dot{\mathbf{Z}}-1),
\end{align}
where we define the function $\mathsf{binarized}(x)$ as $\mathsf{binarized}(x) = + 1$ if $x\geq 0$ and $\mathsf{binarized}(x) = - 1$ otherwise.
Since our objective function $\tilde{S}_{\mathcal{T}}(\mathbf{A})$ depends on $\mathbf{A}$, we can easily rewrite it as a function relying on the decision variables $\dot{\mathbf{Z}}$ and $\mathbf{Z}$ as $\tilde{S}_{\mathcal{T}}(\dot{\mathbf{Z}},\mathbf{Z})$.
We proceed to handle budget constraint \eqref{eq:const22}. Our goal is to transform this constraint as part of the objective function so that we can thoroughly optimize the objective beyond $B$ steps. To this end, we impose a LASSO penalty~\cite{Tibshirani94regressionshrinkage} on the continuous soft decision variables $\dot{\mathbf{Z}}$. Our choice of LASSO comes from the fact that LASSO can obtain sparser solutions compared with the L2 penalty~\cite{Hoerl1}. Based on Eqn.~\eqref{eqn-z-z}, we can observe that a larger $\dot{\mathbf{Z}}_{ij}$ indicates that it is more likely to modify the entry $\mathbf{A}_{ij}$ (i.e., $\mathbf{Z}_{ij} = -1$). As a result, in the optimization process, while the LASSO penalty term pushes the entries in $\dot{\mathbf{Z}}$ to zero, it is also restricting the modifications made to $\mathbf{A}$, achieving a similar effect of the budget constraint.
\subsubsection{Attacking $\mathsf{OddBall}$}
Now, we can reformulate the attack problem as an optimization problem with $\dot{\mathbf{Z}}$ and $\mathbf{Z}$ as decision variables:
\begin{subequations}
\label{bin-model}
\begin{align}
&\mathbf{\dot{\mathbf{Z}}}^* = \argmin_{\mathbf{\dot{Z}}} \sum_{a=1}^{\tau}(E_{a}-e^{\rho})^{2}+\lambda||\dot{\mathbf{Z}}||_{1}^{1}, \label{eq:goal3}\\
&\text{s.t.}\quad \rho = (1,\ln N_{a})^{T}([1,\ln N]^{T}[1,\ln N])^{-1}[1,\ln N]^{T}\ln E; \\
&\quad \quad N_{i}=\sum_{j=1}^{n}\mathbf{A}_{ij}, \quad E_{i}=N_{i}+\frac{1}{2}\mathbf{A}_{ii}^{3}; \label{eq:const31} \\
&\quad \quad \mathbf{A}=(\mathbf{A}_{0}-0.5\cdot\mathbf{1}^{n\times n})\odot \mathbf{Z}+0.5; \label{eq:const32} \\
&\quad \quad \mathbf{Z}=-\mathsf{binarized}(2\cdot{\mathbf{\dot{Z}}}-1). \label{eq:const33}
\end{align}
\end{subequations}
Note that we used a parameter $\lambda$ to tune the relative importance of the adversarial objective and the penalty term.
Now, we can solve \eqref{bin-model} through iteration. Specifically, in the forward pass, we will evaluate the objective by the discrete dummy variables $\mathbf{Z}$. Intuitively, this will more accurately measure the effect of discrete updates of the graph structure on the objective. In the backward pass, we can compute the gradients with respect to $\dot{\mathbf{Z}}$, and update $\mathbf{Z}$ from~\eqref{eq:const33}.
In this way, by observing $\dot{\mathbf{Z}}$, we can obtain the entries in $\mathbf{A}$ that the attacker needs to modify. We note that the discrete variables $\mathbf{Z}$ are only used to evaluate the objective function for optimization; the final decisions are made from the soft variables $\dot{\mathbf{Z}}$, thus the reason why $\mathbf{Z}$ are called dummy variables.
\subsubsection{Attacking $\mathsf{LGCN}$}
As for attacking $\mathsf{LGCN}$, similarly we deploy the interaction between a discrete decision variable $\mathbf{Z}\in\{-1,+1\}^{n\times n}$ and continuous random variable $\mathbf{\dot{Z}}\in[0,1]^{n\times n}$ by Eqn.~\eqref{eqn-z-z}. Then the objective function of $\mathsf{BinarizedAttack}$ is:
\begin{subequations}
\begin{align}
\mathbf{\dot{Z}}^{*}=&\argmin_{\mathbf{\dot{Z}}} -\mathcal{L}_{\text{R-BCE}}(\mathbf{A},\mathbf{X},\mathbf{W}^{*},\mathbf{Y})+\lambda||\mathbf{\dot{Z}}||_{1}^{1} ;\\
\text{s.t.} \quad &\tilde{\mathbf{A}}=\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}(\mathbf{A+I})\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}},\\ &\mathbf{W}^{*}=((\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\tilde{\mathbf{A}}^{2}\mathbf{X}+\xi \mathbf{I}_{p\times p})^{-1}(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\mathbf{Y}^{train},\\
&\mathbf{A}=(\mathbf{A}_{0}-0.5\cdot\mathbf{1}^{n\times n})\odot \mathbf{Z}+0.5,\\
&\mathbf{Z}=-\mathsf{binarized}(2\cdot{\mathbf{\dot{Z}}}-1).
\end{align}
\end{subequations}
The training procedure of the $\mathsf{BinarizedAttack}$ against $\mathsf{LGCN}$ is shown in Alg.~\ref{alg-binarizedattack-LGCN}. We leave the model comparison between the three attack methods based on the surrogate model in the experiment part.
\begin{algorithm}[t]
\caption{$\mathsf{BinarizedAttack}$ for $\mathsf{LGCN}$}
\label{alg-binarizedattack-LGCN}
\textbf{Input}: clean graph $\mathbf{A}^{0}$, budget $B$, surrogate model $\mathsf{LGCN}$, training nodal labels $\mathbf{Y}^{train}$, $\Lambda=\{\lambda_{k}\}_{k=1}^{K}$, iteration number $T$, learning rate $\eta$.\\
\textbf{Parameter}: Perturbation $\dot{\mathbf{Z}}$.\\
\begin{algorithmic}[1]
\STATE Pre-train $\mathsf{LGCN}$ and get prediction $\hat{\mathbf{Y}}^{test}$ based on the prediction score: $\mathbf{Z}^{*}_{test}=\text{sigmoid}(\mathbf{A}_{0}^{2}\mathbf{X}\mathbf{W}^{*}).$
\STATE Let $t=0$ and initialize $\dot{\mathbf{Z}}$.
\FOR {$k\leftarrow 1,2,...,K$}
\WHILE{$t\leq T$}
\STATE \textbf{Forward Pass}:
\STATE \quad Calculate $\mathbf{Z} =- \mathsf{binarized}(2\cdot \dot{\mathbf{Z}}-1).$
\STATE \quad Calculate $\mathbf{A}=(\mathbf{A}_0 - 0.5 \cdot \mathbf{1}^{n\times n})\odot \mathbf{Z}+0.5$.
\STATE \quad Calculate the symmetric normalized Laplacian $\tilde{\mathbf{A}}=\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}(\mathbf{A+I})\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}$.
\STATE \quad Obtain RWLS point estimate $\mathbf{W}^{*}=((\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\tilde{\mathbf{A}}^{2}\mathbf{X}+\xi \mathbf{I}_{p\times p})^{-1}(\tilde{\mathbf{A}}^{2}\mathbf{X})^{T}\mathbf{D}\mathbf{Y}^{train}$.
\STATE \quad Obtain goal function $\mathcal{L}_{\text{R-BCE}}+\lambda_{k}||\dot{Z}||_{1}^{1}$.
\STATE \textbf{Backward Pass}:
\STATE \ \ \ \ $\forall i,j\in{1,2,...,n},$ calculate the gradient of the goal function $\mathcal{L}_{\text{R-BCE}}$ w.r.t $\dot{\mathbf{Z}}_{ij}$, i.e., $\frac{\partial \mathcal{L}_{\text{R-BCE}}}{\partial \dot{\mathbf{Z}}_{ij}}.$
\STATE \textbf{Projection Gradient Descent}:
\STATE \ \ \ \ $\dot{\mathbf{Z}}\rightarrow\prod_{[0,1]}(\dot{\mathbf{Z}}-\eta\frac{\partial \mathcal{L}_{\text{R-BCE}}}{\partial \dot{\mathbf{Z}}_{ij}})$
\ENDWHILE
\STATE \textbf{return} $\dot{\mathbf{Z}}$
\ENDFOR
\FOR {$b\leftarrow 1,2,...,B$}
\STATE Pick out $\dot{\mathbf{Z}}=min \ \mathcal{L}_{\text{R-BCE}}$ satisfies $\sum \mathbf{Z}=-b$.
\STATE Get poisoned graph $\mathbf{A}^{b}=(\mathbf{A}_0 - 0.5 \cdot \mathbf{1}^{n\times n})\odot \mathbf{Z}+0.5$.
\STATE \textbf{return} $\mathbf{A}^{b}$.
\ENDFOR
\end{algorithmic}
\end{algorithm}
After the training phase of the structural attacks against the surrogate model, we ideally feed the poisoned graphs under different attack power into six popular GAD systems with the $\mathsf{GCN}$ as their backbone in a black-box manner: GCN-reweight \cite{kipf2017semisupervised}, GAT-reweight \cite{GAT}, FdGars \cite{FdGars}, GEM \cite{GEM}, Player2vec \cite{Player2vec}, GraphSMOTE \cite{GraphSMOTE}. We leave the model description and the black-box attack results in the experiment part.
\section{Background on GAD Systems}
In this section, we provide the necessary background on the two families of GAD systems -- one based on feature extractions and the other based on GNNs.
\subsection{FeXtra-Based GAD Systems}
\label{sec-GAD}
We introduce $\mathsf{OddBall}$~\cite{oddball} as one of the representative Feature eXtraction (FeXtra for short) based GAD systems that we aim to attack. At a high level, $\mathsf{OddBall}$ extracts some carefully crafted structural features for each node in the graph, and computes an anomaly score for each node based on those features, where a larger score indicates that the node is more likely to be anomalous.
To formalize, we denote a graph as $\mathcal{G}=(V, E)$, where $V$ denotes a set of $n$ nodes and $E$ represents the edges. We consider a simple unweighted graph with the adjacency matrix $\mathbf{A} \in \{0,1\}^{n \times n}$.
$\mathsf{OddBall}$ focuses on the local structural information of the nodes for anomaly detection. Specifically, for a node $v_i \in V$, $\mathsf{OddBall}$ examines an Egonet $\mathsf{ego}_i$ centered at $v_i$, where $\mathsf{ego}_i$ is the reduced sub-graph contains $v_i$ and its one-hop neighbors.
An important finding of $\mathsf{OddBall}$ is that the Egonets for anomalous nodes tend to appear in either a \textit{near-clique} or \textit{near-star} structure (as shown in Fig.~\ref{fig-anomalous-pattern}). To detect such anomalous structures, $\mathsf{OddBall}$ identifies two critical features $E_i$ and $N_i$ among others from the Egonet $\mathsf{ego}_i$, where $E_i$ and $N_i$ denote the number of edges and nodes in $\mathsf{ego}_i$, respectively. It was observed that $E_i$ and $N_i$ follow an \textit{Egonet Density Power Law}~\cite{oddball}: $E_i \propto N_i^\alpha, 1 \leq \alpha \leq 2$. The nodes that significantly deviate from this law are thus flagged as anomalous.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth,height=2.5cm]{figures/anomaly_structure.pdf}
\caption{Anomalous patterns}
\label{fig-anomalous-pattern}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth,height=2.5cm]{figures/OLS_plot.pdf}
\caption{Feature distribution}
\label{fig-feature}
\end{subfigure}
\caption{(a) The \textit{near-clique} (left) and \textit{near-star} (right) structural patterns. (b) The distribution of node features and the regression line. }
\end{figure}
$\mathsf{OddBall}$ uses a regression-based approach to quantify the deviation. Specifically, let vectors $\mathbf{E} = (E_1,E_2, \cdots, E_n)$ and $\mathbf{N} = (N_1, N_2, \cdots, N_n)$ be the collected features of all nodes. Based on the power law observation, one can use the following linear model to fit the pair of features $(E_i, N_i)$ for a node $i$:
\begin{equation}
\label{power law}
\ln E_{i}=\beta_{0}+\beta_{1} \ln N_{i}+\epsilon.
\end{equation}
In the above equation, the model parameters $\beta_{0}$ and $\beta_{1}$ are given by the Ordinary Least Square (OLS) \cite{Zdaniuk2014} estimation:
\begin{equation}
\label{eqn-parameter}
[\beta_{0},\beta_{1}] = ([\mathbf{1},\ln \mathbf{N}]^{T}[\mathbf{1},\ln \mathbf{N}])^{-1}[\textbf{1},\ln \mathbf{N}]^{T}\ln \mathbf{E},
\end{equation}
where $\mathbf{1}$ is an $n$-dimensional vector of all $1$'s.
The anomaly score $S_i(\mathbf{A})$ for the node $v_i$ is then computed as
\begin{equation}
S_i(\mathbf{A})=\frac{\max(E_{i},e^{\beta_{0}}N_{i}^{\beta_{1}})}{\min(E_{i},e^{\beta_{0}}N_{i}^{\beta_{1}})}\ln(|E_{i}-e^{\beta_{0}}N_{i}^{\beta_{1}}|+1).
\end{equation}
Note that we made the dependency of $S_i(\mathbf{A})$ on $\mathbf{A}$ explicit as all of $\beta_0, \beta_1, E_i, N_i$ rely on $\mathbf{A}$.
As illustrated in Fig.~\ref{fig-feature}, the anomaly score $S_i(\mathbf{A})$ intuitively measures the distance between the point $(N_{i},E_{i})$ and the regression line along the vertical axis. Finally, nodes with high anomaly scores (e.g., by ranking) are determined as anomalous by $\mathsf{OddBall}$.
\subsection{GCN-Based GAD Systems}
\label{sec-GCN-GAD}
With the advances of graph neural networks, an increasing number of GAD systems are using GCN~\cite{kipf2017semisupervised} as the backbone for automatically learning useful anomalous patterns. In essence, anomaly detection is cast as a supervised classification process where the labels of a portion of nodes are provided for training.
Specifically, GCN-based methods will take as input an attributed graph where nodes are associative with feature vectors, and generate node embeddings through several convolutional layers as follows:
\begin{subequations}
\label{eqn-convolutional-layer}
\begin{align}
&\mathbf{H}^{t+1}=\sigma(\tilde{\mathbf{A}}\mathbf{H}^{t}\mathbf{W}), \text{where }\\
&\tilde{\mathbf{A}}=\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}}(\mathbf{A+I})\text{diag}(\sum_{i=1}^{n}\mathbf{A}_{i})^{-\frac{1}{2}} \text{ and }\\ &\mathbf{H}^{0}=\sigma(\tilde{\mathbf{A}}\mathbf{X}\mathbf{W}).
\end{align}
\end{subequations}
In the above, $\mathbf{A}$ and $\mathbf{X}$ are the adjacency matrix and feature matrix, respectively; $\mathbf{W}$ summarizes the learned model parameters; $\mathbf{H}^i$ are the learned embeddings through the layers. Finally, the embeddings are fed into a fully connected layer for classification. The specific methods in this family integrate some minor modifications into this framework to adapt to the various application scenarios that they are designed for. In the following, we introduce six typical GCN-based GAD systems and highlight their differences:
\begin{itemize}
\item[$\bullet$] \textbf{GCN-reweight}~\cite{kipf2017semisupervised}. This is the $\mathsf{GCN}$ model with class-specific loss weight to mitigate the imbalance problem. It assigns higher weight loss for the minority class.
\item[$\bullet$] \textbf{GAT-reweight}~\cite{GAT}. This is the $\mathsf{GCN}$ model augmented with the graph attention mechanism to automatically distinguish the contributions from different neighbors in the aggregation phase. Similar to the GCN-reweight, it assigns higher weight loss for the minority class.
\item[$\bullet$] \textbf{FdGars}~\cite{FdGars}. It utilizes $\mathsf{GCN}$ for fraudulent detection in the online APP review system to distinguish the malicious users out of benign users.
\item[$\bullet$] \textbf{GEM}~\cite{GEM}. It is the first heterogeneous graph neural network model to detect anomalous accounts at Alipay. It also augments the attention mechanism to control the contributions from different nodes, and utilizes EM algorithm to iteratively update the node embeddings and model parameters.
\item[$\bullet$] \textbf{Player2vec}~\cite{Player2vec}. It is specially designed to detect the cybercriminals in the e-commercial system by adopting $\mathsf{GCN}$ augmented with the attention mechanism to automatically capture the anomalous behaviors in the environment.
\item[$\bullet$] \textbf{GraphSMOTE}~\cite{GraphSMOTE}. GraphSMOTE adopts synthetic minority oversampling techniques (SMOTE) to generate an augmented graph to mitigate the class imbalance problem in the bot detection field by using the synthetic node generator and edge generator to interpolate new synthetic nodes and edges for the minority class. Then the $\mathsf{GCN}$ classifier is implemented on the augmented graphs for imbalanced node classification.
\end{itemize}
\section{Black-box Attacks Against GCN-Based Anomaly Detection}
\label{sec-black-box-attack}
\subsection{Attack Model}
\section{Attack Transferability}
\label{sec-transfer}
In this section, we investigate the attack transferability of $\mathsf{BinarizedAttack}$ to two other GAD systems that are based on representation learning.
\subsection{Representation-learning-based GAD}
Recently, graph analysis based on representation learning has attracted extensive research attention. Graph representation learning~\cite{hamilton2020graph} aims to learn a low-dimensional latent vector for each node (called embedding) that could capture the feature as well as the structural information of the graph. Those learned embeddings are then used in a wide range of down-stream tasks such as node classification and link prediction. Following this trend, graph representation learning has also been used as the core technique for anomaly detection. These techniques entitle the defender the freedom to choose different GAD systems in practice, which raises a practical question: could an attack method that is specifically designed for a target GAD system be effective to other GAD systems? We investigate this problem of attack transferability by using $\mathsf{BinarizedAttack}$ to attack two other GAD systems (GAL~\cite{10.1145/3340531.3411979} and ReFeX~\cite{ReFeX}) based on graph representation learning.
At a high level, representation-learning-based GAD systems have two components: representation learning and classification. First, given a graph as input, the system learns the embeddings of nodes using various methods (e.g., graph neural networks, random walk, etc.). These embeddings are then fed into classifiers such as Multi-Layer Perceptron (MLP) which will classify nodes as anomalous or benign. These GAD systems differ mainly in the methods used for learning the node embeddings. We focus on two representative systems as below.
\subsubsection{GAL~\cite{10.1145/3340531.3411979}} GAL utilizes Graph Neural Networks \cite{kipf2017semisupervised} (GNNs) to learn the node embeddings. In particular, GAL replaced the loss function in typical GNNs with a \textit{graph anomaly loss} that is specially designed for anomaly detection. It is a class-distribution-aware margin loss which solves the imbalanced problem in anomaly detection via automatically adjusting the margins for the minority class (i.e., anomaly). Specifically, the loss on node $u$ is computed as:
\begin{align}
\mathcal{L}(u)=&E_{u_{+}\sim \mathcal{U}_{u+},u_{-}\sim \mathcal{U}_{u-}} max\{0,g(u,u_{-})-g(u,u_{+})+\Delta_{y_{u}}\}, \nonumber \\
&\text{where} \quad \Delta_{y_{u}}=\frac{C}{n_{y_{u}}^{1/4}}. \label{eqn-gal}
\end{align}
Here $\mathcal{U}_{u+}$ denotes the set of nodes share the same label with $u$, and vice versa. $g(u,u^{\prime})=f(u)^{T}f(u^{\prime})$ measures the similarity of the representation of two nodes $u$ and $u^{\prime}$, $f$ is the GNN. $C$ is the constant hyperparameter to be tuned. $\Delta_{y_{u}}$ is proved to best weigh up between the improvement of the generalization of minority class and the sub-optimal margin for the frequent class.
\subsubsection{ReFeX~\cite{ReFeX}}
ReFeX (Recursive Feature eXtraction) is a novel algorithm that combines node-based local features with neighborhood (ego-based) features recursively, and output regional features which can capture the behavior information in large social networks. ReFeX aggregates the local and ego-based features and use them to create recursive features. Local features are essentially node degree measures. Egonet features are $\mathbf{N}$ and $\mathbf{E}$ as mentioned in Section III. ReFeX recurses the local and ego-based features by calculating their means and sums. Next ReFeX implements the pruning method using \textit{vertical logarithmic binning}. At last, ReFeX transforms the pruned recursive features to binary-valued embeddings. The pruned recursive features proved to efficiently capture the inner structural information provided by the graph. In this paper, we make use of these features for node anomaly detection.
\subsection{Transfer attack methodology}
\label{sec-transfer-method}
Our transfer attack to both GAL and ReFeX consists of four steps: \textit{data pre-processing}, \textit{targets identification}, \textit{graph poisoning} and \textit{evaluation}.
\subsubsection{Data pre-processing}
$\mathsf{OddBall}$ operates in an unsupervised setting while both GAL and ReFeX are supervised methods. We thus pre-process the data by assigning anomaly labels to a set of nodes. Specifically, given a graph, we first use $\mathsf{OddBall}$ to compute the anomaly scores for all the nodes and label those nodes with high anomaly scores as anomalous. We then randomly split the nodes into training and testing sets.
\subsubsection{Targets identification}
As $\mathsf{BinarizedAttack}$ is a targeted attack, the goal of our transfer attack is not to decrease the prediction accuracies of the GAD systems. Instead, we are interested in changing the prediction results of a set of nodes -- those nodes that are initially predicted as anomalous by the GAD systems.
Thus, we feed the pre-processed data into GAL or ReFeX and obtain the predicted labels of the test nodes and pick those nodes \textit{predicted as anomalous} as our attack targets.
\subsubsection{Graph poisoning} With the targeted nodes identified from the previous step, we directly use $\mathsf{BinarizedAttack}$ to generate a poisoned graph. We emphasize that the attack occurred in a black-box setting, where we do not need any information from GAL nor ReFeX.
\subsubsection{Evaluation} We provide the clean graph and poisoned graph separately as input to GAL and ReFeX. We evaluate the performance of transfer attack mainly from two aspects: 1) whether the target nodes can successfully evade the detection of GAL or ReFeX; 2) the changes of node embeddings generated by GAL and ReFeX before and after attack.
We present the detailed experiment settings and results in Section~VIII.
|
1,314,259,995,829 | arxiv |
\section{Introduction}\label{sec:intro}
Metamaterials are drawing increased attention for their ability to achieve a variety of non-intuitive properties that stem from their intentionally hierarchical structures~\cite{Schumacher2015}. While they traditionally consist of one unit cell that is repeated everywhere, multiple unit cells can also be assembled to create \textit{aperiodic} mechanical metamaterials with, {\em e.g.}, spatially-varying or functionally-gradient properties~\cite{Schumacher2015,Maskery2018tpms}. Over the past few years, conventional computational methods have been adapted to design these complex structures, including topology optimization (TO) of the microscale unit cells within a fixed macroscale structure~\cite{Choi2019preallocate,Vogiatzis2018conformal}, and hierarchical and concurrent multiscale TO that design both the macrostructure and a pre-specified number of unique unit cells~\cite{Deng2017concurrent,Du2018connectivity,Liu2019subdomain}. However, as the desire to attain even more intricate behaviors grows, so too does the complexity of the design process, which must account for the expensive physical simulations and, in aperiodic structures, the vast combinatorial design space and disconnected neighboring unit cells~\cite{Schumacher2015,Coulais2016combinatorial}.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{Fig_1.pdf}
\caption{A high-level overview of data-driven metamaterials design, and how our proposed method, METASET, fits in. As an example, we show $C^H$, the homogenized elastic tensor, as the unit cell properties.}
\label{fig:flow}
\end{figure}
Capitalizing on advances in computing power, data-driven metamaterials design can be a more efficient and therefore enticing solution to those challenges. Its success hinges on precomputed unit cell libraries or datasets, which can avoid costly on-the-fly physical simulations and multiscale TO in huge design spaces, as well as provide candidate unit cells that are better connected to their neighbors. Fig.~\ref{fig:flow} shows an overview of two common approaches in data-driven design: global optimization methods, and machine learning (ML) based methods. In the first case, combinatorial optimization algorithms can be used to directly search for the set of unit cells that realize a target macroscale behavior while minimizing or constraining the boundary mismatch between neighboring cells ~\cite{Schumacher2015,Coulais2016combinatorial,Zhu2017twoscale}. From another perspective, data-driven methods can use the dataset to train ML models that further accelerate design. For example, they have been used to rapidly predict homogenized physical properties as part of the optimization loop~\cite{Wang2020smo,Bostanabad2019gagp,White2019neural,Chen2015coarse}. Additionally, deep generative models inspired by the computer vision field can learn embedded geometric descriptors that act as reduced dimensional design variables, and construct new designs, {\em e.g.}{}, optical 2D metamaterials~\cite{Ma2019vae,Liu2018nano}, almost instantaneously. Accelerated by data-driven techniques, challenging designs such as spatially-varying displacement profiles and nonlinear behavior that are prohibitively expensive via conventional methods are now tangible.
The efficacy of data-driven methods, however, relies highly on the size and coverage of the datasets. The search space of global optimization methods can quickly explode when the number of unit cells increases. Meanwhile, imbalanced datasets with skewed data distributions can reduce the chance of meeting certain property or compatibility requirements, and hobble the performance of ML models since they may not learn a less frequent property or shape as well~\cite{Haixiang2017imbalance}. Therefore, due to the importance of the data on downstream tasks, in this work we focus on the first step of data-driven design: dataset selection.
In existing literature, metamaterial datasets are often built using heuristics or the designer's intuition, with the assumption that the unit cells will offer sufficient coverage for the desired application. Many employ TO to inversely design unit cells that meet pre-specified target properties~\cite{Schumacher2015,Zhu2017twoscale,Wang2020smo}, and some expand the dataset by morphing the shapes~\cite{Schumacher2015,Wang2020smo} or randomly flipping pixels or voxels~\cite{Zhu2017twoscale}. Alternatively, Panetta~{\em et~al.}{} developed graph-based rules to create truss-like unit cells~\cite{Panetta2015}. Although these are more feasible than enumerating over all possibilities, bias toward particular properties or shapes can be unintentionally introduced, deteriorating the performance of the design algorithm or the design itself.
Moreover, the point at which to stop generating new unit cells has thus far been heuristic with the same goal in mind: to cover a broad property space. The range of this space is sometimes restricted for specific applications~\cite{Choi2019preallocate}, or strict symmetry and manufacturability constraints are implemented to limit the possible shapes~\cite{Panetta2015}. More often, the property space is allowed to grow at will, {\em e.g.}{}, TO and shape perturbation are repeated until the change in the density of the property space is less than a given tolerance~\cite{Zhu2017twoscale,Wang2020smo}. While efficient, all of the works to date have only considered coverage in the property space alone, which can produce similar shapes or overlook those that might benefit the design with regards to boundary connectivity. In contrast, our work explores coverage in both property and shape spaces.
Improving imbalance arising from data with multiple classes has been extensively researched in computer science. The most relevant to our application are the data preprocessing strategies such as undersampling to remove data from majority classes, oversampling to replicate data from minority classes, or combinations thereof~\cite{Haixiang2017imbalance}. However, the former can accidentally remove samples with important features, {\em i.e.}{}, decrease the diversity, and the latter can lead to model overfitting and increased training overhead~\cite{Branco2016imbalance2}. Nor are they made to consider the diversity of data with features that have drastically different representations, like shape and property. The issue of downsampling a metamaterial database was addressed by Chen~{\em et~al.}{}~\cite{Chen2015coarse}, who compressed the size of their database by selecting the samples that are farthest from each other with respect to properties (not shape), allowing them to more efficiently fit a property prediction model. As far as we know, there is currently no method to assess or select a diverse set of unit cells that can simultaneously cover the shape and property spaces.
Despite the dearth in the metamaterials field, measuring and ranking items based on their quality as well as their contribution to the diversity of a whole set or subset is an ongoing research area. In computer science, for example, recommender systems rank diverse items such as online products to match users' preferences. These are based on the concept of diminishing marginal utility~\cite{Coombs1977diminishing}, wherein lower ranking items bestow less additional value onto the users. In design, too, researchers have developed methods to help designers sift through large sets of ideas by ranking them. In particular, to balance diversity against quality of designs, Ahmed~{\em et~al.}{} introduced the idea of clustering items into groups for subset selection~\cite{ahmed2016discovering} by employing submodular functions that follow the property of diminishing marginal utility. Additionally, Ahmed~{\em et~al.}{}~\cite{Ahmed2017ranking} showed the application of Determinantal Point Processes (DPPs)~\cite{kulesza2012determinantal}, which model the likelihood of selecting a subset of diverse items as the determinant of a kernel matrix, to the diverse ranking task. The latter, in particular, are elegant probabilistic models that capture the trade-off between competing ideas like quality and diversity. While the goal of maximizing the determinant is similar to the optimality criterion used in generating D-optimal designs~\cite{de1995d} in design of experiments, DPPs are not restricted to linear kernels, and have advantages in that calculating marginals, computing certain conditional probabilities and sampling can all be done in polynomial time. This paper shows that DPPs can also be used for coverage in multiple spaces defined over the shapes and properties of unit cells.
\textbf{Our contributions:}
We propose METASET, an automated methodology that simultaneously considers the diversity of shape and property to select subsets of unit cells from existing datasets. By doing so, we can achieve scalable data-driven design of metamaterials using smaller yet diverse subsets and eliminate bias in imbalanced datasets to improve any downstream task in the data-driven framework. As a part of METASET, we introduce similarity metrics to efficiently assess the diversity of the shapes and properties of 2D and 3D metamaterials. We also propose that a weighted sum of Determinantal Point Process (DPP) kernels based on the shape and property similarities can measure and allow the maximization of the joint diversity of both spaces.
For the first time in data-driven metamaterials design~---~to our knowledge~---~we reveal through 2D case studies that diverse subsets can expedite and even enhance the design performance and connectivity of aperiodic metamaterials.
Finally, applying METASET to 3D unit cells, we identify diverse families of isosurface unit cells and discover that these extend beyond the ones commonly considered in the design of functionally-graded structures~\cite{Li2019tpms,Maskery2018tpms}.
The components of our methodology are detailed in Sec.~\ref{sec:methods}.
In our 2D case studies (Sec.~\ref{sec:2dcase}), we explore the effects of diversity and subset size on 2D metamaterial designs with non-intuitive target displacement profiles. In a 3D example (Sec.~\ref{sec:3dcase}), we compare the impact of different shape similarity metrics on diverse unit cell families and demonstrate that METASET can diversify datasets regardless of the chosen metric.
\section{METASET: Assessing and Optimizing Diversity}\label{sec:methods}
The inner workings of METASET consist of three main steps:
\begin{enumerate*}
\item[1)] Defining similarity metrics for metamaterials that quantify the difference between pairs of 2D or 3D shapes and mechanical properties (Sec.~\ref{sec:similarity});
\item[2)] Using a DPP-based submodular objective function to measure the joint coverage of a set of unit cells in shape and property spaces via pairwise similarity kernel matrices (Sec.~\ref{sec:dpp});
\item[3)] Maximizing the joint diversity with an efficient greedy algorithm while allowing trade-off in the two spaces to be tuned to suit the desired application (Sec.~\ref{sec:ranking}).
\end{enumerate*}
In this section, we describe these components and summarize the methodology with Algorithm \ref{alg:alg_metaset}.
\subsection{Similarity Metrics for Metamaterials}\label{sec:similarity}
A diverse metamaterial dataset should ideally contain unit cells that are sufficiently different, {\em i.e.}{}, dissimilar, such that they cover the shape and property spaces. To measure the diversity of a set, then, the similarities between the shapes and properties of unit cells first need to be quantified. We do so by defining metrics independently in each space, based on the observation that a set of unit cells dissimilar in shape space is not necessarily also dissimilar in property space, and vice versa.
This can be illustrated by a simple example. Say we wish to distill diverse values from $x$ and $y$, which we assume to be sets of integers: $x = \{0, 1, 2, 4, 5\}$ and $y = \{0, 2, 10, 20, 10\}$. We assume that $y = x*k$, where $k = \{3, 2, 5, 5, 2\}$ is a transformation function. If we were to select three diverse values of $x$, {\em i.e.}{}, the values that most cover its space, we would select $\{0, 2, 5\}$. For $y$, however, we would choose $\{0, 10, 20\}$ rather than $\{0, 10, 10\}$, the ones corresponding to the diverse $x$ values. Hence, though some relationship between two spaces may exist, {\em e.g.}{}, an intrinsic function between shape and property, there is a need to model their coverage separately. This observation is validated in our later design experiments (Sec.~\ref{sec:2ddpp}), where the correlation coefficient between shape and property coverage shows that no link exists between the two.
\subsubsection{Property Similarity}
Since mechanical properties are generally scalar values that can be expressed as a vector, {\em e.g.}{}, by flattening the elastic tensor, we can use any similarity metric between vectors. In this work, we use the Euclidean distance. We note that the properties do not need to be the tensor components; rather, they can be other values of interest such as elastic or shear moduli, or Poisson's ratios. Neither do they need to be limited to scalar mechanical properties. For instance, dynamic acoustic dispersion curves or bandgaps could be considered if the pairwise similarity can be quantified.
\subsubsection{Shape Similarity}
Shape similarity metrics are key in many computer vision and graphics applications, {\em e.g.}, facial recognition and object retrieval from databases. In these methods, the shapes are usually first represented by structural descriptors extracted from individual shapes~\cite{Bustos2005featurebased}, or by embedded features learned via data-driven methods such as clustering or deep learning~\cite{Rostami2018embedded,achlioptas18ae}. The distances between features can then be measured in Euclidean~\cite{Bustos2005featurebased} or Riemmanian space~\cite{Sharon2006conformal2d,Su2015conformal3d}. Since Riemannian metrics are based on geodesic distances, they are suitable if one needs invariance to deformation, {\em i.e.}{}, if one considers a shape to be the same after bending.
For metamaterials, however, we must rule out deformation and rotation invariant metrics since any transformation of a unit cell impacts its properties. Additionally, we seek techniques that are efficient but still able to discriminate fine details and form positive semi-definite similarity matrices for the next step involving DPPs. Thus, we introduce the following Euclidean metrics based on structural features: a descriptor-based distance for 2D, and two point cloud-based metrics for 3D, namely, the Hausdorff distance and embedded cosine similarity utilizing deep learning. While we elected for separate metrics in 2D and 3D by bearing in mind their respective computational efficiencies, shape analysis is a wide and ever-growing topic of research in computer science; many other metrics are available. As we later show in Sec.~\ref{sec:3ddpp}, METASET selects diverse subsets regardless of the metric used, as long as the requirements for DPPs are met.
\vspace{6pt}
\textbf{2D Descriptor-Based Euclidean Distance: }\label{sec:2dsimilarity}
For 2D unit cells, which are typically binary images resulting from TO, we propose using a descriptor-based approach by first extracting division-point-based descriptors~\cite{vamvakas2010handwritten} to reduce the images into vectors that capture salient features at different levels of granularity. This has been applied to the field of optical character recognition~\cite{das2012statistical,sarkhel2017multi}. The binary image of a unit cell is recursively divided into sub-regions that contain an equal number of solid pixels. The coordinates of all division points, {\em i.e.}{}, points at the intersection of two division lines between each sub-region, are then obtained as descriptors of the unit cell. This process is repeated until the desired level of detail is captured, constructing a \textit{k}-d tree of the distribution of solid materials. In our 2D case study (Sec.~\ref{sec:2dcase}), we obtain a sufficient amount of detail by performing the division seven times for each unit cell, resulting in 62 division points that constitute a 124-dimensional shape descriptor.
Using the above method, we can represent each 2D unit cell as a vector, then use the Euclidean norm to find the distance between any pair. However, the input for a DPP is a positive semi-definite similarity matrix, $L$, so we transform the distance to a similarity metric through a radial basis function kernel with unit bandwidth, {\em i.e.}, $L_{i,j}=\exp(-0.5~d(i,j)^2)$, where $d(i,j)$ is the distance between $i$-th and $j$-th unit cells. In practice, the choice of an appropriate transformation is equivalent to choosing the right distance metric between items. Our empirical study on other common transformations showed that different choices mainly affect the distribution of similarity values but do not significantly affect the final outcome or the key findings of our work.
\vspace{6pt}
\textbf{3D Hausdorff Distance: }\label{sec:3dsim_hausd}
As for 3D unit cells, mesh formats such as STL are commonly used so that the metamaterials can be manufactured through additive manufacturing. However, since performing analysis on 3D shapes is undoubtedly more computationally intense due to the curse of dimensionality, we suggest representing each unit cell as points on the surface of the original mesh, {\em i.e.}{}, point clouds, which are more efficient for extracting and processing 3D features~\cite{Kobbelt2004pointcloud}. This extra conversion can take little computation with well-established sampling methods, {\em e.g.}{}, randomly sampling the surface of a mesh with the probability of choosing a point weighted by the area of the triangular faces.
We then use a distance metric commonly utilized to measure the distance between sets of points, the Hausdorff distance. In essence, it computes the difference between two clouds as the maximum of the nearest neighbor distances of each point. This is expressed as \cite{Huttenlocher1993hausdorff}:
\begin{equation}\label{eq:hausdorff}
h(A,B) = \max_{a \in A}{\big[\min_{b \in B}{\lVert \cdot \rVert}\big]},
\end{equation}
where $a$ is a point within cloud $A$ and $b$ is a point in the second cloud $B$. The notation $\lVert \cdot \rVert$ indicates that any distance can be used; for example, we can use the Euclidean norm or the cosine distance between two points. In our implementation, we computed the nearest neighbor Euclidean norms using a GPU-enabled code by Fan {\em et~al.}{}~\cite{Fan2017nndist}. Then, to obtain a symmetric distance, we take the maximum as follows:
\begin{equation}\label{eq:hausdorff2}
d_\text{H}(A,B) = d_\text{H}(B,A)= \max{\big[h(A,B), h(B,A)\big]}.
\end{equation}
Finally, we convert the pairwise distances into a DPP similarity kernel, $L$, using the following transformation: $L_{ij} = \frac{1}{1+d(i,j)}$.
\vspace{6pt}
\textbf{3D Embedded Cosine Similarity:}\label{sec:3dsim_ae}
Alternatively, the embedded features of the unit cells in a given dataset can be extracted using deep learning models as simple as an autoencoder, a dimension reduction technique that compresses, {\em i.e.}{}, encodes, complex shapes into vectors. Once such a model has been trained, an embedding-based shape similarity metric can be defined as the similarity between the vector representations of unit cells, much like the 2D descriptor-based distance earlier.
Here we also leverage point clouds, which are growing as a scalable and powerful representation for 3D deep learning~\cite{Guo2020deeppoints}. We utilize a point cloud autoencoder provided by Achlioptas~{\em et~al.}{}~\cite{achlioptas18ae} with the Earth Mover's distance as the reconstruction loss. Our 3D dataset (described in Sec.~\ref{sec:3dgen}) is split into training, test and validation sets by 70\%, 15\%, and 15\%, respectively, and a grid search is performed to decide the hyperparameters: 64-dimensional embedded vectors for each unit cell, a learning rate of 0.0005 and batch size of 32. After training the model for 120 epochs, we can then take the cosine similarity between the embedded vector representations of any two unit cells as the shape metric. In our 3D experiment (Sec.~\ref{sec:3ddpp}), we compare the diverse subsets obtained using this embedded feature approach against those using the Hausdorff distance.
\subsection{Determinantal Point Processes for Joint Diversity in Two Spaces}\label{sec:dpp}
With a similarity kernel matrix $L$, we can now measure the diversity of a dataset using Determinantal Point Processes (DPPs), which are models of the likelihood of choosing a diverse set of items. They have been used for set selection in ML, {\em e.g.}{}, diverse pose detection and information retrieval~\cite{kulesza2012determinantal,kulesza2011k}, and recently in ranking design ideas based on diversity and quality~\cite{Ahmed2017ranking}.
Viewed as joint distributions over the binary variables that indicate item selection, DPPs capture negative correlations.
This means that, intuitively, the determinant of $L$ is related to the volume that the set covers in a continuous space. In other words, the larger the determinant, the more diverse the set.
To model our data, we construct DPPs through L-ensembles~\cite{borodin2009determinantal}, using a positive semi-definite matrix $L$ to define a DPP. Hence, given the full unit cells dataset of size $N$, which we denote as ground set $G$, DPPs allow us to find the probability of selecting any possible subset $M$ of unit cells as:
\begin{equation} \label{eq:dppk}
\mathbb{P}(M) = \frac{det(L_M)}{det(L+I)},
\end{equation}
where $L_M \equiv [L_{ij}]_{ij \in M}$ is the submatrix of $L$ with entries indexed by elements of the subset $M$, and $I$ is a $N \times N$ identity matrix. The probability of a set containing two items increases as the similarity between them decreases. Therefore, the most diverse subset of any size has the maximum likelihood $\mathbb{P}(M)$, {\em i.e.}{}, the largest determinant.
For a fixed subset size, the denominator can be ignored when maximizing the diversity via an algorithm such as the one described in Sec.~\ref{sec:ranking}.
Unlike submodular clustering approaches, DPPs only require the similarity kernel matrix $L$ as an input, and do not explicitly need the data to be clustered or a function that models diversity to be defined. This also makes them more flexible, since we only need to provide a valid similarity kernel, rather than an underlying Euclidean space or clusters.
For METASET, we calculate two different similarity values~---~one in shape space and another in property space~---~between any two unit cells. Hence, for all the unit cells combined, we have one kernel matrix corresponding to each of the two spaces.
In order to measure the joint coverage in both spaces, we take a weighted sum of the two matrices, thus also allowing the trade-off between diversifying in shape or property space:
\begin{equation} \label{eq:jointkernel}
L = (1-w) \cdot L_P + w \cdot L_S,
\end{equation}
where $L$, $L_P$ and $L_S$ are, respectively, the joint, property and shape similarity kernels, and $w$ is a weight parameter can be varied between 0 and 1. By adding the two kernels, we assume that the total similarity between two unit cells is the weighted average of how similar they are in the shape and property spaces.
While it is possible to combine two kernel matrices in many ways, we choose this formulation for two reasons. First, the weighted sum of two positive semi-definite matrices is also positive semi-definite, which is a pre-requisite for a DPP kernel. Second, it allows us to control the amount of diversity in both spaces, as well as to frame the later subset selection problem as multi-objective one, using a single tuning parameter $w$. We conducted multiple experiments on simulated data with easy-to-verify coverage metrics and found that this approach is effective in capturing diversity in both spaces. For brevity, we have not included these experiments here but directly report and discuss the results using joint kernels for metamaterials in Secs.~\ref{sec:2ddpp} and~\ref{sec:3ddpp}.
\input{algo}
\subsection{Algorithm for Optimizing Diversity}\label{sec:ranking}
Optimizing the diversity of a subset $M$ in two spaces is an inherently multi-objective problem that can be accomplished by maximizing the log determinant of the joint similarity kernel, {\em i.e.}{}, $f=\log [det(L_M)]$. Note that the log determinant of a positive semi-definite matrix is monotonically non-decreasing and submodular.
In general, finding the set of items that maximizes a submodular diversity function is NP-Hard. When solving such problems, a well-known limit due to Feige~\cite{feige2011maximizing} is that any polynomial-time algorithm can only approximate the solution up to $1-\frac{1}{e}\approx 67$\% of the optimal.
However, this is where choosing a submodular function $f$ as the objective comes in handy. It turns out that greedily maximizing this function is guaranteed to achieve the optimality bound~\cite{feige2011maximizing}.
We use this property to substantially accelerate diversity optimization using a scalable greedy algorithm~\cite{nemhauser1978analysis}, which has theoretical approximation guarantees and is widely used in practice.
At each step, the algorithm picks an item, {\em i.e.}, a unit cell, that provides the maximum marginal gain in the objective function (lines 5-8 in Algorithm \ref{alg:alg_metaset}). This makes greedy maximization of diversity the best possible polynomial-time approximation to an otherwise NP-Hard problem.
\section{METASET in Data-Driven 2D Metamaterials Design}\label{sec:2dcase}
Selecting a diverse and economical dataset prior to design can augment the performance and results of any data-driven algorithm. In this section, we demonstrate that this improvement can be achieved by adding METASET to existing data-driven frameworks with little extra cost (Fig.~\ref{fig:flow}) by designing 2D aperiodic mechanical metamaterials that meet desired displacement profiles and constraints on the connectivity of neighboring unit cells.
Given a 2D dataset of unit cells from our previous work (briefly described in Sec.~\ref{sec:2dgen}), we use METASET to select several subsets with differing sizes and diversity scores (Sec.~\ref{sec:2ddpp}).
By employing these subsets to assemble full structures, we study the effects of subset size and diversity on the search process and final designs (Sec.~\ref{sec:2ddesign}). To emphasize that our diverse selection methodology is an advantageous addendum to any data-driven method, we perform the designs with two existing approaches~---~genetic algorithm for an illustrative example, and a two-stage method for a more complex design motivated by practical applications (Sec.~\ref{sec:2ddesign_MRF}).
The design settings, a classic MBB beam and a cantilever, along with the boundary conditions and target displacement profiles (red curves) are shown in Fig.~\ref{fig:2dsettings}. The design objective for both is to minimize the mean squared error (MSE) between the target and achieved displacement profiles. These types of structures, which require spatially varying elastic behavior and therefore benefit from aperiodic configurations and data-driven methods, have been a growing focus in recent research, with applications such as soft robotic grippers and biomedical devices~\cite{Schumacher2015,Mirzaali2018softdevice}.
We deliberately choose these since spatially varying properties are difficult to obtain using conventional methods, particularly when the objective is dependent on the relative spatial distribution of properties rather than an absolute performance value like compliance.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.25\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_2a.png}
\caption{Classic MBB beam (Sec.~\ref{sec:2ddesign})}
\label{fig:2dsetting_mbb}
\end{subfigure}\qquad
\begin{subfigure}[b]{0.6\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_2b.png}
\caption{Cantilever (Sec.~\ref{sec:2ddesign_MRF})}
\label{fig:2dsetting_sine}
\end{subfigure}
\caption{Problem settings of the 2D examples, both of which should achieve the target displacement profiles shown in red.}
\label{fig:2dsettings}
\end{figure}
To support this claim, we attempted to benchmark the performance of a conventional TO approach based on the Solid Isotropic Material with Penalization (SIMP) scheme~\cite{Bendse2004simp} for the MBB problem (Eq.~\ref{eq:ga_problem}), whose sensitivities can be derived using adjoint analysis. Each unit cell is discretized into $50\times 50$ quadrilateral finite elements, and the density of each element is treated as a design variable, $\rho_e \in [0,1]$, where the goal is to converge as close to 0 (void) or 1 (solid) as possible. To eliminate mesh dependency, a sensitivity filter with a radius of 2 is applied. For combinations of different penalty factors, $p\in \{1,3\}$, and volume fraction constraints, $V\in \{0.50, 0.75, 1.0\}$, we minimize the MSE using the Method of Moving Asymptotes (MMA)~\cite{Svanberg1987mma} and the same stopping criteria. All results are infeasible, however, with high MSE ranging from 4.69 to 6137.08 and numerous intermediate densities (more than 95\% of the elements). This underscores the need for more advanced approaches like data-driven design, which have successfully achieved target spatially varying behavior~\cite{Schumacher2015,Coulais2016combinatorial,Zhu2017twoscale,Wang2020smo}. We will leverage two such approaches in the following sections, since the goal of this paper is not to propose new design methods but to select diverse subsets which provide salient advantages to any existing data-driven design framework.
\subsection{Generation of 2D Unit Cells via Topology Optimization and Perturbation}\label{sec:2dgen}
In~\cite{Bostanabad2019gagp,Wang2020smo}, we previously proposed using a combination of TO and stochastic shape perturbation to generate a large dataset of 2D unit cells. To initialize the dataset, we ran density-based TO for each uniformly sampled target property, the components of homogenized elastic tensors, and then iteratively perturbed the shape of the unit cells with the most extreme or uncommon properties. By doing so, we created a dataset of 88,000 unit cells that covered a relatively large property space within reasonable computational cost. Note that we did not build this dataset with geometry in mind, leading to many similar shapes. Also, even though we aimed to fill the less populated regions of the property space by perturbing unit cells in those locations, there is a higher concentration of final unit cells with lower property values (the lower left corners in Fig.~\ref{fig:2d_project_prop}), indicating that the dataset is somewhat imbalanced. For details, please see~\cite{Wang2020smo}.
Before applying METASET, we preprocess the data by randomly sampling unit cells from the original dataset that have a volume fraction greater than 0.70, resulting in 17,380 unit cells. This fraction was chosen so that the chosen unit cells are less likely to have very thin features, which makes them more feasible for manufacturing.
Additionally, when computing shape diversity, if unit cells occupy very different volume fractions, a diverse subset is more likely to be dominated by flimsy, low density structures, whose shapes have the least probability of overlap with other unit cells.
However, as we will show with the design examples, this preprocessing does not impede the chances of designing well-connected structures that met the targets quite well.
\subsection{Diverse 2D Unit Cells}\label{sec:2ddpp}
For the dataset of 17,380 2D unit cells, which we now refer to as the full or ground set $G$, we calculate the property and shape similarity matrices, $L_P$ and $L_S$, respectively, as described in Sec.~\ref{sec:similarity}. Taking their weighted sum forms the joint DPP kernel matrix $L$ (Sec.~\ref{sec:dpp}), whose determinant, $det(L_M)$, scores the diversity in both spaces. To explore this, we rank several subsets using the greedy algorithm from Sec.~\ref{sec:ranking} by varying their sizes, $N_M$, and kernel weights, $w$. From the results, we can make three observations:
\begin{enumerate}
\item By increasing $w$, we shift from ranking a subset based on diversity in the property space alone, to a mixture of both spaces, and to the shape space only. In essence, the trade-off between shape and property diversity can be easily controlled.
\item The correlation coefficient between the shape and property diversity scores of 1,000 random subsets of size five is 0.0047. Similar near-zero correlation is found for other set sizes too. In addition, the correlation between the shape and property similarity values of 100,000 random pairs of unit cells is $-0.0024$. Therefore, our assumption that the joint similarity can be modeled as a weighted sum is appropriate.
\item By observing the joint diversity score of the subsets as more items, {\em i.e.}{} unit cells, are added, we find that the gains in shape and property diversities saturate at approximately $N_M=20$. Thus, a very small number of unit cells are sufficient to cover both spaces.
\end{enumerate}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_3a.png}
\caption{Subset diverse in property space ($w=0$)}
\label{fig:2d_0}
\end{subfigure}
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_3b.png}
\caption{Subset diverse in shape and property spaces ($w=0.5$)}
\label{fig:2d_05}
\end{subfigure}
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_3c.png}
\caption{Subset diverse in shape space ($w=1$)}
\label{fig:2d_10}
\end{subfigure}
\caption{Examples of 2D unit cells from the diverse subsets used in the cantilever and MBB design problems.}
\label{fig:2d_sets}
\end{figure}
Ten example unit cells from the subsets with $w\in\{0,0.5,1\}$ are shown in Fig.~\ref{fig:2d_sets}, where the subset optimized for only shape diversity (Fig.~\ref{fig:2d_10}) displays the most variety of topologies compared to the subset diverse in only properties (Fig.~\ref{fig:2d_0}). Meanwhile, the balanced subset contains a mixture of unit cells akin to both extreme sets (Fig.~\ref{fig:2d_05}). This may be counter-intuitive since similar shapes should have similar mechanical properties. However, note that upon close inspection, the property diverse unit cells exhibit tiny features that lead to low effective elastic property values. Such small details in the shape may lead to a larger change according to the physical simulations and the property similarity metric, {\em i.e.}{}, the Euclidean norm.
Comparing the properties of the unit cells in diverse subsets to the ground and randomly sampled sets (Fig.~\ref{fig:2d_project_prop}), we can confirm that the property diverse subsets cover all regions of the original property space, even the sparsely populated areas. As expected, the shape diverse subset does not do as well, and the random subset contains tight clusters in certain areas. Along with the observation that the diversity scores as well as the similarity values in the shape and property spaces are essentially uncorrelated, these findings confirm that the formulation of the joint kernel $L_M$ as a weighted linear sum (Eq.~\ref{eq:jointkernel}) is effective for controlling the amount of diversity in either space.
Finally, the result that only 20 unit cells is needed to cover the shape and property spaces is quite interesting since a main tenet of data-driven design thus far is that ''more is better''~---~larger datasets provide more candidates from which we can choose compatible unit cells. So, to explore the impact of the subset size on the data-driven approach, we selected the top 20 as well as top 100 ranking unit cells from each subset to move on to the next step: full structure assembly.
\begin{figure}[hb]
\centering
\begin{subfigure}{0.3\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_4a.pdf}
\caption{Property diverse samples}
\label{fig:2d_project_prop_a}
\end{subfigure}
\begin{subfigure}{0.3\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_4b.pdf}
\caption{Random samples}
\label{fig:2d_project_prop_b}
\end{subfigure}
\begin{subfigure}{0.3\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_4c.pdf}
\caption{Shape diverse samples}
\label{fig:2d_project_prop_c}
\end{subfigure}
\caption{The property space of the 2D unit cell subsets optimized for property and shape diversity, and a randomly sampled set, plotted against the full dataset. We observe that property diverse subsets cover the space well, hence it is more likely to have unit cells near any target property combination.}
\label{fig:2d_project_prop}
\end{figure}
\subsection{Illustrative Study on the Effects of Size and Diversity}\label{sec:2ddesign}
We begin by designing a relatively simple classical example from the TO field, the MBB beam, such that its horizontal centerline conforms to the red curve when loaded with a vertical force $F$ (Fig.~\ref{fig:2dsetting_mbb}). Due to the structural symmetry, we only need to design the right half of the beam with $4\times 4$ unit cells, outlined by the solid black lines. The full structure can then be obtained by reflecting over the vertical centerline.
Using subsets of unit cells with varying sizes and levels of diversity for metamaterials design using global optimization, we can elucidate 1) the effect of subset size on the search algorithm's efficiency, and 2) the impact of diversity on the final design performance as well as the compatibility of neighboring unit cells.
We choose the following diverse subsets using METASET:
\begin{itemize}
\setlength{\itemsep}{0pt}
\item $P_{20}$: Property diverse subset of size 20
\item $SP_{20}$: Shape and property diverse subset of size 20
\item $S_{20}$: Shape diverse subset of size 20 diverse
\item $P_{100}$: Property diverse subset of size 100
\item $SP_{100}$: Shape and property diverse subset of size 100
\item $S_{100}$: Shape diverse subset of size 100.
\end{itemize}
In addition, we utilize these sets, which are not diverse and are not selected by our method, as baselines:
\begin{itemize}
\setlength{\itemsep}{0pt}
\item $R_{20}$: Random subset of size 20
\item $R_{100}$: Random subset of size 100
\item $G$: Full dataset of size 17,380.
\end{itemize}
To design the MBB beam, we pass each of the datasets to a global optimization method, which for this example is a single objective genetic algorithm. Although the approach is simple, we chose it to focus on illustrating the effects of subset size and diversity on the final results. It also allows us to restrict our design to the discrete choice of unit cells in our subsets, whereas most gradient-based algorithms for data-driven metamaterials design map continuous design variables to the nearest existing, or interpolated, unit cell in dense databases \cite{Schumacher2015,Zhu2017twoscale}.
Specifically, the genetic algorithm is used to select the combination of unit cells from each given dataset that minimizes the MSE between the achieved and target displacement profiles. In addition, since detached neighbours are not desirable, we add a compatibility constraint by requiring that the number of disconnected unit cells, $N_{dc}$, in the full structure be equal to zero. The optimization problem is formulated as:
\begin{equation}\label{eq:ga_problem}
\begin{aligned}
& \underset{\bm{l}}{\text{minimize}}
& & \frac{1}{n} \lVert \bm{u}(\bm{l})-\bm{u}_t \rVert _2^2 \\
& \text{subject to}
& & \bm{K}(\bm{l})\bm{U} = \bm{F},\\
& & & N_{dc}(\bm{l})=0,\\
& & & l_i \in \{1,2,\cdots,N_M\}, \quad i=1,2,\ldots,N_f,
\end{aligned}
\end{equation}
where $\bm{u}$ is the displacement of $n$ nodes located on the centerline of the structure, $\bm{u}_t$ is the discretized target displacements, $\bm{K}$ is the global stiffness matrix, and $\bm{U}$ and $\bm{F}$ are global displacement and loading vectors, respectively. The number of unit cells in the given dataset is $N_M$ while the number in the full structure is $N_f$, and $\bm{l}=[l_1,l_2,\ldots,l_{N_f}]^T$ is a vector of the indices of the chosen unit cells.
Due to the stochasticity of genetic algorithms, we run the optimization ten times for each dataset and report the MSE of the final topologies in Fig.~\ref{fig:2d_case_bar}. In addition, we show a measure of the connectivity of the final structure: the mean ratio of disconnected pixels on the boundaries of touching unit cells, $r_{dc}$. Similar to $N_{dc}$ in the constraint (Eq.~\ref{eq:ga_problem}), a fully compatible structure should have $r_{dc}$ as zero. The averages of these results are also disclosed in Table~\ref{table:2d_mean}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.55\columnwidth]{Fig_5.pdf}
\caption{The final objective values (MSE) and ratios of disconnectivity ($r_{dc}$) of 10 runs per subset. Lower values are better. The best overall MSE is obtained by $SP_{20}$ and $S_{20}$, and the best $r_{dc}$ by $S_{20}$ and $SP_{20}$.}
\label{fig:2d_case_bar}
\end{figure}
\begin{table}[ht]
\caption{Means of the final results for the MBB example, with the lowest values in \textbf{bold}.}
\centering
\begin{tabular}{@{}llllllllll@{}}
\toprule
& $G$ & $R_{100}$ & $S_{100}$ & $SP_{100}$ & $P_{100}$ & $R_{20}$ & $S_{20}$ & $SP_{20}$ & $P_{20}$ \\ \midrule
MSE & 1.3E+18 & 1.5341 & 0.4278 & 0.6454 & 1.6648 & 1.2395 & 0.2865 & \textbf{0.2017} & 0.4926 \\
$r_{dc}$ & 0.5184 & 0.4770 & 0.3406 & 0.3347 & 0.4653 & 0.4836 & \textbf{0.2488} & 0.2578 & 0.3996 \\ \bottomrule
\end{tabular}
\label{table:2d_mean}
\end{table}
When given the baseline full dataset, $G$, the genetic algorithm is overwhelmed and not able to find any designs with satisfactory MSE (see the high values in Table~\ref{table:2d_mean}), even failing to meet the compatibility constraint in one run. This can be attributed to a vast search space since the number of possible unit cell combinations grows exponentially as the size of the dataset increases. A larger set may also contain more redundant shapes or properties that contribute little to diversity, exacerbating the search challenge and possibility of local optima.
Conversely, every run using the 20- and 100-item subsets satisfy the design requirements (Fig.~\ref{fig:2d_case_bar} and Table~\ref{table:2d_mean}). These include the baseline random subsets selected without our method, which obtain reasonable performance and connectivity due to the reduced search space. The values of MSE and $r_{dc}$ using random subsets, however, vary widely.
In fact, our results highlight that \textit{smaller yet diverse} subsets more consistently outperform all other sets under the same search algorithm and termination criteria. Notably, the lowest mean MSE is reached by the small $SP_{20}$ and $S_{20}$ sets.
Moreover, the best connected structures, {\em i.e.}{}, those with lowest $r_{dc}$, result from the diverse subsets that consider shape, {\em i.e.}{}, $S_{20}$ and $SP_{20}$.
We remark that our optimization problem only constrains the number of disconnected unit cells and does not explicitly minimize $r_{dc}$. Therefore, the shape diverse results naturally attain higher connectivity.
\begin{figure}[ht]
\centering
\begin{subfigure}{0.7\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_6a.pdf}
\caption{Using property diverse subset of size 20 ($P_{20}$), the unit cells are connected, but by small features.}
\label{fig:cons_p20}
\end{subfigure}
\begin{subfigure}{0.7\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_6b.pdf}
\caption{Using shape diverse subset of size 20 ($S_{20}$), we observe superior connectivity between neighboring unit cells.}
\label{fig:cons_s20}
\end{subfigure}
\caption{Final topologies and displacement profiles of the classic MBB beam example with the lowest MSE out of 10 runs using 20-item diverse sets. The full structure after symmetry is shown.}
\label{fig:2dphi}
\end{figure}
Fig.~\ref{fig:2dphi} shows the final topologies and optimal displacement profiles of the runs that achieve the minimum MSE for two datasets. As expected from the worse performance and compatibility, the designs using the full dataset $G$ (not pictured) contain disconnected and oddly matched unit cells.
In a similar vein, the high $r_{dc}$ for property diverse sets correspond to mediocre connectivity, as shown by the $P_{20}$ result in Fig.~\ref{fig:cons_p20}, where neighbors are linked by tiny features.
This can be associated with the observation in Sec.~\ref{sec:2ddpp} that METASET tends to include unit cells with small features as it maximizes property diversity, leading to subsets with less compatible unit cells.
With shape diverse subsets, however, the final designs possess excellent compatibility, such as in Fig.~\ref{fig:cons_s20}, further enforcing the advantages of shape diversity.
Although our constrained genetic algorithm provides satisfactory designs, we must point out that our goal is not to introduce new design methods; this global method was implemented to showcase the impact of subset size and diversity. While more elegant optimization techniques would be better suited for practical applications, we nevertheless believe that the insights gained from this study~---~that selecting diverse subsets can accelerate and benefit metamaterial design~---~can be generalized to other data-driven methods, such as the one in the next section.
\subsection{Additional Study with a Complex Metamaterial Structure}\label{sec:2ddesign_MRF}
In the previous section, a simple example using genetic algorithm demonstrated that data-driven metamaterials design can benefit from small and diverse subsets of unit cells. To validate that this is also true for more sophisticated algorithms and designs, we now test the same hypothesis by combining our diverse subsets with an advanced optimization method we proposed in~\cite{Wang2020smo}, which is described briefly below.
Here we design a cantilever composed of $4 \times 30$ unit cells to achieve a sine-wave shape when a prescribed displacement boundary condition is imposed (Fig.~\ref{fig:2dsetting_sine}). As opposed to the MBB beam, the spatially varying behavior of the cantilever is designed to deform in opposite directions in the left and right halves, and we expect that different regions in the structure will require distinctly contrasting properties.
The prescribed boundary instead of a point load poses an additional challenge. The closest problem to this that has been addressed by traditional TO methods is the compliant mechanism design, which aims to control the ratios between output and input displacements or forces by minimizing the displacement at a particular node. To obtain feasible mechanism designs, however, Deepak~{\em et~al.}{} found in~\cite{deepak2009comparative} that it is necessary to assume a force-displacement relationship, {\em i.e.}{}, a spring, at that output node. In contrast, our problem minimizes the MSE over all nodes along the centerline. Since adding a spring at each of those would significantly deviate from our problem setting, conventional design methods are not plausible.
Due to the difficulty of this problem, or indeed any realistic metamaterials design, searching over larger datasets to locate compatible unit cells while meeting the desired performance is also expensive or even intractable. In our case, we are only able to use the smaller diverse subsets $S_{20}$, $SP_{20}$ and $P_{20}$ introduced earlier, as well as baseline random subsets $R_{20}$.
Since there are 120 macro-elements in the cantilever, this still means that there are $120^{20}$ possible combinations of unit cells for each subset.
For this example, we follow our two-stage optimization framework~\cite{Wang2020smo}, wherein inverse TO is utilized in the first stage to determine the macroscale property distribution, and combinatorial optimization based on weighted graphs is used in the second stage to assemble unit cells that meet the target properties with compatible boundaries.
Specifically, we define the following optimization problem for the first stage:
\begin{equation}\label{eq:Inverse_problem}
\begin{aligned}
& \underset{\bm{C_{e}}}{\text{minimize}}
& & \frac{1}{n} \lVert \bm{u}(\bm{l})-\bm{u}_t \rVert _2^2 \\
& \text{subject to}
& & \bm{K}(\bm{C_{e}})\bm{U} = \bm{F},\\
& & & -\phi(\bm{C_{e}}) \leq 0.\\
\end{aligned}
\end{equation}
Compared to the problem solved via genetic algorithm in the previous section (Eq.~\ref{eq:ga_problem}), this inverse property design directly uses the element stiffness matrix $\bm{C_{e}}$ as design variables, which are constrained by the signed L2 distance field $\phi$ of the property space of the full subset $G$. This inverse problem can be efficiently solved with MMA~\cite{Svanberg1987mma}.
After obtaining the optimized macro-property distribution, we construct a grid-like weighted graph with each node representing an element in the macrostructure, and with edges connecting neighbouring unit cells. We can then view the assembly problem as selecting an index from the given subset to label each node in the graph. The Euclidean distance to the target property is assigned as the nodal weight during this process, and the ratio of disconnectivity, $r_{dc}$ defined in the last section, is assigned as the edge weight for each pair of neighboring nodes. With this graph, we can use a dual decomposition Markov random field (DD-MRF) method~\cite{Komodakis2010} to efficiently find the optimal labels of the graph with the lowest sum of nodal and edge weights, thereby designing a full structure that meets the target properties and is well-connected.
\begin{figure}[ht]
\centering
\includegraphics[width=0.85\columnwidth]{Fig_7.png}
\caption{Optimized structures using different subsets, and their associated displacement profiles, for the cantilever example.}
\label{fig:SineResult}
\end{figure}
Since the labeling problem for the graph is a complex combinatorial optimization process where a large candidate set of unit cells equates to an immense search space, a small subset is required for a higher efficiency. As aforementioned, we use three diverse subsets, $S_{20}$, $SP_{20}$, $P_{20}$, and five subsets randomly selected without METASET, $R_{20}$, each with 20 unit cells as the candidate sets for the second stage. The resulting full structures and their respective MSE values and displacement profiles are shown in Fig.~\ref{fig:SineResult}. We repeat the design using random subsets five times, then plot the mean displacement profile and depict the fluctuation of the results with the shaded area.
By virtue of our weighted graph method, all optimized designs have compatible boundaries. However, the subsets which account for shape diversity, {\em i.e.}{}, $S_{20}$ and $SP_{20}$, include a wider variety of unit cells in the full structure. This can be credited to an observation we made in the previous MBB beam example, that a shape diverse set can provide more compatible pairs, rendering a larger feasible design space for the assembly problem. In addition, we note that although some random subsets can achieve relatively low MSE, this performance is not guaranteed; the mean MSE is still the worst overall. In constrast, the shape and property diverse subset $SP_{20}$ has the lowest MSE value. The reason is that, even with small subsets, shape diversity provides better compatibility while property diversity helps to achieve the target property distribution. This is again in line with our findings that a small yet diverse subset considering shape and properties is a boon for data-driven metamaterials design, and has exciting implications for future works.
\section{METASET for Discovery of Diverse 3D Unit Cell Families}\label{sec:3dcase}
Beyond selecting diverse subsets for direct use in design, another advantage of METASET is eliminating inherent bias by optimizing the diversity of a dataset. We demonstrate this with a 3D study, first introducing a new method based on periodic functions to generate families of unit cells with the same underlying structure but varying densities, which although fast creates a great number of overlapping shapes. Our goal in applying METASET to this 3D data is to sift through the overlaps to discover diverse sets of unique isosurface families, which can subsequently be leveraged for data-driven design or ML of, {\em e.g.}{}, property prediction or generative models (Fig.~\ref{fig:flow}).
Triply periodic isosurface unit cells, whose symmetries follow those of crystal structures~\cite{Wohlgemuth2001bicontinuous}, are often used in 3D mechanical metamaterials design due to excellent surface area-to-performance ratios and manufacturability~\cite{Maskery2018tpms}. In addition, their representation as level-set functions allows the density of the unit cells to be easily manipulated for functionally-graded structures~\cite{Maskery2018tpms,Li2019tpms} and tailorable acoustic bandgaps~\cite{Abueidda2018bandgap}. A level-set function $f(x,y,z) = t$ is an implicit representation of geometry where the $t$-isocontour, {\em i.e.}{}, the points where $f=t$, describes the surface of the structure, while the locations where $f<t$ are solid material, and void where $f>t$. Thus, by varying the isovalue $t$, an entire family of isosurface unit cells with graded densities can be extracted from one level-set function.
The most prevalent type of isosurfaces used in metamaterials design is a special subset known as Triply Periodic Minimal Surfaces (TPMS). However, only a few TPMS families have been used since their functions are complex to derive~\cite{Wohlgemuth2001bicontinuous}. For example, Maskery~{\em et~al.}{} use six families in their design work~\cite{Maskery2018tpms}, while Li~{\em et~al.}{} use four~\cite{Li2019tpms}.
Moreover, it has not been investigated whether these few families cover the gamut of shapes and properties needed for design applications.
Suppose a researcher wishes to design a new functionally-graded 3D metamaterial by tuning the densities of isosurface functions, but does not know beforehand which families would best suit their application. Due to the computational expense of design in 3D, they may desire to select a smaller set of families that can then be used in their optimization method. In this section, we present METASET as a procedure to choose those families such that the resultant subset has large coverage over different properties and shapes. In doing so, we also demonstrate that METASET removes bias in datasets by maximizing diversity.
\subsection{Generation 3D Unit Cell Families using Level-Set Functions}\label{sec:3dgen}
Before selecting diverse families, we must first generate an initial pool to choose from. Thus, to build a large 3D dataset, we propose a new method to create isosurface families based on the level-set functions of crystallographic structure factors, which describe how particles are arranged in a crystal unit cell~\cite{ITC2010crystaltables}. In contrast to most unit cell generation methods, our approach here does not set targets in the property space or use TO, and different from TPMS functions, a larger variety of shapes can be found without complex derivations.
In crystallography, structures that are invariant under the same symmetry operations belong to the same space group, of which there are 230 for 3D structures. For the purposes of our work, we will focus on the 36 cubic groups, No. 195 through 230, to obtain our level-set functions. Experimentally, the space group of a crystal can be determined through, {\em e.g.}{}, X-ray techniques, by scattering radiation off a lattice plane denoted by $(hkl)$, and then observing the diffraction pattern. These symmetric patterns have been analytically modeled as \textit{structure factors}, which are periodic functions of the form:
\begin{equation}\label{eq:SF_analytical}
f_{group,(hkl)}(X,Y,Z) = A + i B,
\end{equation}
where $A = \cos{\big(hX+kY+lZ\big)}$, $B = \sin{\big(hX+kY+lZ\big)}$, $X = 2\pi x$, $Y = 2\pi y$, and $Z = 2\pi z$. The equations of these structure factors are listed in~\cite{ITC2010crystaltables} for all space groups and their allowable $(hkl)$.
We can split each structure factor into six isosurface families by separating $A$ and $B$ in Eq.~\ref{eq:SF_analytical} (inspired by~\cite{Wohlgemuth2001bicontinuous}), and converting them into level-set functions as follows:
\begin{equation}\label{eq:SF_forms}
\begin{aligned}
A_{group,(hkl)}(X,Y,Z) &\leq t, \\
A_{group,(hkl)}(X,Y,Z) &\geq t, \\
A^2_{group,(hkl)}(X,Y,Z) &\leq t^2,
\end{aligned}
\end{equation}
and similarly for $B_{group,(hkl)}$. These, respectively, correspond to setting as solid material the function values that are less than $t$ (Fig.~\ref{fig:iso_demo_tm}), greater than $t$ (Fig.~\ref{fig:iso_demo_tp}), and in between $-t$ and $t$ (leading to a ''thin-walled'' structure; Fig.~\ref{fig:iso_demo_tt}).
\begin{figure}[htb]
\centering
\begin{subfigure}{0.4\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_8a.png}
\caption{Family $A_{229,(001)}(X,Y,Z) \leq t$}
\label{fig:iso_demo_tm}
\end{subfigure}
\begin{subfigure}{0.4\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_8b.png}
\caption{Family $A_{229,(001)}(X,Y,Z) \geq t$}
\label{fig:iso_demo_tp}
\end{subfigure}
\begin{subfigure}{0.4\columnwidth}
\centering
\includegraphics[width=1\columnwidth]{Fig_8c.png}
\caption{Family $A^2_{229,(001)}(X,Y,Z) \leq t^2$}
\label{fig:iso_demo_tt}
\end{subfigure}
\caption{Examples of unit cells from isosurface families generated by the structure factor for space group No. 229 and $(hkl)=(001)$. The effect of increasing $t$ to create a family is shown from left to right.}
\label{fig:iso_demo}
\end{figure}
Thus, instead of using the limited TPMS functions, we can use the structure factors of all 36 cubic space groups and their corresponding $(hkl)$ to generate a greater number of isosurface families for data-driven design. To ensure manufacturability, we also identify the feasible density range of each family by prohibiting internal voids and disconnected features, and eliminate families whose feasible range is $\rho_{max} - \rho_{min} < 0.2$. In this way, we quickly created 294 families without performing property-driven optimization.
Although efficient, this method also causes an imbalance in geometry, since several structure factors differ only by a coefficient and lead to overlapping families. For example, the equations for space groups No. 195 and 196 listed in~\cite{ITC2010crystaltables} are related as $A_{195,(hkl)} = 4 \cdot A_{196,(hkl)}$, and therefore generate the same structures. Next, we demonstrate the prowess of METASET in systematically removing such overlaps when selecting diverse subsets.
\subsection{Diverse 3D Families and Comparison of Shape Similarity Metrics}\label{sec:3ddpp}
While applying METASET to discover unique isosurface families, we also test the impact of the two proposed 3D shape similarity metrics (Sec.~\ref{sec:3dsim_ae}): the Hausdorff distance and the cosine similarity between deep learning-based embeddings.
As the families are comprised of a range of densities and thus shapes and properties, we need to capture the similarities of individual unit cells while assessing the similarities between families. Therefore, we generate 100 samples from each family covering the feasible range identified in the previous section, giving 29,400 unit cells total. Each unit cell is represented as a 4096-dimensional point cloud by first converting its level-set field into a triangle mesh~\cite{Vogiatzis2017mesh}, and then sampling on the triangular faces~\cite{sample_mesh}. We also remove any small disconnected features during post-processing, and find the homogenized elastic tensors of each unit cell using a code modified from~\cite{Dong2018homogenization}.
To quantify the similarity between two families, we assume each family is a collection of points, where each point corresponds to a unit cell. This reduces the problem of finding similarity between two families to one between two point sets using the Hausdorff distance (Eq.~\ref{eq:hausdorff}).
We calculate the similarity between families $C$ and $D$ in two steps: first using one of the 3D metrics to calculate the distance between individual unit cells $c \in C$ and $d \in D$, and then substituting this into the Hausdorff distance to obtain the \textit{inter-familial} distance, $h(C,D)$.
Intuitively, this means that the shape similarity between two families is the maximum of the similarities between closest-in-shape pairs of unit cells.
In property space, the similarity between families is related to the maximum of the pairwise Euclidean distances between each unit cell's properties.
Therefore, rather than simply averaging the features of each family, the inter-familial similarities also consider the diversity of individuals within each family.
In short, we apply METASET to our 3D dataset using two approaches to measure shape similarity:
\begin{itemize}
\setlength{\itemsep}{0pt}
\item H-H: Hausdorff distance between unit cells, followed by Hausdorff distance between families
\item E-H: embedded cosine similarity between unit cells, followed by Hausdorff distance between families.
\end{itemize}
Utilizing both of our shape similarity metrics, along with the property metric, we find diverse subsets of $10$ isosurface families. In addition, we vary the joint diversity weight $w$ between $0$ and $1$ (Eq.~\ref{eq:jointkernel}).
Some example subsets of diverse families are shown in Figs.~\ref{fig:3d_sets} and~\ref{fig:3d_sets_cosine}, where the median sample from each family are pictured. Like the 2D diverse subsets (Sec.~\ref{sec:2ddpp}), the property-only and shape-only sets (Figs.~\ref{fig:3d_0} and \ref{fig:3d_10}, respectively) share very few of the same families.
Intriguingly, the shape diverse sets obtained from either metric contain families generated from the same space group and $(hkl)$, but different level-set forms (Eq.~\ref{eq:SF_analytical}). For example, with the H-H approach, the fourth and fifth items in Fig.~\ref{fig:3d_10} have the equations $A_{213,(011)}\geq t$ and $A_{213,(011)}\leq t$. The same families appear with the E-H approach as the eighth and sixth items. One could think of these as completely different shapes with almost no overlaps, which is further validation of the shape diversity chosen by METASET.
\begin{figure}[htb]
\centering
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_9a.png}
\caption{Families diverse in property space ($w=0$)}
\label{fig:3d_0}
\end{subfigure}
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_9b.png}
\caption{Families diverse in shape and property spaces ($w=0.5$)}
\label{fig:3d_05}
\end{subfigure}
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=1\columnwidth]{Fig_9c.png}
\caption{Families diverse in shape space ($w=1$)}
\label{fig:3d_10}
\end{subfigure}
\caption{Examples of subsets of 3D isosurface families selected by METASET using the H-H shape metric.}
\label{fig:3d_sets}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\columnwidth]{Fig_10.png}
\caption{Shape diverse subset ($w=1$) selected by METASET using the embedding-based E-H shape metric.}
\label{fig:3d_sets_cosine}
\end{figure}
\begin{table}[ht]
\centering
\caption{Shape diversity scores of subsets of 10 isosurface families, evaluated using either the Hausdorff (H-H) or embedded (E-H) shape metrics. The first two columns are for shape diverse subsets selected by METASET; the last shows the maximum of 10,000 random sets. The highest scores of each row in \textbf{bold} indicate that METASET always maximizes the diversity score with respect to the metric used during selection.}
\label{tab:3d_max}
\begin{tabular}{@{}llll@{}}
\toprule
& METASET (H-H) & METASET (E-H) & Random Sampling \\ \midrule
Score (H-H) & \textbf{1.0554E-04} & 6.3690E-05 & 2.8504E-05 \\
Score (E-H) & 1.6250E-13 & \textbf{6.4271E-12} & 5.4262E-14 \\
\bottomrule
\end{tabular}
\end{table}
Comparing the subsets obtained via either similarity metric (Figs.~\ref{fig:3d_10} and~\ref{fig:3d_sets_cosine}), we observe that 6 out of 10 shape diverse families overlap, indicating that the choice of metric does not drastically impact diversification.
This is supported by a correlation coefficient of 0.836 between the H-H and E-H shape similarity kernels, $L_S$, of the shape diverse families.
Additionally, we cross-examine these results by applying the E-H approach to score ($det(L_S)$) the shape diverse subset chosen using H-H, and vice versa. As a baseline, we also randomly sample 10,000 sets of 10 families without METASET and measure their diversity with respect to each metric. The results are reported in Table~\ref{tab:3d_max}, where the greatest (most diverse) scores across each row reveal that the greedy algorithm will maximize the diversity score regardless of the similarity metric employed. Moreover, the subsets chosen by METASET have higher diversity than the random ones no matter which metric is used to evaluate the score.
The high diversity of our subsets can also be seen in Fig.~\ref{fig:3d_dpp_tradeoff}, where their property and shape scores using H-H are plotted against those of the 10,000 random subsets.
Here, 99.74\% of the random sets (which are representative of the distribution of pairwise similarity values for our dataset) still fall short of the optimized subset with the lowest shape diversity score. This is compelling evidence that 1) the original dataset was severely imbalanced, and 2) METASET is able to combat such bias and select more diverse subsets.
Fig.~\ref{fig:3d_dpp_tradeoff} additionally visualizes the trade-off between diversity in the shape and property spaces. Although our greedy algorithm maximizes the joint diversity score, the independent shape and property scores illustrate that, in general, the diversity in one space drops as we select sets that are more diverse in the other.
This trade-off might raise a question as to whether a set of families that are quite diverse in property space can have low diversity in shapes, even though similar shapes are expected to possess similar properties. Our previous observation emerges as an answer: the sets of families with higher diversity in property space and seemingly ``low'' diversity in shape space actually have larger shape scores than the majority of the random sets. Therefore, the highest diversity in property space is achieved by a set of families which are also very diverse in shape.
\begin{figure}[htb]
\centering
\includegraphics[width=0.55\columnwidth]{Fig_11.pdf}
\caption{Trade-off between diversity in property vs. shape (using the H-H approach) spaces. The minimum diversity in shape space for optimized sets has a diversity score greater than 99.74\% of random samples.}
\label{fig:3d_dpp_tradeoff}
\end{figure}
Finally, we note that our diversified sets include isosurface families beyond the common TPMS used in existing metamaterials design, such as the Primitive, Gyroid and Diamond (see~\cite{Maskery2018tpms,Li2019tpms}).
We provide the data of the METASET results publicly so that our diverse families can be employed by any designer in their work as well.
For example, these can be directly utilized in existing functionally-graded design methods such as~\cite{Li2019tpms}. Data-driven design with diverse isosurface families will be investigated in future works.
\section{Discussion}\label{sec:discussion}
Although we illustrated the benefits of METASET with several case studies, there are nevertheless some topics worthy of examining in the future. From our design of 2D aperiodic structures, we saw that shape diverse subsets may increase the chance to find compatible neighboring unit cells, while property diverse sets might enhance problems that require a wider range of target properties at the cost of connectivity. This dependence on shape vs. property diversity extends to ML tasks in the data-driven design framework (Fig.~\ref{fig:flow}) as well. To train property prediction models, one may need a property diverse dataset, while for a deep generative model that learns geometric features, a shape diverse set might be more appropriate. Along these lines, it would be interesting to further validate the improved performance of design and ML tasks using our subsets of diverse 3D unit cell families in a future work.
In the 2D examples, we also observed that smaller subsets led to designs with performance closer to the targets; in fact, we found using METASET that only 20 unit cells were enough to form a diverse subset. In most cases, the benefits of reducing the search space, model training time, or storage requirement of the dataset could outweigh any loss of data. However, certain applications such as ML may need large datasets.
A key benefit of using a METASET, even for large subset sizes, is that it reduces bias by rank ordering all items in the dataset. The items with the highest redundancy in shape or property (like duplicates) are pushed toward the end of the rank-ordered list, so that ML algorithms trained on any subset will be less biased.
While it is not difficult to increase the size of the set, determining how much data is enough is more challenging since this too is contingent on the application and any limits on computational cost.
The effectiveness of size and diversity on specific tasks in metamaterials design is an important question for future studies.
Fortunately, the ease at which a subset's size as well as the weight of shape and property diversity can be explored is yet another advantage of METASET.
Lastly, we remark that the capability of METASET depends on the choice of similarity metrics as well as the definition of the joint similarity kernel, both of which are avenues of further research. Our 3D study demonstrated that METASET will maximize diversity regardless of the metric adopted. Although the results indicated that different shape similarity metrics can be highly correlated and slightly change the diverse subsets, there are a wealth of other choices that may provide different results. Extending METASET to more complex properties, like dynamic ones, may necessitate new metrics. For the joint DPP kernel, we chose a simple weighted sum to join the shape and property matrices, thereby casting the greedy selection as a multi-objective problem. We found in Sec.~\ref{sec:2ddpp} that this was a valid assumption, but other methods to combine kernels while preserving submodularity are also possible. However, swapping these to best suit the application is easily done since the input of the DPPs-based greedy algorithm in METASET is a positive semi-definite similarity kernel that can be obtained from any appropriate metric or definition.
\section{Conclusion}\label{sec:conclusion}
In this paper, we proposed a methodology, METASET, that incorporates joint diversity in the shape and property spaces into data selection to improve the downstream tasks in data-driven design. As an enhancement to any existing data-driven framework, METASET is efficient and flexible, allowing the emphasis on either shape or property to be easily traded by measuring and maximizing the joint diversity of subsets through a weighted DPP similarity kernel. To calculate this kernel matrix, we introduced similarity metrics that cater specifically to 2D and 3D metamaterials.
By way of our 2D aperiodic metamaterial design examples, we demonstrated that small yet diverse subsets of unit cells can boost the scalability of search algorithms while leading to designs with greater performance and enhanced boundary compatibility. This revelation shakes a common belief in the field of data-driven mechanical metamaterials design that a larger and denser dataset is required to design well-connected structures while still meeting the target behavior. To our knowledge, this is the first time that such a result has been studied and presented.
In our 3D case study, we not only proposed a new method to generate triply periodic isosurface unit cells using crystallographic structure factors, but also verified that METASET can effectively discover unique unit cell families in order to build diverse, unbiased and economical datasets for design regardless of the shape similarity metric employed. Different from well-known TPMS unit cells, our dataset of families are optimized for shape and property diversity rather than arbitrarily chosen. In future works, we will explore the use these diverse families for data-driven metamaterials design and ML.
Although this paper focused on showcasing METASET through the design of mechanical metamaterials, the methods we proposed are broadly applicable to other metamaterial domains, or indeed any other design problems that need to balance design space against some performance or quality space. In design ideation, our method can be used to select ideas that are functionally different from each other while achieving different performance goals. It can also be integrated with existing multi-objective optimization algorithms as a niching method. To contribute to the growth and capability of data-driven metamaterials design methods and other fields, we have shared diversified subsets of 2D and 3D unit cells, as well as the corresponding equations of isosurface families. These unit cells can be directly plugged into the application of any metamaterials designer.
\section*{Acknowledgements}
We are grateful for support from the National Science Foundation (NSF) CSSI program (Grant No.~OAC-1835782). Yu-Chin Chan thanks the NSF Graduate Research Fellowship (Grant No.~DGE-1842165). Liwei Wang acknowledges support from the Zhiyuan Honors Program for Graduate Students of Shanghai Jiao Tong University for his predoctoral visiting study at Northwestern University.
\section*{Data Availability}
The diverse datasets autonomously selected by the METASET method can be found at \url{https://github.com/lychan110/metaset}.
\bibliographystyle{unsrtnat}
|
1,314,259,995,830 | arxiv | \section{Introduction}
With the advent of digital antenna arrays, massive multiple input multiple output (MIMO) wireless communications system have become a reality. Multiple user equipments (UEs) can operate on the same frequencies through spatial multiplexing techniques achieved through multi-user detection on the uplink (UL) and multi-user precoding on the downlink (DL). A previous work has shown that when the number of base station (BS) antennas is much larger than the number of UEs, detection and precoding are successfully implemented using linear processing techniques \cite{Hoydis:2013}. It was also noted in \cite{Hoydis:2013} that minimum mean squared error (MMSE) detection more rapidly approaches the massive MIMO UL capacity bound as the number of BS antennas increases, compared to matched filter (MF) detection. Similar results were shown in \cite{Hoydis:2013} for zero-forcing (ZF) precoding compared to time-reversal (TR) precoding for the DL.
Massive MIMO networks with widely dispersive channels have traditionally used orthogonal frequency division multiplexing (OFDM) modulation. OFDM is particulary attractive in these environments since it eliminates inter-symbol interference (ISI) by dividing up the signal into a large number of subcarriers. Despite OFDM's popularity in current wireless standards, this modulation scheme has some drawbacks and is not optimized for every scenario. For example, OFDM is often criticized for its high peak-to-average power ratio (PAPR), which results in a high power back-off to maintain linearity. Consequently, OFDM systems have low power efficiency \cite{Tse:2005}, which is particularly detrimental to battery operated UEs. It also affects DL operations at the BS.
To alleviate the power efficiency problem with OFDM, single carrier modulation (SCM) has been reintroduced in a number of scenarios \cite{Benvenuto:2010}. When framed with a cyclic prefix (CP) similar to OFDM, SCM can be effectively processed in a massive MIMO scenario using frequency domain (FD) processing, achieving the same capacity as OFDM. While MF detection and TR precoding have been shown to be optimal for SCM for operation at low input signal-to-noise ratio (SNR) \cite{Ngo:2013} \& \cite{Larsson:2012}, multi-user interference (MUI) cannot be overcome at moderate to high input SNR unless SCM uses techniques such as MMSE detection \cite{Kenney:Globecom:2020} or ZF precoding \cite{Dinis:2018}. The details of ZF precoding in the FD were presented in \cite{Dinis:2018}. The implementation of the ZF solution is efficient given that the computational complexity, which is dominated by a matix inversion calculation, scales with the number of UEs rather than by the number of BS antennas.
This paper provides the details for a regularized zero-forcing (RZF) precoder in the FD that reuses the MMSE-based matrix inverses that are calculated for the UL detector in \cite{Kenney:Globecom:2020}. Because of the regularization factor in the matrix inverses, there are three special accommodations that are unique to the RZF solution. First, a RZF-specific scale factor must be calculated to maintain the desired power at the BS, depending on the SNR operating point. Second, on the UE detection side, another specialized scale factor is calculated to ensure an unbiased estimate. Both of these scale factors can be pre-computed and accessed through table look-up. The third accommodation results from the fact that RZF precoding does not perfectly cancel MUI, which becomes an issue when there are large variations in large-scale path loss for different UEs. This issue is solved by optimizing the transmit power to each UE such that the output signal-to-interference-plus-noise ratio (SINR) is equal for all UEs. The resulting performance of the RZF precoder exceeds that of the ZF approach for low and moderate input SNR and has the same performance at high SNR. The improved performance of the RZF precoder may seem counterintuitive, but it will be shown that when the regularization term starts to dominate the matrix inverse, the precoder converges to TR precoding, which is optimal at low input SNR. These results are accentuated by the fact that the computational complexity required for the precoder is reduced by an order of magnitude or more compared to the ZF precoder in \cite{Dinis:2018} for relevant numbers of simultaneous UEs.
The remainder of this paper is organized as follows: Section \ref{System_Model} introduces the system model that will be used to discuss DL precoding in the FD; Section \ref{Downlink_Precoding} details the calculations for precoding the multi-user signal at the BS and detecting the signal at each UE; Section \ref{Complexity_Comparison} compares the computational complexity of RZF precoding to ZF precoding; the RZF performance is analyzed in Section \ref{Performance_Analysis}; Section \ref{Simulation_Results} presents the results of the DL simulation to verify the analysis; and concluding remarks are in Section \ref{Conclusion}.
\section{System Model} \label{System_Model}
The time domain duplex (TDD) scenario modeled in this paper assumes that the UEs do not have any channel state information (CSI). Each UE is assumed to have knowledge of the cell parameters such as the number of BS antennas and number of simultaneous users. The BS has the CSI between each UE and each of the $M$ BS antennas, which can be obtained by transmitting pilot signals from the UEs. This paper assumes perfect CSI at the BS. The effects of channel estimation are left to a future work.
The DL transmission is divided into frames, where the BS transmits $N$ unit variance symbols for each user. The symbols designated for the $k^{\textrm{th}}$ UE are represented by the vector $\mathbf{s}_k$. After precoding, a CP is added to the front of each frame to preserve circular convolution. Since each user's frame is transmitted simultaneously with the other users, the frames must be precoded to limit MUI. The precoding generates a unique transmit vector for each of the BS antennas. The received DL signal for user $k$ after CP removal is expressed as
\begin{equation}
\mathbf{r}_k = \frac{ 1 }{ \sqrt{p_k} } \sum_{ m=1}^M \mathbf{H}_{m,k} \mathbf{x}_m + \mathbf{w}_k,
\label{eq:01}
\end{equation}
where $1 / \sqrt{p_k}$ is the scalar path loss between the BS to user $k$, $\mathbf{H}_{m,k}$ is the $N \times N$ circulant convolutional channel matrix between antenna $m$ and user $k$, $\mathbf{x}_m$ is the precoded signal vector for antenna $m$, and $\mathbf{w}_k$ is the receiver noise vector at the UE with variance $\sigma^2_w$. Let $\mathbf{h}_{m,k}$ represent the channel impulse response vector between antenna $m$ and user $k$, which is of length $L_h$. $\mathbf{H}_{m,k}$ is formed by taking $\mathbf{h}_{m,k}$, appending $N-L_h$ zeros to form ${\mathbf{h}_{m,k} }_{(0)}$, and then taking downward cyclic shifts of ${ \mathbf{h}_{m,k} }_{(0)}$ to create
\begin{equation}
\mathbf{H}_{m,k}=
\begingroup
\setlength\arraycolsep{4pt}
\begin{bmatrix}
{ \mathbf{h}_{m,k} }_{(0)} & { \mathbf{h}_{m,k} }_{(1)} \dots & { \mathbf{h}_{m,k} }_{(N-2)} & { \mathbf{h}_{m,k} }_{(N-1)}
\end{bmatrix}
\endgroup,
\label{eq:02}
\end{equation}
where the parenthetical subscript represents the number of downward cyclical shifts applied to the base vector.
For convenience and without loss of generality, the average channel power for each user over all BS antennas, $\frac{1} {M} \sum_{m=1}^M \mathbf{h}_{m,k}^{\textrm{H}} \mathbf{h}_{m,k}$, is normalized to unity. Consequently, the input SNR is ${ 1 }/ {\sigma^2_w}$. Note that the individual channel power values between each of the $M$ antennas and user $k$ are allowed to vary widely.
The approach taken in this paper is to perform precoding in the FD. Taking the $N$-point Discrete Fourier Transform (DFT) of \eqref{eq:01} results in
\begin{equation}
\tilde{ \mathbf{r} }_k = \frac{ 1 }{ \sqrt{p_k} } \sum_{m=1}^M \mathbf{\Lambda}_{m,k} \tilde{ \mathbf{x} }_m + \tilde{ \mathbf{w} }_k,
\label{eq:03}
\end{equation}
where the tilde represents the FD representation of the vectors. The diagonal matrix $\mathbf{\Lambda}_{m,k}$ contains the eigenvalues of $\mathbf{H}_{m,k}$. This results from the fact that the circulant matrix $\mathbf{H}_{m,k}$ is diagonalized by the DFT matrix, $\mbox{\boldmath$\cal F$}$, where $\mbox{\boldmath$\cal F$}$ is scaled such that $\mbox{\boldmath$\cal F$}^{-1} = \mbox{\boldmath$\cal F$}^{\textrm{H}}$ (i.e., $\mathbf{H}_{m,k} = \mbox{\boldmath$\cal F$}^{-1} \mathbf{\Lambda}_{m,k} \mbox{\boldmath$\cal F$}$) \cite{Farhang:Adaptive_Filters}. It follows that taking the $N$-point DFT results in $\mbox{\boldmath$\cal F$} \mathbf{H}_{m,k} = \mathbf{\Lambda}_{m,k} \mbox{\boldmath$\cal F$}$. Let $\lambda_{m,k,i}$ represent the $i^{\textrm{th}}$ value along the diagonal of $\mathbf{\Lambda}_{m,k}$. The eigenvalues can be obtained by taking the $N$-point DFT of the channel impulse response $\mathbf{h}_{m,k}$, which is used to form $\mathbf{H}_{m,k}$. For more efficient computation, it is noted that all of the FD conversions can be performed with the Fast Fourier Transform (FFT) instead of the DFT.
The precoding is performed on a frequency bin basis. We represent the $n^{\textrm{th}}$ bin of the received signal for each user as
\begin{equation} \small
\begin{bmatrix}
\tilde{r}_{1,n} \\
\tilde{r}_{2,n} \\
\vdots \\
\tilde{r}_{K,n}
\end{bmatrix} = \mathbf{P}^{ - \frac{1}{2}}
\begin{bmatrix}
\lambda_{1,1,n} & \lambda_{2,1,n} & \dots & \lambda_{M,1,n} \ \\
\lambda_{2,2,n} & \lambda_{2,2,n} & \dots & \lambda_{M,2,n} \ \\
\vdots \\
\lambda_{1,K,n} & \lambda_{2,K,n} & \dots & \lambda_{M,K,n} \ \\
\end{bmatrix}
\begin{bmatrix}
\tilde{x}_{1,n} \\
\tilde{x}_{2,n} \\
\vdots \\
\tilde{x}_{M,n}
\end{bmatrix} +
\begin{bmatrix}
\tilde{w}_{1,n} \\
\tilde{w}_{2,n} \\
\vdots \\
\tilde{w}_{K,n}
\end{bmatrix},
\label{eq:04}
\end{equation}
where $\mathbf{P}$ is a $K \times K$ diagonal matrix with values of $p_k$ and $ \tilde{x}_{m,n}$ is the precoded value in the FD for antenna $m$ and bin $n$. This can be succinctly represented as
\begin{equation}
\tilde{ \mathbf{r} }_{:,n} = \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \tilde{ \mathbf{x} }_{:,n} + \tilde{ \mathbf{w} }_{:,n}.
\label{eq:05}
\end{equation}
\section{Downlink Precoding and Detection} \label{Downlink_Precoding}
DL precoding is effectively performed in the FD for SCM waveforms with a CP in a massive MIMO scenario. By processing in the FD, the precoding can simultaneously cancel the interference and pre-compensate for ISI. A detailed analysis of DL precoding for SCM with a CP has been presented in \cite{Dinis:2018}, where the zero-forcing (ZF) algorithm is used to calculate the precoded vectors for each antenna. The complexity of the ZF solution is mainly driven by a matrix inverse with dimension $K \times K$. In this section, we take a different approach to the precoding, which is based on reusing the matrix inverse that is calculated for the UL detection as presented in \cite{Kenney:Globecom:2020}. This solution drastically reduces the number of computations for the DL, compared to ZF precoding, without sacrificing performance. Specifically, performance is maintained for high SNR operation and exceeds the ZF performance in the low SNR regime.
One aspect to consider for DL precoding is that the distance from the BS may vary considerably from user to user, resulting in a wide range of large-scale path loss values. Traditionally, the BS transmits the signal intended for each UE at a power corresponding to the inverse of the large-scale path loss in order to achieve the same SNR at each UE. This power variation is not a consideration for UL detection, where power control is assumed. However, we will show that the matrix inverse calculated for the UL under power control conditions is still applicable to DL precoding, and performance can be optimized for large variations in the large-scale path loss by carefully allocating the power for each UE.
\subsection{Zero-Forcing Precoding}
The expression in \eqref{eq:05} shows the FD representation of the $n^{\textrm{th}}$ bin of the received signal vector for each of the UEs. Each UE applies a scale factor to the received signal to obtain the FD symbol estimate (i.e., $\hat{ \mathbf{s} }_k = \mathbf{r}_k / \beta_{\textrm{ZF}}$ in this case). We can express the signal in the FD without the addition of noise as
\begin{equation}
\tilde{ \mathbf{s} }_{:,n} = \frac{1}{ \beta_{\textrm{ZF}} } \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \tilde{ \mathbf{x} }_{:,n}.
\label{eq:06}
\end{equation}
It follows that the ZF precoding presented in \cite{Dinis:2018} is given as
\begin{equation}
\tilde{ \mathbf{x} }_{:,n}^{\textrm{ZF}} = \beta_{\textrm{ZF}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* )^{-1} \mathbf{ P }^{ \frac{1}{2}} \tilde{ \mathbf{s} }_{:,n},
\label{eq:07}
\end{equation}
where $(\ )^*$ represents the complex conjugate operator. In order to maintain the transmission power defined by $\textrm{tr} \left[ \mathbf{P} \right]$, $\beta_{\textrm{ZF}}$ must be set to the square root of $K$ divided by the gain associated with $\mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* )^{-1}$, namely
\begin{align}
\beta_{\textrm{ZF}} &=\sqrt{ \frac{ K } { \mathbb{E} \{ \textrm{tr} [ ( \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* )^{-1} )^{ \textrm{H} } \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* )^{-1} ] \} } } \nonumber \\
&= \sqrt{ \frac{ K } { \mathbb{E} \{ \textrm{tr} [ ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* )^{-1} ] \} } } = \sqrt{ \frac{ K } { \frac{ K } { M - K } } } \nonumber \\
&= \sqrt{ M - K }.
\label{eq:08}
\end{align}
The first part of the second expression of \eqref{eq:08} results from the fact that $ ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* )^{-1}$ is Hermitian symmetric, and the second part follows from Lemma 2.10 of \cite{Verdu:Random_Matrix_Theory}. The ZF precoding only requires a $K \times K$ matrix inversion, which is about the same computational complexity as the detector used in \cite{Kenney:Globecom:2020} for the UL.
\subsection{Regularized ZF Precoding with Matrix Inverse Reuse} \label{Regularized ZF Precoding with Matrix Inverse Reuse}
An alternative approach to calculating the precoded vector is to solve \eqref{eq:06} for $\tilde{\mathbf{x}}_{:,n}$ in the following way (note that $\beta_{\textrm{ZF}}$ is replaced by $\beta_{\textrm{RZF}}$). We start by multiplying both sides of \eqref{eq:06} by $\beta_{\textrm{RZF}} \mathbf{A}_n^* \mathbf{P}^{1/2}$. The next step is to invert the $M$-dimensional square matrix $ \mathbf{A}_n^* \mathbf{A}_n^{\textrm{T}}$. Since $\mathbf{A}_n^*$ is only of rank $K$ and $K < M$, the matrix is singular. By adding a regularization factor to the matrix prior to inverting, we can obtain the following solution:
\begin{equation}
{\tilde{\mathbf{x}}}_{:,n}^{\textrm{RZF}}= \beta_{\textrm{RZF}} \left( \mathbf{A}_n^* \mathbf{A}_n^{\textrm{T}} + \sigma^2_w \mathbf{I}_M \right)^{-1} \mathbf{A}_n^* \mathbf{ P }^{\frac{1}{2}} \tilde{ \mathbf{s} }_{:,n},
\label{eq:09}
\end{equation}
where $\beta_{\textrm{RZF}}$ is a scale factor and $\sigma^2_w \mathbf{I}_M$ is chosen as the regularization term. This regularization term selection will be shown to reuse the matrix inversion calculation performed for the UL. To reduce the size of the matrix inverse, we next apply the matrix inversion lemma from \cite{Woodbury:1950}, which yields
\begin{equation}
\tilde{\mathbf{x}}_{:,n}^{\textrm{RZF}} = \beta_{\textrm{RZF}} \mathbf{A}_n^* \left( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K \right)^{-1} \mathbf{ P }^{\frac{1}{2}} \tilde{ \mathbf{s} }_{:,n}.
\label{eq:10}
\end{equation}
Note that the RZF precoder defined in \eqref{eq:10} is based on a $K \times K$ matrix inversion, which is the complex conjugate of the matrix inversion used for the UL detector in \cite{Kenney:Globecom:2020}. This result is highly desirable because reusing the matrix inverse that was calculated for the UL drastically reduces the computational complexity of the DL precoding.
As in the ZF case, a transmission scale factor is used to maintain an average transmission power of $\textrm{tr} [ \mathbf{P} ]$. Using the same form as \eqref{eq:08}, we have the following equation:
\begin{align}
\beta_{\textrm{RZF}} &=\sqrt{ \frac{ K } { \mathbb{E} \{ \textrm{tr} [ ( \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} )^{ \textrm{H} } \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} ] \} } } \nonumber \\
&= \frac{ 1 } { \sqrt{ \mathbb{E} \left\{ \lambda_{\beta} \right\} } },
\label{eq:11}
\end{align}
where $\lambda_{\beta}$ is the average eigenvalue of the matrix shown in the first expression of \eqref{eq:11}. The second expression in \eqref{eq:11} results from the fact that the trace of a matrix is equal to the sum of its eigenvalues \cite{Strang:1988}. Using the value of $K$ from the numerator, the trace can be replaced by the average eigenvalue of the matrix. The average is dropped, since the expectation is taken. Since $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$ is a Wishart matrix, its eigenvalues have the following probability distribution function (PDF) based on Theorem 2.17 of \cite{Verdu:Random_Matrix_Theory}:
\begin{equation}
f_{ \lambda } (z) = \frac{1}{K} \sum_{ k=0 }^{ K-1 } \left( \frac{ k! \left( L_k^{ M-K } (z) \right)^2 }{ \left( k + M - K \right)! } \right) z^{ M-K } e^{-z}, \label{eq:12}
\end{equation}
where $ L_k^{ M-K } (z)$ is the Laguerre polynomial of order $k$.
After accounting for the differences between the matrix from \eqref{eq:11} and the Wishart matrix $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$, the value of $\lambda_{\beta}$ is calculated as
\begin{equation}
\lambda_{\beta} = \int_0^{ \infty} f_{\lambda} (z) \frac{ z }{ \left( z + \sigma^2_w \right)^2 } dz. \label{eq:13}
\end{equation}
There is no closed form solution to the integral above, but the $\lambda_{\beta}$ value can be calculated for a given value of $\sigma^2_w$ using numerical integration techniques. The resulting value of $\beta_{\textrm{RZF}}$ for values of $K$ and $\sigma^2_w$ can be stored in a 2D look-up table at the base station. An equivalent set of values is also needed at each UE to properly scale the received signal.
\subsection{UE Detection} \label{UE Detection}
As shown in \eqref{eq:05}, the precoded signal is multiplied by $ \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}}$ as it traverses the channel. In order to produce an unbiased estimate of the FD symbols, the received signal must be properly scaled at the UE. The expression for the detected symbols is
\begin{align}
\hat{ \tilde{ \mathbf{s} } }_{:,n} &= \frac{ \alpha } { \beta_{\textrm{RZF}} } \left( \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \tilde{\mathbf{x}}_{:,n}^{\textrm{RZF}} + \mathbf{w}_{:,n} \right) \nonumber \\
&= \alpha \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \mathbf{ P }^{\frac{1}{2}} \tilde{ \mathbf{s} }_{:,n} + \frac{ \alpha \tilde{ \mathbf{w} }_{:,n} }{ \beta_{\textrm{RZF}} },
\label{eq:14}
\end{align}
where the scalar $\alpha$ is included to compensate for the scaling that results from the matrix $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1}$. The unbiased condition is met if $\alpha$ is set such that
\begin{equation}
\tilde{ \mathbf{s} }_{:,n} = \alpha \mathbb{E} \left\{ \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \mathbf{ P }^{\frac{1}{2}} \right\} \tilde{ \mathbf{s} }_{:,n}. \label{eq:15}
\end{equation}
The expectation in \eqref{eq:15} must be set to $\frac{ 1 }{ \alpha } \mathbf{I}_K$ in order for \eqref{eq:15} to be satisfied. This result can be simplified as
\begin{align}
\frac{ 1 }{ \alpha } \mathbf{I}_K &= \mathbb{E} \left\{ \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \mathbf{ P }^{\frac{1}{2}} \right\} \nonumber \\
&= \mathbb{E} \left\{ \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \right\} \nonumber \\
&= \mathbf{I}_K - \sigma_w^2 \mathbb{E} \left\{ ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \right\}. \label{eq:16}
\end{align}
Since $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$ is a Wishart matrix, the expected value of the off-diagonal values of $( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1}$ is zero. The expected value of the diagonal entries is a constant. Hence, \eqref{eq:16} can be further reduced to
\begin{equation}
\frac{ 1 }{ \alpha } \mathbf{I}_K = \mathbf{I}_K - \sigma_w^2 \lambda_{\alpha} \mathbf{I}_K, \label{eq:16a}
\end{equation}
where $\lambda_{\alpha}$ is equal to $\mathbb{E} \left\{ \textrm{tr} \left[ ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \right] \right\} / K$, which is equal to the average eigenvalue of $( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1}$.
Although the expression for $\lambda_{\alpha}$ has no closed form solution, it can be calculated by computing the integral of the probability distribution function (PDF). We note that $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$ is a Wishart matrix, which has the PDF shown in \eqref{eq:12} for its eigenvalues. Using the PDF in \eqref{eq:12}, we can express $\lambda_{\alpha}$ as
\begin{equation}
\lambda_{\alpha} = \int_0^{ \infty} \frac{ f_{\lambda} (z) }{ z + \sigma^2_w } dz. \label{eq:17}
\end{equation}
The value in \eqref{eq:17}, can be readily calculated for a given value of $\sigma^2_w$, $K$, and $M$ using numerical integration techniques. Like the $\beta_{\textrm{RZF}}$ scalar, the values for $\alpha$ can be pre-computed and stored in a 2D table based on $K$ and $\sigma_w^2$. The value of $\lambda_{\alpha}$ is simulated by taking $1/K$ times the average value of the trace of several random instantiations of a Wishart matrix, $\mathbf{W} = \mathbf{A}^{\textrm{T}} \mathbf{A}^*$, where the elements of $\mathbf{A}$ are i.i.d. complex Gaussian, zero-mean, and unit variance. For example, simulated results with $10^4$ iterations match very closely with the results obtained through numerical integration.
Once $\lambda_{\alpha}$ is calculated, it can be used to compute $\alpha$ based on \eqref{eq:16a} as follows:
\begin{equation}
\alpha = \frac{ 1 }{ 1 - \sigma^2_w \lambda_{\alpha} }. \label{eq:18}
\end{equation}
Asymptotes for the value of $\alpha$ are easily attained by examining \eqref{eq:14} in the low and high SNR regimes. In the low SNR case, $\sigma^2_w$ dominates the inverse in \eqref{eq:14} such that it converges to $\frac{1}{ \sigma^2_w } \mathbf{I}_K$. Since the expected value of each diagonal element of $\mathbf{A}_n^{\textrm{H}} \mathbf{A}_n $ is $M$, the asymptotic value for $\alpha$ in the low SNR regime is $\frac{ \sigma^2_w }{ M }$. In the high SNR case, the matrix inverse converges to $( \mathbf{A}_n^{\textrm{H}} \mathbf{A}_n )^{-1}$, resulting in a unity scaling for each symbol. As a result, the asymptotic value for $\alpha$ in the high SNR regime is unity.
\subsection{Power Optimization} \label{Power Optimization}
When the large-scale path loss varies greatly for different UEs, the UEs that are close to the BS will see higher levels of MUI than the UEs that are at the edge of the cell. Based on \eqref{eq:14}, the output SINR for user $k$ is
\begin{equation}
\gamma_k = \frac{ 1 }{ \alpha^2 \sigma^2_{\!\textrm{od}} \frac{ \left( p_{\textrm{total} } - p_k \right) }{ p_k }+ \frac{ \alpha^2 }{ \beta^2_{\textrm{RZF}} } \sigma_w^2 }, \label{eq:19}
\end{equation}
where $\sigma^2_{\!\textrm{od}}$ is the variance for all of the off-diagonal elements of $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1}$, and $p_{\textrm{total} } = \sum_{k=1}^K p_k$. The numerator of \eqref{eq:19} is unity because the scaling previously described results in an unbiased estimate. The first term in the denominator is the MUI, which is based on the power of the other users (i.e., $p_{\textrm{total} } - p_k$). The MUI is divided by $p_k$ due to the large-scale path loss for user $k$. The second term in the denominator is the receiver noise with its appropriate scaling. For the case of equal power with equal large-scale path loss, we find that each UE achieves an output SINR of
\begin{equation}
\gamma_{\textrm{eq}} = \frac{ 1 }{ \alpha^2 \sigma^2_{\!\textrm{od}} \left( K - 1 \right) + \frac{ \alpha^2 }{ \beta^2_{\textrm{RZF}} } \sigma_w^2 }. \label{eq:20}
\end{equation}
In order to achieve the same SINR for each UE when large-scale path loss is not equal, the amplitude scaling of $\mathbf{P}^{1/2}$ in \eqref{eq:10} and \eqref{eq:14} is replaced by the diagonal matrix $\mathbf{Q}^{1/2}$, where $\mathbf{Q}$ has diagonal elements of $q_k$. In order to not introduce any bias from the new scaling, each UE multiplies its received signal by an additional scale factor of $\sqrt{p_k / q_k}$. Consequently, each UE must have knowledge of the channel statistics as well as the amplitude scaling factor applied by the BS. The estimates of the FD symbols in \eqref{eq:14} now change to
\begin{equation}
\hat{ \tilde{ \mathbf{s} } }_{:,n} = \alpha \mathbf{ Q }^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* ( \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* + \sigma^2_w \mathbf{I}_K )^{-1} \mathbf{ Q }^{\frac{1}{2}} \tilde{ \mathbf{s} }_{:,n} + \frac{ \alpha \mathbf{ Q }^{- \frac{1}{2}} \mathbf{ P }^{\frac{1}{2}} \tilde{ \mathbf{w} }_{:,n} }{ \beta_{\textrm{RZF}} }. \label{eq:21}
\end{equation}
The values for $q_k$ are set to achieve the same SINR for each user, $\gamma$, while abiding by the power constraint of $\sum_{k=1}^K q_k = p_{\textrm{total} }$. The output SINR for each user with power optimization is expressed as
\begin{equation}
\gamma = \frac{ 1 }{ \alpha^2 \sigma^2_{\!\textrm{od}} \frac{ \left( p_{\textrm{total} } - q_k \right) }{ q_k } + \frac{ \alpha^2 p_k }{ \beta^2_{\textrm{RZF}} q_k } \sigma_w^2 }. \label{eq:22}
\end{equation}
An expression for the achieved value of $\gamma$ is obtained by rearranging \eqref{eq:22} and summing over all $K$ UEs, which yields
\begin{align}
\sum_{k=1}^K \gamma \left( \alpha^2 \sigma^2_{\!\textrm{od}} \left( p_{\textrm{total} } - q_k \right) + \frac{ \alpha^2 }{ \beta^2_{\textrm{RZF}} } p_k \sigma_w^2 \right) &= \sum_{k=1}^K q_k \nonumber \\
\gamma \left( \alpha^2 \sigma^2_{\!\textrm{od}} \left( K p_{\textrm{total} } - p_{\textrm{total} } \right) + \frac{ \alpha^2 }{ \beta^2_{\textrm{RZF}} } p_{\textrm{total} } \sigma_w^2 \right) &= p_{\textrm{total} } \nonumber \\
\gamma = \frac{ 1 }{ \alpha^2 \sigma^2_{\!\textrm{od}} \left( K - 1 \right) + \frac{ \alpha^2 }{ \beta^2_{\textrm{RZF}} } \sigma_w^2 } &= \gamma_{\textrm{eq}}. \label{eq:23}
\end{align}
The result in \eqref{eq:23} shows that the power optimization produces the same performance that is achieved in the case of equal power with equal large-scale path loss. Equating the expressions in \eqref{eq:22} and \eqref{eq:23} and solving for $q_k$ we arrive at
\begin{equation}
q_k = \frac{ \sigma^2_{\!\textrm{od}} p_{\textrm{total}} + p_k \frac{ \sigma_w^2 }{ \beta^2_{\textrm{RZF}} } }{ K \sigma^2_{\!\textrm{od}} + \frac{ \sigma_w^2 }{ \beta^2_{\textrm{RZF}} } }.
\end{equation}
\section{Complexity Comparison} \label{Complexity_Comparison}
\begin{figure}[!t]
\centering
\includegraphics[width=3.6in, clip=true, trim=4cm 8.5cm 4cm 9cm ]{Complexity_Comparison_DL}
\caption{The number of complex multiplies per bin is plotted versus the number of users for the two precoding implementations and for different numbers of BS antennas (i.e., $M=32$, $64$, and $128$). The ZF precoder, here, follows the formulation presented in \cite{Dinis:2018}. The RZF approach that reuses the matrix inversion has significant savings, especially as $K$ grows large. }
\label{complexity_comp_DL}
\end{figure}
By reusing the matrix inversion from the UL detector, the RZF precoder is much more computationally efficient than the ZF precoder of \cite{Dinis:2018}. A comparison in terms of the number of complex multiplies is presented here. Since both algorithms require conversion to and from the frequency domain, this operation is not considered in the comparison. Likewise, since the amplitude scaling by $\mathbf{P}^{1/2}$ or $\mathbf{Q}^{1/2}$ is the same, it is also excluded even though its contribution is insignificant. As before, we assign $K^3$ complex multiplies for a $K \times K$ matrix inversion. The number of complex multiples involved in a matrix (or vector) multiplication is the product of the outer dimensions times the inner dimension.
The following operations are required for each bin of the ZF precoding in \eqref{eq:07}, resulting in a total of $K^2 + KM + K^3 + K^2 M$ complex multiplies per bin:
\begin{description}
\item[$K^2 M$] Multiply $(K \times M)$ matrix by $(M \times K)$ matrix
\item[$K^3$] Invert the resulting $(K \times K)$ matrix
\item[$K^2$] Multiply $(K \times K)$ matrix by $(K \times 1)$ vector
\item[$KM$] Multiply $(M \times K)$ matrix by $(K \times 1)$ vector
\end{description}
To reuse the matrix inversion from the UL detector, the complex conjugate is taken, which does not require any complex multiplies. Thus, the computational complexity of the RZF precoder in \eqref{eq:10} is reduced to the a total of $K^2 + KM$ multiplies per bin:
\begin{description}
\item[$K^2$] Multiply $(K \times K)$ matrix by $(K \times 1)$ vector
\item[$KM$] Multiply $(M \times K)$ matrix by $(K \times 1)$ vector
\end{description}
Fig. \ref{complexity_comp_DL} shows a comparison of the ZF and the RZF precoder.
\section{Performance Analysis} \label{Performance_Analysis}
The DL performance of the RZF precoder is very similar to the UL detector performance reported in \cite{Kenney:Globecom:2020}. The estimated symbol expression in \eqref{eq:14} will be analyzed for the low-SNR and the high-SNR cases, using the expressions for $\beta_{\textrm{RZF}}$ presented previously for each case.
\subsection{Low SNR Operation}
When the noise variance dominates the expression for the matrix inverse in \eqref{eq:14}, the matrix inverse will converge to the identity matrix scaled by the inverse of the noise variance. The value of $\alpha$ converges to $\sigma^2_w / M$, and $\beta_{\textrm{RZF}} = \sigma^2_w / \sqrt{M}$ in the low-SNR regime as shown below:
\begin{align}
\beta_{\textrm{RZF,Low-SNR}} &=\sqrt{ \frac{ K } { \frac{1}{ \sigma^4_w } \mathbb{E} \left\{ \textrm{tr} \left[ \mathbf{A}_n^{ \textrm{T} } \mathbf{A}_n^* \right] \right\} } } \nonumber \\
&= \sigma^2_w \sqrt{ \frac{ K }{ MK } } \nonumber \\
&= \frac{ \sigma^2_w }{ \sqrt{ M } },
\label{eq:24}
\end{align}
where the second expression results from Lemma 2.9 of \cite{Verdu:Random_Matrix_Theory}.
The resulting expression for the Low-SNR DL symbol estimates in the FD is
\begin{equation}
\hat{\tilde{\mathbf{s}}}_{:,n}^{\textrm{Low-SNR,DL}} = \frac{1}{M} \mathbf{P}^{-\frac{1}{2}} \mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^* \mathbf{P}^{\frac{1}{2}} \tilde{ \mathbf{s} }_{:,n} + \frac{ \tilde{ \mathbf{w} }_{:,n} }{ \sqrt{M}}.
\label{eq:25}
\end{equation}
The effect of the $\mathbf{P}$ matrices were discussed at the end of Section \ref{Power Optimization}. With the aforementioned method of mitigating the effect of large disparities in the path loss between users, we can set $\mathbf{P} = \mathbf{I}_K$ for the performance analysis. With this convention, we see that the expected value of the diagonals of $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$ are equal to $M$, which results in an unbiased estimate. The noise variance of $\sigma^2_w$ is divided by $M$. The resulting SINR is equal to $M/ \sigma^2_w$ (i.e., performance gain of $M$), since the noise dominates the MUI caused by the off-diagonal elements of $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$ in the low-SNR regime. This is an identical result to the low SNR UL case in \cite{Kenney:Globecom:2020}.
\subsection{High SNR Operation}
At high SNR the noise variance becomes vanishingly small. As a result, the matrix inverse in \eqref{eq:14} reduces to $(\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*)^{-1}$, which cancels the $\mathbf{A}_n^{\textrm{T}} \mathbf{A}_n^*$ matrix. Then the $\mathbf{P}^{-\frac{1}{2}}$ term cancels with the $\mathbf{P}^{\frac{1}{2}}$ term. At high SNR, $\alpha$ converges to unity, and $\beta_{\textrm{RZF}}$ converges to $\beta_{\textrm{ZF}}$ in \eqref{eq:08}. The expression for the symbol estimates in the FD is simplified to
\begin{equation}
\hat{\tilde{\mathbf{s}}}_{:,n}^{\textrm{High-SNR,DL}} = \tilde{\mathbf{s}}_{:,n} + \frac{ \tilde{ \mathbf{ w } }_{:,n} } { \sqrt{ M - K } }.
\label{eq:26}
\end{equation}
Based on \eqref{eq:26}, the high-SNR performance has a performance gain of $M-K$ due to the scaling of the noise. Unlike the low SNR case, the near-far effect does not contribute to the interference at high SNR. This is an identical result to the high SNR UL case in \cite{Kenney:Globecom:2020}.
\section{Simulation Results} \label{Simulation_Results}
\begin{figure}[!t]
\centering
\includegraphics[width=3.6in, clip=true, trim=4cm 8.5cm 4cm 9cm ]{FD-MUD_In-Out_Comparison_with_64_Antennas_DL}
\caption{Ratio of output SINR to input SNR (i.e., performance gain) is plotted versus the input SNR for the DL scenario where $M=64$ and $K=16$. Both RZF and ZF performance gains match the values in the performance analysis. RZF has a sizeable advantage over the ZF precoder at low input SNR values. }
\label{DL_performance}
\end{figure}
A single-cell DL scenario was simulated to show the performance of the FD multi-user precoding techniques detailed in this section. During each frame, $N=2048$ samples are transmitted. The BS transmits to $K$ UEs simultaneously. In order to reach each UE with the same power, the BS must transmit more power to more distant UEs with higher large-scale path losses. In this simulation, the excess path loss for each UE is distributed between $0$ dB and $20$ dB. Because of the additional MUI created by the large variation in transmit power of the different signal components, the power optimization method presented in Section \ref{Power Optimization} was simulated. The resulting performance matches the case where the large-scale path losses are equal.
Each channel impulse response between BS antenna $m$ and UE $k$ is randomly selected with an exponential power delay profile and a roll-off factor of 25 samples. The length of the channel impulse response, $L_h$, is set to 130 samples. It should be noted that the performance of this algorithm is independent of the value of $L_h$ as long as $L_h \leq L_{\textrm{CP}} - 1$, where $L_{\textrm{CP}}$ is the length of the CP. The average power of each channel for each user is set to unity, but the individual power values are uniformly distributed between $0.1$ and $2.0$. Slight modifications to the scaling are made to maintain the unity average power constraint after the individual power values are chosen. These results assume perfect CSI is available at the BS.
The main objective of the simulation is to measure the resulting SINR after being detected at the receiver and then averaged across all UEs. Because the BS has $M$ antennas, an ideal antenna gain of $M$ would be expected for a single-user case (i.e., the output SINR could be as high as $M$ times the input SNR), which is also the predicted performance at low input SNR. Based on the value of $\beta_{\textrm{ZF}}$, the ZF precoder performance gain is expected to be $M-K$ over all input SNR values. This is also the predicted performance for the RZF precoder case at high input SNR. Fig. \ref{DL_performance} shows the results of the simulation.
\section{Conclusion} \label{Conclusion}
This paper presented a RZF precoding solution that reuses the matrix inverses calculated as part of the UL detection as presented in \cite{Kenney:Globecom:2020}. Derivations for the scale factors necessary to achieve the transmit power target and to obtain unbiased symbol estimates at the UE were provided. In addition, a transmit power optimization was presented that compensates for MUI effects that reduce the average SINR at the UEs when large variations in the large-scale path loss exist. Because of the reuse of the matrix inverse for each processing bin, it was shown that the RZF precoding solution requires drastically fewer complex multiplies than the ZF precoder. The performance analysis showed that the achievable output SINR from the RZF precoder is greater than that of the ZF precoder for moderate and low input SNR values. It was also shown that the RZF precoder performance converged to that of the ZF precoder for high SNR values. The analysis was verified via simulation of a single-cell massive MIMO scenario with $64$ BS antennas and $16$ UEs. With the low computational complexity and superior performance at moderate input SNR values, we conclude that RZF is the preferred approach to massive MIMO DL precoding for SCM waveforms with a CP.
|
1,314,259,995,831 | arxiv | \section{Introduction}
The transmission eigenvalue problem is at the heart of inverse scattering theory for inhomogeneous media. This eigenvalue problem is a late arrival in scattering theory with its first appearance in \cite{old}, \cite{k22}, in connection with injectivity of the relative scattering operator. Transmission eigenvalues are related to interrogating frequencies for which there is an incident field that doesn't scatterer by the medium. The transmission eigenvalue problem has a deceptively simple formulation, namely two elliptic PDEs in a bounded domain (one governs the wave propagation in the scattering medium and the other in the background that occupies the support of the medium) that share the same Cauchy data on the boundary, but presents a perplexing mathematical structure. In particular, it is a non-selfadjoint eigenvalue problem for a non-strongly elliptic operator, hence the investigation of its spectral properties becomes challenging. Roughly, the spectral properties depend on the assumptions on the contrasts in the media (i.e. the difference of the respective coefficients in each of the equations) near the boundary. Questions central to the inverse scattering theory include: {\it discreteness} of the spectrum that is closely related to the determination of the support of inhomogeneity from scattering data using linear sampling and factorization methods \cite{CCH16}, {\it location} of transmission eigenvalues in the complex plane that is essential to the development of the time domain linear sampling method \cite{CMV20}, and the {\it existence} of transmission eigenvalues as well as the {\it accurate determination} of real transmission eigenvalues from scattering data, which has became important since real transmission eigenvalues could be used to obtain information about the material properties of the scattering media. We refer the reader to \cite{CCH16} for a recent and self-contained introduction to the topic.
This paper concerns the discreteness and location of transmission eigenvalues in the scattering of time-harmonic electromagnetic waves by an inhomogeneous (possibly anisotropic) medium of bounded support. Let us introduce the mathematical formulation of the electromagnetic transmission eigenvalue problem. To this end, let $\Omega$ be an open, bounded subset of $\mathbb{R}^3$ representing the support of the inhomogeneity, which we assume to be of class $C^2$. Let $\varepsilon,\, \mu, \, \hat \varepsilon, \, \hat \mu$ be ($3 \times 3$) symmetric, uniformly elliptic, matrix-valued functions defined in $\Omega$ with $L^\infty(\Omega)$ entries. A complex number $\omega$ is called an eigenvalue of the transmission eigenvalue problem, or a {\it transmission eigenvalue}, associated with $\varepsilon, \, \mu, \,\hat \varepsilon, \, \hat \mu$ in $\Omega$ if there exists a non-zero solution $(E, H, \hat E, \hat H) \in [L^2(\Omega)]^{12}$ of the following system
\begin{equation}\label{sys-ITE}
\left\{\begin{array}{c}
\nabla \times E = i \omega \mu H \\[6pt]
\nabla \times H = - i \omega \varepsilon E
\end{array}\right. \, \mbox{ in } \Omega,\qquad \quad \left\{\begin{array}{c}
\nabla \times \hat E = i \omega \hat \mu \hat H \\[6pt]
\nabla \times \hat H = - i \omega \hat \varepsilon \hat E
\end{array}\right. \, \mbox{ in } \Omega,
\end{equation}
\begin{equation}\label{bdry-ITE}
(\hat E - E) \times \nu = 0\, \mbox{ on } \partial \Omega, \quad \mbox{ and } \quad (\hat H - H) \times \nu = 0 \, \mbox{ on } \partial \Omega,
\end{equation}
where $\nu$ denotes the outward unit normal vector to $\partial \Omega$.
\medskip
The main result that we prove in this paper is stated in \Cref{thm-main} below. For the reader convenience we first must clarify some terminology used in the formulation of this theorem. A $3 \times 3$ matrix-valued function $M$ defined in a subset $O \subset \mathbb{R}^3$ is called isotropic at $x \in O$ if it is proportional to the identity matrix at $x$, i.e., $M(x) = m I$ for some scalar $m = m(x)$ where $I$ denotes the $3 \times 3$ identity matrix. In this case, for the notational ease, we also denote $m(x)$ by $M(x)$. If $M$ is isotropic for $x \in O$, then $M$ is said to be isotropic in $O$. Condition~\eqref{thm-main-cond} below is understood under the convention $m(x) = M(x)$.
\begin{theorem}\label{thm-main} Assume that
\begin{enumerate}
\item[i)] $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$ are of class $C^1$ in some neighborhood of $\partial \Omega$,
\item[ii)] $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$ are isotropic on $\partial \Omega$,
\item[iii)]
\begin{equation}\label{thm-main-cond}
\varepsilon \neq \hat \varepsilon, \quad \mu \neq \hat \mu, \quad \varepsilon/ \mu \neq \hat \varepsilon/ \hat \mu \quad \mbox{ on } \partial \Omega.
\end{equation}
\end{enumerate}
The set of the transmission eigenvalues of \eqref{sys-ITE} and \eqref{bdry-ITE} is discrete with $\infty$ as the only possible accumulation point.
\end{theorem}
The analysis used in the proof of \Cref{thm-main} also allows us to obtain the following result on the transmission eigenvalue free region of the complex plane ${\mathbb C}$.
\begin{proposition}\label{pro} Assume that $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$ are of class $C^1$ in some neighborhood of $\partial \Omega$, isotropic on $\partial \Omega$, and
\begin{equation}\label{thm-main-cond2}
\varepsilon \neq \hat \varepsilon, \quad \mu \neq \hat \mu, \quad \varepsilon/ \mu \neq \hat \varepsilon/ \hat \mu \quad \mbox{ on } \partial \Omega.
\end{equation}
For $\gamma >0$, there exists $\omega_0 > 0$ such that if $\omega \in \mathbb{C}$ with $|\Im(\omega^2)| \ge \gamma |\omega|^2$ and $|\omega| \ge \omega_0$, then $\omega$ is not a transmission eigenvalue.
\end{proposition}
Here and and in what follows, for $z \in \mathbb{C}$, let $\Im(z)$ denote the imaginary part of $z$.
\begin{remark}
\em{Since $\gamma>0$ can be chosen arbitrary small, the result of Proposition \ref{pro} together the fact that $\infty$ is the only accumulation point of the transmission eigenvalues proven in \Cref{thm-main}, imply that all the transmission eigenvalues $\omega$, but finitely many, lie in a wedge of arbitrary small angle.}
\end{remark}
The structure of spectrum of the transmission eigenvalue problem is better understood in the case of scalar inhomogeneous Helmoltz equations. In this case, the transmission eigenvalue problem can be stated as follows. Let $d \ge 2$ and $\Omega$ be an open, bounded Lipschitz subset of $\mathbb{R}^d$ . Let $A_1, A_2$ be two ($d \times d$) symmetric, uniformly elliptic, matrix-valued bounded function defined in $\Omega$ and $\Sigma_1$ and $\Sigma_2$ be two bounded positive functions defined in $\Omega$. A complex number $\omega$ is called an eigenvalue of the transmission eigenvalue problem, or a transmission eigenvalue, if there exists a non-zero solution $(u_1, u_2)$ of the system
\begin{equation}\label{pro1a}
\left\{\begin{array}{lll}
\operatorname{div}(A_1 \nabla u_1) + \omega^2 \Sigma_1 u_1= 0 ~~&\text{ in}~\Omega, \\[6pt]
\operatorname{div}(A_2 \nabla u_2) + \omega^2 \Sigma_2 u_2= 0 ~~&\text{ in}~\Omega,
\end{array} \right.
\end{equation}
\begin{equation}\label{pro1b}
u_1 =u_2, \quad A_1 \nabla u_1\cdot \nu = A_2 \nabla u_2\cdot \nu ~\text{ on } \partial \Omega.
\end{equation}
The discreteness of transmission eigenvalues for the Helmholtz equation has been investigated extensively in the literature. The first discreteness result appeared in \cite{first}, whereas \cite{MinhHung17} proves the state-of-the-art results on the discreteness of transmission eigenvalues for anisotropic background and inhomogeneity under most general assumptions on the coefficients using Fourier and multiplier approaches. More specifically, it is shown in \cite{MinhHung17} that the transmission eigenvalue problem has a discrete spectrum if the coefficients are smooth only near the boundary, and
\begin{enumerate}
\item[i)] $A_1(x), \, A_2(x)$ satisfy the complementing boundary condition with respect to $\nu(x)$ for all $x \in \partial \Omega$, i.e.,
for all $x \in \partial \Omega$ and for all $\xi \in \mathbb{R}^d \setminus \{0\}$ with $\xi \cdot \nu = 0$, we have
\begin{equation*}
\langle A_2 \nu, \nu \rangle \langle A_2 \xi, \xi \rangle - \langle A_2 \nu, \xi \rangle^2 \neq \langle A_1 \nu, \nu \rangle \langle A_1 \xi, \xi \rangle - \langle A_1 \nu, \xi \rangle^2,
\end{equation*}
\item[ii)] $\big\langle A_1(x) \nu(x), \nu(x) \big\rangle \Sigma_1(x) \neq \big\langle A_2(x) \nu(x), \nu(x) \big\rangle \Sigma_2(x)$ for all $x \in \partial \Omega$.
\end{enumerate}
Additional results in \cite{MinhHung17}, also include various combinations of the sign of contrasts $A_1-A_2$ and $\Sigma_1-\Sigma_2$ on the boundary. Previous results on discreteness can be found in \cite{BCH11, LV12, Sylvester12} and references therein. We must emphasize that the conditions i) and ii) are more general than
simply one sign contrasts $A_2 - A_1$ and/or $\Sigma_2 - \Sigma_1$ near the boundary. To complete the picture on the transmission eigenvalue problem in the scalar case, we remark that the first answer to the existence of transmission eigenvalues for one sign contrast in $\Omega$ was given in \cite{PS06} where the authors showed the existence of a few real transmission eigenvalues for the index of refraction sufficiently large, followed by \cite{exist}, \cite{kirsch}, which prove the existence of infinite real transmission eigenvalues removing the size restriction on the index. Completeness of transmission eigenfunctions and first estimates on the counting function are shown in \cite{Robiano13}, \cite{Robbiano16} for $C^\infty$ boundary and coefficients since they use semiclassical analysis and pseudo-differential calculus. Again in $C^\infty$ setting, \cite{Vodev15}, \cite{Vodev18} prove the sharpest known results in the scalar case on eigenvalue free zones and Weyl's law for the scalar case improving an earlier result by \cite{lassi}.
The story of the transmission eigenvalue problem for Maxwell's equations is not as complete as for the scalar case discussed above. Of the first results on discreteness is given by Haddar in \cite{Haddar04} where it considers the case of $\mu = \hat \varepsilon = \hat \mu = I $, and $\varepsilon - I $ invertible in $\Omega$. Chesnel in \cite{Chesnel12} employs the so-called $T$-coercivity to prove discreteness when $\hat \varepsilon = \hat \mu = I $, and $\varepsilon - I $ and $\mu^{-1} - I $ are both greater than $cI$ or both less $-c I$ in $\Omega$ for some positive constant $c$ in a neighborhood of $\partial \Omega$. Cakoni et al. in \cite{CHM15} use an integral equation approach to study discreteness for the case when $\mu = \hat \varepsilon = \hat \mu = I $ and the matrix valued $\varepsilon$ becomes constant not equal to $1$ near the boundary. \Cref{thm-main} therefore adds to this list quite general conditions on the coefficients for which the discreteness holds. To our knowledge, our paper is the first to establish discreteness of transmission eigenvalues for Maxwell's equations without assuming any restrictions on the sign combination of the contrasts $\varepsilon-\hat \varepsilon$ and $\mu - \hat \mu$ near the boundary (as long as they do not change sign up the boundary), and allowing for all the electromagnetic parameters to be inhomogeneous and anisotropic, except for on the boundary where they are isotropic but not necessarily constant as it is often assumed in the literature. For the case of electromagnetic transmission eigenvalue problems, other type of results are rather limited, and we refer the reader to \cite{exist} for the existence of real transmission eigenvalues and \cite{HM18} for the completeness of eigenfunctions for the setting related to the one in \cite{CHM15} mentioned above.
The analysis in this paper is inspired by the concept of complementary conditions suggested by Agmon, Douglis, and Nirenberg in their celebrated papers \cite{ADNI, ADNII} for elliptic systems. For Maxwell's equations, the complementary condition for the Cauchy problems has been recently investigated in \cite{NgSil} for general anisotropic coefficients in the context of negative index metamaterials. To be able to apply the theory of complementing conditions to the Maxwell equations, various forms of the Poincar\'e lemma and Helmholtz decomposition are used with a suitable implementation of local charts. The analysis in this paper is in the spirit of the one developed in \cite{MinhHung17}. The idea is to show that the following system
\begin{equation}\label{sys-ITE-E}
\left\{\begin{array}{c}
\nabla \times E = i \omega \mu H + J_e\mbox{ in } \Omega, \\[6pt]
\nabla \times H = - i \omega \varepsilon E + J_m \mbox{ in } \Omega,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = i \omega \hat \mu \hat H + \hat J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times \hat H = - i \omega \hat \varepsilon \hat E + \hat J_m \mbox{ in } \Omega,
\end{array}\right.
\end{equation}
\begin{equation}\label{bdry-ITE-E}
(\hat E - E) \times \nu = 0 \mbox{ on } \partial \Omega, \quad \mbox{ and } \quad (\hat H - H) \times \nu = 0 \mbox{ on } \partial \Omega,
\end{equation}
is well-posed for some $\omega \in \mathbb{C}$ where $(J_e, J_m, \hat J_e, \hat J_m)$ is the input, which belongs to an appropriate functional space. Moreover, a key fact is to prove that the corresponding transformation which maps the input $(J_e, J_m, \hat J_e, \hat J_m)$ to the output $(E, H, \hat E, \hat H)$ is compact. It is worth mentioning that the compactness is one of the crucial/critical difference between the study of the Maxwell equations and the Helmholtz equation. In our analysis, the functional space for the input is well-chosen so that the compactness property holds (see \eqref{def-Hspace}) for $\omega$ in some domain. For example, these facts hold under the assumptions of \Cref{thm-main} provided that $i \omega = |\omega| e^{i \pi/4}$, i.e., $\omega = |\omega| e^{- i \pi/4}$ and $|\omega|$ is large. To this end, we analyze the corresponding Cauchy problem with constant coefficients in a half-space (\Cref{pro-HS}). Using the decay of Maxwell equations (\Cref{lem-decay}), we can prove the uniqueness for \eqref{sys-ITE-E} and \eqref{bdry-ITE-E}. To establish the existence of a solution, the limiting absorption principle is used, and several processes involving the Fredholm theory of compact operator are applied. Again the choice of the functional space to ensure the compactness plays an important role here in order to use the Fredholm theory. Deriving \eqref{thm-main-cond} and handling the compactness are the key difference in the analysis of this paper and the one for the scalar case \cite{MinhHung17}.
The Cauchy problem also naturally appears in the context of negative index materials after using reflections as initiated in \cite{Ng-Complementary}. The well-posedness and the limiting absorption principle for the Helmholtz equations with sign-changing coefficients
was developed in \cite{Ng-WP} using the Fourier and multiplier approach. Recently, with Sil, the second author investigate these problems for the Maxwell equations \cite{NgSil}. Both papers \cite{Ng-WP}, \cite{NgSil} deal with the stability question of negative index materials and are the starting point for the analysis of the discreteness of transmission eigenvalues for the Helmholtz equation \cite{MinhHung17} and Maxwell's equations in this work. Other aspects and applications of negative index materials involving the stability and instability the Cauchy problem \eqref{sys-ITE-E} and \eqref{bdry-ITE-E} are discussed in \cite{Ng-Superlensing-Maxwell, Ng-CALR-O-M, Ng-Negative-Cloaking-M} and the references therein.
The paper is organized as follows. In \Cref{sect-notation}, we introduce several notations used frequently in this paper. \Cref{sect-HS} is devoted to the analysis in the half space. The main result in this section is \Cref{pro-HS}. Condition~\eqref{thm-main-cond} will appear very naturally there. Finally, we present the proof of \Cref{thm-main} in \Cref{sect-proof}.
The choice of the right functional space plays an important role there so that the Fredholm theory can be applied. The proof of \Cref{pro} is also given in this section.
\section{Notations}\label{sect-notation}
The following notations are used frequently throughout the paper. Denote
$$
\mathbb{R}^3_+ = \Big\{x = (x_1, x_2, x_3) \in \mathbb{R}^3; \; x_3 > 0 \Big\}
$$
and
$$
\mathbb{R}^3_0 = \Big\{x = (x_1, x_2, x_3) \in \mathbb{R}^3; \; x_3 = 0 \Big\}.
$$
Let $\Omega$ be a bounded, open subset of $\mathbb{R}^3$ and of class $C^2$, or $\Omega = \mathbb{R}^3_+$. We define the spaces
\begin{equation*}
H(\operatorname{curl}, \Omega) = \Big\{u \in [L^2(\Omega)]^3; \nabla \times u \in [L^2(\Omega)]^3 \Big\},
\end{equation*}
\begin{equation*}
H_0(\operatorname{curl}, \Omega) = \Big\{u \in H(\operatorname{curl}, \Omega) ; u \times \nu = 0 \mbox{ on } \partial \Omega \Big\},
\end{equation*}
\begin{equation*}
H(\operatorname{div}, \Omega) = \Big\{u \in [L^2(\Omega)]^3 ; \operatorname{div} u \in L^2(\Omega) \Big\}.
\end{equation*}
Set $\Gamma = \partial \Omega$, and for $s = -1/2$, or $1/2$, define the trace space
\begin{equation*}
H^{s}_{\operatorname{div}} (\Gamma) = \Big\{u \in [H^{s}(\Gamma)]^3; u \cdot \nu = 0 \mbox{ and } \operatorname{div}_{\Gamma} u \in H^{s}(\partial \Omega) \Big\}.
\end{equation*}
For a vector field $u$ defined in a subset of $\mathbb{R}^3$, $u_j$ denotes its $j$-th component for $1 \le j \le 3$.
We also denote, for $s > 0$,
\begin{equation}\label{def-Os}
\Omega_s = \Big\{x \in \Omega; \mbox{dist}(x, \partial \Omega) < s \Big\}.
\end{equation}
\section{Analysis on a half space} \label{sect-HS}
In order to simplify presentation, we let $k \in \mathbb{C}$ be $k:= i \omega$. Let $\varepsilon$, $\mu$, $\hat \varepsilon$, $\hat \mu$ be four symmetric, uniformly elliptic matrix-valued functions defined in $\mathbb{R}^3_+$. In this section,
we are interested in the following Cauchy problem for Maxwell's equations in $\mathbb{R}^3_+$, with
$J_e, J_m, \hat J_e, \hat J_m \in L^2(\mathbb{R}^3_+)$ and $f_e, f_m \in H^{-1/2}_{\operatorname{div}} (\mathbb{R}^3_0)$,
\begin{equation}\label{sys-C}
\left\{\begin{array}{c}
\nabla \times E = k \mu H + J_e \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \mathbb{R}^3_+,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = k \hat \mu \hat H + \hat J_e \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times \hat H = - k \hat \varepsilon \hat E + \hat J_m \mbox{ in } \mathbb{R}^3_+,
\end{array}\right.
\end{equation}
and \footnote{$e_3 = (0, 0, 1) \in \mathbb{R}^3$.}
\begin{equation}\label{bdry-C}
(\hat E - E) \times e_3 = f_e \mbox{ on } \mathbb{R}^3_0, \quad \mbox{ and } \quad (\hat H - H) \times e_3 = f_m \mbox{ on } \mathbb{R}^3_0 .
\end{equation}
We begin with proving the following lemma.
\begin{lemma}\label{lem-HS} Let $\gamma > 0$ and $k \in \mathbb{C}$ with $\big| \Im (k^2) \big| \ge \gamma |k|^2 $ and $|k| \ge 1$. Furthermore, let $\Lambda \ge 1$ and $\varepsilon$, $\mu$ be two positive constants such that $\Lambda^{-1} \le \varepsilon, \, \mu \le \Lambda$. For $J_e, J_m \in [L^2(\mathbb{R}^3_+)]^3$, there exists a unique solution $(E, H) \in [L^2(\mathbb{R}^3)]^{6}$ of the system
\begin{equation}\label{lem-HS-sys}
\left\{\begin{array}{c}
\nabla \times E= k \mu H \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \mathbb{R}^3_+, \\[6pt]
E \times e_3 = 0 \mbox{ on } \mathbb{R}^3_0.
\end{array}\right.
\end{equation}
Moreover, for some positive constant $C$ depending only on $\Lambda$ and $\gamma$,
\begin{equation}\label{lem-HS-est1}
\| (E, H) \|_{L^2(\mathbb{R}^3_+)} \le \frac{C}{|k|} \| J_m \|_{L^2(\mathbb{R}^3_+)},
\end{equation}
and if $\operatorname{div} J_m \in L^2(\Omega)$, then
\begin{equation}\label{lem-HS-est2}
\| (E, H) \|_{H^1(\mathbb{R}^3_+)} \le C \left( \| J_m\|_{L^2(\mathbb{R}^3_+)} + \frac{1}{|k|} \| \operatorname{div} J_m \|_{L^2(\mathbb{R}^3_+)} \right).
\end{equation}
\end{lemma}
\begin{remark} \rm We emphasize here that the constant $C$ appearing in \eqref{lem-HS-est1} and \eqref{lem-HS-est2} is independent of $k$.
\end{remark}
\begin{proof} We have, from the system of $(E, H)$,
\begin{equation}\label{lem-HS-sysE}
\nabla \times (\nabla \times E) + k^2 \varepsilon \mu E = k \mu J_m \mbox{ in } \mathbb{R}^3_+.
\end{equation}
Multiplying \eqref{lem-HS-sysE} by $\bar \varphi$ (the conjugate of $\varphi$) with $\varphi \in H_0(\operatorname{curl}, \mathbb{R}^3_+)$, and integrating by parts yields
\begin{equation}\label{lem-HS-bilinear}
\int_{\mathbb{R}^3_+} \langle \nabla \times E, \nabla \times \varphi \rangle + k^2 \varepsilon \mu \int_{\mathbb{R}^3} \langle E, \varphi \rangle
= \int_{\mathbb{R}^3} k \mu \langle J_m, \varphi \rangle.
\end{equation}
Take $\varphi = E$. Since $\big| \Im (k^2) \big| \ge \gamma |k|^2 $ and $|k| \ge 1$, after considering the imaginary part and the real part of \eqref{lem-HS-bilinear}, we obtain
$$
\int_{\mathbb{R}^3_+} |\nabla \times E|^2 + |k|^2 |E|^2 \le C \int_{\mathbb{R}^3_+} |J_m|^2,
$$
which implies \eqref{lem-HS-est1} since $\nabla \times E = k \mu H$ in $\mathbb{R}^3_+$. The uniqueness of $(E, H)$ follows.
To derive \eqref{lem-HS-est2}, we note that
$$
\| \nabla \times E\|_{L^2(\mathbb{R}^3_+)} \le C \| J_m \|_{L^2(\mathbb{R}^3_+)},
$$
$$
\| \operatorname{div} E\|_{L^2(\mathbb{R}^3_+)} \le \frac{C}{|k |} \| \operatorname{div} J_m \|_{L^2(\mathbb{R}^3_+)},
$$
$$
\| E\|_{L^2(\mathbb{R}^3_+)} \le C \| J_m \|_{L^2(\mathbb{R}^3_+)},
$$
$$
E \times e_3 = 0 \mbox{ on } \mathbb{R}^3_+.
$$
It follows from the Gaffney inequality, see e.g. \cite[Theorems 3.7]{GR86}, \cite[Theorem 1]{CDS18}, that $E \in H^1(\mathbb{R}^3_+)]^3$ and
\begin{equation}\label{lem-HS-e1}
\| E \|_{H^1(\mathbb{R}^3_+)} \le C \| J_m \|_{L^2(\mathbb{R}^3_+)} + \frac{C}{|k|} \| \operatorname{div} J_m \|_{L^2(\mathbb{R}^3_+)}.
\end{equation}
We also have
$$
\| \nabla \times H \|_{L^2(\mathbb{R}^3_+)} \le C \| J_m \|_{L^2(\mathbb{R}^3_+)},
$$
$$
\| \operatorname{div} H\|_{L^2(\mathbb{R}^3_+)} = 0,
$$
$$
\| H\|_{L^2(\mathbb{R}^3_+)} \le C \| J_m \|_{L^2(\mathbb{R}^3_+)},
$$
and, since $E \times e_3 = 0 $ on $\mathbb{R}^3_+$,
$$
H \cdot e_3 = 0 \mbox{ on } \mathbb{R}^3_+.
$$
It follows from Gaffney inequality again, see e.g. \cite[Theorems 3.9]{GR86}, \cite[Theorem 1]{CDS18}, that $H \in H^1(\mathbb{R}^3_+)$ and
\begin{equation}\label{lem-HS-e2}
\| H \|_{H^1(\mathbb{R}^3_+)} \le C \| J_m \|_{L^2(\mathbb{R}^3_+)} + \frac{C}{|k|} \| \operatorname{div} J_e \|_{L^2(\mathbb{R}^3_+)}.
\end{equation}
Combining \eqref{lem-HS-e1} and \eqref{lem-HS-e2} yields \eqref{lem-HS-est2}.
To prove the existence of $(E, H)$, we first apply the Lax-Milgram theory for variational formula given in \eqref{lem-HS-bilinear} where the bilinear form is defined by the LHS and the linear functional is defined by the RHS in the Hilbert space $H_0(\operatorname{curl}, \mathbb{R}^3_+)$. We then derive that there exists a solution $E \in H_0(\operatorname{curl}, \mathbb{R}^3_+)$ of \eqref{lem-HS-sysE}. Set
$$
H = \frac{1}{k \mu} \nabla \times E \mbox{ in } \mathbb{R}^3_+.
$$
Then
$$
\nabla \times E = k \mu H \mbox{ in } \mathbb{R}^3_+,
$$
and
$$
\nabla \times H = \frac{1}{k \mu} \nabla \times \big(\nabla \times E \big) = - k \varepsilon E + J_m \mbox{ in } \mathbb{R}^3_+.
$$
In other words, $(E, H) \in [L^2(\mathbb{R}^3_+)]^6$ is a solution of \eqref{lem-HS-sys}. The proof is complete.
\end{proof}
We now state the main result of this section, which plays a key role in the proof of \Cref{thm-main}.
\begin{proposition}\label{pro-HS} Let $\alpha \in \mathbb{C}$ with $\alpha^2 \in \mathbb{R}$ and $|\alpha| = 1$, $\gamma > 0$, $k \in \mathbb{C}$ with $|\Im(k^2)| \ge \gamma |k|^2$, $|\Im(\alpha^2 k^2)| \ge \gamma |k|^2$, and $|k| \ge 1$, and let $\Lambda \ge 1$ and $\varepsilon$, $\mu$, $\hat \varepsilon$, $\hat \mu$ be four positive constants such that
$$
\Lambda^{-1} \le \varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \le \Lambda.
$$
Assume that, for some $\Lambda_1 > 0$
$$
|\varepsilon - \hat \varepsilon| \ge \Lambda_1, \quad |\mu - \hat \mu| \ge \Lambda_1, \quad \mbox{ and } \quad |\varepsilon/ \mu - \hat \varepsilon/ \hat \mu| \ge \Lambda_1.
$$
Let $J_e, J_m, \hat J_e, \hat J_m \in H(\operatorname{div}, \mathbb{R}^3_+)$ with $
(J_{e, 3} - \hat J_{e, 3}, J_{m, 3} - \hat J_{m, 3}) \in [H^{1/2}(\mathbb{R}^3_0)]^2$,
and let $f_e, f_m \in H^{1/2}(\operatorname{div}, \mathbb{R}^3_0)$.
There exists a unique solution $(E, H, \hat E, \hat H) \in [L^2(\mathbb{R}^3)]^{12}$ of the system
\begin{equation}\label{sys-C*}
\left\{\begin{array}{c}
\nabla \times E = k \mu H + J_e \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \mathbb{R}^3_+,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = \alpha k \hat \mu \hat H + \hat J_e \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times \hat H = - \alpha k \hat \varepsilon \hat E + \hat J_m \mbox{ in } \mathbb{R}^3_+,
\end{array}\right.
\end{equation}
\begin{equation}\label{bdry-C*}
(\hat E - E) \times e_3 = f_e \mbox{ on } \mathbb{R}^3_0, \quad \mbox{ and } \quad (\hat H - H) \times e_3 = f_m \mbox{ on } \mathbb{R}^3_0.
\end{equation}
Moreover, we have
\begin{align}\label{pro-HS-est}
C \Big( \| (E, H, \hat E, \hat H) & \|_{H^1(\mathbb{R}^3_+)} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)} \Big) \\[6pt]
\le \quad & \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)}
+ \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \nonumber \\[6pt]
& + \frac{1}{|k|} \| (J_{e, 3} - \hat J_{e, 3}, J_{m, 3} - \hat J_{m, 3}) \|_{H^{1/2}(\mathbb{R}^3_0)}
+ |k|^{1/2} \|(f_e , f_m) \|_{L^2(\mathbb{R}^3_0)} \nonumber\\[6pt]
& + \|(f_e , f_m) \|_{H^{1/2}(\mathbb{R}^3_0)} +
\frac{1}{|k|} \|(\operatorname{div}_{\Gamma} f_e , \operatorname{div}_{\Gamma} f_m) \|_{H^{1/2} (\mathbb{R}^3_0)}, \nonumber
\end{align}
for some positive constant $C$ depending only on $\gamma$, $\Lambda$, and $\Lambda_1$.
\end{proposition}
Recall that, by our convention, $J_{e, 3}, \, J_{m, 3}, \, \hat J_{e, 3}, \hat J_{m, 3}$ denote the third component of $J_{e}, \, J_{m}, \, \hat J_{e}, \hat J_{m}$. It is worth noting that the constant $C$ is independent of $k$.
\begin{proof} Let $(E^1, H^1), \, (E^2, H^2), \, (\hat E^1, \hat H^1), \, (\hat E^2, \hat H^2) \in [L^2(\mathbb{R}^3_+)]^6$ be respectively the unique solutions of the following systems
\begin{equation*}
\left\{\begin{array}{c}
\nabla \times E^1= k \mu H^1 \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H^1 = - k \varepsilon E^1 + J_m \mbox{ in } \mathbb{R}^3_+, \\[6pt]
E^1 \times e_3 = 0 \mbox{ on } \mathbb{R}^3_0,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E^1 = \alpha k \hat \mu \hat H^1 \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times \hat H^1 = - \alpha k \hat \varepsilon \hat E^1 + \hat J_m \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\hat E^1 \times e_3 = 0 \mbox{ on } \mathbb{R}^3_0,
\end{array}\right.
\end{equation*}
\begin{equation*}
\left\{\begin{array}{c}
\nabla \times E^2= k \mu H^2 + J_e \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H^2 = - k \varepsilon E^1 \mbox{ in } \mathbb{R}^3_+, \\[6pt]
H^2 \times e_3 = 0 \mbox{ on } \mathbb{R}^3_0,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E^2 = \alpha k \hat \mu \hat H^2 + \hat J_e \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times \hat H^2 = - \alpha k \hat \varepsilon \hat E^2 \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\hat H^2 \times e_3 = 0 \mbox{ on } \mathbb{R}^3_0.
\end{array}\right.
\end{equation*}
Applying \Cref{lem-HS}, we obtain
\begin{multline}\label{pro-HS-e1}
\| (E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{H^1(\mathbb{R}^3_+)} \\[6pt]
+ |k| \, \| (E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{L^2(\mathbb{R}^3_+)} \\[6pt]
\le C \Big( \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)} +
\frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \Big).
\end{multline}
From the trace theory, we derive from \eqref{pro-HS-e1} that
\begin{multline}\label{pro-HS-e2}
\| (E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{H^{1/2}(\mathbb{R}^3_0)} \\[6pt] \le C \| (E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{H^1(\mathbb{R}^3_+)} \\[6pt]
\le C \Big( \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)} +
\frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \Big).
\end{multline}
We have
\begin{align*}
\|\operatorname{div}_{\Gamma} (H^1 \times e_3) -& \operatorname{div}_{\Gamma} (\hat H^1 \times e_3)\|_{H^{1/2}(\mathbb{R}^3_0)} \\[6pt]
= & \; \|(\nabla \times H^1) \cdot e_3 - (\nabla \times \hat H^1) \cdot e_3 \|_{H^{1/2}(\mathbb{R}^3_0)} \\[6pt]
= &\;
\| (- k \varepsilon E^1 + J_m) \cdot e_3 - (- \alpha k \hat \varepsilon \hat E^1 + \hat J_m) \cdot e_3 \|_{H^{1/2}(\mathbb{R}^3_0)} \\[6pt]
\le & \;C |k| \, \| (E^1, \hat E^1) \|_{H^{1/2}(\mathbb{R}^3_0)} + C \|J_{m, 3} - \hat J_{m, 3} \|_{H^{1/2}(\mathbb{R}^3_0)}.
\end{align*}
It follows from \eqref{pro-HS-e2} that
\begin{multline}\label{pro-HS-e3}
\|\operatorname{div}_{\Gamma} (H^1 \times e_3) - \operatorname{div}_{\Gamma} (\hat H^1 \times e_3)\|_{H^{1/2}(\mathbb{R}^3_0)}
\le C |k|
\Big( \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \\[6pt]
+ \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} + \frac{1}{|k|} \|J_{m, 3} - \hat J_{m, 3} \|_{H^{1/2}(\mathbb{R}^3_0)} \Big).
\end{multline}
Similarly, we obtain
\begin{multline}\label{pro-HS-e4}
\|\operatorname{div}_{\Gamma} (E^1 \times e_3) - \operatorname{div}_{\Gamma} (\hat E^1 \times e_3)\|_{H^{1/2}(\mathbb{R}^3_0)} \le
C |k|
\Big( \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \\[6pt] +
\frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} + \frac{1}{|k|} \|J_{e, 3} - \hat J_{e, 3} \|_{H^{1/2}(\mathbb{R}^3_0)} \Big).
\end{multline}
Using the fact, for $u \in H^1(\mathbb{R}^3_+)$,
\begin{equation}\label{ineq-interpolation}
\int_{\mathbb{R}^3_0} |u|^2 \le 2 \int_{\mathbb{R}^3_+} |u| |\partial_{x_3} u| \le 2 \| u\|_{L^2(\mathbb{R}^3_+)} \| \nabla u\|_{L^2(\mathbb{R}^3_+)},
\end{equation}
we have
\begin{multline*}
\|(E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{L^2(\mathbb{R}^3_0)} \\[6pt]\le C \| (E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{L^2(\mathbb{R}^3_+)}^{1/2} \| (E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{H^1(\mathbb{R}^3_+)}^{1/2}.
\end{multline*}
This yields
\begin{multline}\label{pro-HS-e5}
k^{1/2}\|(E^1, H^1, E^2, H^2, \hat E^1, \hat H^1, \hat E^2, \hat H^2) \|_{L^2(\mathbb{R}^3_0)} \\[6pt]
\le C |k| \, \| (E^1, H^1, \hat E^1, \hat H^1) \|_{L^2(\mathbb{R}^3_+)} + C \| (E^1, H^1, \hat E^1, \hat H^1) \|_{H^1(\mathbb{R}^3_+)}.
\end{multline}
By considering $(E - E^1 - E^2 , H - H^1 - H^2, \hat E - \hat E^1 - \hat E^2, \hat H - \hat H^1 - \hat H^2)$, from \eqref{pro-HS-e1}, \eqref{pro-HS-e2}, \eqref{pro-HS-e3}, \eqref{pro-HS-e4}, and \eqref{pro-HS-e5}, w.l.o.g. one might assume that
$$
J_e = J_m = \hat J_e = \hat J_m = 0 \mbox{ in } \mathbb{R}^3_+.
$$
This will be assumed later on. Thus
\begin{equation}\label{pro-HS-sys-C-*}
\left\{\begin{array}{c}
\nabla \times E = k \mu H \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H = - k \varepsilon E \mbox{ in } \mathbb{R}^3_+,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = \alpha k \hat \mu \hat H \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times \hat H = - \alpha k \hat \varepsilon \hat E \mbox{ in } \mathbb{R}^3_+,
\end{array}\right.
\end{equation}
\begin{equation}\label{pro-HS-bdry-C-*}
(\hat E - E) \times e_3 = f_e \mbox{ on } \mathbb{R}^3_0, \quad \mbox{ and } \quad (\hat H - H) \times e_3 = f_m \mbox{ on } \mathbb{R}^3_0.
\end{equation}
Using the identity for a vector field $A$
$$
\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \Delta A,
$$
we obtain the following equations for $E$ and $\hat E$
\begin{equation}\label{pro-HS-EE}
\left\{\begin{array}{c} \Delta E - k^2 \varepsilon \mu E = 0 \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\Delta \hat E - \alpha^2 k^2 \hat \varepsilon \hat \mu \hat E = 0 \mbox{ in } \mathbb{R}^3_+
\end{array}\right.
\end{equation}
(recall that here the coefficients are all constants). In the following, we denote the Fourier transform with respect to $(x_1, x_2) \in \mathbb{R}^2$ of an appropriate function $u: \mathbb{R}^3_+ \to \mathbb{C}$ by $u^{{\mathcal F}}$, i.e.,
$$
u^{{\mathcal F}}(\xi, x_3) = \frac{1}{2 \pi} \int_{\mathbb{R}^2} u(x_1, x_2, x_3) e^{ - i (x_1 \xi_1 + x_2 \xi_2)} \, d x_1 \, d x_2 \mbox{ for } (\xi, x_3) = (\xi_1, \xi_2, x_3) \in \mathbb{R}^3_+.
$$
Similar notation is used for an appropriate function defined on $\mathbb{R}^3_0$.
Consider the first two equations of the system for $E$ and the first two equations of the system for $\hat E$ in \eqref{pro-HS-EE}. Solving these equations using the Fourier transform with respect to $(x_1, x_2)$ yields
\begin{equation}\label{pro-HS-E}
E^{{\mathcal F}}_j(\xi, x_3) = a_j(\xi) e^{- x_3 \sqrt{|\xi|^2 + k^2 \varepsilon \mu }} \quad \mbox{ in } \mathbb{R}^3_+,
\end{equation}
\begin{equation}\label{pro-HS-hE}
\hat E^{{\mathcal F}}_j(\xi, x_3) = \hat a_j(\xi) e^{- x_3 \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu }} \quad \mbox{ in } \mathbb{R}^3_+,
\end{equation}
for $j=1, 2$, where
$$
a_j(\xi) = E^{{\mathcal F}}_j(\xi, 0) \quad \mbox{ and } \quad \hat a_j(\xi) = \hat E^{{\mathcal F}}_j(\xi, 0) \mbox{ for } \xi \in \mathbb{R}^2.
$$
We then have, with $a = (a_1, a_2)$ and $\hat a = (\hat a_1, \hat a_2)$,
\begin{equation}\label{pro-HS-ab1}
\hat a(\xi) - a(\xi) = h (\xi) \mbox{ where } h(\xi) = - f_e^{{\mathcal F}} (\xi, 0) \times e_3.
\end{equation}
Here and in what follows, we identity a vector $(y_1, y_2, 0) \in \mathbb{R}^3_0$ with $(y_1, y_2) \in \mathbb{R}^2$.
Since $\operatorname{div} E = 0$ in $\mathbb{R}^3_+$, it follows that
$$
\partial_{x_3} E_3 = - (\partial_{x_1} E_1 + \partial_{x_2} E_2) \quad \mbox{ in } \mathbb{R}^3_+.
$$
This implies
$$
\partial_{x_3} E_3^{{\mathcal F}} (\xi, x_3) = - i \xi_1 E_1^{{\mathcal F}} (\xi, x_3) - i \xi_2 E_2^{{\mathcal F}} (\xi, x_3) \quad \mbox{ in } \mathbb{R}^3_+.
$$
Using \eqref{pro-HS-E}, we obtain
\begin{equation*}
\partial_{x_3} E_3^{{\mathcal F}}(\xi, x_3) = - i \xi \cdot a(\xi) e^{- x_3 \sqrt{|\xi|^2 + k^2 \varepsilon \mu}} \quad \mbox{ in } \mathbb{R}^3_+.
\end{equation*}
We thus get
\begin{equation}\label{pro-HS-E3}
E_3^{{\mathcal F}}(\xi, x_3) = - \int_{x_3}^\infty i \xi \cdot a(\xi) e^{- s \sqrt{|\xi|^2 + k^2 \varepsilon \mu}} \, d s = \frac{ i \xi \cdot a(\xi) e^{- x_3 \sqrt{|\xi|^2 + k^2 \varepsilon \mu}}}{\sqrt{|\xi|^2 + k^2 \varepsilon \mu}} \quad \mbox{ in } \mathbb{R}^3_+.
\end{equation}
Similarly, we have
\begin{equation}\label{pro-HS-hE3}
\hat E_3^{{\mathcal F}}(\xi, x_3) = \frac{ i \xi \cdot \hat a(\xi) e^{- x_3 \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} }}{\sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu}} \quad \mbox{ in } \mathbb{R}^3_+.
\end{equation}
Since $\hat H \times e_3 - H \times e_3 = f_m$ on $\mathbb{R}^3_0$, and $\nabla \times H = - k \varepsilon E$ and $\nabla \times \hat H = - \alpha k \hat \varepsilon \hat E$ in $\mathbb{R}^3_+$, it follows that
$$
\alpha \hat \varepsilon \hat E_3 - \varepsilon E_3 = - \frac{1}{k} \operatorname{div}_{\mathbb{R}^3_0} f_m \quad \mbox{ on } \mathbb{R}^3_0.
$$
Using \eqref{pro-HS-E3} and \eqref{pro-HS-hE3}, we derive that
\begin{equation}\label{pro-HS-ab2}
\frac{\alpha \hat \varepsilon \xi \cdot \hat a(\xi)}{\sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu}} - \frac{\varepsilon \xi \cdot a(\xi)}{\sqrt{|\xi|^2 + k^2 \varepsilon \mu}} = g: = \big( \frac{i}{k} \operatorname{div}_{\mathbb{R}^3_0} f_m \big)^{{\mathcal F}} \quad \mbox{ on } \mathbb{R}^2.
\end{equation}
Combining \eqref{pro-HS-ab1} and \eqref{pro-HS-ab2}, and noting $a = \hat a - h$, yield, on $\mathbb{R}^2$,
\begin{equation*}
\xi \cdot \hat a \left( \frac{\alpha \hat \varepsilon}{\sqrt{|\xi|^2 + \alpha k^2 \hat \varepsilon \hat \mu}} - \frac{ \varepsilon }{\sqrt{|\xi|^2 + k^2 \varepsilon \mu}} \right) = - \frac{\varepsilon}{\sqrt{|\xi|^2 + k^2 \varepsilon \mu}} \xi \cdot h + g,
\end{equation*}
which implies
\begin{multline*}
\xi \cdot \hat a = \frac{ \sqrt{|\xi|^2 +k^2 \varepsilon \mu } \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu } \Big(\varepsilon \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} + \alpha \hat \varepsilon \sqrt{|\xi|^2 + k^2 \varepsilon \mu} \Big)}{(\alpha^2 \hat \varepsilon^2 - \varepsilon^2) |\xi|^2 + \alpha^2 k^2 \varepsilon \hat \varepsilon \mu \hat \mu (\hat \varepsilon/ \hat \mu - \varepsilon/ \mu)} \times \\[6pt]
\times \left( - \frac{\varepsilon}{\sqrt{|\xi|^2 + k^2 \varepsilon \mu}} \xi \cdot h + g \right).
\end{multline*}
Since $\varepsilon \neq \hat \varepsilon$, $\varepsilon/\mu \neq \hat \varepsilon/ \hat \mu$, $\alpha^2 = \pm 1$, and $|\Im (k^2)| \ge \gamma |k|^2$, $|k| \ge 1$, we get
\begin{equation}\label{pro-HS-em}
|(\alpha^2 \hat \varepsilon^2 - \varepsilon^2) |\xi|^2 + \alpha^2 k^2 \varepsilon \hat \varepsilon \mu \hat \mu (\hat \varepsilon/ \hat \mu - \varepsilon/ \mu)| \ge C (|\xi|^2 + |k|^2).
\end{equation}
We deduce that
\begin{equation*}
|\xi \cdot \hat a(\xi) | \le C \Big(|\xi \cdot h(\xi)| + \sqrt{|\xi|^2 + |k|^2} |g(\xi)| \Big),
\end{equation*}
which yields, since $a = \hat a - h$,
\begin{equation}\label{pro-HS-est1-1}
|\xi \cdot a(\xi) | + |\xi \cdot \hat a(\xi) | \le C \Big(|\xi \cdot h(\xi)| + \sqrt{|\xi|^2 + |k|^2} |g(\xi)| \Big).
\end{equation}
We have, in $\mathbb{R}^3_+$,
\begin{equation*}
k \mu H_1 = \partial_{x_2} E_3 - \partial_{x_3} E_2, \quad \alpha k \hat \mu \hat H_1 = \partial_{x_2} \hat E_3 - \partial_{x_3} \hat E_2.
\end{equation*}
Since $\hat H_1 - H_1 = f_{m, 2} : = f_m \cdot e_2$ with $e_2 = (0, 1, 0)$ on $\mathbb{R}^3_0$, it follows from \eqref{pro-HS-E}, \eqref{pro-HS-hE}, \eqref{pro-HS-E3}, \eqref{pro-HS-hE3} that
\begin{multline*}
\frac{1}{\alpha \hat \mu} \left(- \frac{ \xi_2 \xi \cdot \hat a(\xi) }{\sqrt{|\xi|^2 +\alpha^2 k^2 \hat \varepsilon \hat \mu}} + \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} \hat a_2(\xi) \right) \\[6pt]
=\frac{1}{\mu} \left( - \frac{ \xi_2 \xi \cdot a(\xi) }{\sqrt{|\xi|^2 + k^2 \varepsilon \mu}} + \sqrt{|\xi|^2 + k^2 \varepsilon \mu} a_2(\xi) \right) + k f_{m, 2}^{{\mathcal F}} (\xi).
\end{multline*}
We derive from \eqref{pro-HS-ab1} that
\begin{multline}
\frac{1}{\alpha \hat \mu} \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} \hat a_2(\xi) - \frac{1}{\mu} \sqrt{|\xi|^2 + k^2 \varepsilon \mu} \hat a_2(\xi) \\[6pt]
= \left(\frac{ \xi \cdot \hat a(\xi) }{\alpha \hat \mu \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} } - \frac{ \xi \cdot a(\xi) }{\mu \sqrt{|\xi|^2 + k^2 \varepsilon \mu} } \right) \xi_2 - \frac{1}{\mu} \sqrt{|\xi|^2 + k^2 \varepsilon \mu} h_2(\xi)
+ k f_{m, 2}^{{\mathcal F}} (\xi).
\end{multline}
We thus obtain
\begin{multline}\label{pro-HS-E*}
\hat a_2(\xi) = \alpha \frac{ \mu \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} + \alpha \hat \mu \sqrt{|\xi|^2 + k^2 \varepsilon \mu}}{ (\mu^2 - \alpha^2 \hat \mu^2) |\xi|^2 + \alpha^2 k^2 \varepsilon \hat \varepsilon \mu \hat \mu (\mu/ \varepsilon - \hat \mu/ \hat \varepsilon) } \\[6pt]
\times
\left\{ \left(\frac{\mu \xi \cdot \hat a (\xi)}{\alpha \sqrt{|\xi|^2 + \alpha^2 k^2 \hat \varepsilon \hat \mu} } - \frac{\hat \mu \xi \cdot a(\xi)}{\sqrt{|\xi|^2 + k^2 \varepsilon \mu} } \right) \xi_2 - \hat \mu \sqrt{|\xi|^2 + k^2 \varepsilon \mu} h_2(\xi) + k \mu \hat \mu f_{m, 2}^{{\mathcal F}}(\xi) \right\}.
\end{multline}
Since $\mu \neq \hat \mu$, $\varepsilon/\mu \neq \hat \varepsilon/ \hat \mu$, $\alpha^2 = \pm 1$, and $|\Im( k^2)| \ge \gamma |k|^2$, $|k| \ge 1$, we get
\begin{equation}\label{pro-HS-em-2}
|(\mu^2 - \alpha^2 \hat \mu^2) |\xi|^2 + \alpha^2 k^2 \varepsilon \hat \varepsilon \mu \hat \mu (\mu/ \varepsilon - \hat \mu/ \hat \varepsilon)| \ge C (|\xi|^2 + |k|^2).
\end{equation}
Using \eqref{pro-HS-em-2}, we derive from \eqref{pro-HS-E*} that
$$
|\hat a_2(\xi)| \le \frac{C |\xi|}{|\xi|^2 + |k|^2} \Big( |\xi \cdot a (\xi)| + |\xi \cdot \hat a (\xi)| \Big) + C \Big( |h_2(\xi)| + |f_{m, 2}^{{\mathcal F}}(\xi)| \Big),
$$
which yields, since $\hat a - a = h$,
\begin{equation}\label{pro-HS-est1-2}
|a_2(\xi)| + |\hat a_2(\xi)| \le \frac{C |\xi|}{|\xi|^2 + |k|^2} \Big( |\xi \cdot a (\xi)| + |\xi \cdot \hat a (\xi)| \Big) + C \Big( |h(\xi)| + |f_{m}^{{\mathcal F}}(\xi)| \Big).
\end{equation}
Combining \eqref{pro-HS-est1-1} and \eqref{pro-HS-est1-2}
yields
\begin{equation*}
(|a_2(\xi)|^2 + |\hat a_2(\xi)|^2 ) \sqrt{|\xi|^{2} + |k|^2}
\le C (|h|^2 + |f_{m}^{{\mathcal F}}|^2\big) \sqrt{|\xi|^2 + |k|^2} + C |g(\xi)|^2 |\xi|.
\end{equation*}
From the definition of $g$ and $h$, we obtain
\begin{multline}\label{pro-HS-est2}
\int_{\mathbb{R}^2} (|a_2(\xi)|^2 + |\hat a_2(\xi)|^2 ) \sqrt{|\xi|^{2} + |k|^2} \\[6pt]
\le C \left( \| (f_e, f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 + |k| \, \| (f_e, f_m) \|_{L^2(\mathbb{R}^3_0)}^2 + \frac{1}{|k|^2} \| (\operatorname{div}_\Gamma f_e, \operatorname{div}_\Gamma f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 \right).
\end{multline}
Similarly, we reach
\begin{multline}\label{pro-HS-est3}
\int_{\mathbb{R}^2} (|a_1(\xi)|^2 + |\hat a_1(\xi)|^2 ) \sqrt{|\xi|^{2} + |k|^2} \\[6pt]
\le C \left( \| (f_e, f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 + |k| \, \| (f_e, f_m) \|_{L^2(\mathbb{R}^3_0)}^2 + \frac{1}{|k|^2} \| (\operatorname{div}_\Gamma f_e, \operatorname{div}_\Gamma f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 \right).
\end{multline}
On the other hand, from \eqref{pro-HS-E}, \eqref{pro-HS-hE}, \eqref{pro-HS-E3}, and \eqref{pro-HS-hE3}, we have
\begin{equation}\label{pro-HS-est4}
\int_{\mathbb{R}^3_+} |(\nabla E, \nabla \hat E)|^2 + |k|^2 |(E, \hat E)|^2 \le C \int_{\mathbb{R}^2} |(a(\xi), \hat a(\xi)|^2 \sqrt{|\xi|^{2} + |k|^2}.
\end{equation}
Combining \eqref{pro-HS-est2}, \eqref{pro-HS-est3}, and \eqref{pro-HS-est4} yields
\begin{multline}\label{pro-HS-est5}
\int_{\mathbb{R}^3_+} |(\nabla E, \nabla \hat E)|^2 + |k|^2 |(E, \hat E)|^2 \\[6pt]
\le C \left( \| (f_e, f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 + |k| \, \| (f_e, f_m) \|_{L^2(\mathbb{R}^3_0)}^2 + \frac{1}{|k|^2} \| (\operatorname{div}_\Gamma f_e, \operatorname{div}_\Gamma f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 \right).
\end{multline}
Similarly, we obtain
\begin{multline}\label{pro-HS-est6}
\int_{\mathbb{R}^3_+} |(\nabla H, \nabla \hat H)|^2 + |k|^2 |(H, \hat H)|^2 \\[6pt]
\le C \left( \| (f_e, f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 + |k| \, \| (f_e, f_m) \|_{L^2(\mathbb{R}^3_0)}^2 + \frac{1}{|k|^2} \| (\operatorname{div}_\Gamma f_e, \operatorname{div}_\Gamma f_m) \|_{H^{1/2}(\mathbb{R}^3_0)}^2 \right).
\end{multline}
The conclusion now follows from \eqref{pro-HS-est5} and \eqref{pro-HS-est6}. The proof is complete.
\end{proof}
As a consequence of \Cref{pro-HS}, we obtain
\begin{corollary}\label{cor-HS}
Let $\alpha \in \mathbb{C}$ with $\alpha^2 \in \mathbb{R}$ and $|\alpha| = 1$, $\gamma > 0$, $k \in \mathbb{C}$ with $|\Im(k^2)| \ge \gamma |k|^2$, $|\Im(\alpha^2 k^2)| \ge \gamma |k|^2$, and $|k| \ge 1$, and let $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \in [L^\infty(\mathbb{R}^3_+)]^{3 \times 3}$ be symmetric, uniformly elliptic, and of class $C^1$. Let $\Lambda \ge 1$ be such that
$$
\Lambda^{-1} \le \varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \le \Lambda \mbox{ in } B_1 \cap \mathbb{R}^3_+ \quad \mbox{ and } \quad \|(\varepsilon, \mu, \hat \varepsilon, \hat \mu) \|_{C^1(\mathbb{R}^3_+ \cap B_1)} \le \Lambda.
$$
Assume that $\varepsilon(0), \, \hat \varepsilon(0), \, \mu(0), \hat \mu(0)$ are isotropic, and for some $\Lambda_1 \ge 0$
$$
|\varepsilon (0) - \hat \varepsilon(0)| \ge \Lambda_1, \quad |\mu(0) - \hat \mu(0)| \ge \Lambda_1, \quad \mbox{ and } \quad |\varepsilon(0)/ \mu (0) - \hat \varepsilon(0)/ \hat \mu(0)| \ge \Lambda_1.
$$
Let $J_e, J_m, \hat J_e, \hat J_m \in H(\operatorname{div}, \mathbb{R}^3_+)$ and assume that $(E, H, \hat E, \hat H) \in [L^2(\mathbb{R}^3)]^{12}$ is a solution of \eqref{sys-C*} and \eqref{bdry-C*} with $f_e = f_m = 0$. There exist $0 < r_0 < 1$ and $k_0 > 1$ depending only on $\gamma$, $\Lambda$, and $\Lambda_1$ such that if the supports of $E, \, H, \, \hat E, \, \hat H$ are in $B_{r_0} \cap \overline{\mathbb{R}^3_+}$ and $|k| \ge k_0$, then
\begin{multline}\label{cor-HS-est}
\| (E, H, \hat E, \hat H) \|_{H^1(\mathbb{R}^3_+)} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)}
\le C \Big( \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \\[6pt] + \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} + \frac{1}{|k|} \| (J_{e, 3} - \hat J_{e, 3}, J_{m, 3} - \hat J_{m, 3})\|_{H^{1/2}(\mathbb{R}^3_0)} \Big),
\end{multline}
for some positive constant $C$ depending only on $\gamma$, $\Lambda$, and $\Lambda_1$.
\end{corollary}
\begin{remark} \rm The constant $C$ in \eqref{cor-HS-est} is independent of $k$. Concerning the isotropic properties of $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$,
we emphasize here that $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$ are not required to be isotropic in $B_1 \cap \overline{\mathbb{R}^3_+}$, we only assume that $\varepsilon(0), \, \mu(0), \, \hat \varepsilon(0), \, \hat \mu(0)$ are.
\end{remark}
Here and what follows $B_r$, for $r>0$ denotes the ball of radius $r$ centered at the origin.
\begin{proof} We rewrite \eqref{sys-C*} under the form
\begin{equation*}
\left\{\begin{array}{c}
\nabla \times E = k \mu (0) H + J_e^{1}
\mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times H = - k \varepsilon (0) E + J_m^{1} \mbox{ in } \mathbb{R}^3_+,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = \alpha k \hat \mu(0) \hat H + \hat J_e^{1} \mbox{ in } \mathbb{R}^3_+, \\[6pt]
\nabla \times \hat H = - \alpha k \hat \varepsilon (0) \hat E + \hat J_m^1 \mbox{ in } \mathbb{R}^3_+,
\end{array}\right.
\end{equation*}
where, in $\mathbb{R}^3_+$,
$$
J_e^{1} (x) = J_e (x) + k (\mu(x) - \mu(0) ) H (x) , \quad J_m^{1} (x) = J_m (x) - k (\varepsilon(x) - \varepsilon(0) ) E (x) ,
$$
$$
\hat J_e^{1} (x) = \hat J_e (x) + \alpha k (\hat \mu(x) - \hat \mu(0) ) \hat H (x) , \quad \hat J_m^{1} (x) = \hat J_m (x) - \alpha k (\hat \varepsilon(x) - \hat \varepsilon(0) ) \hat E (x) .
$$
From \Cref{pro-HS}, we obtain
\begin{multline}\label{cor-HS-p1}
C \Big( \| (E, H, \hat E, \hat H) \|_{H^1(\mathbb{R}^3_+)} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)} \Big) \\[6pt]
\le \| (J_e^1, J_m^1, \hat J_e^1, \hat J_m^1)\|_{L^2(\mathbb{R}^3_+)}
+ \frac{1}{|k|} \| (\operatorname{div} J_e^1, \operatorname{div} J_m^1, \operatorname{div} \hat J_e^1, \operatorname{div} \hat J_m^1)\|_{L^2(\mathbb{R}^3_+)} \\[6pt]
+ \frac{1}{|k|} \| (J_{e, 3}^1 - \hat J_{e, 3}^1, J_{m, 3}^1 - \hat J_{m, 3}^1) \|_{H^{1/2}(\mathbb{R}^3_0)}.
\end{multline}
On the other hand, from the definition of $(J_e^1, J_m^1, \hat J_e^1, \hat J_m^1)$, one has
\begin{equation}\label{cor-HS-p2}
\| (J_e^1, J_m^1, \hat J_e^1, \hat J_m^1)\|_{L^2(\mathbb{R}^3_+)}
\le C \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)} + C r_0 |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)}
\end{equation}
\begin{multline}\label{cor-HS-p3}
\frac{1}{|k|} \| (\operatorname{div} J_e^1, \operatorname{div} J_m^1, \operatorname{div} \hat J_e^1, \operatorname{div} \hat J_m^1)\|_{L^2(\mathbb{R}^3_+)}
\le \frac{C}{|k|} \| \big(\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m \big) \|_{L^2(\mathbb{R}^3_+)} \\[6pt]
+ C \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)}
+ C r_0 \left\| \left(\nabla E, \nabla H, \nabla \hat E, \nabla \hat H \right) \right\|_{L^2(\mathbb{R}^3_+)},
\end{multline}
and
\begin{multline}\label{cor-HS-p4}
\frac{1}{|k|} \| (J_{e, 3}^1 - \hat J_{e, 3}^1, J_{m, 3}^1 - \hat J_{m, 3}^1) \|_{H^{1/2}(\mathbb{R}^3_0)} \le \frac{C}{|k|} \| (J_{e, 3} - \hat J_{e, 3}, J_{m, 3} - \hat J_{m, 3}) \|_{H^{1/2}(\mathbb{R}^3_0)} \\[6pt]
+ C r_0 \| (E, H, \hat E, \hat H) \|_{H^{1}(\mathbb{R}^3_+)} + C \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)}.
\end{multline}
Here in the last inequality, we involved the trace theory and used
\begin{multline*}
\left\| \left((\mu (x) - \mu(0)) H , (\varepsilon (x) - \varepsilon(0)) E, (\hat \mu (x) - \hat \mu(0)) \hat H, (\hat \varepsilon (x) - \hat \varepsilon(0)) \hat E \right) \right\|_{H^{1}(\mathbb{R}^3_+)} \\[6pt]
\le C r_0 \| (E, H, \hat E, \hat H) \|_{H^{1}(\mathbb{R}^3_+)} + C \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)}.
\end{multline*}
Combining \eqref{cor-HS-p1} - \eqref{cor-HS-p4} yields
\begin{multline}\label{cor-HS-p5}
C \Big( \| (E, H, \hat E, \hat H) \|_{H^1(\mathbb{R}^3_+)} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)} \Big) \\[6pt]
\le \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\mathbb{R}^3_+)}
+ \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\mathbb{R}^3_+)} \\[6pt] + \frac{1}{|k|} \| (J_{e, 3} - \hat J_{e, 3}, J_{m, 3} - \hat J_{m, 3})\|_{H^{1/2}(\mathbb{R}^3_0)} \\[6pt]
(|k| r_0 + 1) \| (E, H, \hat E, \hat H) \|_{L^2(\mathbb{R}^3_+)}
+ r_0 \| ( E, H, \hat E, \hat H ) \|_{H^1(\mathbb{R}^3_+)}.
\end{multline}
Fix $r_0 = \min\{C / 4, 1/4\}$ where $C$ is the constant in \eqref{cor-HS-p5}. Take $k_0$ such that $k_0 r_0 \ge 1$. One then can absorb the last two terms of the RHS of \eqref{cor-HS-p5} by the LHS. The conclusion then follows.
\end{proof}
\section{Proof of \Cref{thm-main}} \label{sect-proof}
In this section, we give the proof of \Cref{thm-main}. We begin with a result which yields the uniqueness and the stability of \eqref{sys-ITE-E} and \eqref{bdry-ITE-E}.
\begin{proposition}\label{pro-main-1} Let $\gamma > 0$, $\alpha \in \mathbb{C}$ with $\alpha^2 \in \mathbb{R}$ and $|\alpha|=1$, and let $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \in [L^\infty(\Omega)]^{3 \times 3}$ be symmetric. Assume that there exist $\Lambda \ge 1$, $\Lambda_1 > 0$, and $s_0 > 0$ such that
\begin{equation*}
\Lambda^{-1} \le \varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \le \Lambda \mbox{ a.e. in } \Omega, \quad \| (\varepsilon, \mu, \hat \varepsilon, \hat \mu) \|_{C^1(\bar \Omega_{s_0})} \le \Lambda,
\end{equation*}
$$
\varepsilon,\, \mu,\, \hat \varepsilon, \, \hat \mu \mbox{ are isotropic on } \partial \Omega,
$$
and, for $x \in \partial \Omega$,
$$
|\varepsilon (x) - \hat \varepsilon(x)| \ge \Lambda_1, \quad |\mu(x) - \hat \mu(x)| \ge \Lambda_1, \quad \mbox{ and } \quad |\varepsilon(x)/ \mu (x) - \hat \varepsilon(x)/ \hat \mu(x)| \ge \Lambda_1.
$$
There exist two positive constants $k_0 \ge 1$ and $C > 0$ depending only on $\Lambda$, $\Lambda_1$, $s_0$, $\gamma$, and $\Omega$ such that for $k \in \mathbb{C}$ with $|\Im{(k^2)}| \ge \gamma |k|^2$ and $|k| \ge k_0$, for every $(J_e, J_m, \hat J_e, \hat J_m) \in [H(\operatorname{div}, \Omega)]^4$ with $(J_{e}\cdot \nu - \hat J_{e} \cdot \nu, J_{m} \cdot \nu - \hat J_{m} \cdot \nu) \in [H^{1/2}(\partial \Omega)]^2$, and for every solution $(E, H, \hat E, \hat H) \in [L^2(\Omega)]^{12}$ of
\begin{equation}\label{sys-ITE-1}
\left\{\begin{array}{c}
\nabla \times E = k \mu H + J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \Omega,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = \alpha k \hat \mu \hat H + \hat J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times \hat H = - \alpha k \hat \varepsilon \hat E + \hat J_m \mbox{ in } \Omega,
\end{array}\right.
\end{equation}
\begin{equation}\label{bdry-ITE-1}
(\hat E - E) \times \nu = 0 \mbox{ on } \partial \Omega, \quad \mbox{ and } \quad (\hat H - H) \times \nu = 0 \mbox{ on } \partial \Omega,
\end{equation}
we have
\begin{multline}\label{pro-main-1-est}
|k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\Omega)} + \| (E, H, \hat E, \hat H) \|_{H^1(\Omega_{s_0/2})}
\le C \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)} \\[6pt]
+ \frac{C}{|k|} \| (\operatorname{div} J_e,\operatorname{div} J_m, \operatorname{div} \hat J_e,\operatorname{div} \hat J_m)\|_{L^2(\Omega)} + \frac{C}{|k|} \| (J_{e} \cdot \nu - \hat J_e \cdot \nu, J_m \cdot \nu - \hat J_m \cdot \nu)\|_{H^{1/2}(\Omega)} .
\end{multline}
\end{proposition}
Recall that $\Omega_s$ is given in \eqref{def-Os}.
\begin{proof} We use local charts for $\Gamma = \partial \Omega$.
In what follows, we denote $Q = (-1, 1)^3$, $Q_{+} = Q \cap \mathbb{R}^3_+$, and $Q_{0}= Q \cap \mathbb{R}^3_0$.
Let $m \ge 1$ and let $\varphi_\ell \in C^2_{c}(\mathbb{R}^3)$, $U_\ell \subset \mathbb{R}^3$ open ball, and $\mathcal T_\ell : U_\ell \to Q$ with $1 \le \ell \le m$ be such that
$ \mathcal T_\ell (U_\ell \cap \Omega) = Q_+$, and $\mathcal T_\ell(U_\ell \cap \Gamma) = Q_0$, $\operatorname{supp} \varphi_\ell \Subset U_\ell$, and $\Phi = 1$ in a neighborhood of $\Gamma$, where
$$
\Phi : = \sum_{\ell=1}^m \varphi_\ell \mbox{ in } \mathbb{R}^3.
$$
In what follows, we also assume that the diameter of the support of $\varphi_\ell$ is sufficiently small and $\nabla \mathcal T_\ell (\varphi_\ell^{-1}(0))$ is a rotation, i.e., $\big( \nabla \mathcal T_\ell \nabla \mathcal T_\ell^T \big) (\varphi_\ell^{-1}(0))= I$.
Set, in $\Omega \cap U_\ell$,
$$
(E^\ell, H^\ell, \hat E^\ell, \hat H^\ell) = (\varphi_\ell E, \varphi_\ell H, \varphi_\ell \hat E, \varphi_\ell \hat H),
$$
and
\begin{equation*}
(J_e^\ell , J_m^\ell , \hat J_e^\ell , \hat J_m^\ell) = (\varphi_\ell J_e + \nabla \varphi_\ell \times E, \varphi_\ell J_m + \nabla \varphi_\ell \times H, \varphi_\ell \hat J_e + \nabla \varphi_\ell \times \hat E, \varphi_\ell \hat J_m + \nabla \varphi_\ell \times \hat H).
\end{equation*}
We have
\begin{equation}
\left\{\begin{array}{c}
\nabla \times E^\ell = k \mu H^\ell + J_e^\ell \mbox{ in } \Omega \cap U_\ell, \\[6pt]
\nabla \times H^\ell = - k \varepsilon E^\ell + J_m^\ell \mbox{ in } \Omega \cap U_\ell,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E^\ell = \alpha k \hat \mu \hat H^\ell + \hat J_e^\ell \mbox{ in } \Omega \cap U_\ell, \\[6pt]
\nabla \times \hat H^\ell = - \alpha k \hat \varepsilon \hat E^\ell + \hat J_m^\ell \mbox{ in } \Omega \cap U_\ell,
\end{array}\right.
\end{equation}
\begin{equation}
(\hat E^\ell - E^\ell) \times \nu = 0 \mbox{ on } \partial \Omega \cap U_\ell, \quad \mbox{ and } \quad (\hat H^\ell - H^\ell) \times \nu = 0 \mbox{ on } \partial \Omega \cap U_\ell.
\end{equation}
Given a diffeomorphism $\mathcal T$ from an open $D$ onto an open $D'$, the following standard notations are used
$$
\mathcal T* u(x') = \nabla \mathcal T(x) u(x),
$$
\begin{equation*}
\mathcal T_*a (x') = \frac{\nabla \mathcal T(x) a(x) \nabla \mathcal T^T(x)}{\det \nabla \mathcal T(x)}, \quad \mbox{ and } \quad \mathcal T_*j (x')= \frac{\nabla \mathcal T(x) j(x)}{\det \nabla \mathcal T (x)},
\end{equation*}
with $x' =\mathcal T(x)$, for a matrix-valued function $a$, and for vector fields $u$ and $j$ defined in $D$.
Set, in $Q_+$,
$$
(\tE^\ell, \mathrm{H}^\ell, \thE^\ell, \hat{\mathrm{H}}^\ell) = \big(\mathcal T_\ell * E^\ell, \mathcal T_\ell * H^\ell, \mathcal T_\ell * \hat E^\ell, \mathcal T_\ell * \hat H^\ell \big),
$$
$$
(\varepsilon^\ell, \mu^\ell, \hat \varepsilon^\ell, \hat \mu^\ell) = ({\mathcal T_\ell}_* \varepsilon, {\mathcal T_\ell}_* \mu, {\mathcal T_\ell}_* \hat \varepsilon, {\mathcal T_\ell}_* \hat \mu),
$$
$$
(\mathrm{J}_e^\ell, \mathrm{J}_m^\ell, \hat{\mathrm{J}}_e^\ell, \hat{\mathrm{J}}_m^\ell) = ({\mathcal T_\ell}_* J_e^\ell, {\mathcal T_\ell}_* J_m^\ell, {\mathcal T_\ell}_* \hat J_e^\ell, {\mathcal T_\ell}_* \hat J_m^\ell).
$$
By a change of variables, see e.g. \cite[Lemma 7]{Ng-Superlensing-Maxwell},
\begin{equation}\label{sys-ITE-2}
\left\{\begin{array}{c}
\nabla \times \tE^\ell = k \mu^\ell \mathrm{H}^\ell + \mathrm{J}_e^\ell \mbox{ in } Q_+, \\[6pt]
\nabla \times \mathrm{H}^\ell = - k \varepsilon^\ell \tE^\ell + \mathrm{J}_m^\ell \mbox{ in } Q_+,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \thE^\ell = \alpha k \hat \mu \hat{\mathrm{H}}^\ell + \hat{\mathrm{J}}_e^\ell \mbox{ in } Q_+, \\[6pt]
\nabla \times \hat{\mathrm{H}}^\ell = - \alpha k \hat \varepsilon \thE^\ell + \hat{\mathrm{J}}_m^\ell \mbox{ in } Q_+,
\end{array}\right.
\end{equation}
\begin{equation}\label{bdry-ITE-2}
(\thE^\ell - \tE^\ell) \times \nu = 0 \mbox{ on } Q_0, \quad \mbox{ and } \quad (\hat{\mathrm{H}}^\ell - \mathrm{H}^\ell) \times \nu = 0 \mbox{ on } Q_0.
\end{equation}
Since $\nabla \mathcal T_\ell (\varphi_\ell^{-1}(0))$ is a rotation, and
$\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$ are isotropic on $\partial \Omega$, one has
$$
\varepsilon^\ell (0), \, \mu^\ell (0), \, \hat \varepsilon^\ell(0), \, \hat \mu^\ell (0) \mbox{ are isotropic}.
$$
By considering the diameter of $\operatorname{supp} \varphi_\ell$ sufficiently small, one can then apply \Cref{cor-HS} to $(\tE^\ell, \mathrm{H}^\ell, \thE^\ell, \hat{\mathrm{H}}^\ell)$. We then obtain
\begin{multline}\label{pro-main-e1}
C \Big( \| ( \tE^\ell, \mathrm{H}^\ell, \thE^\ell, \hat{\mathrm{H}}^\ell) \|_{H^1(Q_+)} + |k| \, \| ( \tE^\ell, \mathrm{H}^\ell, \thE^\ell, \hat{\mathrm{H}}^\ell) \|_{L^2(Q_+)} \Big) \\[6pt] \le \| (\mathrm{J}_e^\ell, \mathrm{J}_m^\ell, \hat{\mathrm{J}}_e^\ell, \hat{\mathrm{J}}_m^\ell)\|_{L^2(Q_+)} + \frac{1}{|k|} \| (\operatorname{div} \mathrm{J}_e^\ell, \operatorname{div} \mathrm{J}_m^\ell, \operatorname{div} \hat{\mathrm{J}}_e^\ell, \operatorname{div} \hat{\mathrm{J}}_m^\ell)\|_{L^2(Q_+)} \\[6pt]
+ \frac{1}{|k|} \| (\mathrm{J}_e^\ell \cdot e_3 - \hat{\mathrm{J}}_e^\ell \cdot e_3, \mathrm{J}_m^\ell \cdot e_3 - \hat{\mathrm{J}}_m^\ell \cdot e_3)\|_{H^{1/2}(Q_0)}.
\end{multline}
We have, by \cite[Corollary 3.59]{Monk03},
$$
\| (\operatorname{div} \mathrm{J}_e^\ell, \operatorname{div} \mathrm{J}_m^\ell, \operatorname{div} \hat{\mathrm{J}}_e^\ell, \operatorname{div} \hat{\mathrm{J}}_m^\ell)\|_{L^2(Q_+)} \le C \| (\operatorname{div} J_e^\ell, \operatorname{div} J_m^\ell, \operatorname{div} \hat J_e^\ell, \operatorname{div} \hat J_m^\ell)\|_{L^2(\Omega \cap U_\ell)}
$$
and we also obtain
$$
\| (\mathrm{J}_e^\ell \cdot e_3 - \hat{\mathrm{J}}_e^\ell \cdot e_3, \mathrm{J}_m^\ell \cdot e_3 - \hat{\mathrm{J}}_m^\ell \cdot e_3)\|_{H^{1/2}(Q_0)} \le C \|(J_e^\ell \cdot \nu - \hat J_e^\ell \cdot \nu, J_m^\ell \cdot \nu - \hat J_m^\ell \cdot \nu \|_{H^{1/2}(\partial \Omega \cap U_\ell)}.
$$
We deduce from \eqref{pro-main-e1} that
\begin{multline}\label{pro-main-e2}
C \Big( \| ( E^\ell, H^\ell, \hat E^\ell, \hat H^\ell) \|_{H^1(\Omega \cap U_\ell)} + |k| \, \| ( E^\ell, H^\ell, \hat E^\ell, \hat H^\ell) \|_{L^2(\Omega \cap U_\ell)} \Big) \\[6pt] \le \| (J_e^\ell, J_m^\ell, \hat J_e^\ell, \hat J_m^\ell)\|_{L^2(\Omega \cap U_\ell)} + \frac{1}{|k|} \| (\operatorname{div} J_e^\ell, \operatorname{div} J_m^\ell, \operatorname{div} \hat J_e^\ell, \operatorname{div} \hat J_m^\ell)\|_{L^2(\Omega \cap U_\ell)} \\[6pt]
+ \frac{1}{|k|} \| (J_e^\ell \cdot \nu - \hat J_e^\ell \cdot \nu , J_m \cdot \nu - \hat J_m \cdot \nu)\|_{H^{1/2}(\partial \Omega \cap U_\ell)}.
\end{multline}
Take the sum with respect to $\ell$. We then have, for some $\tau_0 < s_0/4$,
\begin{multline}\label{pro-main-p1}
C \Big( \| ( E, H, \hat E, \hat H) \|_{H^1(\Omega_{\tau_0})} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\Omega_{\tau_0})} \Big) \\[6pt] \le \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)} + \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\Omega)} \\[6pt]+
\frac{1}{|k|} \| (J_e \cdot \nu - \hat J_e \cdot \nu, J_m \cdot \nu - \hat J_m \cdot \nu)\|_{H^{1/2}(\partial \Omega)} \\[6pt]
+ \| (E, H, \hat E, \hat H) \|_{L^2(\Omega_{s_0/2})} + \frac{1}{|k|} \| ( E, H, \hat E, \hat H) \|_{H^1(\Omega_{s_0/2})}.
\end{multline}
Applying \Cref{lem-decay} below, we have
\begin{equation}\label{pro-main-p2}
\| (E, H, \hat E, \hat H) \|_{L^2(\Omega \setminus \Omega_{\tau_0})} \le c_1 e^{- c_2|k|} \| (E, H, \hat E, \hat H) \|_{L^2(\Omega_{\tau_0})},
\end{equation}
for some positive constants $c_1, c_2$ depending only on $\Lambda$, $\gamma$, $\tau_0$, and $\Omega$.
Since $(\varepsilon, \mu, \hat \varepsilon, \hat \mu) \in C^1(\Omega_{3\tau_0} \setminus \Omega_{\tau_0/2})$, it follows from \eqref{sys-ITE-1} that
\begin{multline}\label{pro-main-p3}
\| (E, H, \hat E, \hat H) \|_{H^1(\Omega_{s_0/2} \setminus \Omega_{\tau_0})} \le C |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\Omega_{s_0} \setminus \Omega_{\tau_0/2})} \\[6pt]
+ C\| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)} + \frac{C}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\Omega)}.
\end{multline}
Taking $k_0$ sufficiently large and $|k| \ge k_0$, from \eqref{pro-main-p2} and \eqref{pro-main-p3}, one can absorb the last two terms of the RHS of \eqref{pro-main-p1} by the LHS of \eqref{pro-main-p1}. We then have
\begin{multline}\label{pro-main-p4}
C \Big( \| (E, H, \hat E, \hat H) \|_{H^1(\Omega_{\tau_0})} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\Omega_{\tau_0})} \Big) \\[6pt]
\le \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)} + \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\Omega)} \\[6pt]
+\frac{1}{|k|} \| (J_e \cdot \nu - \hat J_e \cdot \nu, J_m \cdot \nu - \hat J_m \cdot \nu)\|_{H^{1/2}(\partial \Omega)}.
\end{multline}
Using \eqref{pro-main-p2} and \eqref{pro-main-p4}, we derive from \eqref{pro-main-p3} that
\begin{multline}\label{pro-main-p5}
C \| (E, H, \hat E, \hat H) \|_{H^1(\Omega_{s_0/2} \setminus \Omega_{\tau_0})}
\le \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)} \\[6pt]
+ \frac{1}{|k|} \| (\operatorname{div} J_e, \operatorname{div} J_m, \operatorname{div} \hat J_e, \operatorname{div} \hat J_m)\|_{L^2(\Omega)}
+\frac{1}{|k|} \| (J_e \cdot \nu - \hat J_e \cdot \nu, J_m \cdot \nu - \hat J_m \cdot \nu)\|_{H^{1/2}(\partial \Omega)}.
\end{multline}
The conclusion now follows from \eqref{pro-main-p4} and \eqref{pro-main-p5}. The proof is complete.
\end{proof}
In the proof of \Cref{pro-main-1}, we used the following decay result on the Maxwell equations:
\begin{lemma}\label{lem-decay} Let $\gamma > 0$, $k \in \mathbb{C}$ with $|\Im(k^2)| \ge \gamma |k|^2$ and $|k| \ge 1$, and let $\varepsilon, \mu \in [L^\infty(\Omega)]^{3\times 3}$ be symmetric and uniformly elliptic, i.e.
$$
\Lambda^{-1} \le \varepsilon, \, \mu, \, \hat \varepsilon, \hat \mu \le \Lambda,
$$
for some $\Lambda \ge 1$. Given $J_e, J_m \in L^2(\Omega)$, let $(E, H) \in [L^2(\Omega)]^{6}$ be a solution of
\begin{equation*}
\left\{\begin{array}{c}
\nabla \times E= k \mu H + J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \Omega.
\end{array}\right.
\end{equation*}
For all $s>0$, there exists two positive constants $c_1$ and $c_2$ depending only on $\Lambda$, $\gamma$, $s$, and $\Omega$ such that
\begin{equation*}
\| (E, H)\|_{L^2(\Omega \setminus \Omega_s)} \le c_1\exp (-c_2 |k| ) \| (E, H)\|_{L^2(\Omega_s)} + c_1 \| (J_e, J_m)\|_{L^2(\Omega)}.
\end{equation*}
\end{lemma}
\begin{proof} Let $(E^1, H^1) \in [L^2(\Omega)]^6$ be the unique solution of
\begin{equation}\label{lem-HS-sys}
\left\{\begin{array}{c}
\nabla \times E^1= k \mu H^1 + J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times H^1 = - k \varepsilon E^1 + J_m \mbox{ in } \Omega, \\[6pt]
E^1 \times e_3 = 0 \mbox{ on } \partial \Omega,
\end{array}\right.
\end{equation}
As in the proof of \Cref{lem-HS}, we have
$$
\| (E^1, H^1) \|_{L^2(\Omega)} \le C \| (J_e, J_m)\|_{L^2(\Omega)}.
$$
Considering $(E- E^1, H-H^1)$, w.l.o.g., one might assume that $J_e = J_m = 0$ in $\Omega$. This is assumed from later on.
Fix $\varphi \in C^2(\Omega)$ such that $\varphi = c s$ in $\Omega \setminus \Omega_s$ and $\varphi = 0$ in $\Omega_{s/2}$, and $|\nabla \varphi| \le c$ in $\Omega$ where $c$ is a small positive constant defined later (the smallness of $c$ depends only on $\Lambda$ and $\Omega$, it is independent of $s$).
Set $\phi(x) = e^{ |k| \varphi(x)}$ and $E^1 (x)= \phi(x)E(x)$ and $H^1(x) = \phi (x) H(x)$ for $x \in \Omega$.
We have
\begin{equation*}
\left\{\begin{array}{c}
\nabla \times E^1= k \mu H^1 + J_e^1 \mbox{ in } \Omega, \\[6pt]
\nabla \times H^1 = - k \varepsilon E^1 + J_m^1 \mbox{ in } \Omega,
\end{array}\right.
\end{equation*}
where
$$
J_e^1 = \nabla \phi \times E \quad \mbox{ and } \quad J_m^1 = \nabla \phi \times H \mbox{ in } \Omega.
$$
Multiplying the first equation with $\bar H^1$, integrating by parts in $\Omega \setminus \Omega_\tau$ for $s/4 < \tau < s/2$, and using the second equation, we have
\begin{multline*}
\left| \int_{\Omega \setminus \Omega_\tau} k \langle \mu H^1, H^1 \rangle + \bar k \int_{\Omega \setminus \Omega_\tau} \langle \varepsilon E^1, E^1 \rangle \right|\\[6pt]
\le \int_{\Omega \setminus \Omega_\tau} |J_e^1| |H^1| + |J_m^1| |E^1| + C \int_{\partial (\Omega \setminus \Omega_\tau)} (|E^1|^2 + |H^1|^2).
\end{multline*}
This yields
\begin{multline}\label{lem-decay-bilinear2}
\left| \int_{\Omega \setminus \Omega_\tau} k^2 \langle \mu H^1, H^1 \rangle + |k|^2 \int_{\Omega \setminus \Omega_\tau} \langle \varepsilon E^1, E^1 \rangle \right|\\[6pt]
\le |k| \int_{\Omega \setminus \Omega_\tau} |J_e^1| |H^1| + |J_m^1| |E^1| + C |k| \int_{\partial (\Omega \setminus \Omega_\tau)} (|E^1|^2 + |H^1|^2).
\end{multline}
By the definition of $J_e^1, \, J_m^1$, and of $E^1$ and $H^1$,
$$
|J_e^1| \le c |k| |E^1|, \quad |J_m^1| \le c |k| |H^1| \mbox{ in } \Omega, \quad \mbox{ and } \quad E^1 - E = H^1 - H = 0 \mbox{ in } \Omega_{s/2},
$$
we derive from \eqref{lem-decay-bilinear2} that, for $c$ sufficiently small,
\begin{equation*}
\left| \int_{\Omega \setminus \Omega_\tau} k^2 \langle \mu H^1, H^1 \rangle + |k|^2 \int_{\Omega \setminus \Omega_\tau} \langle \varepsilon E^1, E^1 \rangle \right| \le C |k| \int_{\partial (\Omega \setminus \Omega_\tau)} (|E|^2 + |H|^2).
\end{equation*}
The conclusion follows by taking $\tau$ such that
\begin{equation*}
\int_{\partial (\Omega \setminus \Omega_\tau)} (|E|^2 + |H|^2) \le C s^{-1} \int_{\Omega_s} (|E|^2 + |H|^2).
\end{equation*}
This yields
\begin{equation*}
\int_{\partial (\Omega \setminus \Omega_\tau)} (|E^1|^2 + |H^1|^2) \le C s^{-1} \int_{\Omega_s} (|E|^2 + |H|^2),
\end{equation*}
and the conclusion follows by the definition of $E^1$ and $H^1$. The proof is complete.
\end{proof}
\begin{remark} \rm The proof of \Cref{lem-decay} is quite standard, see e.g. \cite[Theorem 2.2]{HJNg1} for a variant dealing with the Helmholtz equation.
\end{remark}
We now apply a limiting absorption principle. To this end, let us set
$$
\mathbf H_1 (\Omega) = \Big\{ (u, v) \in [H(\operatorname{curl}, \Omega)]^2; (u -v) \times \nu = 0 \mbox{ on } \partial \Omega \Big\}.
$$
One can check that $\mathbf H_1(\Omega)$ is a Hilbert space equipped with the natural scalar product induced from the one of $[H(\operatorname{curl}, \Omega)]^2$.
Consider $(J_e, J_m, \hat J_e, \hat J_m) \in [L^2(\Omega)]^{12}$. Let $\gamma > 0$, $k \in \mathbb{C}$ with $|\Im(k^2)| \ge \gamma |k|^2$ and $|k| \ge 1$. Set $a = \mbox{ sign } \Im(k^2)$. For $\delta > 0$ sufficiently small with respect to $\gamma$, we claim that there exists a unique solution $(E^\delta, H^\delta, \hat E^\delta, \hat H^\delta) \in [L^2(\Omega)]^{12}$ of the system
\begin{equation}\label{sys-ITE-d}
\left\{\begin{array}{c}
\nabla \times E^\delta = (1 - i a \delta ) k \mu H ^\delta+ J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times H^\delta = - (1 - i a \delta ) k \varepsilon E^\delta + J_m \mbox{ in } \Omega,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = - (1 - i a\delta) k \hat \mu \hat H^\delta + \hat J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times \hat H^\delta = (1 - i a \delta) k \hat \varepsilon \hat E^\delta + \hat J_m \mbox{ in } \Omega,
\end{array}\right.
\end{equation}
\begin{equation}\label{bdry-ITE-d}
(\hat E^\delta - E^\delta) \times \nu = 0 \mbox{ on } \partial \Omega, \quad \mbox{ and } \quad (\hat H^\delta - H^\delta) \times \nu = 0 \mbox{ on } \partial \Omega.
\end{equation}
Indeed, consider the following equation
\begin{multline}\label{Step2-bilinear}
\int_{\Omega} \langle (1 - i a \delta)^{-1}\mu^{-1} \nabla \times E^\delta, \nabla \times \varphi \rangle + k^2 (1 - i a \delta) \langle \varepsilon E^\delta, \varphi \rangle \\[6pt] + \int_{\Omega} \langle (1 - i a \delta)^{-1}\hat \mu^{-1} \nabla \times \hat E^\delta, \nabla \times \hat \varphi \rangle + k^2 (1 - i a \delta) \langle \hat \varepsilon \hat E^\delta, \hat \varphi \rangle \\[6pt] = \int_{\Omega} \langle (1 - i a\delta)^{-1} \mu^{-1} J_e, \nabla \times \varphi \rangle + k \langle J_m, \varphi \rangle + \int_{\Omega} \langle (1 - i a\delta)^{-1} \hat \mu^{-1} \hat J_e, \hat \varphi \rangle - k \langle \hat J_m, \hat \varphi \rangle,
\end{multline}
for all $(\varphi, \hat \varphi) \in \mathbf H_1(\Omega)$.
Note that, the absolute value of the imaginary part of the LHS of \eqref{Step2-bilinear} with $(\varphi, \hat \varphi) = (E^\delta, \hat E^\delta) $ is greater than $C \delta \int_{\Omega} |(E^\delta, \nabla \times E^\delta, \hat E^\delta, \nabla \times \hat E^\delta)|^2$ for $\delta$ sufficiently small. By Lax-Milgram's theory, there exists a unique solution $(E^\delta, \hat E^\delta) \in \mathbf H_1(\Omega)$ for \eqref{Step2-bilinear}. Set, in $\Omega$,
$$
H^\delta = k^{-1}(1 - i a \delta)^{-1} \mu^{-1} \Big( \nabla \times E^\delta - J_e\Big) \quad \mbox{ and } \quad \hat H^\delta = - k^{-1}(1 - i a \delta) \hat \mu^{-1} \Big( \nabla \times \hat E^\delta - \hat J_e\Big).
$$
Considering $\varphi, \hat \varphi \in C^1_{c}(\Omega)$ in \eqref{Step2-bilinear}, we obtain
$$
\nabla \times H^\delta = - (1 + i \delta) k \varepsilon E^\delta + J_m \mbox{ in } \Omega \quad \mbox{ and } \quad \nabla \times \hat H^\delta = - (1 - i \delta) ik \hat \varepsilon \hat E^\delta + \hat J_m \mbox{ in } \Omega.
$$
This in turn implies that
$$
(\hat H^\delta - H^\delta ) \times \nu = 0 \mbox{ on } \partial \Omega.
$$
Therefore, the existence and the uniqueness of $(E_\delta, H_\delta)$ are established.
\medskip
Next denote
\begin{multline}\label{def-Hspace}
\mathbf H (\Omega) = \Big\{ (u, v, \hat u, \hat v) \in [L^2(\Omega)]^{12}; \operatorname{div} (\varepsilon u) = \operatorname{div} (\mu v) = \operatorname{div} (\hat \varepsilon \hat u) = \operatorname{div}(\hat \mu \hat v) = 0 \mbox{ in } \Omega, \\[6pt] \mbox{ and }
\varepsilon u \cdot \nu - \hat \varepsilon \hat u \cdot \nu = \mu v \cdot \nu - \hat \mu \hat v \cdot \nu = 0 \mbox{ on } \partial \Omega \Big\},
\end{multline}
and let
$$
\| (u, v, \hat u, \hat v) \|_{\mathbf H} = \| (u, v, \hat u, \hat v) \|_{L^2(\Omega)}.
$$
One can check that $\mathbf H(\Omega)$ is a Hilbert space with the corresponding scalar product.
\medskip
We finally have
\begin{proposition}\label{pro-main} Let $\gamma > 0$, $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \in [L^\infty(\Omega)]^{3 \times 3}$ be symmetric. Assume that there exist $\Lambda \ge 1$, $\Lambda_1 > 0$, and $s_0 > 0$ such that
\begin{equation*}
\Lambda^{-1} \le \varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu \le \Lambda \mbox{ in } \Omega, \quad \| (\varepsilon, \mu, \hat \varepsilon, \hat \mu) \|_{C^1(\Omega_{s_0})} \le \Lambda,
\end{equation*}
$$
\varepsilon,\, \mu,\, \hat \varepsilon, \, \hat \mu \mbox{ are isotropic on } \partial \Omega,
$$
and, for $x \in \partial \Omega$,
$$
|\varepsilon (x) - \hat \varepsilon(x)| \ge \Lambda_1, \quad |\mu(x) - \hat \mu(x)| \ge \Lambda_1, \quad \mbox{ and } \quad |\varepsilon(x)/ \mu (x) - \hat \varepsilon(x)/ \hat \mu(x)| \ge \Lambda_1.
$$
There exist two positive constants $k_0 \ge 1$ and $C > 0$ depending only on $\Lambda$, $\Lambda_1$, $\gamma$, $s_0$, and $\Omega$ such that for $k \in \mathbb{C}$ with $|\Im{(k^2)}| \ge \gamma |k|^2$ and $|k| \ge k_0$, and for every $(J_e^1, J_m^1, \hat J_e^1, \hat J_m^1) \in \mathbf H(\Omega)$, there exists a unique solution $(E, H, \hat E, \hat H) \in \mathbf H(\Omega)$ of
\begin{equation}\label{sys-ITE-m}
\left\{\begin{array}{c}
\nabla \times E = k \mu H + J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \Omega,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = k \hat \mu \hat H + \hat J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times \hat H = - k \hat \varepsilon \hat E + \hat J_m \mbox{ in } \Omega,
\end{array}\right.
\end{equation}
\begin{equation}\label{bdry-ITE-m}
(\hat E - E) \times \nu = 0 \mbox{ on } \partial \Omega, \quad \mbox{ and } \quad (\hat H - H) \times \nu = 0 \mbox{ on } \partial \Omega,
\end{equation}
where $(J_e, J_m, \hat J_e, \hat J_m) = (\mu J_m^1, \varepsilon J_e^1, \hat \mu \hat J_m^1, \hat \varepsilon \hat J_e^1)$. Moreover,
\begin{equation}\label{pro-main-est}
|k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\Omega)} + \| (E, H, \hat E, \hat H) \|_{H^1(\Omega_{s_0/2})}
\le C \| (J_e^1, J_m^1, \hat J_e^1, \hat J_m^1)\|_{L^2(\Omega)}.
\end{equation}
\end{proposition}
\begin{proof}
For $\delta > 0$, let $(E^\delta, H^\delta, \hat E^\delta, \hat H^\delta) \in [L^2(\Omega)]^{12}$ be the unique solution of the system \eqref{sys-ITE-d} and \eqref{bdry-ITE-d}. Applying \Cref{pro-main-1}, we have
\begin{equation}
|k| \, \| (E^\delta, H^\delta, \hat E^\delta, \hat H^\delta) \|_{L^2(\Omega)} + \| (E^\delta, H^\delta, \hat E^\delta, \hat H^\delta) \|_{H^1(\Omega_{s_0/2})}
\le C \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)}.
\end{equation}
Letting $\delta \to 0$, we obtain the existence of a solution of \eqref{sys-ITE-d} and \eqref{bdry-ITE-d} with $\delta = 0$. Moreover, by \Cref{pro-main-1} again, this solution is unique and it holds, for some $s_0 > 0$,
\begin{equation}\label{thm-main-p1}
\| (E, H, \hat E, \hat H) \|_{H^1(\Omega_{s_0})} + |k| \, \| (E, H, \hat E, \hat H) \|_{L^2(\Omega)}
\le C \| (J_e, J_m, \hat J_e, \hat J_m)\|_{L^2(\Omega)}.
\end{equation}
Define the operator
\begin{eqnarray}
\begin{array}{rcccc}
T_1: & \mathbf H (\Omega) &\to& \mathbf H (\Omega) \\[6pt]
& (J_e^1, J_m^1, \hat J_e^1, \hat J_m^1) & \mapsto & (E, H, \hat E, \hat H),
\end{array}
\end{eqnarray}
It is clear that system \eqref{sys-ITE-m} can be rewritten under the form
\begin{equation}
\left\{\begin{array}{c}
\nabla \times E = k \mu H + J_e \mbox{ in } \Omega, \\[6pt]
\nabla \times H = - k \varepsilon E + J_m \mbox{ in } \Omega,
\end{array}\right. \quad \left\{\begin{array}{c}
\nabla \times \hat E = - k \hat \mu \hat H + \hat J_e + 2k \hat \mu \hat H \mbox{ in } \Omega, \\[6pt]
\nabla \times \hat H = k \hat \varepsilon \hat E + \hat J_m - 2 k \hat \varepsilon \hat E \mbox{ in } \Omega,
\end{array}\right.
\end{equation}
Thus, for $(E, H, \hat E, \hat H) \in \mathbf H(\Omega)$, system \eqref{sys-ITE-m} and \eqref{bdry-ITE-m} is equivalent to
$$
(E, H, \hat E, \hat H) = T_1 (J_e^1, J_m^1, \hat J_e^1 - 2 k \hat E, \hat J_m^1 + 2k \hat H) = T_1 (J_e^1, J_m^1, \hat J_e^1, \hat J_m^1 ) + T_1 (0, 0, - 2 k \hat E, 2k \hat H).
$$
Since this equation has at most one solution by \Cref{pro-main-1} and from the fact that $T_1$ is compact, this equation has a unique solution. The proof is completed by applying again \Cref{pro-main-1} to obtain \eqref{pro-main-est}.
\end{proof}
We are ready to give the proof of our main theorem.
\begin{proof}[Proof of \Cref{thm-main}]
Define the operator
\begin{eqnarray}
\begin{array}{rcccc}
T: & \mathbf H (\Omega) &\to& \mathbf H (\Omega)\\[6pt]
& (J_e^1, J_m^1, \hat J_e^1, \hat J_m^1) & \mapsto & (E, H, \hat E, \hat H),
\end{array}
\end{eqnarray}
where $(E, H, \hat E, \hat H) \in \mathbf H(\Omega)$ is the unique solution of \eqref{sys-ITE-m} and \eqref{bdry-ITE-m} for $k\in {\mathbb C}$ satisfying the assumptions in Proposition \ref{pro-main}. We claim that $T$ is compact. Indeed, this follows from
$$
\operatorname{div}(\varepsilon E) = \operatorname{div} (\mu H) = \operatorname{div} (\hat \varepsilon \hat E) = \operatorname{div}(\hat \mu \hat H) = 0.
$$
and \eqref{pro-main-est}. By the theory of compact operator see, e.g., \cite{Brezis-FA}, the spectrum of $T$ is discrete. It is clear that an eigenfunction pair of the ITE problem corresponding to the eigenvalue $\omega$ is an eigenfunction pair of $T$ corresponding to the eigenvalue $k=i \omega $. Hence, the spectrum of the ITE problem is discrete, and the only possible accumulation point of the transmission eigenvalues is $\infty$ since they coincide with the eigenvalues of the inverse of $T$.
\end{proof}
Finally, we present
\begin{proof}[Proof of \Cref{pro}] \Cref{pro} is just a consequence of \Cref{pro-main} by noting that the solution given there is 0 if $(J_e, J_m, \hat J_e, \hat J_m) = 0$.
\end{proof}
\section*{Acknowledgments}
{The research of F. Cakoni is partially supported by the AFOSR Grant FA9550-20-1-0024 and NSF Grant DMS-1813492. H.-M. Nguyen thanks Fondation des Sciences Math\'ematiques de Paris (FSMP) for the Chaire d'excellence which allows him to visit Laboratoire Jacques Louis Lions and Mines ParisTech. This work is completed during this visit.}
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1,314,259,995,832 | arxiv | \section{Introduction}
Let $f$ be a non-CM cuspidal eigenform and let $\ell$ be a prime integer. By the work of Ribet (\cite{ribet1}, \cite{ribet3}) and Momose (\cite{momose}), it is known that the $\ell$-adic Galois representation $\rho_{f,\ell}$ associated with $f$ has large image for every $\ell$ and that for almost every $\ell$ it satisfies
\medskip
\begin{center}
(cong${}_\ell$) $\im\,\rho_{f,\ell}$ contains the conjugate of a principal congruence subgroup $\Gamma(\ell^m)$ of $\SL_2(\Z_\ell)$.
\end{center}
\medskip
\noindent For instance if $\im\,\rho_{f,\ell}$ contains an element with eigenvalues in $\Z_\ell^\times$ distinct modulo $\ell$ then (cong${}_\ell$) holds.
\noindent In \cite{hida}, Hida proved an analogous statement for $p$-adic families of non-CM ordinary cuspidal eigenforms, where $p$ is any odd prime integer. We fix once and for all an embedding $\overline{\Q}\into\overline{\Q}_p$, identifying $\Gal(\overline{\Q}_p/\Q_p)$ with a decomposition subgroup $G_p$ of $\Gal(\overline{\Q}/\Q)$. We also choose a topological generator $u$ of $\Z_p^\times$.
Let $\Lambda=\Z_p[[T]]$ be the Iwasawa algebra and let $\m=(p,T)$ be its maximal ideal. A special case of Hida's first main theorem (\cite[Th.I]{hida}) is the following.
\begin{theorem} Let ${\bf f}$ be a non-CM Hida family of ordinary cuspidal eigenforms defined over a finite extension $\I$ of $\Lambda$ and let $\rho_{\bf f}\colon \Gal(\overline{\Q}/\Q)\to\GL_2(\I)$ be the associated Galois representation. Assume that $\rho_{\bf f}$ is residually irreducible and that there exists an element $d$ in its image with eigenvalues $\alpha,\beta\in\Z_p^\times$ such that $\alpha^2\not\equiv\beta^2\pmod{p}$. Then there exists a nonzero ideal ${\mathfrak l}\subset \Lambda$ and an element $g\in\GL_2(\I)$ such that
$$ g\Gamma({\mathfrak l})g^{-1}\subset \im\,\rho_{\bf f}, $$
where $\Gamma({\mathfrak l})$ denotes the principal congruence subgroup of $\SL_2(\Lambda)$ of level ${\mathfrak l}$.
\end{theorem}
Under mild technical assumptions it is also shown in \cite[Th. II]{hida} that if the image of the residual representation of $\rho_{\bf f}$ contains a conjugate of $\SL_2(\F_p)$ then ${\mathfrak l}$ is trivial or $\m$-primary, and if the residual representation is dihedral ``of CM type'' the height one prime factors $P$ of ${\mathfrak l}$ are exactly those of the g.c.d. of the adjoint $p$-adic $L$ function of ${\bf f}$ and the anticyclotomic specializations of Katz' $p$-adic $L$ functions associated with certain Hecke characters of an imaginary quadratic field. This set of primes is precisely the set of congruence primes between the given non-CM family and the CM families.
In her PhD dissertation (see \cite{lang}), J. Lang improved on Hida's Theorem I. Let $\T$ be Hida's big ordinary cuspidal Hecke algebra; it is finite and flat over $\Lambda$. Let $\Spec\,\I$ be an irreducible component of $\T$. It corresponds to a surjective $\Lambda$-algebra homomorphism $\theta\colon \T\to \I$ (a $\Lambda$-adic Hecke eigensystem). We also call $\theta$ a Hida family. Assume that it is not residually Eisenstein. It gives rise to a residually irreducible continuous Galois representation $\rho_\theta\colon G_\Q\to\GL_2(\I)$ that is $p$-ordinary.
We suppose for simplicity that $\I$ is normal. Consider the $\Lambda$-algebra automorphisms $\sigma$ of $\I$ for which there exists a finite order character $\eta_\sigma\colon G_\Q\to\I^\times$
such that for every prime $\ell$ not dividing the level, $\sigma\ccirc\theta(T_\ell)=\eta_\sigma(\ell) \theta(T_\ell)$
(see \cite{ribet3} and \cite{lang}). These automorphisms form a finite abelian $2$-group $\Gamma$.
Let $\I_0$ be the subring of $\I$ fixed by $\Gamma$. Let $H_0=\bigcap_{\sigma\in\Gamma}\ker\,\eta_\sigma$;
it is a normal open subgroup of $G_\Q$. One may assume, up to conjugation by an element of $\GL_2(\I)$,
that $\rho_\theta\vert_{H_0}$ takes values in $\GL_2(\I_0)$.
\begin{theorem} \cite[Th. 2.4]{lang}
Let $\theta\colon\T\to\I$ be a non-CM Hida family such that $\overline{\rho}_\theta$ is absolutely irreducible. Assume that $\overline{\rho}_\theta\vert_{H_0}$ is an extension of two distinct characters. Then there exists a nonzero ideal ${\mathfrak l}\subset \I_0$ and an element $g\in\GL_2(\I)$ such that
$$g\Gamma({\mathfrak l})g^{-1}\subset \im\,\rho_{\theta},$$
where $\Gamma({\mathfrak l})$ denotes the principal congruence subgroup of $\SL_2(\I_0)$ of level ${\mathfrak l}$.
\end{theorem}
For all of these results it is important to assume the ordinarity of the family, as it implies
the ordinarity of the Galois representation and in particular that some element of the image of inertia at $p$ is conjugate to the matrix
$$C_T=\left(\begin{array}{cc} u^{-1}(1+T)&\ast\\0&1\end{array}\right).$$
Conjugation by the element above defines a $\Lambda$-module structure on the Lie algebra
of a pro-$p$ subgroup of $\im\,\rho_{\bf f}$ and this is used to produce the desired ideal ${\mathfrak l}$.
Hida and Lang use Pink's theory of Lie algebras of pro-$p$ subgroups of $\SL_2(\I)$.
In this paper we propose a generalization of Hida's work to the finite slope case.
We establish analogues of Hida's Theorems I and II. These are Theorems \ref{betalevel}, \ref{comparison}
and \ref{exponents} in the text.
Moreover, we put ourselves in the more general setting considered in Lang's work.
In the positive slope case the existence of a normalizing matrix analogous to $C_T$ above is obtained by
applying relative Sen theory (\cite{sen1} and \cite{sen2}) to the expense of extending scalars to
the completion $\C_p$ of an algebraic closure of $\Q_p$.
More precisely, for every $h\in (0, \infty)$, we define an Iwasawa algebra $\Lambda_h=\cO_h[[t]]$
(where $t=p^{-s_h}T$ for some $s_h\in\Q\cap ]0,\frac{1}{p-1}[$ and $\cO_h$
is a finite extension of $\Z_p$ containing $p^{s_h}$ such that its fraction field is Galois over $\Q_p$) and a finite torsion free
$\Lambda_h$-algebra $\T_h$ (see Section \ref{subsectnest}), called an adapted
slope $\leq h$ Hecke algebra.
Let $\theta\colon\T_h\to\I^\circ$ be an irreducible component; it is finite torsion-free over $\Lambda_h$.
The notation $\I^\circ$ is borrowed from the theory of Tate algebras, but $\I^\circ$ is not a Tate or an affinoid algebra. We write $\I=\I^\circ[p^{-1}]$.
We assume again for simplicity that $\I^\circ$ is normal.
The finite slope family $\theta$ gives rise to a continuous Galois representation
$\rho_\theta\colon G_\Q\to\GL_2(\I^\circ)$. We assume that the residual representation $\overline{\rho_\theta}$
is absolutely irreducible.
We introduce the finite abelian $2$-group $\Gamma$ as above, together with its fixed ring $\I_0$
and the open normal subgroup $H_0\subset G_\Q$.
In Section \ref{liealg} we define a ring $\B_{r}$ (with an inclusion $\I_0\into\B_r$) and a Lie algebra $\fH_{r}\subset\fsl_2(\B_{r})$ attached to the image of $\rho_\theta$.
In the positive slope case CM families do not exist (see Section \ref{cmforms}) hence no ``non-CM'' assumption is needed in the following. As before we can assume, after conjugation by an element of $\GL_2(\I^\circ)$, that $\rho_\theta(H_0)\subset\GL_2(\I_0^\circ)$. Let $P_1\subset\Lambda_h$ be the prime $(u^{-1}(1+T)-1)$.
\begin{theorem} \textnormal{(Theorem \ref{betalevel})}
Let $\theta\colon\T_h\to\I^\circ$ be a positive slope family such that $\overline{\rho}_\theta\vert_{H_0}$ is absolutely irreducible. Assume that there exists $d\in \rho_\theta(H_0)$ with eigenvalues $\alpha,\beta\in\Z_p^\times$ such that $\alpha^2\not\equiv\beta^2\pmod{p}$. Then there exists a nonzero ideal ${\mathfrak l}\subset \I_0[P_1^{-1}]$ such that
$$ \fl\cdot\fsl_2(\B_{r})\subset\fH_r. $$
\end{theorem}
\noindent The largest such ideal $\fl$ is called the Galois level of $\theta$.
We also introduce the notion of fortuitous CM congruence ideal for $\theta$ (see Section \ref{congrid}). It is the ideal $\fc\subset \I$
given by the product of the primary ideals modulo which a congruence between $\theta$ and a slope $\leq h$ CM form occurs.
Following the proof of Hida's Theorem II we are also able to show (Theorem \ref{comparison}) that the set of primes of $\I_0=\I_0^\circ[p^{-1}]$
containing $\fl$ coincides
with the set of primes containing $\fc\cap\I_0$, except possibly for the primes of $\I_0$ above
$P_1$ (the weight $1$ primes).
Several generalizations of the present work are currently being studied by one of the authors\footnote{A. Conti}. They include a generalization of \cite{hidatil}, where the authors treated the ordinary case for $\GSp_4$ with a residual representation induced from the one associated with a Hilbert modular form, to the finite slope case and to bigger groups and more types of residual representations.
\bigskip
\textbf{Acknowledgments.} This paper owes much to Hida's recent paper \cite{hida}. We also thank Jaclyn Lang for making her dissertation \cite{lang} available to us and for some very useful remarks pertaining to Section \ref{imgfinslope}.
We thank the referee of this article for the careful reading of the manuscript and for useful suggestions which hopefully led to improvements.
\bigskip
\section{The eigencurve}
\subsection{The weight space}
Let us recall that we have fixed $p>2$ be a prime integer.
We call \textit{weight space} the rigid analytic space over $\Q_p$, $\cW$, canonically associated with the formal scheme over $\Z_p$, ${\rm Spf}(\Z_p[[\Z_p^\times]])$. The $\C_p$-points of $\cW$ parametrize continuous homomorphisms $\Z_p^\times\to\C_p^\times$.
Let $X$ be a rigid analytic space defined over some finite extension $L/\Q_p$. We say that a subset $S$ of $X(\C_p)$ is Zariski-dense if the only closed analytic subvariety $Y$ of $X$ satisfying $S\subset Y(\C_p)$ is $X$ itself.
For every $r>0$, we denote by $\cB(0,r)$, respectively $\cB(0,r^-)$, the closed, respectively open, disc in $\C_p$ of centre $0$ and radius $r$.
The space $\cW$ is isomorphic to a disjoint union of
$p-1$ copies of the open unit disc $\cB(0,1^-)$ centered in $0$ and indexed by the group $\Z/(p-1)\Z=\widehat{\mu}_{p-1}$. If $u$ denotes a topological generator of $1+p\Z_p$, then an isomorphism is given by
$$ \Z/(p-1)\Z\times \cB(0,1^-)\to\cW,\quad (i,v)\mapsto\chi_{i,v}, $$
where $\chi_{i,v}((\zeta,u^x))=\zeta^i (1+v)^x$. Here we wrote an element of $\Z_p^\times$ uniquely as a pair $(\zeta, u^x)$ with $\zeta\in \mu_{p-1}$ and $x\in \Z_p$. We make once and for all the choice $u=1+p$.
We say that a point $\chi\in\cW(\C_p)$ is classical if there exist $k\in\N$ and a finite order character $\psi\colon\Z_p^\times\to\C_p^\times$ such that $\chi$ is the character $z\mapsto z^k\psi(z)$. The set of classical points is Zariski-dense in $\cW(\C_p)$
If $\Spm\, R\subset\cW$ is an affinoid open subset, we denote by $\kappa=\kappa_R\colon\Z_p^\times\to R^\times$ its tautological character
given by $\kappa(t)(\chi)=\chi(t)$ for every $\chi\in\Spm\, R$.
Recall (\cite[Prop. 8.3]{buzzard}) that $\kappa_R$ is $r$-analytic for every sufficiently small radius $r>0$ (by which we mean that it extends to a rigid analytic function on $\Z_p^\times \cB(1,r)$).
\subsection{Adapted pairs and the eigencurve}\label{sectadapt}
Let $N$ be a positive integer prime to $p$. We recall the definition of the spectral curve $Z^N$ and of the cuspidal eigencurve $C^N$ of
tame level $\Gamma_1(N)$. These objects were constructed in \cite{colmaz} for $p>2$ and $N=1$ and in \cite{buzzard} in general. We follow the presentation of \cite[Part II]{buzzard}.
Let $\Spm\,R \subset\cW$ be an affinoid domain and let $r=p^{-s}$ for $s\in\Q$ be a radius smaller than the radius of analyticity of $\kappa_R$.
We denote by $M_{R,r}$ the $R$-module of $r$-overconvergent modular forms of weight $\kappa_R$.
It is endowed it with a continuous action of the Hecke operators $T_\ell$, $\ell\nmid Np$, and $U_p$.
The action of $U_p$ on $M_{R,r}$ is completely continuous, so we can consider its associated Fredholm series
$F_{R,r}(T)=\det(1-U_pT\vert M_{R,r})\in R\{\{T\}\}$.
These series are compatible when $R$ and $r$ vary, in the sense that there exists $F\in\Lambda\{\{T\}\}$ that restricts to $F_{R,r}(T)$ for every $R$ and $r$.
The series $F_{R,r}(T)$ converges everywhere on the $R$-affine line $\Spm\,R\times \A^{1,\an}$, so it defines
a rigid curve $Z^{N}_{R,r}=\{F_{R,r}(T)=0\}$ in $\Spm\,R\times\A^{1,\an}$.
When $R$ and $r$ vary, these curves glue into a rigid space $Z^N$ endowed with
a quasi-finite and flat morphism $w_Z\colon Z^N\to\cW$. The curve $Z^N$ is called the spectral curve
associated with the $U_p$-operator.
For every $h\geq 0$, let us consider
$$ Z_R^{N,\leq h}=Z^N_R\cap\left(\Spm\,R\times B(0,p^h)\right). $$
By \cite[Lemma 4.1]{buzzard} $Z_R^{N,\leq h}$ is quasi-finite and flat over $\Spm\,R$.
We now recall how to construct an admissible covering of $Z^N$.
\begin{definition}
We denote by $\calC$ the set of affinoid domains $Y\subset Z$ such that:
\begin{itemize}
\item there exists an affinoid domain $\Spm\,R\subset\cW$ such that $Y$ is a union of connected components of $w_Z^{-1}(\Spm\,R)$;
\item the map $w_Z\vert_Y\colon Y\to\Spm\,R$ is finite.
\end{itemize}
\end{definition}
\begin{proposition} \cite[Th. 4.6]{buzzard}
The covering $\calC$ is admissible.
\end{proposition}
Note in particular that an element $Y\in\calC$ must be contained in $Z_R^{N,\leq h}$ for some $h$.
For every $R$ and $r$ as above and every $Y\in\calC$ such that $w_Z(Y)=\Spm\,R$, we can associate with $Y$ a direct factor $M_Y$ of $M_{R,r}$ by the construction in \cite[Sec. I.5]{buzzard}.
The abstract Hecke algebra $\calH=\Z[T_\ell]_{\ell\nmid Np}$ acts on $M_{R,r}$ and $M_Y$ is stable with respect to this action.
Let $\T_Y$ be the $R$-algebra generated by the image of $\calH$ in $\End_R(M_Y)$ and let $C_Y^N=\Spm\,\T_Y$.
Note that it is reduced as all Hecke operators are self-adjoint for a certain pairing and mutually commute
For every $Y$ the finite covering $C_Y^N\to\Spm\,R$ factors through $Y\to\Spm\,R$.
The eigencurve $C^N$ is defined by gluing the affinoids $C_Y^N$ into a rigid curve, endowed with a finite morphism $C^N\to Z^N$.
The curve $C^N$ is reduced and flat over $\cW$ since it is so locally.
We borrow the following terminology from Bella\"\i che.
\begin{definition}\label{adapt} \cite[Def. II.1.8]{bell}
Let $\Spm\,R\subset\cW$ be an affinoid open subset and $h>0$ be a rational number.
The couple $(R,h)$ is called adapted if $Z_R^{N,\leq h}$ is an element of $\calC$.
\end{definition}
\noindent By \cite[Cor. II.1.13]{bell} the sets of the form $Z_R^{N,\leq h}$ are sufficient to admissibly cover the spectral curve.
Now we fix a finite slope $h$. We want to work with families of slope $\leq h$ which are finite over a wide open subset of the weight space.
In order to do this it will be useful to know which pairs $(R,h)$ in a connected component of $\cW$ are adapted.
If $\Spm\,R^\prime\subset\Spm\,R$ are affinoid subdomains of $\cW$ and $(R,h)$ is adapted then $(R^\prime,h)$ is also adapted
by \cite[Prop. II.1.10]{bell}. By \cite[Lemma 4.3]{buzzard}, the affinoid $\Spm\,R$ is adapted to $h$ if and only if the weight map $Z_R^{N,\leq h}\to\Spm\, R$ has fibres of constant degree.
\begin{remark}
Given a slope $h$ and a classical weight $k$, it would be interesting to have a lower bound for the radius of a disc of centre $k$ adapted to $h$. A result of Wan \textnormal{(\cite[Th. 2.5]{wan})} asserts that for a certain radius $r_h$ depending only on $h,N$ and $p$, the degree of the fibres of $Z_{\cB(k,r_h)}^{N,\leq h}\to\Spm\, \cB(k,r_h)$ at classical weights is constant. Unfortunately we do not know whether the degree is constant at all weights of $\cB(k,r_h)$, so this is not sufficient to answer our question. Estimates for the radii of adapted discs exist in the case of eigenvarieties for groups different than $\GL_2$; see for example the results of Chenevier on definite unitary groups \textnormal{(\cite[Sec. 5]{chenevier})}.
\end{remark}
\subsection{Pseudo-characters and Galois representations}
Let $K$ be a finite extension of $\Q_p$ with valuation ring $\cO_K$.
Let $X$ be a rigid
analytic variety defined over $K$.
We denote by $\cO(X)$ the ring of global analytic functions on $X$ equipped with the coarsest locally convex topology making the restriction map $\cO(X)\to\cO(U)$ continuous for every affinoid $U\subset X$. It is a Fr\'echet space isomorphic to the inverse limit over all affinoid domains $U$ of the $K$-Banach spaces $\cO(U)$. We denote by $\cO(X)^\circ$ the $\cO_K$-algebra of functions bounded by $1$ on $X$, equipped with the topology induced by that on $\cO(X)$. The question of the compactness of this ring is related to the following property of $X$.
\begin{definition}\label{defnested}
\cite[Def. 7.2.10]{bellchen}
We say that a rigid
analytic variety $X$ defined over $K$ is nested if there is an admissible covering $X=\bigcup X_i$ by open affinoids $X_i$
defined over $K$ such that the maps $\cO(X_{i+1})\to\cO(X_i)$ induced by the inclusions are compact.
\end{definition}
We equip the ring $\cO(X)^\circ$ with the topology induced by that on $\cO(X)=\varprojlim_i \cO(X_i)$.
\begin{lemma}\label{nestcpt}
\cite[Lemma 7.2.11(ii)]{bellchen}
If $X$ is reduced and nested, then $\cO(X)^\circ$ is a compact (hence profinite) $\cO_K$-algebra.
\end{lemma}
We will be able to apply Lemma \ref{nestcpt} to the eigenvariety thanks to the following.
\begin{proposition}
\cite[Cor. 7.2.12]{bellchen}
The eigenvariety $C^N$ is nested for $K=\Q_p$.
\end{proposition}
Given a reduced nested subvariety $X$ of $C^N$ defined over a finite extension $K$ of $\Q_p$ there is a pseudo-character on $X$ obtained by interpolating the classical ones.
\begin{proposition}\label{pseudochar}
\cite[Th. IV.4.1]{bell}
There exists a unique pseudo-character
$$ \tau\colon G_{\Q,Np}\to \cO(X)^\circ $$
\noindent of dimension $2$ such that for every $\ell$ prime to $Np$, $\tau(\Frob_\ell)=\psi_X(T_\ell)$, where $\psi_X$ is the composition of $\psi\colon\calH\to\cO(C^N)^\circ$ with the restriction map $\cO(C^N)^\circ\to\cO(X)^\circ$.
\end{proposition}
\begin{remark} One can take as an example of $X$ a union of irreducible components of $C^N$ in which case $K=\Q_p$.
Later we will consider other examples where $K\neq\Q_p$.
\end{remark}
\bigskip
\section{The fortuitous congruence ideal}\label{sectcong}
In this section we will define families with slope bounded by a finite constant and coefficients in a suitable profinite ring.
We will show that any such family admits at most a finite number of classical specializations which are CM modular forms.
Later we will define what it means for a point (not necessarily classical) to be CM and we will associate with a family a congruence ideal describing its CM points.
Contrary to the ordinary case, the non-ordinary CM points do not come in families so the points detected by the congruence ideal do not correspond to a crossing between a CM and a non-CM family.
For this reason we call our ideal the ``fortuitous congruence ideal''.
\subsection{The adapted slope $\leq h$ Hecke algebra}\label{subsectnest}
Throughout this section we fix a slope $h>0$ .
Let $C^{N,\leq h}$ be the subvariety of $C^N$ whose points have slope $\leq h$. Unlike the ordinary case treated in \cite{hida} the weight map $w^{\leq h}\colon C^{N,\leq h}\to\cW$ is not finite which means that a family of slope $\leq h$ is not in general defined by a finite map over the entire weight space. The best we can do in the finite slope situation is to place ourselves over the largest possible wide open subdomain $U$ of $\cW$ such that the restricted weight map $w^{\leq h}\vert_U\colon C^{N,\leq h}\times_{\cW}U\to U$ is finite. This is a domain ``adapted to $h$'' in the sense of Definition \ref{adapt} where only affinoid domains were considered.
The finiteness property will be necessary in order to apply going-up and going-down theorems.
Let us fix a rational number $s_h$ such that for $r_h=p^{-s_h}$ the closed disc $B(0,r_h)$ is adapted for $h$.
We assume that $s_h>\frac{1}{p-1}$ (this will be needed later to assure the convergence of the exponential map). Let $\eta_h\in\overline{\Q}_p$ be an element
of $p$-adic valuation $s_h$. Let $K_h$ be the Galois closure (in $\C_p$) of $\Q_p(\eta_h)$ and let $\cO_h$ be its valuation ring. Recall that $T$ is the variable on the open disc of radius $1$. Let $t=\eta_h^{-1}T$ and $\Lambda_h=\cO_h[[t]]$.
This is the ring of analytic functions, with $\cO_h$-coefficients and bounded by one, on the wide open disc $\cB_h$ of radius $p^{-s_h}$.
There is a natural map $\Lambda\to\Lambda_h$ corresponding to the restriction of analytic functions on the open disc of radius $1$, with $\Z_p$ coefficients and bounded by $1$, to the open disc of radius $r_h$. The image of this map is the ring $\Z_p[[\eta t]]\subset\cO_h$.
For $i\geq 1$, let $s_i=s_h+1/i$ and $\cB_i=\cB(0,p^{-s_i})$. The open disc $\cB_h$ is the increasing union of the affinoid discs $\cB_i$.
For each $i$ a model for $\cB_i$ over $K_h$ is given by Berthelot's construction of $\cB_h$ as
the rigid space associated with the $\cO_h$-formal scheme $\Spf\,\Lambda_h$. We recall it briefly following \cite[Sec. 7]{dejong}.
Let
$$ A_{r_i}^\circ=\cO_h\langle t,X_i\rangle/(pX_i-t^i). $$
We have $\cB_i=\Spm\,A_{r_i}^\circ[p^{-1}]$ as rigid space over $K_h$.
For every $i$ we have a morphism $A_{r_{i+1}}^\circ\to A_{r_i}^\circ$ given by
$$ X_{i+1}\mapsto X_i t $$
$$ t\mapsto t $$
We have induced morphisms $A_{r_{i+1}}^\circ[p^{-1}]\to A_{r_i}^\circ[p^{-1}]$, hence open immersions $\cB_i\to\cB_{i+1}$ defined over $K_h$. The wide open disc $\cB_h$ is defined as the inductive limit of the affinoids $\cB_i$ with these transition maps.
We have $\Lambda_h=\varprojlim_i A_{r_i}^\circ$.
Since the $s_i$ are strictly bigger than $s_h$ for each $i$, $\cB(0,p^{-s_i})=\Spm\, A_{r_i}^\circ[p^{-1}]$ is adapted to $h$. Therefore for every $r>0$ sufficiently small and for every $i\geq 1$ the image of the abstract Hecke algebra acting on $M_{A_{r_i},r}$ provides a finite affinoid $A_{r_i}^\circ$-algebra $\T_{A_{r_i}^\circ,r}^{\leq h}$. The morphism $w_{A_{r_i}^\circ,r}\colon\Spm\,\T_{A_{r_i}^\circ,r}^{\leq h}\to\Spm\, A_{r_i}^\circ$ is finite.
For $i<j$ we have natural inclusions $\Spm\,\T_{A_{r_j}^\circ,r}^{\leq h}\to\Spm\,\T_{A_{r_i}^\circ,r}^{\leq h}$ and corresponding restriction maps $\T_{A_{r_i}^\circ,r}^{\leq h}\to\T_{A_{r_j}^\circ,r}^{\leq h}$. We call $C_h$ the increasing union $\bigcup_{i\in\N,r>0}\Spm\,\T_{A_{r_i}^\circ,r}^{\leq h}$; it is a wide open subvariety of $C^{N}$. We denote by $\T_h$ the ring of rigid analytic functions bounded by $1$ on $C_h$. We have $\T_h=\cO(C_h)^\circ=\varprojlim_{i,r}\T_{A_{r_i}^\circ,r}^{\leq h}$. There is a natural weight map $w_h\colon C_h\to \cB_h$ that restricts to the maps $w_{A_{r_i}^\circ,r}$. It is finite because the closed ball of radius $r_h$ is adapted to $h$.
\subsection{The Galois representation associated with a family of finite slope}
Since $\cO(B_h)^\circ=\Lambda_h$, the map $w_h$ gives $\T_h$ the structure of a finite $\Lambda_h$-algebra; in particular $\T_h$ is profinite.
Let $\fm$ be a maximal ideal of $\T_h$. The residue field $k=\T_h/\fm$ is finite.
Let Let $\T_\fm$ denote the localization of $\T_h$ at $\fm$. Since $\Lambda_h$ is henselian,
$\T_\fm$ is a direct factor or $\T_h$, hence it is finite over $\Lambda_h$;
it is also local noetherian and profinite. It is the ring of functions bounded by $1$ on a connected component of $C_h$.
Let $W=W(k)$ be the ring of Witt vectors of $k$. By the universal property of $W$, $\T_\fm$ is a $W$-algebra.
$\Spm\,\T_\fm$ contains points $x$ corresponding to cuspidal eigenforms
$f_x$ of weight $w(x)=k_x\geq 2$ and level $Np$. Let $\Q^{Np}$ be the maximal extension of $\Q$ unramified
outside $Np$ and let $G_{\Q,Np}=\Gal(\Q^{Np}/\Q)$. The Galois representations $\rho_{f_x}$
associated with $f_x$ give rise to a residual representation $\overline{\rho}\colon G_{\Q,Np}\to \GL_2(k)$
that is independent of $f_x$. By Proposition \ref{pseudochar}, we have a pseudo-character
$$ \tau_{\T_\fm}\colon G_{\Q,Np}\to\T_\fm $$
such that for every classical point $x\colon\T_\fm\to L$, defined over some finite extension $L/\Q_p$,
the specialization of $\tau_{\T_\fm}$ at $x$ is the trace of the usual representation $G_{\Q,Np}\to\GL_2(L)$ attached to $x$.
\begin{proposition}\label{pseudocharrep}
If $\overline{\rho}$ is absolutely irreducible there exists a unique continuous irreducible Galois representation
$$ \rho_{\T_\fm}\colon G_{\Q,Np}\to\GL_2(\T_\fm), $$
lifting $\overline{\rho}$ and whose trace is $\tau_{\T_\fm}$.
\end{proposition}
\noindent This follows from a result of Nyssen and Rouquier (\cite{nyssen}, \cite[Cor. 5.2]{rouquier}), since $\T_\fm$ is local henselian.
Let $\I^\circ$ be a finite torsion-free $\Lambda_h$-algebra. We call \textit{family} an irreducible component of $\Spec\,\T_h$ defined by a surjective morphism $\theta\colon\T_h\to\I^\circ$ of $\Lambda_h$-algebras. Since such a map factors via $\T_\fm\to\I^\circ$ for some maximal ideal $\fm$ of $\T_h$, we can define a residual representation $\overline{\rho}$ associated with $\theta$ as above. Suppose that $\overline{\rho}$ is irreducible. By Proposition \ref{pseudocharrep} we obtain a Galois representation $\rho\colon G_\Q\to\GL_2(\I^\circ)$ associated with $\theta$.
\begin{remark}
If $\eta_h\notin\Q_p$, the open disc $\cB_h$ is not defined over $\Q_p$. In particular $\Lambda_h$ is not a power series ring over $\Z_p$.
\end{remark}
\subsection{Finite slope CM modular forms}\label{cmforms}
In this section we study non-ordinary finite slope CM modular forms. We say that a family is CM if all its classical points are CM.
We prove that for every positive slope $h>0$ there are no CM families with positive slope $\leq h$.
However, contrary to the ordinary case, every family of finite positive slope may contain classical CM points of weight $k\geq 2$.
Let $F$ be an imaginary quadratic field, $\ff$ an integral ideal in $F$, $I_{\ff}$ the group of fractional ideals prime to ${\ff}$.
Let $\sigma_1,\sigma_2$ be the embeddings of $F$ into $\C$ (say that $\sigma_1=\Id_F$) and let $(k_1,k_2)\in\Z^2$.
A Gr\"ossencharacter $\psi$ of infinity type $(k_1,k_2)$ defined modulo $\ff$ is a homomorphism
$\psi\colon I_{\ff}\to\C^\ast$ such that $\psi((\alpha))=\sigma_1(\alpha)^{k_1}\sigma_2(\alpha)^{k_2}$ for all $\alpha\equiv 1$ $(\mathrm{mod}^\times \ff)$ .
Consider the $q$-expansion
$$ \sum_{\fa\subset\cO_F, (\fa,\ff)=1}\psi(\fa)q^{N(\fa)}, $$
where the sum is over ideals $\fa\subset\cO_F$ and $N(\fa)$ denotes the norm of $\fa$.
Let $F/\Q$ be an imaginary quadratic field of discriminant $D$ and let $\psi$ be a Gr\"ossencharacter of exact conductor $\ff$ and infinity type $(k-1,0)$.
By \cite[Lemma 3]{shim} the expansion displayed above defines a cuspidal newform $f(F,\psi)$ of level $N(\ff)D$.
Ribet proved in \cite[Th. 4.5]{ribet2} that if a newform $g$ of weight $k\geq 2$ and level $N$ has CM by a quadratic imaginary field $F$, one has
$g=f(F,\psi)$ for some Gr\"ossencharacter $\psi$ of $F$ of infinity type $(k-1,0)$.
\begin{definition}\label{classcmform}
We say that a classical modular eigenform $g$ of weight $k$ and level $Np$ has CM by an imaginary quadratic field $F$
if its Hecke eigenvalues for the operators $T_\ell$, $\ell\nmid Np$, coincide with those of $f(F,\psi)$
for some Gr\"ossencharacter $\psi$ of $F$ of infinity type $(k-1,0)$. We also say that $g$ is CM without specifying the field.
\end{definition}
\begin{remark}
For $g$ as in the definition the Galois representations $\rho_g,\rho_{f(F,\psi)}\colon G_\Q\to\GL_2(\overline{\Q}_p)$ associated with $g$ and $f(F,\psi)$ are isomorphic, hence the image of the representation $\rho_g$ is contained in the normalizer of a torus in $\GL_2$, if and only if the form $g$ is CM.
\end{remark}
\begin{proposition}\label{cmslopes}
Let $g$ be a CM modular eigenform of weight $k$ and level $Np^m$ with $N$ prime to $p$ and $m\geq 0$.
Then its $p$-slope is either $0$, $\frac{k-1}{2}$, $k-1$ or infinite.
\end{proposition}
\begin{proof}
Let $F$ be the quadratic imaginary field and $\psi$ the Gr\"ossencharacter of $F$ associated with the CM form $g$ by Definition \ref{classcmform}.
Let $\ff$ be the conductor of $\psi$.
We assume first that $g$ is $p$-new, so that $g=f(F,\psi)$. Let $a_p$ be the $U_p$-eigenvalue of $g$.
If $p$ is inert in $F$ we have $a_p=0$, so the $p$-slope of $g$ is infinite.
If $p$ splits in $F$ as $\fp\bar{\fp}$, then $a_p=\psi(\fp)+\psi(\bar{\fp})$. We can find an integer $n$ such that $\fp^n$ is
a principal ideal $(\alpha)$ with $\alpha\equiv 1\,(\mathrm{mod}^\times\ff)$. Hence $\psi((\alpha))=\alpha^{k-1}$. Since $\alpha$
is a generator of $\fp^n$ we have $\alpha\in\fp$ and $\alpha\notin\bar{\fp}$; moreover $\alpha^{k-1}=\psi((\alpha))=\psi(\fp)^n$,
so we also have $\psi(\fp)\in\fp-\bar{\fp}$. In the same way we find $\psi(\bar{\fp})\in\bar{\fp}-\fp$. We conclude that $\psi(\fp)+\psi(\bar{\fp})$ does not belong to $\fp$, so its $p$-adic valuation is $0$.
If $p$ ramifies as $\fp^2$ in $F$, then $a_p=\psi(\fp)$. As before we find n such that $\fp^n=(\alpha)$ with $\alpha\equiv 1\,(\mathrm{mod}^\times\ff)$. Then $(\psi(\fp))^n\psi(\fp^n)=\psi((\alpha))=\alpha^{k-1}=\fp^{n(k-1)}$. By looking at $p$-adic valuations we find that the slope is $\frac{k-1}{2}$.
If $g$ is not $p$-new, it is the $p$-stabilization of a CM form $f(F,\psi)$ of level prime to $p$. If $a_p$ is the $T_p$-eigenvalue of $f(F,\psi)$, the $U_p$-eigenvalue of $g$ is a root of the Hecke polynomial $X^2-a_pX+\zeta p^{k-1}$ for some root of unity $\zeta$. By our discussion of the $p$-new case, the valuation of $a_p$ belongs to the set $\left\{0,\frac{k-1}{2},k-1\right\}$. Then it is easy to see that the valuations of the roots of the Hecke polynomial belong to the same set.
\end{proof}
We state a useful corollary.
\begin{corollary}
There are no CM families of strictly positive slope.
\end{corollary}
\begin{proof}
We show that the eigencurve $C_h$ contains only a finite number of points corresponding to classical CM forms. It will follow that almost all classical points of a family in $C_h$ are non-CM.
Let $f$ be a classical CM form of weight $k$ and positive slope. By Proposition \ref{cmslopes} its slope is at least $\frac{k-1}{2}$. If $f$ corresponds to a point of $C_h$ its slope must be $\leq h$, so we obtain an inequality $\frac{k-1}{2}\leq h$. The set of weights $\calK$ satisfying this condition is finite.
Since the weight map $C_h\to B_h$ is finite, the set of points of $C_h$ whose weight lies in $\calK$ is finite. Hence the number of CM forms in $C_h$ is also finite.
\end{proof}
We conclude that, in the finite positive slope case, classical CM forms can appear only as isolated points
in an irreducible component of the eigencurve $C_h$. In the ordinary case, the congruence ideal
of a non-CM irreducible component is defined as the intersection ideal of the CM irreducible components
with the given non-CM component.
In the case of a positive slope family $\theta\colon\T_h\to\I^\circ$, we need to define the congruence ideal in a different way.
\subsection{Construction of the congruence ideal}\label{congrid}
Let $\theta\colon\T_h\to\I^\circ$ be a family. We write $\I=\I^\circ[p^{-1}]$
Fix an imaginary quadratic field $F$ where $p$ is inert or ramified; let $-D$ be its discriminant.
Let $\fQ$ be a primary ideal of $\I$; then $\fq=\fQ\cap\Lambda_h$ is a primary ideal of $\Lambda_h[p^{-1}]$.
The projection $\Lambda_h\to\Lambda_h/\fq$ defines a point of $\cB_h$ (possibly non-reduced) corresponding to a weight $\kappa_\fQ\colon\Z_p^\ast\to(\Lambda_h/\fq)^\ast$.
For $r>0$ we denote by $\cB_r$ the ball of centre $1$ and radius $r$ in $\C_p$. By \cite[Prop. 8.3]{buzzard} there exist $r>0$ and a character $\kappa_{\fQ,r}\colon\Z_p^\times\cdot\cB_r\to(\Lambda_h/\fq)^\times$ extending $\kappa_\fQ$.
Let $\sigma$ be an embedding $F\into\C_p$. Let $r$ and $\kappa_{\fQ,r}$ be as above.
For $m$ sufficiently large $\sigma(1+p^m\cO_F)$ is contained in $\Z_p^\times\cdot\cB_r$, the domain of definition of $\kappa_{\fQ,r}$.
For an ideal $\ff\subset\cO_F$ let $I_{\ff}$ be the group of fractional ideals prime to ${\ff}$.
For every prime $\ell$ not dividing $Np$ we denote by $a_{\ell,\fQ}$ the image of the Hecke operator $T_\ell$ in $\I^\circ/\fQ$.
We define here a notion of non-classical CM point of $\theta$ (hence of the eigencurve $C_h$) as follows.
\begin{definition}
Let $F,\sigma,\fQ,r,\kappa_{\fQ,r}$ be as above. We say that $\fQ$ defines a CM point of weight $\kappa_{\fQ,r}$ if there exist an integer $m>0$, an ideal $\ff\subset\cO_F$ with norm $N(\ff)$ such that $DN(\ff)$ divides $N$, a quadratic extension $(\I/\fQ)^\prime$ of $\I/\fQ$ and a homomorphism $\psi\colon I_{\ff p^m}\to (\I/\fQ)^{\prime\times}$ such that:
\begin{enumerate}
\item $\sigma(1+p^m\cO_F)\subset\Z_p^\times\cdot\cB_r$;
\item for every $\alpha\in\cO_F$ with $\alpha\equiv 1\, (\mathrm{mod}^\times\ff p^m)$, $\psi((\alpha))=\kappa_{\fQ,r}(\alpha)\alpha^{-1}$;
\item $a_{\ell,\fQ}=0$ if $l$ is a prime inert in $F$ and not dividing $Np$;
\item $a_{\ell,\fQ}=\psi(\fl)+\psi(\bar{\fl})$ if $\ell$ is a prime splitting as $\fl\bar{\fl}$ in $F$ and not dividing $Np$.
\end{enumerate}
\end{definition}
\noindent Note that $\kappa_{\fQ,r}(\alpha)$ is well defined thanks to condition $1$.
\begin{remark}
If $\fP$ is a prime of $\I$ corresponding to a classical form $f$ then $\fP$ is a CM point if and only if $f$ is a CM form in the sense
of Section \ref{cmforms}
\end{remark}
\begin{proposition}\label{finiteCM}
The set of CM points in $\Spec\,\I$ is finite.
\end{proposition}
\begin{proof}
By contradiction assume it is infinite. Then we have an injection $\I\into\prod_\fP\I/\fP$ where $\fP$ runs over the set
of CM prime ideals of $\I$. One can assume that the imaginary quadratic field of complex multiplication is constant along $\I$. We can also assume that the ramification
of the associated Galois characters
$\lambda_{\fP}\colon G_F\to (\I/\fP)^\times$ is bounded (in support and in exponents).
On the density one set of primes of $F$ prime to $\ff p$ and of degree one, they take value in the image of $\I^\times$ hence they define
a continuous Galois character $\lambda\colon G_F\to\I^\times$ such that $\rho_\theta=\Ind^{G_\Q}_{G_F}\lambda$, which is absurd (by specialization at non-CM classical points which do exist).
\end{proof}
\begin{definition}
The (fortuitous) congruence ideal $\fc_\theta$ associated with the family $\theta$ is defined as the intersection of all the primary ideals of $\I$ corresponding to CM points.
\end{definition}
\begin{remark}\label{cmlocus} (Characterizations of the CM locus)
\begin{enumerate}[leftmargin=*]
\item Assume that $\overline{\rho}_\theta=\Ind^{G_\Q}_{G_K}\overline{\lambda}$ for a unique imaginary quadratic field $K$.
Then the closed subscheme $V(\fc_\theta)=\Spec\,\I/\fc_\theta\subset \Spec\,\I$ is the largest subscheme on which
there is an isomorphism of Galois representations $\rho_\theta\cong\rho_\theta\otimes\left(\frac{K/\Q}{\bullet}\right)$.
Indeed, for every artinian $\Q_p$-algebra $A$, a CM point $x \colon\I\to A$ is characterized by the conditions
$x(T_\ell)=x(T_\ell)\left(\frac{K/\Q}{\ell}\right)$ for all
primes $\ell$ prime to $Np$.
\item Note that $N$ is divisible by the discriminant $D$ of $K$. Assume that $\I$ is $N$-new and that $D$ is prime to $N/D$.
Let $W_D$ be the Atkin-Lehner involution associated with $D$. Conjugation by $W_D$ defines an automorphism $\iota_D$
of $\T_h$ and of $\I$.
Then $V(\fc_\theta)$ coincides with the (schematic) invariant locus $(\Spec\,\I)^{\iota_D=1}$.
\end{enumerate}
\end{remark}
\bigskip
\section{The image of the representation associated with a finite slope family}\label{imgfinslope}
It is shown by J. Lang in \cite[Th. 2.4]{lang} that, under some technical hypotheses,
the image of the Galois representation $\rho\colon G_\Q\to\GL_2(\I^\circ)$ associated with a non-CM ordinary family
$\theta\colon\T\to\I^\circ$ contains a congruence subgroup of $\SL_2(\I^\circ_0)$, where $\I^\circ_0$ is the subring
of $\I^\circ$ fixed by certain ``symmetries'' of the representation $\rho$.
In order to study the Galois representation associated with a non-ordinary family we will
adapt some of the results in \cite{lang} to this situation. Since the crucial step (\cite[Th. 4.3]{lang})
requires the Galois ordinarity of the representation (as in \cite[Lemma 2.9]{hida}), the results of this section
will not imply the existence of a congruence subgroup of $\SL_2(\I^\circ_0)$ contained in the image of $\rho$.
However, we will prove in later sections the existence of a ``congruence Lie subalgebra'' of $\fsl_2(\I_0^\circ)$ contained in a suitably defined Lie algebra
of the image of $\rho$ by means of relative Sen theory.
For every ring $R$ we denote by $Q(R)$ its total ring of fractions.
\subsection{The group of self-twists of a family}\label{selftwists}
We follow \cite[Sec. 2]{lang} in this subsection.
Let $h\geq 0$ and $\theta\colon\T_h\to\I^\circ$ be a family of slope $\leq h$ defined over a finite torsion free $\Lambda_h$-algebra $\I^\circ$.
Recall that there is a natural map $\Lambda\to\Lambda_h$ with image $\Z_p[[\eta t]]$.
\begin{definition}
We say that $\sigma\in\Aut_{Q(\Z_p[[\eta t]])}(Q(\I^\circ))$ is a conjugate self-twist for $\theta$ if there exists a Dirichlet character $\eta_\sigma\colon G_\Q\to\I^{\circ,\times}$ such that
$$ \sigma(\theta(T_\ell))=\eta_\sigma(\ell)\theta(T_\ell) $$
for all but finitely many primes $\ell$.
\end{definition}
Any such $\sigma$ acts on $\Lambda_h=\cO_h[[t]]$ by restriction, trivially on $t$ and by a Galois automorphism on $\cO_h$.
The conjugates self-twists for $\theta$ form a subgroup of $\Aut_{Q(\Z_p[[\eta t]])}(Q(\I^\circ))$.
We recall the following result which holds without assuming the ordinarity of $\theta$.
\begin{lemma}\label{gammaabel} \cite[Lemma 7.1]{lang}
$\Gamma$ is a finite abelian $2$-group.
\end{lemma}
We suppose from now on that $\I^\circ$ is normal. The only reason for this hypothesis is that in this case $\I^\circ$ is stable under the action of $\Gamma$ on $Q(\I^\circ)$, which is not true in general. This makes it possible to define the subring $\I^\circ_0$ of elements of $\I^\circ$ fixed by $\Gamma$.
\begin{remark}
The hypothesis of normality of $\I^\circ$ is just a simplifying one.
We could work without it by introducing the $\Lambda_h$-order $\I^{\circ,\prime}=\Lambda_h[\theta(T_\ell),\ell\nmid Np]\subset\I^\circ$: this is an analogue of the $\Lambda$-order $\I^\prime$ defined in \cite[Sec. 2]{lang}, and it is stable under the action of $\Gamma$.
We would define $\I^\circ_0$ as the fixed subring of $\I^{\circ,\prime}$ and the arguments in the rest of the article could be adapted to this setting.
\end{remark}
We denote by $\cO_{h,0}$ the subring of $\cO_h$ fixed by $\Gamma$ and we put $\Lambda_{h,0}=\cO_{h,0}[[t]]$. We also denote by $K_{h,0}$ the field of fractions of $\cO_{h,0}$.
\begin{remark}\label{arithprimes}
By definition $\Gamma$ fixes $\Z_p[[\eta t]]$, so we have $\Z_p[[\eta t]]\subset\Lambda_{h,0}$. In particular it makes sense to speak about the ideal $P_k\Lambda_{h,0}$ for every arithmetic prime $P_k=(1+\eta t-u^k)\subset\Z_p[[\eta t]]$. Note that $P_k\Lambda_h$ defines a prime ideal of $\Lambda_h$ if and only if the weight $k$ belongs to the open disc $B_h$, otherwise $P_k\Lambda_h=\Lambda_h$. We see immediately that the same statement is true if we replace $\Lambda_h$ by $\Lambda_{h,0}$.
\end{remark}
Note that $\I^\circ_0$ is a finite extension of $\Lambda_{h,0}$ because $\I^\circ$ is a finite $\Lambda_h$-algebra.
Moreover, we have $K_h^\Gamma=K_{h,0}$ (although the inclusion $\Lambda_h\cdot\I_0^\circ\subset \I^\circ$
may not be an equality).
We define two open normal subgroups of $G_\Q$ by:
\begin{itemize}
\item $H_0=\bigcap_{\sigma\in\Gamma}\ker\eta_\sigma$;
\item $H=H_0\cap\ker(\det\overline{\rho})$.
\end{itemize}
\noindent Note that $H_0$ is an open normal subgroup of $G_\Q$ and that $H$ is
a pro-$p$ open normal subgroup of $H_0$ and of $G_\Q$.
\subsection{The level of a general ordinary family}
We recall the main result of \cite{lang}. Denote by $\T$ the big ordinary Hecke algebra, which is finite over $\Lambda=\Z_p[[T]]$. Let $\theta\colon\T\to\I^\circ$ be an ordinary family with associated Galois representation $\rho\colon G_\Q\to\GL_2(\I^\circ)$. The representation $\rho$ is $p$-ordinary, which means that its restriction $\rho\vert_{D_p}$ to a decomposition subgroup $D_p\subset G_\Q$ is reducible. There exist two characters $\varepsilon,\delta\colon D_p\to\I^{\circ,\times}$, with $\delta$ unramified, such that $\rho\vert_{D_p}$ is an extension of $\varepsilon$ by $\delta$.
Denote by $\F$ the residue field of $\I^\circ$ and by $\overline{\rho}$ the representation $G_\Q\to\GL_2(\F)$ obtained by reducing $\rho$ modulo the maximal ideal of $\I^\circ$. Lang introduces the following technical condition.
\begin{definition}
The $p$-ordinary representation $\overline{\rho}$ is called $H_0$-regular if $\overline{\varepsilon}\vert_{D_p\cap H_0}\neq\overline{\delta}\vert_{D_p\cap H_0}$.
\end{definition}
The following result states the existence of a Galois level for $\rho$.
\begin{theorem}\label{ordlevel} \cite[Th. 2.4]{lang}
Let $\rho\colon G_\Q\to\GL_2(\I^\circ)$ be the representation associated with an ordinary, non-CM family $\theta\colon \T\to\I^\circ$.
Assume that $p>2$, the cardinality of $\F$ is not $3$ and the residual representation $\overline{\rho}$
is absolutely irreducible and $H_0$-regular. Then there exists $\gamma\in\GL_2(\I^\circ)$ such that $\gamma\cdot \im\,\rho\cdot \gamma^{-1}$ contains a congruence subgroup of $\SL_2(\I^\circ_0)$.
\end{theorem}
\noindent The proof relies on the analogous result proved by Ribet (\cite{ribet1}) and Momose (\cite{momose})
for the $p$-adic representation associated with a classical modular form.
\subsection{An approximation lemma}\label{sectapprox}
In this subsection we prove an analogue of \cite[Lemma 4.5]{hidatil}.
It replaces in our approach the use of Pink's Lie algebra theory, which is relied upon in the case of ordinary representations
in \cite{hida} and \cite{lang}.
Let $\I_0^\circ$ be a domain that is finite torsion free over $\Lambda_h$. It does not need to be related to a Hecke algebra for the moment.
Let $N$ be an open normal subgroup of $G_\Q$ and let $\rho\colon N\to\GL_2(\I_0^\circ)$ be an arbitrary continuous representation.
We denote by $\fm_{\I_0^\circ}$ the maximal ideal of $\I_0^\circ$ and by $\F=\I_0^\circ/\fm_{\I_0^\circ}$ its residue field of cardinality $q$.
In the lemma we do not suppose that $\rho$ comes from a family of modular forms.
We will only assume that it satisfies the following technical condition:
\begin{definition}\label{Z_p-reg}
Keep notations as above.
We say that the representation $\rho\colon N\to\GL_2(\I_0^\circ)$ is $\Z_p$-regular if there exists $d\in\im\,\rho$ with
eigenvalues $d_1,d_2\in\Z_p$ such that $d_1^2\not\equiv d_2^2\pmod{p}$.
We call $d$ a $\Z_p$-regular diagonal element.
If $N^\prime$ is an open normal subgroup of $N$ then we say that $\rho$ is $(N^\prime,\Z_p)$-regular if $\rho\vert_{N^\prime}$ is $\Z_p$-regular.
\end{definition}
Note that $\rho(\delta)\in\im\,\rho$ is of finite order dividing $p-1$.
Let $B^{\pm}$ denote the Borel subgroups consisting of upper, respectively lower, triangular matrices in $\GL_2$.
Let $U^{\pm}$ be the unipotent radical of $B^{\pm}$.
\begin{proposition}\label{approx}
Let $\I_0^\circ$ be a finite torsion free $\Lambda_{h,0}$-algebra, $N$ an open normal subgroup of $G_\Q$ and $\rho$ a continuous representation $N\to\GL_2(\I_0^\circ)$ that is $\Z_p$-regular. Suppose (upon replacing $\rho$ by a conjugate) that the $\Z_p$-regular element is diagonal.
Let $\bP$ be an ideal of $\I_0^\circ$ and $\rho_\bP\colon N\to\GL_2(\I_0^\circ/\bP)$ be the representation given by the reduction of $\rho$ modulo $\bP$. Let $U^\pm(\rho)$, respectively $U^\pm(\rho_\bP)$ be the upper and lower unipotent subgroups of the image $\im\,\rho$, respectively $\im\,\rho_\bP$.
Then the natural maps $U^+(\rho_\theta)\to U^+(\rho_\bP)$ and $U^-(\rho_\theta)\to U^-(\rho_\bP)$ are surjective
\end{proposition}
\begin{remark}
The ideal $\bP$ in the proposition is not necessarily prime. At a certain point we will need to take $\bP=P\I_0^\circ$ for a prime ideal $P$ of $\Lambda_{h,0}$.
\end{remark}
As in \cite[Lemma 4.5]{hidatil} we need two lemmas.
Since the argument is the same for $U^+$ and $U^-$,
we will only treat here the upper triangular case $U=U^+$ and $B=B^+$.
For $\ast=U, B$ and every $j\geq 1$ we define the groups
$$ \Gamma_{\ast}(\bP^j)=\{x\in \SL_2(\I_0^\circ)\, |\, x\,\, (\mathrm{mod}\,\,\bP^j)\in \ast (\I_0^\circ/\bP^j)\}.$$
Let $\Gamma_{\I_0^\circ}(\bP^j)$
be the kernel of the reduction morphism $\pi_j\colon \SL_2(\I_0^\circ)\to \SL_2(\I_0^\circ/\bP^j)$.
Note that $\Gamma_{U}(\bP^j)=\Gamma_{\I_0^\circ}(\bP^j)U(\I_0^\circ)$ consists of matrices
$\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ such that $a,d\equiv 1 \pmod{\bP^j}$, $c\equiv 0\pmod{\bP^j}$.
Let $K=\im\,\rho$ and
$$ K_{U}(\bP^j)=K\cap\Gamma_{U}(\bP^j),\quad
K_{B}(\bP^j)=K\cap\Gamma_{B}(\bP^j). $$
Since $U(\I_0^\circ)$ and $\Gamma_{\I_0^\circ}(\bP)$
are $p$-profinite, the groups $\Gamma_{U}(\bP^j)$ and $K_{U}(\bP^j)$ for all $j\geq 1$ are also $p$-profinite.
Note that
$$ \left[\left(\begin{smallmatrix}a&b\\
c&-a\end{smallmatrix}\right),\left(\begin{smallmatrix}e&f\\
g&-e\end{smallmatrix}\right)\right]
=\left(\begin{smallmatrix}bg-cf&2(af-be)\\
2(ce-ag)&cf-bg\end{smallmatrix}\right). $$
\noindent From this we obtain the following.
\begin{lemma}\label{ijk}
If $X,Y\in\fsl_2(\I_0^\circ)\cap\left(\begin{smallmatrix}\bP^j&\bP^k\\
\bP^i&\bP^j\end{smallmatrix}\right)$ with $i\ge j\ge k$, then
$[X,Y]\in \left(\begin{smallmatrix}\bP^{i+k}&\bP^{j+k}\\
\bP^{i+j}&\bP^{i+k}\end{smallmatrix}\right)$.
\end{lemma}
We denote by $\mathrm{D}\Gamma_{U}(\bP^j)$ the topological commutator subgroup $(\Gamma_{U}(\bP^j),\Gamma_{U}(\bP^j))$. Lemma \ref{ijk} tells us that
\begin{equation}\label{BN} \mathrm{D}\Gamma_{U}(\bP^j)\subset\Gamma_{B}(\bP^{2j})\cap\Gamma_{U}(\bP^j). \end{equation}
By the $\Z_p$-regularity assumption, there exists a diagonal element $d\in K$ whose eigenvalues are in $\Z_p$ and distinct modulo $p$.
Consider the element $\delta=\lim_{n\to\infty}d^{p^n}$, which belongs to $K$ since this is $p$-adically complete.
In particular $\delta$ normalizes $K$. It is also diagonal with coefficients in $\Z_p$, so it normalizes $K_{U}(\bP^j)$ and $\Gamma_{B}(\bP^j)$. Since $\delta^p=\delta$, the eigenvalues $\delta_1$ and $\delta_2$ of $\delta$ are roots of unity of order dividing $p-1$. They still satisfy $\delta_1^2\neq\delta_2^2$ as $p\neq 2$.
Set $\alpha=\delta_1/\delta_2\in\F_p^\times$ and let $a$ be the order of $\alpha$ as a root of unity.
We see $\alpha$ as an element of $\Z_p^\times$ via the Teichm\"uller lift.
Let $H$ be a $p$-profinite group normalized by $\delta$.
Since $H$ is $p$-profinite, every $x\in H$ has a unique $a$-th root.
We define a map $\Delta\colon H\to H$ given by
$$ \Delta(x)=[x\cdot \ad(\delta) (x)^{\alpha^{-1}}\cdot\ad(\delta^2)(x)^{\alpha^{-2}}\cdot\,\cdots\,\cdot
\ad(\delta^{a-1})(x)^{\alpha^{1-a}}]^{1/a} $$
\begin{lemma}\label{2jlem}
If $u\in \Gamma_{U}(\bP^j)$ for some $j\ge 1$,
then $\Delta^2(u)\in \Gamma_{U}(\bP^{2j})$
and $\pi_j(\Delta(u))=\pi_j(u)$.
\end{lemma}
\begin{proof}
If $u\in \Gamma_{U}(\bP^j)$, we have $\pi_j(\Delta(u))=\pi_j(u)$
as $\Delta$ is the identity map on $U(\I_0^\circ/\bP^j)$.
Let $\mathrm{D}\Gamma_{U}(\bP^j)$ be the topological commutator subgroup of $\Gamma_{U}(\bP^j)$.
Since $\Delta$ induces the projection of the $\Z_p$-module $\Gamma_{U}(\bP^j)/\mathrm{D}\Gamma_{U}(\bP^j)$ onto its $\alpha$-eigenspace for $\ad(d)$,
it is a projection onto $U(\I_0^\circ) \mathrm{D}\Gamma_{U}(\bP^j)/\mathrm{D}\Gamma_{U}(\bP^j)$.
The fact that this is exactly the $\alpha$-eigenspace comes from the Iwahori
decomposition of $\Gamma_{U}(\bP^j)$, hence
a similar direct sum decomposition holds in the abelianization $\Gamma_{U}(\bP^j)/\mathrm{D}\Gamma_{U}(\bP^j)$.
By \eqref{BN}, $\mathrm{D}\Gamma_{U}(\bP^j)\subset \Gamma_{B}(\bP^{2j})\cap \Gamma_{U}(\bP^j)$.
Since the $\alpha$-eigenspace
of $\Gamma_{U}(\bP^j)/\mathrm{D}\Gamma_{U}(\bP^j)$ is inside $\Gamma_{B}(\bP^{2j})$,
$\Delta$ projects $u\Gamma_{U}(\bP^j)$ to
$$ \overline{\Delta}(u)\in (\Gamma_{B}(\bP^{2j})\cap \Gamma_{U}(\bP^j))/\mathrm{D}\Gamma_{U}(\bP^j). $$
In particular, $\Delta(u)\in \Gamma_{B}(\bP^{2j})\cap \Gamma_{U}(\bP^j)$.
Again apply $\Delta$. Since $\Gamma_{B}(\bP^{2j})/\Gamma_{\I_0^\circ}(\bP^{2j})$ is sent
to $\Gamma_{U}(\bP^{2j})/\Gamma_{\I_0^\circ}(\bP^{2j})$ by $\Delta$,
we get $\Delta^2(u)\in \Gamma_{U}(\bP^{2j})$ as desired.
\end{proof}
\begin{proof}
We can now prove Proposition \ref{approx}.
Let $\overline{u}\in U(\I_0^\circ/\bP)\cap\im(\rho_\bP)$.
Since the reduction map $\im(\rho)\to\im(\rho_\bP)$ induced by $\pi_1$
is surjective, there exists $v \in\im(\rho)$
such that $\pi_1(v)=\overline{u}$.
Take $u_1\in U(\I_0^\circ)$ such that $\pi_1(u_1)=\overline{u}$ (this is possible since $\pi_1\colon U(\Lambda_h)\to U(\Lambda_h/P)$ is surjective).
Then $v u_1^{-1}\in\Gamma_{\I_0^\circ}(\bP)$,
so $v\in K_{U}(\bP)$.
By compactness of $K_{U}(\bP)$ and by Lemma \ref{2jlem}, starting with $v$ as above, we see that
$\lim_{m\to \infty}\Delta^m(v)$ converges $\bP$-adically
to $\Delta^\infty(v)\in U(\I_0^\circ)\cap K$
with $\pi_1(\Delta^\infty(v))=\overline{u}$.
\end{proof}
\begin{remark}
Proposition \ref{approx} is true with the same proof if we replace $\Lambda_{h,0}$ by $\Lambda_h$ and $\I_0^\circ$ by a finite torsion free $\Lambda_h$-algebra.
\end{remark}
As a first application of Proposition \ref{approx} we give a result that we will need in the next subsection.
Given a representation $\rho\colon G_\Q\to\GL_2(\I^\circ)$ and every ideal $\bP$ of $\I^\circ$ we define $\rho_\bP$, $U^{\pm}(\rho)$ and $U^{\pm}(\rho_\bP)$ as above, by replacing $\I_0^\circ$ by $\I^\circ$.
\begin{proposition}\label{nontrivunip}
Let $\theta\colon\T_h\to\I^\circ$ be a family of slope $\leq h$ and $\rho_\theta\colon G_\Q\to\GL_2(\I^\circ)$
be the representation associated with $\theta$.
Suppose that $\rho_\theta$ is $(H_0,\Z_p)$-regular and let $\rho$ be a conjugate of $\rho_\theta$ such that $\im\,\rho\vert_{H_0}$ contains a diagonal $\Z_p$-regular element.
Then $U^+(\rho)$ and $U^-(\rho)$ are both nontrivial.
\end{proposition}
\begin{proof}
By density of classical points in $\T_h$ we can choose a prime ideal $\bP\subset\I^\circ$ corresponding to a classical modular form $f$.
The modulo $\bP$ representation $\rho_\bP$ is the $p$-adic representation classically associated with $f$.
By the results of \cite{ribet1} and \cite{momose}, there exists an ideal $\fl_\bP$ of $\Z_p$ such that $\im\,\rho_\bP$ contains the congruence subgroup $\Gamma_{\Z_p}(\fl_\bP)$.
In particular $U^+(\rho_\bP)$ and $U^-(\rho_\bP)$ are both nontrivial.
Since the maps $U^+(\rho)\to U^+(\rho_\bP)$ and $U^-(\rho)\to U^-(\rho_\bP)$ are surjective we find nontrivial elements in $U^+(\rho)$ and $U^-(\rho)$.
\end{proof}
We adapt the work in \cite[Sec. 7]{lang} to show the following.
\begin{proposition}\label{propI0}
Suppose that the representation $\rho\colon G_\Q\to\GL_2(\I^\circ)$ is $(H_0,\Z_p)$-regular.
Then there exists $g\in\GL_2(\I^\circ)$ such that the conjugate representation $g\rho g^{-1}$ satisfies the following two properties:
\begin{enumerate}
\item the image of $g\rho g^{-1}\vert_{H_0}$ is contained in $\GL_2(\I^\circ_0)$;
\item the image of $g\rho g^{-1}\vert_{H_0}$ contains a diagonal $\Z_p$-regular element.
\end{enumerate}
\end{proposition}
\begin{proof}
As usual we choose a $\GL_2(\I^\circ)$-conjugate of $\rho$ such that a $\Z_p$-regular element $d$ is diagonal.
We still write $\rho$ for this conjugate representation and we show that it also has property (1).
Recall that for every $\sigma\in\Gamma$ there is a character $\eta_\sigma\colon G_\Q\to(\I^\circ)^\times$ and an equivalence $\rho^\sigma\cong\rho\otimes\eta_\sigma$.
Then for every $\sigma\in\Gamma$ there exists $\bt_\sigma\in\GL_2(\I^\circ)$ such that, for all $g\in G_\Q$,
\begin{equation}\label{rhoequiv}
\rho^\sigma(g)=\bt_\sigma\eta_\sigma(g)\rho(g)\bt_\sigma^{-1}.
\end{equation}
We prove that the matrices $\bt_\sigma$ are diagonal.
Let $\rho(t)$ be a non-scalar diagonal element in $\im\,\rho$ (for example $d$).
Evaluating (\ref{rhoequiv}) at $g=t$ we find that $\bt_\sigma$ must be either a diagonal or an antidiagonal matrix.
Now by Proposition \ref{nontrivunip} there exists a nontrivial element $\rho(u^+)\in\im\,\rho\cap U^+(\I^\circ)$.
Evaluating (\ref{rhoequiv}) at $g=u^+$ we find that $\bt_\sigma$ cannot be antidiagonal.
It is shown in \cite[Lemma 7.3]{lang} that there exists an extension $A$ of $\I^\circ$, at most quadratic, and a function $\zeta\colon\Gamma\to A^\times$ such that $\sigma\to\bt_\sigma\zeta(\sigma)^{-1}$ defines a cocycle with values in $\GL_2(A)$.
The proof of this result does not require the ordinarity of $\rho$. Eq. (\ref{rhoequiv}) remains true if we replace $\bt_\sigma$ with $\bt_\sigma\zeta(\sigma)^{-1}$, so we can and do suppose from now on that $\bt_\sigma$ is a cocycle with values in $\GL_2(A)$. In the rest of the proof we assume for simplicity that $A=\I^\circ$, but everything works in the same way if $A$ is a quadratic extension of $\I^\circ$ and $\F$ is the residue field of $A$.
Let $V=(\I^\circ)^2$ be the space on which $G_\Q$ acts via $\rho$. As in \cite[Sec. 7]{lang} we use the cocycle $\bt_\sigma$ to define a twisted action of $\Gamma$ on $(\I^\circ)^2$.
For $v=(v_1,v_2)\in V$ we denote by $v^\sigma$ the vector $(v_1^\sigma,v_2^\sigma)$.
We write $v^{[\sigma]}$ for the vector $\bt_\sigma^{-1}v^\sigma$.
Then $v\to v^{[\sigma]}$ gives an action of $\Gamma$ since $\sigma\mapsto \bt_\sigma$ is a cocycle. Note that this action is $\I^\circ_0$-linear.
Since $\bt_\sigma$ is diagonal for every $\sigma\in\Gamma$, the submodules $V_1=\I^\circ(1,0)$ and $V_2=\I^\circ(0,1)$ are stable under the action of $\Gamma$.
We show that each $V_i$ contains an element fixed by $\Gamma$.
We denote by $\F$ the residue field $\I^\circ/\fm_\I^\circ$.
Note that the action of $\Gamma$ on $V_i$ induces an action of $\Gamma$ on the one-dimensional $\F$-vector space $V_i\otimes\I^\circ/\fm_{\I^\circ}$.
We show that for each $i$ the space $V_i\otimes\I^\circ/\fm_{\I^\circ}$ contains a nonzero element $\overline{v}_i$ fixed by $\Gamma$.
This is a consequence of the following argument, a form of which appeared in an early preprint of \cite{lang}.
Let $w$ be any nonzero element of $V_i\otimes\I^\circ/\fm_{\I^\circ}$ and let $a$ be a variable in $\F$. The sum
$$ S_{aw}=\sum_{\sigma\in\Gamma}(aw)^{[\sigma]} $$
is clearly $\Gamma$-invariant. We show that we can choose $a$ such that $S_{aw}\neq 0$. Since $V_i\otimes\I^\circ/\fm_{\I^\circ}$ is one-dimensional, for every $\sigma\in\Gamma$ there exists $\alpha_\sigma\in\F$ such that $w^{[\sigma]}=\alpha_\sigma w$. Then
$$ S_{aw}=\sum_{\sigma\in\Gamma}(aw)^{[\sigma]}=\sum_{\sigma\in\Gamma}a^\sigma w^{[\sigma]}=\sum_{\sigma\in\Gamma}a^\sigma\alpha_\sigma w=\left(\sum_{\sigma\in\Gamma}a^\sigma\alpha_\sigma a^{-1}\right)aw. $$
By Artin's lemma on the independence of characters, the function $f(a)=\sum_{\sigma\in\Gamma}a^\sigma\alpha_\sigma a^{-1}$ cannot be identically zero on $\F$. By choosing a value of $a$ such that $f(a)\neq 0$ we obtain a nonzero element $\overline{v}_i=S_{aw}$ fixed by $\Gamma$.
We show that $\overline{v}_i$ lifts to an element $v_i\in V_i$ fixed by $\Gamma$.
Let $\sigma_0\in\Gamma$. By Lemma \ref{gammaabel} $\Gamma$ is a finite abelian $2$-group, so the minimal polynomial $P_m(X)$ of $[\sigma_0]$ acting on $V_i$ divides $X^{2^k}-1$ for some integer $k$. In particular the factor $X-1$ appears with multiplicity at most $1$. We show that its multiplicity is exactly $1$.
If $\overline{P_m}$ is the reduction of $P_m$ modulo $\fm_{\I^\circ}$ then $\overline{P_m}([\sigma_0])=0$ on $V_i\otimes\I^\circ/\fm_{\I^\circ}$.
By our previous argument there is an element of $V_i\otimes\I^\circ/\fm_{\I^\circ}$ fixed by $\Gamma$ (hence by $[\sigma_0]$) so we have $(X-1)\mid\overline{P_m(X)}$. Since $p>2$ the polynomial $X^{2^k}-1$ has no double roots modulo $\fm_{\I^\circ}$, so neither does $\overline{P_m}$. By Hensel's lemma the factor $X-1$ lifts to a factor $X-1$ in $P_m$ and $\overline{v}_i$ lifts to an element $v_i\in V_i$ fixed by $[\sigma_0]$. Note that $\I^\circ\cdot v_i=V_i$ by Nakayama's lemma since $\overline{v}_i\neq 0$.
We show that $v_i$ is fixed by all of $\Gamma$.
Let $W_{[\sigma_0]}=\I^\circ v_i$ be the one-dimensional eigenspace for $[\sigma_0]$ in $V_i$. Since $\Gamma$ is abelian $W_{[\sigma_0]}$ is stable under $\Gamma$.
Let $\sigma\in\Gamma$. Since $\sigma$ has order $2^k$ in $\Gamma$ for some $k\geq 0$ and $v_i^{[\sigma]}\in W_{[\sigma_0]}$, there exists a root of unity $\zeta_\sigma$ of order $2^k$ satisfying $v_i^{[\sigma]}=\zeta_\sigma v_i$. Since $\overline{v}_i^{[\sigma]}=\overline{v}_i$, the reduction of $\zeta_\sigma$ modulo $\fm_{\I^\circ}$ must be $1$. As before we conclude that $\zeta_\sigma=1$ since $p\neq 2$.
We found two elements $v_1\in V_1$, $v_2\in V_2$ fixed by $\Gamma$. We show that every element of $v\in V$ fixed by $\Gamma$ must belong to the $\I^\circ_0$-submodule generated by $v_1$ and $v_2$. We proceed as in the end of the proof of \cite[Th. 7.5]{lang}.
Since $V_1$ and $V_2$ are $\Gamma$-stable we must have $v\in V_1$ or $v\in V_2$. Suppose without loss of generality that $v\in V_1$. Then $v=\alpha v_1$ for some $\alpha\in\I^\circ$. If $\alpha\in\I^\circ_0$ then $v\in\I^\circ_0 v_1$, as desired. If $\alpha\notin\I^\circ_0$ then there exists $\sigma\in\Gamma$ such that $\alpha^\sigma\neq\alpha$. Since $v$ is $[\sigma]$-invariant we obtain $(\alpha v_1)^{[\sigma]}=\alpha^\sigma v_1^{[\sigma]}=\alpha^\sigma v_1\neq \alpha v$, so $\alpha v_1$ is not fixed by $[\sigma]$, a contradiction.
Now $(v_1,v_2)$ is a basis for $V$ over $\I^\circ$, so the $\I^\circ_0$ submodule $V_0=\I^\circ_0v_1+\I^\circ_0v_2$ is an $\I^\circ_0$-lattice in $V$.
Recall that $H_0=\bigcap_{\sigma\in\Gamma}\ker\eta_\sigma$.
We show that $V_0$ is stable under the action of $H_0$ via $\rho\vert_{H_0}$, i.e. that if $v\in V$ is fixed by $\Gamma$, so is $\rho(h)v$ for every $h\in H_0$.
This is a consequence of the following computation, where $v$ and $h$ are as before and $\sigma\in\Gamma$:
$$ (\rho(h)v)^{[\sigma]}=\bt_\sigma^{-1}\rho(h)^\sigma v^\sigma=\bt_\sigma^{-1}\eta_\sigma(h)\rho(h)^\sigma v^\sigma=\bt_\sigma^{-1}\bt_\sigma\rho(h)\bt_\sigma^{-1}v^\sigma=\rho(h)v^{[\sigma]}. $$
Since $V_0$ is an $\I^\circ_0$-lattice in $V$ stable under $\rho\vert_{H_0}$, we conclude that $\im\,\rho\vert_{H_0}\subset\GL_2(\I^\circ_0)$.
\end{proof}
\subsection{Fullness of the unipotent subgroups}
From now on we write $\rho$ for the element in its $\GL_2(\I^\circ)$ conjugacy class such that $\rho\vert_{H_0}\in\GL_2(\I^\circ_0)$.
Recall that $H$ is the open subgroup of $H_0$ defined by the condition $\det\overline{\rho}(h)=1$ for every $h\in H$.
As in \cite[Sec. 4]{lang} we define a representation $H\to\SL_2(\I^\circ_0)$ by
$$ \rho_0=\rho\vert_H\otimes\left(\det\rho\vert_H\right)^{-\frac{1}{2}}. $$
We can take the square root of the determinant thanks to the definition of $H$.
We will use the results of \cite{lang} to deduce that the $\Lambda_{h,0}$-module generated by the unipotent subgroups of the image of $\rho_0$ is big.
We will later deduce the same for $\rho$.
We fix from now on a height one prime $P\subset\Lambda_{h,0}$ with the following properties:
\begin{enumerate}
\item there is an arithmetic prime $P_k\subset\Z_p[[\eta t]]$ satisfying $k>h+1$ and $P=P_k\Lambda_{h,0}$;
\item every prime $\fP\subset\I^\circ$ lying above $P$ corresponds to a non-CM point.
\end{enumerate}
Such a prime always exists. Indeed, by Remark \ref{arithprimes} every classical weight $k>h+1$ contained in the disc $B_h$ defines a prime $P=P_k\Lambda_{h,0}$ satisfying (1), so such primes are Zariski-dense in $\Lambda_{h,0}$, while the set of CM primes in $\I^\circ$ is finite by Proposition \ref{finiteCM}.
\begin{remark}\label{etale}
Since $k>h+1$, every point of $\Spec\,\T_h$ above $P_k$ is classical by \cite[Th. 6.1]{coleman}. Moreover the weight map is \'etale at every
such point by \cite[Th. 11.10]{kisin}. In particular the prime $P\I_0^\circ=P_k\I_0^\circ$ splits as a product of distinct primes of $\I_0^\circ$.
\end{remark}
Make the technical assumption that the order of the residue field $\F$ of $\I^\circ$ is not $3$.
For every ideal $\bP$ of $\I^\circ_0$ over $P$ we let $\pi_\bP$ be the projection $\SL_2(\I^\circ_0/\bP)\to\SL_2(\I^\circ_0/\bP)$.
We still denote by $\pi_\bP$ the restricted maps $U^\pm(\I^\circ_0/\bP)\to U^\pm(\I^\circ_0/\bP)$.
Let $G=\im\,\rho_0$. For every ideal $\bP$ of $\I^\circ_0$ we denote by $\rho_{0,\bP}$ the representation $\pi_\bP(\rho_0)$ and by $G_\bP$ the image of $\rho_\bP$. Clearly $G_\bP=\pi_\bP(G)$.
We state two results from Lang's work that come over unchanged to the non-ordinary setting.
\begin{proposition}\label{openproj}\cite[Cor. 6.3]{lang}
Let $\fP$ be a prime of $\I^\circ_0$ over $P$. Then $G_{\fP}$ contains a congruence subgroup $\Gamma_{\I^\circ_0/\fP}(\fa)\subset\SL_2(\I^\circ_0/\fP)$. In particular $G_\fP$ is open in $\SL_2(\I^\circ_0/\fP)$.
\end{proposition}
\begin{proposition}\label{openprod}\cite[Prop. 5.1]{lang}
Assume that for every prime $\fP\subset\I^\circ_0$ over $P$ the subgroup $G_\fP$ is open in $\SL_2(\I^\circ_0/\fP)$.
Then the image of $G$ in $\prod_{\fP|P}\SL_2(\I^\circ_0/\fP)$ through the map $\prod_{\fP|P}\pi_\fP$ contains a product of congruence subgroups $\prod_{\fP|P}\Gamma_{\I^\circ_0/\fP}(\fa_\fP)$.
\end{proposition}
\begin{remark}
The proofs of Propositions \ref{openproj} and \ref{openprod} rely on the fact that the big ordinary Hecke algebra is \'etale over $\Lambda$ at every arithmetic point. In order for these proofs to adapt to the non-ordinary setting it is essential that the prime $P$ satisfies the properties above Remark \ref{etale}.
\end{remark}
We let $U^\pm(\rho_0)=G\cap U^{\pm}(\I^\circ_0)$ and $U^\pm(\rho_\bP)=G_\bP\cap U^{\pm}(\I^\circ_0/\bP)$.
We denote by $U(\rho_\bP)$ either the upper or lower unipotent subgroups of $G_{\bP}$ (the choice will be fixed throughout the proof).
By projecting to the upper right element we identify $U^+(\rho_0)$ with a $\Z_p$-submodule of $\I^\circ_0$ and $U^+(\rho_{0,\bP})$ with a $\Z_p$-submodule of $\I^\circ_0/\bP$. We make analogous identifications for the lower unipotent subgroups.
We will use Proposition \ref{openprod} and Proposition \ref{approx} to show that, for both signs, $U^{\pm}(\rho)$ spans $\I^\circ_0$ over $\Lambda_{h,0}$.
First we state a version of \cite[Lemma 4.10]{lang}, with the same proof. Let $A$ and $B$ be Noetherian rings with $B$ integral over $A$. We call $A$-\textit{lattice} an $A$-submodule of $B$ generated by the elements of a basis of $Q(B)$ over $Q(A)$.
\begin{lemma}\label{lattice}
Any $A$-lattice in $B$ contains a nonzero ideal of $B$. Conversely, every nonzero ideal of $B$ contains an $A$-lattice.
\end{lemma}
We prove the following proposition by means of Proposition \ref{approx}. We could also use Pink theory as in \cite[Sec. 4]{lang}.
\begin{proposition}\label{lambdaspan_0}
Consider $U^\pm(\rho_0)$ as subsets of $Q(\I^\circ_0)$. For each choice of sign the $Q(\Lambda_{h,0})$-span of $U^\pm(\rho_0)$ is $Q(\I^\circ_0)$. Equivalently the $\Lambda_{h,0}$-span of $U^\pm(\rho_0)$ contains a $\Lambda_{h,0}$-lattice in $\I^\circ_0$.
\end{proposition}
\begin{proof}
Keep notations as above. We omit the sign when writing unipotent subgroups and we refer to either the upper or lower ones (the choice is fixed throughout the proof).
Let $P$ be the prime of $\Lambda_{h,0}$ chosen above.
By Remark \ref{etale} the ideal $P\I_0^\circ$ splits as a product of distinct primes in $\I_0^\circ$.
When $\fP$ varies among these primes, the map $\bigoplus_{\fP|P}\pi_\fP$ gives embeddings of $\Lambda_{h,0}/P$-modules $\I^\circ_0/P\I^\circ_0\into\bigoplus_{\fP|P}\I^\circ_0/\fP$ and $U(\rho_{P\I_0^\circ})\into\bigoplus_{\fP|P}U(\rho_\fP)$.
The following diagram commutes:
\begin{equation}\label{immersions}
\begin{tikzcd}[baseline=(current bounding box.center)]
U(\rho_{P\I_0^\circ}) \arrow[hook]{r}{\bigoplus_{\fP|P}\pi_\fP}\arrow[hook]{d}
& \bigoplus_{\fP|P}U(\rho_\fP)\arrow[hook]{d}\\
\I^\circ_0/P\I^\circ_0 \arrow[hook]{r}{\bigoplus_{\fP|P}\pi_\fP}
& \bigoplus_{\fP|P}\I^\circ_0/\fP
\end{tikzcd}
\end{equation}
By Proposition \ref{openprod} there exist ideals $\fa_\fP\subset\I^\circ_0/\fP$ such that $(\bigoplus_{\fP|P}\pi_\fP)(G_{P\I_0^\circ})\supset\bigoplus_{\fP|P}\Gamma_{\I^\circ_0/\fP}(\fa_\fP)$.
In particular $(\bigoplus_{\fP|P}\pi_\fP)(U(\rho_{P\I_0^\circ}))\supset\bigoplus_{\fP|P}(\fa_\fP)$.
By Lemma \ref{lattice} each ideal $\fa_\fP$ contains a basis of $Q(\I^\circ_0/\fP)$ over $Q(\Lambda_{h,0}/P)$, so that the $Q(\Lambda_{h,0}/P)$-span of $\bigoplus_{\fP|P}\fa_\fP$ is the whole $\bigoplus_{\fP|P}Q(\I^\circ_0/\fP)$.
Then the $Q(\Lambda_{h,0}/P)$-span of $(\bigoplus_{\fP|P}\pi_\fP)(G_\fP\cap U(\rho_{\fP}))$ is also $\bigoplus_{\fP|P}Q(\I^\circ_0/\fP)$.
By commutativity of diagram (\ref{immersions}) we deduce that the $Q(\Lambda_{h,0}/P)$-span of $G_P\cap U(\rho_{P\I_0^\circ})$ is $Q(\I^\circ_0/P\I^\circ_0)$.
In particular $G_{P\I_0^\circ}\cap U(\rho_{P\I_0^\circ})$ contains a $\Lambda_{h,0}/P$-lattice, hence by Lemma \ref{lattice} a nonzero ideal $\fa_P$ of $\I^\circ_0/P\I^\circ_0$.
Note that the representation $\rho_0\colon H\to\SL_2(\I_0^\circ)$ satisfies the hypotheses of Proposition \ref{approx}.
Indeed we assumed that $\rho\colon G_\Q\to\GL_2(\I)$ is $(H_0,\Z_p)$-regular, so the image of $\rho\vert_{H_0}$ contains a diagonal $\Z_p$-regular element $d$.
Since $H$ is a normal subgroup of $H_0$, $\rho(H)$ is a normal subgroup of $\rho(H_0)$ and it is normalized by $d$.
By a trivial computation we see that the image of $\rho_0=\rho\vert_H\otimes(\det\rho\vert_H)^{-1/2}$ is also normalized by $d$.
Let $\fa$ be an ideal of $\I^\circ_0$ projecting to $\fa_P\subset U(\rho_{0,P\I_0^\circ})$. By Proposition \ref{approx} applied to $\rho_0$ we obtain that the map $U(\rho_0)\to U(\rho_{0,P\I_0^\circ})$ is surjective, so the $\Z_p$-module $\fa\cap U(\rho_0)$ also surjects to $\fa_P$.
Since $\Lambda_{h,0}$ is local we can apply Nakayama's lemma to the $\Lambda_{h,0}$-module $\Lambda_{h,0}(\fa\cap U(\rho_0)$ to conclude that it coincides with $\fa$. Hence $\fa\subset\Lambda_{h,0}\cdot U(\rho_0)$, so the $\Lambda_{h,0}$-span of $U(\rho_0)$ contains a $\Lambda_{h,0}$-lattice in $\I^\circ_0$.
\end{proof}
We show that Proposition \ref{lambdaspan_0} is true if we replace $\rho_0$ by $\rho\vert_H$.
This will be a consequence of the description of the subnormal sugroups of $\GL_2(\I^\circ)$ presented in \cite{taz}, but
we need a preliminary step because we cannot induce a $\Lambda_{h,0}$-module structure on the unipotent subgroups of $G$.
For a subgroup $\cG\subset\GL_2(\I^\circ_0)$ define $\cG^p=\{g^p,\, g\in G\}$ and $\widetilde{\cG}=\cG^p\cap(1+p\Mat_2(\I^\circ_0))$. Let $\widetilde{\cG}^{\Lambda_{h,0}}$ be the subgroup of $\GL_2(\I^\circ)$ generated by the set $\{g^\lambda\colon g\in\widetilde{\cG}, \lambda\in\Lambda_{h,0}\}$ where $g^\lambda=\exp(\lambda\log g)$. We have the following.
\begin{lemma}\label{uniplattice}
The group $\widetilde{\cG}^{\Lambda_{h,0}}$ contains a congruence subgroup of $\SL_2(\I^\circ_0)$ if and only if both of the unipotent subgroups $\cG\cap U^+(\I^\circ_0)$ and $\cG\cap U^-(\I^\circ_0)$ contain a basis of a $\Lambda_{h,0}$-lattice in $\I^\circ_0$.
\end{lemma}
\begin{proof}
It is easy to see that $\cG\cap U^+(\I^\circ_0)$ contains the basis of a $\Lambda_{h,0}$-lattice in $\I^\circ_0$ if and only if the same is true for $\widetilde{\cG}\cap U^+(\I^\circ_0)$. The same is true for $U^-$.
By a standard argument, used in the proofs of \cite[Lemma 2.9]{hida} and \cite[Prop. 4.2]{lang}, $\cG^{\Lambda_{h,0}}\subset\GL_2(\I^\circ_0)$ contains a congruence subgroup of $\SL_2(\I^\circ_0)$ if and only if both its upper and lower unipotent subgroup contain an ideal of $\I^\circ_0$.
We have $U^+(\I^\circ_0)\cap\cG^{\Lambda_{h,0}}=\Lambda_{h,0}(\cG\cap U^+(\I^\circ_0))$, so by Lemma \ref{lattice} $U^+(\I^\circ_0)\cap\cG^{\Lambda_{h,0}}$ contains an ideal of $\I^\circ_0$ if and only if $\cG\cap U^+(\I^\circ_0)$ contains a basis of a $\Lambda_{h,0}$-lattice in $\I^\circ_0$. We proceed in the same way for $U^-$.
\end{proof}
Now let $G_0=\im\,\rho\vert_H$, $G=\im\,\rho_0$. Note that $G_0\cap\SL_2(\I^\circ_0)$ is a normal subgroup of $G$.
Let $f\colon\GL_2(\I^\circ_0)\to\SL_2(\I^\circ_0)$ be the homomorphism sending $g$ to $\det(g)^{-1/2}g$.
We have $G=f(G_0)$ by definition of $\rho_0$. We show the following.
\begin{proposition}\label{unipbasis}
The subgroups $G_0\cap U^{\pm}(\I^\circ_0)$ both contain the basis of a $\Lambda_{h,0}$-lattice in $\I^\circ_0$ if and only if $G\cap U^{\pm}(\I^\circ_0)$ both contain the basis of a $\Lambda_{h,0}$-lattice in $\I^\circ_0$.
\end{proposition}
\begin{proof}
Since $G=f(G_0)$ we have $\widetilde{G}=f(\widetilde{G_0})$.
This implies that $\widetilde{G}^{\Lambda_{h,0}}=f(\widetilde{G_0}^{\Lambda_{h,0}})$.
We remark that $\widetilde{G_0}^{\Lambda_{h,0}}\cap\SL_2(\I_0^\circ)$ is a normal subgroup of $\widetilde{G}^{\Lambda_{h,0}}$.
Indeed $\widetilde{G_0}^{\Lambda_{h,0}}\cap\SL_2(\I_0^\circ)$ is normal in $\widetilde{G_0}^{\Lambda_{h,0}}$, so its image $f(G_0^{\Lambda_{h,0}}\cap\SL_2(\I_0^\circ))=G_0^{\Lambda_{h,0}}\cap\SL_2(\I_0^\circ)$ is normal in $f(G_0^{\Lambda_{h,0}})=\widetilde{G}^{\Lambda_{h,0}}$.
By \cite[Cor. 1]{taz} a subgroup of $\GL_2(\I^\circ_0)$ contains a congruence subgroup of $\SL_2(\I^\circ_0)$ if and only if it is subnormal in $\GL_2(\I^\circ_0)$ and it is not contained in the centre. We note that $\widetilde{G_0}^{\Lambda_{h,0}}\cap\SL_2(\I^\circ_0)=(\widetilde{G_0}\cap\SL_2(\I^\circ_0))^{\Lambda_{h,0}}$ is not contained in the subgroup $\{\pm 1\}$. Otherwise also $\widetilde{G_0}\cap\SL_2(\I^\circ_0)$ would be contained in $\{\pm 1\}$ and $\im\,\rho\cap\SL_2(\I^\circ_0)$ would be finite, since $\widetilde{G_0}$ is of finite index in $G_0^p$.
This would give a contradiction: indeed if $\fP$ is an arithmetic prime of $\I^\circ$ of weight greater than $1$ and $\fP^\prime=\fP\cap\I^\circ_0$, the image of $\rho$ modulo $\fP^\prime$ contains a congruence subgroup of $\SL_2(\I_0^\circ/\fP^\prime)$ by the result of \cite{ribet1}.
Since $\widetilde{G_0}^{\Lambda_{h,0}}\cap\SL_2(\I^\circ_0)$ is a normal subgroup of $\widetilde{G}^{\Lambda_{h,0}}$, we deduce by \cite[Cor. 1]{taz} that $\widetilde{G_0}^{\Lambda_{h,0}}\cap\SL_2(\I^\circ_0)$ (hence $\widetilde{G_0}^{\Lambda_{h,0}}$) contains a congruence subgroup of $\SL_2(\I^\circ_0)$ if and only if $\widetilde{G}^{\Lambda_{h,0}}$ does.
By applying Lemma \ref{uniplattice} to $\cG=G_0$ and $\cG=G$ we obtain the desired equivalence.
\end{proof}
By combining Propositions \ref{lambdaspan_0} and \ref{unipbasis} we obtain the following.
\begin{corollary}\label{lambdaspan}
The $\Lambda_{h,0}$-span of each of the unipotent subgroups $\im\,\rho\cap U^{\pm}$ contains a $\Lambda_{h,0}$-lattice in $\I^\circ_0$.
\end{corollary}
Unlike in the ordinary case we cannot deduce from the corollary that $\im\,\rho$ contains a congruence subgroup of $\SL_2(\I^\circ_0)$, since we are working over $\Lambda_h\neq\Lambda$ and we cannot induce a $\Lambda_h$-module structure (not even a $\Lambda$-module structure) on $\im\,\rho\cap U^{\pm}$.
The proofs of \cite[Lemma 2.9]{hida} and \cite[Prop. 4.3]{lang} rely on the existence, in the image of the Galois group, of an element inducing by conjugation a $\Lambda$-module structure on $\im\,\rho\cap U^{\pm}$. In their situation this is predicted by the condition of Galois ordinarity of $\rho$.
In the non-ordinary case we will find an element with a similar property via relative Sen theory.
In order to do this we will need to work with a suitably defined Lie algebra rather than with the group itself.
\bigskip
\section{Relative Sen theory}\label{sentheory}
We recall the notations of Section \ref{subsectnest}.
In particular $r_h=p^{-s_h}$, with $s_h\in\Q$, is the $h$-adapted radius (which we also take smaller than $p^{-\frac{1}{p-1}})$, $\eta_h$ is an element in $\C_p$ of norm $r_h$, $K_h$ is the Galois closure in $\C_p$ of $\Q_p(\eta_h)$ and $\cO_h$ is the ring of integers in $K_h$.
The ring $\Lambda_h$ of analytic functions bounded by $1$ on the open disc $\cB_h=\cB(0,r_h^-)$ is identified to $\cO_h[[t]]$. We take a sequence of radii $r_i=p^{-s_h-1/i}$ converging to $r_h$ and denote by $A_{r_i}=K_h\langle t,X_i\rangle/(pX_i-t^i)$ the $K_h$-algebra defined in Section \ref{subsectnest} which is a form over $K_h$ of the $\C_p$-algebra of analytic functions on the closed ball $\cB(0,r_i)$ (its Berthelot model). We denote by $A_{r_i}^\circ$ the $\cO_h$-subalgebra of functions bounded by $1$.
Then $\Lambda_h=\varprojlim_{i} A_{r_i}^\circ$ where $A_{r_j}^\circ\to A_{r_i}^\circ$ for $i<j$ is the restriction of analytic functions.
We defined in Section \ref{selftwists} a subring $\I_0^\circ\subset\I^\circ$, finite over $\Lambda_{h,0}\subset\Lambda_h$. For $r_i$ as above, we write $A_{0,r_i}^\circ=\cO_{h,0}\langle t,X_i\rangle/(pX_i-t^i)$ with maps $A_{0,r_j}^\circ\to A_{0,r_i}^\circ$ for $i<j$, so that $\Lambda_{h,0}=\varprojlim_iA_{0,r_i}^\circ$.
Let $\I_{r_i}^\circ=\I^\circ\widehat{\otimes}_{\Lambda_h} A_{r_i}^\circ$ and $\I_{0,r_i}^\circ=\I_0^\circ\widehat{\otimes}_{\Lambda_{h,0}} A_{0,r_i}^\circ$, both endowed with their $p$-adic topology. Note that $(\I_{r_i}^\circ)^\Gamma=\I_{r_i,0}^\circ$.
Consider the representation $\rho\colon G_\Q\to\GL_2(\I^\circ)$ associated with a family $\theta\colon \T_h\to\I^\circ$.
We observe that $\rho$ is continuous with respect to the profinite topology of $\I^\circ$ but not with respect to the $p$-adic topology.
For this reason we fix an arbitrary radius $r$ among the $r_i$ defined above and consider the representation $\rho_r\colon G_\Q\to\GL_2(\I_{r}^\circ)$ obtained by composing $\rho$ with the inclusion $\GL_2(\I^\circ)\into\GL_2(\I_{r}^\circ)$.
This inclusion is continuous, hence the representation $\rho_r$ is continuous with respect to the $p$-adic topology on $\GL_2(\I_{0,r}^\circ)$.
Recall from Proposition \ref{propI0} that, after replacing $\rho$ by a conjugate, there is an open normal subgroup $H_0\subset G_\Q$ such that the restriction $\rho\vert_{H_0}$ takes values in $\GL_2(\I_0^\circ)$ and is $(H_0,\Z_p)$-regular. Then the restriction $\rho_r\vert_{H_0}$ gives a representation $H_0\to\GL_2(\I_{0,r}^\circ)$ which is continuous with respect to the $p$-adic topology on $\GL_2(\I_{0,r}^\circ)$.
\subsection{Big Lie algebras}\label{liealg}
Recall that $G_p\subset G_\Q$ denotes our chosen decomposition group at $p$.
Let $G_r$ and $G_r^\loc$ be the images respectively of $H_0$ and $G_p\cap H_0$ under the representation $\rho_r\vert_{H_0}\colon H_0\to\GL_2(\I_{0,r}^\circ)$.
Note that they are actually independent of $r$ since they coincide with the images of $H_0$ and $G_p\cap H_0$ under $\rho$.
For every ring $R$ and ideal $I\subset R$ we denote by $\Gamma_{\GL_2(R)}(I)$ the $\GL_2$-congruence subgroup consisting of elements $g\in\GL_2(R)$ such that $g\equiv\Id_2\pmod{I}$.
Let $G_r^{\prime}=G_r\cap\Gamma_{\GL_2(\I_{0,r}^\circ)}(p)$ and $G_r^{\prime,\loc}=G_r^\loc\cap\Gamma_{\GL_2(\I_{0,r}^\circ)}(p)$, so that $G_r^{\prime}$ and $G_{r}^{\prime,\loc}$ are pro-$p$ groups.
Note that the congruence subgroups $\Gamma_{\GL_2(\I_{0,r})}(p^m)$ are open in $\GL_2(\I_{0,r})$ for the $p$-adic topology.
In particular $G_r^\prime$ and $G_{r}^{\prime,\loc}$ can be identified with the images under $\rho$ of the absolute Galois groups of finite extensions of $\Q$ and respectively $\Q_p$.
\begin{remark}\label{propsubgr}
We remark that we can choose an arbitrary $r_0$ and set, for every $r$, $G_r^{\prime}=G_r\cap\Gamma_{\GL_2(\I_{0,r_0}^\circ)}(p)$.
Then $G_r^\prime$ is a pro-$p$ subgroup of $G_r$ for every $r$ and it is independent of $r$ since $G_r$ is.
This will be important in Section \ref{comparison} where we will take projective limits over $r$ of various objects.
\end{remark}
We set $A_{0,r}=A_{0,r}^\circ[p^{-1}]$ and $\I_{0,r}=\I_{0,r}^\circ[p^{-1}]$.
We consider from now on $G_r^{\prime}$ and $G_r^{\prime,\loc}$ as subgroups of $\GL_2(\I_{0,r})$ through the inclusion $\GL_2(\I_{0,r}^\circ)\into\GL_2(\I_{0,r})$.
We want to define big Lie algebras associated with the groups $G_r^{\prime}$ and $G_r^{\prime,\loc}$.
For every nonzero ideal $\fa$ of the principal ideal domain $A_{0,r}$, we denote by $G_{r,\fa}^\prime$ and $G_{r,\fa}^{\prime,\loc}$ the images respectively of $G_r^{\prime}$ and $G_r^{\prime,\loc}$ under the natural projection $\GL_2(\I_{0,r})\to\GL_2(\I_{0,r}/\fa \I_{0,r})$. The pro-$p$ groups $G_{r,\fa}^\prime$ and $G_{r,\fa}^{\prime,\loc}$
are topologically of finite type
so we can define the corresponding $\Q_p$-Lie algebras $\fH_{r,\fa}$ and $\fH_{r,\fa}^\loc$ using the $p$-adic logarithm map: $\fH_{r,\fa}=\Q_p\cdot\Log\, G_{r,\fa}^{\prime}$ and $\fH_{r,\fa}^\loc=\Q_p\cdot\Log\, G_{r,\fa}^{\prime,\loc}$.
They are closed Lie subalgebras of the finite dimensional $\Q_p$-Lie algebra $\Mat_2(\I_{0,r}/\fa\I_{0,r})$.
Let $B_r=\varprojlim_{(\fa,P_1)=1}A_{0,r}/ \fa A_{0,r}$ where the inverse limit is taken over nonzero ideals $\fa\subset A_{0,r}$ prime to
$P_1=(u^{-1}(1+T)-1)$ (the reason for excluding $P_1$ will become clear later).
We endow $B_r$ with the projective limit topology coming from the $p$-adic topology on each quotient.
We have a topological isomorphism of $K_{h,0}$-algebras
$$ B_r\cong\prod_{P\neq P_1} \widehat{(A_{0,r})}_{P}, $$
where the product is over primes $P$ and $\widehat{(A_{0,r})}_P=\varprojlim_{m\geq 1}A_{0,r}/P^mA_{0,r}$
denotes the $K_{h,0}$-Fr\'echet space inverse limit of the finite dimensional $K_{h,0}$-vector spaces $A_{0,r}/P^mA_{0,r}$.
Similarly, let $\B_r=\varprojlim_{(\fa,P_1)=1}\I_{0,r}/\fa\I_{0,r}$, where as before $\fa$ varies over
all nonzero ideals of $A_{0,r}$ prime to $P_1$. We have
$$ \B_r\cong\prod_{P\neq P_1} \widehat{(\I_{0,r})}_{P\I_{0,r}}\cong\prod_{\fP\nmid P_1} \widehat{(\I_{0,r})}_{\fP}\cong\varprojlim_{(\fQ,P_1)=1}\I_{0,r}/\fQ, $$
where the second product is over primes $\fP$ of $\I_{0,r}$ and the projective limit is over primary ideals $\fQ$ of $\I_{0,r}$. Here $\widehat{(\I_{0,r})}_\fP$ denotes the projective limit of finite dimensional $K_{h,0}$-algebras (endowed with the $p$-adic topology).
The last isomorphism follows from the fact that $\I_{0,r}$ is finite over $A_{0,r}$, so that there is an isomorphism $\I_{0,r}\otimes\widehat{(A_{0,r})}_P=\prod_\fP\widehat{(\I_{0,r})}_\fP$ where $P$ is a prime of $A_{0,r}$ and $\fP$
varies among the primes of $\I_{0,r}$ above $P$.
We have natural continuous inclusions $A_{0,r}\into B_r$ and $\I_{0,r}\into \B_r$, both with dense image. The map $A_{0,r}\into\I_{0,r}$ induces an inclusion $B_r\into\B_r$ with closed image. Note however that $\B_r$ is not finite over $B_r$.
We will work with $\B_r$ for the rest of this section, but we will need $B_r$ later.
For every $\fa$ we have defined Lie algebras $\fH_{r,\fa}$ and $\fH_{r,\fa}^\loc$ associated with the finite type Lie groups $G_{r,\fa}^\prime$ and $G_{r,\fa}^{\prime,\loc}$. We take the projective limit of these algebras to obtain Lie subalgebras of $\Mat_2(\B_r)$.
\begin{definition}\label{blalg}
The Lie algebras associated with $G_{r}^\prime$ and $G_{r}^{\prime,\loc}$ are the closed $\Q_p$-Lie subalgebras of $\Mat_2(\B_r)$ given respectively by
$$ \fH_{r}=\varprojlim_{(\fa,P_1)=1}\fH_{r,\fa} $$
and
$$ \fH_{r}^\loc=\varprojlim_{(\fa,P_1)=1}\fH_{r,\fa}^\loc, $$
where as usual the products are taken over nonzero ideals $\fa\subset A_{0,r}$ prime to $P_1$.
\end{definition}
For every ideal $\fa$ prime to $P_1$, we have continuous homomorphisms $\fH_r\to\fH_{r,\fa}$ and $\fH_r^\loc\to\fH_{r,\fa}^\loc$.
Since the transition maps are surjective these homomorphisms are surjective.
\begin{remark}\label{primlim}
The limits in Definition \ref{blalg} can be replaced by limits over primary ideals of $\I_{0,r}$. Explicitly, let $\fQ$ be a primary ideal of $\I_{0,r}$. Let $G_{r,\fQ}^\prime$ be the image of $G_r^\prime$ via the natural projection $\GL_2(\I_{0,r})\to\GL_2(\I_{0,r}/\fQ)$ and let $\fH_{r,\fQ}$ be the Lie algebra associated with $G_{r,\fQ}^\prime$ (which is a finite type Lie group). We have an isomorphism of topological Lie algebras
$$ \fH_{r}=\varprojlim_{(\fQ,P_1)=1}\fH_{r,\fQ}, $$
where the limit is taken over primary ideals $\fQ$ of $\I_{0,r}$. This is naturally a subalgebra of $\Mat_2(\B_r)$ since $\B_r\cong\varprojlim_{(\fQ,P_1)=1}\I_{0,r}/\fQ$. The same goes for the local algebras.
\end{remark}
\subsection{The Sen operator associated with a Galois representation}\label{senconstr}
Recall that there is a finite extension $K/\Q_p$ such that $G_r^{\prime,\loc}$ is the image of $\rho\vert_{\Gal(\overline{K}/K)}$
and, for an ideal $P\subset A_{0,r}$ and $m\geq 1$, $G_{r,P^m}^{\prime,\loc}$ is the image of $\rho_{r,P^m}\vert_{\Gal(\overline{K}/K)}$. Following \cite{sen1} and \cite{sen2} we can define a Sen operator
associated with $\rho_r\vert_{\Gal(\overline{K}/K)}$ and $\rho_{r,P^m}\vert_{\Gal(\overline{K}/K)}$
for every ideal $P\subset A_{0,r}$ and every $m\geq 1$. We will see that these operators satisfy a compatibility property.
We write for the rest of the section $\rho_r$ and $\rho_{r,P^m}$ while implicitly taking the domain to be $\Gal(\overline{K}/K)$.
We begin by recalling the definition of the Sen operator associated with a representation $\tau\colon\Gal(\overline{K}/K)\to\GL_m(\cR)$
where $\cR$ is a Banach algebra over a $p$-adic field $L$. We follow \cite{sen2}. We can suppose $L\subset K$;
if not we just restrict $\tau$ to the open subgroup $\Gal(\overline{K}/KL)\subset\Gal(\overline{K}/K)$.
Let $L_{\infty}$ be a totally ramified $\Z_p$-extension of $L$.
Let $\gamma$ be a topological generator of $\Gamma=\Gal(L_{\infty}/L)$, $\Gamma_n\subset\Gamma$ the subgroup generated by $\gamma^{p^n}$ and $L_{n}=L_{\infty}^{\gamma^{p^n}}$, so that $L_{\infty}=\cup_n L_{n}$.
Let $L_n^\prime=L_{n}K$ and $G_n^\prime=\Gal(\overline{L}/L_n^\prime)$.
If $\cR^m$ is the $\cR$-module over which $\Gal(\overline{K}/K)$ acts via $\tau$, define an action of $\Gal(\overline{K}/K)$ on $\cR\widehat{\otimes}_L \C_p$ by letting $\sigma\in\Gal(\overline{K}/K)$ map $x\otimes y$ to $\tau(\sigma)(x)\otimes\sigma(y)$.
Then by the results of \cite{sen1} and \cite{sen2} there is a matrix
$M\in \GL_m\left(\cR\widehat{\otimes}_L \C_p\right)$, an integer $n\ge 0$ and a representation
$\delta\colon\Gamma_n\to \GL_m(\cR\otimes_L L_{n}^\prime)$ such that for all $\sigma\in G_n^\prime$
$$ M^{-1}\tau(\sigma)\sigma(M)=\delta(\sigma). $$
\begin{definition}
The Sen operator associated with $\tau$ is
$$ \phi=\lim_{\sigma\to 1}\frac{\log(\delta\bigl(\sigma)\bigr)}{\log(\chi(\sigma))}\in \Mat_m(\cR\widehat{\otimes}_L \C_p). $$
\end{definition}
The limit exists as for $\sigma$ close to $1$ the map $\displaystyle \sigma\mapsto
\frac{\log(\delta\bigl(\sigma)\bigr)}{\log(\chi(\sigma))}$ is constant.
It is proved in \cite[Sec. 2.4]{sen2} that $\phi$ does not depend on the choice of $\delta$ and $M$.
If $L=\cR=\Q_p$, we define the Lie algebra $\fg$ associated with $\tau(\Gal(\overline{K}/K))$ as the $\Q_p$-vector space generated by the image of the $\Log$ map in $\Mat_m(\Q_p)$. In this situation the Sen operator $\phi$ associated with $\tau$ has the following property.
\begin{theorem}\label{liealgsen}\cite[Th. 1]{sen1}
For a continuous representation $\tau\colon G_K\to\GL_m(\Q_p)$, the Lie algebra $\fg$ of the group $\tau(\Gal(\overline{K}/K))$ is the smallest $\Q_p$-subspace of $\Mat_m(\Q_p)$ such that $\fg{\otimes} \C_p$ contains $\phi$.
\end{theorem}
\noindent This theorem is valid in the absolute case above, but relies heavily on the fact that the image of the Galois group is a finite dimensional Lie group.
In the relative case it is doubtful that its proof can be generalized.
\subsection{The Sen operator associated with $\rho_r$}
Set $\I_{0,r,\C_p}=\I_{0,r}\widehat{\otimes}_{K_{h,0}}\C_p$. It is a Banach space for the natural norm.
Let $\B_{r,\C_p}=\B_r\widehat{\otimes}_{K_{h,0}}\C_p$; it is the topological $\C_p$-algebra completion of $\B_r\otimes_{K_{h,0}} \C_p$ for the (uncountable)
set of nuclear seminorms $p_{\fa}$ given by the norms on $\I_{0,r,\C_p}/\fa\I_{0,r,\C_p}$ via the specialization morphisms
$\pi_\fa\colon \B_r\otimes_{K_{h,0}}\C_p\to \I_{0,r,\C_p}/\fa\I_{0,r,\C_p}$.
Let $\fH_{r,\fa,\C_p}=\fH_{r,\fa}\otimes_{K_{h,0}}\C_p$ and $\fH_{r,\fa,\C_p}^\loc=\fH_{r,\fa,}^\loc\otimes_{K_{h,0}}\C_p$.
Then we define $\fH_{r,\C_p}=\fH_r\widehat{\otimes}_{K_{h,0}}\C_p$ as the topological $\C_p$-Lie algebra completion of
$\fH_r\otimes_{K_{0,h}}\C_p$ for the (uncountable)
set of seminorms $p_\fa$ given by the norms on $\fH_{r,\fa,\C_p}$ and similar specialization morphisms $\pi_\fa\colon\fH_{r,}\otimes_{K_{h,0}}\C_p\to\fH_{r,\fa,\C_p}$. We define in the same way $\fH_{r,\C_p}^\loc$ in terms of the norms on $\fH^{\loc}_{r,\fa,\C_p}$. Note that by definition we have
$$\fH_{r,\C_p}=\varprojlim_{(\fa,P_1)=1}\fH_{r,\fa,\C_p},\,\,\mathrm{and}\,\, \fH_{r,\C_p}^\loc=\varprojlim_{(\fa,P_1)=1}\fH_{r,\fa,\C_p}^\loc.$$
We apply the construction of the previous subsection to $L=K_{h,0}$, $\cR=\I_{0,r}$ which is a Banach $L$-algebra with the $p$-adic topology, and $\tau=\rho_r$. We obtain an operator $\phi_r\in \Mat_2(\I_{0,r,\C_p})$.
Recall that we have a natural continuous inclusion $\I_{0,r}\into \B_r$, inducing inclusions $\I_{0,r,\C_p}\into \B_{r,\C_p}$ and $\Mat_2(\I_{0,r,\C_p})\into \Mat_2(\B_{r,\C_p})$. We denote all these inclusions by $\iota_{\B_r}$ since it will be clear each time to which we are referring to.
We will prove in this section that $\iota_{\B_r}(\phi_r$) is an element of $\fH_{r,\C_p}^\loc$.
Let $\fa$ be a nonzero ideal of $A_{0,r}$. Let us apply Sen's construction to $L=K_{h,0}$, $\cR=\I_{0,r}/\fa\I_{0,r}$ and $\tau=\rho_{r,\fa}\colon\Gal(\overline{K}/K)\to\GL_2(\I_{0,r}/\fa\I_{0,r})$; we obtain an operator $\phi_{r,\fa}\in \Mat_2(\I_{0,r}/\fa\I_{0,r}\widehat{\otimes}_{K_{h,0}}\C_p)$.
Let
$$\pi_\fa\colon\Mat_2(\I_{0,r}\widehat{\otimes}_{K_{h,0}}\C_p)\to \Mat_2(\I_{0,r}/\fa\I_{0,r}\widehat{\otimes}_{K_{h,0}}\C_p)$$
and
$$\pi_\fa^\times\colon\GL_2(\I_{0,r}\widehat{\otimes}_{K_{h,0}}\C_p)\to\GL_2(\I_{0,r}/\fa\I_{0,r}\widehat{\otimes}_{K_{h,0}}\C_p)$$
be the natural projections.
\begin{proposition}\label{senproj}
We have $\phi_{r,\fa}=\pi_\fa(\phi_r)$ for all $\fa$.
\end{proposition}
\begin{proof}
Recall from the construction of $\phi_r$ that there exist $M\in\GL_2\left(\I_{0,r,\C_p}\right)$, $n\geq 0$ and
$\delta\colon\Gamma_n\to\GL_2(\I_{0,r}\widehat{\otimes}_{K_{h,0}}K_{h,0,n}^\prime)$ such that for all $\sigma\in G_n^\prime$ we have
\begin{equation}\label{sigmadelta}
M^{-1}\rho_r(\sigma)\sigma(M)=\delta(\sigma)
\end{equation}
and
\begin{equation}\label{eqsen}
\phi_r=\lim_{\sigma\to 1}\frac{\log(\delta\bigl(\sigma)\bigr)}{\log(\chi(\sigma))}.
\end{equation}
Let $M_\fa=\pi^\times_\fa(M)\in\GL_2(\I_{0,r,\C_p}/\fa\I_{0,r,\C_p})$ and $\delta_\fa=\pi_\fa^\times\circ\delta\colon\Gamma_n\to\GL_2((\I_{0,r}/\fa\I_{0,r})\widehat{\otimes}_{K_{h,0}}K_{h,0,n}^\prime)$
Denote by $\phi_{r,\fa}\in \Mat_2((\I_{0,r}/\fa\I_{0,r})\widehat{\otimes}_{K_{h,0}}K_{h,0,n}^\prime)$ the Sen operator associated with $\rho_{r,\fa}$.
Now (\ref{sigmadelta}) gives
\begin{equation}\label{sigmadeltaQ}
M_\fa^{-1}\rho_{r,\fa}(\sigma)\sigma(M_\fa)=\delta_\fa(\sigma)
\end{equation}
so we can calculate $\phi_{r,\fa}$ as
\begin{equation}\label{eqsenQ}
\phi_{r,\fa}=\lim_{\sigma\to 1}\frac{\log(\delta_\fa\bigl(\sigma)\bigr)}{\log(\chi(\sigma))}\in \Mat_2(\cR\widehat{\otimes}_L \C_p).
\end{equation}
\noindent By comparing this with (\ref{eqsen}) we see that $\phi_{r,\fa}=\pi_\fa(\phi_r)$.
\end{proof}
Let $\phi_{r,\B_r}=\iota_{\B_r}(\phi_r)$. For a nonzero ideal $\fa$ of $A_{0,r}$ let $\pi_{\B_r,\fa}$ be the natural projection $\B_r\to\I_{0,r}/\fa\I_{0,r}$. Clearly $\pi_{\B_r,\fa}(\phi_{r,\B_r})=\pi_{\fa}(\phi_r)$ and $\phi_{r,\fa}=\pi_{\fa}(\phi_r)$ by Proposition \ref{senproj}, so we have $\phi_{r,\B_r}=\varprojlim_{(\fa,P_1)=1}\phi_{r,\fa}$.
We apply Theorem \ref{liealgsen} to show the following.
\begin{proposition}\label{liealgsenQ}
Let $\fa$ be a nonzero ideal of $A_{0,r}$ prime to $P_1$. The operator $\phi_{r,\fa}$ belongs to the Lie algebra $\fH_{r,\fa,\C_p}^\loc$.
\end{proposition}
\begin{proof}
Let $n$ be the dimension over $\Q_p$ of $\I_{0,r}/\fa\I_{0,r}$; by choosing a $\Q_p$-basis $(\omega_1,\ldots,\omega_n)$ of this algebra,
we can define an injective ring morphism $\alpha\colon\Mat_2(\I_{0,r}/\fa\I_{0,r})\into \Mat_{2n}(\Q_p)$ and an injective group morphism $\alpha^\times\colon\GL_2(\I_{0,r}/Q\I_{0,r})\into\GL_{2n}(\Q_p)$. In fact, an endomorphism $f$ of the $(\I_{0,r}/\fa\I_{0,r})$-module
$(\I_{0,r}/\fa\I_{0,r})^2=(\I_{0,r}/\fa\I_{0,r})\cdot e_1\oplus (\I_{0,r}/\fa\I_{0,r})\cdot e_2$ is $\Q_p$-linear, so it induces an endomorphism $\alpha(f)$ of the $\Q_p$-vector space $(\I_{0,r}/\fa\I_{0,r})^2=\bigoplus_{i,j} \Q_p\cdot \omega_i e_j$; furthermore if $\alpha$ is an automorphism then $\alpha(f)$ is one too.
In particular $\rho_{r,\fa}$ induces a representation $\rho_{r,\fa}^\alpha=\alpha^\times\circ\rho_{r,\fa}\colon\Gal(\overline{K}/K)\to\GL_{2n}(\Q_p)$. The image of $\rho_{r,\fa}^\alpha$ is the group $G_{r,\fa}^{\loc,\alpha}=\alpha^\times(G_{r,\fa}^\loc)$.
We consider its Lie algebra $\fH_{r,\fa}^{\loc,\alpha}=\Q_p\cdot\Log\,(G_{r,\fa}^{\loc,\alpha})\subset \Mat_{2n}(\Q_p)$.
The $p$-adic logarithm commutes with $\alpha$ in the sense that $\alpha(\Log\, x)=\Log\,(\alpha^\times(x))$ for every $x\in\Gamma_{\I_{0,r}/\fa\I_{0,r}}(p)$, so we have $\fH_{r,\fa}^{\loc,\alpha}=\alpha(\fH_{r,\fa}^\loc)$ (recall that $\fH_{r,\fa}^\loc=\Q_p\cdot \Log\, G_{r,\fa}^\loc)$.
Let $\phi_{r,\fa}^\alpha$ be the Sen operator associated with $\rho_{r,\fa}^\alpha\colon\Gal(\overline{K}/K)\to\GL_{2n}(\Q_p)$. By Theorem \ref{liealgsen} we have $\phi_{r,\fa}^\alpha\in\fH_{r,\fa,\C_p}^{\loc,\alpha}=\fH_{r,\fa}^{\loc,\alpha}\widehat{\otimes}\C_p$.
Denote by $\alpha_{\C_p}$ the map $\alpha\widehat{\otimes}1\colon\Mat_2(\I_{0,r,\C_p}/\fa\I_{0,r,\C_p})\into \Mat_{2n}(\C_p)$.
We show that $\phi_{r,\fa}^{\alpha_{\C_p}}=\alpha_{\C_p}(\phi_{r,\fa})$, from which it follows that $\phi_{r,\fa}\in\fH_{r,\fa,\C_p}^\loc$ since $\fH_{r,\fa,\C_p}^{\loc,\alpha_{\C_p}}=\alpha_{\C_p}(\fH_{r,\fa,{\C_p}}^\loc)$ and $\alpha_{\C_p}$ is injective. Now let $M_\fa$, $\delta_\fa$ be as in (\ref{sigmadeltaQ}) and $M_\fa^{\alpha_{\C_p}}=\alpha_{\C_p}(M_\fa)$, $\delta_\fa^{\alpha_{\C_p}}=\alpha_{\C_p}\circ\delta_\fa$.
By applying $\alpha_C$ to (\ref{sigmadelta}) we obtain $(M_\fa^{\alpha_{\C_p}})^{-1}\rho_{r,\fa}^{\alpha_{\C_p}}(\sigma)\sigma(M_\fa^{\alpha_{\C_p}})=\delta_\fa^{\alpha_{\C_p}}(\sigma)$ for every $\sigma\in G_n^\prime$, so we can calculate
$$ \phi_{r,\fa}^{\alpha_{\C_p}}=\lim_{\sigma\to 1}\frac{\log(\delta_\fa^{\alpha_{\C_p}}\bigl(\sigma)\bigr)}{\log(\chi(\sigma))}, $$
which coincides with $\alpha_{\C_p}(\phi_{r,\fa})$.
\end{proof}
\begin{proposition}\label{seninalg}
The element $\phi_{r,\B_r}$ belongs to $\fH_{r,{\C_p}}^\loc$, hence to $\fH_{r,{\C_p}}$.
\end{proposition}
\begin{proof}
By definition of the space $\fH_{r,{\C_p}}^\loc$ as completion
of the space $\fH_{r}^\loc\otimes_{K_{h,0}}\C_p$ for the seminorms $p_\fa$
given by the norms on $\fH_{r,\fa,{\C_p}}^\loc$, we have
$\fH_{r,{\C_p}}^\loc=\varprojlim_{(\fa,P_1)=1}\fH_{r,\fa,{\C_p}}^\loc$.
By Proposition \ref{senproj}, we have $\phi_{r,\B_r}=\varprojlim_{\fa}\phi_{r,\fa}$ and by Proposition \ref{liealgsenQ} we have for every $\fa$, $\phi_{r,\fa}\in\fH_{r,\fa,{\C_p}}$. We conclude that $\phi_{r,\B_r}\in\fH_{r,{\C_p}}^\loc$.
\end{proof}
\begin{remark}\label{primlimsen}
In order to prove that our Lie algebras are ``big'' it will be useful to work with primary ideals of $A_r$, as we did in this subsection. However, in light of Remark \ref{primlim}, all of the results can be rewritten in terms of primary ideals $\fQ$ of $\I_{0,r}$. This will be useful in the next subsection, when we will interpolate the Sen operators corresponding to the classical modular representations.
\end{remark}
From now on we identify $\I_{0,r,{\C_p}}$ with a subring of $\B_{r,{\C_p}}$ via $\iota_{\B_r}$, so we also identify $\Mat_2(\I_{0,r})$ with a subring of $\Mat_2(\B_r)$ and $\GL_2(\I_{0,r,{\C_p}})$ with a subgroup of $\GL_2(\B_{r,{\C_p}})$. In particular we identify $\phi_r$ with $\phi_{r,\B_r}$ and we consider $\phi_r$ as an element of $\fH_{r,{\C_p}}\cap\Mat_2(\I_{0,r,{\C_p}})$.
\subsection{The characteristic polynomial of the Sen operator}\label{charpolyn}
Sen proved the following result.
\begin{theorem}\label{charpolsen}
Let $L_1$ and $ L_2$ be two $p$-adic fields.
Assume for simplicity that $L_2$ contains the normal closure of $L_1$. Let
$\tau\colon\Gal(\overline{L}_1/L_1)\to\GL_m(L_2)$ be a continuous representation.
For each embedding $\sigma\colon L_1\to L_2$,
there is a Sen operator $\phi_{\tau,\sigma}\in \Mat_m(\C_p\otimes_{L_1,\sigma}L_2)$ associated with $\tau$ and $\sigma$.
If $\tau$ is Hodge-Tate and its Hodge-Tate weights with respect to $\sigma$ are $h_{1,\sigma},\ldots,h_{m,\sigma}$ (with multiplicities, if any), then the characteristic polynomial of $\phi_{\tau,\sigma}$ is $\prod_{i=1}^m(X-h_{i,\sigma})$.
\end{theorem}
Now let $k\in\N$ and $P_k=(u^{-k}(1+T)-1)$ be the corresponding arithmetic prime of $A_{0,r}$.
Let $\fP_f$ be a prime of $\I_r$ above $P$, associated with the system of Hecke eigenvalues of a classical modular form $f$.
Let $\rho\colon\G_Q\to\GL_2(\I_r)$ be as usual. The specialization of $\rho$ modulo $\fP$ is the representation $\rho_f\colon G_\Q\to\GL_2(\I_r/\fP)$ classically associated with $f$, defined over the field $K_f=\I_r/\fP\I_r$. By a theorem of Faltings (\cite{faltings}), when the weight of the form $f$ is $k$, the representation $\rho_f$ is Hodge-Tate of Hodge-Tate weights $0$ and $k-1$. Hence by Theorem \ref{charpolsen} the Sen operator $\phi_f$ associated with $\rho_f$ has characteristic polynomial $X(X-(k-1))$.
Let $\fP_{f,0}=\fP_f\cap\I_{0,r}$. With the notations of the previous subsection, the specialization of $\rho_r$ modulo $\fP_{f,0}$ gives a representation $\rho_{r,\fP_{f,0}}\colon\Gal(\overline{K}/K)\to\GL_2(\I_{0,r}/\fP_{f,0})$, which coincides with $\rho_f\vert_{\Gal(\overline{K}/K)}$. In particular the Sen operator $\phi_{r,\fP_{f,0}}$ associated with $\rho_{r,\fP_{f,0}}$ is $\phi_f$.
By Proposition \ref{senproj} and Remark \ref{primlimsen}, the Sen operator $\phi_r\in \Mat_2(\I_{0,r,\C_p})$ specializes modulo $\fP_{f,0}$ to the Sen operator $\phi_{r,\fP_{f,0}}$ associated with $\rho_{r,\fP_{f,0}}$, for every $f$ as above.
Since the primes of the form $\fP_{f,0}$ are dense in $\I_{0,r,\C_p}$, the eigenvalues of $\phi_{r,Q}$ are given by the unique interpolation of those of $\rho_{r,\fP_{f,0}}$.
This way we will recover an element of $\GL_2(\B_{r,\C_p})$ with the properties we need.
Given $f\in A_{0,r}$ we define its $p$-adic valuation by $v_p^\prime(f)=\inf_{x\in\cB(0,r)}v_p(f(x))$, where $v_p$ is our chosen valuation on $\C_p$.
Then if $v^\prime(f-1)\leq p^{-\frac{1}{p-1}}$ there are well-defined elements $\log(f)$ and $\exp(\log(f))$ in $A_{0,r}$, and $\exp(\log(f))=f$.
Let $\phi^\prime_r=\log(u)\phi_r$.
Note that $\phi^\prime_r$ is a well-defined element of $\Mat_2(\B_{r,\C_p})$ since $\log(u)\in\Q_p$.
Recall that we denote by $C_T$ the matrix $\diag(u^{-1}(1+T),1)$.
We have the following.
\begin{proposition}\label{sendiag}
\begin{enumerate}[leftmargin=*]
\item The eigenvalues of $\phi^\prime_r$ are $\log(u^{-1}(1+T))$ and $0$.
In particular the exponential $\Phi_r=\exp(\phi^\prime_r)$ is defined in $\GL_2(\B_{r,\C_p})$.
Moreover $\Phi^\prime_r$ is conjugate to $C_T$ in $\GL_2(\B_{r,\C_p})$.
\item The element $\Phi^\prime_r$ of part (1) normalizes $\fH_{r,{\C_p}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
For every $\fP_{f,0}$ as in the discussion above, the element $\log(u)\phi_r$ specializes to $\log(u)\phi_{r,\fP_{f,0}}$ modulo $\fP_{f,0}$. If $\fP_{f,0}$ is a divisor of $P_k$, the eigenvalues of $\log(u)\phi_{r,\fP_{f,0}}$ are $\log(u)(k-1)$ and $0$.
Since $1+T=u^k$ modulo $\fP_{f,0}$ for every prime $\fP_{f,0}$ dividing $P_k$, we have $\log(u^{-1}(1+T))=\log(u^{k-1})=(k-1)\log(u)$ modulo $\fP_{f,0}$. Hence the eigenvalues of $\log(u)\phi_{r,\fP_{f,0}}$ are interpolated by $\log(u^{-1}(1+T))$ and $0$.
Recall that in Section \ref{subsectnest} we chose $r_h$ smaller than $p^{-\frac{1}{p-1}}$.
Since $r<r_h$, $v_p^\prime(T)<p^{-\frac{1}{p-1}}$. In particular $\log(u^{-1}(1+T))$ is defined and $\exp(\log(u^{-1}(1+T)))=u^{-1}(1+T)$, so $\Phi_r=\exp(\phi^\prime_r)$ is also defined and its eigenvalues are $u^{-1}(1+T)$ and $1$.
The difference between the two is $u^{-1}(1+T)-1$; this belongs to $P_1$, hence it is invertible because of our definition of $\B_r$. This proves (1).
By Proposition \ref{seninalg}, $\phi_r\in\fH_{r,{\C_p}}$. Since $\fH_{r,{\C_p}}$ is a $\Q_p$-Lie algebra, $\log(u)\phi_r$ is also an element of $\fH_{r,{\C_p}}$. Hence its exponential $\Phi^\prime_r$ normalizes $\fH_{r,{\C_p}}$.
\end{proof}
\bigskip
\section{Existence of the Galois level for a family with finite positive slope}
Let $r_h\in p^\Q\cap]0,p^{-\frac{1}{p-1}}[$ be the radius chosen in Section \ref{sectcong}. As usual we write $r$ for any one of the radii $r_i$ of Section \ref{subsectnest}.
Recall that $\fH_{r}\subset \Mat_2(\B_r)$ is the Lie algebra attached to the image of $\rho_{r}$ (see Definition \ref{blalg})
and $\fH_{0,r,\C_p}=\fH_{r}\widehat{\otimes}\C_p$.
Let $\fu^\pm$, respectively $\fu^\pm_{\C_p}$, be the upper and lower nilpotent subalgebras of $\fH_{r}$, respectively $\fH_{r,\C_p}$.
\begin{remark} \label{remind_r} The commutative Lie algebra $\fu^\pm$ is independent of $r$ because it is equal to $\Q_p\cdot\Log(U(\I_0^\circ)\cap G_r^\prime)$ which is independent of $r$, provided $r_0\leq r<r_h$.
\end{remark}
We fix $r_0\in p^\Q\cap]0,r_h[$ arbitrarily and we work from now on with radii $r$ satisfying $r_0\leq r<r_h$.
As in Remark \ref{propsubgr} this fixes a finite extension of $\Q$ corresponding to the inclusion $G_r^\prime\subset G_r$.
For $r<r^\prime$ we have a natural inclusion $\I_{0,r^\prime}\into\I_{0,r}$.
Since $\B_r=\varprojlim_{(\fa P_1)=1}\I_{0,r}/\fa \I_{0,r}$ this induces an inclusion $\B_{r^\prime}\into\B_r$.
We will consider from now on $\B_{r^\prime}$ as a subring of $\B_r$ for every $r<r^\prime$. We will also consider $\Mat_2(\I_{0,r^\prime,\C_p})$ and $\Mat_2(\B_{r^\prime})$ as subsets of $\Mat_2(\I_{0,r,\C_p})$ and $\Mat_2(\B_r)$ respectively.
These inclusions still hold after taking completed tensors with $\C_p$.
Recall the elements $\phi^\prime_r=\log(u)\phi_r\in \Mat_2(\B_{r,\C_p})$ and $\Phi^\prime_r=\exp(\phi^\prime_r)\in\GL_2(\I_{0,r,\C_p})$ defined at the end of the previous section.
The Sen operator $\phi_r$ is independent of $r$ in the following sense: if $r<r^\prime<r_h$ and $\B_{r^\prime,\C_p}\to\B_{r,\C_p}$ is the natural inclusion then the image of $\phi_{r^\prime}$ under the induced map $\Mat_2(\B_{r^\prime,\C_p})\to \Mat_2(\B_{r,\C_p})$ is $\phi_r$.
We deduce that $\phi^\prime_r$ and $\Phi^\prime_r$ are also independent of $r$ (in the same sense).
By Proposition \ref{sendiag}, for every $r<r_h$
there exists an element $\beta_r\in\GL_2(\B_{r,\C_p})$ such that $\beta_r\Phi^\prime_r\beta_r^{-1}=C_T$. Since $\Phi^\prime_r$ normalizes $\fH_{r,\C_p}$, $C_T=\beta_r\Phi^\prime_r\beta_r^{-1}$ normalizes $\beta_r\fH_{r,\C_p}\beta_r^{-1}$ .
We denote by $\fU^{\pm}$ the upper and lower nilpotent subalgebras of $\fsl_2$. The action of $C_T$ on $\fH_{r,\C_p}$ by conjugation is semisimple, so we can decompose $\beta_r\fH_{r,\C_p}\beta_r^{-1}$ as a sum of eigenspaces for $C_T$:
$$ \beta_r\fH_{r,\C_p}\beta_r^{-1}=\left(\beta_r\fH_{r,\C_p}\beta_r^{-1}\right)[1]\oplus \left(\beta_r\fH_{r,\C_p}\beta_r^{-1}\right)[u^{-1}(1+T)]\oplus \left(\beta_r\fH_{r,\C_p}\beta_r^{-1}\right)[u(1+T)^{-1}] $$
with
$$ \left(\beta_r\fH_{r,\C_p}\beta_r^{-1}\right)[u^{-1}(1+T)]\subset \fU^+(\B_{r,\C_p}) \mbox{\qquad and \qquad} \left(\beta_r\fH_{r,\C_p}\beta_r^{-1}\right)[u(1+T)^{-1}]\subset\fU^-(\B_{r,\C_p}). $$
\noindent Moreover, the formula $$\left(\begin{array}{cc}u^{-1}(1+T)&0\\0&1\end{array}\right)
\left(\begin{array}{cc}1&\lambda\\0&1\end{array}\right)
\left(\begin{array}{cc}u^{-1}(1+T)&0\\0&1\end{array}\right)^{-1}=
\left(\begin{array}{cc}1&u^{-1}(1+T)\lambda\\0&1\end{array}\right)$$
shows that the action of $C_T$ by conjugation coincides with multiplication by $u^{-1}(1+T)$.
By linearity this gives an action of the polynomial ring $\C_p[T]$ on $\beta_r\fH_{r,\C_p}\beta_r^{-1}\cap\fU^+(\B_{r,\C_p})$,
compatible with the action of ${\C_p}[T]$ on $\fU^+(\B_{r,\C_p})$ given by the inclusions
$\C_p[T]\subset\Lambda_{h,0,{\C_p}}\subset B_{r,\C_p}\subset\B_{r,\C_p}$. Since ${\C_p}[T]$ is dense in $A_{h,0,{\C_p}}$
for the $p$-adic topology, it is also dense in $B_{r,\C_p}$. Since $\fH_{r,\C_p}$ is a closed Lie subalgebra of $\Mat_2(\B_{r,\C_p})$,
we can define by continuity a $B_{r,\C_p}$-module structure on
$\beta_r\fH_{r,\C_p}\beta_r^{-1}\cap\fU^+(\B_{r,\C_p})$, compatible with that on $\fU^+(\B_{r,\C_p})$. Similarly
we have
$$ \left(\begin{array}{cc}u^{-1}(1+T)&0\\0&1\end{array}\right)\left(\begin{array}{cc}1&0\\ \mu &1\end{array}\right)\left(\begin{array}{cc}u^{-1}(1+T)&0\\0&1\end{array}\right)^{-1}=
\left(\begin{array}{cc}1&0\\u(1+T)^{-1}\mu &1\end{array}\right). $$
We note that $1+T$ is invertible in $A_{0,r}$ since $T=p^{s_h}t$ where $r_h=p^{-s_h}$. Therefore $C_T$ is invertible and by twisting by $(1+T)\mapsto (1+T)^{-1}$ we can also give $\beta_r\fH_{r,\C_p}\beta_r^{-1}\cap\fU^-(\B_{r,\C_p})$
a structure of $B_{r,\C_p}$-module compatible with that on $\fU^-(\B_{r,\C_p})$.
By combining the previous remarks with Corollary \ref{lambdaspan}, we prove the following ``fullness'' result for the big Lie algebra $\fH_r$.
\begin{theorem}\label{betalevel}
Suppose that the representation $\rho$ is $(H_0,\Z_p)$-regular. Then there exists a nonzero ideal $\fl$ of $\I_0$, independent of $r<r_h$, such that for every such $r$ the Lie algebra $\fH_r$ contains $\fl\cdot\fsl_2(\B_r)$.
\end{theorem}
\begin{proof}
Since $U^\pm(\B_r)\cong \B_r$, we can and shall identify $\fu^+=\Q_p\cdot\Log\, G_r^\prime\cap\fU^+(\B_r)$ with a $\Q_p$-vector subspace of $\B_r$ (actually of $\I_0$), and $\fu^+_{\C_p}$ with a $\C_p$-vector subspace of $\B_{r,\C_p}$. We repeat that these spaces are independent of $r$ since
$G_r^\prime$ is, provided that $r_0\leq r<r_h$ (see Remark \ref{propsubgr}).
By Corollary \ref{lambdaspan}, $\fu^\pm\cap\I_0$ contains a basis $\{e_{i,\pm}\}_{i\in I}$ for $Q(\I_0)$ over $Q(\Lambda_{h,0})$.
The set $\{e_{i,+}\}_{i\in I}\subset\fu^+$ is a basis for $Q(\I_{0})$ over $Q(\Lambda_{h,0})$, so $\fu^+$ contains the basis of a $\Lambda_{h,0}$-lattice in $\I_0$.
By Lemma \ref{lattice} we deduce that $\Lambda_{h,0}\fu^+$ contains a nonzero ideal $\fa^+$ of $\I_0$.
Hence we also have $B_{r,\C_p}\fu^+_{\C_p}\supset B_{r,\C_p} \fa^+$.
Now $\fa^+$ is an ideal of $\I_0$ and $B_{r,\C_p}\I_{0,\C_p}=\B_{r,\C_p}$, so $B_{r,\C_p}\fa^+=\fa^+\B_{r,\C_p}$
is an ideal in $\B_{r,\C_p}$.
We conclude that $B_{r,\C_p}\cdot\fu^+\supset\fa^+\B_{r,\C_p}$ for a nonzero ideal $\fa^+$ of $\I_0$.
We proceed in the same way for the lower unipotent subalgebra, obtaining $B_{r,\C_p}\cdot\fu^-\supset\fa^-\B_{r,\C_p}$ for some nonzero ideal $\fa^-$ of $\I_0$.
Consider now the Lie algebra $B_{r,\C_p}\fH_{\C_p}\subset \Mat_2(\B_{r,\C_p})$.
Its nilpotent subalgebras are $B_{r,\C_p}\fu^+$ and $B_{r,\C_p}\fu^-$, and we showed $B_{r,\C_p}\fu^+\supset\fa^+\B_{r,\C_p}$ and $B_{r,\C_p}\fu^-\supset\fa^-\B_{r,\C_p}$.
Denote by $\ft\subset\fsl_2$ the subalgebra of diagonal matrices over $\Z$. By taking the Lie bracket, we see that
$[\fU^+(\fa^+\B_{r,\C_p}),\fU^-(\fa^-\B_{r,\C_p})]$ spans $\fa^+\cdot\fa^-\cdot\ft(\B_{r,\C_p})$ over $B_{r,\C_p}$.
We deduce that $B_{r,\C_p}\fH_{\C_p}\supset\fa^+\cdot\fa^-\cdot\fsl_2(\B_{r,\C_p})$.
Let $\fa=\fa^+\cdot\fa^-$. Now $\fa\cdot\fsl_2(\B_{r,\C_p})$ is a $\B_{r,\C_p}$-Lie subalgebra of $\fsl_2(\B_{r,\C_p})$.
Recall that $\beta_r\in\GL_2(\B_{r,\C_p})$; hence by stability by conjugation we have
$\beta_r\left(\fa\cdot\fsl_2(\B_{r,\C_p})\right)\beta_r^{-1}=\fa\cdot\fsl_2(\B_{r,\C_p})$.
Thus, we constructed $\fa$ such that $B_{r,\C_p}\left(\beta_r\fH_{r,\C_p}\beta_r^{-1}\right)\supset\fa\cdot\fsl_2(\B_{r,\C_p})$.
In particular, if $\fu_{\C_p}^{\pm,\beta_r}$ denote the unipotent subalgebras of $\beta_r\fH_{r,\C_p}\beta_r^{-1}$, we have $B_{r,\C_p}\fu_{\C_p}^{\pm,\beta_r}\supset\fa\B_{r,\C_p}$ for both signs. By the discussion preceding the proposition the subalgebras $\fu_{\C_p}^{\pm,\beta_r}$ have a structure of $B_{r,\C_p}$-modules, which means that $\fu_{\C_p}^{\pm,\beta_r}=B_{r,\C_p}\fu_{\C_p}^{\pm,\beta_r}$.
We conclude that $\fu_{\C_p}^{\pm,\beta_r}\supset \beta_r\left(\fa\cdot\fU^\pm(\B_{r,\C_p})\right)\beta_r^{-1}$ for both signs.
By the usual argument of taking the bracket, we have $\beta_r\fH_{r,\C_p}\beta_r^{-1}\supset\fa^2\cdot\fsl_2(\B_{r,\C_p})$.
We can untwist by the invertible matrix $\beta_r$ to conclude that, for $\fl=\fa^2$, we have
$\fH_{r,\C_p}\supset\fl\cdot\fsl_2(\B_{r,\C_p})$.
Let us get rid of the completed extension of scalars to $\C_p$.
For every ideal $\fa\subset\I_{0,r}$ not dividing $P_1$, let $\fH_{r,\fa}$ be the image of $\fH_r$ in $\Mat_2(\I_{0,r}/\fa\I_{0,r})$.
Consider the two finite dimensional $\Q_p$-vector spaces $\fH_{r,\fa}$ and $\fl\cdot\fsl_2(\I_{0,r}/\fa\I_{0,r})$. Note that they are both subspaces of the finite dimensional $\Q_p$-vector space $\Mat_2(\I_{0,r}/\fa\I_{0,r})$. After extending scalars to $\C_p$, we have
\begin{equation}\label{cpincl} \fl\cdot\fsl_2(\I_{0,r}/\fa \I_{0,r})\otimes\C_p\subset\fH_{r,\fa}\otimes\C_p. \end{equation}
Let $\{e_i\}_{i\in I}$ be an orthonormal basis of the Banach space $\C_p$ over $\Q_p$, with $I$ some index set, such that $1\in\{e_i\}_{i\in I}$. Let $\{v_j\}_{j=1,...,n}$ be a $\Q_p$-basis of $\Mat_2(\I_{0,r}/\fa\I_{0,r})$ such that, for some $d\leq n$, $\{v_j\}_{j=1,...,d}$ is a $\Q_p$-basis of $\fH_{r,\fa}$.
Let $v$ be an element of $\fl\cdot\fsl_2(\I_{0,r}/\fa\I_{0,r})$.
Then $v\otimes 1\in\fl\cdot\fsl_2(\I_{0,r}/\fa \I_{0,r})\otimes\C_p$ and by (\ref{cpincl}) we have $v\otimes 1\in\fH_{r,\fa}\otimes\C_p$. As $\{v_j\otimes e_i\}_{1\le j\le d, i\in I}$, respectively
$\{v_j\otimes e_i \}_{1\le j\le n,i\in I}$ is an orthonormal basis of $\fH_{r,\fa}\otimes\C_p$, respectively of $\Mat_2(\I_{0,r}/\fa\I_{0,r})\otimes \C_p$ over $\Q_p$, there exist $\lambda_{j,i}\in\Q_p, (j,i)\in \{1,2,...,d\}\times I$ converging to $0$ in the filter of complements of finite subsets of $\{1,2,...,d\}\times I$ such that $v\otimes 1=\sum_{j=1,...,d;\, i\in I}\lambda_{j,i}(v_j\otimes e_i)$.
But $v\otimes 1\in \Mat_2(\I_{0,r}/\fa\I_{0,r})\otimes 1\subset \Mat_2(\I_{0,r}/\fa\I_{0,r})\otimes \C_p$ and therefore $v\otimes 1=\sum_{1\le j\le n} a_j(v_j\otimes 1)$, for some $a_j\in \Q_p$, $j=1,...,n$. By the uniqueness of a representation of an element in a
$\Q_p$-Banach space in terms of a given orthonormal basis we have
$$
v\otimes 1=\sum_{j=1}^d a_j(v_j\otimes 1),\mbox{\quad i.e.\quad} v=\sum_{j=1}^da_jv_j\in \fH_{r,\fa}.
$$
By taking the projective limit over $\fa$, we conclude that
$$\fl\cdot \fsl_2(\B_r)\subset \fH_r.$$
\end{proof}
\begin{definition}
The Galois level of the family $\theta\colon\T_h\to\I^\circ$ is the largest ideal $\fl_\theta$ of $\I_{0}[P_1^{-1}]$ such that $\fH_{r}\supset\fl_\theta\cdot\fsl_2(\B_r)$ for all $r<r_h$.
\end{definition}
\noindent It follows by the previous remarks that $\fl_\theta$ is nonzero.
\bigskip
\section{Comparison between the Galois level and the fortuitous congruence ideal}\label{compar}
Let $\theta\colon\T_h\to\I^\circ$ be a family. We keep all the notations from the previous sections.
In particular $\rho\colon G_\Q\to\GL_2(\I^\circ)$ is the Galois representation associated with $\theta$.
We suppose that the restriction of $\rho$ to $H_0$ takes values in $\GL_2(\I^\circ_0)$.
Recall that $\I=\I^\circ[p^{-1}]$ and $\I_0=\I_0^\circ[p^{-1}]$.
Also recall that $P_1$ is the prime of $\Lambda_{h,0}$ generated by $u^{-1}(1+T)-1$.
Let $\fc\subset\I$ be the congruence ideal associated with $\theta$.
Set $\fc_0=\fc\cap\I_0$ and $\fc_1=\fc_0\I_0[P_1^{-1}]$.
Let $\fl=\fl_\theta\subset\I_0[P_1^{-1}]$ be the Galois level associated with $\theta$.
For an ideal $\fa$ of $\I_0[P_1^{-1}]$ we denote by $V(\fa)$ the set of prime ideals of $\I_0[P_1^{-1}]$ containing $\fa$.
We prove the following.
\begin{theorem}\label{comparison}
Suppose that
\begin{enumerate}
\item $\rho$ is $(H_0,\Z_p)$-regular;
\item there exists no pair $(F,\psi)$, where $F$ is a real quadratic field and $\psi\colon\Gal(\overline{F}/F)\to\F^\times$
is a character, such that $\overline{\rho}\colon G_\Q\to\GL_2(\F)\cong\Ind_F^\Q\psi$.
\end{enumerate}
Then we have $V(\fl)=V(\fc_1)$.
\end{theorem}
Before giving the proof we make some remarks.
Let $P$ be a prime of $\I_0[P_1^{-1}]$ and $Q$ be a prime factor of $P\I[P_1^{-1}]$. We consider $\rho$ as a representation $G_\Q\to\GL_2(\I[P_1^{-1}])$ by composing it with the inclusion $\GL_2(\I)\into\GL_2(\I[P_1^{-1}])$.
We have a representation $\rho_{Q}\colon G_\Q\to\GL_2(\I[P_1^{-1}]/Q)$ obtained by reducing $\rho$ modulo $Q$. Its restriction $\rho_{Q}\vert_{H_0}$ takes values in $\GL_2(\I_0[P_1^{-1}]/(Q\cap\I_0[P_1^{-1}]))=\GL_2(\I_0[P_1^{-1}]/P)$ and coincides with the reduction $\rho_P$ of $\rho\vert_{H_0}\colon H_0\to\GL_2(\I_0[P_1^{-1}])$ modulo $P$.
In particular $\rho_{Q}\vert_{H_0}$ is independent of the chosen prime factor $Q$ of $P\I[P_1^{-1}]$.
We say that a subgroup of $\GL_2(A)$ for some algebra $A$ finite over a $p$-adic field $K$ is \textit{small} if it admits a finite index abelian subgroup.
Let $P$, $Q$ be as above, $G_P$ be the image of $\rho_P\colon H_0\to\GL_2(\I_0[P_1^{-1}]/P)$ and $G_{Q}$ be the image of $\rho_{Q}\colon G_\Q\to\GL_2(\I[P_1^{-1}]/Q)$.
By our previous remark $\rho_P$ coincides with the restriction $\rho_{Q}\vert_{H_0}$, so $G_P$ is a finite index subgroup of $G_{Q}$ for every $Q$.
In particular $G_P$ is small if and only if $G_{Q}$ is small for all prime factors $Q$ of $P\I[P_1^{-1}]$.
Now if $Q$ is a CM point the representation $\rho_{Q}$ is induced by a character of $\Gal(F/\Q)$ for an imaginary quadratic field $F$.
Hence $G_{Q}$ admits an abelian subgroup of index $2$ and $G_P$ is also small.
Conversely, if $G_P$ is small, $G_{Q^\prime}$ is small for every prime $Q^\prime$ above $P$.
Choose any such prime $Q^\prime$; by the argument in \cite[Prop. 4.4]{ribet2} $G_{Q^\prime}$ has an abelian subgroup of index $2$.
It follows that $\rho_{Q^\prime}$ is induced by a character of $\Gal(\overline{F}_{Q^\prime}/F_{Q^\prime})$ for a quadratic field $F_{Q^\prime}$.
If $F_{Q^\prime}$ is imaginary then ${Q^\prime}$ is a CM point.
In particular, if we suppose that the residual representation $\bar{\rho}\colon G_\Q\to\GL_2(\F)$ is not induced by a character of $\Gal(\overline{F}/F)$ for a real quadratic field $F/\Q$, then $F_{Q^\prime}$ is imaginary and ${Q^\prime}$ is CM.
The above argument proves that $G_P$ is small if and only if all points ${Q^\prime}\subset\I[P_1^{-1}]$ above $P$ are CM.
\begin{proof}
We prove first that $V(\fc_1)\subset V(\fl)$. Fix a radius $r<r_h$.
By contradiction, suppose that a prime $P$ of $\I_0[P_1^{-1}]$ contains $\fc_0$ but $P$ does not contain $\fl$.
Then there exists a prime factor $Q$ of $P\I[P_1^{-1}]$ such that $\fc\subset Q$.
By definition of $\fc$ we have that $Q$ is a CM point in the sense of Section \ref{congrid}, hence the representation $\rho_{\I[P_1^{-1}],Q}$ has small image in $\GL_2(\I[P_1^{-1}]/Q)$.
Then its restriction $\rho_{\I[P_1^{-1}],Q}\vert_{H_0}=\rho_P$ also has small image in $\GL_2(\I_0[P_1^{-1}]/P)$.
We deduce that there is no nonzero ideal $\frakI_P$ of $\I_0[P_1^{-1}]/P$ such that the Lie algebra $\fH_{r,P}$ contains $\frakI_P\cdot\fsl_2(\I_0[P_1^{-1}]/P)$.
Now by definition of $\fl$ we have $\fl\cdot\fsl_2(\B_r)\subset\fH_r$.
Since reduction modulo $P$ gives a surjection $\fH_r\to\fH_{r,P}$, by looking at the previous inclusion modulo $P$ we find $\fl\cdot \fsl_2(\I_{0,r}[P_1^{-1}]/P\I_{0,r}[P_1^{-1}])\subset\fH_{r,P}$.
If $\fl\not\subset P$ we have $\fl/P\neq 0$, which contradicts our earlier statement.
We deduce that $\fl\subset P$.
We prove now that $V(\fl)\subset V(\fc_1)$.
Let $P\subset\I_0[P_1^{-1}]$ be a prime containing $\fl$. Recall that $\I_0[P_1^{-1}]$ has Krull dimension one, so $\kappa_{P}=\I_0[P_1^{-1}]/P$ is a field.
Let $Q$ be a prime of $\I[P_1^{-1}]$ above $P$. As before $\rho$ reduces to representations $\rho_Q\colon G_\Q\to\GL_2(\I[P_1^{-1}]/Q)$ and $\rho_P\colon H_0\to\GL_2(\I_0[P_1^{-1}]/P)$.
Let $\fP\subset\I_0[P_1^{-1}]$ be the $P$-primary component of $\fl$ and let $\fA$ be an ideal of $\I_0[P_1^{-1}]$ containing $\fP$ such that
the localization at $P$ of $\fA/\fP$ is one-dimensional over $\kappa_P$.
Choose any $r<r_h$.
Let $\fs=\fA/\fP\cdot\fsl_2(\I_{0,r}[P_1^{-1}]/\fP)\cap\fH_{r,\fP}\subset
\fA/\fP\cdot\fsl_2(\I_{0,r}[P_1^{-1}]/\fP)$.
We show that $\fs$ is stable under the adjoint action $\Ad(\rho_Q)$ of $G_\Q$. Let $\fQ$ be the $Q$-primary component of $\fl\cdot\I[P_1^{-1}]$.
Recall that $\fH_{r,\fP}$ is the Lie algebra associated with the pro-$p$ group $\im\,\rho_{r,\fQ}\vert_{H_0}\cap\Gamma_{\GL_2(\I_{0,r}[P_1^{-1}]/\fP)}(p)\subset\GL_2(\I_{0,r}[P_1^{-1}]/\fP)$. Since this group is open in $\im\,\rho_{r,\fQ}\subset\GL_2(\I_r[P_1^{-1}]/\fQ)$, the Lie algebra associated with $\im\,\rho_{r,\fQ}$ is again $\fH_{r,\fP}$.
In particular $\fH_{r,\fP}$ is stable under $\Ad(\rho_Q)$.
Since $\fH_{r,\fP}\subset\fsl_2(\I_{0,r}[P_1^{-1}]/\fP)$ we have $\fA/\fP\cdot\fsl_2(\I_{0,r}[P_1^{-1}]/\fP)\cap\fH_{r,\fP}=\fA/\fP\cdot\fsl_2(\I_r[P_1^{-1}]/\fQ)\cap\fH_{r,\fP}$. Now $\fA/\fP\cdot\fsl_2(\I_r[P_1^{-1}]/\fQ)$ is clearly stable under $\Ad(\rho_Q)$, so the same is true for $\fA/\fP\cdot\fsl_2(\I_r[P_1^{-1}]/\fQ)\cap\fH_{r,\fP}$, as desired.
We consider from now on $\fs$ as a Galois representation via $\Ad(\rho_Q)$.
By the proof of Theorem \ref{betalevel} we can assume, possibly considering a sub-Galois
representation, that $\fH_r$ is a $\B_r$-submodule of $\fsl_2(\B_r)$
containing $\fl\cdot\fsl_2(\B_r)$ but not $\fa\cdot\fsl_2(\B_r)$ for any $\fa$ strictly bigger than $\fl$.
This allows us to speak of the localization $\fs_P$ of $\fs$ at $P$.
Note that, since $\fP$ is the $P$-primary component of $\fl$ and $\fA_P/\fP_P\cong\kappa_P$, when $P$-localizing we find $\fH_{r,P}\supset\fP_{P}\cdot\fsl_2(\B_{r,P})$ and $\fH_{r,P}\not\supset\fA_P\cdot\fsl_2(\B_{r,P})$.
The localization at $P$ of $\fa/\fP\cdot\fsl_2(\I_{0,r}[P_1^{-1}]/\fP)$ is $\fsl_2(\kappa_P)$, so $\fs_P$ is contained in $\fsl_2(\kappa_P)$. It is a $\kappa_P$-representation $G_\Q$ (via $\Ad(\rho_Q)$) of dimension at most $3$. We distinguish various cases following its dimension.
We cannot have $\fs_P=0$. By exchanging the quotient with the localization we would obtain $(\fA_{P}\cdot\fsl_2(\B_{r,P})\cap\fH_{r,P})/\fP_{P}=0$. By Nakayama's lemma $\fA_{P}\cdot\fsl_2(\B_{r,P})\cap\fH_{r,P}=0$, which is absurd since $\fA_{P}\cdot\fsl_2(\B_{r,P})\cap\fH_{r,P}\supset\fP_{P}\cdot\fsl_2(\B_{r,P})\neq 0$.
We also exclude the $3$-dimensional case. If $\fs_{P}=\fsl_2(\kappa_{P})$, by exchanging the quotient with the localization we obtain $(\fA_{P}\cdot\fsl_2(\B_{r,P})\cap\fH_{r,P})/\fP_{P}=(\fA_P\cdot\fsl_2(\I_{0,r,P}[P_1^{-1}]))/\fP_{P}\I_{0,r,P}[P_1^{-1}]$, because
$\fA_{P}\I_{0,r,P}[P_1^{-1}]/\fP_P\I_{0,r,P}[P_1^{-1}]=\left(\I_{0,r,P}[P_1^{-1}]/\fP_{P}\I_{0,r,P}[P_1^{-1}]\right)$ and this is isomorphic to $\kappa_{P}$.
By Nakayama's lemma we would conclude that $\fH_{r,P}\supset\fA\cdot\fsl_2(\B_{r,P})$, which is absurd.
We are left with the one and two-dimensional cases. If $\fs_P$ is two-dimensional we can always replace it by its orthogonal
in $\fsl_2(\kappa_P)$ which is one-dimensional; indeed the action of $G_\Q$ via $\Ad(\rho_Q)$ is isometric with respect to the scalar product $\Tr(XY)$ on $\fsl_2(\kappa_P)$.
Suppose that $\fsl_2(\kappa_P)$ contains a one-dimensional stable subspace.
Let $\phi$ be a generator of this subspace over $\kappa_P$.
Let $\chi\colon G_\Q\to\kappa_P$ denote the character satisfying $\rho_Q(g)\phi\rho_Q(g)^{-1}=\chi(g)\phi$ for all $g\in G_\Q$.
Now $\phi$ induces a nontrivial morphism of representations $\rho_Q\to\rho_Q\otimes\chi$. Since $\rho_P$ and $\rho_Q\otimes\chi$ are irreducible, by Schur's lemma $\phi$ must be invertible. Hence we obtain an isomorphism $\rho_Q\cong\rho_Q\otimes\chi$.
By taking determinants we see that $\chi$ must be quadratic. If $F_0/\Q$ is the quadratic extension fixed by $\ker\chi$, then $\rho_Q$ is induced by a character $\psi$ of $\Gal(\overline{F_0}/F_0)$.
By assumption the residual representation $\rho_{\fm_\I}\colon G_\Q\to\GL_2(\F)$ is not of the form $\Ind_F^\Q\psi$ for a real quadratic field $F$ and a character $\Gal(\overline{F}/F)\to\F^\times$. We deduce that $F_0$ must be imaginary, so $Q$ is a CM point by Remark \ref{cmlocus}(1). By construction of the congruence ideal $\fc\subset Q$ and $\fc_0\subset Q\cap\I_0[P_1^{-1}]=P$.
\end{proof}
We prove a corollary.
\begin{corollary}
If the residual representation $\overline{\rho}\colon G_\Q\to\GL_2(\F)$ is not dihedral then $\fl=1$.
\end{corollary}
\begin{proof}
Since $\overline{\rho}$ is not dihedral there cannot be any CM point on the family $\theta\colon\T_h\to\I^\circ$.
By Theorem \ref{comparison} we deduce that $\fl$ has no nontrivial prime factor, hence it is trivial.
\end{proof}
\begin{remark}
Theorem \ref{comparison} gives another proof of Proposition \ref{finiteCM}. Indeed the CM points of a family $\theta\colon\T_h\to\I^\circ$ correspond to the prime factors of its Galois level, which are finite in number.
\end{remark}
We also give a partial result about the comparison of the exponents of each prime factor in $\fc_1$ and $\fl$.
This is an analogous of what is proved in \cite[Th. 8.6]{hida} for the ordinary case; our proof also relies on the strategy there.
For every prime $P$ of $\I_0[P_1^{-1}]$ we denote by $\fc_1^P$ and $\fl^P$ the $P$-primary components respectively of $\fc_1$ and $\fl$.
\begin{theorem}\label{exponents}
Suppose that $\overline{\rho}$ is not induced by a character of $G_F$ for a real quadratic field $F/\Q$.
We have $(\fc_1^P)^2\subset\fl^P\subset\fc_1^P$.
\end{theorem}
\begin{proof}
The inclusion $\fl^P\subset\fc_1^P$ is proved in the same way as the first inclusion of Theorem \ref{comparison}.
We show that the inclusion $(\fc_1^P)^2\subset\fl^P$ holds.
If $\fc_1^P$ is trivial this reduces to Theorem \ref{comparison}, so we can suppose that $P$ is a factor of $\fc_1$.
Let $Q$ denote any prime of $\I[P_1^{-1}]$ above $P$. Let $\fc_1^Q$ be a $Q$-primary ideal of $\I[P_1^{-1}]$ satisfying $\fc_1^Q\cap\I_0[P_1^{-1}]=\fc_1^P$.
Since $P$ divides $\fc_1$, $Q$ is a CM point, so we have an isomorphism $\rho_P\cong\Ind_F^\Q\psi$ for an imaginary quadratic field $F/\Q$ and a character $\psi\colon G_F\to\C_p^\times$.
Choose any $r<r_h$.
Consider the $\kappa_P$-vector space $\fs_{\fc_1^P}=\fH_r\cap\fc_1^P\cdot\fsl_2(\I_{0,r})/\fH_r\cap\fc_1^PP\cdot\fsl_2(\I_{0,r})$.
We see it as a subspace of $\fsl_2(\fc_1^P/\fc_1^PP)\cong\fsl_2(\kappa_{P})$. By the same argument as in the proof of Theorem \ref{comparison}, $\fs_{\fc_1^P}$ is stable under the adjoint action $\Ad(\rho_{\fc_1^QQ})\colon G_\Q\to\Aut(\fsl_2(\kappa_P))$.
Let $\chi_{F/\Q}\colon G_\Q\to\C_p^\times$ be the quadratic character defined by the extension $F/\Q$.
Let $\varepsilon\in G_\Q$ be an element projecting to the generator of $\Gal(F/\Q)$. Let $\psi^\varepsilon\colon G_F\to\C_p^\times$ be given by $\psi^\varepsilon(\tau)=\psi(\varepsilon\tau\varepsilon^{-1})$. Set $\psi^-=\psi/\psi^\varepsilon$.
Since $\rho_Q\cong\Ind_F^\Q\psi$, we have a decomposition $\Ad(\rho_Q)\cong\chi_{F/\Q}\oplus\Ind_F^\Q\psi^-$, where the two factors are irreducible.
Now we have three possibilities for the Galois isomorphism class of $\fs_{\fc_1^P}$: it is either that of $\Ad(\rho_P)$ or that of one of the two irreducible factors.
If $\fs_{\fc_1^P}\cong\Ad(\rho_Q)$, then as $\kappa_P$-vector spaces $\fs_{\fc_1^P}=\fsl_2(\kappa_P)$.
We proceed as in the proof of Theorem \ref{comparison} to obtain $\fs_{\fc_1^P}=\fsl_2(\kappa_P)$.
By Nakayama's lemma $\fH_r\supset\fc_1^P\cdot\fsl_2(\B_{r})$.
This implies $\fc_1^P\subset\fl^P$, hence $\fc_1^P=\fl^P$ in this case.
If $\fs_{\fc_1^P}$ is one-dimensional then we proceed as in the proof of Theorem \ref{comparison} to show that $\rho_{\fc_1^QQ}\colon G_\Q\to\GL_2(\I_r[P_1^{-1}]/\fc_1^QP\I_r[P_1^{-1}])$ is induced by a character $\psi_{\fc_1^QQ}\colon G_F\to\C_p^\times$. In particular the image of $\rho_{\fc_1^PP}\colon H\to\GL_2(\I_{0,r}[P_1^{-1}]/\fc_1^PP\I_{0,r})$ is small.
This is a contradiction, since $\fc_1^P$ is the $P$-primary component of $\fc_1$, hence it is the smallest $P$-primary ideal $\fA$ of $\I_{0,r}[P_1^{-1}]$ such that the image of $\rho_\fA\colon G_\Q\to\GL_2(\I_r[P_1^{-1}]/\fA\I_r[P_1^{-1}])$ is small.
Finally, suppose then that $\fs_{\fc_1^P}\cong\Ind_F^\Q\psi^-$.
Let $d=\diag(d_1,d_2)\in\rho(G_\Q)$ be the image of a $\Z_p$-regular element.
Since $d_1$ and $d_2$ are nontrivial modulo the maximal ideal of $\I_0^\circ$, the image of $d$ modulo $\fc_1^QQ$ is a nontrivial diagonal element $d_{\fc_1^QQ}=\diag(d_{1,\fc_1^QQ},d_{2,\fc_1^QQ})\in\rho_{\fc_1^QQ}(G_\Q)$.
We decomposte $\fs_{\fc_1^P}$ in eigenspaces for the adjoint action of $d_{\fc_1^QQ}$: we write $\fs_{\fc_1^P}=\fs_{\fc_1^P}[a]\oplus\fs_{\fc_1^P}[1]\oplus\fs_{\fc_1^P}[a^{-1}]$, where $a=d_{1,\fc_1^QQ}/d_{2,\fc_1^QQ}$.
Now $\fs_{\fc_1^P}[1]$ is contained in the diagonal torus, on which the adjoint action of $G_\Q$ is given by the character $\chi_{F/\Q}$.
Since $\chi_{F/\Q}$ does not appear as a factor of $\fs_{\fc_1^P}$, we must have $\fs_{\fc_1^P}[1]=0$.
This implies that $\fs_{\fc_1^P}[a]\neq 0$ and $\fs_{\fc_1^P}[a^{-1}]\neq 0$.
Since $\fs_{\fc_1^P}[a]=\fs_{\fc_1^P}\cap\fu^+(\kappa_P)$ and $\fs_{\fc_1^P}[a^{-1}]=\fs_{\fc_1^P}\cap\fu^-(\kappa_P)$, we deduce that $\fs_{\fc_1^P}$ contains nontrivial upper and lower nilpotent elements $\overline{u^+}$ and $\overline{u^-}$.
Then $\overline{u^+}$ and $\overline{u^-}$ are the images of some elements $u^+$ and $u^-$ of $\fH_r\cap\fc_1^P\cdot\fsl_2(\I_{0,r}[P_1^{-1}])$ nontrivial modulo $\fc_1^PP$.
The Lie bracket $t=[u^+,u^-]$ is an element of $\fH_r\cap\ft(\I_{0,r}[P_1^{-1}])$ (where $\ft$ denotes the diagonal torus) and it is nontrivial modulo $(\fc_1^P)^2P$.
Hence the $\kappa_P$-vector space $\fs_{(\fc_1^P)^2}=\fH_r\cap(\fc_1^P)^2\cdot\fsl_2(\I_{0,r,\C_p}[P_1^{-1}])/\fH_r\cap(\fc_1^P)^2P\cdot\fsl_2(\I_{0,r,\C_p}[P_1^{-1}])$ contains nontrivial diagonal, upper nilpotent and lower nilpotent elements, so it is three-dimensional.
By Nakayama's lemma we conclude that $\fH_r\supset(\fc_1^P)^2\cdot\fsl_2(\I_{0,r}[P_1^{-1}])$, so $(\fc_1^P)^2\subset\fl^P$.
\end{proof}
\bigskip
|
1,314,259,995,833 | arxiv | \section*{Introduction}
This paper describes a third-generation proof of the
slope filtration theorem for Frobenius modules over the Robba ring
(Theorem~\ref{T:slope filt} herein).
This proof is more expedient than what one finds
in our original paper \cite{kedlaya-local} or its sequel
\cite{kedlaya-slope}.
In addition, we
generalize the slope filtration theorem by allowing for ring endomorphisms
which do not act as Frobenius lifts on scalars, only on the series
variable. This is intended as a prelude to a theory of Frobenius
modules in families; we will not develop such a theory here, but
see the next section for reasons one might want to do so, from the
realm of $p$-adic Hodge theory.
(Note that \cite{kedlaya-slope} itself generalizes \cite{kedlaya-local} in
a different direction, replacing the power series rings by somewhat more
general
objects; we do not treat that generalization here.)
For
an alternate perspective on this theorem and some related results in
$p$-adic differential equations and $p$-adic Hodge theory,
we also recommend Colmez's Bourbaki notes \cite{colmez-bourbaki}.
\subsection{Context}
The slope filtration theorem
\cite[Theorem~6.10]{kedlaya-local} (also exposed in \cite{kedlaya-slope})
gives a partial classification of
Frobenius-semilinear transformation on finite free
modules over the Robba ring (a certain ring of univariate formal Laurent
series with $p$-adic coefficients).
It is loosely analogous to the
eigenspace decomposition of a linear transformation in ordinary
linear algebra; it is also closely related to Manin's classification
of rational Dieudonn\'e modules.
The slope filtration theorem was originally introduced in the context of
Berthelot's rigid
cohomology, a $p$-adic Weil cohomology
for varieties in characteristic $p$.
There, one obtains a analogue of the $\ell$-adic local monodromy theorem,
originally conjectured by Crew \cite{crew};
this analogue can be used to establish
various structural results such as finiteness of cohomology
\cite{kedlaya-finite} and purity in the sense of Deligne \cite{kedlaya-weilii}.
The effect of the slope
filtration theorem on $p$-adic Hodge theory has perhaps
been even more acute: it enables one
to study $p$-adic Galois representations via their
associated $(\phi,\Gamma)$-modules over the Robba ring. This point of view
has been put forth chiefly by Berger with striking consequences:
he has proved Fontaine's conjecture that de Rham representations
are potentially semistable \cite{berger-cst}, and given
an alternate proof of the Colmez-Fontaine theorem
on admissibility of filtered $(\phi,N)$-modules \cite{berger-weak}.
(A useful variant of the latter argument has been given by Kisin
\cite{kisin}.)
More recently Colmez \cite{colmez-tri} used this viewpoint to define a class
of \emph{trianguline representations} of a $p$-adic Galois group;
these play an important role in the $p$-adic
local Langlands correspondence for $\GL_2(\mathbb{Q}_p)$ \cite{colmez-local}.
The trianguline are also important in the theory of $p$-adic modular forms, as
most local Galois representations attached to overconvergent $p$-adic
modular forms (namely, those of noncritical slope)
are trianguline.
The $p$-adic local Langlands correspondence in turn
has touched off a flurry of activity, which this
introduction is not the right place to summarize; we merely note
the resolution of Serre's conjecture by
Khare-Wintenberger \cite{kw, kw2},
and progress on the
Fontaine-Mazur conjecture by Kisin \cite{kisin-fm}
and Emerton (in preparation).
In both rigid cohomology and $p$-adic Hodge theory,
one is led to study Frobenius modules in families,
i.e., over the Robba ring with coefficients not in a $p$-adic field but
in, say, an affinoid algebra.
In either situation, the first step to studying
Frobenius modules in families is to pass from a family to a generic point,
which on rings amounts to replacing an integral affinoid algebra with a
complete field containing it.
In the rigid cohomology version of this
argument, the resulting field is itself acted on by Frobenius, so the
slope filtration theorem as
presented in \cite{kedlaya-local, kedlaya-slope}
is immediately applicable; indeed, the key technique in
\cite{kedlaya-finite} is to extend the application of the local monodromy
theorem on the generic point to a large enough subspace of the base space.
However, in the
$p$-adic Hodge theory version, one might like to allow ``Frobenius'' to
act in some fashion on the base of the family other than simply a lift of
the $p$-power map; in fact, one natural situation is where the base is not
moved at all.
One goal of this paper, and in fact the principal reason for its existence,
is to generalize the slope filtration theorem to
modules over the Robba ring with an action of a ``relative Frobenius'', which
may do whatever one wishes to coefficients as long as it acts like a
Frobenius lift on the series parameter. We hope this will lead to some
study of $p$-adic Hodge theory in families; some of the corresponding
analysis in equal characteristics has been initiated by Hartl \cite{hartl},
using an equal-characteristic analogue of the slope filtration theorem
based on the work of Hartl and Pink \cite{hartl-pink}.
In mixed characteristics,
Hartl \cite{hartl2} has set up part of a corresponding theory,
which addresses a conjecture of Rapoport and Zink \cite{rapoport-zink}
from their work on period spaces for $p$-divisible groups; results are
presently quite fragmentary, but a good theory of $(\phi, \Gamma)$-modules
in families may help. Another potential application would be to
analysis of the local geometry of the Coleman-Mazur eigencurve
\cite{coleman-mazur}, which parametrizes the Galois representations
attached to certain $p$-adic modular forms, or of higher-dimensional
``eigenvarieties'' associated to automorphic representations on
groups besides $\GL_2$. An initial step in this direction has already
been taken by Bella\"\i che-Chenevier \cite{bellaiche-chenevier},
who study deformations of trianguline representations; however, this
involves only a zero-dimensional base, so they can already apply the
usual slope filtration theory after
a restriction of scalars. For other questions, e.g., properness, one would
want to consider positive-dimensional bases like a punctured disc.
In this direction,
Berger and Colmez have introduced
a theory of \'etale $(\phi, \Gamma)$-modules associated
to $p$-adic Galois representations in families
\cite{berger-colmez},
which relativizes some of the results of Cherbonnier-Colmez
\cite{cherbonnier-colmez} and
Berger \cite{berger-weak} for a single $p$-adic Galois representation.
\subsection{About the results}
For the sake of introduction, we give here a very brief description of
what the original slope filtration theorem says, how the main result of
this paper extends it, and what novelties in the argument are
introduced in this paper. Start with a complete
discretely valued field $K$ of mixed characteristics $(0,p)$.
Let $\mathcal{R}$ be the ring of formal Laurent series $\sum_{n \in \mathbb{Z}} c_n u^n$
convergent on some annulus with outer radius 1 (but whose inner radius may
depend on which series is being considered). Let $\phi_K: K \to K$ be
an endomorphism lifting the absolute $q$-power Frobenius
on the residue field
of $K$, for some power $q$ of $p$,
and define a map $\phi: \mathcal{R} \to \mathcal{R}$ by the formula
$\phi(\sum c_n u^n) = \sum \phi_K(c_n) \phi(u)^n$,
where $\phi(u) - u^q$ has all coefficients of norm less than 1.
Let $M$ be a finite free $\mathcal{R}$-module equipped with a
$\phi$-semilinear map $F: M \to M$ which takes any basis of $M$ to another basis
of $M$ (it is enough to check for a single basis). Then
\cite[Theorem~6.10]{kedlaya-slope} asserts that
$M$ admits an exhaustive
filtration whose successive quotients are each pure of
some slope (i.e., some power of $F$ times some scalar acts on some
basis via an invertible matrix over the subring of $\mathcal{R}$ of series with
\emph{integral} coefficients),
and the slopes increase as you go up the filtration; moreover,
those requirements uniquely characterize the filtration.
As noted earlier,
the slope filtration should be thought of as analogous to what one might get
from a linear transformation over $K$ by grouping eigenspaces, interpreting
the slope of an eigenspace as the valuation of its eigenvalue. One can
in fact deduce
an analogous such result for semilinear transformations over $K$,
which also follows from the Dieudonn\'e-Manin classification theorem.
One might then expect that the slope filtration can be generalized so
as to allow \emph{any} isometric
action on $K$, not just a Frobenius lift;
that is what is established in this paper
(Theorem~\ref{T:slope filt}).
As promised earlier in this introduction, one happy side effect of this
generalization is the introduction of
some technical simplifications.
We give a development of the theory of slopes which does not
depend on already having established the Dieudonn\'e-Manin-style
classification; this follows up on a suggestion made in
\cite{kedlaya-slope}. We give a much simplified version of the descent
argument that deduces the filtration theorem from the DM classification,
based on the idea of replacing the Galois descent used previously
with faithfully flat descent; this avoids the use of comparison between
generic and special Newton polygons, and of some intricate approximation
arguments. (In particular, there is no longer any need to deal with
finite extensions of the Robba ring, which allows for some notational
and expository simplifications.)
That substitution creates some flexibility in what we may take
as the ``extended Robba ring'' for the DM classification; here we use
a ring made from generalized power series, some of whose properties are
a bit more transparent than for the corresponding ``big rings'' in
\cite{kedlaya-local} and \cite{kedlaya-slope}.
\subsection{Structure of the paper}
The structure of this paper is a bit unusual, as we have attempted
to make the paper more friendly to the novice reader by fronting some of the
key assertions and pushing back more technical aspects. (This
assertion applies both to the paper as a whole, and to
Sections~\ref{sec:dm} and~\ref{sec:descend} individually.)
The consequence is that
the logical structure is a bit loopy: results are stated, and sometimes
used, before having been proved. However, we hope that it is not too hard
to see that there are indeed no vicious circles in the reasoning.
In Section~\ref{sec:filt thm}, we introduce the Robba ring, the category
of $\phi$-modules, the notions of degree and slope, the subcategories
of pure $\phi$-modules of various slopes, and the statement of the
filtration theorem.
In Section~\ref{sec:dm}, we introduce an extended Robba ring (whose elements
are modeled on Hahn-Mal'cev-Neumann generalized power series rather than
ordinary power series), state a classification theorem for $\phi$-modules
over the extended Robba ring, then perform the calculations required to prove
this theorem.
In Section~\ref{sec:descend}, we deduce the slope filtration theorem from
the classification theorem over the extended Robba ring. The key tool here
is an invocation of faithfully flat descent for modules.
\subsection*{Acknowledgments}
Thanks to Laurent Berger for the original suggestion to consider
relative Frobenius and for subsequent
discussions, to Lucia di Vizio for providing the reference to
Praagman's work, and to Peter Schneider for additional comments.
The author was supported by NSF grant DMS-0400727,
NSF CAREER grant DMS-0545904, and a Sloan Research Fellowship.
\section{Statement of the filtration theorem}
\label{sec:filt thm}
\subsection{The Robba ring}
\begin{defn} \label{D:initial}
Let $K$ be a field complete for a discrete valuation,
with residue field $k$; let $\mathfrak{o}_K$ denote the
valuation subring of $K$ and let $\mathfrak{m}_K$ denote the maximal ideal of
$\mathfrak{o}_K$.
(We need not make any restriction on the characteristics of $K, k$.)
Write $|\cdot|$ for some fixed norm corresponding
to the valuation (the normalization does not matter).
For $r>0$, let $\mathcal{R}^r$ be the ring of rigid analytic functions on the
annulus $e^{-r} \leq |t| < 1$ (these are just Laurent series in
the variable $t$ convergent on this
region), and let $\mathcal{R}$ be the union of the $\mathcal{R}^r$. The ring $\mathcal{R}$
is called the \emph{Robba ring} over $K$.
It follows from the work of Lazard
\cite{lazard}
that $\mathcal{R}$ is a \emph{B\'ezout domain},
that is, an integral domain in which every finitely generated ideal is principal.
\end{defn}
\begin{remark}
Any B\'ezout domain $R$ enjoys a number of nice properties generalizing
properties of principal ideal domains,
including the following.
Some of these are actually properties of \emph{Pr\"ufer domains}, in which
every finitely generated ideal is projective; these generalize Dedekind
domains to the non-noetherian setting.
\begin{itemize}
\item
Any finite locally free $R$-module is free
\cite[Proposition~2.5]{kedlaya-local}.
\item
Any torsion-free $R$-module is flat; this holds for any Pr\"ufer domain
\cite[VII Proposition~4.2]{cartan-eilenberg}.
\item
Any finitely presented projective $R$-module is free
\cite[Proposition~4.8]{crew}.
\item
If $M$ is a finite free $R$-module and $N$ is a submodule of $M$ which is
saturated,
i.e., $N = M \cap (N \otimes_R \Frac R)$, then $N$ and $M/N$ are both free
\cite[Proposition~4.8]{crew}, \cite[Lemma~2.4]{kedlaya-local}.
\end{itemize}
\end{remark}
\begin{defn}
Let $\mathcal{R}^{\inte}$ be the subring of $\mathcal{R}$ consisting of series with
coefficients in $\mathfrak{o}_K$; this ring is a discrete valuation ring with
residue field $k ((t ))$, which is not complete but is
henselian \cite[Lemma~3.9]{kedlaya-local}.
Let $\mathcal{R}^{\bd}$ be the subring of $\mathcal{R}$ consisting
of series with bounded coefficients; it is the fraction field
of $\mathcal{R}^{\inte}$.
\end{defn}
\begin{remark} \label{R:bounded by 1}
Note that for $x \in \mathcal{R}$, one has
$x \in \mathcal{R}^{\inte}$ if and only if there exists an integer $n$
such that the function $t^n x$ is bounded by 1 on some annulus
$e^{-r} \leq |t| < 1$.
\end{remark}
\begin{remark} \label{R:same units}
Lazard's work \cite{lazard}
includes a theory of Newton polygons for elements of $\mathcal{R}$,
using which one can read off numerous structural properties. One key
example is that
the units in $\mathcal{R}$ are precisely the nonzero elements of $\mathcal{R}^{\bd}$
\cite[Corollary~3.23]{kedlaya-local}.
\end{remark}
\begin{remark}
One can also define the Robba ring even if the valuation on $K$ is not
discrete, but its properties are very different. For instance,
$\mathcal{R}^{\bd}$ is no longer the fraction field of $\mathcal{R}^{\inte}$.
This makes even the formulation of a slope theory over such $K$,
let alone any proofs, somewhat more delicate than the approach we take here.
\end{remark}
\subsection{Frobenius lifts on the Robba ring}
\begin{defn}
Fix an integer $q>1$. (To see why we forbid $q=1$, see
Remark~\ref{R:no identity}.)
A \emph{relative ($q$-power)
Frobenius lift} on the Robba ring is a homomorphism
$\phi: \mathcal{R} \to \mathcal{R}$ of the form $\sum_i c_i t^i \mapsto \sum_i \phi_K(c_i)
u^{i}$, where $\phi_K$ is an isometric field
endomorphism of $K$ and $u \in \mathcal{R}^{\inte}$ is such that $u - t^q$ is in the
maximal ideal of $\mathcal{R}^{\inte}$. If $k$ has characteristic $p>0$ and
$q$ is a power of $p$,
we define an \emph{absolute ($q$-power)
Frobenius lift} as a
relative Frobenius lift in which $\phi_K$ is itself a $q$-power
Frobenius lift.
\end{defn}
\begin{remark}
The treatments in \cite{kedlaya-local, kedlaya-slope} only allow
absolute Frobenius lifts, and the approaches do not carry over easily to the
general case because of the use of Galois descent at some key moments.
See the introduction for discussion of why one needs the relative case.
\end{remark}
\begin{defn}
For $r > 0$, let $|\cdot|_r$ denote the supremum norm on the
circle $|t| = e^{-r}$, as applied to elements of $\mathcal{R}^r$; one
easily verifies that
\[
\left| \sum_{i \in \mathbb{Z}} c_i t^i \right|_r = \sup_i \{|c_i| e^{-ri}\}.
\]
We extend the definition to vectors by taking the maximum over entries.
\end{defn}
\begin{remark} \label{R:entire}
Note that for $f$ analytic on the entire open unit disc (i.e., represented
by an ordinary power series rather than a Laurent series),
we have $|f|_r \leq |f|_s$
whenever $0 < s \leq r$; in other words, the supremum of $f$ over the entire
disc $|t| \leq e^{-s}$ occurs on the circle $|t| = e^{-s}$.
In fancier language, the circle $|t| = e^{-s}$ is the \emph{Shilov boundary}
of the disc $|t| \leq e^{-s}$, as in \cite[Corollary~2.4.5]{berkovich}.
\end{remark}
\begin{remark} \label{R:annuli}
Let $\phi$ be a relative Frobenius lift; then for some $r_0 > 0$, we have
$|\phi(t)/t^q - 1|_{r_0/q} < 1$. It follows that for $r \in (0, r_0)$ and
$f \in \mathcal{R}^r$, $\phi(f) \in \mathcal{R}^{r/q}$ and
$|f|_r = |\phi(f)|_{r/q}$.
In geometric terms, $\phi$ induces a surjective map
from the annulus $e^{-r/q} < |t| < 1$ to the annulus $e^{-r} < |t| < 1$.
(Compare \cite[Lemma~3.7]{kedlaya-local}.)
\end{remark}
The following is both a typical example of how to make calculations on
Robba rings and a crucial ingredient in what follows.
\begin{prop} \label{P:h1 both ways}
Let $\phi$ be a relative Frobenius lift, and
let $A$ be an $n \times n$ matrix over $\mathcal{R}^{\inte}$. Then the map
$\mathbf{v} \mapsto \mathbf{v} - A \phi(\mathbf{v})$ on column vectors induces a bijection
on $(\mathcal{R}/\mathcal{R}^{\bd})^n$.
\end{prop}
\begin{proof}
The problem is unaffected if we replace $\mathbf{v}, A$ by $t^m \mathbf{v}, (t^m/\phi(t^m))
A$, so by Remark~\ref{R:bounded by 1}, we may
reduce to the case where the entries of $A$ are bounded by 1
on some annulus with outer
radius 1.
Choose $r_0$ as in Remark~\ref{R:annuli}.
To check injectivity, we must argue that if
$\mathbf{w} = \mathbf{v} - A \phi(\mathbf{v})$ is bounded,
then so is $\mathbf{v}$.
Choose
$r \in (0,r_0)$ such that $A, \mathbf{w}, \phi(\mathbf{v})$ have entries which are defined
on the annulus $e^{-r} \leq |t| < 1$, and the entries of $A$ are bounded
by 1 there.
Choose $c>0$ such that $|\mathbf{w}|_s \leq c$ for $0<s \leq r$, and such that
$|\phi(\mathbf{v})|_s \leq c$ for
$r/q \leq s \leq r$. (The latter is possible because every
analytic function on a closed annulus is bounded.)
Then $|\mathbf{v}|_s = |\mathbf{w} + A \phi(\mathbf{v})|_s \leq c$ for $r/q \leq s \leq r$,
so $|\phi(\mathbf{v})|_s \leq c$ for $r/q^2 \leq s \leq r/q$.
Repeating the argument, we see that $|\mathbf{v}|_s \leq c$ for $0 < s \leq r$,
proving the claim.
(Compare \cite[Lemma~3.3.3]{kedlaya-slope}.)
To check surjectivity, take $\mathbf{w} \in \mathcal{R}^n$.
Choose
$r \in (0,r_0)$ such that $A, \mathbf{w}$ have entries which are defined
on the annulus $e^{-r} \leq |t| < 1$, and the entries of $A$ are bounded by 1 there.
Define the sequence $\{\mathbf{w}_l\}_{l=0}^\infty$ as follows. Start with
$\mathbf{w}_0 = \mathbf{w}$. Given $\mathbf{w}_l$, write $\mathbf{w}_l = \sum_{i \in \mathbb{Z}} \mathbf{w}_{l,i} t^i$,
put $\mathbf{w}_l^+ = \sum_{i>0} \mathbf{w}_{l,i} t^i$ and $\mathbf{w}_l^- = \mathbf{w}_l - \mathbf{w}_l^+$,
and put $\mathbf{w}_{l+1} = A \phi(\mathbf{w}_l^+)$.
Since the entries of $t^{-1} \mathbf{w}_l^+$ are analytic on the entire open unit
disc,
by Remark~\ref{R:entire} we have
\[
|\mathbf{w}_l^+|_{r} \leq e^{-r+r/q} |\mathbf{w}_l^+|_{r/q} \leq e^{-r+r/q} |\mathbf{w}_l|_{r/q};
\]
consequently, $|\mathbf{w}_{l+1}|_{r/q} \leq e^{-r+r/q} |\mathbf{w}_l|_{r/q}$.
Thus the sequence $\mathbf{w}_l^+$
converges to zero under $|\cdot|_{r/q}$, and hence also
under $|\cdot|_s$ for $s \geq r/q$ by Remark~\ref{R:entire}.
On the other hand, for $0 < s \leq r/q$,
applying Remark~\ref{R:entire} after substituting $t \mapsto t^{-1}$ gives
\[
|\mathbf{w}_l^-|_s \leq |\mathbf{w}_l^-|_{r/q} \leq |\mathbf{w}_l|_{r/q}.
\]
Now set $\mathbf{v} = \sum_{l=0}^\infty \mathbf{w}_l^+$; then $\mathbf{v}$ has entries analytic on
the closed disc of radius $e^{-r/q}$, and $\mathbf{w} - \mathbf{v} + A \phi(\mathbf{v}) = \sum_{l=0}^\infty
\mathbf{w}_l^-$ is bounded on $e^{-r/q} \leq |t| < 1$. Since $\phi(\mathbf{v})$ is
analytic on the closed disc of radius $e^{-r/q^2}$, we can write
$\mathbf{v} = \mathbf{w} + A \phi(\mathbf{v}) - \sum_{l=0}^\infty \mathbf{w}_l^-$
and thus extend $\mathbf{v}$ across the annulus
$e^{-r/q} \leq |t| \leq e^{-r/q^2}$; by induction, $\mathbf{v}$ extends to
the entire open unit disc.
This proves the desired surjectivity.
\end{proof}
One can also prove the following, as in \cite[Lemma~5.4.1]{kedlaya-slope}.
\begin{prop} \label{P:cherbonnier1}
Let $\mathcal{E}$ denote the $\mathfrak{m}_K$-adic completion of
$\mathcal{R}^{\bd}$.
Let $\phi$ be a relative Frobenius lift on $\mathcal{R}$, and
let $A$ be an $n \times n$ matrix over $\mathcal{R}^{\inte}$. If $\mathbf{v}
\in \mathcal{E}^n$ is a column vector such that
$A \mathbf{v} = \phi (\mathbf{v})$, then $\mathbf{v} \in (\mathcal{R}^{\bd})^n$.
\end{prop}
\begin{proof}
This will follow later from Proposition~\ref{P:cherbonnier2};
we will not use it in the interim.
\end{proof}
\begin{remark} \label{R:cherbonnier1}
In the case where $A$ is invertible,
Proposition~\ref{P:cherbonnier1}
was proved independently by Cherbonnier (unpublished,
but see \cite[Th\'eor\`eme~III.1.1]{cherbonnier-colmez})
and Tsuzuki \cite[Proposition~4.1.1]{tsuzuki-etale}. Tsuzuki's
underlying argument
can be used even when $A$ is not invertible; see
\cite[Proposition~2.2.2]{tsuzuki-etale}.
\end{remark}
\begin{remark}
It should be possible to carry everything in this paper over to the
case where one only assumes $\phi(t) = \sum_i c_i t^i$ such that
$c_q \in \mathfrak{o}_K^*$ and $c_i \in \mathfrak{m}_K$ for $i < q$.
(For instance, in the theory of $(\phi, \Gamma)$-modules, the composition
of the usual $\phi$ with any nontrivial $\gamma \in \Gamma$ would have this
property.) The proof of Proposition~\ref{P:h1 both ways} extends to this
setting, but the embedding of $\mathcal{R}$ into the extended Robba ring
$\tilde{\mathcal{R}}$ of Section~\ref{sec:dm} must be modified, as accordingly
must the projection construction of Section~\ref{sec:descend}.
\end{remark}
\subsection{$\phi$-modules}
\begin{defn}
Define a \emph{$\phi$-(ring/field)}
to be a ring/field $R$
equipped with an endomorphism $\phi$; we say $R$ is \emph{inversive}
if $\phi$ is bijective.
Define a \emph{(strict) $\phi$-module} over a $\phi$-ring $R$
to be a finite free $R$-module $M$
equipped with an isomorphism $\phi^* M \to M$, which we also think of as a
semilinear $\phi$-action on $M$; the semilinearity means that for $r \in R$
and $m \in M$, $\phi(rm) = \phi(r) \phi(m)$.
Note that the category of $\phi$-modules admits tensor products,
symmetric and exterior
powers, and duals.
\end{defn}
\begin{remark}
The definition of $\phi$-module used here is somewhat more restrictive than
one sees in other contexts, hence the optional modifier ``strict''.
For instance, in some cases one allows
modules which are projective but not free, or worse.
In other cases, one allows the $\phi$-action to take kernel and cokernel in
some $\phi$-stable Serre category of $R$-modules; we will do this ourselves
shortly.
\end{remark}
\begin{remark}
It will be convenient for us to describe $\phi$-modules in terms of bases
and matrices. If $M$ is a $\phi$-module and $\mathbf{e}_1, \dots, \mathbf{e}_n$ is a
basis of $M$, we can completely describe the $\phi$-action on $M$ by specifying
the invertible $n \times n$ matrix $A$ which satisfies $\phi(\mathbf{e}_j) = \sum_i
A_{ij} \mathbf{e}_i$. Note that the semilinearity skews conjugation:
if $\mathbf{e}'_1, \dots, \mathbf{e}'_n$ is another basis and
the change of basis matrix $U$ is defined by $\mathbf{e}'_j = \sum_i U_{ij} \mathbf{e}_i$,
then the $\phi$-action on the new basis is via the matrix
$U^{-1} A \phi(U)$.
\end{remark}
It is also useful to think of
$\phi$-modules as modules for a twisted polynomial ring.
\begin{defn}
For $R$ a $\phi$-ring,
define the \emph{twisted polynomial ring} $R\{T\}$ to be the set of
finite formal sums $\sum_{i=0}^\infty a_i T^i$ with $a_i \in R$,
equipped with the noncommutative
ring structure in which $T a = \phi(a) T$ for $a \in R$.
If $R$ is a field, then all left ideals
of $R\{T\}$ are principal, by the division algorithm \cite[Theorem~6]{ore}.
If $R$ is inversive,
one may similarly define a \emph{twisted Laurent polynomial ring}
$R\{T^{\pm}\}$.
\end{defn}
\begin{remark}
In general, a $\phi$-module over $R$ can be interpreted as a
left $R\{T\}$-module which is finite free over $R$, but one must remember
the condition that $\phi$ carries some basis to another basis.
On the other hand, if $R$ is inversive,
then the data of a $\phi$-module over $R$ is equivalent to the data
of a left $R\{T^{\pm}\}$-module which is finite free over $R$.
If $R$ is an inversive $\phi$-field, then
irreducible $\phi$-modules over $R$ all have the form
$R\{T^{\pm}\}/R\{T^{\pm}\}P$ for some irreducible twisted polynomial $P$.
\end{remark}
When talking about pure slopes, it will be helpful to switch from
working with $\phi$ to working with a power of $\phi$; the following
definition facilitates this switch.
\begin{defn} \label{D:pull push}
View $\phi$-modules as left modules for the twisted polynomial ring
$R\{T\}$. For $a$ a positive integer, define the
\emph{$a$-pushforward functor} $[a]_*$ from $\phi$-modules to $\phi^a$-modules
to be the restriction along the inclusion $R\{T^a\} \to R\{T\}$.
Define the \emph{$a$-pullback functor} $[a]^*$ from $\phi^a$-modules
to $\phi$-modules to be the extension of scalars functor
\[
M \mapsto R\{T\} \otimes_{R\{T^a\}} M.
\]
The following are easily verified (as in
\cite[\S 3.2]{kedlaya-slope}):
\begin{itemize}
\item
The functors $[a]^*$ and $[a]_*$ form an adjoint pair.
\item
The functors $[a]_*$ and $[a]^*$ are exact and commute with duals; consequently,
$[a]_*$ and $[a]^*$ also form an adjoint pair (i.e., in the other order).
\item
The functor $[a]_*$ commutes with tensor products over $R$ (but
$[a]^*$ does not).
\item
If $M$ is a $\phi$-module and $N$ is a $\phi^a$-module, then
$M \otimes [a]^* N \cong [a]^* ([a]_* M \otimes N)$.
\item
If $M$ is a $\phi$-module, then $\rank([a]_* M) = \rank(M)$.
\item
If $N$ is a $\phi^a$-module, then $\rank([a]^* N) = a \rank(N)$.
\item
If $N$ is a $\phi^a$-module, then
$[a]_* [a]^* N \cong N \oplus \phi^*(N) \oplus
\cdots \oplus (\phi^{a-1})^*(N)$.
\end{itemize}
\end{defn}
\begin{defn} \label{D:h0 h1}
For $M$ a $\phi$-module, put
\[
H^0(M) = \ker(\phi-1: M \to M), \qquad
H^1(M) = \coker(\phi-1: M \to M).
\]
One easily checks that in the category of $\phi$-modules over $R$,
\[
\Hom(M, N) \cong H^0(M^\vee \otimes N), \qquad
\Ext(M, N) \cong H^1(M^\vee \otimes N).
\]
Moreover, for $N$ a $\phi^a$-module, there are natural bijections
\[
H^i(N) \cong H^i([a]^* N) \qquad (i=0,1).
\]
\end{defn}
\begin{remark}
Beware that although the pullback/pushforward terminology was inspired
by a related construction in \cite{hartl-pink}, the two do not agree in
that context.
\end{remark}
\subsection{Degrees, slopes, and stability}
For the rest of this section, we will put ourselves in the following
situation. Note that Hypothesis~\ref{hypo:robba}
has a weak form and a strong form; we will
assume only the weak form unless otherwise specified.
(Thanks to Peter Schneider for suggesting this dichotomy.)
\begin{hypo} \label{hypo:robba}
Let $R^{\inte} \subseteq R^{\bd} \subseteq R$ be inclusions of B\'ezout domains
such that $R^* \subset R^{\bd}$.
Let $\phi$ be an endomorphism of $R$ acting also on $R^{\bd}$ and $R^{\inte}$.
Let $w: R^{\bd} \to \mathbb{Z} \cup \{+\infty\}$ be a $\phi$-equivariant
valuation such that $w(R^*) = \mathbb{Z}$ and
$R^{\inte} = \{r \in R^{\bd}: w(r) \geq 0\}$.
Suppose in addition that for any $n \times n$ matrix $A$ over $R^{\inte}$,
the map $\mathbf{v} \mapsto \mathbf{v} - A \phi(\mathbf{v})$ on column vectors induces an injection
(weak form) or bijection (strong form)
on $(R/R^{\bd})^n$. Note that the analogous hypothesis for $\phi^a$ also holds,
since one can identify the kernel and cokernel of $\mathbf{v} \mapsto \mathbf{v} - A
\phi^a(\mathbf{v})$ on $(R/R^{\bd})^n$ with the kernel and cokernel of
\[
(\mathbf{v}_0, \mathbf{v}_1, \dots, \mathbf{v}_{a-1}) \mapsto (\mathbf{v}_0 - A \phi(\mathbf{v}_{a-1}),
\mathbf{v}_1 - \phi(\mathbf{v}_0), \dots, \mathbf{v}_{a-1} - \phi(\mathbf{v}_{a-2}))
\]
on $(R/R^{\bd})^{na}$. (Compare the last remark in Definition~\ref{D:h0 h1}.)
\end{hypo}
\begin{example} \label{exa:Robba}
For our purposes, the principal example of strong
Hypothesis~\ref{hypo:robba} is
as follows. We take $R, R^{\bd}, R^{\inte} = \mathcal{R}, \mathcal{R}^{\bd}, \mathcal{R}^{\inte}$
to be the Robba ring and variants over $K$; note that $\mathcal{R}^{\bd} =
\mathcal{R}^* \cup \{0\}$.
We take $\phi$ to be a relative Frobenius lift, and $w$ to be
the valuation on $\mathcal{R}^{\bd}$
for which $\mathcal{R}^{\inte}$ is the valuation subring.
The last condition in strong
Hypothesis~\ref{hypo:robba} holds by virtue of
Proposition~\ref{P:h1 both ways}.
We will construct a variation of this example, the extended Robba ring $\tilde{\mathcal{R}}$,
in Section~\ref{sec:dm}; using the axiomatic approach avoids some repetition.
\end{example}
\begin{example} \label{ex:hartl-pink}
Besides the Robba ring, additional examples of strong Hypothesis~\ref{hypo:robba}
are also possible.
Here is one from the work of Hartl and Pink \cite{hartl-pink}:
take $\mathbb{C}$ to be the
completed algebraic closure of a local field of equal characteristic $p$,
$R$ to be the Laurent series over $\mathbb{C}$
convergent on the punctured open unit disc,
$R^{\bd}$ to be the series which are meromorphic at zero,
$\phi$ to be the map $\sum c_i t^i \mapsto \sum c_i^q t^i$ for $q$
a power of $p$, and
$w$ to be the order of vanishing at 0.
See Remark~\ref{R:hartl-pink}
and Question~\ref{Q:q-analogue} for further discussion around this example.
\end{example}
\begin{defn}
For $M$ a $\phi$-module over $R$ of rank $n$, the top
exterior power $\wedge^n M$ has rank 1 over $R$; let $\mathbf{v}$ be a generator,
and write $\phi(\mathbf{v}) = r\mathbf{v}$ for some $r \in R^*$. Define
the \emph{degree} of $M$ by setting $\deg(M) = w(r)$; note that this does not
depend on the choice of the generator
by virtue of the $\phi$-equivariance of $w$.
If $M$ is nonzero, define the \emph{slope} of $M$ by setting
$\mu(M) = \deg(M)/\rank(M)$.
\end{defn}
\begin{remark} \label{R:pull push}
Keeping in mind that degree is analogous to the valuation of the determinant (of
a linear transformation on a finite dimensional vector space over a valued field),
the following formal properties are easily verified
(as in \cite[\S 3.4]{kedlaya-slope}).
\begin{itemize}
\item
If $0 \to M_1 \to M \to M_2 \to 0$ is exact, then
$\deg(M) = \deg(M_1) + \deg(M_2)$; hence $\mu(M)$ is a weighted average
of $\mu(M_1)$ and $\mu(M_2)$.
\item
We have $\mu(M_1 \otimes M_2) = \mu(M_1) + \mu(M_2)$.
\item
We have $\mu(\wedge^i M) = i \mu(M)$.
\item
We have $\deg(M^\vee) = -\deg(M)$ and $\mu(M^\vee) = - \mu(M)$.
\item
If $M$ is a $\phi$-module, then $\mu([a]_* M) = a \mu(M)$.
\item
If $N$ is a $\phi^a$-module, then $\mu([a]^* N) = a^{-1} \mu(N)$.
\end{itemize}
\end{remark}
By analogy with the theory of vector bundles, we make the following definition.
\begin{defn}
We say a $\phi$-module $M$ is \emph{(module-)semistable}
if for any nontrivial $\phi$-submodule $N$, we have $\mu(N) \geq \mu(M)$.
We say $M$ is \emph{(module-)stable}
if for any proper nontrivial $\phi$-submodule $N$, we have $\mu(N) > \mu(M)$.
Note that both properties are preserved under \emph{twisting}
(tensoring with a rank 1 module).
\end{defn}
\begin{remark}
In \cite{kedlaya-slope}, the terms ``stable'' and ``semistable'' were
used without the ``module'' modifier; here we will usually retain the
modifier in statements and drop it in proofs.
The modifier is meant to emphasize the difference between this notion of
semistability and the concept of a
``semistable $(\phi, \Gamma)$-module'' in the sense of $p$-adic Hodge theory,
meaning one which appears to come from a semistable Galois representation.
In the end, over the Robba ring the term ``module-semistable'' will be shown to
be synonymous with ``pure'', so the terminological overload will cease to
be a problem.
\end{remark}
\begin{remark}
Those familiar with stability of vector bundles (or with
\cite{hartl-pink}) will notice that our definitions
differ from the usual convention
by an overall minus sign. The sign convention here (which is also the
one used in \cite{kedlaya-local, kedlaya-slope}) seems to be
more consistent with usage in the theory of crystalline cohomology.
\end{remark}
\begin{prop} \label{P:rank 1 stable}
Any $\phi$-module of rank $1$ is module-stable.
\end{prop}
\begin{proof}
This is a consequence of the assumptions built into weak
Hypothesis~\ref{hypo:robba}.
Namely, by twisting, it suffices to show that the trivial $\phi$-module
$M \cong R$ is
stable. If $N$ is a $\phi$-submodule of $M$,
we may write $N = Rx$ for some $x \in M$
such that $\lambda = \phi(x)/x \in R^*$, and by definition
$\mu(N) = w(\lambda)$. If $\mu(N) \leq 0$, then
$x - \lambda^{-1} \phi(x) = 0$ implies $x \in R^{\bd}$ by
weak Hypothesis~\ref{hypo:robba}; hence $N = M$ and $\mu(N) = w(\phi(x)) - w(x)
= 0$. In other words, $\mu(N) > 0$ unless
$N = M$, as desired.
\end{proof}
\begin{cor} \label{C:same rank}
If $N \subseteq M$ is an inclusion of $\phi$-modules of the same rank, then
$\mu(N) \geq \mu(M)$, with equality if and only if $N = M$.
\end{cor}
\begin{proof}
Put $n = \rank M$
and apply Proposition~\ref{P:rank 1 stable} to the inclusion $\wedge^n N
\subseteq \wedge^n M$.
\end{proof}
\begin{lemma} \label{L:least slope}
Let $M$ be a $\phi$-module over $R$. Then the slopes of nonzero
$\phi$-submodules of $M$
are bounded below.
\end{lemma}
\begin{proof}
We proceed by induction on $\rank(M)$. By Corollary~\ref{C:same rank}, the slopes
of $\phi$-submodules of $M$ of full rank are bounded below by $\mu(M)$.
If $M$ has no nontrivial $\phi$-submodules of lower rank,
then there is nothing more to check.
Otherwise, let $N$ be a saturated $\phi$-submodule of lower rank; then by
hypothesis, the slopes of nonzero $\phi$-submodules of both $N$ and $M/N$ are
bounded below. If now $P$ is any nonzero $\phi$-submodule of $M$, then the sequence
\[
0 \to N \cap P \to P \to P/(N \cap P) \to 0
\]
is exact. If both factors
are nonzero, we have $\mu(N \cap P) \geq \mu(N)$
and $\mu(P/(N \cap P)) \geq \mu(M/N)$,
and $\mu(P)$ is a weighted average of $\mu(N \cap P)$ and $\mu(P/(N \cap P))$, so
it is bounded below. If one factor vanishes, then $\mu(P)$ simply equals the slope
of the other factor, so the same conclusion holds.
\end{proof}
\begin{lemma} \label{L:least slope2}
Let $M$ be a nonzero $\phi$-module over $R$. Then there is a largest
$\phi$-submodule of $M$ of least slope, which is module-semistable.
\end{lemma}
\begin{proof}
The fact that there is a least slope $s$ holds by Lemma~\ref{L:least slope}
and the fact that the denominators of slopes are bounded above by the rank
of $M$; clearly any $\phi$-submodule of slope $s$ must be semistable.
If $N_1$ and $N_2$ are two such submodules,
then the kernel of the surjection $N_1 \oplus N_2
\to N_1 + N_2$ must have slope at least $s$, so $\mu(N_1 + N_2) \leq s$. On the other hand,
$\mu(N_1 + N_2) \geq s$ because $N_1 + N_2 \subseteq M$, so $\mu(N_1 + N_2) = s$.
Hence the $\phi$-submodules of $M$ of slope $s$ are closed under sum, yielding
the existence of a largest such submodule.
\end{proof}
\begin{cor} \label{C:semi push}
Let $M$ be a $\phi$-module over $R$. Then for any positive integer $a$,
$M$ is module-semistable if and only if $[a]_* M$ is module-semistable.
\end{cor}
\begin{proof}
If $[a]_* M$ is semistable, evidently $M$ is too. Conversely, if
$[a]_* M$ is not semistable, then its largest $\phi^a$-submodule of least slope
is a $\phi^a$-submodule $M_1$ of lower rank. By the uniqueness in
Lemma~\ref{L:least slope2}, $M_1$ must in fact be
preserved by $\phi$, so $M$ is not semistable either.
\end{proof}
\begin{defn}
Let $M$ be a $\phi$-module over $R$.
A \emph{module-semistable filtration} of $M$ is a
filtration $0 = M_0 \subset M_1 \subset \cdots \subset M_l = M$ by saturated
$\phi$-submodules such that each quotient $M_i/M_{i-1}$ is module-semistable. A
\emph{Harder-Narasimhan (HN) filtration} is a module-semistable filtration in which
\[
\mu(M_1/M_0) < \cdots < \mu(M_l/M_{l-1}).
\]
\end{defn}
\begin{prop}
Every $\phi$-module over $R$ admits a unique HN filtration, whose first step
is the submodule defined in Lemma~\ref{L:least slope2}.
\end{prop}
\begin{proof}
This is a formal consequence of Lemma~\ref{L:least slope2}; see
\cite[Proposition~4.2.5]{kedlaya-slope}.
\end{proof}
\begin{defn}
Define the \emph{slope multiset} of a module-semistable filtration of a
$\phi$-module of $M$ as the multiset in which each slope of a successive
quotient occurs with multiplicity equal to the rank of that quotient.
These assemble into the lower boundary of a convex region in
the $xy$-plane as follows: start at $(0,0)$, then take each slope $s$
in increasing order and append to the polygon a segment with slope $s$
and width equal to the multiplicity of $s$. The result is called the
\emph{slope polygon} of the filtration; for the HN filtration,
we call the result the \emph{HN polygon}.
\end{defn}
\begin{prop} \label{P:on or above}
The HN polygon lies on or above the slope
polygon of any module-semistable filtration, with the same endpoint.
\end{prop}
\begin{proof}
This is a formal consequence of the definition of an HN filtration: see
\cite[Proposition~3.5.4]{kedlaya-slope}.
\end{proof}
\begin{prop}
Let $M_1, M_2$ be $\phi$-modules over $R$ such that each slope of the
HN polygon of $M_1$ is less than each slope of the HN polygon of $M_2$.
Then $\Hom(M_1,M_2) = 0$.
\end{prop}
\begin{proof}
Choose $f \in \Hom(M_1,M_2)$.
Let $N_1$ be the first step in the HN filtration of $M_1$; then
either $f(N_1) = 0$, or $\mu(f(N_1)) \leq \mu(N_1)$. The latter is impossible
because $\mu(f(N_1))$ is no less than the least slope of $M_2$, whereas
$\mu(N_1)$ is no greater than the greatest slope of $M_1$. Hence
$f$ factors through $M_1/N_1$; repeating,
we obtain $f = 0$.
\end{proof}
\subsection{\'Etale $\phi$-modules}
\begin{defn}
A $\phi$-module $M$ over $R$ or $R^{\bd}$
is said to be \emph{\'etale} (or \emph{unit-root})
if it can be obtained by base extension from
a (strict) $\phi$-module over $R^{\inte}$; that is, $M$ must admit an
$R^{\inte}$-lattice $N$ such that
$\phi$ induces an isomorphism $\phi^* N \to N$. We call such an $N$ an \emph{\'etale
lattice} of $M$. Note that $N$ is not in general
unique; for instance, it may be rescaled. Note also that the dual of
an \'etale $\phi$-module is again \'etale.
\end{defn}
\begin{remark}
The term ``unit-root'' is standard in applications to crystalline cohomology, where
it refers to the process of extracting the unit roots (roots of valuation 0)
of a $p$-adic polynomial. By contrast,
the term ``\'etale'' is standard in applications to
$p$-adic Hodge theory.
\end{remark}
One of the basic results about \'etale $\phi$-modules is that in a certain
sense, they do not lose
information when base-changed from $R^{\bd}$ to $R$. This can be deduced from
a slightly more general result, which we already used once (to justify that
the Robba ring satisfies Hypothesis~\ref{hypo:robba}) and will use again shortly
(in the proof of Theorem~\ref{T:pure semi}).
\begin{defn}
Define an \emph{isogeny $\phi$-module} over $R^{\inte}$
to be a finite free $R^{\inte}$-module $M$ equipped with an injection
$\phi^* M \to M$ whose cokernel is killed by some power of a uniformizer of
$R^{\inte}$.
Such an object becomes a strict $\phi$-module upon tensoring with
$R^{\bd}$ or $R$.
\end{defn}
\begin{prop} \label{P:h0}
Let $M$ be an isogeny $\phi$-module over $R^{\inte}$.
Then the natural maps $H^i(M \otimes R^{\bd}) \to H^i(M \otimes R)$ for
$i=0$ (under weak Hypothesis~\ref{hypo:robba})
or $i=0,1$ (under strong Hypothesis~\ref{hypo:robba}) are bijective.
\end{prop}
\begin{proof}
This is an immediate consequence of the
final clause of Hypothesis~\ref{hypo:robba}.
\end{proof}
\begin{prop} \label{P:etale equiv}
The base change functor from \'etale $\phi$-modules over $R^{\bd}$
to \'etale $\phi$-modules over $R$ is an equivalence of categories.
\end{prop}
\begin{proof}
The essential surjectivity holds by definition, so we need only check
full faithfulness. That is, for any \'etale $\phi$-modules $M_1, M_2$
over $R^{\bd}$, we must check that the natural map
\[
H^0(M_1^\vee \otimes M_2) \to H^0(M_1^\vee \otimes M_2 \otimes R)
\]
is a bijection; this follows from Proposition~\ref{P:h0}.
\end{proof}
\begin{prop} \label{P:etale extend}
Let $0 \to M_1 \to M \to M_2 \to 0$ be a short exact sequence of $\phi$-modules
over $R$. If any two of $M_1, M_2, M$ are \'etale
(except
possibly $M_1,M_2$ in the case of weak Hypothesis~\ref{hypo:robba}),
then so is the third.
\end{prop}
\begin{proof}
First, suppose that $M$ and $M_2$ are \'etale.
By Proposition~\ref{P:etale equiv},
the $\phi$-modules $M, M_2$ and the morphism $M \to M_2$ all descend to
$R^{\bd}$. By Lemma~\ref{L:etale lattice} below,
we can then produce an \'etale lattice in $M_1$ by taking the
kernel of the map from an \'etale lattice of $M$ to $M_2$.
Next, suppose that $M$ and $M_1$ are \'etale. We then dualize to obtain a second
exact sequence in which $M^\vee$ and $M_1^\vee$ are \'etale. By the previous paragraph,
$M_2^\vee$ is then \'etale, as then is $M_2$.
Finally, suppose that $M_1$ and $M_2$ are \'etale
and that strong Hypothesis~\ref{hypo:robba} holds.
By applying Proposition~\ref{P:h0},
$M_1$, $M_2$, and the exact sequence $0 \to M_1 \to M \to M_2 \to 0$ all descend
to $R^{\bd}$; by rescaling appropriately, we can descend the sequence to
$R^{\inte}$. We can then produce an \'etale lattice in $M$ by lifting
an \'etale lattice from $M_2$, then adding an
\'etale lattice from $M_1$.
\end{proof}
\begin{lemma} \label{L:etale lattice}
Let $M$ be an \'etale $\phi$-module over $R^{\bd}$. Then any finitely generated
$\phi$-stable $R^{\inte}$-submodule of $M$ is a $\phi$-module over $R^{\inte}$.
\end{lemma}
\begin{proof}
Let $M_0$ be an \'etale lattice of $M$, and let $N$ be a finitely generated
$\phi$-stable $R^{\inte}$-submodule of $M$; by rescaling, we may assume
$N \subseteq M_0$. Then $N$ is already an isogeny $\phi$-module,
and it suffices to check that $\deg(N) = 0$; we may do this after
replacing $M$ by $\wedge^{\rank(N)} M$, i.e., we may assume $\rank(N) = 1$.
Let $\mathbf{e}_1, \dots,\mathbf{e}_n$ be a basis of $M_0$, let
$\mathbf{v} = \sum_{i=1}^n c_i \mathbf{e}_i$ be a generator of $N$,
and put $\phi(\mathbf{v}) = \sum_{i=1}^n d_i \mathbf{e}_i$. Then
$\deg(N) = \min_i \{w(d_i)\} - \min_i\{w(c_i)\}$, but this
difference is zero because $M_0$ is an \'etale lattice.
\end{proof}
We can also show that \'etale $\phi$-modules are module-semistable, but it will be convenient to do that more generally for pure $\phi$-modules in the
next subsection.
\subsection{Pure $\phi$-modules}
\begin{defn} \label{D:pure}
Let $M$ be a $\phi$-module over $R^{\bd}$ or $R$
of slope $s = c/d$, where $c,d$ are coprime integers
with $d>0$. We say $M$ is \emph{pure} (or \emph{isoclinic}, or sometimes
\emph{isocline}) of slope $s$
if for some $\phi$-module $N$ of rank $1$ and degree $-c$,
$([d]_* M)\otimes N$ is \'etale (the same then holds for any such $N$).
It will follow from Lemma~\ref{L:pure push} below that
it is equivalent to impose this condition for any one pair $c,d \in \mathbb{Z}$ with
$s = c/d$ and $d>0$.
Note that:
\begin{itemize}
\item
any $\phi$-module of rank 1 is pure;
\item
a $\phi$-module is pure of slope 0 if and only if it is \'etale;
\item
the dual of a pure $\phi$-module of slope $s$ is itself pure of slope $-s$.
\end{itemize}
\end{defn}
\begin{remark}
This definition is not that of
\cite[Definition~6.3.1]{kedlaya-slope}, but it is equivalent to
it by \cite[Proposition~6.3.5]{kedlaya-slope}. It has the advantage that it
can be stated without reference to any sort of Dieudonn\'e-Manin
classification; the downside is that one must expend a bit of effort to check
some natural-looking properties, as we do below.
\end{remark}
\begin{lemma} \label{L:pure push}
Let $M$ be a $\phi$-module over $R^{\bd}$ or $R$, and let $a$
be a positive integer. Then $M$ is pure of some slope $s$ if and only if
$[a]_* M$ is pure of slope $as$.
\end{lemma}
\begin{proof}
We first check the case where $s=0$. If $M$ is \'etale, then clearly
$[a]_* M$ is too. Conversely, if $[a]_* M$ is \'etale,
then $\phi$ induces isomorphisms $(\phi^{i+1})^* [a]_* M \to (\phi^{i})^* [a]_* M$
over $R$; by Proposition~\ref{P:etale equiv}, these isomorphisms
descend to $R^{\bd}$. That is, we may reduce to working over $R^{\bd}$.
In this case, let $N_0$ be an \'etale lattice of $[a]_* M$.
Let $N$ be the $R^{\inte}$-span of
$N_0, \phi(N_0), \dots, \phi^{a-1}(N_0)$; then $N$ is an \'etale
lattice of $M$.
Hence $M$ is \'etale.
In the general case, write $s = c/d$ in lowest terms, and put $b =
\gcd(a,d)$; then in lowest terms, $as = (ac/b)/(d/b)$.
Let $N$ be a $\phi^d$-module of rank 1 and degree $-c$;
then $[a/b]_* N$ has rank 1 and degree $-ac/b$.
The following are equivalent:
\begin{itemize}
\item
$M$ is pure of slope $s$;
\item
$([d]_* M) \otimes N$ is \'etale (definition);
\item
$[a/b]_* (([d]_* M) \otimes N) \cong
([ad/b]_* M) \otimes ([a/b]_* N) \cong
([d/b]_* ([a]_* M)) \otimes ([a/b]_* N)$ is \'etale (by above);
\item
$[a]_* M$ is pure of slope $as$ (definition).
\end{itemize}
This yields the claim.
\end{proof}
\begin{cor} \label{C:tensor pure}
If $M_1, M_2$ are pure $\phi$-modules of slopes $s_1, s_2$, then
$M_1 \otimes M_2$ is pure of slope $s_1+s_2$.
\end{cor}
\begin{proof}
By Lemma~\ref{L:pure push}, we may reduce to the case where $s_1, s_2 \in \mathbb{Z}$.
By twisting, we may then reduce to the case where $s_1 = s_2 = 0$. In this case the
result follows from the fact that $\phi$-modules over
$R^{\inte}$ admit tensor products.
\end{proof}
We can thus generalize Propositions~\ref{P:etale equiv} and~\ref{P:etale extend}
as follows.
\begin{theorem} \label{T:pure equiv}
For any rational number $s$,
the base change functor from pure $\phi$-modules of slope $s$ over $R^{\bd}$
to pure $\phi$-modules of slope $s$ over $R$ is an equivalence of categories.
\end{theorem}
\begin{proof}
If $M_1, M_2$ are pure of slope $s$, then $M_1^\vee \otimes M_2$ is \'etale.
Hence the proof of Proposition~\ref{P:etale equiv} goes through unchanged.
\end{proof}
\begin{theorem}\label{T:pure ext}
Let $0 \to M_1 \to M \to M_2 \to 0$ be a short exact sequence of
$\phi$-modules over $R$. If any two of $M_1, M_2, M$ are pure of
slope $s$ (except
possibly $M_1,M_2$ in the case of weak Hypothesis~\ref{hypo:robba}),
then so is the third.
\end{theorem}
\begin{proof}
By Lemma~\ref{L:pure push}, we may apply $[a]_*$ to reduce to the case where
$s \in \mathbb{Z}$; by twisting, we may force $s=0$. The result now follows from
Proposition~\ref{P:etale extend}.
\end{proof}
\begin{remark}
In a short exact sequence $0 \to M_1 \to M \to M_2 \to 0$ over $R$,
the fact that $M$ is pure of slope $s$ does not by itself
imply the same for $M_1$
and $M_2$, unless the sequence splits (see Corollary~\ref{C:split exact}).
For example, if $M$ is pure of rank 2 and slope $0$, it can happen
that $M_1$ is pure of rank 1 and slope 1, while $M_2$ is pure of rank 1
and slope $-1$.
This sort of example arises naturally from $p$-adic Hodge theory,
as in the theory of trianguline representations introduced by Colmez
\cite{colmez-tri}.
\end{remark}
\begin{lemma} \label{L:h0 pos slope}
Let $M$ be a pure $\phi$-module over $R$ of positive slope. Then
$H^0(M) = 0$.
\end{lemma}
\begin{proof}
By replacing $M$ with $[a]_* M$ for $a = \rank(M)$, we can reduce to the case
where $\mu(M) \in \mathbb{Z}_{>0}$.
By Theorem~\ref{T:pure equiv}, there exists a pure $\phi$-module $M_0$
over $R^{\bd}$ with $M \cong M_0 \otimes R$.
By Proposition~\ref{P:h0}, we have $H^0(M_0) = H^0(M)$.
Choose a basis $\mathbf{e}_1, \dots, \mathbf{e}_n$ of $M_0$ such that the matrix $A$
defined by
$\phi(\mathbf{e}_j) = \sum_i A_{ij} \mathbf{e}_i$ has all entries of valuation
at least $\mu(M)$. If $\mathbf{v} = \sum c_i \mathbf{e}_i \in H^0(M)$ is nonzero, then
$c_i = \sum_j A_{ij} \phi(c_j)$ implies that $\min_i \{w(c_i)\}
> \min_j\{w(c_j)\}$,
contradiction. Hence $H^0(M) = 0$.
\end{proof}
\begin{cor} \label{C:distinct slopes}
If $M$ and $N$ are pure $\phi$-modules over $R$ with
$\mu(M) < \mu(N)$, then $\Hom(M,N) = 0$.
\end{cor}
\begin{proof}
The conditions ensure that $M^\vee \otimes N$ is pure of positive slope;
by Lemma~\ref{L:h0 pos slope}, $\Hom(M,N) = H^0(M^\vee \otimes N) = 0$.
\end{proof}
\begin{theorem} \label{T:pure semi}
Let $M$ be a pure $\phi$-module over $R$ of slope $s$.
\begin{enumerate}
\item[(a)]
$M$ is module-semistable.
\item[(b)]
If $N$ is a $\phi$-submodule of $M$ with $\mu(N) = s$,
then $N$ is saturated, and both $N$ and $M/N$ are pure of slope $s$.
\end{enumerate}
\end{theorem}
\begin{proof}
For (a),
let $N$ be a $\phi$-submodule of $M$; we wish to show that $\mu(N) \geq s$.
By replacing $M$ by $\wedge^{\rank(N)} M$,
we may assume that $\rank(N) = 1$.
By Lemma~\ref{L:pure push}, we may assume further that $s \in \mathbb{Z}$.
By twisting, we may assume further that $N$ is trivial, so that $H^0(M) \neq 0$.
To avoid contradicting Lemma~\ref{L:h0 pos slope}, we must then have
$s \leq 0 = \mu(N)$, yielding semistability.
For (b), by applying $[a]_*$ and twisting,
we may again reduce to the case $s=0$.
Let $M_0$ be an \'etale lattice in $M$; by Lemma~\ref{L:etale lattice},
the kernel of $M_0 \to M/N$ is a $\phi$-module over $R^{\inte}$,
so the image is as well. Let $P$ be the $R$-span of this
image; it is an \'etale $\phi$-submodule of $M/N$ of the same rank.
Since $\mu(N) = \mu(M) = 0$, we also have
$\mu(M/N) = 0$, so $M/N = P$ by Corollary~\ref{C:same rank}. Hence
$M/N$ is \'etale; the same logic applied after dualizing implies that
$N^\vee$ is \'etale, as then is $N$.
\end{proof}
\begin{cor} \label{C:split exact}
If $M_1,M_2$ are $\phi$-modules, then $M = M_1 \oplus M_2$ is pure of
slope $s$ if and only if both $M_1$ and $M_2$ are pure of slope $s$.
\end{cor}
\begin{proof}
If $M_1$ and $M_2$ are pure of the same slope, then visibly so is $M$.
Conversely, if $M$ is pure of slope $s$, then $M$ is semistable by
Theorem~\ref{T:pure semi}(a), so the $\phi$-submodules $M_1$ and $M_2$
each have slope at least $s$.
Since $\mu(M)$ is a weighted average of $\mu(M_1)$ and $\mu(M_2)$,
we must in fact have $\mu(M_1) = \mu(M_2) = s$; by
Theorem~\ref{T:pure semi}(b), $M_1$ and $M_2$ are both pure of slope $s$.
\end{proof}
\begin{cor} \label{C:pure pull}
Let $M$ be a $\phi^a$-module over $R$. Then $M$ is pure of some slope $s$
if and only if $[a]^* M$ is pure of slope $s/a$.
\end{cor}
\begin{proof}
By Lemma~\ref{L:pure push}, $[a]^* M$ is pure of slope $s/a$ if and only if
$[a]_* [a]^* M$ is pure of slope $s$. If $M$ is pure of slope $s$,
then so are $(\phi^i)^* M$ for $i=0, \dots, a-1$; since
\begin{equation} \label{eq:push pull}
[a]_* [a]^* M \cong \oplus_{i=0}^{a-1} (\phi^i)^* M
\end{equation}
by Definition~\ref{D:pull push}, $[a]_* [a]^* M$ is pure of slope $s$.
Conversely, if $[a]_* [a]^* M$ is pure of slope $s$, then
\eqref{eq:push pull} shows that $M$ is a direct summand of $[a]_* [a]^* M$,
and hence is pure by Corollary~\ref{C:split exact}.
\end{proof}
\subsection{The slope filtration theorem}
So far all of our work has been formal modulo the assumption of
an appropriate analogue of Proposition~\ref{P:h1 both ways}.
We now restrict attention from general rings $R$ as in
strong Hypothesis~\ref{hypo:robba}
to the Robba ring $\mathcal{R}$ (as in Example~\ref{exa:Robba}), where one
can make the description of $\phi$-modules much more precise.
We have already described a natural filtration on $\phi$-modules over
$\mathcal{R}$, namely the Harder-Narasimhan filtration. The trouble is that
the construction is so formal that one cannot deduce any useful
properties about the resulting filtration or its associated slopes; for
instance, it is not clear that module-semistability is preserved by
tensor product. (The fact that the analogous statement is true for vector
bundles on smooth varieties in characteristic 0 is highly nontrivial:
it reduces to the case of tensoring two semistable
vector bundles of slope 0 on curves \cite{mr},
in which case it follows from an analytic classification of stable
bundles due to Narasimhan-Seshadri
\cite{ns1, ns2}.)
The slope filtration theorem, which is the main result of
this paper, asserts that in fact the
steps of the Harder-Narasimhan
filtration are much more structured than one might have
otherwise predicted.
\begin{theorem}[Slope filtration theorem] \label{T:slope filt}
Every module-semistable $\phi$-module over the Robba ring $\mathcal{R}$ is
pure. In particular, every $\phi$-module $M$ over $\mathcal{R}$ admits a unique
filtration $0 = M_0 \subset M_1 \subset \cdots \subset M_l = M$ by
saturated $\phi$-modules whose successive quotients are pure with
$\mu(M_1/M_0)< \cdots < \mu(M_l/M_{l-1})$.
\end{theorem}
This theorem is stated as a forward reference, as its proof will occupy
most of the rest of the paper; here we give only a top-level summary.
\begin{proof}[Proof of Theorem~\ref{T:slope filt}]
The proof of Theorem~\ref{T:slope filt} will be obtained by constructing
(in Subsection~\ref{subsec:extended})
an extended Robba ring $\tilde{\mathcal{R}}$ which also satisfies strong
Hypothesis~\ref{hypo:robba}, and then establishing the following facts.
\begin{itemize}
\item
If $M$ is a semistable $\phi$-module over $\mathcal{R}$, then
$M \otimes \tilde{\mathcal{R}}$ is also semistable (Theorem~\ref{T:ascend semistable}).
\item
If $\tilde{M}$ is a semistable $\phi$-module over $\tilde{\mathcal{R}}$, then
$\tilde{M}$ is pure (Theorem~\ref{T:semistable pure}).
\item
If $M$ is a $\phi$-module over $\mathcal{R}$ and
$M \otimes \tilde{\mathcal{R}}$ is pure, then $M$ is pure
(Theorem~\ref{T:descend pure}).
\end{itemize}
These together yield the claim.
\end{proof}
\begin{remark}
Theorem~\ref{T:slope filt} implies that the tensor product of
module-semistable $\phi$-modules is pure (by Corollary~\ref{C:tensor
pure}) and hence module-semistable (by Theorem~\ref{T:pure semi}). This
formally implies that the slopes of $\phi$-modules behave like
valuations of eigenvalues, or like Deligne's weights in \'etale
cohomology. That is, if $M$ has slopes $c_1, \dots, c_m$
and $M'$ has slopes $c'_1, \dots, c'_n$, both counted with multiplicity,
then:
\begin{itemize}
\item
the slopes of $M \oplus M'$ are
$c_1, \dots, c_m, c'_1, \dots, c'_n$;
\item
the slopes of $M \otimes M'$ are
$c_i c'_j$ for $i=1, \dots, m$ and $j = 1, \dots, n$;
\item
the slopes of $\wedge^d M$ are
$c_{i_1} + \cdots + c_{i_d}$ for $1 \leq i_1 < \cdots < i_d \leq m$;
\item
the slopes of $[a]_* M$ are $ac_1, \dots, ac_m$;
\item
the slopes of $M(b)$ are $c_1 + b, \dots, c_m + b$;
\item
the slopes of $[a]^* M$ are $c_1/a, \dots, c_m/a$, each repeated $a$
times.
\end{itemize}
In some sense, the slope filtration theorem is thus playing a role in this
theory analogous to Deligne's analysis of determinantal weights in his
second proof of the Weil conjectures \cite{deligne}.
\end{remark}
\begin{remark}
The uniqueness in Theorem~\ref{T:slope filt} means that the slope filtration
inherits any additional group action on the original $\phi$-module. In
particular, if $M$ is a $(\phi, \Gamma)$-module, then the steps of the
slope filtration are $(\phi, \Gamma)$-submodules of $M$. As shown by Berger
\cite[Th\'eor\`eme~V.2.1]{berger-weak},
this leads to a
proof of the Colmez-Fontaine theorem that $(\phi, N)$-modules over a $p$-adic
field which are \emph{weakly admissible}, in the sense of satisfying a
necessary numerical criterion, indeed arise
from Galois representations via $p$-adic Hodge theory.
(See also the variant of Berger's argument given by Kisin \cite{kisin}.)
\end{remark}
\begin{remark} \label{R:cherbonnier}
The \'etale $(\phi, \Gamma)$-modules attached to Galois representations of a
$p$-adic field were originally defined by Fontaine over the $p$-adic completion
of $\mathcal{R}^{\bd}$; the fact that they can be descended to
$\mathcal{R}^{\bd}$ is a theorem of Cherbonnier and Colmez
\cite[Corollaire~III.5.2]{cherbonnier-colmez}.
The fact that the descent is unique follows from the fact that
the base change from \'etale $\phi$-modules over $\mathcal{R}^{\bd}$ to its
completion is fully faithful, which in turn follows from
Proposition~\ref{P:cherbonnier1}.
\end{remark}
\begin{remark}
In the context of $p$-adic differential equations and rigid cohomology,
Theorem~\ref{T:slope filt} arises with $M$ carrying the extra structure of
a connection $\nabla: M \to M \otimes \Omega^1_{\mathcal{R}/K}$ compatible with
the $\phi$-action; that is, $M$ is a \emph{($\phi, \nabla$)-module}.
One can see that the steps of the slope filtration are
$(\phi, \nabla)$-submodules by using Corollary~\ref{C:distinct
slopes} as follows.
The map $\nabla$ induces a homomorphism $M_1 \to (M/M_1) \otimes
\Omega^1_{\mathcal{R}/K}$
of $\phi$-modules. Since $\Omega^1_{\mathcal{R}/K}$
is a rank $1$ $\phi$-module of nonnegative
slope (the slope is actually positive, but we don't need this here), each slope
of $(M/M_1) \otimes \Omega^1_{\mathcal{R}/K}$ is strictly greater than $\mu(M_1)$. Repeated
application of Corollary~\ref{C:distinct slopes} yields the claim.
Given that the slope filtration is a filtration by $(\phi,\nabla)$-submodules,
one may prove the local monodromy theorem for $p$-adic differential equations
as in \cite{kedlaya-local},
by showing each successive quotient in the slope filtration becomes trivial
as a $\nabla$-module after tensoring with a suitable finite unramified extension
of $\mathcal{R}^{\inte}$. This reduces easily to the \'etale case, which is
a theorem of Tsuzuki \cite[Theorem~4.2.6]{tsuzuki}. Beware, however, that this
last step only applies for $\phi_K$ absolute; in particular, this approach
cannot be used to prove \cite[Proposition~6.2.1]{berger-colmez}.
\end{remark}
\begin{remark} \label{R:hartl-pink}
By \cite[Theorem~11.1]{hartl-pink},
the conclusion of Theorem~\ref{T:slope filt} also holds
in the situation of Example~\ref{ex:hartl-pink};
indeed, what one obtains is an analogue of the classification
of $\phi$-modules over the extended Robba ring $\tilde{\mathcal{R}}$ to be introduced
in Section~\ref{sec:dm}.
That result is not covered by this paper, though (as \cite{hartl-pink}
already points out) there are very strong parallels between the ensuing
calculations. However, Theorem~\ref{T:slope filt} itself
does address a related
situation: if we take $K = k((z))$ with $k$ of characteristic $p>0$,
and $\phi_K$ to be a power of the absolute Frobenius,
then $\mathcal{R}$ consists of Laurent series
in $t$ over $z$ which converge for $|z|^c < |t| < 1$ for some $c>0$.
Since the valuation on $k$ is trivial, it is equivalent to require
convergence when $0 < |z| < |t|^{1/c}$; that is, we are considering series in
$z$ over $k((t))$ convergent on some punctured disc around the origin.
In this case (assuming $q$ is a power of $p$), Theorem~\ref{T:slope filt}
is a result of Hartl \cite[Theorem~1.7.7]{hartl}.
\end{remark}
It would be interesting to know about the following $q$-analogue of
Remark~\ref{R:hartl-pink}; it may be related to the formal classification
of linear difference operators \cite{praagman}, in much the same way that
the construction of the canonical lattice of an irregular meromorphic
connection
\cite{malgrange} reduces to the formal classification of linear
differential operators \cite{levelt}.
\begin{question} \label{Q:q-analogue}
Let $K$ be a complete field, either archimedean or nonarchimedean.
Take $R$ to be the ring of germs of
analytic functions over $K$ on punctured discs
around the origin, $R^{\bd}$ to be the germs meromorphic at zero,
$w$ to be the order of vanishing at zero, and
$\phi$ to be the map $\sum c_i t^i \mapsto \sum c_i q^i t^i$ for some $q \in
K^*$ with $|q| < 1$.
Does the analogue of Theorem~\ref{T:slope filt} hold in this setting?
\end{question}
\begin{remark} \label{R:dieudonne-manin}
The conclusion of Theorem~\ref{T:slope filt} also holds for
$\phi$-modules over $K$ itself; this is a straightforward consequence of
Proposition~\ref{P:factor twisted}.
In addition, if $\phi$ is bijective on $K$, then it is easy to check that
$H^1(M) = 0$ for $M$ pure of nonzero slope, so the slope filtration
splits uniquely. This gives a semilinear
analogue of the eigenspace decomposition
of a vector space equipped with a linear transformation.
If $k$ is algebraically closed of
characteristic $p>0$ and $\phi$ is an absolute Frobenius lift,
this recovers the Dieudonn\'e-Manin classification of rational Dieudonn\'e
modules \cite{manin}.
\end{remark}
\begin{remark} \label{R:no identity}
The conclusion of Theorem~\ref{T:slope filt} does not hold for $\phi$
equal to the identity map on $\mathcal{R}$. In fact, the conclusion is equivalent
to the condition that the characteristic polynomial of $\phi$ have all
coefficients in $\mathcal{R}^{\bd}$, whereas the definition of a $\phi$-module
only forces the determinant to belong to $\mathcal{R}^{\bd}$.
\end{remark}
\section{Classification over an extended Robba ring}
\label{sec:dm}
In this section and the next, we give a proof of
Theorem~\ref{T:slope filt}. Although somewhat simplified in some technical
aspects,
the argument follows the same arc as in
\cite{kedlaya-local} and \cite{kedlaya-slope}, with two basic stages.
In the first stage, performed in this section, we show that $\phi$-modules
over a suitable overring of $\mathcal{R}$ admit a very simple classification
(analogous to the Dieudonn\'e-Manin classification alluded to
in Remark~\ref{R:dieudonne-manin}),
and in particular admit a slope filtration.
In the second stage, we show that the slope filtration descends back to
$\mathcal{R}$.
On a first reading, we recommend reading only Subsection~\ref{subsec:overview}
for an overview, then returning later for the technical details
in the rest of the section.
\subsection{Overview}
\label{subsec:overview}
\begin{hypo} \label{H:field size}
Throughout this section, assume that $\phi$ is a relative Frobenius lift
on $\mathcal{R}$ such that $\phi_K$ is an \emph{automorphism}
of $K$. Also assume that
any \'etale $\phi$-module over $K$ is trivial; this is equivalent
to asking that any $\phi$-module over the residue field
$k$ be trivial. It also implies
that $H^1$ vanishes for any \'etale $\phi$-module over $K$ or any
$\phi$-module over $k$.
Using Definition~\ref{D:h0 h1},
we deduce the same conclusions with $\phi$ replaced by
$\phi^a$ for any positive integer $a$.
\end{hypo}
\begin{remark}
In the absolute Frobenius case, Hypothesis~\ref{H:field size}
can be satisfied by taking $k$ to be algebraically closed.
In general,
one must work a bit harder; see Proposition~\ref{P:splitting field}.
\end{remark}
We will define (Definition~\ref{D:extended Robba})
an \emph{extended Robba ring} $\tilde{\mathcal{R}}$ which has
the following properties:
\begin{itemize}
\item
$\tilde{\mathcal{R}}$ is a B\'ezout domain containing $\mathcal{R}$, and admits an automorphism
$\phi$ extending the given Frobenius lift on $\mathcal{R}$
(see Remark~\ref{R:analytic ring} and Proposition~\ref{P:embedding}).
\item
The units in $\tilde{\mathcal{R}}$ are the nonzero elements of a subfield
$\tilde{\mathcal{R}}^{\bd}$, which is the fraction field of a discrete valuation
ring $\tilde{\mathcal{R}}^{\inte}$ for which $\tilde{\mathcal{R}}^{\inte} \cap \mathcal{R} = \mathcal{R}^{\inte}$
(see Remark~\ref{R:analytic ring}).
\item
The strong form of Hypothesis~\ref{hypo:robba} holds for $R = \tilde{\mathcal{R}}$
(see Proposition~\ref{P:h1 both ways2}).
\end{itemize}
The classification of $\phi$-modules over $\tilde{\mathcal{R}}$
rests on a sequence of structural results, which we state in roughly
increasing order of difficulty; their proofs occupy the remainder of this
section.
\begin{prop} \label{P:h0 nonzero}
Let $M,N$ be pure $\phi$-modules over $\tilde{\mathcal{R}}$ obtained by base change
from $K$, with $\mu(M) > \mu(N)$. Then $\Hom(M,N) \neq 0$.
\end{prop}
\begin{proof}
See Subsection~\ref{subsec:extended}.
\end{proof}
\begin{notation}
Choose a uniformizer $\pi$ of $K$, and let $\tilde{\mathcal{R}}(1)$ be the
$\phi$-module of rank 1 and degree 1 on which $\phi$ acts on some
generator via multiplication by $\pi$. We use $\tilde{\mathcal{R}}(1)$ as a
twisting sheaf, writing
$M(n) = M \otimes \tilde{\mathcal{R}}(1)^{\otimes n}$.
\end{notation}
\begin{prop} \label{P:eigenvector}
Let $M$ be a nonzero $\phi$-module over $\tilde{\mathcal{R}}$. Then for all sufficiently
large integers $n$, $H^0(M(-n)) \neq 0$ and $H^1(M(-n)) = 0$.
\end{prop}
\begin{proof}
See Subsection~\ref{subsec:fixed vec}.
\end{proof}
\begin{prop} \label{P:pure equiv2}
For any rational number $s$, the base change functor from pure
$\phi$-modules of slope $s$ over $K$ to pure $\phi$-modules of
slope $s$ over $\tilde{\mathcal{R}}$ is an equivalence of categories.
\end{prop}
\begin{proof}
See Subsection~\ref{subsec:classif}.
\end{proof}
\begin{prop} \label{P:local calc}
Let $n$ be a positive integer,
let $N'$ be a pure $\phi^n$-module over $\tilde{\mathcal{R}}$ of rank $1$ and degree $1$,
let $P$ be a pure $\phi$-module over $\tilde{\mathcal{R}}$ of rank $1$ and degree $-1$,
and suppose
\[
0 \to [n]^* N' \to M \to P \to 0
\]
is a short exact sequence of $\phi$-modules. Then
$H^0(M) \neq 0$.
\end{prop}
\begin{proof}
See Subsection~\ref{subsec:local calc}.
\end{proof}
These assemble to give the following classification theorem.
\begin{theorem} \label{T:semistable pure}
Any module-semistable $\phi$-module over $\tilde{\mathcal{R}}$ is pure.
Consequently, the
successive quotients of the HN filtration of a $\phi$-module over $\tilde{\mathcal{R}}$
are all pure.
\end{theorem}
\begin{remark} \label{R:any hom}
Before proving Theorem~\ref{T:semistable pure}, we make an observation
which figures prominently in the argument.
If one knows Theorem~\ref{T:semistable pure} for $\phi$-modules
of rank $\leq n$, it follows from
Propositions~\ref{P:h0 nonzero} and~\ref{P:pure equiv2}
(and the assumption that \'etale $\phi$-modules over $K$ are trivial,
as built into Hypothesis~\ref{H:field size})
that
for $M$ a pure $\phi$-module over $\tilde{\mathcal{R}}$ and $N$ an \emph{arbitrary}
$\phi$-module over $\tilde{\mathcal{R}}$ with $\rank(N) \leq n$ and
$\mu(M) \geq \mu(N)$, we have
$\Hom(M,N) \neq 0$; in particular, if $\rank(M) = 1$, we would have
an injection of $M$ into $N$. This is because the first step of the HN
filtration of $N$ always has slope $\leq \mu(N)$.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{T:semistable pure}]
We proceed by induction on rank, the case of rank 1 being evident.
Assume that $n \geq 1$ and that for every positive integer $a$,
every semistable $\phi^a$-module of rank $\leq n$
is pure.
Suppose that $M$ is a semistable $\phi^a$-module of rank $n+1$ over $\tilde{\mathcal{R}}$;
we wish to show that $M$ is pure.
We may reduce to the case where
$\mu(M) \in \mathbb{Z}$ by applying $[d]_*$ and invoking
Corollary~\ref{C:semi push} (to see that semistability is preserved)
and Lemma~\ref{L:pure push} (to see that purity is reflected);
we may then twist to ensure $\mu(M) = 0$.
For ease of notation, we will assume hereafter that $M$ is a $\phi$-module
(at the expense of replacing $\phi$ by a power, which does not
disturb Hypothesis~\ref{H:field size}).
Put $M' = [n]_* M$; then $M'$ is semistable by Corollary~\ref{C:semi
push} again.
By Proposition~\ref{P:eigenvector}, there exists a nonnegative integer $c$
such that $M'$ admits a pure $\phi^{n}$-submodule $N'$
of rank 1 and slope $c$; choose $c$ as small as possible.
Suppose that $c \geq 2$; since $\mu(M'/N') < 0 \leq c-2$, we may apply
Remark~\ref{R:any hom} to produce a
$\phi^{n}$-submodule of $M'/N'$ isomorphic to
$\tilde{\mathcal{R}}(c-2)$. Let $Q'$ be the inverse image of that submodule in $M'$;
applying Proposition~\ref{P:local calc} (in the case $n=1$)
to the exact sequence
\[
0 \to N'(1-c) \to Q'(1-c) \to \tilde{\mathcal{R}}(-1) \to 0,
\]
we see that $H^0(Q'(1-c)) \neq 0$ and hence $H^0(M'(1-c)) \neq 0$, contradicting
the minimality of $c$.
Suppose that $c = 1$. Put $N = [n]^* N'$;
then $N$ is pure of slope $1/ n$ by Corollary~\ref{C:pure pull}.
The adjunction between $[n]^*$ and $[n]_*$
converts the inclusion $N' \hookrightarrow M'$ into a nonzero map
$f: N \to M$. Since $N$ is semistable by Theorem~\ref{T:pure semi},
$\mu(f(N)) \leq 1/n$;
moreover, the denominator of $\mu(f(N))$ is at most $\rank(f(N)) \leq n$.
Consequently, either $\mu(f(N)) \leq 0$, in which case
Remark~\ref{R:any hom} implies that $H^0(f(N)) \neq 0$; or $\mu(f(N)) =
1/n$,
in which case $f$ must be injective and we have an exact sequence
\[
0 \to N \to M \to P \to 0
\]
with $P$ pure of rank 1 and slope $-1$, to which we apply
Proposition~\ref{P:local calc} to deduce that $H^0(M) \neq 0$.
In either case, we contradict the minimality of $c$.
We deduce that $c=0$, i.e., $M'$ admits a nontrivial \'etale $\phi$-submodule
$N'$; the quotient $M'/N'$ is also semistable, hence pure by
the induction hypothesis. By Theorem~\ref{T:pure ext}, $M'$ is pure, as then
is $M$ by Lemma~\ref{L:pure push}. This completes the proof.
\end{proof}
\begin{remark}
In the proof of Theorem~\ref{T:semistable pure},
the passage from $M$ to $M'$ is made in order to simplify the statement
of Proposition~\ref{P:local calc}. One can do some extra work
to prove a version of Proposition~\ref{P:local calc} in which $[n]^* N$
is replaced by any pure $\phi$-module of rank $n$
and degree 1; however, the internal
improvement is immaterial in the end, as even this stronger
form of Proposition~\ref{P:local calc}
is itself an immediate consequence of Theorem~\ref{T:semistable pure}.
\end{remark}
\subsection{The extended Robba ring}
\label{subsec:extended}
We now go back and construct the extended Robba ring $\tilde{\mathcal{R}}$.
\begin{defn}
Let $R$ be a ring and let $G$ be a totally ordered abelian group. The
ring of \emph{Hahn series} (or \emph{Mal'cev-Neumann series}, or
\emph{generalized power series}) over $R$ with value group $G$ is the
set of functions $f: G \to R$ with well-ordered support, with pointwise
addition and multiplication
given by convolution; it is a standard calculation
\cite[Chapter~13]{passman} to verify that these operations give
a well-defined ring, which is a field if $R$ is.
We typically represent elements of this ring as formal
series $\sum_{g \in G} r_g u^g$ in some dummy variable $u$ with powers
indexed by $g \in G$, and the ring is correspondingly denoted $R((u^G))$.
For $G \subseteq \mathbb{R}$,
we view $R((u^G))$ as being equipped with the $u$-adic valuation $v$ sending
$\sum_g r_g u^g$ to the smallest $g$ for which $r_g \neq 0$ (i.e., the
least element of the support).
\end{defn}
\begin{lemma} \label{L:kernel cokernel}
Let $\phi: R((u^\mathbb{Q})) \to R((u^\mathbb{Q}))$ be an automorphism of the form
$\sum_i a_i u^i \mapsto \sum_i \phi_R(a_i)
u^{qi}$, with $\phi_R$ an automorphism
of $R$. Then the map $1 - \phi$ is bijective on the set of series with
zero constant term.
\end{lemma}
\begin{proof}
If $x \in R((u^\mathbb{Q}))$ and $v(x) < 0$, then
$v(x - \phi(x)) = q v(x)$, whereas if $v(x)> 0$, then $v(x -
\phi(x)) = v(x)$.
This proves injectivity.
Given $x \in R((u^\mathbb{Q}))$, write $x = \sum_i x_i u^i$, and put
\begin{align*}
y_+ &= \sum_{j=0}^\infty \sum_{i>0} \phi^j_R(x_i) u^{iq^j} \\
y_- &= \sum_{i<0} \left( \sum_{j=0}^\infty
-\phi^{-j-1}_R(x_{iq^{j+1}}) \right) u^{i}.
\end{align*}
Since both sums give well-defined elements of $R((u^\mathbb{Q}))$ (in the definition of $y_-$, the
sum over $j$ is finite for each $i$), we may
put $y = y_+ + y_-$, which has zero constant term and satisfies
$y - \phi(y) = x - x_0$. This proves surjectivity.
\end{proof}
\begin{cor} \label{C:kernel cokernel}
With $k$ as in Hypothesis~\ref{H:field size},
for any $c \in k^*$, the map $1 - c \phi$ on $k((u^\mathbb{Q}))$ is surjective.
\end{cor}
\begin{proof}
By Hypothesis~\ref{H:field size}, there exists $a \in k^*$ such that
$\phi(a) = c a$, so we can always write
\[
(1 - c\phi)(x) = a^{-1} (ax - \phi(ax)).
\]
It thus suffices to check the case $c=1$; this follows from
Lemma~\ref{L:kernel cokernel} and the fact that $1-\phi$ is surjective
on $k$, which again is a consequence of Hypothesis~\ref{H:field size}.
\end{proof}
Corresponding to the extension from power series to generalized power series,
we define an enlargement of the Robba ring. We first construct the ring,
then the embedding of the original Robba ring into it.
\begin{defn} \label{D:extended Robba}
For $r>0$,
let $\tilde{\mathcal{R}}^r$ be the set of formal sums $\sum_{i \in \mathbb{Q}} a_i u^i$
with $a_i \in K$, satisfying the following conditions.
\begin{itemize}
\item
For each $c > 0$, the set of $i \in \mathbb{Q}$ such that $|a_i| \geq c$ is well-ordered.
\item
We have $|a_i| e^{-ri} \to 0$ as $i \to -\infty$.
\item
For all $s>0$, we have $|a_i| e^{-si} \to 0$ as $i \to +\infty$.
\end{itemize}
Then $\tilde{\mathcal{R}}^r$ can be shown to form a ring.
We call the union $\tilde{\mathcal{R}} = \tilde{\mathcal{R}}_K = \cup_{r} \tilde{\mathcal{R}}^r$ the
\emph{extended Robba ring} over $K$.
Let $\tilde{\mathcal{R}}^{\bd}$ and $\tilde{\mathcal{R}}^{\inte}$ be the subrings
of $\tilde{\mathcal{R}}$ consisting of series
with bounded and integral coefficients, respectively.
We equip $\tilde{\mathcal{R}}^r$ with the norm
\[
\left| \sum_i a_i u^i \right|_r = \sup_i \{|a_i| e^{-ri}\}
\]
and $\tilde{\mathcal{R}}$ with the automorphism
\[
\phi \left(\sum_i a_i u^i \right) = \sum_i \phi_K(a_i) u^{qi}.
\]
\end{defn}
\begin{remark} \label{R:analytic ring}
The ring $\tilde{\mathcal{R}}$
can be viewed as an example of an ``analytic ring'' in the sense of
\cite[\S 2.4]{kedlaya-slope}, by taking $\phi_K$ to be an absolute Frobenius lift
on $K$.
Thus the results of \cite[Chapter~2]{kedlaya-slope} apply to show that
$\tilde{\mathcal{R}}$ shares many of the nice properties of $\mathcal{R}$, as follows.
\begin{itemize}
\item
The ring $\tilde{\mathcal{R}}$ is a B\'ezout domain \cite[Theorem~2.9.6]{kedlaya-slope}.
\item
The ring $\tilde{\mathcal{R}}^{\inte}$ is
a henselian discrete valuation ring, and its fraction field
is $\tilde{\mathcal{R}}^{\bd}$
\cite[Lemma~2.1.12]{kedlaya-slope}.
\item
The units of $\tilde{\mathcal{R}}$ are the nonzero elements of $\tilde{\mathcal{R}}^{\bd}$
\cite[Lemma~2.4.7]{kedlaya-slope}.
\end{itemize}
\end{remark}
\begin{prop} \label{P:embedding}
There exists a $\phi$-equivariant embedding $\psi: \mathcal{R} \hookrightarrow
\tilde{\mathcal{R}}$ such that for any $r_0$ as in Remark~\ref{R:annuli} and any
$r \in (0,r_0)$, $\mathcal{R}^r$ maps to $\tilde{\mathcal{R}}^r$ preserving $|\cdot|_r$.
\end{prop}
\begin{proof}
We inductively construct homomorphisms $\psi_l: \mathcal{R} \to \tilde{\mathcal{R}}$,
each of the form $\psi_l(\sum c_i t^i) = \sum c_i u_l^i$ for some
$u_l \in \tilde{\mathcal{R}}^{\inte}$ with $|u_l|_r = |t|_r$ for $r \in (0,r_0)$, satisfying
\[
\psi_l(\phi(x)) \equiv \phi(\psi_l(x)) \pmod{\pi^l}
\qquad (x \in \mathcal{R}^{\inte}),
\]
starting with $u_1 = u$.
Given $\psi_l$, we may repeatedly invoke
Corollary~\ref{C:kernel cokernel} (if $q \neq 0$ in $k$)
or the fact that $\phi$ is surjective on $\tilde{\mathcal{R}}^{\inte}$
(if $q =0$ in $k$)
to construct $\Delta \in \tilde{\mathcal{R}}^{\inte}$ with
\begin{equation} \label{eq:Delta}
\phi(\pi^l \Delta/u ) - q (\pi^l \Delta/u)
= (\psi_l(\phi(t)) - \phi(u_l))/u^q.
\end{equation}
For any $r \in (0,r_0)$,
\[
|\psi_l(\phi(t))|_{r/q}, |\phi(u_l)|_{r/q} \leq |t^q|_{r} = |u^q|_{r}
\]
and so the right side of \eqref{eq:Delta} has $(r/q)$-norm at most 1.
{}From this plus either
the proof of Lemma~\ref{L:kernel cokernel} (if $q \neq 0$ in $k$)
or direct inspection (if $q=0$ in $k$),
we deduce that $|\pi^l \Delta/u|_{r} \leq 1$.
We may thus set $u_{l+1} = u_l + \pi^l \Delta$ to construct $\psi_{l+1}$;
this has the desired effect because
\[
\psi_{l+1}(\phi(t))
\equiv \psi_l(\phi(t)) + q \pi^l \Delta u^{q-1} \pmod{\pi^{l+1}}.
\]
The property $|u_l|_r = |t|_r$ implies that
each $\psi_l$ carries
$\mathcal{R}^r$ to $\tilde{\mathcal{R}}^r$ preserving $|\cdot|_r$.
By continuity, we obtain a map $\psi$ with the same property,
as desired.
\end{proof}
\begin{lemma} \label{L:where fixed}
The fixed elements of $\tilde{\mathcal{R}}$ under $\phi$ all belong to $K$.
\end{lemma}
\begin{proof}
For $x =\sum_i a_i u^i \in\tilde{\mathcal{R}}$, we have $\phi(x) = \sum_i \phi_K(a_i) u^{qi}$.
If $\phi(x) = x$ and $a_i \neq 0$ for some $i \neq 0$, then $|a_{i q^n}| = |a_i|$
for all $n \in \mathbb{Z}$; but this contradicts the fact that for any $c>0$, the
set of $i \in\mathbb{Q}$ with $|a_i| \geq c$ is well-ordered.
Hence $a_i = 0$ for all $i \neq 0$, proving the claim.
\end{proof}
We now notice that strong Hypothesis~\ref{hypo:robba} holds for $\tilde{\mathcal{R}}$.
\begin{prop} \label{P:h1 both ways2}
Let $A$ be an $n \times n$ matrix over $\tilde{\mathcal{R}}^{\inte}$. Then the map
$\mathbf{v} \mapsto \mathbf{v} - A \phi(\mathbf{v})$ on column vectors induces a bijection
on $(\tilde{\mathcal{R}}/\tilde{\mathcal{R}}^{\bd})^n$.
\end{prop}
\begin{proof}
The proof proceeds as in Proposition~\ref{P:h1 both ways}, using the
definition of $|\cdot|_r$ given in Definition~\ref{D:extended Robba}.
\end{proof}
\begin{remark}
As a reminder, here are some key properties of
$\tilde{\mathcal{R}}$ which we will use going forward.
\begin{itemize}
\item
Given a relative Frobenius lift $\phi$ on $\mathcal{R}$, we can define an
action of $\phi$ on $\tilde{\mathcal{R}}$ and an
equivariant embedding $\psi: \mathcal{R} \hookrightarrow
\tilde{\mathcal{R}}$ which preserves $|\cdot|_r$ for $r \in (0,r_0)$
(Proposition~\ref{P:embedding}).
\item
The map $\phi$ is bijective on $\tilde{\mathcal{R}}$.
\item
The map $1 - \phi$ is bijective on
$\tilde{\mathcal{R}}^{\inte}/ \mathfrak{o}_K$ (easy consequence of
Lemma~\ref{L:kernel cokernel}).
\item
There is a natural direct limit topology, restricting to the direct limit
of Fr\'echet topologies on $\mathcal{R}$, under which $\tilde{\mathcal{R}}$ is complete.
\end{itemize}
In \cite{kedlaya-local} and \cite{kedlaya-slope}, the role of our $\tilde{\mathcal{R}}$
is played by the ring $\Gamma^{\alg}_{\an,\con}$, which is constructed
to be minimal for the
above properties; that ring
coincides with the ring
denoted $\tilde{\mathbf{B}}^\dagger_{\mathrm{an}}$
(as in \cite[\S II]{berger-cst}) or
more commonly $\tilde{\mathbf{B}}^\dagger_{\mathrm{rig}}$
(as in \cite{colmez-bourbaki}).
We opt here for the ring $\tilde{\mathcal{R}}$ instead in hopes that
the construction using generalized power series makes the analogy to
$\mathcal{R}$ a bit more apparent.
\end{remark}
To conclude this section, we prove Proposition~\ref{P:h0 nonzero}:
if $M,N$ are pure $\phi$-modules over $\tilde{\mathcal{R}}$ obtained by base
change from $K$, with $\mu(M) > \mu(N)$, then $\Hom(M,N) \neq 0$.
\begin{proof}[Proof of Proposition~\ref{P:h0 nonzero}]
It is equivalent to show that if $M$ is pure with $\mu(M) < 0$, obtained
by base change from $K$, then
$H^0(M) \neq 0$. Write
$M = M_0 \otimes_K \tilde{\mathcal{R}}$ for $M_0$ a pure $\phi$-module over $K$.
Take any nonzero $\mathbf{w} \in M_0$ and any $i>0$;
the sum
\[
\mathbf{v} = \sum_{n \in \mathbb{Z}} \phi^n (u^i \mathbf{w})
\]
will converge to a nonzero element of $H^0(M)$.
(Compare \cite[Proposition~3.3.4(c2)]{kedlaya-slope}.)
\end{proof}
\subsection{Construction of fixed vectors}
\label{subsec:fixed vec}
We next treat Proposition~\ref{P:eigenvector}:
if $M$ is a nonzero $\phi$-module over $\tilde{\mathcal{R}}$, then for all sufficiently
large integers $n$, $H^0(M(-n)) \neq 0$ and $H^1(M(-n)) = 0$.
(Also compare \cite[Theorem~4.1]{hartl-pink}.)
\begin{proof}[Proof of Proposition~\ref{P:eigenvector}]
We follow \cite[Proposition~4.2.2]{kedlaya-slope}.
View $M$ as a space of column vectors with the action of $\phi$ given by
multiplication by the matrix $A$ times the componentwise action.
Choose $r>0$ so that $A$ and $A^{-1}$ have entries in $\tilde{\mathcal{R}}^{qr}$.
For $d \in \mathbb{Q}_{>0}$ to be specified below,
define the ``splitting functions''
$f_d^+, f_d^-$ as follows: given $x = \sum a_i u^i$, put
\[
f_d^+(x) = \sum_{i \geq d} a_i u^i, \qquad
f_d^-(x) = \sum_{i<d} a_i u^i,
\]
then extend to vectors componentwise. For $\mathbf{w}$ a vector, we write
$\mathbf{w}^{\pm}$ for $f_d^{\pm}(\mathbf{w})$.
Define the map $g: M \to M$ by
\[
g(\mathbf{w}) = \pi^{-n} A \phi(\mathbf{w}^+) + \phi^{-1}(\pi^n A^{-1} \mathbf{w}^-)
\]
and note that
\begin{equation} \label{eq:bound g}
|g(\mathbf{w})|_r \leq \max\{|\pi|^{-n} |A|_r e^{-rd(q-1)},
|\pi|^{n} |A^{-1}|_{qr} e^{-rd(q^{-1}-1)} \} |\mathbf{w}|_r.
\end{equation}
If we can choose $d$ such that
the two quantities in the maximum in \eqref{eq:bound g} are both
strictly less than 1, then $g$ will be contractive towards zero. This happens
if
\begin{equation} \label{eq:bound d}
d \in \left( \frac{n(-\log |\pi|) + \log |A|_r}{r(q-1)},
\frac{qn(-\log |\pi|) - q \log |A^{-1}|_{qr}}{r(q-1)} \right);
\end{equation}
for $n$ sufficiently large the interval is nonempty.
(Note that
consistently with Proposition~\ref{P:h0 nonzero}, if $M$ is \'etale over $K$
we can take any $n>0$.)
Fix $n,d$ satisfying \eqref{eq:bound d}.
Given $\mathbf{w}$ with entries in $\tilde{\mathcal{R}}^r$, we define the sequence
$\mathbf{w}_0 = \mathbf{w}, \mathbf{w}_{l+1} = g(\mathbf{w}_l)$, then set
\begin{equation} \label{eq:sum}
\mathbf{v} = \sum_{l=0}^\infty (\mathbf{w}_l^+ - \phi^{-1} (\pi^n A^{-1} \mathbf{w}_l^-)),
\end{equation}
so that $|\mathbf{v}|_r \leq |\mathbf{w}|_r$ and $\mathbf{v} - \pi^{-n} A \phi(\mathbf{v}) = \mathbf{w}$.
We only know \emph{a priori} that the sum defining $\mathbf{v}$ converges under
$|\cdot|_r$, but using
the equation $\mathbf{v} = \pi^{-n} A \phi(\mathbf{v}) + \mathbf{w}$, we may deduce that
the sum converges under $|\cdot|_{r/q}$, $|\cdot|_{r/q^2}$, and so on.
Hence $\mathbf{v}$ has entries in $\tilde{\mathcal{R}}^r$, yielding
$H^1(M(-n)) = 0$.
To deduce $H^0(M(-n)) \neq 0$,
we modify the previous construction slightly.
Put $\mathbf{w} = (u^d, 0, \dots, 0)$ and construct $\mathbf{v}$ as in
\eqref{eq:sum}.
Then put $\mathbf{w}'_0 = \mathbf{w}$, $\mathbf{w}'_1 =
\phi^{-1}(\pi^n A^{-1} \mathbf{w}'_0)$, and
$\mathbf{w}'_{l+1} = g(\mathbf{w}'_l)$ for $l \geq 1$.
(That is, at the first step, transfer
the boundary term $u^d$ from the plus part to the minus part.)
If we now define
\[
\mathbf{v}' = -\phi^{-1}(\pi^n A^{-1}\mathbf{w}'_0) +
\sum_{l=1}^\infty ((\mathbf{w}_l')^+ - \phi^{-1} (\pi^n A^{-1} (\mathbf{w}_l')^-)),
\]
we obtain $\mathbf{v}' - \pi^{-n} A \phi(\mathbf{v}') = \mathbf{w}$ as before. However,
$|\mathbf{v}|_r = |u^d|_r$ whereas $|\mathbf{v}'|_r < |u^d|_r$, so
$\mathbf{v}- \mathbf{v}'$ is a nonzero element of $H^0(M(-n))$, as desired.
\end{proof}
\subsection{Twisted polynomials and their Newton polygons}
Before continuing, we need to analogize, to the realm of twisted polynomials
over $k((u^\mathbb{Q}))$, some facts about polynomials over
valued fields and their Newton polygons. With a bit of care, we can obtain
at the same time some results over $K$ which we will need later
(see Proposition~\ref{P:splitting field}).
\begin{notation} \label{N:valued field}
Throughout this subsection only, fix a real number $s \geq 1$,
and
let $F$ be a field equipped with an automorphism $\phi = \phi_F$
and a valuation $v_F$ with the properties that $F$ is complete under $v_F$
and $v_F(\phi_F(x)) = s v_F(x)$ for all $x \in F$.
Let $\mathfrak{o}_F$ and $\mathfrak{m}_F$
denote the valuation subring of $F$ and the maximal ideal of $\mathfrak{o}_F$,
respectively.
\end{notation}
\begin{defn} \label{D:Newton poly}
For $i \in \mathbb{Z}$, write $[i] = \sum_{j=0}^{i-1} s^j$, so that
$[0] = 0$, $[1] = 1$, and $[i+j] = [i] + s^i[j]$.
For $r \in \mathbb{R}$ and $P(T) \in F\{T^{\pm}\}$,
write $P(T) = \sum_{i \in \mathbb{Z}} a_i T^i$, and write
\[
v_r(P) = \min_i \{ v_F(a_i) + r[i]\}.
\]
Define the \emph{homogeneous Newton polygon} of $P$ as the lower convex
hull of the set
\[
\{(-[i], v_F(a_i)): i \in \mathbb{Z}\};
\]
we refer to the slopes of this polygon as the \emph{(Newton) slopes of $P$}.
\end{defn}
\begin{lemma} \label{L:scale product}
For $P(T) \in F\{T\}$ and $Q(T) \in F\{T^{-1}\}$
such that $v_r(Q) \geq 0$, we have
$v_r(PQ) \geq v_r(P) + v_r(Q)$.
\end{lemma}
\begin{proof}
Write $P(T) = \sum_{i \geq 0} a_i T^i$ and $Q(T) =
\sum_{j \leq 0} b_j T^j$.
We have
\[
(PQ)(T) = \sum_k \sum_{i+j=k} a_i \phi^i(b_j) T^k,
\]
and
\begin{equation} \label{eq:newton}
v_F(a_i \phi^i(b_j)) + [i+j]r =
v_F(a_i) + [i]r + s^i (v_F(b_j) + [j]r).
\end{equation}
The right side of \eqref{eq:newton} is at least
$v_r(P) + s^i v_r(Q)$. Since $i \geq 0$ and $s \geq 1$, if $v_r(Q) \geq 0$, then
the right side of \eqref{eq:newton} is at least $v_{r}(P) + v_{r}(Q)$.
This yields the claim.
\end{proof}
\begin{prop}
Let $r_0 \in \mathbb{R}$ be a real number, and suppose that
$P(T) \in F\{T\}$ and $Q(T) \in F\{T^{-1}\}$ are such that
$P$ has constant term $1$ and all slopes $\leq r_0$,
and $Q$ has constant term $1$ and all slopes $\geq r_0$.
Then
the slopes of $PQ$ are obtained by taking the union (with multiplicities)
of the sets of slopes
of $P$ and $Q$.
\end{prop}
\begin{proof}
The conditions on the slopes of $P$ and $Q$ imply that
\begin{align*}
r \geq r_0 &\implies v_r(P) = 0, v_r(Q) \leq 0 \\
r \leq r_0 &\implies v_r(P) \leq 0, v_r(Q) = 0.
\end{align*}
It thus suffices to check that
\[
v_{r}(PQ) = \begin{cases} v_{r}(Q) & r > r_0 \\
0 & r = r_0 \\
v_{r}(P) & r < r_0.
\end{cases}
\]
Retain notation as in Lemma~\ref{L:scale product}.
If $r \geq r_0$, take the smallest $j$ that minimizes
$v_F(b_j) + [j]r$; then \eqref{eq:newton} equals $v_{r}(Q)$ for $i=0$ but not
for any other pair $i,j$ with the same sum.
If $r \leq r_0$, take the largest $i$ that
minimizes $v_F(a_i) + [i]r$; then \eqref{eq:newton} equals $v_{r}(P)$ for $j=0$
but not for any other pair $i,j$ with the same sum.
This yields the desired result.
\end{proof}
\begin{prop} \label{P:factor twisted}
Let $r \in \mathbb{R}$ be a real number, and suppose that
$R \in F\{T^{\pm}\}$
satisfies $v_r(R-1) > 0$.
Then there exist $c \in F$, $P(T) \in F\{T\}$,
$Q(T) \in F\{T^{-1}\}$ such that
$v_F(c-1) > 0$,
$P$ has constant term $1$ and all slopes $< r$,
$Q$ has constant term $1$ and all slopes $> r$,
and $cPQ = R$.
\end{prop}
\begin{proof}
Put $c_0 = P_0 = Q_0 = 1$. Given $c_i, P_i, Q_i$, write
$R - c_i P_i Q_i = \sum_j r_j T^j$, and put
\begin{align*}
c_{i+1} &= c_i + r_0 \\
P_{i+1} &= P_i + \sum_{j>0} r_j T^j \\
Q_{i+1} &= Q_i + \sum_{j<0} r_j T^j.
\end{align*}
Suppose that $\min\{v(c-1), v_r(P_i-1), v_r(Q_i - 1)\} \geq v_r(R-1)$.
By Lemma~\ref{L:scale product}, $v_r(R- c_i P_i Q_i) \geq v_r(R-1)$, and
\[
v_r(R - c_{i+1} P_{i+1} Q_{i+1}) \geq v_r(R - c_i P_i Q_i) + v_r(R-1).
\]
It follows that $c_i, P_i, Q_i$ converge to
limits $c,P,Q$ with the desired properties.
\end{proof}
\begin{cor} \label{C:one slope}
If $R(T) \in F \{T^{\pm} \}$ is irreducible, then it has only one slope.
\end{cor}
\subsection{Classification of pure $\phi$-modules}
\label{subsec:classif}
We next classify the $\phi$-modules over $k((u^\mathbb{Q}))$, then
classify the pure $\phi$-modules over $\tilde{\mathcal{R}}$ (Proposition~\ref{P:pure
equiv2}).
\begin{notation}
Throughout this subsection only, write $F = k((u^\mathbb{Q}))$;
note that this is consistent with Notation~\ref{N:valued field} if we
put $s = q$,
take $v_F$ to be the $u$-adic valuation, and take $\phi_F$ of the form
$\sum c_i u^i \mapsto \sum \phi_k(c_i) u^{qi}$ for some automorphism
$\phi_k$ of $k$.
\end{notation}
\begin{lemma} \label{L:slope 0 root}
Let $P(T) \in F\{T\}$ be a twisted polynomial over $F$
with all Newton slopes equal to $0$. Then there exists
$x \in \mathfrak{o}_F^*$ such that $P(\phi)(x) = 0$.
\end{lemma}
\begin{proof}
We may assume that $P$ has constant term 1.
Since $\phi$-modules over $k$
are trivial (by Hypothesis~\ref{H:field size}), we can find
$z \in \mathfrak{o}_F^*$ with $P(\phi)(z) \in \mathfrak{m}_F$.
Since $(P-1)(\phi)$ is contractive towards 0 on $\mathfrak{m}_F$, we
can find $y \in \mathfrak{m}_F$ such that $P(\phi)(y) = P(\phi)(z)$.
Put $x = z - y$; then $P(\phi)(x) = 0$.
\end{proof}
\begin{lemma} \label{L:factor slope 0}
Let $P(T) \in F\{T\}$ be a monic twisted polynomial over $F$
with all Newton slopes equal to $0$. Then $P(T)$ factors as a
product $\prod_j (T - a_j)$ for some $a_j \in \mathfrak{o}_F^*$.
\end{lemma}
\begin{proof}
By Lemma~\ref{L:slope 0 root}, there exists $x \in \mathfrak{o}_F^*$
such that $P(\phi)(x) = 0$. By the division algorithm for
twisted polynomials,
$P(T)$ is right divisible by $T - a$ for $a = \phi(x)/x$; the claim
then follows by induction.
\end{proof}
\begin{lemma} \label{L:irreducible over k}
Every irreducible $\phi$-module over $F$
is trivial.
\end{lemma}
\begin{proof}
Let $V$ be an irreducible $\phi$-module over $F$;
we can then write $V$ as $F\{T^{\pm}\}/F\{T^{\pm}\}P$ for some
monic irreducible twisted polynomial $P(T)$.
By Corollary~\ref{C:one slope},
$P$ has only one slope, which we can force to be 0 by rescaling.
By Lemma~\ref{L:factor slope 0}, $P$ must equal
$T - a$ for some $a \in \mathfrak{o}_F^*$. But the equation $\phi(x) = ax$ has
a solution $x \in \mathfrak{o}_F^*$ by Lemma~\ref{L:slope 0 root}, yielding
the triviality of $V$.
\end{proof}
\begin{prop} \label{P:essential over k}
Every $\phi$-module over $F = k((u^\mathbb{Q}))$ is trivial.
\end{prop}
\begin{proof}
Any $\phi$-module over $F$ can be written as a
successive extension of irreducibles, which are all trivial by
Lemma~\ref{L:irreducible over k}.
By Corollary~\ref{C:kernel cokernel}, the extensions between trivial
$\phi$-modules all split, yielding the claim.
\end{proof}
\begin{defn}
For $P(T) = \sum_i a_i T^i
\in F\{T^{\pm}\}$ nonzero and $z \in F$, define the \emph{inhomogeneous
Newton polygon} of the pair $(P,z)$ as the lower convex hull of the set
\[
\{(-q^i, v_F(a_i)): i \in \mathbb{Z}\} \cup \{(0, v_F(z))\};
\]
note that any slope of this polygon not involving the point $(0, v_F(z))$ is
equal to $q-1$ times a slope of the homogeneous Newton polygon.
\end{defn}
\begin{prop} \label{P:inhomogeneous}
Given $P(T) \in F\{T^{\pm}\}$ nonzero and $z\in F$, for each $r \in \mathbb{R}$ occurring
as a slope of the inhomogeneous Newton polygon of $(P,z)$, there exists
$x \in F$ with $v_F(x) = r$ such that $P(\phi)(x) = z$.
\end{prop}
\begin{proof}
By applying Proposition~\ref{P:factor twisted}, we may reduce to the case
where $P$ has a single homogeneous Newton slope; by twisting, we may force
that slope to be 0. By Lemma~\ref{L:factor slope 0}, we may reduce to
the case $P(T) = T - a$ for $a \in \mathfrak{o}_F^*$. By Lemma~\ref{L:slope 0 root},
we may assume that $a = 1$; in this case, the claim follows from
Corollary~\ref{C:kernel cokernel}.
\end{proof}
Before proving Proposition~\ref{P:pure equiv2},
we need one
more calculation, which includes Proposition~\ref{P:cherbonnier1}
(see also Remark~\ref{R:cherbonnier1}).
\begin{prop} \label{P:cherbonnier2}
Let $\tilde{\mathcal{E}}$ denote the $\mathfrak{m}_K$-adic completion of
$\tilde{\mathcal{R}}^{\bd}$. Let $A$ be an $n \times n$ matrix over $\tilde{\mathcal{R}}^{\inte}$.
If $\mathbf{v}
\in \tilde{\mathcal{E}}^n$ is a column vector such that
$A \mathbf{v} = \phi (\mathbf{v})$, then $\mathbf{v} \in (\tilde{\mathcal{R}}^{\bd})^n$.
\end{prop}
\begin{proof}
By rescaling by a factor of $u$ (as in the proof of
Proposition~\ref{P:h1 both ways}), we may reduce to the case where
the entries of $A$ are bounded by 1 under $|\cdot|_r$;
we may also assume $\mathbf{v}$ has entries in the completion of
$\tilde{\mathcal{R}}^{\inte}$.
Write $\mathbf{v} = \sum_{j=1}^n \sum_{i \in \mathbb{Q}} c_{ij} u^i \mathbf{e}_j$,
where $\mathbf{e}_1, \dots, \mathbf{e}_n$ are the standard basis vectors;
it suffices to show that $|c_{ij}u^i|_r \leq 1$ for all $i,j$,
as then $\mathbf{v}$ will have entries in $\tilde{\mathcal{R}}^s$ for any $s \in (0,r)$.
Suppose the contrary; note that
$|c_{ij}| \leq 1$ for all $i,j$ by our normalization of $\mathbf{v}$,
so any pair $i,j$ with $|c_{ij} u^i|_r > 1$ must have $i<0$,
and hence
\begin{equation} \label{eq:cherbonnier2}
|\phi^{-1}(c_{ij} u^i)|_r = |c_{ij} u^{i/q}|_r < |c_{ij} u^{i}|_r.
\end{equation}
Let $h$ be the maximum of $|c_{ij}|$ over all pairs
$i,j$ with $|c_{ij} u^i|_r > 1$. Then there is a pair $(i_0,j_0)$
with $|c_{i_0,j_0}| = h$ which maximizes $|c_{i_0,j_0} u^{i_0}|_r$.
However, if we expand $A \mathbf{v} =
\sum_{j=1}^n \sum_{i \in \mathbb{Q}} d_{ij} u^i \mathbf{e}_j$,
then for each pair $i,j$ with $|d_{ij}| = h$,
we have $|\phi^{-1}(d_{ij} u^i)|_r < |c_{i_0,j_0} u^{i_0}|_r$
by \eqref{eq:cherbonnier2}. This contradicts the equality
$\mathbf{v} = \phi^{-1}(A \mathbf{v})$, proving the claim.
\end{proof}
We now prove Proposition~\ref{P:pure equiv2}: the categories of
pure $\phi$-modules over $K$ and over $\tilde{\mathcal{R}}$ of a given slope $s$
are equivalent.
\begin{proof}[Proof of Proposition~\ref{P:pure equiv2}]
We first check full faithfulness.
By Lemma~\ref{L:pure push} and twisting, it suffices to check this for $s=0$;
that is, we must check that given an
\'etale $\phi$-module $M_0$ over $K$, we must have
$H^0(M_0) \cong H^0(M_0 \otimes_K \tilde{\mathcal{R}})$.
By Hypothesis~\ref{H:field size}, we may assume that $M_0$ is trivial;
then Lemma~\ref{L:where fixed} yields the claim.
We next check essential surjectivity;
we may proceed as in the proof of Theorem~\ref{T:pure equiv}
to reduce to the case $s=0$. Let $M$ be an \'etale $\phi$-module
over $\tilde{\mathcal{R}}$, and choose an \'etale lattice $M_0$ of $M$.
By repeated application of
Proposition~\ref{P:essential over k}, after tensoring with the
$\mathfrak{m}_K$-adic completion of $\tilde{\mathcal{R}}^{\inte}$, we can find a basis of $M_0$
fixed by $\phi$. By Proposition~\ref{P:cherbonnier2}, this basis is in fact
contained in $M_0$ itself, yielding the claim.
\end{proof}
\subsection{The local calculation}
\label{subsec:local calc}
We now perform
the explicit calculation that proves Proposition~\ref{P:local
calc}, thus completing the proof of Theorem~\ref{T:semistable pure}. To avoid
notational overload, we elide a few routine calculations that can be found
in \cite{kedlaya-local}. (Also compare \cite[\S 9,10]{hartl-pink}.)
\begin{defn}
Let $\tilde{\mathcal{R}}^{\tr}$ (for ``truncated'') denote the set of elements of $\tilde{\mathcal{R}}$
whose support is bounded below. This forms a subring of $\tilde{\mathcal{R}}$ carrying
a $u$-adic
valuation $v$. Note that a unit in $\tilde{\mathcal{R}}^{\tr}$ is precisely an element
$x = \sum_i a_i u^i$ for which the support of $x$ has a least element $j$,
and for which $|a_i| \leq |a_j|$ for all $i \in \mathbb{Q}$; in particular,
such elements belong to $\tilde{\mathcal{R}}^{\bd}$, so we can apply
the valuation $w$ to them.
\end{defn}
\begin{lemma} \label{L:positioning}
Let $P$ be a $\phi$-module over $K$ of rank $1$ and degree $n > 0$, and
fix a generator $\mathbf{v}$ of $P$.
\begin{enumerate}
\item[(a)]
For any $x \in \tilde{\mathcal{R}}^{\tr}$ with support in $[0, +\infty)$,
the class of $x \mathbf{v}$ in $H^1(P \otimes \tilde{\mathcal{R}})$ vanishes.
\item[(b)]
Each class in $H^1(P \otimes \tilde{\mathcal{R}})$ has a representative of the form
$\sum_{j=0}^{n-1} u_j \mathbf{v}$, where for each $j$, either $u_j = 0$,
or $u_j \in (\tilde{\mathcal{R}}^{\tr})^*$, $w(u_j) = j$, and $v(u_j) < 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (a), we first use Hypothesis~\ref{H:field size} to eliminate
constant terms, then note that if $x$ has no constant term,
the sum $\sum_{i=0}^\infty \phi^i(x \mathbf{v})$ converges
and its limit $\mathbf{w}$ satisfies $\mathbf{w} - \phi(\mathbf{w}) = x \mathbf{v}$.
We deduce (b) from (a) plus a direct calculation;
see also \cite[Lemmas~4.13 and~4.14]{kedlaya-local}
or \cite[Lemma~4.3.2]{kedlaya-slope}.
\end{proof}
We now prove Proposition~\ref{P:local calc}: if
$N'$ is a pure $\phi^n$-module over $\tilde{\mathcal{R}}$ of rank
1 and degree 1, $P$ is a pure $\phi$-module over $\tilde{\mathcal{R}}$ of rank 1 and
degree -1, and
\begin{equation} \label{eq:exact seq}
0 \to [n]^* N' \to M \to P \to 0
\end{equation}
is a short exact sequence of $\phi$-modules, then $H^0(M) \neq 0$.
\begin{proof}[Proof of Proposition~\ref{P:local calc}]
The snake lemma gives an exact sequence
\[
H^0(M) \to H^0(P) \to H^1([n]^* N'),
\]
where the second map is pairing with the class $\alpha \in H^1(P^\vee
\otimes [n]^* N')$ corresponding to the extension \eqref{eq:exact seq};
it suffices to show that this second map has nonzero kernel.
Note that $P^\vee \otimes [n]^* N' \cong [n]^* ([n]_* P^\vee \otimes N')$
as in Definition~\ref{D:pull push}, so we may view $\alpha$ as an element of
$H^1([n]^*([n]_* P^\vee \otimes N')) \cong H^1([n]_* P^\vee \otimes N')$.
Similarly,
we may view the pairing with $\alpha$ as the composition of the map
$H^0(P) \to H^0([n]_* P)$ with the map $H^0([n]_* P) \to
H^1(N')$ given by pairing with the class in $H^1([n]_* P^\vee \otimes N')$.
If the class vanishes, there is nothing to check, so we may assume that
it does not vanish.
By Proposition~\ref{P:pure equiv2}, $P$ and $N'$ are obtained by base
change from certain $\phi$- and $\phi^n$-modules $P_0$ and $N'_0$,
respectively, over $K$;
choose generators $\mathbf{v}$ and $\mathbf{w}$ of $P_0$ and $N'_0$, and define
$\lambda, \mu \in K^*$ by $\phi(\mathbf{v}) = \lambda \mathbf{v}$ and $\phi^n(\mathbf{w})
= \mu \mathbf{w}$.
Put $Q_0 = [n]_* P_0^\vee \otimes N'_0$ and $Q = [n]_* P^\vee \otimes N'
\cong Q_0 \otimes_K \tilde{\mathcal{R}}$; let $\mathbf{x}$ be the generator $\mathbf{v}^\vee \otimes \mathbf{w}$
of $Q_0$ (where $\mathbf{v}^\vee$ is the generator of $P^\vee$ dual to $\mathbf{v}$).
By Lemma~\ref{L:positioning}, we can then represent
the class $\alpha \in H^1(Q)$ by a nonzero element of $Q$ of the form
$\sum_{j=0}^n u_j \mathbf{x}$, where each
$u_j$ is either zero or a unit in $\tilde{\mathcal{R}}^{\tr}$ with $w(u_j) = j$
and $v(u_j) < 0$.
We now follow \cite[Lemma~4.12]{kedlaya-local}.
For $j \in \{0, \dots, n\}$ such that $u_j \neq 0$,
$l \in \mathbb{Z}$, and $m \in (0, +\infty)$, define
\[
e(j,l,m) = (v(u_j) + m q^{-l})q^{-n(j+l)}.
\]
For fixed $j$ and $m$, $e(j,l,m)$ approaches 0 from below as $l \to +\infty$,
and tends to $+\infty$ as $l \to -\infty$. Hence the minimum
$h(m) = \min_{j,l} \{e(j,l,m)\}$ is well-defined; we observe that
$h$ is a continuous, piecewise linear, and
increasing map from
$(0,+\infty)$ to $(-\infty,0)$, and that
$h(qm) = q^{-n} h(m)$ because $e(j,l+1,qm) = q^{-n} e(j,l,m)$. Another interpretation
is that the lower convex hull of the set $H$ of points
\[
(-q^{-nj-(n+1)l}, q^{-nj-nl} v(u_j)) \qquad (j=0,\dots,n; \quad l \in \mathbb{Z})
\]
has all slopes positive, and all segments finite.
Pick $r \in (0,+\infty)$ at which $h$ changes slope; that is,
$r$ is a slope of the lower convex hull of $H$. Let $S$ denote the set
of ordered pairs $(j,l)$ for which $e(j,l,r) < q^{-n} h(r)$; this set
is finite. Let $T$ be the set of ordered pairs $(j,l)$ for which
$e(j,l,r) < 0$; this set (which contains $S$) is infinite, but the values
of $l$ for pairs $(j,l) \in T$ are bounded below. For each pair $(j,l)$,
put $s(j,l) = \lfloor \log_{q^n}(h(r)/e(j,l,r)) \rfloor$. Then
the following properties hold.
\begin{enumerate}
\item[(a)]
For $(j,l) \in T$, $s(j,l) \geq 0$.
\item[(b)]
For $(j,l) \in T$, $e(j,l,r) q^{ns(j,l)} \in [h(r), q^{-n}h(r))$.
\item[(c)]
We have $(j,l) \in S$ if and only if $(j,l) \in T$ and $s(j,l) = 0$.
\item[(d)]
For any $c > 0$, there are only finitely many pairs $(j,l) \in T$ with
$s(j,l) \leq c$.
\end{enumerate}
Define the twisted powers $\lambda^{\{m\}}$ and $\mu^{\{m\}}$ of $\lambda$
and $\mu$ by the two-way recurrences
\begin{gather*}
\lambda^{\{0\}} = 1, \qquad
\lambda^{\{m+1\}} = \phi(\lambda^{\{m\}}) \lambda \\
\mu^{\{0\}} = 1, \qquad
\mu^{\{m+1\}} = \phi^n(\mu^{\{m\}}) \mu.
\end{gather*}
For $c \in \mathbb{R}$, let $U_c$ be the set of $z \in \tilde{\mathcal{R}}^{\tr} \cap \tilde{\mathcal{R}}^{\inte}$
with $v(z) \geq c$. Then the function
\[
R(z) = \sum_{(j,l) \in T}
\mu^{\{-j-l+s(j,l)\}} \phi^{-nj-nl+ns(j,l)} (u_j \lambda^{\{-l\}} \phi^{-l}(z))
\]
carries $U_r$ into $U_{h(r)}$ by a direct calculation.
Modulo $\pi$, we have
\begin{equation} \label{eq:puiseux poly}
R(z) \equiv \sum_{(j,l) \in S}
\mu^{\{-j-l\}} \phi^{-nj-nl} (u_j \lambda^{\{-l\}} \phi^{-l}(z));
\end{equation}
note that the values $-nj-(n+1)l$ are distinct for all $(j,l) \in S$,
since $j$ only runs over $\{0,\dots,n\}$.
Write the reduction modulo $\pi$
of the right side of \eqref{eq:puiseux poly} as
$Q(\phi)(z)$ for some twisted Laurent polynomial
$Q(T) \in F\{T^{\pm}\}$ with $F = k((u^{\mathbb{Q}}))$.
By Proposition~\ref{P:inhomogeneous} applied repeatedly, we
can construct a nonzero $z \in U_r$ such that $R(z) = 0$.
One now calculates using Lemma~\ref{L:positioning}(a)
(see \cite[Lemma~4.12]{kedlaya-local} for the full calculation) that
the element
\[
\sum_{l \in \mathbb{Z}} \phi^{-l}(z \mathbf{v}) = \sum_{l \in \mathbb{Z}}
\lambda^{\{-l\}} \phi^{-l}(z) \mathbf{v} \in H^0(P)
\]
pairs to
zero with the class of $\alpha$. This yields the desired result.
\end{proof}
\section{Descending the slope filtration}
\label{sec:descend}
As noted at the beginning of the previous section, the proof of the slope
filtration theorem (Theorem~\ref{T:slope filt})
consists of two stages, the first of which (classifying
$\phi$-modules over the overring $\tilde{\mathcal{R}}$ of $\mathcal{R}$) has been accomplished in
the previous section. In this section, we explain how to descend the
resulting slope filtration from $\tilde{\mathcal{R}}$ back to $\mathcal{R}$.
As was done in the previous section, we recommend on a first reading
to read only the overview (Subsection~\ref{subsec:descend overview}),
then return later for the technical details.
\subsection{Overview}
\label{subsec:descend overview}
\begin{defn} \label{D:descent setup}
We now revert to allowing $K$ to be an arbitrary field as in
Definition~\ref{D:initial}. Choose a complete extension $L$ of $K$
with the same value group,
admitting an extension $\phi$ to an automorphism, such that
every \'etale $\phi$-module
over $L$ is trivial. More precisely, form such an $L$ by first
taking the completed direct limit of $K \stackrel{\phi}{\to}
K \stackrel{\phi}{\to} \cdots$ and then applying Proposition~\ref{P:splitting
field} below.
Under these conditions, we can embed $\mathcal{R}_K$ into $\mathcal{R}_L$, and then
embed $\mathcal{R}_L$ into $\tilde{\mathcal{R}}_L$ as in Proposition~\ref{P:embedding}.
\end{defn}
Recall that we are trying to prove Theorem~\ref{T:slope filt}, which
states that every module-semistable $\phi$-module over $\mathcal{R}$ is pure.
As noted earlier, this result
follows from Theorem~\ref{T:semistable pure} (which asserts
that module-semistable $\phi$-modules over $\tilde{\mathcal{R}}_L$ are pure)
plus the following assertions.
\begin{theorem} \label{T:ascend semistable}
Let $M$ be a module-semistable $\phi$-module over $\mathcal{R}$. Then
$M \otimes \tilde{\mathcal{R}}_L$ is module-semistable.
\end{theorem}
\begin{theorem} \label{T:descend pure}
Let $M$ be a $\phi$-module over $\mathcal{R}$ such that
$M \otimes \tilde{\mathcal{R}}_L$ is pure. Then $M$ is pure.
\end{theorem}
The proofs of Theorems~\ref{T:ascend semistable} and~\ref{T:descend pure}
amount to faithfully flat descent:
Theorem~\ref{T:ascend semistable} relies on the fact that the first
step of the HN filtration of $M \otimes \tilde{\mathcal{R}}_L$ descends to $\mathcal{R}$,
while
Theorem~\ref{T:descend pure} depends on the fact that the
pure $\phi$-module over $\tilde{\mathcal{R}}_L^{\bd}$ obtained by descending
$M \otimes \tilde{\mathcal{R}}_L$ itself descends to $\mathcal{R}^{\bd}$.
The rest of this section will be occupied with setting up the descent
formalism, then making the calculations that allow the use of faithfully
flat descent.
\subsection{Splitting \'etale $\phi$-modules}
We now construct the field $L$ demanded by Definition~\ref{D:descent setup}.
\begin{defn}
Suppose that $\phi_K$ is bijective.
By an \emph{admissible extension} of $K$, we will mean a field $L$ containing
$K$, complete for a nonarchimedean absolute value extending the one on $K$
with the same value group, and equipped with an isometric field automorphism
$\phi_L$ extending $\phi_K$.
\end{defn}
\begin{lemma} \label{L:split h1}
For any $z \in K^*$,
there exists an admissible extension $L$ of $K$ such that the equation
$\phi(x) - x = z$ has a solution $x \in L$.
\end{lemma}
\begin{proof}
Let $L$ be the completion of the rational function field $K(x)$ for the
Gauss norm with $|x| = |z|$. Extend $\phi_K$ to an automorphism $\phi_L$ of
$L$ by setting
$\phi_L(x) = x + z$.
\end{proof}
\begin{lemma} \label{L:split poly}
Let $P(T) = T^n + a_{n-1} T^{n-1} + \cdots + a_0$ be a twisted polynomial
over $\mathfrak{o}_K$ with $|a_0| = 1$. Then there exists an admissible extension
$L$ of $K$ such that the equation $P(\phi)(x) = 0$ has a
solution $x \in \mathfrak{o}_L^*$.
\end{lemma}
\begin{proof}
Let $L$ be the completion of the rational function field
$K(y_0, \dots, y_{n-1})$
under the Gauss norm normalized with $|y_0| = \cdots = |y_{n-1}| = 1$.
Extend $\phi_K$ to an automorphism $\phi_L$ of $L$ by setting
$\phi_L(y_i) = y_{i+1}$ for $i=0, \dots, n-2$ and
$\phi_L(y_{n-1}) = -a_{n-1}y_{n-1} - \cdots - a_0 y_0$, then
take $x = y_0$.
\end{proof}
\begin{prop} \label{P:splitting field}
There exists a complete extension $L$ of $K$ with the same value group,
equipped with an extension of $\phi_K$, such that
any \'etale $\phi$-module over $L$ is trivial.
\end{prop}
\begin{proof}
It suffices to construct $L$
trivializing a single irreducible \'etale $\phi$-module
$M$ over $K$, as we can construct the desired field by transfinitely iterating
this construction and completing at all limit stages.
Since $M$ is irreducible, we must have $M \cong K\{T^{\pm}\}/K\{T^{\pm}\}P(T)$
for some irreducible monic twisted polynomial $P(T)$. If we write
$P(T) = T^n + a_{n-1} T^{n-1} + \cdots + a_0$, then $|a_0| = 1$ because $\deg(M) = 0$.
By Corollary~\ref{C:one slope} (in the case $s=1$),
$P$ can only have one Newton slope, which must be 0;
hence $P(T)$ has coefficients in $\mathfrak{o}_K$. We can then apply
Lemma~\ref{L:split poly} to construct $L$ over which the equation
$P(\phi)(x) = 0$ has a solution $x \in \mathfrak{o}_L^*$;
that solution
gives rise to a nontrivial $\phi$-submodule of $M$.
Repeating
the construction, we obtain a field over which $M$ becomes a successive
extension of trivial \'etale $\phi$-modules of rank 1. By repeated use of
Lemma~\ref{L:split h1}, we can split this filtration
by passing to a suitably large $L$.
This yields the claim.
\end{proof}
\begin{remark}
Note that the field $L$ constructed above is not a Picard-Vessiot extension
of $K$ in the sense of the Galois theory of difference fields;
this Galois theory is a bit
complicated because it cannot be carried out within the category of fields,
as examples like the difference equation $\phi(x) = -x$ show.
See \cite[Chapter~1]{singer-vanderput} for more discussion of this point,
and a development of difference Galois theory in a restricted setting;
see also \cite{andre-diff} for a more general development. (Thanks
to Michael Singer for pointing out this reference.)
\end{remark}
\subsection{The use of faithfully flat descent}
In this subsection, we set up faithfully flat descent and illustrate
how we will use it to prove
Theorems~\ref{T:ascend semistable}
and~\ref{T:descend pure}.
\begin{defn} \label{D:descent}
Let $R \to S$ be a faithfully flat morphism of rings
equipped with
compatible endomorphisms $\phi$. Let $M$ be a $\phi$-module over $R$,
put $M_S = M \otimes_R S$, and let $N_S$ be a $\phi$-submodule of $M_S$.
We say that $N_S$ \emph{descends to $R$} if there exists a
$\phi$-submodule $N$ of $M$ such that the image of
$N \otimes_R S$ in $M_S$ coincides with $N_S$.
We say a filtration descends to $R$ if each
term does so.
\end{defn}
\begin{prop} \label{P:descent}
Let $R \to S$ be a faithfully flat morphism of domains
equipped with compatible endomorphisms $\phi$.
Put $S_2 = S \otimes_R S$ and define $i_1, i_2: S \to S_2$ by
$i_1(s) = s \otimes 1$ and $i_2(s) = 1 \otimes s$.
Let $M$ be a $\phi$-module over $R$, put $M_S = M \otimes_R S$, and let
$N_S$ be a $\phi$-submodule of $M_S$.
Then $N_S$ descends to $R$ if and only if $N \otimes_{i_1} S_2 =
N \otimes_{i_2} S_2$ within $M \otimes_R S_2$; moreover, if this occurs,
then there is a \emph{unique} $\phi$-submodule $N$ of $M$ such that
$N_S = N \otimes_R S$ within $M_S$.
\end{prop}
\begin{proof}
The equality $N \otimes_{i_1} S_2 =
N \otimes_{i_2} S_2$ implies that the effective descent datum obtained from $M$
induces a descent datum on $N$ (the cocycle condition can be checked on $M$).
We may thus apply faithfully flat descent for modules
\cite[Expos\'e~VIII, Corollaire~1.3]{grothendieck} to conclude.
\end{proof}
We use faithfully flat descent as follows.
\begin{defn}
Define
\begin{align*}
\mathcal{S} &= \tilde{\mathcal{R}}_L \otimes_{\mathcal{R}} \tilde{\mathcal{R}}_L \\
\mathcal{S}^{\bd} &= \tilde{\mathcal{R}}_L^{\bd} \otimes_{\mathcal{R}^{\bd}} \tilde{\mathcal{R}}_L^{\bd} \\
\mathcal{S}^{\inte} &= \tilde{\mathcal{R}}_L^{\inte} \otimes_{\mathcal{R}^{\inte}} \tilde{\mathcal{R}}_L^{\inte}.
\end{align*}
We will show later that $\mathcal{R} \to \tilde{\mathcal{R}}_L$, $\mathcal{R}^{\bd} \to \tilde{\mathcal{R}}_L^{\bd}$
are faithfully flat and that
$\mathcal{S}^{\bd} \to \mathcal{S}$ is injective
(Remark~\ref{R:faithfully flat}).
\end{defn}
The following weak analogue of Proposition~\ref{P:h1 both ways}
will be proved in Subsection~\ref{subsec:tensor}.
\begin{prop} \label{P:tensor bounded}
Let $A$ be an $n \times n$ matrix over $\mathcal{S}^{\inte}$, and let
$\mathbf{v}$ be a column vector over $\mathcal{S}$
such that $\mathbf{v} = A \phi(\mathbf{v})$. Then
$\mathbf{v}$ has entries in $\mathcal{S}^{\bd}$.
\end{prop}
We now demonstrate how Proposition~\ref{P:tensor bounded} can be used
to establish the theorems asserted at the start of this section.
\begin{proof}[Proof of Theorem~\ref{T:ascend semistable}]
Suppose that $M \otimes \tilde{\mathcal{R}}_L$ is not semistable.
Let $0 = M_{L,0} \subset M_{L,1} \subset \cdots \subset M_{L,l} = M_L$
denote the HN filtration of $M_L = M \otimes \tilde{\mathcal{R}}_L$. We will show
that $M_{L,1} \otimes_{i_2} S_2 \subseteq M_{L,j} \otimes_{i_1} S_2$ for
$j = l,l-1,\dots,1$ by descending induction; the base case $j=l$ is trivial.
Given that $M_{L,1} \otimes_{i_2} S_2 \subset M_{L,j} \otimes_{i_1} S_2$
for some $j > 1$,
we get a homomorphism
\[
M_{L,1} \otimes_{i_2} S_2 \to (M_{L,j}/M_{L,j-1}) \otimes_{i_1} S_2.
\]
Since $M_{L,1}$ and $M_{L,j}/M_{L,j-1}$ are pure and
$\mu(M_{L,1}) < \mu(M_{L,j}/M_{L,j-1})$, this homomorphism is forced to
vanish: otherwise,
by Proposition~\ref{P:tensor bounded} the morphism would be defined
over $\mathcal{S}^{\bd}$, but in that case it would have to preserve slopes
because $\mathcal{S}^{\bd}$ carries an $\mathfrak{m}_K$-adic valuation.
Hence $M_{L,1} \otimes_{i_2} S_2 \subseteq M_{L,j-1} \otimes_{i_1} S_2$,
completing the induction.
The induction shows that $M_{L,1}$ satisfies the condition for
faithfully flat descent (Proposition~\ref{P:descent}), so it descends
to $\mathcal{R}$. Hence $M$ cannot be semistable either.
\end{proof}
\begin{proof} [Proof of Theorem~\ref{T:descend pure}]
By applying $[a]_*$ (invoking
Lemma~\ref{L:pure push})
and twisting, we may reduce to the case $\mu(M) = 0$,
so $M \otimes \tilde{\mathcal{R}}_L$ is \'etale.
Choose a basis $\mathbf{v}_1, \dots, \mathbf{v}_n$ of an \'etale lattice of
$M \otimes \tilde{\mathcal{R}}_L$, so that the matrix
$A$ defined by $\phi(\mathbf{v}_j) = \sum_i A_{ij} \mathbf{v}_i$ is
invertible over $\tilde{\mathcal{R}}_L^{\inte}$.
There exists an invertible change-of-basis matrix $U$
over $\mathcal{S}$ such that
\[
\mathbf{v}_j \otimes_{i_1} 1 = \sum_i U_{ij} (\mathbf{v}_i \otimes_{i_2} 1).
\]
Upon applying $\phi$ to both sides, we deduce that
$U(A \otimes_{i_1} 1) = (A \otimes_{i_2} 1)\phi(U)$.
By Proposition~\ref{P:tensor bounded},
$U$ has entries in $\mathcal{S}^{\bd}$, as does its inverse by the same
argument with $M$ replaced by $M^\vee$.
Hence by Proposition~\ref{P:descent}, $M$ descends to $\mathcal{R}^{\bd}$;
let $N$ be the resulting $\phi$-module over $\mathcal{R}^{\bd}$.
Choose any basis of $N$ and let $P$ be the $\mathcal{R}^{\inte}$-span of the images
of the basis elements under powers of $\phi$.
By computing in terms of $\mathbf{v}_1, \dots,
\mathbf{v}_n$, we see that $P$ is bounded, hence is a $\phi$-stable
$\mathcal{R}^{\inte}$-lattice
in $M$. By Lemma~\ref{L:etale lattice}, $P \otimes \tilde{\mathcal{R}}_L^{\inte}$
is a $\phi$-module, as then must be $P$. Thus $M$ is \'etale, as desired.
\end{proof}
It now remains to prove the faithful flatness results and to make the
calculation to check Proposition~\ref{P:tensor bounded}; these occupy the
remainder of the chapter.
\subsection{Interlude: tensoring over B\'ezout domains}
In order to use faithfully flat descent for our purposes,
it will help to gather a few facts about
tensoring over B\'ezout domains.
\begin{prop} \label{P:faithfully flat1}
Let $R \hookrightarrow S$ be an inclusion of domains with $R$
B\'ezout. Then $S$ is faithfully flat over $R$ if and only if
$S^* \cap R = R^*$.
\end{prop}
\begin{proof}
Recall that $S$ is flat (resp.\ faithful) over $R$ if and only if for each
finitely generated proper ideal $I$ of $R$, the multiplication
map $I \otimes S \to S$ is injective (resp.\ not surjective).
Since $R$ is B\'ezout, $I$ admits a single generator $r \notin R^*$, and
$I \otimes S = rR \otimes S \cong rS$, so the map $I \otimes S \to S$
is injective, and it is surjective if and only if $r \in S^*$.
This yields the claim.
\end{proof}
\begin{lemma} \label{L:rewrite tensor}
Let $M,N$ be modules over a B\'ezout domain $R$. Given a presentation
$\sum_{i=1}^n y_i \otimes z_i$ of $x \in M \otimes_{R} N$ and elements
$u_1, \dots, u_n \in R$ generating the unit ideal,
there exists another presentation
$\sum_{j=1}^n y'_j \otimes z'_j$ of $x$ with $y'_1 = \sum_{i=1}^n u_i y_i$.
\end{lemma}
\begin{proof}
By \cite[Lemma~2.3]{kedlaya-local},
we can construct an invertible matrix $U$ over $R$ with $U_{i1} = u_i$
for $i=1,\dots,n$.
Then
\begin{align*}
\sum_i y_i \otimes z_i &=
\sum_{i,j,l} U_{ij} (U^{-1})_{jl} y_i \otimes z_l \\
&= \sum_j \left( \sum_i U_{ij} y_i \right) \otimes
\left( \sum_l (U^{-1})_{jl} z_l \right),
\end{align*}
so we may take $y'_j = \sum_{i=1}^n U_{ij} y_i$ and $z'_j = \sum_{l=1}^n (U^{-1})_{jl} z_l$.
\end{proof}
\begin{cor} \label{C:lin ind}
Let $M,N$ be modules over a B\'ezout domain $R$.
If $\sum_{i=1}^n y_i \otimes z_i$ is a presentation of
some $x \in M \otimes_R N$ with $n$ minimal,
then $y_1, \dots, y_n$ are linearly independent over $R$.
\end{cor}
\begin{proof}
If on the contrary $y_1, \dots, y_n$ are linearly dependent over $R$,
then we can find $u_1, \dots, u_n \in R$ such that $u_1 y_1 + \cdots + u_n y_n = 0$.
By the B\'ezout property, $u_1, \dots, u_n$ generate a principal ideal, so we
can divide through by a generator to reduce to the case where $u_1, \dots, u_n$
generate the unit ideal. Applying Lemma~\ref{L:rewrite tensor} now yields
a contradiction to the minimality of $n$.
\end{proof}
\subsection{Projections}
\label{subsec:tensor}
The key to the descent argument is the construction of a certain projection
from $\tilde{\mathcal{R}}_L$ back to $\mathcal{R}$, sectioning the inclusion going the other way
that was constructed by Proposition~\ref{P:embedding}.
We now construct this projection, then use it to resolve all the outstanding
statements needed to complete the proof of Theorem~\ref{T:slope filt}.
\begin{defn}
Let $\ell$ be the residue field of $L$,
fix a basis $\overline{B}$ of $\ell$ over $k$ containing 1,
lift $\overline{B}$ to a subset $B$ of $\mathfrak{o}_L$ containing 1,
and fix a uniformizer $\pi$ of $K$.
Then as in \cite[Proposition~4.1]{kedlaya-full}, one sees that every
element $x \in \tilde{\mathcal{R}}^{\inte}_L/\mathfrak{m}_K^n \tilde{\mathcal{R}}_L^{\inte}$ can be written
uniquely as a formal sum
\[
\sum_{\alpha \in [0,1) \cap \mathbb{Q}} \sum_{b \in B} x_{\alpha,b} u^\alpha b
\qquad (x_{\alpha,b} \in \mathcal{R}^{\inte}/\mathfrak{m}_K^n \mathcal{R}^{\inte})
\]
in which:
\begin{itemize}
\item
for each $\alpha
\in [0,1) \cap \mathbb{Q}$, there are only finitely many $b$ for which $x_{\alpha,b}
\neq 0$;
\item
if we write $S_c$ for the
set of $\alpha \in [0,1) \cap \mathbb{Q}$ for
which the $t$-adic valuation of any $x_{\alpha,b}$
(which is well-defined because
$x_{\alpha,b}$ is truncated modulo $\pi^n$) is less than $c$, then
$S_c$ is well-ordered for all $c$ and empty for sufficiently small $c$.
\end{itemize}
Given $x$ thusly presented, write $f(x) = x_{0,1}$; then
again as in \cite[Proposition~4.1]{kedlaya-full},
one checks that for $r_0$ as in Remark~\ref{R:annuli} and $r \in (0,r_0)$,
$f$ induces a continuous map $\tilde{\mathcal{R}}_L^r \to \mathcal{R}^r$ with the
property that for $x \in \tilde{\mathcal{R}}_L^r$,
\begin{equation} \label{eq:compare bounds}
|x|_r = \sup_{\alpha \in [0,1) \cap \mathbb{Q}, a \in L^*} \{
|a|^{-1} e^{-\alpha r} |f(a u^{-\alpha} x)|_r\}.
\end{equation}
(Compare also
\cite[Proposition~8.1]{dejong} and \cite[Lemma~2.2.19]{kedlaya-slope}.)
\end{defn}
\begin{prop} \label{P:mult}
The multiplication map $\tilde{\mathcal{R}}_L^{\bd} \otimes_{\mathcal{R}^{\bd}} \mathcal{R} \to
\tilde{\mathcal{R}}_L$ is injective.
\end{prop}
\begin{proof}
Suppose the contrary; choose $x \neq 0$
in the kernel of the multiplication map, and
choose a presentation $x = \sum_{i=1}^n y_i \otimes z_i$ with $n$ minimal.
Then $z_1, \dots, z_n$ are linearly independent over $\mathcal{R}^{\bd}$ by
Corollary~\ref{C:lin ind}. On the other hand, as a
corollary of \eqref{eq:compare bounds}, we may choose $\alpha \in
[0,1) \cap \mathbb{Q}$ and $a \in L^*$ such that
$f(a u^{-\alpha} y_1) \neq 0$; we then obtain the nontrivial dependence
relation
$0 = \sum_{i=1}^n f(a u^{-\alpha} y_i) z_i$, contradiction.
\end{proof}
\begin{remark} \label{R:faithfully flat}
We now have a number of faithfully flat inclusions.
For one, $\mathcal{R}^{\bd} \to \mathcal{R}$ is faithfully flat by
Proposition~\ref{P:faithfully flat1} and the fact that
$\mathcal{R}^* = (\mathcal{R}^{\bd})^*$ (Remark~\ref{R:same units}).
For another, $\mathcal{R} \to \tilde{\mathcal{R}}_L$ is faithfully flat by
Proposition~\ref{P:faithfully flat1} and the fact that
$\tilde{\mathcal{R}}_L^* = (\tilde{\mathcal{R}}_L^{\bd})^*$ (Remark~\ref{R:analytic ring});
similarly, $\mathcal{R}^{\bd} \to \tilde{\mathcal{R}}_L^{\bd}$ is faithfully flat.
Putting these together and using Proposition~\ref{P:mult}
yields injections
\[
\tilde{\mathcal{R}}_L^{\bd} \otimes_{\mathcal{R}^{\bd}} \tilde{\mathcal{R}}_L^{\bd}
\hookrightarrow \tilde{\mathcal{R}}_L^{\bd} \otimes_{\mathcal{R}^{\bd}} \tilde{\mathcal{R}}_L
\cong (\tilde{\mathcal{R}}_L^{\bd} \otimes_{\mathcal{R}^{\bd}} \mathcal{R}) \otimes_{\mathcal{R}} \tilde{\mathcal{R}}_L
\hookrightarrow \tilde{\mathcal{R}}_L \otimes_{\mathcal{R}} \tilde{\mathcal{R}}_L;
\]
that is, $\mathcal{S}^{\bd} \to \mathcal{S}$ is injective.
\end{remark}
In order to calculate on $\mathcal{S}$, we use the following two-variable
analogue of \eqref{eq:compare bounds}.
\begin{lemma} \label{L:bounded}
For $x \in \mathcal{S}$, we have $x \in \mathcal{S}^{\bd}$ if and only if
for some $r>0$,
the quantities
\begin{equation} \label{eq:bounded}
|ab|^{-1} e^{-\alpha s - \beta s} |(f \otimes f)((a u^{-\alpha}
\otimes b u^{-\beta})x)|_s
\end{equation}
are bounded over all $s \in (0,r]$, all $a,b \in L^*$, and all
$\alpha,\beta \in [0,1) \cap \mathbb{Q}$.
\end{lemma}
\begin{proof}
If $x \in \mathcal{S}^{\bd}$, then we can bound the quantity
\eqref{eq:bounded} by bounding each term in a presentation of $x$.
Conversely, suppose the quantity \eqref{eq:bounded} is bounded.
Choose a presentation $x = \sum_{i=1}^n y_i \otimes z_i$
with $y_i, z_i \in \tilde{\mathcal{R}}_L$ and $n$ minimal.
We proceed by induction on $n$; we may assume
$x \neq 0$. Then $y_1 \neq 0$, so we can choose
$a, \alpha$ with $f(a u^{-\alpha} y_1) \neq 0$.
By \eqref{eq:compare bounds}, $\sum_{i=1}^n f(a u^{-\alpha} y_i) z_i
\in \tilde{\mathcal{R}}_L^{\bd}$; in particular, the ideal generated by
the $f(a u^{-\alpha} y_i)$ in $\mathcal{R}$
extends to the unit ideal in $\tilde{\mathcal{R}}_L$. Since the ideal in $\mathcal{R}$
is finitely generated, it is principal, and since
$\tilde{\mathcal{R}}_L^* = (\tilde{\mathcal{R}}_L^{\bd})^*$, the generator in $\mathcal{R}$ must already
be a unit. That is, the $f(a u^{-\alpha} y_i)$ generate the unit ideal
in $\mathcal{R}$; by Lemma~\ref{L:rewrite tensor}, we can choose another presentation
$x = \sum_{i=1}^n y'_i \otimes z'_i$ with $z'_1 = \sum_{i=1}^n
f(a u^{-\alpha} y_i) z_i \in \tilde{\mathcal{R}}_L^{\bd}$. We must have $z'_1 \neq 0$
to avoid contradicting the minimality of $n$.
Pick $b, \beta$ so that $f(b u^{-\beta} z'_1)$ is nonzero and hence is a unit
in $\mathcal{R}$ (since it must lie in $\mathcal{R}^{\bd}$).
Put $c_i = f(b u^{-\beta} z'_i) / f(b u^{-\beta} z'_1)$ for $i=2, \dots, n$,
then set
\[
y''_i =
\begin{cases}
y'_1 + c_2 y'_2 + \cdots + c_n y'_n & i = 1 \\
y'_i & i > 1,
\end{cases}
\qquad
z''_i =
\begin{cases}
z'_i & i = 1 \\
z'_i - c_i z'_1 & i > 1,
\end{cases}
\]
so that $x = \sum_{i=1}^n y_i'' \otimes z_i''$.
Then $f(b u^{-\beta} z''_i) = 0$ for $i=2, \dots, n$,
so $y''_1 f(b u^{-\beta} z''_1) = \sum_{i=1}^n y''_i f(b u^{-\beta} z''_i) \in
\tilde{\mathcal{R}}_L^{\bd}$ by \eqref{eq:compare bounds}.
Since already $f(b u^{-\beta}
z''_1) \in
\mathcal{R}^{\bd}$, we have $y''_1 \in \tilde{\mathcal{R}}_L^{\bd}$. Applying the induction
hypothesis to $x - y''_1 \otimes z''_1 = \sum_{i=2}^n y''_i \otimes z''_i$ yields the claim.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{P:tensor bounded}]
For each entry $\mathbf{v}_i$ of $\mathbf{v}$, choose a presentation
$\sum_{j} y_{ij} \otimes z_{ij}$ with $y_{ij}, z_{ij} \in \tilde{\mathcal{R}}_L$.
As in the proof of Proposition~\ref{P:h1 both ways},
after possibly rescaling by a power of $u$,
we may choose $r \in (0,r_0)$ such that each term in a presentation of
$A$ has entries in $\tilde{\mathcal{R}}_L^r$ and
is bounded by 1 on the annulus $e^{-r} \leq |u| < 1$;
we may also ensure that
$y_{ij},z_{ij} \in \tilde{\mathcal{R}}_L^r$ for all $i,j$.
Choose $c>0$ such that for $s \in [r/q,r]$ and all $i,j$,
$|y_{ij}|_s \leq c$ and $|z_{ij}|_s \leq c$ (possible because we are
picking $s$ in a closed interval);
then for all nonnegative integers $m$, we have
$|\phi^m(y_{ij})|_{s/q^m} \leq c$ and $|\phi^m(z_{ij})|_{s/q^m} \leq c$.
{}From the equation
\[
\mathbf{v} = A \phi(A) \cdots \phi^{m-1}(A) \phi^m(\mathbf{v}),
\]
we deduce that for all $\alpha,\beta \in [0,1)$ and all $a,b \in L^*$,
\[
|ab|^{-1} e^{-\alpha s-\beta s} |(f \otimes f)((a u^{-\alpha} \otimes
b u^{-\beta}) \mathbf{v})|_s \leq c
\]
for all $s \in [r/q^{m+1}, r/q^m]$; by varying $m$, we get the same
conclusion for all $s \in (0,r]$. By Lemma~\ref{L:bounded},
$\mathbf{v}$ has entries in $\mathcal{S}^{\bd}$, as desired.
\end{proof}
|
1,314,259,995,834 | arxiv | \section{Introduction}
Digital VLSI designs are getting much larger and more complex as manufacturing technologies are making rapid progress. Standard cell-based designs exploit libraries of small predefined building blocks called standard cells, which facilitate the productivity and reliability of the VLSI design flow. Accordingly, standard cell libraries have a fundamental impact on the quality of VLSI designs, e.g., power, performance, area, and cost (PPAC)~\cite{PD}.
\subsection{Background of Standard Cell Merging}
Since commonly-used standard cell libraries cannot meet all the requirements in some special scenarios~\cite{domainASIC,domainFPGA,ALPINE,pattern}, as an alternative solution, academia~\cite{CELLERITY,cellTK,ASTRAN,Libra,ASAP7,gateMerging,iTPlace,li2020mcell,DTCO,SPR,NCTUcell,7nm2019,Bonncell,PROBE2} and industry~\cite{postlayoutPattern,specificComplex,pattern,IBM,NVCell} take continuous efforts to extend standard cell libraries with custom standard cells for their technology nodes and domain-specific designs. One of the potential sources of these custom standard cells is \textbf{\emph{standard cell merging}}, which merges several existing standard cells into a new one with an optimized layout, as a simplified example shown in Fig.~\ref{merging}. It can benefit the design flow from two aspects:
\begin{itemize}
\item It enlarges the back-end solution space of the VLSI design, e.g., placement and routing, and hence enables some design goals which cannot be achieved based on the original standard cell library, via transistor-level optimizations like diffusion sharing and in-cell signal routing~\cite{ASTRAN}.
\item It shrinks the problem scale of the VLSI back-end tools by reducing the number of gates in the post-technology mapping gate-level netlist and enforcing pre-defined relative position constraints. These factors facilitate the tools to converge at better results~\cite{jianliPlacer}.
\end{itemize}
Standard cell merging provides noticeable optimization potentials but it is critically challenging since numerous factors, e.g., the design context, transistor layout, design rules and expected PPAC metrics, should be considered to realize a beneficial library. First, as for the design context, designers should identify the characteristics of the target design to locate the optimization opportunities and design pattern mining is one of the promising approaches. Second, the transistor network and layout should be designed under the constraints of design rules and PPAC metrics, which is usually called standard cell layout synthesis. In this paper, we propose a fully automatic standard-cell library extension framework, AutoCellLibX which can analyze characteristics of the target gate-level netlist and extent an initial standard cell library with custom complex standard cells to minimize the area cost. Related works and challenges are discussed in Section~\ref{background} and ~\ref{challenges} respectively.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{merging.pdf}
\caption{Simplified example of standard cell merging: it shows the advantages of standard cell merging which enables transistor-level optimization.}
\label{merging}
\end{figure}
\subsection{Related Works}\label{background}
\subsubsection{Standard Cell Layout Synthesis}
This workflow takes in three types of inputs~\cite{CELLERITY,PROBE2}: (A) cell architecture settings, e.g. the number of routing tracks, and the location of power rails; (B) design rules, e.g., width/spacing rules for all mask layers depending on cell architecture; and (C) transistor netlist indicating the interconnection of sized library cells. With these inputs, standard cell layout synthesis will conduct transistor placement, in-cell routing, and design rule checking. The general objectives of this synthesis flow are to improve the performance and lower the consumption of area/power.
Cellerity~\cite{CELLERITY} was proposed in 1997 as a comprehensive standard cell layout synthesis flow. It consists of simulated annealing transistor placement, Echelon in-cell router~\cite{Echelon}, and SQUEEZE layout compactor~\cite{SQUEEZE}. In 2014, ASTRAN~\cite{ASTRAN}, an open-source standard cell layout synthesis framework, was released and it realized promising results based on a cell layout compaction methodology using MILP. Moreover, some solutions~\cite{ye2015standard,xu2016parr,yu2020pin} related to layout regularity and pin access optimization have been done. Recently, a series of researches targeting sub-10nm technology nodes~\cite{SPR,NCTUcell,Bonncell,7nm2019,PROBE2} were presented, most of which utilized SMT solvers to optimize the layout of cell designs and, ~\cite{SimultaneousLayout} realized efficient simultaneous transistor folding and placement. These frameworks focus on the transistor layouts, and the analysis of the overall gate-level netlist and the input of customization are left to the manual effort.
\subsubsection{Design Pattern Mining}
\textbf{\emph{Isomorphic design patterns}} are defined as the identical or functionally equivalent logic structures which recur frequently in the entire design, as an example shown in Fig.~\ref{FSM_intro}. Most styles of the manual design flow, design architecture, synthesis, or mapping have a bias to generate recurring regular patterns in the gate-level netlists~\cite{pattern}. Such regularity can guide the standard cell design since frequently-recurring patterns imply the high coverage and reuse of the corresponding custom standard cell designs.
Accordingly, design pattern mining is to identify which part of the gate-level netlist should be replaced by custom standard cell designs. It's hard since the netlists are large and complex and related solutions were not provided in the aforementioned previous works~\cite{CELLERITY,cellTK,ASTRAN,Libra,ASAP7,gateMerging,iTPlace,DTCO,SPR,NCTUcell,7nm2019,Bonncell,PROBE2,postlayoutPattern,specificComplex,pattern,IBM,NVCell}. In some works on synthesis, placement, and routing, researchers have tried to extract template-based regularity from gate-level netlists, e.g., datapath~\cite{datapath1,datapath2,datapath3,datapath4} or array~\cite{array1}, but it is hard for these template-based solutions to handle the general gate-level netlists with complex interconnections. Moreover, they did not check the isomorphism between the patterns.
Frequent subgraph (pattern) mining (FSM) is a hot topic of computer science with some matured solutions like ~\cite{apriori, gspan,FFSM,grami}. These techniques have been widely applied in many domains to extract patterns~\cite{jiang2013survey}, such as chemistry, web, social network, and biology. Only a few works~\cite{AFSEM, FSMVLSI} have been done for VLSI gate-level netlist. Moreover, they mainly focused on directly applying FSM algorithms like ~\cite{gspan} on gate-level netlists to find frequent sub-circuits and the consideration of VLSI characteristics was absent in their solutions.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{FSM_intro.pdf}
\caption{Example of Frequent Subgraph Mining: It can find the frequently-recurring local logic structures (i.e., design patterns) in a large gate-level design for standard cell customization}
\label{FSM_intro}
\end{figure}
\subsection{Challenges of Automated Standard Cell Library Extension}\label{challenges}
To realize the co-optimization of designs and standard cell library and close the loop of netlist pattern mining and standard cell layout synthesis, we need to overcome the following challenges:
\begin{itemize}
\item Without identifying the promising pattern subgraphs as the inputs of standard cell layout synthesis, the resultant custom standard cell might not benefit the VLSI design. Meanwhile, simply selecting one or two patterns from the customization candidates to create new standard cells cannot realize the optimal solution, e.g., minimizing area cost. Therefore, we need to find the proper combination of patterns.
\item Without the consideration of the requirements of VLSI design flow, FSM algorithms cannot find the optimal candidates for standard cell layout synthesis. For example, in previous works, their FSM solutions simply focused on the frequency of pattern recurrence in the gate-level netlist. However, pattern coverage and area saving after customization should be considered. Moreover, the overlapped pattern subgraphs in the netlist should not be counted more than once since each node in the netlist can be mapped to only one standard cell during the mapping flow in logic synthesis. Finally, it is difficult to identify the patterns from large and complex gate-level netlist since the general FSM algorithms are timing-consuming and memory-intensive.
\end{itemize}
With the consideration of the scenarios in real applications discussed above, the contributions of AutoCellLibX are highlighted as follows:
\begin{itemize}
\item A practical vertex encoding algorithm, which can find a proper set of neighbor vertex for pattern growth with the consideration of standard cell characteristics, as a part of high-efficiency FSM solution;
\item A pattern growth algorithm that can expand the sizes of gate-level patterns while preserving their high recurrence frequencies. Compared to previous FSM approaches, our pattern growth solution carefully handle the overlaps between pattern subgraphs to meet the technology mapping constraint and maximize area reduction;
\item A pattern combination algorithm which can iteratively find a set of gate-level patterns from numerous candidates as the extension part of the initial standard cell library to maximize the area reduction of the entire VLSI design;
\item To the best of our knowledge, it is the first automated standard cell extension framework that closes the optimization loop between the analysis of gate-level netlist and standard cell library customization. AutoCellLibX can generate SPICE netlists and GDSII layouts of the custom standard cells for downstream VLSI design flow.
\end{itemize}
\section{Preliminaries}\label{Preliminaries}
In this section, we formulate the pattern mining problem in the scenarios of standard cell library extension and present an overview of our proposed framework.
\subsection{Problem Formulation}\label{formulation}
The post-technology mapping gate-level netlist can be formulated as a directed graph $G = (V, E, Lv, Le)$ with the following definitions:
\begin{itemize}
\item vertices $V(G) = \{v_1 , v_2, \dots , v_n \}$ represent cells after technology mapping in the gate-level netlist with flip-flops removed. The reason for such cell removal is that sequential circuits consist of combinational logic gates and sequential memory elements, and our targets of customization are the combinational logic gates.
\item edges $E(G) =\{e_1, e_2, \dots , e_m\}$ represent $m$ pin-to-pin nets according to the driver-sink relationships in the gate-level netlist
\item $L{v_i}$ represents the label for vertex $v_i$, which is the standard cell type of $v_i$
\item $L{e_i}$ represents the label for edge $e_i$, which is a tuple $({Output}_i, {Input}_i)$, indicating that the edge $e_i$ connects an output pin named ${Output}_i$ of its driver cell and an input pin named ${Input}_i$ of its sink cell.
\item A pattern group $PGS_i = \{G'_1, G'_2, \dots , G'_{Ns} \}$ is a set of $N_s$ subgraphs of $G$. All the vertices in each subgraph in $PGS_i$ are connected. The subgraphs in $PGS_i$ are isomorphic, and it means that the sub-circuits corresponding to these subgraphs have the same circuit functionality. They are not overlapped with each other, i.e., when $ p\neq q$ and $ G'_p, G'_q\in PGS_i $, we have $V(G'_p) \cap V(G'_q)=\emptyset $
\item The coverage of a pattern group $Cov(PGS_i)$ is the number of vertices in it, i.e., $Cov(PGS_i) = | V(G'_1) \cup V(G'_2) \cup \dots \cup V(G'_{Ns}) |$, where $G'_j \in PGS_i$ \label{Coverage}
\item A combination of pattern candidates $C = \{PGS_1 , PGS_2, \dots , PGS_{Np} \}$ is a set of ${N_p}$ pattern groups. The subgraphs in different pattern groups are not isomorphic, and they are not overlapped with each other, i.e., $\forall PGS_i,PGS_j \in C$, when $i\neq j$, $\forall G' \in PGS_i$, $\forall H' \in PGS_j$, $G'$ is not overlapped with $H'$.
\item A reward function $R(C,G)$ indicates how much area can be saved for $G$ when each subgraph $G'_i$ in the pattern groups of $C$ is replaced by a corresponding custom standard cell generated by standard cell layout synthesis flow.
\end{itemize}
According to the definitions above, the problem of standard cell library extension can be formulated as follows: given a gate-level netlist $G$ and the initial standard cell library $LIB$,
\begin{equation}
\begin{aligned}
\max_{C} \quad & R(C,G)\\
\textrm{s.t.} \quad & |C|<N_p \\
& \forall G'\in PGS_i\in C, |G'|<S_p
\end{aligned}
\end{equation}
where $N_p$ is the maximum number of custom standard cell types and $S_p$ is the maximum number of vertices in each pattern subgraph. These two parameters are determined by designers to limit the size of the extended library and the size of the custom standard cell. Empirically, in our implementation, $N_p$ and $S_p$ are set to be 5 and 10 respectively, since when these parameters become larger, the area benefit is marginal for the benchmarks. More discussion about these parameters can be found in Section~\ref{TerminationCriteriaofPatternMining} The resultant $C$ will be used to complement $LIB$ and generate a new extended $LIB'$.
\begin{figure}[b]
\centering
\includegraphics[width=\linewidth]{outline.pdf}
\caption{The outline of AutoCellLibX's workflow with 5 process phases }
\label{outline}
\end{figure}
\subsection{The Framework of AutoCellLibX}
The workflow of AutoCellLibX is shown in Fig.~\ref{outline}. The inputs of AutoCellLibX are the post-technology mapping gate-level netlist (BLIF file) of the target VLSI design generated by Yosys~\cite{yosys} and an initial standard cell library used during the logic synthesis. The outputs are the SPICE netlists, GDSII layouts and the area reduction report of the combination of patterns. The workflow overview of AutoCellLibX is described in this subsection.
\subsubsection{Initial Pattern Seed Identification} Initial pattern seed identification generates a set of initial 2-level tree-based patterns, $\mathbf{L}_{init}$, which will be the starting point of later iterative pattern growth for standard cell customization.
\subsubsection{Neighbor Encoding} In the iterative loop, the first stage is to efficiently enumerate and encode the neighbor vertices of subgraphs in the target pattern group $PGS_{tgt}$ with the highest coverage in a list of sorted patterns, $\mathbf{L}_{iter}$ (or $\mathbf{L}_{init}$), to find the promising direction of the pattern growth.
\subsubsection{Pattern Growth} Inspired by ~\cite{gspan}, some of the encoded neighbor vertices will be absorbed into the subgraphs in $PGS_{tgt}$ and a new pattern group $PGS_{tgt}'$ with subgraphs of larger size is generated. Meanwhile, the global pattern information will be updated to improve the efficiency of later iterations of pattern mining.
\subsubsection{Generation of SPICE Netlist and GDSII Layout} According to the subgraph topology and label information in the new pattern group $PGS_{tgt}'$, a SPICE netlist indicating the interconnection of sized transistors for a potential custom standard cell will be generated. With the SPICE netlist and predefined design rules of the initial standard cell library, ASTRAN~\cite{ASTRAN}, the open-source standard cell layout synthesis tool, will be called to generate the GDSII layout of the custom cell.
\subsubsection{Pattern Combination Evaluation} According to the generated GDSII layouts and the recurrence frequencies of the pattern groups, the overall benefit of a potential combination of patterns are evaluated by replacing corresponding subgraphs in $G$ with the custom standard cells. In this stage, AutoCellLibX will analyze the improvement potential of the patterns and make the decision whether further pattern mining iterations should be terminated.
As shown in Fig.~\ref{outline}, neighbor encoding, pattern growth, generation of SPICE netlist and GDSII layout, and pattern combination evaluation are the five successive stages in the iterative pattern mining loop.
\section{Implementation of AutoCellLibX}\label{implementation}
\subsection{Initial Pattern Seed Identification}\label{InitialPatternSeedIdentification}
The pattern growth procedure is to let some frequent subgraphs gradually absorb their neighbor vertices and initial patterns are the seeds of pattern growth in FSM algorithms.
In conventional solutions, like ~\cite{gspan,FFSM,grami}, which mainly count the frequencies of various subgraph patterns without other constraints, the initial patterns are simply all the edges with two interconnected vertices in $G$. For the general scenarios of standard cell customization without considering logic duplication, each vertex in the gate-level netlist should be mapped to only one standard cell during technology mapping. Edge-based initial patterns might lead to the result that some vertices will be covered by multiple pattern subgraphs, such as high-fanin cells and broadcast signals. For example, the NOR3X1 cell with index 0 is connected with two AND2X2 cells in Fig.~\ref{treePattern} but the NOR3X1 cell can be mapped to only one custom standard cell with one of the two AND2X2 cells.
To consider the constraint aforementioned, our pattern seed identification employs tree encoding as illustrated in Algorithm~\ref{initialPatternAlgorithm}.
\subsubsection{Initial Tree Encoding} In $G$, each vertex will be regarded as a root, and its 1-hop predecessors will construct a 2-level tree with the root vertex. Each of these trees will be encoded with a tree code, according to the steps shown in line 1-13 of Algorithm~\ref{initialPatternAlgorithm}. The tree code for a tree consists of two parts: the code of root (root code) and the code of leaves (leaf codes). The root code is simply the standard cell type of the root, like "[NOR3X1]". The leaf code for each leaf in the tree will be a code including its standard cell type, and the label of the edge interconnecting the root and it, i.e., $Le_i$ in Section~\ref{formulation}. For example, "(AND2X2,Y,A)" for AND2X2 cell with index 1 in Fig.~\ref{treePattern} means that the output pin "Y" of an AND2X2 cell is connected to the input pin "A" of the root NOR3X1 cell. The tree code is the root code followed by the leaf codes of leaves, where the leaf codes are concatenated in the lexicographical order (line 5 in Algorithm~\ref{initialPatternAlgorithm}). For example, the tree code for the initial tree in red dash curve in Fig.~\ref{treePattern} is "[NOR3X1](AND2X2,Y,A)(AND2X2,Y,B)(NOR3X1,Y,C)".
It can be noticed that the tree code includes all the information of the corresponding tree, e.g., the vertices, the edges between them, and related labels. Meanwhile, due to the lexicographical order of the codes of leaves in tree code, each of these initial tree subgraphs will be mapped to only one tree code. Therefore, trees with the same tree code are isomorphic.
As described in lines 8-13 in Algorithm~\ref{initialPatternAlgorithm}, AutoCellLibX will enumerate the vertices in $G$, construct corresponding pattern trees, encode the trees and record the recurrences of the tree patterns based on the codes. According to the recurrence record, we can rank the initial patterns in descending order of their recurrence frequencies.
\subsubsection{Overlap Elimination}\label{OverlapElimination} Please note that at this stage, the pattern subgraphs could overlap with some other patterns at some of the vertices. As described in lines 14-21 in Algorithm~\ref{initialPatternAlgorithm}, to eliminate these overlaps, AutoCellLibX will enumerate the patterns from those most frequent ones to those less frequent to conduct checking. If all the vertices in a subgraph have not been occupied, these vertices will be marked as occupied. Otherwise, the subgraph will be abandoned from the record. As an example shown in Fig.~\ref{patternOverlap}, if two pattern subgraphs are overlapped, one of the subgraphs will be abandoned if it is visited later or belong to a pattern with a lower frequency. In this way, AutoCellLibX removes the overlaps among the subgraphs in the patterns and preserves those high-frequency subgraph patterns for the reuse of custom standard cells. After the elimination of the overlaps, the pattern groups with different tree codes will be recorded in $\mathbf{L}_{init}$ and ranked in descending order of their coverage in $G$, i.e. $Cov(PGS_i)$ defined in Section~\ref{Coverage}, for later iterative pattern growth, since it is more likely for the patterns with high coverage to realize a noticeable impact on the entire design according to Section~\ref{BenefitEvaluationFunction}.
\begin{algorithm}[t]
\small
\DontPrintSemicolon
\KwIn{the gate-level netlist after technology mapping $G$}
\KwOut{A pattern list in descending order of the frequencies of the initial patterns $\mathbf{L}_{init}=\{{PGS}_i\}$}
\SetKwFunction{FMain}{TreeEncode}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\FMain{root, predecessors}}{
predCodes = \{\};
\ForEach{$v_i$ $\in$ predecessors}
{
predCodes.append(stdCellType($v_i$)+$Le$(root,$v_i$));
}
\textbf{return} concat(stdCellType(root), lexSort(predCodes));
}
code2Trees = record of the trees with the same code;
code2Freq = record of the code frequency;
\tcc{construct trees and encode them}
\ForEach{root $\in$ $V$}
{
preds = getPreds(root); // get direct 1-hop predecessors
treeCode = TreeEncode(root,preds);
code2Freq[treeCode] += 1;
code2Trees[treeCode].append(getTree(root));
}
sortedCodes = sortPatternCodes(code2Freq);
\tcc{eliminate overlaps}
\ForEach{code $\in$ sortedCodes}
{
\ForEach{tree $\in$ code2Trees[code]}
{
\If{allVerticesAreUnoccupied(tree)}
{
OccupyAndIndexAllVerticesIn(tree);
}
\Else
{
code2Freq[code] -= 1;
code2Trees[code].remove(tree);
}
}
}
$\mathbf{L}_{init}$ = sortedPatternListAccordingToCoverage(code2Trees);
\Return{$\mathbf{L}_{init}$}
\caption{Initial Pattern Seed Identification}
\label{initialPatternAlgorithm}
\end{algorithm}
\begin{figure*}[!t]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.5cm}
\centering
\subfigure[Details of an example pattern]{
\begin{minipage}[b]{0.25\linewidth}
\includegraphics[width=\linewidth]{encoding_zoomIn}
\label{treePattern}
\end{minipage}
}
\subfigure[Example of pattern overlap ]{
\begin{minipage}[b]{0.20\linewidth}
\includegraphics[width=\linewidth]{encoding_zoomoutC}
\label{patternOverlap}
\end{minipage}
}
\subfigure[Example of pattern recurrences: subgraphs in red dash curve are 5 separate isomorphic subgraphs in 1 $PGS_{tgt}$ ]{
\begin{minipage}[b]{0.50\linewidth}
\includegraphics[width=\linewidth]{encoding_zoomout}
\label{patternForest}
\end{minipage}
}
\caption{Example for the pattern mining procedure: initial pattern seed identification, neighbor encoding and pattern growth.}
\label{patternMining}
\end{figure*}
\subsubsection{Post-process for Iterative Pattern Mining} Each vertex in a tree will be assigned an index in the tree to facilitate isomorphism checking among subgraphs with the same pattern in later procedures. Root vertex will have an index of 0 while the predecessor vertices will be indexed from 1 to the number of predecessors, according to the lexicographical order of their leaf codes. For example, as shown in Fig.~\ref{treePattern}, the bottom NOR3X1 cell has an index of 0, and the two AND2X2 cells are assigned index 1 and 2 respectively. This index will be used in neighbor encoding in Section~\ref{NeighborEncodingBasedonInterconnectionFeatures}.
\begin{algorithm}[!t]
\small
\DontPrintSemicolon
\KwIn{the top-1 pattern group $PGS_{tgt}$ from $\mathbf{L}_{init}$ or $\mathbf{L}_{iter}$ }
\KwOut{the record of the neighbor vertices of $PGS_{tgt}$ and their pattern codes}
\SetKwFunction{FMain}{NeighborEncode}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\FMain{neighbor, patternSubG}}{
codes = \{\};
\tcc{ get the edge codes}
\ForEach{$v_i$ $\in$ patternSubG}
{
viIndex = getIndex($v_i$, patternSubG);
\If {neighbor is driver of $v_i$}
{
codes.append($Le$(neighbor,$v_i$,)+viIndex);
}
\Else{codes.append(viIndex+$Le$( $v_i$, neighbor));}
}
\textbf{return} concat(stdCellType(neighbor), lexSort(codes));
}
code2Neighbors=record of the neighbors with the same code;
neighbor2Subgraph = record of the owner subgraph of a neighbor;
code2Freq = record of the code frequency;
encoded = record of encoded vertices;
\tcc{enumerate the neighbors of the subgraphs in $PGS_{tgt}$ and encode them}
\ForEach{subG $\in PGS_{tgt}$}
{
\ForEach {neighbor $\in$ 1-hop neighbors of subG}
{
\If {NOT encoded[neighbor]}
{
code = NeighborEncode(neighbor, subG);
neighbor2Subgraph[neighbor] = subG;
encoded[neighbor] = True; // avoid overlaps
code2Neighbors[code].append(neighbor);
}
}
}
\Return{code2Neighbors, neighbor2Subgraph}
\caption{Neighbor Encoding}
\label{neighborEncodingAlgorithm}
\end{algorithm}
\subsection{Neighbor Encoding}\label{NeighborEncoding}
Our pattern mining procedure is based on pattern growth which gradually merges some selected vertices into the subgraphs of existing patterns to enlarge the pattern size while trying to preserve the recurrence frequencies of the patterns. Therefore, selecting proper neighbors is a critical part of pattern mining, which will be illustrated in this subsection.
AutoCellLibX will take the top-1 pattern group $PGS_{tgt}$ which has the highest coverage in the gate-level netlist, from $\mathbf{L}_{init}$ or $\mathbf{L}_{iter}$, as the input of neighbor encoding as depicted in Fig.~\ref{outline}. Correspondingly, the outputs of neighbor encoding are a set of neighbor vertices of $PGS_{tgt}$ and their pattern codes. The overall steps of neighbor encoding are presented in Algorithm~\ref{neighborEncodingAlgorithm}.
\subsubsection{Neighbor Enumeration} As shown in lines 14-20 of Algorithm~\ref{neighborEncodingAlgorithm}, in this stage, AutoCellLibX will enumerate the neighbors of all the subgraphs in $PGS_{tgt}$ to conduct encoding. Here, a neighbor of a subgraph is defined as a vertex that is not a vertex in the subgraph but is a 1-hop neighbor (predecessor or successor) of a vertex in the subgraph. The output of NeiborEncode function is determined by both the neighbor and the subgraph. A neighbor might be shared by several subgraphs of the same pattern but it will be encoded once when it is visited for the first time, as indicated by line 16 of Algorithm~\ref{neighborEncodingAlgorithm}. This constraint can ensure that there will be no overlap among the later extended pattern subgraphs and AutoCellLibX will record the owner subgraph of each neighbor as shown in line 18 of Algorithm~\ref{neighborEncodingAlgorithm}.
\subsubsection{Neighbor Encoding with Interconnection Features}\label{NeighborEncodingBasedonInterconnectionFeatures} The code of a neighbor (neighbor code) consists of its standard cell type, and the labels of edges between the neighbor and the subgraph, as demonstrated in line 3-9 of Algorithm~\ref{neighborEncodingAlgorithm}. Each edge between the neighbor and the subgraph will be enumerated and a corresponding edge code of the edge will be generated. As mentioned in Section~\ref{InitialPatternSeedIdentification} and ~\ref{PatternGrowth}, each vertex in an existing subgraph of a specific pattern has its index in the subgraph. Accordingly, the edge code includes the index of the vertex in the subgraph and the label of edges. The order of these two parts in the edge code depends on whether the neighbor is the driver of the edge. For example, if the subgraph in the red dash curve is the target subgraph, the edge between OR2X1 cell and NOR3X1 in Fig.~\ref{treePattern} will be encoded as "(Y,B,3)" while the edge between OR2X1 cell and AND2X2 in Fig.~\ref{treePattern} will be encoded as "(2,Y,A)". The neighbor code is constructed as the standard cell type of the neighbor vertex followed by the edge codes of edges between the neighbor vertex and the corresponding subgraph, where the edge codes are concatenated in the lexicographical order. For example, as shown in Fig.~\ref{treePattern}, the neighbor code of the OR2X1 cell will be "[OR2X1](2,Y,A)(Y,B,3)" if the subgraph in the red dash curve is the target subgraph for neighbor encoding. Consequently, the recurrence frequencies of the OR2X1 cell (encoded as "[OR2X1](2,Y,A)(Y,B,3)") and OR3X1 cell (encoded as "[OR3X1](Y,A,1)") is 1 and 3 respectively in Fig.~\ref{patternForest}.
Please be aware that some input pins of some standard cells, e.g., the input pin "A" and "B" of OR2X1 cell in Fig.~\ref{treePattern}, have equivalent functionality and highly similar parasitics. Switching the interconnection edges among these pins will not change the logic functionality. For the sake of this factor, during the actual implementation of edge code generation, equivalent pins will be encoded to the same name. For example, pin "B" of OR2X1 will be encoded as "A".
Since the neighbor code encodes all the necessary information of the interconnection between the neighbor and the vertices in the subgraph in a unique form, those neighbors with the same neighbor code have the isomorphic interconnection egdes with their corresponding subgraphs. Therefore, the neighbor codes can classify the neighbor vertices into different sets for later pattern growth.
In our proposed solution, the time complexity of neighbor encoding is O($|E|$), determined by lines 14-20 of Algorithm~\ref{neighborEncodingAlgorithm}, where neighbor vertices of the subgraphs are enumerated for encoding.
\newtheorem{mytheorem}{Theorem}
\subsection{Pattern Growth}\label{PatternGrowth}
In the stage of pattern growth, AutoCellLibX merges some neighbor vertices into the subgraphs of $PGS_{tgt}$ to enlarge the pattern size while trying to preserve the recurrence frequencies of the original pattern to realize a high coverage of the new pattern group.
\subsubsection{Fundamentals of The Selection of Neighbors} Selecting neighbors for pattern growth is based on two theorems verified in classic subgraph mining solutions~\cite{apriori, gspan}.
\begin{mytheorem}\label{absorbVertex}
Given two isomorphic subgraph $G_A(V_A,E_A)$ and $G_B(V_B,E_B)$ of $G$, and they have neighbor vertex $v_a$ and $v_b$ respectively. If $v_a$ and $v_b$ have isomorphic interconnection edge set $E_a$ and $E_b$ connected to $G_A$ and $G_B$ respectively (i.e., $v_a$ and $v_b$ have the same neighbor codes), $G'_A(V'_A,E'_A)$ and $G'_B(V'_B,E'_B)$ are isomorphic, where $V'_A = v_a \cup V_a, V'_B = v_b \cup V_b, E'_A = E_a \cup E_A, E'_B = E_b \cup E_B$ ~\cite{gspan}.
\end{mytheorem}
\begin{mytheorem}\label{frequencyDescending}
(Recurrence Frequency Monotonicity): If a subgraph pattern $G_s$ recurs $R$ times in $G$, then any subgraph of $G_s$ recurs no less than $R$ time in $G$ ~\cite{apriori}.
\end{mytheorem}
Based on Theorem~\ref{absorbVertex}, we can merge those neighbor vertices with the same neighbor code, as well as the interconnection edges between them and their owner subgraphs, into the owner subgraphs in $PGS_{tgt}$ to obtain a new pattern group $PGS_{new}$, where isomorphic subgraphs get larger compared to those in $PGS_{tgt}$. However, the size of $PGS_{new}$ will not be greater than the size of $PGS_{tgt}$, since some of the subgraphs in $PGS_{tgt}$ do not have a neighbor with the specific neighbor code. This result is supported by Theorem~\ref{frequencyDescending}.
As the example shown in Fig.~\ref{patternForest}, suppose the subgraphs in the red dash curve are those in $PGS_{tgt}$, the size of which is 5. Only 3 subgraphs have a neighbor OR3X1 cell encoded as "[OR3X1](Y,A,1)" and it means that if we merge these OR3X1 cells into the corresponding subgraphs in $PGS_{tgt}$, we will get a new pattern group $PGS_{new}$, the size of which will be 3.
\begin{algorithm}[!t]
\small
\DontPrintSemicolon
\KwIn{$\mathbf{L}_{iter}$ (or $\mathbf{L}_{init}$), the top-1 pattern group $PGS_{tgt}$, code2Neighbors, neighbor2Subgraph in Algorithm~\ref{neighborEncodingAlgorithm} }
\KwOut{updated list of pattern groups $\mathbf{L'}_{iter}$}
targetCode = getFreqNeighhborCode(code2Neighbors);
targetNeighbors = code2Neighbors[targetCode];
$PGS_{new}$ = \{\}
\tcc{enumerate the neighbors with the target neighbor code}
\ForEach{$v_i$ $\in$ targetNeighbors}
{
ownerSubG = neighbor2Subgraph[neighbor];
\If {ownerSubG is NOT abandoned}
{
\tcc{deal with neighbor occupied by other pattern subgraph }
\If {$\exists PGS'$, $v_i$ $\in$ $PGS'$ $\in \mathbf{L}_{iter}$}
{
overlapSubG = getSubgraphContain($v_i$);
overlapSubG.setAbandoned();
$PGS'$.remove(overlapSubG);
}
setIndexInSubGragh($v_i$,ownerSubG);
newSubG = ownerSubG $\cup$ $v_i$ $\cup$ E($v_i$,ownerSubG);
ownerSubG.setAbandoned();
$PGS_{tgt}$.remove(ownerSubG);
$PGS_{new}$.append(newSubG);
}
}
$\mathbf{L}_{supSeed}$ = patternSeedSupplement($G$);
$\mathbf{L}'_{iter}$ = sortByCoverage($\mathbf{L}_{iter}$+$PGS_{new}$+$L_{supSeed}$);
removeLowCoveragePatternGroups($\mathbf{L}'_{iter}$);
\Return{$\mathbf{L}'_{iter}$}
\caption{Pattern Growth}
\label{patternGrowthAlgorithm}
\end{algorithm}
\subsubsection{Acquisition of New Pattern Group}
To realize high coverage of $PGS_{new}$, AutoCellLibX will select the neighbor code which has the most recurrences among the subgraphs in $PGS_{tgt}$, e.g., "[OR3X1](Y,A,1)" for the subgraphs in Fig.~\ref{patternForest}, and merge the neighbors with the selected neighbor code into their owner subgraphs. When merging a neighbor into a pattern subgraph, the vertex will be assigned an index of the number of vertices of the original subgraph. This index indicates the "posistion" of the vertex in the pattern subgraph and will be used in neighbor encoding in Section~\ref{NeighborEncodingBasedonInterconnectionFeatures}.
Those extended subgraphs will be removed from $PGS_{tgt}$ and added to $PGS_{new}$. Meanwhile, similar to the example in Fig.~\ref{patternOverlap} for initial pattern seed generation, if those involved neighbors originally belong to the subgraphs in some pattern groups in $\mathbf{L}_{iter}$ (or $\mathbf{L}_{init}$), the subgraphs will be abandoned by the corresponding pattern groups and vertices in those subgraphs will be marked as unoccupied to meet the no-overlap constraint. These steps are demonstrated in lines 4-18 of Algorithm~\ref{patternGrowthAlgorithm}. In this procedure, AutoCellLibX tries to raise the coverage of $PGS_{new}$ and shrink the low-coverage pattern groups.
\subsubsection{Post-process for Later Pattern Mining}\label{PostprocessforLaterPatternMining}
Since some subgraphs are eliminated from their pattern groups due to overlaps during pattern growth, some vertices in $G$ become unoccupied. These vertices will be fed into a function for pattern seed supplement, which will generate the seed patterns with these unoccupied vertices, similar to initial pattern seed generation in Section~\ref{InitialPatternSeedIdentification}.
The pattern groups in $\mathbf{L}'_{iter}$ will be sorted in descending order of their coverage in $G$ to maximize area reduction with standard cell customization. Details for the relationship between pattern coverage and area benefit are illustrated in Section~\ref{BenefitEvaluationFunction}. Meanwhile, pattern groups with coverage lower than 2.5\% of the number of vertices in $G$ will be removed from $\mathbf{L}'_{iter}$ to prune unnecessary exploration.
The time complexity of pattern growth is O($|V|$), determined by lines 4-15 of Algorithm~\ref{patternGrowthAlgorithm}, where neighbor vertices with the same neighbor code of the subgraphs are enumerated.
\subsection{Generation of SPICE Netlist and GDSII Layout}\label{GenerationofSPICENetlistandGDSIILayout}
AutoCellLibX needs to generate SPICE netlists and GDSII layouts for the patterns, so it can evaluate their area benefits.
\subsubsection{Selection of Patterns for Generation} \label{SelectionofPatternsforGeneration}
While there are many pattern groups in $\mathbf{L}_{iter}$ and standard cell layout synthesis is time-consuming, just a few pattern groups should be considered for customization. According to the empirical observation on our benchmarks, the pattern groups outside the top-5 in $\mathbf{L}_{iter}$ cover less than 10\% of vertices in $G$ during the runtime of AutoCellLibX. Therefore, AutoCellLibX will only generate the SPICE netlist and GDSII layout for the top-$N_p$ pattern groups in $\mathbf{L}_{iter}$ for each mining iteration, where $N_p$ is the maximum number of custom standard cell types defined in Section~\ref{Preliminaries}.
\subsubsection{SPICE Netlist Generation}
As mentioned in Section~\ref{Preliminaries}, $G$ is a gate-level netlist after technology mapping, where each vertex is actually a standard cell. In the standard cell library, the SPICE netlist for each cell is provided. For standard cell merging, a pattern subgraph represents a set of interconnected standard cells. Therefore, AutoCellLibX can merge the SPICE netlists of the cells by connecting the pins of sized transistors according to the topology of the subgraph. To make the most of standard cell customization, AutoCellLibX will check all the I/O signals in the pattern subgraph, and if an I/O signal has no interconnection outside the subgraph in $G$ and becomes a completely internal signal of the pattern subgraph, the I/O pin for this signal on the SPICE interface of this custom standard cell will be removed. This can lower the pressure of I/O pin placement and provide more flexibility for the other parts of placement and routing during layout synthesis.
\subsubsection{GDSII Layout Generation}
Based on the SPICE netlist and the design rules of the standard cell library (e.g., number of routing tracks), AutoCellLibX will involve ASTRAN~\cite{ASTRAN} to conduct layout synthesis for a specified pattern subgraph.
\subsection{Pattern Combination Evaluation}\label{PatternCombinationEvaluation}
After previous stages of the iterative mining loop, at this stage, AutoLibCellX obtains a list of sorted pattern groups which do not overlap with each other and the area benefit of a combination of pattern groups should be evaluated.
\subsubsection{Reward Function}\label{BenefitEvaluationFunction} The overall reward function $R(C,G)$ indicating how much area could be saved can be formulated as:
\begin{eqnarray}
R(C,G)=\sum_{PGS_i \in C} [|PGS_i| * F(PatternG_i)]
\end{eqnarray}
where $PGS_i$ are the pattern groups in pattern combination $C$, $PatternG_i$ is one of the subgraphs in $PGS_i$, and $F$ is a function evaluating the area benefit of the custom standard cell generated according to $PatternG_i$ compared to the combination of original standard cells.
\begin{figure}[b]
\centering
\includegraphics[width=0.8\linewidth]{patternSize.pdf}
\caption{Example of 50 custom standard cells showing the correlation between the pattern size and the area saved with the custom cell for the pattern}
\label{patternSize}
\end{figure}
To find out the factors impacting $ F(PatternG_i)$, a series of experiments have been conducted and the empirical results of 50 custom standard cells generated by ASTRAN~\cite{ASTRAN} based on FreePDK45 process~\cite{freepdk45} are shown in Fig.~\ref{patternSize}. According to the results, the area benefit of a custom standard cell is nearly proportional to $PatternSize_i$, the number of original standard cells in the pattern subgraph $PatternG_i$. Therefore, we can get $R_{approx}(C,G)$, an approximation of $R(C,G)$ as follows:
\begin{eqnarray}
R(C,G)&&\approx K\sum_{PGS_i \in C} [|PGS_i| * PatternSize_i] \\
&& = K\sum_{PGS_i \in C} Cov(PGS_i) = R_{approx}(C,G)
\end{eqnarray} where $K$ is a linear approximation factor. Accordingly, for a specific iteration, selecting the top-$N_p$ pattern groups from $\mathbf{L}_{iter}$, which is sorted according to pattern coverage (i.e., $R_{approx}(C,G)$), as the pattern combination $C$ to generate custom standard cells is the near-optimal solution from the perspective of area benefit. Such a relationship between pattern coverage and area benefit explains why AutoCellLibX sorts the pattern groups by coverage in Section~\ref{PostprocessforLaterPatternMining} and ~\ref{OverlapElimination}.
\subsubsection{Termination Criteria of Pattern Mining}\label{TerminationCriteriaofPatternMining} The area benefit of the top-$N_p$ pattern groups could change as the pattern mining loop iterates. Therefore, AutoCellLibX should stop the loop at a proper iteration to reach the optimal combination of custom standard cells.
According to Theorem~\ref{frequencyDescending} (recurrence frequency monotonicity), it can be noticed that the values of $|PGS_i|$ tend to have a descending trend, while the values of $PatternSize_i$ are increasing, during the iterative pattern mining. As indicated in Section~\ref{Preliminaries}, $PatternSize_i$ will be limited within $S_p$ so $R(C,G)$ has an upper bound for each benchmark. Moreover, the values of $PatternSize_i$, starting from 1, increase by at most 1 in each iteration while the declines of $|PGS_i|$ are usually greater than 20\% in each iteration for the benchmarks in our experiments. It means that when $PatternSize_i$ is greater than 5, it is much harder for the increase of $PatternSize_i$ to compensate for the decline of $|PGS_i|$.
Based on these analyses, AutoCellLibX will terminate the pattern mining loop if $R(C,G)$ declines in the last two iterations since it has little probability to realize higher area benefit if the loop continues.
\section{Experimental Results}
In this section, experiments to comprehensively evaluate the performance of AutoCellLibX are presented.
\subsection{Target Process, Benchmarks and Environment}\label{experimentSetting}
Our evaluation experiments are based on FreePDK45~\cite{freepdk45}, an open-source generic process design kit developed by Oklahoma State University and North Carolina State University, which uses a predictive 45nm CMOS technology process. Concretely, in the experiments, the benchmarks are synthesized by Yosys~\cite{yosys} based on the FreePDK45 technology library, SPICE netlists of custom standard cells are obtained according to the SPICE netlists of FreePDK45 standard cells, and ASTRAN generates the layout of custom standard cells according to the design rules of FreePDK45.
To comprehensively evaluate AutoCellLibX in different scenarios, we collect 31 open-source benchmarks from various domains, including the EPFL combinational benchmark suite~\cite{epfl}, BOOM~\cite{boom}, a high-performance RISC-V CPU design, and Gemmini~\cite{gemmini}, a systolic array accelerator design for deep learning. For larger designs like BOOM and Gemmini which include multiple submodules with significantly different functionality and logic characteristics, e.g., DCache and ALU in BOOM, it is impractical for AutoCellLibX to find common patterns among these unique submodules. AutoCellLibX provides a hierarchical partitioning tool that can partition the design to a given depth so designers can select some of the partitions and find the custom standard cells for each of them. As mentioned in Section~\ref{formulation}, $N_p$ and $S_p$ are set to be 5 and 10 respectively.
Experiments are conducted on Ubuntu 20.04 with Intel i7-9850H CPU (2.60 GHz, 12 logic cores) and 32GB DDR4.
\begin{table}[!t]
\caption{Experimental Results}
\centering
\smaller
\setlength\tabcolsep{2pt}
\begin{tabular}{ccccclcc}
\hline
\multicolumn{1}{|c|}{Benchmark} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Original\\ Area(um2)\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Optimized\\Area(um2)\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Reduce\\Area(\%)\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Run\\-time(s)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}c@{}}FSM\\Time(s)\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Pattern\\ Sizes\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Pattern\\Cov(\%)\end{tabular}} \\ \hline
\multicolumn{8}{c}{EPFL Benchmark Suite~\cite{epfl}} \\ \hline
\multicolumn{1}{|c|}{adder} & \multicolumn{1}{c|}{1890} & \multicolumn{1}{c|}{1654} & \multicolumn{1}{c|}{12.51} & \multicolumn{1}{c|}{1314} & \multicolumn{1}{c|}{0.02} & \multicolumn{1}{c|}{4/3/-/-/-} & \multicolumn{1}{c|}{56.86} \\ \hline
\multicolumn{1}{|c|}{priority} & \multicolumn{1}{c|}{1111} & \multicolumn{1}{c|}{983} & \multicolumn{1}{c|}{11.53} & \multicolumn{1}{c|}{475} & \multicolumn{1}{c|}{0.02} & \multicolumn{1}{c|}{3/3/4/-/-} & \multicolumn{1}{c|}{39.22} \\ \hline
\multicolumn{1}{|c|}{multiplier} & \multicolumn{1}{c|}{40326} & \multicolumn{1}{c|}{35976} & \multicolumn{1}{c|}{10.79} & \multicolumn{1}{c|}{1554} & \multicolumn{1}{c|}{0.24} & \multicolumn{1}{c|}{5/3/3/4/2} & \multicolumn{1}{c|}{47.34} \\ \hline
\multicolumn{1}{|c|}{div} & \multicolumn{1}{c|}{45283} & \multicolumn{1}{c|}{40669} & \multicolumn{1}{c|}{10.19} & \multicolumn{1}{c|}{2222} & \multicolumn{1}{c|}{0.38} & \multicolumn{1}{c|}{5/2/3/2/4} & \multicolumn{1}{c|}{45.74} \\ \hline
\multicolumn{1}{|c|}{dec} & \multicolumn{1}{c|}{630} & \multicolumn{1}{c|}{593} & \multicolumn{1}{c|}{5.85} & \multicolumn{1}{c|}{158} & \multicolumn{1}{c|}{0.02} & \multicolumn{1}{c|}{4/-/-/-/-} & \multicolumn{1}{c|}{15.69} \\ \hline
\multicolumn{1}{|c|}{voter} & \multicolumn{1}{c|}{17303} & \multicolumn{1}{c|}{16339} & \multicolumn{1}{c|}{5.57} & \multicolumn{1}{c|}{2407} & \multicolumn{1}{c|}{0.09} & \multicolumn{1}{c|}{2/3/3/3/3} & \multicolumn{1}{c|}{35.33} \\ \hline
\multicolumn{1}{|c|}{square} & \multicolumn{1}{c|}{31115} & \multicolumn{1}{c|}{29389} & \multicolumn{1}{c|}{5.55} & \multicolumn{1}{c|}{1228} & \multicolumn{1}{c|}{0.13} & \multicolumn{1}{c|}{3/4/3/3/3} & \multicolumn{1}{c|}{26.98} \\ \hline
\multicolumn{1}{|c|}{max} & \multicolumn{1}{c|}{4298} & \multicolumn{1}{c|}{4079} & \multicolumn{1}{c|}{5.1} & \multicolumn{1}{c|}{415} & \multicolumn{1}{c|}{0.03} & \multicolumn{1}{c|}{2/3/4/4/2} & \multicolumn{1}{c|}{42.67} \\ \hline
\multicolumn{1}{|c|}{arbiter} & \multicolumn{1}{c|}{16990} & \multicolumn{1}{c|}{16403} & \multicolumn{1}{c|}{3.45} & \multicolumn{1}{c|}{3323} & \multicolumn{1}{c|}{0.15} & \multicolumn{1}{c|}{7/2/4/4/2} & \multicolumn{1}{c|}{15.29} \\ \hline
\multicolumn{1}{|c|}{cavlc} & \multicolumn{1}{c|}{1086} & \multicolumn{1}{c|}{1055} & \multicolumn{1}{c|}{2.87} & \multicolumn{1}{c|}{266} & \multicolumn{1}{c|}{0.02} & \multicolumn{1}{c|}{4/2/3/-/-} & \multicolumn{1}{c|}{11.64} \\ \hline
\multicolumn{1}{|c|}{sqrt} & \multicolumn{1}{c|}{45259} & \multicolumn{1}{c|}{43996} & \multicolumn{1}{c|}{2.79} & \multicolumn{1}{c|}{3869} & \multicolumn{1}{c|}{0.25} & \multicolumn{1}{c|}{4/3/4/4/3} & \multicolumn{1}{c|}{29.39} \\ \hline
\multicolumn{1}{|c|}{int2float} & \multicolumn{1}{c|}{370} & \multicolumn{1}{c|}{360} & \multicolumn{1}{c|}{2.49} & \multicolumn{1}{c|}{344} & \multicolumn{1}{c|}{0.02} & \multicolumn{1}{c|}{3/4/-/-/-} & \multicolumn{1}{c|}{13.42} \\ \hline
\multicolumn{1}{|c|}{router} & \multicolumn{1}{c|}{332} & \multicolumn{1}{c|}{325} & \multicolumn{1}{c|}{2.31} & \multicolumn{1}{c|}{1031} & \multicolumn{1}{c|}{0.01} & \multicolumn{1}{c|}{4/-/-/-/-} & \multicolumn{1}{c|}{8.7} \\ \hline
\multicolumn{1}{|c|}{sin} & \multicolumn{1}{c|}{9155} & \multicolumn{1}{c|}{8976} & \multicolumn{1}{c|}{1.95} & \multicolumn{1}{c|}{707} & \multicolumn{1}{c|}{0.05} & \multicolumn{1}{c|}{5/3/-/-/-} & \multicolumn{1}{c|}{7.55} \\ \hline
\multicolumn{1}{|c|}{mem\_ctrl} & \multicolumn{1}{c|}{66114} & \multicolumn{1}{c|}{64881} & \multicolumn{1}{c|}{1.86} & \multicolumn{1}{c|}{475} & \multicolumn{1}{c|}{0.40} & \multicolumn{1}{c|}{2/2/3/3/3} & \multicolumn{1}{c|}{9.89} \\ \hline
\multicolumn{1}{|c|}{i2c} & \multicolumn{1}{c|}{1818} & \multicolumn{1}{c|}{1793} & \multicolumn{1}{c|}{1.38} & \multicolumn{1}{c|}{105} & \multicolumn{1}{c|}{0.01} & \multicolumn{1}{c|}{3/2/-/-/-} & \multicolumn{1}{c|}{6.59} \\ \hline
\multicolumn{1}{|c|}{ctrl} & \multicolumn{1}{c|}{195} & \multicolumn{1}{c|}{193} & \multicolumn{1}{c|}{0.79} & \multicolumn{1}{c|}{22} & \multicolumn{1}{c|}{0.01} & \multicolumn{1}{c|}{2/-/-/-/-} & \multicolumn{1}{c|}{7.41} \\ \hline
\multicolumn{1}{|c|}{Average} & \multicolumn{2}{c|}{} & \multicolumn{1}{c|}{5.11} & \multicolumn{1}{c|}{1171} & \multicolumn{1}{c|}{0.11} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{24.69} \\ \hline
\multicolumn{8}{c}{BOOM~\cite{boom}} \\ \hline
\multicolumn{1}{|c|}{PTW} & \multicolumn{1}{c|}{394413} & \multicolumn{1}{c|}{375985} & \multicolumn{1}{c|}{4.67} & \multicolumn{1}{c|}{1446} & \multicolumn{1}{c|}{1.62} & \multicolumn{1}{c|}{2/3/3/3/5} & \multicolumn{1}{c|}{52.25} \\ \hline
\multicolumn{1}{|c|}{ALU} & \multicolumn{1}{c|}{3983} & \multicolumn{1}{c|}{3835} & \multicolumn{1}{c|}{3.73} & \multicolumn{1}{c|}{1765} & \multicolumn{1}{c|}{0.03} & \multicolumn{1}{c|}{4/2/-/-/-} & \multicolumn{1}{c|}{17.32} \\ \hline
\multicolumn{1}{|c|}{DCache} & \multicolumn{1}{c|}{2175977} & \multicolumn{1}{c|}{2098008} & \multicolumn{1}{c|}{3.58} & \multicolumn{1}{c|}{2225} & \multicolumn{1}{c|}{7.8} & \multicolumn{1}{c|}{6/2/3/3/4} & \multicolumn{1}{c|}{20.19} \\ \hline
\multicolumn{1}{|c|}{FPU} & \multicolumn{1}{c|}{105770} & \multicolumn{1}{c|}{102140} & \multicolumn{1}{c|}{3.43} & \multicolumn{1}{c|}{1004} & \multicolumn{1}{c|}{0.51} & \multicolumn{1}{c|}{3/4/3/3/3} & \multicolumn{1}{c|}{18.75} \\ \hline
\multicolumn{1}{|c|}{RegFile} & \multicolumn{1}{c|}{110269} & \multicolumn{1}{c|}{106788} & \multicolumn{1}{c|}{3.16} & \multicolumn{1}{c|}{1384} & \multicolumn{1}{c|}{0.47} & \multicolumn{1}{c|}{5/3/2/3/3} & \multicolumn{1}{c|}{24.27} \\ \hline
\multicolumn{1}{|c|}{FTQueue} & \multicolumn{1}{c|}{191581} & \multicolumn{1}{c|}{185752} & \multicolumn{1}{c|}{3.04} & \multicolumn{1}{c|}{823} & \multicolumn{1}{c|}{2.51} & \multicolumn{1}{c|}{2/3/3/3/3} & \multicolumn{1}{c|}{29.83} \\ \hline
\multicolumn{1}{|c|}{Rob} & \multicolumn{1}{c|}{84443} & \multicolumn{1}{c|}{82188} & \multicolumn{1}{c|}{2.67} & \multicolumn{1}{c|}{816} & \multicolumn{1}{c|}{0.3} & \multicolumn{1}{c|}{2/3/3/3/3} & \multicolumn{1}{c|}{31.32} \\ \hline
\multicolumn{1}{|c|}{BrPred} & \multicolumn{1}{c|}{2126189} & \multicolumn{1}{c|}{2070836} & \multicolumn{1}{c|}{2.6} & \multicolumn{1}{c|}{1085} & \multicolumn{1}{c|}{6.69} & \multicolumn{1}{c|}{4/2/3/2/2} & \multicolumn{1}{c|}{20.29} \\ \hline
\multicolumn{1}{|c|}{Rename} & \multicolumn{1}{c|}{89945} & \multicolumn{1}{c|}{89297} & \multicolumn{1}{c|}{0.72} & \multicolumn{1}{c|}{325} & \multicolumn{1}{c|}{0.6} & \multicolumn{1}{c|}{2/3/3/-/-} & \multicolumn{1}{c|}{4.43} \\ \hline
\multicolumn{1}{|c|}{Average} & \multicolumn{2}{c|}{} & \multicolumn{1}{c|}{3.07} & \multicolumn{1}{c|}{1208.11} & \multicolumn{1}{c|}{2.28} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{24.29} \\ \hline
\multicolumn{8}{c}{Gemmini~\cite{gemmini}} \\ \hline
\multicolumn{1}{|c|}{Transposer} & \multicolumn{1}{c|}{2024} & \multicolumn{1}{c|}{1807} & \multicolumn{1}{c|}{10.72} & \multicolumn{1}{c|}{228} & \multicolumn{1}{c|}{0.43} & \multicolumn{1}{c|}{3/2/-/-/-} & \multicolumn{1}{c|}{75.18} \\ \hline
\multicolumn{1}{|c|}{LoopConv} & \multicolumn{1}{c|}{291632} & \multicolumn{1}{c|}{277801} & \multicolumn{1}{c|}{4.74} & \multicolumn{1}{c|}{1220} & \multicolumn{1}{c|}{3.27} & \multicolumn{1}{c|}{4/3/3/3/3} & \multicolumn{1}{c|}{23.45} \\ \hline
\multicolumn{1}{|c|}{LpMatmul} & \multicolumn{1}{c|}{76819} & \multicolumn{1}{c|}{74058} & \multicolumn{1}{c|}{3.6} & \multicolumn{1}{c|}{572} & \multicolumn{1}{c|}{0.36} & \multicolumn{1}{c|}{2/3/3/3/3} & \multicolumn{1}{c|}{20.92} \\ \hline
\multicolumn{1}{|c|}{Spad} & \multicolumn{1}{c|}{24771641} & \multicolumn{1}{c|}{24010083} & \multicolumn{1}{c|}{3.07} & \multicolumn{1}{c|}{1940} & \multicolumn{1}{c|}{99.68} & \multicolumn{1}{c|}{4/2/5/3/2} & \multicolumn{1}{c|}{22.62} \\ \hline
\multicolumn{1}{|c|}{Mesh} & \multicolumn{1}{c|}{147649} & \multicolumn{1}{c|}{144038} & \multicolumn{1}{c|}{2.45} & \multicolumn{1}{c|}{994} & \multicolumn{1}{c|}{0.83} & \multicolumn{1}{c|}{3/3/3/3/3} & \multicolumn{1}{c|}{15.53} \\ \hline
\multicolumn{1}{|c|}{Average} & \multicolumn{2}{c|}{} & \multicolumn{1}{c|}{4.91} & \multicolumn{1}{c|}{991} & \multicolumn{1}{c|}{20.91} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{31.54} \\ \hline
\end{tabular}
\label{resultTab}
\end{table}
\subsection{Effectiveness of Proposed Pattern Mining Techniques}
The experimental results are presented in Table~\ref{resultTab}. The major submodules of BOOM~\cite{boom} and Gemmini~\cite{gemmini} are evaluated individually. The submodules included in the table cover more than 75\% logic vertices in the gate-level netlist of BOOM or Gemmini. According to the "Pattern Cov(\%)" in Table~\ref{resultTab}, a ratio showing how many vertices in the gate-level netlist are covered by the selected regular patterns, the existence of patterns is noticeable for most of the benchmarks since the average pattern coverage of the 31 benchmarks is 25.67\%.
\subsubsection{Area Benefit} The resultant data of area are collected from the post-technology mapping reports before placement and routing. The area reduction ratio indicates how much area is reduced for a corresponding benchmark when pattern subgraphs in its gate-level netlist are replaced by the selected custom standard cells generated by AutoCellLibX. For benchmarks in EPFL benchmark suite~\cite{epfl}, submodules of BOOM~\cite{boom}, and submodules of Gemmini~\cite{gemmini}, their average area reduction ratios are 5.11\%, 3.07\%, and 4.91\% respectively. The average area reduction ratio for all the benchmarks in the table is 4.49\%. AutoLibCellX achieves significant area reduction compared to the FreePDK45 library while overlap constraint, pattern selection and area reduction are not considered in the previous works~\cite{AFSEM,FSMVLSI} of pattern mining for VLSI design.
The optimization of area usage is noticeable but the customization benefit varies among benchmarks. Here, we analyze the related factors as follows:
\begin{itemize}
\item Design Hierarchy Depth: Benchmarks in EPFL benchmark suite~\cite{epfl} can get better results because they are designed specifically for a local circuit function so the patterns can have a higher coverage. In contrast, if AutoCellLibX consumes the entire gate-level netlist of BOOM or Gemmini without partitioning, each of the pattern groups will have a coverage lower than 10\%, and the resultant area reduction ratio will be lower than 1.5\% since some patterns are not shared by different submodules. Even submodules of these designs have multiple hierarchical levels. This indicates that proper partitioning of the input design can help to find out suitable custom standard cells.
\item Pattern Coverage and Pattern Size: Benchmarks like "ctrl", "i2c", and "Rename" benefits little from standard cell customization, because their netlists mainly consist of random logic so the pattern coverage is low. From another perspective, benchmarks like "mem\_ctrl" reach a lower area reduction ratio because the sizes of their high-frequent patterns are too small.
\end{itemize}
Compared to conventional frequent subgraph mining, for standard cell customization, AutoCellLibX emphasizes the overlaps between pattern subgraphs. For example, 3 of the 5 selected patterns for benchmark "DCache" overlap with each other on 49107 vertices in the gate-level netlist. Direct frequency-driven pattern mining, ignoring the overlaps, will lead to false frequency results while AutoCellLibX analyzes each vertex during neighbor encoding and pattern growth, avoiding overlaps and optimizing area usage.
\subsubsection{Algorithm Efficiency} Considering neighbor encoding and pattern growth, the time complexity of pattern mining is reduced to $O(M(|E|+|V|))$, where $M$ is the maximum number of pattern mining iterations. According to the analysis of the termination criteria of pattern mining illustrated in Section~\ref{TerminationCriteriaofPatternMining}, for most scenarios, $M$ is less than 10. Therefore, our FSM algorithm is highly efficient, as shown in the "FSM time" in Table~\ref{resultTab}, and the time complexity of our FSM algorithm is much lower than the one of the gSpan algorithm~\cite{gspan} utilized in previous works~\cite{AFSEM,FSMVLSI}, which is proportional to the number of subgraphs in the gate-level netlist. However, currently, the timing-consuming layout synthesis, especially the simulated-annealing transistor placer and MILP compactor, takes up more than 95\% of the runtime of AutoCellLibX for the benchmarks, although AutoCellLibX has already pruned most of the minor patterns to avoid unnecessary layout generation and reduce the runtime. The average runtime is 1152 seconds for all the benchmarks while the longest runtime is 3869 seconds for benchmark "voter".
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{largePattern.pdf}
\caption{A large pattern subgraph with 7 standard cells for benchmark "router" and corresponding custom standard cell layout}
\label{largePattern}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{sharedPatterns.pdf}
\caption{Some patterns shared across benchmarks in Table~\ref{resultTab}}
\label{sharedPatterns}
\end{figure}
\subsubsection{Characteristics of Custom Standard Cells}
The "Pattern Size" column of Table~\ref{resultTab} shows the sizes of the selected pattern subgraphs for corresponding benchmarks. The largest pattern is the one with 7 vertices for benchmark "arbiter", as shown in Fig.~\ref{largePattern}, which it is hard for template-based algorithms~\cite{datapath1,datapath2,datapath3,datapath4,array1} to detect. Although we set $S_p$ to 10, the pattern mining procedures for all the benchmarks do not hit this bound which is predicted by Section~\ref{TerminationCriteriaofPatternMining}. Some benchmarks have less than 5 (i.e., $N_p$) custom standard cells since some candidates have been pruned due to low coverage during pattern growth. The patterns of benchmarks are various but some patterns are shared by some of the benchmarks, as shown in Fig.~\ref{sharedPatterns}, which shows the potential of reusing custom standard cells for different designs, improving productivity of VLSI design flow.
\subsection{Existing Limitations and Future Works}
While AutoCellLibX shows the power of efficient pattern mining for standard cell customization, it has several limitations, which are listed as follows:
\begin{itemize}
\item Currently, AutoCellLibX cannot the finish complete characterization of standard cells, e.g., the extraction of RC parasitics and timing information. While the results of area reduction are promising, the reward function $R(C,G)$ in Section~\ref{Preliminaries} can be replaced by a function of other factors like delay and power.
\item Standard cell layout synthesis is the bottleneck of AutoCellLibX runtime.
Moreover, the open-source layout generator ASTRAN~\cite{ASTRAN} does not support some of the design rules of high-end sub-10nm technology nodes currently.
Furthermore, factors considered in manual designs like pin access, and via number reduction, could be included for practical application.
\end{itemize}
Since it is our initial attempt to explore the potentials of design-aware pattern mining of VLSI netlist and standard cell customization, these limitations are out of the scope of this proposed work, which presents an interesting and concrete approach to customizing standard cells, and we will consider more factors in our future works.
In the next development stage, AutoCellLibX will include the optimization of routability and timing and layout synthesis framework, to realize the practicality of AutoCellLib for sub-10nm process technology. Currently, we are adapting our algorithms to the technology design rules in ASAP7~\cite{ASAP7} and integrating the existing solutions from ~\cite{iTPlace,li2020mcell,DTCO,SPR,NCTUcell,7nm2019,Bonncell,PROBE2} into AutoCellLibX. Estimation model of timing, routability and DRC will be included in further evaluation. Any criticism and suggestions will be appreciated!
\section{Conclusion}
In this paper, we presents an automatic standard-cell library extension framework, AutoCellLibX. According to the post-technology mapping gate-level netlist of the design and the initial standard cell library, AutoCellLibX can find a set of standard cell cluster pattern candidates that occur frequently based on our proposed high-efficiency sub-graph mining algorithm for gate-level netlist. Meanwhile, to maximize the area benefit of standard cell customization for the given gate-level netlist, AutoCellLibX includes our proposed pattern combination algorithm which can iteratively find a set of gate-level patterns from numerous candidates as the extension part of the given initial library. AutoCellLibX closes the optimization loop between the analysis of gate-level netlist and standard cell library customization for VLSI design productivity. The experiments with FreePDK45 library and benchmarks from various domains show that AutoCellLibX can generate the library extension with up to 5 custom standard cells in 1184 seconds on average for the benchmarks and the resultant extension of the standard cell library can save design area by 4.49\% averagely.
\bibliographystyle{IEEEtran}
|
1,314,259,995,835 | arxiv | \section{\label{sec:intro}Introduction}
In the course of the observational campaign on the bright Supernova
2010jl we obtained spectropolarimetry of this object using the Calar
Alto Faint Object Spectrograph (CAFOS), mounted at the 2.2 m telescope
in Calar Alto, Spain (Meisenheimer \cite{cafos}). The results were
published in Patat et al. (\cite{patat10}). The polarimetric mode of
CAFOS has not been used very extensively, and mostly in imaging mode
(see Greiner et al. \cite{greiner} for an example). As we could not
find a proper characterisation of the instrumental effects in the
literature, during the campaign on SN~2010jl we ran a full analysis of
the instrument. This is presented here with the aim of making it
available to a wider community, who might find it useful for future
spectropolarimetric observations with this instrument.
Dual-beam polarimeters like CAFOS are composed by a half-wave retarder
plate (HWP) followed by the analyzer, which is a Wollaston prism (WP)
producing two beams with orthogonal directions of polarization,
usually indicated as ordinary (O) and extraordinary (E) beams. With
this instrumental setup, the Stokes parameters $Q$ and $U$ are derived
measuring the intensities in the O and E beams ($f_{O,i}$, $f_{E,i}$)
at a given set of HWP angles $\theta_i$ (for a general overview see
Patat \& Romaniello \cite{patat06}, and references therein).
This is typically achieved through the normalized flux differences
$F_i$,
\begin{displaymath}
F_i=\frac{f_{O,i}-f_{E,i}}{f_{O,i}+f_{E,i}}.
\end{displaymath}
For an ideal polarimeter, the normalized flux differences obey to the
following relation: $F_i=P \cos(4\theta_i - 2\chi)$, where
$P=\sqrt{Q^2+U^2}$ is the polarization degree, and
$\chi=\frac{1}{2}\arctan(U/Q)$ is the polarization position
angle. Although any set of angles $\theta_i$ is in principle suitable
for obtaining $Q$ and $U$, the optimal choice is
$\theta_i=\frac{\pi}{8} i$. In these conditions one has that:
\begin{displaymath}
Q=\frac{2}{N}\sum_{i=0}^{N-1} F_i \cos \left ( \frac{\pi}{2} i\right ) \\
\end{displaymath}
\begin{displaymath}
U=\frac{2}{N}\sum_{i=0}^{N-1} F_i \sin \left ( \frac{\pi}{2} i\right ),
\end{displaymath}
\noindent where $N$ is the number of HWP angles. As $F_i$ can be
thought as a co-sinusoidal signal modulated by the rotation of the HWP
with a fundamental period 2$\pi$, these can be rewritten as a Fourier
series (see Fendt et al. \cite{fendt}):
\begin{equation}
F_i = a_0 +
\sum_{k=1}^{N/2} \left [ a_k \cos \left( k \frac{2\pi i}{N}\right) +
b_k \sin \left( k \frac{2\pi i}{N}\right) \right],
\end{equation}
\noindent where $a_k$ and $b_k$ are the Fourier coefficients. The
Fourier analysis is particularly useful when $N$=16; under these
circumstances, the polarization signal is carried by the $k$=4
component. In an ideal system, all other components are rigorously
zero. Therefore, non-null Fourier coefficients for $k\neq$4 signal
possible problems in the polarimeter. For the meaning of the various
components the reader is referred to Fendt et al. (\cite{fendt}).
\begin{figure}
\centerline{ \includegraphics[width=90mm]{unpol.jpg} }
\caption{\label{fig:unpol}Fourier analysis applied to the unpolarized
standard star HD~14069 at 5500 \AA\/ (200 \AA\/ bin). {\bf Top:} normalized
flux differences. The curves trace the partial reconstruction using
8 harmonics (solid) and the fourth harmonic only (dotted). The
dashed horizontal line is placed at the average of the $F$ values
($a_0$). {\bf Bottom:} harmonics power spectrum. The dashed line indicates
the 5-sigma level of the uncertainty.}
\end{figure}
\section{\label{sec:obs}Observations and data reduction}
The observations were carried out with CAFOS (Meisenheimer
\cite{cafos}). In this multi-mode instrument, equipped with a
2K$\times$2K SITe-1d CCD (24$\mu m$ pixels, 0.53 arcsec/pixel),
polarimetry is performed by introducing into the optical path a WP
(18$^{\prime\prime}$ throw) and a super-achromatic HWP, between the
collimator and the grism. For our study we observed the polarized star
BD+59d389 ($P(V)$=6.70$\pm$0.02\%, $\chi$=98.1 degrees; Schmidt et
al. \cite{schmidt}), and the unpolarized star HD~14069
($P(V)$=0.02$\pm$0.02\%; Schmidt et al. \cite{schmidt}) on 2010,
November 18.8 UT. All spectra were obtained with the low-resolution
B200 grism coupled with a 1.0 arcsec slit, giving a spectral range
3300-8900 \AA, a dispersion of $\sim$4.7 \AA\/ px$^{-1}$, and a FWHM
resolution of 14.0 \AA. The slit was aligned along the N-S direction.
To enable the Fourier analysis up to the 8-th harmonic we used $N$=16
half-wave plate angles (0, 22.5, ..., 337.5). The exposure times were
180 seconds per HWP angle for both standard stars.
Data were bias and flat-field corrected, and wavelength calibrated
using standard tasks within IRAF\footnote{IRAF is distributed by the
National Optical Astronomy Observatories, which are operated by the
Association of Universities for Research in Astronomy, under
contract with the National Science Foundation.}. The Fourier
analysis was carried out using specific routines written by us.
\section{\label{sec:inspol}Instrumental polarization}
\begin{figure}
\centerline{ \includegraphics[width=90mm]{inspol.jpg} }
\caption{\label{fig:inspol} CAFOS instrumental polarization. {\bf Upper
panel}: instrumental polarization position angle. The dashed line
indicates the average value. {\bf Mid panel}: instrumental polarization
degree. {\bf Lower panel}: instrumental Stokes parameters. The dashed
lines indicate the average value of $Q$ and $U$ in the wavelength
range 4000--8600 \AA, while the dotted lines mark the $\pm$0.05\%
deviations from the average value.}
\end{figure}
To characterize the instrumental polarization of CAFOS, we first
analyzed the data obtained for the unpolarized standard star. The
result of the Fourier analysis is presented in Fig.~\ref{fig:unpol}
for a 200 \AA\/ wide bin centered at 5500 \AA. The normalized flux
differences show a marked modulation (upper panel), well reproduced by
a sinusoidal function. The power spectrum (lower panel) displays a
neat peak at the $k$=4 overtone, corresponding to a linear
polarization signal (see Sec.~\ref{sec:intro}), reaching $P$=0.26\%
(the $k$=0 term is also non null, but we will discuss this in
Sect.~\ref{sec:fourier}). The fact that the signal is modulated by the
retarder plate rotation implies that the source of instrumental
polarization precedes the HWP along the optical path. Therefore, most
likely the observed polarization arises within the collimator and/or
the telescope mirrors. For instance, inhomogeneities in the mirror
coatings can break circular symmetry, and lead to an incomplete
cancellation of the linear polarization generated by reflections (see
Tinbergen \cite{tinbergen} and Leroy \cite{leroy} for general
introductions to the subject). Such a system would behave as a partial
polarizer, characterized by a certain position angle ($\chi_{ins}$)
that does not depend on wavelength, but only on the geometry of the
system asymmetry. In general, the effect of the instrumental
polarization depends on the Stokes vector that characterizes the input
signal, and this makes the correction for instrumental polarization
particularly difficult. However, when the instrumental polarization is
much smaller than 1, the effect is additive, and the spurious signal
can be removed subtracting it vectorially from the measured one (see
for instance Patat \& Romaniello \cite{patat06} for the case of
VLT-FORS1).
\begin{figure}
\centerline{ \includegraphics[width=90mm]{polstd.jpg} }
\caption{\label{fig:polstd}Fourier analysis applied to the polarized
standard star BD+59d389 at 5500 \AA\/ (200 \AA\/ bin). {\bf Top:} normalized
flux differences. The curves trace the partial reconstruction using
8 harmonics (solid) and the fourth harmonic only (dotted). The
dashed horizontal line is placed at the average of the $F$ values
($a_0$). {\bf Middle:} residuals from the reconstruction using the $k$=4
harmonic. {\bf Bottom:} harmonics power spectrum. The dashed line
indicates the 5-sigma level uncertainty.}
\end{figure}
The presence of a constant position angle is confirmed by the Fourier
analysis run across the whole wavelength range covered by our
observations. In Fig.~\ref{fig:inspol} we present the values of
$Q_{ins}$ and $U_{ins}$ derived within 200 \AA\/ wide bins between
3400 and 8600 \AA\/ (lower panel), and the implied position angle
$\chi_{ins}$ (upper panel). The average value of $\chi_{ins}$ is 166.3
degrees, and the RMS deviation of the single measurements is 3.6
degrees. The smooth oscillation seen in the position angle is related
to the chromatism of the HWP retardance (see Sect.~\ref{sec:hwp}).
As far as the polarization is concerned, this reaches 0.74$\pm$0.08\%
at 3400 \AA, and it rapidly decreases to 0.33$\pm$0.01\% at 4000 \AA,
to remain constant to within 0.05\% up to 8600 \AA. The reason for the
marked increase seen bluewards of 4000 \AA\/ is not clear, but it
might be related to the decrease of efficiency in the anti-reflexion
coatings of the collimator lenses.
The average values of the instrumental Stokes parameters above 4000
\AA\/ are $\langle Q_{ins}\rangle$=+0.25$\pm$0.03\%, and $\langle
U_{ins}\rangle$=$-$0.13$\pm$0.03\% respectively, leading to an average
polarization $P_{ins}$=0.28$\pm$0.03\%. The wavelength range below
3800 \AA\/ is affected by other instrumental problems which make it
hardly usable with the typical set of 4 HWP angles (see next section).
Therefore, this constant correction is sufficient to guarantee the
removal of the instrumental polarization with a maximum error of
0.05\%, which is comparable to the maximum accuracy one can reach with
CAFOS with 4 HWP angles (see next section).
We remark that the instrumental polarization correction derived here
is strictly valid only for an object placed on the CAFOS reference
pixel used for the acquisition onto the 1.0 arcsec slit. With the
present analysis we cannot exclude position-dependent effects,
similarly to what happens in the FORS instruments (Patat \& Romaniello
\cite{patat06}).
\begin{figure}
\centerline{ \includegraphics[width=90mm]{fullanal.jpg} }
\caption{\label{fig:fullanal}Power spectrum of the first 6 harmonics
as a function of wavelength (ordinate scale is in \%). The solid
thin lines trace the 5-$\sigma$ confidence level. The filled squares
in the $k$=4 plot are the broad-band polarization measurements by
Schmidt et al. (\cite{schmidt}).}
\end{figure}
\section{\label{sec:fourier}Fourier analysis}
For the Fourier analysis of the CAFOS polarimetric performances we
have used the data obtained for the polarized standard
star. Fig.~\ref{fig:polstd} shows an example for a 200 \AA\/ wide bin
centered at 5500 \AA. The only components which show a statistically
significant power are $k$=0 and $k$=4; there is a hint of a nun-null
$k$=2 component, which is related to the so-called pleochroism (Fendt
et al. \cite{fendt}; Patat \& Romaniello \cite{patat06}), but this is
only marginally significant at the 5-$\sigma$ level. The original
signal can be reconstructed using only the $k$=4 harmonic, with
maximum residuals $\Delta F_i$ of $\sim$0.1\%. This implies that 4 HWP
angles are sufficient to the derive the Stokes parameters with a
maximum error of this order. The polarization degree derived using 16
HWP angles at 5500 \AA\/ is 6.43$\pm$0.01\%. After applying the
instrumental polarization correction described in the previous section
this value becomes 6.6$\pm$0.1\%. This is fully consistent with
the reference value 6.70$\pm$0.02\% measured in the V passband
(Schmidt et al. \cite{schmidt}).
In the example illustrated in Fig.~\ref{fig:polstd} we find
$a_0$=1.88$\pm$0.01\% (the corresponding value derived from the
unpolarized standard is 1.93$\pm$0.01\%; see also
Fig.~\ref{fig:unpol}, upper panel). This indicates that the WP
deviates from the ideal case, in that an unpolarized incoming beam is
not exactly split into two identical fractions (see Patat \&
Romaniello \cite{patat06}, their Sect.~7). As a consequence, using
only 2 HWP angles (which is the minimum set needed to fully
reconstruct the Stokes vector) would lead to a very significant error
on the final result.
\begin{figure}
\centerline{ \includegraphics[width=90mm]{chromat.jpg} }
\caption{\label{fig:chromat} Instrumental polarization corrected
Stokes parameters $Q$ (top panel) and $U$ (mid panel) for
BD+59d389. The bottom panel shows the phase retardance variation as
a function of wavelength.}
\end{figure}
To study the instrumental performance as a function of wavelength, we
have run the same analysis within 200 \AA\/ wide bins between 3400 and
8600 \AA. The result for the first 6 harmonics is shown in
Fig.~\ref{fig:fullanal}. The $k$=0 component is always significant,
exceeding $\sim$3\% at 7500 \AA, but this is fairly well corrected if
the data set includes at least 4 HWP positions. As for components
$k$=1 and 2, these are detected at a significant level below 3800
\AA\/ and above 7000 \AA. At 3600 \AA\/ the usage of 4 HWP angles
leads to errors larger than 0.3\%, making data bluewards of 3800 \AA\/
hardly usable. At the red edge, deviations are below 0.2\% bluewards of
7400 \AA, while they can exceed 0.3\% above 8200 \AA.
As the $k$=4 component of the power spectrum is the linear
polarization degree, its wavelength dependence can be directly
compared to the broad band values available in the literature (Schmidt
et al. \cite{schmidt}). These are overplotted in the $k$=4 panel of
Fig.~\ref{fig:fullanal} (filled squares). As expected based on the
estimates of the instrumental polarization (see
Sect.~\ref{sec:inspol}), there is a difference of about 0.3\% above
4000 \AA. The value corresponding to the $U$ passband shows a larger
deviation (0.7\%), which is consistent with the increase of the
instrumental polarization seen below 4000 \AA\/ (see
Fig.~\ref{fig:inspol}). It is worth noting that, as the
polarization signals of the star and the instrument are close to
orthogonal, the corrected value is higher than the measured one.
\section{\label{sec:hwp}HWP chromatism}
Although the retarder plate deployed in CAFOS is super-achromatic, the
phase retardance is expected to deviate from an ideal behavior as a
function of wavelength. To quantify this effect we have used the
polarized standard as reference. For this star the polarization
position angle is constant to within 0.1 degrees in the UBVRI domain,
the average value being $\chi_0$=98.2$\pm$0.1 degrees (Schmidt et
al. \cite{schmidt}). Therefore, if $Q_{obs}$ and $U_{obs}$ are the
measured Stokes parameters, the phase retardance variation across the
wavelength range can be computed as $\Delta \chi=\chi-\chi_0$, where
$\chi=\frac{1}{2}\arctan [(U_{obs}-U_{ins})/(Q_{obs}-Q_{ins})]$. The
result is plotted in Fig.~\ref{fig:chromat}, and the values listed in
Table~\ref{tab:chromat}.
\begin{table}
\tabcolsep 3 mm
\begin{tabular}{ccccccc}
\hline
$\lambda$ & $\Delta \chi$ & $\sigma$ & & $\lambda$ & $\Delta \chi$ & $\sigma$\\
(\AA) & (deg) & (deg) & & (\AA) & (deg) & (deg)\\
\hline
3600 & 7.44 & 0.71 & & 6200 & 4.06 & 0.08 \\
3800 & 10.55 & 0.20 & & 6400 & 3.22 & 0.09\\
4000 & 11.90 & 0.12 & & 6600 & 2.46 & 0.09\\
4200 & 12.76 & 0.09 & & 6800 & 2.02 & 0.10\\
4400 & 12.51 & 0.07 & & 7000 & 1.82 & 0.10\\
4600 & 12.00 & 0.07 & & 7200 & 1.91 & 0.10\\
4800 & 11.31 & 0.07 & & 7400 & 1.93 & 0.12\\
5000 & 10.00 & 0.07 & & 7600 & 2.06 & 0.13\\
5200 & 8.91 & 0.07 & & 7800 & 2.09 & 0.13\\
5400 & 7.95 & 0.06 & & 8000 & 2.19 & 0.13\\
5600 & 6.57 & 0.07 & & 8200 & 2.30 & 0.15\\
5800 & 5.42 & 0.07 & & 8400 & 2.72 & 0.16\\
6000 & 4.63 & 0.08 & & 8600 & 3.11 & 0.16\\
\hline
\end{tabular}
\caption{\label{tab:chromat}HWP retardance variation as a function of
wavelength.}
\end{table}
Having these values at hand, the corrected Stokes parameters $Q_c$ and
$U_c$ can be obtained by the following rotation:
\begin{displaymath}
Q_c = Q \cos 2\Delta \chi + U \sin 2\Delta \chi
\end{displaymath}
\begin{displaymath}
U_c = U \cos 2\Delta \chi - Q \sin 2\Delta \chi ,
\end{displaymath}
\noindent where $Q$ and $U$ are the instrumental polarization
corrected Stokes parameters. Alternatively, the position angle
obtained from $Q$ and $U$ can be corrected subtracting $\Delta \chi$.
Usually the zero-point of the HWP angle is set so that $\theta$=0
corresponds to a null astronomical position angle in the plane of the
sky around the central wavelength. This is not the case in CAFOS, as
the deviation at 6000 \AA\/ is about 5.5 degrees, and it is never zero
between 3600 and 8600 \AA\/ (Fig.~\ref{fig:chromat}, bottom panel).
However, given the way we have computed $\Delta \chi$, this correction
will give position angles in the plane of the sky, with $\chi$=0
corresponding to the N-S direction.
\section{\label{sec:conc}Conclusions}
In this note we presented a full analysis of the linear polarization
properties of CAFOS. Although the instrument appears to suffer from a
significant spurious polarization, this can be removed to within
$\sim$0.1\%. The effect appears to be additive, and can be therefore
easily corrected by vectorially subtracting the instrumental
component on the Stokes $Q,U$ plane.
As is typical of other dual-beam polarimeters (see for instance the
case of FORS1, Patat \& Romaniello \cite{patat06}), the Wollaston
prism departs from the ideal case. In the worst case the fraction of
light in the ordinary and extraordinary beams for an unpolarized
incoming signal deviates by $\sim$2\% from the theoretical 50/50
ratio. However, this defect is largely removed by the adoption of 4
retarder plate angles during the observations. Using the minimum set
(2 HWP angles) leads to large errors, especially in the case of low
polarizations ($\sim$1\%), and it is therefore strongly discouraged.
The Fourier analysis shows that all harmonics with $k\neq$0,4 are
negligible in the wavelength range 3800--7400 \AA, where a rms accuracy
of 0.1\% can be reached with a sufficient signal-to-noise. This can be
considered as the instrumental limit attainable with CAFOS with 4 HWP
angles, and within this spectral range. Below 3800 \AA\/ the
polarimetric properties rapidly degrade, requiring a larger number of
HWP angles. The same applies, though to a smaller extent, to the
region redwards of 8200 \AA.
For this work we have used data obtained with the B200 grism. Because
of its tilted surfaces, the grism can act as a poor linear
polarizer. Since in CAFOS this is placed after the analyzer, the
spurious polarization produced by transmission is not modulated by the
HWP rotation, and hence the redundancy in the retarder-plate position
effectively removes it (see Patat \& Romaniello \cite{patat06}). The
exact effect produced by the grism depends on its properties. However,
the conclusions reached in this paper do not depend on the grism,
provided that the data are obtained using at least 4 HWP angles.
In general, CAFOS appears to be perfectly suitable for linear
polarization studies aiming at accuracies of a few 0.1\%, making it a
valid instrument for bright objects. As a term of reference, an
accuracy of $\sim$0.1\% per resolution element ($\sim$50 \AA) was
reached for SN~2010jl ($V\sim$13.5), with 4 exposures of 40 minutes
each (Patat et al. \cite{patat10}).
\begin{acknowledgements}
We wish to thank the personnel at Calar Alto Observatory for obtaining
the extra calibrations required for the full polarimetric
characterisation of CAFOS. This work has been done in the framework of
the European Large Collaboration {\it Supernova Variety and
Nucleosynthesis Yields} (http://graspa.oapd.inaf.it/),
whose members are acknowledged.
\end{acknowledgements}
|
1,314,259,995,836 | arxiv | \section{Introduction}
In recent years, semiconductor compounds based on gallium arsenide (GaAs) containing small amounts of nitrogen (N) have proven to be promising materials for all kinds of optoelectronic applications. One of the first applications were (GaIn)(NAs) laser diodes. \cite{Kondow1996} More recently, silicon-matched Ga(NAsP) structures as laser active materials have become a highly active area of research. \cite{Kunert2006} Since the band gap of GaN was known to be larger than that of GaAs, a blueshift of the photoluminescence (PL) edge was expected for Ga(NAs). However, the first experimental PL of dilute nitride Ga(NAs) measured by \textcite{Weyers1992} showed a redshift of the PL. Further studies showed, that the underlying band gap bowing was unusually strong. \cite{Kondow1994,Uesugi1999} In contrast to other III/V-semiconductors, the description of bowing within the commonly used virtual crystal approximation agrees less well with experimental band gaps than the slightly more involved conduction band anti-crossing (CBAC) model. \cite{Vurgaftman2003} While this dependence allows to specifically manipulate the band gap by controlling the composition, the unexpected strength of this effect also called for an explanation.
Based on density functional theory (DFT) and quasiparticle (GW) calculations, \textcite{Rubio1995} were able to attribute this effect to an increase of the volume caused by the lattice mismatch, lowering the energy of the conduction band (CB) edge, which is strongly localized on the N atoms.
Further studies, mainly based on DFT within the local density approximation (LDA) were performed in the following years. \cite{Wei1996,Bellaiche1996,Bellaiche1997,Zhang2003}
A study based on the hybrid functional HSE06 showed an improved agreement with the observed band gap bowing. \cite{Virkkala2012}
Regarding the origin of the strong bowing, further studies found that the CB states centered on the N atoms are extended especially along the zigzag lines in the zinc blende structure. \cite{Kent2001,Virkkala2013}
In the context of our work, it is noteworthy that none of the cited \textit{ab initio} results reproduced the band gaps accurately.
To correct this, Zunger \textit{et al.} constructed empirical pseudopotentials fitted to GW band structures, experimentally determined band gaps and LDA deformation potentials.
This allowed them to perform calculations with huge supercells (13824 atoms). \cite{Bellaiche1996,Bellaiche1997,Kent2001}
Thereby, they showed that the large lattice mismatch between As and N leads to long range strain fields. \cite{Kent2001} This, and the extension of N-centered CB states indicates the need for large supercells. \cite{Zunger1999}
In order to further the development of optoelectronic devices based on dilute nitrides, it is desirable to predict the optical properties of new materials.
This requires an accurate \textit{ab initio} description of the band structure and the band gap without using empirical or experimental parameters.
While DFT is most often the method of choice in material science, there are two factors that usually prevent a quantitative description using DFT supercell calculations:
Firstly, commonly used density functionals mostly underestimate the band gap of semiconductors considerably (LDA,\cite{Perdew1992,Perdew1985} PBE\cite{PBE}).
Hybrid functionals including exact exchange like HSE06\cite{Krukau2006} (more recently used with tailored parameters\cite{Moussa2012}) can solve this issue but are computationally more demanding and hardly applicable to supercells with several hundred atoms in a routine fashion.
Secondly, the enormous lattice mismatch between GaN and GaAs due to the size difference of As and N (the lattice constant of GaN is roughly $80\,\%$ that of GaAs) in combination with the periodicity of the finite size supercells cause an artificial reduction of the band gap. \cite{Kent2001,Virkkala2013}
Overcoming the latter problem requires large supercells which adds to the former by increasing the computational cost.
An alternative approach is to use special quasirandom structures (SQS) as proposed by Zunger \textit{et al.}, which aim to emulate a random distribution in a semiconductor alloy.\cite{Zunger1990} This approach has often been successfully used in the past but requires convergence with respect to cell size nonetheless.
Therefore, a functional capable of accurately describing the band gap while still being computationally efficient is needed.
The recently developed meta-GGA functional TB09 (also known as mBJLDA or TB-mBJ)\cite{Tran2009} is a promising candidate to fulfill these requirements for Ga(NAs).
It has been shown that TB09 allows to obtain band gaps commonly in good agreement with experimental data\cite{Tran2009,Kim_PRB_82_2010,Jiang_JCP_138_2013} while being computationally very efficient in comparison to HSE06 and GW calculations. \cite{Tran2009}
Thus, in this work we will use the TB09 functional in combination with fairly large supercells (up to 432 atoms) to solve the long standing issue of the \textit{ab initio }prediction of accurate band gaps in dilute nitrides. In Section \ref{Sec:methods} we introduce our computational methods and parameters.
Thereafter, we present our results, starting with a brief validation of the approach used followed by a discussion of the structural effects of nitrogen incorporation and finally the effect on the band gap for different supercells.
\section{Computational Method\label{Sec:methods}}
DFT calculations were performed using the Vienna \textit{ab initio} Simulation Package (VASP 5.3.5)\cite{VASP1993, VASP1994, VASP1996a, VASP1996b} with a plane wave basis in conjunction with the projector-augmented wave method. \cite{Blochl1994, Kresse1999}
The basis set energy cut-off was set to $450\,\text{eV}$ for optimizations and $350\,\text{eV}$ for large supercell band gap computations.
The reciprocal space was sampled with a $\Gamma$-centered Monkhorst-Pack grid with six intersections per direction for primitive unit cells and an accordingly reduced set for supercells. \cite{monkhorstpack}
Cell relaxations of primitve cells and all lattice relaxations were done with the PBE functional, \cite{PBE} corrected for dispersion interactions with the DFT-D3 scheme. \cite{GrimmeD3, GrimmeD3BJ}
Lattice parameters for binary materials were derived from fitting the Vinet equation of state, yielding a theoretically optimized value.\cite{Vinet1986}
The lattice parameters for ternary cells were linearly interpolated from the constituents (Vegard's rule).
For $6^3$ supercells with more than one N atom, sampling was performed by randomly selecting As atoms to be replaced by N atoms and averaging over few arrangements.
For a small number of N atoms, more qualitatively distinct arrangements (e.g. different average distances) are possible; thus, more arrangements were sampled for smaller concentrations.
Calculations of band gaps and band structures used the TB09 functional\cite{Tran2009} as well as PBE0\cite{Perdew1996}, HSE06\cite{Krukau2006}, and LDA-1/2\cite{Ferreira2008,Ferreira2011} functionals for comparison.
Potentials for the LDA-1/2 calculations were prepared using the ATOM code with a cut-off (in a.u.) of 0.9875 for Ga, 3.7725 for As, 3.6550 for P, and 2.9275 for N. \cite{Ferreira2008,atomico}
In addition, G$_0$W$_0$ calculations were carried out based on the PBE orbitals.
For all band gap computations, spin-orbit coupling was considered, with the exception of G$_0$W$_0$. For relaxations, the electronic energy and forces were converged to $10^{-6}\,\text{eV}$ and $10^{-2}\,\text{eV}\cdot \text{\AA}^{-1}$, respectively. The electronic energy and eigenvalues for band gap computations in supercells were converged to $10^{-4}\,\text{eV}$.
A supercell calculation yields an energy dispersion $E(\mathbf{K})$ where $\mathbf{K}$ is the wave vector in the reciprocal space of the supercell.
Often, it is desirable to project the supercell eigenstates $|\mathbf{K},n\rangle$ on the eigenstates $|\mathbf{k}_j,m\rangle$ of the respective primitive cell.
The index $j$ accounts for the fact that every supercell eigenstate matches $N^3$ primitive cell eigenstates where $N$ depends on the size of the supercell.
Thus, a so called \textit{effective} band structure (EBS) $E(\mathbf{k})$ can be extracted.
For this, we calculate the spectral weight that is a measure for the Bloch character of a specific eigenstate.
We follow the steps outlined in the appendix of Ref.~\cite{Popescu2012}.
The spectral weight $w_{n,\mathbf{K}}(\mathbf{k})$ is given by the square sum of the relevant plane wave coefficients
\begin{equation}
w_{n,\mathbf{K}}(\mathbf{k}_j) =
\sum_{\mathbf{g}}|C_{n,\mathbf{K}}(\mathbf{g}+\mathbf{G}_j)|^2
\end{equation}
where $n$ is the band index and $\mathbf{g}$ and $\mathbf{G}_j$ are reciprocal lattice vectors of the primitive cell and the supercell, respectively.
Apart from cubic supercells, SQS cells have been used for ternary cells. \cite{Zunger1990} SQS cells have been generated with the Alloy-Theoretic Automated Toolkit (ATAT), \cite{VanDeWalle2013} with correlations between two atoms up to the third sphere of the same lattice site, three atoms up to the second sphere and four atoms in the first sphere only, corresponding to four distinct two-atom clusters, ten three-atom clusters and two four-atom clusters.
The $(N\times N \times N)$ supercells of the primitive zinc blende unit cell will be referred to as $N^3$ supercells, SQS cells with $N$ atoms as SQS-$N$.
Structures and band gaps for the used cells are shown in the supporting information.
\section{Results and Discussion}
\subsection{Method benchmark}
First, we revisit the question of band gap computation for common functional classes.
TABLE~\ref{tab:functionals} shows a comparison of band gaps from PBE, LDA-1/2, PBE0, HSE06, and TB09 together with the quasiparticle approach G$_0$W$_0$ for binary III/V-semiconductors GaAs, GaN and GaP at theoretically optimized lattice constants.
While PBE shows the well-known shortcoming of GGA functionals for band gap computations, the LDA-1/2 approach and the hybrid functionals achieve a decent agreement both with the many-body approach and experimental values.
However, the TB09 functional clearly outperforms the other density functionals and competes very well with the much more demanding GW-method, indicating its suitability for the band gap calculation of large supercells. Thus, we selected this functional to compute the electronic structure of the compounds.
\begin{table}
\caption{Direct band gap (eV) of III/V-semiconductors in zinc blende structure with various methods. G$_0$W$_0$ is based on PBE and without spin-orbit coupling. Lattice constants (GaAs: 5.689~\AA, GaN: 4.580~\AA, GaP: 5.477~\AA) were theoretically optimized (PBE-D3) as described in the method section. The root mean square deviation (RMSD) w.r.t. experimental reference values is given for each method.}\label{tab:functionals}
\begin{ruledtabular}
\begin{tabular}{lccccccc}
& PBE & LDA-1/2 & PBE0 & HSE06 & TB09 & G$_0$W$_0$ & Ref.\cite{Vurgaftman2001} \\
& & & & & & (PBE) & \\
\hline
GaAs & 0.32 & 1.17 & 1.71 & 1.11 & 1.44 & 1.41 & 1.52 \\
GaN & 1.58 & 3.16 & 3.48 & 2.78 & 3.03 & 3.09 & 3.28 \\
GaP & 1.74 & 2.58 & 3.31 & 2.66 & 2.95 & 2.86 & 2.86 \\
RMSD & 1.36 & 0.27 & 0.30 & 0.39 & 0.16 & 0.13 & \\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Structural effect of N-incorporation compared to other group V elements}
It has been shown before that from the available group V atoms in the GaAs lattice, incorporation of N results in the by far strongest displacement of nearest neighbor Ga atoms. The N incorporation leads to roughly twice the displacement from ideal lattice position ($-0.379$\AA) compared to the other extreme in that group, Bi ($+0.190$~\AA).
This lattice distortion enhances the extension of the nitrogen state in real space, causing the interaction with translational images of the atom. This is the underlying cause for the artificial band gap reduction described above.
Beyond nearest neighbors, the distortion propagates further and is highly anisotropic.
We show this for a $5^3$ supercell with one As-atom replaced by N and Bi, respectively.
Both cells have the lattice constant of the respective compound and the atomic positions have been relaxed.
The distortion is shown in TABLE~\ref{tab:distortions} in terms of displacement for an atom at a given position relative to atom E (E = N, Bi).
The labeling of these positions is explained in FIG.~\ref{fig:positionscheme} and is based on the connectivity to atom E along chemical bonds rather than spatial distance.
For the first two coordination spheres, i.e. the nearest neighbors (NN) and next-nearest neighbors (NNN), the distortion is isotropic.
At the third-nearest neighbors (3N) positions, a distinction can be made between the Ga-atoms along the \{110\} zig-zag chains ($\mathrm{Ga_{3N}(on)}$) and those sitting off these chains ($\mathrm{Ga_{3N}(off)}$).
While the former are further away from atoms E in terms of spatial distance than the latter, their displacement is about four times as large.
Along the \{110\}-directions, every step beyond the next-nearest neighbor spheres approximately halves the distortion.
Comparing Ga(NAs) and Ga(AsBi), the distance over which the GaAs lattice surrounding atom E is relaxed is twice as large for the dilute nitride as for the bismide, in accordance with their difference in atomic radius.
This explains, why dilute bismides can be described well with moderately sized supercells\cite{Bannow2016} in contrast to dilute nitrides.
\begin{figure}
\includegraphics[width=.6\textwidth]{positions}
\caption{Schematic representation of the environment of an atom E in GaAs lattice with E=N, Bi.}\label{fig:positionscheme}
\end{figure}
\begin{table}
\caption{Displacement relative to ideal lattice site of selected atoms in Ga$_{125}$(As$_{124}$N$_1$) compared to Ga$_{125}$(As$_{124}$Bi$_1$) in \AA. Labeling of position relative to atom E and according to FIG.~\ref{fig:positionscheme}}\label{tab:distortions}
\begin{ruledtabular}
\begin{tabular}{lcc}
& Ga$_{125}$(As$_{124}$N$_1$) & Ga$_{125}$(As$_{124}$Bi$_1$) \\
\hline
Ga$_\text{NN}$ & $-0.379$ & $0.190$ \\
As$_\text{NNN}$ & $-0.110$ & $0.053$ \\
Ga$_\text{3N}$(on) & $-0.067$ & $0.036$ \\
Ga$_\text{3N}$(off) & $-0.017$ & $0.008$ \\
As$_\text{4N}$(on) & $-0.032$ & $0.019$ \\
Ga$_\text{5N}$(on) & $-0.005$ & $0.003$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
Since the displacement is halved with every step to the next position, it should decay to sub-pm magnitude with the 6N-position in the labeling used here.
In the $5^3$ supercell, the 5N-position is already halfway between the N atom and its periodic image and is as such balanced by those (the displacement is perpendicular to the connection between the N-atom and its periodic image and a result of the displacement of neighboring atoms).
By extrapolation to the next largest cell size, the strain field can be expected to be sufficiently decayed.
\subsection{Band gap evolution for small and large supercells}
In order to study the effect of the cell size on the band gap of a dilute nitride, it is instructive to compare properties of two cubic supercells with the same N-concentration.
Thus, two supercells of different size for $3.7\,\%$ N have been taken to study different contributions to the band gap reduction due to N incorporation.
The smaller one is a $3^3$-supercell (54 atoms in total, one N), the larger one a $6^3$-supercell (432 atoms in total, eight N).
For the larger supercell, four different random arrangements of the N atoms have been sampled (see Section \ref{Sec:methods}).
One of these showed excellent agreement with the CBAC reference and will be used as an example in the remainder of this paragraph (see FIG.~\ref{fig:ganas-chgdens}b for the structure).
Following \textcite{Kent2001}, different contributions to the band gap change can be distinguished.
In the first step, the lattice constant of GaAs is changed to that of the Ga(NAs) compound, which increases the band gap to $1.61\,\text{eV}$, see TABLE~\ref{tab:gap-evolution}.
An appropriate number of As atoms in each cell is replaced by N atoms in the second step, without relaxing the structure.
This changes the band gap by different amounts for each supercell size: by $-0.42\,\text{eV}$ for the $3^3$ supercell and by $-0.22\,\text{eV}$ for the $6^3$ supercell, showing the smaller finite size effect for the latter.
In the third step, the lattice positions are relaxed, further reducing the band gap to $0.58\,\text{eV}$ for the $3^3$ and to $1.06\,\text{eV}$ for the $6^3$ supercell.
Especially this last step clearly shows the spurious interactions of translational images in small supercells.
Finally, it is instructive to replace the N atoms in the relaxed structure by As atoms again, in order to separate electronic and structural effects.
In this case, the band gap of the $3^3$ supercell is $0.12\,\text{eV}$ smaller than that of the $6^3$ supercell, which emphasizes the dampened effect of the strain field in the latter.
It should be stressed, that these contributions are not additive in either case and a synergy between both effects can be observed.
Both the single and combined effect is smaller for the larger supercell.
The relaxation step in the procedure described above can be further divided in two different ways.
First, following the band gap along the relaxation by computing it for partly relaxed structures, it can be seen that the band gap reduction is linear with relaxation, showing a simple relationship between structural deformation and change in band gap.
This emphasizes the stronger effect on the band gap when combining strongly mismatched materials.
Secondly, relaxing only the four nearest neighbors of the N atom accounts for $97\,\%$ of the band gap reduction, showing that the overlap between N and nearest neighbor-Ga atoms is most influential for the stabilization of the states at the CB edge.
\begin{table}
\caption{Band gaps of the $3^3$ and one $6^3$ supercells with $3.7\,\%$ N for the unrelaxed and relaxed structure and the relaxed structure, where N has been replaced with As. All values in eV.}\label{tab:gap-evolution}
\begin{ruledtabular}
\begin{tabular}{cccccc}
Supercell & GaAs & GaAs($a_\text{Ga(NAs)}$) & unrelaxed & relaxed & As-for-N \\
\hline
$3^3$ & \multirow{2}{*}{$1.44$} & \multirow{2}{*}{1.61} & 1.19 & 0.58 & 1.39 \\
$6^3$ & & & 1.39 & 1.06 & 1.51 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Extension of N states in real space}
The partial charge densities of the first conduction band of both cells discussed in the previous section are shown in FIG.~\ref{fig:ganas-chgdens}.
As was already described previously, \cite{Kent2001} for the too small $3^3$ supercell the N-atom interacts with its own translational image (FIG.~\ref{fig:ganas-chgdens}a).
Also, the charge density spreads along the \{110\}-zigzag chains.
In the case of the larger $6^3$ supercell, the partial charge density is spread mainly over three N-atoms and the chains in between those (FIG.~\ref{fig:ganas-chgdens}b).
While some artificial periodicity clearly remains, there is no such spurious interaction between translational images as for the smaller cell.
For an unrelaxed lattice, the N state is localized at the N atom itself, without spreading through the lattice (not shown).
Thus, the strong combined relaxation effect for too small supercells described above is well reflected in the real space electronic structure.
The corresponding partial charge density for the $5^3$ supercell used to study structural effects is shown as well (FIG.~\ref{fig:ganas-chgdens}c).
It is sufficiently localized at the N-atom and contained within the unit cell to avoid artificial interaction over the cell boundaries.
These observations agree well with the structural finding above, since the CB states are extended mainly along the same zigzag lines, on which the strain field propagates.
In a real space picture, the reduced interatomic distance along these lines increases the orbital overlap, thereby increasing the extension of the N-centered state.
\begin{figure}
\includegraphics[width=.5\textwidth]{chgdenses-ganas-sc}
\caption{Partial charge density of the first conduction band of Ga(NAs) with $3.7\,\%$ N (a) $3^3$ supercell, b) $6^3$ supercell) and $0.8\,\%$ N (c) $5^3$ supercell). Isosurface at $10\,\%$ of the respective maximal value. Ga: light blue, As: orange, N: blue; Ga and As atoms are shown in smaller size.}\label{fig:ganas-chgdens}
\end{figure}
\subsection{Band gaps of dilute nitrides for concentrations up to 11\%}
While the $6^3$ supercell clearly is better suited than the smaller $3^3$ supercell, it remains to be determined if it is large enough to reproduce accurate band gaps for nitrides over a relevant concentration range.
To this end, both simple supercells and SQS cells of increasing size are compared for various concentrations with reference values from an established CBAC\cite{Vurgaftman2003} model in TABLE~\ref{tab:gap-compare}.
For all concentrations, the $3^3$ supercells perform not only considerably worse than the $6^3$ cells, but show deficits for increasing concentrations of N, especially for specific arrangements of atoms in the cell (a particular arrangement of three N atoms bound to one Ga atom almost closes the band gap for $11.1\,\%$ N).
An average deviation from the reference of $-0.50\,\text{eV}$ demonstrates the effect of the spurious interaction between electronic states of periodic images of N atoms for the $3^3$ supercells.
The larger supercells on the other side agree with the reference values within the accuracy that can be expected from the functional, on average underestimating the reference by approximately $0.06\,\text{eV}$, and are thus large enough to overcome the artificial band gap reduction caused by the periodic boundary conditions.
This shows, that the combination of an accurate, yet feasible functional with moderately large unit cells allows quantitative, predictive computations of dilute nitride band gaps.
Furthermore, SQS cells were investigated regarding their suitability for predictive band gap computations.
This approach is designed to efficiently model random structures with periodic boundary conditions.
As such it saves the need for sampling and generally reduces the required cell size.
Given the notorious difficulty in the prediction of dilute nitride band gaps, however, a control is indicated.
The SQS cells of different sizes yield a range of band gaps that are generally close to one another for a given concentration.
For the lower concentrations of $3.7\,\%$ and $7.4\,\%$~N, the tested SQS cells agree well with the reference and clearly outperform the $3^3$ supercells.
This is especially noteworthy for the SQS-54 of the same size.
For $7.4\,\%$~N in particular, the agreement to the reference is remarkable for the SQS-54 ($+0.02\,\mathrm{eV}$) and the SQS-216 ($+0.01\,\mathrm{eV}$).
For $11.1\,\%$ N the picture changes somewhat, as the SQS cells strongly underestimate the band gap compared to the reference, yielding only slightly higher band gaps than the $3^3$ supercells.
This holds for all tested SQS cells, thus increasing the size does not mitigate this effect.
Based on the observations described above, it appears that the unavoidable periodicity prevents the strain field and the N-centered CB states from decaying sufficiently for high concentrations such as $11.1\,\%$ N even for larger SQS cells.
The $6^3$ supercells on the other hand seems to fulfill this requirement even for larger concentrations.
However, for $N^3$ supercells, scattering of band gap values is inevitable, making sampling a necessity.
Thus, for smaller concentrations of N in Ga(NAs), SQS cells remain a viable, very efficient alternative, especially since small SQS cells are sufficient in the cases where the SQS approach is applicable.
\begin{table}
\caption{Band gaps of Ga(NAs) with various N-concentrations and supercell types and sizes. Cubic $3^3$ and $6^3$ supercells as well as SQS-54, -108, -162, and -216 cells have been used for N concentrations of $3.7$, $7.4$, and $11.1\,\%$. Values were obtained by averaging over several arrangements for the $N^3$ supercells with the exception of the $3^3$ cell for $3.7\,\%$, for which only one arrangement exists. Direct band gaps in eV.}\label{tab:gap-compare}
\begin{ruledtabular}
\begin{tabular}{rrrrrrrr}
{\% N} & $3^3$ & $6^3$ & {S-54} & {S-108} & {S-162} & {S-216} & {Ref.\cite{Vurgaftman2003}} \\
\hline
3.7 & 0.58 & 1.00 & 0.91 & 0.99 & 1.30 & 1.17 & 1.06 \\
7.4 & 0.31 & 0.77 & 0.87 & 0.92 & 0.93 & 0.86 & 0.85 \\
11.1 & 0.20 & 0.63 & 0.25 & 0.35 & 0.28 & 0.24 & 0.68 \\
RMSD & 0.50 & 0.06 & 0.26 & 0.20 & 0.27 & 0.26 & \\
\end{tabular}
\end{ruledtabular}
\end{table}
FIG.~\ref{fig:gaps-sc6} shows band gaps we calculated using the $6^3$ supercell and the band gap decrease as predicted by the CBAC model. For the three lower concentrations ($1.9\,\%$, $3.4\,\%$, and $5.6\,\%$), four arrangements have been averaged, three for $7.4\,\%$ and $9.3\,\%$ and two for $11.1\,\%$.
Since different local motifs are more likely to be repeated for higher concentrations, the scattering of values tends to decrease (indicated as error bars in FIG.~\ref{fig:gaps-sc6}).
Using the experimental band gap for GaAs $E_g = 1.52\,$eV and the CBAC model parameters from Ref.~\cite{Vurgaftman2003}, the predicted band gaps are overall larger than our calculated ones.
However, since our supercell calculations are based on the TB09 functional, which yielded a band gap of $E_g = 1.44\,$eV (see TABLE~\ref{tab:functionals}), this theoretical (th.) value should be used with the CBAC model in order to allow for comparability.
And indeed, an excellent agreement between CBAC (th.) and the supercell band gaps is found in FIG.~\ref{fig:gaps-sc6}.
\begin{figure}
\includegraphics[width=.8\textwidth]{bandgaps-sc6}
\caption{Band gaps of Ga(NAs) for a $6^3$ supercell averaged over four arrangements each for the three concentrations up to $5.6\,\%$ N, three arrangements each for $7.4$ and $9.3\,\%$ N and two for $11.1\,\%$. The references are based on the CBAC model parameters as given by Ref.~\cite{Vurgaftman2003} with the experimentally derived band gap (exp.) and with the TB09 band gap (th.) of GaAs (see TABLE~\ref{tab:functionals}).}
\label{fig:gaps-sc6}
\end{figure}
It can be concluded, that a $6^3$ supercell is indeed large enough to counteract the artificial interaction with periodic images that plagues smaller supercells, while SQS of smaller size can be used for computing band gaps at lower N-concentrations, yet they seem too small at higher concentration.
\subsection{Effective band structure of Ga(NAs)}
In order to predict optical properties from \textit{ab initio} calculations, it is necessary to go beyond the computation of band gaps alone.
To explore the possibility for the system at hand, a part of the band structure along the $\Delta$ and $\Lambda$ high symmetry axes in the immediate vicinity of the $\Gamma$-point was computed and unfolded using a custom unfolding routine for the single arrangement of $\mathrm{Ga}_{216}\mathrm{N}_8\mathrm{As}_{208}$ already used above.
The effective band structure (EBS) is shown and compared with a 10-band CBAC $\mathbf{k}\cdot\mathbf{p}$-band structure\cite{Vurgaftman2003,Shan1999,Lindsay1999,OReilly2002} in FIG.~\ref{fig:ebs-sc6}.
Despite the simplistic CBAC model with a single nitrogen level, the overall agreement between both approaches is rather good, especially for the valence bands.
The two conduction bands described by the CBAC-model can be identified in the DFT derived EBS, however, one additional band with a spectral weight of roughly $0.2$ and several other bands with lower spectral weight occur in the EBS.
The $\mathrm{E}_{+}$-band of the $\mathbf{k}\cdot\mathbf{p}$-band structure can be considered as an average of the higher bands in this context.
However, the additional band right above the lowest conduction band is not accounted for by the CBAC-model.
Since the $\mathrm{E}_{-}$-band has a higher spectral weight and is lower in energy, it is the most relevant for the computation of optical properties.
Thus, this issue does not invalidate the CBAC-$\mathbf{k}\cdot\mathbf{p}$-model as a starting point for this purpose.
Nevertheless, the existence of an intermediate band may be relevant for higher excitations.
\begin{figure}
\includegraphics[width=.6\textwidth]{ebs-sc6}
\caption{Effective band structure of a $6^3$ supercell of Ga(N$_{0.037}$As$_{0.963}$) along $\Delta$ and $\Lambda$ high symmetry axes (top row) contrasted to a $\mathbf{k}\cdot\mathbf{p}$-band structure obtained from a 10 band conduction band anti-crossing Hamiltonian (bottom row).}
\label{fig:ebs-sc6}
\end{figure}
\section{Conclusions and Outlook}
We revisited the long standing issue of \textit{ab initio} calculations of the electronic structure of dilute nitrides by combining a density functional producing quantitative band gaps (TB09) with a converged supercell approach.
Our main finding is, that with this approach band gaps in excellent agreement with the established CBAC model can be achieved for the Ga(NAs) system using feasible supercell sizes.
The required supercell size is determined by the extension of the distortion field the N atom introduces in the GaAs lattice and its electronic states.
Even with a functional that produces good band gaps by itself, this criterion must be met to avoid spurious interactions of the N atom with its translational image, which manifests itself in the strong overestimation of relaxation effects on the band gap.
Comparison with the SQS-approach showed, that this design principle allows us to reach superior results compared to simple supercells of the same size also for this material system.
At higher concentrations, however, simple supercells of sufficient size allow for an easy and fast access to the electronic properties of dilute nitrides and allow to systematically study the effect of N atom arrangements on these.
Building on the results presented in this paper, it is now possible to obtain further data on the electronic structure of dilute nitrides, especially predict the properties of new, unknown materials, by computing effective band structures and extracting carrier effective masses and other band structure parameters.
\begin{acknowledgments}
This work was supported by the DFG in the framework of the Research Training Group ``Functionalization of Semiconductors'' (GRK~1782). The authors thank the HRZ Marburg, CSC-LOEWE Frankfurt and HLR Stuttgart for providing computational resources.
\end{acknowledgments}
|
1,314,259,995,837 | arxiv | \section{Introduction}
Crystalline solids can undergo structural phase transitions in
which there is a spontaneous change in the shape or internal
geometry of their unit cells
\cite{FolkSch1976,Cowley1980,Bruce1980,Fujimoto2005}. These
transitions are signalled by the softening of certain elastic
moduli or of phonon modes at a discrete set of points in the
Brillouin zone. Lattices with coordination number $z=z_c=2d$ in
$d$ spatial dimensions, which we will call Maxwell lattices
\cite{Maxwell1864}\footnote{The term \emph{isostatic} is
often incorrectly used to describe any system with $z=z_c$.
Finite Isostatic lattices have $z=z_c^N=z_c -d(d+1)/N$ and no
states of self stress - See~\cite{Calladine1978} for example.
There is no universally accepted definition of \emph{isostatic}
in lattices with periodic boundary conditions, but one such as
the square lattice with many states of self-stress is surely
not isostatic.}, exist at the edge of mechanical instability,
and they are critical to the understanding of systems as
diverse as engineering structures
\cite{Heyman1999,Kassimali2005}, diluted lattices near the
rigidity threshold \cite{Feng1984,FengLob1984,Jacobs1995},
jammed systems \cite{Liu1998,Wyart2005a,LiuNag2010a},
biopolymer networks
\cite{Elson1988,Kasza2007,Alberts2008,Janmey1990,BroederszMac2014},
and network glasses \cite{Phillips1981,Thorpe1983}. Hypercubic
lattices in $d$ dimensions and the kagome lattice and its
generalization to higher dimensions with nearest-neighbor (NN)
Hookean springs of spring constant $k$ are a special type of
Maxwell lattice whose phonon spectra have harmonic-level zero
modes not at a discrete set of points but at all $N^{(d-1)/d}$
points on $(d-1)$-dimensional hyperplanes oriented along
symmetry directions and passing through the origin
\cite{Souslov2009}. A question that arises naturally is whether
these lattices can be viewed as critical lattices at the
boundary between phases of different symmetry and, if so, what
is the nature of the two phases and the phase transition
between them.
Here we introduce and study, both analytically and with
Monte-Carlo simulations, a square-lattice model (easily
generalized to higher dimensions) in which
next-nearest-neighbors (NNNs) are connected via an anharmonic
potential consisting of a harmonic term with a
spring constant $\kappa$ tuned from
positive to negative and a quartic stabilizing term. When
$\kappa>0$, the square lattice is stable even at zero
temperature. When $\kappa=0$, NNN springs contribute only at
anharmonic order, and the harmonic phonon spectrum is identical
to that of the NN-lattice. When $\kappa<0$, the NNN potential
has two minima, and the ground state of an individual plaquette
is a rhombus that can have any orientation (Fig.~\ref{Fig:Model}c).
Plaquettes in the same row (or column), however, are
constrained to have the same orientation, but plaquettes in
adjacent rows can either tilt in the same direction or in the
mirror-image direction as shown in Figs.~\ref{Fig:Model}d-f,
leading to $2\times 2^{\sqrt{N}}$ equivalent ground states and
a subextensive but divergent entropy of order $\sqrt{N}\ln2$.
The properties of this model, including the subextensive
entropy at zero temperature, are very similar to those of
colloidal particles confined to a low-height cells
\cite{Pieranski1983,Han2008} and to the anti-ferromagnetic
Ising model on a deformable triangular lattice
\cite{Shokef2011}. In addition, the scaling of the shear
modulus near the zero-temperature critical point is analogous
to that observed in finite-temperature simulations of randomly
diluted lattices near the rigidity-percolation threshold
\cite{DennisonMac2013} and to finite-temperature scaling near
the jamming transition \cite{IkedaBir2013}, suggesting that
generalizations of our model and approach may provide useful
insight into the thermal properties of other systems near the
Maxwell rigidity limit.
\begin{figure}
\centering
\includegraphics[width=0.43\textwidth]{models.eps}
\caption{(a) The square lattice model with NN (blue thick) bonds and NNN (brown thin)
bonds. White disks showing a shift of the second row is one example of a floppy
modes of the lattice with no NNN bond. (b) Density plot of the phonon spectrum
of the NN square lattice showing lines of zero modes (darker color corresponds to lower frequency).
(c) The $T=0$ ground states of a plaquette when $\kappa < 0$, with the undeformed
reference state shown in gray. (d-f) Examples of $T=0$ ground states of the
whole lattice when $\kappa < 0$. (d) shows the uniformly sheared rhombic lattice,
which we show to be the preferred configuration at small $T$ in the thermodynamic
limit. (e) is a randomly zigzagging configuration, and (f) is the ordered maximally
zigzagging configuration, which has a unit cell consisting of two particles.
}
\label{Fig:Model}
\end{figure}
\section{Results}
Strong fluctuations arising from the large number of zero modes
lead to interesting physics at $T>0$ in this square lattice model. We show the following:
\begin{itemize}
\item Among all the equal-energy zigzagging configurations
at $\kappa<0$, the uniformly sheared rhombic lattice,
shown in Fig.~\ref{Fig:Model}d, has the lowest free
energy: the ground-state degeneracy is broken by
thermal fluctuations through an order-by-disorder
effect~\cite{Villain1980,Shender1982,Henley1987,Henley1989,
Chubukov1992,Reimers1993, Bergman2007, Shokef2011}.
\item Thermal fluctuations lead to a negative coefficient
of thermal expansion and corrections to the shear
rigidity
\cite{RubinsteinBas1992,Barriere1995,Plischke1998,TessierDis2003}
that enable the square lattice state to remain
thermodynamically stable in a region of the phase
diagram at $\kappa\leq0$ (Fig.~\ref{FIG:PD}).
\item Fluctuations drive the transition from the rhombic to
the square phase first order and lead to the phase
diagram shown in Fig. \ref{FIG:PD} in which the
temperature of the transition approaches zero as
$\kappa \rightarrow 0^-$.
\item The low-$T$ shear modulus $G$ (Fig.~\ref{FIG:PD})
in both phases is proportional to $|\kappa|$ at low temperature, and there
is a critical regime with $T > \text{const.}
\times\kappa^{3/2}$ in which $G\sim T^{3/2}$.
In addition, there is a region in the square phase (mostly metastable with
respect to the rhombic phase) with $T < \text{const.} \times
|\kappa|^{3/2}$ in which $G \sim (T/|\kappa|)^2$.
This behavior near the $T=0$ critical point is analogous to
that found in the randomly diluted triangular lattice
\cite{DennisonMac2013} near the central-force rigidity
threshold. Interestingly, the critical regime in our
model is fundamentally a consequence of nonlinearity as
is the case for dynamical scaling near the jamming
transition \cite{IkedaBir2013}.
\end{itemize}
These predictions are supported by our Monte-Carlo simulations
and by direct calculations of entropic contributions to the
free energy phonon fluctuations in different arrangements of
kinks.
\begin{figure}
\centering
\includegraphics[width=.42\textwidth]{PD1s.eps}
\caption{(a) An example of the phase diagram of the model square lattice at $k=1$,
$g=10$, $a=1$. The black solid line (the red dots) shows the boundary obtained from analytic theory (Monte Carlo simulation) between the square phase on the right and the rhombic phase on the left. The square phase is stabilized by thermal fluctuations even for $\kappa<0$ and the shear modulus $G$ of the square lattice exhibit different scaling regimes (separated by black dashed lines) determined by Eq.~\eqref{EQ:hsscaling}. (b) and (c) show Monte Carlo snapshots of the square and rhombic phases respectively. A small number of zigzags exist in the rhombic phase Monte Carlo snapshots, resulting from finite size effects as discussed in Sec.~\ref{sec:model}.
}
\label{FIG:PD}
\end{figure}
\section{Discussion}
Our model of the square-to-rhomic transition is very similar to
a model, studied by Brazovskii \cite{Brazovskii1975}, for the
transition from an isotropic fluid to a crystal and later
applied to the Rayleigh-B\'{e}nard instability
\cite{SwiftHoh1977} and to the nematic-to-smectic-$C$
transitions in liquid crystals \cite{ChenLub1976,Swift1976}. In
all of these systems, there is a subextensive but infinite
manifold of zero modes [a $(d-1=2)$-dimensional spherical shell
in the Brazovskii case, a $(d-1=1)$-dimensional circle in the
Rayleigh-B\'{e}nard case, and two $(d-2=1)$-dimensional circles
in the liquid crystal case] in the disordered phase leading to
a singular contribution to the free energy. We use the
Brazovskii theory to calculate the temperature of the
first-order square-to-rhomic transition as a function of
$\kappa<0$ and negative thermal expansion in the square phase.
In applying the Brazovskii approach to our problem, we develop
an expansion of the free energy that maintains rotational
invariance of elastic distortions in the target space. Previous
treatments \cite{FolkSch1976} of structural transitions tend to
mix up nonlinear terms arising from nonlinearities in the
strain tensor required to ensure rotational invariance and
nonlinear terms in the elastic potential itself. This advance
should be useful for the calculation of renormalized free
energies and critical exponents in standard structural phase
transitions.
The number and nature of zero modes of the critical NN square
lattice has a direct impact on the properties of the lattices
with $\kappa>0$ and $\kappa<0$, and it is instructive to review
their origin. A powerful index theorem \cite{Calladine1978}
relates the number of zero modes of a frame consisting of $N$
points and $N_B \equiv \frac{1}{2} z N$ bonds, where $z$ is the
average coordination number, via the relation
\begin{equation}
N_0 = d N - N_B + S ,
\end{equation}
where $S$ is the number of independent states of self-stress in
which bonds are under tension or compression and in which the
net force on each point is zero. In his seminal $1864$ paper
\cite{Maxwell1864}, Maxwell considered the case with $S=0$ that
yields the Maxwell relation for the critical coordination
number at which $N_0$ is equal to the number $n(d)$ ($=d$ for
periodic and $d(d+1)/2$ for free boundary conditions) of zero
modes of rigid translation and rotation:
\begin{equation}
z_c^N = 2 d - \frac{2 n(d)}{N} .
\end{equation}
In the limit of large $N$, $z_c^N \rightarrow z_c^\infty = 2
d$.
There are many small unit cell periodic Maxwell lattices with
$N_0 = S$. The NN square and kagome lattices in two dimensions
and, the cubic and pyrochlore lattices in three dimensions are
special examples of these lattices that have sample-spanning
straight lines of bonds that support states of self-stress
under periodic boundary conditions. They, therefore, have of
order $N^{(d-1)/d}$ states of self-stress and the same number
of zero modes, which are indicators of buckling instabilities
of the lines when subjected to compression. Geometrical
distortions of theses lattices that remove straight lines, as
is the case with the twisted kagome lattice \cite{Sun2012},
remove states of self-stress and associated zero modes. When
subjected to free rather than periodic boundary conditions,
these distorted lattices have a deficiency of order
$N^{(d-1)/d}$ bonds and as a result the same number of surface
zero modes (there are no bulk zero mode other than the trivial
ones of uniform translation), which can have a topological
character \cite{Kane2014} or be described in the
long-wavelength limit by a conformal field theory
\cite{Sun2012}. Unlike the infinitesimal zero modes of
hypercubic lattices, those of the kagome and pyrochlore do not
translate into finite zero modes of the lattices when
finite sections are cut from a lattice under periodic boundary
conditions. Thus, it is not yet clear whether the ground state
of the latter lattices will are highly degenerate or not.
Nevertheless, Brasovskii theory should provide a sound
description of thermal properties of these lattices in the
vicinity of the point $T=0$, $\kappa=0$.
\section{Model and order-by-disorder in the low-symmetry phase}
\label{sec:model}
The model we consider is a square lattice with two different
types of springs -- those connecting nearest neighbors and
those connecting next-nearest neighbors, as shown in
Figure~\ref{Fig:Model}(a). The NN springs
are Hookian, with potential
\begin{equation}
V_{\textrm{NN}}\left(x\right) = \frac{k}{2} x^2
\end{equation}
where $k > 0$. The NNN springs are introduced with an
anharmonic potential
\begin{equation}
V_{\textrm{NNN}}\left(x\right) = \frac{\kappa}{2} x^2 + \frac{g}{4!} x^4,
\end{equation}
where $\kappa$ can be either positive or negative and $g$,
introduced for stability, is always positive. The Hamiltonian
of the whole lattice is thus,
\begin{align}\label{EQ:Hami}
H =& \sum_{\langle i,j\rangle\in\textrm{NN}}
V_{\textrm{NN}}\left(\vert \vec{R}_i - \vec{R}_j \vert- a\right) \nonumber\\
&+ \sum_{\langle i,j\rangle\in\textrm{NNN}}
V_{\textrm{NNN}}\left(\vert \vec{R}_i - \vec{R}_j \vert-
\sqrt{2}a\right),
\end{align}
where $\vec{R}_i$ is the positions of the node $i$ and $a$ is
the lattice constant. In what follows, we will use the reduced
variables
\begin{equation}
\kr\equiv \kappa/k \qquad \text{and} \qquad \gr\equiv g a^2/k
\end{equation}
to measure the strength of couplings in $V_{\textrm{NNN}}$.
For $\kappa > 0$, $V_{\textrm{NNN}}\left(x\right)$ has a unique
minimum at $x = 0$, and the ground state of $H$ is the square
lattice with lattice spacing $a$. All elastic moduli of this
state are nonzero, and it is stable with respect to thermal
fluctuations, though, as we shall see, it does undergo thermal
contraction at nonzero temperature. When $\kappa<0$,
$V_{\textrm{NNN}}\left(x\right)$ has two minima at $x = \pm
\sqrt{6 \kappa/g}$, corresponding to stretch and compression,
respectively. This change in length of NNN springs is resisted
by the NN springs, and in minimum energy configurations one NNN
bond in each plaquette will stretch and the other will
contract. The alternative of having both stretch or contract
would cost too much NN energy. A reasonable assumption, which
is checked by our direct calculation, is that the stretching
and contraction will occur symmetrically about the center so
that the resulting equilibrium shape is a rhombus rather than a
more general quadrilateral (see Supplementary Information Sec. I). The shape of a rhombus is uniquely
specified by the lengths $d_1$ and $d_2$ of its diagonals
(which are perpendicular to each other), whose equilibrium
values are obtained by minimizing the sum over plaquettes of
\begin{eqnarray}
V_{\textrm{PL}}& = &2 V_{\textrm{NN}} (\frac{1}{2}\sqrt{d_1^2 +
d_2^2} - a) + \nonumber\\
& & V_{\textrm{NNN}} (d_1 - \sqrt{2}a) +
V_{\textrm{NNN}} (d_2 - \sqrt{2}a) .
\end{eqnarray}
The grounds state of the entire lattice when $\kappa <0$ must
correspond to a tiling of the plane by identical rhombi each of
whose vertices are four-fold coordinated. It is clear that
zigzag arrangements of rows (or columns) of rhombi in which
adjacent rows tilt in either the same or opposite directions
constitute a set of ground states. A derivation showing that
this is the complete set can be found in
Ref.~\cite{Shokef2011}, which considered packing of isosceles
triangles, which make up half of each rhombus. The ground state
energy per site, $\epsilon_0$, is simply $V_{\textrm{PL}}$
evaluated at the equilibrium values of $d_1$ and $d_2$.
Each ground-state configuration of a system with $N_x$ vertical
columns and $N_y$ horizontal rows has $K=0, \cdots , N_y$
horizontal zigzags or $K=0, \cdots , N_x$ vertical zigzags.
Thus, the ground state entropy diverges in the thermodynamic
limit, though it is sub-extensive and proportional to $N_x +
N_y$ in a system of $N = N_x N_y$ particles. Such ground state
configurations are found in other systems, most notably the
zigzagging phases seen in suspensions of confined colloidal
particles~\cite{Han2008,Pieranski1983}. The confined colloidal
system has a phase diagram that depends only on the planar
density and the height of confinement of the colloids. For
sufficiently large heights, the colloids form a phase of two
stacked square lattices. In a neighboring region of the phase
diagram, explored in simulations of
Refs.~\cite{Schmidt1996,Schmidt1997}, this square lattice
symmetry is broken through a weakly discontinuous transition
and a rhombic phase is observed. This region of the phase
diagram of confined colloids thus provides a physical
realization of the Hamiltonian~(\ref{EQ:Hami}).
At low but nonzero temperatures, the degeneracy of the ground
state is broken by thermal fluctuations through the
order-by-disorder mechanism~\cite{Villain1980, Henley1987,
Henley1989, Chubukov1992, Reimers1993, Bergman2007,
Shokef2011}. This splitting of degeneracy due to small phonon
fluctuations around the ground state may be calculated using
the dynamical matrix in the harmonic approximation to the
Hamiltonian~(\ref{EQ:Hami}). For each ground state
configuration $\vec{R}_i$, we write the deformation as $\vec{R}_i \to \vec{R}_i
+ \vec{u}_i$, and expand to quadratic order in $\vec{u}$, $H =
\frac{1}{2} \sum_{\langle i j\rangle}\vec{u}_i \mathbf{D}_{ij}
\vec{u}_j$. The Fourier transform of $\mathbf{D}_{ij}$,
$\mathbf{D}_q$, is block-diagonal and the phonon free energy,
which is purely entropic, for that configuration is
\begin{equation}
F_p(N_x,N_y,K) = \frac{1}{2} k_B T \sum_\mathbf{q} \ln \det \mathbf{D}_\mathbf{q}
\equiv N k_B T w_p,
\label{eq:sp}
\end{equation}
where $w_p$ is the free-energy per site in units of $k_B T$. In
general, $F_p$ depends not only on $K$, but also on the
particular sequence of zigzags. We numerically calculated this
free energy for all periodically zigzagged configurations with
up to 10 sites per unit cell. We found that the
lowest-free-energy state is the uniformly sheared state
[Fig.~\ref{Fig:Model}(c)] with $K = 0$, and the
highest-free-energy state is the maximally zigzagged sate
[Fig.~\ref{Fig:Model}(e)] with $K = N_y$ and two sites per
unit cell. Both of these energies are extensive in the number
of sites $N$, and we define
\begin{eqnarray}
\Delta F_0 (N_x,N_y)& = & F_p(N_x,N_y,N_y) - F_p(N_x,N_y, 0 )\nonumber \\
&\equiv & N_x N_y k_b T \Delta w_0 \label{Eq:DeltaF}\\
\Delta F(N_x,N_y,K) & = & F_p(N_x,N_y,K) - F_p(N_x,N_y,0) . \nonumber
\end{eqnarray}
Figure~\ref{fig:o-by-diso}(a) displays our calculation of $\Delta w_0
(\tau)$, which vanishes, as expected, at $\tau = 0$ and also at
large $\tau$ at which the rhombus collapses to a line. Figure
\ref{fig:o-by-diso}(b) plots $\Delta F/\Delta F_0$ as a
function of $\phi = K/N_y$ for different values of $\tau$. By
construction, this function must vanish at $\phi = 0$ and be
equal to one at $\phi = 1$. All of the points lie
approximately on a straight line of slope one.
Thus, we can
approximate $F_p(N_x,N_y,K)$ by
\begin{equation}
F_p(N_x,N_y,K) = F_p(N_x,N_y,0)+ N_x K k_B T \Delta w_0 .
\label{eq:Fp}
\end{equation}
Note that for each $K$, this energy is extensive in $N=N_x
N_y$ as long as $\phi \neq 0$.
These calculations were carried out in the thermodynamics
limit, $N \rightarrow 0$, in which the sum over $\mathbf{q}$ is
replaced by the continuum limit by an integral. In order to
compare these results with the Monte Carlo results of the next
section, it is necessary to study finite size effects. The
first observation is that for any finite $N_x$, the system will
be effectively one-dimensional for a sufficiently large $N_y$,
and as a result, we would expect the number of zigzags to
fluctuate. To proceed, we continue to use the continuum limit
to evaluate Eq.~(\ref{eq:sp}), and we use Eq.~(\ref{eq:Fp}) for
$F_p(N_x,N_y,K)$. The partition function for this energy is
\begin{eqnarray}
Z&= &\sum_{K=0}^{N_y} e^{-\beta[F_p(N_x,N_y,0)+E_0(N_x,N_y)]}
\binom{N_y}{K}e^{-N_x K \Delta w_0} \nonumber\\
& = & e^{-\beta[F_p(N_x,N_y,0)+E_0(N_x,N_y)]}
\left(1+ e^{-N_x \Delta w_0}\right)^{N_y}
\end{eqnarray}
where $E_0(N_x,N_y)$ denotes the potential energy which is independent of $K$, and the full free energy is
\begin{align}
& F(N_x,N_y,T) = -T \ln Z \nonumber \\
& = N_x N_y (f_0 + e_0) - N_y k_B T \ln (1 + e^{- N_x \Delta w_0}) ,
\end{align}
where $e_0$ and $f_0$ are the potential energy and phonon free energy per site of the uniformly sheared state.
Thus, as expected, when $N_x \Delta w_0 \gg 1$, zigzag
configurations make only a very small, subextensive
contribution to the free energy. On the other hand, in the
opposite limit, they make an extensive contribution of $N_x N_y
k_B T \Delta w_0$ to the energy. Therefore, at a given $\tau$, the zigzag configurations are favored when the system is small, and in thermodynamic limit, the rhombic configuration is always favored. Our Monte Carlo simulation verified this [see inset of Fig.~\ref{fig:o-by-diso}(b)].
\begin{figure}
\centering
\includegraphics{o-by-diso-2.eps}
\caption{ The phonon contribution to the free energy for
various zigzagging configurations. (a) shows the free energy
difference $\Delta w_0$ between the maximally zigzagging
configuration (Fig.~\ref{Fig:Model}e) and the uniformly sheared
square lattice configuration (Fig.~\ref{Fig:Model}c). For
sufficiently large $\tau$, the lattice collapses onto a line,
at which point the free energy difference goes to zero. (b)
shows the free energy of the phonons as a ratio $\Delta F /
\Delta F_0$ for $\tau = -0.02$, $-0.1$ and $-1$ ($\lambda =
10$), where $\Delta F$ and $\Delta F_0$ are defined in
Eq.~(\ref{Eq:DeltaF}) and $\Delta F$ is evaluated for all
possible configurations with unit cells with at most $10$ sites
as a function of the zigzag fraction $\phi = K/N_x$. The
lattice without zigzags is entropically favored for any any
value of $\kr$ in our calculations. $\Delta F/\Delta F_0$ is
well approximated by the line $\phi$, which corresponds to
non-interacting zigzags. This interaction results in the
dispersion of values of $\Delta F/\Delta F_0$.
The inset of (b) shows Monte Carlo simulation results
for the zigzagging fraction $\phi$ plotted against the linear
system size $\sqrt{N}$.}
\label{fig:o-by-diso}
\end{figure}
\section{Simulation}
We simulate the system using a Monte Carlo (MC) algorithm
inside a periodic box whose shape and size are allowed to
change in order to maintain zero pressure. In this version of
the Metropolis algorithm, also used in
Refs.~\cite{Shokef2009,Shokef2011}, for each MC step a particle
is picked at random and a random trial displacement is
performed. The trial displacement is initially uniformly
distributed within a radius of $0.1 a$, but throughout the
simulation the radius is adjusted to keep the
acceptance probability between $0.35$ and $0.45$. Given the
initial configuration energy $E_i$ and the trial configuration
energy $E_j$, the trial configuration is accepted with
probability $[1 + \exp(E_i - E_j)/T]^{-1}$, i.e., Glauber
dynamics is used. After initializing
the system using the square lattice configuration with lattice
constant $a$, the simulation is first run at a high temperature
and is then annealed to the final low temperature. For each
intermediate temperature, an equilibration cycle in a
sample of $N$ sites consists of at least $N^{4}\times10^{2}$
MC steps. To accommodate areal and shear distortions in the
different phases we encounter, the simulation box area and
shape are changed using a similar acceptance algorithm, with
the trial deformation adjusted to keep the acceptance
probability between $0.35$ and $0.45$, such that the simulation
box retains the shape of a parallelogram~\cite{Frenkel2001}. The simulation is
thus performed at zero pressure, and a range of temperatures
measured in units of $k a^2$ for up to $N=3600$ sites.
We use these simulations to investigate the phase diagram
corresponding to the
Hamiltonian~(\ref{EQ:Hami})~\cite{Binder1981a} and to
investigate the properties of the phases we encounter, such as
ground state degeneracy, order-by-disorder and negative thermal
expansion. As all simulations involve a finite lattice and are
run for a finite time, we took care to make sure that the
system is sufficiently large to capture the thermodynamic
behavior and that the simulation time is sufficiently long for
the system to relax to equilibrium. To capture the
subtlety of the order by disorder effect for a finite system,
we simulated the model for a range of sizes and times and
calculated the average fraction of zigzags $n$ in equilibrium
[inset of Fig.~\ref{fig:o-by-diso}(b)]. While for small systems, $\phi
\approx \frac{1}{2}$, as in a disordered zigzagging
configuration, for large systems, $\phi$ approaches $0$,
suggesting that the system prefers the configuration of a
uniformly sheared square lattice. Thus, we find good agreement
with theoretical results from Sec.~\ref{sec:model}.
We calculated the shape of the phase boundary shown in
Fig.~\ref{FIG:PD}, the behavior of the order parameter across
the phase boundary shown in Fig.~\ref{fig:OPNTE}a, and the
negative of the thermal expansion coefficient $(L-L_0)/L_0$
shown in Fig.~\ref{fig:OPNTE}(b), where $L$ is the length at
$T>0$ and $L_0$ that at $T=0$. The phase boundary is obtained
by calculating the heat capacity of the system as a function of
temperature at fixed $\lambda$ and $\tau$. The location of the
peak of the heat capacity corresponds to the location of the
phase transition in the thermodynamic limit, and in our
simulations, the locations of the peak converge to the values
seen in Fig.~\ref{FIG:PD}. The order parameter values are
calculated locally, i.e., $t$ is calculated for each
plaquette and each configuration via the angle between the two
adjacent nearest-neighbor bonds and then averaged over all
plaquettes in the system and over all 100
configurations. In this approach, $t$ is independent of the
particular zigzag configuration, and its evaluation does not
exhibit the long relaxation process to the uniformly sheared
square lattice. The behavior of the order parameter in
Fig.~\ref{fig:OPNTE} is consistent with a weakly-discontinuous
transition.
\begin{figure}
\centering
\includegraphics{NTEnew-2.eps}
\caption{(a) Order parameter $t$ calculated from theory (lines) and simulation (data points),
at $\gr=10$, and from right to left, $T/(ka^2)=0.0001, 0.001, 0.003, 0.005, 0.007$.
(b)Negative thermal expansion. Shown in the figure are normalized size change $(L_0-L)/L_0$ as
a function of $T$ at $\kappa=0,g=10$ (red triangles: MC data; red upper line: theory),
and at $\kappa=0.1, g=0$ (black circles: MC data; black lower line: theory).}
\label{fig:OPNTE}
\end{figure}
\section{Analytic theory and the Phase Diagram}\label{sec:theory}
The special feature of our model is its large but subextensive
number of soft modes living on the $q_x$ and $q_y$ axes in the
first Brillouin zone, as shown in Fig.~\ref{Fig:Model}(b) in the
limit $\kappa\to 0$)~\cite{Souslov2009,Mao2010}. As we discuss
below, these floppy modes provide a divergent fluctuation
correction to the rigidity of the square lattice and render the
transition from the square to the rhombic phase first order.
Thus, this model is analogous to the one introduced by
Brazovskii \cite{Brazovskii1975} for a liquid-solid transition
in which mode frequencies of the form
$\omega=\Delta+(q-q_c)^2/m$ vanish on a $(d-1)$-dimensional
hypersphere when $\Delta \rightarrow 0$ and contribute terms to
the free energy singular in $\Delta$.
To study the square-to-rhombic transition, we take the $T=0$
square lattice, with site $i$ at position $\vec{R}_{0,i}$, as
the reference state. We then represent positions in the
distorted lattices as the sum of a part arising from uniform
strain characterized by a deformation tensor $\dg$ and
deviations $\uf_{i}$ from that uniform strain:
\begin{align}
\vec{R}_{i} = \mathbf{\dg} \cdot \vec{R}_{0,i} + \uf_{i} ,
\label{eq:R0i}
\end{align}
The deviations $\uf_{i}$ are constrained to satisfy periodic
boundary conditions, and their average $\langle \uf_{i}
\rangle$ is constrained to be zero. The former condition
ensures that the sum over all bond stretches arising from these
deviations vanishes for every configuration. Without loss of
generality, we take $\Lambda_{yx}$ to be zero, leaving three
independent parameters to parameterize the three independent
strains.
As we detail in Supplementary Information Sec. IV, the strain parameter
characterizing pure shear with axes along the $x$ and $y$ axes
of the reference lattice is of order $t^2$ and can be
ignored near $\tau = 0$, and we set
\begin{align}\label{EQ:Lambda}
\mathbf{\dg} = \left( \begin{array}{cc}
1+s & t \\
0 & 1+s
\end{array} \right) .
\end{align}
Note that $\dg$ is invertible even though it is not symmetric.
$t$ is the order parameter that distinguishes the rhombic phase
from the square phase. Thermal fluctuations lead to $s<0$ in
both phases.
Expanding the Hamiltonian [Eq.~(\ref{EQ:Hami})] in to second
order $\uf_{i}$ about the homogeneously deformed state $\langle
\vec{R}_{i} \rangle=\mathbf{\dg} \cdot \vec{R}_{0,i}$, we
obtain
\begin{align}\label{EQ:HExpa}
H = H_0(\mathbf{\dg}) + \frac{1}{V} \sum_{q} \uf_{q} \cdot \mathbf{D}_q(\mathbf{\dg}) \cdot \uf_{-q}
+ O((\uf)^3) ,
\end{align}
where $H_0$ is the energy of the uniformly deformed state, and
\begin{align}\label{EQ:DML}
\mathbf{D}_q(\mathbf{\dg}) = v_q(\mathbf{\dg}^T \mathbf{\dg}) \mathbf{I}
+ \mathbf{\dg}\cdot \mathbf{M}_q(\mathbf{\dg}^T \mathbf{\dg}) \cdot \mathbf{\dg}^{T}
\end{align}
is the $d\times d$ dimensional ($d=2$ being spatial dimension)
dynamical matrix with scalar $v_q$ and second rank tensor
$\mathbf{M}_q$ determined by the potentials. There is no term
linear in $\uf_{i}$ in Eq.~(\ref{EQ:HExpa}) because of the
periodicity constraint (see Supplementary Information Sec. II).
Integrating out the fluctuations $\uf$ from the
Hamiltonian~\eqref{EQ:HExpa}, we obtain the free energy of the
deformed state~\footnote{This form is similar to
Eq.~\ref{eq:sp}, except that here we expand around uniformly
deformed lattice of continuously varying $\dg$, rather than
about a zigzagging state.}
\begin{align}
\label{eq:f}
F(\mathbf{\dg}) = H_0(\mathbf{\dg}^T \mathbf{\dg}) +
\frac{T}{2}\ln\textrm{Det}\tilde{\mathbf{D}}(\mathbf{\dg}) ,
\end{align}
where $\tilde{\mathbf{D}}=v_q + M_q \mathbf{\dg}^T
\mathbf{\dg}$ depends only on $\mathbf{\dg}^T \mathbf{\dg}=1+2
\mathbf{\st0}$, where $\mathbf{\st0}$ is the full nonlinear
strain. Thus the one-loop free energy of Eq.~(\ref{eq:f}) is a
function of the nonlinear, rather than the linear strain, so
that rotational invariance in the target space is guaranteed
and there is a clean distinction between nonlinear terms in
linearized deformations arising from nonlinearities in
$\mathbf{\st0}$ in from nonlinear terms in the expansion in
powers of $\mathbf{\st0}$.
To analyze the transition between the square and the rhombic
phases at low temperature we expand $F$ as a series in
$\mathbf{\st0}$, by expanding the transformed dynamical matrix
as $\tilde{\mathbf{D}} = \mathbf{D}_0
+\mathbf{A}(\mathbf{\st0})$, where $ \mathbf{D}_0 =
\mathbf{D}\vert_{\mathbf{\st0} = 0}$ is the dynamical matrix of
the undeformed state. The free energy is then
\begin{align}\label{eq:Fexpansion}
F(\mathbf{\st0}) = H_0(\mathbf{\st0}) + \frac{T}{2}\textrm{Tr} \ln
\mathbf{D}_0 \left\lbrack \mathbf{I}+\mathbf{G}_0 \cdot \mathbf{A}(\mathbf{\st0})\right\rbrack
\equiv V f(\mathbf{\st0}) ,
\end{align}
where $\mathbf{G}_0 \equiv \mathbf{D}_0 ^{-1}$ is the phonon
Green's function in the undeformed state, $V=Na^2$ and $f$ is
the free energy density. The expansion of $F$ at small
$\mathbf{\st0}$ follows from this.
Close to the transition, $F$ is dominated by fluctuations
coming from the floppy modes as we discussed above. As $\kappa\to
0$, the frequency of these floppy modes vanishes as
$\omega\sim\sqrt{\kappa}$, and the corresponding phonon Green's
function diverges, leading to divergent fluctuation corrections
to the coefficients of $\st0$ in Eq.~\eqref{eq:Fexpansion} as
detailed in Supplementary Information Sec.~III. Keeping leading order terms as
$\kr \rightarrow 0$, we can identify the two phases through the
equations of state,
\begin{align}
\frac{\partial f(\mathbf{\st0})}{\partial s}
&\simeq2 ka^2\left(\frac{\tilde{T}}{\sqrt{\kr}} +s \right)=0 \label{EQ:EOSxx}\\
\frac{\partial f(\mathbf{\st0})}{\partial t}
&\simeq ka^2t
\left(\kr + \frac{\lambda\tilde{T}}{\sqrt{\kr}} +\frac{1}{12} \lambda t^{2} \right)=0 ,
\label{EQ:EOSxy}
\end{align}
where
\begin{equation}
\tilde{T} = \frac{\pi T }{8 k a^2}
\end{equation}
is a unitless reduced temperature. Eq.~(\ref{EQ:EOSxy}) has
three solution for $t$: $t = 0$ corresponding to the square
phase, and two solutions for $t \ne 0$ corresponding to the two
orientations of the rhombic phase. There is only a single
solution for $s$, with $s < 0$, from which we conclude that
both phases exhibit \emph{negative thermal expansion}. The
elastic rigidity and thus the stability of the two phases is
determined by the second derivatives of $F$ with respect to $s$
and $t$. In particular, the reduced shear modulus ($G/k$
where $G$ is the shear modulus) is
\begin{align}\label{EQ:Rordered}
r =\frac{1}{k} \frac{\partial^2 f(\mathbf{\st0})}{\partial t^2}
\simeq \kr + \frac{\tilde{T}}{\sqrt{\kr}} + \frac{1}{4} \lambda t^2.
\end{align}
To obtain these leading order equations we (i) assume low
$T$, so
only terms singular in $\tau$ as $\tau\to 0$, such as
$t/\sqrt{\tau}$, in the integral of
$\frac{T}{2}\ln\textrm{Det}\tilde{\mathbf{D}}(\mathbf{\dg})$,
and (ii) assume that
\begin{align}\label{EQ:Assu}
|s| \sim t^2 \ll \kr \ll 1 ,
\end{align}
the validity of which will be verified below.
As observed in the simulation (Fig.~\ref{FIG:PD}), thermal
fluctuations at $T >0$ stabilize the square relative to the
rhombic phase even for $\kappa<0$. To understand this phenomenon
within the analytic approach, we use a self-consistent-field
approximation in which $\kr$ is replaced by with its
renormalized value $r$ in the phonon Green's function
$\mathbf{G}_0$ and thus in the denominators on the right hand
sides of Eqs.~(\ref{EQ:EOSxx}), (\ref{EQ:EOSxy}), and
(\ref{EQ:Rordered}). In this approximation, the shear rigidity
of the square ($t=0$) and rhombic ($t\neq 0$) satisfy
\begin{align}\label{EQ:rsquare}
r =
\begin{cases}
\begin{array}{ll}
\kr + \frac{\tilde{T}\lambda}{\sqrt{r}} &\qquad\text{square} \\
-2 \kr - 2 \frac{\tilde{T}\lambda}{\sqrt{r}} & \qquad \text{rhombic}
\end{array},
\end{cases}
\end{align}
where we used the equation of state, Eq.~(\ref{EQ:EOSxy}) to
eliminate $t^2$ from Eq.~(\ref{EQ:Rordered}). In the square
phase, Eq.~(\ref{EQ:rsquare}) has a solution $r > 0$, implying
local stability, everywhere except at $\tilde{T} = 0,
\kr < 0$ and in the uninteresting limit, $z \rightarrow -
\infty$. This local stability implies that the transition to
the rhombic phase must be first order. In the rhombic phase,
solutions $r>0$ only exist for $\kr < \kr_{c1}$ ($z < z_{c1}$),
where
\begin{equation}
\kr_{c1} = -\frac{3}{2} (\tilde{T}\lambda)^{2/3}
\label{eq:krc1}
\end{equation}
The solutions to Eq.(\ref{EQ:rsquare}) can conveniently be
expressed as scaling functions in the two phases:
\begin{equation}
\frac{|\kr|}{r} = h_{\nu} (|z|), \qquad z = \frac{\kr}{(\tilde{T}\lambda)^{2/3}} ,
\label{Eq:scaling1}
\end{equation}
where $\nu=s,\,\,r$ for the square and rhombic phases,
respectively. The scaling functions $h_s(|z|)$ and $h_r(|z|)$
depicted in Fig.~\ref{fig:scalefn} have the following limits
\begin{eqnarray}
h_s(|z|) &\sim &
\begin{cases}
1 & \qquad z \rightarrow +\infty \\
|z| & \qquad z \rightarrow 0 \\
|z|^3 & \qquad z \rightarrow - \infty
\end{cases} \\ \label{EQ:hsscaling}
h_r(|z|) & \sim &
\begin{cases}
(3/2)-\sqrt{6|z-z_{c1}|} & \qquad z\rightarrow z_{c1}^- \\
1/2 & \qquad z\rightarrow z \rightarrow - \infty,
\end{cases}\label{EQ:hrscaling}
\end{eqnarray}
where $z_{c1}=-3/2$.
The $z^3$ regimes of $h_s$ is in the metastable regime where
the rhombic phase is stable. These results yield the scaling
phase diagram of Fig.~\ref{FIG:PD}.
\begin{figure}
\centering
\includegraphics{scale-fn.eps}
\caption{(color online) Plots of $h_s(|z|)$ (black) and $h_r(|z|)$ (red)
as a function of $z$. Note the singular behavior of $h_r(|z|$ in the
vicinity of $z_{c1}$ and the large difference between $h_s$ and $h_r$ at the
first order transition at $z=z_{c2}$.}
\label{fig:scalefn}
\end{figure}
The phase boundary of this discontinuous transition occurs
along the coexistence line (i.e., equal free-energy line) of
the two phases. Following Brazovskii, we have already
calculated the limit of metastability of the rhombic phase,
i.e., for the value of $\kappa=\kr_{c1}$ [Eq.~(\ref{eq:krc1})]
at the local free energy minimum where that phase first
appears. We then calculate the free energy difference between
the two phases, which is evaluated through the following
integral for a given $\kr$,
\begin{align}\label{EQ:DeltaF}
\Delta F = \int_{0}^{t_{\textrm{rhombic}}}
\frac{d F(\mathbf{\st0})}{d t} dt =\int_{r_0}^{r_1}
\frac{d F(\mathbf{\st0})}{d t} \frac{dt}{dr} dr ,
\end{align}
where we have substituted $r(t)$ for the integral measure. Here
$r_0$ and $r_1$ are the values of $r$ at the minima of $F$
corresponding to the square and the rhombic phases determined
by Eq.~(\ref{EQ:rsquare}). Along the path of this integral,
Eq.~\eqref{EQ:Rordered} is valid, but the equations of
state~(\ref{EQ:EOSxx}) and (\ref{EQ:EOSxy}) and the equation
[Eq.~(\ref{EQ:rsquare})] for $r$ in the rhombic phase are not
satisfied, because they only apply to equilibrium states. The
phase boundary corresponds to the curve of $\kr=\kr_{c2}$ along
which $\Delta F\vert_{\kr = \kr_{c2}}=0$.
As shown in Supplementary Information Sec. IV,
an asymptotic solution valid at low $\kr$, can be obtained
by expanding the equation around $\kr=\kr_{c1}$, assuming that
$\kr_1$ and $\kr_2$ are of the same order of magnitude
(verified below). This yields
\begin{align}\label{EQ:PD}
\kr_{c2} =- \left(\frac{3}{2}+c \right) (\tilde{T}\lambda)^{2/3} = - 1.716 \tilde{T}^{2/3} ,
\end{align}
for $\tilde{T}\ll 1$ where $c\simeq 0.216$ is a constant. This
transition line is shown in Fig.~\ref{FIG:PD}. Excellent
agreement between theory and simulation is obtained without any
fitting parameter.
Along the phase boundary, $r_0,r_1\sim \tilde{T}^{2/3}>0$, so that
both phases are locally stable. The order parameter for the
transition, $t$, jumps from $0$ to
\begin{align}
t_{c2} = 3.4 \tilde{T}^{1/3}\lambda^{-1/6}
\end{align}
at the transition. As $T\to 0 $ this discontinuity vanishes,
consistent with the continuous nature of the transition at $T =
0$. A good agreement between $t$ values in theory and in
simulation is shown in Fig.~\ref{fig:OPNTE}(a).
From Eq.~(\ref{EQ:EOSxx}), the negative thermal expansion
coefficient in both the square and rhombic phases,
\begin{equation}
s= -\frac{\tilde{T}}{\sqrt{r}} ,
\end{equation}
is determined by the equation of state, Eq.~\eqref{EQ:EOSxx}.
Equation (\ref{EQ:rsquare}) for $r$ then implies the following
behavior for $s$ in different regions of the phase diagram: In
the critical region $0<|\kr|<\tilde{T}^{2/3}$ of the square phase,
\begin{equation}
s \simeq - \tilde{T}^{2/3} \lambda^{-1/3}.
\end{equation}
Deep in the square and rhombic phases, where $0<\tilde{T}^{2/3} \ll
|\kr|$,
\begin{equation}
s =
\begin{cases}
\begin{array}{ll}
- \tilde{T}/\sqrt{|\tau|} & \qquad \text{square} \\
-\tilde{T}/\sqrt{2|\tau|} & \qquad \text{rhombic}
\end{array}
\end{cases} .
\end{equation}
Finally along the coexistence curve in both phases,
$s\sim-\tilde{T}/\sqrt{|\kr|} \sim \tilde{T}^{2/3}$. These results agree
well with simulation measurement of negative thermal expansion,
as shown in Fig.~\ref{fig:OPNTE}b. In this lattice the negative
thermal expansion behavior results from strong transverse
fluctuations associated with soft modes.
These solutions for $s$ and $t$ verifies that our assumptions
in Eq.~(\ref{EQ:Assu}) are satisfied, provided that $\gr \gg
1$.
\section{Review}
We have presented an analysis of a model based on the
square lattice with NN harmonic and NNN anharmonic springs that
can be tuned at zero temperature from a stable square lattice
through the mechanically unstable NN square lattice to a
highly degenerate zigzag state by changing the coefficient
$\kappa$ of the harmonic term in the NNN spring from positive
through zero to negative. Using analytic theory, including a
generalization of the Brazovskii theory for the
liquid-to-crystal transition, we investigated the phase diagram
and mechanical properties of this model at $T>0$. The
degeneracy of the zero-$T$ zigzag state is broken by an
order-by-disorder effect, thermal fluctuations drive the
square-to-rhombic phase transition first order, and the
elastic modulus of the square phase a crossover from being
proportional to $\kappa$ for $\kappa\gg k(Tg/k^2)^{2/3} >0$ to
$T^{2/3}$ for $|\kappa| \ll k(Tg/k^2)^{2/3}$ to $T^2$ for
$\kappa \ll - k(Tg/k^2)^{2/3}<0$ as function of the scaling
variable $z=\tau/(\tilde{T} \lambda)^{2/3} = \sim
(\kappa/k)/(Tg/k^2)^{2/3}$. This behavior arises because the
spectrum of the NN square lattice with $N$ sites exhibits
$\sqrt{N}$ zero modes on a one-dimensional manifold in the
Brillouin Zone. Other lattices such as the 2D kagome lattice,
the 3D simple cubic lattice, and the 3D pyrochlore and
$\beta$-cristobalite \cite{Hammonds1996} lattices have similar
spectra, and it is our expectation that generalizations of our
model to these lattices will exhibit similar behavior. It is
also likely that our model can inform us about more physically
realistic models in which interactions lead to spectra with a
large set of modes with small but not zero frequency.
\textit{Acknowledgments -- } A.S. gatefully acknowledges discussions with P. A. Rikvold, Gregory Brown, Shengnan Huang and Andrea J. Liu. This work was supported in part by the NSF under grants DMR-1104707 and DMR-1120901 (TCL), DMR-1207026 (AS), grant DGAPA IN-110613 (CIM) and by the Georgia Institute of Technology (AS).
\bibliographystyle{naturemag}
|
1,314,259,995,838 | arxiv | \section{Introduction}
\subsection{Main results}
In \cite{G_asym} Gromov proposed a rough classification of periodic CAT(0) spaces modulo hyperbolic ones.
It vaguely states that these spaces decompose into pieces such that every piece either displays
almost hyperbolic behavior or else has uniformly distributed flats. One may want to think of
a non-positively curved compact 3-manifold and its decomposition into Seifert and atoroidal pieces.
Each Seifert piece admits an $\mathbb R^3$- or
$\mathbb{H}^2\times\mathbb R$-structure whereas the atoroidal pieces admit
$\mathbb{H}^3$-structures.
Gromov's expectation roots in the general principle that the failure of hyperbolic behavior is caused by (unbounded parts of) flats --
isometric embeddings of higher dimensional Euclidean spaces. For instance, an axial isometry exhibits north-south dynamics
at infinity unless one (and then every) of its axes bounds a flat half-plane.
One concrete interpretation of Gromov's vision is provided by the following two conjectures due to Ballmann and Buyalo \cite{BaBu_periodic}
which classify CAT(0) spaces by means of their {\em rank}.
A complete geodesic is said to have {\em rank 1} if it does not bound a flat half-plane.
Accordingly, a CAT(0) space is called {\em rank 1} if it contains a rank 1 geodesic,
otherwise we say it has {\em higher rank}.
Recall that if a group $\Gamma$ acts on a CAT(0) space $X$, then its {\em limit set} $\Lambda(\Gamma)$ is the set of accumulation points
of a $\Gamma$-orbit in the ideal boundary $\partial_{\infty} X$.
\begin{conj}[Closing Lemma]\label{conj_cl}
Let $X$ be a locally compact CAT(0) space which admits a properly discontinuous action $\Gamma\curvearrowright X$ with $\Lambda(\Gamma)=\partial_{\infty} X$.
If $X$ contains a geodesic of rank 1, then it also contains a $\Gamma$-periodic
geodesic of rank 1.
\end{conj}
Conjecture~\ref{conj_cl} is known to hold in each of the following settings.
\begin{itemize}
\item If $\Gamma$ satisfies the duality condition of Chen
and Eberlein, compare \cite[Chapter~III]{ballmannbook}.
\item If $X$ is a piecewise
smooth complex of dimension two and $\Gamma$ acts {\em geometrically} -- properly discontinuously with compact quotient, see \cite{BaBr_orbi}.
\item If $X$ is a finite-dimensional CAT(0) cube complex and $\Gamma$ acts geometrically \cite{CS_rr}.
\end{itemize}
\begin{conj}[Diameter Rigidity]\label{conj_rr}
Let $X$ be a locally compact and geodesically complete CAT(0) space which admits a properly discontinuous action $\Gamma\curvearrowright X$ with $\Lambda(\Gamma)=\partial_{\infty} X$.
If $X$ has higher rank, then $X$ is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a
metric product.
\end{conj}
Recall that in a CAT(0) space $X$, every complete geodesic bounds a flat half-plane if and only if its Tits boundary $\partial_{T} X$
has diameter equal to $\pi$, compare Section~\ref{sec_catzero}. This explains the name of the conjecture.
The following special cases of Conjecture~\ref{conj_rr} are known to hold.
\begin{itemize}
\item If $X$ is a Hadamard manifold and $\Gamma$ satisfies the duality condition \cite{B_higher,BS_higher,EH_diff}; for a comprehensive proof, see Chapter IV in
\cite{ballmannbook}.
\item If $X$ is a homogeneous Hadamard manifold \cite{Heber}.
\item If $X$ is a piecewise
smooth complex of dimension two or a piecewise Euclidean complex of dimension 3 and
$\Gamma$ acts geometrically \cite{BaBr_orbi,BB_rr}.
\item If $X$ is a finite-dimensional
CAT(0) cube complex and $\Gamma$ acts geometrically \cite{CS_rr}.
\end{itemize}
We confirm Conjectures~\ref{conj_cl} and~\ref{conj_rr} for CAT(0) spaces with geometric group actions and without 3-flats.
By \cite[Theorem~C]{Kleiner}, the absence of 3-flats is equivalent to $X$ not containing 3-dimensional quasi-flats.
\begin{introthm}\label{thm_mainA}
Let $\Gamma$ be a group acting geometrically on a locally compact CAT(0) space without $3$-flats.
If $X$ contains a geodesic of rank 1, then it also contains a
$\Gamma$-periodic geodesic of rank 1.
\end{introthm}
\begin{introthm}\label{thm_mainB}
Let $\Gamma$ be a group acting geometrically on a locally compact and geodesically complete CAT(0) space without $3$-flats.
If $X$ has higher rank,
then $X$ is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product.
\end{introthm}
A consequence of Theorems~\ref{thm_mainA} and~\ref{thm_mainB} is that all such groups satisfy the Tits Alternative:
\begin{introcor}\label{cor_tits}
Let $\Gamma$ be a group acting geometrically on a locally compact and geodesically complete CAT(0) space without $3$-flats.
Then either $\Gamma$ contains a free non-abelian subgroup or else $X$ is flat and $\Gamma$ is a Bieberbach group.
\end{introcor}
Another application of Theorem~\ref{thm_mainA} gives the following.
\begin{introcor}\label{cor_main}
Let $X$ and $X'$ be irreducible locally compact geodesically complete CAT(0) spaces. Suppose that $X$ does not contain a 3-flat.
If the same group $\Gamma$ acts geometrically on $X$ and $X'$,
then either $X$ and $X'$ are isometric after possibly rescaling or there exists an element $\gamma\in\Gamma$ with $\gamma$-axes $c\subset X$
and $c'\subset X'$ such that neither of them bounds a flat half-plane.
\end{introcor}
The present article is part of a series motivated by the Diameter Rigidity Conjecture (sometimes also referred to
as Higher Rank Rigidity). The main results obtained in the other parts are as follows.
\begin{thm}[{\cite[Theorem~A]{St_rrI}}]\label{thm_rrI}
Let $X$ be a locally compact CAT(0) space whose Tits boundary has dimension $n-1\geq 1$.
Suppose that every geodesic in $X$ lies in an $n$-flat.
If $X$ contains a periodic $n$-flat, then $X$ is a Riemannian symmetric space or a Euclidean building, or $X$ non-trivially splits as a metric product.
\end{thm}
\begin{thm}[{\cite[Main Theorem]{St_rrII}}]\label{thm_rrII}
Let $X$ be a locally compact CAT(0) space with a geometric group action $\Gamma\curvearrowright X$.
Suppose that there exists $n\geq 2$ such that every geodesic in $X$ lies in an $n$-flat.
If $X$ contains a periodic $n$-flat which does not bound a flat half-space, then $X$ is a Riemannian symmetric space or a Euclidean building or $X$
non-trivially splits as a metric product.
\end{thm}
\subsection{Organization}
In Section~\ref{sec_pre}, we introduce notation and recall background on the geometry of spaces with upper curvature bounds.
Apart from the standard material, we introduce the notion {\em essential Tits boundary} of a CAT(0) space without 3-flats.
In Section~\ref{sec_cl} we proof the Closing Lemma by showing that if a group $\Gamma$ acts geometrically on a CAT(0) space without 3-flats
and preserves a proper closed subset at infinity, then the space has higher rank (Theorem~\ref{thm_diambound}).
Section~\ref{sec_rank_2} builds up to a proof of Theorem~\ref{thm_mainB} assuming our key technical result, the \hyperref[lem_key_tech]{Half-Plane Lemma}.
We proceed by first showing that the essential Tits boundary is a spherical building and then argue that the essential Tits boundary
already agrees with the ordinary Tits boundary.
Appenix~\ref{sec_app} is devoted to geometric measure theory with the aim to provide a proof of the \hyperref[lem_key_tech]{Half-Plane Lemma}.
We also prove some folklore results, including monotonicity and volume rigidity of minimizing currents in CAT(0) spaces.
However, our approach is to find a streamlined path to the \hyperref[lem_key_tech]{Half-Plane Lemma} and we do not try to prove
required ingredients in their most general form.
\subsection{Acknowledgments}
It's my pleasure to thank several people for their support.
I want to thank Alexander Lytchak for answering many questions, for reading a late version of this article and
for his critical and very helpful feedback.
I want to thank Bruce Kleiner for several inspiring discussions.
This paper draws heavily from his ideas and his work.
I want to thank Bernhard Leeb for valuable comments.
I also want to thank Anton Petrunin for a helpful discussion on
Proposition~\ref{prop_mon}.
I was supported by DFG grant SPP 2026.
\section{Preliminaries}\label{sec_pre}
General references for this section are \cite{AKP, Ballmann, BH, KleinerLeeb}.
\subsection{Metric spaces}
Euclidean $n$-space with its flat metric will be denoted by $\mathbb R^n$. The unit sphere $S^{n-1}\subset\mathbb R^n$ equipped with the induced metric will be referred to as a
{\em round sphere}. Its intersection with a half-space $\mathbb R^{n-1}\times[0,\infty)$ is called a {\em round hemisphere}.
We denote the distance between two points $x$ and $y$ in a metric space $X$ by $|x,y|$.
If $A\subset X$ denotes a subset, then $|x,A|$ refers to the greatest lower bound for distances from points in $A$ to $x$.
For $x\in X$ and $r>0$, we denote by $B_r(x)$ and $\bar B_r(x)$ the open and closed $r$-ball around $x$, respectively.
Similarly, $N_r(A)$ and $\bar N_r(A)$ denote the open and closed $r$-neighborhood of a subset $A\subset X$, respectively.
Moreover, $S_r(x)$ denotes the $r$-sphere around $x$ and by $\dot B_r(x)$ we denote the punctured $r$-ball $B_r(x)\setminus\{x\}$.
A \emph{geodesic}
is an isometric embedding of an interval. It is called a {\em geodesic segment}, if it is compact.
The {\em endpoints} of a geodesic segment $c$ are denoted by $\partial c$.
A geodesic segment $c$ {\em branches} at an endpoint $y\in \partial c=\{x,y\}$, if there are geodesics $c^\pm$ starting in $x$ which strictly contain $c$
and such that $c^-\cap c^+=c$. The point $y$ is then called a {\em branch point}.
A \emph{triangle} is a union of three geodesics connecting three points.
$X$ is \emph{a geodesic metric space} if
any pair of points of $X$
is connected by a geodesic.
It is \emph{geodesically complete} if every geodesic segment is contained in a complete local geodesic.
\subsection{Spaces with an upper curvature bound}
For $\kappa \in \mathbb R $, let $D_{\kappa} \in (0,\infty] $ be the diameter of the complete, simply connected surface $M^2_{\kappa}$
of constant curvature $\kappa$. A complete metric space is called a \emph{CAT($\kappa$) space}
if any pair of its points with distance less than $D_{\kappa}$ is connected by a geodesic and if
all triangles
with perimeter less than $2D_{\kappa}$
are not thicker than
the \emph{comparison triangle} in $M_{\kappa} ^2$. In particular, geodesics between points of distance less than $D_{\kappa}$
are unique. Hence, $X$ is a CAT($\kappa$) space, then we can define for every $p\in X$ and every subset $A\subset B_{D_{\kappa}}(p)$
the {\em geodesic cone $C_p(A)$}.
\subsection{Directions and angles}
For any CAT($\kappa$) space $X$,
the angle between each pair of geodesics starting at the same point
is well defined.
Let $x,y,z$ be three points at pairwise distance less than $D_{\kappa}$ in a CAT($\kappa$) space $X$.
Whenever $x\neq y$, the geodesic between $x$ and $y$ is unique and will be denoted
by $xy$. For $y,z \neq x$, the angle at $x$ between $xy$ and $xz$
will be denoted by $\angle_x(y,z)$. It is defined by
\[\angle_x(y,z)=\lim\limits_{y',z'\to x}\tilde\angle_x(y',z')\]
where $\tilde\angle_x(y',x')$ denotes the angle of the comparison triangle at the vertex corresponding to $x$;
and the points $y',z'$ converge to $x$ along the geodesic $xy$ and $xz$, respectively.
A comparison argument shows that this is well defined as a monotonic limit.
The \emph{space of directions} or {\em link}
at a point $x\in X$ is the completion of the space of equivalence classes of geodesic germs at $x$ with respect to the angle metric.
The resulting space $(\Sigma_x X,\angle)$
is a CAT(1) space. Its elements are called {\em directions (at $x$)}.
\subsection{Dimension}
A natural notion of dimension $\dim (X)$ for a CAT($\kappa$) space $X$ was
introduced by
Kleiner in \cite{Kleiner}, originally referred to as {\em geometric dimension}.
It vanishes precisely when the space is discrete.
In general, it is defined inductively:
\[\dim (X)= \sup _{x\in X} \{ \dim (\Sigma _xX) +1 \}.\]
For instance, in a 1-dimensional CAT($\kappa$) space every link is discrete.
The geometric dimension coincides with the supremum of
topological dimensions\footnote{Here topological dimension corresponds to Lebesgue covering dimension.} of compact subsets in $X$ \cite{Kleiner}.
If the dimension of $X$ is finite, then $\dim(X)$ agrees with the largest number $k$
such that $X$ admits a bilipschitz embedding of an open subset in $\mathbb R^k$ \cite{Kleiner}.
The dimension of a locally compact and geodesically complete space is finite and agrees with the topological dimension
as well as the Hausdorff dimension \cite{OT_cba, LN_gcba}.
\subsection{CAT(1) spaces}
For two CAT(1) spaces $Z_1$ and $Z_2$ we denote by $Z_1\circ Z_2$ their {\em spherical join}.
It is a CAT(1) space of diameter $\pi$.
Note that every round subsphere $\sigma'$ in a round sphere $\sigma$ yields a join decomposition $\sigma=\sigma'\circ\sigma''$
where $\dim(\sigma)=\dim(\sigma')+\dim(\sigma'')+1$. Also, every round hemisphere $\tau$ decomposes as $\tau=\partial\tau\circ\zeta$
where $\zeta$ denotes the center of $\tau$. A CAT(1) space $Z$ is called {\em irreducible}, if it does not admit a non-trivial
spherical join decomposition.
Recall that two points in a CAT(1) space $Z$ are called {\em antipodes}, if their distance is at least $\pi$.
In case $C$ consists of a pair of {\em antipodes} $\xi^\pm$, $|\xi^-,\xi^+|=\pi$, then the convex hull of
$\{\xi^-,\xi^+\}$ and $\operatorname{pol}(\{\xi^-,\xi^+\})$ is isometric to a spherical suspension of $\operatorname{pol}(\{\xi^-,\xi^+\})$.
For a subset $M\subset Z$ we denote by $\operatorname{Ant}(M)\subset Z$ the set of antipodes of $M$.
\begin{definition}\label{def_reg_point}
Let $Z$ be a CAT(1) space of dimension $k$. We call a point $\xi\in Z$ {\em regular}, if
it has a neighborhood homeomorphic to an open set in $\mathbb R^k$.
\end{definition}
\begin{lem}[{\cite[Lemma~2.1]{BL_building}}]\label{lem_antintop}
Let $Z$ be a CAT(1) space of dimension $n$. If $\sigma\subset Z$ is a round $n$-sphere, then every point $\xi\in Z$ has an antipode in $\sigma$.
\end{lem}
This implies the following well-known criterion for geodesic completeness.
\begin{lem}[{\cite[Lemma~2.4]{St_rrI}}]\label{lem_gc}
Let $Z$ be a CAT(1) space of dimension $n$. If every point $\xi\in Z$ is contained in a round $n$-sphere in $Z$,
then $Z$ is geodesically complete.
\end{lem}
\subsection{CAT(0) spaces}\label{sec_catzero}
The {\em ideal boundary} of a CAT(0) space $X$, equipped with the cone topology, is denoted by $\partial_{\infty} X$.
If $X$ is locally compact, then $\partial_{\infty} X$ is compact. If $X$ is a Hadamard $n$-manifold, then $\partial_{\infty} X$ is homeomorphic to an $(n-1)$-sphere.
The {\em Tits boundary} of $X$ is denoted by $\partial_{T} X$, it is the ideal boundary equipped with
the Tits metric $|\cdot,\cdot|_T$. Recall that the Tits metric is the intrinsic metric associated to the {\em Tits angle}. For a point $p$ in $X$
and ideal points $\xi,\eta$ in $\partial_{\infty} X$ the Tits angle is defined by
\[\angle_T(\xi,\eta):=\lim\limits_{x\to\xi, y\to\eta}\tilde\angle_p(x,y).\]
The Tits boundary of a CAT(0) space is a CAT(1) space. It decodes the intersection patterns of asymptotically flat subspaces.
For instance, the Euclidean $n$-space has the round $(n-1)$-sphere as Tits boundary, while any hyperbolic space has a discrete Tits boundary.
A subset in a CAT(0) space is convex, if it contains the geodesic between any pair of its points.
If $C$ is a closed convex subset, then it is CAT(0) with respect to the induced metric. In this case, $C$ admits a 1-Lipschitz retraction $\pi_C:X\to C$.
If $C_1$ and $C_2$ are closed convex subsets, then the distance function $d(\cdot,C_1)|_{C_2}$ is convex; and constant if
and only if $\pi_{C_1}$ restricts to an isometric embedding on $C_2$. We call $C_1$ and $C_2$ {\em parallel}, $C_1\| C_2$, if
and only if $d(\cdot,C_1)|_{C_2}$ and $d(\cdot,C_2)|_{C_1}$ are constant.
Let $Y\subset X$ be a geodesically complete closed convex subset. Then we define the {\em parallel set} $P(Y)$
as the union of all closed convex subsets parallel to $Y$. The parallel set is closed, convex and splits canonically as a metric product
\[P(Y)\cong Y\times CS(Y)\]
where the {\em cross section} $CS(Y)$ is a closed convex subset.
The Tits boundary is given by
\[\partial_{T} P(Y)\cong\partial_{T} Y\circ\operatorname{pol}(\partial_{T} Y).\]
If $X_1$ and $X_2$ are CAT(0) spaces, then their metric product $X_1\times X_2$ is again a CAT(0) space.
We have $\partial_{T} (X_1\times X_2)=\partial_{T} X_1\circ \partial_{T} X_2$ and $\Sigma_{(x_1,x_2)}(X_1\times X_2)=\Sigma_{x_1} X_1\circ \Sigma_{x_2} X_2$.
If $X$ is a geodesically complete CAT(0) space, then any join decomposition of $\partial_{T} X$ is induced by a metric product decomposition of $X$ \cite[Proposition~2.3.7]{KleinerLeeb}.
A CAT(0) space $X$ is called {\em irreducible}, if it does not admit a non-trivial splitting as a metric product.
The Tits boundary of a closed convex subset $Y\subset X$ embeds canonically $\partial_{T} Y\subset\partial_{T} X$.
If two closed convex subsets $Y_1$ and $Y_2$ intersect in $X$, then
\[\partial_{T}(Y_1\cap Y_2)=\partial_{T} Y_1\cap \partial_{T} Y_2.\]
Let $X_1$ and $X_2$ be CAT(0) spaces containing closed convex subsets $C_1$ and $C_2$, respectively.
If there is an isometry $f:C_1\to C_2$, then by \cite[8.9.1 Reshetnyak's gluing theorem]{AKP}, the glued space
\[X_1\cup_f X_2:=(X_1\dot\cup X_2)/x_1\sim f(x_1)\]
is CAT(0) with respect to the induced length metric.
In particular, for every CAT(0) space $X$ which contains a closed convex subset $C$ we can construct a CAT(0) space $\hat X$, the {\em double}
of $X$ along $C$, by
\[\hat X=X^-\cup_C X^+\]
where $X^\pm$ are two isometric copies of $X$. The Tits boundary of the double is given by
\[\partial_{T}\hat X=\partial_{T} X^-\cup_{\partial_{T} C} \partial_{T} X^+.\]
For points $\xi\in\partial_{\infty} X$ and $x\in X$ we denote the {\em Busemann function centered at $\xi$ based at $x$} by $b_{\xi,x}$.
If $\rho:[0,\infty)\to X$ denotes the geodesic ray asymptotic to $\xi$ and with $\rho(0)=x$, then
\[b_{\xi,x}(y)=\lim\limits_{t\to\infty}(|y,\rho(t)|-t).\]
It is a 1-Lipschitz convex function whose negative gradient at a point $y\in X$ is given by $\log_y(\xi)$.
We denote the {\em horoball} centered at a point $\xi\in\partial_{\infty} X$ and based at the point $x\in X$ by
\[HB(\xi,x):=b^{-1}_{\xi,x}((-\infty,0]).\]
It is a closed convex subset with
\[\partial_{T} HB(\xi,x)=\bar B_{\frac{\pi}{2}}(\xi)\subset\partial_{T} X.\]
A {\em $n$-flat} $F$ in a CAT(0) space $X$ is a closed convex subset isometric to $\mathbb R^n$.
In particular, $\partial_{T} F\subset\partial_{T} X$ is a round $(n-1)$-sphere.
On the other hand, if $X$ is locally compact and $\sigma\subset \partial_{T} X$ is a round $(n-1)$-sphere, then either there exists an
$n$-flat $F\subset X$ with $\partial_{T} F=\sigma$, or there exists a round $n$-hemisphere $\tau^+\subset\partial_{T} X$
with $\sigma=\partial\tau^+$ \cite[Proposition~2.1]{Leeb}. Consequently, if $\partial_{T} X$ is $(n-1)$-dimensional, then any round $(n-1)$-sphere in $\partial_{T} X$
is the Tits boundary of some $n$-flat in $X$. Moreover, a round $n$-hemisphere $\tau^+\subset\partial_{T} X$ bounds a flat $(n+1)$-half-space in $X$ if and only if
its boundary $\partial\tau^+$ bounds an $n$-flat in $X$. A {\em flat ($n$-dimensional) half-space} $H\subset X$ is a closed convex subset isometric to a Euclidean half-space $\mathbb R_+^n$.
Its boundary $\partial H\subset X$ is an $(n-1)$-flat and its Tits boundary is a round $(n-1)$-hemisphere $\partial_{T} H\subset\partial_{T} X$. Flat half-spaces will play a certain role in our arguments later, and we agree to denote them by $H$
and their boundaries by $\partial H=h$.
We define the {\em parallel set} of a round sphere $\sigma\subset \partial_{T} X$ as
\[P(\sigma)=P(F)\]
where $F$ is a flat in $X$ with $\partial_{T} F=\sigma$, if such a flat exists.
\subsection{The essential Tits boundary}
\begin{definition}
For a CAT(0) space without 3-flats, we define its {\em essential Tits boundary} $\partial^{ess}_{T} X$ as the subset of $\partial_{T} X$ given by the union of all simple closed local geodesics.
\end{definition}
Note that if $X$ is a CAT(0) space with isolated flats and 1-dimensional Tits boundary which supports a geometric group action, then $\partial^{ess}_{T} X$ is dense in $\partial_{\infty} X$ but not closed.
However, we do have the following.
\begin{lem}\label{lem_diameter_etits}
Let $X$ be a CAT(0) space with $1$-dimensional Tits boundary which contains a 2-flat. If the diameter of $\partial_{T} X$ is equal to $\pi$, then
$\partial^{ess}_{T} X\subset\partial_{T} X$ is a convex subset of diameter $\pi$. In particular, $\partial^{ess}_{T} X$ is a geodesically complete CAT(1) space.
\end{lem}
\proof
Since $X$ contains a 2-flat, the essential Tits boundary is non-empty.
Let $\xi$ and $\hat\xi$ be points in $\partial^{ess}_{T} X$. Then there is a geodesic $\alpha$ of length at most $\pi$ between them in $\partial_{T} X$.
We will show that this geodesic is contained in $\partial^{ess}_{T} X$. Since $\partial^{ess}_{T} X$ is geodesically complete, we may assume that $\alpha$ has length $\pi$
to begin with. Let $\eta$ be a point on $\alpha$ and let $\epsilon>0$ be such that $\eta\notin N_\epsilon(\{\xi,\hat\xi\})$.
Now we extend $\alpha$ beyond $\xi$ to a point $\xi'$ by a geodesic $\alpha'$ of length $\epsilon$ in $\partial^{ess}_{T} X$.
Let $\beta$ be a geodesic in $\partial_{T} X$ from $\xi'$ to $\hat \xi$.
Then $\xi\notin\beta$ and since $\partial_{T} X$ is CAT(1), we must have
$\beta\cap(\alpha\setminus B_\epsilon(\hat\xi))=\emptyset$.
The union $\alpha\cup\alpha'\cup\beta$ contains a simple closed local geodesic $c$.
By definition, $c$ lies in $\partial^{ess}_{T} X$, and by construction, $c$ contains $\alpha\setminus B_\epsilon(\hat\xi)$ and therefore $\eta$.
Thus, all of $\alpha$ is contained in $\partial^{ess}_{T} X$ as required.
Now, the convexity implies that $\partial^{ess}_{T} X$ is geodesic. Since $\partial^{ess}_{T} X$ cannot contain a closed local geodesic of length less then $2\pi$
it is a CAT(1) space of dimension 1. By Lemma~\ref{lem_gc}, it is geodesically complete.
\qed
\section{Closing Lemma}\label{sec_cl}
\subsection{Asymptotically minimal actions}
\begin{definition}
We call an action $\Gamma\curvearrowright X$ on a CAT(0) space {\em asymptotically minimal}, if the induced action $\Gamma\curvearrowright\partial_{\infty} X$ is {\em minimal}, i.e.
$\partial_{\infty} X$ does not contain
closed $\Gamma$-invariant proper subsets.
\end{definition}
The following is our source for periodic rank 1 geodesics.
\begin{prop}[{\cite[Proposition~1.10]{BaBu_periodic}}]\label{prop_BB}
Suppose $\Gamma$ acts geometrically and asymptotically minimal on a CAT(0) space $X$. If the diameter of the Tits boundary $\partial_{T} X$ is strictly larger than
$\pi$, the $X$ contains a $\Gamma$-periodic geodesic of rank one.
\end{prop}
In order to deal with actions which are not asymptotically minimal, we will rely on work of Russell Ricks \cite{Ricks_1D}.
Recall the following construction due to Guralnik-Swenson \cite{GS_trans}.
Let $\Gamma$ be a discrete group acting on
a compact Hausdorff space $Z$. Denote by $\beta\Gamma$ the Stone–Čech compactification of $\Gamma$. By compactness, for each $z\in Z$,
we can extend
the orbit map $\rho_z:z\mapsto\gamma z$ to a map $\beta\rho_z$ and, for $\omega\in\beta\Gamma$ we define
\[T^\omega:Z\to Z,\ z\mapsto \beta\rho_z(\omega).\]
For fixed $z\in Z$, the
map $\beta G\to Z$ which maps $\omega$ to $T^\omega(z)$ is continuous.
The family $\{T^\omega\}_{\omega\in\beta\Gamma}$
of operators is closed under composition.
Now let $\Gamma\curvearrowright X$ be a geometric action on a locally compact CAT(0) space.
Since $\bar X= X\cup\partial_{\infty} X$ is a compact Hausdorff space, the above construction applies.
Guralnik and Swenson observed that every operator $T^\omega$ is 1-Lipschitz on $\partial_{T} X$ by the semi-continuity of the Tits metric.
Following \cite{Ricks_1D}, if $\sigma\subset\partial_{T} X$ is a round sphere and $\omega\in\beta \Gamma$ is such that $T^\omega:\partial_{T} X\to T^\omega\sigma$
is a retraction which restricts to an isometry $\sigma\to T^\omega \sigma$, we say $\omega$ folds $\partial_{T} X$ (and $\sigma$) onto $T^\omega\sigma$.
We call $T^\omega\sigma$ a {\em folded round sphere}.
For every top-dimensional round sphere $\sigma\subset\partial_{T} X$ there exists $\omega\in\beta\Gamma$ and a folded sphere $\sigma'$ such that $T^\omega$
restricts to an isometry $\sigma\to\sigma'$ \cite[Corollary~3.2]{Ricks_1D}.
Let $X$ be a locally compact CAT(0) space with a geometric group action $\Gamma\curvearrowright X$ and without $3$-flats.
Further, let $M\subset X$ be a minimal closed $\Gamma$-invariant set. If $M$ intersects a round circle $\sigma\subset\partial_{T} X$ in an infinite set,
then the action $\Gamma\curvearrowright\partial_{\infty} X$ is minimal, $M=\partial_{\infty} X$ \cite[Lemma~4.1]{Ricks_1D}.
\begin{lem}[{\cite[Lemma~4.8, Theorem~5.2]{Ricks_1D}}]\label{lem_ricks}
Let $\Gamma\curvearrowright X$ be a geometric action on a locally compact CAT(0) space without $3$-flats.
Suppose that $\sigma\subset\partial_{T} X$ is a folded round circle.
Let $M\subset\partial_{\infty} X$ be a minimal closed $\Gamma$-invariant proper subset which minimizes $l=\# (M\cap\sigma)$
among all such sets. If $l\geq 3$ is an odd number, then the following holds.
\begin{enumerate}
\item $\partial^{ess}_{T} X\subset\partial_{T} X$ is a closed convex subset.
\item $\partial^{ess}_{T} X$ is a union of round circles.
\item Every round circle in $\partial^{ess}_{T} X$ contains precisely $2l$ branch points, evenly spaced around the circle;
these branch points coincide with the points of $M\cup\operatorname{Ant}(M)$ on the circle.
\item $\partial_{T} X$ is the union of $\partial^{ess}_{T} X$ with a (possibly empty) collection of trees.
Each such tree $T$ intersects $\partial^{ess}_{T} X$ at a single point $m=m(T)\in M$ and $T\subset B_{\frac{\pi}{l}}(m)$.
\end{enumerate}
If $l<3$ or $l$ even, then $\partial_{T} X$ is a spherical building.
Furthermore, if $\operatorname{diam}(\partial_{T} X)=\pi$, then $\partial_{T} X$ is also a spherical building.
\end{lem}
\subsection{Buildings at infinity}
It is well-known that the Tits boundary of a higher rank symmetric space or higher dimensional Euclidean building is a spherical building
\cite[Proposition~4.2.1]{KleinerLeeb}.
Vice versa, Bernhard Leeb proved the following striking result.
\begin{thm}[{\cite[Main Theorem]{Leeb}}]\label{thm_leeb}
Let $X$ be a locally compact, geodesically complete CAT(0) space. If $\partial_{T} X$ is a connected thick irreducible spherical building,
then $X$ is either a symmetric space or a Euclidean building.
\end{thm}
In the presence of a geometric group action one can relax the assumptions slightly, as we are now going to show.
\begin{prop}\label{prop_buil}
Let $X$ be a locally compact CAT(0) space with a geometric group action $\Gamma\curvearrowright X$. Further, let
$B\subset \partial_{T} X$ be a spherical building with $\dim(B)=\dim(\partial_{T} X)\geq 1$.
If $\partial_{T} X\subset N_{\frac{\pi}{2}}(B)$, then $\partial_{T} X=B$.
\end{prop}
\medskip
The proof of Proposition~\ref{prop_buil} relies on the following observation.
\begin{lem}\label{lem_closed}
Let $X$ be a locally compact CAT(0) space and let
$B\subset \partial_{T} X$ be a spherical building with $\dim(B)=\dim(\partial_{T} X)\geq 1$ and $\partial_{T} X\subset N_{\frac{\pi}{2}}(B)$.
Then $B$ is closed in $\partial_{\infty} X$.
\end{lem}
\proof
Let $(x_k)$ be a sequence in $B$ which converges to a point $\xi$ in $\partial_{\infty} X$.
Since $B$ is top-dimensional, $\xi$ has an antipode $\hat\xi$ in $B$.
We extend geodesics $\hat\xi x_k$ inside $B$ to antipodes $\xi_k$ of $\hat\xi$.
Choose geodesics $c_k\subset B$ from $\hat\xi$ to $\xi_k$ which all have the same starting direction and therefore agree up to a certain point $\eta\in B$.
By compactness of $\partial_{\infty} X$ and semi-continuity of the Tits metric, we can pass to a subsequence such that $c_k$ converges to a geodesic $c$ from $\hat\xi$
to $\xi$. Note that $c$ contains the segment $\hat\xi\eta$.
Now extend the segment $\eta\hat\xi$ up to a point $\hat\zeta\in B$ such that
$\hat\xi$ and $\hat\eta$ lie in the same chamber $\hat\Delta\subset B$.
Denote by $\zeta_k$ the antipode of $\hat\zeta$ on $c_k$. Then $\xi_k$ and $\zeta_k$ lie in a chamber $\Delta_k\subset B$.
Let $\alpha_k\subset B$ be an apartment which contains the chambers $\hat\Delta$ and $\Delta_k$.
Then there is a round circle $\sigma_k\subset \alpha_k$
which contains the four points $\hat\zeta,\hat\xi,\zeta_k,\xi_k$.
Moreover, since $|\hat\xi,\zeta_k|<\pi$, we also have $\eta\in\sigma_k$.
Let $f_k:S^1\to\partial_{T} X$ be an isometric embedding with image $\sigma_k$. Using compactness of $\partial_{\infty} X$ and semi-continuity of the Tits metric again,
we obtain a 1-Lipschitz limit map $f_\infty:S^1\to\partial_{T} X$. Since $\hat\xi\hat\zeta\subset\sigma_k$, $\hat\xi \eta\subset\sigma_k$ and $\xi_k\to\xi$, we see that the image $\sigma_\infty$ of $f_\infty$ contains the geodesic $c$ and the segment $\hat\xi\hat\zeta$. It follows that $\sigma_\infty$
is a round circle. Since $\partial_{T} X\subset N_{\frac{\pi}{2}}(B)$, $\sigma_\infty$ has to lie in $B$. Thus $\xi\in B$ as required.
\qed
\begin{rem}
Note that the strict $\frac{\pi}{2}$-density above is necessary.
\end{rem}
\proof[Proof of Proposition~\ref{prop_buil}]
Every round sphere $\sigma\subset\partial_{T} X$ intersects $B$ in a convex set $C$.
By assumption, the set $C$ cannot be contained in a hemisphere of $\sigma$.
Thus $\sigma$ lies entirely in $B$. If $\sigma$ is top-dimensional, then, by \cite[Corollary 2.5]{BMS_affine}, the orbit $\Gamma\sigma$
is dense in $\partial_{\infty} X$ \footnote{\cite[Corollary 2.5]{BMS_affine} only states that the family of top-dimensional round spheres
is dense in $\partial_{\infty} X$ but their argument proves the stronger form.}. Since $B$ is closed by Lemma~\ref{lem_closed}, the conclusion follows.
\qed
\subsection{Diameter bound}
By Proposition~\ref{prop_BB}, in order to prove the Closing Lemma it is enough to verify the following.
\begin{thm}\label{thm_diambound}
Let $X$ be a locally compact CAT(0) space without 3-flats. Suppose that
$\Gamma\curvearrowright X$ is a geometric action which is not asymptotically minimal.
Then the Tits boundary $\partial_{T} X$ is a spherical join or a spherical building.
In particular, the diameter of $\partial_{T} X$ is equal to $\pi$.
\end{thm}
For the rest of this section, the assumptions of Theorem~\ref{thm_diambound}
are in place. Let $\sigma\subset \partial_{T} X$ be a folded round circle.
We choose a minimal closed $\Gamma$-invariant proper subset $M\subset\partial_{\infty} X$ which minimizes $l=\# (M\cap\sigma)$
among all such sets. Recall that the number $l$ has to be finite, otherwise $\Gamma$ acts minimally \cite[Lemma~4.1]{Ricks_1D}.
Moreover, we may assume $l$ is odd and $\geq 3$ \cite[Theorem~D]{Ricks_1D}.
Recall that the {\em essential Tits boundary} $\partial^{ess}_{T} X$ of $X$ is the subset of $\partial_{T} X$
given by the union of all simple closed local geodesics.
\begin{lem}\label{lem_esstits}
The essential Tits boundary $\partial^{ess}_{T} X$ is an irreducible spherical building.
\end{lem}
\proof
We show that the diameter $D$ of $\partial^{ess}_{T} X$ is equal to $\pi$.
By Lemma~\ref{lem_ricks}, $\partial^{ess}_{T} X$ has the structure of a simplicial complex with edge length $\frac{\pi}{l}$ and diameter at most $\frac{l+1}{l}\pi$.
Moreover, every edge has precisely one boundary vertex in $M$
and one boundary vertex in $\operatorname{Ant}(M)$. Thus, if $D$ is strictly larger than $\pi$,
then $M\cap\operatorname{Ant}(M)\neq\emptyset$. But then $M\subset\partial_{T} X$ is a proper closed subset with $\operatorname{Ant}(M)\subset M$ \cite[Lemma~3.17]{Ricks_1D}.
In turn, $\partial^{ess}_{T} X$ would have to be a spherical join or spherical building \cite{Ly_rigidity}. Contradiction.
Thus $\partial^{ess}_{T} X$ has diameter $\pi$. The claim follows from \cite[Theorem~6.1]{CL_metric}.
\qed
\proof[Proof of Theorem~\ref{thm_diambound}]
By Lemma~\ref{lem_esstits}, $B:=\partial^{ess}_{T} X$ is an irreducible spherical building.
By Lemma~\ref{lem_ricks}, $B$ is $\frac{\pi}{3}$-dense in $\partial_{T} X$.
Thus, $B\subset\partial_{\infty} X$ is closed by Lemma~\ref{lem_closed}. Therefore
we conclude $\partial_{T} X=B$ from Proposition~\ref{prop_buil}.
\qed
\section{Diameter Rigidity}\label{sec_rank_2}
\subsection{Projecting ideal points onto parallel sets}
\begin{lem}\label{lem_minimum}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary.
Let $\xi^-$ and $\xi^+$
be a pair of antipodes in $\partial_{T} X$ and let $\eta$ be a pole of $\{\xi^-,\xi^+\}$.
Further, let $P\subset X$ be a closed convex subset with $\eta\notin\partial_{T} P$ and $\{\xi^-,\xi^+\}\subset\partial_{T} P$.
If $b_\eta$ is a horofunction associated to $\eta$, then $b_\eta$ attains a minimum on $P$. Moreover, the minimum set
contains a complete geodesic $c$ with $\partial_{\infty} c=\{\xi^+,\xi^-\}$.
\end{lem}
\proof
Let us first show that the parallel set $P(\xi^+,\xi^-)$ is non-empty. Denote by $\tau^+\subset \partial_{T} X$
the round hemisphere with $\partial\tau^+=\{\xi^+,\xi^-\}$ and center $\eta$.
If there is no round hemisphere $\tilde\tau^+$ in $\partial_{T} P$ with $\partial\tilde\tau^+=\{\xi^+,\xi^-\}$, then by \cite[Proposition~2.1]{Leeb},
there exists a complete geodesic $c$ in $P$ with $\partial_{\infty} c=\{\xi^+,\xi^-\}$. On the other hand, if there does exist such a
$\tilde\tau^+$, then, since $\eta\notin\partial_{T} P$, the union $\sigma:=\tilde\tau^+\cup\tau^+$ forms a round 1-sphere in $\partial_{T} X$.
Since $\dim(\partial_{T} X)=1$, \cite[Proposition~2.1]{Leeb} ensures the existence of a 2-flat $F$ with $\partial_{T} F=\sigma$ and therefore $F\subset P(\xi^+,\xi^-)$.
Now it follows from \cite[Sublemma~2.3]{Leeb}, if $HB_\eta$ is a horoball based at $\eta$ such that $HB_\eta\cap P$ is non-empty, then
$HB_\eta\cap P$ contains a complete geodesic $c$ with $\partial_{\infty} c=\{\xi^+,\xi^-\}$.
Since $b_\eta$ is constant on such geodesics, in order to find the required minimum, it is enough to restrict $b_\eta$
to $P\cap CS(\xi^+,\xi^-)$. Note that $|\eta',\eta|=\pi$ for every point $\eta'\in\partial_{T} CS(\xi^+,\xi^-)\setminus\{\eta\}$.
Hence $\lim\limits_{x\to\eta'}b_\eta(x)=+\infty$ and
$b_\eta$ does attain a minimum at a point $p$ in $P$.
By assumption, we have
\[\{\xi^-,\xi^+\}\subset\partial_{T}(HB(\eta,p)\cap P).\]
Thus, by \cite[Sublemma~2.3]{Leeb}, there is a complete geodesic $c$ in $HB(\eta,p)\cap P$ with $\partial_{\infty} c=\{\xi^+,\xi^-\}$.
Since $b_\eta$ is bounded above on $HB(\eta,p)$ by its value at $p$ the claim follows.
\qed
\begin{lem}\label{lem_proj}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary. Let $\sigma\subset\partial_{T} X$ be a round 1-sphere.
Let $\tau^+\subset\partial_{T} X$ be a round hemisphere with $\tau^+\cap\sigma=\partial\tau^+$.
Then there exists a 2-flat $F\subset X$ with $\partial_{T} F=\sigma$ and a flat half-plane $H$, orthogonal to $F$ and with
$\partial_{T} H=\tau^+$. Moreover, for any geodesic $c\subset F$ which is not parallel to $\partial H$, holds $\angle(H,P(c))\geq\frac{\pi}{2}$.
\end{lem}
\proof
Since $\dim(\partial_{T} X)=1$, the parallel set $P(\sigma)$ is a non-empty closed convex subset of $X$
which splits isometrically as $P(\sigma)\cong\mathbb R^2\times C$ where $C$ is a compact CAT(0) space.
Let $\eta$ be the center of $\tau^+$ and let $b_\eta$ be an associated horofunction.
By Lemma~\ref{lem_minimum},
$b_\eta$ attains a minimum on $P(\sigma)$ and the minimum set $Z\subset P(\sigma)$ contains a complete geodesic $l$ with $\partial_{\infty} l=\partial\tau^+$.
Let $H$ be the flat half-plane with $\partial_{T} H=\tau^+$ and $\partial H=l$.
Because $l\subset P(\sigma)$, there exists a 2-flat $F\subset X$ with $\partial_{T} F=\sigma$ which contains $l$.
Since $l$ lies in the minimum set $Z$, the half-plane $H$ has to be orthogonal to $F$.
Let $c\subset F$ be a geodesic not parallel to $l$. Choose a point $x\in l$ and a geodesic ray $\rho$ starting in $x$ and asymptotic to $\eta$.
In particular, $\rho\subset H$.
We first claim that $\rho(t)\notin P(c)$ for $t>0$. Indeed, if $\rho(t)\in P(c)$, then, since $\rho(t)\in P(l)$, there exists lines $c_t\| c$ and $l_t\| l$ through
$\rho(t)$. These span a flat $F_t\subset P(\sigma)$. Hence $t=0$ because $\rho(0)$ minimizes $b_\eta$.
Now suppose $\angle(\rho,P(c))<\frac{\pi}{2}$. Then there exists a geodesic $c_1$ in $P(c)$ starting in $x$ and realizing the angle,
$\angle(\rho,c_1)=\angle(\rho,P(c))$. In particular, $b_\eta(c_1(t))<b_\eta(x)$.
Note that since $c$ is not parallel to $l$, the point $\eta$ is not in $\partial_{T} P(c)$. Moreover, we have $\partial\tau^+\subset\partial_{T} P(c)\cap\bar B_{\frac{\pi}{2}}(\eta)$.
Hence, by Lemma~\ref{lem_minimum}, $b_\eta$ attains a minimum on $P(c)$, and the minimum set $Z_c$ contains a complete geodesic
$\tilde l$ with $\partial \tilde l=\{\xi, \hat\xi\}$.
But then $\tilde l$ lies in $P(c)$ and therefore, arguing as above, there is a 2-flat $\tilde F\subset P(\sigma)$ with $\tilde l\subset \tilde F$.
This contradicts the fact that $Z$ minimizes $b_\eta$ on $P(\sigma)$. Therefore $\angle(H,P(c))=\angle(\rho,P(c))\geq\frac{\pi}{2}$.
\qed
\subsection{Parallel sets do not accumulate in the Tits metric}\label{subsec_no_acc}
\begin{lem}\label{lem_no_acc}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary.
Let $g$ be an axis of an axial isometry $\gamma$.
Suppose that there exists $a>0$ and a sequence of 2-flats $(F_k)$ in $P(g)$
with $g\subset N_a(F_k)$ for every $k\in\mathbb N$. Further, suppose that there is a sequence $(H_k)$
of flat half-planes with boundaries $h_k:=\partial H_k\subset F_k$ and $\partial_{\infty} H_k\cap\partial_{\infty} F_k=\partial_{\infty} h_k$.
If $\partial_{\infty} h_k\to\partial_{\infty} g$ with respect to the Tits metric, then $\partial_{\infty} h_k=\partial_{\infty} g$
for almost all $k$.
\end{lem}
\proof
Suppose for contradiction that the convergence $\partial_{\infty} h_k\to\partial_{\infty} g$ is non-trivial.
By Lemma~\ref{lem_proj}, there exist 2-flats $\tilde F_k\subset P(g)$ parallel to $F_k$;
and flat half-planes $\tilde H_k$ orthogonal to $\tilde F_k$ and with $\partial_{\infty} \tilde H_k=\partial_{\infty} H_k$; and such that
$\tilde H_k$ is orthogonal to $P(g)$. Set $\tilde h_k:=\partial \tilde H_k$. Since $X$ has rank 2 and a cocompact isometry group, there exists $D>0$
such that $|F_k,\tilde F_k|_H\leq D$ for all $k\in\mathbb N$. In particular,
there exist complete geodesics $\tilde g_k\subset \tilde F_k$ which are parallel to $g$ and
satisfy $|g, \tilde g_k|\leq a+D$ for all $k\in\mathbb N$. Denote by $\tilde x_k\in \tilde F_k$ the intersection point of $\tilde g_k$ and $ \tilde h_k$.
Then there are powers of $\gamma$, denoted by $\gamma_k$, such that the points $\gamma_k(\tilde x_k)$ lie in a fixed compact set.
Since $\gamma$ preserves $P(g)$, $\gamma_k \tilde H_k$ is still orthogonal to $P(g)$. Since $X$ is locally compact, after choosing a subsequence,
the flat half-planes $\gamma_k \tilde H_k$ converge to a flat half-plane $\tilde H_\infty$.
Note that by assumption $\gamma_k (\partial_{\infty} \tilde h_k)\to\partial_{\infty} g$ with respect to the
Tits metric.
Hence $\tilde H_\infty\subset P(g)$. On the other hand, by upper semi-continuity of angles, $\tilde H_\infty$ has to be orthogonal to $P(g)$. Contradiction.
\qed
\subsection{Regular points at infinity}
In this section we will prove that if the Tits boundary of a cocompact CAT(0) space has dimension $1$ and diameter $\pi$, then it
contains regular points.
To achieve this, we will use the following technical result in an essential way.
\begin{namedlemma}[Half-Plane]\label{lem_key_tech}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary. Let $F\subset X$ be a 2-flat with $\partial_{\infty} F=\sigma$.
Further, let $g$ be an axis of an axial isometry $\gamma$ and assume $\partial_{\infty} g\subset\sigma$.
Let $\xi^\pm\in\sigma$ be antipodes disjoint from $\partial_{\infty} g$.
Suppose that there is a sequence of local geodesics $\alpha^+_k\subset \partial_{T} X$ such that
\begin{itemize}
\item $\partial\alpha^+_k=\{\xi^-_k,\xi^+_k\}$ and $\xi^\pm_k\to\xi^\pm$ with respect to the Tits metric;
\item the length of $\alpha_k^+$ converges to $\pi$ as $k\to\infty$.
\end{itemize}
Then there exists a 2-flat $\tilde F\subset P(g)$, and a flat half-plane $\tilde H$ with boundary $\tilde h:=\partial \tilde H\subset\tilde F$ such that the following properties hold.
\begin{enumerate}
\item $g\subset N_a(\tilde F)$ if $g\subset N_a(F)$ for $a>0$;
\item $|\partial_{\infty} \tilde h,\partial_{\infty} g|=|\{\xi^-,\xi^+\},\partial_{\infty} g|$;
\item $\partial_{\infty}\tilde H\cap\partial_{\infty}\tilde F=\partial_{\infty}\tilde h$.
\end{enumerate}
\end{namedlemma}
The proof of the \hyperref[lem_key_tech]{Half-Plane Lemma} requires methods from geometric measure theory.
Since these techniques do not play a role in the rest of the paper,
we defer their discussion, as well as the proof of the \hyperref[lem_key_tech]{Half-Plane Lemma}, to Appendix~\ref{sec_app}.
\medskip
The following is certainly well-known, we include it for completeness.
\begin{lem}\label{lem_per2fl}
Let $X$ be a locally compact CAT(0) space with a geometric group action $\Gamma\curvearrowright X$.
Let $F\subset X$ be a $\Gamma$-periodic $k$-flat which bounds a flat $(k+1)$-half-space $H$.
Suppose that $G\cong\mathbb Z^k$ is a subgroup in $\Gamma$ which acts geometrically on $F$.
If $G$ preserves $H$, then there exists a $\Gamma$-periodic $(k+1)$-flat $\hat F\subset P(F)$.
\end{lem}
\proof
Let $\rho\subset H$ be a geodesic ray orthogonal to $F$. Choose points $x_k\in\rho$ with $|x_k,F|\to\infty$.
We find a sequence $(\gamma_k)\subset\Gamma$ such that $\gamma_k(x_k)$ stays in a fixed compact set $K$.
We can pass to a subsequence such that for all $k<l$ the element $\gamma_l^{-1}\gamma_k$
is axial (cf. proof of \cite[Theorem~11]{Sw_cut}). Now note that since $G$ preserves $H$, each of its elements $g$ has constant
displacement on $H$, $|x,g x|=a_g>0$ for every $x\in H$.
Thus $|\gamma_k x_k,(\gamma_k g\gamma_k^{-1})\gamma_k x_k|=|x_k, g x_k|=a_g$. Since $\Gamma$ acts properly discontinuously and $\gamma_k x_k\in K$, we may pass to a further subsequence
such that $\gamma_1 g\gamma_1^{-1}=\gamma_k g\gamma_k^{-1}$ holds for all $k\in \mathbb N$.
Hence $\beta_k=\gamma_k^{-1}\gamma_1$ is an axial isometry which commutes with $g$. Letting $g$ vary through a basis of $G$ implies the claim.
\qed
\begin{prop}\label{prop_regular_points}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary and a geometric group action $\Gamma\curvearrowright X$.
Suppose that the diameter of $\partial_{T} X$ is equal to $\pi$.
Let $g$ be an axis of an axial isometry $\gamma\in\Gamma$. Then there exists a positive $\epsilon$ and a round 1-sphere $\sigma\subset\partial_{T} P(g)$
such that $\dot B_\epsilon(g(+\infty))\cap\sigma$
does not contain a branch point. In particular, $\sigma\subset\partial_{T} X$ contains a non-empty open relatively compact subset.
\end{prop}
\proof
Since the diameter of $\partial_{T} X$ is $\pi$, the axis $g$ bounds a flat half-plane $H\subset X$.
If $\gamma$ preserves $H$, then by Lemma~\ref{lem_per2fl}, the parallel set $P(g)$ contains a periodic 2-flat.
In this case, the claim follows from \cite[Corollary~5.7]{St_rrI}.
If $H$ is not preserved, then $\sigma=\tau^+\cup\gamma\tau^+$ is a round sphere in $\partial_{T} X$ where $\tau^+:=\partial_{\infty} H$.
Let $F$ be a 2-flat with $\partial_{\infty} F=\sigma$. Let $a>0$ be such that $g\subset N_a(F)$.
Now suppose that there is a sequence $(\xi_k^+)$ of pairwise distinct branch points in $\sigma$ with $\xi_k^+\to g(+\infty)$.
Denote by $\xi_k^-$ the antipode of $\xi_k^+$ in $\sigma$.
Since $\partial_{T} X$ has diameter $\pi$, we find for each $k\in\mathbb N$ a sequence of local geodesics $\alpha_{kl}\subset \partial_{T} X$ such that
\begin{itemize}
\item $\partial\alpha_{kl}=\{\xi^-_{kl},\xi^+_{kl}\}$ and $\xi^\pm_{kl}\to\xi_k^\pm$ with respect to the Tits metric;
\item the lengths of $\alpha_{kl}$ converges to $\pi$ as $l\to\infty$.
\end{itemize}
Hence by the \hyperref[lem_key_tech]{Half-Plane Lemma}, there exists a sequence of 2-flats $(\tilde F_{k})$ in $P(g)$,
and a sequence of flat half-planes $(\tilde H_{k})$ with boundaries $\tilde h_k:=\partial\tilde H_k\subset\tilde F_k$ and such that
the following properties hold.
\begin{enumerate}
\item $g\subset N_a(\tilde F_k)$ for every $k\in\mathbb N$;
\item $|\partial_{\infty} \tilde h_k,\partial_{\infty} g|=|\{\xi_k^-,\xi_k^+\},\partial_{\infty} g|$;
\item $\partial_{\infty}\tilde H_k\cap\partial_{\infty}\tilde F_k=\partial_{\infty}\tilde h_k$.
\end{enumerate}
In particular, $\partial_{\infty} \tilde h_k\to \partial_{\infty} g$ with respect to the Tits metric.
Hence Lemma~\ref{lem_no_acc} implies that $\tilde h_k$ is parallel to $g$ for almost all $k$.
This is a contradiction since the points $\xi_k^+$ are pairwise distinct.
\qed
\subsection{The Tits boundary is a building}
\begin{lem}\label{lem_reg_1D}
Let $Z$ be a 1-dimensional CAT(1) space of diameter $\pi$.
Then the subset $O\subset Z$ of regular points is closed under taking antipodes.
\end{lem}
\proof
Let $\xi\in O$ be a regular point and let $\hat\xi\in Z$ be an antipode.
Then there exists $s>0$ such that $\bar B_s(\xi)$ is isometric to a closed interval.
Denote by $\xi^\pm$ the two endpoints of $\bar B_s(\xi)$.
Since the diameter of $Z$ is equal to $\pi$, we see that
$\xi$ is the unique antipode of $\hat\xi$ in $B_s(\xi)$.
Thus, $|\xi,\xi^\pm|+|\xi^\pm,\hat\xi|=\pi$ and $\xi,\hat\xi$ lie in a round 1-sphere $\sigma$.
We claim that $B_s(\hat\xi)\subset\sigma$. Let $\hat\eta\in Z$ be a point in $B_s(\hat\xi)$.
Then $\hat\eta$ has an antipode $\eta\in B_s(\xi)$ and as before, it has to be unique and we find a round 1-sphere $\sigma'$
which contains the points $\eta$ and $\hat\eta$. By regularity, $\bar B_s(\xi)\subset\sigma\cap\sigma'$.
Since $|\hat\xi,\eta|<s$, we conclude $\sigma=\sigma'$ and therefore $\eta\in\sigma$ as required.
\qed
\begin{lem}\label{lem_etits_build}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary and a geometric group action.
Suppose that the diameter of $\partial_{T} X$ is equal to $\pi$. Then $\partial^{ess}_{T} X$ is a spherical join or a spherical building.
\end{lem}
\proof
We may assume that $\partial^{ess}_{T} X$ is not a round sphere. By \cite[Theorem~C]{Kleiner}, $X$ contains a 2-flat.
Thus, by Lemma~\ref{lem_diameter_etits}, the diameter of $\partial^{ess}_{T} X$ is equal to $\pi$. By \cite[Theorem~11]{Sw_cut}, the group $\Gamma$
contains an axial element.
Hence, the open subset $O\subset\partial^{ess}_{T} X$ of regular points is non-empty by Proposition~\ref{prop_regular_points}.
We infer from Lemma~\ref{lem_reg_1D} that $\partial^{ess}_{T} X\setminus O$ is a proper closed subset of $\partial^{ess}_{T} X$ which contains with every point all of its antipodes.
The claim follows from \cite[Main Theorem]{Ly_rigidity}.
\qed
\begin{cor}\label{cor_rank2_build}
Let $X$ be a locally compact CAT(0) space with $1$-dimensional Tits boundary and a geometric group action.
Suppose that the diameter of $\partial_{T} X$ is equal to $\pi$. Then $\partial_{T} X$ is a spherical join or a spherical building.
\end{cor}
\proof
By Proposition~\ref{prop_buil} and Lemma~\ref{lem_etits_build}, it is enough to show that every point in $\partial_{T} X$ has distance less than $\frac{\pi}{2}$
from $\partial^{ess}_{T} X$. Suppose for contradiction that there exists a point $\xi\in\partial_{T} X\setminus\partial^{ess}_{T} X$ at
distance $\frac{\pi}{2}$ from $\partial^{ess}_{T} X$. Choose $\eta\in\partial^{ess}_{T} X$ such that
$|\xi,\eta|<\frac{\pi}{2}+\delta$ for a small positive $\delta$.
Extend the geodesic $\xi\eta$ inside $\partial^{ess}_{T} X$ up to a local geodesic $\alpha$ of length $\pi+\delta$ and let $\eta'$ denote its endpoint.
Now choose a geodesic $\beta$ from $\eta'$ to $\xi$. Since $\partial_{T} X$ is CAT(1), $\beta$ avoids the interior of a subsegment $\alpha_0\subset\alpha$
of length $\pi$. In particular, $\alpha\cup\beta$ contains a simple closed local geodesic $c$ of length at least $2\pi$.
Thus, the geodesic $\xi\eta$ intersects $\partial^{ess}_{T} X$ in a segment of length at least $\frac{\pi}{2}-\delta$ and distance between $\xi$
and $\partial^{ess}_{T} X$ is less then $\frac{\pi}{2}$. Contradiction.
\qed
\proof[Proof of Theorem~\ref{thm_mainB}]
By Corollary~\ref{cor_rank2_build}, the Tits boundary $\partial_{T} X$ is a spherical join or a building.
Since $X$ is geodesically complete and locally compact, the claim follows from Theorem~\ref{thm_leeb} and \cite[Proposition~2.3.7]{KleinerLeeb}.
\qed
\section{Applications}
In this section we provide the proofs of Corollaries~\ref{cor_tits} and~\ref{cor_main}.
\proof[Proof of Corollary~\ref{cor_tits}]
If the Tits diameter is strictly larger than $\pi$, then by Theorem~\ref{thm_mainA}, $X$ contains a $\Gamma$-periodic geodesic of rank one.
Thus $\Gamma$ contains a non-abelian free subgroup \cite[Theorem~3.5]{ballmannbook}. On the other hand, if the Tits diameter is equal to $\pi$,
then $X$ is either an irreducible symmetric space, an irreducible Euclidean building or splits metrically as a product.
The first case follows from Tits original theorem \cite{Ti_free} and the second case is covered by \cite[Theorem~F]{BaBr_orbi}.
We are left with the product case, $X\cong X_1\times X_2$. By assumption, the factors $X_i$ do not contain $2$-flats.
After passing to an index two subgroup, we may assume that $\Gamma$ preserves the factors.
We will argue that there exist two $\Gamma$-axes such that their projections to one of the factors
are not asymptotic to one another and therefore lead to a non-abelian free subgroup \cite[Theorem~3.5]{ballmannbook}.
Suppose for contradiction that any pair of $\Gamma$-axes project to (one-sided) asymptotic geodesics in both factors.
Then every axis has to be regular, i.e. not parallel to a factor.
Since $\Gamma$ acts geometrically, this means that every axial element lies in a subgroup $\mathbb Z^2<\Gamma$.
Every such subgroup preserves a $2$-flat. Two different $\Gamma$-periodic $2$-flats cannot be asymptotic to the same Weyl chamber because
$\Gamma$ acts properly discontinuous. Thus, we either find the required $\Gamma$-axes and hence the non-abelian subgroup, or $\Gamma$ is
a Bieberbach group.
\qed
\proof[Proof of Corollary~\ref{cor_main}]
Under the assumptions, we find a $\Gamma$-equivariant quasi-isometry $\Phi:X\to X'$.
Hence for every axial element $\gamma\in\Gamma$ a pair of $\gamma$-axes $c\subset X$ and $c'\subset X'$
has to have the same rank. Indeed, $\Phi(c)$ lies at finite Hausdorff distance from $c'$
and therefore $c'$ has quadratic growth if $c$ does. Thus the claim follows from Theorems~\ref{thm_mainA} and~\ref{thm_mainB}
together with \cite[Theorem~1.1.3]{KleinerLeeb}.
\qed
|
1,314,259,995,839 | arxiv | \section{Introduction}
The approximation properties of the multivariate Bernstein-Durrmeyer linear operator defined with respect to a nonnegative, bounded Borel measure $\mu:{\cal{B}}_{S^{d}}\to \mathbb{R}_{+}$, by
$$M_{n, \mu}(f)(x)$$
\begin{equation}\label{form1}
=\sum_{|\alpha|=n}\frac{\int_{S^{d}}f(t)B_{\alpha}(t)d\mu(t)}{\int_{S^{d}}B_{\alpha}(t)d\mu(t)}\cdot B_{\alpha}(x):=\sum_{|\alpha|=n}c(\alpha, \mu)\cdot B_{\alpha}(x), \, x\in S^{d},\, n\in \mathbb{N},
\end{equation}
where ${\cal{B}}_{S^{d}}$ denotes the sigma algebra of all Borel measurable subsets in the power set ${\cal{P}}(S^{d})$ and $f$ is supposed to be $\mu$-integrable on the standard simplex
$$S^{d}=\{(x_{1}, ..., x_{d}) ; 0\le x_{1}, ..., x_{d}\le 1, \, 0\le x_{1}+ ... +x_{d}\le 1\},$$
were studied in, e.g., the recent papers \cite{BJ}, \cite{Berd1}, \cite{Berd2}, \cite{Berd3} and \cite{Li}.
Note that in (\ref{form1}), it is used the notation
$$B_{\alpha}(x)=\frac{n!}{\alpha_{0}! \cdot \alpha_{1}! \cdot ... \cdot \alpha_{d}!}(1-x_{1}-x_{2}- ... - x_{d})^{\alpha_{0}}\cdot x_{1}^{\alpha_{1}} \cdot ... \cdot x_{d}^{\alpha_{d}}$$
$$=:\frac{n!}{\alpha_{0}! \cdot \alpha_{1}! \cdot ... \cdot \alpha_{d}!}\cdot P_{\alpha}(x),$$
where $\alpha=(\alpha_{0}, \alpha_{1}, ..., \alpha_{d})$, $\alpha_{j}\in \mathbb{N}\bigcup\{0\}$, $j=0, ..., d$, $|\alpha|=\alpha_{0}+\alpha_{1}+ ... +\alpha_{d}=n$.
In the very recent paper \cite{Gal-Opris}, we have proved that the approximation results in the above mentioned papers remain valid for the more general case when
$\mu$ is a monotone, normalized and submodular set function on $S^{d}$ and the integrals used in (\ref{form1}) are the nonlinear Choquet integrals with respect to $\mu$.
The main goal of this paper is to obtain quantitative estimates in terms of the modulus of continuity and in terms of some $K$-functionals, for
the pointwise and uniform approximation obtained in \cite{Gal-Opris} and for the univariate $L^{p}$-approximation, $p\ge 1$, in the case of more general multivariate Bernstein-Durrmeyer polynomial operators defined by
\begin{equation}\label{form1-bis}
M_{n, \Gamma_{n, x}}(f)(x)=\sum_{|\alpha|=n}c(\alpha, \mu_{n, \alpha, x})\cdot B_{\alpha}(x), \, x\in S^{d},\, n\in \mathbb{N},
\end{equation}
where
$$c(\alpha, \mu_{n, \alpha, x})=\frac{(C)\int_{S^{d}}f(t)B_{\alpha}(t)d\mu_{n, \alpha, x}(t)}{(C)\int_{S^{d}}B_{\alpha}(t)d\mu_{n, \alpha, x}(t)}
=\frac{(C)\int_{S^{d}}f(t)P_{\alpha}(t)d\mu_{n, \alpha, x}(t)}{(C)\int_{S^{d}}P_{\alpha}(t)d\mu_{n, \alpha, x}(t)}$$
and for every $n\in \mathbb{N}$ and $x\in S^{d}$, $\Gamma_{n, x}=(\mu_{n, \alpha, x})_{|\alpha|=n}$ is a family of bounded, monotone, submodular and strictly positive set functions on ${\cal{B}}_{S^{d}}$.
Note that if $\Gamma_{n, x}$ reduces to one element (i.e. $\mu_{n, \alpha, x}=\mu$ for all $n$, $x$ and $\alpha$), then the operator given by (\ref{form1-bis}) reduces to the operator considered in \cite{Gal-Opris}.
The plan of the paper is as follows. Section 2 contains some preliminaries on possibility theory and on Choquet integral. In Section 3, general quantitative estimates in terms of the modulus of continuity and in terms of a $K$-functional for the pointwise and uniform approximation by the operators $M_{n, \Gamma_{n, x}}(f)(x)$ defined by (\ref{form1-bis}) are obtained. Also, when $\Gamma_{n, x}$ reduces to two elements, one a Choquet submodular set function and the other one a Borel measure, for suitable modified Bernstein-Durrmeyer-Choquet operators, univariate $L^{p}$-approximations, $p\ge 1$, with quantitative estimates in terms of a $K$-functional are presented. Finally, in Section 4, in the particular case when $d=1$ and the Choquet integrals are taken with respect to some concrete possibility measures, the pointwise estimate in terms of the modulus of continuity is detailed. Also, some concrete example of operators improving the classical error estimates are presented and potential applications to practical methods dealing with data are mentioned.
\section{Preliminaries}
Firstly, we present a few known concepts in possibility theory useful for the next considerations.
For details, see, e.g., \cite{DubPrad}.
{\bf Definition 2.1.} For the non-empty set $\Omega$, denote by ${\cal{P}}(\Omega)$ the family of all subsets of $\Omega$.
(i) A function $\lambda : \Omega \to [0, 1]$ with the property $\sup\{\lambda(s) ; s\in \Omega\}=1$, is called possibility distribution on $\Omega$.
(ii) A possibility measure is a set function $P:{\cal{P}}(\Omega)\to [0, 1]$, satisfying the axioms $P(\emptyset)=0$, $P(\Omega)=1$ and $P(\bigcup_{i\in I}A_{i})=\sup\{P(A_{i}) ; i\in I\}$ for all $A_{i}\subset \Omega$, and any $I$, an at most countable family of indices. Note that if $A, B\subset \Omega$, $A\subset B$, then the last property easily implies that $P(A)\le P(B)$ and that $P(A\bigcup B)\le P(A)+P(B)$.
Any possibility distribution $\lambda$ on $\Omega$, induces the possibility measure $P_{\lambda}:{\cal{P}}(\Omega)\to [0, 1]$, given by the formula $P_{\lambda}(A)=\sup \{\lambda(s) ; s\in A\}$, for all $A\subset \Omega$ (see, e.g., \cite{DubPrad}, Chapter 1).
Some known concepts and results concerning the Choquet integral can be summarized by the following.
{\bf Definition 2.2.} Suppose $\Omega\not=\emptyset$ and let ${\cal{C}}$ be a $\sigma$-algebra of subsets in $\Omega$.
(i) (see, e.g., \cite{WK1}, p. 63) The set function $\mu:{\cal{C}}\to [0, +\infty]$ is called a monotone set function (or capacity) if $\mu(\emptyset)=0$ and $\mu(A)\le \mu(B)$ for all $A, B\in {\cal{C}}$, with $A\subset B$. Also,
$\mu$ is called submodular if
$$\mu(A\bigcup B)+\mu(A\bigcap B)\le \mu(A)+\mu(B), \mbox{ for all } A, B\in {\cal{C}}.$$
$\mu$ is called bounded if $\mu(\Omega)<+\infty$ and normalized if $\mu(\Omega)=1$.
(ii) (see, e.g., \cite{WK1}, p. 233, or \cite{Choquet}) If $\mu$ is a monotone set function on ${\cal{C}}$
and if $f:\Omega \to \mathbb{R}$ is ${\cal{C}}$-measurable (that is, for any Borel subset $B\subset \mathbb{R}$ it follows $f^{-1}(B)\in {\cal{C}}$), then for any $A\in {\cal{C}}$, the concept of Choquet integral is defined by
$$(C)\int_{A} f d\mu=\int_{0}^{+\infty}\mu\left (F_{\beta}(f)\bigcap A\right )d\beta+\int_{-\infty}^{0}\left [\mu\left (F_{\beta}(f)\bigcap A\right )-\mu(A)\right ]d \beta,$$
where we used the notation $F_{\beta}(f)=\{\omega\in \Omega; f(\omega)\ge \beta\}$.
Notice that if $f\ge 0$ on $A$, then in the above formula we get $\int_{-\infty}^{0}=0$.
The function $f$ will be called Choquet integrable on $A$ if $(C)\int_{A}f d\mu\in \mathbb{R}$.
In what follows, we list some known properties of the Choquet integral.
{\bf Remark 2.3.} If $\mu:{\cal{C}}\to [0, +\infty]$ is a monotone set function, then the following properties hold :
(i) For all $a\ge 0$ we have $(C)\int_{A}af d\mu = a\cdot (C)\int_{A}f d\mu$ (if $f\ge 0$ then see, e.g., \cite{WK1}, Theorem 11.2, (5), p. 228 and if $f$ is of arbitrary sign, then see, e.g., \cite{Denn}, p. 64, Proposition 5.1, (ii)).
(ii) For all $c\in \mathbb{R}$ and $f$ of arbitrary sign, we have (see, e.g., \cite{WK1}, pp. 232-233, or \cite{Denn}, p. 65) $(C)\int_{A}(f+c)d \mu = (C)\int_{A}f d\mu + c\cdot \mu(A)$.
If $\mu$ is submodular too, then for all $f, g$ of arbitrary sign and lower bounded, we have
(see, e.g., \cite{Denn}, p. 75, Theorem 6.3)
$$(C)\int_{A}(f + g) d\mu \le (C)\int_{A}f d\mu + (C)\int_{A}g d\mu.$$
(iii) If $f\le g$ on $A$ then
$(C)\int_{A}f d\mu \le (C)\int_{A}g d\mu$ (see, e.g., \cite{WK1}, p. 228, Theorem 11.2, (3) if $f, g\ge 0$ and p. 232 if $f, g$ are of arbitrary sign).
(iv) Let $f\ge 0$. If $A\subset B$ then $(C)\int_{A}f d \mu \le (C)\int_{B}f d\mu.$
In addition, if $\mu$ is finitely subadditive, then
$$(C)\int_{A\bigcup B}f d\mu \le (C)\int_{A}f d\mu + (C)\int_{B}f d\mu.$$
(v) It is immediate that $(C)\int_{A}1\cdot d\mu(t)=\mu(A)$.
(vi) The formula $\mu(A)=\gamma(M(A))$, where
$\gamma :[0, 1]\to [0, 1]$ is an increasing and concave function, with $\gamma(0)=0$, $\gamma(1)=1$ and
$M$ is a probability measure (or only finitely additive) on a $\sigma$-algebra on $\Omega$ (that is, $M(\emptyset)=0$, $M(\Omega)=1$ and $M$ is countably additive), gives simple examples of normalized, monotone and submodular set functions (see, e.g., \cite{Denn}, pp. 16-17, Example 2.1). For example, we can take $\gamma(t)=\sqrt{t}$.
If the above $\gamma$ function is increasing, concave and satisfies only $\gamma(0)=0$, then for any bounded Borel measure $m$, $\mu(A)=\gamma(m(A))$ gives a simple example of bounded, monotone and submodular set function.
Note that any possibility measure $\mu$ is normalized, monotone and submodular. Indeed, the axiom $\mu(A\bigcup B)=\max\{\mu(A), \mu(B)\}$ implies the monotonicity, while the property $\mu(A\bigcap B)\le \min\{\mu(A), \mu(B)\}$ implies the submodularity.
(vii) If $\mu$ is a countably additive bounded measure, then the Choquet integral $(C)\int_{A}f d\mu$ reduces to the usual Lebesgue type integral (see, e.g., \cite{Denn}, p. 62, or \cite{WK1}, p. 226).
\section{Quantitative estimates for general Bernstein-Durrmeyer-Choquet operators}
Recall that $\mu:{\cal{B}}_{S^{d}}\to [0, +\infty)$ is said strictly positive if for every open set $A\subset \mathbb{R}^{n}$ with $A\cap S^{d}\not=\emptyset$, we have $\mu(A\cap S^{d})>0$.
The support of $\mu$ is defined by
$$supp(\mu)=\{x\in S^{d} ; \mu(N_{x})>0 \mbox{ for every open neighborhood } N_{x}\in {\cal{B}}_{S^{d}} \mbox{ of } x\}.$$
Note that the strict positivity of $\mu$, evidently implies the condition $supp(\mu)\setminus \partial S^{d}\not=\emptyset$, which guarantees that $(C)\int_{S^{d}}B_{\alpha}(t)d\mu(t)>0$, for all $B_{\alpha}$.
Let us consider $C_{+}(S^{d})=\{f : S^{d}\to \mathbb{R}_{+} ; f \mbox{ is continuous on } S^{d}\}$, endowed with the norm $\|F\|_{C(S^{d})}=\sup\{|F(x)| ; x\in S^{d}\}$.
The first main result of this section consists in the following general quantitative estimates in pointwise and uniform approximation.
{\bf Theorem 3.1.} {\it For each fixed $n\in \mathbb{N}$ and $x\in S^{d}$, let $\Gamma_{n, x}=\{\mu_{n, \alpha, x}\}_{|\alpha|=n}$ be a family of bounded, monotone, submodular and strictly positive set functions on ${\cal{B}}_{S^{d}}$.
(i) For every $f\in C_{+}(S^{d})$, $x=(x_{1}, ..., x_{d})\in S^{d}$, $n\in \mathbb{N}$, we have
$$|M_{n, \Gamma_{n, x}}(f)(x)-f(x)|\le 2\omega_{1}(f ; M_{n, \Gamma_{n, x}}(\varphi_{x})(x))_{S^{d}},$$
where $M_{n, \Gamma_{n, x}}(f)(x)$ is given by (\ref{form1-bis}), $\|x\|=\sqrt{\sum_{i=1}^{d}x_{i}^{2}}$, $\varphi_{x}(t)=\|t-x\|$ and $\omega_{1}(f ; \delta)_{S^{d}}=\sup\{|f(t)-f(x)| ; t, x\in S^{d}, \|t-x\|\le \delta\}$.
(ii) Suppose that the family $\Gamma_{n, x}$ does not depend on $x$. Then, for any $f\in C_{+}(S^{d})$ and $n\in \mathbb{N}$, we get
$$\|M_{n, \Gamma_{n}}(f)-f\|_{C(S^{d})}\le 2 K\left (f ; \frac{\Delta_{n}}{2}\right ),$$
where $\Delta_{n}=\sum_{i=1}^{d}\|M_{n, \Gamma_{n}}(|\varphi_{e_{i}}-x_{i}{\bf 1}|)\|_{C(S^{d})}$,
$$K(f; t)=\inf_{g\in C^{1}_{+}(S^{d})}\{\|f-g\|_{C(S^{d})} + t \|\nabla g\|_{C(S^{d})}\},$$
$C^{1}_{+}(S^{d})$ is the subspace of all functions $g\in C_{+}(S^{d})$ with continuous partial derivatives $\partial_{i} g$, $i \in \{1, ..., d\}$ and
$$\|\nabla g\|_{C(S^{d})}=\max_{i=\{1, ..., d\}}\{\|\partial_{i} g\|_{C(S^{d})}\},$$
$\varphi_{e_{i}}(x)=x_{i}$, $i\in \{1, ..., d\}$, $x=(x_{1}, ..., x_{d})$, ${\bf 1}(x)=1$, for all $x\in S^{d}$.}
{\bf Proof.} (i) For $x\in S^{d}$, $n\in \mathbb{N}$ and $|\alpha|=n$ arbitrary fixed, let us consider $T_{n, \alpha, x}:C_{+}(S^{d})\to \mathbb{R}_{+}$ defined by
$$T_{n, \alpha, x}(f)=(C)\int_{S^{d}}f(t) P_{\alpha}(t)d \mu_{n, \alpha, x}(t), f\in C_{+}(S^{d}).$$
Based on the above Remark 2.3, (i), (ii), (iii) and reasoning exactly as in the proof of Lemma 3.1 in \cite{Gal-Opris}, we get
$|T_{n, \alpha, x}(f)-T_{n, \alpha, x}(g)|\le T_{n, \alpha, x}(|f-g|)$. Then, since $T_{n, \alpha, x}$ is positively homogeneous, sublinear and monotonically increasing, we immediately get that $M_{n, \Gamma_{n, x}}$ keeps the same properties and as a consequence it follows
\begin{eqnarray}\label{eq1}
|M_{n, \Gamma_{n, x}}(f)(x)-M_{n, \Gamma_{n, x}}(g)(x)|\le M_{n, \Gamma_{n, x}}(|f-g|)(x),
\end{eqnarray}
$M_{n, \Gamma_{n, x}}(\lambda f)=\lambda M_{n, \Gamma_{n, x}}(f)$, $M_{n, \Gamma_{n, x}}(f+g)\le M_{n, \Gamma_{n, x}}(f)+M_{n, \Gamma_{n, x}}(g)$ and that $f\le g$ on $S^{d}$ implies $M_{n, \Gamma_{n, x}}(f)\le M_{n, \Gamma_{n, x}}(g)$ on $S^{d}$,
for all $\lambda\ge 0$, $f, g\in C_{+}(S^{d})$, $n\in \mathbb{N}$, $|\alpha|=n$, $x\in S^{d}$.
Denoting $e_{0}(t)=1$ for all $t\in S^{d}$, since obviously $M_{n, \Gamma_{n, x}}(e_{0})(x)=1$ for all $x\in S^{d}$ and taking into account the properties in Remark 2.3, (i) and (\ref{eq1}), for any fixed $x$ we obtain
\begin{eqnarray}\label{eq2}
|M_{n, \Gamma_{n, x}}(f)(x)-f(x)|&=&|M_{n, \Gamma_{n, x}}(f(t))(x)-M_{n, \Gamma_{n, x}}(f(x))(x)|\nonumber \\
&\le& M_{n, \Gamma_{n, x}}(|f(t)-f(x)|)(x).
\end{eqnarray}
But taking into account the properties of the modulus of continuity, for all $t,x\in S^{d}$ and $\delta>0$, we get
\begin{eqnarray}\label{eq3}
|f(t)-f(x)|\leq\omega_{1}(f;\|t-x\|)_{S^{d}}\leq\left[ \frac{1}{\delta }\|t-x\|+1
\right] \omega_{1}(f;\delta)_{S^{d}}.
\end{eqnarray}
Now, from (\ref{eq2}) and applying $M_{n, \Gamma_{n, x}}$ to (\ref{eq3}), by the properties of $M_{n, \Gamma_{n, x}}$ mentioned after the inequality (\ref{eq1}), we immediately get
$$|M_{n, \Gamma_{n, x}}(f)(x)-f(x)|\leq \left[ \frac{1}{\delta}M_{n, \Gamma_{n, x}}(\varphi_{x})(x)+1\right] \omega_{1}(f;\delta)_{S^{d}}.$$
Choosing here $\delta=M_{n, \Gamma_{n, x}}(\varphi_{x})(x)$, we obtain the desired estimate.
(ii) Let $f, g\in C_{+}(S^{d})$. We have
$$f(x)-M_{n, \Gamma_{n}}(f)(x)$$
$$=f(x)-g(x)+M_{n, \Gamma_{n}}(g)(x)-M_{n, \Gamma_{n}}(f)(x)+g(x)-M_{n, \Gamma_{n}}(g)(x),$$
which, by using (\ref{eq1}) too, implies
$$|f(x)-M_{n, \Gamma_{n}}(f)(x)|$$
$$\le |f(x)-g(x)|+|M_{n, \Gamma_{n}}(g)(x)-M_{n, \Gamma_{n}}(f)(x)|+|g(x)-M_{n, \Gamma_{n}}(g)(x)|$$
$$\le |f(x)-g(x)|+M_{n, \Gamma_{n}}(|g - f|)(x)+|g(x)-M_{n, \Gamma_{n}}(g)(x)|$$
$$\le 2 \|f-g\|_{C(S^{d})}+|g(x)-M_{n, \Gamma_{n}}(g)(x)|.$$
By following the lines in the proof of Theorem 4.5 in \cite{BJ}, since from the lines after relation (\ref{eq1}) in the above point (i), the operator $M_{n, \Gamma_{n}}$ is monotone and subadditive, for all $g\in C^{1}_{+}(S^{d})$, $x\in S^{d}$,
we immediately get
$$|g(x)-M_{n, \Gamma_{n}}(g)(x)|$$
$$\le M_{n, \Gamma_{n}}(|g- g(x){\bf 1}|)(x)\le \|\nabla g\|_{C(S^{d})}\cdot M_{n, \Gamma_{n}}\left (\sum_{i=1}^{d}|\varphi_{e_{i}}-x_{i}{\bf{1}}|\right )(x)$$
$$\le \|\nabla g\|_{C(S^{d})}\cdot \sum_{i=1}^{d} M_{n, \Gamma_{n}}\left (|\varphi_{e_{i}}-x_{i}{\bf{1}}|\right )(x)
\le \|\nabla g\|_{C(S^{d})}\cdot \Delta_{n}.$$
Concluding, it follows
$$\|f-M_{n, \Gamma_{n}}(f)\|_{C(S^{d})}\le 2 \left [\|f-g\|_{C(S^{d})} + \frac{\Delta_{n}}{2} \|\nabla g\|_{C(S^{d})}\right ],$$
which immediately implies the required estimate in (ii). $\hfill \square$
{\bf Remark 3.2.} The positivity of function $f$ in Theorem 3.1, (i) and (ii), is necessary because of the positive homogeneity of the Choquet integral used in their proofs. However, if $f$ is of arbitrary sign and lower bounded on $S^{d}$ with $f(x)-m\ge 0$, for all $x\in S^{d}$, then the statement of Theorem 3.1, (i), (ii), can be restated for the slightly modified Bernstein-Durrmeyer operator defined by
$$M_{n, \Gamma_{n, x}}^{*}(f)(x)=M_{n, \Gamma_{n, x}}(f-m)(x)+m.$$
Indeed, in the case of Theorem 3.1, (i), this is immediate from $\omega_{1}(f-m;\delta)_{S^{d}}=\omega_{1}(f;\delta)_{S^{d}}$ and from
$M_{n,\Gamma_{n, x}}^{*}(f)(x)-f(x)=M_{n,\Gamma_{n, x}}(f-m)(x)-(f(x)-m)$.
Note that in the case of Theorem 3.1, (ii), since we may consider here that $m<0$, we immediately get the relations
$$K(f - m ; t)=\inf_{g\in C^{1}_{+}(S^{d})}\{\|f-(g+m)\|_{C(S^{d})}+t \|\nabla g\|_{C(S^{d})}\}$$
$$=\inf_{g\in C^{1}_{+}(S^{d})}\{\|f-(g+m)\|_{C(S^{d})}+t \|\nabla (g+m)\|_{C(S^{d})}\}$$
$$=\inf_{h\in C^{1}(S^{d}), \, h\ge m}\{\|f-h\|_{C(S^{d})}+t \|\nabla h\|_{C(S^{d})}\}.$$
In the particular case when the family $\Gamma_{n, x}$ does not depend on $x$ and $n$, it is natural to ask for quantitative estimates in the $L^{1}$-approximation of the Choquet integrable functions (not necessarily continuous).
If, for example, $\Gamma_{n, x}=\{\mu\}$, $d=1$ and $\mu$ is a bounded, monotone and submodular set function, then for the Bernstein-Durrmeyer-Choquet operators
$$D_{n, \mu}(f)(x)=\sum_{k=0}^{n}p_{n, k}(x)\cdot \frac{(C)\int_{0}^{1}f(t)p_{n, k}(t)d \mu(t)}{(C)\int_{0}^{1}p_{n, k}(t)d \mu(t)}, \, p_{n, k}(x)={n\choose k} x^{k}(1-x)^{n-k},$$
with $f\in L^{1}_{\mu}$ meaning $f$ is ${\cal{B}}_{[0, 1]}$ measurable and $\|f\|_{L^{1}_{\mu}}=(C)\int_{0}^{1}|f(t)|d\mu(t)<\infty$, we get
$$\|D_{n, \mu}(f)\|_{L^{1}_{\mu}} \le \sum_{k=0}^{n}(C)\int_{0}^{1}p_{n, k}(t)|f(t)|d\mu(t)\le n \cdot \|f\|_{L^{1}_{\mu}}, \, n\in \mathbb{N}.$$
This is due to the fact that $(C)\int f d\mu$ is is not, in general, additive as function of $f$ (it is only subadditive).
Therefore, quantitative estimates in $L^{p}_{\mu}$-approximation by Bernstein-Durrmeyer-Choquet operators, remains, in the general case, an open question.
However, in the particular case when the family of set functions $\Gamma_{n, x}$ reduces, for example, to two elements (one being a Choquet submodular set function $\mu$ and the other one a Borel measure $\delta$), for suitable defined Bernstein-Durrmeyer-Choquet operators, quantitative $L^{p}_{\mu}$-approximation results, $p\ge 1$ hold. For this purpose, let us make the following notations :
$$L^{p}_{\mu}=\{f:[0, 1]\to \mathbb{R} ; f \mbox{ is } {\cal{B}}_{[0, 1]} \mbox{measurable and} (C)\int_{0}^{1}|f(t)|^{p}d\mu(t)<+\infty\},$$
$$L^{p}_{\mu, +}=L^{p}_{\mu}\bigcap \{f:[0, 1]\to \mathbb{R}_{+}\},$$
$$\overline{K}\left (f ; t\right )_{L^{p}_{\mu, \delta}}=\inf_{g\in C^{1}_{+}([0, 1])}\{\|f-g\|_{L^{p}_{\mu}}+\|f-g\|_{L^{p}_{\delta}} + t \|g^{\prime}\|_{C([0, 1])}\}.$$
It is easy to see that if $\mu\le \delta$ then for all $f\in L^{1}_{\delta}$ and $t\ge 0$, we have $2K(f; t/2)_{L^{p}_{\mu}}\le \overline{K}\left (f ; t\right )_{L^{p}_{\mu, \delta}}\le 2 K(f; t)_{L^{p}_{\delta}}$, where $K$ is of usual form and with the infimum taken for $g\in C^{1}_{+}([0, 1])$.
For $p=1$, we have :
{\bf Theorem 3.3.} {\it Let $\mu$ be a bounded, monotone, submodular and strictly positive set function on ${\cal{B}}_{[0, 1]}$ and $\delta$ a bounded, strictly positive Borel measure on ${\cal{B}}_{[0, 1]}$, such that $\mu(A)\le \delta(A)$ for all $A\in {\cal{B}}_{[0, 1]}$. Then, denoting $L^{1}_{\delta, +}\subset L^{1}_{\mu, +}$ and defining
the Bernstein-Durrmeyer-Choquet operators
$$\overline{D}_{n, \delta, \mu}(f)(x)=\sum_{k=0}^{n-1}p_{n, k}(x)\cdot \frac{\int_{0}^{1}f(t)p_{n, k}(t)d \delta(t)}{\int_{0}^{1}p_{n, k}(t)d \delta(t)}+x^{n}\cdot \frac{(C)\int_{0}^{1}f(t)t^{n}d \mu(t)}{(C)\int_{0}^{1}t^{n}d \mu(t)},$$
for all $f\in L^{1}_{\delta, +}$, $n\in \mathbb{N}$ and denoting $\varphi_{x}(t)=|t-x|$, we have
$$\|f-\overline{D}_{n, \delta, \mu}(f)\|_{L^{1}_{\mu}}\le 2 \overline{K}\left (f ; \frac{\|\overline{D}_{n, \delta, \mu}(\varphi_{x})\|_{L^{1}_{\mu}}}{2}\right )_{L^{1}_{\mu, \delta}}.$$}
{\bf Proof.} Firstly, note that $\delta$ is monotone, submodular (in fact modular, i.e. submodular with equality)
strictly positive and that for all $f\in L^{1}_{\delta, +}$ we have $\int_{0}^{1}f(t)d \delta (t)=(C)\int_{0}^{1}f(t)d \delta(t)$ (see Remark 2.3, (vii)). From here, from $\mu\le \delta$ and from Definition 2.2, (ii), we immediately get $(C)\int_{0}^{1}f(t)d \mu(t) \le \int_{0}^{1}f(t)d \delta(t)$, which means $L^{1}_{\delta, +}\subset L^{1}_{\mu, +}$ and for all $f\in L^{1}_{\delta, +}$ implies
$$\|\overline{D}_{n, \delta, \mu}(f)\|_{L^{1}_{\mu}}\le \sum_{k=0}^{n-1}\frac{(C)\int_{0}^{1}p_{n, k}(x)d\mu(x)}{\int_{0}^{1}p_{n, k}(t)d\delta(t)}\cdot \int_{0}^{1}f(t)p_{n, k}(t)d \delta+(C)\int_{0}^{1}f(t)t^{n}d \mu(t)$$
\begin{equation}\label{eq111}
\le \int_{0}^{1}f(t)\left [\sum_{k=0}^{n-1}p_{n, k}(t)\right ]d \delta +(C)\int_{0}^{1}f(t)t^{n}d \mu(t)\le \|f\|_{L^{1}_{\delta}}+\|f\|_{L^{1}_{\mu}}.
\end{equation}
Let $f, g\in L^{1}_{\delta, +}$. From
$$|f(t)-\overline{D}_{n, \delta, \mu}(f)(t)|$$
$$\le |f(t)-g(t)|+|\overline{D}_{n, \delta, \mu}(g)(t)-\overline{D}_{n, \delta, \mu}(f)(t)|+|g(t)-\overline{D}_{n, \delta, \mu}(g)(t)|,$$
integrating with respect to $\mu$, from the properties of the Choquet integral, of the operator $\overline{D}_{n, \delta, \mu}$ (similar with those of $M_{n, \Gamma_{n, x}}$ in the proof of Theorem 3.1, (i)) and from (\ref{eq111}), we obtain
$$\|f-\overline{D}_{n, \delta, \mu}\|_{L^{1}_{\mu}}=(C)\int_{0}^{1}|f(t)-\overline{D}_{n, \delta, \mu}(f)(t)|d \mu(t)$$
$$\le (C)\int_{0}^{1}|f(t)-g(t)|d \mu(t) + (C)\int_{0}^{1}|\overline{D}_{n, \delta, \mu}(g)(t)-\overline{D}_{n, , \delta, \mu}(f)(t)|d \mu(t)$$
$$+(C)\int_{0}^{1}|g(t)-\overline{D}_{n, \delta, \mu}(g)(t)| d \mu(t)$$
$$\le \|f-g\|_{L^{1}_{\mu}} + (C)\int_{0}^{1} \overline{D}_{n, \delta, \mu}(|g-f|)(t)d \mu(t)+\|g - \overline{D}_{n, \delta, \mu}(g)\|_{L^{1}_{\mu}}$$
$$\le \|f-g\|_{L^{1}_{\mu}}+(\|f-g\|_{L^{1}_{\mu}}+\|f-g\|_{L^{1}_{\delta}})+\|g - \overline{D}_{n, \delta, \mu}(g)\|_{L^{1}_{\mu}}.$$
It remains to estimate $\|g-D_{n, \mu}(g)\|_{L^{1}_{\mu}}$.
But from
$$|g(x)-\overline{D}_{n, \mu}(g)(x)|=|\overline{D}_{n, \mu}(g(x))(x)-\overline{D}_{n, \mu}(g(t))(x)|\le \overline{D}_{n, \mu}(|g(x)-g(t)|)(x)$$
and since for $g\in C^{1}_{+}([0, 1])$, we get $|g(x)-g(t)|\le \|g^{\prime}\|_{C([0, 1])}|x-t|$, applying $\overline{D}_{n, \delta, \mu}$, it follows
$\overline{D}_{n, \delta, \mu}(|g(x)-g(t)|)(x)\le \|g^{\prime}\|_{C([0, 1])}\cdot \overline{D}_{n, \mu}(\varphi_{x})(x)$.
Therefore, integrating above with respect to $x$ and $\mu$, we obtain
$$\|g-\overline{D}_{n, \delta, \mu}(g)\|_{L^{1}_{\mu}}\le \|g^{\prime}\|_{C([0, 1])}\cdot \|\overline{D}_{n, \delta, \mu}(\varphi_{x})\|_{L^{1}_{\mu}},$$
which immediately leads to
$$\|f-\overline{D}_{n, \delta, \mu}(f)\|_{L^{1}_{\mu}}\le 2\|f-g\|_{L^{1}_{\mu}}+\|f-g\|_{L^{\delta}}+\|g^{\prime}\|_{C([0, 1])}\cdot \|\overline{D}_{n, \delta, \mu}(\varphi_{x})\|_{L^{1}_{\mu}}$$
$$\le 2\left (\|f-g\|_{L^{1}_{\mu}}+\|f-g\|_{L^{1}_{\delta}}+\|g^{\prime}\|_{C([0, 1])}\cdot \frac{\|\overline{D}_{n, \delta, \mu}(\varphi_{x})\|_{L^{1}_{\mu}}}{2}\right )$$
and to the conclusion of the theorem. $\hfill \square$
In what follows, because of some difference with respect to the case $p=1$, we extend separately Theorem 3.3 to the $L^{p}_{\mu}$ space with $p>1$.
{\bf Theorem 3.4.} {\it Let $\mu$ be a bounded, monotone, submodular, strictly positive set function on ${\cal{B}}_{[0, 1]}$, which also is continuous by increasing sequences of sets, that is if $A_{n}\in {\cal{B}}_{[0, 1]}$, $n\in \mathbb{N}$, with $A_{n}\subset A_{n+1}$, for all $n$ and $A:=\bigcup _{n=1}^{\infty}A_{n}\in {\cal{B}}_{[0, 1]}$, then $\lim_{n\to \infty}\mu(A_{n})=\mu(A)$.
Also, let $\delta$ be a bounded, strictly positive Borel measure on ${\cal{B}}_{[0, 1]}$, such that $\mu(A)\le \delta(A)$ for all $A\in {\cal{B}}_{[0, 1]}$. Then, for any $p>1$, $L^{p}_{\delta, +}\subset L^{p}_{\mu, +}$ and for
the Bernstein-Durrmeyer-Choquet operators $\overline{D}_{n, \delta, \mu}(f)(x)$ defined by Theorem 3.3,
for all $f\in L^{p}_{\delta, +}=L^{p}_{\delta}\bigcap \{f:[0, 1]\to [0, +\infty)\}$ and $n\in \mathbb{N}$, we have
$$\|f-\overline{D}_{n, \delta, \mu}(f)\|_{L^{p}_{\mu}}\le 2 \overline{K}\left (f ; \frac{\|\overline{D}_{n, \delta, \mu}(\varphi_{x})\|_{L^{p}_{\mu}}}{2}\right )_{L^{p}_{\mu, \delta}}.$$}
{\bf Proof.} The proof for $L^{p}_{\delta, +}\subset L^{p}_{\mu, +}$ follows exactly as in the proof of Theorem 3.3.
By the convexity of $t^{p}$ on $[0, +\infty)$, by $\sum_{k=0}^{n}p_{n, k}(x)=1$, we easily arrive at the inequalities (exactly as, for example, in the proof of Lemma 2.2 in \cite{Li})
$$\|\overline{D}_{n, \delta, \mu}(f)\|^{p}_{L^{p}_{\mu}}\le (C)\int_{0}^{1}\left [\sum_{k=1}^{n-1}p_{n, k}(x)\cdot
\frac{\left (\int_{0}^{1}f(t)p_{n, k}(t)d \delta(t)\right )^{p}}{\left (\int_{0}^{1}p_{n, k}(t)d \delta(t)\right )^{p}}\right .$$
$$\left . + x^{n}\cdot \frac{\left ((C)\int_{0}^{1}f(t)t^{n}d \mu(t)\right )^{p}}{\left ((C)\int_{0}^{1}t^{n}d \mu(t)\right )^{p}}\right ]d \mu(x).$$
Applying the H\"older's inequality for the integrals from the denominators (in the case of Choquet integrals with respect to $\mu$, the inequality is the same with that for the integrals with respect to the Borel measure $\delta$, see. e.g., Theorem 3.5 in \cite{RSWang} or Theorem 2 in \cite{Cerda1})
and reasoning as in the proof of Lemma 2.2 in \cite{Li} and as for formula (\ref{eq111}) in the proof of Theorem 3.3,
we easily arrive at
$$\|\overline{D}_{n, \delta, \mu}(f)\|^{p}_{L^{p}_{\mu}}\le \|f\|_{L^{p}_{\mu}}+\|f\|_{L^{p}_{\delta}}, \mbox{ for all } f\in L^{p}_{\delta, +}.$$
Then, since H\"older's inequality for the Choquet integral with respect to $\mu$ implies the Minkowski inequality
(see, e.g., Theorem 3.7 in \cite{RSWang} or Theorem 2 in \cite{Cerda1}), using the above inequality and exactly the reasonings in the proof of Theorem 2.1 in \cite{Li}, we arrive at the desired inequality in the statement.
It remains to discuss the requirement on $\mu$ to be continuous by increasing sequences of sets. This is due
to the fact that for Choquet integrals, the H\"older's inequality hold only if both integrals from its right-hand side are not equal to zero (see the proofs of Theorem 3.5 in \cite{RSWang} or of Theorem 2 in \cite{Cerda1}).
To have valid the H\"older's inequality in its full generality, we need that for $F\ge 0$, $(C)\int_{0}^{1}F(t) d\mu=0$ if and only if $F(t)=0$, $\mu$ almost everywhere on $[0,1]$. But according to Theorem 11.3, p. 228 in \cite{WK1},
if $\mu$ is continuous by increasing sequences of sets, then the above mentioned property holds. $\hfill \square$
{\bf Remark 3.5.} Concrete choices for $\mu$ and $\delta$ in Theorem 3.3 can be, for example, $\delta(A)=m(A)$ and $\mu(A)=\sin[m(A)]$, where $m$ is the Lebesgue measure on ${\cal{B}}_{[0, 1]}$. Indeed, $\mu(A)\le m(A)$ for all $A\in {\cal{B}}_{[0, 1]}$ and since $\sin$ is concave on $[0,1]$ and $sin(0)=0$, by Remark 2.3, (vi) it follows that $\mu$ is bounded, monotone and submodular.
{\bf Remark 3.6.} It is easy to see that another Bernstein-Durrmeyer-Choquet operator satisfying the estimates in Theorems 3.3 and 3.4, can be defined by
$$\tilde{D}_{n, \delta, \mu}(f)(x)=(1-x)^{n}\frac{(C)\int_{0}^{1}f(t)(1-t)^{n}d \mu(t)}{(C)\int_{0}^{1}(1-t)^{n}d \mu(t)}+\sum_{k=1}^{n}p_{n, k}(x)\frac{\int_{0}^{1}f(t)p_{n, k}(t)d \delta(t)}{\int_{0}^{1}p_{n, k}(t)d \delta(t)}.$$
Also, defining
$$D^{\star}_{n, \delta, \mu}(f)(x)=(1-x)^{n}\frac{(C)\int_{0}^{1}f(t)(1-t)^{n}d \mu(t)}{(C)\int_{0}^{1}(1-t)^{n}d \mu(t)}+\sum_{k=1}^{n-1}p_{n, k}(x)\frac{\int_{0}^{1}f(t)p_{n, k}(t)d \delta(t)}{\int_{0}^{1}p_{n, k}(t)d \delta(t)}$$
$$+x^{n}\cdot \frac{(C)\int_{0}^{1}f(t)t^{n}d \mu(t)}{(C)\int_{0}^{1}t^{n}d \mu(t)},$$
by similar reasonings with those in the proofs of Theorems 3.3 and 3.4, for any $p\ge 1$ we immediately obtain the estimate
$$\|f-D^{\star}_{n, \delta, \mu}(f)\|_{L^{p}_{\mu}}\le 3 \overline{K}\left (f ; \frac{\|D^{\star}_{n, \delta, \mu}(\varphi_{x})\|_{L^{p}_{\mu}}}{3}\right ).$$
{\bf Remark 3.7.} For $\delta$ a bounded Borel measure on ${\cal{B}}_{[0, 1]}$, denote by $D_{n, \delta}$ the classical Bernstein-Durrmeyer operator (i.e. with all the integrals in terms of $\delta$). By Theorem 4.5 in \cite{BJ},
we have the estimate
$$\|D_{n, \delta}(f)-f\|_{L^{1}_{\delta}}\le 2K\left (f ; \frac{\|D_{n, \delta}(\varphi_{x})\|_{L^{1}_{\delta}}}{2}\right )_{L^{1}_{\delta}}, \mbox{ for all } f\in L^{1}_{\delta},$$
where $K(f ; t)_{L^{1}_{\delta}}=\inf_{g\in C^{1}([0, 1])}\{\|f-g\|_{L^{1}_{\delta}} + t \|g^{\prime}\|_{C([0, 1])}\}$.
Comparing with the estimate for $\|\overline{D}_{n, \delta, \mu}(f)-f\|_{L^{1}_{\mu}}$ in Theorem 3.3 and taking into account that $\|f-g\|_{L^{1}_{\mu}}\le \|f-g\|_{L^{1}_{\delta}}$ and $\|D_{n, \delta}(\varphi_{x})\|_{L^{1}_{\mu}}\le
\|D_{n, \delta}(\varphi_{x})\|_{L^{1}_{\delta}}$, it follows that it is possible that in some cases, $\overline{D}_{n, \delta, \mu}(f)$ in Theorem 3.3, approximates better $f\in L^{1}_{\delta, +}$ in the $L^{1}_{\mu}$-"norm" than approximate $D_{n, \delta}(f)$ the same function $f$ but in the $L^{1}_{\delta}$-norm.
\section{Concrete Bernstein-Durrmeyer-Choquet operators}
Since the estimates in Theorem 3.1 are of very general and abstract form, involving the apparently difficult to be calculated Choquet integrals, it is of interest to obtain in some particular cases, concrete expressions for the order of approximation.
In this sense, we will apply Theorem 3.1, (i), for $d=1$ and for some special choices of the submodular set functions.
Thus, we will consider the case of the measures of possibility. Denoting $p_{n, k}(x)={n \choose k}x^{k}(1-x)^{n-k}$, let us define $\lambda_{n, k}(t)=\frac{p_{n, k}(t)}{k^{k}n^{-n}(n-k)^{n-k}{n\choose k}}=\frac{t^{k}(1-t)^{n-k}}{k^{k}n^{-n}(n-k)^{n-k}}$, $k=0, ..., n$. Here, by convention we consider $0^{0}=1$, so that the cases $k=0$ and $k=n$ have sense.
By considering the root $\frac{k}{n}$ of $p^{\prime}_{n, k}(x)$, it is easy to see that $\max\{p_{n, k}(t) ; t\in [0, 1]\}=k^{k}n^{-n}(n-k)^{n-k}{n\choose k}$, which implies that each $\lambda_{n, k}$ is a possibility distribution on $[0, 1]$. Denoting by $P_{\lambda_{n, k}}$ the possibility measure induced by $\lambda_{n, k}$ and $\Gamma_{n, x}:=\Gamma_{n}=\{P_{\lambda_{n, k}}\}_{k=0}^{n}$ (i.e. $\Gamma$ is independent of $x$), the nonlinear Bernstein-Durrmeyer polynomial operators given by (\ref{form1-bis}), in terms of the Choquet integrals with respect to the set functions in $\Gamma_{n}$, will become
\begin{eqnarray}\label{eq8}
D_{n, \Gamma_{n}}(f)(x)=\sum_{k=0}^{n}p_{n, k}(x)\cdot \frac{(C)\int_{0}^{1}f(t)t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)}{(C)\int_{0}^{1}t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)}.
\end{eqnarray}
It is easy to see that any possibility measure $P_{\lambda_{n, k}}$ is bounded, monotone, submodular and strictly positive, $n\in \mathbb{N}$, $k=0, 1, ..., n$, so that we are under the hypothesis of Theorem 3.1, (i).
We can state the following result.
{\bf Theorem 4.1.} {\it If $D_{n, \Gamma_{n}}(f)(x)$ is given by (\ref{eq8}), then for every $f\in C_{+}([0, 1])$, $x\in [0, 1]$ and $n\in \mathbb{N}$, $n\ge 2$, we have
$$|D_{n, \Gamma_{n}}(f)(x)-f(x)|\le 2\omega_{1}\left (f ; \frac{(1+\sqrt{2})\sqrt{x(1-x)}+\sqrt{2}\sqrt{x}}{\sqrt{n}}+\frac{1}{n}\right )_{[0, 1]}.$$}
For its proof, we need the following auxiliary result.
{\bf Lemma 4.2.} {\it Let $n\in \mathbb{N}$, $n\ge 2$ and $x\in [0, 1]$. Denoting
$$A_{n, k}(x):=\sup\{|t-x|t^{k}(1-t)^{n-k};t\in [0, 1]\}=$$
$$\max\{\sup\{(t-x)t^{k}(1-t)^{n-k};t\in [x, 1]\}, \, \sup\{(x-t)t^{k}(1-t)^{n-k};t\in [0, x]\}\},$$
with the convention $0^{0}=1$, for all $k=0, ..., n$ we have
$$A_{n, k}(x)=\max\{(t_{2}-x)t_{2}^{k}(1-t_{2})^{n-k}, \, (x-t_{1})t_{1}^{k}(1-t_{1})^{n-k}\},$$
with $t_{1}, t_{2}$ given by
\begin{eqnarray}\label{eq5}
t_{1}=\frac{nx+k+1-\sqrt{\Delta}}{2(n+1)}, \, \, t_{2}=\frac{nx+k+1+\sqrt{\Delta}}{2(n+1)},
\end{eqnarray}
where
$$\Delta=(nx+k+1)^{2}-4k x(n+1)=n^{2}\left [(x+(k+1)/n)^{2}-4x\frac{k}{n}\cdot \frac{n+1}{n}\right ]$$
$$=(nx-k)^{2}+2x(n-k)+2k(1-x)+1\ge 1.$$}
{\bf Proof.} Let us denote $H_{n, k}(t)=t^{k}(1-t)^{n-k}|t-x|$, with $k\in \{0, ..., n\}$. We have two cases :
(i) $1\le k\le n-1$ and (ii) $k=0$ or $k=n$.
Case (i). For $t\in [x, 1]$ we obtain
$H_{n, k}(t)=(t-x)t^{k}(1-t)^{n-k}$ and from $H^{\prime}_{n, k}(t)=t^{k-1}(1-t)^{n-k-1}[-t^{2}(n+1)+t(nx+k+1)-kx]=0$,
it follows $-t^{2}(n+1)+t(nx+k+1)-kx=0$, which has the discriminant
$$\Delta=(nx+k+1)^{2}-4k x(n+1)=(nx-k)^{2}+2x(n-k)+2k(1-x)+1\ge 1.$$
Therefore, the quadratic equation has two real distinct solutions $t_{1}<t_{2}$
$$t_{1}=\frac{nx+k+1-\sqrt{\Delta}}{2(n+1)}, \, \, t_{2}=\frac{nx+k+1+\sqrt{\Delta}}{2(n+1)},$$
where by simple calculation we derive $0\le t_{1}<t_{2}\le 1$. Also, since $H_{n, k}(0)=H_{n, k}(x)=H_{n, k}(1)=0$ and $H_{n, k}(t)\ge 0$ for $t\in [x, 1]$, simple graphical reasonings show that the only possibility is $0\le t_{1}\le x\le t_{2}\le 1$, with $t=t_{2}$ maximum point on $[x, 1]$ for $H_{n, k}(t)$.
Similarly, for $t\in [0, x]$, since $H_{n, k}(t)=(x-t)t^{k}(1-t)^{n-k}$, using the above reasonings we obtain
$H^{\prime}_{n, k}(t)=t^{k-1}(1-t)^{n-k-1}[t^{2}(n+1)-t(nx+k+1)+kx]$ and that $t_{1}$ is a maximum point of $H_{n, k}(t)$ on $[0, x]$.
In conclusion, with $t_{1}, t_{2}$ given by (\ref{eq5}), we get
$$A_{n, k}(x)=\max\{(t_{2}-x)t_{2}^{k}(1-t_{2})^{n-k}, \, (x-t_{1})t_{1}^{k}(1-t_{1})^{n-k}\}.$$
Case (ii). Suppose first that $k=0$. By the calculation from the case (i), for $t\in [x, 1]$ we get $0=t_{1}\le x\le t_{2}=\frac{nx +1}{n+1}\le 1$, $H_{n, 0}(t)\ge 0$ and $H_{n, 0}(x)=H_{n, 0}(1)=0$, which by similar graphical reasonings leads to the fact that the maximum of $H_{n, 0}(t)$ on $[x, 1]$ is $H_{n, 0}(t_{2})=(t_{2}-x)(1-t_{2})^{n}$. Therefore, we recapture the case (i) with the convention that $0^{0}=1$. Similarly, for $t\in [0, x]$, we get that
the maximum of $H_{n, 0}(t)$ is $H_{n, 0}(t_{1})=(x-t_{1})(1-t_{1})^{n}$
The subcase $k=n$ is similar, which proves the lemma.
$\hfill \square$
{\bf Proof of Theorem 4.1.} According to Theorem 3.1, (i), we have to estimate
$$D_{n, \Gamma_{n}}(\varphi_{x})(x)=\sum_{k=0}^{n}p_{n, k}(x)\cdot \frac{(C)\int_{0}^{1}|t-x|t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)}{(C)\int_{0}^{1}t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)}.$$
First of all, by Definition 2.2, (ii), we get
$$(C)\int_{0}^{1}t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)=\int_{0}^{+\infty}P_{\lambda_{n, k}}(\{t\in [0, 1] ;
t^{k}(1-t)^{n-k}\ge \beta\})d \beta$$
$$=\int_{0}^{1}P_{\lambda_{n, k}}(\{t\in [0, 1] ; t^{k}(1-t)^{n-k}\ge \beta\})d \beta$$
$$=\int_{0}^{1}\sup\{\lambda_{n, k}(s) ; s\in \{t\in [0, 1] ; t^{k}(1-t)^{n-k}\ge \beta\}\}d \beta$$
$$=\frac{1}{k^{k}n^{-n}(n-k)^{n-k}}\cdot
\int_{0}^{1}\sup\{s^{k}(1-s)^{n-k} ; s\in \{t\in [0, 1] ; t^{k}(1-t)^{n-k}\ge \beta\}\}d \beta.$$
For simplicity, denote $E_{n, k}=k^{k}n^{-n}(n-k)^{n-k}$, where again we take $0^{0}=1$. Since for $\beta > E_{n, k}$ we have
$\{t\in [0, 1] ; t^{k}(1-t)^{n-k}\ge \beta\}=\emptyset$ and since we can take $\sup\{s^{k}(1-s)^{n-k} ; s\in \emptyset\}=0$, it follows
$$(C)\int_{0}^{1}t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)$$
$$=\frac{1}{E_{n, k}}\cdot \int_{0}^{E_{n, k}}\sup\{s^{k}(1-s)^{n-k} ; s\in \{t\in [0, 1] ; t^{k}(1-t)^{n-k}\ge \beta\}\}d \beta$$
\begin{eqnarray}\label{eq11}
=\frac{1}{E_{n, k}}\cdot \int_{0}^{E_{n, k}}E_{n, k}d \beta=E_{n, k}.
\end{eqnarray}
On the other hand, denoting $A_{n, k}(x)=\sup\{|t-x|t^{k}(1-t)^{n-k};t\in [0, 1]\}$, by Remark 2.3, (iii), (v) and by Lemma 4.2, for $t_{1}< t_{2}$ in (\ref{eq5}) we obtain
$$(C)\int_{0}^{1}|t-x|t^{k}(1-t)^{n-k}d P_{\lambda_{n, k}}(t)\le (C)\int_{0}^{1}A_{n, k}(x)d P_{\lambda_{n, k}}(t)$$
$$=A_{n, k}(x)(C)\int_{0}^{1}1 d P_{\lambda_{n, k}}(t) = \max\{(t_{2}-x)t_{2}^{k}(1-t_{2})^{n-k}, \, (x-t_{1})t_{1}^{k}(1-t_{1})^{n-k}\}$$
$$\le (t_{2}-x)t_{2}^{k}(1-t_{2})^{n-k} + (x-t_{1})t_{1}^{k}(1-t_{1})^{n-k}.$$
Since $\frac{t_{2}^{k}(1-t_{2})^{n-k}}{k^{k}n^{-n}(n-k)^{n-k}}\le 1$,
$\frac{t_{1}^{k}(1-t_{1})^{n-k}}{k^{k}n^{-n}(n-k)^{n-k}}\le 1$ and
by Lemma 4.2 we get
$$\frac{A_{n, k}(x)}{k^{k}n^{-n}(n-k)^{n-k}}\le (t_{2}-x)\cdot \frac{t_{2}^{k}(1-t_{2})^{n-k}}{k^{k}n^{-n}(n-k)^{n-k}} + (x-t_{1})\cdot \frac{t_{1}^{k}(1-t_{1})^{n-k}}{k^{k}n^{-n}(n-k)^{n-k}}$$
$$\le t_{2}-t_{1}=\frac{\sqrt{\Delta}}{n+1}\le \frac{\sqrt{(nx-k)^{2}+2x(n-k)+2k(1-x)+1}}{n}$$
$$\le \sqrt{(x-k/n)^{2}+2x/n+(2k/n)\cdot (1-x)/n +1/n^{2}}$$
$$\le |x-k/n|+\sqrt{2x}/\sqrt{n}+(\sqrt{2k}/\sqrt{n})\cdot \sqrt{(1-x)/n}+1/n,$$
this immediately implies
$$D_{n, \Gamma_{n}}(\varphi_{x})(x)\le \sum_{k=0}^{n}p_{n, k}(x)(|x-k/n|+\sqrt{2x}/\sqrt{n}+\sqrt{2k/n}\cdot \sqrt{(1-x)/n}+1/n)$$
$$\le \frac{\sqrt{x(1-x)}}{\sqrt{n}}+\frac{\sqrt{2 x}}{\sqrt{n}}+\frac{\sqrt{2}\sqrt{x(1-x)}}{\sqrt{n}}+\frac{1}{n}
=\frac{(1+\sqrt{2})\sqrt{x(1-x)}+\sqrt{2}\sqrt{x}}{\sqrt{n}}+\frac{1}{n}.$$
Above we have used the well-known estimate
$\sum_{k=0}^{n}p_{n, k}(x)|x-k/n|\le \frac{\sqrt{x(1-x)}}{\sqrt{n}}$ and the Cauchy-Schwarz inequality for Bernstein polynomials, $B_{n}(f)(x)\le \sqrt{B_{n}(f^{2})(x)}$, applied for $f(t)=\sqrt{t}$.
Finally, applying Theorem 3.1, (i), the proof of Theorem 4.1 follows. $\hfill\square$
{\bf Remark 4.3.} For $\mu=\sqrt{m}$ with $m$ denoting the Lebesgue measure on $[0, 1]$, another particular case of the Bernstein-Durrmeyer operators for which the quantitative estimates in Theorem 3.1 are applicable would be, for example, $$D_{n, \mu}(f)(x)=\sum_{k=0}^{n}p_{n, k}(x)\cdot \frac{(C)\int_{0}^{1}f(t)t^{k}(1-t)^{n-k}d \mu}{(C)\int_{0}^{1}t^{k}(1-t)^{n-k}d \mu}.$$
It is worth noting that the uniform convergence of this $D_{n, \mu}(f)$ to $f$, follows directly from the general Theorem 3.2 in \cite{Gal-Opris}. Also, it can be obtained by using the nonlinear Feller kind scheme expressed by Theorem 3.1 in \cite{Gal-Ann} (combined with Remark 3.2 there), since by direct calculation we can show that $D_{n, \mu}(e_{1})$ converges uniformly to $e_{1}$ and $D_{n, \mu}((\cdot - x)^{2})$ converges uniformly to $0$, on $[0, 1]$.
{\bf Remark 4.4.} Since the Bernstein-Durrmeyer-Choquet operators in this paper can be defined with respect to a family of Borel or Choquet measures, combined in various ways, this fact offers a very high flexibility and generality, allowing to construct operators having even better approximation properties.
A first example for this flexibility is shown by Theorem 3.4 and Remark 3.6.
For the second example, let us replace in formula (\ref{eq8}) the family $\Gamma_{n}$ of measures of possibilities $P_{\lambda_{n, k}}$, $k=0, ..., n$, by the family consisting in the Dirac measures $\delta_{k/n}$, $k=0, 1, ..., n-1$, (which are Borel measures and therefore with the corresponding Choquet integrals reducing to the classical ones) together with a monotone, submodular, strictly positive set function $\mu$. Then, denoting by $B_{n}(f)(x)$ the classical Bernstein operators, for $D_{n, \Gamma_{n}}$ in (\ref{eq8}) we get
$$D_{n, \Gamma_{n}}(f)(x)-f(x)=\left [\sum_{k=0}^{n-1}p_{n, k}(x)f\left (\frac{k}{n}\right )+x^{n}\cdot \frac{(C)\int_{0}^{1}f(t)t^{n}d\mu(t)}{(C)\int_{0}^{1}t^{n}d \mu(t)}\right ] -f(x)$$
$$=B_{n}(f)(x)-f(x) +x^{n}\left [\frac{(C)\int_{0}^{1}f(t)t^{n} d \mu(t)}{(C)\int_{0}^{1}t^{n} d \mu(t)}-f(1)\right ].$$
Suppose now that $f\ge 0$ is strictly increasing and strictly convex on $[0, 1]$ and, for example, that $\mu(A)=\sqrt{m(A)}$ or $\mu(A)=\sin[m(A)]$, with $m$ the Lebesgue measure. The strict convexity implies $B_{n}(f)(x)-f(x)>0$ for all $x\in (0, 1)$ and the property of $f$ to be strictly increasing easily implies
$$\frac{(C)\int_{0}^{1}f(t)t^{n}d \mu(t)}{(C)\int_{0}^{1}t^{n}d \mu(t)}-f(1)<\frac{f(1)\cdot (C)\int_{0}^{1}t^{n}d \mu(t)}{(C)\int_{0}^{1}t^{n}d \mu(t)}-f(1)=0.$$
Therefore, in this case we get
$$|D_{n, \Gamma_{n}}(f)(x)-f(x)|<\max\left \{|B_{n}(f)(x)-f(x)|, x^{n}\left |\frac{(C)\int_{0}^{1}f(t)t^{n}d \mu(t)}{(C)\int_{0}^{1}t^{n}d \mu(t)}-f(1)\right |\right \},$$
i.e. for $x\in (0, 1)$, $D_{n, \Gamma_{n}}(f)(x)$ approximates better than $B_{n}(f)(x)$.
Here it is clear that $B_{n}(f)(x)$ can also be viewed as the Bernstein-Durrmeyer operators in the case when $\Gamma_{n}$ is composed by the Dirac measures $\delta_{k/n}$, $k=0, ..., n$, where we note that although the Dirac measures are not strictly positive, however the Bernstein-Durrmeyer operators attached to them are well defined. This fact contrasts with the classical case in \cite{Berd1} when $\Gamma_{n}$ is composed by only one set function, independent of $n$, and when the strict positivity of the set function is necessarily for the convergence (see Theorem 1 in \cite{Berd1}). In other words, the strict positivity of the set functions in Theorem 3.1 is not always necessary.
For another example, let us consider the genuine Bernstein-Durrmeyer-Cho\-qu\-et operators given by
$$U_{n, \Gamma_{n}}(f)(x)=p_{n, 0}(x)\cdot \frac{(C)\int_{0}^{1}f(t)(1-t)^{n}d \nu_{n, 0}}{(C)\int_{0}^{1}(1-t)^{n}d \nu_{n, 0}}+ p_{n, n}(x)\cdot \frac{(C)\int_{0}^{1}f(t)t^{n}d \nu_{n, n}}{(C)\int_{0}^{1}t^{n}d \nu_{n, n}}$$
$$+\sum_{k=1}^{n-1}p_{n, k}(x)\cdot \frac{(C)\int_{0}^{1}f(t)p_{n-2,k-1}(t)d \mu_{n-2, k-1}(t)}{(C)\int_{0}^{1}p_{n-2,k-1}(t)d \mu_{n-2, k-1}(t)},$$
where $\Gamma_{n}=\{\nu_{n, 0}, \nu_{n, n}, \mu_{n-2, k-1}, k=1, ..., n-1\}$.
Let us denote by $G_{n}(f)(x)$, the classical genuine Bernstein-Durmeyer operator (see, e.g., \cite{Gonska}).
Choosing in $\Gamma_{n}$ the set functions $\mu_{n-2, k-1}, k=1, ..., n-1$ as the Lebesgue measure, $\nu_{n, 0}=\delta_{0}$ (as Dirac measure) and $\nu_{n, n}$ as a monotone, submodular and strictly positive set function, we immediately obtain
$$U_{n, \Gamma_{n}}(f)(x)-f(x)=G_{n}(f)(x)-f(x)+x^{n}\left [\frac{(C)\int_{0}^{1}f(t)t^{n} d \nu_{n, n}(t)}{(C)\int_{0}^{1}t^{n} d \nu_{n, n}(t)}-f(1)\right ].$$
Since the strict convexity of $f$ implies $G_{n}(f)(x)-f(x)>0$ for all $x\in (0, 1)$ (see, e.g., Lemma 2.1, (iv) in \cite{Gonska}), similar reasonings with those for the previous example show that if $f$ is strictly convex and strictly increasing on $[0, 1]$ (and, for example, $\nu_{n, n}(A)=\sqrt{m(A)}$ or $\nu_{n, n}(A)=\sin[m(A)]$), then $U_{n, \Gamma_{n}}(f)(x)$ approximates better $f$ on on $(0, 1)$ than the classical genuine operator, $G_{n}(f)(x)$.
{\bf Remark 4.5.} Recall that in \cite{Gal-Ann}, Example 4.2, for the nonlinear Picard-Choquet operators we have obtained a general estimate similar to that for the classical Picard operators, while for particular functions of the form $f(x)=Me^{-Ax}$, $M, A >0$, we got there essentially better error estimates.
{\bf Remark 4.6.} In \cite{Li} applications of the classical Bernstein-Durrmeyer operators to learning theory are presented. Taking onto account the very recent applications of the Choquet integral to learning theory (see, e.g., \cite{Hull1} and the references therein), it becomes of interest to see for potential applications of the Bernstein-Durrmeyer-Choquet operators to learning theory. Also, taking into account the applications of the classical Bernstein-Durrmeyer operators in regression estimation in, e.g., \cite{Raf} and the very recent applications of the Choquet integral to regression model, see, e.g., \cite{Grab}, it would be of interest to see for possible applications of the Bernstein-Durrmeyer-Choquet operators to the regression model.
|
1,314,259,995,840 | arxiv | \section{Introduction}\label{intro}
It has long been known that for a locally compact group $G$ there are many $C^*$-algebras between the full group $C^*$-algebra $C^*(G)$ and the reduced algebra $C^*_r(G)$ (see \cite{eym}).
However, little study has been made regarding the extent to which these intermediate algebras can be called group $C^*$-algebras.
This paper is inspired by recent work of Brown and Guentner \cite{BrownGuentner},
which studies such intermediate algebras for discrete groups,
and \cite{Okayasu}, which shows that in fact there can be a continuum of such intermediate algebras.
We shall consider a general locally compact group $G$, and show that
by elementary harmonic analysis
there is a one-to-one correspondence between $G$-invariant weak*-closed subspaces $E$ of the Fourier-Stieltjes algebra $B(G)$ containing $B_r(G)$ and
quotients
$C^*_E(G)$
of $C^*(G)$ which are intermediate between $C^*(G)$ and the reduced group algebra $C^*_r(G)$.
We are primarily interested in the following results:
\begin{itemize}
\item $E$ is an ideal if and only if there is a coaction $C^*_E(G)\to M(C^*_E(G)\otimes C^*(G))$.
\item $E$ is a subalgebra if and only if there is a comultiplication $C^*_E(G)\to M(C^*_E(G)\otimes C^*_E(G))$.
\end{itemize}
(See Propositions~\ref{coaction} and \ref{comultiplication} for more precise statements.)
These $C^*$-algebras can be used to describe various properties of $G$,
e.g., if $G$ is discrete and $E=\bar{B(G)\cap c_0(G)}$,
then $G$ has the Haagerup property if and only if $C^*_E(G)=C^*(G)$
(see \cite[Corollary~3.4]{BrownGuentner}).
Brown and Guentner also prove that (again, in the discrete case)
$C^*_E(G)$ is a compact quantum group, because it carries a comultiplication, and this caught our attention since it makes a connection with noncommutative crossed-product duality.
If we have a $C^*$-dynamical system $(B,G,\alpha)$,
one can form the full crossed product $B\rtimes_\alpha G$
or the reduced crossed product $B\rtimes_{\alpha,r} G$.
We show in \secref{exotic coaction} that for $E$ as above there is an ``$E$-crossed product'' $B\rtimes_{\alpha,E} G$,
and we speculate that these ``intermediate'' crossed products satisfy an ``exotic'' version of crossed-product duality involving $C^*_E(G)$.
After a short section on preliminaries, in \secref{certain quotients} we prove the above-mentioned results concerning the existence of a coaction or comultiplication on $C^*_E(G)$ .
In \secref{classical} we briefly explore the analogue for arbitrary locally compact groups of the construction used in \cite{BrownGuentner}, where for discrete groups they construct group $C^*$-algebra s starting with ideals of $\ell^\infty(G)$.
In \secref{discrete} we specialize (for the only time in this paper) to the discrete case, showing that a quotient $C^*_E(G)$ is a group $C^*$-algebra\ if and only if it is \emph{topologically graded} in the sense of \cite{ExelAmenability}.
Finally, in \secref{exotic coaction} we outline a possible application of our exotic group algebras to noncommutative crossed-product duality.
After this paper was circulated in preprint form, we learned that Buss and Echterhoff \cite{BusEch} have
given counterexamples to \conjref{E-coaction}
and have proven \conjref{E dual}.
We thank the referee for helpful comments.
\section{Preliminaries}\label{prelim}
All ideals of $C^*$-algebras will be closed and two-sided.
If $A$ and $B$ are $C^*$-algebras, then $A\otimes B$ will denote the minimal tensor product.
For one of our examples we will need the following elementary fact, which is surely folklore.
\begin{lem}\label{onto}
Let $A$ be a $C^*$-algebra, and let $I$ and $J$ be ideals of $A$.
Let $\phi:A\to A/I$ and $\psi:A\to A/J$ be the quotient maps,
and define
\[
\pi=\phi\oplus \psi:A\to (A/I)\oplus (A/J).
\]
Then $\pi$ is surjective if and only if $A=I+J$.
\end{lem}
\begin{proof}
First assume that $\pi$ is surjective,
and let $a\in A$.
Choose $b\in A$ such that
\[
\pi(b)=\bigl(\phi(a),0\bigr),
\]
i.e.,
$\phi(b)=\phi(a)$ and $\psi(b)=0$.
Then $a-b\in I$, $b\in J$, and $a=(a-b)+b$.
Conversely, assume that $A=I+J$,
and let $a\in A$.
Choose $b\in I$ and $c\in J$ such that $a=b+c$.
Then $\psi(c)=0$, and $\phi(c)=\phi(a)$ since $a-c\in I$.
Thus
\[
\pi(c)=\bigl(\phi(a),0\bigr).
\]
It follows that $\pi(A)\supset (A/I)\oplus \{0\}$, and similarly $\pi(A)\supset \{0\}\oplus (A/J)$,
and hence $\pi$ is onto.
\end{proof}
A point of notation: for a homomorphism between $C^*$-algebras, or for a bounded linear functional on a $C^*$-algebra, we use a bar to denote the unique strictly continuous extension to the multiplier algebra.
We adopt the conventions of \cite{enchilada} for actions and coactions of a locally compact group $G$ on a $C^*$-algebra $A$. In particular, we use \emph{full} coactions
$\delta:A\to M(A\otimes C^*(G))$,
which are nondegenerate injective homomorphisms satisfying the \emph{coaction-nondegeneracy} property
\begin{equation}\label{coaction nondegenerate}
\clspn\{\delta(A)(1\otimes C^*(G))=A\otimes C^*(G)
\end{equation}
and the \emph{coaction identity}
\begin{equation}\label{identity}
\bar{\delta\otimes\text{\textup{id}}}\circ\delta=\bar{\text{\textup{id}}\otimes\delta_G}\circ\delta,
\end{equation}
where $\delta_G$ is the canonical coaction on $C^*(G)$,
determined by $\bar{\delta_G}(x)=x\otimes x$ for $x\in G$
(and where $G$ is identified with its canonical image in $M(C^*(G))$).
Recall that $\delta$ gives rise to a
right $B(G)$-module structure on $A^*$ given by
\[
\omega\cdot f=\bar{\omega\otimes f}\circ\delta
\quad\text{for $\omega\in A^*$ and $f\in B(G)$,}
\]
and also
to a left $B(G)$-module structure on $A$ given by
\[
f\cdot a=\bar{\text{\textup{id}}\otimes f}\circ\delta(a)\quad\text{for $f\in B(G)$ and $a\in A$},
\]
and that moreover
\[
(\omega\cdot f)(a)=\omega(f\cdot a)\quad\text{for all $\omega\in A^*$, $f\in B(G)$, and $a\in A$.}
\]
Further recall that
$1_G\cdot a=a$ for all $a\in A$, where $1_G$ is the constant function with value $1$.
In fact, suppose we have a homomorphism
$\delta:A\to M(A\otimes C^*(G))$ satisfying all the conditions of a coaction except perhaps injectivity. Then $\delta$ is in fact a coaction, because injectivity follows automatically,
by the following folklore trick:
\begin{lem}\label{injective}
Let $\delta:A\to M(A\otimes C^*(G))$ be a homomorphism satisfying
\eqref{coaction nondegenerate} and
\eqref{identity}.
Then for all $a\in A$ we have
\[
\bar{\text{\textup{id}}\otimes 1_G}\circ\delta(a)=a,
\]
where $1_G\in B(G)$ is the constant function with value $1$.
In particular, $\delta$ is injective and hence a coaction.
\end{lem}
\begin{proof}
First of all,
\begin{align*}
A
&=\clspn\Bigl\{(\text{\textup{id}}\otimes g)\bigl(\delta(a)(1\otimes c)\bigr):g\in B(G),a\in A,c\in C^*(G)\Bigr\}
\\&=\clspn\bigl\{\bar{\text{\textup{id}}\otimes c\cdot g}\circ\delta(a):g\in B(G),a\in A,c\in C^*(G)\bigr\}
\\&=\clspn\bigl\{\bar{\text{\textup{id}}\otimes f}\circ\delta(a):f\in B(G),a\in A\bigr\}.
\end{align*}
Now the following computation suffices: for all $a\in A$ and $f\in B(G)$ we have
\begin{align*}
&\bar{\text{\textup{id}}\otimes 1_G}\circ\delta\bigl(\bar{\text{\textup{id}}\otimes f}\circ\delta(a)\bigr)
\\&\quad=\bar{\text{\textup{id}}\otimes 1_G}\circ \bar{\text{\textup{id}}\otimes\text{\textup{id}}\otimes f}\circ (\delta\otimes\text{\textup{id}})\circ\delta(a)
\\&\quad=\bar{\text{\textup{id}}\otimes 1_G\otimes f}\circ (\text{\textup{id}}\otimes\delta_G)\circ\delta(a)
\\&\quad=\bar{\text{\textup{id}}\otimes 1_Gf}\circ\delta(a)
\\&\quad=\bar{\text{\textup{id}}\otimes f}\circ\delta(a)
\qedhere
\end{align*}
\end{proof}
\section{Exotic quotients of $C^*(G)$}\label{certain quotients}
Let $G$ be a locally compact group,.
We are interested in certain quotients $C^*_E(G)$ (see \defnref{E quotient} for this notation).
We will always assume that ideals of $C^*$-algebras are closed and two-sided.
Let $B(G)$ denote the Fourier-Stieltjes algebra, which we identify with the dual of $C^*(G)$.
We give $B(G)$ the usual $C^*(G)$-bimodule structure: for $a,b\in C^*(G)$ and $f\in B(G)$ we define
\[
\<b,a\cdot f\>=\<ba,f\>\midtext{and}\<b,f\cdot a\>=\<ab,f\>.
\]
This bimodule structure extends to an $M(C^*(G))$-bimodule structure, because for $m\in M(C^*(G))$ and $f\in B(G)$ the linear functionals $a\mapsto \<am,f\>$ and $a\mapsto \<ma,f\>$ on $C^*(G)$ are bounded.
Regarding $G$ as canonically embedded in $M(C^*(G))$, the associated $G$-bimodule structure on $B(G)$ is given by
\[
(x\cdot f)(y)=f(yx)\midtext{and}(f\cdot x)(y)=f(xy)
\]
for $x,y\in G$ and $f\in B(G)$.
A quotient $C^*(G)/I$ is uniquely determined by the annihilator $E=I^\perp$ in $B(G)$,
which is a weak*-closed subspace.
We find it convenient to work in terms of $E$ rather than $I$, keeping in mind that we will have $I={}^\perp E$, the preannihilator in $C^*(G)$.
First we record the following well-known property:
\begin{lem}
\label{invariant}
For any weak*-closed subspace $E$ of $B(G)$, the following are equivalent:
\begin{enumerate}
\item ${}^\perp E$ is an ideal;
\item $E$ is a $C^*(G)$-subbimodule;
\item $E$ is $G$-invariant.
\end{enumerate}
\end{lem}
\begin{proof}
(1)$\Leftrightarrow$(2) follows from, e.g., \cite[Theorem~3.10.8]{ped}, and (2)$\Leftrightarrow$(3) follows by integration.
\end{proof}
\begin{defn}\label{E quotient}
If $E$ is a weak*-closed $G$-invariant subspace of $B(G)$, let $C^*_E(G)$ denote the quotient $C^*(G)/{}^\perp E$.
\end{defn}
Note that the above definition makes sense, by \lemref{invariant}.
\begin{ex}
Of course we have
\[
C^*(G)=C^*_{B(G)}(G).
\]
Also,
\[
C^*_r(G)=C^*_{B_r(G)}(G),
\]
where $B_r(G)$ is the regular Fourier-Stieltjes algebra of $G$,
because if $\lambda:C^*(G)\to C^*_r(G)$ denotes the regular representation of $G$ then
\[
(\ker\lambda)^\perp=B_r(G).
\]
Recall for later use that the intersection $C_c(G)\cap B(G)$ is norm-dense in the Fourier algebra $A(G)$ (for the norm of functionals on $C^*(G)$),
and is weak*-dense in $B_r(G)$ \cite{eym}.
\end{ex}
\begin{rem}
If $E$ is a weak*-closed $G$-invariant subspace of $B(G)$, and $q:C^*(G)\to C^*_E(G)$ is the quotient map, then the dual map $q^*:C^*_E(G)^*\to C^*(G)^*=B(G)$ is an isometric isomorphism onto $E$, and we identify $E=C^*_E(G)^*$ and regard $q^*$ as an inclusion map.
\end{rem}
Inspired in part by \cite{BrownGuentner}, we pause here to give another construction of the quotients $C^*_E(G)$:
\begin{enumerate}
\item
Start with a
$G$-invariant, but \emph{not necessarily weak*-closed}, subspace $E$ of $B(G)$.
\item
Call a representation $U$ of $G$ on a Hilbert space $H$ an \emph{$E$-representation} if there is a dense subspace $H_0$ of $H$ such that the matrix coefficients
\[
x\mapsto \<U_x\xi,\eta\>
\]
are in $E$ for all $\xi,\eta\in H_0$.
\item
Define a $C^*$-seminorm $\|\cdot\|_E$ on $C_c(G)$ by
\[
\|f\|_E=\sup\{\|U(f)\|:\text{$U$ is an $E$-representation of $G$}\}.
\]
\end{enumerate}
The following lemma is presumably well-known, but we include a proof for the convenience of the reader.
\begin{lem}\label{DJ}
With the above notation, let $I$
be the ideal of $C^*(G)$
given by
\begin{equation}
\label{kernel}
I=\{a\in C^*(G):\|a\|_E=0\}.
\end{equation}
Then:
\begin{enumerate}
\item$I={}^\perp E$.
\item The weak*-closure $\bar E$ of $E$ in $B(G)$ is $G$-invariant,
and
$C^*_{\bar E}(G)=C^*(G)/I$ is the Hausdorff completion of $C_c(G)$ in the seminorm $\|\cdot\|_E$.
\item If $E$ is an ideal or a subalgebra of $B(G)$, then so is $\bar E$.
\end{enumerate}
\end{lem}
\begin{proof}
(1)
To show that $I\subset {}^\perp E$, let $a\in I$
and $f\in E$.
Since $f\in B(G)$, we can choose a representation $U$ of $G$ on a Hilbert space $H$ and vectors $\xi,\eta\in H$ such that
\[
f(x)=\<U_x\xi,\eta\>\quad\text{for $x\in G$.}
\]
Let $K_0$ be the smallest $G$-invariant subspace of $H$ containing both $\xi$ and $\eta$,
and let $K=\bar{K_0}$.
Then $K$ is a closed $G$-invariant subspace of $H$, so determines a subrepresentation $\rho$ of $G$.
For every $\zeta,\kappa\in K_0$, the function $x\mapsto \<U_x\zeta,\kappa\>$ is in $E$
because $E$
is $G$-invariant.
Thus $\rho$ is an $E$-representation.
We have
\begin{align*}
|\<a,f\>|
&=|\<\rho(a)\xi,\eta\>|
\\&\le \|\rho(a)\|\|\xi\|\|\eta\|
\\&\le \|a\|_E\|\xi\|\|\eta\|
\\&=0.
\end{align*}
Thus $a\in {}^\perp E$.
For the opposite containment, suppose by way of contradiction that we can find
$a\in {}^\perp E\setminus I$.
Then $\|a\|_E\ne 0$, so we can also choose an $E$-representation $U$ of $G$ on a Hilbert space $H$ such that $U(a)\ne 0$.
Let $H_0$ be a dense subspace of $H$ such that for all $\xi,\eta\in H_0$ the function
$x\mapsto\<U_x\xi,\eta\>$
is in $E$.
By density we can choose $\xi,\eta\in H_0$ such that $\<U(a)\xi,\eta\>\ne 0$.
Then $g(x)=\<U_x\xi,\eta\>$ defines an element $g\in E$,
and we have
\[
\<a,g\>
=\<U(a)\xi,\eta\>
\ne 0,
\]
which is a contradiction.
Therefore ${}^\perp E\subset I$,
as desired.
(2)
Since $I={}^\perp E$ we have $\bar E=I^\perp$, which is $G$-invariant because $I$ is an ideal, by \lemref{invariant}.
We have $I={}^\perp \bar E$, so $C^*_{\bar E}(G)=C^*(G)/I$ by \defnref{E quotient}.
Since $C_c(G)$ is dense in $C^*(G)$, the result now follows by the definition of $I$ in \eqref{kernel}.
(3)
This follows immediately from separate weak*-continuity of multiplication in $B(G)$.
This is a well-known property of $B(G)$, but we include the brief proof here for completeness:
the bimodule action of $B(G)$ on the enveloping algebra $W^*(G)=B(G)^*$,
given by
\[
\<a\cdot f,g\>=\<a,fg\>=\<f\cdot a,g\>
\quad\text{for $a\in W^*(G),f,g\in B(G)$,}
\]
leaves $C^*(G)$ invariant, because it satisfies the submultiplicativity condition $\|a\cdot f\|\le \|a\|\|f\|$ on norms and leaves $C_c(G)\subset C^*(G)$ invariant.
Thus, if $f_i\to 0$ weak* in $B(G)$ and $g\in B(G)$, then for all $a\in C^*(G)$ we have
\[
\<a,f_ig\>=\<a\cdot g,f_i\>\to 0.
\qedhere
\]
\end{proof}
\begin{cor}\label{E rep}\
\begin{enumerate}
\item A representation $U$ of $G$ is an $E$-representation if and only if, identifying $U$ with the corresponding representation of $C^*(G)$, we have $\ker U\supset {}^\perp E$.
\item A nondegenerate homomorphism $\tau:C^*(G)\to M(A)$, where $A$ is a $C^*$-algebra, factors through a homomorphism of $C^*_E(G)$ if and only if
\[
\bar\omega\circ\tau\in \bar E\quad\text{for all $\omega\in A^*$,}
\]
where again $\bar E$ denotes the weak*-closure of $E$.
\end{enumerate}
\end{cor}
\begin{proof}
This follows readily from \lemref{DJ}.
\end{proof}
\begin{rem}
In light of \lemref{DJ}, if we have a $G$-invariant subspace $E$ of $B(G)$ that is not necessarily weak*-closed, it makes sense to, and we shall, write $C^*_E(G)$ for $C^*_{\bar E}(G)$.
However, whenever convenient we can replace $E$ by its weak*-closure, giving the same quotient $C^*_E(G)$.
\end{rem}
\begin{obs}\label{immed}
By \lemref{DJ}, if $E$ is a $G$-invariant subspace of $B(G)$ then:
\begin{enumerate}
\item
$C^*_E(G)=C^*(G)$ if and only if $E$ is weak*-dense in $B(G)$.
\item
$C^*_E(G)=C^*_r(G)$ if and only if $E$ is weak*-dense in $B_r(G)$.
\end{enumerate}
\end{obs}
We record an elementary consequence of our definitions:
\begin{lem}\label{ccg}
For a weak*-closed $G$-invariant subspace $E$ of $B(G)$, the following are equivalent:
\begin{enumerate}
\item
${}^\perp E\subset \ker\lambda$;
\item
$E\supset B_r(G)$;
\item
$E\supset A(G)$;
\item
$E\supset (C_c(G)\cap B(G))$;
\item
there is a \(unique\) homomorphism $\rho:C^*_E(G)\to C^*_r(G)$
making the diagram
\[
\xymatrix{
C^*(G) \ar[dr]^q \ar[dd]_\lambda
\\
&C^*_E(G) \ar@{-->}[dl]^\rho_{!}
\\
C^*_r(G)
}
\]
commute.
\end{enumerate}
\end{lem}
\begin{defn}\label{group algebra}
For a weak*-closed $G$-invariant subspace $E$ of $B(G)$, we say the quotient $C^*_E(G)$ is a \emph{group $C^*$-algebra\ of $G$} if the above equivalent conditions (1)--(4) are satisfied.
If $B_r(G)\subsetneq E\ne B(G)$ we say the group $C^*$-algebra\ is \emph{exotic}.
\end{defn}
We will see
in \propref{graded}
that if $G$ is discrete then a quotient $C^*_E(G)$ is
a group $C^*$-algebra\ if and only if it is
topologically graded in Exel's sense \cite[Definition~3.4]{ExelAmenability}.
We are especially interested in group $C^*$-algebra s that carry a coaction or a comultiplication.
We will need the following result, which is folklore among coaction cognoscenti:
\begin{lem}\label{quotient coaction}
If $\delta:A\to M(A\otimes C^*(G))$
is a coaction of $G$ on a $C^*$-algebra $A$ and $I$ is an ideal of $A$, then
the following are equivalent:
\begin{enumerate}
\item
there is a coaction $\tilde\delta$ on $A/I$ making the diagram
\begin{equation}\label{quotient}
\xymatrix{
A \ar[r]^-\delta \ar[d]_q
&M(A\otimes C^*(G)) \ar[d]^{\bar{q\otimes\text{\textup{id}}}}
\\
A/I \ar[r]_-{\tilde\delta}
&M(A/I\otimes C^*(G))
}
\end{equation}
commute \(where $q$ is the quotient map\);
\item \label{kernel condition}
$I\subset \ker \bar{q\otimes \text{\textup{id}}}\circ \delta$.
\item $I^\perp$ is a $B(G)$-submodule of $A^*$.
\end{enumerate}
\end{lem}
\begin{proof}
This is well-known, but difficult to find in the literature, so we include the brief proof for the convenience of the reader.
There exists a \emph{homomorphism} $\tilde\delta$ making the diagram~\eqref{quotient} commute if and only if
(2)
holds,
and in that case $\tilde\delta$ will satisfy
the coaction-nondegeneracy \eqref{coaction nondegenerate}
and the coaction identity \eqref{identity}.
By \lemref{injective} this implies that $\tilde\delta$ is a coaction.
Thus (1)$\Leftrightarrow$(2), and (2)$\Leftrightarrow$(3) follows from a routine calculation using the fact that
$\{\psi\otimes f:\psi\in (A/I)^*,f\in B(G)\}$ separates the elements of $M(A/I\otimes C^*(G))$.
\end{proof}
Recall that the multiplication in $B(G)$ satisfies
\[
\<a,fg\>=\<\delta_G(a),\bar{f\otimes g}\>\quad\text{for $a\in C^*(G)$ and $f,g\in B(G)$,}
\]
where here we use the notation $f\otimes g$ to denote the functional in $(C^*(G)\otimes C^*(G))^*$ determined by
\[
\<x\otimes y,\bar{f\otimes g}\>=f(x)g(y)\quad\text{for $x,y\in G$.}
\]
\begin{rem}\label{max}
Note that we need to explicitly state the above convention for $f\otimes g$, since we are using the minimal tensor product:
if $G$ is a group for which the canonical surjection
\[
C^*(G)\otimes_{\max} C^*(G)\to C^*(G)\otimes C^*(G)
\]
is noninjective\footnote{e.g., any infinite simple group with property T --- see \cite[Theorem~6.4.14 and Remark~6.4.15]{brownozawa}},
then
\begin{align*}
C^*(G)\otimes C^*(G)&\ne C^*(G\times G)
\\
(C^*(G)\otimes C^*(G))^*&\ne B(G\times G),
\end{align*}
because $C^*(G\times G)=C^*(G)\otimes_{\max} C^*(G)$.
\end{rem}
\begin{cor}\label{coaction}
Let $E$ be a weak*-closed $G$-invariant subspace of $B(G)$, and let $q:C^*(G)\to C^*_E(G)$ be the quotient map.
Then there is a coaction $\delta_G^E$
of $G$ on $C^*_E(G)$
such that
\[
\bar{\delta_G^E}(q(x))=q(x)\otimes x\quad\text{for $x\in G$}
\]
if and only if $E$ is an ideal of $B(G)$.
\end{cor}
\begin{proof}
Since $E$ is the annihilator of $\ker q$, this follows immediately from \lemref{quotient coaction}.
\end{proof}
Recall that in \defnref{group algebra} we called $C^*_E(G)$ a group $C^*$-algebra if $E$ is a weak*-closed $G$-invariant subspace of $B(G)$ containing $B_r(G)$;
this latter property is automatic if $E$ is an ideal (as long as it's nonzero):
\begin{lem}\label{smallest}
Every nonzero
norm-closed $G$-invariant ideal of $B(G)$ contains $A(G)$, and hence
every nonzero weak*-closed $G$-invariant ideal of $B(G)$ contains $B_r(G)$.
\end{lem}
\begin{proof}
Let $E$ be the ideal.
It suffices to show that $E\cap A(G)$ is norm dense in $A(G)$.
There exist $t\in G$ and $f\in E$ such that $f(t)\ne 0$.
By \cite[Lemma~3.2]{eym} there exists
$g\in A(G)\cap C_c(G)$
such that $g(t)\ne 0$, and then $fg\in E\cap C_c(G)$ is nonzero at $t$.
By $G$-invariance of $E$, for all $x\in G$ there exists $f\in E$ such that $f(x)\ne 0$.
Then for any $y\ne x$ we can find $g\in A(G)\cap C_c(G)$ such that $g(x)\ne 0$ and $g(y)=0$, and so $fg\in E$ is nonzero at $x$ and zero at $y$. Thus $E\cap A(G)$ is an ideal of $A(G)$ that is nowhere vanishing on $G$ and separates points, so by \cite[Corollary~3.38]{eym} $E\cap A(G)$ is norm dense in $A(G)$, so we are done.
\end{proof}
Recall that a \emph{comultiplication} on
a $C^*$-algebra $A$
is a homomorphism (which we do \emph{not} in general require to be injective) $\Delta:A\to M(A\otimes A)$ satisfying the \emph{co-associativity} property
\[
\bar{\Delta\otimes\text{\textup{id}}}\circ\Delta=\bar{\text{\textup{id}}\otimes\Delta}\circ\Delta
\]
and the \emph{nondegeneracy properties}
\[
\clspn\{\Delta(A)(1\otimes A)\}=A\otimes A=\clspn\{(A\otimes 1)\Delta(A)\}.
\]
A $C^*$-algebra with a comultiplication is called a \emph{$C^*$-bialgebra} (see \cite{kawamura} for this terminology).
A comultiplication $\Delta$ on $A$ is used to make the dual space $A^*$ into a Banach algebra in the standard way:
\[
\omega\psi:=\bar{\omega\otimes\psi}\circ\Delta\quad\text{for $\omega,\psi\in A^*$.}
\]
The following is another folklore result, proved similarly to \lemref{quotient coaction}:
\begin{lem}\label{quotient comultiplication}
If $\Delta:A\to M(A\otimes A)$
is a comultiplication on a $C^*$-algebra $A$ and $I$ is an ideal of $A$, then
the following are equivalent:
\begin{enumerate}
\item
there is a comultiplication $\tilde\Delta$ on $A/I$ making the diagram
\[
\xymatrix{
A \ar[r]^-\Delta \ar[d]_q
&M(A\otimes A) \ar[d]^{\bar{q\otimes q}}
\\
A/I \ar[r]_-{\tilde\Delta}
&M(A/I\otimes A/I)
}
\]
commute \(where $q$ is the quotient map\);
\item
$I\subset \ker \bar{q\otimes q}\circ \Delta$.
\item $I^\perp$ is a subalgebra of $A^*$.
\end{enumerate}
\end{lem}
We apply this to
the canonical comultiplication $\delta_G$ on $C^*(G)$:
\begin{prop}\label{comultiplication}
Let $E$ be a weak*-closed $G$-invariant subspace of $B(G)$, and let $q:C^*(G)\to C^*_E(G)$ be the quotient map.
Then the following are equivalent:
\begin{enumerate}
\item
there is a comultiplication $\Delta$ making the diagram
\[
\xymatrix@C+30pt{
C^*(G) \ar[r]^-{\delta_G} \ar[d]_q
&M(C^*(G)\otimes C^*(G)) \ar[d]^{\bar{q\otimes q}}
\\
C^*_E(G) \ar[r]_-\Delta
&M(C^*_E(G)\otimes C^*_E(G))
}
\]
commute;
\item
${}^\perp E\subset \ker\bar{q\otimes q}\circ\delta_G$;
\item
$E$ is a subalgebra of $B(G)$.
\end{enumerate}
\end{prop}
\begin{rem}\label{hopf}
\propref{comultiplication} tells us that if $E$ is a weak*-closed $G$-invariant subalgebra of $B(G)$, then
the group algebra $C^*_E(G)$ is a $C^*$-bialgebra.
However, this
probably
does not make $C^*_E(G)$ a
locally compact quantum group,
since this would require an antipode.
It might be difficult to investigate the general question of whether there exists \emph{some} antipode on $C^*_E(G)$ that is compatible with the comultiplication;
it seems more reasonable to ask whether the quotient map $q:C^*(G)\to C^*_E(G)$ takes the canonical antipode on $C^*(G)$ to an antipode on $C^*_E(G)$.
This requires $E$ to be closed under inverse
i.e., if $f\in E$ then so is the function $f^\vee$ defined by $f^\vee(x)=f(x^{-1})$.
Now, $f^\vee(x)=\bar{f^*(x)}$
where $f^*$ is defined by $f^*(a)=\bar{f(a^*)}$ for $a\in C^*(G)$.
Since $f\in E$ if and only if $f^*\in E$,
we see that $E$ is invariant under $f\mapsto f^\vee$ if and only if
it is invariant under complex conjugation.
In all our examples (in particular \secref{classical}) $E$ has this property.
Note that $C^*_E(G)$ always has a Haar weight, since we can compose the canonical Haar weight on $C^*_r(G)$ with the quotient map $C^*_E(G)\to C^*_r(G)$.
However, this Haar weight on $C^*_E(G)$ is faithful if and only if $E=B_r(G)$.
\end{rem}
\begin{rem}\label{closed}
By \lemref{DJ},
if $E$ is
a $G$-invariant
ideal of $B(G)$
and $I={}^\perp E$,
then
$\bar E$ is
also a $G$-invariant ideal,
so by \propref{coaction}
there is a coaction $\delta_G^E$ of $G$ on
$C^*_E(G)$
such that
\[
\bar{\delta_G^E}(q(x))=q(x)\otimes x\quad\text{for $x\in G$,}
\]
where
$q:C^*(G)\to C^*_E(G)$ is the quotient map.
Similarly, if $E$ is
a $G$-invariant
subalgebra of $B(G)$
then
$\bar E$ is
also a $G$-invariant subalgebra,
so by \propref{comultiplication}
there is a comultiplication $\Delta$ on
$C^*_E(G)$
such that
\[
\bar{\Delta}(q(x))=q(x)\otimes q(x)\quad\text{for $x\in G$.}
\]
\end{rem}
\begin{ex}
Note that if the quotient $C^*_E(G)$ is a group $C^*$-algebra, then the quotient map $q:C^*(G)\to C^*_E(G)$ is faithful on $C_c(G)$,
and so by \lemref{DJ} $C^*_E(G)$ is the completion of $C_c(G)$ in the associated norm $\|\cdot\|_E$.
However,
$q$ being faithful on $C_c(G)$
is not sufficient for $C^*_E(G)$ to be a group $C^*$-algebra.
The simplest example of this is in \cite[Exercise~XI.38]{FellDoran2} (which we modify only slightly): let $0\le a<b<2\pi$, and define a surjection
\[
q:C^*(\mathbb Z)\to C[a,b]
\]
by
\[
q(n)(t)=e^{int}.
\]
Then the unitaries $q(n)$ are linearly independent,
so $q$ is faithful on $c_c(\mathbb Z)$,
but $q(C^*(\mathbb Z))$ is not a group $C^*$-algebra\ because $\ker q$ is a nontrivial ideal of $C^*(\mathbb Z)$ and $\mathbb Z$ is amenable, so that $\ker\lambda=\{0\}$.
\end{ex}
\begin{ex}
The paper
\cite{EQInduced} shows how to construct exotic group $C^*$-algebra s $C^*_E(G)$
(see also \cite[Remark~9.6]{Kyed} for similar exotic quantum groups)
with no coaction:
let
\[
q=\lambda\oplus 1_G,
\]
where $1_G$ denotes the trivial $1$-dimensional representation of $G$.
The quotient $C^*_E(G)$ is a group $C^*$-algebra\ since $\ker q=\ker\lambda\cap \ker 1_G$.
On the other hand, we have
\[
E=(\ker q)^\perp=B_r(G)+\C1_G,
\]
which is not an ideal of $B(G)$ unless it is all of $B(G)$, i.e., unless $q$ is faithful;
as remarked in \cite{EQInduced}, this behavior would be quite bizarre,
and in fact we do not know of any discrete nonamenable group with this property.
However, these quotients $C^*_E(G)$ are $C^*$-bialgebras, because $B_r(G)+\mathbb C 1_G$ is a subalgebra of $B(G)$.
Thus, these quotients give
examples of exotic group $C^*$-bialgebras that are different from those in \cite[Proposition~4.4 and Remark~4.5]{BrownGuentner}. It is interesting to note that these quotients of $C^*(G)$ are of a decidedly elementary variety:
by \lemref{onto} we have
\[
C^*_E(G)=C^*_r(G)\oplus \mathbb C,
\]
because $C^*(G)=\ker\lambda+\ker 1_G$ since $G$ is nonamenable.
To see this latter implication, recall that if $G$ is nonamenable
then $1_G$ is not weakly contained in $\lambda$,
so $\ker 1_G\not\supset \ker\lambda$,
and hence $C^*(G)=\ker\lambda+\ker 1_G$ since $\ker 1_G$ is a maximal ideal.
Valette has a similar example in \cite[Theorem~3.6]{valetteT} where he shows that if $N$ is a closed normal subgroup of $G$ that has property (T), then
$C^*(G)$ is the direct sum of $C^*(G/N)$ and a complementary ideal.
For a different source of exotic group $C^*$-bialgebras, see \exref{maxmin}.
\end{ex}
\begin{ex}
We can also find examples of group $C^*$-algebra s with no comultiplication: modify the preceding example by taking
\[
q=\lambda\oplus \gamma,
\]
where $\gamma$ is a nontrivial character of $G$ (assuming that $G$ has such characters).
Then
\[
(\ker q)^\perp=B_r(G)+\mathbb C \gamma,
\]
which is not a subalgebra of $B(G)$ when $G$ is nonamenable.
\end{ex}
\begin{ex}\label{maxmin}
Let $G$ be a locally compact group for which
the canonical surjection
\begin{equation}\label{max onto}
C^*(G)\otimes_{\max} C^*(G)\to C^*(G)\otimes C^*(G)
\end{equation}
is not injective,
where in the second tensor product we use the minimal $C^*$-tensor norm as usual (see \remref{max}).
Let $I$ denote the kernel of this map.
Since the algebraic product $B(G)\odot B(G)$ is weak*-dense in $(C^*(G)\otimes C^*(G))^*$, the annihilator $E=I^\perp$ is the weak*-closed span of functions of the form
\[
(x,y)\mapsto f(x)g(y)\quad\text{for $f,g\in B(G)$.}
\]
This is clearly a subalgebra, but not an ideal, because it contains $1$.
Also, $E\supset B_r(G\times G)$ because the surjection \eqref{max onto} can be followed by
\[
C^*(G)\otimes C^*(G)\to C^*_r(G)\otimes C^*_r(G)\cong C^*_r(G\times G).
\]
Thus the canonical coaction $\delta_{G\times G}$ of $G\times G$ on $C^*(G\times G)$ descends to a comultiplication on the group $C^*$-algebra\ $C^*_E(G\times G)\cong C^*(G)\otimes C^*(G)$, but not to a coaction of $G\times G$.
\end{ex}
\section{Classical ideals}\label{classical}
We continue to let $G$ be an arbitrary locally compact group.
We will apply the theory of the preceding sections to
group $C^*$-algebra s $C^*_E(G)$ with $E$
of the form
\[
E=D\cap B(G),
\]
where $D$ is some familiar
$G$-invariant
set of functions on $G$.
\begin{notn}
If $D$ is a $G$-invariant set of functions on $G$, we write
$\|f\|_D=\|f\|_{D\cap B(G)}$, and similarly $C^*_D(G)=C^*_{D\cap B(G)}(G)$.
\end{notn}
So, for instance, we can consider $C^*_{C_c}(G)$, $C^*_{C_0(G)}(G)$, and $C^*_{L^p(G)}(G)$.
In each of these cases the intersection $E=D\cap B(G)$ is a $G$-invariant ideal of $B(G)$,
so by
\remref{closed} and \lemref{smallest}
these quotients are all group $C^*$-algebras carrying coactions of $G$, and hence by \propref{comultiplication} they carry comultiplications.
In the case that $G$ is discrete, $c_c(G)$, $c_0(G)$, and $\ell^p(G)$ could be regarded as classical ideals of $\ell^\infty(G)$;
this is the context of Brown and Guentner's ``new completions of discrete groups'' \cite{BrownGuentner}.
We have
\[
C^*_{C_c(G)}(G)=C^*_{A(G)}(G)=C^*_r(G),
\]
because $C_c(G)\cap B(G)$ is norm dense in $A(G)$, and hence weak*-dense in $B_r(G)$.
However, the quotients $C^*_{C_0(G)}(G)$ and $C^*_{L^p(G)}(G)$ are more mysterious.
Nevertheless, we have the following
(which, for the case of discrete $G$, is \cite[Proposition~2.11]{BrownGuentner}):
\begin{prop}\label{l2}
For all $p\le 2$ we have
$C^*_{L^p(G)}(G)=C^*_r(G)$.
\end{prop}
\begin{proof}
Since $L^p(G)\cap B(G)$ consists of bounded functions,
for $p\le 2$ we have
\[
C_c(G)\cap B(G)\subset L^p(G)\cap B(G)\subset L^2(G)\cap B(G).
\]
Now,
if $U$ is a representation of $G$ having a cyclic vector $\xi$ such that the function $x\mapsto \<U_x\xi,\xi\>$ is in $L^2(G)$, then $U$ is contained in $\lambda$
(see, e.g., \cite{carey}),
and consequently $L^2(G)\cap B(G)\subset A(G)$.
Thus
\begin{align*}
B_r(G)
&=\wkstcl{C_c(G)\cap B(G)}
\\&\subset \wkstcl{L^p(G)\cap B(G)}
\\&\subset \wkstcl{L^2(G)\cap B(G)}
\\&\subset \wkstcl{A(G)}
\\&=B_r(G),
\end{align*}
and the result follows.
\end{proof}
\begin{rem}
\begin{enumerate}
\item
The proof of \propref{l2} is much easier when $G$ is discrete,
because
then for $\xi\in \ell^2(G)$ we have
\[
\xi(x)=\<\lambda_x\raisebox{2pt}{\ensuremath{\chi}}_{\{e\}},\bar\xi\>,
\]
so $\ell^2(G)\subset A(G)$.
\item
In general,
$\wkstcl{C_0(G)\cap B(G)}\supset B_r(G)$,
and the containment can be proper
(for perhaps the earliest result along these lines, see \cite{menchoff}).
When $G$ is discrete, this phenomenon occurs precisely when $G$ is a-T-menable but nonamenable, by the result of \cite{BrownGuentner} mentioned in the introduction.
\item
Using the method outlined in this section, if we start with a $G$-invariant ideal $D$ of $L^\infty(G)$ and put $E=\wkstcl{D\cap B(G)}$, we get many weak*-closed ideals of $B(G)$, but probably not all.
For example, if we let $z_F$ be the supremum in the universal enveloping von Neumann algebra $W^*(G)=C^*(G)^{**}$ of the support projections of finite dimensional representations of $G$, then
it follows from
\cite[Proposition~1, Theorem~2, Proposition~8]{walterstructure}
that $(1-z_F)\cdot B(G)$ is an ideal of $B(G)$
and $z_F\cdot B(G)=AP(G)\cap B(G)$ is a subalgebra.
It seems unlikely that for all locally compact groups $G$ the ideal $(1-z_F)\cdot B(G)$ arises as an intersection $D\cap B(G)$ for an ideal $D$ of $L^\infty(G)$.
\end{enumerate}
\end{rem}
\section{Graded algebras}\label{discrete}
In this
short
section we impose the condition that the group $G$ is discrete.
We made this a separate section for the purpose of clarity --- here the assumptions on $G$ are different from everywhere else in this paper.
\cite[Definition~3.1]{ExelAmenability} and \cite[VIII.16.11--12]{FellDoran2} define $G$-graded $C^*$-algebras as certain quotients of Fell-bundle algebras\footnote{\cite{ExelAmenability, FellDoran2} would require the images of the fibres to be linearly independent.}. When the fibres of the Fell bundle are $1$-dimensional, each one consists of scalar multiplies of a unitary. When these unitaries can be chosen to form a representation of $G$,
the $C^*$-algebra is a quotient $C^*_E(G)$.
The following can be regarded as a special case of \cite[Theorem~3.3]{ExelAmenability}:
\begin{prop}\label{graded}
Let $E$ be a weak*-closed $G$-invariant subspace of $B(G)$,
and let $q:C^*(G)\to C^*_E(G)$ be the quotient map.
Then the following are equivalent:
\begin{enumerate}
\item
$C^*_E(G)$ is a group $C^*$-algebra\ in the sense of \defnref{group algebra};
\item
there is a bounded linear functional $\omega$ on $C^*_E(G)$ such that
\[
\omega(q(x))=\begin{cases}
1& \text{if } x=e\\
0& \text{if } x\ne e;
\end{cases}
\]
\item
$E$ contains the canonical trace $\text{\textup{tr}}$ on $C^*(G)$;
\item
$E\supset B_r(G)$;
\item
there is a \(unique\) homomorphism $\rho:C^*_E(G)\to C^*_r(G)$
making the diagram
\[
\xymatrix{
C^*(G) \ar[dr]^q \ar[dd]_\lambda
\\
&C^*_E(G) \ar@{-->}[dl]^\rho_{!}
\\
C^*_r(G)
}
\]
commute.
\end{enumerate}
\end{prop}
\begin{proof}
Assuming (2),
the composition $\omega\circ q$
coincides with $\text{\textup{tr}}$,
so $\text{\textup{tr}}\in E$, and conversely if $\text{\textup{tr}}\in E$ then we get a suitable $\omega$. Thus (2) $\Leftrightarrow$ (3).
For the rest, just note that $B_r(G)=(\ker\lambda)^\perp$ is the weak*-closed $G$-invariant subspace generated by $\text{\textup{tr}}=\raisebox{2pt}{\ensuremath{\chi}}_{\{e\}}$, and appeal to \lemref{ccg}.
\end{proof}
\begin{rem}
Condition (2) in \propref{graded} is precisely what
Exel's
\cite[Definition~3.4]{ExelAmenability}
would require to say that $C^*_E(G)$ is \emph{topologically graded}.
\end{rem}
\section{Exotic coactions}\label{exotic coaction}
We return to the context of an arbitrary locally compact group $G$.
The coactions appearing in noncommutative crossed-product duality come in a variety of flavors: \emph{reduced} vs. \emph{full} (see \cite[Appendix]{enchilada} or \cite{boiler}, for example), and, among the full ones, a spectrum with \emph{normal} and \emph{maximal} coactions at the extremes (see \cite{ekq}, for example).
In this concluding section we briefly propose a new program in crossed-product duality: ``exotic coactions'', involving the exotic group $C^*$-algebra s $C^*_E(G)$ in the sense of \defnref{group algebra}.
From now until \propref{coaction bialgebra}
we are concerned with nonzero $G$-invariant weak*-closed ideals $E$ of $B(G)$.
By Lemmas~\ref{ccg} and \ref{smallest}
the quotient $C^*_E(G)=C^*(G)/{{}^\perp E}$ is a group $C^*$-algebra.
By \propref{coaction}, there is a coaction $\delta_G^E$ of $G$ on $C^*_E(G)$
making the diagram
\[
\xymatrix{
C^*(G) \ar[r]^-{\delta_G} \ar[d]_q
&M(C^*(G)\otimes C^*(G)) \ar[d]^{\bar{q\otimes\text{\textup{id}}}}
\\
C^*_E(G) \ar[r]_-{\delta_G^E}
&M(C^*_E(G)\otimes C^*(G))
}
\]
commute, where $q$ is the quotient map,
and by \propref{comultiplication} there is a quotient comultiplication $\Delta$ on $C^*_E(G)$.
Recall that we defined the \emph{exotic}
group $C^*$-algebra s
to be the ones strictly between the two extremes $C^*(G)$ and $C^*_r(G)$, corresponding to $E=B(G)$ and $E=B_r(G)$, respectively.
On one level, we could try to study coactions of Hopf $C^*$-algebras associated to the locally compact group $G$ other than $C^*(G)$ and $C^*_r(G)$.
But there is an inconvenient subtlety here (see \remref{hopf}).
However, there is a deeper level to this program, relating more directly to crossed-product duality. At the deepest level, we aim for a characterization of \emph{all} coactions of $G$ in terms of the quotients $C^*_E(G)$.
We hasten to emphasize that at this time
some of the following is speculative, and is intended merely to outline a program of study.
From now on, the unadorned term ``coaction'' will refer to a full coaction of $G$ on a $C^*$-algebra $A$.
Let $\psi:(A^m,\delta^m)\to (A,\delta)$ be the maximalization of $\delta$,
so that $\delta^m$ is a maximal coaction, $\psi:A^m\to A$ is an equivariant surjection, and the crossed-product surjection
\[
\psi\times G:A^m\rtimes_{\delta^m} G\to A\rtimes_\delta G
\]
(for the existence of which, see \cite[Lemma~A.46]{enchilada}, for example)
is an isomorphism.
Since $\delta^m$ is maximal, the canonical surjection
\[
\Phi:A^m\rtimes_{\delta^m} G\rtimes_{\widehat{\delta^m}} G\to A^m\otimes\mathcal K(L^2(G))
\]
is an isomorphism (this is ``full-crossed-product duality'').
Blurring the distinction between $A^m\rtimes_{\delta^m} G$ and the isomorphic crossed product $A\rtimes_\delta G$,
and recalling that $\psi\times G:A^m\rtimes_{\delta^m} G\to A\rtimes_\delta G$ is $\widehat{\delta^m}-\widehat\delta$ equivariant,
we can regard $\Phi$ as an isomorphism
\[
\xymatrix{
A\rtimes_{\delta} G\rtimes_{\widehat\delta} G \ar[r]^-\Phi_-\cong
&A^m\otimes\mathcal K(L^2(G)).
}
\]
We have a surjection
\[
\psi\otimes\text{\textup{id}}:A^m\otimes\mathcal K(L^2(G))\to A\otimes\mathcal K(L^2(G)),
\]
whose kernel is $(\ker\psi)\otimes\mathcal K(L^2(G))$ since $\mathcal K(L^2(G))$ is nuclear.
Let $K_\delta$ be the inverse image under $\Phi$ of this kernel, giving an ideal of $A\rtimes_{\delta} G\rtimes_{\widehat\delta} G$ and an isomorphism $\Phi_\delta$ making the diagram
\begin{equation}\label{Q}
\xymatrix@C+30pt{
A\rtimes_\delta G\rtimes_{\widehat\delta} G
\ar[r]^-\Phi_-\cong \ar[d]_Q
&A^m\otimes \mathcal K(L^2(G))
\ar[d]^{\psi\otimes\text{\textup{id}}}
\\
(A\rtimes_\delta G\rtimes_{\widehat\delta} G)/K_\delta
\ar[r]_-{\Phi_\delta}^-\cong
&A\otimes \mathcal K(L^2(G))
}
\end{equation}
commute, where $Q$ is the quotient map.
Adapting the techniques of \cite[Theorem~3.7]{eq:full}\footnote{This is a convenient place to correct a slip in the last paragraph of the proof of \cite[Theorem~3.7]{eq:full}: ``contains'' should be replaced by ``is contained in'' (both times).},
it is not hard to see that $K_\delta$ is contained in the kernel of the regular representation
$\Lambda:A\rtimes_\delta G\rtimes_{\widehat\delta} G\to A\rtimes_\delta G\rtimes_{\widehat\delta,r} G$.
If $\delta$ is maximal, then
diagram~\ref{Q} collapses to a single row.
On the other hand, if $\delta$ is normal, then $Q$ is the regular representation $\Lambda$ and in particular
\[
(A\rtimes_\delta G\rtimes_{\widehat\delta} G)/K_\delta=A\rtimes_\delta G\rtimes_{\widehat\delta,r} G.
\]
(In this case the isomorphism $\Phi_\delta$ is ``reduced-crossed-product duality''.)
With the ultimate goal
(which at this time remains elusive
--- see Conjectures~\ref{E-coaction} and \ref{E dual})
of achieving an ``$E$-crossed-product duality'',
intermediate between full- and reduced-crossed-product dualities,
below we will propose tentative definitions of
``$E$-crossed-product duality''
and ``$E$-crossed products'' $B\rtimes_{\alpha,E} G$ by actions $\alpha:G\to \aut B$,
and we will prove that they have the following properties:
\begin{enumerate}
\item a coaction satisfies $B(G)$--crossed-product duality\ if and only if it is maximal.
\item a coaction satisfies $B_r(G)$--crossed-product duality\ if and only if it is normal.
\item $B\rtimes_{\alpha,B(G)} G=B\rtimes_\alpha G$.
\item $B\rtimes_{\alpha,B_r(G)} G=B\rtimes_{\alpha,r} G$.
\item The dual coaction $\hat\alpha$ on the full crossed product $B\rtimes_\alpha G$
satisfies $B(G)$-crossed-product duality.
\item The dual coaction $\hat\alpha^n$ on the reduced crossed product $B\rtimes_{\alpha,r} G$
satisfies $B_r(G)$-crossed-product duality.
\item In general, $B\rtimes_{\alpha,E} G$ is a quotient of $B\rtimes_\alpha G$ by an ideal contained in the kernel of the regular representation
\[
\Lambda:B\rtimes_\alpha G\to B\rtimes_{\alpha,r} G.
\]
\item There is a dual coaction $\hat\alpha_E$ of $G$ on $B\times_{\alpha,E} G$.
\end{enumerate}
\begin{defn}\label{J_E}
Define an ideal $J_{\alpha,E}$ of the crossed product $B\rtimes_\alpha G$ by
\[
J_{\alpha,E}=\ker \bar{\text{\textup{id}}\otimes q}\circ \hat\alpha,
\]
and define the \emph{$E$-crossed product} by
\[
B\rtimes_{\alpha,E} G=(B\rtimes_\alpha G)/J_{\alpha,E}.
\]
\end{defn}
Note that the above properties (1)--(7) are obviously satisfied (because $\hat\alpha$ is maximal and $\hat\alpha^n$ is normal), and we now verify that (8) holds as well:
\begin{thm}\label{dual}
Let $E$ be a nonzero weak*-closed $G$-invariant ideal of $B(G)$, and
let $Q:B\rtimes_\alpha G\to B\rtimes_{\alpha,E} G$ be the quotient map.
Then there is a coaction $\hat\alpha_E$
making the diagram
\[
\xymatrix{
B\rtimes_\alpha G \ar[r]^-{\hat\alpha} \ar[d]_{Q}
&M((B\rtimes_\alpha G)\otimes C^*(G)) \ar[d]^{\bar{Q\otimes\text{\textup{id}}}}
\\
B\rtimes_{\alpha,E} G \ar[r]_-{\hat\alpha_E}
&M((B\rtimes_{\alpha,E} G)\otimes C^*(G))
}
\]
commute.
\end{thm}
\begin{proof}
By \lemref{coaction}, we must show that
\[
J_{\alpha,E}\subset \ker \bar{Q\otimes\text{\textup{id}}}\circ \hat\alpha.
\]
Let $a\in J_{\alpha,E}$, $\omega\in (B\rtimes_{\alpha,E} G)^*$, and $g\in B(G)$.
Then
\begin{align*}
\bar{\omega\otimes g}\circ \bar{Q\otimes\text{\textup{id}}}\circ\hat\alpha(a)
&=\bar{Q^*\omega\otimes g}\circ\hat\alpha(a)
\\&=Q^*\omega\circ \bar{\text{\textup{id}}\otimes g}\circ\hat\alpha(a)
\\&=Q^*\omega(g\cdot a).
\end{align*}
Now, since $Q^*\omega\in J_{\alpha,E}^\perp$, it suffices to show that $g\cdot a\in J_{\alpha,E}$.
For $h\in E$ we have
\[
h\cdot (g\cdot a)=(hg)\cdot a=(gh)\cdot a=g\cdot (h\cdot a)=0,
\]
because $h\cdot a=0$ by \lemref{kill} below.
\end{proof}
\begin{lem}\label{kill}
With the above notation, we have:
\begin{enumerate}
\item $J_{\alpha,E}=\{a\in B\rtimes_\alpha G:E\cdot a=\{0\}\}$, and
\item $J_{\alpha,E}^\perp=\clspn\{(B\rtimes_\alpha G)^*\cdot E\}$,
where the closure is in the weak*-topology.
\end{enumerate}
\end{lem}
\begin{proof}
(1)
For $a\in B\rtimes_\alpha G$, we have
\begin{align*}
&a\in J_{\alpha,E}
\\&\quad\Leftrightarrow \bar{\text{\textup{id}}\otimes q}\circ\hat\alpha(a)=0
\\&\quad\Leftrightarrow \bar{\omega\otimes h}\circ\bar{\text{\textup{id}}\otimes q}\circ\hat\alpha(a)=0
\\&\hspace{1in}\text{for all $\omega\in (B\rtimes_{\alpha,E} G)^*$ and $h\in C^*_E(G)^*$}
\\&\quad\Leftrightarrow \bar{\omega\otimes q^*h}\circ\hat\alpha(a)=0
\\&\hspace{1in}\text{for all $\omega\in (B\rtimes_{\alpha,E} G)^*$ and $h\in C^*_E(G)^*$}
\\&\quad\Leftrightarrow \bar{\omega\otimes g}\circ\hat\alpha(a)=0
\\&\hspace{1in}\text{for all $\omega\in (B\rtimes_{\alpha,E} G)^*$ and $g\in E$}
\\&\quad\Leftrightarrow \bar\omega\circ\bar{\text{\textup{id}}\otimes g}\circ\hat\alpha(a)=0
\\&\hspace{1in}\text{for all $\omega\in (B\rtimes_{\alpha,E} G)^*$ and $g\in E$}
\\&\quad\Leftrightarrow \omega(g\cdot a)=0
\quad\text{for all $\omega\in (B\rtimes_{\alpha,E} G)^*$ and $g\in E$}
\\&\quad\Leftrightarrow g\cdot a=0
\quad\text{for all $g\in E$.}
\end{align*}
(2)
If $a\in J_{\alpha,E}$, $\omega\in (B\rtimes_\alpha G)^*$, and $f\in E$,
\[
(\omega\cdot f)(a)=\omega(f\cdot a)=0,
\]
so $\omega\cdot f\in J_{\alpha,E}^\perp$, and hence the left-hand side contains the right.
For the opposite containment, it suffices to show that
\[
J_{\alpha,E}\supset {}^\perp\bigl((B\rtimes_\alpha G)^*\cdot E\bigr).
\]
If $a\in {}^\perp((B\rtimes_\alpha G)^*\cdot E)$, then for all $\omega\in (B\rtimes_\alpha G)^*$ and $f\in E$ we have
\begin{align*}
0
&=(\omega\cdot f)(a)
=\omega(f\cdot a),
\end{align*}
so $f\cdot a=0$, and therefore $a\in J_{\alpha,E}$.
\end{proof}
\begin{rem}
We could define a covariant representation $(\pi,U)$ of the action $(B,\alpha)$ to be an \emph{$E$-representation} if the representation $U$ of $G$ is an $E$-representation,
and we could define an ideal $\tilde J_{\alpha,E}$ of $B\rtimes_\alpha G$ by
\begin{equation}\label{tilde ideal}
\tilde J_{\alpha,E}=\{a:\pi\times U(a)=0\text{ for every $E$-representation $(\pi,U)$}\},
\end{equation}
similarly to what is done in \cite[Definition~5.2]{BrownGuentner}.
It follows from \corref{E rep} that $(\pi,U)$ is an $E$-representation in the above sense if and only if
\[
\bar\omega\circ U\in E\quad\text{for all $\omega\in \bigl(\pi\times U(B\rtimes_\alpha G)\bigr)^*$,}
\]
where $i_G:C^*(G)\to M(B\rtimes_\alpha G)$ is the canonical nondegenerate homomorphism,
and consequently
\[
\tilde J_{\alpha,E}^\perp=\{\omega\in (B\rtimes_\alpha G)^*:\bar\omega\circ i_G\in E\}.
\]
In the following lemma we show one
containment that always holds between \eqref{tilde ideal} and the ideal of \defnref{J_E},
after which we explain why
these ideals do \emph{not} coincide in general.
\end{rem}
\begin{lem}\label{ideals}
With the above notation, we have
\[
\tilde J_{\alpha,E}\subset J_{\alpha,E}.
\]
\end{lem}
\begin{proof}
If $\omega\in (B\rtimes_\alpha G)^*$ and $f\in E$, then
\begin{align*}
\bar{\omega\cdot f}\circ i_G
&=\bar{\omega\otimes f}\circ\bar{\hat\alpha}\circ i_G
\\&=\bar{\omega\otimes f}\circ\bar{i_G\otimes\text{\textup{id}}}\circ \delta_G
\\&=\bar{\bar\omega\circ i_G\otimes f}\circ \delta_G
\\&=\bigl(\bar\omega\circ i_G\bigr)f,
\end{align*}
which is in $E$ because $f\in E$ and $E$ is an ideal of $B(G)$.
Thus $\omega\cdot f\in \tilde J_{\alpha,E}^\perp$.
\end{proof}
\begin{ex}
To see that the inclusion of \lemref{ideals} can be proper,
consider the extreme case $E=B_r(G)$, so that $B\rtimes_{\alpha,E} G=B\rtimes_{\alpha,r} G$.
In this case $J_{\alpha,E}$ is the kernel of the regular representation $\Lambda:B\rtimes_\alpha G\to B\rtimes_{\alpha,r} G$.
On the other hand, $\tilde J_{\alpha,E}$ comprises the elements that are killed by every representation $\pi\times U$ for which $U$ is weakly contained in the regular representation $\lambda$ of $G$.
\cite[Example~5.3]{QS} gives an example of an action $(B,\alpha)$
having a covariant representation $(\pi,U)$ for which $U$ is weakly contained in $\lambda$ but $\pi\times U$ is not weakly contained in $\Lambda$.
Thus $\ker \pi\times U$ contains $\tilde J_{\alpha,E}$ and $J_{\alpha,E}$ has an element not contained in $\ker \pi\times U$, so $\tilde J_{\alpha,E}$ is properly contained in $J_{\alpha,E}$ in this case.
\end{ex}
\begin{defn}
We say that $G$ is \emph{$E$-amenable} if there are positive definite functions $h_n$ in $E$ such that $h_n\to 1$ uniformly on compact sets.
\end{defn}
\begin{lem}
If $G$ is $E$-amenable and $(A,G,\alpha)$ is an action, then $J_{\alpha,E}=\{0\}$,
so
\[
A\rtimes_\alpha G\cong A\rtimes_{\alpha,E} G.
\]
\end{lem}
\begin{proof}
By \lemref{kill}, we have $h_n\cdot a=0$ for all $a\in J_{\alpha,E}$.
Since $h_n\to 1$ uniformly on compact sets,
it follows that $h_n\cdot a\to a$ in norm.
To see this, note that since the $h_n$ are positive definite and $h_n\to 1$, the sequence $\{h_n\}$ is bounded in $B(G)$, and certainly for $f\in C_c(G)$ we have
\[
h_n\cdot \bigl(fa\bigr)=(h_n f)a\to fa
\]
in norm, because the pointwise products $h_n f$ converge to $f$ uniformly and hence in the inductive limit topology since $\supp f$ is compact.
Therefore $J_{\alpha,E}=\{0\}$.
\end{proof}
\begin{rem}\label{amenable}
In \cite[Section~5]{BrownGuentner}, Brown and Guentner study actions of a discrete group $G$ on a unital abelian $C^*$-algebra $C(X)$,
and introduce the concept of a $D$-amenable action, where $D$ is a $G$-invariant ideal of $\ell^\infty(G)$.
In particular, if $G$ is $D$-amenable then every action of $G$ is $D$-amenable.
They show that if the action is $D$-amenable then
$\tilde J_{\alpha,E}=\{0\}$,
i.e.,
\[
C^*_D(X\rtimes G)\cong C(X)\rtimes_\alpha G.
\]
Here we have used the notation of \cite{BrownGuentner}:
$C^*_D(X\rtimes G)$ denotes the quotient of the crossed product $C(X)\rtimes_\alpha G$ by the ideal $\tilde J_{\alpha,E}$ (although Brown and Guentner give a different, albeit equivalent, definition).
\begin{q}
With the above notation,
form a weak*-closed $G$-invariant ideal $E$ of $B(G)$ by taking the weak*-closure of $D\cap B(G)$.
Then is the stronger statement $J_{\alpha,E}=\{0\}$ true?
(One easily checks it for $E=B_r(G)$,
and it is trivial for $E=B(G)$.)
\end{q}
Note that the techniques of \cite{BrownGuentner} rely heavily on the fact that they are using ideals of $\ell^\infty(G)$, whereas our methods require ideals of $B(G)$.
\end{rem}
\begin{defn}
A coaction $(A,\delta)$ \emph{satisfies $E$-\duality} if
\[
K_\delta=J_{\widehat\delta,E},
\]
where $K_\delta$ is the ideal from \eqref{Q} and $J_{\widehat\delta,E}$ is the ideal associated to the dual action $\widehat\delta$ in \defnref{J_E}.
\end{defn}
Thus $(A,\delta)$ satisfies $E$-\duality\ precisely when we have an isomorphism
$\Phi_E$ making the diagram
\[
\xymatrix{
A\rtimes_\delta G\rtimes_{\widehat\delta} G \ar[r]^-\Phi \ar[d]_Q
&A\otimes \mathcal K(L^2(G))
\\
A\rtimes_\delta G\rtimes_{\widehat\delta,E} G \ar[ur]_{\Phi_E}^\cong
}
\]
commute,
where $Q$ is the quotient map.
\begin{conj}\label{E-coaction}
Every coaction satisfies $E$-\duality\ for some $E$.
\end{conj}
\begin{obs}\label{trivial}
If $E$ is an ideal of $B(G)$, then every
group $C^*$-algebra\ $C^*_E(G)$ is an $E$-crossed product:
\[
C^*_E(G)=\mathbb C\rtimes_{\iota,E} G,
\]
where $\iota$ is the trivial action of $G$ on $\mathbb C$,
because the kernel of the quotient map $C^*(G)\to C^*_E(G)$ is ${}^\perp E$.
This generalizes the extreme cases
\begin{enumerate}
\item $C^*(G)=\mathbb C\rtimes_\iota G$;
\item $C^*_r(G)=\mathbb C\rtimes_{\iota,r} G$.
\end{enumerate}
\end{obs}
\begin{conj}\label{E dual}
If $(B,\alpha)$ is an action, then the dual coaction $\hat\alpha_E$ on the $E$-crossed product $B\rtimes_{\alpha,E} G$ satisfies $E$-\duality.
\end{conj}
\begin{rem}
In particular, by \obsref{trivial},
\conjref{E dual} would imply as a special case
that the canonical coaction $\delta_G^E$ on the group algebra $C^*_E(G)$ satisfies $E$-\duality.
\end{rem}
For our final result, we
only require that $E$ be
a weak*-closed $G$-invariant subalgebra of $B(G)$ (but not necessarily an ideal).
By \propref{comultiplication}, $C^*_E(G)$ carries a comultiplication $\Delta$ that is a quotient of the canonical comultiplication $\delta_G$ on $C^*(G)$.
Techniques similar to those used in the proof of \thmref{dual},
taking $g\in E$ rather than $g\in B(G)$,
can be used to show:
\begin{prop}\label{coaction bialgebra}
Let $E$ be a weak*-closed $G$-invariant subalgebra of $B(G)$,
and let $(B,\alpha)$ be an action.
Then there is a coaction $\Delta_\alpha$ of the $C^*$-bialgebra $C^*_E(G)$ making the diagram
\[
\xymatrix{
B\rtimes_\alpha G \ar[r]^-{\hat\alpha} \ar[d]_{Q}
&M((B\rtimes_\alpha G)\otimes C^*(G)) \ar[d]^{\bar{Q\otimes q}}
\\
B\rtimes_{\alpha,E} G \ar[r]_-{\Delta_\alpha}
&M((B\rtimes_{\alpha,E} G)\otimes C^*_E(G))
}
\]
commute, where we use notation from \thmref{dual}.
\end{prop}
We close with a rather vague query:
\begin{q}
What are the relationships among $E$-crossed products, $E$-coactions, and coactions of the $C^*$-bialgebra $C^*_E(G)$?
\end{q}
We hope to investigate this question, together with Conjectures~\ref{E-coaction} and \ref{E dual}, in future research.
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|
1,314,259,995,841 | arxiv | \subsection{Prime Denominators}
The equation~$B_P=p$ (in other words~$f=1$) occupies something of
a middle ground between Theorems \ref{genuinetheorem} and
\ref{primepower}. The possibilities do seem to be more subtle. In
\cite{chuds}, \cite{eds}, \cite{primeds} and \cite{pe}, this
apparently more difficult question has been considered. In
\cite{primeds} and \cite{pe} we argued that, for rank-1
subgroups~$G$ (in other words, multiples of a single point) only
finitely many points $P\in G$ should yield prime values~$B_P$. In
certain cases, we could prove this and we conjectured that, for a
rank-1 subgroup~$G$, the number of rational points~$P\in G$ as in
(\ref{genericrationalpoint}), with~$B_P$ a prime is uniformly
bounded if the defining equation (\ref{weierstrassequation}) is in
minimal form. Many calculations suggest that for an elliptic curve
in minimal form, a little over $30$ prime values $B_P$ seems to be
the limit. The following example may well provide the limiting case.
It is taken from Elkies' table of small height points in
\cite{elkies}. Note that the curves in Elkies' table are not
presented in minimal form; the example following is the second on
the list, with the equation rendered in minimal form. The point
shown has the second smallest known height amongst all rational
points on elliptic curves.
\begin{example}Let $P$ denote the point $P=(-386,-3767)$ on the
elliptic curve
$$
y^2+xy=x^3-141875x+13893057.
$$
The values $x(nP)$ have prime square denominators for~$31$ values
of~$n$, ending with (apparently)~$n=613$. The first example on
Elkies' list yields (apparently)~$30$ such primes. These computations
were performed using PARI-GP
\end{example}
In a short subsection (\S\ref{uniformity}) we will give an outline of a
proof of uniformity assuming an unproven (yet generally believed
conjecture) as well as an improvement to known results in elliptic
transcendence theory.
On the other hand, in \cite{rew}, many examples of rank-2 curves
were considered where, apparently, there are infinitely many
rational points~$P$ with~$B_P$ prime.
\begin{example}\label{with}The curve
$$y^2=x^3-28x+52
$$
has rank 2, with generators $P_1=(-2,10)$ and $P_2=(-4,10)$. In
\cite{rew} we considered the possibility that asymptotically $\rho
\log T$ values $$x(n_1P_1+n_2P_2) \mbox{ with } \max
\{|n_1|,|n_2|\}<T$$ possess a prime square denominator, where
$\rho
>0$ is a constant depending upon $E$.
The table in \cite{2deds}, constructed by Peter Rogers, exhibits
many more rank-2 curves which are thought to yield infinitely many
rational points~$P$ as in (\ref{genericrationalpoint}) with~$B_P$
a prime.
\end{example}
Theorem~\ref{primepower} allows many examples which yield only
finitely many primes to be constructed, when a descent is possible.
\begin{example}\label{morewithout}The curve
$$
y^2=x^3-6400x+192000
$$
has rank 2, with independent points $P_1=(65,225)$ and
$P_2=(56,96)$. Theorem \ref{primepower} guarantees there are
finitely many prime square denominators amongst the points
$n_1P_1+n_2P_2$ because the points shown are the images of the
points $(0,100)$ and $(6,104)$ under a 2-isogeny from the curve
$$
y^2 = x^3+100x+10000.
$$
\end{example}
In Section \ref{primepowers} we give a proof of Theorem
\ref{genuinetheorem}: since it relies on Faltings' Theorem, it is
not effective. Following this, we prove Theorem \ref{primepower};
the effectiveness statement uses local heights and elliptic
transcendence theory whilst the explicit bound for the exceptional
points relies upon a strong version of Siegel's Theorem
\cite{grosssilverman}, \cite{silvermansiegel}. The final section
proves Theorem~\ref{eveneasierfiniteness}, using a strong form of
Thue's Theorem to rid us of the dependence upon the rank.
\section{Proof of Theorem \ref{genuinetheorem}}\label{primepowers}
Let $K$ be a finite field extension of $\mathbb{Q}$. The
valuations on $K$, written~$M_K$, consist of the usual archimedean
absolute values together with the non-archimedean, $\wp$-adic
valuations, one for each prime ideal $\wp$ of~$K$. Write $K_v$ for
the completion of $K$ with respect to $v$. Let $S \subset M_K$
denote a finite set of valuations containing the archimedean
valuations. The ring $O_S$ of $S$-integers is given by
\[
O_S = \{x\in K : \nu(x) \ge 0 \textrm{ for all } \nu \in M_K, \nu
\notin S \}
\]
and the unit group $O_S^*$ of $O_S$ is given by
\[
O_S^* = \{x\in K : \nu(x) = 0 \textrm{ for all } \nu \in M_K, \nu
\notin S \}.
\]
Completing the square in (\ref{weierstrassequation}), it is
sufficient to consider an equation
\begin{equation} \label{nop}
y^2=x^3+a_2x^2+a_4x+a_6
\end{equation}
where $x^3+a_2x^2+a_4x+a_6 \in \mathbb{Q}[x]$ has distinct zeros
$\alpha_1, \alpha_2, \alpha_3$ in some finite extension $K$ of
$\mathbb Q$. Of course we might have introduced powers of $2$ into the
denominators in \eqref{nop}; we will see that this does not matter.
Let $S$ be a sufficiently large (finite) subset of $M_K$ so that $O_S$
is a \PID and $2,\alpha_i-\alpha_j \in O_S^*$ for all $i \ne j$.
Now let $L/K$ be the extension of $K$ obtained by adjoining to $K$
the square root of every element of $O_S^*$. Note that $L/K$ is a
finite extension, since $O_S^*/(O_S^*)^2$ is finite from
Dirichlet's $S$-unit theorem. Further let $T \subset M_L$ be a
finite set containing the places of $L$ lying over elements of
$S$ and such that $O_T$ is a \PID, where, by abuse of notation, $O_T$
denotes the ring of $T$-integers in $L$.
Now we turn to the proof of the Theorem. Replacing $(x,y)$ by
$(x/q^2,y/q^3)$ in \eqref{nop}, we are searching for
solutions in $\Q\cap O_S$ to
\begin{equation} \label{p}
y^2=x^3+a_2q^2x^2+a_4q^4x+a_6q^6,\qquad \gcd(xy,q)=1.
\end{equation}
We will show that, for fixed $f>1$, there are only finitely many
prime powers $q=p^f\in \mathbb{Z}$ for which \eqref{p} has a solution. Note
that, since $f$ is fixed and $T$ is finite, we may assume that $p$ is
large enough so that no valuation of $L$ dividing $p$ lies in $T$.
Let $x,y\in \mathbb Q\cap O_S$ be a
solution to \eqref{p}; then
\begin{equation} \label{sq}
y^2=(x-q^2\alpha_1)(x-q^2\alpha_2)(x-q^2\alpha_3).
\end{equation}
Let $\wp$ be a
prime ideal of $O_S$ dividing $x-q^2\alpha_i$; then $\wp$ cannot
divide $q$, since $(x,q)=1$. Hence $\wp$ can divide at most one term
$x-q^2\alpha_i$, since if it divides both $x-\alpha_iq^2$ and
$x-\alpha_jq^2$ then it divides also $(\alpha_i-\alpha_j)q^2$. From
\eqref{sq} it follows that there are elements $z_i \in O_S$ and
units $b_i \in O_S^*$ so that
\[
x-\alpha_iq^2=b_iz_i^2.
\]
We have $b_i= \beta_i^2$, for some $\beta_i\in O_T$ so
\begin{equation} \label{3}
x-\alpha_iq^2=(\beta_iz_i)^2.
\end{equation}
Taking the difference of any two of these equations yields
\[
(\alpha_j-\alpha_i)q^2=(\beta_iz_i-\beta_jz_j)(\beta_iz_i+\beta_jz_j).
\]
Note that $\alpha_j-\alpha_i \in O_T^*$ while each of the two
factors on the right is in $O_T$. It follows that each of these
factors is made from primes $\pi | p$ in $O_T$. Further we may
assume these factors are coprime, since if $\pi| p$ divides
$2\beta_iz_i$ and $p>2$ then from (\ref{3}) $\pi$ divides $x$.
Without loss of generality, we assume that
there is a prime $\pi \in O_T$ dividing $\beta_1z_1+\beta_2z_2$.
If $\pi$ does not divide $\beta_2z_2+\beta_3z_3$ then $\pi$ does
not divide $\beta_1z_1-\beta_3z_3$. So Siegel's identity
\[
\frac{\beta_1z_1+\beta_2z_2}{\beta_1z_1-\beta_3z_3}-
\frac{\beta_2z_2+\beta_3z_3}{\beta_1z_1-\beta_3z_3}=1
\]
gives
\begin{equation} \label{4}
a_p^2u+b_p^2v=c_p^2
\end{equation}
where $a_p,b_p,c_p \in O_T$ divide $q$, are not all $T$-units and
are pairwise coprime. If $\pi$ divides $\beta_2z_2+\beta_3z_3$
then $\pi$ divides $\beta_1z_1-\beta_3z_3$ so $\pi$ does not
divide $\beta_1z_1+\beta_3z_3$. Then Siegel's identity
\[
\frac{\beta_2z_2+\beta_1z_1}{\beta_2z_2-\beta_3z_3}-\frac{\beta_1z_1+\beta_3z_3}{\beta_2z_2-\beta_3z_3}=1
\]
gives (\ref{4}).
Since $q=p^f$ is a prime power with~$f>1$, each of $a_p$ and
$b_p$ is itself an $f^{\rm th}$ power. Finally, we note that the
group~$O_T^*/(O_T^*)^{2f}$ is finite so we fix once and for
all a set of coset representatives. Then, by \eqref{4}, each solution
to \eqref{p} gives us a solution of an equation:
$$
ux^{2f}+vy^{2f}=1, \qquad x,y\in L,
$$
with $2f \ge 4$, where~$u$ and~$v$ belong to this finite set of
representatives, which
depends only upon~$T$ and~$f$. Since each such curve has genus
$(2f-1)(f-1)\ge 3$, Faltings' Theorem \cite{ft} guarantees there are
only finitely many solutions. Since we have only finitely many such
equations, we are done.
\hfill$\square$
\section{Proof of Theorem \ref{primepower}}
Let $E$ and $E'$ be two elliptic curves, defined over
$\mathbb Q$. An {\em isogeny} is a non-zero homomorphism
\begin{equation*}
\sigma :E'\rightarrow E
\end{equation*}
taking the zero of $E'$ to the zero of $E$. There is a dual
isogeny $\sigma^*:E\rightarrow E'$ and the composite homomorphisms
$\sigma\sigma^*$ and $\sigma^*\sigma$ are multiplication by $d$ on $E$
and $E'$ respectively, for some integer $d$, which is said to be
the \textit{degree} of the isogeny. The curves $E$ and $E^{\prime
}$ are said to be {\em $d$-isogenous} if there is an isogeny of
degree $d$ between them.
\begin{definition}\label{defofS}For any
point $P\in E(\mathbb Q)$ let $S(P)$ denote the non-archimedean
valuations $v$ in $M_{\mathbb Q}$ for which $|x(P)|_v>1$. The
definition of $S(P)$ depends upon the Weierstrass equation so we
assume this equation has been fixed.
\end{definition}
\begin{theorem}\label{finite} Let $E$ denote an elliptic curve
which is defined over $\mathbb Q$ and let $G \subset E(\mathbb Q)$
denote a subset contained in the image of a subset~$G'$ of
rational points under a non-trivial isogeny. Then there are only finitely
many $P \in G$ for which $S(P)$ consists of a single valuation:
these points are effectively computable and they are at most
$c^{(r_{G}+1)\omega(\Delta_{E})}$ in number.
\end{theorem}
Theorem \ref{finite} will be proved in Section \ref{proof}
following a section with some basic properties of heights under
isogeny.
\subsection{Heights}\label{hival}
Write
\begin{equation}\label{localheight}
h_v(\alpha)=\log \max \{ 1, |\alpha|_v\},
\end{equation}
for the local (logarithmic) height at $v$. The na\" ive global
logarithmic height of $Q$ is defined to be
$$h(\alpha)=\sum_{v\in M_{\mathbb Q}} h_v(\alpha)=
\sum_{v\in M_{\mathbb Q}} \log \max \{ 1, |\alpha|_v\},
$$
the sum running over all the valuations of $\mathbb Q$. If $P$
denotes any rational point on an elliptic curve, we write
$h_v(P)=h_v(x(P))$ and $h(P)=h(x(P))$. Usually, in the
literature, the global height is further normalized by dividing by
$2$.
Suppose $P$ denotes a rational point of $E$. The theory of heights
gives an estimate for
$$
h(P)=\widehat h(P)+O(1),
$$
where $\widehat h(P)$ denotes the canonical height of $P$. The
canonical height enjoys the additional property that $\widehat
h(mP)=m^2\widehat h(P)$ for any $m\in \mathbb Z$. More generally,
if $\sigma:E'\rightarrow E$ denotes a $d$-isogeny then for $P'\in
E'(\mathbb Q)$
\begin{equation}\label{goingup}\widehat h(\sigma(P'))=d\widehat h(P').
\end{equation}
\begin{lemma}\label{growth}
Suppose $P_1,\dots, P_r \in E(\mathbb Q)$ are independent
rational points and $T$ is a torsion point. Then
$$
h_v(n_1P_1+\dots +n_rP_r+T)=
O(\log |\underline n| (\log \log |\underline n|)^{r+2}),
$$
for any valuation $v$\ where $|\underline n|=\max \{|n_1|,\dots
,|n_r|\}$ for $\underline n \in \mathbb Z^r$. This can
be written
\begin{equation}\label{growthagain}
\log |x(P)|_v =O
\log \widehat h(P) (\log \log \widehat
h(P))^{r+2}),
\end{equation}
for any $P\in E(\mathbb Q)$.
\end{lemma}
\begin{corollary}\label{fixedset}
With the same notation as Lemma~\ref{growth}, let
$S$ be any fixed, finite set of valuations of
$\mathbb Q$. Then
$$
\sum_{v\in S}h_v(n_1P_1+\dots +n_rP_r+T)=O((\log |\underline n|)^2).
$$
This can be written
\begin{equation}\label{finitegrowth}
\sum_{v\in S}h_v(P)=O((\log \widehat h(P))^2),
\end{equation}
for any $P\in E(\mathbb Q)$.
\end{corollary}
\begin{proof}[Proof of Lemma~\ref{growth}.]
We will detail a proof for the archimedean valuation only. The
proof for non-archimedean valuations is similar. The estimate in
Lemma~\ref{growth} follows from an appropriate upper bound for
$|x(n_1P_1+\dots +n_rP_r)|_v$. Putting the given model of the
curve into a short Weierstrass equation only translates $x$ by at
most a constant. Let $z_{P_i}$ correspond to $P_i$ under an
analytic isomorphism $E(\mathbb C)\simeq \mathbb C/L$, for some
lattice $L$, with~$z_T$ corresponding to~$T$. Thus we may assume
that the $x$-coordinate of a point is given using the Weierstrass
$\wp$-function with Laurent expansion in even powers of $z$,
$$x=\wp_L(z)=\frac{1}{z^2}+c_0+c_2z^2+\dots
$$
Write $\{n_1z_{P_1}+\dots +n_rz_{P_r}+z_T\}$ for $n_1z_{P_1}+\dots
+n_rz_{P_r}+z_T$ modulo $L$. When the quantity $|x(n_1P_1+\dots
+n_rP_r+T)|$ is large it means $n_1P_1+\dots +n_rP_r+T$ is close
to zero modulo $L$, thus the quantities $|x(n_1P_1+\dots
+n_rP_r+T)|$ and $1/|\{n_1z_{P_1}+\dots +n_rz_{P_r}+z_T\}|^2$ are
commensurate. On the complex torus, this means the elliptic
logarithm is close to zero. So it is sufficient to supply a lower
bound for $|\{n_1z_{P_1}+\dots +n_rz_{P_r}+z_T\}|$ and this can be
given by elliptic transcendence theory (see \cite{dav}). We use
Th\'eor\`eme 2.1 in~\cite{dav} but see also~\cite{ST} where an
explicit version of David's Theorem appears on page 20. The nature
of the bound is
\begin{equation}\label{b2}
\log |x(n_1P_1+\dots +n_rP_r+T)| \ll \log |\underline n| (\log
\log |\underline n|)^{r+2} ,
\end{equation}
where the implied constant depends upon $E$, the valuation $v$ and
the points $P_1,\dots ,P_r$.
For the final assertion, we need only note that the global canonical height
$\hat h(n_1P_1+\dots +n_rP_r+T)$ is a positive definite quadratic form
in $n_1,...,n_r$, and hence comensurate with $|\underline n|^2$.
\end{proof}
We will need some more theory of elliptic curves over local
fields, see~\cite{MR87g:11070}. For every non-archimedean
valuation $v$, write ord$_v$ for the corresponding order function.
There is a subgroup of the group of $\mathbb Q_{v}$-rational
points:
$$
E_1(\mathbb Q_{v})=\{O\} \cup \{P\in E(\mathbb Q_{v}):\mbox{ord} _{v}(x(P))
\leq -2\}.
$$
In \cite{MR87g:11070}, Silverman proves the
following.
\begin{proposition}For all $P\in E_1(\mathbb Q_{v})$ and all $d\in \mathbb Z$:
\begin{equation}\label{ordthing}
\log|x(mP)|_v=\log |x(P)|_v-\log |m|_v.
\end{equation}
\end{proposition}
For finitely many (bad) primes $p$, the reduction of $E$ is not an elliptic
curve because the reduced curve is singular. We write $S_E$ for the
set of valuations in $M_\Q$ corresponding to all such primes. The
equation~\eqref{ordthing} then yields the following corollary:
\begin{corollary}\label{bigOthing} Suppose $\sigma:E'\rightarrow E$
is a $d$-isogeny and $P'\in E'_1(\mathbb Q_v)$. If $v \notin \SE$
then $h_v(\sigma(P'))\ge h_v(P')$. The local heights are related
by the formula
\begin{equation}\label{newordthing}
h_v(\sigma(P'))=h_v(P')+O(1),
\end{equation}
where the implied constant depends only upon the isogeny and is
independent of $P'$.
\end{corollary}
\begin{proof}
Suppose $v$ corresponds to the prime $p$. Provided $v \notin \SE$,
both curves and the isogeny can be reduced modulo powers of $p$
and the first statement in the Corollary follows. Applying the
dual isogeny $\sigma^*$ gives a similar inequality $h_v(\sigma^*(P))\ge
h_v(P)$, for all $P\in E_1(\mathbb Q_v)$. However, composing
$\sigma$ with its dual gives multiplication by $d$ on $E'$. Now
(\ref{ordthing}) applies to prove (\ref{newordthing}).
\end{proof}
\subsection{Proof of Theorem \ref{finite}}\label{proof}
\hfill\break
Suppose $G'$ is a subset of $E'(\mathbb Q)$ and $\sigma:E'\rightarrow
E$ is a $d$-isogeny with $\sigma(G')=G$. From
(\ref{goingup}),
\begin{equation}\label{compareheights}
\widehat h(P)=d\widehat h(P'),
\end{equation}
where $\widehat h$ denotes the canonical heights of rational
points on~$E$ and~$E'$.
Suppose $S(P)$ consists of the single valuation $v$.
If $v\in\SE$ then, by~\eqref{finitegrowth}, we have
$$
\widehat h(P')=O((\log \widehat h(P'))^2).
$$
If $v\not\in\SE$ then $P\in E_1(\Q_v)$ and, by reduction,
$S(P')\subset S(P)$. Again by~\eqref{finitegrowth},
\begin{equation}\label{heightdifference}
h(P)-h_v(P)=O((\log \widehat h(P'))^2)=h(P')-h_v(P').
\end{equation}
Now the canonical height differs from the na\"ive height by a bounded
amount so we are justified in using the canonical height
in~\eqref{heightdifference}. It follows from~\eqref{compareheights}
that
$$
d\widehat h(P')-h_v(P)=O((\log \widehat h(P'))^2).
$$
However, from~\eqref{newordthing}
$$
h_v(P')-h_v(P)=O(1).
$$
Subtracting these last two formulae, and dividing by $d-1>0$, gives
\begin{equation}\label{heightlessthanlogheight}
\widehat h(P')=O((\log \widehat h(P'))^2),
\end{equation}
with an implied constant which is effectively computable. In
particular, we have obtained the same inequality for any valuation
$v$. Equation~\eqref{heightlessthanlogheight} bounds the height
$\widehat h(P')$ so can only be satisfied
by finitely many points~$P'$, which can be computed effectively.
\medskip
Finally, we estimate the number of exceptional points, using a
strong form of Siegel's Theorem, proved by Gross and Silverman.
There is a non-trivial torsion point~$T'$ in the kernel
of~$\sigma:E'\rightarrow E$. Any isogeny factorizes as a product
of isogenies of prime degree so, since the rank $r_G$ and the
discriminant $\Delta_E$ are preserved under isogeny, we may assume from
now that~$\sigma$ has prime degree. Let~$K$ denote a number field over
which~$T'$ is
defined. By Mazur's famous result (\cite[Theorem 1]{mazur}), only
finitely many primes can occur as degrees of prime degree isogenies
which map onto rational points (the largest of which is
$163$). Let~$S$ denote the subset of $M_\Q$ consisting of all such
prime degrees, together with the primes of bad reduction (in $\SE$).
Let $\SK$ denote the subset of $M_K$ consisting of all places above
those in $S$, together with the places dividing the denominator of
$T'$.
Now the points~$P'$ and~$P'+T'$ both map to~$P$ under~$\sigma$. It
follows that if ~$P'$ (respectively ~$P'+T'$) has denominator divisible
by a place of good reduction $w$ (respectively $w'$), then $w$
(respectively $w'$) is guaranteed to
appear in the denominator of~$P$. Moreover, if~$w,w'\not\in\SK$, they
are guaranteed to be distinct: indeed, any
good reduction place dividing the denominator of both
points will divide the denominator of~$T'$ and hence lie in
$\SK$. Further, since $P'$ is a $\Q$-rational
point, $w'$ is coprime to the prime of $\Q$ below $w$ so there are two
distinct primes of $\Q$ dividing~$B_P$.
This ensures~$B_P$ cannot be a prime power unless either~$P'$
or~$P'+T'$ is an~$\SK$-integral point on~$E'$. By the
theorem of Gross and Silverman \cite[Theorem 0.1]{grosssilverman},
there are at most
$$
d\eta^{r\delta (j_{E'})+|\SK|}
$$
$\SK$-integral points in~$E'(K)$, for an explicit constant~$\eta$,
where~$d=[K:\mathbb Q]$ and~$\delta (j_{E'})$ denotes the number
of primes dividing the denominator of the $j$-invariant of~$E'$.
In their paper,~$r$ denotes the rank of the group~$E(K)$; however,
their results are valid for the number of $\SK$-integral points
inside a subgroup of rank~$r$. Since the primes dividing the
denominator of~$j_{E'}$ divide~$\Delta_{E'}=\Delta_E$, since~$|\SK|$ is a
bounded multiple of~$d\omega(\Delta_{E})$, and since~$d$ is
uniformly bounded by Mazur's Theorem, the bound in
(\ref{numberofexceptionalpoints}) follows.
\hfill$\square$
\subsection{Uniformity}\label{uniformity}
Under the assumption that $G$ consists of a rank-1 subgroup
of~$E(\mathbb Q)$, we may harness the ideas in Section \ref{proof}
to explain how these might be used as part of a proof of
uniformity where a descent is possible. In the rank-1 case, the
inequality (\ref{heightlessthanlogheight}) can be stated more
explicitly. Let~$P$ denote a generator of~$G$ and~$\sigma(P')=P$.
Writing~$h'=\widehat h(P')$ and invoking the explicit form of
David's Theorem in \cite{ST} shows that the integers~$n$ for
which~$S(nP)$ consists of a single valuation must satisfy
\begin{equation}\label{beexplicit}h'n^2<\tau \log n(\log \log n+\log
\Delta_{E'})^3,
\end{equation}
where $\tau$ denotes a uniform bound. Lang's Conjecture asserts a
uniform upper bound for~$(\log \Delta_{E'})/h'$. If the dependence
upon~$\log \Delta_{E'}$ on the right hand side of
(\ref{beexplicit}) were linear, then Lang's Conjecture would
guarantee a uniform upper bound for the number of prime square
denominators in the sequence~$x(nP)$ when descent is possible.
The reason Example \ref{jonthesisexample} works is that Lang's
Conjecture can be proved in an explicit manner for 1-parameter
families such as this \cite{silvariation}. Also, it is possible to
obtain a stronger form of Lemma \ref{growth} without transcendence
theory: since~$P$ lies off the connected component of the identity
so do all its odd multiples and for these an upper bound for the
$x$-coordinate exists. Also, (\ref{ordthing}) allows an easy
treatment of the bad reduction primes by showing the local
heights~$h_v(nP)$ are each~$O(\log n)$.
\section{The curve $u^3+v^3=D$}\label{othermodels}
In \cite{primeds}, it was proved that the integers $B_P$ are prime
powers for only finitely many $\mathbb Q$-points $P$. The proof
used the well-known bi-rational equivalence of (\ref{homogform})
with the curve
$$
y^2=x^3-432D^2.
$$
{\bf Example} As Ramanujan famously pointed out, the taxi-cab
equation
\begin{equation}\label{taxicab}
u^3+v^3=1729,
\end{equation}
has two distinct integral solutions. These give rise to points
$P=[1,12]$ and $Q=[9,10]$ on the elliptic curve (\ref{taxicab}).
The only rational points on (\ref{taxicab}) which seem to yield
prime power denominators are $2Q$ and $P+Q$ (and their inverses).
\begin{proof}[Proof of Theorem \ref{eveneasierfiniteness}]
We assume that $D$ is integral; if $D$ is not integral then we can
scale by a non-zero integer to reduce to this case.
Firstly, we recall the easy proof of Siegel's Theorem for curves
in homogeneous form~(\ref{homogform}). The equation factorizes as
$$
(u + v)(u^2 - uv + v^2) = D
$$
so
$$
|u^2 - uv + v^2| \leq |D|.
$$
But the left hand side of this is $(u - v/2)^2 + 3v^2/4$ so $|v|^2
\le 4|D|/3$. Hence the result with an explicit bound for $|v|$.
The bound for $|u|$ is identical.
This bounds the number of solutions as $O(|D|^{1/2})$ but we can
easily improve
on this. First we have $u+v=m$, and $u^2-uv+v^2=n$, for some
integers $m,n$ such that $mn=D$ and $\gcd(m,n)|3$.
If $\z$ is a non-trivial cube root of unity, then we get
$$
(u+\z v)(u-\z v)= n,
$$
where the factors on the left hand side
have greatest common divisor dividing $2$. The number of ways of
factorizing $n$ in this way is $O(2^{2\omega(n)})$ so the total number
of solution $(u,v)$ is $O(\mu^{\omega(D)})$, for some (explicit)
uniform constant $\mu$.
\medskip
Now suppose
$$
(u + v)(u^2 - uv + v^2) = u^3 + v^3 = Dq^3
$$
where $q=p^f$ is a prime power and $u,v$ are integers coprime to
$q$. We consider first the case $p>3$.
Then $q$ cannot possess a factor in common with both
brackets. Hence one bracket is $m$ and the other is $nq^3$, where
$mn=D$ and $\gcd(m,n)|3$. If the quadratic bracket
is bounded then we bound the number of solutions as before, so we
assume
$$
u + v = m \mbox{ and } (u+\z v)(u-\z v) = nq^3.
$$
Since $u+\z v$ and $u-\z v$ are coprime, we get
$$
u + v = m \mbox{ and } u - \z v = k\rho^3
$$
where $k$ divides $n$ and $\rho$ divides $q$ in $\mathbb Z[\z]$. If we write $k
= c + d\z$ and $\rho = a + b\z$, with $a,b,c,d \in \mathbb Z$, then we
can write $u$ and $v$ explicitly in terms of $a,b,c,d$.
Substituting into the equation $u + v = m$, we get
$$
(c + d)a^3 + (3c - 6d)a^2b + (3d - 6c)ab^2 + (c + d)b^3 = m.
$$
For each of the finitely many values of $c$ and $d$, this is a
Thue Equation and it is non-singular -- that is, for any
non-zero $c$ and $d$, the cubic
$$(c + d)X^3 + (3c - 6d)X^2 + (3d - 6c)X + (c + d)
$$
does not have repeated roots, since its discriminant is
$$
729c^4 - 1458dc^3 + 2187d^2c^2 - 1458 d^3c + 729d^4 =729(c^2 - cd
+ d^2)^2,
$$
which does not vanish unless $c = d = 0$. Thus each of
the finitely many Thue equations has finitely many solutions so
there were only finitely many values of $q$.
By~\cite[Theorem 1]{ev} (see also \emph{op. cit.} page 122), a
non-singular integral cubic Thue Equation
$$
F(x,y)=m,
$$
with $m$ cube-free,
has a number of integral solutions which is bounded by
$\mu^{\omega(m)}$, for some (explicit) uniform constant $\mu$. This
must be multiplied by the total number of
equations, which depends only upon the number of factorizations of $mn=D$
with $\gcd(m,n)|3$, so does not change the shape of the bound claimed
by the theorem.
Finally, the cases $p=2,3$ are dealt with similarly, with minor
alterations. The point is that the greatest common divisors of the
various brackets are uniformly bounded, so the shape of the final number
of solutions is unchanged.
\end{proof}
|
1,314,259,995,842 | arxiv |
\section{Algorithmic Results}\label{sec:algo}
In this section we give an algorithm to approximate the profit of $OPT_{on}$, for any joint distributions over edge weights of each ball $t$.
\thmalg*
\paragraph{\underline{An LP Relaxation.}}
Our starting point is a linear program (LP) called LP-Match, which we show upper bounds the gain of any online algorithm for $\ensuremath{\textsc{RideHail}}\xspace$. Below, the variables we optimize over are $\{x_{i,t}\}$, which we think of as ``the probability that the online algorithm matches ball $t$ to bin $i$''. Recall that ball $t$ arrives with probability $p_t$.
\begin{align}
\textbf{LP-Match:} \hspace{2em} \qquad \max \enspace \enspace & \sum_{i, t} w_{i,t} \cdot x_{i,t} &&& \nonumber \\
\text{s.t.} \enspace \enspace
& \sum_{t} x_{i,t} \le 1 & & \text{ for all } i \label{eqn:lpatmost1}
\\
& \sum_{i} x_{i,t} \le p_{t} & & \text{ for all } t
\label{eqn:lpatmostqi}
\\
& x_{i,t} \hfill \le p_{t} \cdot \left( 1 - \sum_{t' < t} x_{i,t'} \right) & & \text{ for all } i,t
\label{eqn:lpthird}
\\
&
x_{i,t}
\ge 0 & & \text{ for all } i,t
\end{align}
Denoting by LP-Match($\mathcal{I})$ the optimal value of LP-Match on Instance $\mathcal{I}$, we have the following.
\begin{lemma}\label{LP-bound}
For any $\ensuremath{\textsc{RideHail}}\xspace$ instance $\mathcal{I}$, we have that $$\textrm{\emph{LP-Match}}(\mathcal{I})\geq OPT_{on}(\mathcal{I}).$$
\end{lemma}
The above lemma is implied by \cite{torrico2017dynamic}.
For completeness, we provide a proof of this lemma below.
\begin{proof}
Let $x^*_{i,t}$ denote the probability that $OPT_{on}$ matches bin $i$ to ball $t$. We note that $x^*$ constitutes a feasible solution for LP-Match because (i) the probability $OPT_{on}$ matches a bin $i$ is at most 1, (ii) the probability $OPT_{on}$ matches a ball $t$
is at most $p_{t}$ (the probability that $t$ arrives),
(iii) the probability $OPT_{on}$ matches a bin $i$ to a ball $t$ is at most $p_{t}$ (the probability $t$ arrives) times $ 1 - \sum_{t' < t} x_{i,t'} $ (the probability that $i$ is not matched by time $t$),\footnote{This uses the fact that arrival of $t$ is independent of the online algorithm's previous choices. Note that this constraint is not valid for the probabilities induced by an \emph{offline} algorithm, so our LP does not upper bound $OPT_{off}(\mathcal{I})$.} and (iv) these probabilities are non-negative. On the other hand, for this $x^*_{i,t}$, the objective of LP-Match is precisely the expected profit of $OPT_{on}$ on this instance, and therefore $\textrm{LP-Match}(\mathcal{I})\geq OPT_{on}(\mathcal{I})$.
\end{proof}
\subsection{The Algorithm}
Given a solution to LP-Match, whose objective upper bounds $OPT_{on}$ by \Cref{LP-bound}, a natural approach to approximate $OPT_{on}$ is to round this solution online.
By simple ``integrality gap'' examples (see \Cref{sec:LP-Match-obs}), this is impossible to do perfectly.
Instead, we show how to do so approximately, by rounding a solution to LP-Match while only incurring a $\nicefrac{1}{2}+c$ multiplicative loss in the rounding, for the constant $c := 0.01$.
For notational simplicity, assume without loss of generality that an optimal solution to LP-Match to the input instance $\mathcal{I}$ satisfies all Constraints \eqref{eqn:lpatmostqi} at equality, i.e., $\sum_i x_{i,t} = p_{t}$ for all balls $t$. This can be guaranteed by adding a dummy bin $i_{t}$ for each ball $t$ with $w_{i,t}=0$, and setting $x_{i_t,t} \leftarrow p_{t} - \sum_{i} x_{i,t}$. These dummy edges do not affect the gain of $OPT_{on}$, nor that of the online algorithm.
After computing a solution to LP-Match as above, our algorithm proceeds iteratively as follows. For each time $t$, if ball $t$ arrives, we pick a single bin $i$ with probability $x_{i,t}/p_{t}$, and if this is bin is vacant (unmatched), we match $(i,t)$ with some probability $q_{i,t}$. (We sometimes refer to this as $i$ \emph{accepts} $t$.) If this did not result in $t$ being matched, we repeat the process a second time, but this time we match $t$ to its picked bin $i$, provided $i$ is vacant, and the edges until time $t$ have nearly saturated Constraint \eqref{eqn:lpatmost1} for $i$. See \Cref{alg:propose-twice}.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\medskip
\State solve LP-Match for $\{ x_{i,t} \}$\label{line:lp-solve}
\State add dummy neighbor for each $t$ so that $\sum_{i} x_{i,t} = p_{t}$
\State $\mathcal{M} \leftarrow \emptyset$
\For{all balls $t=1,2,\dots$}
\State pick a single bin $i$ with probability $\frac{x_{i,t}}{p_{t}}$ \label{line:pick1}
\If{$i$ is unmatched in $\mathcal{M}$}
\With{\textbf{probability} $q_{i,t} := \min \left( 1, \frac{\nicefrac{1}{2} + c}{1 - \sum_{t' < t} x_{i, t'} \cdot (\nicefrac{1}{2} + c) } \right)$}
\State $\mathcal{M} \leftarrow \mathcal{M} \cup \{ (i,t)\}$ \label{line:probacceptfirstpick} \label{line:updateMfirstpick} \label{line:acceptfirstproposal}
\EndWith
\EndIf
\If{$t$ is still unmatched in $\mathcal{M}$}\label{line:second-pick-start}
\State pick a single bin $i$ with probability $\frac{x_{i,t}}{p_{t}}$ \label{line:pick2}
\If{$i$ is unmatched in $\mathcal{M}$ \textbf{and} $\sum_{t' < t} x_{i, t'} > \frac{\nicefrac{1}{2}-c}{\nicefrac{1}{2}+c}$}
\State $\mathcal{M} \leftarrow \mathcal{M} \cup \{ (i,t)\}$ \label{line:acceptsecondproposal}
\EndIf
\EndIf
\EndFor \State \textbf{Output} $\mathcal{M}$
\end{algorithmic}
\caption{Rounding LP-Match Online}\label{alg:propose-twice}
\end{algorithm}
By Constraint \eqref{eqn:lpatmostqi}, Lines \ref{line:pick1} and \ref{line:pick2} are well-defined. Also, by Constraint \eqref{eqn:lpatmost1}, \Cref{line:probacceptfirstpick} is well-defined since $c < 1/2$.
We also note that the algorithm clearly outputs a matching.
As we shall show, our \Cref{alg:propose-twice} fares well in comparison to $OPT_{on}(\mathcal{I})$. In particular, we will show the following per-edge guarantees.
\begin{theorem}\label{edge-marginals}
Each edge $(i,t)\in E$ is matched by \Cref{alg:propose-twice} with probability at least
$$\Pr[(i,t)\in \mathcal{M}] \geq x_{i,t}\cdot (\nicefrac{1}{2} + c).$$
\end{theorem}
\Cref{edge-marginals} implies that our algorithm is a polynomial-time $0.51$-approximation of the optimal online algorithm, thus proving \Cref{thm:alg}.
\begin{proof}[Proof of \Cref{thm:alg}]
All steps of \Cref{alg:propose-twice}, including solving the polynomially-sized LP in \Cref{line:lp-solve}, can be implemented in polynomial time.
The approximation ratio follows directly from linearity of expectation, together with \Cref{LP-bound} and \Cref{edge-marginals}.
\end{proof}
The remainder of this section is dedicated to proving \Cref{edge-marginals}. To this end, we consider two events for edge $(i,t)$ being matched---depending on whether it was matched as a first pick or second pick, in \Cref{line:acceptfirstproposal} or \Cref{line:acceptsecondproposal}, respectively.
We bound the probability of an edge being matched either as a first pick or as a second pick in the following sections.
\subsection{Analysis: First Pick}\label{sec:first-pick}
In this section we bound the probability of an edge being matched as a first pick. That is, the probability that edge $(i,t)$ is added to $\mathcal{M}$ in \Cref{line:acceptfirstproposal}. We start with the following useful definition.
\begin{definition}
Ball $t$ is \emph{early} for bin $i$ if $\sum_{t'<t} x_{i,t'} \leq \frac{\nicefrac{1}{2}-c}{\nicefrac{1}{2}+c}$. Otherwise, it is \emph{late}. Edge $(i,t)$ is early (late) if $t$ is early (late) for $i$. We use $E_i$ and $L_i$ to denote the early and late balls for $i$, respectively.
\end{definition}
Intuitively, a ball is late for bin $i$ if most balls $t'$ (weighted by $x_{i,t'}$-value) precede $t$. Note that the early/late distinction determines whether or not the probability $q_{i,t}$ in \Cref{line:probacceptfirstpick} is 1. In particular, this probability is less than 1 only if $(i,t)$ is early, and equal to 1 when $(i,t)$ is late. We will use this observation frequently in the subsequent analysis.
For every $(i,t)$, we let $V_{i,t}$ be an indicator random variable for the event that bin $i$ is \emph{vacant} (i.e., unmatched) at time $t$. We additionally let $\mathcal{M}_1 \subseteq \mathcal{M}$ denote the edges in $\mathcal{M}$ added as a result of a bin $i$ accepting a ball's first pick (i.e., in Line \ref{line:acceptfirstproposal}), and $\mathcal{M}_2 \subseteq \mathcal{M}$ denote the edges in $\mathcal{M}$ added as a result of a bin $i$ accepting a ball's second pick (i.e., in Line \ref{line:acceptsecondproposal}). Note that $\mathcal{M} = \mathcal{M}_1 \sqcup \mathcal{M}_2$.
The next lemma bounds the probability of an edge $(i,t)$ being matched as a first pick (in \Cref{line:acceptfirstproposal}).
\begin{lemma}\label{first-pick-marginals}
If edge $(i,t)\in E$ is early, then
$$\Pr[(i,t)\in \mathcal{M}_1] = x_{i,t}\cdot (\nicefrac{1}{2}+c).$$
In addition, for any edge $(i,t)\in E$,
$$x_{i,t}\cdot (\nicefrac{1}{2}-3c)\leq \Pr[(i,t)\in \mathcal{M}_1] \leq x_{i,t}\cdot (\nicefrac{1}{2}+c).$$
\end{lemma}
\begin{proof}
Fix $i$. We prove by strong induction that these bounds hold for all edges $(i,t')$ with $t' < t$. The base case, for $t=1$, is vacuously true. Assume the claim holds for all $t' < t$; we will prove it holds for $t$ as well.
The event $(i,t) \in \mathcal{M}_1$ requires that ball $t$ arrives and bin $i$ is picked in Line \ref{line:pick1}, that bin $i$ is vacant at time $t$, and that bin $i$ accepts the offer. Note that $i$ being vacant at time $t$ is independent from the arrival of $t$, and the first pick of $t$. Therefore,
\begin{equation}\label{eqn:probitinM1}
\Pr[(i,t) \in \mathcal{M}_1] = x_{i,t} \cdot \Pr[V_{i,t}] \cdot q_{i,t}.
\end{equation}
For this reason, we turn our attention to bounding the probability of $i$ being vacant at time $t$,
\begin{equation}\label{eqn:probVit}
\Pr[V_{i,t}] = 1 - \sum_{t' < t} \Pr[(i,t') \in \mathcal{M}] = 1 - \sum_{t' < t} \Pr[(i,t') \in \mathcal{M}_1] - \sum_{t' < t} \Pr[(i,t') \in \mathcal{M}_2].
\end{equation}
First, the inductive hypothesis and the definition of $x_{i,t}$ imply the following upper bound on $\Pr[V_{i,t}]$.
\begin{align}\label{Prvit-UB}
\Pr[V_{i,t}] \le 1 - \sum_{t' < t, \atop t' \in E_i} \Pr[(i,t') \in \mathcal{M}_1] = 1 - \sum_{t' < t, \atop t' \in E_i} x_{i,t'} \cdot (\nicefrac{1}{2} + c).
\end{align}
If $(i,t)$ is early, this bound is tight because $(i,t')$ is early for any $t' < t$; hence, for early $(i,t)$ we have that $\Pr[V_{i,t}] = 1 - \sum_{t' < t} x_{i,t'} \cdot (\nicefrac{1}{2} + c)$. Recalling that $q_{i,t} = \frac{\nicefrac{1}{2} + c}{1 - \sum_{t' < t} x_{i, t'} \cdot (\nicefrac{1}{2} + c) }$ for early $(i,t)$, \Cref{eqn:probitinM1} then implies that $\Pr[(i,t) \in \mathcal{M}_1] = x_{i,t} \cdot (\nicefrac{1}{2} + c)$ for early edges $(i,t)$.
If $(i,t)$ is late, then $\sum_{t' < t, t' \in E_i} x_{i,t'} = \sum_{t' \in E_i} x_{i,t'} \ge \frac{\nicefrac{1}{2} - c}{\nicefrac{1}{2} + c}$. Hence, by \Cref{Prvit-UB} we have that
\begin{align}\label{vacancy-ub}
\Pr[V_{i,t}] \le 1 - \left( \frac{\nicefrac{1}{2} - c}{\nicefrac{1}{2} + c} \right) \cdot (\nicefrac{1}{2} + c) = \nicefrac{1}{2} + c.
\end{align}
Again, \Cref{eqn:probitinM1} then implies that $\Pr[(i,t) \in \mathcal{M}_1] \le x_{i,t} \cdot (\nicefrac{1}{2} + c)$ for late edges $(i,t)$.
Finally, we lower bound $\Pr[(i,t) \in \mathcal{M}_1]$ for late $(i,t)$. To do so, we lower bound $\Pr[V_{i,t}]$; here, our analysis must account for the fact that late edges can be matched in either $\mathcal{M}_1$ or $\mathcal{M}_2$. Hence, we first note that for any $t'<t$ that is late for $i$, we have, similarly to \Cref{Prvit-UB} that the probability of edge $(i,t')$ being matched as a second pick is at most
\begin{equation}\label{eqn:it'inM2upperbound}
\Pr[(i,t')\in \mathcal{M}_2] \leq x_{i,t'} \cdot \Pr[V_{i,t'}] \leq x_{i,t'}\cdot ( \nicefrac{1}{2} + c).
\end{equation}
Now, combining equations \eqref{eqn:probVit} and \eqref{eqn:it'inM2upperbound}, we lower bound $\Pr[V_{i,t}]$ as follows:
\begin{align*}
\Pr[V_{i,t}]
&\ge 1 - \sum_{t' < t} x_{i,t'} \cdot (\nicefrac{1}{2} + c) - \sum_{t' < t, \atop t' \in L_i} x_{i,t'}\cdot(\nicefrac{1}{2}+c)
\ge 1 - (\nicefrac{1}{2} + c) - \left( 1 - \frac{\nicefrac{1}{2} - c}{\nicefrac{1}{2} + c} \right)\cdot (\nicefrac{1}{2} + c),
\end{align*}
which simplifies to
\begin{align}\label{vacancy-lb}
\Pr[V_{i,t}] \ge \nicefrac{1}{2} - 3c.
\end{align}
Again, \Cref{eqn:probitinM1} then implies that $\Pr[(i,t)\in \mathcal{M}_1]\geq x_{i,t}\cdot (\nicefrac{1}{2}-3c)$.
\end{proof}
The proof of \Cref{first-pick-marginals} yields upper and lower bounds on $\Pr[V_{i,t}]$ (equations \eqref{Prvit-UB}, \eqref{vacancy-ub} and \eqref{vacancy-lb}), which will prove useful later. For convenience, we extract these bounds in the following corollary.
\begin{corollary}\label{VitBounds}
For any edge $(i,t)$, we have that $\Pr[V_{i,t}] \ge \nicefrac{1}{2} - 3c$. For any late $(i,t)$, we have that $\Pr[V_{i,t}] \le \nicefrac{1}{2} +c$.
For any early $(i,t)$, we have that $\Pr[V_{i,t}] = 1 - \sum_{t' < t} x_{i,t'} \cdot (\nicefrac{1}{2} + c).$
\end{corollary}
Given \Cref{first-pick-marginals}, in order to prove \Cref{edge-marginals}, we wish to prove that the second attempt of $t$ to match will ensure late edges $(i,t)$ a probability of at least $x_{i,t}\cdot 4c$ of being matched. This is the meat of our analysis, and the next section is dedicated to its proof.
\subsection{Analysis: Second Pick}\label{sec:second-pick}
In this section we prove that the second pick of ball $t$, in Lines \ref{line:second-pick-start}-\ref{line:acceptsecondproposal}, does indeed increase the probability of late edges $(i,t)$ to be matched.
In particular, we prove the following theorem.
\begin{theorem}\label{second-pick-marginals}
For any late edge $(i,t)\in E$,
$$\Pr[(i,t)\in \mathcal{M}_2] \geq x_{i,t}\cdot 4c.$$
\end{theorem}
Before proving the above theorem, we provide some useful intuition and outline the challenges the proof of \Cref{second-pick-marginals} needs to overcome.
By \Cref{first-pick-marginals}, the probability of a late edge $(i,t)$ being matched as a first pick is at least
\begin{align}\label{prob-it-first-pick}
\Pr[(i,t)\in \mathcal{M}_1] \geq x_{i,t}\cdot (\nicefrac{1}{2}-3c).
\end{align}
Moreover, by the same lemma, each edge $(i,t)\in E$ (whether early or late) is matched as a first pick with probability at most $\Pr[(i,t)\in \mathcal{M}_1] \leq x_{i,t}\cdot (\nicefrac{1}{2}+c)$.
Denote by $A_t$ the event that $t$ arrives and denote by $U_1(t)$ the event that $t$ is unmatched after its first pick of $i_1=j$. Then, we have $$ \Pr[U_1(t) \mid A_t, i_1 = j] = 1 - \Pr[V_{j, t}] \cdot q_{j,t}.$$
If $(j,t)$ is late, then because $\Pr[V_{j,t}] \le \nicefrac{1}{2} + c$ by \Cref{VitBounds}, the above quantity is at least $\nicefrac{1}{2} - c$. If $(j,t)$ is early, then because $\Pr[V_{j,t}] = 1 - \sum_{t' < t} x_{j,t'} (\nicefrac{1}{2} + c)$, by \Cref{VitBounds}, combined with the definition of $q_{j,t}$, we have that the above quantity is exactly equal to $\nicefrac{1}{2} - c$.
In summary,
\begin{align}\label{prob-t-rejected}
\Pr[U_1(t) \mid A_t, i_1 = j] \ge \nicefrac{1}{2}-c.
\end{align}
Now, we recall that for late edges $(i,t)$, we have that $q_{i,t} = 1$. So, a late edge $(i,t)$ is matched iff $i$ is vacant by time $t$ and $i$ is picked in \Cref{line:pick1} or \Cref{line:pick2}.
One might then be tempted to guess that
$\Pr[(i,t)\in \mathcal{M}_2 \mid U_1(t)]$ is equal to $\Pr[(i,t)\in \mathcal{M}_1]$, which by \eqref{prob-it-first-pick} and \eqref{prob-t-rejected} would imply that
$\Pr[(i,t)\in \mathcal{M}_2] \geq x_{i,t} \cdot(\nicefrac{1}{2}-c)\cdot(\nicefrac{1}{2}-3c)\geq x_{i,t}\cdot 4c$ (the last inequality using $c\leq 0.01$), as desired.
\subsubsection{The Key Challenges}
There are two key issues with the simplistic argument above.
\paragraph{\underline{Challenge 1: Re-drawing $i$.}}
Unfortunately, conditioning on $U_1(t)$ does not result in the probability of $(i,t)$ being matched in the second pick equalling that of it being matched in the first pick. To see this, suppose a ball $t$ was late for a single bin $i$, and $x_{i,t}/p_{t}=1$. In that case, conditioning on $U_1(t)$ is equivalent to conditioning on $i$ being occupied (matched) before time $t$. Consequently, for this late edge $(i,t)$, we have that $\Pr[(i,t)\in \mathcal{M}_1] \geq x_{i,t}\cdot (\nicefrac{1}{2}-3c)$ by \Cref{first-pick-marginals}, while
$\Pr[(i,t)\in \mathcal{M}_2 \mid U_1(t)] = 0$, which implies that the second pick does not increase the probability of $(i,t)$ to be matched \emph{at all}, as $\Pr[(i,t)\in \mathcal{M}_2] = 0$(!).
This is where Constraint \eqref{eqn:lpthird} of LP-Match comes in: This constraint implies that if $t$ is late for bin $i$, then the probability that $i$ was picked in \Cref{line:pick1} at time $t$ conditioned on arrival of $t$ is at most
$$\frac{x_{i,t}}{p_{t}} \leq 1-\sum_{t'<t} x_{i,t'} \leq 1-\frac{\nicefrac{1}{2}-c}{\nicefrac{1}{2}+c} = \frac{2c}{\nicefrac{1}{2}+c} \leq 4c.$$
This implies that there is a (high) constant probability of $i$ not being picked in \Cref{line:pick1}.
\begin{lemma}\label{i1!=i}
For any late edge $(i,t)$, for $i_1$ the bin picked in \Cref{line:pick1} at time $t$,
$$\Pr[i_1 \neq i \mid A_t] \geq 1-4c.$$
\end{lemma}
\paragraph{\underline{Challenge 2: Positive Correlation Between Bins.}}
\Cref{i1!=i} alone does not resolve our problems.
Suppose that ball $t$ is late for all bins for which $x_{i,t}\neq 0$, and all these bins have perfectly positively correlated matched status, i.e., $V_{i,t} = V_{j,t}$ for all bins $i,j$ always. If this were the case, then we would have that $\Pr[V_{i,t} \mid U_1(t)] = 0$, since if $t$ is not matched to its first $i_1$, then $i_1$ and $i$ must both have been matched before. This again would result in
$\Pr[(i,t)\in \mathcal{M}_2] = 0$.
To overcome the above, we show that the above scenario does not occur. In particular, we show that while positive correlations between different bins' matched statuses are possible, such correlations cannot be too large. More formally, we show the following.
\begin{lemma}\label{nearly-neg-corr}
For any time $t$ and bins $i\neq j$, we have that
$$\mathrm{Cov}(V_{i,t},V_{j,t}) \leq 12c.$$
\end{lemma}
The crux of our analysis is proving \Cref{nearly-neg-corr}. Using it, we will be able to argue that for any late edge $(i,t)$, the probability that $i$ is free at time $t$, conditioned on $U_1(t)$ and on the first pick satisfying $i_1\neq i$ (a likely event, by \Cref{i1!=i}), is not changed much compared to the unconditional probability of $i$ being free at time $t$.
In particular, this implies that the probability of $(i,t)$ being matched as a second pick, conditioned on $U_1(t)$, is not too much smaller compared to its probability of being matched as a first pick.
In particular, we will show that $\Pr[(i,t)\in \mathcal{M}_2]\geq x_{i,t}\cdot 4c$, for sufficiently small $c>0$, as stated in \Cref{second-pick-marginals}.
We prove that lemmas \ref{i1!=i} and \ref{nearly-neg-corr} indeed imply \Cref{second-pick-marginals}, as outlined above, in \Cref{sec:second-pick-wrapup}. But first, we turn to proving our key technical lemma, namely \Cref{nearly-neg-corr}.
\subsubsection{Bounding Correlations of Occupancies}
To bound the correlation of vacancy indicators, it is convenient to define the indicator random variable $O_{i,t} := 1 - V_{i,t}$, which indicate whether $i$ is occupied (i.e., matched) at time $t$. We additionally decompose the variables $O_{i,t}$ into two variables, based on whether $i$ was matched (became occupied) along an early or late edge. In particular, we let $O_{i,t}^E \leq O_{i,t}$ be an indicator for the event that $i$ is matched along an early edge before $t$ arrives. Similarly, we let $O_{i,t}^L := O_{i,t} - O^E_{i,t}$ be an indicator for the event that $i$ is matched along a late edge before $t$ arrives.
To bound the pairwise correlations of variables $O_{i,t}$, we will show that $O^E_{i,t}$ contributes most of the probability mass of $O_{i,t}$, and that the variables $O_{i,t}^E$ and $O_{j,t}^E$ are negatively correlated. To prove this negative correlation, we will prove the following, stronger statement.
\begin{lemma}\label{requisite-NA-lemma}
For any time $t$, the variables $\{O^E_{i,t}\}_i$ are negatively associated (NA).
\end{lemma}
\begin{proof}
For every edge $(i,t)$, let $X_{i,t}$ be the indicator random variable for the event that ball $t$ arrives and picks bin $i$ as its first pick.
Let $Y_{i,t} \sim \text{Ber} \left( q_{i,t} \right)$ be an indicator for the event that bin $i$ accepts, i.e., it will be matched to ball $t$ if it arrives and picks $i$ as its first pick and $i$ is free.
For fixed $t$, the variables $\{X_{i,t} \}$ are 0/1 random variables whose sum is at most 1 always, so they are NA by the 0-1 Principle (\Cref{0-1-NA}).
On the other hand, the variables $\{Y_{i,t}\}_{i}$ are independent, and hence NA. Moreover, $\{X_{i,t}\}_{i}$, $\{Y_{i,t}\}_{i}$ are mutually independent distributions, and so by closure of NA under independent union (\Cref{NA-closure}), we also have that $\{X_{i,t}, Y_{i,t}\}_{i}$ is NA. Likewise, the lists $\{X_{i,t}, Y_{i,t}\}_{i}$ are mutually independent as we vary $t$; again using closure of NA under independent union we find that $\{X_{i,t}, Y_{i,t}\}_{i,t}$ are also NA.
Fix $t$. For each bin $i$, let $t_i$ denote the largest $t' < t$ so that $(i,t')$ is early. We note that bin $i$ cannot be matched as a second pick to any $t' \le t_i$. So, it is matched along an early edge before $t$ arrives if and only if there are some $t' \le t_i$ and $r$ such that ball $t'$ arrives and picks bin $i$, and bin $i$ accepts the proposal (for the smallest such $t'$, bin $i$ is guaranteed to be free). Therefore, we have that
$$O_{i,t}^E = \bigvee_{t' \le t_i} (X_{i,t'} \wedge Y_{i,t'}).$$
Note that we have written $\{O_{i,t}^E\}_i$ as the output of monotone non-decreasing functions defined on disjoint subsets of the variables in $\{X_{i,t}, Y_{i,t}\}_{i,t}$. Hence, by closure of NA under monotone function composition (\Cref{NA-closure}), we have that $\{O_{i,t}^E\}_i$ are NA.
\end{proof}
By \Cref{NA:neg-corr}, the above lemma implies that any $O^E_{i,t}$ and $O^E_{j,t}$ are negatively correlated.
\begin{corollary}\label{ML-neg-corr}
For any time $t$ and bins $i\neq j$, we have that
$\mathrm{Cov}(O^E_{i,t}, O^E_{j,t})\leq 0.$
\end{corollary}
We are now ready to prove \Cref{nearly-neg-corr}.
\begin{proof}[Proof of \Cref{nearly-neg-corr}]
First, we show that the probability of a bin $i$ being matched along a late edge before time $t$ is small, which we later use to bound the covariance of $O^L_{i,t}$ and other binary variables.
First, if $(i,t)$ is not late, then trivially,
$\Pr[O^L_{i,t}]=0\leq 4c.$
Otherwise, we have that $\sum_{t'<t: (i,t') \textrm{ early }} x_{i,t'} \geq \frac{\nicefrac{1}{2}-c}{\nicefrac{1}{2}+c}$.
Thus, by \Cref{first-pick-marginals}, we have that $\Pr[O_{i,t}^E] > \frac{\nicefrac{1}{2} - c}{\nicefrac{1}{2} + c} \cdot ( \nicefrac{1}{2} + c) = \nicefrac{1}{2} - c$. On the other hand, by \Cref{VitBounds}, we also have that $\Pr[V_{i,t}] \ge \nicefrac{1}{2} - 3c$. Therefore, we find that here, too, the probability of $O^L_{i,t}$ is small.
\begin{align*}
\Pr[O_{i,t}^L] &= \Pr[O_{i,t}] - \Pr[O_{i,t}^E] < (\nicefrac{1}{2} + 3c) - (\nicefrac{1}{2} - c) = 4c.
\end{align*}
From the above, we find that regardless of whether or not $(i,t)$ is late, we have that
\begin{align}\label{eqn:OitL}
\Pr[O_{i,t}^L]\leq 4c.
\end{align}
Therefore, using the additive law of covariance for $\mathrm{Cov}(O_{i,t}, O_{j,t}) = \mathrm{Cov}(1 - O_{i,t}, 1 - O_{j,t}) = \mathrm{Cov}(V_{i,t}, V_{j,t}) $, we obtain the desired bound,
\begin{align*}
\mathrm{Cov}(V_{i,t}, V_{j,t}) &=
\mathrm{Cov}(O^E_{i,t} + O^L_{i,t}, O^E_{j,t} + O^L_{j,t}) \\
& = \mathrm{Cov}(O^E_{i,t}, O^E_{j,t}) + \mathrm{Cov}(O^E_{i,t}, O^L_{j,t}) + \mathrm{Cov}(O^L_{i,t}, O^E_{j,t}) + \mathrm{Cov}(O^L_{i,t}, O^L_{j,t}) \\
& \leq 0 + \Pr[O^E_{i,t}, O^L_{j,t}] + \Pr[O^L_{i,t}, O^E_{j,t}] + \Pr[O^L_{i,t}, O^L_{j,t}] & \textrm{Cor. \ref{ML-neg-corr}} \\
& \leq 0 + \Pr[O^L_{j,t}] + \Pr[O^L_{i,t}] + \Pr[O^L_{i,t}] \\
&\le 12c. & \textrm{Eq. \eqref{eqn:OitL}} & \qedhere \end{align*}
\end{proof}
\subsubsection{Putting it All Together}\label{sec:second-pick-wrapup}
We are now ready to use weak positive correlation (if any) between vacancy indicators $V_{i,t}$ and $V_{j,t}$.
In particular, we will show that the probability of bin $i$ to be occupied a time $t$ is not changed much when conditioning on $A_t$ (arrival of $t$), the first picked bin at time $t$ being $i_1\neq i$, and $U_1(t)$ (ball $t$ bot being matched to its first pick).
\begin{lemma} \label{Oitconditionedonrejected} For any late edge $(i,t)$, we have that
$$\Pr[ O_{i,t} \mid A_t, i_1 \neq i, U_1(t)] \le \Pr[ O_{i, t}] \cdot \left( 1 + \frac{12c}{(\nicefrac{1}{2} - c)^2} \right).$$
\end{lemma}
\begin{proof}
To analyze the conditional probability above, we first look at $\Pr[O_{i,t}, A_t, i_1 = j, U_1(t)]$. This is the probability of bin $i$ being occupied at time $t$, ball $t$ arriving and picking $j$ as its first pick, and not being matched due to this first pick. Note that $A_t$ and the first pick is independent of bins' occupancy statuses at time $t$. Additionally, we notice that with probability $1 - q_{j,t}$ bin $j$ will deterministically reject. With probability $q_{j,t}$, it rejects if and only if $j$ is occupied. So, for any $j \neq i$,
\begin{equation}\label{eqn:Oitjoint}
\Pr[O_{i,t}, A_t, i_1 = j, U_1(t)] = \Pr[O_{i,t}] \cdot \Pr[A_t, i_1 = j] \cdot \left( (1 - q_{j,t}) + q_{j,t} \cdot \Pr[O_{j,t} \mid O_{i,t}] \right).
\end{equation}
We now turn to relating the last term in the above product, namely $(1 - q_{j,t}) + q_{j,t} \cdot \Pr[O_{j,t} \mid O_{i,t}]$, to its "unconditional" counterpart, $\Pr[U_1(t) \mid A_t , i_1 =j] = (1 - q_{j,t}) + q_{j,t} \cdot \Pr[O_{j,t}]$. For notational convenience, we which we abbreviate by $$z_{i,j,t} := (1 - q_{j,t}) + q_{j,t} \cdot \Pr[O_{j,t} \mid O_{i,t}].$$ Recalling that $\mathrm{Cov}(O_{i,t}, O_{j,t}) = \mathrm{Cov}(V_{i,t}, V_{j,t})\leq 12c$, by \Cref{nearly-neg-corr}, we have
\begin{equation}\label{eqn:Oitjt}
\Pr[O_{j,t} \mid O_{i,t}] = \frac{\Pr[O_{j,t}, O_{i,t}]}{\Pr[O_{i,t}]} = \frac{\Pr[O_{j,t}] \cdot \Pr[O_{i,t}] + \text{Cov}(O_{j,t}, O_{i,t}) }{\Pr[O_{i,t}]} \le \Pr[O_{j,t}] + \frac{12c}{\Pr[O_{i,t}]}.
\end{equation}
Hence,
\begin{align}
\nonumber z_{i,j,t} &\le (1 - q_{j,t}) + q_{j,t} \cdot \left( \Pr[O_{j,t}] + \frac{12c}{\Pr[O_{i,t}]} \right) & & \text{(Eq. (\ref{eqn:Oitjt}))}\\
\nonumber &\le (1 - q_{j,t}) + q_{j,t} \cdot \left( \Pr[O_{j,t}] + \frac{12c}{\nicefrac{1}{2}-c} \right) && \text{(Cor. \ref{VitBounds}, } c<\nicefrac{1}{2}) \\
\nonumber &= \Pr[U_1(t) \mid A_t, i_1 = j] + q_{j,t} \cdot \frac{12c}{\nicefrac{1}{2} - c} \\
&\le \Pr[U_1(t) \mid A_t, i_1 = j] \cdot \left( 1 + \frac{12c}{(\nicefrac{1}{2} - c)^2} \right) &&\text{(Eq. (\ref{prob-t-rejected}), } q_{j,t} \le 1)
\end{align}
Using this bound in \Cref{eqn:Oitjoint} and summing over all $j \neq i$, we have $$\Pr[O_{i,t}, A_t, i_1 \neq i, U_1(t)] \le \Pr[O_{i,t}] \cdot \Pr[A_t, i_1 \neq i, U_1(t)] \cdot \left( 1 + \frac{12c}{(\nicefrac{1}{2} - c)^2} \right).$$ The desired inequality therefore follows by Bayes' theorem.
\end{proof}
With this lemma in place, we are ready to conclude this section by proving \Cref{second-pick-marginals}, i.e. that $\Pr[(i,t) \in \mathcal{M}_2] \ge x_{i,t} \cdot 4c$ for any late edge $(i,t)$.
\begin{proof}[Proof of \Cref{second-pick-marginals}]
We start by bounding
\begin{equation}\label{eqn:itinM2firstbound}
\Pr[(i,t) \in \mathcal{M}_2] \ge \Pr[(i,t) \in \mathcal{M}_2 \mid A_t, i_1 \neq i, U_1(t)] \cdot \Pr[A_t, i_1 \neq i, U_1(t)].
\end{equation}
In words, the probability $(i,t)$ is matched as a second pick is at least the probability of the same event and $i_1\neq i$. By \Cref{i1!=i} we know that $\Pr[A_t, i_1 \neq i] \ge p_{t}\cdot (1 - 4c)$; by \Cref{prob-t-rejected}, we know that $ \Pr[U_1(t) \mid A_t, i_1 = j] \ge \nicefrac{1}{2} - c$ for any $j \neq i$. As a consequence, by Bayes' theorem and our choice of $c < \nicefrac{1}{4}$, we have that
\begin{equation}\label{eqn:13}
\Pr[A_t, i_1 \neq i, U_1(t)] = \Pr[A_t, i_1 \neq i] \cdot \Pr[ U_1(t) \mid A_t, i_1 \neq i] \ge p_{t} \cdot (1-4c) \cdot (\nicefrac{1}{2} - c).
\end{equation}
Next, we note that
\begin{equation}\label{eqn14}
\Pr[(i,t) \in \mathcal{M}_2 \mid A_t, i_1 \neq i, U_1(t)] = \frac{x_{i,t}}{p_{t}} \cdot \Pr[V_{i,t} \mid A_t, i_1 \neq i, U_1(t)]
\end{equation}
because conditioned on $A_t$, picking someone other than $i$ first, and being rejected, we will match $(i,t)$ exactly when $t$'s second pick is $i$ and $i$ is vacant.
\Cref{Oitconditionedonrejected} yields the following lower bound on the probability of $[V_{i,t} \mid A_t$, $i_1 \neq i, U_1(t)]$:
\begin{align}
\nonumber \Pr[V_{i,t} \mid A_t, i_1 \neq i, U_1(t)] &= 1 - \Pr[O_{i,t} \mid A_t, i_1 \neq i, U_1(t)] \\
\nonumber &\ge 1 - \Pr[O_{i,t}] \cdot \left( 1 + \frac{12c}{(\nicefrac{1}{2} - c)^2} \right) \\
\nonumber &= \Pr[V_{i,t}] - \frac{12c}{(\nicefrac{1}{2} - c)^2} \cdot (1 - \Pr[V_{i,t}] ) \\
&\ge \nicefrac{1}{2} - 3c - \frac{12c}{(\nicefrac{1}{2} - c)^2} \cdot (\nicefrac{1}{2} + 3c) &(\text{Cor. \ref{VitBounds}}) \label{eqn15}
\end{align}
Combining equations \ref{eqn14} and \ref{eqn15} we thus have
\begin{equation}\label{eqn:16}
\Pr[(i,t) \in \mathcal{M}_2 \mid A_t, i_1 \neq i, U_1(t)] \ge \frac{x_{i,t}}{p_{t}} \cdot \left( \nicefrac{1}{2} - 3c - \frac{12c}{(\nicefrac{1}{2} - c)^2} \cdot (\nicefrac{1}{2} + 3c) \right).
\end{equation}
Putting it all together, equations (\ref{eqn:itinM2firstbound}), (\ref{eqn:13}), and (\ref{eqn:16}) and our choice of (sufficiently small) $c=0.01$ imply the desired inequality,
\begin{align*}
\Pr[(i,t) \in \mathcal{M}_2] & \ge \frac{x_{i,t}}{p_{t}} \cdot \left( \nicefrac{1}{2} - 3c - \frac{12c}{(\nicefrac{1}{2} - c)^2} \cdot (\nicefrac{1}{2} + 3c) \right) \cdot p_{t} \cdot (1 - 4c) \cdot (\nicefrac{1}{2} - c) \geq x_{i,t}\cdot 4c. \qedhere
\end{align*}
\end{proof}
\section{Generalizing the Algorithm}\label{sec:general-algo}
Our algorithm and its analysis of \Cref{sec:algo} generalize seamlessly to a setting in which weights of each online node $t$ are drawn from discrete joint distributions. For brevity, we only outline the small changes in the LP, algorithm and analysis here.
\paragraph{Problem Statement.}
We are given a complete bipartite graph, with vertices of one side (bins) give up front, and vertices of the other side (balls) arriving sequentially, with ball $t$ arriving at time $t$ (with probability one).
The vector of edge weights of any ball $t$, denoted by $w^t:=(w_{1,t},w_{2,t},\dots)$, is drawn from some discrete joint distribution, $w^t \sim \mathcal{D}_t$.
The vector of all edge weights, $w:=(w^1,w^2,\dots)$, is drawn from the product distribution, $w\sim \mathcal{D} := \prod_t \mathcal{D}_t$. That is, the weights of any ball's edges may be arbitrarily correlated, but weights of different balls' edges are independent.
We assume that these discrete distributions are given explicitly, e.g., via a list of tuples of the form $(v_{t,j}, p_{t,j})$ with $p_{t,j} := \Pr_{\mathcal{D}_t}[w^t=v_{t,j}]$.
We note that the problem considered in previous sections is a special instance of this problem with each $\mathcal{D}_t$ consisting of two-point distributions, with one of the possible realizations of $w^t\sim \mathcal{D}_t$ being the all-zeros vector.
\paragraph{Generalizing LP-Match.}
The generalization of LP-Match now has decision variables $y_{i,t,j}$, which we think of as proxies for the probability of edge $(i,t)$ being matched by the optimal online algorithm when ball $t$'s edge weights are $w^t=v_{t,j}$.
Generalizing the argument behind Constraint \eqref{eqn:lpthird}, we note that $w^t$ is independent of bin $i$ not being matched by the optimal online algorithm by time $t$. From this we obtain Constraint \eqref{LP-gen:newconstraint} below. The remaining constraints of the obtained LP (below) are matching constraints.
\begin{figure}[h]
\begin{align}
\textbf{LP-Match-Gen:} \hspace{2em} \qquad \max \enspace \enspace & \sum_{i, t, j} w_{i,t,j} \cdot y_{i,t,j} &&& \nonumber \\
\text{s.t.} \enspace \enspace
& \sum_{t}\sum_{j} y_{i,t,j} \le 1 & & \text{ for all } i \nonumber
\\
& \sum_{i} y_{i,t,j} \le p_{t,j} & & \text{ for all } t,j
\nonumber
\\
& y_{i,t,j} \hfill \le p_{t,j} \cdot \left( 1 - \sum_{t' < t} \sum_{j'} y_{i,t',j'} \right) & & \text{ for all } i,t,j
\label{LP-gen:newconstraint}
\\
&
y_{i,t,j} \ge 0 & & \text{ for all } i,t,j\nonumber
\end{align}
\end{figure}
\paragraph{Generalizing the algorithm.}
Our general algorithm will match each edge $(i,j)$ when $w^t=v_{t,j}$ with marginal probability at least probability
\begin{align*}
\Pr[(i,t)\in \mathcal{M}, w^t=v_{t,j}] \geq y_{i,t,j}\cdot (\nicefrac{1}{2}+c).
\end{align*}
To do so, when ball $t$ arrives, we first observe the realization of the edge weight vector $w^t=v_{t,j}$. Then, When picking a bin $i$ (either as first or second pick) at time $t$, we now do so with probability $\frac{y_{i,t,r}}{p_{t,j}}$.
Moreover, we take $q_{i,t}:=\min\left(1,\frac{\nicefrac{1}{2}+c}{1-\sum_{t'<t}\sum_{j'} y_{i,t',j'}\cdot (\nicefrac{1}{2}+c)}\right)$ to be the probability of a vacant picked bin $i$ to be matched to ball $t$ by the algorithm.
The dummy nodes $i_t$ are now assigned values $y_{i_t,t,j}\leftarrow p_{t,j}-\sum_{i} y_{i,t,j}$ for each $j$.
Apart from this, the algorithm is unchanged.
We note that this algorithm can be implemented in polynomial time in the size of the input (the representation of $\mathcal{D}$).
\paragraph{Generalizing the Analysis.}
Extending the analysis of \Cref{alg:propose-twice} to this more general problem is a rather simple syntactic generalization. We therefore only outline the changes in the analysis.
Broadly, all changes needed for the analysis require us to refine our claims as follows. Denote by $R_t$ a random variable denoting the random index of the weight vector of edges of $t$. That is, $R_t=j \iff w^t=v_{t,j}$. Then, all our bounds for the probability of $(i,t)$ being matched (as a first or second pick, or either) now need to refer to $R_t=j$, and relate to $y_{i,t,j}$. So, for example, \Cref{first-pick-marginals} will be restated to show that for each early edge $(i,t)$ and index $j$, we have that $\Pr[(i,t)\in \mathcal{M}_1, R_t=j] = y_{i,t,r}\cdot (\nicefrac{1}{2}+c)$, and for any edge $(i,t)$, we have that $y_{i,t,r}\cdot (\nicefrac{1}{2}-3c)\leq \Pr[(i,t)\in \mathcal{M}_1, R_t=j] \leq y_{i,t,r}\cdot (\nicefrac{1}{2}+c)$.
\Cref{requisite-NA-lemma} requires some care in setting up the NA variables to prove that $O^E_{i,t}$ are NA, by also accounting for the realization of $R_t$, with indicators $[R_t=j]$, which are NA by the 0-1 Principle (\Cref{0-1-NA}).
Apart from that, the proofs are essentially unchanged, except for replacing occurrences of $A_t$ by $R_t=j$ in every probability conditioned on arrival of $t$, and appropriately replacing $\frac{x_{i,t}}{p_t}$ by $\frac{y_{i,t,j}}{p_{t,j}}$.
\section{LP-Match: Additional Observations}\label{sec:LP-Match-obs}
Here we make a few additional observations concerning the usefulness of Constraint \eqref{eqn:lpthird} and LP-Match in general, as well as some natural limits to this LP.
First, we note that LP-Match captures the optimal online algorithm \emph{precisely} for the classic single-item prophet inequality problem. That is, for \ensuremath{\textsc{RideHail}}\xspace instances with a single bin $i$, solutions to this LP can be rounded online losslessly.
\begin{observation}
LP-Match$(\mathcal{I})= OPT_{on}(\mathcal{I})$ for any \ensuremath{\textsc{RideHail}}\xspace instance $\mathcal{I}$ with a single bin $i$.
\end{observation}
\begin{proof}
Consider the following online algorithm, which starts by computing a solution $\vec{y}$ to LP-Match. Next, upon arrival of ball $t$ with with $w_{i,t}=w_{i,t,r}$ (i.e., $R_t=r$), match $(i,t)$ with probability
\begin{align*}
\frac{y_{i,t,r}}{p_{t,r}\cdot \left(1-\sum_{t'<t}\sum_{r'} y_{i,t',r'}\right)}.
\end{align*}
This last quantity is indeed a probability, by Constraint \ref{eqn:lpthird}.
A simple proof by induction shows that for each $t$ and $r$, we have that $\Pr[(i,t)\in \mathcal{M}, R_t=r] = y_{i,t,r}$, and consequently $\Pr[F_{i,t}]=1-\sum_{t'<t}\sum_{r'} y_{i,t',r'}$, from which we obtain the inductive step, as
\begin{align*}
\Pr[(i,t)\in \mathcal{M}, R_t=r] & = p_{t,r}\cdot \frac{y_{i,t,r}}{p_{t,r}\cdot \left(1-\sum_{t'<t}{r'} y_{i,t',r'}\right)}\cdot \left(1-\sum_{t'<t}\sum_{r'} y_{i,t',r'}\right) = y_{i,t,r}.
\end{align*}
By linearity of expectation, this online algorithm for instance $\mathcal{I}$ has expected reward precisely
\begin{align*}
\sum_{i,t,r} w_{i,t,r}\cdot y_{i,t,r}= \textrm{LP-Match}(\mathcal{I}).
\end{align*}
Consequently, $OPT_{on}(\mathcal{I})\geq \textrm{LP-Match}(\mathcal{I})$. The opposite inequality follows from \Cref{LP-bound}.
\end{proof}
On the other hand, for general \ensuremath{\textsc{RideHail}}\xspace instances, there is a limit to the approximation guarantees obtainable using LP-Match.
In particular, simple examples show that there is a gap between the upper bound given by LP-Match and the expected profit of $OPT_{on}$, appropriately restricting the approximation guarantees provable using this LP. This is to be expected, given our work in \Cref{sec:lowerbound}. We present a simple example of such a gap instance below.
\begin{observation}
There exists a $\ensuremath{\textsc{RideHail}}\xspace$ instance $\mathcal{I}$ with $w_{i,t}\in \{0,1\}$ for all $(i,t)\in E$ for which LP-Match$(\mathcal{I})\geq \nicefrac{8}{7}\cdot OPT_{on}(\mathcal{I})$.
\end{observation}
\begin{proof}
We consider an instance $\mathcal{I}$ with three balls and two bins. For $k=1,2$, ball $t=k$ has with probability $p_{k,0}=1/2$ edge weights $w_{i,t}=0$ for all $i$. With the remaining probability $p_{k,1}=1/2$, its edges have weights $w_{k,k}=1$ and $w_{k,3-k}=0$.
The last ball has weights $w_{3,k}=1$ for all bins $k=1,2$ with probability one.
An optimal solution to LP-Match on this Instance $\mathcal{I}$ assigns $y_{k,k,1}=1/2$ for $k=1,2$, and $y_{3,k,1}=1/2$ for $k=1,2$, achieving an objective value of $\sum_{i,t,r} y_{i,t,r} = 2$.
However, with probability $\nicefrac{1}{4}$, both of the first two balls have all their edge weights zero, and so an online algorithm can at most achieve an expected value of $7/4$. That is, $OPT_{on}(\mathcal{I})\leq \nicefrac{7}{8}\cdot \textrm{LP-Match}(\mathcal{I})$.
\end{proof}
\section{Unweighted Hardness}
\input{unweighted-hardness}
\section{Omitted Proofs of \Cref{sec:prelims}}\label{app:ommittedproofs}
In this section we provide proofs deferred from \Cref{sec:prelims}, restated below for ease of reference.
\approxfact*
\begin{proof}
As $f(x)=\frac{\alpha+x}{1+x} = 1-\frac{1-\alpha}{1+x}$ is monotone increasing in $x\geq -1$ for $\alpha\in (0,1)$, we have that
$(\frac{\alpha + \beta}{1 + \beta}) \geq \frac{\alpha + Q'/Q}{1+Q'/Q} = \frac{\alpha\cdot Q+Q'}{Q+Q'}$. Thus,
An $(\frac{\alpha + \beta}{1 + \beta})$-approximation to $Q + Q'$ yields a number $T$ in the range $$\Big[ \frac{\alpha + \beta}{1 + \beta} \cdot \left( Q + Q' \right) , Q + Q' \Big] \subseteq [\alpha\cdot Q + Q', Q+Q'].$$
Subtracting $Q'$ from $T$ then yields a number $T-Q'$ in the range $[\alpha \cdot Q, Q]$.
\end{proof}
Next, we provide a proof of the underlying PSPACE-hardness result of \citet{condon1997random} used in our reductions.
\randomdebaters*
\begin{proof}
This lemma follows from the proof in \cite{condon1997random}; here, we briefly explain why.
In that paper, the authors prove their main result that $\text{RPCD}(\log n, 1) = \text{PSPACE}$ in Theorem 2.4. Using this theorem, they prove that it is PSPACE-hard to approximate MAX-SSAT in Theorem 3.1. In their proof, they start with a language $L$ in PSPACE and an input $x$, and construct an RPCDS for $L$ flipping $O(\log n)$ coins and reading $O(1)$ bits of the debate. From this, they construct a MAX-SSAT instance $\phi$ such that if $x \in L$, all clauses of $\phi$ can be satisfied with probability 1, while if $x \notin L$ there is no way to satisfy more than an $\alpha < 1$ fraction of the clauses of $\phi$. Their construction of $\phi$ builds a constant-size 3CNF for each possible realization of the $O(\log n)$ coin flips, and takes the conjunction of these 3CNFs. Each constant-size 3CNF has variables corresponding to the bits of the debate that $V$ queries for a specific realization of the coin-flips. Hence, to show that $\phi$ only has each random variable appear in $O(1)$ clauses, it suffices to show that each random-bit in the RPCDS constructed is queried for only $O(1)$ realizations of the coin flips.
To show this, we turn to the construction of the RPCDS used to prove Theorem 2.4. Via Lemma 2.1, the authors first show that it is sufficient to consider RPCDSs where the verifier can read a constant number of \emph{rounds} of Player 1 (and not just a constant number of bits).
In Lemma 2.3, the authors describe their protocol for a verifier $V$ which can read $O(1)$ rounds of Player 1. Note that the random coins in this protocol are used to select a ``random odd-numbered round $k > 1$" and a ``random bit of round $k-1$ of Player 0." In fact, this is the \emph{only} time that the verifier reads a random bit of Player 0. So, in this construction, each random bit is only queried in $O(1)$ realizations of the coin flips. With Lemma 2.1, the authors transform this RPCDS to one that only reads a constant number of bits. We note that this transformation only impacts the strings that player 1 writes, and does not affect the coin flips or the bits of player $0$ read.
From this, it holds that the MAX-SSAT instance $\phi$ constructed in Theorem 3.1 has each random variable appear in $O(1)$ clauses. That instance does not yet satisfy the property that random variables only appear non-negated. Condon et al. give a fix for this in the proof of Theorem 3.3; we briefly note that after the modification provided in this proof, it will still hold that random variables appear in $O(1)$ clauses.
\end{proof}
\section{Hardness of Computing Approximately-Optimal Online Policies}\label{app:policyhardnesschernoff}
In this section we justify our claim that a hardness result for approximating the \emph{value} achieved by the optimal online algorithm implies a hardness result for the computation of the \emph{decisions} made by an (approximately) optimal online algorithm.
Let $\alpha$ be as in \Cref{thm:ridehailpspacehard}.
\begin{claim}
No polynomial-time algorithm computes the decisions made of an online algorithm which $\left( \frac{\alpha + 1}{2} \right)$-approximates the optimal online \ensuremath{\textsc{RideHail}}\xspace algorithm, unless $PSPACE=BPP$.\footnote{BPP denotes the decision problems solvable in polynomial times by \emph{randomized} algorithms which fail with probability at most $\nicefrac{1}{3}$.}
\end{claim}
\begin{proof}
We reduce from the problem of computing an $\alpha$-approximation to the profit obtained by $OPT_{on}$ for a fixed input $\mathcal{I}$, with polynomially bounded weights and inverse arrival probabilities. Let $\textsc{OPT}$ denote this profit. Let $P$ denote the maximum possible profit for $\mathcal{I}$ for any realization of the randomness.
Assume we could compute the decisions made by an algorithm $\mathcal{A}$ which achieves an $\left( \frac{\alpha + 1}{2} \right)
$-approximation. For some parameter $T$, use these decisions to run the algorithm on $T$ independent instantiations of a given input and record the profits as $X_1$, $X_2$, $\ldots$, $X_T$. Let $\bar{X}$ denote the sample average $\bar{X} := \frac{1}{T} \sum_{i=1}^T X_i$. Using the Chernoff-Hoeffding bound, we can bound the probability $\bar{X}$ deviates from its expectation as
$$\Pr \left[ \Big| \bar{X} - \mathbb E[\bar{X}] \Big| \ge \left( \frac{1 - \alpha}{4} \cdot \textsc{OPT} \right) \right] \le 2\cdot \exp \left( - \frac{2T^2 ( \frac{1-\alpha}{4} \cdot \textsc{OPT} )^2}{T \cdot P^2} \right) \le \exp \left(- \Theta \left( T \cdot \frac{\textsc{OPT}^2}{P^2} \right) \right).$$
Take $T = \Theta(n \cdot P^2/\textsc{OPT}^2)$; note this is polynomial in the size of the input, as long as all weights and inverse arrival probabilities of $\mathcal{I}$ are polynomially bounded. Then,
$$\Pr \left[ \Big| \bar{X} - \mathbb E[\bar{X}] \Big| \ge \left( \frac{1 - \alpha}{4} \cdot \textsc{OPT} \right) \right] \le \exp \left(- \Theta \left( n \right) \right).$$ so
we can clearly in polynomial time compute a $\bar{X}$ that is, w.h.p., at most $\left( \frac{1-\alpha}{4} \right) \cdot \textsc{OPT}$ far away from a $\left( \frac{\alpha + 1}{2} \right)$-approximation to $\textsc{OPT}$. In particular, $$\bar{X} \in \left[ \textsc{OPT} \left( \frac{3 \alpha + 1}{4} \right), \textsc{OPT} \left( \frac{5 - \alpha}{4} \right) \right].$$ We immediately observe that the quantity $\bar{X} - \textsc{OPT}\left(\frac{1 - \alpha}{4}\right)$ is hence in the interval $[ \textsc{OPT} \cdot \alpha, \textsc{OPT}]$. Hence, w.h.p., we have given an $\alpha$-approximation to $\textsc{OPT}$. As we demonstrated this problem to be PSPACE-complete, if we can do this in polynomial time w.h.p. then $\text{PSPACE} = \text{BPP}$.
\end{proof}
\section{Conclusions and Open Questions}\label{sec:conclusion}
We studied the online stochastic max-weight bipartite matching problem through the lens of approximation algorithms, rather than that of competitive analysis.
In particular, we study the efficient approximability of the optimal online algorithm on any given input.
On the one hand, we show that the optimal online algorithm cannot be approximated beyond some constant (barring shocking developments in complexity theory). On the other hand, we present a polynomial-time online algorithm which yields a $0.51$ approximation of the optimal online algorithm's gain---surpassing the approximability threshold of $\nicefrac{1}{2}$ of the optimal offline algorithm.
Many intriguing research questions remain.
First, it is natural to further study the efficient approximability of our problem. We suspect that much better approximation guarantees are achievable; in particular, \cite{torrico2017dynamic} suggests a family of additional constraints strengthening our LP relaxation, possibly leading to improved approximation.
One might also ask if our general algorithmic approach can be extended to \emph{implicitly} represented weight distribution $\mathcal{D}$.
For example, what can one show if $\mathcal{D}_t$ is itself a product distribution, $\mathcal{D}_t = \prod_i \mathcal{D}_{i,t}$, with $w_{i,t}\sim \mathcal{D}_{i,t}$?
A related interesting question is to obtain better approximation for the widely-studied special case of balls drawn from some i.i.d distribution (see, e.g., \cite{manshadi2012online,haeupler2011online,huang2018online,mahdian2011online,karande2011online}).
More broadly, one might ask how well one can approximate the optimal online algorithm of online Bayesian selection problems under the numerous constraints studied in the literature, including matroid and matroid intersections, knapsack constraints, etc.
For which of these problems is the online optimum easy to compute? Which admit a PTAS? Which admit constant approximations? Which are hard to approximate? We are hopeful that the ideas developed here, both algorithmic, as well as our new hardness gadgets, will prove useful when exploring this promising research agenda.
\paragraph{Acknowledgements.} We thank the anonymous EC'21 reviewers and Neel Patel for useful comments which helped improve the presentation of this manuscript, and we thank the authors of \cite{torrico2017dynamic} for drawing our attention to their work.
\section{PSPACE-Hardness} \label{sec:lowerbound}
In this section, we prove our PSPACE-hardness result.
\pspacehard*
\subsection{Extending Stochastic SAT Hardness}\label{sec:extend-SSAT}
We first extend hardness of approximation for MAX-SSAT instances as in \Cref{thm:k-max-ssat-hard} to instances which in addition satisfy that \emph{deterministic} variables appear in at most $k$ clauses.
\begin{lemma}\label{lem:ssatm}
There exist constants $k \in \mathbb N$ and $\alpha \in (0,1)$ so that it is PSPACE-hard to compute an $\alpha$-approximation to $OPT_{on}(\phi)$ for a MAX-SSAT instance $\phi$ satisfying
\begin{enumerate}[(1)]
\item no random variable appears negated in any clause of $\phi$, and \label{item:rand-non-negated}
\item \emph{each} variable (both random and deterministic) appears in at most $k$ clauses of $\phi$.\label{item:both-det-and-rand}
\end{enumerate}
\end{lemma}
We give a polynomial-time reduction from $\alpha$-approximating $OPT_{on}(\phi)$ for a MAX-SSAT instance $\phi$
as in \Cref{thm:k-max-ssat-hard} to $\alpha'$-approximating $OPT_{on}(\phi')$ on a MAX-SSAT instance $\phi'$ satisfying both properties \ref{item:rand-non-negated} and
\ref{item:both-det-and-rand} for some $k'=O(1)$ and constant $\alpha'\in (0,1)$.
\paragraph{The reduction.}
For odd (deterministic) $i$, if the variable $x_i$ appears in $a(i)$ clauses in $\phi$, we replace the $j$\textsuperscript{th} occurrence of $x_i$ with a new variable $x_{i, j}$ for $1 \le j \le a(i)$. Let $\phi'$ denote the new 3CNF formula after these replacements.
We also add clauses to force the optimal online algorithm to set all of $(x_{i, 1}, x_{i, 2}, \ldots x_{i, a(i)})$ equal to each other, without increasing their number of occurrences by more than a constant. Specifically,
for each odd $i$, we construct via \Cref{lem:explicitexpanders} an expander graph $G_i$ on $a(i)$ vertices with maximum degree at most $d = O(1)$ and expansion at least 1. Associate the vertices of $G_i$ with the literals $(x_{i,1}, x_{i,2}, \ldots, x_{i,a(i)})$ arbitrarily. For any edge in $G_i$ between $x_{i, j}$ and $x_{i, j'}$, add the following two clauses to $\phi'$:
\begin{align}\label{additional-clauses}
(x_{i,j} \vee \overline{x_{i, j'}}) \wedge (\overline{x_{i,j}} \vee x_{i, j'}).
\end{align}
Note that if $x_{i, j} \neq x_{i, j'}$, we satisfy exactly one of these two clauses, while if $x_{i, j} = x_{i, j'}$ we satisfy both.
The order of variables $x_{i,j}$ and $x_i$ in $\phi$ is some arbitrary order such that variables in $\phi'$ corresponding to (copies of) variables $x_i$ and $x_j$ in $\phi$ appear in an order consistent with the variables $x_i$ and $x_j$ in $\phi$. By adding dummy random variables, we further guarantee that copies of deterministic/random variables in $\phi$ are likewise deterministic/random in $\phi'$.
The following lemma relates the maximum expected number of satisfiable clauses in $\phi$ and $\phi'$, needed to complete our reduction's analysis.
\begin{lemma}\label{opt-phi-opt-phi'}
Let $E_n := \sum_{ \text{odd } i \le n } 2|E(G_{i})|$. Then, the MAX-SSAT instances $\phi$ and $\phi'$ satisfy
\begin{equation*}
OPT_{on}(\phi') = OPT_{on}(\phi) + E_n.
\end{equation*}
\end{lemma}
\begin{proof}
We first prove $OPT_{on}(\phi')\geq OPT_{on}(\phi)+E_n$.
Consider an online algorithm $\mathcal{A}$ which for odd $i$ sets $x_{i,1} = x_{i,2} = \ldots = x_{i, a(i)} = b_i$, where $b_i$ is the assignment for $x_i$ of $OPT_{on}$ on $\phi$ given the induced history. This algorithm for $\phi'$ is clearly implementable.
Moreover, this algorithm satisfies each of the $E_n$ clauses of form \eqref{additional-clauses}, and satisfies $OPT_{on}(\phi)$ of the original clauses in expectation. Hence $OPT_{on}(\phi')\geq \mathcal{A}(\phi') = OPT_{on}(\phi) + E_n.$
We now prove that $OPT_{on}(\phi') \le OPT_{on}(\phi) + E_n.$
Assume that for some odd $i$, and some fixed history for all variables before $(x_{i,1}, \ldots, x_{i,a(i)})$, an SSAT algorithm $\mathcal{A}$ sets $(x_{i,1}, x_{i,2}, \ldots, x_{i, a(i)})$ such that they do not all take the same value (with some positive probability). Consider the minimum size subset $S \subseteq \{1, 2, \ldots, a(i)\}$ such that flipping all $\{x_{i, j}\}_{j \in S}$ would result in all variables being set to the same value (so, $1 \le |S| \le a(i)/2$). Since the expansion of $G_i$ is at least 1, we know that $|E(S, V \setminus S)| \ge |S|$; flipping all the $\{x_{i, j}\}_{j \in S}$ would hence let us satisfy at least $|S|$ additional clauses of the form \eqref{additional-clauses}, and possibly satisfy $|S|$ fewer clauses corresponding to clauses in $\phi$ containing $x_i$. Thus, $\mathcal{A}$ would satisfy at least as many clauses in expectation by flipping the sign of $\{x_{i,j} \}_{j\in S}$.
Repeatedly applying this transformation results in an improved online algorithm $\mathcal{A}'$ as stated in the previous paragraph, from which we find that $OPT_{on}$ satisfies at most $OPT_{on}(\phi) \leq \mathcal{A}'(\phi') \leq OPT_{on}(\phi') + E_n$ clauses in expectation. The lemma follows.
\end{proof}
We now show that $E_n$ is bounded from above by a constant times $OPT_{on}(\phi)$.
\begin{observation}\label{Enupperboundobs}
$E_n\leq 12d\cdot OPT_{on}(\phi).$
\end{observation}
\begin{proof}
Since for each odd $i$, the expander graph $G_i$ contains at most $d$ edges per each of the $a(i)$ occurrences of $i$ in $\phi$, we have that $E_n = \sum_{\textrm{odd }i\leq n} 2|E(G_i)|\leq \sum_{\textrm{odd }i\leq n} 2d\cdot a(i)$.
Next, for $m$ the number of clauses in $\phi$, since $\phi$ is a 3-CNF formula, $\sum_{\text{odd } i \le n} a(i) \le 3m$.
Finally, we note that, since
setting each variable randomly satisfies at least half of the clauses in expectation,
$ m/2\leq OPT_{on}(\phi)$. Combining these observations, we find that
\begin{align}
E_n &= \sum_{\text{odd } i \le n} 2|E(G_i)| \le \sum_{\text{odd } i \le n} d \cdot a(i) \le 6dm \le 12d \cdot OPT_{on}(\phi).\qedhere \nonumber
\end{align}
\end{proof}
Given the above, we are now ready to prove \Cref{lem:ssatm}.
\begin{proof}[Proof of \Cref{lem:ssatm}]
Let $\alpha\in (0,1)$ and $k$ be the constants in the statement of \Cref{thm:k-max-ssat-hard}.
Let $\phi$ be a MAX-SSAT instance as in the statement of that lemma and $\phi'$ be the obtained instance from the reduction of this section, which is polynomial-time, by \Cref{lem:explicitexpanders}.
By construction, no random variable appears negated in any clause, and each variable appears in at most $k'=\max(d+2,k) = O(1)$ clauses.
By \Cref{opt-phi-opt-phi'}, $OPT_{on}(\phi') = OPT_{on}(\phi) + E_n$.
Next, we let $Q = OPT_{on}(\phi)$, $Q' = E_n$, and $\beta = 12d$, and note that
$Q'/Q \le \beta$, by
\Cref{Enupperboundobs}.
Thus, by \Cref{lem:approxlem}, for the constant $\alpha' := \left( \frac{\alpha + 12d}{1 + 12d} \right)\in (0,1)$, an $\alpha'$-approximation to $OPT_{on}(\phi') = OPT_{on}(\phi) + E_n = Q+Q'$ yields an $\alpha$-approximation of $Q=OPT_{on}(\phi)$, which is PSPACE-hard, by \Cref{thm:k-max-ssat-hard}.
\end{proof}
\subsection{Hardness of Algorithms for $\ensuremath{\textsc{RideHail}}\xspace$}\label{sec:pspace-reduction}
We are now ready to prove our main theorem about the hardness of $\ensuremath{\textsc{RideHail}}\xspace$. Throughout this proof, we will let $k = O(1)$ be the constant in the statement of \Cref{lem:ssatm}.
Denote the variables in an SSAT instance $\phi$ as in \Cref{lem:ssatm} by $(x_1, x_2, \ldots, x_n)$ and the number of clauses of $\phi$ by $m$. Without loss of generality, suppose $n$ is even.
From $\phi$, we construct a $\ensuremath{\textsc{RideHail}}\xspace$ instance $\mathcal{I}_{\phi}$, with weights $w_{i,t} = w_t$ for each pair $(i,t)\in E$, where we refer to $w_t$ as the weight of ball $t$.
The instance has $2n$ bins, corresponding to the literals $\{x_i, \overline{x_i} \mid i\in [n]\}.$
The instance $\mathcal{I}_{\phi}$ has $n+m$ balls; we will refer to the first $n$ balls as ``literal balls'' and the final $m$ balls as ``clause balls'' (for reasons that will become clear shortly). For odd $t\leq n$, ball $t$ arrives with probability 1, has weight $1$, and has an edge only to bins $x_{t}$ and $\overline{x_{t}}$. For even $t\leq n$, ball $t$ arrives with probability $1/2$, has weight $1$, and has an edge only to bin $x_{t}$. The last $m$ clause balls $t=n+1,\dots,n+m$ each have weight $\frac{m^4}{2k}$ and arrive with probability $m^{-4}$. The clause ball $t=n+r$ corresponding to clause $C_r$ neighbors only the bins corresponding to literals in $C_r$. (See \Cref{fig:binreduction}.)
\begin{figure}[h]
\centering
\begin{comment}
\begin{asy}
void drawOffNode(pair c, string s) {
label(s, c);
filldraw(circle(c, 0.4), white, fontsize(10pt));
}
void drawHorizDots(pair c){
dot(c - (0.3, 0));
dot(c);
dot(c+(0.3, 0));
}
void drawRedHorizDots(pair c){
dot(c - (0.3, 0.2), linewidth(1.7));
dot(c + (0, -0.2), linewidth(1.7));
dot(c+(0.3, -0.2), linewidth(1.7));
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void drawOnNode(pair c, string s, string f) {
label(s, c, fontsize(11pt));
filldraw(circle(c, 0.35), white);
label(f, c + (0, -0.8), fontsize(8pt));
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label(s, c, fontsize(9pt));
filldraw(circle(c, 0.35), white);
label(f, c + (0, -0.8), fontsize(8pt));
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label(s, c, fontsize(9pt));
filldraw(circle(c, 0.35), white);
label(f, c + (0, -0.8), fontsize(8pt));
}
size(13.5cm);
defaultpen(fontsize(12pt));
//code to draw lines goes here
draw((0,0)--(0.5, -2));
draw((1,0)--(0.5, -2));
draw((2,0)--(2, -2));
draw((3,0)--(3.5, -2));
draw((4,0)--(3.5, -2));
draw((5,0)--(5,-2));
draw((7,0)--(7,-2));
draw((7.75,-1.75)--(1,-0.3));
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draw((7.9,-1.8)--(7,0));
draw((10,-2)--(8.7, -1.5));
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label("Bins:", (-1.5, 0), fontsize(10pt));
drawOffNode((0,0), "$x_1$");
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drawOffNode((2,0), "$x_2$");
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label("Balls:", (-1.5, -2), fontsize(10pt));
drawOnNode((0.5, -2), "1", "1");
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drawOnNode((5,-2), "1", "0.5");
drawHorizDots((6, -2));
drawOnNode((7, -2), "1", "0.5");
drawOnClauseNode((8, -2), "$\frac{m^4}{2k}$", "$m^{-4}$");
drawHorizDots((9, -2));
drawOnClauseNodeFinal((10,-2), "$\frac{m^4}{2k}$", "$m^{-4}$");
label("$p_t$:", (-1.5, -2.8), fontsize(10pt));
drawRedHorizDots((6, -2.7));
drawRedHorizDots((9, -2.7));
label("(for clause $C_1 = \overline{x_1} \vee \overline{x_3} \vee x_n$)", (10, -0.1), fontsize(8pt));
draw((9, -0.3) -- (8.3, -1.7), EndArrow);
\end{asy}
\end{comment}
\includegraphics{fig1.pdf}
\caption{The $\ensuremath{\textsc{RideHail}}\xspace$ instance $\mathcal{I}_{\phi}$}
\vspace{0.2cm}
\footnotesize{Bins are labeled by their corresponding literal, while balls are labeled by their weight.}
\label{fig:binreduction}
\end{figure}
We shall see that $OPT_{on}(\mathcal{I}_\phi)$ and $OPT_{on}(\phi)$ are, up to a negligible error term, related by a simple linear relation.
In particular, we will show that
\begin{align} \label{relation-phi-calIphi}
OPT_{on}(\mathcal{I}_{\phi}) = 0.75n + \frac{ (1 - m^{-4})^{m-1} }{2k}\cdot OPT_{on}(\phi)+ o(1).
\end{align}
We prove \Cref{relation-phi-calIphi} in the following two lemmas. The first proves that $OPT_{on}$ run on $\mathcal{I}_\phi$ matches all arriving literal balls.
\begin{lemma}\label{match-literal-balls}
Algorithm $OPT_{on}$ matches all arriving literal balls of $\mathcal{I}_\phi$.
\end{lemma}
\begin{proof}
Suppose that there is some history $h$ (occurring with probability $q > 0$) after which $OPT_{on}$ does not match a literal ball $t$ which arrives; let $\mathcal{A}'$ be the algorithm that follows exactly what $OPT_{on}$ does, with the exception that it will match $t$ if $t$ arrives after the history $h$. Then, $$\mathcal{A}'(\mathcal{I}_{\phi}) - OPT_{on}(\mathcal{I}_{\phi}) \ge q \cdot \left( 1 - k \cdot \frac{m^4}{2k} \cdot m^{-4} \right) = q/2 > 0.$$ Indeed, if the history $h$ occurs, $\mathcal{A}'$ gets a guaranteed profit of 1 from matching $t$ that $OPT_{on}$ does not receive. The expected profit $OPT_{on}$ gets from having the additional bin available to be potentially matched to clause balls is at most $k \cdot \frac{m^4}{2k} \cdot m^{-4}$, since each literal bin has at most $k$ clause balls adjacent to it, each of which has value $\frac{m^4}{2k}$ and arrives with probability $m^{-4}$.
As the above would imply $\mathcal{A}(\mathcal{I}_\phi) > OPT_{on}(\mathcal{I}_\phi)$,
we conclude that $OPT_{on}$ must match each literal ball that arrives.
\end{proof}
A simple corollary of the above is that $OPT_{on}$ gets value of $0.75n$ in expectation from the literal balls it matches.
Moreover, this lemma gives a natural correspondence between $OPT_{on}$ on $\mathcal{I}_\phi$ and algorithms for $\phi$.
The following lemma relies on \Cref{match-literal-balls} to bound the value $OPT_{on}$ obtains from the clause balls in terms of the expected number of clauses of $\phi$ satisfied by $OPT_{on}$.
\begin{lemma}\label{value-clause-balls}
Let $B$ be the gain of $OPT_{on}$ from clause balls of $\mathcal{I}_\phi$.
Then, for some $\delta\in [0,2m^{-1}]$,
$$\mathbb E[B] = \frac{ (1 - m^{-4})^{m-1} }{2k}\cdot OPT_{on}(\phi) + \delta.$$
\end{lemma}
\begin{proof}
By \Cref{match-literal-balls}, $OPT_{on}$ matches each arriving literal ball.
We consider the following natural mapping between MAX-SSAT algorithms $\mathcal{A}$ on $\phi$ and families of algorithms $\mathcal{F}_\mathcal{A}$ which match each literal ball in $\mathcal{I}_\phi$.
For odd $t\leq n$, an algorithm $\mathcal{A}'\in \mathcal{F}_\mathcal{A}$
matches ball $t$ to bin $\overline{x_t}$ ($x_t$) iff
algorithm $\mathcal{A}$ sets $x_t$ to \texttt{True} (\texttt{False}). For even $t\leq n$, if ball $t$ arrives, an algorithm $\mathcal{A}'\in \mathcal{F}_\mathcal{A}$ matches ball $t$ to bin $x_t$; this corresponds to nature setting $x_t = \texttt{False}$. Otherwise, bin $x_t$ is unmatched up to time $m+1$, and we will think of this as nature setting $x_t = \texttt{True}$. (Note that ball $t$ arrives with probability 50\%, so the variables are set to \texttt{True}/\texttt{False} with the correct probability.)
Finally, algorithms $\mathcal{A}'\in \mathcal{F}_\mathcal{A}$ match each arriving clause ball to some available neighboring bin when possible. A simple exchange argument shows that $OPT_{on}(\mathcal{I}_\phi)\in \mathcal{F}_\mathcal{A}$ for some algorithm $\mathcal{A}$.
Let $C$ be the number of clause balls of $\mathcal{I}_\phi$ that arrive.
Then, with probability $\Pr[C=1]=m \cdot m^{-4} \cdot (1 - m^{-4})^{m-1}$, exactly one such clause ball arrives, equally likely to correspond to any of the $m$ clauses in $\phi$.
On the other hand, a literal $x_t$ (respectively, $\overline{x_t}$) is unmatched by $\mathcal{A}'\in \mathcal{F}_\mathcal{A}$ immediately prior to time $m+1$ iff $\mathcal{A}(\phi)$ or nature set $x_t$ to \texttt{True} (respectively, \texttt{False}).
We conclude that Algorithm $\mathcal{A}'\in \mathcal{F}_{\mathcal{A}}$ gains $\frac{\mathcal{A}(\phi)}{m}\cdot \frac{m^4}{2^k}$ expected value from conditioned on a single clause ball arriving.
Thus, the expected gain $\mathbb E[B]$ of $OPT_{on}(\mathcal{I}_\phi)$ from clause balls is at least
\begin{align}\label{clause-gain-lb}
\mathbb E[B]\geq \mathbb E[B \mid C=1]\cdot \Pr[C=1] = \frac{(1 - m^{-4})^{m-1}}{2k}\cdot OPT_{on}(\phi).
\end{align}
Let $\mathcal{A}$ be the MAX-SSAT algorithm for which $OPT_{on}(\mathcal{I}_\phi)\in \mathcal{F}_\mathcal{A}$.
By the above argument yielding \Cref{clause-gain-lb}, the expected gain of $OPT_{on}(\mathcal{I}_\phi)$ from clause balls conditioned on $C=1$ is precisely
\begin{align}\label{gain-single-clause-ball}
\Pr[B \mid C=1] = \frac{\mathcal{A}(\phi)}{m} \cdot \frac{m^4}{2k} \le \frac{OPT_{on}(\phi)}{m} \cdot \frac{m^4}{2k}.
\end{align}
Next, we note that the probability that multiple clause balls arrive is inverse polynomial in $m$. \begin{align}\label{prob-two-balls-or-more}
\Pr[C \ge 2] = \sum_{t=2}^m \binom{m}{t} m^{-4t} (1 - m^{-4})^{m-t} \le \sum_{t=2}^m m^t \cdot m^{-4t} \le m^{-6} + m \cdot m^{-9} \leq 2m^{-6}.
\end{align}
On the other hand, conditioned on at multiple clause balls arriving, the expected profit of $OPT_{on}$ from clause balls is at most
\begin{align}\label{opton-given-two-balls-or-more}
\mathbb E[B \mid C\geq 2 ] \le m\cdot \frac{m^4}{2k} \le m^5.
\end{align}
Combining equations \eqref{gain-single-clause-ball}, \eqref{prob-two-balls-or-more} and \eqref{opton-given-two-balls-or-more}, we find that the expected gain of $OPT_{on}(\mathcal{I}_\phi)$ from matching clause balls is at most
\begin{align*}
\mathbb E[B] & = \mathbb E[B \mid C=1]\cdot \Pr[C=1] + \mathbb E[B \mid C\geq 2]\cdot \Pr[C\geq 2] \\
& \leq \frac{OPT_{on}(\phi)}{m} \cdot \frac{m^4}{2k}\cdot m\cdot m^{-4} (1 - m^{-4})^{m-1} + m^5\cdot 2m^{-6} \\
& = \frac{(1 - m^{-4})^{m-1}}{2k}\cdot OPT_{on}(\phi) + 2m^{-1}.\qedhere
\end{align*}
\end{proof}
We now conclude the reduction, and obtain the proof of our hardness result.
\begin{proof}[Proof of \Cref{thm:ridehailpspacehard}]
Let $\alpha\in (0,1)$ be the constant from the statement of \Cref{lem:ssatm} and $\phi$ be a MAX-SSAT instance as in the statement of that lemma. Without loss of generality, we assume that $\phi$ has no pairs of consecutive variables $x_{2k-1}$ and $x_{2k}$ which appear in no clauses. (Else, we remove these variable pairs and relabel the remaining variables while preserving parity of indices. This does not change the clauses, nor does it change the expected number of clauses satisfied by $OPT_{on}$.)
Next, let $\mathcal{I}_\phi$ be the obtained $\ensuremath{\textsc{RideHail}}\xspace$ instance from the (clearly polynomial-time) reduction of this section; note furthermore than $\mathcal{I}_\phi$ has all weights and inverse arrival probabilities bounded above by some polynomial in the size of the input. From \Cref{match-literal-balls}, the expected gain of $OPT_{on}(\mathcal{I}_\phi)$ from literal balls is $0.75n$. Combining this with \Cref{value-clause-balls} we find that for $\gamma := \frac{ (1 - m^{-4})^{m-1} }{2k}$ and some $\delta\in [0,2m^{-1}]$,
$$OPT_{on}(\mathcal{I}_\phi) = 0.75n + \gamma\cdot OPT_{on}(\phi)+\delta.$$
Next, since $\phi$ is a 3-CNF formula with at least half its variables appear in at least one clause, the number of variables is at most $n\leq 6m$. Moreover, since setting all variables randomly satisfies at least half of the clauses in expectation, we have $m/2\leq OPT_{on}(\phi)$. Combining these two observations, we get
\begin{equation}\label{eqn:0.75natmost8}
0.75n < n\le 12 \cdot OPT_{on}(\phi),
\end{equation}
Next, let
$Q = \gamma \cdot OPT_{on}(\phi) + \delta$, $Q' = 0.75n$, and $\beta = \frac{12}{\gamma}$. Note that $Q'/Q \le \beta$ by \Cref{eqn:0.75natmost8}, and that $\beta = O(1)$, since $k=O(1)$.
Therefore, by \Cref{lem:approxlem}, for the constant $\alpha' := ( \frac{\alpha\cdot (\gamma+2m^{-1})/\gamma + \beta}{1 + \beta})$, which is in the range $(0,1)$ for sufficiently large $m$, an $\alpha'$-approximation to $OPT_{on}(\mathcal{I}_\phi) = OPT_{on}(\phi) + 0.75n = Q+Q'$ yields an $\alpha\cdot (\gamma+2m^{-1})/\gamma$-approximation of $Q \in [\gamma\cdot OPT_{on}(\phi), (\gamma+2m^{-1}) \cdot OPT_{on}(\phi)]$. By scaling appropriately, this yields an $\alpha$-approximation to $OPT_{on}(\phi)$, which is PSPACE-hard to obtain, by \Cref{lem:ssatm}. The theorem follows.
\end{proof}
\section{Introduction}
\iffalse
\color{blue}
We study the online Bayesian bipartite matching or $\ensuremath{\textsc{RideHail}}\xspace$ problem.
The input to this problem is a bipartite graph, whose (random) edge weights are revealed over time.
Initially, the $m$ nodes on one side of the graph, termed taxis or bins, are present.
The $n$ nodes on the other side, termed passengers or balls, are revealed over time, together with their edge weights. Specifically, at time $t$, ball $t$ arrives, and the weights of all of its edges $(i,t)$ are revealed.
The weights of edges of each ball $t$ can follow any joint distribution.
These distributions are given a priori in explicit form as lists of tuples $(p_{t,r},\{w_{i,t,r}\}_i)$, where with probability $p_{t,r}$, every edge $(i,t)$ of ball $t$ has weight $w_{i,t}=w_{i,t,r}$.
When ball $t$ arrives we observe the realizations its edge weights (i.e., the $r$ for which $w_{i,t}=w_{i,t,r}$ for all $i$) and we can choose to match $t$ irrevocably to some unmatched neighbor $i$ before time $t+1$, yielding a profit $w_{i,t,r}$.
Our goal is to maximize the overall expected profit, where we have a priori knowledge of the bipartite graph, the weights $w_{i,t,r}$ and the probabilities $p_{t,r}$ (but not the realizations of the latter, which are revealed over time).
\color{black}
\fi
Decision-making in an uncertain, dynamic environment influenced by one's decisions has arguably
always been the essence of life, and yet it appears to have been first confronted mathematically
by Herbert Robbins and Richard Bellman, from different perspectives, in the late 1940s and early 1950s.
Decision theory initially focused on instantaneous decisions, but later gave us stopping rules
and the gem of prophet inequalities \cite{krengel1978semiamarts}. Later, the Internet age brought us new business models
relying exclusively on stochastic decision making --- online advertising, ride hailing,
kidney exchanges --- in which the changing environment affected by
the agents' decisions can often be abstracted as an evolving weighted bipartite graph.
Here we study one such problem,
the online Bayesian bipartite matching, or $\ensuremath{\textsc{RideHail}}\xspace$, problem.
The input to this problem is a random bipartite graph, revealed over time.
Initially, the $m$ nodes on one side of the graph, termed taxis or bins, are present.
The $n$ nodes on the other side, termed passengers or balls, are revealed over time, in a fixed order known to us.
Initially, we know for each ball $t$ the probability $p_t$ of it actually arriving, as well
as the non-negative weight $w_{i,t}$ of the edge connecting it to any bin $i$ --- if it arrives.
If ball $t$ does not arrive, we do nothing at time $t$; if it does arrive, we can choose to match it, irrevocably, to some unmatched neighbor $i$ before time $t+1$, yielding a profit $w_{i,t}$.
Our goal is to maximize the overall expected profit.
$\ensuremath{\textsc{RideHail}}\xspace$ generalizes the classic single-item online Bayesian stopping rule problem --- the so-called \emph{prophet inequality} problem.
In particular, our problem with a single offline node already captures the worst-case instances of the prophet inequality, for which no online algorithm is better than $\nicefrac{1}{2}$-competitive against the optimal offline algorithm.
On the other hand, $\ensuremath{\textsc{RideHail}}\xspace$ is a special case of the unit-demand combinatorial auctions problem and online stochastic maximum-weight matching, for both of which $\nicefrac{1}{2}$-competitive algorithms are known \cite{feldman2015combinatorial, dutting2020prophet,ezra2020online}.
There is an extensive literature on numerous variations of online Bayesian selection problems, relating the performance of online algorithms with the omniscient prophet of inequality fame --- that is to say, with the offline optimum (see \Cref{sec:related-work}).
In particular, these works study achievable competitive ratios: the worst-case ratio over all inputs between the online algorithm and the best offline algorithm.
While this may be the right thing to do when the input is adversarial, when the input is generated {\em stochastically} one perhaps could do better.
In particular, in the stochastic case, the optimum online algorithm {\em for the given input} is a well-defined benchmark that can be computed in exponential time. Suddenly we are in the realm of {\em approximation algorithms,} rather than of competitive analysis.
In approximation algorithms, typically one explores two interesting questions: First, is approximation hard? And second, what is the best approximation ratio achievable in polynomial time?
In this paper we address both questions. First, we show that for some $\alpha<1$ it is \textsc{PSPACE}-hard to approximate the $\ensuremath{\textsc{RideHail}}\xspace$ problem within a factor of $\alpha$.
\begin{wrapper}
\begin{restatable}{theorem}{pspacehard}\label{thm:ridehailpspacehard}
It is \textsc{PSPACE}-hard to approximate the optimal online $\ensuremath{\textsc{RideHail}}\xspace$ algorithm within a factor of $\alpha$, for some absolute constant $\alpha<1$. This remains true even when all weights and inverse arrival probabilities are bounded by some polynomial in the size of the input.
\end{restatable}
\end{wrapper}
Here, $1-\alpha$ is small, limited by the current status of expander constructions and approximation hardness of MAX-SSAT (see \Cref{sec:prelims}). To our knowledge, no past work on variants of online matching had demonstrated such level of hardness.
(We briefly note that PSPACE is the ``right'' complexity class for this problem, which can be solved in polynomial space by standard techniques.)
Finally, we note (see \Cref{app:policyhardnesschernoff}) that our hardness of approximation result directly implies hardness of computing an approximately-optimal online algorithm, and not just its expected value.\footnote{Note that for some problems, although computing the expected value of the optimal policy is hard, computing the optimal policy itself is actually easy. For example, computing the probability that a random graph containing each edge $e$ with probability $p_e$ is connected is the
\#P-hard the network reliability problem \cite{karger2001randomized,valiant1979complexity,provan1983complexity}, while computing connectivity in the realized graph (even in an online setting) is trivial.}
We then develop an approximation algorithm, and a technique to bound the (online) optimum. To our knowledge, all past work on approximating the large family of online Bayesian selection problems, with the exception of \cite{anari2019nearly}, has used the prophet inequality benchmark of the offline optimum, which necessarily limits the approximation ratio for many variations to be at most $\nicefrac{1}{2}$.
{\em We go for bounding the online optimum.} We achieve this by identifying a new constraint which separates online from offline algorithms. In particular, we note that online algorithms cannot match an edge $(i,t)$ with probability greater than the probability of ball $t$ arriving, times the probability of bin $i$ not being matched by the online algorithm beforehand, due to the independence of these events.
This constraint, which is not true of offline algorithms, poses restrictions on the marginal probabilities of edges to be matched by the optimal online algorithm.
Combining this constraint with the natural matching constraints we obtain a new LP which bounds the optimal online algorithm's gain.
Using this new LP bound (and a number of further ideas, see \Cref{sec:techniques}), we design a new algorithm which recovers at least $51\%$ of the online optimum, i.e., a
ratio strictly better than the optimal competitive ratio of $\nicefrac{1}{2}$.
\begin{wrapper}
\begin{restatable}{theorem}{thmalg}\label{thm:alg}
There exists a polynomial-time online algorithm which is a $0.51$-approximation of the optimal online algorithm for the $\ensuremath{\textsc{RideHail}}\xspace$ problem.
\end{restatable}
\end{wrapper}
We further generalize our algorithm and achieve the same approximation bound for the more general problem in which weights of any given ball's edges can follow any joint distribution, but weights of different ball's edges are independent. That is, we extend our positive results to the more general bipartite weighted matching problem studied by prior work~\cite{ezra2020online,dutting2020prophet,feldman2015combinatorial}. (See \Cref{sec:general-algo}.)
\subsection{Techniques}\label{sec:techniques}
Here we give a very brief overview of the key ideas used to obtain our main results.
\subsubsection{Hardness} For our \textsc{PSPACE}-hardness result, we first refine the result of \citet{condon1997random} for maximum satisfiability of stochastic SAT instances.
In the stochastic SAT (SSAT) problem, introduced by \citet{papadimitriou1985games},
a 3CNF formula is given, and variables $x_1,x_2,\dots,x_n$ are alternatingly set by an (online) algorithm and randomly set by nature.
\citet{condon1997random} proved that approximating the maximum expected number of satisfied clauses of an SSAT instance is \textsc{PSPACE}-hard.
Using an \emph{expander graph} construction, we extend this result to SSAT instances in which each variable appears in at most $O(1)$ clauses.
We then give a polynomial-time reduction from approximating maximum
satisfiability of a bounded-occurrence SSAT instance to approximating the optimal online algorithm for the \ensuremath{\textsc{RideHail}}\xspace problem, implying our claimed \textsc{PSPACE}-hardness.
\subsubsection{Algorithm} Our algorithmic results involve a number of ideas. We outline the key ones here.
\paragraph{\underline{Our LP Benchmark.}}
We recall that we want to approximate the optimal online algorithm within a factor strictly greater than $\nicefrac{1}{2}$ (which is tight against the optimal offline algorithm). Hence, our first objective is to identify a property which separates online from offline algorithms.
To this end, we note (as did \cite{torrico2017dynamic}) that for any online algorithm $\mathcal{A}$, the event of the arrival of ball $t$ is independent of the event that bin $i$ is not matched by Algorithm $\mathcal{A}$ prior to time $t$. (Note that this constraint does not necessarily hold for the prophetic optimum offline algorithm, which makes its matching choices based on both past and future balls' arrivals.)
Consequently, the probability that edge $(i,t)$ is matched by online algorithm $\mathcal{A}$ is at most the product of these two events' probabilities.
Combining this constraint with natural matching constraints, we obtain an LP which bounds the expected gain of the optimal online algorithm (but not its offline counterpart).
In \Cref{sec:LP-Match-obs} we note that this LP completely characterizes the optimal online algorithm for instances with a single offline node, equivalent to the single-item online Bayesian selection problem. This is not true for general instances (as we would expect due to \Cref{thm:ridehailpspacehard}); we therefore use this LP to \emph{approximate} the optimal online policy.
\paragraph{\underline{A Second Chance Algorithm.}}
We present an efficient online algorithm for approximately rounding a solution to the above LP.
Let $x_{i,t}$ be the decision variables of this LP.
Intuitively, these $x_{i,t}$ serve as proxies for the probability of $(i,t)$ to be matched by the optimal online algorithm.
Our online algorithm matches each edge $(i,t)$ with probability at least $x_{i,t}\cdot (\nicefrac{1}{2}+c)$ for $c=\nicefrac{1}{100}$.
Our algorithm can be seen as a generalization and extension of the $\nicefrac{1}{2}$-competitive algorithm of Ezra et al.~\cite{ezra2020online} for our problem.
Their algorithm can be thought of as approximately rounding the above LP (without the new constraint) as follows. After each arrival of ball $t$, pick a bin $i$ with probability proportional to $x_{i,t}$, and then, if bin $i$ is unmatched, match edge $(i,t)$ with some probability $q_{i,t}$. These $q_{i,t}$ are set to guarantee that each edge $(i,t)$ is matched with marginal probability $x_{i,t}\cdot \nicefrac{1}{2}$, which can be thought of as applying an online contention resolution scheme as in \cite{feldman2016online}.
To improve on this,
we first note that modifying these $q_{i,t}$ appropriately results in each edge $(i,t)$ being matched with probability precisely $x_{i,t}\cdot (\nicefrac{1}{2}+c)$ if $\sum_{t'<t} x_{i,t'}$ is small, and
at least $x_{i,t}\cdot (\nicefrac{1}{2}-O(c))$ otherwise.
To increase these marginal probabilities to $x_{i,t}\cdot (\nicefrac{1}{2}+c)$ for each edge $(i,t)$, we repeat the above process if $t$ is unmatched, letting $t$ pick a second bin $i'$ and possibly matching edge $(i',t)$. For this second pick to achieve its desired effect, bin $i$ should not be matched too often when picked by ball $t$ in its second pick.
That is, conditioning on $t$ not being matched after its first pick should not decrease the probability of $i$ being free by too much. This is the core of our analysis.
\paragraph{\underline{Analysis.}}
To prove that conditioning on ball $t$ not being matched after its first pick indeed does not decrease the probability of bin $i$ being free by much, we show that (i) the bins' matched statuses by time $t$ have low correlation, and (ii) bin $i$ is unlikely to be picked twice by ball $t$.
To prove Property (i), we show that most of the probability of a bin to be matched by this algorithm is accounted for by variables which are negatively correlated, and even \emph{negatively associated} (see \Cref{sec:prelims}). For our proof of Property (ii), we finally reap the rewards from our new LP constraint. In particular, this constraint implies that for bins $i$ with $\sum_{t'<t} x_{i,t'}$ large, as above, $x_{i,t}$ must be low, implying that bin $i$ is unlikely to be picked by ball $t$ as its first pick.
Properties (i) and (ii) together imply that conditioning on ball $t$ not being matched after its first pick does not decrease the probability of bin $i$ to be unmatched much. This then implies that the second pick is not too unlikely to result in a match of edge $(i,t)$. We thus find that each edge $(i,t)$ is matched by our algorithm with probability $(\nicefrac{1}{2}+c)\cdot x_{i,t}$, from which our $(\nicefrac{1}{2}+c)$-approximation follows.
\begin{comment}
\subsection{Previous Techniques: Scrap}
For our algorithm, we draw inspiration from the recent prophet inequality and batched OCRS of Ezra et al.~\cite{ezra2020online}, and extend it.
In our context, we think of the algorithm of \cite{ezra2020online} as solving and rounding an LP bounding an offline algorithm's expected gain. This LP has decision variables $y_{i,t,r}$ corresponding to the marginal probabilities of edges $(i,t)$ being matched in the max-weight matching with their $r^{\textrm{th}}$ weight realization.
For any realization $r$ of the random edge weights revealed at time $t$, with probability proportional to $y_{i,t,r}$, bin $i$ is picked. If this bin is unmatched, then with some probability $q_{i,t}$, we match $(i,t)$. Choosing $q_{i,t}$ appropriately \cite{ezra2020online} results in edge $(i,t)$ being matched when its weight is $w_{i,t,r}$ with marginal probability precisely $y_{i,t,r}\cdot \nicefrac{1}{2}$, resulting in a $\nicefrac{1}{2}$-competitive algorithm.
Generalizing the above, we note that modifying $q_{i,t}$ appropriately results in an increase of this marginal probability to $y_{i,t,r}\cdot (\nicefrac{1}{2}+c)$ if bin $i$ is far from being fully fractionally matched by $y$, and only decreases the same marginal probability to $y_{i,t,r}\cdot (\nicefrac{1}{2}-O(c))$ if bin $i$ is nearly fully fractionally matched.
To increase these latter edges' marginal probabilities to $y_{i,t,r}\cdot(\nicefrac{1}{2}+c)$, we let ball $t$ repeat the above process (making a second pick), if $t$ is still unmatched after its first pick.
For this second pick to achieve its desired goal, conditioning on $t$ not being matched after its first pick must not decrease the probability of $i$ being free by too much, as the converse would imply that $(i,t)$ are unlikely to be matched due to a second pick.
Fittingly, we prove that (i) any nearly-fully fractionally matched bin $i$ is unlikely to be picked both times by $t$, and (ii) the bins' matched statuses by time $t$ have very low positive correlation (if any).
To prove Property (ii), we prove that most of the probability of a node to be matched is contributed by variables which are negatively correlated, and even \emph{negatively associated} (see \Cref{sec:prelims}). To prove Property (i), unfortunately, we run into problems, as this property simply does not hold for $y$ as above.\footnote{This is to be expected, as the converse would contradict the lower bound of $\nicefrac{1}{2}$ for prophet inequalities (competitive ratios) for single-item online Bayesian selection.}
To overcome this challenge, we identify a constraint which applies to the marginal matching probabilities of \emph{online} algorithms (but crucially, not those of offline algorithms) which in particular implies Property (i) for $y$ satisfying this additional constraint. Roughly, this constraint states that the probability of an online algorithm matching $(i,t)$, conditioned on its weight being $w_{i,t,r}$, is at most the unconditional probability that $i$ is not matched before time $t$.
Using this new constraint, we show that this second pick increases the marginal probability of edge $(i,t)$ being matched when it has weight $w_{i,t,r}$ to $y_{i,t,r}\cdot (\nicefrac{1}{2}+c)$, for $c>0$ sufficiently small and for $y$ the solution of the stronger LP which bounds the gain of the optimal online algorithm. By linearity of expectation, this yields our $(\nicefrac{1}{2}+c)$ approximation of this optimal online algorithm.
Conceptually for this problem, this implies that our new constraint indeed separates online from offline algorithms for our problem, as the latter cannot be approximated within a factor better than $\nicefrac{1}{2}$ \cite{krengel1978semiamarts}.
\end{comment}
\begin{comment}
\subsection{Techniques: Too detailed (to delete)}
Here we give a brief overview of the key ideas used to obtain our main results.
\paragraph{\textsc{PSPACE}-hardness.}
For our \textsc{PSPACE}-hardness result, we start by refining the result of \citet{condon1997random} for maximum satisfiability of stochastic SAT instances.
In the stochastic SAT (SSAT) problem, introduced by \citet{papadimitriou1985games},
a 3CNF formula is given, and variables are alternatingly set by an (online) algorithm and randomly set by nature.
\citet{condon1997random} proved that approximating the maximum expected number of satisfiable clauses of an SSAT instance is \textsc{PSPACE}-hard.
Using \emph{expander graph} constructions, we extend this result to SSAT instances in which each variable appears in at most a constant number of clauses.
We then reduce the problem of approximating maximum
satisfiability of a bounded-occurrence SSAT instance to the \ensuremath{\textsc{RideHail}}\xspace problem.
At a high level, this reduction results in a \ensuremath{\textsc{RideHail}}\xspace instance which forces an optimal online algorithm to first simulate the variable assignment phase of an algorithm for the input SSAT instance, gaining some predictable expected value, and then accrue a further gain which is proportionate to the fraction of satisfied clauses under this assignment, and also proportionate to the value gained from the first part of the input.
This then implies that a sufficiently good approximation of the optimal online \ensuremath{\textsc{RideHail}}\xspace algorithm for this instance yields a good approximation of the maximum satisfiability of the input SSAT instance, and is therefore \textsc{PSPACE}-hard.
\paragraph{Algorithm.}
For our algorithm, we draw inspiration from the recent optimally-competitive algorithm of Ezra et al.~\cite{ezra2020online}, extending it to a family of algorithms with a hyper-parameter $c\in \mathbb{R}$. Letting $y_{i,t,r}$ denote the probability of edge $(i,t)$ appearing in the max-weight matching when $w_{j,t} = w_{j,t,r}\,\,\forall j$, this family of algorithms can be stated briefly as follows: at time $t$, if $w_{j,t} = w_{j,t,r}\,\,\forall j$, ball $t$ picks a single bin $i$ with probability $y_{i,t,r}/p_{t,r}$. Then, if the picked bin $i$ is unmatched, $(i,t)$ is matched with probability
\begin{align}
q_{i,t}:=\min\left(1, \frac{\nicefrac{1}{2}+c}{1-\sum_{t'<t}\sum_{r'} y_{i,t',r'}\cdot (\nicefrac{1}{2}+c)}\right).\nonumber
\end{align}
For $c=0$, which is the algorithm of \cite{ezra2020online}, these probabilities are carefully chosen precisely to guarantee that each edge $(i,t)$ be matched when $w_{j,t}=w_{j,t,r} \,\,\forall j$ with marginal probability $y_{i,t,r}\cdot \nicefrac{1}{2}$ (easily provable by induction). From this, the competitive ratio of $\nicefrac{1}{2}$ follows directly by linearity of expectation.
To approximate the optimal online algorithm within a factor of $\nicefrac{1}{2}+c$, for $c > 0$, we first note that
for bins $i$ for which the minimizer in the expression for $q_{i,t}$ is not one, the edge $(i,t)$ is matched when $w_{j,t}=w_{j,t,r} \,\,\forall j$ with marginal probability $y_{i,t,r}\cdot (\nicefrac{1}{2}+c)$.
For the remaining problematic bins, for which $\sum_{t'<t} \sum_{r'} y_{i,t',r'}$ must be high, the marginal probability of this event is not much lower, and in particular is at least $y_{i,t,r}\cdot (\nicefrac{1}{2}-O(c))$. The question, then, is how to increase this marginal probability to $y_{i,t,r}\cdot (\nicefrac{1}{2}+c)$.
Our approach is simple to describe: if $t$ is not matched by the above process, we have $t$ make a second pick in an attempt to match to one of these problematic bins.
The challenge in analyzing this approach is to show that such a second pick achieves its intended goal of increasing the marginal probability of matching such problematic bins $i$. In particular, if conditioning on
the first pick not resulting in a match increases the probability of bin $i$ being previously matched, then this second pick may not increase the marginal probability of edge $(i,t)$ to be matched sufficiently (or indeed, at all).
Such a bad scenario happens if (i) bin $i$ is likely to be picked in both attempts, and/or (ii) there is strong positive correlation between the matched status of bins.
To overcome issue (i), we identify a new constraint which applies to online algorithms but not to offline ones, and add this constraint to our LP with which we upper bound the online optimum's expected value, now letting $y_{i,t,r}$ be the decision variables of this LP.
Specifically, we note that, conditioned on any realization of the edge weights of $t$, an edge $(i,t)$ is matched by an online algorithm with probability at most equal to the probability of the online algorithm not matching this bin before time $t$.\footnote{Note that this constraint does not apply to offline algorithms, which may determine their matches at time $t-1$ based on the realized weights at time $t$.} Formally, this gives the following constraint.
\begin{align}\label{new-constraint}
y_{i,t,r} \leq p_{t,r} \cdot \left(1-\sum_{t'<t}\sum_{r'} y_{i,t',r'}\right).
\end{align}
Constraint \eqref{new-constraint} plays a key role in our proof, as it implies that the probability of picking a problematic bin $i$ in the first attempt is small, specifically at most a small $y_{i,t,r}/p_{t,r} \leq 1-\sum_{t'<t}\sum_{r'} y_{i,t',r'}$.
To overcome issue (ii), we show, relying on the theory of negative association (see \Cref{sec:prelims}), that our algorithm results in bins having only very weakly positively correlated matched statuses.\todo{should address relation to \cite{gamlath2019online}, which might be asked. see comment}
While we do not elaborate on this part of our analysis here due to the required background, this is the heart of our proof.\todo{say something more here}
Combining the above, we find that each edge $(i,t)$ is matched when $w_{j,t} = w_{j,t,r}\,\,\forall j$ either as a first pick or as a second pick with marginal probability at least $(\nicefrac{1}{2}+c)\cdot x_{i,t}$. Our desired $(\nicefrac{1}{2}+c)$-approximation of the optimal online \ensuremath{\textsc{RideHail}}\xspace algorithm then follows by linearity of expectation.
\end{comment}
\begin{comment}
\paragraph{Algorithm.}
For our polynomial-time approximation of the optimal online algorithm, we rely on two key ideas.
The first is strengthening the natural LP benchmark capturing the expected \emph{offline} optimum matching to also include constraints applicable only to online algorithms.
In particular, we add a constraint which states that a bin $i$ and ball $t$ can be matched with probability at most $p_t$, the arrival probability of $t$, times the probability that $i$ is not matched prior to the arrival of $t$.
That is, denoting by $x_{i,t}$ the probability with which an online algorithm matches edge $(i,t)$, we add the constraint
\begin{align}\label{new-constraint}
x_{i,t}\leq p_t\cdot \left(1-\sum_{t'<t}x_{i,t'}\right).
\end{align}
This constraint naturally applies to all online algorithms, but not to offline ones, which may use information about realization of arrivals of balls arriving after $t$. For example, an offline algorithm can decide to always match edge $(i,t)$ if $t$ appears, and match $i$ to some ball $t'<t$ if it does not.
The second idea, which relies crucially on Constraint \eqref{new-constraint}, is based on an extension of the recent batch prophet inequality algorithm of \citet{ezra2020online}, which can be used to match each edge $(i,t)$ with probability $\nicefrac{1}{2}\cdot x_{i,t}$. For our setting, this algorithm has an arriving online node $t$ pick a single bin $i$ with probability $x_{i,t}/p_t$, and if this bin is free, match $(i,t)$ with probability $1/(1-\sum_{t'<t} \nicefrac{1}{2}\cdot x_{i,t'})$.
A simple proof by induction shows that $1/(1-\sum_{t'<t} \nicefrac{1}{2}\cdot x_{i,t'}) = 1/\Pr[i \textrm{ free at time } t]$, and so this algorithm matches each edge $(i,t)$ with probability precisely $\nicefrac{1}{2}\cdot x_{i,t}$.
We note that increasing the probability of matching ball $t$ to its picked bin $i$ if free to $\min\left(1, (\nicefrac{1}{2}+c)/\Pr[i \textrm{ free at time }t]\right)$ will even result in edges $(i,t)$ for which $(\nicefrac{1}{2}+c)/\Pr[i \textrm{ free at time }t]\leq 1$ to be matched with probability $(\nicefrac{1}{2}+c)\cdot x_{i,t}$.
On the other hand, for sufficiently small (but positive constant) $c>0$, this results in bins $i$ for which $(\nicefrac{1}{2}+c)/\Pr[i \textrm{ free at time }t]> 1$, i.e., bins with low probability of being free at time $t$, which must also have $\sum_{t'<t}x_{i,t'}$ large, the probability of $(i,t)$ being matched is at least $(\nicefrac{1}{2}-O(c))\cdot x_{i,t}$.
In order to increase these edges' probability of being matched to $(\nicefrac{1}{2}+c)\cdot x_{i,t}$, we repeat the above process if $t$ was not matched following its first pick.
The problem here is to show that such a second pick increases the probability of such high-(fractional-)degree bins $i$ to be matched. In particular, if conditioning on
the first pick not resulting in a match increases the probability of bin $i$ being previously matched, this second pick may not increase the probability of edge $(i,t)$ to be matched at all.
This may happen if, for example (i) bin $i$ is picked in both attempts, and (ii) there being strong positive correlation between the matched status of bins.
Constraint \eqref{new-constraint} plays a key role in avoiding issue (i), as it implies that the probability of picking bin $i$ in the first attempt is small, specifically at most $x_{i,t}/p_t \leq 1-\sum_{t'<t}x_{i,t'}$.
To overcome issue (ii), we show, relying on the theory of negative association (see \Cref{sec:prelimNA}), that our algorithm results in bins having only very weakly positively correlated matched statuses.
From this we find that each edge $(i,t)$ is matched with probability at least $(\nicefrac{1}{2}+c)\cdot x_{i,t}$, yielding our desired $(\nicefrac{1}{2}+c)$-approximation of the optimal online \ensuremath{\textsc{RideHail}}\xspace algorithm.
\end{comment}
\begin{comment}
We show how this constraint allows us to match ly, and in each edge $(i,t)$ with probability $(\nicefrac{1}{2}+c)\cdot x_{i,t}$, by: (1) matching each edge $(i,t)$
This constraint implies, in particular, that if we let a ball $t$ pick a single bin $i$ with probability $x_{i,t}/p_t$, the probability $i$ is picked is rather small if $\sum_{t'<t}x_{i,t'}$ is large.
This allows us to design, by extending the batched-OCRS approach of \cite{ezra2020online}, an algorithm which matches all edge $(i,t)$ with $\sum_{t'<t} x_{i,t'}$ small to an extent of $(\nicefrac{1}{2}+c)\cdot x_{i,t}$, while
Our second idea is to round the above LP solution $x$ so as to obtain an online algorithm which matches each pair $(i,t)$ with probability $(\nicefrac{1}{2}+c)\cdot x_{i,t}$ for some constant $c>0$.
Here $x_{i,t}$ are the decision variables of the LP, which we think of as the probability of the edge $(i,t)$ to be matched by the optimal online algorithm.
Broadly, to guarantee the above, our approach will be at each time $t$, if ball $t$ arrives, to pick a single random bin $i$ with probability $x_{i,t}/p_t$, and then, if bin $i$ is free (unmatched), match $(i,t)$ with probability $(\nicefrac{1}{2}+c)/\Pr[i \textrm{ free at time }t]$. Unfortunately, this expression is not always a probability. In particular, this expression exceeds one if (and only if) $\sum_{t'<t} x_{i,t'}$ is sufficiently large.
We therefore match the free picked bin $i$ to ball $t$ with probability $\min\left(1,(\nicefrac{1}{2}+c)/\Pr[i \textrm{ free at time }t]\right)$.
This makes edges $(i,t)$ with $\sum_{t'<t} x_{i,t'}$ sufficiently small be matched with probability at least $(\nicefrac{1}{2}+c)\cdot x_{i,t}$, while other edges are matched with probability at least $(\nicefrac{1}{2} - O(c))\cdot x_{i,t}$.
To increase these edges' probability of being matched, we repeat the process of $t$ picking a bin $i$ and the bin accepting if it is free one more time.
Our new LP constraint, $x_{i,t}\leq p_t\cdot (1-\sum_{t'<t}x_{i,t'})$, implies that if edge $(i,t)$ is not matched due to $\sum_{t'<t} x_{i,t'}$ being large, then bin $i$ is likely to not have been picked by $t$ in its first attempt.
Crucially, we also show that this algorithm results in the matching status of different bins having bounded positive correlation.
Consequently, conditioning on the event that $t$ is not matched after its first pick does not decrease the probability of $i$ being free much, and therefore a second pick increases the probability of matching such an edge $(i,t)$ to a $(\nicefrac{1}{2}+c)\cdot x_{i,t}$.
\end{comment}
\subsection{Related Work}\label{sec:related-work}
The literature on online Bayesian selection problems is a long and illustrious one. We briefly outline some of the most relevant work here. See also surveys on the topic \cite{correa2019recent,hill1992survey,lucier2017economic,hartline2012approximation}.
A seminal result in the stopping theory literature, the first prophet inequality, a $\nicefrac{1}{2}$-competitive algorithm for the single-item online Bayesian selection problem, was first given in the late 70s \cite{krengel1978semiamarts}. Multiple algorithms achieving this bound are known \cite{samuel1984comparison,alaei2014bayesian,kleinberg2019matroid,ezra2020online,alaei2012online}.
On the other hand, better bounds are known for various special cases, most prominently for i.i.d.~distributions \cite{correa2017posted,hill1982comparisons,abolhassani2017beating}.
Numerous \emph{multiple-item} online Bayesian selection problems were studied over the years.
Generalizations of the classic $\nicefrac{1}{2}$-competitive prophet inequality of \cite{krengel1978semiamarts} for single-items were given for matroid constraints \cite{kleinberg2019matroid},
for multiple items \cite{alaei2014bayesian},
for bipartite matching under one-sided vertex arrivals \cite{alaei2012online,feldman2015combinatorial},
and for general matching under vertex arrivals \cite{ezra2020online}, with positive results known for many other constraints \cite{dutting2020prophet,feldman2016online,feldman2015combinatorial,kleinberg2019matroid, dutting2020log}.
For matching under \emph{edge} arrivals, a number of positive results are known \cite{feldman2016online,kleinberg2019matroid,gravin2019prophet}, and a competitive ratio of $\nicefrac{1}{2}$ is impossible for this stochastic problem \cite{gravin2019prophet,ezra2020online}. This mirrors a similar separation between vertex arrivals and edge arrivals for this problem's (unweighted) deterministic counterpart \cite{gamlath2019online}.
Much of this work on approximating the optimal offline algorithm (prophet inequalities) for online Bayesian selection problems was motivated by connections discovered between prophet inequalities and algorithmic mechanism design \cite{hajiaghayi2007automated,chawla2010multi,correa2019pricing,feldman2015combinatorial}.
The computational complexity of approximating the \emph{online} optimal algorithm, however, was significantly less well studied.
The only previous positive result for approximating the online optimum algorithm (better than offline optimum) for an online Bayesian selection problem is due to Anari et al.~\cite{anari2019nearly}, who gave a PTAS for a special class of matroid constraints.
On the computational complexity front, the only hardness for such problems we are aware of is the recent result of Agrawal et al.~\cite{agrawal2020optimal}, who show that computing the optimal \emph{ordering} of the random variables for a single-item problem is \textsc{NP}-hard (with an EPTAS for this ordering problem due to \cite{segev2020efficient}).
The (in)approximability of the optimum online algorithm was studied for other stochastic online optimization problems recently, including probing problems \cite{goel2010probe, chen2016combinatorial, fu2018ptas,segev2020efficient},
stochastic matching problems in infinite-horizon settings under Poisson arrivals and departures \cite{aouad2020dynamic}, two-stage stochastic matching problems \cite{feng2021two}, and stochastic dynamic programming problems \cite{fu2018ptas}.
The computational complexity of approximating $OPT_{on}$ for these and other problems remains an intriguing open problem.
We are hopeful that the tools we develop here will prove useful in extending the literature on computational complexity and approximability of such problems of decision-making under uncertainty.
\paragraph{\textbf{Follow-up work:}} Following this work, the last two authors have extended this paper's algorithm to obtain improved algorithms for the (seemingly unrelated) online edge coloring problem \cite{saberi2021greedy}. Their ideas can be used to improve our approximation ratio from $0.51$ to $0.526$. In another work, Kessel et al.~\cite{kessel2021stationary} study a stationary version of the prophet inequality problem, and obtain optimal competitive ratios, and improved approximation of the optimal online algorithm. Whether other online Bayesian selection problems admit better (efficient) approximation of their optimal online algorithms compared to the optimal prophet inequality remains to be seen. (See \Cref{sec:conclusion}.)
\section{Preliminaries}\label{sec:prelims}
For any algorithm $\mathcal{A}$ and instance $\mathcal{I}$ of a problem $\Pi$, we let $\mathcal{A}(\mathcal{I})$ denote the value of the output of algorithm $\mathcal{A}$ on instance $I$. We use $OPT_{on}^{\Pi}(\mathcal{I})$ to denote an optimal online algorithm for $\Pi$ on $\mathcal{I}$. Since the problem $\Pi$ will be clear from context, we will usually just write $OPT_{on}(\mathcal{I})$.
Our interest is in understanding how well this value can be approximated by efficient online algorithms.
Throughout, we say an algorithm gives an $\alpha$-approximation to a quantity $Q$, for $\alpha\in (0,1)$, if it outputs a number in the range $[\alpha Q, Q]$.
The following simple fact, whose proof is deferred to \Cref{app:ommittedproofs}, is useful for reductions involving hardness of approximation.
\begin{restatable}{fact}{approxfact}\label{lem:approxlem}
Let $Q, Q' \geq 0$ be positive quantities, such that $Q'/Q\leq \beta$, and let $\alpha \in (0,1)$. Then, an $\big(\frac{\alpha+\beta}{1 + \beta }\big)$-approximation to $Q + Q'$ yields an $\alpha$-approximation to $Q$.
\end{restatable}
We now turn to providing background on problems and tools used in this work.
\paragraph{\underline{Stochastic SAT.}}\label{sec:SSAT}
The stochastic SAT (SSAT) problem was first defined by Papadimtriou \cite{papadimitriou1985games}.
In this work, we will consider the maximization variant of this problem, defined below.
\begin{definition} The input to the MAX-SSAT problem is a 3CNF formula $\phi$ over an ordered list of variables $(x_1, x_2, \ldots, x_n)$. We choose a value of either \texttt{True} or \texttt{False} for $x_1$, nature chooses a value of either \texttt{True} or \texttt{False} for $x_2$ uniformly at random, we choose a value of either \texttt{True} or \texttt{False} for $x_3$, and so on. Our goal is to maximize the expected number of satisfied clauses in $\phi$ after all the variables have been assigned a value. We will refer to $\{x_1, x_3, \ldots \}$ as the ``deterministic variables'' and $\{x_2, x_4, \ldots \}$ as the ``random variables.''
\end{definition}
In his work introducing SSAT, \citet{papadimitriou1985games} proved PSPACE-hardness of determining the probability of satisfiability of an SSAT instance.
Over a decade later, this was improved to a
\emph{hardness of approximation} result by \citet{condon1997random}, via extensions of the PCP theorem \cite{arora1998proof}.
In particular, they prove the following hardness of approximation result.
\begin{restatable}{lemma}{randomdebaters}(\cite[Theorem 3.3]{condon1997random}) \label{thm:k-max-ssat-hard}
There exist constants $k \in \mathbb N$ and $\alpha \in (0,1)$ so that it is PSPACE-hard to compute an $\alpha$-approximation to $OPT_{on}(\phi)$ for a MAX-SSAT instance $\phi$ satisfying:
\begin{enumerate}
\item no random variable appears negated in any clause of $\phi$, and
\item each random variables appears in at most $k$ clauses of $\phi$.
\end{enumerate}
\end{restatable}
It is worth noting that Theorem 3.3 in \cite{condon1997random} only includes the statement about random variables being non-negated. The second property is a direct consequence of the proof of the theorem. In \Cref{app:ommittedproofs} we explain the necessary modifications to the proof to add this guarantee.
\paragraph{\underline{Expander Graphs.}}
Define the expansion of a graph $G$ as
$$h(G) := \min_{S \subseteq V, |S| \le |V|/2} \frac{|E(S, V \setminus S)|}{|S|},$$
where $E(X,Y):=\{(x,y)\in E\mid e\in X, y\in Y\}$ denotes the edges with one endpoint in $X$ and the other in $Y$.
We will utilize results providing explicit, deterministic constructions of graphs with constant degree and constant expansion (e.g. \cite{gabber1981explicit,lubotzky1988ramanujan}).
\begin{lemma} \label{lem:explicitexpanders}
There exists a deterministic, polynomial-time construction of a graph on $n$ vertices with expansion at least 1 and maximum degree at most some constant $d$.
\end{lemma}
\paragraph{\underline{Negative Association.}}
\label{sec:prelimNA}
We briefly review some notions of negative dependence we need in this work, in particular, the notion of \emph{Negatively Associated} random variables.
\begin{definition}[\cite{khursheed1981positive,joag1983negative}]\label{def:NA}
Random variables $X_1,\dots,X_n$ are \emph{negatively associated (NA)}, if every two monotone non-decreasing functions $f$ and $g$ defined on disjoint subsets of the variables in $\vec{X}$ are negatively correlated. That is,
\begin{equation}\label{eq:NA}
\mathbb E[f\cdot g] \leq \mathbb E[f]\cdot \mathbb E[g].
\end{equation}
\end{definition}
A family of independent random variables are trivially negatively associated. A more interesting example of negatively associated random variables is the following.
\begin{proposition}[0-1 Principle \cite{dubhashi1996balls}]\label{0-1-NA}
Let $X_1,\dots,X_n\in \{0,1\}$ be binary random variables such that $\sum_i X_i\leq 1$ always. Then, the joint distribution $(X_1,\dots,X_n)$ is negatively associated.
\end{proposition}
More elaborate NA distributions can be obtained via the following closure properties.
\begin{proposition}[NA Closure Properties \cite{khursheed1981positive,joag1983negative,dubhashi1996balls}]\label{NA-closure}
$\phantom{a}$
\begin{enumerate}
\item \label{P7_union} \underline{Independent union.}
Let $(X_1,\dots,X_n)$ and $(Y_1,\dots,Y_m)$ be
two mutually independent negatively associated joint distributions. Then, the joint distribution $(X_1,\dots,X_n,Y_1,\dots,Y_m)$ is also NA.
\item \label{P6_inc_funs} \underline{Function composition.}
Let\, $\mathbf{X}= (X_1,\dots,X_n)$ be NA, and let $f_1,\dots,f_k$ be monotone non-decreasing functions defined on disjoint subsets of\, $\mathbf{X}$. Then the joint distribution $(f_1,\dots,f_k)$ is also NA.
\end{enumerate}
\end{proposition}
Negative association implies many powerful concentration inequalities and other useful properties (see e.g., \cite{dubhashi1996balls,khursheed1981positive,joag1983negative,asadpour2017log}). For our purposes we will use the pairwise negative correlation of NA variables, implied by \Cref{eq:NA} with the disjoint functions $f(\vec{X})=X_i$ and $g(\vec{X})=X_j$ for $i\neq j$.
\begin{proposition}\label{NA:neg-corr}
Let $X_1,\dots,X_n$ be NA random variables. Then, for all $i\neq j$,
$\mathrm{Cov}(X_i,X_j)\leq 0$.
\end{proposition}
|
1,314,259,995,843 | arxiv | \section{Introduction}
Cryptographic algorithms and protocols are the indispensable building blocks for modern distributed computing systems and smartphone apps. Cryptographic primitives, implemented as cryptographic APIs in typical libraries, provide various functionalities such as keys/certificate management, encryption, decryption, digital signature, message digest, and utilization of security protocols. Such cryptographic primitives have been reported vulnerable to many attacks and misused seriously on different platforms \cite{DBLP:conf/apsys/LazarCWZ14, DBLP:conf/ccs/EgeleBFK13, DBLP:conf/nss/LiZLG14, DBLP:conf/codaspy/AlghamdiALM18}.
Previous detection approaches of cryptographic misuses fall into three categories: static analysis \cite{DBLP:conf/ccs/EgeleBFK13, DBLP:conf/ccs/MuslukhovBB18, DBLP:conf/ccs/MaLLD16, DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/ccs/RahamanXASTFKY19, DBLP:conf/ccs/FahlHMSBF12}, dynamic analysis \cite{DBLP:conf/csfw/FocardiS17}, or the combination of both \cite{DBLP:conf/ndss/SounthirarajSGLK14, DBLP:conf/nss/LiZLG14}. In general, static analysis approaches tend to be more scalable. Unlike studies that take the common rules of misuses as a threat model \cite{DBLP:conf/ccs/EgeleBFK13, DBLP:conf/ccs/MuslukhovBB18, DBLP:conf/ccs/MaLLD16}, recent studies \cite{DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/ccs/RahamanXASTFKY19} focused on more fine-grained vulnerability types of cryptographic misuses. To create and maintain the up-to-date rules of misuses from the developers' perspective, the research community has investigated the evolutionary characteristics of cryptographic misuses and their security fixes \cite{DBLP:conf/pldi/PaletovTRV18, DBLP:conf/msr/GaoKLBK19}. Because of the pervasiveness and diversity of cryptographic misuses, one-hop mitigation is impractical. For example, weak hashing may be used to check the integrity of inconsequential local data or the password delivered to some network interface. The developer tends to fix the later case in priority. We have to estimate the real threat of the misuses and decide how we react to the related vulnerabilities, e.g., tolerating, correcting, or applying runtime resistance on them. Currently, the way used to estimate the threat level of cryptographic misuses is simple. The state-of-the-art approaches, e.g., in Table 1 of \cite{DBLP:conf/ccs/RahamanXASTFKY19}, attach the severity level to the vulnerability type itself.
This kind of assessment is coarse-grained and imprecise for measuring the real threat caused by cryptographic misuses. In Android apps, the risk of cryptographic misuse is highly dependent on the data flows affected by the misuse. Suppose a data source operated by some cryptographic misuse flows to a dangerous sink, such as a sink of the network. In that case, we believe this kind of cryptographic misuse is riskier than those not involved in any sensitive data flow.
This paper proposes an extensible framework for assessing the risk of cryptographic misuses in Android apps. The framework consists of two main components. The first is an adapter-based detection that assembles three state-of-the-art misuse detectors into a detector chain. Each detector in the chain has an adapter to parse the output of the detector, interpret the detected cryptographic misuses into a unified format, and map each misuse to a concrete type in a more comprehensive list of vulnerabilities. The second component is a data-flow-driven risk assessment to quantify each app's threat level of cryptographic misuses and profile the threat summary to guide the app vetting on a large scale. This component uses a misuse-originating data-flow analysis to obtain the data flows between misuse and possible information-leaking channels. Then we quantify the overall security risk for an app based on such data flows. A clustering-based approach is used in our evaluation to predict the most significant threats of apps on a large scale. We highlight our contributions as follows:
\begin{compactitem}[\textbullet]
\item We propose a new detection scheme to improve the precision and recall of static cryptographic misuse detection. The new scheme aims to assemble multiple detectors as a toolchain to detect a more comprehensive list of cryptographic vulnerabilities, which is a more complete threat model in the cryptographic misuse analysis of Android apps.
\item We propose a risk assessment of cryptographic misuses based on misuse-originating data-flow analysis with precise sink category identification. With the identified data flows between cryptographic misuses and the related data leakages, the risk level of misuses is quantified for each app, and the most significant threats of misuses are predicted on a large scale with clustering for the app vetting.
\item With an instantiated implementation of our framework, we evaluate the effectiveness of our threat model, the accuracy of the adapter-based detection, and the effect of data-flow-driven risk assessment on over 40,000 real-world apps, including several popular apps. Our security observations indicate several important risk patterns.
\end{compactitem}
\section{Motivating Example}\label{sec:example}
The example in Figure~\ref{fig:example} shows that different data flows originating from cryptographic misuse may lead to different impacts and severities to the application. In method \textsf{encrypt}, we firstly derive the encryption key with \textsf{KeyGenerator}. Then we use \textsf{Cipher.getInstance} to construct and initialize \textsf{Cipher} object. Finally, we call \textsf{Cipher.doFinal} to encrypt the sensitive string and return the ciphertext. The parameter of \textsf{Cipher.getInstance} should consist of three parts: \textit{algorithm}, \textit{mode of operation}, and \textit{padding scheme}. When we only specify the algorithm ``\textsf{AES}'', the method will apply the ECB mode of operation by default, which has been proved insecure. The state-of-the-art approach~\cite{DBLP:conf/ccs/RahamanXASTFKY19} simply identifies the risk level of this misuse as \emph{medium} severity. In contrast, we want to further understand how this misuse may be exploited to characterize its severity better. If we treat this misused call to \textsf{Cipher.getInstance} as a data source, the program has two sensitive data flows: \ding{182}$\rightarrow$\ding{183}$\rightarrow$\ding{184}$\rightarrow$\ding{185}$\rightarrow$\ding{187} and \ding{182}$\rightarrow$\ding{183}$\rightarrow$\ding{184}$\rightarrow$\ding{186}$\rightarrow$\ding{188}. We want to classify the sinks of these flows into different categories to distinguish the severity of the two sensitive data flows. However, identifying the precise category of some sink might not be straightforward. For example, the category of \textsf{DataOutputStream.write}, i.e., \textsf{NETWORK} in our categorization, is related to the type of \textsf{urlConn}. We apply an intra-procedural data-source tracking, i.e., \ding{187}$\rightarrow$\ding{189}, to find such correlation and help categorize the sink. As we realized, the two sensitive flows serve as potential attack surfaces to exploit the program's misuse. However, the flow to \textsf{NETWORK}(\ding{187}) is generally riskier than the flow to \textsf{FILE}(\ding{188}). Thus, we assign risk weights to different categories of sinks according to their intuitive severity so that the overall risk of the app can be quantified in a more precise way based on the identified \emph{misuse-originating data-flows}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{example/example}
\caption{Motivating Example}\label{fig:example}
\vspace{-3ex}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{fig/CryptoEval-overview}
\caption{Framework of CryptoEval}\label{fig:framework}
\vspace{-3ex}
\end{figure}
\section{CryptoEval Design}\label{sec:approach}
This section describes our framework for assessing the threat level of cryptographic misuses in Android apps.
\subsection{Overview}
We present the framework of CryptoEval in Figure~\ref{fig:framework}. Firstly, our approach assembles the state-of-the-art cryptographic misuse detectors, e.g., \cite{DBLP:conf/ccs/RahamanXASTFKY19, DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/ccs/MuslukhovBB18}, as a toolchain to detect a more comprehensive list of vulnerability types and to achieve high accuracy of detection, which is accomplished by developing a series of adapters. A typical adapter translates the output of a detector into a unified format whose elements will be formally defined in Section~\ref{subsec:adapter}.
Then, according to the results of cryptographic misuse detection, we perform a misuse-originating data-flow analysis to establish the dependencies between the vulnerabilities caused by the misuses and different sink categories. This analysis can take advantage of the state-of-the-art context-sensitive and object-sensitive taint analysis on Android apps, e.g., FlowDroid\cite{DBLP:conf/pldi/ArztRFBBKTOM14} and Amandroid\cite{DBLP:conf/ccs/WeiROR14}. Their sources and sinks are easily configurable. For each app, we update the list of taint sources based on the result of misuse detection. Then, a customized data-flow analysis is conducted from both parameter-related misuses and data-related misuses to find all the specific sinks that may leak data affected by the misuses. Then we use an intra-procedural data-source tracking started from these sinks to infer the exact sink categories. We attach the sink categories to each misuse to build the explicit correlations and quantify the overall threat level of each app.
\begin{table*}
\renewcommand{\arraystretch}{1.1}
\caption{Comprehensive list of vulnerabilities caused by cryptographic misuses. Abbreviation of each approach: CG=CryptoGuard\cite{DBLP:conf/ccs/RahamanXASTFKY19}, CL=CryptoLint\cite{DBLP:conf/ccs/EgeleBFK13}, BS=BinSight\cite{DBLP:conf/ccs/MuslukhovBB18}, CR=CDRep\cite{DBLP:conf/ccs/MaLLD16}, CC=CogniCrypt$_\text{SAST}$\cite{cognicrypt}, FD=FixDroid\cite{DBLP:conf/ccs/NguyenWA0WF17}, DC=DiffCode\cite{DBLP:conf/pldi/PaletovTRV18}}
\label{tab:rules}\footnotesize
\centering
\begin{tabular}{rllllll}
\hline
ID & Vulnerability Description of CryptoEval & \textsf{CG} & \textsf{CL}/\textsf{BS}/\textsf{CR}$^\dagger$ & \textsf{CC} & Table~1,\textsf{FD} & Fig~9,\textsf{DC} \\
\hline
1 & Predictable/constant cryptographic keys & vul.1 & Rule~3 & RequiredPredicateError & Pitfall~3 & R10 \\
2 & Predictable/constant passwords for PBE & vul.2 & N/A & NeverTypeOfError & N/A & N/A \\
3 & Predictable/constant passwords for KeyStore & vul.3 & N/A & NeverTypeOfError & N/A & N/A \\
4 & Custom Hostname verifiers to accept all hosts & vul.4 & N/A & N/A & Pitfall~6 & N/A \\
5 & Custom TrustManager to trust all certificates & vul.5 & N/A & N/A & Pitfall~11 & N/A \\
6 & Custom SSLSocketFactory w/o manual Hostname verification & vul.6 & N/A & N/A & N/A & N/A \\
7 & Occasional use of HTTP & vul.7 & N/A & N/A & Pitfall~8,9 & N/A \\
8 & Usage of expired protocol by SSLContext & N/A & N/A & ConstraintError & N/A & N/A \\
9 & Predictable/constant PRNG seeds & vul.8 & Rule~6 & TypestateError & Pitfall~7 & R12 \\
10 & Cryptographically insecure PRNGs (e.g., java.util.Random) & vul.9 & N/A & RequiredPredicateError & N/A & R3,R6 \\
11 & Static Salts in PBE & vul.10 & Rule~4 & RequiredPredicateError & N/A & R11 \\
12 & ECB mode in symmetric ciphers & vul.11 & Rule~1 & ConstraintError & Pitfall~1,5 & R7 \\
13 & Static IVs in CBC mode symmetric ciphers & vul.12 & Rule~2 & RequiredPredicateError & Pitfall~2 & R9 \\
14 & Fewer than 1,000 iterations for PBE & vul.13 & Rule~5 & ConstraintError & Pitfall~4 & R2 \\
15 & 64-bit block ciphers (e.g., DES, IDEA, Blowfish, RC4, RC2) & vul.14 & N/A & ConstraintError & N/A & R8 \\
16 & Insecure asymmetric ciphers (e.g, RSA, ECC) & vul.15 & N/A & ConstraintError & N/A & N/A \\
17 & Insecure cryptographic hash (e.g., SHA1, MD5, MD4, MD2) & vul.16 & Rule~7$^\dagger$ & ConstraintError & N/A & R1 \\
18 & Incorrect sequence of cryptographic API calls & N/A & N/A & TypestateError & N/A & N/A \\
19 & Usage of forbidden APIs & N/A & N/A & ForbiddenMethodError & N/A & R4 \\
20 & Incomplete usage of cryptographic API & N/A & N/A & IncompleteOperationError & N/A & N/A \\
21 & Suspected usage needs further testing & N/A & N/A & ImpreciseValueExtractionError & N/A & N/A \\
\hline
\end{tabular}
\vspace{-3ex}
\end{table*}
To extend the usage of our per-app assessment, we investigate the potential threat patterns of apps on a large scale for the subsequent app vetting at app stores. We extract the features from the misuse tuples with sink information, use unsupervised learning to predict the potential representative threats. An association rule mining is conducted to infer the correlations between different sensitive data flows.
\subsection{Adapter Construction}\label{subsec:adapter}
Firstly, we investigate the state-of-the-art approaches to analyzing cryptographic misuses \cite{DBLP:conf/ccs/RahamanXASTFKY19, DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/pldi/PaletovTRV18, DBLP:conf/ccs/NguyenWA0WF17, DBLP:conf/ccs/EgeleBFK13, DBLP:conf/ccs/MaLLD16, DBLP:conf/ccs/MuslukhovBB18}. In these works, the typical features of the vulnerabilities caused by cryptographic misuses and the principles of cryptographic usage are defined. Based on the investigation, we summarized a more comprehensive list of vulnerabilities caused by cryptographic misuses, as shown in Table~\ref{tab:rules}. These vulnerabilities are reported as an extension to the vulnerabilities in \cite{DBLP:conf/ccs/RahamanXASTFKY19}, i.e., the same vulnerability has the same description. The new list of vulnerabilities covers most of the cryptographic misuses claimed by these works. We find that CryptoGuard and \textsc{CogniCrypt}$_\textsc{sast}$ address more vulnerability types in this list than the other approaches. Meanwhile, several approaches have addressed vulnerabilities caused by other reasons. For example, FixDroid \cite{DBLP:conf/ccs/NguyenWA0WF17} also mentioned the security coding pitfalls of SQL injection and local HTML loading. Some rules elicited by DiffCode \cite{DBLP:conf/pldi/PaletovTRV18}, e.g., using BouncyCastle library (R5) and key integrity check (R13), are too restrictive to be a common type of cryptographic misuses. Therefore we do not mention them in our vulnerability list.
According to the ability of different detection approaches presented in Table~\ref{tab:rules}, we found that integrating existing detection tools as a detector chain is sufficient to detect all the vulnerability types in our list.
Two aspects decide the extensibility of this detector chain: 1) the misuse types covered by the detection approaches, 2) the informativeness of the output of detectors. To represent the cryptographic misuse in a unified form, we formally define the sink-related cryptographic misuse as follow.
\begin{definition}[sink-related cryptographic misuse]
A \emph{sink-related cryptographic misuse} is a tuple $\langle m,id,p,d,t,\textsf{S}\rangle$. It consists of
\begin{compactitem}[\textbullet]
\item $m$: the signature of misused cryptographic API. Generally, the definition of this method is provided by the cryptographic libraries.
\item $id$: the $id$ of vulnerability caused by the misuse, as defined in Table~\ref{tab:rules}.
\item $p$: the signature of the parent method, i.e., the user-defined method where the misuse is located in.
\item $d$: the descriptions of misuse, including the information about the reason and actual parameters of the misuse.
\item $t$: the tool used to detect the misuse, e.g., \textsf{CG}, \textsf{CC}, \textsf{BS}, etc.
\item \textsf{S}: the list of sink categories. Each of the sink categories contains at least one sink that is detected to release sensitive data affected by the misused cryptographic API call.
\end{compactitem}
\end{definition}
Each adapter we construct is a batch-job tool that takes the output of a cryptographic misuse detector as input and produces a set of these tuples. The tuples parsed from the output of detectors hold empty \textsf{S} until the sink categories are detected in the data-flow analysis in Section~\ref{subsec:taint-analysis}.
\begin{corollary}[valid detector chain]\label{corollary1}
A set of cryptographic misuse detectors is \emph{valid} to our approach, iff the following conditions are satisfied:
\begin{compactitem}[\textbullet]
\item the union of vulnerability types addressed by each detector contains our comprehensive list of vulnerability types.
\item the output of each detector can be parsed to a set of misuse tuples.
\end{compactitem}
\end{corollary}
We use CryptoGuard and \textsc{CogniCrypt}$_\textsc{sast}$ as the minimal valid detector chain and discuss to use an extension of the detector chain in RQ1 of Section~\ref{sec:evaluation}. Deciding the element $id$ of each tuple involves mapping from the misuse types of existing approaches to the vulnerabilities in our list. For the detection output of some approaches, e.g., CryptoGuard and BinSight, such mapping is an easy one-to-one case, see \textsf{CG} and \textsf{BS} in Table~\ref{tab:rules}.
However, for the output of \textsc{CogniCrypt}$_\textsc{sast}$, the misuse types are more coarse-grained than our vulnerability list. Therefore, mapping the misuses in their types, e.g., the misuse with type constantError$\in Rule_{\textsf{CC}}$, to our vulnerability types needs more work which will be presented in Section~\ref{sec:implementation}.
\subsection{Misuse-originating Data-Flow Analysis}\label{subsec:taint-analysis}
This section introduces a misuse-originating data-flow analysis to find the sensitive data flows originating from the cryptographic misuses and the related sink categories. Firstly, we investigate the functionality of cryptographic APIs and divide them into two categories:
\begin{compactitem}[\textbullet]
\item Data-related APIs (DAPI): the APIs used for direct data processing, e.g., encryption, message digest, MAC, SSL/TLS operation.
\item Parameter-related APIs (PAPI): the APIs used for constructing the parameters required by DAPI, e.g., PRNG, key generation, certificate management.
\end{compactitem}
\noindent We manually classify the APIs in JCA cryptographic libraries through a careful study of the JCA API document and the results are given in Table~\ref{tab:api-classification}. The usage of data-related API usually depends on the usage of some parameter-related APIs. For example, in Figure~\ref{fig:example}, if we initialize a 64-bit AES \textsf{KeyGenerator} by the PAPI \textsf{init} on Line 12 which leads to a misuse of weak key, the DAPI \textsf{Cipher.doFinal} on Line 15 and its encrypted data are affected to be insecure. Our data-flow analysis should address such dependency and take the DAPI as misuse too. Specifically, the sources used in our misuse-originating data-flow analysis are firstly expanded based on the relation of PAPIs and DAPIs in the specific app. Firstly, we state several related definitions as follows.
\begin{definition}[taint connection analysis]
Given the predefined taint sources $O$ and sinks $S$, a taint connection analysis $TA_{O,S}$ is a procedure deriving a set of call-site pairs from an app. $TA_{O,S}: X\mapsto P(Loc\times Loc)$ such that for each app $x\in X$, $TA_{O,S}(x)=\{(l_o,l_s)|l_o,l_s$ are call sites of $o\in O$ and $s\in S$ respectively, and $l_s$ is tainted by setting $l_o$ as the taint source$\}$, where $Loc$ holds all the code locations of $x\in X$.
\end{definition}
\begin{definition}[misuse refinement for source]
For a misused cryptographic API, the refinement procedure $\mathcal{R}$ of misuse derives a list of cryptographic APIs on each app $x\in X$:
\begin{displaymath}
\mathcal{R}(m)=
\begin{cases}
\{m\} ,\hfill \text{iff } m\in \text{DAPI}, \\
\{m\}\cup \{s\mid (\_,l_s)\in TA_{\{m\},\text{DAPI}}(x)\}, \hfill \text{iff }m\in \text{PAPI}.\\
\end{cases}
\end{displaymath}
\end{definition}
\begin{algorithm}[t]
\small
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{1) $\Gamma_x$: The misuse tuples of application $x$ generated by the adapters of detector chain. 2) The sink list $S$.}
\Output{The updated misuse tuples $\Gamma_x$.}
\Begin{
\ForEach{$(\gamma\equiv\langle m,id,p,d,t,\emph{\textsf{S}}\rangle)\in \Gamma_x$}{
$O\leftarrow \mathcal{R}(\gamma.m)$\;
$S_{tmp}\leftarrow \{l_s\mid (l_o,l_s)\in TA_{O,S}(x) \wedge l_o\text{ in }\gamma.p\}$\;
\tcp{Here $\gamma.\textsf{S} = \emptyset$}
\ForEach{$l_s\in S_{tmp}$}{
$\gamma.\textsf{S}\leftarrow \gamma.\textsf{S}\cup DSTrack(l_s)$\;
}
}
}
\caption{Misuse-originating Data-Flow Analysis}\label{algo:dfa}
\end{algorithm}
The procedure of our data-flow analysis is presented in Algorithm~\ref{algo:dfa}. When the misused API is a parameter-related API, we further consider the data-related APIs tainted by the misused parameter as taint sources. Note that the taint analysis tool can only start taint tracking from all call sites of a specified API method. However, not all call sites are misuses. Thus, after the taint analysis, we rely on the parent method $p$ of each misuse tuple to filter out taint sources that are not reported as vulnerabilities. Finally, we get all the related sink locations in $S_{tmp}$. Each sink location should fall into a sink category. To identify the precise category for each sink location, we perform a backward intra-procedural data-source tracking $DSTrack$ initiating from each call site of the affected sink to find all the related data types and calculate the exact sink category. For example, from the call site of \textsf{write} in method \textsf{send} of Figure~\ref{fig:example}, we track the data type \textsf{DataOutputStream}, \textsf{byte[]}, \textsf{HttpURLConnection}, \textsf{String}. Because we have tagged several related classes with sink category, e.g., \textsf{HttpURLConnection} with \textsf{NETWORK}, we merge the tracked sink categories with the default sink category of \textsf{write}, i.e., \textsf{OUT\_STREAM}. We take the most sensitive category, i.e., \textsf{NETWORK} in \{\textsf{NETWORK, OUT\_STREAM}\}, as the category of this sink and add this sink category to the misuse tuple.
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\caption{Classification of Cryptographic APIs}\small
\label{tab:api-classification}
\centering
\begin{tabular}{lrr}
\hline
Package Name & \# DAPI & \# PAPI\\
\hline
\textsf{java.security} & 202 & 806 \\
\textsf{javax.crypto} & 96 & 179 \\
\textsf{javax.net.ssl} & 99 & 162 \\
\textsf{javax.xml.crypto} & 23 & 169 \\
\hline
\end{tabular}
\vspace{-2ex}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{Sink Category and Statistics}\small
\label{tab:sink-category}
\centering
\begin{tabular}{lrrr}
\hline
Sink Category ($sc$) & \# FlowDroid(default) & \# Added & \#Total\\
\hline
\textsf{FILE} & 3 & 0 & 3 \\
\textsf{LOG} & 14 & 0 & 14\\
\textsf{NETWORK} & 17 & 1 & 18 \\
\textsf{SMS\_MMS} & 3 & 0 & 3 \\
\textsf{SYNC} & 5 & 0 & 5\\
\textsf{NC\_STORAGE} & 36 & 53 & 89 \\
\textsf{NC\_ICC} & 71 & 0 & 71\\
\textsf{NC\_OUT\_STREAM} & 10 & 1 & 11 \\
\textsf{NC\_OTHER} & 4 & 1 & 5 \\
\hline
\end{tabular}
\vspace{-3ex}
\end{table}
The taint connection analysis can be further parameterized by different static data-flow analyses, e.g., \cite{DBLP:conf/ccs/WeiROR14, DBLP:conf/pldi/ArztRFBBKTOM14, DBLP:conf/ndss/GordonKPGNR15, DBLP:conf/eurosp/CalzavaraGM16}. In contrast to the tightly embedded annotations to label sources and sinks \cite{DBLP:conf/ndss/GordonKPGNR15}, the sources and sinks of \cite{DBLP:conf/ccs/WeiROR14, DBLP:conf/pldi/ArztRFBBKTOM14} are easier to be configured and more suitable for our analysis. Deciding on the predefined sink list $S$ for the analysis in Algorithm~\ref{algo:dfa} is critical to define the impact of misuses. Specifically, we extend the default sinks of FlowDroid 2.7.1 (163 sinks) with 56 Android APIs involving network and serialization operations (e.g., the ones of \textsf{Parcel}/\textsf{Parcelable}, \textsf{ObjectOutputStream}, and \textsf{SQLiteDatabase}), see the number of sinks in Table~\ref{tab:sink-category}. Beyond this, we follow the categorization strategy of SuSi \cite{DBLP:conf/ndss/RasthoferAB14} to categorize the sinks for the efficiency of learning-based risk assessment. Moreover, for the sinks in \textsf{NO\_CATEGORY}(\textsf{NC}), we further divide these sinks into four sub-categories: \emph{ICC}, \emph{storage component}, \emph{output stream}, and \emph{others}. The ICC sinks include the APIs for inter-component communications. The sinks of storage components are the APIs of storage-related components, e.g., \textsf{SharedPreferences} and \textsf{ContentResolver}.
\subsection{Risk Assessment of Misuses}\label{subsec:clustering}
In order to assess the quantitative risk of cryptographic misuses, we assign severity weight $w_{id}$ to each vulnerability type in Table~\ref{tab:vul-weight}. The severity weights are compatible with CryptoGuard (Severity in Table~1 of \cite{DBLP:conf/ccs/RahamanXASTFKY19}), and we only compensate the weights for $id=8,18\sim 21$. We also need to assign a risk weight $w_{sc}$ to each sink category as explained in more detail in Section~\ref{sec:implementation}. For each app $x$, from the set of misuse tuples $\Gamma_x$ derived by Algorithm~\ref{algo:dfa}, we define two parameters for quantifying the risk. (1) $b_{\textsf{tool},i}$ stands for whether some misuse caused by the $i$-th vulnerability is detectable by the detector \textsf{tool}, i.e., $b_{\textsf{tool},i} \equiv (|\Gamma_x(t=\textsf{tool}\wedge id=i)|>0?1:0$, where $\textsf{tool}\in \{\textsf{CG},\textsf{CC}\}$ for the minimal valid detector chain. (2) $n_{sc,i}$ stands for the number of sensitive flows that the data affected by the $i$-th vulnerability can flow to a sink in category $sc$, i.e., $n_{sc,i}\equiv |\Gamma_x(\textsf{S}=sc\wedge id=i)|$, where $sc\in\{\textsf{FILE},\textsf{LOG},\ldots\}$. When calculating $n_{sc,i}$, we removed the duplicates of the same misuse detected by multiple detectors so that each $n_{sc,i}$ can reflect the number of access points on the attack surface of each app. We define the app-level risk value of cryptographic misuses:
\begin{displaymath}
R_x = \sum_{i=1}^{21} w_i\cdot \bigvee_{\textsf{tool}\in\{\textsf{CG},\textsf{CC}\}}b_{\textsf{tool},i} \cdot(\sum_{sc\in\{\textsf{FILE},\ldots\}} w_{sc}\cdot n_{sc,i})
\end{displaymath}
Such quantification is sufficient to assess the overall risk level of the individual app caused by the potential cryptographic misuses, which is helpful for the developers when the in-development assessment of cryptographic APIs is required.
To extend our approach to support app vetting at app stores, we investigate concise but concrete features of misuse threats to a large scale of apps and try to capture the potential threat patterns of apps. Due to the lack of ground truth of cryptographic misuses, we propose a clustering-guided risk prediction for this purpose. In general, we use the top-ranking labels, i.e., (\textit{vulnerability type}, \textit{sink category}) relations attached to each cluster, to predict the most significant threats caused by the cryptographic misuses in the apps of each cluster. Such top-ranking labels of each cluster compensate the risk value of the app with more concrete knowledge of the tendency of misuse threats. In detail, the feature vector for each app is of the form $(\mu,\nu)$, where $\mu$ is $\overline{(b_{\textsf{CG},i}, b_{\textsf{CC},i})}_{i=1..21}$ and $\nu$ is $\overline{(n_{\textsf{FILE},i}, n_{\textsf{LOG},i}, \ldots)}_{i=1..21}$. The dimension of the feature vector is $231=2\times 21+9\times 21$ where 2 for the number of misuse detectors, 9 for the number of sink categories, and 21 for the number of vulnerability types. To figure out the top-ranking labels for each cluster, we sum up the values of each dimension of $\nu$ over all apps in the cluster to get the total number of each type of data flows. Then we calculate the average number of each type of data flows per app in the cluster. A more significant average number represents a broader attack surface on such data flows. We use the $(id,sc)$ pair of the most significant average numbers to label the cluster and predict the most significant threats to the apps in this cluster. The top-label-based prediction is adept in presenting an overall threat summary of each cluster than only counting the most significant numbers of sensitive flows from the feature vector of each app.
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\caption{Severity Weight of Vulnerability}\small
\label{tab:vul-weight}
\centering
\begin{tabular}{cc}
\hline
Vulnerability ID ($id$) & Severity Weight ($w_{id}$)\\
\hline
1$\sim$8, 17 & high (10)\\
9$\sim$13, 19 & medium (7)\\
14$\sim$16, 18 & low (4)\\
20, 21 & very low (1) \\
\hline
\end{tabular}
\vspace{-2ex}
\end{table}
\section{Implementation Settings}\label{sec:implementation}
With the framework proposed in Section~\ref{sec:approach}, our implementation can be parameterized at least on the following aspects: (1) the valid detector chains for the misuse detection, (2) the choice of sinks and the risk weight assignment $w_{sc}$ for each sink category, (3) the static data-flow analyses for Algorithm~\ref{algo:dfa}, and (4) the clustering algorithms. To develop an instance of the framework to demonstrate our evaluations, we used FlowDroid \cite{DBLP:conf/pldi/ArztRFBBKTOM14} as the taint-connection analysis of the misuse-originating data-flow analysis. We developed $DSTrack$ with Androguard \cite{androguard}. Unlike the data-flow analysis used in identifying the misuse patterns \cite{DBLP:conf/ccs/RahamanXASTFKY19,cognicrypt}, our data-flow analysis is mainly used to capture the dependencies of data leakage to the cryptographic misuses, and $DSTrack$ takes an ad-hoc approach backward through the control-flow paths. $DSTrack$ is intra-procedural, while the taint-connection analysis with FlowDroid is inter-procedural, context-, flow-, and object-sensitive. Then, we use the $k$-means clustering \cite{macqueen1967some} for Euclidean distance to group the apps. Next, we discuss the choices for the first two parameters in detail.
Our framework is extensible to adapt more state-of-the-art detectors. In general, when adding or removing a detector, we must preserve Corollary~\ref{corollary1} of the valid detector chain. This is the main criteria for the selection of detectors used in CryptoEval. Even if the misuse rules addressed by one detector are only a subset of the vulnerabilities of another detector, adapting more detector into the detector chain is still meaningful. First, the new detector may detect more vulnerability sites and sensitive flows. Second, the new adapter provides more features for the clustering-guided assessment, which affects the top-ranking labels of each cluster. We have developed three adapters to assemble CryptoGuard, \textsc{CogniCrypt}$_\text{SAST}$, and BinSight \cite{DBLP:conf/ccs/MuslukhovBB18} into the detector chain. The adapters implement a unified interface for extracting detector output to build the misuse tuples, converting misuse information from \textsf{smali} to Java, and mapping each rule of misuse to one vulnerability $id$ in our threat model. Our implementation consists of 3.2 kLoC combining Java and Python, including adapters, data-flow analysis, feature extraction, learning, and glue code.
Among all the parameters, the most flexible one is the choice of sinks and the assignment of their risk weights, since the sinks may be derived and classified into different categories by different strategies \cite{DBLP:conf/ndss/RasthoferAB14, DBLP:conf/trust/GiblerCEC12}. Different categorizations of sinks lead to different assessment results because they may affect the dimension of the feature vector and the number of sensitive flows falling into each category. The sinks of our implementation are an extension of the default sinks of FlowDroid. The rationality of risk weights for the sink category is another issue impacting the quantitative risk of each app. To decide the risk weights $w_{sc}$, we first use the keywords in the signature of sink APIs to search for the Android-relevant CVEs and the corresponding CVSS metrics in NVD. Based on the average value of the CVSS scores correlated to the sinks of each sink category, we find a partial order of the sink categories. Then we approximate and normalize the risk ranking to get the values in Table~\ref{tab:sink-weight}. As another choice, risk ranks may be derived by making statistics on the results of the sink ranking approach \cite{DBLP:conf/ccs/TianTYR17} on the permission-based features and a ground-truth dataset. While the risk ranking of sinks is important for per-app assessment, we choose to use the detectability by detectors ($b_{\textsf{tool},i}$) and the number of sensitive flows ($n_{sc,i}$) to decide the features for clustering. This choice can avoid the threat summary of each cluster being affected by the uncertainty of risk weights.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{Risk Weight of Sink Category}\small
\label{tab:sink-weight}
\centering
\begin{tabular}{lc}
\hline
Sink Category ($sc$) & Risk Weight ($w_{sc}$)\\
\hline
\textsf{SMS\_MMS}, \textsf{NC\_OTHER} & 1 \\
\textsf{LOG} & 3 \\
\textsf{SYNC}, \textsf{NC\_STORAGE} & 4\\
\textsf{FILE}, \textsf{NC\_OUT\_STREAM} & 5\\
\textsf{NC\_ICC} & 7\\
\textsf{NETWORK} & 10\\
\hline
\end{tabular}
\vspace{-2ex}
\end{table}
Besides setting parameters, constructing an adapter for a detector reporting some coarse-grained misuse type, e.g., constantError of \textsc{CogniCrypt}$_\textsc{sast}$, requires more effort. From the raw output of \textsf{CC}, we extract $m, p, d$ of the misuse tuple as well as the corresponding error type $err\in Rule_{\textsf{CC}}$.
We conducted a manual analysis by two independent security experts over a small subset $P_0$ of our dataset $P$ to build a mapping from each detected triple $(m,d,err)$ to some $id$ of our vulnerability list. At each step, we pick an app $x\in P$, analyze it with the detector, label the output misuses manually with our vulnerability $id$s, develop the parsing strategies for these output patterns, and apply the strategies on all the detection results of $P$. The picked app will be added to $P_0$ until the results of $P$ are completely covered. Specifically, we developed our mapping with only 198 apps in $P_0$ (175 from CryptoAPI-Bench \cite{DBLP:conf/secdev/AfroseRY19} and 23 from AndroZoo \cite{Allix:2016:ACM:2901739.2903508}), which is less than 0.5\% of all the app samples.
\input{evaluation}
\section{Discussion}\label{sec:discussion}
In this section, we discuss several further potential improvements and the threats to the validity of our approach.
\paragraph{Impact of the granularity of vulnerability types.} In Section~\ref{sec:implementation}, we resort to the manual analysis over a small set $P_0$ of apps to build the mapping from the misuse types of existing approaches to the vulnerabilities in our comprehensive list. Although evaluated to be sufficient for our datasets and the empirical study, it should not be complete to deal with all the possible cases. Moreover, if we arrange more vulnerability types, e.g., use different vulnerability types to distinguish R3 and R6 of DiffCode, see column \textsf{DC} in Table~\ref{tab:rules}, we will have to do more such analysis even on the output of CryptoGuard. Therefore, the granularity of the vulnerability types matters to our approach.
\paragraph{False positives of analysis.}
We have introduced the voting mechanism of the detector chain to mitigate the impact of false positives of misuse detectors on real-world apps. However, the false positive of data-flow analysis is another issue. The accuracy of data-flow analysis impacts the assessment of the risk level of cryptographic misuses. In the current implementation, we faithfully adapt the results of FlowDroid into the risk assessment without addressing the inaccuracy of the data-flow analysis. Due to the homogeneous nature of state-of-the-art data-flow analyses on the taint-connection analysis, we may also bring in the voting mechanism based on multiple data-flow analyses, e.g., \cite{DBLP:conf/ccs/WeiROR14, DBLP:conf/pldi/ArztRFBBKTOM14}, to reduce the false positives of individual analysis.
\paragraph{Possible missing of flows.} When sensitive data threatened by cryptographic misuses first flow to an intermediate location with one sink and then read by another source to leak at another sink, such flow through intermediate location should have been identified by the taint-connection analysis of CryptoEval because we enumerate both sinks for the taint-connection analysis. This issue becomes difficult when the data-flow analysis is only modularly performed. Therefore CryptoEval requires a global data-flow analysis, and such analysis should be re-conducted when the app changes in evolution. Besides, there is no guarantee that the data-flow analysis can report all critical flows. Such false negatives cannot be solved by the voting mechanism since voting is generally to decide the positives. Consequently, our approach is up against a potential underestimation of the cryptographic misuse risks.
\input{relatedwork}
\section{Conclusion}\label{sec:conclusion}
To mitigate the struggle on common sense that all cryptographic misuses should be fixed mandatorily, we present CryptoEval, a framework combining adapter-based misuse detection and data-flow-driven risk assessment to estimate the threat of cryptographic misuses in Android apps. CryptoEval adapts different misuse detectors to achieve detections against more comprehensive vulnerability types, quantifies the risk of cryptographic misuses based on the result of the misuse-originating data-flow analysis, and predicts the potential threats of cryptographic misuses for the app vetting with unsupervised learning. We implemented an instance of CryptoEval and conducted evaluations on the accuracy of detection and the effect of risk assessment. With the help of CryptoEval, we observed security facts on the attack surface of apps and the severity of vulnerabilities caused by cryptographic misuses. We also found vulnerable popular real-world app whose data affected by cryptographic misuses tend to be leaked. Future work includes extending the sink categories and investigating the tendency of misusing different cryptographic libraries to benefit the development of cryptographic libraries.
\bibliographystyle{IEEEtran}
\section{Evaluation}\label{sec:evaluation}
This section investigates the accuracy of CryptoEval and the effect of risk assessment over the flow-related features. Moreover, we try to make useful security observations based on an empirical study and the analysis of popular real-world apps.
We evaluate CryptoEval on two datasets of Android apps, as shown in Table~\ref{tab:dataset}. We use a large dataset S1 for the empirical study. Dataset S2 contains CryptoAPI-Bench, a public crypto-misuse benchmark with ground truths to evaluate accuracy. S2 also contains the apps to elaborate the ability of our framework on revealing threats to real-world apps. Our experiments are conducted on an elastic compute service with 2.5GHz$\times$ 4 Intel Xeon(R) Platinum 8269CY CPU, 16GB RAM, Linux 4.15.0-88-generic kernel (Ubuntu 18.04), and JDK 1.8. For the detector chain, we use CryptoGuard 03.04.00 (commit id 1d520e4) \cite{DBLP:conf/ccs/RahamanXASTFKY19}, \textsc{CogniCrypt}$_\textsc{sast}$-Android 1.0.0 (commit id 856b1da) \cite{cognicrypt-android}, and BinSight (commit id cd8b680) \cite{DBLP:conf/ccs/MuslukhovBB18}. For the accuracy investigation, we also use \textsc{CogniCrypt}$_\textsc{sast}$ 2.7.1 (commit id 98eccd4) for Java \cite{DBLP:conf/ecoop/KrugerS0BM18, cognicrypt}. For data-flow analysis, we use FlowDroid 2.7.1 (commit id 72734bd) \cite{DBLP:conf/pldi/ArztRFBBKTOM14} and Androguard (commit id 22849b6) \cite{androguard}. We aim to answer the following research questions.
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\caption{Summary of Datasets}\scriptsize
\label{tab:dataset}
\begin{tabular}{crl}
\hline
Dataset & \#App & Description\\
\hline
S1 & 40604 & Released Jan.$\sim$Dec. 2019 on Google Play, Anzhi, and other\\
& & popular app markets, and downloaded through AndroZoo \cite{Allix:2016:ACM:2901739.2903508} \\
S2 & 190 & 175 from CryptoAPI-Bench (commit id 759622f) \cite{DBLP:conf/secdev/AfroseRY19}, 15 \\
& & popular real-world apps (Chrome, PayPal, etc) \\
\hline
\end{tabular}
\end{table}
\paragraph{RQ1. What's the accuracy of the valid detector chains of CryptoEval?}
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\centering
\caption{Severity-classified metrics comparison of CryptoEval, CryptoGuard and \textsc{CogniCrypt}$_\text{SAST}$. TP/FP/FN are the number of true positives/false positives/false negatives respectively}
\label{tab:severity-metrics}
\scriptsize
\begin{tabular}{c|c|ccc|ccc|ccc}
\hline
Severity & & \multicolumn{3}{c|}{CryptoGuard} & \multicolumn{3}{c|}{\textsc{CogniCrypt}$_\text{SAST}$} & \multicolumn{3}{c}{CryptoEval} \\
Weight & GTP & \multicolumn{3}{c|}{(\textsf{CG})} & \multicolumn{3}{c|}{(\textsf{CC})} & \multicolumn{3}{c}{(\textsf{CG}+\textsf{CC})} \\\cline{3-11}
($w_{id}$) & & TP & FP & FN & TP & FP & FN & TP & FP & FN \\
\hline
1 & 67 & 0 & 0 & 67 & 67 & 0 & 0 & 67 & 0 & 0 \\
4 & 42 & 39 & 7 & 3 & 13 & 7 & 29 & 40 & 7 & 2 \\
7 & 36 & 32 & 5 & 4 & 14 & 4 & 22 & 34 & 5 & 2 \\
10 & 63 & 57 & 8 & 6 & 26 & 10 & 37 & 63 & 11 & 0 \\
\hline
Total & 208 & 128 & 20 & 80 & 120 & 21 & 88 & 204 & 23 & 4 \\
\hline
\multicolumn{2}{c|}{Precision(\%)} & \multicolumn{3}{c|}{86.5} & \multicolumn{3}{c|}{85.1} & \multicolumn{3}{c}{89.9} \\
\multicolumn{2}{c|}{Recall(\%)} & \multicolumn{3}{c|}{61.5} & \multicolumn{3}{c|}{57.7} & \multicolumn{3}{c}{98.1} \\
\multicolumn{2}{c|}{F1(\%)} & \multicolumn{3}{c|}{71.9} & \multicolumn{3}{c|}{68.8} & \multicolumn{3}{c}{93.8} \\
\hline
\end{tabular}
\end{table}
To evaluate the accuracy of the adapter-based detection of CryptoEval, firstly, we compare the minimal valid detector chain (\textsf{CG}+\textsf{CC}) with the standalone CryptoGuard and \textsc{CogniCrypt}$_\text{SAST}$ on CryptoAPI-Bench of dataset S2. The statistics and comparison of metrics (precision, recall, and F1) are presented in Table~\ref{tab:severity-metrics}. Because we are dealing with a superset of vulnerability types compared with CryptoAPI-Bench, we refine the original ground truth of CryptoAPI-Bench (in Table VI,\cite{DBLP:conf/secdev/AfroseRY19}) manually to build the ground truth with respect to the new comprehensive vulnerability list, i.e., the 208 ground truth positives (GTPs) in Table~\ref{tab:severity-metrics}. Then the true positives (TP), false positives (FP), and false negatives (FN) of different approaches are identified by checking the detection results against the GTPs. We summarize the numbers of TP/FP/FN on different severity levels according to the severity weights of vulnerability types in Table~\ref{tab:vul-weight}. When addressing our new threat types, the test cases introduce 67 incomplete usages of cryptographic API (the 67 misuses of the vulnerability type 20), which can be detected by \textsc{CogniCrypt}$_\text{SAST}$. Assembling multiple detectors into CryptoEval, though tending to introduce more false positives (FP$_{\textsf{CG+CC}}$= FP$_{\textsf{CG}} \cup$FP$_{\textsf{CC}}$), can significantly suppress the false negatives (FN$_{\textsf{CG+CC}}$=(FN$_{\textsf{CG}}$-TP$_{\textsf{CC}}$)$\cup$(FN$_{\textsf{CC}}$-TP$_{\textsf{CG}}$)) and improve the recall of detection. Since we also merge the true positives, more detectors can improve the precision.
The metrics based on ground truths do not help decide the proper sources for the misuse-originating data-flow analysis of real-world apps because we do not have the ground truths for real-world apps given the absence of source code. We propose a voting mechanism to estimate the \emph{expected true positives}. The principle is, among all the misuses detected by the detectors (i.e., positives), we determine a misuse to be an expected true positive only if $>$50\% of detectors capable of dealing with the related vulnerability type report it as a misuse. Then, the precision of the detector chain is decided by its detection results against the expected true positives. The voting leads to a fair metric for estimating the true positives without human aid, assuming that each detector in the detector chain is independent.
Relying on this voting system, we conduct an evaluation on the 15 popular real-world apps in S2 (their names listed in Table~\ref{tab:real-world-findings}) to derive the expected true positives and demonstrate the effect of augmenting the detector chain. We introduce a third detector, BinSight (\textsf{BS}), over the minimal detector chain (\textsf{CG}+\textsf{CC}). After adding BinSight, we found the precision of the new detector chain (\textsf{CG}+\textsf{CC}+\textsf{BS}) decided by voting increases from 77.6\% to 78.7\% compared with the case that we only have \textsf{CG} and \textsf{CC} in the detector chain. Intuitively, introducing more detectors into the voting procedure will improve the estimation. The improvement is slight because BinSight only votes for the positives of vulnerability types 1,9,11$\sim$14. Moreover, the expected true positives derived by the three-detector chain are used in the evaluation of RQ3. Note that \textsf{BS} is adapted only to decide the expected true positive misuses. We do not merge it into the definition of $R_x$ in Section~\ref{subsec:clustering}.
\framebox[0.95\linewidth][l]{
\parbox{3.1in}{
The detector chain of CryptoEval outperforms the individual detectors on precision and recall based on CryptoAPI-Bench. The expected true positives voted by the detectors of the detector chain serve as a proper source of data-flow-based risk assessment.
}
}
\paragraph{RQ2. Is our comprehensive vulnerability list better than the existing vulnerability types and rules to specify the misuse?}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{fig/misuse}
\caption{Distribution of cryptographic misuses on different vulnerability types}\label{fig:misuse}
\end{figure}
We performed the adapter-based misuse detection of CryptoEval with the minimal valid detector chain (\textsf{CG}+\textsf{CC}) on dataset S1. We found there are 33410 apps detected to have some cryptographic misuse. Among these apps, CryptoGuard reports misuses on 30466 apps, and \textsc{CogniCrypt}$_\text{SAST}$ reports misuses on 19626 apps. The distribution of misuses on each vulnerability type is presented in Figure~\ref{fig:misuse}. The insecure PRNGs (type 10, reported 211283 cases) and insecure cryptographic hash operations (type 17, reported 175450 cases) are the most prominent vulnerabilities in the apps. The predictable/constant PRNG seeds (type 9, reported 389 cases) and insecure asymmetric ciphers (type 16, reported 411 cases) rarely appear.
The SSL/TLS related vulnerabilities (type 4$\sim$8) involve 86693 cases and account for 14.3\% of the detected misuses. From the statistics, we found that with intensive attention to the traditional vulnerability types (type 1,9,11$\sim$14), such misuses have become less pervasive (only 10.3\%) in real-world apps. If we compare with recent threat models, the threat model of CryptoGuard \cite{DBLP:conf/ccs/RahamanXASTFKY19} (lacking vulnerability type 8,18$\sim$21) will miss 8.6\% cryptographic misuses. The rules of \textsc{CogniCrypt}$_\text{SAST}$ \cite{DBLP:conf/ecoop/KrugerS0BM18} (lacking vulnerability type 4$\sim$7) will miss 11.7\% cryptographic misuses. Therefore, the threat models proposed by such approaches will cause considerable missing on important vulnerability types.
\framebox[0.95\linewidth][l]{
\parbox{3.1in}{
The insecure PRNGs (type 10) and insecure cryptographic hash operations (type 17) are the most prominent vulnerabilities. Traditional misuses (type 1,9,11$\sim$14) become less pervasive (10.3\%), and more recent approaches \cite{DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/ccs/RahamanXASTFKY19} still miss a considerable number of misuses, which validates our direction to synthesize a more comprehensive threat model.
}
}
\paragraph{RQ3. What can we find on the attack surface of real-world apps with our framework?}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{fig/sink}
\caption{Distribution of data flows triggered by cryptographic misuses on different sink categories}\label{fig:sink}
\end{figure}
We conducted the data-flow analysis of CryptoEval on the 33410 apps with cryptographic misuses and found 24055 apps (72.0\%) containing data flows originating from some cryptographic misuse. On the misuse level, we found 152305 misuse-originating sensitive data flows. We classify these sensitive data flows into different sink categories, as shown in Figure~\ref{fig:sink}. We observed that for the two most risky sink categories, there are 57909 (38.0\%) sensitive flows to \textsf{NETWORK} sinks and 17347 (11.4\%) sensitive flows to \textsf{NC\_ICC} sinks. Meanwhile, there are only two sensitive flows to \textsf{SMS\_MMS} sinks, and none flows to \textsf{NC\_OTHER} sinks from cryptographic misuses. For each sink method in different sink categories, we find the \textsf{NETWORK} sinks and \textsf{LOG} sinks are the top-2 common access points on the attack surface to exploit the vulnerabilities caused by cryptographic misuses.
\begin{table*}
\caption{Threats on Real-world Apps identified by CryptoEval. \textit{other} stands for the misuses with vulnerability types which trigger no sensitive flows on these apps. Sink categories abbreviated, e.g. \text{O\_S} = \text{NC\_OUT\_STREAM}}
\label{tab:real-world-findings}
\scriptsize
\begin{tabular}{p{1.5cm}|p{0.6cm}|p{1cm}|p{1cm}|p{1cm}|p{0.6cm}|p{1cm}|p{0.6cm}|p{1.1cm}|p{1cm}|p{1cm}|p{1cm}|p{0.4cm}|r}
\hline
& \multicolumn{12}{c|}{Vulnerability $id$} & \\ \cline{2-13}
App & 1 & 3 & 8 & 10 & 12 & 13 & 15 & 17 & 18 & 20 & 21 & other & $R_x$ \\
\hline
Amazon Shopping & 5 (\textsf{STO}:2) & 2 & 2 & 13 & 2 (\textsf{STO}:4) & & 1 (\textsf{STO}:4) & 9 (\textsf{STO}:1) & & 6 (\textsf{NET}:2, \textsf{STO}:24, \textsf{SYNC}:3) & 1 & 6 & 424 \\\hline
Chrome & 2 & & & 11 & & & & 10 & & 5 & & 1 & 0 \\\hline
Booking & 1 & 1 (\textsf{NET}:1) & 1 (\textsf{NET}:1) & 7 & & 6 (\textsf{O\_S}:6) & 1 & 3 & 1 (\textsf{O\_S}:1) & 2 (\textsf{O\_S}:1) & & 22 & 435 \\\hline
Moneycontrol & & & & 7 & & & & 14 & & 1 & & 1 & 0 \\\hline
eBay & 24 & 1 (\textsf{O\_S}:1, \textsf{NET}:1) & 1 (\textsf{O\_S}:1, \textsf{NET}:1) & 3 & & & & & & & & 18 & 300 \\\hline
PayPal & 1 & 1 & 4 & 4 & & & 2 & 7 & 1 & 1 & 2 & 1 & 0 \\\hline
Uber & 1 & 1 (\textsf{ICC}:1) & 7 (\textsf{NET}:2, \textsf{O\_S}:1, \textsf{ICC}:5) & 1 & & & & 6 & & & 2 (\textsf{ICC}:1) & 2 & 677 \\\hline
Lyft & & 1 & 3 (\textsf{NET}:2, \textsf{O\_S}:1) & 9 & & & & 7 & & & 1 & & 250 \\\hline
Facebook Lite & & 2 (\textsf{ICC}:1) & 5 (\textsf{ICC}:3) & 11 & & & & 9 & 1 & & 5 (\textsf{ICC}:3) & 3 & 301 \\\hline
WeChat & 7 & & & 13 & 1 & 1 & 7 & 24 & 2 & 1 & 2 & 2 & 0 \\\hline
TikTok & 2 & & & 11 & 2 & & & 9 (\textsf{ICC}:2) & & & & 1 & 140 \\\hline
PNC Mobile & 2 & 3 (\textsf{LOG}:1) & 3 & 4 & 1 & & & 6 & & & 4 (\textsf{LOG}:1) & 2 & 33\\\hline
CIBC Mobile Banking & 1 & 1 (\textsf{STO}:1, \textsf{NET}:6, \textsf{LOG}:2, \textsf{FILE}:1) & 3 (\textsf{STO}:2, \textsf{NET}:13, \textsf{LOG}:4, \textsf{FILE}:2) & 5 & 1 (\textsf{ICC}:3, \textsf{O\_S}:1) & & & 10 (\textsf{STO}:3, \textsf{NET}:2, \textsf{LOG}:2, \textsf{O\_S}:1, \textsf{ICC}:1) & 1 & 1 & 3 (\textsf{STO}:1, \textsf{NET}:6, \textsf{LOG}:2, \textsf{FILE}:1) & 4 & 3107 \\\hline
Starbucks & 2 & 2 (\textsf{STO}:2) & 4 (\textsf{STO}:2, \textsf{O\_S}:1) & & & 1 & & 1 & & & 4 (\textsf{STO}:6, \textsf{O\_S}:1) & & 239 \\\hline
Walmart & & 1 (\textsf{NET}:2, \textsf{LOG}:1) & 2 & 8 (\textsf{NET}:4, \textsf{LOG}:2) & & & & 17 (\textsf{STO}:1) & & & 1 & & 730 \\
\hline
\end{tabular}
\end{table*}
To show our framework's ability to discover threats in real-world apps, we investigate the vulnerabilities caused by cryptographic misuses and the related sensitive flows in the 15 popular real-world apps of dataset S2. The results are presented in Table~\ref{tab:real-world-findings}. For each app, we count the expected true positives of misuses identified by the valid detector chain (\textsf{CG}+\textsf{CC}+\textsf{BS}). Then we report the sensitive flows originating from these misuses in the brackets in Table~\ref{tab:real-world-findings} in terms of \textit{sink\_category:num\_flows}. We also calculate the risk value $R_x$ for each app. We figure out among all risky misuses ($w_{id}=7,10$), type 3,8,17 are more likely to trigger sensitive data flows. Moreover, the misuses of type 3 and 8 are likely to trigger flows to more risky sinks (\textsf{NETWORK} and \textsf{NC\_ICC}). It means that in typical apps, vulnerable passwords for KeyStore (type 3) and usage of the expired protocol by SSLContext (type 8) are more likely to be responsible for the data leakages caused by the misuses, which we should pay close attention to in the development of apps. Also, we have found risky apps (with bigger $R_x$), including a financial app, CIBC mobile banking. We manually confirmed that the threat caused by the type 8 misuses detected in this app have also been reported by CVE-2014-5594 \cite{CVE-2014-5594}.
\framebox[0.95\linewidth][l]{
\parbox{3.1in}{
The \textsf{NETWORK} and \textsf{LOG} sinks are the most common access points on the attack surface to exploit the vulnerabilities caused by the misuses. The quantitative risk value $R_x$ of the app effectively assesses the risk of cryptographic misuses. Our per-app assessment reports severe real-world threat in a financial app confirmed by a CVE record.
}
}
\paragraph{RQ4. How can the risk assessment of cryptographic misuses benefit the app store-side vetting?}
To show the effect of our framework in the app-vetting process for app stores, we apply the clustering-based assessment on the 33410 apps that contain cryptographic misuses. We label each cluster with the most frequently occurred (\textit{vul\_type}, \textit{sink\_category}) pairs. We take into account both the Davies-Bouldin Index (DBI) \cite{4766909} and the discrimination of top-ranking labels of each cluster to decide $k$, the proper number of clusters. On dataset S1, we found a proper prediction model when $k=7$. The top-3 labels of each cluster are presented in Table~\ref{tab:clustering}. In cluster $c_1$, 53.1\% of detected sensitive flows are from customized \textsf{TrustManager} (vulnerability type 5) to \textsf{NETWORK} sink. All the apps in this cluster contain at least one sensitive flow with this label, and the average number of this kind of flow is 2.42 in each app. The small clusters $c_2$ and $c_5$ address the impact of static initialization vectors in CBC-mode ciphers (type 13) on inter-component communications. The difference is that in $c_2$, the static IVs also affect the storage components remarkably, while in $c_5$, the impacts are more from insecure 64-bit block ciphers. Around 50\% of sensitive flows in $c_3$ result in leakages to log, either from 64-bit block ciphers (type 15) or insecure keys (type 1). Over 95\% of sensitive flows in $c_4$ lead to the storage components, mainly from suspected usages under more testing (type 21) and vulnerable passwords for \textsf{KeyStore} (type 3). The insecure cryptographic hash operations (type 17) lead the source of sensitive flows in cluster $c_6$ and $c_7$. In detail, the insecure hash operations trigger over 23\% of sensitive flows in $c_6$ and over 32\% in $c_7$. We also found that over 30\% of sensitive flows in $c_6$ affect the storage components from the insecure hash, static IVs, and constant keys. Moreover, all the apps with no sensitive flows (33410 - 24055 = 9355) are grouped into $c_7$. However, the amount of top-ranking sensitive flows in $c_6$ and $c_7$ is far from dominating all detected flows in these clusters, which indicates a richer diversity of sensitive flows in these clusters.
One interesting discovery is that the insecure PRNGs (type 10), though detected most frequently (see Figure~\ref{fig:misuse}), are in general unlikely to trigger sensitive flow, which means more occurrences of the misuses do not confirm a broad attack surface. Besides, an app falling into a specific cluster will be predicted to have a higher chance of triggering some specific kinds of sensitive flows. Moreover, the clustering result gives a more detailed summary of the most representative threats found in each cluster. This kind of summary can further guide the app-store-side vetting strategies. For example, hybrid analysis \cite{DBLP:conf/ndss/SounthirarajSGLK14} on MITM attacks may be applied with priority on the apps in clusters $c_1, c_5$, and $c_6$ because the SSL/TLS related vulnerabilities (type 4$\sim$8) tend to trigger sensitive flows to network in these clusters.
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\caption{Clustering results on Dataset S1. \#Labels per app = (\#specific flows)/(\#apps in each cluster). \% of labels = (\#specific flows)/(\#all detected flows in each cluster)}
\label{tab:clustering}
\scriptsize
\centering
\begin{tabular}{c|r|c|r|r|r}
\hline
Cluster & \#App & Label($id\rightsquigarrow sc$) & \#App & \#Label per & \% of \\
ID & & & with label & app (avg.) & labels \\
\hline
& & 5$\rightsquigarrow$\textsf{NETWORK} & 13821 & 2.42 & 53.1 \\\cline{3-6}
$c_1$ & 13821 & 4$\rightsquigarrow$\textsf{NETWORK} & 4606 & 0.46 & 10.2 \\\cline{3-6}
& & 8$\rightsquigarrow$\textsf{NETWORK} & 859 & 0.19 & 4.1 \\\hline
& & 13$\rightsquigarrow$\textsf{NC\_ICC} & 26 & 210.85 & 98.2 \\\cline{3-6}
$c_2$ & 26 & 13$\rightsquigarrow$\textsf{NC\_STORAGE} & 25 & 3.35 & 1.6 \\\cline{3-6}
& & 13$\rightsquigarrow$\textsf{LOG} & 12 & 0.58 & 0.3 \\\hline
& & 15$\rightsquigarrow$\textsf{LOG} & 1531 & 4.01 & 31.7 \\\cline{3-6}
$c_3$ & 1542 & 1$\rightsquigarrow$\textsf{LOG} & 900 & 2.30 & 18.2 \\\cline{3-6}
& & 15$\rightsquigarrow$\textsf{NC\_STORAGE} & 1061 & 1.47 & 11.6 \\\hline
& & 21$\rightsquigarrow$\textsf{NC\_STORAGE} & 1668 & 3.69 & 47.5 \\\cline{3-6}
$c_4$ & 1671 & 3$\rightsquigarrow$\textsf{NC\_STORAGE} & 1649 & 3.63 & 46.8 \\\cline{3-6}
& & 8$\rightsquigarrow$\textsf{NC\_STORAGE} & 25 & 0.09 & 1.1 \\\hline
& & 13$\rightsquigarrow$\textsf{NC\_ICC} & 16 & 90.00 & 86.9 \\\cline{3-6}
$c_5$ & 16 & 15$\rightsquigarrow$\textsf{NC\_ICC} & 8 & 4.50 & 4.3 \\\cline{3-6}
& & 8$\rightsquigarrow$\textsf{NETWORK} & 10 & 1.38 & 1.3 \\\hline
& & 17$\rightsquigarrow$\textsf{NC\_STORAGE} & 335 & 5.50 & 16.6 \\\cline{3-6}
$c_6$ & 401 & 7$\rightsquigarrow$\textsf{NETWORK} & 96 & 2.68 & 8.1 \\\cline{3-6}
& & 13$\rightsquigarrow$\textsf{NC\_STORAGE} & 194 & 2.62 & 7.9 \\\hline
& & 17$\rightsquigarrow$\textsf{LOG} & 1854 & 0.27 & 11.9 \\\cline{3-6}
$c_7$ & 15933 & 17$\rightsquigarrow$\textsf{NC\_ICC} & 1260 & 0.24 & 10.7 \\\cline{3-6}
& & 20$\rightsquigarrow$\textsf{LOG} & 413 & 0.16 & 7.0 \\\hline
\end{tabular}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{fig/risks}
\caption{Quantitative risk levels of each cluster}\label{fig:quantitative-risk}
\end{figure}
Figure~\ref{fig:quantitative-risk} presents the app-level risk values of the apps in each cluster. For the convenience of the demonstration, the 9935 apps with no sensitive flow, whose risk value $R_x = 0$, are assigned with a minimal positive $10^{-4}$. We found the small clusters $c_2$ and $c_5$ are with high risk because the major sink category \textsf{NC\_ICC} is risky, and the average numbers of major sensitive flows (i.e., 210.85 per app in $c_2$ and 94.50 per app in $c_5$) in each app are big.
Then followed by $c_6$, whose top-6 sensitive flows account for 55\% of all detected flows, and 41\% of them flow to the riskiest \textsf{NETWORK} sinks.
The apps in cluster $c_7$ are generally at low risk because of the large portion of apps with no sensitive flows. In general, the ranges of risk values for different clusters discriminate well, which justifies the design of $R_x$ as well as the choices on severity weights in Table~\ref{tab:vul-weight} and risk weights in Table~\ref{tab:sink-weight}.
\begin{table}[!ht]
\renewcommand{\arraystretch}{1.1}
\caption{Association rules generated with FP-growth, selected by (\#app instance $>500$) and (confidence $>0.8$). Gray rules support associations between top-ranking labels in Table~\ref{tab:clustering}}
\label{tab:association}
\footnotesize
\centering
\begin{tabular}{l|r}
\hline
Association rule & Conf \\
\hline
$[3\rightsquigarrow\textsf{NETWORK}]$:1089 $\Rightarrow [8\rightsquigarrow \textsf{NETWORK}]$:1076 & 0.99 \\
\cellcolor{lightgray}$[21\rightsquigarrow\textsf{NC\_STORAGE}]$:1929 $\Rightarrow [3\rightsquigarrow \textsf{NC\_STORAGE}]$:1769 & 0.92 \\
$[15\rightsquigarrow\textsf{NC\_OUT\_STREAM}]$:1300 $\Rightarrow [15\rightsquigarrow\textsf{LOG}]$:1186 & 0.91 \\
$[8\rightsquigarrow\textsf{NC\_STORAGE}]$:1002 $\Rightarrow [5\rightsquigarrow \textsf{NETWORK}]$:903 & 0.90 \\
$[15\rightsquigarrow\textsf{NC\_OUT\_STREAM}]$:1300 $\Rightarrow [15\rightsquigarrow \textsf{NC\_STORAGE}]$:1161 & 0.89 \\
\cellcolor{lightgray}$[8\rightsquigarrow\textsf{NC\_STORAGE}]$:1002 $\Rightarrow [3\rightsquigarrow \textsf{NC\_STORAGE}]$:884 & 0.88 \\
\cellcolor{lightgray}$[15\rightsquigarrow\textsf{NC\_STORAGE}]$:1387 $\Rightarrow [15\rightsquigarrow \textsf{LOG}]$:1204 & 0.87 \\
$[18\rightsquigarrow\textsf{NC\_STORAGE}]$:1021 $\Rightarrow [17\rightsquigarrow \textsf{LOG}]$:868 & 0.85 \\
\cellcolor{lightgray}$[13\rightsquigarrow\textsf{LOG}]$:687 $\Rightarrow [13\rightsquigarrow\textsf{NC\_STORAGE}]$:582 & 0.85 \\
\cellcolor{lightgray}$[1\rightsquigarrow\textsf{LOG}]$:1277 $\Rightarrow [15\rightsquigarrow\textsf{LOG}]$:1070 & 0.84 \\
$[15\rightsquigarrow\textsf{NC\_STORAGE}]$:1387 $\Rightarrow [15\rightsquigarrow \textsf{NC\_OUT\_STREAM}]$:1161 & 0.84 \\
$[13\rightsquigarrow\textsf{LOG}]$:687 $\Rightarrow [15\rightsquigarrow\textsf{LOG}]$:572 & 0.83 \\
\cellcolor{lightgray}$[4\rightsquigarrow\textsf{NETWORK}]$:5832 $\Rightarrow [5\rightsquigarrow\textsf{NETWORK}]$:4826 & 0.83 \\
$[3\rightsquigarrow\textsf{NETWORK}]$:1089 $\Rightarrow [5\rightsquigarrow\textsf{NETWORK}]$:900 & 0.83 \\
\cellcolor{lightgray}$[8\rightsquigarrow\textsf{NETWORK}]$:1388 $\Rightarrow [5\rightsquigarrow\textsf{NETWORK}]$:1130 & 0.81 \\
$[13\rightsquigarrow\textsf{LOG}]$:687 $\Rightarrow [13\rightsquigarrow\textsf{NC\_OUT\_STREAM}]$:556 & 0.81 \\
\hline
\end{tabular}
\end{table}
To figure out more evidence for the relations between important (\textit{vul\_type}, \textit{sink\_category}) pairs, we use the $\nu$ part of feature vectors to perform an association rule mining with the FP-growth method \cite{DBLP:conf/sigmod/HanPY00} provided by Weka \cite{DBLP:journals/sigkdd/HallFHPRW09}.
The generated rules are presented in Table~\ref{tab:association}. We choose the associations with more significant numbers of app instances ($> 500$) since we want to find associations between the top-ranking labels of each cluster.
From the results, we found informative co-occurrences for the top-ranking labels of $c_1\sim c_4$. Moreover, the sensitive flows from insecure passwords for \textsf{KeyStore} (type 3) to \textsf{NETWORK} sinks are highly correlated with the sensitive flows caused by SSL/TLS related vulnerabilities (type 4,5,8).
The sensitive flows from SSL/TLS misuses, especially customizing \textsf{HostnameVerifier} and \textsf{TrustManager}, tend to happen simultaneously to trigger network leakages in many cases. The popular apps in Table~\ref{tab:real-world-findings} also evidence Such correlations (type 3,8 to network).
\vspace{1ex}
\framebox[0.95\linewidth][l]{
\parbox{3.1in}{
Clustering-based assessment explains the most representative threats and estimates the risk level for each cluster, which can guide the vetting strategies of app stores. The frequently detected insecure PRNGs generally do not broaden the attack surface. The certificate-related and SSL/TLS related misuses tend to trigger sensitive flows to the network simultaneously. Such aspects should be paid close attention to in developing apps.
}
}
\section{Related Work}\label{sec:related}
Some works have addressed cryptographic misuse analysis through different means. The case study by Lazar et al. \cite{DBLP:conf/apsys/LazarCWZ14} has analyzed the cryptographic vulnerabilities in CVE from 2011 to 2014. They found that the misuse of cryptographic libraries caused over 80\% of cryptographic vulnerabilities. In Android apps, the proportion of vulnerable apps containing cryptographic API misuse was 88\%, even under a straightforward threat model \cite{DBLP:conf/ccs/EgeleBFK13}. Other statistics show that over 65\% of iOS apps contain various defects caused by cryptographic misuses \cite{DBLP:conf/nss/LiZLG14}. Although people tend to simplify the design of cryptographic libraries by limiting the decision space of parameters, comprehensive documentation and code samples are critical \cite{DBLP:conf/sp/Acar0FGKMS17}, and the misleading knowledge should be filtered out \cite{DBLP:conf/icse/MengNYZA18,DBLP:conf/icse/ChenFMWG19} on using the cryptographic APIs correctly. This intricate situation prompted us to investigate the vulnerability rules and the severity of cryptographic misuses in Android apps.
CryptoLint \cite{DBLP:conf/ccs/EgeleBFK13} is a pioneer work that proposes a static analysis to detect flows between improper cryptographic parameters and cryptographic operations. The authors proposed six common rules of cryptographic API usage against IND-CPA. BinSight \cite{DBLP:conf/ccs/MuslukhovBB18} considers the same set of rules, identifies the violations of these rules, and tracks the sources of misuses with static program slicing. The results show that most of the misused call-sites of cryptographic APIs originate from third-party libraries. CDRep \cite{DBLP:conf/ccs/MaLLD16} attempts to fix the cryptographic misuses identified with CryptoLint by instrumenting the app's bytecode with several patch templates. Their security rules slightly extend the rules of CryptoLint, i.e., \cite{DBLP:conf/ccs/EgeleBFK13, DBLP:conf/dasc/ShaoDGYS14}.
It has been realized that more fine-grained vulnerability specifications should correlate to the cryptographic API misuses \cite{DBLP:conf/pldi/PaletovTRV18, DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/ccs/RahamanXASTFKY19, DBLP:conf/msr/GaoKLBK19}. \textsc{CogniCrypt}$_\textsc{sast}$ \cite{DBLP:conf/ecoop/KrugerS0BM18, DBLP:conf/kbse/KrugerNRAMBGGWD17, cognicrypt, cognicrypt-android} takes the rules for correct usages of cryptographic APIs as input, translates them into a flow- and context-sensitive static analysis that checks apps for compliance with these rules. The rules are specified in the \textsc{CrySL} language. By defining the correct usages as a safelist, \textsc{CogniCrypt}$_\textsc{sast}$ classifies the violations to these rules into seven misuse types \cite{cognicrypt}. CryptoGuard \cite{DBLP:conf/ccs/RahamanXASTFKY19} extends the threat model to include the misuses of SSL/TLS APIs. The authors developed an on-demand flow-, context- and field-sensitive program slicing and provided contextual refinements for false-positive reduction. The data-flow analysis used in these works aims at identifying the misuse itself. In contrast, our misuse-originating data-flow analysis profiles the attack surface on the data threatened by the cryptographic misuses. On the temporal dimension, Paletov et al. \cite{DBLP:conf/pldi/PaletovTRV18} argued that cryptographic APIs and their exploits evolve over time. They derived the usage changes of cryptographic APIs from paired directed acyclic graphs based on the mining of code commit messages. A comprehensive set of security rules were elicited and evaluated over a dataset of Java projects. Gao et al. \cite{DBLP:conf/msr/GaoKLBK19} leverage the report of \textsc{CogniCrypt}$_\textsc{sast}$ to find pairwise misuse update among successive releases of the application. They found several evolutionary features indicating the inadequacy of the developer's fix behaviors. The types and rules of cryptographic misuse in these approaches are more general than the threat model of CryptoLint. Compared with these cryptographic misuse detection approaches, our work aims at demonstrating the risk of dependency between cryptographic misuse and related data leakage.
Our work focuses on the impact of cryptographic misuses in general uses. In contrast, several works have addressed the impact of API-level cryptographic vulnerabilities on very specialized aspects, e.g., in ad libraries \cite{DBLP:conf/ccs/0001BD16} and SSL/TLS \cite{DBLP:conf/ccs/FahlHMSBF12, DBLP:conf/ccs/GeorgievIJABS12, DBLP:conf/ndss/SounthirarajSGLK14, DBLP:conf/sp/FischerBXSA0F17}. MalloDroid \cite{DBLP:conf/ccs/FahlHMSBF12} performs static analysis on Android apps to detect different types of flaws related to the misuses of SSL/TLS, e.g., accepting all certificates or all hostnames. Georgiev et al. \cite{DBLP:conf/ccs/GeorgievIJABS12} analyzed the certificate validations in security-critical apps and libraries, and found the incorrect validations were caused by some critical design flaws of SSL/TLS APIs. SMV-Hunter \cite{DBLP:conf/ndss/SounthirarajSGLK14} focused on the custom validation code, i.e., overriding of \textsf{X509TrustManager} and \textsf{HostNameVerifier}. After detecting the custom validations and the UI entry points leading to these validations statically, its dynamic analysis emulates user interaction by feeding the app with customized inputs to trigger the vulnerable code under MitM attacks. Even in the vulnerability categories proposed by code smell and secure coding researches, e.g., \cite{DBLP:conf/scam/GhafariGN17, DBLP:conf/ccs/NguyenWA0WF17, DBLP:conf/sp/FischerBXSA0F17}, the cryptographic and SSL/TLS misuses are considered prominent security vulnerabilities.
|
1,314,259,995,844 | arxiv | \section{Introduction}
Opinion and group formation in human societies have been investigated
in the last years as complex phenomena, well described with the
methods of non-linear dynamics and statistical physics (for a review
see \cite{castellano2009statistical}). It has become clear that the
evolutionary dynamics of
societies, based on
social pressure and imitation, must be understood together with the
intricate network of contacts established among single agents. Individuals
are represented by the nodes of a graph and interact with their
neighbors under a given set of rules to form their own
opinion. Interactions may take place in several ways, and many models
have dealt with various mechanisms in which individuals might meet as well as
with different rules to update their opinions.
In the simplest \emph{voter model}~\cite{clifford1973model,holley1975ergodic,de1993non} individuals on the nodes of a regular lattice may hold two opinions encoded by a binary spin variable $\sigma=\pm 1$. Two randomly selected individuals interact with each other (\emph{one-to-one} interaction) in which the second simply adopts the opinion of the first. In one dimension, the voter model is identical to the kinetic Ising model with zero-temperature Glauber dynamics~\cite{liggett1985interacting,castellano2006zero,castellano2005comparison}. The latter is
exactly solvable for regular lattices in any dimension~\cite{frachebourg1996exact,krapivsky1992kinetics}.
This simple dynamics leads to consensus in finite systems, which is
the only stable solution: all spins are aligned, while both states
are equally likely to be the consensus opinion---provided they are equally abundant in the initial state---i.e.\ ensemble magnetization
is conserved \cite{liggett1985interacting}.
On heterogeneous networks, special updating mechanisms like the \emph{link-update}~\cite{suchecki2005conservation,sood2008voter,sood2005voter} approach are needed to ensure conservation of the ensemble magnetization. For the \emph{reverse-voter}~\cite{castellano2005comparison,sood2008voter}, where a node is randomly selected and its opinion copied to a randomly selected neighbor, it was found that the time to consensus differs from the direct voter model. A common feature of the voter dynamics on various complex networks, such as in small-world~\cite{castellano2003incomplete}, scale-free~\cite{suchecki2005conservation,suchecki2005voter,sood2005voter}, and random~\cite{vazquez2008analytical} graphs, is the absence of consensus in the thermodynamic limit, which has also been supported by analytical investigations on uncorrelated
graphs~\cite{vazquez2008analytical}. However, consensus is systematically reached on finite graphs~\cite{sood2005voter,castellano2003incomplete}, even though metastable states of coexisting opinions are observed for long periods before the system evolves towards consensus for a time which scales with the system's size.
In the voter model, the inclusion of more than two opinions~\cite{sire1995coarsening,vazquez2003constrained}, non-confident vacillating voters~\cite{lambiotte2007dynamics}, zealots with a fanatic position~\cite{mobilia2007role,mobilia2003does}, or a threshold number of successful encounters~\cite{dall2007effective} allows the
coexistence of opinions. Encounters among many agents have been considered in the \emph{majority model}~\cite{krapivsky2001organization,chen2005consensus},
in which a group of individuals is randomly selected and all of them adopt the opinion of the majority (\emph{all-to-all} interactions), or in the \emph{rumor spreading model}~\cite{galam2002minority,galam2007role}, in which more than one group-encounters take place simultaneously. In both of these models consensus on one or the other opinion depending on the initial conditions is achieved for various structures. Nevertheless, if noise is introduced in this group dynamics, as in the \emph{majority-minority}~\cite{mobilia2003majority} or the \emph{majority-vote}~\cite{liggett1985interacting} models, the system evolves either towards a consensus or towards a state in which both opinions are equally represented (zero magnetization state in the Ising model analogy). The same behavior is observed in the \emph{Sznajd model}~\cite{sznajd2000opinion,slanina2003analytical} where two individuals sharing the same opinion impose it on their neighbors.
Another approach for describing social interactions that accounts for the influence of the whole neighborhood on a single individual (\emph{one-to-all} interactions) has been discussed in \emph{social impact theory}~\cite{latane1981psychology,latane1990private,lewenstein1992statistical}. An agent changes her opinion if the pressure in favour of the opinion change overcomes the support to keep the current position. This model yields stable coexistence of opinions, unless the presence of external fields is taken into account, for which metastable states are observed in which domains with the minority opinion successively shrink~\cite{lewenstein1992statistical}.
The \emph{Abrams-Strogatz} (AS) model describes the competition between two languages in a population, with non-linear transition rates proportional to $x^a$, where $x$ is the population fraction of one particular language, and $a$ the volatility characterizing the tendency to change state~\cite{abrams2003}. For high volatilities, $a<1$, coexistence of both languages has been found, whereas for $a>1$ one of the two languages becomes dominant~\cite{abrams2003}. Studying the AS model on networks, it has been shown that a decrease in network connectivity leads to a reduced parameter regime with language coexistence~\cite{vazquez2010agent}. In the same study, the AS model was also compared with a variation including a bilingual state, which has been observed to hinder coexistence.
Here, we propose a model which aims to capture the mechanisms of community formation and how these depend on the specific structure of the contact network. It describes the dynamics of a society in which individuals may either be neutral (hold a default opinion) or belong to a minority with radical positions on a current topic. Examples of human behavior with such a dynamics are the membership and involvement in political parties or organizations, belonging to religious communities, the interest in leisure activities as online games, the behavior of consumers with new technologies, or the typical cycle of addiction to chemicals. Neutral individuals may at some point in time make acquaintance with a new product or idea and subsequently become customers or adopt it as their default position. Hereafter, they can either like it and consolidate as their default state or not and return to neutral behavior. Additionally, if agents in a default state are in contact with fanatics (radical agents)
they can also turn into radicals. Radical agents, in turn, may get bored or disappointed after a while reducing their level of commitment and turn back to a default position or even completely leave the community and become neutral agents again.
There are two characteristic features of our model, defined mathematically in the next chapter, which are essential for its dynamics. First, we account for the fact that the population density may vary over the contact network, i.e. in addition to individuals with different states (opinions) we also consider empty sites. Second, the probability of a given individual to change its state is taken to be proportional to both the number of like and alike peers in its neighborhood. This product form of the transition rates is the main difference to previously studied non-linear voter-like models~ \cite{schweitzer2009non-linear,vazquez2008systems} and the AS model~\cite{abrams2003}. This form implies, that transition rates may change from a linear to a non-linear form upon changing the composition of the population.
The outlined scheme bears similarities with the formation of mafias, in which normal citizens are the analog to default opinions, while mafias represent the radical minority. Neutral individuals are those who do not take active part in the society and that are represented by empty sites. Allegorically these sites may enter the community as citizens by a \emph{birth event} and join the mafia later due to the social pressure of their vicinity. In this paper, for illustration we will use the metaphoric language of mafias.
We will show that the \emph{mafia model} yields a rich phase behavior which contains both coexistence of opinions and consensus on the default position in well-mixed societies. In scale-free networks, coexistence appears in a larger regime of the parameter space due to the effect of local interactions: the stability of the absorbing state (extinction) is lost. We will discuss the role played by the nature of interactions---\emph{one-to-all}, i.e. the whole neighborhood determines the state of single agents, as compared to one-to-one interactions like in the voter model---for the loss of stability of the absorbing state. Furthermore, we will show that the inclusion of external fields in the model, representing the presence of elements controlling the interactions (which we refer to as police individuals), gives rise to micro-phase separation where regions with different dynamics are observed to coexist in the system.
This paper is organized as follows: We first describe the dynamics of the mafia model and then we analytically discuss the phase diagram expected for well-mixed populations. Then, we analyze the behavior on structured societies showing different degrees of heterogeneity, which we investigate through extensive stochastic simulations. We finally address the role of external control elements in the system and explore how specific distributions of those into highly or sparsely connected nodes modifies the system's behavior. An extended mean-field theory accounting for the local structure of networks is given in the Appendix.
\section{The mafia model}
We model human interactions in social networks by graphs whose nodes allocate interacting agents and whose edges describe the relations between pairs of individuals (Fig.~\ref{fig: sfn cartoon}). The nodes of a network can either be empty ($\Phi$) or occupied by an individual. In the simplest case agents belong to one of two existing groups identified with what we refer to as two different \emph{strategies}, namely they either belong to the mafia ($M$) or to the group of lawful citizens ($C$). The network's size is assumed to be fixed and, therefore, the sum of the number of citizens, mafiosi, and empty places (or their corresponding fractions) remains constant: $C+ M + \Phi =N$.
\begin{figure}[t]
\centering
\resizebox{0.25\textwidth}{!}{%
\includegraphics{figures/sfn_paper.eps}}
\caption{[color online] Individuals (red mafiosi, green citizens, white empty sites) are distributed in the nodes of a scale-free network. The composition of the neighborhood of the agent who makes a decision, marked with a question mark, reads $(c, m,\phi) = (3/5,1/5,1/5)$.}
\label{fig: sfn cartoon}
\end{figure}
We employ an agent-based model in which the society evolves according to the following stochastic dynamics: (i) at empty nodes citizens are born with a site-independent fixed rate $b$, (ii) at occupied nodes individuals die with a constant rate $d$, and (iii) strategy changes take place in which citizens join the mafia at rate $w_{c\to m} (=: w_{cm})$ while mafiosi leave it at a rate $w_{m\to c} (=: w_{mc})$. Since individuals who reconsider their status, namely whether leaving or joining the mafia, are influenced by their surroundings, these rates are determined by the composition of the neighborhood of a given individual.
\begin{displaymath}
\centering
\xymatrix{
& {\Phi}\ar@<0.5ex>[dr]^{b}\\
M\ar@<0.5ex>[rr]^{w_{mc}}\ar[ur]_{d}&& C\ar@<0.5ex>[ll]^{w_{cm}}\ar@<0.5ex>[ul]^{d}
}
\end{displaymath}
As the neighborhood of a given agent we define the set of all its nearest neighbors, i.e.\ the set of all other agents connected directly to it through an edge of the network. In Fig.~\ref{fig: sfn cartoon}, for instance, the neighborhood of the citizen with the question mark contains three citizens (green), one mafioso (red), and one empty place (white). The composition of this neighborhood is characterized by the tuple $(c, m,\phi)$, where $c$, $m$ and $\phi$ denote the fraction of mafiosi, citizens and empty sites in the neighborhood of a given individual at a given instant of time.
Single agents feel both the pressure from neighbors holding a different strategy and the support of neighbors who are members of their peer group, motivating the following choice for the transition rates:
\begin{eqnarray}
w_{c\to m} = w_{cm} & = & m\, s_m \,(1-c),\label{eq: wcm}\\
w_{m\to c} = w_{mc} & = & c\, s_c\, (1-m)\label{eq: wmc}.
\end{eqnarray}
In this way, the pressure exerted on a given agent by individuals belonging to the adversary group is proportional to their fraction among the nearest neighbors, i.e.\ $m$ and $c$, respectively. The strength of this pressure, $s_m$ or $s_c$, which we also call persuasiveness, is taken to be the same and constant for all individuals. At the same time, the supportiveness of the alike individuals attenuates this pressure through a multiplicative factor, $(1-c)$ and $(1-m)$, respectively. This product form of the transition rates implies that individuals do not reach a stable state but keep on changing their strategies unless the full neighborhood holds the same opinion and the agent is fully surrounded by alike agents. As a consequence the relative strength of the linear and non-linear term in the transition rates change with species frequencies, cf. $\omega_{cm}\sim m (1-c) = m (m+\phi)$: If the abundance of the adversary species is much smaller than the fraction of empty places $(m \ll \phi$), the leading term to change an indiviuals' state is linear, whereas for species abundances significantly larger than the fraction of empty places the leading term is non-linear and the dynamics changes accordingly. This is distinct from previously studied non-linear opinion models like the Abrams-Strogatz model where the type of non-linearity remains fixed independent of the composition of the populations~\cite{abrams2003,vazquez2010agent}. In our notation the rates of the AS model read $\omega_{cm} = s_m m^a$ and $\omega_{cm} = s_c c^a$ with volatility $a$, and prestiges $s_{m/c}$ of ``languages'' $m$ and $c$.
Moreover, since the tuple $(c, m,\phi)$ denotes the overall composition of an individual's neighborhood, a given agent interacts with its neighbors in a \emph{one-to-all} scheme. As it will turn out, the ensuing dynamics and stationary states on networks are genuinely different from those obtained for a \emph{one-to-one} scheme where a given individual interacts pairwise with a single randomly chosen neighboring site. The main difference is that in \emph{one-to-all} schemes one has to account for a discrete set of possible neighborhood compositions. This feature invalidates standard mean-field theories even in models where one adds strong mixing. As shown in Appendix A, an extended mean-field theory accounting for these discrete set of possible neighborhood compositions explains why a one-to-all interaction in structured societies induces changes in the phase diagram.
In this paper, we will focus on the \emph{asymmetric case} in which mafiosi are more persuasive (much stronger) than citizens, $s_m \gg s_c \approx 0$. For this case, the transition rate $w_{mc}$ vanishes and individuals can leave the mafia only rather indirectly, namely via a death-birth process. The more general case yields less interesting dynamics; a full discussion may be found in \cite{balbas2010diss}.
Decision processes may, in general, also be influenced by many external factors, such as mass media, independent of the interacting actors. More particularly, societies may provide means to regulate their conflicts and protect citizens against the damage of mafias. One would like to account thus for the role of control elements such as police in actual societies. Control elements do not participate directly in the social dynamics by interacting with the agents and evolving in time, but specifically control the relations between pairs of individuals. They are located at the edges connecting nodes between two individuals of a social network. The fraction of total edges allocating control elements is $p$. The model includes their two-fold catalytic role activating the transition $m\to c$ and inhibiting the transition $c\to m$:
\begin{eqnarray}
w_{cm} & = & m s_m (1-c) (1-p),\\
w_{mc} & = & (c s_c + s_p p)(1-m).
\end{eqnarray}
The control elements protect citizens from the mafia's pressure, attenuating their strength proportional to the factor $(1-p)$. Simultaneously they persuade individuals to leave the mafia proportional to their fraction $p$ and strength $s_p$. The persuasiveness adds to that of the citizens, both being still attenuated by the presence of mafiosi as before.
\section{Mean-field approximation}
For well-mixed populations in which every individual interacts with all other agents in the system, the problem we pursue to solve is topologically equivalent to having the agents allocated on the nodes of a complete graph. Then the \emph{local field} which every agent experiences is the same as the global field, given by the average relative abundance in the whole population. This simplification leads to a mean-field approach for well-mixed societies which allows an analytical characterization of the problem. We note that this type of mean-field limit does not correspond to the limit of fast diffusion on networks if the interaction is \emph{one-to-all} as is the case in our mafia model. Then, the dynamics of the network-structured society is actually better described within an extended mean-field theory which accounts for the fact that for each given node there is a finite set of neighborhood compositions, cf. Appendix A. In the following we discuss the standard mean-field approach in order to highlight the novel effects introduced by the network structure and the
\emph{one-to-all} interaction scheme.
We non-dimensionalize by choosing the inverse death rate, $d^{-1}$, as our basic time scale and define dimensionless time and all other parameters accordingly: $\tau = td$, $\beta = b/d$, $\sigma_i = s_i/d$, $\omega_{ij} = w_{ij}/d$, with $i,j \in \{c,m\}$. Then, the mean-field equations for the time evolution of the different population fractions read:
\begin{eqnarray}
\dot{c} & = & \phi\,\beta + m\,\omega_{mc} - c\,\omega_{cm} - c,\label{eq: dotc}\\
\dot{m} & = & -m\,\omega_{mc} + c\,\omega_{cm} - m,\label{eq: dotm}\\
\dot{\phi} & = & -\phi\,\beta + (1-\phi ),\label{eq: dotphi}
\end{eqnarray}
where a dot signifies a time derivative with respect to $\tau$. From Eq.~\eqref{eq: dotphi} we immediately see that the stationary fraction of empty places is independent of the social dynamics encoded in the transition rates $\omega_{cm}$ and $\omega_{mc}$, but only depends on the dimensionless birth rate $\beta$:
$$\phi=\frac{1}{1+\beta}\,.$$
In other words, the birth rates determines how densely populated the system is. Inserting this result together with the constraint $c+m+\phi=1$ into Eq.~\eqref{eq: dotm} yields an implicit equation, whose solutions are the fixed points or stationary states of the population dynamics:
$$
m\,\omega_{mc}(c,m,p) - \left(\frac{\beta}{1+\beta} - m\right)\,\omega_{cm}(c,m,p) + m = 0.
\label{eq:master eq}
$$
In the following, as noted above, we focus on the asymmetric case where mafiosi are much stronger than citizens, $\sigma_m \gg \sigma_c\approx 0$, and use the simplified notation $\sigma:= \sigma_m$. Then, in the absence of control elements, the mean-field equations reduce to:
\begin{eqnarray}
\dot{c} & = & \phi\,\beta -\sigma\,m\,(1-c)\,c -c,\label{eq: dotc fam}\\
\dot{m} & = & \sigma\, m\,(1-c)\,c - m,\label{eq: dotm fam}
\end{eqnarray}
whose fixed points are:
\begin{equation}
m^0=0, \qquad m^{\pm} = \frac{1}{2}\left(\frac{2\beta}{1+\beta}-1\pm \sqrt{1-4 / \sigma}\right).\end{equation}
The ensuing phase diagram (Fig.~\ref{fig: stability diagram}) shows three
different regimes: coexistence, bistability, and mafia extinction.
For a given birth rate $\beta$, mafiosi unavoidably get extinct below the threshold strength
$\sigma^*_{\text{sn}}$, while for strengths larger than $\sigma^*_{\text{tc}}$
coexistence is the only stable solution. In the intermediate
regime, $\sigma^*_{\text{sn}} < \sigma <\sigma^*_{\text{tc}}$, the
system is bistable and the stationary solution critically depends on
the initial conditions. It turns out that the minimal strength of
mafiosi to ensure that they do not get extinct increases with the
birth rate of citizens, although very weak mafiosi die out
independently of the birth rate.
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/stability-diag-asymmetric.eps}}
\caption{Phase diagram as a function of the birth rate $\beta$ and persuasiveness $\sigma$. Three regions with different regimes (coexistence, bistability, extinction) are shown, separated by a transcritical (solid) and a saddle-node (dashed) bifurcation.}
\label{fig: stability diagram}
\end{figure}
For a particular value of the birth rate $\beta\neq 0$ (Fig.~\ref{fig: bd}), the absorbing state $m^0$ becomes unstable via a transcritical bifurcation for a persuasiveness larger than the threshold value $\sigma^*_{\text{tc}} = (1+\beta)^2/\beta$, while the coexistence state $m^+$ is real only above the saddle-node bifurcation at $\sigma^*_{\text{sn}}=4$ (for $\beta >1$). In the regime between these two values the system is bistable. The third solution $m^-$ is unstable for all meaningful values. The stationary fraction of mafiosi in the coexistence state increases with their strength up to the asymptotic value $m^+ =\rho = \beta/(1+\beta)$ in the limit of infinite strength, $\sigma \to \infty$, where $\rho$ is the fraction of populated nodes.
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/bd-asymm-sigma_xtics.eps}
}
\caption{Bifurcation diagram for a fixed birth rate $\beta=10$ as a
function of the strength parameter $\sigma$. The solid and dashed
lines represent stable and unstable solutions, respectively. The two
solid circles stand for the saddle-node bifurcation, at which the solutions
$m^{\pm}$ become complex, and the transcritical bifurcation, above which the
absorbing state $m^0$ becomes unstable, respectively. In the dark grey and white areas the
system is monostable---extinction and coexistence are the stable
states respectively---and bistable in the light grey region.}
\label{fig: bd}
\end{figure}
\section{Structured social networks}
Individuals in actual societies do not interact with the whole population, but rather only with agents in their immediate vicinity. A given agent who updates her strategy is influenced by her nearest neighbors, so that the fraction of citizens, mafiosi, and control elements entering the transition rates \eqref{eq: wcm} and \eqref{eq: wmc} are now given by the instantaneous composition of her neighborhood. The population frequencies $(c,m, \phi)$ are restricted to a \emph{finite set} of possible combinations determined by the size of the neighborhood. They are thus no longer continuous as in the mean-field case discussed in the previous section, but take a \emph{discrete} set of values determined by the composition of the neighborhood of a given node.
We investigate the behavior of the mafia model on scale-free networks (SFN) whose degree distribution scales as a power law $p(k)\propto k^{-\gamma}$. The parameter $\gamma$ is related to the heterogeneity of the network and increases for more homogeneous structures. Scale-free networks~\cite{albert2002statistical,cohen2003scale} exhibit very interesting features suitable for the modelling of social systems, such as large heterogeneity---with a non-negligible fraction of highly connected nodes---and relative small average path-lengths, i.e.\ the average over the shortest distance between all pairs of nodes. The latter scales logarithmically with the network's size, as it has been observed in human communities~\cite{albert2002statistical}.
The scale-free networks used in this work have been generated following
the uncorrelated model \cite{catanzaro2005generation}, for which the resulting average degree $\langle k \rangle$ is a function of $\gamma$. This method imposes a maximal cutoff for the degree, automatically yielding the average degree for a given network size; see \cite{catanzaro2005generation} for further details. In this work, we have used networks with increasing $\gamma$ ranging from $2$ to $4$, and corresponding decreasing average connectivities $\langle k \rangle$ varying from $7.6$ to $2.4$.
We have performed stochastic simulations employing an agent-based model following the dynamics described above. Individuals belonging to both populations were taken as randomly distributed with specific initial fractions $m_0$ and $c_0$. We used random sequential updating according
to the dynamical rules defined previously. The results discussed in this paper were averaged over $1000$ runs. Starting from a given initial state, the system evolved until it reached a quasi-stationary state; since the system has an absorbing state, $m^0=0$, it will be unavoidably reached if one waits long enough. The population remained in this state for a long enough time window where the relevant observables were measured. We have carried out a finite size analysis and assessed that the systems reaches a quasi-stationary state at a time $\tau '$ proportional to the system's size, $\tau '= \lambda N$, with the proportionality factor $\lambda$ ranging from $0.01$ to $0.015$ (not shown).
We characterize the stationary state of the system in terms of the extinction probability $p_{\text{ext}}$, which is the probability that mafiosi got extinct at some time $\tau '$ within the quasi-stationary time window. The ensuing phase diagrams, as shown in Fig.~\ref{fig: FAM p_ext}, are qualitatively different for well-mixed graphs and scale-free networks. In particular, the topology of the underlying network strongly matters in the parameter regime where mean-field theory predicts bistability, i.e. a dependence of the stationary state on the initial state of the population.
\begin{figure}[b]
\centering
\subfigure[complete graph]
\includegraphics{figures/mf_asymmetric_99.9_0.05_N_6000.eps}
\subfigure[$\gamma=3$ SFN]
\includegraphics{figures/sfn_300_asymmetric_99.9_0.05_N_8000.eps}
\caption{[color online] Extinction probability---as encoded in the side bar---for initial conditions with a very small fraction of mafiosi, $m_0=5\cdot 10^{-4}$, for (a) a complete graph of size $N = 6000$ and (b) a scale-free network (SFN) with $\gamma=3$ and $N=8000$ sites ($\langle k \rangle=3.16$). The solid black line represents the deterministic mean-field prediction separating a region of coexistence above it from a regime in which mafiosi get extinct below it.
}
\label{fig: FAM p_ext}
\end{figure}
We have specifically analyzed the limiting case in which the initial population of mafiosi vanishes $(m_0\to 0)$. In this limit, the standard mean-field theory asserts that mafiosi go extinct for all of the bistable area. This is indeed the case for well-mixed societies (complete graphs), where the results of our stochastic simulations closely resemble the prediction of the (deterministic) mean-field theory. There are some minor differences in the actual position of the phase boundary which we attribute to stochastic effects; see Fig.~\ref{fig: FAM p_ext}a. In contrast, coexistence is systematically found in the bistable regime (of mean-field theory) for all studied networks (scale-free networks with $\gamma$ ranging from $2$ to $4$)\footnote{We have also performed simulations on square grids which show similar behavior as scale-free networks~\cite{balbas2010diss}.}. An exemplary phase diagram for a network with $\gamma = 3$ and a small initial mafiosi fraction is shown in Fig.~\ref{fig: FAM p_ext}b.
We attribute this interesting feature of what one could call an ``evolutionary stability'' of mafiosi to the locality of interactions on networks or regular graphs. In structured societies, the one-to-all character of interactions becomes relevant, as averaged frequencies suitably reproduce interactions between two individuals (one-to-one) but fail to describe interactions with a whole vicinity. An extended mean-field analysis, accounting for the discrete set of possible neighborhood compositions is thoroughly discussed in Appendix A. This extended mean-field theory gives a phase diagram where the bistable regime is drastically reduced in favour of the coexistence regime, in accordance with the above numerical results.
What is the micro-dynamics giving rise to these results? A vanishing small fraction of mafiosi gives rise to negligible transition rates $\omega_{cm}$ within a standard mean-field approach and consequently the mafia dies out in a short period of time. However, if interactions are local, citizens' vicinities containing any mafioso yield considerable transition rates for updating citizens to become mafiosi. Under which circumstances can a few isolated mafiosi invade a citizen population? The following discussion gives a set of heuristic arguments which are not meant to be quantitative but illustrate the basic mechanism behind the observed evolutionary stability.
Starting from some isolated mafiosi, the dynamics may be separated into three phases as illustrated in Fig.~\ref{fig: invasion-expansion-saturation}: invasion-expansion-saturation.
\begin{figure}[h]
\centering
\subfigure[isolated]{\resizebox{0.2\textwidth}{!}{%
\includegraphics{figures/isolated_mafiosi.eps}}}
\subfigure[invasion]{\resizebox{0.2\textwidth}{!}{%
\includegraphics{figures/invasion.eps}}}\\
\subfigure[expansion]{\resizebox{0.2\textwidth}{!}{%
\includegraphics{figures/expansion.eps}}}
\subfigure[saturation]{\resizebox{0.2\textwidth}{!}{%
\includegraphics{figures/saturation.eps}}}
\caption{[color online] Invasion-expansion-saturation
process. \emph{Invasion}: A few isolated mafiosi invade their nearest
neighbors instead of dying, which leads to the formation of very small clusters of mafiosi. \emph{Expansion}: these small clusters expand
as more and more citizens successively join the mafia. \emph{Saturation}: the
society reaches a stationary state with a dynamic balance between birth and death processes.}
\label{fig: invasion-expansion-saturation}
\end{figure}
First, the isolated mafiosi must \emph{invade} some neighboring cell before dying, i.e.\ the probability for any citizen in the mafioso's neighborhood to change her strategy must be larger than her death probability. The probability for a mafioso at site $i$ to invade any neighboring cell is the sum over the probabilities for all her $k_i$ neighbors (assumed to be citizens) to change strategy, i.e.\ $p_{\text{inv}}=\sum_{j=0}^{k_i} p^j_{c\to m} =\sum_{j=0}^{k_i} \sigma m (1-c)\Delta\tau$, which yields $k_i \sigma (1/k_j)^2 \Delta\tau$ considering that every citizen has a single mafioso in her vicinity. If the structure is homogeneous enough, so that one may consider similar connectivities, i.e.\ $k_i\sim k_j$,\footnote{Note that this approximation is formally valid only for SFN with large $\gamma$ or random graphs.} then the invasion probability becomes $ \sigma / k_i\,\Delta\tau$. The invasion probability is larger than the death probability for mafia's strength larger than the degree $\sigma_{\text{tinv}}> k_i$. This heuristic argument nicely explains the results obtained from simulations which show that the extinction probability decreases with the network's average degree $\left< k \right>$ (not shown). Say now the mafioso has not died out but managed to invade some neighboring cells and thereby created small mafia clusters. In order for these clusters to survive they have to \emph{expand} as groups to achieve a stable coexistence state in the population. Mafia clusters will grow if the fraction of citizens becoming mafiosi at their interfaces is larger than the fraction of those citizens gained via birth process. Assuming similar fractions of citizens and mafiosi in the vicinity of an agent located at the interface, $m \sim c \sim 1/2$, and large enough birth rates\footnote{This assumption implies that one can neglect empty sites and approximate the fraction of citizens and mafiosi at both sides of the cluster boundary by $1$. The approximation becomes better with an increasing size of the vicinities. For small neighborhoods, taking empty sites into account would make it easier for the mafiosi to expand their territory as citizens have less peer protection. Therefore, accounting for empty sites would slightly lower the threshold strength such that our approximation can be regarded as an upper bound.} $\beta$, the fraction of born citizens is given by $l_{\text{int}} \phi \beta$ and those becoming mafiosi $l_{\text{int}} \omega_{cm}\sim l_{\text{int}} \sigma m (1-c) \sim l_{\text{int}} \sigma /4$, where $l_{\text{int}}$ is the length of the cluster interface. Mafia clusters will thus grow if their strength becomes larger than some threshold value, $\sigma > \sigma_{\text{texp}} =4$. Finally, mafia clusters will expand to the point in which there are no citizens left at which expense they can grow. This \emph{saturation} takes places for a residual fraction of citizens, which equals the fraction of born citizens at empty places. A dynamic stationary state has thus been reached at which both populations coexist due to the asymmetric birth of citizens---otherwise citizens would get extinct. These heuristic arguments for the growth process in structures, supported by the results of the stochastic simulations, nicely evidences that above some threshold value for the mafia's strength, the mafia can always invade a citizens population. This threshold is given by the larger of the following two values: the mafia strength required to invade neighboring sites, $\sigma_{\text{tinv}}$, or for small clusters to expand, $\sigma_{\text{texp}}$, i.e. $\sigma > \mathrm{max}(\sigma_{\text{tinv}}, \sigma_{\text{texp}}) = \mathrm{max}(\langle k \rangle, 4)$.
In the above line of reasoning, we have assumed that very node has the same connectivity, which is true only for random networks. However, since mafia's expansion is a process which depends on the \emph{local} dynamics, we suppose that our heuristic approach should remain valid for scale-free networks. Moreover, the existence of regions with a connectivity significantly smaller than the average in scale-free networks suggests that coexistence is more likely to be found there than for the corresponding random graph with the same average degree.
Another interesting result is that in the limit of complete random graphs, $\langle k \rangle\to N$, the above heuristic approach gives a threshold strength $\sigma \geq 1+\beta$ for mafiosi to survive, which agrees well with the standard mean-field analysis---see Fig.~\ref{fig: FAM p_ext}a.
\section{The role of control elements}
Societies can regulate themselves by introducing control elements which fight the expansion of mafias. Here, we consider control elements (police) which are randomly attached to the edges of a given graph, such that a fraction $p$ of the total number of edges---and thus of pairs of individual relations---are controlled. With an increasing number of police in the asymmetric model the mafiosi also need to enhance their strength/persuasiveness in order to survive. For simplicity, we consider parameters such that $\sigma_m=\beta = \sigma$ corresponding to the diagonal in Fig.~\ref{fig: stability diagram} where only two regimes are
observed in the mean-field limit, namely extinction and bistability.
A standard mean-field analysis -- equivalent to a well-mixed population or alternatively a complete graph -- including control elements with strength $\sigma_p = \sigma$ shows the same type of regimes as for the model without police: For small police fractions the system is bistable and coexistence or extinction is obtained depending on the initial composition of the population. Above a certain police fraction, the mean-field analysis predicts extinction of mafiosi as the only stable fixed point; see Fig.~\ref{fig:policed_structures}a for a phase diagram with initial condition $m_0 = 0.5$. The mafia's strength required to achieve coexistence is increased with the presence of control elements.
\begin{figure}[t]
\centering
\subfigure[complete graph]
\includegraphics{figures/mf_policed_50_50_N_6000.eps}
\subfigure[SFN $\gamma=4$]
\includegraphics{figures/sfn_400_policed_50_50_N_8000.eps}
\caption{[color online] Extinction probability for a policed system on (a) a complete graph (well-mixed population) with $N=6000$ and on (b) the nodes of a scale-free network with $\gamma=4$ ($\langle k \rangle=2.45$) and $N=8000$. The solid line in (a) represents the separatrix predicted by the deterministic mean-field theory for the given initial conditions $m_0=1/2$.}
\label{fig:policed_structures}
\end{figure}
On scale-free networks, the local character of interactions again leads to an enlarged coexistence region as compared to the well-mixed case, cf. Fig.~\ref{fig:policed_structures}b. However, there is a significant difference to networks without control elements. In the present case, the coexistence regime is enlarged beyond the bistable regime of mean-field theory and now also covers an area in parameter space where the system was monostable for well-mixed populations (i.e.\ where extinction was the only stable state); cf. Fig.~\ref{fig:policed_structures}a and b. The reason for this anomalous behavior is that the control elements introduce a kind of \emph{quenched disorder}, which from thermodynamic systems is known to strongly affect phase behavior. Indeed, we have checked by numerical simulations (data not shown) that coexistence is found only in a parameter regime where mean-field theory predicts bistability, if the control elements are allowed to diffuse on the network (annealed disorder). Analyzing the population dynamics on a local scale, we observe that the presence of control elements leads to a kind of micro-phase separation in the society: while citizens preferentially populate policed regions, mafiosi aggregate in unpoliced areas. The unpoliced areas display the same dynamics as the fully asymmetric model, in which species coexist for mafias' strength larger than some threshold, $\sigma > \sigma_t$. Fig.~\ref{fig: snapshot policed lattice} shows a snapshot of a typical population structure on a square lattice, where both regimes are clearly recognizable.
\begin{figure}[b]
\centering
\resizebox{0.45\columnwidth}{!}{%
\includegraphics{figures/snapshot_b_25_s_25_f_50_50_30_t_10.eps}}
\caption{[color online] Snapshot of the population structure on a lattice with a police fraction $p=0.3$ and size $N=4900$. Red cells are occupied with mafiosi, green with citizens, and white cells correspond to empty nodes. The black segments represent policed connections between adjacent cells. Citizens and mafiosi cluster preferentially around policed and unpoliced areas, respectively.}
\label{fig: snapshot policed lattice}
\end{figure}
Micro-phase separation results in an increase of the respective minority's population size with increasing network homogeneity $\gamma$, or, equivalently, decreasing average degree $\langle k \rangle$, cf. Fig.~\ref{fig: stationary population policed gamma}. This can be understood as follows: Increasing network homogeneity implies smaller neighborhood sizes and larger average paths between pairs of nodes, and thereby increasingly hinders mixing of mafiosi and citizens. As a consequence, phase separation into isolated mafia and citizen clusters becomes more pronounced. Then, the respective minority species can find niches where they can grow to population sizes larger than in a corresponding well-mixed environment.
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/policed_gamma_pop_N_10000_paper_1.eps}
}
\caption{[color online] Stationary fractions of mafiosi and citizens as a function of the network homogeneity $\gamma$ for two different values of the police fraction $p$. The values of $p$ were chosen such that for $\gamma = 2$ the stationary state corresponds either to coexistence ($p=0.1$, red), or mafia extinction ($p=0.2$, blue). Hence, for $p=0.1$ citizens are the minority species, while for $p=0.2$ mafiosi are the minority. Solid and open symbols represent mafia and citizens fractions, respectively. In both cases the respective minority species increases in population size with increasing network homogeneity $\gamma$, or, equivalently, decreasing average degree $\langle k \rangle$. The average degrees with increasing $\gamma$ ticks read $\langle k \rangle= 7.63, 4.31, 3.16, 2.69, 2.45$. Data are obtained for the following parameters: $N=10 000$, $m_0=c_0=1/2$, $\sigma=\beta=10$. }
\label{fig: stationary population policed gamma}
\end{figure}
A finite size analysis of the extinction probability with the increasing heterogeneity of graphs, for a fixed police fraction ($p=0.3$), is shown in Fig.~\ref{fig: extinction policed gamma}. The increasing slope of the extinction probability with $N$ suggests that as $N \to \infty$ there might be a phase transition separating an extinction from a coexistence regime. To make this point conclusive, however, would require to simulate even larger system sizes. Our main point is that for sufficiently homogeneous networks (large $\gamma$) the phase separation between policed and unpoliced areas allows survival of mafiosi in the unpoliced areas for police fractions which otherwise would suffice for their extinction in
well-mixed and heterogeneous populations.
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/pol_gamma_N_wb.eps}
}
\caption{Extinction probability of mafiosi for a fixed police fraction $p=0.3$ as a function of network homogeneity $\gamma$. With increasing overall population size $N$ the transition between certain extinction and certain survival of mafiosi becomes very pronounced indicating that there might be a phase transition as a function of $\gamma$. Data were obtained for the following set of parameters: $\sigma = \beta =10$ and $m_0=c_0=1/2$. The average degrees with increasing $\gamma$ ticks read $\langle k \rangle= 7.63, 4.31, 3.16, 2.69, 2.45$.}
\label{fig: extinction policed gamma}
\end{figure}
\subsection*{Targeted police distribution}
In the same way that vaccination during an epidemic is more effective if targeted to highly connected nodes or hubs \cite{pastor2002immunization,anderson1982directly}, one might suppose that the distribution of control elements also matters in eradication of mafias.
To explore this question we investigate the effect of two specific distributions $q_\alpha (e_{ij})$ of control elements:
\begin{eqnarray}
q_1(e_{ij}) & =& \frac{d_i d_j}{\sum_{ij}d_i d_j},\\
q_2(e_{ij}) & = & \frac{(d_i d_j)^{-1}}{\sum_{ij}(d_i d_j)^{-1}}.
\label{eq: police probability distribution}
\end{eqnarray}
These probability distributions for a control element to be allocated at an edge $e_{ij}$ connecting nodes $i$ and $j$ are taken as either directly ($q_1$) or inversely ($q_2$) proportional to the product of the degrees of the nodes connected by the edge, $d_id_j$. Therefore, the first distribution favours police attachment close to highly connected nodes, whereas the second promotes control elements to surround sparsely connected nodes.
The network topology should be the key element for understanding the relevance of a specific police distribution. Following the analogy with the problem of vaccination, one would
expect that surrounding network hubs with control elements should hinder the proliferation of mafias, since individuals in hubs are expected to be more influential. Although highly connected individuals influence the decision process of a larger number of agents, their influence on the targeted individual is as important as that of sparsely connected agents.
Therefore, controlling the population in hubs is not decisive for the system's dynamics. Actually, due to the concentration of control elements around hubs, the fact that a large number of small nodes are left unpoliced plays a more important role in the time evolution of the system than blocking the hubs. A large number of not so well connected agents are free to influence the society and drive it to a stationary state in which mafias do not get extinct.
The micro-phases induced by the presence of control elements are
now localized around highly and sparsely connected nodes depending
on the police distribution. We have observed
that policed areas are quickly occupied by citizens during the initial stages of the
dynamics, while at later times the dynamics evolves in the unpoliced regions. In the stationary state, there is also a clear division in the degree of the nodes occupied by citizens
and mafiosi: the population distribution follows the police distribution with citizens clustering around policed edges; see Fig.~\ref{fig: population deg-dist} for an example with control
elements drawn from the $q_1$-distribution.
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/sfn_deg_dist_gamma_2.5_pd_1_p_0.2.eps}}
\caption{[color online] Stationary fraction of nodes of degree $k$ populated by citizens (free squares) and mafiosi (red circles) for the case where control elements preferentially attach to hubs ($q_1$). The simulations were run for $\sigma =10$, $p=0.2$ in a $\gamma =2.5$ scale-free network of size $N=8000$.}
\label{fig: population deg-dist}
\end{figure}
What is the system's behavior when control elements attach to specific targets as compared to the random case? The relation between the fraction of edges accommodating control elements and the fraction of nodes which are protected by those edges seems to be the key factor in the analysis of the problem. Figures \ref{fig: police distributions} and \ref{fig: ext police distributions} show the fraction of both populations and the extinction probability for the three distributions, $q_1$, $q_2$, and random, as a function of network homogeneity, $\gamma$.
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/sfn_gamma_pol_dist_p_0.1_wb.eps}}
\caption{Stationary population fractions for different distributions
of the control elements and fixed police fraction $p=0.1$ as a
function of the increasing homogeneity of the scale-free network ($\gamma$). The solid
symbols represent the mafia fraction, the empty ones the citizen
population in the stationary state. The average degrees with increasing $
\gamma$ ticks read $\langle k \rangle= 7.63, 4.31, 3.16, 2.69, 3.45$.}
\label{fig: police distributions}
\end{figure}
The resulting fraction of controlled nodes, if control elements attach to large nodes, $q_1$, whose degree is close to the maximal value $k_M$, is smaller than in a random distribution,
$p/k_M < p/\left<k\right>$. Similarly, the effective fraction of protected nodes is larger than in the random case if control elements target sparsely connected nodes with the minimal degree $k_m$, $p/k_m > p/\left< k\right>$. Thus, for a fixed police fraction, the population of mafiosi in equilibrium is larger for the $q_1$-distribution than for the random case, and this population is larger for the latter than for the $q_2$ distribution: $m(q_1) > m(\text{rand}) > m(q_2)$, as illustrated in Fig.~\ref{fig: police distributions}. The difference in the stationary population fractions induced by the specific distributions naturally increases with the heterogeneity of networks (small $\gamma$).
\begin{figure}[h]
\centering
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figures/sfn_pol_dist_ext.eps}}
\caption{The police fraction needed to achieve extinction of mafiosi
on a $\gamma=2.5$ ($\langle k \rangle = 4.31$) scale-free network of size $N=10000$ crucially
depends on the distribution of control elements on the network. A
larger mafia fraction is needed when control elements are located
around hubs ($q_1$) compared with the random distribution, whereas
the required fraction to fight the mafia
depends on the initial conditions for control elements attached to
small nodes. Initial conditions are $m_0=c_0 =1/2$.}
\label{fig: ext police distributions}
\end{figure}
Fig.~\ref{fig: ext police distributions} illustrates that the fraction
of control elements at which extinction takes place increases if
the control elements concentrate around hubs ($q_1$). In this case, an
effective homogeneous structure is left unpoliced and, as we have
discussed before, even small fractions of mafiosi manage to invade a
citizens' population. In contrast, if the control elements preferentially surround
small nodes ($q_2$), the unpoliced region consists of
highly connected nodes, whose dynamics is similar to
that of the well-mixed population for the unpoliced asymmetric
model. In this case, the stationary state critically depends on the
initial conditions, so that we cannot draw any general conclusions
on the effect of the distribution on the system's behavior.
Whether a larger or smaller fraction of control elements than for the
random distribution is needed to drive the mafia to extinction depends
thus on the particular
initial conditions in the unpoliced areas.
\section{Conclusions and outlook}
We have proposed a novel way to model the influence of the neighborhood in social dynamics which accounts for the protection of alike peers proportional to their fraction. In this way, both coexistence of opinions and consensus on the neutral position (citizens, in the language used in the introduction) emerge as possible solutions in a natural way. We find that the dynamics of the model in structured societies drastically differs from that predicted for well-mixed populations on complete graphs. In the parameter window where the population dynamics exhibits bistability on complete graphs, we find coexistence between citizens and mafiosi on scale-free networks (and square lattices). This is traced back to the local character of the interaction which allows mafiosi to invade societies in structured societies with small neighborhood sizes. We have rationalized this effect employing both heuristic arguments on the population dynamics as well as an extended mean-field analysis, which accounts for the local network structure. Moreover, stochastic simulations for scale-free networks with increasing average degree $\langle k \rangle$ show that the coexistence region shrinks and the bistable regime is recovered in the limit of a complete graph~\cite{balbas2010diss}.
The presence of control elements (police) splits up the social structure into unconnected
subnetworks ruled by different dynamics. Neutral individuals (citizens) populate regions influenced by the control elements (policed areas), while the remaining (unpoliced) areas develop the same dynamics as in the model without police, yielding a stationary state which depends on the network homogeneity. We have discussed the positive effect of this micro-phase separation on the population size of the minority group. In addition, we report a markedly sharp crossover from extinction of the minority to coexistence of species as a function of the heterogeneity of the underlying scale-free network. By preferential attachment
of the control elements to highly (sparsely) connected nodes, smaller (larger) effective fractions of the control elements emerge. Depending on the kind of nodes the control elements are preferentially attached to, the areas free of external influence (no police) exhibit now a degree of heterogeneity different from that of the whole network. We have specifically investigated the role of two different distributions of the external elements on the given structure, showing that the attachment rules for the control elements determine the threshold fraction needed to bring the minority group (mafia) to extinction.
We conclude with a comparison of the mafia model with other non-linear opinion model and generalizations thereof. The mafia model can be viewed as an extension of the Abrams-Strogatz (AS) model \cite{abrams2003} in the following sense. In the limit $d\to 0$ and $\phi\to 0$, where there are no more empty places in the mafia model, it reduces to a two-state reaction scheme $c\leftrightarrow m$ with transtion rates $\omega_{cm}= s_m m^2$ and $\omega_{mc}= s_c c^2$, i.e.\ an AS model with a low volatility of $a=2$. In the limit of the asymmetric model, $s_c\to 0$, considered in this manuscript, the only stable fixed point is $(m,c)=(1,0)$, i.e. consensus on one of the two opinions as in the AS model. For general prestige values, $s_{m/c}\neq0$, the coexistence fixed point is unstable and both consensus states become stable, i.e.\ the dynamics exhibits bistability.
These results highlight the importance of empty places (a third state) for the observed dynamics in the mafia model. Note, however, that the presence of empty places is only a necessary but not a sufficient condition for the emergence of bistability and coexistence. The functional form of the transition rates, i.e. the fact that citizens protect themselves, also matters in an important and interesting way. The transition rates of our model, e.g. $\omega_{cm} = s_m m (1-c)$, weaken the tendency to switch to the opposite state by a factor $(1-c)$ as compared to neutral voter-like models. Instead of these transition rates, we could have adopted an AS scheme to investigate the effect of peer support: $\omega_{cm} = s_m m^a$ with $1<a<2$. However, these non-linear transition rates are qualitatively different from the mafia model: for a fixed fraction of empty sites, $\phi$, the rates in the mafia model are linear or non-linear depending on the relative fractions of species and empty sites, while they are always non-linear in the AS model.
Indeed, if one adopts transition rates like those of the AS model $\omega_{cm}=\sigma_m m^a$ and $\omega_{mc}=0$, i.e.\ generalizes the AS model to contain empty places, one does not find all the regimes of the mafia model. For $a=1$, there is a transition between two monostable regimes, extinction $m=0$ and coexistence at $\sigma = 1/(1+\beta)$. For $a=2$ there is a saddle-node bifurcation with a transition from a monostable regime with extinction of mafiosi to a bistable regime, where extinction and coexistence are stable fixed points. However, it does not show a monostable regime with coexistence of both species. The mafia model exhibits richer behavior than its corresponding limits in terms of the generalized AS model, i.e.\ $a=1$ and $a=2$. In a sense, its dynamics combines all the regimes found for the two limiting cases of the AS model with empty sites: monostable regimes, with coexistence and extinction of mafiosi, and a bistable regime.
Our model still lacks the inclusion of the evolution of the network of contacts which takes place in actual societies. Considering adaptive networks which also account for topological features is, in our view, the most compelling extension of the model.
\section{Acknowledgements}
We thank Heiko Hotz for the implementation of the code to generate scale-free networks. Financial support by the Deutsche Forschungsgemeinschaft through the SFB TR12 ``Symmetries and Universalities in Mesoscopic Systems" is gratefully acknowledged.
|
1,314,259,995,845 | arxiv | \section{Introduction}
Open domain semantic parsing aims to map natural language utterances to structured meaning representations. Recently, seq2seq based approaches have achieved promising performance by structure-aware networks, such as sequence-to-action\cite{chen2018sequence} and STAMP\cite{sun-etal-2018-semantic}.
However, this kind of approach mixes up low-level entities, predicates and high-level structures together, which loses precision at each level to some extent. So the sketch-based method may be an another choice for disentangling high-level structures from low-level details. In this work, we conduct our sketch-based approach on MSParS, a large hand-annotated semantic dataset mapping questions to logical forms. We argue there are at least two advantages to sketch-based method. Firstly, basic attention based seq2seq network\cite{bahdanau2014neural,luong-etal-2015-effective} does not perform well in semantic parsing because logical forms are structured sequences and it fails to incorporate structure information of logical forms. Then sequence-to-tree(seq2tree)\cite{dong-lapata-2016-language} proposes a structure-aware decoder to utilize the information. But its architecture also becomes much more complex. Instead of using intricate decoders, we can extract high-level sketches for logical forms and classify samples into several sketch classes. Logical forms of a certain sketch class have a fixed pattern which is shown in Table~\ref{table1}. So the structure problem is finally simplified to a classification task. Secondly, logical forms often need to copy a span of questions. Although Copynet\cite{gu-etal-2016-incorporating} and Pointer\cite{see-etal-2017-get} implement the copy mechanism, it is still difficult to achieve the expected effect. But for the sketch-based method, this problem becomes an individual entity labeling task which is easier than generating entities. Generally speaking, the seq2seq way decodes the entire meaning representation in one go while we deal with different parts at different levels of granularity just like coarse2fine\cite{dong-lapata-2018-coarse}. Although we increase the number of stages, the network architecture of each stage is much simpler without sacrificing the accuracy. In this way, we are able to locate the errors and optimize according parts.
\begin{table}
\caption{Examples demonstrating sketches of logical forms. $P$ represents predicate and $E$ represents entity. Subscripts are applied to distinguish different ones.}\label{table1}
\centering
\begin{tabular}{l|p{250pt}}
\hline
Class & Sketch of Logical Form \\ \hline
aggregation & count ( lambda ?x ( $P_1\;E_1$ ?x ) ) \\ \hline
cvt & ( lambda ?x exist ?y ( and ( $P_1\;E_1$ ?y ) ( $P_2$ ?y $E_2$ ) ( $P_3$ ?y ?x ) ) ) \\ \hline
multi-turn-entity & ( lambda ?x ( $P_1\;E_1$ ?x ) ) $|||$ ( lambda ?x ( $P_2\;E_1$ ?x ) ) \\ \hline
multi-turn-answer$\ $ & ( lambda ?x ( $P_1\;E_1$ ?x ) ) $|||$ ( lambda ?x exist ?y ( and ( $P_1\;E_1$ ?y ) ( $P_2$ ?y ?x ) ) ) \\ \hline
single-relation & ( lambda ?x ( $P_1\;E_1$ ?x ) ) \\ \hline
yesno & ( $P_1\;E_1\;E_2$ ) \\ \hline
\end{tabular}
\end{table}
We propose to decompose the process into three stages. In the first stage, we deal with a sketch classification task. Then, we find the entities in the questions through an entity labeling task. Actually, we combine the two stages through the multi-task model for both accuracy and efficiency\cite{chen2019bert}. The last stage is the most difficult part since the knowledge base of MSParS is not available. We define question pattern-logical form pattern pair and use the matching network to rank all these pairs. Seq2seq based approach is one of the two methods we adopted here to help rescore on the whole. We also incorporate state-of-art pre-trained work, Bert\cite{devlin-etal-2019-bert}, in above tasks to incorporate more priori knowledge.
The error rate of our multi-task model is lower than 2\%, which ensures the right sketch and entities. So the last stage actually determines the accuracy to a large extent. Our accuracy achieves 77.42\% after above three stages. Seq2seq based approach and co-occurrence relationship improve the accuracy to 86.86\% in validation set. Our final accuracy in full test set reaches 84.47\%. And the accuracy on hard test subset has been promoted to 63.08\% finally which is higher than the best model on the submission list by 5.65\%.
In the rest of our paper, we first analyze the special features of MSParS for this task in section 2. Afterwords, we discuss our system in detail in section 3. Then in section 4, we demonstrate our experimental setup, results and analyzation. Related works are mentioned in section 5. At last, we make a conclusion of the whole paper and propose our future work.
\section{Data Analyzation}
The dataset MSParS is published by NLPCC 2019 evaluation task. The whole dataset consists of 81,826 samples annotated by native English speakers. 80\% of them are used as training set. 10\% of them are used as validation set while the rest is used as test set. 3000 hard samples are selected from the test set. Metric for this dataset is the exactly matching accuracy on both full test set and hard test subset. Each sample is composed of the question, the logical form, the parameters(entity/value/type) and question type as the Table~\ref{table5} demonstrates.
\begin{table}
\caption{An sample of MSParS.}\label{table5}
\centering
\begin{tabular}{l|l}
\hline
question & what is birth date for chris pine \\ \hline
logical form & ( lambda ?x ( mso:people.person.date\_of\_birth chris\_pine ?x ) ) \\ \hline
parameters & chris\_pine (entity) [5,6] \\ \hline
question type$\ $ & single-relation \\ \hline
\end{tabular}
\end{table}
Samples are classified to 12 classes originally at a coarse level while we reclassify them at a finer level, which is the basis of our sketch-based method. We replace the predicate in the triple as $P_i$, the entity in the triple as $E_i$ and distinguish different ones with subscripts. The number in superlative class and comparative class is replaced as $V$ while the type in the triple begin with special predicate ``isa" is replaced as $T$ as well. In this way, we get the sketch of the logical form. Finally, we produce 15 classes of sketches.
We believe the features of questions highly correlate with the sketch of logical forms. For instance, the sketch must begin with ``argmore" or ``argless" if there are comparative words such as ``higher", ``more" and ``before" in questions. Therefore, we take questions as input to classify samples to different sketch classes.
As the Table~\ref{table5} suggests, entities are concatenated tokens from the question. So we implement entity labeling to label every token in the questions.
Nonetheless, cases are tough when there are more than one entities in the logical form. Suppose that we have labeled $E_1$ and $E_2$ from the question. We do not know which one we should choose to fill in the first entity slot in the sketch. We solve this problem and pick out the suitable predicate simultaneously. The entities in the questions are replaced by label ``entity'' with subscipts suggesting the order they appear in questions to get question patterns. When it comes to logical form patterns, the entities in logical forms are substituted as well while predicates are split to small tokens. Table~\ref{table2} gives an example of these two patterns. In this way, we combine the entity collocations with predicates successfully. Another reason for label ``entity'' used here is generalization. For instance, ``what is birth date for barack obama" shares the same question pattern ``what is birth date for entity1" with ``what is birth date for donald trump". The predicate used in these logical forms is ``mso:people.person.date\_of\_birth''. So we can draw the conclusion that the predicate for this question pattern is likely to be ``mso:people.person.date\_of\_birth''. If ``what is birth date for george bush" appears in the test set, we are able to find the right predicate even if we do not see ``george bush'' before. Without the impact of specific entities, our model learns the mapping from question patterns to logical form patterns more accurately. Since we do not have a knowledge base, we can only extract logical form patterns in training set. And we find 90.34\% of logical form patterns in validation set are covered by that in training set, which ensures the feasibility of our method.
\begin{table}
\caption{An example for question pattern and logical form pattern.}\label{table2}
\centering
\begin{tabular}{l|p{220pt}}
\hline
question & travels in the interior districts of africa has how many pages? $|||$ when is the date of publication of the book edition? \\ \hline
question pattern & entity1 has how many pages? $|||$ when is the date of publication of the book edition? \\ \hline
logical form & ( lambda ?x ( mso:book.edition.number\_of\_pages travels\_in\_the\_interior\_districts\_of\_africa ?x ) ) $|||$ ( lambda ?x ( mso:book.edition.publication\_date travels\_in\_the\_interior\_districts\_of\_africa ?x ) ) \\ \hline
logical form pattern$\ $ & book edition number of pages entity1 ?x $|||$ book edition publication date entity1 ?x \\ \hline
\end{tabular}
\end{table}
We take question patterns paired with logical form patterns as input. Then, we get logical form candidates through combining sketches and entities with logical form patterns. The ones with higher scores are more likely to be right.
\section{Proposed Approach}
\subsection{Sketch Classification}
The single sentence classification fine-tuned task in Bert is applied in this stage. A special classification embedding ([CLS]) is added to the beginning. We use the final hidden state corresponding to this token as the aggregate sequence representation for classification task denoted as $C_s \in \mathbb{R}^h$, so the probability of class $c_i$ can be computed as:
\begin{equation}
p(c_i|x) = softmax_i(W_sC_s + b_s)
\end{equation}
where $W_s \in \mathbb{R}^{k_s \times h}$ and $b_s \in \mathbb{R}^{k_s}$, $k_s$ is the number of sketch classes here. $W_s$, $b_s$ and all the parameters of Bert are fine-tuned jointly to maximize the log likelihood probability of the correct label.
\subsection{Entity Labeling}
We use the single sentence tagging fine-tuned task in Bert here to label every token in the question whether it is an entity token that appears in the logical form as well. To simplify the problem, we use 3 labels for the tokens in the questions. Label ``b" represents the first token in an entity while label ``i'' for the rest ones. And label ``o'' represents those tokens which are not in any entities. Because of the lexical rules in Bert, we also label the special token ([CLS]) at the beginning of the sentence and the special token ([SEP]) at the ending of the sentence as ``o''. The last label ``p'' is for all the padding tokens added to reach max\_length. Besides, some tokens in the questions are split into several smaller tokens by Bert. For the split ones, they are labeled as ``i'' if they are in the entities and ``o'' otherwise. In this stage, we use all the final hidden states denoted as $D \in \mathbb{R}^{h \times m}$ where m is the max\_length of the input tokens we set. The hidden state is mapped into dimension $k_e$ via $E = W_eD + b_e$ where $W_e \in \mathbb{R}^{k_e \times h}$ and $b_e \in \mathbb{R}^{k_e \times m}$, $k_e$ is the number of labels here. We employ the CRF on the top of the network taking $E$ as input representations. The objective is to minimize the loss of CRF layer.
\subsection{Multi-Task Model}
We combine sketch classification and entity labeling to share information together, which means sketches of samples can help label entities while the labeled entities can help sketch classification conversely. The architecture of our model is shown in Fig.~\ref{fig1} where the parameters of Bert model is fine-tuned together for two tasks. Since the scale of dataset is large, we can save lots of time through multi-task model instead of training two different models. Finally, it contributes to both accuracy and efficiency. In this way, our loss to minimize is the weighted sum of the cross-entropy loss in sketch classification task and the CRF loss in entity labeling task.
\begin{figure}
\includegraphics[width=\textwidth]{multi-task.png}
\caption{An overview of multi-task model proposed. The original input question is ``chemical compound of citric acid". It becomes ``chemical compound of ci \#\#tric acid" after the tokenization of Bert.} \label{fig1}
\end{figure}
\subsection{Pattern Pair Matching Network}
Besides the single sentence tasks, Bert provides sentence pair classification tasks as well. We implement the matching network taking question patterns and logical form patterns as input. The right pattern pairs are regarded as positive samples. We select negative samples only from the logical form patterns in the same sketch class for fixed question patterns. The sketch mentioned is from the multi-task model. Just like sketch classification, we denote the final hidden state corresponding to token ([CLS]) as $C_p \in \mathbb{R}^h$, so the probability can be computed as:
\begin{equation}
p(c_j|x) = softmax_j(W_pC_p + b_p)
\end{equation}
where $W_p \in \mathbb{R}^{2 \times h}$, $b_p \in \mathbb{R}^{2}$ and $c_j \in \{0, 1\}$. $W_p$, $b_p$ and all the parameters of bert are fine-tuned jointly to maximize the log likelihood probability of the correct class.
In the prediction stage, the candidates for a question pattern are from logical form patterns in the same sketch class as well. The probabilities of class ``1" are scores we get for these pattern pairs. From logical form patterns, we get not only right predicates, but right orders as well in which entities should appear. So with the sketch and entities we aquire in the multi-task model, we can already generate complete logical form candidates with scores between 0 and 1.
\subsection{Predicate-Entity Pair Matching Network}
To alleviate the absence of knowledge base, we incorporate the co-occurrence relationship between predicates and entities to evaluate the candidates. We create the second matching network based on Bert as well. This time, the pairs we take as input are predicate-entity ones. We label the predicate-entity pair as ``1'' if they have ever appeared in one triple in training set. For a certain entity, we select predicates that never appear with this entity as negetive samples. In the prediction stage, we score the predicate-entity pairs in logical form candidates. However, this network does not take questions into account. The predicate for a certain entity can differ a lot according to various questions. For instance, the predicate for ``what is birth date for barack obama" is apparently different from that for ``what is birth place for barack obama". But the entity ``barack obama" has only one predicate with highest score. Although this matching network only considers the co-occurrence relationship regardless of the information from questions, scores produced by it do work as an auxiliary.
\subsection{Pointer Network}
Although it is not easy for a complex network to generate the whole logical form, such networks do reflect the mapping from an overall perspective. So we adopt Pointer\cite{see-etal-2017-get} here to rerank. We take the questions as input. For logical forms, entities and predicates are composed of words concatenated by ``\_'' or ``.''. In order to utilize the information of words, we split all entities and predicates and take split logical form candidates as output. For a fixed question, we calculate cross-entropy losses of different pairs with split logical forms. Then every loss is divided by the max one and subtracted by 1 to be normalized between 0 and 1. The higher the score is, the more the logical form candidate is likely to be true.
\subsection{Ranking}
A linear combination of the three intermediate scores from pattern pair matching network, predicate-entity pair matching network and Pointer is used to rerank logical form candidates. Weights are roughly adjusted in validation set.
\section{Experiment}
The number of sketch classes is 15 and the number of labels is 4 in the multi-task model. The Bert model we applied is ``BERT-Base, Uncased" with 12-layer, 768-hidden, 12-heads and 110M parameters\footnote{https://github.com/google-research/bert}. All the parameters are fine-tuned in validation set. In the multi-task model, we train the model for 10 epoches. We set batch size to 32 and learning rate to 2e-5. The weight of the loss in sketch classification is 1 while that in entity labeling is 2. We train 3 models in pattern pair matching network with different epoches. As for predicate-entity pair matching network, the number of epoch we use is 3. In Pointer, word embeddings were initialized by Glove\cite{pennington2014glove}. The hidden dim of LSTM is set to 256. More details will be released in our source codes later.
Because of the instability of the performance of neural network over training epoches,
ensemble learning is incorporated both in pattern pair matching network and Pointer. Scores of Pointer is the simple average of scores from 3 models with different epoches. When it comes to pattern pair matching net, it is a little complex. We make a prediction for training set with our ``best" model. We apply ranking sampling here. From those labeled as ``0'' but with probabilities larger than 0.0001, we select 20 of them while 5 of those whose probabilities are smaller than 0.0001 as new negative samples. We train new models with new training data resampled before every epoch based on one ``best" model and base model of Bert. After several epoches, we average the probabilities of new models and original models for ensemble.
We demonstrate the detailed performance in Table~\ref{table3}. All samples are classified into 15 classes. We show the results for every class and the overall weighted average preformance in validation set. Because the complete test set is not open yet, we only provide the overall results in test set returned after submission.
It can be seen the overall error rate of our multi-task model is only 1.93\% which means this task is successful. In sketch classification, $Err_s$ scores of all classes are lower than 1\% except multi-turn-answer. Its recall is 100.00\% while its precision is 91.38\%. 0.92\% of samples in multi-turn-entity are misclassified to multi-turn-answer in validation set. We find there are separator ``$|||$'' in logical forms from three classes of multi-turn questions. Multi-turn-predicate questions have two different entities while both multi-turn-entity and multi-turn-answer questions have only one. This kind of entity information is passed to sketch classification through shared parameters. So our system makes some mistakes while distinguishing multi-turn-entity samples from multi-turn-answer samples. As for entity labeling, the overall error rate is 1.72\%. We check the wrong samples and find our model is not so good at recognizing entity boundaries especially while encountering some special tokens such as articles, genitive ``s'' and quotation mark. Actually, it is not easy for human to define an entity in these cases as well.
\begin{table}
\caption{Performances of our best model. $F1_s$ represents the $F1$ score of sketch classification in multi-task model. We compute $Err_s$ by $1 - F1_s$. $Err_e$ represents the error rate of entity labeling part in multi-task model(an sample is regarded as right only when all of its entities are labeled correctly). $Err_m$ represents the error rate of the whole multi-task model(an sample is regarded as right only when both sketch classification subtask and entity labeling subtask are correct). $Acc_l$ is the exactly matching accuracy of logical forms. We compute $Err_l$ by $1 - Acc_l$.}\label{table3}
\centering
\begin{tabular}{l|l|cccc}
\hline
\multicolumn{2}{l|}{Dataset} & $\ Err_s\ $ & $\ Err_e\ $ & $\ Err_m\ $ & $\ Err_l\ $ \\ \hline
\multirow{16}{*}{Dev} & aggregation & 0.22\% & 1.99\% & 2.10\% & 23.84\% \\
& comparative & 0.00\% & 0.00\% & 0.00\% & 0.00\% \\
& cvt & 0.57\% & 4.04\% & 4.04\% & 13.41\% \\
& multi-choice & 0.37\% & 6.72\% & 7.46\% & 52.24\% \\
& multi-constraint & 0.17\% & 1.71\% & 1.71\% & 6.83\% \\
& multi-hop & 0.26\% & 1.28\% & 1.54\% & 3.85\% \\
& multi-turn-answer & 4.50\% & 0.00\% & 0.00\% & 5.66\% \\
& multi-turn-entity & 0.51\% & 1.37\% & 2.29\% & 16.59\% \\
& multi-turn-predicate$\ \ $ & 0.50\% & 2.00\% & 2.00\% & 12.00\% \\
& single-relation & 0.27\% & 1.18\% & 1.31\% & 9.26\% \\
& superlative0 & 0.40\% & 3.04\% & 3.04\% & 24.21\% \\
& superlative1 & 0.00\% & 0.00\% & 0.00\% & 6.90\% \\
& superlative2 & 0.00\% & 0.00\% & 0.00\% & 0.00\% \\
& superlative3 & 0.00\% & 0.00\% & 0.00\% & 0.00\% \\
& yesno & 0.17\% & 1.33\% & 1.33\% & 9.67\% \\ \cline{2-6}
& overall & 0.36\% & 1.72\% & 1.93\% & 13.14\% \\ \hline
\multicolumn{2}{l|}{Test(full)} & - & - & - & 15.53\% \\ \hline
\multicolumn{2}{l|}{Test(hard)} & - & - & - & 36.92\% \\ \hline
\end{tabular}
\end{table}
At last, $Err_f$ of our best model is 13.14\% in validation set, 15.53\% in full test set and 36.92\% in hard test subset. We inspect the output of our model in order to identify the causes of errors. The entity error takes up 20.43\% not only because of wrong entities but also right entities in wrong order. 79.57\% of incorrect samples have wrong predicates although their entities are right. Our accuracy is extremely low for multi-choice. We look into this class and find 50.72\% of errors are because of right entities with wrong order. Actually, there are three different entities in sketch of multi-choice class and two of them are semantically exchangeable in the form $( or ( equal\ ?x\ E_1 ) ( equal\ ?x\ E_2 ) )$. So it is not easy for our pattern pair matching network to deal with this problem. In the meantime, our model achieves error rate of 0\% for 3 classes in validation set.
\begin{table}
\caption{Performance comparison. Metric is exactly matching accuracy of logical forms. ``Cover" represents our covered words supplementry. ``Point" represents the application of Pointer losses. ``Pep'' represents the predicate-entity pair matching network.}\label{table4}
\centering
\begin{tabular}{l|ccc}
\hline
\multirow{2}{*}{System} & \multicolumn{3}{c}{Acc} \\ \cline{2-4}
& Dev & Test(full) & Test(hard) \\ \hline
$Soochow\_SP(1st)$ & - & {\bfseries 85.68\%} & 57.43\% \\
$NP$-$Parser(2nd)$ & - & 83.73\% & 51.93\% \\
$WLIS(Ours)(3rd)$ & 82.86\% & 82.53\% & 47.83\% \\
$BBD(4th)$ & - & 68.82\% & 35.41\% \\ \hline
$WLIS_{NEW}(Our\ New\ Baseline)\ $ & 77.42\% & - & - \\
$WLIS_{NEW} + point$ & 84.02\% & - & - \\
$WLIS_{NEW} + pep$ & 85.99\% & - & - \\
$WLIS_{NEW} + point + pep\ $ & {\bfseries 86.86\%} & 84.47\% & {\bfseries 63.08\%} \\ \hline
\end{tabular}
\end{table}
Our system is compared with that of other teams in NLPCC 2019 Shared Task 2. The top 4 results are shown in Table~\ref{table4}. Our system on the submission list is $WLIS$ which achieves the 3rd place. After some optimizations for parameters, seq2seq network structure and sampling, the performance of our new system has been improved a lot. The accuracy of our new baseline reaches 77.42\%. By incorporating two auxiliary scores, the accuracy is improved to 86.86\% in validation set. Accuracy achieves 84.47\% in full test set and 63.08\% in hard test subset. Our accuracy in full test set supasses the 2nd place but is still lower than the 1st place by 1.21\% while the accuracy on hard subset is higher than that of the 1st place by 5.65\%.
\section{Related Work}
Semantic parsing is a long-standing problem in NLP mapping natural language utterances to logical forms\cite{berant2014semantic,kwiatkowski2011lexical,liang2013learning,pasupat2016inferring,wong2007learning,zettlemoyer2005learning}. Since it is not easy for semantic parsing to label data manually, reinforcement learning\cite{liang2018memory} and transfer\cite{wang2018multi,xiong2019transferable} are applied when data is not enough. But in most cases, we are studying how to improve the results when enough data is available for supervised learning. Basic seq2seq network\cite{NIPS2014_5346} enables the model to be trained in an end-to-end mode. Later, structure-aware models are designed to generate logical forms more elaborately. Seq2tree\cite{dong-lapata-2016-language} is equipped with a tree-structured decoder to parse hierarchical logical forms while STAMP\cite{sun-etal-2018-semantic} adopts a switching gate in the decoder to control the generation of SQL. The models mentioned above all generate the whole logical form in one go.
There are also some works that applied sketch-based approach to solve the problem. It has already been explored in the field of program synthesis\cite{solar2008program}. Coarse2fine\cite{dong-lapata-2018-coarse} decomposes the decoding process to 2 stages. Sketches are generated in the first stage while model in the second stage fills in missing details. SQL generating is especially suitable for this method because of its easy sketches. Sqlnet\cite{xu2017sqlnet} divides the task into 6 subtasks to generate different part of SQL. SQLova\cite{hwang2019comprehensive} also inherits this idea and incorporate Bert\cite{devlin-etal-2019-bert} in his model. The idea of our system is similar to that of SQLova. We do not use complex decoders to make our network structure-aware. The architectures of models are easy in every stage. We first determine sketches as the high-level structure. Low-level details are added in later stages. The losses of seq2seq network is applied here to rerank from an overall perspective. So we actually combine both seq2seq method and sketch-based method to some extent.
\section{Conclusion}
In this paper, we presented a sketch-based system for semantic parsing which disentangles high-level structures from low-level details. Due to the absence of knowledge base, we propose to collect question patterns and logical form patterns to capture the implicit relationship between questions and predicates, which can then be used to perform reranking in a Pointer network within a seq2seq framework. Our previous submitted system achieves the 3rd place while our new system outperforms the 1st place for accuracy in hard test subset. Since the knowledge base will be released later, in future work we would like to incorporate new knowledge to improve our system. We will extend our system to other semantic parsing tasks as well.
\section{Acknowledgements}
This work is supported in part by the NSFC (Grant No.61672057, 61672058, 61872294), the National Hi-Tech R\&D Program of China (No. 2018YFB1005100). For any correspondence, please contact Yansong Feng.
\bibliographystyle{splncs04}
\section{Introduction}
Open domain semantic parsing aims to map natural language utterances to structured meaning representations. Recently, seq2seq based approaches have achieved promising performance by structure-aware networks, such as sequence-to-action\cite{chen2018sequence} and STAMP\cite{sun-etal-2018-semantic}.
However, this kind of approach mixes up low-level entities, predicates and high-level structures together, which loses precision at each level to some extent. So the sketch-based method may be an another choice for disentangling high-level structures from low-level details. In this work, we conduct our sketch-based approach on MSParS, a large hand-annotated semantic dataset mapping questions to logical forms. We argue there are at least two advantages to sketch-based method. Firstly, basic attention based seq2seq network\cite{bahdanau2014neural,luong-etal-2015-effective} does not perform well in semantic parsing because logical forms are structured sequences and it fails to incorporate structure information of logical forms. Then sequence-to-tree(seq2tree)\cite{dong-lapata-2016-language} proposes a structure-aware decoder to utilize the information. But its architecture also becomes much more complex. Instead of using intricate decoders, we can extract high-level sketches for logical forms and classify samples into several sketch classes. Logical forms of a certain sketch class have a fixed pattern which is shown in Table~\ref{table1}. So the structure problem is finally simplified to a classification task. Secondly, logical forms often need to copy a span of questions. Although Copynet\cite{gu-etal-2016-incorporating} and Pointer\cite{see-etal-2017-get} implement the copy mechanism, it is still difficult to achieve the expected effect. But for the sketch-based method, this problem becomes an individual entity labeling task which is easier than generating entities. Generally speaking, the seq2seq way decodes the entire meaning representation in one go while we deal with different parts at different levels of granularity just like coarse2fine\cite{dong-lapata-2018-coarse}. Although we increase the number of stages, the network architecture of each stage is much simpler without sacrificing the accuracy. In this way, we are able to locate the errors and optimize according parts.
\begin{table}
\caption{Examples demonstrating sketches of logical forms. $P$ represents predicate and $E$ represents entity. Subscripts are applied to distinguish different ones.}\label{table1}
\centering
\begin{tabular}{l|p{250pt}}
\hline
Class & Sketch of Logical Form \\ \hline
aggregation & count ( lambda ?x ( $P_1\;E_1$ ?x ) ) \\ \hline
cvt & ( lambda ?x exist ?y ( and ( $P_1\;E_1$ ?y ) ( $P_2$ ?y $E_2$ ) ( $P_3$ ?y ?x ) ) ) \\ \hline
multi-turn-entity & ( lambda ?x ( $P_1\;E_1$ ?x ) ) $|||$ ( lambda ?x ( $P_2\;E_1$ ?x ) ) \\ \hline
multi-turn-answer$\ $ & ( lambda ?x ( $P_1\;E_1$ ?x ) ) $|||$ ( lambda ?x exist ?y ( and ( $P_1\;E_1$ ?y ) ( $P_2$ ?y ?x ) ) ) \\ \hline
single-relation & ( lambda ?x ( $P_1\;E_1$ ?x ) ) \\ \hline
yesno & ( $P_1\;E_1\;E_2$ ) \\ \hline
\end{tabular}
\end{table}
We propose to decompose the process into three stages. In the first stage, we deal with a sketch classification task. Then, we find the entities in the questions through an entity labeling task. Actually, we combine the two stages through the multi-task model for both accuracy and efficiency\cite{chen2019bert}. The last stage is the most difficult part since the knowledge base of MSParS is not available. We define question pattern-logical form pattern pair and use the matching network to rank all these pairs. Seq2seq based approach is one of the two methods we adopted here to help rescore on the whole. We also incorporate state-of-art pre-trained work, Bert\cite{devlin-etal-2019-bert}, in above tasks to incorporate more priori knowledge.
The error rate of our multi-task model is lower than 2\%, which ensures the right sketch and entities. So the last stage actually determines the accuracy to a large extent. Our accuracy achieves 77.42\% after above three stages. Seq2seq based approach and co-occurrence relationship improve the accuracy to 86.86\% in validation set. Our final accuracy in full test set reaches 84.47\%. And the accuracy on hard test subset has been promoted to 63.08\% finally which is higher than the best model on the submission list by 5.65\%.
In the rest of our paper, we first analyze the special features of MSParS for this task in section 2. Afterwords, we discuss our system in detail in section 3. Then in section 4, we demonstrate our experimental setup, results and analyzation. Related works are mentioned in section 5. At last, we make a conclusion of the whole paper and propose our future work.
\section{Data Analyzation}
The dataset MSParS is published by NLPCC 2019 evaluation task. The whole dataset consists of 81,826 samples annotated by native English speakers. 80\% of them are used as training set. 10\% of them are used as validation set while the rest is used as test set. 3000 hard samples are selected from the test set. Metric for this dataset is the exactly matching accuracy on both full test set and hard test subset. Each sample is composed of the question, the logical form, the parameters(entity/value/type) and question type as the Table~\ref{table5} demonstrates.
\begin{table}
\caption{An sample of MSParS.}\label{table5}
\centering
\begin{tabular}{l|l}
\hline
question & what is birth date for chris pine \\ \hline
logical form & ( lambda ?x ( mso:people.person.date\_of\_birth chris\_pine ?x ) ) \\ \hline
parameters & chris\_pine (entity) [5,6] \\ \hline
question type$\ $ & single-relation \\ \hline
\end{tabular}
\end{table}
Samples are classified to 12 classes originally at a coarse level while we reclassify them at a finer level, which is the basis of our sketch-based method. We replace the predicate in the triple as $P_i$, the entity in the triple as $E_i$ and distinguish different ones with subscripts. The number in superlative class and comparative class is replaced as $V$ while the type in the triple begin with special predicate ``isa" is replaced as $T$ as well. In this way, we get the sketch of the logical form. Finally, we produce 15 classes of sketches.
We believe the features of questions highly correlate with the sketch of logical forms. For instance, the sketch must begin with ``argmore" or ``argless" if there are comparative words such as ``higher", ``more" and ``before" in questions. Therefore, we take questions as input to classify samples to different sketch classes.
As the Table~\ref{table5} suggests, entities are concatenated tokens from the question. So we implement entity labeling to label every token in the questions.
Nonetheless, cases are tough when there are more than one entities in the logical form. Suppose that we have labeled $E_1$ and $E_2$ from the question. We do not know which one we should choose to fill in the first entity slot in the sketch. We solve this problem and pick out the suitable predicate simultaneously. The entities in the questions are replaced by label ``entity'' with subscipts suggesting the order they appear in questions to get question patterns. When it comes to logical form patterns, the entities in logical forms are substituted as well while predicates are split to small tokens. Table~\ref{table2} gives an example of these two patterns. In this way, we combine the entity collocations with predicates successfully. Another reason for label ``entity'' used here is generalization. For instance, ``what is birth date for barack obama" shares the same question pattern ``what is birth date for entity1" with ``what is birth date for donald trump". The predicate used in these logical forms is ``mso:people.person.date\_of\_birth''. So we can draw the conclusion that the predicate for this question pattern is likely to be ``mso:people.person.date\_of\_birth''. If ``what is birth date for george bush" appears in the test set, we are able to find the right predicate even if we do not see ``george bush'' before. Without the impact of specific entities, our model learns the mapping from question patterns to logical form patterns more accurately. Since we do not have a knowledge base, we can only extract logical form patterns in training set. And we find 90.34\% of logical form patterns in validation set are covered by that in training set, which ensures the feasibility of our method.
\begin{table}
\caption{An example for question pattern and logical form pattern.}\label{table2}
\centering
\begin{tabular}{l|p{220pt}}
\hline
question & travels in the interior districts of africa has how many pages? $|||$ when is the date of publication of the book edition? \\ \hline
question pattern & entity1 has how many pages? $|||$ when is the date of publication of the book edition? \\ \hline
logical form & ( lambda ?x ( mso:book.edition.number\_of\_pages travels\_in\_the\_interior\_districts\_of\_africa ?x ) ) $|||$ ( lambda ?x ( mso:book.edition.publication\_date travels\_in\_the\_interior\_districts\_of\_africa ?x ) ) \\ \hline
logical form pattern$\ $ & book edition number of pages entity1 ?x $|||$ book edition publication date entity1 ?x \\ \hline
\end{tabular}
\end{table}
We take question patterns paired with logical form patterns as input. Then, we get logical form candidates through combining sketches and entities with logical form patterns. The ones with higher scores are more likely to be right.
\section{Proposed Approach}
\subsection{Sketch Classification}
The single sentence classification fine-tuned task in Bert is applied in this stage. A special classification embedding ([CLS]) is added to the beginning. We use the final hidden state corresponding to this token as the aggregate sequence representation for classification task denoted as $C_s \in \mathbb{R}^h$, so the probability of class $c_i$ can be computed as:
\begin{equation}
p(c_i|x) = softmax_i(W_sC_s + b_s)
\end{equation}
where $W_s \in \mathbb{R}^{k_s \times h}$ and $b_s \in \mathbb{R}^{k_s}$, $k_s$ is the number of sketch classes here. $W_s$, $b_s$ and all the parameters of Bert are fine-tuned jointly to maximize the log likelihood probability of the correct label.
\subsection{Entity Labeling}
We use the single sentence tagging fine-tuned task in Bert here to label every token in the question whether it is an entity token that appears in the logical form as well. To simplify the problem, we use 3 labels for the tokens in the questions. Label ``b" represents the first token in an entity while label ``i'' for the rest ones. And label ``o'' represents those tokens which are not in any entities. Because of the lexical rules in Bert, we also label the special token ([CLS]) at the beginning of the sentence and the special token ([SEP]) at the ending of the sentence as ``o''. The last label ``p'' is for all the padding tokens added to reach max\_length. Besides, some tokens in the questions are split into several smaller tokens by Bert. For the split ones, they are labeled as ``i'' if they are in the entities and ``o'' otherwise. In this stage, we use all the final hidden states denoted as $D \in \mathbb{R}^{h \times m}$ where m is the max\_length of the input tokens we set. The hidden state is mapped into dimension $k_e$ via $E = W_eD + b_e$ where $W_e \in \mathbb{R}^{k_e \times h}$ and $b_e \in \mathbb{R}^{k_e \times m}$, $k_e$ is the number of labels here. We employ the CRF on the top of the network taking $E$ as input representations. The objective is to minimize the loss of CRF layer.
\subsection{Multi-Task Model}
We combine sketch classification and entity labeling to share information together, which means sketches of samples can help label entities while the labeled entities can help sketch classification conversely. The architecture of our model is shown in Fig.~\ref{fig1} where the parameters of Bert model is fine-tuned together for two tasks. Since the scale of dataset is large, we can save lots of time through multi-task model instead of training two different models. Finally, it contributes to both accuracy and efficiency. In this way, our loss to minimize is the weighted sum of the cross-entropy loss in sketch classification task and the CRF loss in entity labeling task.
\begin{figure}
\includegraphics[width=\textwidth]{multi-task.png}
\caption{An overview of multi-task model proposed. The original input question is ``chemical compound of citric acid". It becomes ``chemical compound of ci \#\#tric acid" after the tokenization of Bert.} \label{fig1}
\end{figure}
\subsection{Pattern Pair Matching Network}
Besides the single sentence tasks, Bert provides sentence pair classification tasks as well. We implement the matching network taking question patterns and logical form patterns as input. The right pattern pairs are regarded as positive samples. We select negative samples only from the logical form patterns in the same sketch class for fixed question patterns. The sketch mentioned is from the multi-task model. Just like sketch classification, we denote the final hidden state corresponding to token ([CLS]) as $C_p \in \mathbb{R}^h$, so the probability can be computed as:
\begin{equation}
p(c_j|x) = softmax_j(W_pC_p + b_p)
\end{equation}
where $W_p \in \mathbb{R}^{2 \times h}$, $b_p \in \mathbb{R}^{2}$ and $c_j \in \{0, 1\}$. $W_p$, $b_p$ and all the parameters of bert are fine-tuned jointly to maximize the log likelihood probability of the correct class.
In the prediction stage, the candidates for a question pattern are from logical form patterns in the same sketch class as well. The probabilities of class ``1" are scores we get for these pattern pairs. From logical form patterns, we get not only right predicates, but right orders as well in which entities should appear. So with the sketch and entities we aquire in the multi-task model, we can already generate complete logical form candidates with scores between 0 and 1.
\subsection{Predicate-Entity Pair Matching Network}
To alleviate the absence of knowledge base, we incorporate the co-occurrence relationship between predicates and entities to evaluate the candidates. We create the second matching network based on Bert as well. This time, the pairs we take as input are predicate-entity ones. We label the predicate-entity pair as ``1'' if they have ever appeared in one triple in training set. For a certain entity, we select predicates that never appear with this entity as negetive samples. In the prediction stage, we score the predicate-entity pairs in logical form candidates. However, this network does not take questions into account. The predicate for a certain entity can differ a lot according to various questions. For instance, the predicate for ``what is birth date for barack obama" is apparently different from that for ``what is birth place for barack obama". But the entity ``barack obama" has only one predicate with highest score. Although this matching network only considers the co-occurrence relationship regardless of the information from questions, scores produced by it do work as an auxiliary.
\subsection{Pointer Network}
Although it is not easy for a complex network to generate the whole logical form, such networks do reflect the mapping from an overall perspective. So we adopt Pointer\cite{see-etal-2017-get} here to rerank. We take the questions as input. For logical forms, entities and predicates are composed of words concatenated by ``\_'' or ``.''. In order to utilize the information of words, we split all entities and predicates and take split logical form candidates as output. For a fixed question, we calculate cross-entropy losses of different pairs with split logical forms. Then every loss is divided by the max one and subtracted by 1 to be normalized between 0 and 1. The higher the score is, the more the logical form candidate is likely to be true.
\subsection{Ranking}
A linear combination of the three intermediate scores from pattern pair matching network, predicate-entity pair matching network and Pointer is used to rerank logical form candidates. Weights are roughly adjusted in validation set.
\section{Experiment}
The number of sketch classes is 15 and the number of labels is 4 in the multi-task model. The Bert model we applied is ``BERT-Base, Uncased" with 12-layer, 768-hidden, 12-heads and 110M parameters\footnote{https://github.com/google-research/bert}. All the parameters are fine-tuned in validation set. In the multi-task model, we train the model for 10 epoches. We set batch size to 32 and learning rate to 2e-5. The weight of the loss in sketch classification is 1 while that in entity labeling is 2. We train 3 models in pattern pair matching network with different epoches. As for predicate-entity pair matching network, the number of epoch we use is 3. In Pointer, word embeddings were initialized by Glove\cite{pennington2014glove}. The hidden dim of LSTM is set to 256. More details will be released in our source codes later.
Because of the instability of the performance of neural network over training epoches,
ensemble learning is incorporated both in pattern pair matching network and Pointer. Scores of Pointer is the simple average of scores from 3 models with different epoches. When it comes to pattern pair matching net, it is a little complex. We make a prediction for training set with our ``best" model. We apply ranking sampling here. From those labeled as ``0'' but with probabilities larger than 0.0001, we select 20 of them while 5 of those whose probabilities are smaller than 0.0001 as new negative samples. We train new models with new training data resampled before every epoch based on one ``best" model and base model of Bert. After several epoches, we average the probabilities of new models and original models for ensemble.
We demonstrate the detailed performance in Table~\ref{table3}. All samples are classified into 15 classes. We show the results for every class and the overall weighted average preformance in validation set. Because the complete test set is not open yet, we only provide the overall results in test set returned after submission.
It can be seen the overall error rate of our multi-task model is only 1.93\% which means this task is successful. In sketch classification, $Err_s$ scores of all classes are lower than 1\% except multi-turn-answer. Its recall is 100.00\% while its precision is 91.38\%. 0.92\% of samples in multi-turn-entity are misclassified to multi-turn-answer in validation set. We find there are separator ``$|||$'' in logical forms from three classes of multi-turn questions. Multi-turn-predicate questions have two different entities while both multi-turn-entity and multi-turn-answer questions have only one. This kind of entity information is passed to sketch classification through shared parameters. So our system makes some mistakes while distinguishing multi-turn-entity samples from multi-turn-answer samples. As for entity labeling, the overall error rate is 1.72\%. We check the wrong samples and find our model is not so good at recognizing entity boundaries especially while encountering some special tokens such as articles, genitive ``s'' and quotation mark. Actually, it is not easy for human to define an entity in these cases as well.
\begin{table}
\caption{Performances of our best model. $F1_s$ represents the $F1$ score of sketch classification in multi-task model. We compute $Err_s$ by $1 - F1_s$. $Err_e$ represents the error rate of entity labeling part in multi-task model(an sample is regarded as right only when all of its entities are labeled correctly). $Err_m$ represents the error rate of the whole multi-task model(an sample is regarded as right only when both sketch classification subtask and entity labeling subtask are correct). $Acc_l$ is the exactly matching accuracy of logical forms. We compute $Err_l$ by $1 - Acc_l$.}\label{table3}
\centering
\begin{tabular}{l|l|cccc}
\hline
\multicolumn{2}{l|}{Dataset} & $\ Err_s\ $ & $\ Err_e\ $ & $\ Err_m\ $ & $\ Err_l\ $ \\ \hline
\multirow{16}{*}{Dev} & aggregation & 0.22\% & 1.99\% & 2.10\% & 23.84\% \\
& comparative & 0.00\% & 0.00\% & 0.00\% & 0.00\% \\
& cvt & 0.57\% & 4.04\% & 4.04\% & 13.41\% \\
& multi-choice & 0.37\% & 6.72\% & 7.46\% & 52.24\% \\
& multi-constraint & 0.17\% & 1.71\% & 1.71\% & 6.83\% \\
& multi-hop & 0.26\% & 1.28\% & 1.54\% & 3.85\% \\
& multi-turn-answer & 4.50\% & 0.00\% & 0.00\% & 5.66\% \\
& multi-turn-entity & 0.51\% & 1.37\% & 2.29\% & 16.59\% \\
& multi-turn-predicate$\ \ $ & 0.50\% & 2.00\% & 2.00\% & 12.00\% \\
& single-relation & 0.27\% & 1.18\% & 1.31\% & 9.26\% \\
& superlative0 & 0.40\% & 3.04\% & 3.04\% & 24.21\% \\
& superlative1 & 0.00\% & 0.00\% & 0.00\% & 6.90\% \\
& superlative2 & 0.00\% & 0.00\% & 0.00\% & 0.00\% \\
& superlative3 & 0.00\% & 0.00\% & 0.00\% & 0.00\% \\
& yesno & 0.17\% & 1.33\% & 1.33\% & 9.67\% \\ \cline{2-6}
& overall & 0.36\% & 1.72\% & 1.93\% & 13.14\% \\ \hline
\multicolumn{2}{l|}{Test(full)} & - & - & - & 15.53\% \\ \hline
\multicolumn{2}{l|}{Test(hard)} & - & - & - & 36.92\% \\ \hline
\end{tabular}
\end{table}
At last, $Err_f$ of our best model is 13.14\% in validation set, 15.53\% in full test set and 36.92\% in hard test subset. We inspect the output of our model in order to identify the causes of errors. The entity error takes up 20.43\% not only because of wrong entities but also right entities in wrong order. 79.57\% of incorrect samples have wrong predicates although their entities are right. Our accuracy is extremely low for multi-choice. We look into this class and find 50.72\% of errors are because of right entities with wrong order. Actually, there are three different entities in sketch of multi-choice class and two of them are semantically exchangeable in the form $( or ( equal\ ?x\ E_1 ) ( equal\ ?x\ E_2 ) )$. So it is not easy for our pattern pair matching network to deal with this problem. In the meantime, our model achieves error rate of 0\% for 3 classes in validation set.
\begin{table}
\caption{Performance comparison. Metric is exactly matching accuracy of logical forms. ``Cover" represents our covered words supplementry. ``Point" represents the application of Pointer losses. ``Pep'' represents the predicate-entity pair matching network.}\label{table4}
\centering
\begin{tabular}{l|ccc}
\hline
\multirow{2}{*}{System} & \multicolumn{3}{c}{Acc} \\ \cline{2-4}
& Dev & Test(full) & Test(hard) \\ \hline
$Soochow\_SP(1st)$ & - & {\bfseries 85.68\%} & 57.43\% \\
$NP$-$Parser(2nd)$ & - & 83.73\% & 51.93\% \\
$WLIS(Ours)(3rd)$ & 82.86\% & 82.53\% & 47.83\% \\
$BBD(4th)$ & - & 68.82\% & 35.41\% \\ \hline
$WLIS_{NEW}(Our\ New\ Baseline)\ $ & 77.42\% & - & - \\
$WLIS_{NEW} + point$ & 84.02\% & - & - \\
$WLIS_{NEW} + pep$ & 85.99\% & - & - \\
$WLIS_{NEW} + point + pep\ $ & {\bfseries 86.86\%} & 84.47\% & {\bfseries 63.08\%} \\ \hline
\end{tabular}
\end{table}
Our system is compared with that of other teams in NLPCC 2019 Shared Task 2. The top 4 results are shown in Table~\ref{table4}. Our system on the submission list is $WLIS$ which achieves the 3rd place. After some optimizations for parameters, seq2seq network structure and sampling, the performance of our new system has been improved a lot. The accuracy of our new baseline reaches 77.42\%. By incorporating two auxiliary scores, the accuracy is improved to 86.86\% in validation set. Accuracy achieves 84.47\% in full test set and 63.08\% in hard test subset. Our accuracy in full test set supasses the 2nd place but is still lower than the 1st place by 1.21\% while the accuracy on hard subset is higher than that of the 1st place by 5.65\%.
\section{Related Work}
Semantic parsing is a long-standing problem in NLP mapping natural language utterances to logical forms\cite{berant2014semantic,kwiatkowski2011lexical,liang2013learning,pasupat2016inferring,wong2007learning,zettlemoyer2005learning}. Since it is not easy for semantic parsing to label data manually, reinforcement learning\cite{liang2018memory} and transfer\cite{wang2018multi,xiong2019transferable} are applied when data is not enough. But in most cases, we are studying how to improve the results when enough data is available for supervised learning. Basic seq2seq network\cite{NIPS2014_5346} enables the model to be trained in an end-to-end mode. Later, structure-aware models are designed to generate logical forms more elaborately. Seq2tree\cite{dong-lapata-2016-language} is equipped with a tree-structured decoder to parse hierarchical logical forms while STAMP\cite{sun-etal-2018-semantic} adopts a switching gate in the decoder to control the generation of SQL. The models mentioned above all generate the whole logical form in one go.
There are also some works that applied sketch-based approach to solve the problem. It has already been explored in the field of program synthesis\cite{solar2008program}. Coarse2fine\cite{dong-lapata-2018-coarse} decomposes the decoding process to 2 stages. Sketches are generated in the first stage while model in the second stage fills in missing details. SQL generating is especially suitable for this method because of its easy sketches. Sqlnet\cite{xu2017sqlnet} divides the task into 6 subtasks to generate different part of SQL. SQLova\cite{hwang2019comprehensive} also inherits this idea and incorporate Bert\cite{devlin-etal-2019-bert} in his model. The idea of our system is similar to that of SQLova. We do not use complex decoders to make our network structure-aware. The architectures of models are easy in every stage. We first determine sketches as the high-level structure. Low-level details are added in later stages. The losses of seq2seq network is applied here to rerank from an overall perspective. So we actually combine both seq2seq method and sketch-based method to some extent.
\section{Conclusion}
In this paper, we presented a sketch-based system for semantic parsing which disentangles high-level structures from low-level details. Due to the absence of knowledge base, we propose to collect question patterns and logical form patterns to capture the implicit relationship between questions and predicates, which can then be used to perform reranking in a Pointer network within a seq2seq framework. Our previous submitted system achieves the 3rd place while our new system outperforms the 1st place for accuracy in hard test subset. Since the knowledge base will be released later, in future work we would like to incorporate new knowledge to improve our system. We will extend our system to other semantic parsing tasks as well.
\section{Acknowledgements}
This work is supported in part by the NSFC (Grant No.61672057, 61672058, 61872294), the National Hi-Tech R\&D Program of China (No. 2018YFB1005100). For any correspondence, please contact Yansong Feng.
\bibliographystyle{splncs04}
|
1,314,259,995,846 | arxiv | \section{Introduction}
\label{sec:introduction}
The Distance-duality (DD) relation, also known as the Etherington's
reciprocity relation \citep{eth33}, is related the angular diameter
distance (ADD, $D_{A}$) to the luminosity distance ($D_{L}$) by
means of a single parameter,
\begin{equation}
\eta \equiv \frac{D_{L}}{D_{A}}{(1+z)}^{-2}=1.
\label{rec1}
\end{equation}
This equation is completely valid for all cosmological models based
on Riemannian geometry \citep{Ellis07}. Therefore, the DD relation
plays an essential role in observational astrophysics and modern
cosmology \citep{Csaki02}, such as galaxy clusters observations
\citep{Lima03,Cunha07}, the anisotropies of cosmic microwave
background (CMB) \citep{Komatsu11}, as well as gravitational lensing
studies \citep{Schneider99,Fu08}.
In principle, if both $D_{A}$ and $D_{L}$ of cosmological sources
at the common redshifts are known, the DD relation ($\eta=1$) could
be directly tested by means of astronomical observations. From
Sunyaev-Zeldovich effect \citep{Sunyaev} and X-ray surface
brightness of galaxy clusters, the observational ADDs of galaxy
clusters can be obtained \citep{Silk78}. By using an isothermal
spherical model for which the hydrostatic equilibrium model and
spherical symmetry assumed, \cite{Reese02} selected 18 galaxy
cluster sample and \cite{Mason01} obtained seven clusters from the X-ray-limited flux sample.
The measurements of the two samples above have been corrected by
using an isothermal elliptical model to get 25 ADDs of galaxy
clusters \citep{DeFilippis05}. Recently, \cite{Boname06} obtained
38 ADD galaxy clusters sample by assuming the spherical model.
\cite{Uzan04} considered ADDs of 18 galaxy cluster sample
\citep{Reese02} to test the DD relation by assuming the
$\Lambda$CDM model via the technique,
$D_A^{\mathrm{cluster}}(z)=D_A^{\Lambda \rm CDM}(z)\eta^2(z)$. They
showed that no violation of the DD relation is only marginally
consistent. \cite{DeBernardis06} considered ADDs of 38 galaxy
cluster for spherical model \citep{Boname06} to test the DD
relation by assuming the $\Lambda$CDM model. Some other works in
context of the $\Lambda$CDM model for astrophysical research of the
DD relation can be found in \cite{Bassett04a}, \cite{Bassett04b},
\cite{More09}, \cite{Avg10}, \cite{HLR11}, and \cite{CaoZ11}.
In order to test the DD relation in a model-independent way, one
should use measurements of $D_L$ such as Type Ia Supernave (SNe Ia)
directly. By binning ADDs from FRIIb radio galaxies
and ultra compact radio sources
and $D_L$ of SN Ia data, \cite{Bassett04} found that the brightening
excess of SNe Ia at $z > 0.5$ could cause a moderate violation at
2$\sigma$ confidence level (CL). \citet{DeBernardis06} binned ADD
data of galaxy clusters \citep{Boname06} and the SNe Ia
data
to find that the validity of $\eta = 1$ is
consistent at 1$\sigma$ CL.
However, it is argued that the above tests may have been influenced
by the particular choice of redshift bin \citep{HLR10}.
Recently, \cite{HLR10} tested the DD relation with two ADD samples
\citep{Boname06,DeFilippis05} and the Constitution set of SNe Ia
data \citep{Hicken09}. For the biggest redshift difference between
clusters and SNe Ia is $\Delta z=|z_{\rm cluters}-z_{\rm
SNe}|\simeq0.01$ for three clusters, a selection criteria ($\Delta
z\le0.005$) for a given pair of data set are used to avoid the
corresponding bias of redshift differences. With the incomplete
spherical model sample \citep{Boname06} in which three ADD data have
been discarded, they found a strong violation ($>3\sigma$) of the DD
relation by using two parameterizations of $\eta$ parameter
[$\eta(z)=1+\eta_1z$, and $\eta(z)=1+\eta_a z/(1+z)$].
More recently, \cite{Li11} used the same selection criteria for
given pairs of observational data to remove more data points of the
galaxy clusters corresponding to the Constitution set and found that
the DD relation could be marginally accommodated at $3\sigma$ CL for
the spherical model if the effect
of the errors of SNe Ia considered.
Additionally, they also examined the DD relation for two more
general parameterizations
[$\eta(z)=\eta_0+\eta_1z$, and $\eta(z)=\eta_0+\eta_1z/(1+z)$] to
show that $\eta(z)=1$ is
compatible with
the spherical model sample and the Union2 set \citep{Amanullah} at
$2\sigma$ CL. Some recent works for testing the DD relation can be
found in, e. g., \cite{Nair11,CaoL11,Meng11,HLR11b,Fu11}.
It is obvious that testing results of the DD relation may be
influenced by the particular choice of the selection criteria for a
given pair of data set. The difference of redshifts between pairs
of galaxy clusters and SNe Ia may cause obvious deviation in testing
the DD relation. In principle, the only strict criterion to form a
given pair is that galaxy clusters and SNe Ia locate at exactly the
same redshift. At other hand, the more stringent selection criteria
are used, the more data points should be removed. In order to avoid
any bias of redshift differences between SNe Ia and galaxy clusters
and ensure the integrity of observational data pairs, we can use the
nearby SNe Ia points to obtain the luminosity distance of SN Ia
point at the same redshift of the corresponding galaxy cluster; this
situation is similar with the cosmology-independent calibration of
GRB relations directly from SNe Ia
\citep{Liang2008,Liang2010,Liang2011}.
In this\textit{ letter}, we test the DD relation
with the Union2 set in which a sub-sample of SNe Ia are corrected to
the same redshifts of the corresponding galaxy clusters sample by
interpolating from the nearby SNe Ia points with the biggest
difference of redshifts $\Delta z_{\rm max}=0.005$ for
a given pair of data set.
We focus on the 38 ADDs from galaxy cluster sample under an
assumption of spherical model \citep{Boname06}. As we well see,
there exists no conceivable evidence for variations in the DD
relation when current observations are confronted, since $\eta(z)=1$
is significant satisfied at $2\sigma$ confidence level for various
parameterizations of $\eta(z)$.
\section{Data Analysis}\label{sec:analysis}
In this work, we test the DD relation with the 38 ADD sample from
galaxy clusters for the spherical model \citep{Boname06} and the
Union2 set which consists of 557 SNe Ia \citep{Amanullah}. It is
easy to find that differences of redshifts between the 38 galaxy
clusters to the Union2 set are more centered around $\Delta z=0$ and
the biggest value at $\Delta z=0.005$ for a given pair of data set;
this situation can provide the accuracy in the interpolating
procedure. Therefore, we can obtain the luminosity distance of SN Ia
at the same redshift of the corresponding galaxy cluster by
interpolating from the nearby SNe Ia points with the biggest
difference of redshifts $\Delta z=0.005$ for a given pair of data
set. Obviously, our method can successfully avoid the systematic
errors brought by redshift incoincidence of the observational data
pairs and ensure the integrity of observational data pairs.
If the DD relation is considered be in a redshift-dependent form,
the observation technique gives
$D_A^{\mathrm{cluster}}(z)=D_A(z)\eta^2(z)$ \citep
Cavaliere}, therefore, $D_A(z)$ must be replaced with
$D_A^{\mathrm{cluster}}(z)\eta^{-2}$ when tested the DD relation
consistently with the SZE+X-ray observations from galaxy clusters
\citep{HLR10}. The observed $\eta_{obs}(z)$ can be determined by
\begin{eqnarray}
\eta_{obs}(z)=(1+z)^2\frac{{D_A^\mathrm{cluster}}(z)}{D_L^{\rm
corrected}(z)}.
\end{eqnarray}
where $D_{A}^{\rm cluster}$ is ADD from galaxy cluster at redshift
$z$ inside the samples, and $D_{L}^{\rm corrected}$ is the corrected
luminosity distance interpolated from the nearby SNe Ia points
$D_{L}^{\rm SNe}$. In the interpolating procedure, we weighted the
SNe data at the same redshifts each other
\begin{equation}
\begin{array}{l}
\bar{\mu}(z)=\frac{\sum\left(\mu_i/\sigma^2_{\mu_i}\right)}{\sum1/\sigma^2_{\mu_i}},
\end{array}
\end{equation} where $\bar{\mu}(z)$ stands for the weighted mean
distance modulus at the same redshift $z$ with its uncertainty
$\sigma_{\bar{\mu}}=(\sum1/\sigma^2_{\mu_i})^{-1/2}$. We note that
the data points of the Union2 set are given in terms of the distance
modulus, which could reduce to the luminosity distance by $
D_L(z)=10^{\mu(z)/5-5}$. Accordingly, the uncertainty of the
luminosity distance could be expressed as $\sigma_{D_L(z)}=(\ln10/5)
D_L(z)\sigma_{\mu(z)}$.
The DD relation can be tested with the combined observational data
by the minimum $\chi^2$ method. The total $\chi^2$ can be given by
\begin{equation}
\chi^2(\mathbf{p})=\sum_z\frac{[\eta(z;\mathbf{p})-\eta_{\rm
obs}(z)]^2}{\sigma^2_{\eta_{\rm obs}}},
\end{equation}
where $\eta(\mathbf{p})$ represents the theoretical value with the
parameter set $\mathbf{p}$, and $\eta_{\rm obs}$ associated with the
observational technique with its error $\sigma_{\eta_{\rm obs}}$,
which comes from the uncertainties of ADDs ($\sigma_{D_A}$) and the
corrected luminosity distances ($\sigma_{D_L^{\rm corrected}}$)
\begin{equation}
\sigma^2_{\eta_{\rm obs}}=\eta_{\rm
obs}^2[(\sigma_{D_A^{\mathrm{cluster}}}/D_A^{\mathrm{cluster}})^2+(\sigma_{D_L^{\rm
corrected}}/D_L^{\rm corrected})^2],
\end{equation}
Following \citet{HLR10, Li11}, we combine the statical and
systematic uncertainties of galaxy clusters in quadrature
\citep{DA04}. The asymmetry uncertainties of galaxy clusters can be
treated by an statistical approach \citep{DA04}, and the topic
systematic uncertainties of galaxy clusters are around $+12.4\%$ and
$-12\%$ \citep{Boname06}. In Figure 1, we plot $D_A$ data from the
galaxy
cluster
and the corresponding corrected $D_L$ data from Union2 sub-sample at
the same redshifts of galaxy clusters.
\begin{figure}
\begin{center}
\includegraphics[width=1.05\hsize]{fig1.eps}
\end {center}
\caption{ Galaxy clusters and the corresponding SNe Ia data with the associated error bars.
The blue open circles and red filled circles stand $(1+z^2)D_A$ from the galaxy clusters for the spherical model and the corresponding corrected $D_L$ from the nearby Union2 sub-sample,
green `$\times$' stand $D_L$ directly from the Union2
sub-sample. \label{fig:hubble}}
\end{figure}
\section{Results}\label{sec:Results}
In Figure \ref{fig:Union2}, we show testing results of the DD
relation with ADDs and the Union2 set by considering one-parameter
parameterizations [$\eta(z)=1+\eta_1z$ (Figure \ref{fig:Union2}
\textit{Left}) and
$\eta(z)=1+\eta_az/(1+z)$ (Figure \ref{fig:Union2} \textit{Right})].
For comparison, the case with the corrected luminosity distance
(${D_L^{\rm corrected}}$) interpolated from the nearby SNe Ia and
the case with the Union2 set directly (${D}^{\rm SNe}_L$) are given
simultaneously. For the case with ${D_L^{\rm corrected}}$, the
best-fit values are $\eta_1=-0.204\pm0.215$ at $2\sigma$ CL with
$\chi^2_{\rm min}=29.00$, and $\eta_a=-0.302\pm{0.328}$ at $2\sigma$
CL with $\chi^2_{\rm min}=29.24$. For the case with the Union2 set
directly, the best-fit values are
$\eta_1=-0.224\pm0.215$ ($2\sigma$),
and $\eta_a=-0.334\pm0.215$ ($2\sigma$).
In order to compare with previous results from the incomplete ADD
sample and the Constitution set, we also show testing results with
complete ADD sample and the Constitution set in Figure
\ref{fig:Constitution}. For the case with ${D_L^{\rm corrected}}$,
the best-fit values are $\eta_1=-0.431\pm0.303$ at $3\sigma$ CL
with $\chi^2_{\rm min}=33.10$, and $\eta_1=-0.664\pm0.457$ at
$3\sigma$ CL with $\chi^2_{\rm min}=32.34$. For the case with the
Constitution set directly, the best-fit values are
$\eta_1=-0.517\pm0.286$
($3\sigma$),
and $\eta_a=-0.793\pm0.436$ ($3\sigma$).
Fitting results with the 38 ADDs of galaxy clusters and the
corrected luminosity distances of the Union2 set and the
Constitution set are summarized in Table 1.
Our results of the case with the Union2 set directly are
consistent with those obtained by \citet{Li11}, where
$\eta_1=-0.22\pm{0.21}$ and $\eta_a=-0.33\pm{0.33} (2\sigma)$. From
comparing to results of the case with the corrected luminosity
distance and the case with SN Ia set (the Union2 set and the
Constitution set) directly, we can see a shift between the best fit
values and the likelihood contours towards the standard DD relation
($\eta=1$) with lower $\chi^2_{\rm min}$ for using of the
interpolating method to obtain ${D_L^{\rm corrected}}$. This
situation shows that the using of the interpolating method tend to
avoid the corresponding bias of redshift differences and make
results be more compatible with the DD relation. Compared to results
of the case with the the Union2 set and the case with Constitution
set, it is shown that that the DD relation of the one-parameter
parameterizations with the Union2 set for the interpolating method
is well satisfied within $2\sigma$ CL; while the DD relation is
inconsistent with the Constitution set for both cases at $3\sigma$
CL. Compared to previous results with the incomplete ADD sample and
the Constitution set, our analyses with the complete spherical model
sample (38 ADDs) and the Constitution set directly are consistent
with previous results obtained by \citet{HLR10} with the incomplete
spherical model sample (35 ADDs, three points removed by selection
criteria) and the Constitution set, where $\eta_1=-0.42\pm0.34$, and
$\eta_a=-0.66\pm0.50$ at $3\sigma$ CL; and inconsistent with those
obtained by \citet{Li11} with the incomplete spherical model sample
(26 ADDs, 12 points removed by selection criteria) and the
Constitution set, where $\eta_1=-0.30\pm0.34$ and
$\eta_a=-0.46\pm0.51(3\sigma)$. This situation shows that the choice
of selection criteria to remove ADD points with large bias of
redshift differences may paly an important role in testing of the DD
relation.
\begin{table
{\small
\begin{tabular}{|l|c|c|c|}
\hline\hline
Parameterization (SN Ia*) ~&$\eta_1{/}\eta_a$ ~&$\chi^2_{\rm min}$&$\chi^2_{\rm dof}
~~\\
\hline
\hline $1$+$\eta_1z$ (Union2*) &$\eta_1=-0.204$$\pm{0.215}(2\sigma)$~&$29.00$~&~$0.78$~\\
\hline $1$+$\eta_1z$ (Union2) &$\eta_1=-0.228$$\pm{0.211}(2\sigma)$~&$29.32$~&~$0.79$~\\
\hline $1$+$\eta_a\frac{z}{1+z}$ (Union2*) &$\eta_a=-0.302$$\pm{0.329}(2\sigma)$~&$29.24$~&~$0.79$~\\
\hline $1$+$\eta_a\frac{z}{1+z}$ (Union2) &$\eta_a=-0.334$$\pm{0.333}(2\sigma)$~&$29.76$~&~$0.80$~\\
\hline
\hline $1$+$\eta_1z$ (Constitution*) &$\eta_1=-0.431$$\pm{0.303}(3\sigma)$~&$33.10$~&~$0.89$~\\
\hline $1$+$\eta_1z$ (Constitution) &$\eta_1=-0.517$$\pm{0.286}(3\sigma)$~&$40.97$~&~$1.10$~\\
\hline $1$+$\eta_a\frac{z}{1+z}$ (Constitution*) &$\eta_a=-0.664$$\pm{0.457}(3\sigma)$~&$32.33$~&~$0.87$~\\
\hline $1$+$\eta_a\frac{z}{1+z}$ (Constitution) &$\eta_a=-0.793$$\pm{0.436}(3\sigma)$~&$40.46$~&~$1.09$~\\
\hline\hline
\end{tabular}
} \tabcolsep 0pt \caption{\label{Tab1} Fitting results with the 38
ADDs of galaxy clusters and the Union2 set and Constitution set, and
$\chi^2_{\rm min}$ (the minimun $\chi^{2}$), $\chi^2_{\rm dof}$
($\chi^{2}_{\rm min}/{\rm dof})$, for $\eta(z)=1+\eta_a z$ and
$\eta(z)=1+\eta_a\frac{z}{1+z}$, respectively. The asterisk
represents the case with the corrected luminosity distance
interpolated from the nearby SNe Ia.
} \vspace*{5pt}
\end{table}
\begin{figure*
\begin{center}
\includegraphics[width=0.33\hsize]{fig2.eps}
\includegraphics[width=0.33\hsize]{fig3.eps}
\end {center}
\caption{ Likelihood contours with the 38
ADDs of galaxy clusters and the corrected luminosity distances of
the Union2 set in the $\eta_1-\Delta\chi^2$ plane (\textit{Left}:
for $\eta(z)=1+\eta_a z$), and in the $\eta_a-\Delta\chi^2$ plane
(\textit{Right}: for $\eta(z)=1+\eta_a\frac{z}{1+z}$ ).
The blue real lines represent the case with the corrected
luminosity distance interpolated from the nearby SNe Ia (Union2*),
the black dashed lines represent the case with the SNe Ia set
(Union2) directly, and the red vertical lines represent $\eta(z)=1$.
\label{fig:Union2}}
\end{figure*}
\begin{figure*
\begin{center}
\includegraphics[width=0.33\hsize]{fig4.eps}
\includegraphics[width=0.32\hsize]{fig5.eps}
\end{center}
\caption{Likelihood contours with the 38 ADDs of galaxy clusters
and the corrected luminosity distances of the Constitution set in
the $\eta_1-\Delta\chi^2$ plane (\textit{Left}: for
$\eta(z)=1+\eta_a z$),
and in the $\eta_a-\Delta\chi^2$ plane (\textit{Right}: for $\eta(z)=1+\eta_a\frac{z}{1+z}$ ).
The blue real lines represent the case with the corrected
luminosity distance interpolated from the nearby SNe Ia
(Constitution*), the black dashed lines represent the case with the
SNe Ia set (Constitution) directly, and the red vertical lines
represent $\eta(z)=1$.
\label{fig:Constitution}}
\end{figure*}
\section{Conclusions}\label{sec:Conclusions}
In this \textit{letter}, we preform a new consistent test for the
distance-duality relation [$\eta(z)\equiv D_{L}(1+z)^{-2}/D_{A}$=1]
in a cosmology-independent way. It is obvious that the redshift
differences of observational samples may cause deviation of the DD
relation. Testing results from given pairs of data set with the
corresponding galaxy clusters and SNe Ia at nearby redshift may be
influenced by the particular choice of the selection criteria; the
more stringent selection criteria are used, the more data points
should be removed. In order to avoid any bias of difference of
redshift and ensure the integrity of the ADD samples, we correct the
luminosity distance of a SN Ia to the same redshift of the
corresponding galaxy cluster directly from the nearby SN Ia points.
With the 38 ADD sample from galaxy clusters under an assumption of
spherical model and the corrected luminosity distances of the
Union2 set, fitting results of the DD relation are
$\eta_1=-0.204$$\pm{0.215}$ at $2\sigma$ CL for parameterization
$\eta(z)=1+\eta_a z$,
and $\eta_a=-0.304$$\pm{0.215}$ at $2\sigma$ CL for
parameterization $\eta(z)=1+\eta_a\frac{z}{1+z}$, respectively
Our results show that there exists no conceivable evidence for
variations in the duality distance relation when the current SNe Ia
and the complete sample of galaxy clusters data are confronted,
since various parameterizations of $\eta(z)$ are significant
satisfied at $2\sigma$ CL by using the interpolating method, which
are more stringent than those obtained in \citet{Li11}, where the DD
relation is only marginally accommodated at $3\sigma$ CL for the
spherical model sample. Compared to previous testing results with
redshift bias and incomplete sample, our results are inconsistent
with those obtained in \citet{HLR10}, where the spherical model
sample give a clear violation of the DD relation ($>3\sigma$). We
conclude that the DD relation is significantly compatible with
current observations of galaxy clusters for the spherical model and
the Union2 set of SNe Ia.
\section*{Acknowledgments}
We thank the anonymous referee for constructive suggestions and
valuable comments. N. L. thanks Puxun Wu, Zhixiang Li and Hao Wang
for discussions and help for calculations. Z-H Z. acknowledges the
National Natural Science Foundation of China under the Distinguished
Young Scholar Grant 10825313 and Grant 11073005, the Ministry of
Science and Technology national basic science Program (Project 973)
under Grant No.2012CB821804 , and ``the Fundamental Research Funds
for the Central Universities".
|
1,314,259,995,847 | arxiv | \section{Direct and inverse problems for representation functions}
Let \ensuremath{ \mathbf N }, $\ensuremath{ \mathbf N }_0,$ and \ensuremath{\mathbf Z}\ denote the sets of positive integers, nonnegative integers, and integers, respectively.
In additive number theory, a classical direct problem is to describe the integers that can be represented as the sum of a bounded number of elements of a fixed set $A$ of integers. For example, describe the integers that are sums of two primes or three squares or four cubes. Given a set $A$ and a positive integer $m,$ we associate various representation functions to the \emph{sumset} $mA = \{a_1 + \cdots + a_m : a_i \in A \text{ for $i=1,\ldots, m$} \}.$ The two most important are the \emph{ordered representation function}
\[
R_{A,m}(n) = \{ (a_1,\ldots, a_m) \in A^m : \sum_{i=1}^m a_i = n \}
\]
and the \emph{unordered representation function}
\[
r_{A,m}(n) = \{ (a_1,\ldots, a_m) \in A^m : \sum_{i=1}^m a_i = n \text{ and } a_1 \leq \cdots \leq a_m \}.
\]
Representation functions\footnote{It is semantically ironic that the unordered representation function counts only linearly ordered representations while the ordered representation function counts unordered representations.}
are functions from \ensuremath{\mathbf Z}\ into the set $\ensuremath{ \mathbf N }_0 \cup \{\infty\}.$
Let $A$ be a set of integers. The \emph{counting function} $A(x_1,x_2)$ counts the number of elements of $A$ between $x_1$ and $x_2,$ that is,
\[
A(x_1,x_2) = \sum_{\substack{a \in A \\ x_1 \leq a \leq x_2} } 1.
\]
The set $A$ has \emph{upper asymptotic density} $d_U(A) = \alpha$ if $\limsup_{x\rightarrow\infty} A(-x,x)/(2x+1) = \alpha.$
The set $A$ has \emph{asymptotic density} $d(A) = \alpha$ if $\lim_{x\rightarrow\infty} A(-x,x)/(2x+1) = \alpha.$
If $S$ is an infinite set of integers and $W$ is a subset of $S$, then the set $W$ has \emph{relative asymptotic density} $d(W,S) = \alpha$ if $\lim_{x\rightarrow\infty} W(-x,x)/S(-x,x) = \alpha.$
The set $A$ is called a \emph{basis of order $m$ for $S$} if $S \subseteq mA,$ that is, if $R_{A,m}(n) \geq 1$ for every element of $S$, and a \emph{basis of order $m$ for almost all $S$} if $\{ n \in S : R_{A,m}(n) = 0\}$ has relative asymptotic density zero.
Nathanson~\cite{nath78d} introduced a class of inverse problems for the representation functions of sets of integers and nonnegative integers. Let $m \geq 2.$ If $A$ and $B$ are sets of integers such that $R_{A,m}(n) = R_{B,m}(n)$ for all $n\in \ensuremath{\mathbf Z},$ then does $A=B$? He proved that the answer is ``yes'' if $A$ and $B$ are sets of nonnegative integers, but if $R_{A,m}(n) = R_{B,m}(n)$ only for all sufficiently large integers $n,$ then $A$ and $B$ are not necessarily equal, but the structures of $A$ and $B$ can be described explicitly.
These results suggest a second class of inverse problems for representation functions. Given any function $f: \ensuremath{\mathbf Z} \rightarrow \ensuremath{ \mathbf N }_0 \cup \{\infty\}$ and an integer $m\geq 2$, does there exist a set $A$ such that $R_{A,m}(n) = f(n)$ for all $n\in \ensuremath{\mathbf Z}$ or $r_{A,m}(n) = f(n)$ for all $n\in \ensuremath{\mathbf Z}$? Describe all sets $A$ such that $R_{A,m} = f$ or $r_{A,m} = f.$
Nathanson~\cite{nath04a,nath05a} proved that \emph{every} function $f: \ensuremath{\mathbf Z} \rightarrow \ensuremath{ \mathbf N }_0 \cup \{\infty\}$ such that $f^{-1}(0)$ is finite is the unordered representation function for infinitely many sets of integers. In particular, there exist \emph{unique representation bases} for the integers, that is, sets $A \subseteq \ensuremath{\mathbf Z}$ such that $r_{A,m}(n) = 1$ for all $n\in \ensuremath{\mathbf Z}$ (For $m=2,$ see {\L}uczak and Schoen~\cite{lucz-scho04}, Nathanson~\cite{nath03a}).
The study of representation functions for sets of nonnegative integers is more complicated. In this case, every integer has only finitely many representations. It is an open problem to describe the set of functions $f: \ensuremath{ \mathbf N }_0 \rightarrow \ensuremath{ \mathbf N }_0$ that are representation functions. A special case of this inverse problem is one of the most famous problems in additive number theory: The conjecture of Erd\H os and Tur\' an that the representation function of an asymptotic basis for the nonnegative integers must be unbounded.
In the last few years there has been considerable work on inverse problems for representation functions (for example,~\cite{borw-choi-chu06, chen07, nath07f, nath07h, grek-hadd-helo-pihk03, hadd-helo04, lee07a, nath04e, nest-serr04, tang-chen07}).
\section{Bases associated to linear forms}
Let $F(x_1,\ldots,x_m) = u_1 x_1 + \cdots + u_mx_m$ be an $m$-ary linear form with nonzero, relatively prime integer coefficients $u_1, \ldots, u_m$. Let $A_1, \ldots, A_h$ be sets of integers. We define the set
\[
F(A_1,\ldots, A_m) = \{F(a_1,\ldots,a_m) : a_i \in A_i \text{ for $i=1,\ldots,m$}\}.
\]
The \emph{representation function} associated with the form $F$ is
\[
R_{A_1,\ldots,A_m,F}(n) = \text{card}\left( \{ (a_1,\ldots,a_m)\in A_1\times \cdots \times A_m: F(a_1,\ldots, a_m) = n \} \right).
\]
This is a function from \ensuremath{\mathbf Z}\ into $\ensuremath{ \mathbf N }_0 \cup \{\infty\}.$
If $A_i=\emptyset$ for some $i=1,\ldots,m,$ then $F(A_1,\ldots, A_m) = \emptyset$ and $R_{A_1,\ldots,A_m,F}(n) = 0$ for all $n \in \ensuremath{\mathbf Z}.$
If $A_i = A$ for all $i=1,\ldots,m,$ then we write
\[
F(A) = F(A,\ldots,A) = \{F(a_1,\ldots, a_m) : a_i \in A \text{ for $i=1,\ldots,m$}\}
\]
and
\[
R_{A,F}(n) = R_{A,\ldots,A,F}(n)
= \text{card}\left( \{ (a_1,\ldots,a_m)\in A^m : F(a_1,\ldots, a_m) = n \} \right).
\]
The sumset $mA$ is a set of the form $F(A),$ where $F$ is the linear form $F(x_1,\ldots,x_m) = x_1 + x_2 + \cdots + x_m$.
Let $S$ be a set of integers and let $F$ be an $m$-ary linear form. The set $A$ is a \emph{basis for $S$ with respect to $F$} if $R_{A,F}(n) \geq 1$ for all integers $n \in S.$
The set $A$ is an \emph{basis with respect to $F$ for almost all $S$} if $\{ n \in S : R_{A,F}(n) = 0\}$ has asymptotic density zero.
Many classical problems about bases in additive number theory have natural analogues for bases with respect to an $m$-ary linear form. For example, a \emph{direct problem} for representation functions is: Given a form $F$ and a set $A$ of integers, compute the representation function $R_{A,F}.$ An \emph{inverse problem} for representation functions is: Given a form $F$ and a function $f:\ensuremath{\mathbf Z} \rightarrow \ensuremath{ \mathbf N }_0 \cup \{\infty\},$ does there exist a set $A$ of integers such that $R_{A,F}(n) = f(n)$ for all $n\in \ensuremath{\mathbf Z}$?
In particular, if $f(n)=1$ for all integers $n$, does there exist a set $A$ such that $R_{A,F}(n) = 1$ for all integers $n$? Such a set is called
a \emph{unique representation basis with respect to the form $F$}.
In this paper we solve the inverse problem for bases with respect to a binary linear form. Let $F(x_1,x_2) = u_1 x_1 + u_2x_2$ be a binary linear form whose coefficients are nonzero, relatively prime integers. We shall prove that if $u_1u_2 \neq \pm 1, -2$ and if $f:\ensuremath{\mathbf Z} \rightarrow \ensuremath{ \mathbf N }_0 \cup\{\infty\}$ is any function such that $f^{-1}(0)$ has zero density, then there exists a set $A$ of integers such that $R_{A,F}=f.$ Equivalently, every nonzero function $f$ such that the set $f^{-1}(0)$ has asymptotic density zero is the representation function of a basis for almost all \ensuremath{\mathbf Z}\ with respect to $F$ for every binary linear form $F(x_1,x_2) \neq x_1 \pm x_2$ or $2x_1-x_2.$
Related work on the additive number theory of finite sets defined by linear forms appears in~\cite{nath07j,nath07d}.
\section{Construction of bases for binary linear forms}
We begin with two simple observations.
\begin{lemma} \label{bfbf:lemma:u}
Let $u_1$ and $u_2$ be nonzero, relatively prime integers with $u_1u_2 \neq \pm 1.$ The following four integers are pairwise distinct:
\[
-u_1^2, -u_1u_2, u_1 u_2, u_2^2
\]
and the following three integers are pairwise distinct:
\[
u_2^2-u_1^2, u_2(u_1+u_2),-u_1(u_1+u_2).
\]
If, in addition, $u_1u_2 \neq -2,$ then these seven integers are pairwise distinct.
\end{lemma}
\begin{proof}
This is a straightforward verification.
\end{proof}
\begin{lemma} \label{bfbf:lemma:density}
Let $A_1,\ldots, A_n$ be sets of integers and let $d_U(A_i)=\alpha_i$ for $i=1,\ldots, n.$ Then $d_U\left( \bigcup_{i=1}^n A_i \right) \leq \sum_{i=1}^n \alpha_i.$ In particular, the union of a finite number of sets of asymptotic density zero has asymptotic density zero.
If $A$ has asymptotic density zero and if $r,s,t$ are integers with $r\neq 0,$ then the set $\{n\in \ensuremath{\mathbf Z} : rn \in \{ sa+t : a\in A\} \}$ has asymptotic density zero.
\end{lemma}
\begin{proof}
For every $\varepsilon > 0$ there is a number $x_0(\varepsilon)$ such that $A_i(-x,x) \leq (\alpha_i + \varepsilon/n)(2x+1)$ for all $x \geq x_0(\varepsilon)$. If $A = \bigcup_{i=1}^n A_i$, then
\[
A(-x,x) \leq \sum_{i=1}^nA_i(-x,x) \leq \sum_{i=1}^n\left( \alpha_i + \varepsilon/n \right) (2x+1) = \left( \sum_{i=1}^n\alpha_i + \varepsilon\right) (2x+1)
\]
for all $x \geq x_0(\varepsilon)$, and so $d_U(A) \leq \sum_{i=1}^n\alpha_i.$ In particular, if $\alpha_i = 0$ for all $i = 1,\ldots, n,$ then $d(A) = d_U(A) = 0.$
Finally, subsets, dilations, and translations of sets with asymptotic density zero also have asymptotic density zero. The last statement of the Lemma follows from this observation.
\end{proof}
The following result is fundamental.
\begin{lemma} \label{bfbf:lemma:fundamental}
Let $F(x_1,x_2) = u_1 x_1+u_2 x_2$ be a binary linear form whose coefficients $u_1, u_2$ are nonzero, relatively prime integers with $u_1u_2 \neq \pm 1$ and $u_1u_2 \neq -2.$ Let $W$ be a set of integers with asymptotic density zero. Let $A'$ be a finite set of integers and let $b$ be any integer such that the sets $W$ and $F(A') \cup \{ b \}$ are disjoint. There exists a set $C$ with $A'\subseteq C $ and $|C \setminus A'| = 2$ such that
\begin{equation} \label{bfbf:rArC}
R_{C,F}(n) =
\begin{cases}
R_{A',F}(b) + 1 & \text{if $n = b$} \\
R_{A',F}(n) & \text{if $n\in F(A')\setminus \{b \}$} \\
1 & \text{if $n\in F(C)\setminus \left( F(A') \cup \{ b\} \right)$} \\
0 & \text{if $n \in W$.}
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
Since $\gcd(u_1,u_2)=1,$ there exist integers $v_1$ and $v_2$ such that $F(v_1,v_2) = u_1v_1+u_2v_2 = 1.$ For every integer $t$ we have
\begin{align*}
F(bv_1+u_2t,bv_2-u_1t) & = u_1(bv_1+u_2t)+u_2(bv_2-u_1t) \\
& = b(u_1v_1+u_2v_2) = b.
\end{align*}
We introduce the sets
\[
B_t = \{ bv_1+u_2t,bv_2-u_1t \}
\]
and
\[
C_t = A' \cup B_t.
\]
Note that the conditions $\gcd(u_1,u_2) = 1$ and $u_1u_2 \neq 0,\pm 1$ imply that $u_1\pm u_2 \neq 0.$
If $(u_1+u_2)t\neq b(v_2-v_1),$ then $bv_1+u_2t \neq bv_2-u_1t$ and $|B_t|=2.$ Similarly, $A' \cap B_t \neq \emptyset$ if and only if
$u_2t = a - bv_1$ or $u_1t = bv_2-a$ for some $a\in A'.$ Since the set $A'$ is finite, it follows that $A' \cap B_t = \emptyset$ and
$|C_t| = |A' \cup B_t| = |A'|+2$ for all but finitely many integers $t$.
We shall prove that there exist infinitely many integers $t$ such that the set $C_t$ also satisfies conditions~\eqref{bfbf:rArC}. Note that $F(C_t)$ is the union of four sets:
\[
F(C_t) = F(A') \cup F(A',B_t) \cup F(B_t,A') \cup F(B_t).
\]
We have
\[
F(A',B_t) = \{ F(a,bv_1+u_2t) : a\in A'\} \cup \{ F(a, bv_2-u_1t ) : a \in A'\}.
\]
If $x \in \{ F(a,bv_1+u_2t) : a\in A'\},$ then there exists $a\in A'$ such that
\[
x = F(a,bv_1+u_2t) = u_1 a + u_2(bv_1+u_2t) = (u_1a + u_2v_1b) + u_2^2t.
\]
If $x \in \{ F(a, bv_2-u_1t ) : a \in A'\},$ then there exists $a\in A'$ such that
\[
x = F(a, bv_2-u_1t ) = (u_1a + u_2v_2b) - u_1 u_2 t.
\]
For every integer $t$, the functions $F(a,bv_1+u_2t)$ and $F(a,bv_2-u_1t )$ are strictly monotonic functions of $a$.
If
\[
\{ F(a,bv_1+u_2t) : a\in A'\} \cap \{ F(a, bv_2-u_1t ) : a \in A'\} \neq \emptyset
\]
then there exist $a,a' \in A'$ such that
\[
(u_1a + u_2v_1b) + u_2^2t = (u_1a' + u_2v_2b) - u_1 u_2 t
\]
or, equivalently,
\[
u_2(u_1+u_2) t = u_1(a' - a) + u_2( v_2 - v_1)b.
\]
Since $u_2(u_1+u_2) \neq 0$ and the set $A'$ is finite, it follows that
for all but finitely many integers $t$ we have
\[
\{ F(a,bv_1+u_2t) : a\in A'\} \cap \{ F(a, bv_2-u_1t ) : a \in A'\} = \emptyset
\]
for all $a,a' \in A',$ and so
\[
R_{A',B_t,F}(n) \leq 1 \qquad\text{for all $n\in \ensuremath{\mathbf Z}.$}
\]
The set $F(A') \cup \{b\} \cup W$ has zero asymptotic density. If
\[
F(A', B_t) \cap \left( F(A') \cup \{b\} \cup W \right) \neq \emptyset
\]
then either
\[
(u_1a + u_2v_1b) + u_2^2t \in F(A') \cup \{b\} \cup W
\]
for some $a\in A'$ or
\[
(u_1a' + u_2v_2b) - u_1 u_2 t \in F(A') \cup \{b\} \cup W
\]
for some $a'\in A'$. In both cases, by Lemma~\ref{bfbf:lemma:density}, the set of integers $t$ for which the membership relation is possible is a set of integers of asymptotic density zero. Equivalently, except for a set of integers $t$ of asymptotic density zero, we have
\[
F(A', B_t) \cap \left( F(A') \cup \{b\} \cup W \right) = \emptyset
\]
and
\[
R_{A',B_t,F}(n) = 0 \qquad\text{for all $n\in F(A') \cup \{b\} \cup W .$}
\]
Similarly,
\begin{align*}
F(B_t, A')
& = \{ F(bv_1+u_2t,a) : a\in A'\} \cup \{ F( bv_2-u_1t, a ) : a \in A'\} \\
& = \{ (u_1v_1 b + u_2 a) + u_1 u_2 t : a\in A'\}
\cup \{ (u_1v_2b + u_2 a) - u_1^2 t : a \in A'\}
\end{align*}
and, except for a set of integers $t$ of asymptotic density zero,
\[
R_{B_t,A',F}(n) \leq 1 \qquad\text{for all $n\in \ensuremath{\mathbf Z}$}
\]
and
\[
R_{B_t, A', F}(n) = 0 \qquad\text{for all $n\in F(A') \cup \{b\} \cup W .$}
\]
If $F(A', B_t) \cap F(B_t, A') \neq \emptyset,$ then there exist integers $a,a' \in A'$ that satisfy at least one of the following four equations:
\begin{align*}
(u_1a + u_2v_1b) + u_2^2t & = (u_1v_1 b + u_2 a') + u_1 u_2 t \\
(u_1a + u_2v_1b) + u_2^2t & = (u_1v_2 b + u_2 a') - u_1^2 t \\
(u_1a + u_2v_2b) - u_1 u_2 t & = (u_1v_1 b + u_2 a') + u_1 u_2 t \\
(u_1a + u_2v_2b) - u_1 u_2 t & = (u_1v_2 b + u_2 a') - u_1^2 t .
\end{align*}
Equivalently,
\begin{align*}
u_2 ( u_2 - u_1) t & = u_2 a' - u_1a + ( u_1- u_2) v_1b \\
(u_1^2 + u_2^2) t & = u_2 a' - u_1a + (u_1v_2 - u_2v_1) b \\
-2 u_1 u_2 t & = u_2 a' - u_1a + (u_1v_1 - u_2v_2)b \\
u_1( u_1 - u_2) t & = u_2 a' - u_1a + (u_1 - u_2) v_2 b .
\end{align*}
Since the coefficients of $t$ are nonzero and the set $A'$ is finite, it follows that there are only finitely many integers $t$ that can satisfy at least one of these equations for some $a,a' \in A$. Except for this finite set of $t$, we have
\[
F(A', B_t) \cap F(B_t, A') = \emptyset
\]
and
\[
R_{A', B_t, F}(n) + R_{B_t, A', F}(n) \leq 1 \qquad\text{for all $n \in \ensuremath{\mathbf Z}.$}
\]
Finally, we consider the set $F(B_t),$ which consists of the integer $b$ and the three integers
\[
(u_1v_2+u_2v_1)b+(u_2^2-u_1^2)t
\]
\[
(u_1+u_2)v_1b+u_2(u_1+u_2)t
\]
\[
(u_1+u_2)v_2b -u_1(u_1+u_2)t .
\]
The coefficients of $t$ in the last three expressions are distinct nonzero integers. This implies that $|F(B_t)|=4$ for all but finitely many $t$, and also that
\[
\left( F(B_t) \setminus \{ b \} \right) \bigcap \left( F(A') \cup W \right) = \emptyset
\]
except for certain integers $t$ belonging to a set of asymptotic density zero.
The coefficients of $t$ in the integers in $F(A',B_t) \bigcup F(B_t,A')$ are $u_2^2$, $\pm u_1u_2, -u_1^2.$ The coefficients of $t$ in the integers in $F(B_t)\setminus \{ b\}$ are $u_2^2-u_1^2$, $u_2(u_1+u_2)$, $-u_1(u_1+u_2).$
Since $u_1u_2\neq -2$, these seven numbers are pairwise distinct by Lemma~\ref{bfbf:lemma:u}, and so the sets $ F(A',B_t) \cup F(B_t,A')$ and $F(B_t)\setminus \{ b\}$ are pairwise disjoint for all but finitely integers $t$. Since the union of a finite number of sets of asymptotic density zero is still a set of asymptotic density zero, it follows that, for almost all integers $t$, the set $C_t$ satisfies the requirements of the Lemma. This completes the proof.
\end{proof}
\begin{theorem}\label{bfbf:theorem:inverse}
Let $F(x_1,x_2) = u_1 x_1+u_2 x_2$ be a binary linear form whose coefficients $u_1, u_2$ are nonzero, relatively prime integers such that $u_1u_2 \neq \pm 1$ and $u_1u_2 \neq -2.$ Let $f: \ensuremath{\mathbf Z} \rightarrow \ensuremath{ \mathbf N }_0 \cup \{\infty\}$ be any function such that the set $f^{-1}(0)$ has asymptotic density zero. There exists a set $A$ of integers such that $R_{A,F}(n) = f(n)$ for all integers $n$.
\end{theorem}
\begin{proof}
Let $W=f^{-1}(0)$ and let $\{b_i\}_{i=1}^{\infty}$ be a sequence of integers that
\[
\left| \{ i\in \ensuremath{ \mathbf N } : b_i = n \}\right| = f(n) \qquad\text{for all $n \in \ensuremath{\mathbf Z}$}.
\]
In particular, $b_i \notin W$ for all $i\in \ensuremath{ \mathbf N }$.
Let $A_0 = \emptyset$. Then $W \cap \left( F(A_0)\cup \{ b_1\}\right) = \emptyset$.
Applying Lemma~\ref{bfbf:lemma:fundamental} with $A'=A_0$ and $b=b_1,$ we obtain a set $C_t$ such that $R_{C_t,F}(n)=1$ if $n\in F(C_t),$ and $F(C_t)\cap W = \emptyset.$ Let $A_1 = C_t.$
Then $R_{A_1, F}(n) \leq f(n)$ for all $n\in \ensuremath{\mathbf Z}.$
Let $i \geq 2$ and suppose that we have constructed sets $A_0 \subseteq A_1 \subseteq \cdots \subseteq A_{i-1}$ with $R_{A_{i-1}, F}(n) \leq f(n)$ for all $n\in \ensuremath{\mathbf Z}.$
If $R_{A_{i-1},F}(b_i)= f(b_i)$, then let $A_i = A_{i-1}$.
Suppose that $R_{A_{i-1},F}(b_i) < f(b_i)$.
Since $W \cap \left( F(A_{i-1})\cup \{ b_i\} \right) = \emptyset$,
we can apply
Lemma~\ref{bfbf:lemma:fundamental}
with $A'=A_{i-1}$ and $b = b_i.$ We obtain a set $C_t$ satisfying conditions~\eqref{bfbf:rArC}. Let $C_t = A_i$. This procedure gives an infinite increasing sequence of finite sets $A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots$
such that, for all $n \in \ensuremath{\mathbf Z},$ we have $R_{A_i,F}(n) \leq f(n)$ for all $i\in \ensuremath{ \mathbf N }$ and $\lim_{i\rightarrow \infty} R_{A_i,F}(n)= f(n)$.
It follows that the set $A = \bigcup_{i=0}^{\infty}A_i$ satisfies $R_{A,F}(n)= f(n)$ for all $n\in \ensuremath{\mathbf Z}.$ This completes the proof.
\end{proof}
\begin{theorem}\label{bfbf:theorem:URB}
Let $F(x_1,x_2) = u_1 x_1+u_2 x_2$ be a binary linear form whose coefficients $u_1, u_2$ are nonzero, relatively prime integers such that $u_1u_2 \neq \pm 1$ and $u_1u_2 \neq -2.$ There exists a unique representation basis with respect to the form $F$.
\end{theorem}
\begin{proof}
Apply Theorem~\ref{bfbf:theorem:inverse} to the function $f(n)=1$ for all $n \in \ensuremath{\mathbf Z}.$
\end{proof}
\section{Sidon sets}
The set $A$ will be called a \emph{Sidon set} with respect to the $m$-ary linear form $F$ if $R_{A,F}(n) \leq 1$ for all integers $n$. More generally, we shall call the set $A$ a $B_F[g]$-set with respect to the form $F$ if $R_{A,F}(n) \leq g$ for all integers $n$. With respect to the classical additive form $F(x_1,\ldots, x_m) = x_1 + \cdots + x_m,$ Erd\H os and Tur\' an conjectured that no $B_F[g]$-set of nonnegative integers is an asymptotic basis for the nonnegative integers. The analogue of this conjecture for arbitrary $m$-ary linear forms is not true. Here is a simple example of a Sidon set of nonnegative integers that is a basis for the nonnegative integers with respect to an $m$-ary form.
\begin{theorem}
For $m \geq 2$ and $g \geq 2$, let
\[
F(x_1,\ldots,x_m) = x_1+gx_2 + g^2 x_3 + \cdots + g^{m-1}x_m.
\]
Let $A$ be the set of all nonnegative integers whose $g^m$-adic representations
use only the digits $\{0,1,\ldots,g-1\},$ that is,
\[
A = \left\{ \sum_{i=0}^{\infty} d_ig^{im} : d_i \in \{0,1,\ldots,g-1\} \text{ and $d_i = 0$ for all sufficiently large $i$}
\right\}.
\]
Then $A$ is a Sidon basis for $\ensuremath{ \mathbf N }_0$ with respect to the form $F$.
\end{theorem}
\begin{proof}
This follows immediately from the uniqueness of the $g$-adic representation of a nonnegative integer.
\end{proof}
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1,314,259,995,848 | arxiv | \section{#1}\setcounter{equation}{0}}
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\title{Energy in the Einstein-Aether Theory}
\author{Christopher Eling}
\email{cteling@physics.umd.edu}
\affiliation{Department of Physics, University of Maryland\\ College Park, MD 20742-4111 USA}
\begin{abstract}
We investigate the energy of a theory with a unit vector field
(the ``aether") coupled to gravity. Both the Weinberg and Einstein
type energy-momentum pseudotensors are employed. In the linearized
theory we find expressions for the energy density of the 5 wave
modes. The requirement that the modes have positive energy is then
used to constrain the theory. In the fully non-linear theory we
compute the total energy of an asymptotically flat spacetime. The
resulting energy expression is modified by the presence of the
aether due to the non-zero value of the unit vector at infinity
and its $1/r$ falloff. The question of non-linear energy
positivity is also discussed, but not resolved.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
Although Lorentz invariance has been a key feature of theoretical
physics for a century, recently there have been a number of
reasons for questioning whether it holds at all energy scales. For
example, some possible quantum gravity effects hint that it may
not be a fundamental symmetry \cite{1}. Thus, it is useful to
construct effective, low energy symmetry breaking models in the
regimes of the Standard Model and General Relativity (GR). The new
effects that appear can then be studied in a familiar context and
compared with observations. In the flat spacetime background used
in the Standard Model, Lorentz invariance can be broken by
background tensor fields \cite{2}. However, when we attempt to
couple such fields to gravity, they will also break the general
covariance of GR, which we regard as fundamental. In order to
bypass this problem, it is straightforward to consider these
fields as dynamical quantities along with the metric. In this
paper, the source of the Lorentz violation (LV) will be modeled as
a unit timelike vector $u^a$. The unit timelike restriction
preserves the well-tested $SO(3)$ group of rotations while
enforcing the breaking of boost symmetry at every point in the
curved spacetime. Therefore, the vector $u^a$ can be said to act
as an ``aether".
Similar work was initiated in the early 1970's by Will, Nordtvedt
and Hellings \cite{3,4,5} who studied a vector-tensor model
without the constraint in the context of alternative theories of
gravity. For a review of more recent work on the subject,
including aspects of observational constraints, waves, cosmology,
and black holes, see \cite{6} and the references therein.
Following these authors, we will refer to this theory as the
``Einstein-Aether" theory. One important open question is whether
the Einstein-Aether theory is energetically viable and stable. The
Will-Nordtvedt-Hellings models, for example, are unstable because
fluctuations of the unconstrained vector can be either timelike or
spacelike, allowing ghost configurations and energy of arbitrary
sign \cite{7}.
Energy in a field theory is defined as the the value of the
Hamiltonian, which acts as the generator of time translations.
Although in diffeomorphism invariant theories there is generally
no preferred notion of time (and thus energy), in asymptotically
flat spacetimes one can naturally define the ADM and Bondi
energies associated with asymptotic time translations at spatial
and null infinity respectively. The ADM and Bondi definitions for
GR have also been shown to satisfy positive energy theorems
\cite{8}.
In this paper we examine energy in the Einstein-Aether theory.
Since it has proven difficult to directly construct the
Hamiltonian for the theory, we instead consider the pseudotensor
method of studying gravitational energy. Such an approach was
first taken in a similiar context by Lee, Lightman, and Ni
\cite{9}, who derived pseudotensors for the unconstrained
vector-tensor models but did not evaluate them on solutions.
Despite the non-covariance of pseudotensors, it is known that they
give well-defined results for the spatially averaged energy
carried by waves in linearized theory and the total energy of
asymptotically flat spacetimes. In gravitational wave physics they
provide a simple and straightforward method for calculating
averaged energy densities and the energy-momentum flux radiated
away from sources. In addition, Chang, Nester, and Chen \cite{10}
have shown that the superpotential associated with every
pseudotensor corresponds to a (albeit non-covariant) quasi-local
Hamiltonian boundary term.
We first discuss the Einstein-Aether theory and then motivate and
construct its modified Weinberg pseudotensor expression. As the
calculational and consistency check we also use a Lagrangian based
method to derive the modified Einstein ``canonical" pseudotensor
and its associated superpotential. We then apply these expressions
to solutions in both the linear and non-linear regimes. In the
linearized theory we find that the Einstein and Weinberg
prescriptions give the same energy densities for the plane wave
modes derived in \cite{11}. Restricting these densities to be
positive yields constraints on the model in terms of the
coefficients of the aether part of the action. These constraints
are also compared to results obtained in the limit where the
metric and aether decouple \cite{12}. In the full non-linear
theory the Einstein-Aether superpotential is used to obtain the
total energy for an asymptotically flat spacetime. This result
agrees with \cite{13}, where the total energy in the
Einstein-Aether theory is derived via the covariant Noether charge
formalism. We conclude with a discussion of the status of positive
energy in the non-linear regime and prospects for a positive
energy theorem.
\section{Einstein-Aether action}
\label{E-A}
We can model gravity with a dynamical preferred frame using a
timelike unit vector $u^{a}$. This vector field breaks local
Lorentz invariance spontaneously in every configuration leaving
behind the 3-D rotation group as the residual symmetry. The unit
norm condition is required to avoid ghosts and also desirable
because we regard the norm as extra information beyond what is
necessary to determine the preferred frame. Taking an effective
field theory point of view we can consider the action as a
derivative expansion, subject to diffeomorphism symmetry. The
result, up to two or fewer derivatives, is
\begin{equation} S = \frac{1}{16\pi G}\int \sqrt{-g}~ L_{\rm ae} ~d^{4}x
\label{action} \end{equation}
where
\begin{equation} L_{\rm ae} = -R-K^{ab}{}_{mn} \nabla_a u^m \nabla_b u^n\\ -
\lambda(g_{ab}u^a u^b - 1). \end{equation}
The ``kinetic" term $K^{ab}{}_{mn}$ is defined as
\begin{equation} K^{ab}{}_{mn} = c_1
g^{ab}g_{mn}+c_2\delta^{a}_{m}\delta^{b}_{n}
+c_3\delta^{a}_{n}\delta^{b}_{m} +c_4u^au^bg_{mn} \end{equation}
and $\lambda$ is a Lagrange multiplier enforcing the unit timelike
constraint. $R$ is the familiar Ricci scalar and the coefficients
$c_i$ in $K^{ab}{}_{mn}$ are dimensionless constants. Note that a
term proportional to $R_{ab} u^a u^b$ is not explicitly included
as it comes about as a combination of the $c_2$ and $c_3$ terms in
(\ref{action}). The metric signature is $({+}{-}{-}{-})$ and the
units are chosen so that the speed of light defined by the metric
$g_{ab}$ is unity.
The field equations from varying the action in (\ref{action})
together with a matter action with respect to $g^{ab}$ and $u^a$
are given by
\begin{eqnarray}
G_{ab} &=& T^{(u)}_{ab}+8\pi G T^{M}{}_{ab}\label{AEE}\\
\nabla_a J^{a}{}_m-c_4 \dot{u}_a \nabla_m u^a &=& \lambda u_m,
\label{ueqn}\\
g_{ab} u^a u^b &=& 1 \label{constraint}.
\end{eqnarray}
where
\begin{equation} J^a{}_{m} = K^{ab}{}_{mn} \nabla_b u^n \label{Jdef}\end{equation}
and
\begin{equation} \dot{u}_a = u^b \nabla_b u_a. \end{equation}
Here we assume that there are no aether-matter couplings in the
matter action. The aether stress tensor is given by \cite{14}
\begin{eqnarray} T^{(u)}{}_{ab}&=&\nabla_m(J_{(a}{}^m u_{b)}- J^m{}_{(a} u_{b)}
- J_{(ab)}u^m) \nonumber\\ &&+ c_1\, \left[(\nabla_m u_a)(\nabla^m u_b)-(\nabla_a u_m)(\nabla_b
u^m) \right]\nonumber\\ &&+ c_4\, \dot{u}_a\dot{u}_b\nonumber\\
&&+\left[u_n(\nabla_m J^{mn})-c_4\dot{u}^2\right]u_a u_b \nonumber\\
&&-\frac{1}{2} L_u g_{ab}, \label{aetherT}\end{eqnarray}
where $L_{u} = -K^{ab}{}_{mn} \nabla_a u^m \nabla_b u^n$ and
$\dot{u}^2 = \dot{u}_a \dot{u}^a$. The Lagrange multiplier $\lambda$
has been eliminated from (\ref{aetherT}) by solving for it via the
contraction of the aether field (\ref{ueqn}) with $u^a$. As we will
see below, the the form of the aether stress tensor and
Einstein-Aether Lagrangian will be important tools in derivation of
the modified Weinberg and Einstein pseudotensors.
\section{Weinberg Pseudotensor}
\label{Weinberg}
Weinberg's pseudotensor construction \cite{15} is based on the
``field theoretic" approach to GR that treats gravity as a spin-2
field on a flat background spacetime. Using Greek indices to
represent coordinate indices, we begin by writing the metric in
coordinates such that $g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$,
where $\eta_{\mu\nu}$ is the flat Minkowski metric and $h_{\mu
\nu}$ is an symmetric tensor field with the asymptotic conditions
$h_{\mu \nu} \sim O(1/r)$, $\partial_\sigma h_{\mu\nu} \sim
O(1/r^2)$, $\partial_\tau \partial_\sigma h_{\mu\nu} \sim
O(1/r^3)$. The Einstein tensor can be expanded into a series of
parts linear, quadratic, and higher order in the field variable
$h_{\mu\nu}$. Following Ch. 20 of Misner, Thorne, and Wheeler
\cite{16} the non-linear corrections to the Einstein tensor are
defined as follows
\begin{equation} 16 \pi G~ t_{\mu\nu} \equiv 2G^{(1)}_{\mu\nu}-2G_{\mu\nu},
\label{split} \end{equation}
where $G^{(1)}_{\mu\nu}$ and $G_{\mu\nu}$ are the linearized and
full non-linear Einstein tensors respectively. Note that this
splitting is non-unique because it depends on the coordinate
system. Since the linearized Einstein tensor is symmetric and
satisfies a linearized Bianchi identity $\partial^\mu
G^{(1)}_{\mu\nu}=0$, it can be rewritten in superpotential form
\begin{equation} 2G^{(1)}_{\mu\nu} = H_{\mu \alpha \nu \beta}{}^{,\alpha \beta}
\label{doublediv}\end{equation}
where $H_{\mu \nu \alpha \beta}$ has the symmetries of the Riemann tensor
$H_{\mu \nu \alpha \beta}=H_{[\mu \nu][\alpha \beta]}=H_{\alpha \beta \mu \nu}$ (see,
for example \cite{17}). Using (\ref{split}) and (\ref{doublediv})
the full Einstein equation becomes
\begin{equation} H_{\mu \alpha \nu \beta}{}^{,\alpha \beta} = 16 \pi G~(t_{\mu \nu}+T_{\mu
\nu}). \label{eesup} \end{equation}
Due to the symmetries of $H_{\mu \alpha \nu \beta}$, this implies that
$\partial^\nu(t_{\mu \nu}+T_{\mu \nu})=0$. Therefore the integral
of $t_{00}+T_{00}$ over a spacelike slice is a conserved quantity.
This conserved quantity
\begin{eqnarray} \int (t_{00}+T_{00}) ~d^3 x &=& \frac{1}{16 \pi G}\int H_{0
\alpha 0 \beta}{}^{,\alpha \beta} d^3x \nonumber\\
&=& \frac{1}{16\pi G}\oint
H_{0\a0\beta}{}^{,\alpha} n^{\beta} ~d^2x, \label{energyeqn}\end{eqnarray}
where $n^\beta$ is the unit normal to the surface at spatial
infinity, is in fact the total energy, with $t_{00}$ acting as the
energy density of the gravitational field alone. To sharpen this
point, consider the case where the gravitational field is weak
everywhere, allowing use of the linearized theory. The leftmost
member of (\ref{energyeqn}) then gives the total matter energy,
which in this case is the total energy. The rightmost member is
insensitive to the interior volume, so replacement by arbitrary
sources and strong fields in the interior will not affect the
identification of (\ref{energyeqn}) as the total energy.
The extension to the Einstein-Aether theory is straightforward.
The metric field equations (\ref{AEE}) take the form
\begin{equation} \widetilde{G}_{\mu\nu} = G_{\mu\nu}-T^{(u)}_{\mu\nu} = 8 \pi G
T_{\mu\nu}.\label{AEE1} \end{equation}
In addition to the metric, we now decompose the aether into
background and dynamical part by writing $u^\mu =
\underline{u}^\mu + v^\mu$. Unlike normal matter fields the aether
stress $T^{(u)}_{\mu\nu}$ contains linear pieces in the
perturbation $v^\mu$ due to the fact that the aether does not
vanish in the background (since it is always a unit vector). These
linear terms will modify the Weinberg pseudotensor and
superpotential. Performing the split of the modified Einstein
tensor $\widetilde{G}_{ab}$ as in (\ref{split}) we find
\begin{equation} 16 \pi G~\widetilde{t}_{\mu \nu} \equiv
2\widetilde{G}^{(1)}_{\mu \nu}-2\widetilde{G}_{\mu \nu}
\label{aeWeinberg}\end{equation}
where $\widetilde{G}^{(1)}_{\mu \nu}=G^{(1)}{}_{\mu
\nu}-T^{(1)(u)}{}_{\mu \nu}$. $\widetilde{G}_{\mu \nu}$ satisfies a
Bianchi identity $\nabla^\mu \widetilde{G}_{\mu \nu}=0$ if the
aether is uncoupled to the matter and if the aether field equation
(\ref{ueqn}) is satisfied. Therefore in the linearized case we can
write
\begin{equation} 2\widetilde{G}^{(1)}{}_{\mu \nu} = \widetilde{H}_{\mu \alpha \nu
\beta}{}^{,\alpha \beta} \end{equation}
along with
\begin{equation} \widetilde{H}_{\mu \alpha \nu \beta}{}^{,\alpha \beta} = 16 \pi
G(\widetilde{t}_{\mu \nu}+T_{\mu \nu}). \label{eesupae} \end{equation}
By the same reasoning as before we could conclude that the total
energy is given by
\begin{equation} E = \frac{1}{16 \pi G}\oint \widetilde{H}_{0\alpha 0 \beta}{}^{,\alpha}
n^\beta d^2x.\end{equation}
However, unlike the GR case (\ref{eesup}), it is not clear whether
the new Weinberg superpotential $\widetilde{H}_{\mu\alpha \nu
\beta}{}^{,\alpha}$ can be expressed as a local function of the fields
$h_{ab}$ and $u^{a}$ \footnote{The author thanks an anonymous
referee for pointing out this fact}. On the other hand, the
pseudotensor $\widetilde{t}_{\mu \nu}$ can be calculated directly
via the non-linear pieces of $T^{(u)}_{\mu \nu}$ and $G_{\mu \nu}$
in $\widetilde{G}_{\mu \nu}$. This will be used to compute the
linearized wave energy densities. Evaluation of the total energy as
a surface integral at spatial infinity requires a locally defined
superpotential. Since we do not have knowledge of the aether
corrections to Weinberg superpotential we shall instead consider the
Einstein superpotential, which can be derived directly from the form
of the Lagrangian. The Einstein formulation of gravitational
energy-momentum will also provide a consistency check when we
evaluate the energy density of the linearized plane wave modes.
\section{Einstein ``canonical" pseudotensor}
\label{Einstein}
The gravitational energy pseudotensor originally derived by
Einstein in 1916 shortly after his discovery of the field
equations of GR is closely related to the familiar canonical
stress tensor of matter fields in flat spacetime. In order to
derive the corresponding expression for the Einstein-Aether
theory, we use a Lagrangian approach based upon the famous work of
Noether relating symmetries to conservation laws. In flat
spacetime, invariance of a Lagrangian under global space and time
translations is associated with the conservation of
energy-momentum expressed by the conservation of the canonical
stress tensor
\begin{equation} T_{\nu}{}^\mu = \frac{\partial {\cal
L}}{\partial(\partial_\mu \psi)}
\partial_\nu\psi-\delta^\mu_{\nu} {\cal L}, \label{flatcan}\end{equation}
where ${\cal L} = {\cal L}(\psi,\partial \psi)$ and $\psi$
represents a general collection of fields with indices suppressed.
In the case of local symmetries, such as the diffeomorphism
invariance of the Einstein-Aether theory, the situation is more
complex. In the Appendix we review a general formalism due to
Julia and Silva \cite{18} for constructing Noether currents and
superpotentials and apply it to the Einstein-Aether theory. Using
these results we show that the pseudotensor and superpotential
have the following general form
\begin{eqnarray}
t_{\nu}{}^\mu &=& \frac{\sqrt{-g}}{16\pi G}\left(\frac{\partial
L}{\partial(\partial_\mu g_{\alpha\beta})} \partial_\nu
g_{\alpha\beta}\right. \nonumber\\&& \left. + \frac{\partial
L}{\partial(\partial_\mu u^{\alpha})}
\partial_{\nu} u^{\alpha} - \delta^\mu_\nu
L\right) \label{Epseudo}\\
U_{\nu}{}^{\mu\gamma} &=& \frac{\sqrt{-g}}{16\pi
G}\left(\frac{\partial L}{\partial(\partial_\mu
g_{\alpha\beta})}(\delta^{\gamma}_{\alpha}g_{\nu\beta}+\delta^{\gamma}_{\beta}g_{\nu\alpha})\right.
\nonumber\\&& \left.-\frac{\partial L}{\partial(\partial_\mu
u^{\alpha})} \delta^{\nu}_{\alpha} u^{\gamma}\right)
\label{Esuper}
\end{eqnarray}
where $L$ is the Lagrangian
\begin{eqnarray} L &=& -
g^{\alpha\beta}(\Gamma^{\eta}_{\alpha\delta}\Gamma^{\delta}_{\eta\beta}
-\Gamma^{\eta}_{\eta\delta}\Gamma^{\delta}_{\alpha\beta})\nonumber\\&&-K^{\alpha\beta}{}_{\mu\nu}
\nabla_\alpha u^\mu \nabla_\beta u^\nu - \lambda(g_{\mu\nu}u^\mu
u^\nu-1). \end{eqnarray}
Note that we have eliminated a surface term in the
Einstein-Hilbert Lagrangian, replacing the Ricci scalar $R$ with
the Einstein-Schrodinger ``$\Gamma^2$'' action, which depends only
on the metric and its first derivatives. When evaluated on-shell
the pseudotensor and superpotential obey the following relations
\begin{eqnarray}
\partial_\mu t_\nu{}^\mu &=& 0 \label{divt}\\
t_\nu{}^\mu &=& -\partial_\gamma U_\nu{}^{\gamma\mu}\label{tsup}.
\end{eqnarray}
To account for the presence of any non-aether matter sources one
only has to make the replacement $t_\nu{}^\mu \rightarrow
t_\nu{}^\mu + T_\nu{}^\mu$ in (\ref{divt}) and (\ref{tsup}). Like
the Weinberg construction, the pseudotensor $t_\nu{}^\mu$ is a
conserved quantity and is related to the divergence of a
superpotential.
The contributions from the pure GR $\Gamma^2$ Lagrangian are the
{\it Einstein} pseudotensor
\begin{eqnarray} ^{\rm einstein}t^{\ \mu}_\rho &=& \frac{\sqrt{-g}}{16\pi G} \left(\delta^{\
\mu}_\rho (\G^\alpha_{\beta\gamma}\def\G{\Gamma}\G^\beta_{\alpha\delta}-\G^\alpha_{\alpha\beta}\G^\beta_{\gamma}\def\G{\Gamma\delta})g^{\gamma}\def\G{\Gamma\delta}
\right. \nonumber\\&&
\left.+\G^\beta_{\rho\alpha}\G^\gamma}\def\G{\Gamma_{\gamma}\def\G{\Gamma\beta}g^{\mu\alpha}-\G^\beta_{\rho\beta}\G^\gamma}\def\G{\Gamma_{\gamma}\def\G{\Gamma\alpha}g^{\mu\alpha}
+\G^\alpha_{\rho\alpha}\G^\mu_{\beta\gamma}\def\G{\Gamma}g^{\beta\gamma}\def\G{\Gamma}\right. \nonumber\\&&
\left.+\G^\mu_{\rho\alpha}\G^\beta_{\beta\gamma}\def\G{\Gamma}g^{\alpha\gamma}\def\G{\Gamma} -2
\G^\mu_{\alpha\beta}\G^\alpha_{\rho\gamma}\def\G{\Gamma}g^{\beta\gamma}\def\G{\Gamma}\right) \label{Einsteinpsu}
\end{eqnarray}
and the {\it von Freud} superpotential (see, e.g. \cite{19})
\begin{equation} ^{\rm fr}U_{\beta}{}^{\lambda\alpha} = \frac{1}{16\pi G}
\frac{1}{\sqrt{-g}}
g_{\beta\tau}\partial_\gamma\{(-g)(g^{\lambda\tau}g^{\alpha\gamma}-
g^{\alpha\tau}g^{\lambda\gamma})\}. \label{Einsteinsup}\end{equation}
To compute the additional aether modifications, we use the
relation
\begin{equation} \frac{\partial L}{\partial(\partial_\mu g_{\alpha \beta})} = {\textstyle{\frac{1}{2}}}
(g^{\alpha \nu}\delta^{\beta}_{\gamma}\def\G{\Gamma} \delta^{\mu}_{\delta}+g^{\alpha \nu}
\delta^{\beta}_{\delta} \delta^{\mu}_{\gamma}\def\G{\Gamma}-g^{\mu \nu} \delta^{\alpha}_{\gamma}\def\G{\Gamma}
\delta^{\beta}_{\delta}) \frac{\partial L}{\partial(\Gamma^{\nu}_{\gamma}\def\G{\Gamma
\delta})} \end{equation}
and $L_u=K^{\alpha\beta}{}_{\mu\nu} \nabla_\alpha u^\mu \nabla_\beta u^\nu$ in
(\ref{Epseudo}) and (\ref{Esuper}) since when evaluated on
solutions any terms related to the unit constraint will vanish. We
find the pseudotensor
\begin{eqnarray} ^{\rm \ae}t_{\nu}{}^{\lambda} &=& \frac{1}{16\pi G}\left(2
{\sqrt{-g}} J^{\lambda}{}_{\rho}\nabla_{\nu} u^{\rho}
-\sqrt{-g}\{(J^{\lambda}{}_{\beta}+J_{\beta}{}^{\lambda})u^{\alpha}
\right. \nonumber\\
&&
\left.-(J^{\alpha}{}_{\beta}+J_{\beta}{}^{\alpha})u^{\lambda}+(J^{\lambda\alpha}
-J^{\alpha\lambda})u_{\beta}\}
\Gamma^{\beta}{}_{\alpha\nu}\right. \nonumber\\
&& \left.+ \delta_{\nu}^{\lambda} {\sqrt{-g}} L_{u}\right).
\label{aepsu} \end{eqnarray}
and the superpotential
\begin{eqnarray} ^{\rm \ae}U_\beta{}^{\lambda\alpha} &=& \frac{1}{16 \pi
G}\sqrt{-g}
\left((J^{\lambda}{}_{\beta}+J_{\beta}{}^{\lambda})u^{\alpha}
\right. \nonumber\\
&& \left.
-(J^{\alpha}{}_{\beta}+J_{\beta}{}^{\alpha})u^{\lambda}+(J^{\lambda\alpha}
-J^{\alpha\lambda})u_{\beta}\right) \label{aesup}\end{eqnarray}
where $J^a{}_\beta$ is defined in (\ref{Jdef}). The above
decompositions of (\ref{Epseudo}) and (\ref{Esuper}) into GR and
aether pieces do not satisfy (\ref{divt}) and (\ref{tsup})
independently. A key requirement when evaluating these
pseudotensorial expressions is that the metric must be written in a
coordinate system where the connection coefficients vanish like
$O(1/r)$ or faster in the asymptotic limit. If the coordinate system
is not chosen properly then these expressions will yield incorrect
energies and momenta \footnote{For example, if one uses the
Schwarzschild metric in spherical polar coordinates $ds^2 =
(1-\frac{2M}{r})dt^2-(1-\frac{2M}{r})^{-1}dr^2-r^2 d\Omega^2$, the
von-Freud superpotential will yield an incorrect total energy. After
re-expressing the metric in Cartesian coordinates $(t,x,y,z)$, the
pseudotensor expression gives $E=M$.}. This condition was not well
understood in the early literature on gravitational energy-momentum,
but can now be explained using an analysis of the boundary terms and
conditions in an action. See the Appendix for further details.
\section{Energy in Linearized Theory}
\label{Linear}
Equipped with the modified Einstein and Weinberg pseudotensors we
can now calculate the energy density of the linearized plane wave
solutions to the Einstein-Aether theory. The plane wave solutions
in the absence of matter are found by linearizing the field
equations above, (\ref{AEE})-(\ref{constraint}), with $g_{\mu\nu}
= \eta_{\mu\nu}+h_{\mu\nu}$ and $u^\mu = \underline{u}^\mu+v^\mu$.
This gives
\begin{eqnarray}
\partial_\alpha J^{(1)\alpha}{}_\beta &=& \lambda \underline{u}_{\beta}\label{linaefield}\\
G^{(1)}_{\alpha\beta} &=& T^{(1)}_{\alpha\beta} \label{linaemetric}\\
v^0 &=& -\frac{1}{2}h_{00}\label{linaeconstr}.
\end{eqnarray}
Cartesian coordinates are used in the flat background,
$\eta_{\mu\nu} = (1,-1,-1,-1)$ and $\underline{u}^\mu = (1,0,0,0)$.
Since the background value of the Lagrange multiplier vanishes,
$\lambda$ in (\ref{linaefield}) represents a perturbation. The
superscript (1) represents quantities written to first order in
the perturbation. Jacobson and Mattingly \cite{11} then proceed to
analyze these equations using the gauge choice
\begin{eqnarray}
h_{0i} &=& 0 \label{gauge1}\\
v_{i,i} &=& 0 \label{gauge2}
\end{eqnarray}
which they prove to be accessible. Inserting plane wave solutions
\begin{eqnarray}
h_{\mu\nu} &=& \epsilon_{\mu\nu}e^{ik_c x^c}\\
v^{\mu} &=& \epsilon^{\mu}e^{ik_c x^c}
\end{eqnarray}
into the equations of motion, imposing the 4 gauge conditions
(\ref{gauge1})-(\ref{gauge2}), and choosing coordinates such that
the wave-vector is $(k_0,0,0,k_3)$ (travelling in the $z$
direction), it is found \cite{11} that the mode polarizations and
speeds are completely determined. The result is a total of 5 wave
modes falling into spin 2, spin 1, and spin 0 types as shown in
Table \ref{speeds} \footnote{Unlike (for example) the Lorentz gauge
in GR, the residual gauge of (\ref{gauge1})-(\ref{gauge2}) is not
compatible with the equations of motion
(\ref{linaefield})-(\ref{linaeconstr}) so it is not clear how to fix
the remaining gauge in a simple way. However, it is possible to
argue for the existence of 5 wave modes by counting gauge
inequivalent degrees of freedom. Consider a theory with $N$ field
variables and $M$ gauge symmetries. One can always use the $M$
constraint equations to solve for the $M$ variables whose time
derivative does not appear in the equations of motion. This reduces
the number of degrees of freedom to $N-M$. Then the remaining $M$
gauge functions can be used to further reduce to $N-2M$. In the case
of the aether theory the metric has 10 degrees of freedom and the
constrained vector has 3, making 13 field variables. There are 4
diffeomorphism symmetries, and 13 - 2x4 = 5.}.
\begin{table*}
\caption{\label{speeds}Wave Mode Speeds and Polarizations}
\begin{ruledtabular}
\begin{tabular}{lll}
Mode&Squared Speed $s^2$ &Polarizations\\
\hline
spin-2&$1/(1-c_{13})$& $h_{12}$,$h_{11}=h_{22}$\\
spin-1&$(c_1-\frac{1}{2}c_1^2+\frac{1}{2}c_3^2)/c_{14}(1-c_{13})$&
$h_{I3}=[c_{13}/(1-c_{13})s]v_{I}$ \\
spin-0&$c_{123}(2-c_{14})/c_{14}(1-c_{13})(2+c_{13}+3c_2)$&$h_{00}=-2v_0,$\\
& &$h_{11}=h_{22}=-c_{14}v_0,$\\
& &$h_{33}=[2c_{14}(c_2+1)/c_{123}]v_0$\\
\end{tabular}
\end{ruledtabular}
\end{table*}
The notation $I$ in subscript refers to the transverse components
of the metric and aether while $c_{14}=c_1+c_4$, etc. The 2 spin-2
TT metric modes look exactly like the usual GR case, except for
the modification of the speed. The 2 spin-1 transverse aether
modes and 1 spin-0 trace mode are new modes coming from the
constrained aether, which is characterized by 3 degrees of
freedom.
In order to determine the energy, note that in the absence of
matter the Weinberg prescription (\ref{energyeqn}) reduces to
\begin{equation} E = \frac{1}{16\pi G} \oint \widetilde{H}_{0\a0\beta}{}^{,\alpha}
n^\beta ~ d^2x = \int \widetilde{t}_{00} ~ d^3 x, \end{equation}
which clearly produces infinite total energy for plane wave modes.
One could reformulate the problem in terms of wavepackets with the
appropriate asymptotic fall-off conditions, but a far more direct
approach is to simply evaluate the plane wave energy density
$\widetilde{t}_{00}$. This quantity is meaningless at a point for
plane waves, but the average over a cycle is well-defined.
Consider a large, but finite region with nearly plane waves. There
are ``surface effects", but the the contribution to $\int
\widetilde{t}_{00} d^3 x$ is dominated by the volume. Thus,
$\widetilde{t}_{00}$ gives an effective energy density.
The 3 general classes of modes were analyzed separately using the
Riemann tensor package \cite{20} in Maple. The package allows the
user to enter the components of the metric and aether vector,
calculate curvature tensors, and to define new tensors involving
both ordinary and covariant derivatives. In this case we entered
the linearized metric and aether, where $h_{\mu\nu}$ and $v^\mu$ take
the plane wave forms. A polarization was written as
\begin{equation} A \exp{ik_3(z-st)} + \overline{A} \exp(-ik_3(z-st)) \end{equation}
where $s$ are speeds shown in Table \ref{speeds} and $A$ is a
complex-valued function. Using this metric we calculated the
explicit form of the Weinberg pseudotensor $\widetilde{t}_{00}$
(\ref{aeWeinberg}) up to quadratic order. Higher order terms will be
small in the linearized theory and oscillatory terms proportional to
$\overline{A}^2$ and $A^2$ can be neglected in the usual time
averaging process. These energy densities were then compared with
the modified Einstein pseudotensor~ $^{\rm
einstein}t_{0}{}^{0}+^{\rm \ae}t_{0}{}^{0}$ from (\ref{Einsteinpsu})
and (\ref{aepsu}) again up to quadratic order in the perturbations.
Note that while (\ref{divt}) holds at quadratic order when the
linearized equations of motion are imposed, (\ref{tsup}) does not.
Therefore, one must use the modified Einstein pseudotensor directly
to compute the energy densities. The results of the Weinberg and
Einstein prescriptions agreed and are displayed below:
\begin{eqnarray}
\mathcal{E}_{\rm spin-2} &=& \frac{1}{8\pi G}~ k_3^2 ~|A|^2\label{TT}\\
\mathcal{E}_{\rm spin-1} &=& \frac{1}{8\pi G}~ k_3^2~
|A|^2\frac{c_3^2-c_1^2+ 2c_1}{1-c_1-c_3}\label{TA}\\
\mathcal{E}_{\rm spin-0} &=& \frac{1}{8\pi G} ~k_3^2 ~|A|^2
c_{14}(2-c_{14})\label{trace}
\end{eqnarray}
These results have been independently verified in \cite{21} using
the Noether charge method and a decomposition of $u^a$ into
irreducible pieces. The lack of $c_i$ dependence in (\ref{TT}) and
the simplicity of (\ref{TA})-(\ref{trace}) is striking considering
the complicated form of the pseudotensor expressions. The energy
of the spin-2 mode is positive definite, like pure GR, while for
the other 2 modes the sign of the energy density depends upon a
combination of $c_1$, $c_3$, and $c_4$. Note that when the $c_i$'s
are zero (\ref{TA}) and (\ref{trace}) are zero as expected. This
set of results for the coefficients also holds for exponentially
growing modes (i.e. when $s^2<0$)
\begin{equation} A \cos(k z+\varphi)\exp(k s t) \end{equation}
when we average over the spatial oscillations. Restricting $s^2
> 0$ in Table \ref{speeds} to eliminate the unstable modes and enforcing
positivity in the energy densities in (\ref{TT})-(\ref{trace})
restricts the $c_i$ values in the Einstein-Aether theory.
In \cite{12}, Lim worked in the limit where the aether and metric
perturbations decouple, with the aether propagating in flat
spacetime. Mathematically this amounts to tuning $c_i, G
\rightarrow 0$ while holding the ratio $c_i/G$ fixed in the action
(\ref{action}). If we then expand the metric as $g = \eta +
\sqrt{G}~ h$ and take the limit, the action reduces to that of
linearized gravity plus aether terms coupled only to $\eta_{ab}$.
In this limit the linearized constraint reduces to $v^0 = 0$ and
we can decompose $v^i$ into spin-0 and spin-1 parts via $v^i =
\partial^i S + N^i$ where $N^i_{,i} = 0$. By examining the
Hamiltonian of these modes they found $c_1 > 0$ for positivity in
both cases, neglecting $c_4$. We can make contact with this result
simply by examining in the small $c_i$ limit of the wave
solutions. The trace and transverse aether energy waves then
correspond to the flat spacetime spin-0 and spin-1 modes. To
lowest order in $c_i/G$ we find that
\begin{eqnarray}
c_{14} > 0\\
c_{1} > 0 \label{smallc}
\end{eqnarray}
for positive energy densities of the spin-0 and spin-1 modes
respectively. Restoring $c_4$ in the flat spacetime analysis
yields complete agreement. Note that for small $c_i$ the $s^2>0$
criteria for stable, non-exponentially growing modes reduce to
$c_1/c_{14}\geq 0$ for the spin 1 aether-metric mode and
$c_{123}/c_{14} \geq 0$ for the spin 0 trace mode. Thus, modes
with positive energy are stable if $c_{123}>0$.
\section{Non-Linear Energy}
\label{Non-linear}
In this section we will attempt to extend the criteria for
positive energy from linearized theory into the non-linear regime.
As a first step, let us consider the total energy of an
asymptotically flat spacetime in the full non-linear theory.
Integrating (\ref{tsup}) over a spacelike slice in the presence of
non-aether matter gives the total energy
\begin{equation} \int T_{\rm eff}{}_0{}^{0} = \int \partial_\lambda ~^{\rm
tot}U_0{}^{\lambda0} = \oint_{\infty} ~^{\rm tot}U_0{}^{\lambda0}
n_\lambda dS \label{ADM} \end{equation}
where $^{\rm tot}U =~ ^{\rm \ae}U+~^{\rm gr}U$ are the aether and
von-Freud superpotentials, (\ref{aesup}) and (\ref{Einsteinsup}),
and $T_{\rm eff} = t+T$ is total matter and gravitational
energy-momentum. The problem now is to calculate the
superpotentials for the asymptotically flat solutions to the
Einstein-Aether theory. We will use Cartesian coordinates
throughout since these have the required asymptotic behavior
discussed at the end of Section \ref{Einstein}. Therefore, the
surface element is $dS = r^2 d\Omega^2$ and the unit normal is
$(\sqrt{2},x/r,y/r,z/r)$ where $r=\sqrt{x^2+y^2+z^2}$. For
asymptotically flat boundary conditions we will assume that as
$r\rightarrow \infty$
\begin{eqnarray} g_{\mu\nu} = \eta_{\mu\nu}+O(1/r)+ \cdots\\
u^\mu = \underline{u}^\mu + O(1/r) + \cdots \end{eqnarray}
where $\underline{u}^\mu=(1,0,0,0)$ with respect to the Minkowksi
metric $\eta_{\mu\nu} = (1,-1,-1,-1)$. Equation (\ref{ADM}) will
only be affected by terms in the metric and aether up to $O(1/r)$.
Using the analysis of the Newtonian limit \cite{22} and applying
the unit constraint, we find that far from the source in any
asymptotically flat solution
\begin{eqnarray}
g_{00} &=& 1-\frac{r_0}{r}+\cdots \label{g00}\\
g_{ij} &=& -1-\frac{r_0}{r}+\cdots \label{gij}\\
g_{0i} &=& O(1/r^2)+\cdots \label{g0i}\\
u^{t} &=& 1+\frac{r_0}{2r}+\cdots \label{timecomp}\\
u^{i} &=& O(1/r^2)+\cdots \label{spacecomp}.
\end{eqnarray}
The constant value at infinity and $1/r$ fall-off term in the
aether are due to the unit timelike constraint. Thus, unlike
ordinary fields, the aether will contribute to the energy
expression directly. Inserting (\ref{g00})-(\ref{spacecomp}) into
the von-Freud superpotential (\ref{Einsteinsup}) and aether
superpotential (\ref{aesup}) yields the usual `ADM mass'' of GR
\begin{equation} E_{\rm GR} = \frac{1}{16\pi G} \oint_{\infty}
(g_{jk,k}-g_{kk,j}) n_j d^2 S = \frac{r_0}{2G} \end{equation}
and the aether modification
\begin{equation} E_{\rm \ae} = \frac{c_{14}}{8\pi G} \oint_{\infty}
\partial^i u^{t}~ n_i d^2 S = -\frac{c_{14}}{2} \frac{r_0}{2G}. \label{aeADM} \end{equation}
Combining, we find
\begin{equation} E_{\rm tot} = \frac{r_0}{2G}(1-\frac{c_{14}}{2})
\label{aetot}.\end{equation}
This shows that the aether contribution effectively renormalizes
the $r_0/2G$ value we usually find for the total energy of an
asymptotically flat spacetime in GR. This renormalization can also
be understood as a rescaling of Newton's constant of the form
$G_N=G/(1-c_{14}/2)$, which agrees with the result of \cite{22}.
Equation (\ref{aetot}) implies that if $c_{14}<2$ then the total
energy of the Einstein-Aether theory is positive if the ADM mass
$r_0/2G$ is positive. However, the positive energy theorem for GR
\cite{8} requires a stress-tensor that satisfies the dominant
energy condition. The aether stress-tensor (\ref{aetherT}) does
not appear to generally satisfy this condition, so proof of total
positive energy remains elusive. For some speculative thoughts on
modifying the positive energy theorem, see \cite{6}.
Despite these difficulties, there are special cases of the
non-linear theory that are simple enough for calculations of the
energy, yet still give important results. One sector of interest
is the non-linear decoupled limit. As discussed above in Section
\ref{Linear} this limit allows one to essentially replace $g_{ab}$
with the flat Minkowski metric $\eta_{ab}$ in the aether parts of
(\ref{action}). One significant example is $c_2=c_3=c_4=0, c_1 \ne
0$ theory. In this case the Lagrangian density for the aether is
\begin{equation} L = c_1 \eta^{ab} \eta_{mn}~ \partial_a u^m \partial _b u^n +
\lambda(u^2-1) \end{equation}
This corresponds to a nonlinear sigma model on the unit
hyperboloid, which has a stress tensor satisfying the dominant
energy condition. A simple way to see this is to note that the
derivatives of the individual scalar components $u^{\mu}$ and are
contracted with $\eta_{\mu \nu}$, which is positive definite on
the unit hyperboloid. Returning to the linearized plane wave
energy densities of Section \ref{Linear} we see that in this
special case of (\ref{TT})-(\ref{trace}), if $0 < c_{1} < 1$
energy is positive in both the linearized and decoupled non-linear
regimes of the theory.
Another important application of the decoupling limit relevant for
our analysis of energy is the work of Clayton \cite{23}. Clayton
examined the Maxwell-like simplified theory where $c_1 = -c_3,
~c_2=c_4=0$ in the decoupled version of non-linear Lagrangian
(\ref{action}), yielding
\begin{equation} L = \int d^3x \{{\textstyle{\frac{1}{2}}}(\partial_t u_i-\partial_i
u_0)^2-{\textstyle{\frac{1}{4}}} F_{ij}^2+{\textstyle{\frac{1}{2}}}
\lambda(u_0^2-\overrightarrow{u}^2-1)\}. \end{equation}
where $F_{\mu\nu} = \partial_\mu u_\nu - \partial_\nu u_\mu$. The
standard calculation of the Hamiltonian and the constraint
equations then produces the following on-shell value for the
Hamiltonian
\begin{equation} H = \int d^3x \{{\textstyle{\frac{1}{2}}} \overrightarrow{P}^2 + P^i\partial_i
u_0 +{\textstyle{\frac{1}{4}}} (F_{ij})^2\}. \end{equation}
Unlike the electromagnetic case, the second term cannot be turned
into a total divergence since now $\nabla \cdot \overrightarrow{P}
= -\lambda u_0$ on-shell. This implies that for some solutions the
value of the Hamiltonian is negative. For example, as initial data
choose $u_i$ to be the gradient of a scalar field and
$P_i=-\partial_i u_0$. Evaluating the Hamiltonian then yields
\begin{equation} E= -1/2 (\partial_i u_0)^2, \label{negener} \end{equation}
which can be made arbitrarily negative by an appropriate choice of
$u_0$.
Moreover, as Clayton points out, the negative energies are not
restricted to this special case. In particular, allowing $c_2 \ne
0$ does not affect the $\overrightarrow{P} \cdot
\overrightarrow{\partial} u_0$ term in the Hamiltonian and even
produces additional questionable terms. The indefinite nature of
the decoupled Hamiltonian contrasts with with the wave energy
densities of Section \ref{Linear}, which clearly can be made
positive definite in the Maxwell-like case. The key point is that
the wave results are in the linearized theory and associated with
quadratic parts of the Hamiltonian, while the indefinite terms
appear at higher orders. For example, the linearized constraint
equation $v^0 = 0$ eliminates $\overrightarrow{P} \cdot
\overrightarrow{\partial} u_0$ from the Maxwell-like Hamiltonian
and forces the $u_0$ in (\ref{negener}) to be quadratic or higher
in the perturbation. Thus, the indefinite pieces begin to appear
at quartic order in the Hamiltonian. This indefiniteness at higher
orders implies that the decoupled, linearized results of Lim and
the ``coupled", linearized analysis of this paper generally do not
detect possible energies of arbitrary sign in the fully non-linear
decoupled Einstein-Aether theory.
\section{Discussion}
In this paper we have derived two energy-momentum pseudotensor
expressions for the Einstein-Aether theory and used them to
compute the energy densities of weak gravitational waves and the
total energy of an asymptotically flat solution. The constraints
of Section \ref{Linear} show that a sector of this LV model
satisfies the important theoretical condition of positive energy
in the linearized case. However, a remaining open question is
whether the energy remains positive when we consider the full
non-linear theory. We have argued that in the decoupled limit the
$c_1 \ne 0 $ non-linear sigma model is immune to the sickness of
energies of indefinite sign. However, other special cases of the
coupling constants yield negative energy solutions even when the
linearized theory has positive energy. A complete answer to the
question of positivity of energy in the non-linear theory is not
yet in hand.
The recipe for the Weinberg pseudotensor discussed in Section
\ref{Weinberg} and the Einstein superpotential and pseudotensor
derived in Section \ref{Einstein} also has applications in
studying the emission of gravitational-aether radiation from
astrophysical sources. In \cite{21} the analog of the quadrupole
formula (which also involves monopole and dipole moments), is
obtained using a pseudotensor expression derived from the related
Noether charge approach. This expression is then used to track
radiative energy losses and study constraints on the model.
\section*{Acknowledgements}
The author thanks Ted Jacobson for valuable discussions and
editing advice and Albert Roura for comments on an earlier draft.
This work was supported in part by the NSF under grant PHY-0300710
at the University of Maryland.
\section*{Appendix}\label{Appendix}
\renewcommand{\arabic{section}.\arabic{equation}}{A.\arabic{equation}}
\setcounter{equation}{0}
\subsection*{Background}
\label{background}
In this appendix we will derive the Einstein psuedotensor and
superpotential using the Noether current formalism of Julia and
Silva \cite{18} applied to Lagrangians that depend on the fields
and their first and second derivatives. We can write a variation
in the Lagrangian as
\begin{equation} \delta L = \frac{\partial L}{\partial \psi} \delta
\psi+\frac{\partial L}{\partial(\partial_\mu \psi)}
\delta(\partial_\mu \psi)+\frac{\partial L}{\partial(\partial_\mu
\partial_\nu \psi)} \delta(\partial_\mu
\partial_\nu \psi) \label{var} \end{equation}
and then integrate by parts to isolate the equations of motion $E$
and a symplectic current $\theta^\mu$,
\begin{equation} \delta L = E \delta \psi + \partial_\mu \theta^\mu,
\end{equation}
where
\begin{eqnarray} \partial_\mu \theta^\mu &=& \partial_\mu \left(\frac{\partial
L}{\partial(\partial_\mu \psi)}\delta
\psi-\partial_\nu(\frac{\partial L}{\partial(\partial_\mu
\partial_\nu \psi)})\delta \psi\right. \nonumber\\&& \left.+ \frac{\partial L}{\partial(\partial_\mu \partial_\nu \psi)}\partial_\nu(\delta
\psi)\right). \label{cons1}\end{eqnarray}
If the action associated with $L$ is invariant under a continuous
transformation of the fields, $\delta L = \partial_\mu S^\mu$.
Thus, we have the equation
\begin{equation} \partial_\mu (S^\mu-\theta^\mu) = E \delta \psi.
\label{varL}\end{equation}
This identifies the on-shell ($E=0$) conserved Noether current,
\begin{eqnarray} J^u &=& \theta^\mu-S^\mu = \frac{\partial
L}{\partial(\partial_\mu \psi)}\delta
\psi-\partial_\nu(\frac{\partial L}{\partial(\partial_\mu
\partial_\nu \psi)})\delta \psi \nonumber\\&&+ \frac{\partial L}
{\partial(\partial_\mu \partial_\nu \psi)}\partial_\nu(\delta
\psi)-S^\mu. \label{consvcurrent} \end{eqnarray}
We now want to consider a gauge transformation of the fields that
involves derivatives of the generator $\xi^A(x)$. Here we will
focus on the special case restricting attention to only the first
derivative. Following the analysis and notation of \cite{18} we
parameterize the gauge transformation as
\begin{equation} \delta \psi = \xi^A \Delta_A + (\partial_\nu \xi^A)
\Delta^\nu_{A} \label{deltapsi}\end{equation}
where $A$ is an internal or spacetime index and $\Delta$ is a
transformation matrix. The quantity $S^\mu$ can be expressed
similarly as
\begin{equation} S^\mu = \xi^A \Sigma^{\mu}_{A}+(\partial_\nu \xi^A)
\Sigma^{\mu \nu}_{A}+(\partial_\tau \partial_\nu \xi^A)
\Sigma^{\mu(\tau \nu)}_{A}.\label{surface}\end{equation}
Inserting these forms into (\ref{consvcurrent}) and combining
terms, we find that on-shell
\begin{equation} \partial_\mu\left(\xi^A J^\mu_{A}+\partial_\nu \xi^A U^{\mu
\nu}_{A}+\partial_\tau \partial_\nu \xi^A V^{\mu(\nu
\tau)}_{A}\right) = 0, \label{localcons} \end{equation}
where
\begin{eqnarray}
J^{\mu}_{A} &=& \left(\frac{\partial L}{\partial(\partial_\mu
\psi)}-\partial_\nu(\frac{\partial L}{\partial(\partial_\mu
\partial_\nu \psi)})\right) \Delta_{A}
\nonumber\\&&+\left(\frac{\partial L}{\partial(\partial_\mu \partial_\nu \psi)}\right)\partial_\nu \Delta_{A}-\Sigma ^{\mu}_A \label{Jeqn}\\
U^{\mu\nu}_{A} &=& \left(\frac{\partial L}{\partial(\partial_\mu
\psi)}-\partial_\tau(\frac{\partial L}{\partial(\partial_\mu
\partial_\tau \psi)})\right)
\Delta^{\nu}_{A}\nonumber\\
&&+\left(\frac{\partial L}{\partial(\partial_\mu
\partial_\tau \psi)}\right)\partial_\tau \Delta^\nu_{A}
+\frac{\partial L}{\partial(\partial_\mu \partial_\nu
\psi)}\Delta_A\nonumber\\&&
-\Sigma ^{\mu\nu}_A \label{Ueqn}\\
V^{\mu(\tau\nu)}_{A} &=& \frac{\partial L}{\partial(\partial_\mu
\partial_\tau \psi)} \Delta^{\nu}_{A}-\Sigma^{\mu(\tau \nu)}_{A}.
\label{Veqn}
\end{eqnarray}
Since $\xi^A$ and its derivatives should be arbitrary and
independent, this single equation decomposes into 4 equations
\begin{eqnarray}
\partial_\mu J^\mu_{A} &\approx& 0 \label{cascade1}\\
J^\mu_{A}+\partial_\nu U^{\nu\mu}_{A} &\approx& 0 \label{cascade2}\\
U^{(\mu\nu)}_{A}+\partial_\tau V^{\tau(\mu \nu)}_{A} &=& 0 \label{cascade3}\\
V^{(\mu\nu\tau)}_{A} &=& 0\label{cascade4}.
\end{eqnarray}
The first two equations hold on-shell, while the last two are
identities (since there are no second or third derivatives of
$\xi^A$ on the right hand side of (\ref{cons1})).
The gauge symmetry implies that $J^\mu_{A}$ is conserved and equal
to the divergence of the superpotential $U^{\mu\nu}_{A}$. Since
$\xi^A = \xi^{A}(x)$, the Noether current $J^{\mu}$
(\ref{consvcurrent}) will now be parameter dependent. Let us
consider a one parameter subgroup of the local gauge or
diffeomorphism symmetry where $\xi^A$ has the decomposition,
\begin{equation} \xi^{A}(x) = \epsilon(x) \xi^{A}_0, \label{Jpsi} \end{equation}
and $\xi^{A}_0$ is fixed. Inserting this form into
(\ref{cascade2}) produces
\begin{equation} \xi^{A}_0 J^{\mu}_{A}+\partial_{\nu}(\xi^{A}_0
U^{\nu\mu}_{A}) = 0, \label{generalcons} \end{equation}
with $J^{\mu}_{\xi_0} = \xi^{A}_0 J^{\mu}_{A}$. The conserved
charge is
\begin{equation} Q = \int J^{0}_{\xi_0} d^3 x = \oint \xi^{A}_0 U^{\nu 0}_{A}
n_\nu d^2 x \end{equation}
$Q$ depends on the choice of $\xi^A_0$ and can be expressed in
terms of the gauge fields up using (\ref{Ueqn}). If $\xi^A_0$ is
an asymptotic translation in an asymptotically flat spacetime,
then the conserved charge will be a total energy or momentum.
Using the variational principle ($\delta S = 0 \Rightarrow$
equations of motion), we will show in the next section that the
choice of $\xi^A_0$ is subject to certain boundary conditions at
infinity. We now apply this type of Lagrangian analysis to the
Einstein-Aether theory.
\subsection*{Application}
Assume that the Lagrangian density ${\cal L}(\psi,\partial
\psi,\partial^{2}\psi)$ is invariant under diffeomorphisms and is
a combination of a scalar density $\widetilde{{\cal L}}$ and a
total divergence $\partial_\mu W^{\mu}$,
\begin{equation} {\cal L}(\psi,\partial \psi,\partial^{2} \psi) =
\widetilde{{\cal L}}(\psi,\partial \psi,\partial^{2}
\psi)+\partial_\mu [W^{\mu}(\psi,\partial \psi)].
\label{genaction}\end{equation}
If $W^\mu$ is a vector density then the total divergence is a
scalar density, but we allow for a non-covariant total divergence.
For a variation that is an infinitesimal diffeomorphism generated
by a vector field $\xi^\mu$, we have $\delta \widetilde{{\cal L}}
=
\partial_\mu(\xi^\mu \widetilde{{\cal L}})$ since $\widetilde{{\cal L}}$
is a scalar density and $\delta(\partial_\mu
W^\mu)=\partial_\mu(\delta W^\mu)$. Therefore the surface term
$S^\mu$ in \ref{varL} has the form
\begin{equation} S^\mu = \xi^\mu \widetilde{{\cal L}} + \delta W^\mu.
\label{Seqn} \end{equation}
Now consider the Einstein-Hilbert action
\begin{equation} S_{\rm EH} = \int \sqrt{-g}~ R \end{equation}
of pure GR. The Ricci scalar $R$ has a dependence on second
derivatives of the metric. In light of this, Einstein exploited a
property of the Hilbert action that allows it to be separated into
a bulk and a surface term
\begin{equation} \int \sqrt{-g}~ R ~d^4x = \int \sqrt{-g}L_{\rm
bulk}+\partial_{\mu}V^{\mu} d^4x. \end{equation}
This decomposition of the Ricci scalar takes the following form
\begin{eqnarray}
L_{\rm bulk} &=&
g^{\alpha\beta}\{\Gamma^{\eta}_{\alpha\delta}\Gamma^{\delta}_{\eta\beta}
-\Gamma^{\eta}_{\eta\delta}\Gamma^{\delta}_{\alpha\beta}\} \\
V^{\mu} &=& \sqrt{-g}\{\Gamma^{\mu}_{\alpha\beta}g^{\alpha\beta}-
\Gamma^{\beta}_{\alpha\beta}g^{\mu\alpha}\}.
\end{eqnarray}
where $\Gamma$ is the Levi-Civita connection. One can eliminate
the total divergence by adding its negative to the
Einstein-Hilbert action
\begin{equation} \int \sqrt{-g}L_{\rm bulk} = \int \sqrt{-g}~ R ~d^4x -
\partial_{\mu}V^{\mu} d^4x. \end{equation}
The elimination does not affect the equations of motion and is
consistent with the general action (\ref{genaction}) with
$\widetilde{{\cal L}} = \sqrt{-g}~R$ and $W^\mu=-V^\mu$. The
result of this is a loss of diffeomorphism invariance since the
remaining $L_{\rm bulk}$ in the ``$\Gamma^2$" action is not a
scalar. We have allowed for this possibility with the
non-covariant $\partial_\mu W^\mu$ term in (\ref{genaction}).
With ${\cal L}(\psi,\partial \psi,\partial^{2}\psi)$ the bulk part
of the Einstein-Hilbert action plus the the aether terms in
(\ref{action}), we arrive at the Einstein-Aether form of
(\ref{varL}) on shell
\begin{equation} \partial_\mu \left(\frac{\partial
\cal{L}}{\partial(\partial_\mu g_{\alpha\beta})}\delta
g_{\alpha\beta}+\frac{\partial \cal{L}}{\partial(\partial_\mu
u^{\alpha})} \delta u^{\alpha} -S^\mu\right) = 0. \label{consg}
\end{equation}
The second derivative terms in (\ref{consvcurrent}) vanish in this
case. Under a diffeomorphism generated by a vector field $\xi^\nu$
the variation of the metric and the aether is simply the Lie
derivative, $\delta g_{\alpha\beta} = \xi^\nu\partial_\nu
g_{\alpha\beta}+
\partial_\alpha \xi^\nu g_{\nu\beta} +
\partial_\beta \xi^\nu g_{\alpha\nu}$ and $\delta u^{\alpha} =
\xi^{\gamma}\partial_{\gamma} u^{\alpha} -
u^{\gamma}\partial_{\gamma}\xi^{\alpha}$. It follows from
(\ref{Seqn}) that $S^\mu = \xi^\mu
L_{bulk}-\sqrt{-g}(\partial_\delta
\partial_\nu \xi^\mu g^{\delta\nu}-\partial_\delta \partial_\nu \xi^\nu
g^{\mu\delta})$. Inserting these forms into (\ref{consg}) produces
\begin{eqnarray}
t_{\nu}{}^\mu &=& \sqrt{-g}\left(\frac{\partial
L}{\partial(\partial_\mu g_{\alpha\beta})} \partial_\nu
g_{\alpha\beta} + \frac{\partial L}{\partial(\partial_\mu
u^{\alpha})} \partial_{\nu} u^{\alpha}\right. \nonumber\\&& \left.
- \delta^\mu_\nu
L\right)\\
U_{\nu}{}^{\mu\gamma} &=& \sqrt{-g}\left(\frac{\partial
L}{\partial(\partial_\mu
g_{\alpha\beta})}(\delta^{\gamma}_{\alpha}g_{\nu\beta}+\delta^{\gamma}_{\beta}g_{\nu\alpha})\right.
\nonumber\\&& \left.-\frac{\partial L}{\partial(\partial_\mu
u^{\alpha})} \delta^{\alpha}_{\nu} u^{\gamma}\right)
\\
V_{\nu}{}^{\mu(\gamma\lambda)} &=& \sqrt{-g}\left(
\delta^{\mu}_{\nu}g^{\lambda\gamma}-\frac{1}{2}\delta^{\gamma}_{\nu}g^{\lambda\mu}-\frac{1}{2}\delta^{\lambda}_{\nu}g^{\gamma\mu}
\right)
\end{eqnarray}
as coefficients of $\xi^{\mu}$, $\partial \xi^{\mu}$ and
$\partial^2 \xi^{\mu}$ respectively. $t_{\nu}{}^{\mu}$ and
$U_{\nu}{}^{\mu\gamma}$, and $V_{\nu}{}^{\mu(\gamma\lambda)}$ are
the analogs of $J^{\mu}_{A}$, $U^{\mu\nu}_{A}$,
$V^{\mu(\gamma\lambda)}_{A}$ in (\ref{Jeqn})-(\ref{Veqn}). The
resulting equations due to the arbitrariness and independence of
the derivatives of $\xi^{\mu}$ are
\begin{eqnarray}
\partial_\mu t_\nu{}^\mu &\approx& 0\\
t_\nu{}^\mu &\approx& -\partial_\gamma U_\nu{}^{\gamma\mu}\\
U_{\nu}{}^{(\gamma\mu)} + \partial_\lambda
V_{\nu}{}^{\lambda(\mu\gamma)} &=& 0\\
V_{\nu}{}^{(\lambda\nu\gamma)} &=& 0.
\end{eqnarray}
Following (\ref{Jpsi}), we can keep the $\xi^\nu$ vector fixed
(and determine it later for each conserved charge) by choosing
$\xi^\nu = \epsilon(x) \xi^\nu_0$. The main result, as before, is
\begin{equation} \xi^\nu_{0} t_{\nu}{}^{\mu} = -\partial_\gamma(\xi^\nu_{0}
U_{\nu}{}^{\gamma\mu}) \label{Eeqn}\end{equation}
showing that a Noether charge is again obtained as a surface term.
Einstein effectively chose the $\xi^\nu_{0}$ vector to be a
constant in (\ref{Eeqn}), reducing the pseudotensor to a form
consistent with the flat spacetime canonical stress tensor
(\ref{flatcan}). However, this choice is not inconsequential. The
variational principle for the $\Gamma^2$ action requires the
vanishing of the surface term in the asymptotic region,
\begin{equation} \int_{S_{\infty}} \frac{{\partial \cal
L}}{\partial(\partial_\mu g_{\alpha \beta})} \delta g_{\alpha \beta} +
\frac{{\partial \cal L}}{\partial(\partial_\mu u^{\alpha})} \delta
u^\alpha . \end{equation}
Therefore we have Dirichlet boundary conditions $\delta
g_{\alpha\beta}=0$ and $\delta u^\alpha = 0$ on the metric and the aether at
infinity. Inserting the Lie derivatives for the variations above,
we see that as $r\rightarrow \infty$
\begin{eqnarray}
\nabla_{(\alpha} \xi_{\beta)} \rightarrow 0\\
{\cal L}_{_\xi} u^{\alpha} \rightarrow 0.
\end{eqnarray}
Since $\xi^\nu$ has been chosen to be constant everywhere and
$u^\alpha$ is asymptotically constant, the connection coefficients
must vanish as one approaches spatial infinity. Thus, one must
compute the pseudotensor and superpotential in a coordinate system
where the connection vanishes asymptotically as $O(1/r)$ or
faster.
|
1,314,259,995,849 | arxiv | \section{Introduction}
Brou\'e, Malle and Michel have shown that the automizer
of an abelian Sylow $p$-subgroup $P$ in a finite simple Chevalley group is an
irreducible complex reflection
group (for $p$ not too small and different from the defining characteristic)
\cite{BrMaMi,BrMi}.
The aim is this note is to show that a suitable version of this property
holds for general finite groups.
\smallskip
We give a simple direct proof, building on
the Lehrer-Springer theory \cite{LeSp}, that
the property above holds for simply connected simple
algebraic groups $G$, provided $p$ is not a torsion prime
(Proposition \ref{pr:reflChevalley}): the automizer
$E=N_G(P)/C_G(P)$ is a reflection group on $\Omega_1(P)$, the largest
elementary abelian subgroup of $P$.
On the other hand, we show that the presence of $p$-torsion in the
Schur multiplier of a finite group $G$ prevents the subgroup of $E$
generated by reflections from being irreducible (Proposition
\ref{pr:notirreducible}).
\smallskip
This suggests considering covering groups of finite simple groups or
equivalently finite simple groups $G$ such that $H^2(G,{\mathbf{F}}_p)=0$.
We also need to allow $p'$-automorphisms and we now
look for a description of the automizer
as an extension of an irreducible reflection group $W$ by
a subgroup of $N_{\operatorname{GL}\nolimits(\Omega_1(P))}(W)/W$.
We actually need a slight generalization:
$\Omega_1(P)$ should be viewed in some cases as a vector space over
a larger finite field (for example in the case of $\operatorname{PSL}\nolimits_2({\mathbf{F}}_{p^n})$)
and we need to allow field automorphisms.
As an example, the automizer of an $11$-Sylow subgroup in the
Monster is
the $2$-dimensional complex reflection group $G_{16}$.
\smallskip
I thank Richard Lyons and Geoff Robinson for useful discussions.
\section{Notation and definitions}
Let $p$ be a prime.
Given $P$ an abelian group, we denote by $\Omega_1(P)$ the
subgroup of $P$ of elements of order $1$ or $p$, {\em i.e.}, the largest elementary
abelian $p$-subgroup of $P$.
Let $V$ be a free module of finite rank over a commutative algebra $K$. A
{\em reflection} is an element $s\in\operatorname{GL}\nolimits_K(V)$ of finite order such that
$V/\ker(s-1)$ is a free $K$-module of rank $1$ (note that we do not
require $s^2=1$). A finite subgroup of
$\operatorname{GL}\nolimits_K(V)$ is a {\em reflection group} if it is generated by reflections.
\section{Main result and remarks}
Let $p$ be a prime and $H$ a simple group such that the $p$-part
of the Schur multiplier of $H$ is trivial, {\em i.e.}\ $H^2(H,{\mathbf{F}}_p)=0$.
Assume $H$ has an abelian Sylow $p$-subgroup $P$.
Let ${\tilde{H}}\le\operatorname{Aut}\nolimits(H)$ be a finite group containing $H$ and such that
${\tilde{H}}/H$ is a Hall $p'$-subgroup of $\operatorname{Out}\nolimits(H)$. Let $E=N_{\tilde{H}}(P)/C_{\tilde{H}}(P)$.
\begin{thm}
\label{th:main}
There is
\begin{itemize}
\item a finite field $K$
\item an ${\mathbf{F}}_p$-subspace $V$ of $\Omega_1(P)$ and
an isomorphism of ${\mathbf{F}}_p$-vector spaces
$K\otimes_{{\mathbf{F}}_p}V\buildrel \sim\over\to \Omega_1(P)$ endowing $\Omega_1(P)$ with a structure
of $K$-vector space
\item a subgroup $N$ of $\operatorname{GL}\nolimits_K\bigl(\Omega_1(P)\bigr)$ and
\item a subgroup $\Gamma$ of $\operatorname{Aut}\nolimits(K)$
\end{itemize}
such that $E=N\rtimes\Gamma$, as subgroups of $\operatorname{Aut}\nolimits\bigl(\Omega_1(P)\bigr)$, and
such that the normal subgroup $W$ of $N$ generated by reflections
acts irreducibly on $\Omega_1(P)$.
\end{thm}
The theorem will be proven in \S \ref{se:proof}.
\begin{rem}
Gorenstein and Lyons have shown that $N_H(P)/C_H(P)$ acts irreducibly
on $\Omega_1(P)$ viewed as a vector space over ${\mathbf{F}}_p$ and, as a consequence,
$P$ is homocyclic \cite[(12.1)]{GoLy}.
Note nevertheless that the subgroup of $N_H(P)/C_H(P)$ generated by reflections
might not be irreducible
in its action on $\Omega_1(P)$: this happens for example in
the case $H={\mathfrak{A}}_{2p}$, $p>3$.
\end{rem}
We can take $K={\mathbf{F}}_p$ in Theorem \ref{th:main}, except for
\begin{itemize}
\item $\operatorname{PSL}\nolimits_2(p^n)$, $n>1$: $K={\mathbf{F}}_{p^n}$
\item $J_1$ and ${^2G}_2(q)$, $p=2$: $K={\mathbf{F}}_8$.
\end{itemize}
In those cases, $V={\mathbf{F}}_p$ and $P=\Omega_1(P)=K$.
\smallskip
Note that the theorem is trivial when $P$ is cyclic: one takes
$K={\mathbf{F}}_p$ and $N=E=W\subset {\mathbf{F}}_p^\times$.
\medskip
Using the classification of finite simple groups, we deduce a statement
about finite groups with abelian Sylow $p$-subgroups.
\begin{cor}
\label{co:general}
Let $G$ be a finite group with an abelian Sylow $p$-subgroup $P$.
Let $H=O^{p'}(G/O_{p'}(G))$.
Assume the $p$-part of the Schur multiplier of $H$ is trivial. Then,
there is a finite group $X$ containing $H$ as a normal subgroup of
$p'$-index and
\begin{itemize}
\item a product $K$ of finite field extensions of ${\mathbf{F}}_p$
\item an ${\mathbf{F}}_p$-subspace $V$ of $\Omega_1(P)$ and
an isomorphism of ${\mathbf{F}}_p$-vector spaces
$K\otimes_{{\mathbf{F}}_p}V\buildrel \sim\over\to \Omega_1(P)$ endowing $\Omega_1(P)$ with a structure
of a free $K$-module
\item a subgroup $N$ of $\operatorname{GL}\nolimits_K\bigl(\Omega_1(P)\bigr)$ and
\item a subgroup $\Gamma$ of $\operatorname{Aut}\nolimits(K)$
\end{itemize}
such that $N_X(P)/C_X(P)=N\rtimes\Gamma$, as subgroups of
$\operatorname{Aut}\nolimits\bigl(\Omega_1(P)\bigr)$, and
such that denoting by $W$ the normal subgroup of $N$ generated by reflections,
we have $\Omega_1(P)^W=1$.
\end{cor}
\begin{proof}
The case where $H$ is simple is Theorem \ref{th:main}.
In general, the classification of finite simple groups shows that
there are finite simple groups $H_1,\ldots,H_r$ such that
$H=H_1\times\cdots\times H_r$ (cf. eg \cite[\S 5]{FoHa}).
Note that $O_p(H)=1$, {\em i.e.}, there is no
non-trivial $p$-group as a direct factor of $H$, since $H^2(H,{\mathbf{F}}_p)=0$.
Now, we take $X=X_1\times\cdots\times X_r$,
where the $X_i$ are associated with $H_i$. We put
$K=K_1\times\cdots\times K_r$, etc.
\end{proof}
\medskip
Following
\cite[Proof of (12.1)]{GoLy},
we give now the list of possible finite simple groups $H$ and primes $p$
such that Sylow $p$-subgroups of $H$ are abelian non-cyclic
and the $p$-part of the Schur multiplier of $H$ is trivial. In the first
case, instead of providing the group $H$, we provide a group $G$
such that $H\le G/O_{p'}(G)\le \operatorname{Aut}\nolimits(H)$ and $p{\not|}\ [G/O_{p'}(G):H]$.
\begin{itemize}
\item $G={\mathbf{G}}^F$ where ${\mathbf{G}}$ is a simply connected simple algebraic group
and $F$ is an endomorphism of ${\mathbf{G}}$, a power of
which is a Frobenius endomorphism
defining a rational structure over a finite field with $q$ elements,
$p{\not|}q$ and $p$ is not a torsion prime for ${\mathbf{G}}$
\item $H={\mathfrak{A}}_n$ and $n<p^2$
\item $H=\operatorname{PSL}\nolimits_2(p^n)$
\item $H={^2G}_2(q)$, $p=2$
\item $H$ is sporadic
\end{itemize}
Assume $K={\mathbf{F}}_p$. We have $V=\Omega_1(P)$ and $\Gamma=1$. Furthermore,
$N=E\subset N_{\operatorname{GL}\nolimits(P)}(W)$. So, in this case, the theorem is equivalent
to the statement that $W$ acts irreducibly on $P$. As a consequence,
in order to show that the theorem holds, it is enough
to prove the statement with ${\tilde{H}}$
replaced by a group $G$ as above.
\begin{rem}
The finite simple groups with an abelian Sylow $p$-subgroup such that
the $p$-part of the Schur multiplier is non-trivial are the following
(cf \cite{Atl}):
\begin{itemize}
\item $H=M_{22},ON,{\mathfrak{A}}_6,{\mathfrak{A}}_7$ and $p=3$
\item $H=\operatorname{PSL}\nolimits_2(q)$, $q\equiv 3,5 \pmod 8$ and $p=2$
\item $H=\operatorname{PSL}\nolimits_3(q)$ and $3|q-1$ or $H=\operatorname{PSU}\nolimits_3(q)$ and $3|q+1$ (here $p=3$)
\end{itemize}
Note that the automizer of a Sylow $3$-subgroup $P$ in $\operatorname{Aut}\nolimits(ON)=ON.2$ does
not contain any reflection (when $P$ is viewed as a vector space over
${\mathbf{F}}_3$). That automizer is not a subgroup of $\operatorname{GL}\nolimits_2(9).2$ (extension
by the Frobenius).
\end{rem}
Note that the presence of $p$-torsion in the Schur multiplier is an
obstruction to the irreducibility of
the subgroup of the automizer generated by reflections on
$\Omega_1(P)$, viewed as a vector space over ${\mathbf{F}}_p$.
\begin{prop}
\label{pr:notirreducible}
Let $G$ be a finite group with an abelian Sylow $p$-subgroup $P$.
Let $E=N_G(P)/C_G(P)$ and let $W$ be the subgroup of $E$ generated by
reflections on $\Omega_1(P)$, viewed as an ${\mathbf{F}}_p$-vector space.
Assume $p>2$.
If $H^2(G,{\mathbf{F}}_p)\not=0$, then $\Omega_1(P)^W\not=0$.
\end{prop}
\begin{proof}
Let $V=\Omega_1(P)^*$. We have
$H^2(G,{\mathbf{F}}_p)\simeq H^2(N_G(P),{\mathbf{F}}_p)\simeq H^2(P,{\mathbf{F}}_p)^E$.
On the other hand, we have an isomorphism of ${\mathbf{F}}_p E$-modules
$H^2(P,{\mathbf{F}}_p)\buildrel \sim\over\to V\oplus\Lambda^2(V)$, so
$H^2(G,{\mathbf{F}}_p)\simeq V^E\oplus \Lambda^2(V)^E\subset V^W\oplus \Lambda^2(V)^W$.
By Solomon's Theorem \cite{So}, we have $\Lambda^2(V)^W\simeq \Lambda^2(V^W)$.
The result follows.
\end{proof}
\begin{rem}
Let $W$ be a reflection group on a complex vector space $L$, with minimal field
of definition $K$.
The subgroup of the outer automorphism group of $W$ of elements fixing
the set of reflections
has always a decomposition as a semi-direct
product $(N_{\operatorname{GL}\nolimits(L)}(W)/W)\rtimes \operatorname{Gal}\nolimits(K/{\mathbf{Q}})$
as shown by Marin and Michel \cite{MaMi}.
\end{rem}
\begin{rem}
It would be interesting to investigate if there is a version of Theorem
\ref{th:main} for non-principal blocks with abelian defect groups.
In a work in progress, we study automizers of maximal elementary abelian
$p$-subgroups in covering groups of simple groups.
\end{rem}
\section{Proof of Theorem \ref{th:main}}
\label{se:proof}
We run through the list of groups $H$ (or $G$) as described above.
\subsection{Chevalley groups}
Let ${\mathbf{G}}$ be a connected and simply connected reductive
algebraic group over an algebraic closure $k$ of a finite field
and endowed with an endomorphism $F$, a power of which is a Frobenius
endomorphism.
Let $G={\mathbf{G}}^F$.
Assume $p$ is invertible in $k$ and $p$ is not a torsion prime for ${\mathbf{G}}$.
\subsubsection{Abelian $p$-subgroups}
Since $p$ is not a torsion prime for ${\mathbf{G}}$,
every abelian $p$-subgroup $Q$ of $G$ is contained in an $F$-stable
maximal torus ${\mathbf{T}}$ of ${\mathbf{G}}$
and ${\mathbf{L}}=C_{{\mathbf{G}}}(Q)$ is a Levi subgroup
(\cite[Corollary 5.10 and Theorem 5.8]{SpSt} and \cite[Proposition 2.1]{GeHi}).
Furthermore, $N_{{\mathbf{G}}}(Q)=N_G(Q)C_{{\mathbf{G}}}(Q)$ \cite[Corollary 5.10]{SpSt}, hence
$N_{{\mathbf{G}}}(Q)/C_{{\mathbf{G}}}(Q)=N_G(Q)/C_G(Q)$.
\smallskip
Let $W=N_{{\mathbf{G}}}({\mathbf{T}})/{\mathbf{T}}$, $X=\operatorname{Hom}\nolimits({\mathbf{T}},{\mathbf{G}}_m)$ and $Y=\operatorname{Hom}\nolimits({\mathbf{G}}_m,{\mathbf{T}})$.
If ${\mathbf{G}}$ is simple, then the action of $W$ on ${\mathbf{C}}\otimes_{\mathbf{Z}} X$ is
irreducible.
We have a canonical map
$N_W(Q)\to N_{{\mathbf{G}}}(Q)/{\mathbf{T}}$. Since
${\mathbf{L}}\subset N_{{\mathbf{G}}}(Q)\subset N_{{\mathbf{G}}}({\mathbf{L}})$, we obtain an isomorphism
$$N_W(Q)/C_W(Q)\buildrel \sim\over\to N_{{\mathbf{G}}}(Q)/C_{{\mathbf{G}}}(Q).$$
\smallskip
Given $L$ an abelian group, we denote by $L_{p^\infty}$ the subgroup of
$p$-elements of $L$.
Let $\mu=k^\times$.
We have an isomorphism
$${\mathbf{T}}_{p^\infty}\buildrel \sim\over\to \operatorname{Hom}\nolimits(X,\mu_{p^\infty}),\
t\mapsto (\chi\mapsto \chi(t)).$$
This provides an isomorphism
$${\mathbf{T}}_{p^\infty}\buildrel \sim\over\to Y\otimes_{{\mathbf{Z}}}\mu_{p^\infty}.$$
These isomorphisms are equivariant for the actions of $W$ and $F$.
\subsubsection{Abelian Sylow $p$-subgroups}
Assume now $P=Q$ is a abelian Sylow $p$-subgroup of $G$.
Let $V=Y\otimes_{\mathbf{Z}}{\mathbf{F}}_p$. We have $V^F\simeq\Omega_1(P)$.
\begin{prop}
\label{pr:reflChevalley}
The group $N_W(P)/C_W(P)$ is a reflection group on $\Omega_1(P)$.
If ${\mathbf{G}}$ is simple, then this reflection group is irreducible.
\end{prop}
\begin{proof}
Note that $N_W(P)/C_W(P)$ is a $p'$-group, since $P$ is an abelian
Sylow $p$-subgroup of $G$ and $N_W(P)/C_W(P)\simeq N_G(P)/C_G(P)$. So,
the canonical map is an isomorphism
$$N_W(P)/C_W(P)\buildrel \sim\over\to N_W(\Omega_1(P))/C_W(\Omega_1(P)).$$
The proposition follows now from the next lemma by Lehrer-Springer theory
\cite{LeSp} extended to positive characteristic \cite{Rou}.
\end{proof}
\begin{lemma}
We have
$\dim V^F\ge \dim V^{wF}$ for all $w\in W$.
\end{lemma}
\begin{proof}
Let $\dot{w}\in N_{\mathbf{G}}({\mathbf{T}})$.
By Lang's Lemma, there is $x\in{\mathbf{G}}$ such that
$\dot{w}=x^{-1}F(x)$. Given $t\in{\mathbf{T}}$, we have $F(xtx^{-1})=x\dot{w}F(t)
\dot{w}^{-1}$. So, $x{\mathbf{T}} x^{-1}$ is $F$-stable and
the isomorphism
$${\mathbf{T}}\buildrel \sim\over\to x{\mathbf{T}} x^{-1},\ t\mapsto xtx^{-1}$$
transfers the action of $wF$ on the left to the action of $F$ on the right.
So,
$$V^{wF}\simeq \bigl(Y(x{\mathbf{T}} x^{-1})\otimes{\mathbf{F}}_p\bigr)^F\simeq
\Omega_1\bigl((x{\mathbf{T}} x^{-1})^F\bigr).$$
The rank of that elementary abelian $p$-subgroup of $G$ is at most the rank
of $P$ and we are done.
\end{proof}
\subsection{Alternating groups}
Let $G={\mathfrak{S}}_n$, $n>7$. Put $n=pr+s$ with $0\le s\le p-1$ and
$r<p$. We have $P\simeq ({\mathbf{Z}}/p)^r$. We put $K={\mathbf{F}}_p$,
$N=W={\mathbf{F}}_p^\times \wr {\mathfrak{S}}_r$.
\begin{rem}
Note that when $n=5$ and $p=2$ or $n=6,7$ and $p=3$,
the $p$-part of the Schur multiplier is not trivial but the description
above is still valid. Note though that when
$n=6$ and $p=3$, then $G$ contains ${\mathfrak{S}}_6$ as a subgroup of index
$2$. We have $K={\mathbf{F}}_3$, $P=K^2$, $N=E$, $W$ is
a Weyl group of type $B_2$ and $[N:W]=2$.
\end{rem}
\subsection{$\operatorname{PSL}\nolimits_2$}
Assume $H=\operatorname{PSL}\nolimits_2(K)$ for a finite field $K$ of characteristic $p$. We have
$W=N=K^\times$ and $\Gamma=\operatorname{Gal}\nolimits(K/{\mathbf{F}}_p)$.
\subsection{${^2G}_2(q)$}
Assume $H={^2G}_2(q)$ and $p=2$. We have
$K={\mathbf{F}}_8$, $W=N=K^\times$ and $\Gamma=\operatorname{Gal}\nolimits(K/{\mathbf{F}}_2)$.
\subsection{Sporadic groups}
We refer to \cite{BrMaRou} for the diagrams for complex reflection groups.
For sporadic groups, we have $P=\Omega_1(P)$.
$$\begin{array}{|c|c|c|c|c|c|c|}
\cline{1-7}
{\tilde{H}} & K & \dim_K(P) & W & N/W & \Gamma & \text{diagram of W} \\
\cline{1-7}
J_1 & {\mathbf{F}}_8 & 1 & {\mathbf{F}}_8^\times & 1 & \operatorname{Gal}\nolimits({\mathbf{F}}_8/{\mathbf{F}}_2) &
{\small \xy (0,0) *++={7} *\frm{o} \endxy} \\
\cline{1-7}
M_{11},M_{23},HS.2 & {\mathbf{F}}_3 & 2 & B_2 & 2 & 1 &
\xy
(0,0) *\cir<6pt>{}="0",
(13,0) *\cir<6pt>{}="1",
"0";"1" **\dir2{-},
\endxy
\\
\cline{1-7}
J_2.2,Suz.2 & {\mathbf{F}}_5 & 2 & G_2 & 2 & 1 &
\xy
(0,0) *\cir<6pt>{}="0",
(13,0) *\cir<6pt>{}="1",
"0";"1" **\dir3{-},
\endxy
\\
\cline{1-7}
He.2,Fi_{22}.2,Fi_{23},Fi_{24} & {\mathbf{F}}_5 & 2 & G_8 & 1 & 1 &
{\small \xy (0,0) *++={4} *\frm{o} ; (13,0) *++={4} *\frm{o} **@{-}
\endxy} \\
\cline{1-7}
Co_1 & {\mathbf{F}}_7 & 2 & G_5 & 1 & 1 &
{\small \xy (0,0) *++={3} *\frm{o} ; (13,0) *++={3} *\frm{o} **@{=} \endxy} \\
\cline{1-7}
Th, BM & {\mathbf{F}}_7 & 2 & G_5 & 2 & 1 &
{\small \xy (0,0) *++={3} *\frm{o} ; (13,0) *++={3} *\frm{o} **@{=} \endxy} \\
\cline{1-7}
M & {\mathbf{F}}_{11} & 2 & G_{16} & 1 & 1 &
{\small \xy (0,0) *++={5} *\frm{o} ; (13,0) *++={5} *\frm{o} **@{-} \endxy} \\
\cline{1-7}
\end{array}$$
|
1,314,259,995,850 | arxiv | \subsection*{Acknowledgements.} This research has been supported by the \emph{École Normale Supérieure}, Paris, by the \emph{``Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni dell’Istituto Nazionale di Alta Matematica F. Severi''} and by the project \emph{Vain-Hopes} within the program \emph{Valere} of \emph{Università degli Studi della Campania ``Luigi Vanvitelli''}. It has also been partially accomplished within the \emph{UMI} Group \emph{TAA ``Approximation Theory and Applications''}.
E.~D’Aniello would like to thank the \emph{École Normale Supérieure}, Paris, for its warm hospitality in March-April, 2022; L.~Moonens would like to thank the \emph{Dipartimento di Matematica e Fisica} of the \emph{Università degli Studi della Campania ``Luigi Vanvitelli''} for its warm hospitality in April-May, 2022.
\bibliographystyle{plain}
\section{Introduction}
Given a measure space $(\Omega,\mathscr{F},\mu)$ and a sequence $\boldsymbol{T}:=(T_k)_{k\in\N}$ of linear operators $T_k:L^p(\Omega,\mathscr{F},\mu)\to L^0(\Omega,\mathscr{F},\mu)$ sending a given $f\in L^p(\Omega,\mathscr{F},\mu)$ to a measurable function $T_kf$ for each $k\in\N$, the question of the \emph{almost everywhere convergence} of $T_kf$ to $f$ \emph{for all} $f\in L^p(\Omega,\mathscr{F},\mu)$ is an important problem in real analysis, especially in the case where $(T_k)$ approximates the identity map in $L^p(\Omega,\mathscr{F},\mu)$ when $k$ grows. Under mild hypotheses on the operators $T_k$, $k\in\N$ (like continuity in measure and, for some results, positivity and their commuting with translations), it follows from general principles, associated to the names of E.~Stein, D.~Burkholder and S.~Sawyer, that the holding of the above a.e. convergence property in $L^p(\Omega, \mathscr{F},\mu)$ is equivalent to the fact that the associated maximal operator $T^*$ defined for a given function $f\in L^p(\Omega,\mathscr{F},\mu)$ by:
$$
T^*f(x):=\sup_{k\in\N} |T_kf(x)|,
$$
either satisfies an inequality of weak type $(p,p)$ (see below for a precise definition) or satisfies the seemingly weaker property that one has $T^*f<+\infty$, $\mu$-a.e. in $\Omega$, for all $f\in L^p(\Omega,f,\mu)$ (see \emph{e.g.} Stein \cite{STEINART}, Burkholder \cite{BURKHOLDER} and Sawyer \cite{SAWYER} for original formulations of this principle, and Garsia \cite{GARSIA} for a beautiful exposition of this way of dealing with a.e. convergence of processes).\\
In the current paper, we will be mainly interested in \emph{Lebesgue differentiation processes} $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}$ on $L^p(\R^2)$ associated to sequences $\boldsymbol{R}=(R_k)$ of rectangles in the plane centered at the origin, the diameters of which tend to zero, \emph{i.e.} in the case where one has, for each $k\in\N$:
$$
T_kf(x):=\frac{\chi_{R_k}}{|R_k|}*f(x)=\frac{1}{|R_k|}\int_{x+R_k} f.
$$
Note that we shall here typically work in the case where the classical Lebesgue differentiation theorem does \emph{not} provide the a.e. convergence of $T_kf$ to $f$ for $f\in L^1(\R^2)$ or even in $L^p(\R^d)$ for $p>1$: one may remember, indeed, that the Lebesgue differentiation theorem provides a.e. convergence of $T_kf$ to $f$ for each $f\in L^1(\R^2)$ in this context if $\boldsymbol{R}$ is a sequence of \emph{squares}, or in case the ratio between the two side-lengths of $R_k$, called its \emph{shape}, remains away (uniformly) from $0$ and $\infty$, and that it follows from Jessen, Marcinkiewicz and Zygmund \cite{JMZ} that this a.e. convergence holds for any $f\in L\log L(\R^2)$ (and hence also in $L^p(\R^2)$ for any $p>1$) if $\boldsymbol{R}$ is a sequence of rectangles parallel to the coordinate axes (see \cite{DM17} and, for example, the introduction in \cite{DMZAA} for a review of the different versions one can formulate of the Lebesgue differentiation with rectangles).
In the following paper, we will hence be mainly dealing with sequences $\boldsymbol{R}$ of rectangles whose shapes are tending to $0$ (or $\infty$) and which will usually \emph{not} be parallel to the coordinate axes.\\
As we mentioned briefly above, and as it will be made precise below, the a.e. convergence of $(T_k f)$ to $f$ for all $f\in L^p(\R^2)$ ($1\leq p<\infty$) is equivalent to having $T^*f<+\infty$ almost everywhere in $\R^2$ for all $f\in L^p(\R^2)$, or the existence of a constant $C>0$ such that for any $f\in L^p(\R^2)$ and any $\lambda>0$, one has:
$$
|\{x\in\R^2 : T^*f(x)>\lambda\}|\leq \frac{C}{\lambda^p}\|f\|_p^p.
$$
If the maximal operator $T^*$ satisfies such an inequality (called weak type $(p,p)$), we shall say that it is \emph{$L^p$-good}; in this case we shall also say that the sequence $\boldsymbol{R}$ is $L^p$-good, keeping in mind that it is equivalent to the a.e. convergence of $T_kf$ to $f$ for all $f\in L^p(\R^2)$. If $T^*$ (or $\boldsymbol{R}$) is not $L^p$-good, we shall call it \emph{$L^p$-bad}.
It also follows from Hagelstein and Parissis \cite{HP} that the a.e. convergence of $(T_kf)$ to $f$ for all $f\in L^\infty(\R^2)$ is equivalent to the following property: for each $0<\lambda <1$, there exists a constant $C>0$ such that for any Borel set $B\subseteq\R^2$ with finite Lebesgue measure, one has:
\begin{equation}\label{eq.linfini-good}
|\{x\in\R^2: T^*\chi_B(x)>\lambda\}|\leq C_\lambda |B|.
\end{equation}
As before, if the above inequality is satisfied, we shall say that $T^*$ and $\boldsymbol{R}$ are \emph{$L^\infty$-good}, meaning in particular that $T_kf$ converges a.e. to $f$ for all $f\in L^\infty(\R^2)$. Again, if $T^*$ or $\boldsymbol{R}$ is not $L^\infty$-good, we shall say it is \emph{$L^\infty$-bad}.\\
Here an interesting comparison with \emph{directional maximal operators} is relevant. Given a set $\Omega\subseteq [0,+\infty)$ (thought of as a set of \emph{slopes}), denote by $M_\Omega$ the maximal operator defined for a given function $f$ on $\R^2$ by:
$$
M_\Omega f(x):=\sup \frac{1}{|R|}\int_R |f|,
$$
where the upper bound is extended on all rectangles containing $x$, \emph{one side of which has a slope $\omega\in\Omega$}. Such a maximal operator is called \emph{directional}.
A deep and beautiful result by Bateman \cite{BATEMAN} states a fundamental dichotomy for directional maximal operators; it can be stated in the following way (we here extend the meaning of ``$L^p$-good'' and ``$L^p$-bad'' to the maximal operator $M_\Omega$ in an obvious way, even though it is not associated to a process):
\begin{enumerate}
\item[(i)] either $M_\Omega$ is $L^p$-good for any $1<p\leq\infty$ (in which case we call $\Omega$ a \emph{good set of directions});
\item[(ii)] or $M_\Omega$ is $L^p$-bad for any $1<p\leq\infty$ (in which case we call $\Omega$ a \emph{bad set of directions}).
\end{enumerate}
Note that Bateman's original dichotomy was stated for finite $1<p<\infty$; the observation that $p=\infty$ could be included was made by three of the current paper's authors in \cite{DMR}. Moreover, Bateman gives a geometric characterization of good sets of directions; we refer to his work \cite{BATEMAN} for more details. Yet, we can say here, for example, that geometric sequences like $\Omega:=\{2^{-k}:k\in\N\}$ are examples of good sets of directions, while sets as simple as $\Omega_s:=\{k^{-s}:k\in\N^*\}$ for $s>0$, are examples of bad sets of directions.\\
In the above setting where $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}$ is the sequence of Lebesgue averages associated to a sequence $\boldsymbol{R}=(R_k)$ of rectangles centered at the origin, the diameters of which tend to zero, we can already formulate an immediate consequence of Bateman's result. To this purpose, denote for each $k\in\N$ by $\omega_{R_k}\in [0,+\infty)$ the slope of the longest side of $R_k$ (in case $R_k$ is a square, denote by $\omega_{R_k}$ the minimum of the two slopes of its sides) and let $\Omega_{\boldsymbol{R}}:=\{\omega_{R_k}:k\in\N\}$ the set of those slopes.
If $\Omega_{\boldsymbol{R}}$ is a good set of directions (in the sense of Bateman recalled above), then we obviously have, using the same notations as before:
\begin{equation}\label{eq.intro1}
T^*\leq M_{\Omega_{\boldsymbol{R}}},
\end{equation}
from which it immediately follows that $T^*$ and $\boldsymbol{R}$ are $L^p$-good for any $1<p\leq\infty$. It is then also the case that $T_kf$ converges a.e. to $f$ for every $f\in L^p(\R^2)$. In this case, the problem we stated initially hence has a trivial (positive) solution.
Yet in case $\Omega_{\boldsymbol{R}}$ is a \emph{bad} set of directions, inequality \eqref{eq.intro1} does not provide any information on the good or bad character of $T^*$; roughly speaking, the process $T^*$ ``extracts'' from $M_{\Omega_R}$ the smallest possible amount of rectangles still providing a differentiation scheme, but may not capture anymore the geometry of rectangles that makes $M_{\Omega_R}$ bad.
It is precisely the main focus of our paper to provide an answer to the following (rather vague) question: assuming $\Omega$ is a bad set of directions like $\Omega_s:=\{k^{-s}:k\in\N^*\}$ for $s>0$, can one still provide examples of sequences $\boldsymbol{R}$ of rectangles as above satisfying $\Omega_{\boldsymbol{R}}=\Omega$ and for which the Lebesgue differentiation process $\boldsymbol{T}_{\boldsymbol{R}}$ is still $L^\infty$-bad?
As we shall explain in the next section, presenting our paper's results, the answer to the latter is positive. On the way to answering it, we shall also present other results concerning $L^1$-good and $L^p$-good processes associated to sequences of rectangles of the above type.\\
A last remark should be made concerning the fact that we insist on working with differentiation \emph{processes} here, in opposition to differentiation \emph{bases} (see de Guzmán \cite{GUZMAN} for the terminology and the precise definitions, on which we do not want to insist here): while to any differentiation process of the type $\boldsymbol{T}_{\boldsymbol{R}}$ for some sequence of rectangles $\boldsymbol{R}$, corresponds a centered, translation-invariant differentiation basis $$
\mathscr{B}_{\boldsymbol{R}}=\bigcup_{x\in\R^2}\mathscr{B}_{\boldsymbol{R}}(x):=\bigcup_{x\in\R^2}\{x+R_k: k\in\N\},
$$
one should insist on the fact that a differentiation basis $\mathscr{B}$ can, at a given point $x\in\R^2$, have an uncountable class of admissible sets $\mathscr{B}(x)$, even more so when $\mathscr{B}$ is a Busemann-Feller-type basis (\emph{i.e.} satisfies that, given $B\in\mathscr{B}$, one has $B\in\mathscr{B}(x)$ if and only if $B\ni x$).
Observe that if $\Omega$ is a set of directions in the plane (see above), the set $\mathscr{B}_\Omega$ of all rectangles oriented along one direction $\omega\in \Omega$ is a translation-invariant Busemann-Feller basis called a \emph{directional basis}.\\
When a translation-invariant differentiation basis $\mathscr{B}=\bigcup_{x\in\R^d}\mathscr{B}(x)$ in $\R^d$ consists of convex sets \emph{and is known to differentiate $L^\infty(\R^2)$} (in which case it is called a \emph{density basis}), it was shown by G.~Oniani (see \cite[Remark~7]{ONIANI}) that for any $1\leq p<\infty$, $\mathscr{B}$ differentiates $L^1(\R^d)\cap L^p(\R^d)$ if and only if $\mathscr{B}^*$ does, where $\mathscr{B}^*=\bigcup_{x\in\R^d}\mathscr{B}^*(x)$ is the Busemann-Feller basis associated to $\mathscr{B}$ defined for $x\in\R^d$ by $\mathscr{B}^*(x):=\{B\in\mathscr{B}:B\ni x\}$.\\
When $\mathscr{B}$ fails to differentiate $L^\infty(\R^d)$ (as is the case when $d=2$ for directional bases $\mathscr{B}_\Omega$ associated to a bad sequence of directions $\Omega=\{\omega_k:k\in\N\}$), the latter equivalence is \emph{not} true anymore (see Hagelstein and Parissis \cite{HP}); our class of examples will show that, under some conditions on the sequence $(\omega_k)$, one can nevertheless construct a sequence $\boldsymbol{R}:=(R_k)$ of rectangles $R_k$ oriented along direction $\omega_k$, for which the associated process $T_{\boldsymbol{R}}$ is $L^\infty$-bad, hence extracting ``ordered'' bad \emph{processes} from the bad directional basis $\mathscr{B}_\Omega$.\\
Let us now formulate our results in a more precise way.
\section{Results}
In this whole section, we use the same notations as in the introduction, and associate to any sequence $\boldsymbol{R}=(R_k)$ of rectangles in the plane centered at the origin, the diameters of which tend to zero, a Lebesgue differentiation process $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}=(T_k)$.
A first observation in the following paper (to which Section~\ref{sec.L1} is devoted) will be to provide a geometrical condition on $\boldsymbol{R}$ ensuring that $\boldsymbol{T}_{\boldsymbol{R}}$ is $L^1$-good. More precisely, we shall prove the following result.
\begin{Theorem}\label{THM1}
If there exists $\alpha>0$ such that for all $k\in\N$ and all $l\in\N$ satisfying $l>k$, one has $$R_k-R_l\subseteq \alpha R_k$$ then the process $\boldsymbol{T}$ is $L^1$-good.
\end{Theorem}
The latter geometric condition appearing in Theorem~\ref{THM1} should be thought of as a kind of \emph{nesting} property of the sequence $(R_k)$. The proof of Theorem~\ref{THM1} will rely on a geometric interpretation of the notion of \textit{correct factors} $Q_k$, $k\in\N$ of the sequence $\boldsymbol{R}$ introduced by Rosenblatt and Wierdl in \cite{RW} and defined for $k\in\N$ by:
$$
Q_k:=\left|\bigcup_{l=k}^\infty R_k-R_l\right|.
$$
The most important property enjoyed by the correct factors is the following: if one has $Q_k<+\infty$ for any $k\in\N$, then the \emph{corrected maximal operator} $\tilde{T}^*$ defined for a given locally integrable $f$ and $x\in\R^2$ by:
$$
\tilde{T}^*f(x):=\max_{k\in\N} \left| \frac{|R_k|}{Q_k} T_kf(x)\right|=\max_{k\in\N} \left|\frac{1}{Q_k}\int_{x+R_k} f\right|,
$$
always has weak type $(1,1)$. It is that property, combined with the geometric hypothesis made on $(R_k)$ in the statement of Theorem~\ref{THM1}, that will ensure $\boldsymbol{T}_{\boldsymbol{R}}$ to be $L^1$-good.\\
In section~\ref{sec.Lp}, we provide a continuous analogous result to the discrete ``corrected'' weak-type $(p,p)$ suggested by Rosenblatt and Wierdl in \cite[p.~551]{RW}. More precisely, we show the following result.
\begin{Theorem}\label{thm.Lp}
Fix $1\leq p<\infty$ and assume that one has $Q_k<\infty$ for each $k\in\N$. Define, for each $k\in\N$, the $k^{\text{th}}$ $L^p$-correct factor (of the sequence $\boldsymbol{R}$) $W_{k,p}$ by letting $W_{k,1}:=Q_k$ and, in case $1<p<\infty$, by letting:
$$
W_{k,p}:=(Q_k)^{\frac 1p }|I_k|^{\frac{1}{p'}},
$$
where $1<p'<\infty$ is the conjugate exponent to $p$ satisfying $\frac 1p + \frac{1}{p'}=1$. Under those assumptions, the corrected maximal operator $\tilde{T}_p^*$ defined for any locally integrable $f$ and any $x\in\R^2$ by:
$$
T_p^*f(x):=\max_{k\in\N} \left|\frac{|R_k|}{W_{k,p}} T_kf(x)\right|=\max_{k\in\N} \left|\frac{1}{W_{k,p}}\int_{x+R_k} f\right|,
$$
has weak type $(p,p)$.
\end{Theorem}
It will also be observed that it is not obvious, in case $p>1$, to get a sufficient geometric condition on $\boldsymbol{R}$ ensuring that $\boldsymbol{T}_{\boldsymbol{R}}$ is a $L^p$-good process, using the latter theorem.\\
In sections~\ref{sec.Linfini1} and \ref{sec.Linfini2}, we will fix an nondecreasing sequence $\boldsymbol{b}=(b_k)$ of positive real numbers satisfying $b_0=0$ and define points $B_k:=(b_k,0)$ for all $k\in\N$ and let $A:=(0,1)$; for each $k\in\N^*$ we then denote by $\Delta_k$ the triangle $AB_{k-1}B_{k}$. For each $k\in\N^*$, we associate to $\Delta_k$ a rectangle $P_k$ centered at the origin, obtained by translating a rectangle oriented along $AB_{k}$ and containing the image of $\Delta_k$ by a homothecy based at $A$ with ratio $\frac 32$ ({see Figure~\ref{fig.Delta_k} for an overview of the situation}, and section~\ref{sec.Linfini2} for a precise description). Finally we fix any sequence $\boldsymbol{\delta}=(\delta_k)_{k\in\N^*}$ of positive real numbers and we consider the sequence of rectangles $\boldsymbol{R}=\boldsymbol{R}(\boldsymbol{b},\boldsymbol{\delta})=(R_k)_{k\in\N^*}$ defined by $R_k:=\delta_n P_k$ if $k\in\N^*$ satisfies $2^n\leq k<2^{n+1}$ for a given $n\in\N$; we call $\boldsymbol{\delta}$ an \emph{admissible sequence} in case one has $\diam R_k\to 0$, $k\to\infty$.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{THM23BIS.pdf}
\caption{The triangle $\Delta_k$ and the rectangle ${P}_k$}\label{fig.Delta_k}
\end{center}
\end{figure}
\begin{Theorem}\label{THM2}
Assume that the sequence $\boldsymbol{b}$ satisfies the following two conditions:
\begin{enumerate}
\item[(i)] there exists a constant $c>0$ such that one has $1 + b_{k-1}^2 \geq c(b_{k} -b_{k-1})^2$ for all $k\in\N^*$;
\item[(ii)] one has: $$G_{ \boldsymbol{b} }:=\sup_{n\in\N}\sup_{1\leq l\leq n} \left(\frac{b_{n+2l}-b_{n+l}}{b_{n+l}-b_n}+\frac{b_{n+l}-b_n}{b_{n+2l}-b_{n+l}}\right)<+\infty.$$
\end{enumerate}
Under those assumptions, the process $\boldsymbol{T}_{\boldsymbol{R}}$ associated to $\boldsymbol{R} = \boldsymbol{R}(\boldsymbol{b},\boldsymbol{\delta})$ is $L^\infty$-bad for any admissible sequence $\boldsymbol{\delta}$.
\end{Theorem}
\begin{Remark}
In the latter statement, the first condition is merely a technical one, while the finiteness of $G_{\boldsymbol{b}}$ is a quantitative way to ensure that the set of directions $\Omega(\boldsymbol{b})$ associated to $\boldsymbol{R}$ (being the set of slopes of the segments $AB_{k}$, $k\in\N^*$) defined by:
$$
\Omega(\boldsymbol{b}):=\left\{\frac{-1}{b_k}:k\in \N^*\right\},
$$
is a \emph{bad set of directions} (see the introduction above for a precise definition of a \emph{bad} set of directions). The quantity $G_{\boldsymbol{b}}$, which one can call the \emph{Perron factor} of $\boldsymbol{b}$, was first introduced by Hare and Rönning \cite{HR}; it was also used by A.~Gauvan in his Master's thesis \cite{GAUVANMEMOIRE} in this precise context, and in \cite{GAUVANPT} for providing concrete examples of homothecy-invariant bases of rectangles differentiating all \emph{vs.} no $L^p$ spaces.
\end{Remark}
\begin{Example}
One can verify that for $s>0$, the sequence $\boldsymbol{b}_s = (k^s)$ satisfies the conditions of Theorem \ref{THM2}.
\end{Example}
Let us now detail how one can obtain the $L^1$-a.e. convergence result stated in Theorem~\ref{THM1}.
\section{Almost everywhere convergence in $L^1$}\label{sec.L1}
We keep the notations used before, and associate to any sequence $\boldsymbol{R}=(R_k)$ of rectangles in the plane centered at the origin, the diameters of which tend to zero, a Lebesgue differentiation process $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}=(T_k)$. Throughout this section we'll omit the index $\boldsymbol{R}$, keeping in mind that we shall always be working with the differentiation process $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}$ associated to $\boldsymbol{R}$.
Recall that one associates to $\boldsymbol{R}$ its \emph{correct factors} $Q_k$, $k\in\N$, defined by Rosenblatt and Wierdl in \cite{RW} by letting, for $k\in\N$:
\begin{equation}\label{eq.corr-fact}
Q_k:=\left|\bigcup_{l=k}^\infty R_k-R_l\right|.
\end{equation}
We define the \emph{corrected maximal function} $\tilde{T}^*$ by letting, for a locally integrable function $f$ and $x\in\R^2$
$$
\tilde{T}^*f(x):=\sup_{k\in\N} \left|\frac{1}{Q_k} \int_{x+I_k} f\right|.
$$
The following theorem is taken from Rosenblatt and Wierdl \cite[Theorem~5.11]{RW} and explains the term ``correct factor''.
\begin{Theorem}\label{thm.RW}
There exists $C>0$ such that for all $f\in L^1(\R^2)$ and all $\lambda>0$, one has:
$$
|\{x\in\R^2: \tilde{T}^*f(x)>\lambda\}|\leq\frac{C}{\lambda} \|f\|_1.
$$
\end{Theorem}
As an easy consequence, this yields a sufficient condition for a sequence of rectangles $\boldsymbol{R}$ to be $L^1$-good.
\begin{Theorem}\label{thm.cond-suff} If there exists a constant $C>0$ such that, for all $k\in\N$, one has $Q_k\leq C|R_k|$, then there exists $C'>0$ so that one has, for all $f$ and $\lambda>0$:
$$
|\{x\in\R^2: T^* f(x)>\lambda\}|\leq\frac{C'}{\lambda} \|f\|_1,
$$
where $T^*$ is the maximal operator associated to $\boldsymbol{T}$.
\end{Theorem}
\begin{Remark}\label{rmq.thm-cond-suff}
It follows immediately from the Sawyer-Stein's principle (see above in the introduction) that, under the hypotheses of the latter theorem, $\boldsymbol{R}$ yields an $L^1$-good process $\boldsymbol{T}$.
\end{Remark}
\begin{proof}
By hypothesis, one has $\eta:=\inf_k \frac{|R_k|}{Q_k}>0$. The theorem then follows immediately from Rosenblatt and Wierdl's result (Theorem~\ref{thm.RW} above).
\end{proof}
We now have a closer look at what the hypothesis in the latter theorem (\emph{i.e.} the growth of the correct factor at a comparable speed to that of the rectangle's areas) means geometrically. To that purpose, a first simple geometrical observation will be useful.
\subsection*{Differences of symmetric rectangles}\label{sec.geom-obs}
Fix real numbers $0<\ell\leq L$, $0< \ell'\leq L'$, assume that one has $L'\leq L$, define a rectangle $R=[-L/2,L/2]\times [-\ell/2,\ell/2]$ parallel to the axes and denote by $R'$ the rectangle obtained by rotating the rectangle $[-L'/2,L'/2]\times [\ell'/2,\ell'/2]$ of an angle $\omega\neq 0$ around the origin. Denote then by $\widehat{R-R'}=[-\hat{L}/2,\hat{L}/2]\times[-\hat{\ell}/2,\hat{\ell}/2]$ the smallest rectangle parallel to the axes that contains the set $R-R'$ (see Figure~\ref{fig.R-R'}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=1.2]{figure10.pdf}\caption{The rectangles $R$, $R'$ and the set $R-R'$}\label{fig.R-R'}
\end{center}
\end{figure}
It is easy to see, studying the coordinates of points in $(L/2,\ell/2)+R'$, that one has:
\begin{equation}\label{eq.R-R'x}
\hat{L}=L+\max_{\eta,\theta\in [-1,1]}(L'\eta\cos\omega-\ell'\theta\sin\omega)= L+L'|\cos\omega|+\ell'|\sin\omega|,
\end{equation}
and:
\begin{equation}\label{eq.R-R'y}
\hat{\ell}=\ell+\max_{\eta,\theta\in [-1,1]}(L'\eta\sin\omega+\ell'\theta\cos\omega)= \ell+L'|\sin\omega|+\ell'|\cos\omega|.
\end{equation}
Moreover, using the fact that $R-R'$ contains the two parallelograms $P_1$ and $P_2$ represented on Figure~\ref{fig.P1}, we also get:
\begin{equation}\label{eq.R-R'min}
|R-R'|\geq \max(\ell L'|\cos\omega|,LL' |\sin\omega|).
\end{equation}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{figure12.pdf}\caption{The parallelograms $P_1$ and $P_2$}\label{fig.P1}
\end{center}
\end{figure}
We are now ready to express in geometrical terms the growth condition on the correct factor appearing in the statement of Theorem~\ref{thm.cond-suff}.
\subsection*{Correct factors and linear growth}
The next lemma expresses is the announced equivalence between the linear growth of the correct factors of $\boldsymbol{R}$ and a geometrical property on the sequence $\boldsymbol{R}$ itself expressing its ``nested'' behavior.
\begin{Lemma}\label{lem.linear}
Assume that $\boldsymbol{R} =(R_k)$ is as before.
The following two properties are equivalent:
\begin{itemize}
\item[(i)] there exists $C>0$ such that one has $Q_k\leq C|R_k|$, where $(Q_k)$ is the sequence of correct factors associated to $(R_k)$ as above;
\item[(ii)] there exists $\alpha>0$ such that for all $k\in\N$ and all $l\in\N$ satisfying $l>k$, one has $R_k-R_l\subseteq \alpha R_k$.
\end{itemize}
\end{Lemma}
\begin{Remark}
In the above statement, property (i) will be referred to by saying that the correct factors of $\boldsymbol{R}$ \emph{have linear growth}; property (ii) will be expressed by saying that $\boldsymbol{R}$ is \emph{almost nested}.
\end{Remark}
\begin{proof}
That (ii) implies (i) is obvious from the definition of $Q_k$ for $k\in\N$.
To prove that (i) implies (ii), start by choosing, for each $k\in\N$,
positive real numbers $L_k$ and $\ell_k$ and angles $\theta_k\in (-\pi/2,\pi/2)$ in such a way that $R_k$ is obtained by rotating $[-L_k/2,L_k/2]\times [-\ell_k/2,\ell_k/2]$ around the origin by an angle $\theta_k$. Assume that (ii) does not hold, and hence that for all $\alpha>0$ there exists integers $k_\alpha\in\N$ and $l_\alpha>k_\alpha$ so that $R_{k_\alpha}-R_{l_\alpha}$ is not included in $\alpha R_{k_\alpha}$.
Fix now $\alpha>2$. If one has $\theta_{k_\alpha}=\theta_{l_\alpha}$ then we see that $R_{k_\alpha}-R_{l_\alpha}$ is a rectangle centered at the origin, parallel to $R_{k_\alpha}$ and of sides $L_{k_\alpha}+L_{l_\alpha}$ and $\ell_{k_\alpha}+\ell_{l_\alpha}$ respectively. Since $R_{k_\alpha}-R_{l_\alpha}$ is not included in $\alpha R_{k_\alpha}$ we get either $L_{l_\alpha}+\ell_{l_\alpha}>\alpha L_{k_\alpha}$ or $\ell_{l_\alpha}+\ell_{k_\alpha}>\alpha\ell_{k_\alpha}$, implying in both cases that:
$$
Q_{k_\alpha}\geq |R_{k_\alpha}-R_{l_\alpha}|=(L_{k_\alpha}+L_{l_\alpha})\cdot (\ell_{k_\alpha}+\ell_{l_\alpha})\geq \alpha L_{k_\alpha}\ell_{k_\alpha}=\alpha | R_{k_\alpha}|.
$$
If one has $\theta_{k_\alpha}\neq \theta_{l_\alpha}$, then, after applying a rotation of angle $-\theta_{k_\alpha}$ around the origin to $R_{k_\alpha}$ and $R_{l_\alpha}$, we are in the situation of section~\ref{sec.geom-obs} with $R=R_{k_\alpha}$, $R'=R_{l_\alpha}$ and $\delta=\theta_{l_\alpha}-\theta_{k_\alpha}$. Since $R_{k_\alpha}-R_{l_\alpha}$ is not contained in $\alpha R_{k_\alpha}$, and since $\widehat{R_{k_\alpha}-R_{l_\alpha}}$ is the smallest rectangle centered at the origin, homothetic to $R_{k_\alpha}$ and containing $R_{k_\alpha}-R_{l_\alpha}$, this implies that $\widehat{R_{k_\alpha}-R_{l_\alpha}}$ is not contained in $\alpha R_{k_\alpha}$ and hence by \eqref{eq.R-R'x} and \eqref{eq.R-R'y} that one has either:
$$
\alpha L_{k_\alpha}<L_{k_\alpha}+L_{l_\alpha}|\cos (\theta_{l_\alpha}-\theta_{k_\alpha})|+\ell_{l_\alpha}|\sin (\theta_{l_\alpha}-\theta_{k_\alpha})|,
$$
or:
$$
\alpha \ell_{k_\alpha}<\ell_{k_\alpha}+L_{l_\alpha}|\sin (\theta_{l_\alpha}-\theta_{k_\alpha})|+\ell_{l_\alpha}|\cos (\theta_{l_\alpha}-\theta_{k_\alpha})|;
$$
since one has $\ell_{l_\alpha}\leq L_{\ell_\alpha}\leq L_{k_\alpha}$, those inequalities imply respectively:
$$
L_{l_\alpha}|\cos (\theta_{l_\alpha}-\theta_{k_\alpha})|\geq (\alpha-2) L_{k_\alpha},
$$
or:
$$
L_{l_\alpha}|\sin (\theta_{l_\alpha}-\theta_{k_\alpha})|\geq (\alpha-2) \ell_{k_\alpha}.
$$
In case the first inequality holds, \eqref{eq.R-R'min} implies that one has:
$$
Q_{k_\alpha}\geq |R_{k_\alpha}-R_{l_\alpha}|\geq \ell_{k_\alpha} L_{l_\alpha} |\cos (\theta_{l_\alpha}-\theta_{k_\alpha})|\geq (\alpha-2) L_{k_\alpha}\ell_{k_\alpha}=(\alpha-2)|R_{k_\alpha}|.
$$
In case it is the second of the above inequalities that holds, we get again using \eqref{eq.R-R'min}:
$$
Q_{k_\alpha}\geq |R_{k_\alpha}-R_{l_\alpha}|\geq L_{k_\alpha} L_{l_\alpha} |\sin (\theta_{l_\alpha}-\theta_{k_\alpha})|\geq (\alpha-2) L_{k_\alpha}\ell_{k_\alpha}=(\alpha-2)|R_{k_\alpha}|.
$$
We hence get $Q_{k_\alpha}\geq (\alpha-2)|R_{k_\alpha}|$ in all cases, contradicting the linear growth of the correct factors, that is property (i), since $\alpha$ can be arbitrarily large. The proof is hence complete.
\end{proof}
We are now ready to (re-)state and prove Theorem~\ref{THM1}.
\begin{Theorem}\label{THM1-v2}
Assume that $\boldsymbol{R}=(R_k)$ is a sequence of rectangles in $\R^2$ centered at the origin, the diameters of which tend to zero.
If $\boldsymbol{R}$ is almost nested, \emph{i.e.} if there exists $\alpha>0$ such that for all $k\in\N$ and all $l\in\N$ satisfying $l>k$, one has $$R_k-R_l\subseteq \alpha R_k$$ then the process $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}$ is $L^1$-good.
\end{Theorem}
\begin{proof}
If $\boldsymbol{R}$ is almost nested, then it follows from Lemma~\ref{lem.linear} that its correct factors $Q_k$, $k\in\N$, have linear growth. But it then follows from Remark~\ref{rmq.thm-cond-suff} that $\boldsymbol{T}_{\boldsymbol{R}}$ is $L^1$-good, as we wanted to show.
\end{proof}
For rectangles parallel to the coordinate axes, the correct factors allow us to rephrase a result by A.~Stokolos \cite{STOKOLOS88}.
\begin{Theorem}[Stokolos]\label{thm.stokolos} Assume that $\boldsymbol{R} = (R_k)$ is as in Lemma~\ref{lem.linear} \emph{and that moreover all rectangles $R_k$ are parallel to the coordinate axes}. Then the following properties are equivalent:
\begin{itemize}
\item[(i)] $\boldsymbol{T}$ is $L^1$-good;
\item[(ii)] $\boldsymbol{R}$ can be decomposed into finitely many subsequences along which the correct factor has linear growth;
\item[(iii)] $\boldsymbol{R}$ can be decomposed into finitely many subsequences which are almost nested.
\end{itemize}
\end{Theorem}
\begin{proof}
That (ii) and (iii) are equivalent is an immediate consequence of Lemma~\ref{lem.linear}. Now that (ii) implies (i) follows easily from Theorem~\ref{thm.cond-suff} and from Sawyer-Stein's principle. Finally, if $(R_k)$ is $L^1$-good, then it follows from a result by A.~Stokolos \cite{STOKOLOS88} (not formulated in this exact way, though: see Moonens and Rosenblatt \cite{MR} where it is explained how Stokolos' 1988 theorem can be rephrased in this fashion~---~or see the more general result in A.~Stokolos' survey \cite[Corollary~1, p.~1448]{STOKOLOS2005}), that the sequence $(R_k^*)$ of all dyadic enlargements of rectangles in $(R_k)$, can be decomposed into finitely many subsequences that are totally ordered by inclusion, from which it results that $(R_k)$ can be decomposed into finitely many almost nested subsequences.
\end{proof}
Let us now state some weak type inequality one can obtain in $L^p$ with our rectangular differentiation processes, using correct factors.
\section{A ``corrected'' weak-type inequality in $L^{p}$}\label{sec.Lp}
As before, we fix a sequence $\boldsymbol{R}=(R_k)$ of rectangles in the plane centered at the origin, the diameters of which tend to zero, and we associate to it the differentiation process $\boldsymbol{T}:=\boldsymbol{T}_{\boldsymbol{R}}$ as above.
Recall that one defines the correct factors $Q_k$, $k\in\N$ associated to $\boldsymbol{R}$ as in \eqref{eq.corr-fact}, p.~\pageref{eq.corr-fact}, and than one defines, for $1\leq p<\infty$, the $L^p$ correct factors $W_{k,p}$, $k\in\N$ by letting $W_{k,1}:=Q_k$ for $k\in\N$, and, in case one has $1<p<\infty$, by letting for $k\in\N$:
\[W_{k,p} : = {\left (Q_{k} \right)}^{\frac{1}{p}} {\vert R_{k} \vert}^{\frac{1}{p'}},\]
where $1<p'<\infty$ is the conjugate exponent to $p$ satisfying the equality $\frac 1p + \frac{1}{p'}=1$.
Assuming that one has $Q_k<+\infty$ for all $k\in\N$ and given $1\leq p<\infty$, we also define a corrected maximal operator $T^*_p$ by letting, for a locally integrable function $f$ on $\R^2$ and $x \in {\mathbb R}^{2}$:
\[T^*_p f(x):= \sup_{k} \left|\frac{1}{W_{k,p}} \int_{x + R_{k}} f\right|.\]
The following theorem is a weak type $(p,p)$ inequality for the corrected maximal operator $T^*_p$; it is an adaptation to the continuous case of the discrete weak $\ell^p$ corrected inequality suggested by Rosenblatt and Wierdl in \cite[Comments and problems, p.~551]{RW}.
\begin{Theorem} \label{RWp}
Assume that one has $Q_k<+\infty$ for all $k\in\N$. Then for
all $f \in L^{p}({\mathbb R}^{2})$ and all $\lambda >0$, one has
\[\vert \{x \in {\mathbb R}^{2}: T^*_p f(x) > \lambda \}\vert \leq \frac{1}{\lambda^p} \|f\|_p^p.\]
\end{Theorem}
\begin{proof}
Fix $N \in {\mathbb N}$. Let $$A_{N} :=
\left\{ x \in {\mathbb R}^{2}: \frac{1}{W_{k,p}} \int_{x + R_{k}} \vert f \vert > \lambda \text{ for some } 0 \leq k \leq N\right\}.$$ It is clearly sufficient to show that one has $\vert A_{N} \vert \leq \frac{1}{\lambda^p} \|f\|_p^p$.
Let $$
E_{N} :=\left\{0\leq k\leq N: \frac{1}{W_{k,p}} \int_{x + R_{k}} \vert f \vert > \lambda \text{ for some } x \in A_{N}\right\}.
$$
For the sake of notational simplicity, we write $A$ and $E$ instead of $A_{N}$ and $E_N$ respectively.
We now construct two finite sequences of sets $\{A_{k} \}$ and $\{E_{k}\}$.
Let $A_{0} = A$ and $E_{0} = E$. Let $k_{0} = \min E_{0}$ and choose $x_{0} \in A_{0}$ such that one has:
\begin{equation}\label{eq.I}
\frac{1}{W_{k_{1},p}} \int_{x_{0} + R_{k_{0}}} \vert f \vert > \lambda
\end{equation}
Define:
\[B_{0} := x_{0} + \bigcup_{l=k_0}^\infty (R_{k_0}-R_l),\]
so that in particular one has $|B_0|=Q_{k_0}$, and:
\begin{multline*}C_{0}:= \bigg\{x: x \in A_{0}, \frac{1}{W_{i,p}} \int_{x + R_{i}} \vert f \vert > \lambda\\ \text{and } (x + R_{i}) \cap (x_{1} + R_{k_{1}}) \not= \emptyset \text{ for some } 0 \leq i \leq N\bigg\}.\end{multline*}
Observe that one has $C_0\subseteq B_0$, which follows from the fact that having $x \in C_{0}$ implies that one has $x \in A_{0}$ and $\left( \left( x - x_{0} \right) + R_{i} \right) \cap R_{k_{0}} \not= \emptyset$ for some $0 \leq i \leq N$.
Notice also that by the minimality of $k_{0}$, one has $i \geq k_{0}$.
Now by definition of $W_{k_{0}, p}$, one has
\[W_{k_{0}, p} = {\left (Q_{k_{0}} \right)}^{\frac{1}{p}} {\vert R_{k_{0}} \vert}^{\frac{1}{p'}}.\]
Hence, one can write \eqref{eq.I} as
\[ \int_{x_{1} + R_{k_{0}}} \vert f \vert > \lambda {\left (Q_{k_{0}} \right)}^{\frac{1}{p}} {\vert R_{k_{0}} \vert}^{\frac{1}{p'}}.\]
By applying H\"older inequality, one thus obtains:
\[{\vert R_{k_{0}} \vert}^{\frac{1}{p'}} {\left ( \int_{x_{1} + R_{k_{0}}} {\vert f \vert}^{p} \right)}^{\frac{1}{p}} \geq \int_{x_{1} + R_{k_{0}}} \vert f \vert > \lambda {\left (Q_{k_{0}} \right)}^{\frac{1}{p}} {\vert R_{k_{0}} \vert}^{\frac{1}{p'}}, \]
so that
\[ \int_{x_{1} + R_{k_{0}}} {\vert f \vert}^{p} > {\lambda}^{p} Q_{k_{0}} = {\lambda}^{p} \vert B_{0} \vert. \]
Define now $A_{1} : = A_{0} \setminus B_{0}$ and:
\[E_{1}:= \left\{0 \leq k \leq N: \frac{1}{W_{k,p}} \int_{x + R_{k}} \vert f \vert > \lambda \text{ for some } x \in A_{1}\right\}.\]
If one has $\vert A_{1} \vert =0$, one stops the procedure. Otherwise, we define $k_{1} = \min E_{1}$ and pick $x_{1} \in A_{1}$.
One then gets:
\begin{equation}\label{eq.II}
\int_{x_{2} + R_{k_{1}}} \vert f \vert > \lambda W_{k_{1},p}.
\end{equation}
Define also: $$B_{1} : = x_{1} + \bigcup_{l=k_1}^\infty (R_{k_1}-R_l),$$so that in particular one has $|B_1|=Q_{k_1}$, and:
\begin{multline*}
C_{1}: = \bigg\{x: x \in A_{1}, \frac{1}{W_{i,p}} \int_{x + R_{i}} \vert f \vert > \lambda\\
\text{and } (x + R_{i}) \cap (x_{1} + R_{k_{1}}) \not= \emptyset \text{ for some }i, 0 \leq i \leq N\bigg\}.
\end{multline*}
Observe that
\[C_{1} \subseteq B_{1}.\]
Notice that by the minimality of $k_{1}$, one has $i \geq k_{1}$.
By definition of the $L^p$ correct factor, \eqref{eq.II} now rewrites:
\[ \int_{x_{2} + R_{k_{1}}} \vert f \vert > \lambda {\left (Q_{k_{1}} \right)}^{\frac{1}{p}} {\vert R_{k_{1}} \vert}^{\frac{1}{p'}}.\]
Applying H\"older inequality, one obtains:
\[{\vert R_{k_{1}} \vert}^{\frac{1}{p'}} {\left ( \int_{x_{2} + R_{k_{1}}} {\vert f \vert}^{p} \right)}^{\frac{1}{p}} \geq \int_{x_{2} + R_{k_{1}}} \vert f \vert > \lambda {\left (Q_{k_{1}} \right)}^{\frac{1}{p}} {\vert R_{k_{1}} \vert}^{\frac{1}{p'}}, \]
and hence:
\[ \int_{x_{2} + R_{k_{1}}} {\vert f \vert}^{p} > {\lambda}^{p} Q_{k_{1}} = {\lambda}^{p} \vert B_{1} \vert. \]
Notice again that, by the definition of $C_{0}$ and using the fact that one has $x_{2} \notin C_{0}$, one can write:
\[\left( x_{1} + R_{k_{0}} \right) \cap \left (x_{2} + R_{k_{1}} \right) = \emptyset.\]
Now we continue this process defining for $i\geq 1$, sets $A_{i} : = A_{i-1} \setminus B_{i-1}$ and:
and
\[E_{i} := \left \{0 \leq k \leq N: \frac{1}{W_{k,p}} \int_{x + R_{k}} \vert f \vert > \lambda \text{ for some } x \in A_{i}\right\},\]
as long as one has $|A_{i}|>0$.
We claim that after finitely many steps, say $M$ steps, this procedure comes to an end. In other words we assert that there is a $M\in\N$ so that, up to a null set,
\[A \subseteq \bigcup_{i=1}^{M} B_{i}\ ;\]
moreover such an $M$ satisfies the following estimate:
\begin{equation}\label{eq.III}
M\leq {\left (\frac{{\Vert f \Vert}_{p}}{{\alpha}^{\frac{1}{p}} \lambda}\right)}^{p},
\end{equation}
where $\alpha$ is defined by $\alpha := \min_{0 \leq k \leq N} Q_{k}$.
To prove this, assume we are at step $M\in\N$ and that the construction has not yet come to an end. We then have, for every $1 \leq i \leq M$:
\[ \int_{x_{i} + R_{k_{i}}} {\vert f \vert}^{p} > {\lambda}^{p} \vert B_{i} \vert \geq {\lambda}^{p} \alpha,\]
and, for every $1 \leq i \leq M$ and $1 \leq j \leq M$, with $i \not=j$:
\[\left( x_{i} + R_{k_{i}} \right) \cap \left (x_{j} + R_{k_{j}} \right) = \emptyset.\]
Therefore, one obtains:
\[M\alpha {\lambda}^{p} \leq {\lambda}^{p} \sum_{i=1}^{M}\vert B_{i} \vert < \sum_{i=1}^{M} \int_{x_{i} + R_{k_{i}}} {\vert f \vert}^{p} = \int_{\bigcup_{i=1}^{M} \left (x_{i}
+ R_{k_{i}}\right)} {\vert f \vert}^{p} \leq \int_{{\mathbb R}^{2}} {\vert f \vert}^{p} = {\Vert f \Vert}_{p}^{p}, \]
concluding the proof of \eqref{eq.III}.
Now let $M\in\N$ satisfy \eqref{eq.III} and, up to a negligible set:
\[A \subseteq \bigcup_{i=1}^{M} B_{i}.\]
Recalling that one has $|B_i|=Q_{k_i}$ for each $i$, we hence get:
\[\vert A \vert \leq \sum_{i=1}^{M}\vert B_{i} \vert \leq M\max_{1\leq i\leq M}Q_{k_i}\leq {\frac{1}{\lambda}{\Vert f \Vert}_{p}^p},\]
and the proof is complete.
\end{proof}
\begin{Remark}
{It is clear, looking at the proof of the above theorem, that at no place it is important to work with sequences of \emph{rectangles} in the \emph{plane}: $\boldsymbol{R}$ could be replaced, in its statement and proof, by any sequence of \emph{bounded Lebesgue-measurable subsets of $\R^d$ having strictly positive Lebesgue measure} and finite correct factors (the definition of which is an immediate generalization of that with rectangles).}
\end{Remark}
\begin{Remark}\label{rmq.equiv}
Clearly, the following two properties are equivalent, by definition of $W_{k,p}$:
\begin{itemize}
\item[(i)] $(Q_k)$ has linear growth (\emph{i.e.} there exists $C>0$ such that one has $Q_k \leq C |R_k|$ for each $k\in\N$);
\item[(ii)] $(W_{k,p})$ has linear growth (\emph{i.e.} there exists $C>0$ such that one has $|W_{k,p}|\leq C |R_k|$ for each $k\in\N$).
\end{itemize}
If one of the two equivalent conditions above is satisfied, then it follows from Theorem \ref{thm.cond-suff} that the associated process $\boldsymbol{T}_{\boldsymbol{R}}$ is $L^1$-good which, by Lemma \ref{lem.linear}, is equivalent to $\boldsymbol{R}$ being almost nested.
\end{Remark}
The following is an example of a sequence $\boldsymbol{R}$ for which $\boldsymbol{T}_{\boldsymbol{R}}$ is $L^p$-good for all $1<p<\infty$, \emph{without} its correct factors $W_{k,p}$, $k\in\N$ having linear growth.
\begin{Example}\label{ex.lpgood}
Let $(\lambda_k)$ be a sequence of positive real numbers tending to $0$.
For any $k\in\N$ and $0\leq i\leq k$, let:
$$
R^k_i:=[-2^{-i-1}\lambda_k,2^{-i-1}\lambda_k]\times [-2^{i-k-1}\lambda_k,2^{i-k-1}\lambda_k],
$$
and observe that $\mathscr{R}_k:=\{R^k_i:0\leq i\leq k\}$ is a collection of $k+1$ rectangles parallel to the coordinate axes, centered at the origin, having the same area $2^{-k}\lambda_k^2$.
Now define a sequence of integers $(n_k)$ by $n_0:=0$ and by requiring that one has $n_k=n_{k-1}+k+1$ for any $k\in\N^*$. One then defines a sequence $\boldsymbol{R}:=(R_n)$ by letting, for $k\in\N$ and $n_k\leq n\leq n_k+k$:
$$
R_n:=R^k_{n-n_k}.
$$
This is just ordering the rectangles in $\bigcup_{k\in\N}\mathscr{R}_k$ by enumerating first rectangles in $\mathscr{R}_0$, then those in $\mathscr{R}_1$ \emph{etc.}, and doing so in a way that in each $\mathscr{R}_k$, $k\in\N$, rectangles are enumerated in decreasing order of their horizontal side-lengths.
Assume now that $\boldsymbol{R}$ is decomposed into finitely many (say $N$) subsequences. At least one of them (call it $(R_{m_l})_{l\in\N}$) has the property that for infinitely many indices $k_0<k_1<\cdots<k_j<\cdots$ ($j\in\N$), the set $$
L_j=\{l\in\N : n_{k_j}\leq m_l\leq n_{k_j}+k_j\}
$$
contains at least $k_j/N$ elements.
Given $j\in\N$, write $\min L_j=m_{l_j}$ for $l_j\in\N$ and define $R_j^-:=\min L_j-n_{k_j}$ and $R_j^+:=\max L_j-n_{k_j}$; we know that $R_j^+-R_j^-\geq \#L_j\geq k_j/N$. Yet we compute, for $j\in\N$ (denoting by $Q_{l_j}$ the $l_j^{\text{th}}$ correct factor of the sequence $(R_{m_l})_{l\in\N}$):
\begin{multline*}
Q_{l_j}\geq |R_{n_{k_j}+R_j^-}-R_{n_{k_j}+R_j^+}|=|R^{k_j}_{R_j^-}-R^{k_j}_{R_j^+}|\\
=(2^{-R_j^-}+2^{-R_j^+})(2^{R_j^-}+2^{R_j^+}) 2^{-k_j}\lambda_{k_j}^2\geq 2^{k_j/N} |R_{l_j}|.
\end{multline*}
Hence $(Q_l)$ does \emph{not} have linear growth.
In summary, we can say that \emph{there is no way to decompose $\boldsymbol{R}$ into finitely many subsequences, the correct factors of which have linear growth}.
\end{Example}
\begin{Remark}
Example~\ref{ex.lpgood} shows that a sequence $\boldsymbol{R}$ can be $L^p$-good for all $1<p<\infty$, without its correct factors $W_{k,p}$, $k\in\N$ having linear growth (the sequence in Example~\ref{ex.lpgood} has all its terms parallel to the coordinate axes).
On the other hand, this is reasonable since in fact the linear growth on $(W_{k,p})_{k\in\N}$ is a condition actually independent of $p$ (it just rephrases the linear growth condition on $(Q_k)$, as was observed in Remark~\ref{rmq.equiv}).
\end{Remark}
Let us now start to develop the tools that will lead us to prove Theorem~\ref{THM2}; among them, the first objects we shall need are (parts of) generalized Perron trees.
\begin{comment}
\color{green}{END OF THE CHECKED PART}
Maybe we could omit this part of section 4?
***
Is there some link to be understood between some mere \emph{behavior} of the $L^p$ correct factor $(W_{k,p})$ and the fact that the sequence $(R_k)$ be $L^p$ good?
\end{Remark}
\begin{Remark}
Assuming moreover that one has $\lambda_k=2^{-k}\lambda_{k-1}$ for all $k\in\N^*$ in the latter example (which guarantees that rectangles in $\mathscr{R}_k$ all lie inside \emph{every} rectangle of $\mathscr{R}_{k-1}$), we can probably also get (using some form of the inclusion-exclusion principle) an upper bound on the correct factor $(Q_{l_j})$ of the form:
$$
|Q_{l_j}|\lesssim 2^{k_j} |R_{l_j}|.
$$
Maybe the above inequality is not exact, but still one should be able to capture the behavior of the correct factors (and hence also of the $L^p$ correct factors) in some fashion.
It is not clear at all how one could exhibit a relation of the type $|W_{k,p}| \leq F((R_k),p)$ in such cases where the behavior of $(R_k)$ is $L^p$-good. Maybe the question is just silly!
***
\end{Remark}
\end{comment}
\section{Dyadic blocks in generalized Perron trees}\label{sec.Linfini1}
In order to prove Theorem \ref{THM2}, we shall need to exploit, in quite a subtle fashion, (blocks of) generalized Perron trees introduced by Hare and Rönning \cite{HR}.
We here recall their construction and properties which will be crucial to our purposes; for the sake of clarity, we first detail the initial simple geometric construction Perron trees are based upon, and then their recursive construction.
\subsection*{The initial construction}\label{par.in-con}
A basic observation from Hare and Rönning \cite[pp.~217-8]{HR} can be formulated as follows: let $\Delta_1$ and $\Delta_2$ be two adjacent triangles lying from left to right on the horizontal axis, denote their respective side-lengths along the horizontal axis by $x$ and $y$, and their common vertical height-length by $h$ (see Figure~\ref{T1T2}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{NEW-figure3.pdf}\caption{The triangles $\Delta_1$ and $\Delta_2$}\label{T1T2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{NEW-figure4.pdf}\caption{The triangles $\Delta_1$ and $\Delta_2^*$}\label{T1T2*}
\end{center}
\end{figure}
Let $0<\alpha<1$ be a real number. Denote by $\Delta_2^*$ the triangle obtained by translating horizontally $\Delta_2$ to the left in such a way that its ``right'' side meets the ``left'' side of $\Delta_1$ at a point of height $\alpha h$ (see Figure~\ref{T1T2*}). It is easy to see that $\Delta_1\cup \Delta_2^*$ is the union of a triangle $\Delta^\alpha_0 (\Delta_1,\Delta_2)$, which is the image of the triangle $\Delta_1\cup \Delta_2$ by a similarity of ratio $\alpha$, and of two ``excess triangles'' $\Delta_i^\alpha(\Delta_1,\Delta_2)$, $i=1,2$ whose areas can be easily computed (see Figure~\ref{SDelta}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{NEW-figure5.pdf}\caption{The triangles $\Delta^\alpha_i(\Delta_1,\Delta_2)$, $i=0,1,2$}\label{SDelta}
\end{center}
\end{figure}
Hence one obtains the following estimate on the area of $\Delta_1\cup \Delta_2^*$:
$$
|\Delta_1\cup \Delta_2^*|=\left[ \alpha^2+(1-\alpha)^2\left(\frac xy + \frac yx \right)\right] |\Delta_1\cup \Delta_2|.
$$
\subsection*{Construction of Perron trees}\label{par.con-pt}
Fix $\boldsymbol{b}:=(b_k)_{k\in\N}$ an increasing sequence of nonnegative real numbers with $b_0=0$. Denote, for $k\in\N$, by $\Delta_k$ the triangle in the plane having vertices $A:=(0,1)$, $B_{k-1}:=(b_{k-1}, 0)$ and $B_k:=(b_k,0)$.
Fix now $n\in\N$ and take a ``dyadic block'' of such triangles $\Delta_{2^n}, \Delta_{2^{n}+1},\dots, \Delta_{2^{n+1}-1}$. The idea is to apply the initial construction described in the previous section to each pair of triangles of the form $(\Delta_{2^n+2j},\Delta_{2^n+{2j+1}})$, $0\leq j\leq 2^{n-1}-1$. Denoting by $\tau_0$ the identity map and by $\tau_j$, $1\leq j\leq 2^{n-1}-1$, translations along the horizontal axis such that the triangles $\Xi_j:=\tau_j \Delta^\alpha_0(\Delta_{2^n+2j},\Delta_{2^n+{2j+1}})$, $0\leq j\leq 2^{n-1}-1$ form a sequence of successively adjacent triangles and defining, for $0\leq j\leq 2^{n-1}-1$, $\Delta_{2^n+2j}':=\tau_j \Delta_{2^n+2j}$ and $\Delta_{2^n+2j+1}':=\tau_j \Delta_{2^n+2j+1}^*$,we get:
$$
\left|\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k'\right|\leq \left[ \alpha^2+G_n(1-\alpha)^2\right]\left|\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k\right|,
$$
where one defined:
$$
G_n:=\sup_{0\leq j\leq 2^{n-1}-1} \left(\frac{x_j}{y_j}+\frac{y_j}{x_j}\right),
$$
using the notations $x_j$ and $y_j$ for $b_{2^{n}+2j+1}-b_{2^n+2j}$ and $b_{2^n+2j+2}-b_{2^n+2j+1}$ respectively.
The idea now is to re-start the initial construction on pairs $(\Xi_j,\Xi_{j+1})$, $0\leq j\leq 2^{n-1}-1$ in order to translate horizontally all initial triangles $\Delta_{2^n},\dots, \Delta_{2^{n+1}-1}$ onto new positions $\Delta_{2^n}'',\dots, \Delta_{2^{n+1}-1}''$ in such a way that the area of $\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k''$ is an (even smaller) proportion of that of $\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k$, and to repeat this procedure $n$ times in total. This, of course, requires to ensure one has a uniform control on the quantities $\frac xy + \frac yx$ that appear along the procedure.
The following statement gives a condition on the sequence $\boldsymbol{b}$ in such a way that the above procedure can be conducted so as to provide translations sending triangles of the $n^{\text{th}}$ dyadic block onto new ones in a way that reduces the area of their union by a factor tending to zero when $n$ tends to infinity. Its proof will be easily supplied by adapting arguments found in Hare and Rönning \cite{HR}.
\begin{Lemma}[Dyadic blocks in generalized Perron trees]\label{lem.pt}
Assume that the sequence $\boldsymbol{b}:=(b_k)$ is as above and satisfies:
\begin{equation}\label{eq.cond-bk}
G_{\boldsymbol{b}}:=\sup_{n\in\N}\sup_{1\leq l\leq n} \left(\frac{b_{n+2l}-b_{n+l}}{b_{n+l}-b_n}+\frac{b_{n+l}-b_n}{b_{n+2l}-b_{n+l}}\right)<+\infty.
\end{equation}
Denote, for $k\in\N^*$, by $\Delta_k$ the triangle in the plane having vertices $A:=(0,1)$, $B_{k-1}:=(b_{k-1}, 0)$ and $B_k:=(b_k,0)$.
Under those assumptions, there exists a sequence $(\epsilon_n)$ of positive real numbers tending to $0$ and horizontal translations $\tau_k$, $k\in\N$ in such a way that, for any $n\in\N$, one has:
$$
\left| \bigcup_{k=2^n}^{2^{n+1}-1} \tau_k \Delta_k\right|\leq \epsilon_n \left| \bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k\right|.
$$
\end{Lemma}
A second important property we will need in order to prove Theorem~\ref{THM2} can be seen as a simple geometric estimate concerning intersections of a triangle with translates of a rectangle containing it.
\section{A simple geometric observation}\label{sec.simple-geom}
In this section we fix real numbers $0\leq b<c$ and define points $A:=(0,1)$, $B:=(b,0)$ and $C:=(c,0)$ and let $\Delta$ denote the (full) triangle $ABC$. Let $\alpha>0$ be such that one has $AB\geq \alpha BC$.
Denote then by $B'$ and $C'$ the points defined by $\overrightarrow{AB'}=\frac 32 \overrightarrow{AB}$ and $\overrightarrow{AC'}=\frac 32 \overrightarrow{AC}$ and let $\tilde{P}$ be the smallest rectangle having $AB'$ as one of its sides, and containing the triangle $AB'C'$ (see figure~\ref{fig.Rtilde}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=1]{NEW-figure6.pdf}\caption{The triangles $ABC$, $AB'C'$ and the rectangle $\tilde{P}$}\label{fig.Rtilde}
\end{center}
\end{figure}
Finally define $P$ to be the rectangle obtained from $\tilde{P}$ by translating it in such a way that its center is the origin, and let $V$ denote the trapezium $BCC'B'$.
The following estimate, although simple, will be crucial in the sequel.
\begin{Lemma}
For any $x \in V$ one has $$\left|\left(x+P\right) \cap \Delta \right| \geq \frac{\min(\alpha,1)}{72}\left|P\right|.$$
\end{Lemma}
\begin{proof}
Denote by $I$ the middle of the segment $AB'$, by $H$ the point on the segment $AB$ having the property that $CH$ is orthogonal to $AB$ and by $J$ the intersection of $BC$ with the line orthogonal to $AB$ and passing through $I$ (see figure~\ref{fig.Rtilde} again). Finally denote by $D'$ the ``lower right'' vertex of $\tilde{P}$ and let $\ell$ denote the length of the segment $B'D'$.
If $\theta_B\in (0,\frac{\pi}{2})$ stands for the (non-oriented) angle between the horizontal line and $AB$, one obviously has $\tan\theta_B=\frac{1}{b}$.
It is also plain to see, using the triangle $B'C'D'$, that one has: $$\ell=B'D'=B'C'\sin\theta_B=\frac{3}{2} BC\cdot\frac{1}{\sqrt{1+\cot^2\theta_B}}=\frac 32 \cdot\frac{BC}{\sqrt{1+b^2}}=\frac{3}{2}\cdot \frac{BC}{AB}.
$$
Using the triangle $BHC$ in a similar fashion, one gets $CH=BC\sin\theta_B$ and $BH=BC\cos\theta_B$. Since one easily gets $BI=\frac 14 AB$, there comes:
$$
IJ=CH\cdot \frac{BI}{BH}=\frac 14 AB\tan\theta_B=\frac{AB}{4b}.
$$
We hence finally compute:
$$
\frac{IJ}{\ell}=\frac{1}{6b}\cdot\frac{(AB)^2}{BC}=\frac{1}{6}\cdot \frac{AB}{b}\cdot \frac{AB}{BC}=\frac 16 \cdot \frac{\sqrt{1+b^2}}{b}\cdot \frac{AB}{BC}\geq \frac 16 \alpha,
$$
using the assumption we made that $AB\geq\alpha BC$.
The previous observations ensure that the intersection of the rectangle $B'+P$ with the triangle $\Delta=ABC$ contains the triangle formed by the points $B$, $I$ and $J':=I+\frac 16 \alpha\overrightarrow{B'D'}$; hence one has, using the obvious equality $BI=\frac 14 AB=\frac 16 AB'$:
$$
|(B'+P)\cap \Delta|\geq \frac 12 BI\cdot \frac{1}{6}\alpha\ell=\frac{1}{72}\alpha AB'\cdot \ell=\frac{1}{72}\alpha |P|.
$$
Denoting now $K$, $L$ and $M$ respectively the intersections of the ``upper'' side of $C'+P$ parallel to $AB'$ with the lines $AC$, $BC$ and $B'C'$ respectively, one computes:
$$
BL=B'M=\frac 12 B'C'=\frac 34 BC.
$$
It hence follows that $CL=\frac 14 BC$ and that the intersection of $\Delta=ABC$ with $C'+P$, which is the triangle $CKL$, is a triangle homothetic to $ABC$ with ratio $\frac 14$, and thus similar to $AB'C'$ with ratio $\frac 16$, yielding in turn:
$$
|(C'+P)\cap \Delta|=\frac{1}{36} |AB'C'|=\frac{1}{72}|P|.
$$
Now denote by $V$ the trapezion $BCC'B$; it is clear that one has:
\begin{equation}\label{eq.inf-x-V}
\inf_{x\in V} |(x+P)\cap \Delta|\geq \min\{|(B'+P)\cap \Delta|,|(C'+P)\cap \Delta|\}\geq \frac{1}{72}\min(\alpha,1)|P|.
\end{equation}
This completes the proof of the lemma.
\end{proof}
We are now ready to prove Theorem~\ref{THM2}.
\section{Proof of Theorem \ref{THM2}}\label{sec.Linfini2}
Fix, as before, a nondecreasing sequence $\boldsymbol{b}=(b_k)$ of positive real numbers satisfying $b_0=0$ and define points $B_k:=(b_k,0)$ for all $k\in\N$ and let $A:=(0,1)$; for each $k\in\N^*$ we then denote by $\Delta_k$ the triangle $AB_{k-1}B_{k}$. For each $k\in\N$, we now associate to $\Delta_k$ a rectangle $P_k$ centered at the origin according to the procedure described in the previous section~\ref{sec.simple-geom}.
Fix also a sequence $\boldsymbol{\delta}=(\delta_k)_{k\in\N^*}$ of positive real numbers and consider the sequence of rectangles $\boldsymbol{R}=\boldsymbol{R}(\boldsymbol{b},\boldsymbol{\delta})=(R_k)_{k\in\N^*}$ defined by $R_k:=\delta_n P_k$ if $k\in\N^*$ satisfies $2^n\leq k<2^{n+1}$ for a given $n\in\N$; remember that we called $\boldsymbol{\delta}$ an \emph{admissible sequence} in case one has $\diam R_k\to 0$, $k\to\infty$.
Let us now (re-)state and prove the main theorem of our paper, referred to before as Theorem~\ref{THM2}, which we recall now.
\begin{Theorem}\label{THM2-v2}
Assume that the sequence $\boldsymbol{b}$ satisfies the following two conditions:
\begin{enumerate}
\item[(i)] there exists a constant $c>0$ such that one has $1 + b_{k-1}^2 \geq c(b_{k} -b_{k-1})^2$ for all $k\in\N^*$;
\item[(ii)] one has: $$G_{ \boldsymbol{b} }:=\sup_{n\in\N}\sup_{1\leq l\leq n} \left(\frac{b_{n+2l}-b_{n+l}}{b_{n+l}-b_n}+\frac{b_{n+l}-b_n}{b_{n+2l}-b_{n+l}}\right)<+\infty.$$
\end{enumerate}
Under those assumptions, the process $\boldsymbol{T}_{\boldsymbol{R}}$ associated to $\boldsymbol{R} = \boldsymbol{R}(\boldsymbol{b},\boldsymbol{\delta})$ is $L^\infty$-bad for any admissible sequence $\boldsymbol{\delta}$.
\end{Theorem}
\begin{proof}
We keep the notations introduced just above and let $(\epsilon_n)_{n\in\N}$ and $(\tau_k)_{k\in\N^*}$ be associated to $\boldsymbol{b}$ (and the corresponding sequence of triangles $\Delta_1$, $\Delta_2$,$\dots$) according to Lemma~\ref{lem.pt}. For $k\in\N^*$, denote also by $V_k$ the trapezium associated to $\Delta_k$ according to the procedure described in section~\ref{sec.simple-geom}. Finally define also, for $n\in\N$:
$$ K^n:=\bigcup_{k=2^n}^{2^{n+1}-1} \tau_k \Delta_k\quad\text{and}\quad V^n:= \bigcup_{k=2^n}^{2^{n+1}-1}\tau_k V_k. $$
Since for any $n\in\N$, $V^n$ contains a similar copy of the triangle $\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k$ with ratio $\frac 13$, it follows one one hand that one has:
\begin{equation}\label{eq.19}
|V^n| \geq\frac 19 |\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k|.
\end{equation}
On the other hand, we have, for each $k\in\N^*$, using hypothesis (i):
$$
AB_{k-1}=\sqrt{1+b_{k-1}^2}\geq \sqrt{c} (b_k-b_{k-1})=\sqrt{c}B_{k-1}B_k.
$$
Hence the estimates obtained in section~\ref{sec.simple-geom} hold for $\Delta_k$, $P_k$ and $V_k$ with $\alpha=\sqrt{c}$.
Given $n\in\N$, it now follows from the definition of $V^n$ and from the observations of the previous section that, for each $x\in V^n$, there exists $2^n\leq k\leq 2^{n+1}-1$ such that one has $x\in \tau_k V_k$ and hence also:
$$
|(x+P_k)\cap \tau_k \Delta_k|\geq \frac{\min(\alpha,1)}{72}|P_k|;
$$
and hence also, defining $t_0:= \frac{\min(\alpha,1)}{72}$:
$$
\frac{1}{|P_k|} |(x+P_k)\cap K^n|\geq \frac{1}{|P_k|} |(x+P_k)\cap \tau_k\Delta_k|\geq t_0.
$$
Fix now an admissible sequence $\boldsymbol{\delta}$ and define $\boldsymbol{R}=\boldsymbol{R}(\boldsymbol{b},\boldsymbol{\delta})$ as above. For $n\in\N$ and $x\in\delta_n V^n$, there comes $\frac{x}{\delta_n}\in V^n$ so that the latter computations ensure the existence of an integer $2^n\leq k\leq 2^{n+1}-1$ for which one has:
$$
\frac{1}{|R_k|} |(x+R_k)\cap \delta_nK^n|\geq \frac{1}{|P_k|} \left|\left(\frac{x}{\delta_n}+P_k\right)\cap K^n\right|\geq t_0,
$$
which rewrites:
$$
\frac{1}{|R_k|} \int_{x+R_k} \chi_{\delta_n K^n}\geq t_0,
$$
and implies that one has $T^*(\chi_{\delta_n K^n})(x)\geq t_0$, where $\boldsymbol{T}=\boldsymbol{T}_{\boldsymbol{R}}$ stands for the process associated to $\boldsymbol{R}=(R_k)_{k\in\N^*}$.
One hence obtains the following inclusion for each $n\in\N$:
$$
\delta_n V^n\subseteq \left\{ T^*\chi_{\delta_n K^n}\geq t_0\right\}.
$$
Using \eqref{eq.19} and Lemma~\ref{lem.pt}, the above inclusion yields in particular, for each $n\in\N$:
$$
\frac{\left|\left\{ T^*\chi_{\delta_n K^n}\geq t_0\right\}\right|}{|\delta_n K^n|}\geq \frac{|\delta_n V^n|}{|\delta_n K^n|}= \frac{|V^n|}{|K^n|}\geq \frac 19 \frac{\left|\bigcup_{k=2^n}^{2^{n+1}-1} \Delta_k\right|}{\left|\bigcup_{k=2^n}^{2^{n+1}-1} \tau_k\Delta_k\right|}\geq \frac{1}{9\epsilon_n}\to\infty.
$$
But then (see equation \eqref{eq.linfini-good}, p.~\pageref{eq.linfini-good}), $\boldsymbol{T}$ cannot be $L^\infty$-good, and the proof is complete.
\end{proof}
\begin{Remark}
In the conditions of the statement of Theorem~\ref{THM2-v2}, the process $\boldsymbol{T}$, $L^\infty$-bad, is also $L^p$-bad for any $1\leq p<\infty$.
It hence follows from an observation we present in the Appendix (see Theorem~\ref{thm.appendix}) that, for any $1\leq p<\infty$, the set of functions $f\in L^p(\R^2)$ for which one has $\mathop{\overline{\lim}}_{k\to\infty} |T_kf|<+\infty$ on a set of positive Lebesgue measure, is \emph{meager} (or a \emph{first category} subset) in $L^p(\R^2)$.
\end{Remark}
\section{Good functions for our bad processes}
As we mentioned before, we will devote this appendix to show that there are few ``good'' functions in $L^p$ for $L^p$-bad processes. Since this fact holds in a much wider generality than the context of rectangular averaging processes we discussed, we formulate it here for convolution processes, since it seemed to us to be worthwile noticing.
More precisely, we prove the following result.
\begin{Theorem}\label{thm.appendix}
Assume that $(\varphi_k)\subseteq L^1_+(\R^d)$ is a collection of nonnegative integrable functions in $\R^d$ and that $1\leq p<\infty$ is a fixed real number.
Associate to $(\varphi_k)$ a process $\boldsymbol{T}=(T_k)$ by letting, for a locally integrable $f$, $T_kf:=\varphi_k*f$. If $\boldsymbol{T}$ is $L^p$-bad, then the set:
$$
\mathscr{G}:=\left\{f\in L^p(\R^d):\mathop{\overline{\lim}}_{k\to\infty} |T_kf|<+\infty\text{ on a set of positive Lebesgue measure}\right\}
$$
is meager in $L^p(\R^d)$.
\end{Theorem}
\begin{proof}
Define, for $n\in\N$, a maximal operator $T^*_n$ by letting, for a locally integrable $f$ and $x\in\R^d$:
$$
T^*_nf(x):=\sup_{k\geq n} |\varphi_k*f(x)|.
$$
Observe that our assumption can be reformulated by saying that $T^*:=T_0^*$ fails to satisfy a weak $(p,p)$-inequality.
For each $n\in\N$, define another operator $S_n$ by letting, for $f$ measurable and $x\in\R^d$:
$$
S_nf(x):=\max_{0\leq k< n} |T_k f(x)|.
$$
Since one has, for each $f\in L^p(\R^d)$:
$$
\|T_kf\|_p\leq \|\varphi_k\|_1\|f\|_p,
$$
it follows that any $T_k$ has weak type $(p,p)$ and hence that the same holds for the operator $S_n$ for any $n\in\N$.
Yet one has, for a measurable $f$ and $x\in\R^d$ and any $n\in\N$:
$$
T^*f(x)=\max\left\{S_nf(x),T_n^*f(x)\right\},
$$
and it hence follows that $T_n^*$ fails to satisfy a weak type $(p,p)$ inequality for any $n\in\N$.
It then follows from \cite[Proposition~1, p.~441]{STEINHA} that for each $n\in\N$ there exists $f_n\in L^p_+(\R^d)$ for which one has $T_n^*f_n=+\infty$ a.e. on $\R^d$. We can of course assume, without loss of generality, that one has $\|f_n\|_p=1$.
Fix now $n\in\N$ and a cube of side $1$, $Q\subseteq\R^d$, and denote by $\mathscr{L}^d\hel Q$ the (outer) Lebesgue measure restricted to $Q$ (\emph{i.e.} defined for a subset $E\subseteq\R^d$ by $\mathscr{L}^d\hel Q(E):=|Q\cap E|$). Consider for each integer $k\in\N$ the operator:
$$
T_k^Q:L^p(\R^d)\to L^0(\R^d,\mathscr{L}^d\hel Q), f\mapsto T_kf.
$$
The operator $T_k^Q$ is continuous in measure since one has, for any $\epsilon$ and any sequence $(f_j)\subseteq L^p(\R^d)$ satisfying $\|f_j-f\|_p\to 0$:
\begin{multline*}
\mathscr{L}^d\hel Q(\{ |T_kf_j-T_kf|>\epsilon\})\leq |\{ |T_kf_j-T_kf|>\epsilon\}|\\
\leq \frac{1}{\epsilon} \|\varphi*(f_j-f)\|_1\leq \frac{1}{\epsilon}\|\varphi_k\|_1\|f_j-f\|_p,
\end{multline*}
implying obviously that one has $\mathscr{L}^d\hel Q(\{ |T_kf_j-T_kf|>\epsilon\})\to 0$, $j\to\infty$.
Applying Del Junco and Rosenblatt's \cite[Theorem~1.1]{DJR} to $\mathscr{B}=L^p(\R^d)$, $\mu=\mathscr{L}^d\hel Q$ and $(T_k)_{k\geq n}$, we find that there is a dense $\mathscr{G}_\delta$ subset $\mathscr{X}_{Q,n}\subseteq L^p(\R^d)$ such that one has $T_n^*f=+\infty$, $(\mathscr{L}^d\hel Q)$-a.e., that is $T_n^*f=+\infty$ a.e. on $Q$, for any $f\in \mathscr{X}_{Q,n}$.
Now let:
$$
\mathscr{X}_n:=\bigcap_{\nu\in\Z^n} \mathscr{X}_{\nu+Q,n}.
$$
Obviously $\mathscr{X}_n$ is a dense $\mathscr{G}_\delta$ in $L^p(\R^d)$; for $f\in \mathscr{X}_n$ we then have $T_n^*f=+\infty$ a.e. on $\nu+Q$ for any $\nu\in\Z^n$, and hence $T_n^*f=+\infty$ for a.e. $x\in\R^d$.
Finally let $\mathscr{X}:=\bigcap_{n\in\N} \mathscr{X}_n$, observe that $\mathscr{X}$ is a dense $\mathscr{G}_\delta$ subset of $L^p(\R^d)$ and that for $f\in \mathscr{X}$ we have, for all $n\in\N$ and a.e. $x\in\R^d$:
$$
T_n^* f(x)=+\infty.
$$
Hence we have, for $f\in\mathscr{X}$ and a.e. $x\in\R^d$:
$$
\mathop{\overline{\lim}}_{k\to\infty} |T_kf(x)|=\inf_{n\in\N}\sup_{k\geq n}|T_kf(x)|=\inf_{n\in\N} T_n^*f(x)=+\infty.
$$
The result now immediately follows.
\end{proof}
|
1,314,259,995,851 | arxiv |
\section*{Acknowledgment}
The authors sincerely thank Aviad Rubinstein for the suggestion of using set-cover to prove hardness. The authors sincerely thank Dana Moshkovitz for pointing out some references about the hardness result of set-cover. The authors would also like to thank Mika G\"{o}\"{o}s, Rasmus Kyng, Zico Kolter, Jelani Nelson, Eric Price, Milan Rubinstein, Jacob Steinhardt, Zhengyu Wang, Eric Wong and David P. Woodruff for useful discussions. Luca Daniel and Tsui-Wei Weng acknowledge the partial support of MIT-Skoltech program and MIT-IBM Watson AI Lab. Huan Zhang and Cho-Jui Hsieh acknowledge the support of NSF via IIS-1719097 and the computing resources provided by Google Cloud and NVIDIA.
\section{Details of Experiments in Section~\ref{sec:exp}}
\label{app:exp}
\subsection{Methods}
Below, we give detailed descriptions on the methods that we compare in Table~\ref{tb:smallnetwork_and_large}, Table~\ref{tb:smallnetwork} and Table~\ref{tb:largenetwork_app}:
\begin{itemize}
\item \textbf{Fast-Lin}\xspace: Our proposed method of directly bounding network output via \textbf{lin}ear upper/lower bounds for ReLU, as discussed in Section~\ref{sec3:convexbnd} and Algorithm~\ref{alg:fast-lin};
\item \textbf{Fast-Lip}\xspace: Our proposed method based on bounding local \textbf{Lip}schitz constant, in Section \ref{sec3:gradbnd} and Algorithm~\ref{alg:fast-lip};
\item \textbf{Reluplex}\xspace: \textbf{Reluplex}\xspace~\cite{katz2017reluplex} is a satisfiability modulo theory (SMT) based solver which delivers a true minimum distortion, but is very computationally expensive;
\item \textbf{LP-Full}\xspace: A linear programming baseline method with formulation borrowed from~\cite{zico17convex}. Note that we solve the primal LP formulation \textit{exactly} to get a best possible bound. This variant solves \textbf{full} relaxed \textbf{LP}\xspace problems at \textit{every} layer to give a final ``adversarial polytope''. Similar to our proposed methods, it only gives a lower bound. We extend this formulation to $p=2$ case, where the input constraint becomes quadratic and requires a quadratic constrained programming (QCP) solver, which is usually slower than LP solvers.
\item \textbf{LP}\xspace: Similar to \textbf{LP-Full}\xspace, but this variant solves only \textbf{one} LP problem for the full network at the output neurons and the layer-wise bounds for the neurons in hidden layers are solved by \textbf{Fast-Lin}\xspace. We also extend it to $p=2$ case with QCP constraints on the inputs. \textbf{LP}\xspace and \textbf{LP-Full}\xspace are served as our baselines to compare with \textbf{Fast-Lin}\xspace and \textbf{Fast-Lip}\xspace;
\item \textbf{Attacks}: Any successful adversarial example gives a valid \textit{upper bound} for the minimum adversarial distortion. For larger networks where \textbf{Reluplex}\xspace is not feasible, we run adversarial attacks and obtain an upper bound of minimal adversarial distortions to compare with. We apply the $\ell_2$ and $\ell_\infty$ variants of Carlini and Wagner's attack (CW)~\cite{carlini2017towards} to find the best $\ell_2$ and $\ell_\infty$ distortions. We found that the CW $\ell_\infty$ attack usually finds adversarial examples with smaller $\ell_\infty$ distortions than using PGD (projected gradient descent). We use EAD~\cite{chen2017ead}, a Elastic-Net regularized attack, to find adversarial examples with small $\ell_1$ distortions. We run CW $\ell_2$ and $\ell_\infty$ attacks for 3,000 iterations and EAD attacks for 2,000 iterations;
\item \textbf{CLEVER}\xspace: \textbf{CLEVER}\xspace~\cite{weng2017evaluating} is an attack-agnostic robustness score based on local Lipschitz constant estimation and provides an estimated lower-bound. It is capable of performing robustness evaluation for large-scale networks but is not a certified lower bound;
\item \textbf{Op-norm}\xspace: Operator norms of weight matrices were first used in~\cite{szegedy2013intriguing} to give a robustness lower bound. We compute the $\ell_p$ induced norm of weight matrices of each layer and use their product as the global Lipschitz constant $L_q^j$. A valid lower bound is given by $g(\bm{x_0})/L_q^j$ (see Section \ref{sec3:gradbnd}). We only need to pre-compute the operator norms once for all the examples.
\end{itemize}
\subsection{Setup}
We use MNIST and CIFAR datasets and evaluate the performance of each method in MLP networks with up to 7 layers or over 10,000 neurons, which is the largest network size for non-trivial and guaranteed robustness verification to date. We use the same number of hidden neurons for each layer and denote a $m$-layer network with $n$ hidden neurons in each layer as $m\times[n]$. Each network is trained with a grid search of learning rates from $\{0.1, 0.05, 0.02, 0.01, 0.005\}$ and weight decays from $\{10^{-4}, 10^{-5}, 10^{-6}, 10^{-7}, 10^{-8}\}$ and we select the network with the best validation accuracy. We consider both targeted and untargeted robustness under $\ell_p$ distortions ($p=1,2,\infty$); for targeted robustness, we consider three target classes: a random class, a least likely class and a runner-up class (the class with second largest probability). The reported average scores are an average of 100 images from the test set, with images classified wrongly skipped. Reported time is per image. We use binary search to find the certified lower bounds in \textbf{Fast-Lin}\xspace, \textbf{Fast-Lip}\xspace, \textbf{LP}\xspace and \textbf{LP-Full}\xspace, and the maximum number of search iterations is set to 15.
We implement our algorithm using Python (with Numpy and Numba)\footnote{\url{https://github.com/huanzhang12/CertifiedReLURobustness}}, while for the LP based method we use the highly efficient Gurobi commercial LP solver with Python Interface. All experiments are conducted in single thread mode (we disable the concurrent solver in Gurobi) on a Intel Xeon E5-2683v3 (2.0 GHz) CPU. Despite the inefficiency of Python, we still achieve two orders of magnitudes speedup compared with \textbf{LP}\xspace, while achieving a very similar lower bound. Our methods are automatically parallelized by Numba and can gain further speedups on a multi-core CPU, but we disabled this parallelization for a fair comparison to other methods.
\subsection{Discussions}
In Table~\ref{tb:smallnetwork_and_large}a (full Table: Table~\ref{tb:smallnetwork}), we compare the lower bound $\beta_L$ computed by each algorithm to the true minimum distortion $r_0$ found by \textbf{Reluplex}\xspace. We are only able to verify 2 and 3 layer MNIST with 20 neurons per hidden layer within reasonable time using \textbf{Reluplex}\xspace. It is worth noting that the input dimension (784) is very large compared to the network evaluated in~\cite{katz2017reluplex} with only 5 inputs. Lower bounds found by \textbf{Fast-Lin}\xspace is very close to \textbf{LP}\xspace, and the gaps are within 2-3X from the true minimum distortion $r_0$ found by \textbf{Reluplex}\xspace. The upper bound given by CW $\ell_\infty$ are also very close to $r_0$.
In Table~\ref{tb:smallnetwork_and_large}b (full Table: Table~\ref{tb:largenetwork_app}), we compare \textbf{Fast-Lin}\xspace, \textbf{Fast-Lip}\xspace with \textbf{LP}\xspace and \textbf{Op-norm}\xspace on larger networks with up to over ten thousands hidden neurons. \textbf{Fast-Lin}\xspace and \textbf{Fast-Lip}\xspace are significantly faster than \textbf{LP}\xspace and are able to verify much larger networks (\textbf{LP}\xspace becomes very slow to solve exactly on 4-layer MNIST with 4096 hidden neurons, and is infeasible for even larger CIFAR models). \textbf{Fast-Lin}\xspace achieves a very similar bound comparing with results of \textbf{LP}\xspace over all smaller models, but being \textit{over two orders of magnitude faster}. We found that \textbf{Fast-Lip}\xspace can achieve better bounds when $p = 1$ in two-layers networks, and is comparable to \textbf{Fast-Lin}\xspace in shallow networks. Meanwhile, we also found that \textbf{Fast-Lin}\xspace scales better than \textbf{Fast-Lip}\xspace for deeper networks, where \textbf{Fast-Lin}\xspace usually provides a good bound even when the number of layers is large. For deeper networks, neurons in the last few layers are likely to have uncertain activations, making \textbf{Fast-Lip}\xspace being too pessimistic. However, \textbf{Fast-Lip}\xspace outperforms the global Lipschitz constant based bound (\textbf{Op-norm}\xspace) which quickly goes down to 0 when the network goes deeper, as \textbf{Fast-Lip}\xspace is bounding the \textit{local} Lipschitz constant to compute robustness lower bound. In Table~\ref{tb:largenetwork_app}, we also apply our method to MNIST and CIFAR models to compare the minimum distortion for \textit{untargeted} attacks. The computational benefit of \textbf{Fast-Lin}\xspace and \textbf{Fast-Lip}\xspace is more significant than $\textbf{LP}\xspace$ because \textbf{LP}\xspace needs to solve $n_m$ objectives (where $n_m$ is the total number of classes), whereas the cost of our methods stay mostly unchanged as we get the bounds for all network outputs simultaneously.
In Table~\ref{tb:distill}, we compute our two proposed lower bounds on neural networks with defending techniques to evaluate the effects of defending techniques (e.g. how much robustness is increased). We train the network with two defending methods, defensive distillation (DD)~\cite{papernot2016distillation} and adversarial training~\cite{madry2017towards} based on robust optimization. For DD we use a temperature of 100, and for adversarial training, we train the network for 100 epochs with adversarial examples crafted by 10 iterations of PGD with $\epsilon=0.3$. The test accuracy for the adversarially trained models dropped from 98.5\% to 97.3\%, and from 98.6\% to 98.1\%, for 3 and 4 layer MLP models, respectively. We observe that both defending techniques can increase the computed robustness lower bounds, however adversarial training is significantly more effective than defensive distillation. The lower bounds computed by \textbf{Fast-Lin}\xspace are close to the desired robustness guarantee $\epsilon=0.3$.
\input{tab_reluplex_compare_app}
\input{tab_mnist_cifar_compare_appendix}
\section{Hardness}\label{app:hardness}In this section we show that finding the minimum adversarial distortion with a certified approximation ratio is hard. We first introduce some basic definitions and theorems in Section~\ref{sec:hardness_definition}. We provide some backgrounds about in-approximability reduction in Section~\ref{sec:hardness_pcp}.
Section~\ref{sec:hardness_warmup} gives a warmup proof for boolean case and then Section~\ref{sec:hardness_main_result} provides the proof of our main hardness result (for network with real inputs).
\subsection{Definitions}\label{sec:hardness_definition}
We provide some basic definitions and theorems in this section. First, we define the classic $\mathsf{3SAT}$ problem.
\begin{definition}[$\mathsf{3SAT}$ problem]\label{def:3sat}
Given $n$ variables and $m$ clauses in a conjunctive normal form $\mathsf{CNF}$ formula with the size of each clause at most $3$, the goal is to decide whether there exists an assignment to the $n$ Boolean variables to make the $\mathsf{CNF}$ formula to be satisfied.
\end{definition}
For the $\mathsf{3SAT}$ problem in Definition~\ref{def:3sat}, we introduce the Exponential Time Hypothesis (ETH), which is a common concept in complexity field.
\begin{hypothesis}[Exponential Time Hypothesis ($\mathsf{ETH}$) \cite{ipz98}]
\label{hypo:eth}
There is a $\delta > 0$ such that the $\mathsf{3SAT}$ problem defined in Definition~\ref{def:3sat} cannot be solved in $O(2^{\delta n})$ time.
\end{hypothesis}
ETH had been used in many different problems, e.g. clustering \cite{abjk18,cmrr18}, low-rank approximation \cite{rsw16,swz17,swz17b,swz18}. For more details, we refer the readers to a survey \cite{lms13}.
Then we define another classical question in complexity theory, the $\mathsf{SET}$-$\mathsf{COVER}$ problem, which we will use in our proof. The exact $\mathsf{SET}$-$\mathsf{COVER}$ problem is one of Karp's 21 NP-complete problems known to be NP-complete in 1972:
\begin{definition}[$\mathsf{SET}$-$\mathsf{COVER}$]
\label{def:set-cover}
The inputs are $U,S$; $U = \{ 1,2, \cdots, n \}$ is a universe, $P(U)$ is the power set of $U$, and $S = \{ S_1, \cdots, S_m \} \subseteq P(U)$ is a family of subsets, $\cup_{j \in [m]} S_j = U$. The goal is to give a {\rm YES/NO} answer to the follow decision problem: \\
\centerline{
Does there exist a set-cover of size $t$, i.e., $\exists C \subseteq [m]$, such that $\cup_{j \in C} S_j = U$ with $|C| = t$?}
\end{definition}
Alternatively, we can also state the problem as finding the minimum set cover size $t_0$, via a binary search on $t$ using the answers of the decision problem in~\ref{def:set-cover}. The Approximate $\mathsf{SET}$-$\mathsf{COVER}$ problem is defined as follows.
\begin{definition}[Approximate $\mathsf{SET}$-$\mathsf{COVER}$]
The inputs are $U,S$; $U = \{ 1,2, \cdots, n \}$ is a universe, $P(U)$ is the power set of $U$, and $S = \{ S_1, \cdots, S_m \} \subseteq P(U)$ is a family of subsets, $\cup_{j \in [m]} S_j = U$. The goal is to distinguish between the following two cases: \\
\rm{(\RN{1})}: There exists a small set-cover, i.e., $\exists C \subseteq [m]$, such that $\cup_{j \in C} S_j = U$ with $|C|\leq t$.\\
\rm{(\RN{2})}: Every set-cover is large, i.e., every $C \subseteq [m]$ with $\cup_{j \in C} S_j = U$ satisfies that $|C| > \alpha t$, where $\alpha >1$.
\end{definition}
An oracle that solves the Approximate $\mathsf{SET}$-$\mathsf{COVER}$ problem outputs an answer $t_U \geq t_0$ but $t_U \leq \alpha t_0$ using a binary search, where $t_U$ is an upper bound of $t_0$ with a guaranteed approximation ratio $\alpha$. For example, we can use a greedy (rather than exact) algorithm to solve the $\mathsf{SET}$-$\mathsf{COVER}$ problem, which cannot always find the smallest size of set cover $t_0$, but the size $t_U$ given by the greedy algorithm is at most $\alpha$ times as large as $t_0$.
In our setting, we want to investigate the hardness of finding the lower bound with a guaranteed approximation ration, but an approximate algorithm for $\mathsf{SET}$-$\mathsf{COVER}$ gives us an upper bound of $t_0$ instead of an lower bound of $t_0$. However, in the following proposition, we show that finding an lower bound with an approximation ratio of $\alpha$ is as hard as finding an upper bound with an approximation ratio of $\alpha$.
\begin{proposition}
\label{prop:lower-higher}
Finding a lower bound $t_L$ for the size of the minimal set-cover (that has size $t_0$) with an approximation ratio $\alpha$ is as hard as finding an upper bound $t_U$ with an approximation ratio $\alpha$.
\end{proposition}
\begin{proof}
If we find a lower bound $t_L$ with $\frac{t_0}{\alpha} \leq t_L \leq t_0 $, by multiplying both sides by $\alpha$, we also find an upper bound $t_U = \alpha t_L$ which satisfies that $t_0 \leq t_U \leq \alpha t_0$. So finding an lower bound with an approximation ratio $\alpha$ is at least as hard as finding an upper bound with an approximation ratio $\alpha$. The converse is also true.
\end{proof}
$\mathsf{SET}$-$\mathsf{COVER}$ is a well-studied problem in the literature. Here we introduce a theorem from ~\cite{rs97,ams06,ds14} which implies the hardness of approximating $\mathsf{SET}$-$\mathsf{COVER}$.
\begin{theorem}[\cite{rs97,ams06,ds14}]\label{thm:approx_set_cover}
Unless $\mathsf{NP}=\mathsf{P}$, there is no polynomial time algorithm that gives a $(1-o(1))\ln n$-approximation to $\mathsf{SET}$-$\mathsf{COVER}$ problem with universe size $n$.
\end{theorem}
We now formally define our neural network robustness verification problems.
\iffalse
\begin{definition}[$\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{R}$)]\label{def:robust_net_real}
Given an $n$ hidden nodes {\rm ReLU} neural network $F(x) : \mathbb{R}^d \rightarrow \mathbb{R}$ where weights are all fixed, for a query input vector $x \in \mathbb{R}^d$ with $F(x) \leq 0$. The goal is to figure out the smallest $r$ such that there exists a vector $y$ with $\| x - y \|_1 \leq r$ and $F(y) > 0$.
\end{definition}
\fi
\begin{definition}[$\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{R}$)]\label{def:robust_net_real}
Given an $n$ hidden nodes {\rm ReLU} neural network $F(x) : \mathbb{R}^d \rightarrow \mathbb{R}$ where all weights are fixed, for a query input vector $x \in \mathbb{R}^d$ with $F(x) \leq 0$. The goal is to give a {\rm YES/NO} answer to the following decision problem:\\
\centerline{
Does there exist a $y$ with $\| x - y \|_1 \leq r$ such that $F(y) > 0$?}
\end{definition}
With an oracle of the decision problem available, we can figure out the smallest $r$ (defined as $r_0$) such that there exists a vector $y$ with $\| x - y \|_1 \leq r$ and $F(y) > 0$ via a binary search.
We also define a binary variant of the $\mathsf{ROBUST}$-$\mathsf{NET}$ problem, denoted as $\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{B}$). The proof for this variant is more straightforward than the real case, and will help the reader understand the proof for the real case.
\iffalse
\begin{definition}[$\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{B}$)]
Given an $n$ hidden nodes {\rm ReLU} neural network $F(x) : \{0,1\}^d \rightarrow \{0,1\}$ where weights are all fixed, for a query input vector $x \in \{0,1\}^d$ with $F(x) = 0$. The goal is to figure out the smallest $r$ such that there exists a vector $y$ with $\| x - y \|_1 \leq r$ and $F(y)=1$.
\end{definition}
\fi
\begin{definition}[$\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{B}$)]
Given an $n$ hidden nodes {\rm ReLU} neural network $F(x) : \{0,1\}^d \rightarrow \{0,1\}$ where weights are all fixed, for a query input vector $x \in \{0,1\}^d$ with $F(x) = 0$. The goal is to give a {\rm YES/NO} answer to the following decision problem:\\
\centerline{
Does there exist a $y$ with $\| x - y \|_1 \leq r$ such that $F(y) = 1$?}
\end{definition}
Then, we define the approximate version of our neural network robustness verification problems.
\begin{definition}[Approximate $\mathsf{ROBUST}$-$\mathsf{NET}$(${\mathbb{B}}$)]\label{def:approx_net_bool}
Given an $n$ hidden nodes {\rm ReLU} neural network $F(x) : \{0,1\}^d \rightarrow \{0,1\}$ where weights are all fixed, for a query input vector $x \in \{0,1\}^d$ with $F(x) = 0$. The goal is to distinguish the following two cases :\\
\rm{(\RN{1}):} There exists a point $y$ such that $\| x - y \|_1 \leq r$ and $F(y)=1$.\\
\rm{(\RN{2}):} For all $y$ satisfies $\| x - y \|_1 \leq \alpha r$, the $F(y) = 0$, where $\alpha >1$.
\end{definition}
\begin{definition}[Approximate $\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{R}$)]\label{def:approx_net_real}
Given an $n$ hidden nodes {\rm ReLU} neural network $F(x) : \mathbb{R}^d \rightarrow \mathbb{R}$ where weights are all fixed, for a query input vector $x \in \mathbb{R}^d$ with $F(x) \leq 0$. The goal is to distinguish the following two cases :\\
\rm{(\RN{1}):} There exists a point $y$ such that $\| x - y \|_1 \leq r$ and $F(y) > 0$.\\
\rm{(\RN{2}):} For all $y$ satisfies $\| x - y \|_1 \leq \alpha r$, the $F(y) \leq 0$, where $\alpha >1$.
\end{definition}
As an analogy to $\mathsf{SET}$-$\mathsf{COVER}$, an oracle that solves the Approximate $\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{R}$) problem can output an answer $r \geq r_0$ but $r \leq \alpha r_0$, which is an upper bound of $r_0$ with a guaranteed approximation ratio $\alpha$. With a similar statement as in Proposition~\ref{prop:lower-higher}, if we divide the answer $r$ by $\alpha$, then we get a lower bound $r^\prime = \frac{r}{\alpha}$ where $r^\prime \geq \frac{r_0}{\alpha}$, which is a lower bound with a guaranteed approximation ratio. If we can solve Approximate $\mathsf{ROBUST}$-$\mathsf{NET}$($\mathbb{R}$), we can get a lower bound with a guaranteed approximation ratio, which is the desired goal of our paper.
\subsection{Background of the PCP theorem}\label{sec:hardness_pcp}
The famous Probabilistically Checkable Proofs (PCP) theorem is the cornerstone of the theory of computational hardness of approximation, which investigates the inherent difficulty in designing efficient approximation algorithms for various optimization problems.\footnote{\url{https://en.wikipedia.org/wiki/PCP_theorem}} The formal definition can be stated as follows,
\begin{theorem}[\cite{as98,almss98}]\label{thm:pcp}
Given a $\mathsf{SAT}$ formula $\phi$ of size $n$ we can in time polynomial in $n$ construct a set of $M$ tests satisfying the following:\\
\rm{(\RN{1})} : Each test queries a constant number $d$ of bits from a proof, and based on the outcome of the queries it either acceptes or reject $\phi$.\\
\rm{(\RN{2})} : (Yes Case / Completeness) If $\phi$ is satisfiable, then there exists a proof so that all tests accept $\phi$.\\
\rm{(\RN{3})} : (No Case / Soundness) If $\phi$ is not satifiable, then no proof will cause more than $M/2$ tests to accept $\phi$.
\end{theorem}
Note that PCP kind of reduction is slightly different from NP reduction, for more examples (e.g. maximum edge biclique, sparsest cut) about how to use PCP theorem to prove inapproximibility results, we refer the readers to \cite{ams11}.
\subsection{Warm-up}\label{sec:hardness_warmup}
We state our hardness result for $\mathsf{ROBUST}$-$\mathsf{NET}$(${\mathbb{B}}$) (boolean inputs case) in this section. The reduction procedure for network with boolean inputs is more straightforward and easier to understand than the real inputs case.
\begin{theorem}
Unless $\mathsf{NP}=\mathsf{P}$, there is no polynomial time algorithm to give a $(1-o(1))\ln n$-approximation to $\mathsf{ROBUST}$-$\mathsf{NET}$$(\mathbb{B})$ problem (Definition~\ref{def:approx_net_bool}) with $n$ hidden nodes.
\end{theorem}
\begin{proof}
Consider a set-cover instance, let $S$ denote a set of sets $\{ S_1, S_2, \cdots, S_d\}$ where $s_j \subseteq [n],\forall j \in [d]$.
For each set $S_j$ we create an input node $u_j$. For each element $i\in [n]$, we create a hidden node $v_i$. For each $i \in [n]$ and $j \in [d]$, if $i \in S_j$, then we connect $u_j$ and $v_i$. We also create an output node $w$, for each $i\in [n]$, we connect node $v_i$ and node $w$.
Let ${\bf 1}_{i \in S_j}$ denote the indicator function that it is $1$ if $i \in S_j$ and $0$ otherwise. Let $T_i$ denote the set that $T_i = \{ j ~|~ i \in S_j, \forall j \in [d] \}$. For each $i\in [n]$, we define an activation function $\phi_i$ satisfies that
\begin{align*}
\phi_i =
\begin{cases}
1, & \text{~if~} \sum_{j\in T_i} u_j \geq 1,\\
0, & \text{~otherwise}.
\end{cases}
\end{align*}
Since $u_j\in\{0,1\}$, $\phi_i$ can be implemented in this way using ReLU activations:
\begin{align*}
\phi_i = 1 - \max \left( 0, 1 - \sum_{j\in T_i} u_j \right).
\end{align*}
Note that $\sum_{j=1}^d {\bf 1}_{i \in S_j} = \sum_{j=1}^d u_j$, because $u_j=1$ indicates choosing set $S_j$ and $u_j=0$ otherwise.
For final output node $w$, we define an activation function $\psi$ satisfies that
\begin{align*}
\psi =
\begin{cases}
1, & \text{~if~} \sum_{i=1}^n v_i \geq n, \\
0, & \text{~otherwise}.
\end{cases}
\end{align*}
Since $v_i\in[n]$, $\psi$ can be implemented as
\begin{align*}
\psi = \max \left( 0, \sum_{i=1}^n v_i -n+1\right).
\end{align*}
We use vector $x$ to denote $\{0\}^d$ vector and it is to easy to see that $F(x) = 0$. Let $\alpha > 1$ denote a fixed parameter. Also, we have $F(y)>0$ if and only if $C=\{j|y_j=1\}$ is a set-cover.
According to our construction, we can have the following two claims,
\begin{comment}
\iffalse
\\\textcolor{blue}{Hongge: I think the logic of Claim A.9. may need to be changed as follows: \\For any SET-COVER problem, we can construct a network as in Theorem A.8. If there exists a point $y \in \{0,1\}^d$ such that $\| x - y \|_1 \leq r$ and $F(y) = 1$, then there exists a set-cover $C \subseteq [d]$ with $\cup_{j\in C} S_j = [n]$ and $|C| \leq r$.\\Since we want to prove ROBUST-NET(B) is NP complete, we need to say that it is "harder" than SET-COVER. For any SET-COVER problem, we can construct a ROBUST-NET(B) problem's special case. Then we can claim that if we can solve ROBUST-NET(B), we can trivially solve SET-COVER. Same as Claim A.10.,Claim A.12. and Claim A.13.}
\fi
\end{comment}
\begin{claim}[Completeness]\label{cla:bool_completeness}
If there exists a set-cover $C \subseteq [d]$ with $\cup_{j\in C} S_j = [n]$ and $|C| \leq r$, then there exists a point $y \in \{0,1\}^d$ such that $\| x - y \|_1 \leq r$ and $F(y) > 0$.
\end{claim}
\begin{claim}[Soundness]\label{cla:bool_soundness}
If for every $C \subseteq [d]$ with $\cup_{j \in C}S_j = U$ satisfies that $|C| > \alpha \cdot t$, then for all $y\in \{0,1\}^d$ satisfies that $\| x - y\|_1 \leq \alpha r$, $F(y) \leq 0$ holds.
\end{claim}
Therefore, using Theorem~\ref{thm:pcp}, Theorem~\ref{thm:approx_set_cover}, Claim~\ref{cla:bool_completeness} and Claim~\ref{cla:bool_soundness} completes the proof.
\end{proof}
\subsection{Main result}\label{sec:hardness_main_result}
With the proof for $\mathsf{ROBUST}$-$\mathsf{NET}$(${\mathbb{B}}$) as a warm-up, we now prove our main hardness result for $\mathsf{ROBUST}$-$\mathsf{NET}$(${\mathbb{R}}$) in this section.
\begin{theorem}\label{thm:robust_net_R}
Unless $\mathsf{NP}=\mathsf{P}$, there is no polynomial time algorithm to give an $(1-o(1))\ln n$-approximation to $\mathsf{ROBUST}$-$\mathsf{NET}(\mathbb{R})$ problem (Definition~\ref{def:approx_net_real}) with $n$ hidden nodes.
\end{theorem}
\begin{proof}
Consider a set-cover instance, let $S$ denote a set of sets $\{ S_1, S_2, \cdots, S_d\}$ where $S_j \subseteq [n],\forall j \in [d]$. For each set $S_j$ we create an input node $u_j$. For each $j \in [d]$, we create a hidden node $t_j$ and connect $u_j$ and $t_j$.
For each element $i\in [n]$, we create a hidden node $v_i$. For each $i \in [n]$ and $j \in [d]$, if $i \in S_j$, then we connect $u_j$ and $v_i$. Finally, we create an output node $w$ and for each $i\in [n]$, we connect node $v_i$ and node $w$.
Let $\delta = 1/ d$. For each $j \in [n]$, we apply an activation function $\phi_{1,j}$ on $t_j$ such that
\begin{align*}
\phi_{1,j} = - \max (0, \delta - u_j ) + \max (0, u_j - 1 + \delta)
\end{align*}
It is easy to see that
\begin{align*}
t_j=\phi_{1,j} =
\begin{cases}
u_j - \delta & \text{~if~} u_j \in [0,\delta] \\
u_j -(1-\delta) & \text{~if~} u_j \in [1-\delta, 1]\\
0 & \text{~otherwise~}.
\end{cases}
\end{align*}
Let $T_i$ denote the set that $T_i = \{ j ~|~ i \in S_j, \forall j \in [d] \}$.
For each $i\in [n]$, we need an activation function $\phi_{2,i}$ on node $v_i$ which satisfies that
\begin{align*}
\phi_{2,i} \in
\begin{cases}
[-\delta,0], & \text{~if~} \forall j \in T_i, t_j \in [-\delta,0], \\
[0,\delta], & \text{~if~} \exists j \in T_i, t_j \in [0,\delta].
\end{cases}
\end{align*}
This can be implemented in the following way,
\begin{align*}
\phi_{2,i} = \max_{j \in T_i} t_j.
\end{align*}
For the final output node $w$, we define it as $$w=\min_{i \in [n]} v_i.$$
\iffalse
\begin{align*}
\psi =
\begin{cases}
1, & \text{~if~} \min_{i \in [n]} (v_i) >0 \\
0, & \text{~if~} \min_{i \in [n]} (v_i) <0
\end{cases}
\end{align*}
\fi
We use vector $x$ to denote $\{0\}^d$ vector and it is to easy to see that $F(x) =-\delta< 0$. Let $\alpha > 1$ denote a fixed parameter.
According to our construction, we can have the following two claims.
\begin{claim}[Completeness]\label{cla:real_completeness}
If there exists a set-cover $C \subseteq [d]$ with $\cup_{j\in C} S_j = [n]$ and $|C| \leq r$, then there exists a point $y \in [0,1]^d$ such that $\| x - y \|_1 \leq r$ and $F(y) > 0$.
\end{claim}
\begin{proof}
Without loss of generality, we let the set cover to be $\{S_1,S_2,...,S_r\}$. Let $y_1=y_2=\cdots=y_r=1$ and $y_{r+1}=y_{r+2}=...=y_{d}=0.$ By the definition of $t_j$, we have $t_1=t_2=\cdots=t_r=\delta.$ Since $\{S_1,S_2,\cdots,S_r\}$ is a set-cover, we know that $v_i=\delta$ for all $i\in[n]$. Then $F(y)=w=\min_{i\in[n ]}v_i=\delta>0.$ Since we also have $\|y\|_1=r,$ the adversarial point is found.
\end{proof}
\begin{claim}[Soundness]\label{cla:real_soundness}
If for every $C \subseteq [d]$ with $\cup_{j \in C}S_j = U$ satisfies that $|C| > \alpha \cdot r$, then for all $y\in [0,1]^d$ satisfies that $\| x - y\|_1 \leq \alpha r (1-1/ d )$, $F(y) \leq 0$ holds.
\end{claim}
\begin{proof}
Proof by contradiction. We assume that there exists $y$ such that $F(y)>0$ and $\| y\|_1 \leq \alpha r (1-1/d)$. Since $F(y)>0$, we have for all $i$, $v_i>0$. Thus there exists $j\in T_i$ such that $t_j>0$. Let $\pi : [n] \rightarrow Q$ denote a mapping ($Q\subseteq[d]$ will be decided later). This means that for each $i \in [n]$, there exists $j\in T_i$, such that $1-\delta<y_{j}\leq 1$, and we let $\pi(i)$ denote that $j$.
We define set $Q \subseteq [d]$ as follows
\begin{align*}
Q = \{ j ~|~ \exists i \in [n], \text{~s.t.~} \pi(i)=j \in T_i \text{~and~} t_j > 0 \}.
\end{align*}
Since $\sum_{j\in[d]}|y_j|=\| y\|_1\leq \alpha r (1-1/d) $, we have
\begin{align*}
\sum_{j \in Q }|y_j|\leq\sum_{ j \in [d] } |y_j| \leq \alpha r (1 - 1/d),
\end{align*}
where the first step follows by $|Q| \leq d$.
Because for all $j \in Q$, $|y_j|>1-\delta=1-1/d$, we have
\begin{align*}
|Q|\leq \frac{ \alpha r ( 1 - 1/d ) }{ ( 1- 1/d )}= \alpha\cdot r.
\end{align*}
So $\{ S_{j} \}_{j \in Q}$ is a set-cover with size less than or equal to $\alpha\cdot r$, which is a contradiction.
\end{proof}
Therefore, using Theorem~\ref{thm:pcp}, Theorem~\ref{thm:approx_set_cover}, Claim~\ref{cla:real_completeness} and Claim~\ref{cla:real_soundness} completes the proof.
\end{proof}
By making a stronger assumption of $\mathsf{ETH}$, we can have the following stronger result which excludes all $2^{o(n^c)}$ time algorithms, where $c>0$ is some fixed constant:
\begin{corollary}
Assuming Exponential Time Hypothesis ($\mathsf{ETH}$, see Hypothesis~\ref{hypo:eth}), there is no $2^{o(n^c)}$ time algorithm that gives a $(1-o(1))\ln n$-approximation to $\mathsf{ROBUST}$-$\mathsf{NET}$ problem with $n$ hidden nodes, where $c>0$ is some fixed constant.
\end{corollary}
\begin{proof}
It follows by the construction in Theorem~\ref{thm:robust_net_R} and \cite{m12,moshkovitz2012projection}.
\end{proof}
Note that in~\cite{m12}, an additional conjecture, Projection Games Conjecture ($\mathsf{PGC}$) is required for the proof, but the result was improved in \cite{moshkovitz2012projection} and $\mathsf{PGC}$ is not a requirement any more.
\section{Proof of Theorem \ref{thm:cvx_bnd}}
\label{app:approach1_explicit_function}
For a $m$-layer ReLU network, assume we know all the pre-ReLU activation bounds $\lwbnd{(k)}$ and $\upbnd{(k)}$, $\forall k \in[ m-1]$ for a $m$-layer ReLU network and we want to compute the bounds of the the $j$ th output at $m$ th layer.
The $j$ th output can be written as
\begin{align}
\label{proof:eq:1stfj}
f_j(\bm{x}) &= \sum_{k=1}^{n_{m-1}} \W{(m)}_{j,k} [\phi_{m-1}(\bm{x})]_k + \bias{(m)}_j, \\
&= \sum_{k=1}^{n_{m-1}} \W{(m)}_{j,k} \sigma(\W{(m-1)}_{k,:} \phi_{m-2}(x)+\bias{(m-1)}_k) + \bias{(m)}_j, \\
&= \sum_{k \in \setIpos{m-1}, \setIneg{m-1}, \setIuns{m-1}} \W{(m)}_{j,k} \sigma(\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x})+\bias{(m-1)}_k) + \bias{(m)}_j.
\end{align}
For neurons belonging to category (i), i.e., $k \in \setIpos{m-1}$,
\begin{equation*}
\sigma(\W{(m-1)}_{k,:} \phi_{m-2}(x)+\bias{(m-1)}_k) = \W{(m-1)}_{k,:} \phi_{m-2}(\bm{x})+\bias{(m-1)}_k.
\end{equation*}
For neurons belonging to category (ii), i.e., $k \in \setIneg{m-1}$,
\begin{equation*}
\sigma(\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x})+\bias{(m-1)}_k) = 0.
\end{equation*}
Finally, for neurons belonging to Category (iii), i.e., $k \in \setIuns{m-1}$, we bound their outputs. If we adopt the linear upper and lower bounds in \eqref{eq:our_cvx_approx} and let $\bm{d}^{(m-1)}_k := \frac{\upbnd{(m-1)}_k}{\upbnd{(m-1)}_k-\lwbnd{(m-1)}_k}$, we have
\begin{equation}
\label{proof:eqcvx}
\bm{d}^{(m-1)}_k (\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k) \leq \sigma(\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k) \leq \bm{d}^{(m-1)}_k (\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k - \lwbnd{(m-1)}_k).
\end{equation}
\subsection{Upper bound}
The goal of this section is to prove Lemma~\ref{lem:approach_1_upper_bound}.
\begin{lemma}[Upper bound with explicit function]\label{lem:approach_1_upper_bound}
Given an $m$-layer ReLU neural network function $f : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$, parameters $p, \epsilon$, there exists two explicit functions $f^L : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ and $f^U :\mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ (see Definition~\ref{def:f_L_f_U}) such that $\forall j \in [n_m]$,
\begin{equation*}
f_{j}(\bm{x}) \leq f_{j}^{U}(\bm{x}), \forall \bm{x} \in B_p(\bm{x}_0,\epsilon).
\end{equation*}
\end{lemma}
Notice that \eqref{proof:eqcvx} can be used to construct an upper bound and lower bound of $f_j(\bm{x})$ by considering the signs of the weights $\W{(m)}_{j,k}$. Let $f_j^{U,m-1}(\bm{x})$ be an upper bound of $f_j(\bm{x})$; $f_j^{U,m-1}(\bm{x})$ can be constructed by taking the right-hand-side (RHS) of \eqref{proof:eqcvx} if $\W{(m)}_{j,k} > 0$ and taking the left-hand-side (LHS) of \eqref{proof:eqcvx} if $\W{(m)}_{j,k} < 0$:
\begin{align}
& \quad f_j^{U,m-1}(\bm{x}) \notag \\
& = \sum_{k\in \setIpos{m-1}} \W{(m)}_{j,k} (\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k) \label{proof:eq:1stfju} \\
& \quad +\sum_{k\in \setIuns{m-1}, \W{(m)}_{j,k}>0} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k (\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k - \lwbnd{(m-1)}_k) \nonumber \\
& \quad +\sum_{k\in \setIuns{m-1}, \W{(m)}_{j,k}<0} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k (\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k) + \bias{(m)}_j \nonumber \\
\label{proof:eq:1stfju1}
& = \sum_{k = 1}^{n_{m-1}} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k (\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \bias{(m-1)}_k) - \sum_{k\in \setIuns{m-1}, \W{(m)}_{j,k}>0} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k \lwbnd{(m-1)}_k + \bias{(m)}_j, \\
\label{proof:eq:1stfju2}
& = \sum_{k = 1}^{n_{m-1}} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k \W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) \\
& \quad +\left(\sum_{k = 1}^{n_{m-1}} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k \bias{(m-1)}_k - \sum_{k\in \setIuns{m-1}, \W{(m)}_{j,k}>0} \W{(m)}_{j,k} \bm{d}^{(m-1)}_k \lwbnd{(m-1)}_k + \bias{(m)}_j \right), \nonumber
\end{align}
where we set $\bm{d}^{(m-1)}_k = 1$ for $k \in \setIpos{m-1}$ and set $\bm{d}^{(m-1)}_k = 0$ for $k \in \setIneg{m-1}$ from \eqref{proof:eq:1stfju} to \eqref{proof:eq:1stfju1} and collect the constant terms (independent of $\bm{x}$) in the parenthesis from \eqref{proof:eq:1stfju1} to \eqref{proof:eq:1stfju2}.
If we let $\A{(m-1)} = \W{(m)}\DD{(m-1)}$, where $\DD{(m-1)}$ is a diagonal matrix with diagonals being $\bm{d}^{(m-1)}_k$, then we can rewrite $f_j^{U,m-1}(\bm{x})$ into the following:
\begin{align}
\label{proof:eq:1stfju3}
f_j^{U,m-1}(\bm{x}) & = \sum_{k = 1}^{n_{m-1}} \A{(m-1)}_{j,k} \W{(m-1)}_{k,:} \phi_{m-2}(\bm{x}) + \left(\A{(m-1)}_{j,:} \bias{(m-1)} - \A{(m-1)}_{j,:} \upbias{(m-1)}_{:,j} + \bias{(m)}_j \right) \\
\label{proof:eq:1stfju4}
& = \sum_{k = 1}^{n_{m-1}} \A{(m-1)}_{j,k} (\sum_{r=1}^{n_{m-2}} \W{(m-1)}_{k,r} [\phi_{m-2}(\bm{x})]_r ) + \left(\A{(m-1)}_{j,:} \bias{(m-1)} - \A{(m-1)}_{j,:} \upbias{(m-1)}_{:,j} + \bias{(m)}_j \right) \\
\label{proof:eq:1stfju5}
& = \sum_{r=1}^{n_{m-2}} \sum_{k = 1}^{n_{m-1}} \A{(m-1)}_{j,k} \W{(m-1)}_{k,r} [\phi_{m-2}(\bm{x})]_r + \left(\A{(m-1)}_{j,:} \bias{(m-1)} - \A{(m-1)}_{j,:} \upbias{(m-1)}_{:,j} + \bias{(m)}_j \right) \\
\label{proof:eq:1stfju6}
& = \sum_{r=1}^{n_{m-2}} \tilde{\bm{W}}^{(m-1)}_{j,r} [\phi_{m-2}(\bm{x})]_r + \tilde{\bm{b}}^{(m-1)}_j.
\end{align}
From \eqref{proof:eq:1stfju2} to \eqref{proof:eq:1stfju3}, we rewrite the summation terms in the parenthesis into matrix-vector multiplications and for each $j \in [n_m]$ let
\begin{equation*}
\upbias{(m-1)}_{k,j} =
\begin{cases}
\lwbnd{(m-1)}_k & \text{if $k \in \setIuns{m-1}, \, \A{(m-1)}_{j,k} > 0$} \\
0 & \text{otherwise}
\end{cases}
\end{equation*}
since $0 \leq \bm{d}^{(m-1)}_k \leq 1$, $\W{(m)}_{j,k} > 0$ is equivalent to $\A{(m-1)}_{j,k} > 0$.
From \eqref{proof:eq:1stfju3} to \eqref{proof:eq:1stfju4}, we simply write out the inner product $\W{(m-1)}_{k,:} \phi_{m-2}(\bm{x})$ into a summation form, and from \eqref{proof:eq:1stfju4} to \eqref{proof:eq:1stfju5}, we exchange the summation order of $k$ and $r$. From \eqref{proof:eq:1stfju5} to \eqref{proof:eq:1stfju6}, we let
\begin{align}
\label{proof:eq:1stfju7}
\tilde{\bm{W}}^{(m-1)}_{j,r} &= \sum_{k = 1}^{n_{m-1}} \A{(m-1)}_{j,k} \W{(m-1)}_{k,r} \\
\label{proof:eq:1stfju8}
\tilde{\bm{b}}^{(m-1)}_j &= \left(\A{(m-1)}_{j,:} \bias{(m-1)} - \A{(m-1)}_{j,:} \upbias{(m-1)}_{:,j} + \bias{(m)}_j \right)
\end{align}
and now we have \eqref{proof:eq:1stfju6} in the same form as \eqref{proof:eq:1stfj}.
Indeed, in \eqref{proof:eq:1stfj}, the running index is $k$ and we are looking at the $m$ th layer, with weights $\W{(m)}_{j,k}$, activation functions $\phi_{m-1}(\bm{x})$ and bias term $\bias{(m)}_j$; in \eqref{proof:eq:1stfju6}, the running index is $r$ and we are looking at the $m-1$ th layer with \textit{equivalent} weights $\tilde{\bm{W}}^{(m-1)}_{j,r}$, activation functions $\phi_{m-2}(\bm{x})$ and \textit{equivalent} bias $\tilde{\bm{b}}^{(m-1)}_j$. Thus, we can use the same technique from \eqref{proof:eq:1stfj} to \eqref{proof:eq:1stfju6} and obtain an upper bound on the $f_j^{U,m-1}(\bm{x})$ and repeat this procedure until obtaining $f_j^{U,1}(\bm{x})$, where
$$f_j(\bm{x}) \leq f_j^{U,m-1}(\bm{x}) \leq f_j^{U,m-2}(\bm{x}) \leq \ldots \leq f_j^{U,1}(\bm{x}).$$
Let the final upper bound $f_j^U(\bm{x}) = f_j^{U,1}(\bm{x})$, and now we have
\begin{equation*}
f_{j}(\bm{x}) \leq f_{j}^{U}(\bm{x}),
\end{equation*}
where $f^U_j(\bm{x}) = [f^{U}(\bm{x})]_j$,
\begin{align*}
f^{U}_j(\bm{x}) & = \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j})
\end{align*}
and for $k = 1, \, \ldots, \, m-1, $
\begin{equation*}
\A{(m-1)} = \W{(m)} \DD{(m-1)}, \, \A{(k-1)} = \A{(k)} \W{(k)} \DD{(k-1)},
\end{equation*}
\begin{align*}
\DD{(0)} & = \bm{I}_{n_0} \nonumber \\
\DD{(k)}_{r,r} & =
\begin{cases}
\frac{\upbnd{(k)}_r}{\upbnd{(k)}_r-\lwbnd{(k)}_r} & \text{if $r \in \setIuns{k}$} \\
1 & \text{if $r \in \setIpos{k}$} \\
0 & \text{if $r \in \setIneg{k}$}
\end{cases}
\\
\upbias{(k)}_{r,j} & =
\begin{cases}
\lwbnd{(k)}_r & \text{if $r \in \setIuns{k}, \, \A{(k)}_{j,r} > 0$} \\
0 & \text{otherwise}
\end{cases}
\end{align*}
\subsection{Lower bound }
The goal of this section is to prove Lemma~\ref{lem:approach_1_lower_bound}.
\begin{lemma}[Lower bound with explicit function]\label{lem:approach_1_lower_bound}
Given an $m$-layer ReLU neural network function $f : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$, parameters $p, \epsilon$, there exists two explicit functions $f^L : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ and $f^U :\mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ (see Definition~\ref{def:f_L_f_U}) such that $\forall j \in [n_m]$,
\begin{equation*}
f_{j}^{L}(\bm{x}) \leq f_{j}(\bm{x}), \forall \bm{x} \in B_p(\bm{x}_0,\epsilon).
\end{equation*}
\end{lemma}
Similar to deriving the upper bound of $f_j(\bm{x})$, we consider the signs of the weights $\W{(m)}_{j,k}$ to derive the lower bound. Let $f_j^{L,m-1}(\bm{x})$ be a lower bound of $f_j(\bm{x})$; $f_j^{L,m-1}(\bm{x})$ can be constructed by taking the right-hand-side (RHS) of \eqref{proof:eqcvx} if $\W{(m)}_{j,k} < 0$ and taking the left-hand-side (LHS) of \eqref{proof:eqcvx} if $\W{(m)}_{j,k} > 0$. Following the procedure in \eqref{proof:eq:1stfju} to \eqref{proof:eq:1stfju6} (except that now the additional bias term is from the set $k \in \setIuns{m-1}, \W{(m)}_{j,k} < 0$), the lower bound is similar to the upper bound we have derived but but replace $\upbias{(m-1)}$ by $\lwbias{(m-1)}$, where for each $j \in [n_m]$,
\begin{equation*}
\lwbias{(m-1)}_{k,j} =
\begin{cases}
\lwbnd{(m-1)}_k & \text{if $k \in \setIuns{m-1}, \, \A{(m-1)}_{j,k} < 0$} \\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
It is because the linear upper and lower bounds in \eqref{eq:our_cvx_approx} has the same slope $\frac{u}{u-l}$ on both sides (i.e. $\sigma(y)$ is bounded by two lines with the same slope but different intercept), which gives the same $\A{}$ matrix and $\DD{}$ matrix in computing the upper bound and lower bound of $f_j(\bm{x})$. This is the key to facilitate a faster computation under this linear approximation \eqref{eq:our_cvx_approx}. Thus, the lower bound for $f_j(\bm{x})$ is:
\begin{equation*}
f_{j}^{L}(\bm{x}) \leq f_{j}(\bm{x}) ,
\end{equation*}
where $f^L_j(\bm{x}) = [f^{L}(\bm{x})]_j$,
\begin{align*}
f^{L}_j(\bm{x}) & = \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j})
\end{align*}
and for $k = 1, \, \ldots, \, m-1, $
\begin{equation*}
\lwbias{(k)}_{r,j} =
\begin{cases}
\lwbnd{(k)}_r & \text{if $r \in \setIuns{k}, \, \A{(k)}_{j,r} < 0$} \\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
\newpage
\section{Proof of Corollary \ref{cor:cvx_bnd}}\label{app:approach1_fixed_value}
By Theorem \ref{thm:cvx_bnd}, for $\bm{x} \in B_p(\bm{x_0},\epsilon)$, we have $ f_j^{L}(\bm{x}) \leq f_j(\bm{x}) \leq f_j^{U}(\bm{x})$. Thus,
\begin{align}
f_j(\bm{x}) &\leq f_j^{U}(\bm{x}) \leq \max_{\bm{x} \in B_p(\bm{x},\epsilon)} f_j^{U}(\bm{x}), \\
f_j(\bm{x}) &\geq f_j^{L}(\bm{x}) \geq \min_{\bm{x} \in B_p(\bm{x},\epsilon)} f_j^{L}(\bm{x}).
\end{align}
Since $f_j^{U}(\bm{x}) = \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j})$,
\begin{align}
\gamma^U_j :=
\max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} f_j^{U}(\bm{x})
&= \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \left( \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j}) \right) \nonumber \\
&= \left( \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \A{(0)}_{j,:}\bm{x} \right) + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j}) \label{proof:cor:max1} \\
&= \epsilon \left( \max_{\bm{y} \in B_p(\bm{0},1)} \A{(0)}_{j,:}\bm{y} \right) + \A{(0)}_{j,:}\bm{x_0} + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j}) \label{proof:cor:max2} \\
&= \epsilon \| \A{(0)}_{j,:}\|_q + \A{(0)}_{j,:}\bm{x_0} + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j}). \label{proof:cor:max3}
\end{align}
From \eqref{proof:cor:max1} to \eqref{proof:cor:max2}, we do a transformation of variable $\bm{y} := \frac{\bm{x}-\bm{x_0}}{\epsilon}$ and therefore $\bm{y} \in B_p(\bm{0},1)$. By the definition of dual norm $\| \cdot \|_*$:
\begin{equation*}
\| \bm{z} \|_* = \{ \sup_{\bm{y}} \bm{z}^\top \bm{y} \mid \| \bm{y} \| \leq 1 \},
\end{equation*}
and the fact that $\ell_q$ norm is dual of $\ell_p$ norm for $p, q \in [1,\infty]$, the term $\left( \max_{\bm{y} \in B_p(\bm{0},1)} \A{(0)}_{j,:}\bm{y} \right)$ in \eqref{proof:cor:max2} can be expressed as $\| \A{(0)}_{j,:}\|_q$ in \eqref{proof:cor:max3}. Similarly,
\begin{align}
\gamma^L_j :=
\min_{\bm{x} \in B_p(\bm{x_0},\epsilon)} f_j^{L}(\bm{x})
&= \min_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \left( \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j}) \right) \nonumber \\
&= \left( \min_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \A{(0)}_{j,:}\bm{x} \right) + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j}) \nonumber\\
&= \epsilon \left( \min_{\bm{y} \in B_p(\bm{0},1)} \A{(0)}_{j,:}\bm{y} \right) + \A{(0)}_{j,:}\bm{x_0} + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j}) \nonumber\\
&= -\epsilon \left( \max_{\bm{y} \in B_p(\bm{0},1)} -\A{(0)}_{j,:}\bm{y} \right) + \A{(0)}_{j,:}\bm{x_0} + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j}) \label{proof:cor:min1}\\
&= -\epsilon \| \A{(0)}_{j,:}\|_q + \A{(0)}_{j,:}\bm{x_0} + \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j}). \label{proof:cor:min2}
\end{align}
Again, from \eqref{proof:cor:min1} to \eqref{proof:cor:min2}, we simply replace $\left( \max_{\bm{y} \in B_p(\bm{0},1)} -\A{(0)}_{j,:}\bm{y} \right)$ by $\| -\A{(0)}_{j,:}\|_q = \| \A{(0)}_{j,:}\|_q$.
Thus, if we use $\nu_j$ to denote the common term $\A{(0)}_{j,:} \bm{x_0} + \bias{(m)}_j + \sum_{k=1}^{m-1}\A{(k)}_{j,:}\bias{(k)}$, we have
\begin{align*}
\gamma^U_j = \epsilon \|\A{(0)}_{j,:}\|_q - \sum_{k=1}^{m-1}\A{(k)}_{j,:}\upbias{(k)}_{:,j} + \nu_j, \quad \quad & \text{(upper bound)}\\
\gamma^L_j = -\epsilon \|\A{(0)}_{j,:}\|_q - \sum_{k=1}^{m-1}\A{(k)}_{j,:}\lwbias{(k)}_{:,j} + \nu_j. \; \quad & \text{(lower bound)}
\end{align*}
\newpage
\section{Algorithms}
\label{sec:app_algs}
We present our full algorithms, \textbf{Fast-Lin}\xspace in Algorithm~\ref{alg:fast-lin} and \textbf{Fast-Lip}\xspace in Algorithm~\ref{alg:fast-lip}.
\input{alg_fastlin}
\input{alg_fastlip}
\newpage
\section{An alternative bound on the Lipschitz constant}\label{app:alternative_bound_on_Lipschitz}
Using the property of norm, we can derive an upper bound of the gradient norm of a $2$-layer ReLU network in the following:
\begin{align}
& \quad \| \nabla f_j(\bm{x}) \|_q \nonumber \\
&= \| \W{(2)}_{j,:} \Lam{(1)} \W{(1)} \|_q \nonumber \\
&= \| \W{(2)}_{j,:} (\Lam{(1)}_a+\Lam{(1)}_u) \W{(1)} \|_q \label{eq:grad_qbnd1} \\
& \leq \| \W{(2)}_{j,:} \Lam{(1)}_a \W{(1)} \|_q + \| \W{(2)}_{j,:} \Lam{(1)}_u \W{(1)} \|_q \label{eq:grad_qbnd2} \\
& \leq \| \W{(2)}_{j,:} \Lam{(1)}_a \W{(1)} \|_q + \sum_{r \in \setIuns{1}} \| \W{(2)}_{j,r} \W{(1)}_{r,:} \|_q \label{eq:grad_qbnd3}
\end{align}
where with a slight abuse of notation, we use $\Lam{(1)}_a$ to denote the diagonal activation matrix for neurons who are always activated, i.e. its $(r,r)$ entry $\Lam{(1)}_{a(r,r)}$ is $1$ if
$r \in \setIpos{1}$ and $0$ otherwise, and we use $\Lam{(1)}_u$ to denote the diagonal activation matrix for neurons whose status are uncertain because they could possibly be active or inactive, i.e. its $(r,r)$ entry $\Lam{(1)}_{u(r,r)}$ is $1$ if $r \in \setIuns{1}$ and $0$ otherwise. Therefore, we can write $\Lam{(1)}$ as a sum of $\Lam{(1)}_a$ and $\Lam{(1)}_u$.
Note that \eqref{eq:grad_qbnd1} to \eqref{eq:grad_qbnd2} is from the sub-additive property of a norm, and \eqref{eq:grad_qbnd2} to \eqref{eq:grad_qbnd3} uses the sub-additive property of a norm again and set the uncertain neurons encoding all to $1$ because $$\| \W{(2)}_{j,:} \Lam{(1)}_u \W{(1)} \| = \| \sum_{r \in \setIuns{1}} \W{(2)}_{j,r} \Lam{(1)}_{u(r,r)} \W{(1)}_{r,:} \| \leq \sum_{r \in \setIuns{1}} \| \W{(2)}_{j,r} \Lam{(1)}_{u(r,r)} \W{(1)}_{r,:} \| \leq \sum_{r \in \setIuns{1}} \| \W{(2)}_{j,r} \W{(1)}_{r,:} \|.$$
Notice that \eqref{eq:grad_qbnd3} can be used as an upper bound of Lipschitz constant and is applicable to compute a certified lower bound for minimum adversarial distortion of a general $\ell_p$ norm attack. However, this bound is expected to be less tight because we simply include all the uncertain neurons to get an upper bound on the norm in \eqref{eq:grad_qbnd3}.
\section{Introduction}
Since the discovery of adversarial examples in deep neural network (DNN) image classifiers \cite{szegedy2013intriguing}, researchers have successfully found adversarial examples in many machine learning tasks applied to different areas, including object detection \cite{xie2017adversarial}, image captioning \cite{chen2017show},
speech recognition \cite{cisse2017houdini}, malware detection \cite{wang2017adversary} and reading comprehension \cite{jia2017adversarial}.
Moreover, black-box attacks have also been shown to be possible, where an attacker can find adversarial examples without knowing the architecture and parameters of the DNN \cite{CPY17_zoo,papernot2017practical,liu2016delving}.
The existence of adversarial examples poses a huge threat to the application of DNNs in mission-critical tasks including security cameras, self-driving cars and aircraft control systems. Many researchers have thus proposed defensive or detection methods in order to increase the robustness of DNNs. Notable examples are defensive distillation \cite{papernot2016distillation},
adversarial retraining/training \cite{kurakin2016adversarial_ICLR,madry2017towards} and model ensembles \cite{tramer2017ensemble,liu2017towards}. Despite many published contributions that aim at increasing the robustness of DNNs, theoretical results are rarely given and there is no guarantee that the proposed defensive methods can reliably improve the robustness. Indeed, many of these defensive mechanism have been shown to be ineffective when more advanced attacks are used \cite{carlini2017towards,carlini2017adversarial,carlini2017magnet,he2017adversarial}.
The robustness of a DNN can be verified by examining a neighborhood (e.g. $\ell_2$ or $\ell_\infty$ ball) near a data point $\bm{x_0}$. The idea is to find the largest ball with radius $r_0$ that guarantees no points inside the neighborhood can ever change classifier decision. Typically, $r_0$ can be found as follows: given $R$, a global optimization algorithm can be used to find an adversarial example within this ball, and thus bisection on $R$ can produce $r_0$. Reluplex \cite{katz2017reluplex} is one example using such a technique but it is computationally infeasible even on a small MNIST classifier. In general, verifying the robustness property of a ReLU network is NP-complete \cite{katz2017reluplex, sinha2017certifiable}.
On the other hand, a lower bound $\beta_L$ of radius $r_0$ can be given, which guarantees that no examples within a ball of radius $\beta_L$ can ever change the network classification outcome. \cite{hein2017formal} is a pioneering work on giving such a lower bound for neural networks that are continuously differentiable, although only a 2-layer MLP network with differentiable activations is investigated.
\cite{weng2017evaluating} has extended theoretical result to ReLU activation functions and proposed a robustness score, CLEVER, based on extreme value theory. Their approach is feasible for large state-of-the-art DNNs but CLEVER is an estimate of $\beta_L$ without certificates. Ideally, we would like to obtain a {\it certified} (which guarantees that $\beta_L \leq r_0$) and {\it non-trivial} (a trivial $\beta_L$ is 0) lower bound $\beta_L$ that is reasonably close to $r_0$ within {\it reasonable amount of computational time}.
In this paper, we develop two fast algorithms for obtaining a \textit{tight} and \textit{certified} lower bound $\beta_L$ on ReLU networks. In addition, we also provide a complementary theoretical result to \cite{katz2017reluplex,sinha2017certifiable} by further showing there does not even exist a polynomial time algorithm that can approximately find the minimum adversarial distortion with a $0.99 \ln n$ approximation ratio. Our contributions are:
\begin{itemize}[nosep,wide,labelindent=0pt,labelwidth=*,align=left]
\item We fully exploit the ReLU networks to give two computationally efficient methods of computing \textit{tighter} and \textit{guaranteed} robustness lower bounds via (1) \textbf{lin}ear approximation on the ReLU units (see Sec~\ref{sec3:convexbnd}, Algorithm~\ref{alg:fast-lin}, \textbf{Fast-Lin})
and (2) bounding network local \textbf{Lip}schitz constant (see Sec~\ref{sec3:gradbnd}, Algorithm~\ref{alg:fast-lip}, \textbf{Fast-Lip}). Unlike the per-layer operator-norm-based lower bounds which are often very loose (close to 0, as verified in our experiments) for deep networks, our bounds are much closer to the upper bound given by the best adversarial examples, and thus can be used to evaluate the robustness of DNNs with theoretical guarantee.
\item We show that our proposed method is at least \textit{four orders of magnitude faster} than finding the exact minimum distortion (with Reluplex), and also around \textit{two orders of magnitude (or more) faster} than linear programming (LP) based methods. We can compute a reasonable robustness lower bound within a minute for a ReLU network with up to 7 layers or over ten thousands neurons, which is so far the best available result in the literature to our best knowledge.
\item We show that there is no polynomial time algorithm that can find a lower bound of minimum $\ell_1$ adversarial distortion with a $(1-o(1))\ln n$ approximation ratio (where $n$ is the total number of neurons) unless $\mathsf{NP}$=$\mathsf{P}$ (see Theorem~\ref{thm:intro_hardness_np_p}).
\end{itemize}
\section{Background and related work}\label{sec:related_work}
\subsection{Solving the minimum adversarial distortion}
\label{sec:bg_mindist}
For ReLU networks, the verification problem can be transformed into a Mixed Integer Linear Programming (MILP) problem \cite{lomuscio2017approach,cheng2017maximum,fischetti2017deep} by using binary variables to encode the states of ReLU activation in each neuron. \cite{katz2017reluplex} proposed Reluplex based on satisfiable modulo theory, which encodes the network into a set of linear constraints with special rules to handle ReLU activations and splits the problem into two LP problems based on a ReLU's activation status on demand. Similarly, \cite{ehlers2017formal} proposed Planet, another splitting-based approach using satisfiability (SAT) solvers. These approaches guarantee to find the exact minimum distortion of an adversarial example, and can be used for formal verification. However, due to NP-hard nature of the underlying problem, these approaches only work on very small networks. For example, in \cite{katz2017reluplex}, verifying a feed-forward network with 5 inputs, 5 outputs and 300 total hidden neurons on a single data point can take a few hours. Additionally, Reluplex can find the minimum distortion only in terms of $\ell_{\infty}$ norm ($\ell_1$ is possible via an extension) and cannot easily generalize to $\ell_p$ norm.
\subsection{Computing lower bounds of minimum distortion}
\cite{szegedy2013intriguing} gives a lower bound on the minimum distortion in ReLU networks by computing the product of weight matrices operator norms, but this bound is usually too loose to be useful in practice, as pointed out in \cite{hein2017formal} and verified in our experiments (see Table~\ref{tb:smallnetwork}). A tighter bound was given by \cite{hein2017formal} using local Lipschitz constant on a network with one hidden layer, but their approach requires the network to be continuously-differentiable, and thus cannot be directly applied to ReLU networks. \cite{weng2017evaluating} further provide the lower bound guarantee to non-differentiable functions by Lipschitz continuity assumption and propose the first robustness score, CLEVER, that can evaluate the robustness of DNNs and scale to large ImageNet networks. As also shown in our experiments in Section \ref{sec:exp}, the CLEVER score is indeed a good robustness estimate close to the true minimum distortion given by Reluplex, albeit without providing certificates. Recently, \cite{zico17convex} propose a convex relaxation on the MILP verification problem discussed in Sec~\ref{sec:bg_mindist}, which reduces MILP to LP when the adversarial distortion is in $\ell_{\infty}$ norm. They focus on adversarial training, and compute layer-wise bounds by looking into the dual LP problem.
\subsection{Hardness and approximation algorithms}
$\mathsf{NP}\neq\mathsf{P}$ is the most important and popular assumption in computational complexity in the last several decades. It can be used to show that the decision of the exact case of a problem is hard. However, in several cases, solving one problem approximately is much easier than solving it exactly. For example, there is no polynomial time algorithm to solve the $\mathsf{MAX}$-$\mathsf{CUT}$ problem, but there is a simple $0.5$-approximation polynomial time algorithm. Previous works \cite{katz2017reluplex,sinha2017certifiable} show that there is no polynomial time algorithm to find the minimum adversarial distortion $r_0$ exactly. A natural question to ask is: does there exist a polynomial time algorithm to solve the robustness problem approximately? In other words, can we give a lower bound of $r_0$ with a guaranteed approximation ratio?
From another perspective, $\mathsf{NP}\neq \mathsf{P}$ only rules out the polynomial running time. Some problems might not even have a sub-exponential time algorithm. To rule out that, the most well-known assumption used is the ``Exponential Time Hypothesis''~\cite{ipz98}. The hypothesis states that $\mathsf{3SAT}$ cannot be solved in sub-exponential time in the worst case. Another example is that while tensor rank calculation is NP-hard~\cite{h90}, a recent work \cite{swz17b} proved that there is no $2^{o(n^{1-o(1)})}$ time algorithm to give a constant approximation of the rank of the tensor. There are also some stronger versions of the hypothesis than ETH, e.g., Strong ETH \cite{ip01}, Gap ETH \cite{d16,mr16}, and average case ETH \cite{f02,rsw16}.
\section{Experiments}\label{sec:exp}
In this section, we perform extensive experiments to evaluate the performance of our proposed two lower-bound based robustness certificates on networks with different sizes and with different defending techniques during training process. Specifically, we compare our proposed bounds\footnote{\url{https://github.com/huanzhang12/CertifiedReLURobustness}} (\textbf{Fast-Lin}\xspace, \textbf{Fast-Lip}\xspace) with Linear Programming (LP) based methods (\textbf{LP}\xspace, \textbf{LP-Full}\xspace), formal verification methods (\textbf{Reluplex}\xspace), lower bound by global Lipschitz constant (\textbf{Op-norm}\xspace), estimated lower bounds (\textbf{CLEVER}\xspace) and attack algorithms (\textbf{Attacks}) for toy networks (2-3 layers with 20 neurons in each layer) and large networks (2-7 layers with 1024 or 2048 neurons in each layer) in Table~\ref{tb:smallnetwork_and_large}. The evaluation on the effects of defending techniques is presented in Table~\ref{tb:distill}. All bound numbers are the average of 100 random test images with random attack targets, and running time (per image) for all methods is measured on a single CPU core.
We include detailed setup of experiments, descriptions of each method, additional experiments and discussions in Appendix~\ref{app:exp} (See Tables~\ref{tb:smallnetwork} and \ref{tb:largenetwork_app}).
The results suggest that our proposed robustness certificates are of high qualities and are computationally efficient even in large networks up to 7 layers or more than 10,000 neurons. In particular, we show that:
\begin{itemize}[nosep,wide,labelindent=0pt,labelwidth=*,align=left]
\item Our certified lower bounds (\textbf{Fast-Lin}\xspace, \textbf{Fast-Lip}\xspace) are close to (gap is only 2-3X) the exact minimum distortion computed by \textbf{Reluplex}\xspace for small networks (\textbf{Reluplex}\xspace is only feasible for networks with less 100 neurons for MNIST), but our algorithm is more than 10,000 times faster than \textbf{Reluplex}\xspace. See Table~\ref{tb:smallnetwork_main} and Table~\ref{tb:smallnetwork}.
\item Our certified lower bounds (\textbf{Fast-Lin}\xspace, \textbf{Fast-Lip}\xspace) give similar quality (the gap is within 35\%, and usually around 10\%; sometimes our bounds are even better) compared with the LP-based methods (\textbf{LP}\xspace, \textbf{LP-Full}\xspace); however, our algorithm is 33 - 14,000 times faster. The LP-based methods are infeasible for networks with more than 4,000 neurons. See Table~\ref{tb:largenetwork_main} and Table \ref{tb:largenetwork_app}.
\item When the network goes larger and deeper, our proposed methods can still give non-trivial lower bounds comparing to the upper bounds founded by attack algorithms on large networks. See Table~\ref{tb:largenetwork_main} and Table \ref{tb:largenetwork_app}.
\item For defended networks, especially for adversarial training~\cite{madry2017towards}, our methods give significantly larger bounds, validating the effectiveness of this defending method. Our algorithms can thus be used for evaluating defending techniques. See Table~\ref{tb:distill}.
\end{itemize}
\ifdef{}{
\input{tab_reluplex_compare_all}
\input{tab_mnist_cifar_compare_appendix}
\input{tab_mnist_cifar_compare_dd}
}
\vspace{-1em}
\section{Conclusions}
In this paper we have considered the problem of verifying the robustness property of ReLU networks. By exploiting the special properties of ReLU networks, we have here presented two computational efficient methods \textbf{Fast-Lin}\xspace and \textbf{Fast-Lip}\xspace for this problem. Our algorithms are two orders of magnitude (or more) faster than LP-based methods, while obtaining solutions with similar quality; meanwhile, our bounds qualities are much better than the previously proposed operator-norm based methods. Additionally, our methods are efficient and easy to implement: we compute the bounds layer-by-layer, and the computation cost for each layer is similar to the cost of matrix products in forward propagation; moreover, we do not need to solve any integer programming, linear programming problems or their duals. Future work could extend our algorithm to handle the structure of convolutional layers and apply our algorithm to evaluate the robustness property of large DNNs such as ResNet on the ImageNet dataset.
\section{The Lipschitz constant of ReLU activation}
\subsection{2-layer NN}
\subsubsection*{Global bound}
The $j_{\text{th}}$ output of a one-hidden-layer neural network can be written as
\begin{equation*}
f_j(\bm{x}) = \sum_{r=1}^{U} \bm{V}_{jr} \cdot \sigma \left ( \sum_{i=1}^{d} \bm{W}_{ri} \cdot x_i + b_r \right ) = \sum_{r=1}^{U} \bm{V}_{jr} \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right ),
\end{equation*}
where $\sigma (z) = \max (z, 0)$ is ReLU activation function, $\bm{W}$ and $\bm{V}$ are the weight matrices of the first and second layer respectively, and $\bm{w}_r$ is the $r_{\text{th}}$ row of $\bm{W}$. Thus, we can compute $g(\bm{x})$ and $\| \nabla g(\bm{x}) \|_q$ below:
\begin{align*}
g(\bm{x}) = f_c(\bm{x}) - f_j(\bm{x})
&= \sum_{r=1}^{U} \bm{V}_{cr} \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right ) - \sum_{r=1}^{U} \bm{V}_{jr} \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right ) \\
&= \sum_{r=1}^{U} ( \bm{V}_{cr} - \bm{V}_{jr} ) \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right )
\end{align*}
and
\begin{align*}
\| \nabla g(\bm{x}) \|_q &= \left \| \sum_{r=1}^{U} \mathbb{I}(\bm{w}_r \bm{x} + b_r) ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \right \|_q,
\end{align*}
where $\mathbb{I}(z)$ is an univariate indicator function:
\[
\mathbb{I}(z) = \Big \{
\begin{tabular}{cc}
1, & \text{if $z>0$,} \\
0, & \text{if $z \leq 0$.}
\end{tabular}
\]
We can also rewrite $g(\bm{x})$ and $\| \nabla g(\bm{x}) \|_q$ as
\begin{align*}
g(\bm{x}) &= \sum_{r \in A(\bm{x})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \cdot (\bm{w}_r \bm{x} + b_r), \\
\| \nabla g(\bm{x}) \|_q &= \| \sum_{r \in A(\bm{x})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \|_q ,
\end{align*}
where $A(\bm{x})$ is an index set of activated nodes, i.e $A(\bm{x}) := \{r \mid \bm{w}_r \bm{x} + b_r > 0 \}$. A global upper bound of $\| \nabla g(\bm{x}) \|_q$ can be computed as follows:
\begin{align}
\| \nabla g(\bm{x}) \|_q &= \| \sum_{r \in A(\bm{x})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \|_q \label{eq:exact_grad_g} \\ \nonumber
&\leq \sum_{r \in A(\bm{x})} \| ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \|_q \\ \nonumber
&= \sum_{r \in A(\bm{x})} |\bm{V}_{cr} - \bm{V}_{jr}| \cdot \| \bm{w}_r^\top \|_q \\
&\leq \sum_{r=1}^{d} |\bm{V}_{cr} - \bm{V}_{jr}| \cdot \| \bm{w}_r^\top \|_q. \label{eq:global_bnd_grad_g}
\end{align}
Therefore, we can actually obtain a global upper bound of the Lipschitz constant $L_q$ by evaluating \eqref{eq:global_bnd_grad_g} and don't have to do sampling.
\subsubsection*{Tighten the bound of $L_{q,x_0}$}
In fact, if we can find out all possible $A(\bm{x})$ around $\bm{x}_0$, e.g. looking at the fixed ball $B_p(\bm{x}_0,R)$ and its corresponding $A(\bm{x})$, then we can obtain the exact value of $L_q$ by computing \eqref{eq:exact_grad_g}. This can be possibly realized in several ways:
\begin{enumerate}
\item[(a)] via sampling on the $B_p(\bm{x}_0,R)$
\item[(b)] construct the neighborhood geometry by computing the distance to the hyperplanes.
\item[(c)] via constructing bounds on the activation status (will be discussed in the next section)
\end{enumerate}
\section{Some Results on 3-layer NN \textcolor{red}{WIP}}
Given $u \in \mathbb{R}^t$, $v\in \mathbb{R}^{k \times t}$, $w\in \mathbb{R}^{d \times k}$, where $u_j \in \mathbb{R}$ is the $j$-th coordinate of $u$, $v_{j,i} \in \mathbb{R}$ is the $j,i$-th entry of $v$, and $w_i \in \mathbb{R}^d$ is the $i$-th column vector of $w$.
Suppose we are using a general activation function $\phi$.
\begin{align*}
f(x) = \sum_{j=1}^t u_j \cdot \phi \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right)
\end{align*}
We can compute the first derivative
\begin{align*}
\frac{\partial f(x)}{\partial x} = & \left[ \sum_{j=1}^t u_j \cdot \phi' \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right) \right. \\
& ~ \left. \sum_{i=1}^k v_{j,i} \phi'(w_i^\top \cdot x) w_i \right]
\end{align*}
For ReLU, the activation depends on the sign of both $w_i^\top \cdot x$ and $\sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x)$ (ignoring a bias). A global bound can be given, but will be even weaker than~\eqref{eq:global_bnd_grad_g} (roughly up to a factor of $\sum_{j=1}^t | u_j |$).
and compute the second derivative (For ReLU, the second derivative is 0)
\begin{align*}
\frac{\partial^2 f(x)}{ \partial x^2 } = & ~ \left[ \sum_{j=1}^t u_j \cdot \phi'' \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right) \right. \\
& ~ \cdot \sum_{i=1}^k v_{j,i} \phi'(w_i^\top \cdot x) \sum_{i'=1}^k v_{j,i'} \phi'(w_{i'}^\top x) w_i w_{i'}^\top \\
& ~ + \sum_{j=1}^t u_j \cdot \phi' \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right) \\
& ~ \left. \cdot \sum_{i=1}^k v_{j,i} \phi''(w_i^\top \cdot x) w_i w_i^\top \right]
\end{align*}
\section{Sufficient condition on activation not change}
\subsection{2-layer NN}
Without loss of generality, let's look at the 1st unit (node) in the hidden layer. If $x+\delta$ has the same activation as $x$, it means that either they are both activated
\begin{align}
& \sum_{i=1}^{d} W_{1i} x_i + b_1 > 0 \\
& \sum_{i=1}^{d} W_{1i} (x_i+\delta_i) + b_1 > 0
\end{align}
or they are both deactivated
\begin{align}
& \sum_{i=1}^{d} W_{1i} x_i + b_1 \leq 0 \\
& \sum_{i=1}^{d} W_{1i} (x_i+\delta_i) + b_1 \leq 0.
\end{align}
If $\delta$ satisfies the following condition, then the activation status will not change (i.e. the sufficient condition of remaining the same activation status):
\begin{equation}
| \sum_{i=1}^{d} W_{1i} \delta_i | < |\sum_{i=1}^{d} W_{1i} x_i + b_1 |.
\end{equation}
Meanwhile, with H\"older's inequality, we can bound $| \sum_{i=1}^{d} W_{1i} \delta_i |$ as follows
\begin{equation}
| \sum_{i=1}^{d} W_{1i} \delta_i | \leq \| W_{1:} \|_q \cdot \| \delta \|_{p} ,
\end{equation}
where $1/p + 1/q = 1$, $p,q \geq 1$. Therefore, if we have
\begin{equation}
\| W_{1:} \|_q \cdot \| \delta \|_{p} < |\sum_{i=1}^{d} W_{1i} x_i + b_1 | \Rightarrow \| \delta \|_{p} < \frac{|\sum_{i=1}^{d} W_{1i} x_i + b_1|}{\| W_{1:} \|_q},
\end{equation}
then we can make sure $x$ and $x+\delta$ have the same activation status on the node 1 in the hidden layer. Similarly, we can derive the sufficient condition for node $j$ and the sufficient condition for all the nodes remaining the same activation status is
\begin{equation}
\label{eq:sufficient_delta_same_act}
\| \delta \|_{p} \leq \inf_{j} \frac{|\sum_{i=1}^{d} W_{ji} x_i + b_j|}{\| W_{j:} \|_q}.
\end{equation}
\section{Combine LP formulation with activation status}
\begin{eqnarray}
\underset{\delta}{\text{minimize }} && \| \delta \|_p \nonumber\\
\text{subject to } && g(x_0 + \delta) < 0 \label{eq:general_opt}
\end{eqnarray}
where $g(x_0 + \delta) = f_c(x_0 + \delta) - f_j(x_0 + \delta)$ and $f(x_0 + \delta) = W^{(n)} \sigma(W^{(n-1)} \sigma(...W^{(1)}(x_0 + \delta)))$, where $W^{(n)}$ is the weight matrix of $n$th layer. The optimization formulation is exact -- if we can solve \eqref{eq:general_opt}, we get an effective attack with minimal norm. However, the constraint set in \eqref{eq:general_opt} is non-convex for general activation function. For ReLU, the constraint set can be non-convex as well. Below, we discuss the cases where the constraint set is linear in 2-layer NN.
\subsection{2-layer NN}
If we add the bound \eqref{eq:sufficient_delta_same_act} as an additional constraint in \eqref{eq:general_opt}, then we are guaranteed that the activation status won't change for $x_0 + \delta$ and we can directly solve the following without using the constraints in Bastani's paper:
\begin{eqnarray}
\underset{\delta}{\text{minimize }} && \| \delta \|_p \nonumber\\
\text{subject to } && \sum_{r \in A(\bm{x_0})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \cdot (\bm{w}_r (\bm{x}_0+\bm{\delta}) + b_r) < 0 \label{eq:lily_bastani_opt} \\
&& \| \delta \|_{p} \leq \inf_{j} \frac{|\sum_{i=1}^{d} W_{ji} x_i + b_j|}{\| W_{j:} \|_q} \nonumber
\end{eqnarray}
If \eqref{eq:lily_bastani_opt} is feasible, then $\delta^*$ is the minimum distortion of $\bm{x}_0$ and we should get the same solution as Bastani's paper; if \eqref{eq:lily_bastani_opt} is infeasible, then $\inf_{j} \frac{|\sum_{i=1}^{d} W_{ji} x_i + b_j|}{\| W_{j:} \|_q}$ is a strict lower bound of minimum distortion (i.e. we won't over-approximate the lower bound as in the calculation in clever).
Alternatively, we can also combine the idea of Lipschitz constant with the activation status and obtain a strict analytic lower bound for $\bm{x}_0$ (to be illustrated later)
\textcolor{red}{Previous draft}
\subsection{Upper bounds of $L_{q,x_0}^j$}
Suppose there are $N$ possible activation patterns $\{ \set{S}^{(1)}, \ldots, \set{S}^{(N)} \}$, where $\set{S}^{(i)} = \prod_{k=1}^{M-1}\mathcal{A}_k(\bm{x}^{(i)})$ with $\bm{x}^{(i)} \in B_p(\bm{x_0},R)$, and let $\set{V} = \bigcap_{i=1}^{N} \set{S}^{(i)}, \set{U} = \bigcup_{i=1}^{N} \set{S}^{(i)}$. The Lipschitz constant can be upper bounded by:
\begin{align}
& \max_{\bm{x} \in B_p(\bm{x_0},R)} \| \nabla g_j(\bm{x}) \| \notag \\
=& \max_{\{ \set{S}^{(1)}, \ldots , \set{S}^{(N)} \}} \| \nabla g_j(\bm{x}) \| \notag \\
=& \max_{r=1 \in [N] } \left\| \sum_{[i_k]_{k=1}^{M-1} \in \set{V} \cup (\set{S}^{(r)} \setminus \set{V}) } \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \right\| \notag \\
\leq & \quad \left\| \sum_{[i_k]_{k=1}^{M-1} \in \set{V} } \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \right\| \notag \\
&+ \max_{r \in [N] } \left\| \sum_{[i_k]_{k=1}^{M-1} \in \set{S}^{(r)} \setminus \set{V} } \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \right\| \notag \\
\leq & \quad \left\| \sum_{[i_k]_{k=1}^{M-1} \in \set{V} } \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \right\| \notag \\
&+ \max_{r \in [N] } \sum_{[i_k]_{k=1}^{M-1} \in \set{S}^{(r)} \setminus \set{V}} \| \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \| \notag \\
\leq & \quad \left\| \sum_{[i_k]_{k=1}^{M-1} \in \set{V} } \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \right\| \notag \\
&+ \sum_{[i_k]_{k=1}^{M-1} \in \set{U} \setminus \set{V}} \| \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top \| \notag
\end{align}
Given the input constraint $\| \bm{x}-\bm{x_0} \|_p \leq R$, we can propagate it layer by layer and check the activation status of neurons in each layer. For neuron $i$ in layer $k$, there are four possible status:
\begin{itemize}
\item[(a)] the neuron is always activated
\begin{itemize}
\item[i.e] $\forall \bm{x} \in B_p(\bm{x_0},R), \bm{z}_i^{(k)}(\bm{x}) = \bm{z}_i^{(k)}(\bm{x}_0) = 1$
\end{itemize}
\item[(b)] the neuron could change from activated to inactivated
\begin{itemize}
\item[i.e] $\exists \bm{x} \in B_p(\bm{x_0},R), \bm{z}_i^{(k)}(\bm{x}) = 0, \bm{z}_i^{(k)}(\bm{x}_0) = 1$
\end{itemize}
\item[(c)] the neuron could change from inactivated to activated
\begin{itemize}
\item[i.e] $\exists \bm{x} \in B_p(\bm{x_0},R), \bm{z}_i^{(k)}(\bm{x}) = 1, \bm{z}_i^{(k)}(\bm{x}_0) = 0$
\end{itemize}
\item[(d)] the neuron is always inactivated
\begin{itemize}
\item[i.e] $\forall \bm{x} \in B_p(\bm{x_0},R), \bm{z}_i^{(k)}(\bm{x}) = \bm{z}_i^{(k)}(\bm{x}_0) = 0$
\end{itemize}
\end{itemize}
The $\set{V}$ defined above is in fact the set of indices of neurons satisfying (a), whereas $\set{U}$ is the set of indices of all neurons satisfying (a) or (b) or (c). Note that neurons in (d) are never activated and thus do not belong to $\set{S}^{(r)}$.
\subsection{Find out all possible activation patterns}
\textcolor{blue}{Describe how to propagate the constraint $\| \bm{x}-\bm{x_0} \|_p \leq R$ to find out the output range of each neuron}
\textcolor{red}{Previous draft}
\subsection{Lipschitz constant of ReLU network}
Let $\mathcal{A}_k(\bm{x}) = \{i \mid \bm{z}_i^{(k)}(\bm{x}) = 1 \}$ denotes the set of activated neurons in layer $k$ with $\bm{z}_i^{(k)}(\bm{x})$ being the indicator variable of activation status for neuron $i$ in layer $k$:
\begin{equation*}
\bm{z}_i^{(k)}(\bm{x}) =
\begin{cases}
1 & \text{if $\W{(k)}\phi_{k-1}(\bm{x})+\bias{(k)} > 0$, (activated)} \\
0 & \text{otherwise, (inactivated)}
\end{cases}
\end{equation*}
we show that the gradient of $i_M$th output in a ReLU network, $f_{i_M}(\bm{x}) = [\phi_M(\bm{x})]_{i_M}$, is only dependent on the weights of activated neurons:
\begin{equation}
\label{eq:grad_Mlayer}
\nabla f_{i_M}(\bm{x}) = \sum_{[i_k]_{k=1}^{M-1} \in \prod_{k=1}^{M-1}\mathcal{A}_k(\bm{x})} \beta_{i_{1}, \cdots, i_{M-1}}^{(i_M)} \W{(1)}(i_1,:)^\top
\end{equation}
where
\begin{equation*}
\beta_{i_{1}, \cdots, i_{M-1}}^{(i_M)} := \prod_{k=1}^{M-1} \W{(k+1)}(i_{k+1},i_k),
\end{equation*}
$\W{}(i,j)$ is the $(i,j)$th entry of matrix $\W{}$, \mbox{$\W{}(i,:)$} is the $i$th row of matrix $\W{}$, $[i_k]_{k=1}^{T}$ is a shorthand for the $T$-tuple $(i_1,i_2,\cdots,i_T)$ and $\prod_{k=1}^{T}\mathcal{A}_k(\bm{x})$ is a shorthand for the Cartesian product $\mathcal{A}_1(\bm{x})\times \cdots \times \mathcal{A}_{T}(\bm{x}) = \{(i_1,i_2,\cdots,i_T) \mid i_k \in \mathcal{A}_k, \forall k \in [T]\}$.
The local Lipschitz constant $L_{q,x_0}^j$ of $g_j(\bm{x}) = f_c(\bm{x})-f_j(\bm{x})$ is the maximum norm of $\nabla g_j(\bm{x})$ where $\bm{x} \in B_p(\bm{x_0},R)$. With \eqref{eq:grad_Mlayer}, $\nabla g_j(\bm{x})$ for a $M$-layer ReLU network can be calculated as
\begin{equation*}
\nabla g_j(\bm{x}) = \sum_{[i_k]_{k=1}^{M-1} \in \prod_{k=1}^{M-1}\mathcal{A}_k(\bm{x})} \alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} \W{(1)}(i_1,:)^\top
\end{equation*}
where
\begin{equation*}
\alpha_{i_{1}, \cdots, i_{M-1}}^{(c,j)} := \beta_{i_{1}, \cdots, i_{M-1}}^{(c)} - \beta_{i_{1}, \cdots, i_{M-1}}^{(j)}.
\end{equation*}\textcolor{blue}{Hongge: Note that since gradient only depends on local information we can assume the sets of activated neurons do not change.}
Notice that $\nabla g_j(\bm{x})$ only depends on the activation patterns $\prod_{k=1}^{T}\mathcal{A}_k(\bm{x})$ and that the number of activation patterns is finite. Thus, given an example $\bm{x_0}$, if we enumerate all the possible activation patterns within the ball $B_p(\bm{x_0},R)$, we can get the exact local Lipschitz constant $L_{q,x_0}^j$ which is the maximum value of $\| \nabla g_j(\bm{x}) \|$. However, the number of all possible patterns grows exponentially as the number of activated neurons increases. In this paper, we aim at finding an approximate upper bound for $\| \nabla g_j(\bm{x}) \|$, which gives us an upper bound for $L_{q,x_0}^j$ and therefore guarantees a \textit{strict} lower bound on the minimum distortion.
In other words, for the inputs having the same activation patterns (i.e. $\bm{x} \neq \bm{x}', \mathcal{A}_k(\bm{x}) = \mathcal{A}_k(\bm{x}'), k = 1, \ldots, M$), the gradient vectors are the same, $\nabla f_{i_M}(\bm{x}) = \nabla f_{i_M}(\bm{x}')$.
\section{The Lipschitz constant of ReLU activation}
\subsection{2-layer NN}
\subsubsection*{Global bound}
The $j_{\text{th}}$ output of a one-hidden-layer neural network can be written as
\begin{equation*}
f_j(\bm{x}) = \sum_{r=1}^{U} \bm{V}_{jr} \cdot \sigma \left ( \sum_{i=1}^{d} \bm{W}_{ri} \cdot x_i + b_r \right ) = \sum_{r=1}^{U} \bm{V}_{jr} \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right ),
\end{equation*}
where $\sigma (z) = \max (z, 0)$ is ReLU activation function, $\bm{W}$ and $\bm{V}$ are the weight matrices of the first and second layer respectively, and $\bm{w}_r$ is the $r_{\text{th}}$ row of $\bm{W}$. Thus, we can compute $g(\bm{x})$ and $\| \nabla g(\bm{x}) \|_q$ below:
\begin{align*}
g(\bm{x}) = f_c(\bm{x}) - f_j(\bm{x})
&= \sum_{r=1}^{U} \bm{V}_{cr} \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right ) - \sum_{r=1}^{U} \bm{V}_{jr} \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right ) \\
&= \sum_{r=1}^{U} ( \bm{V}_{cr} - \bm{V}_{jr} ) \cdot \sigma \left ( \bm{w}_r \bm{x} + b_r \right )
\end{align*}
and
\begin{align*}
\| \nabla g(\bm{x}) \|_q &= \left \| \sum_{r=1}^{U} \mathbb{I}(\bm{w}_r \bm{x} + b_r) ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \right \|_q,
\end{align*}
where $\mathbb{I}(z)$ is an univariate indicator function:
\[
\mathbb{I}(z) = \Big \{
\begin{tabular}{cc}
1, & \text{if $z>0$,} \\
0, & \text{if $z \leq 0$.}
\end{tabular}
\]
We can also rewrite $g(\bm{x})$ and $\| \nabla g(\bm{x}) \|_q$ as
\begin{align*}
g(\bm{x}) &= \sum_{r \in A(\bm{x})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \cdot (\bm{w}_r \bm{x} + b_r), \\
\| \nabla g(\bm{x}) \|_q &= \| \sum_{r \in A(\bm{x})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \|_q ,
\end{align*}
where $A(\bm{x})$ is an index set of activated nodes, i.e $A(\bm{x}) := \{r \mid \bm{w}_r \bm{x} + b_r > 0 \}$. A global upper bound of $\| \nabla g(\bm{x}) \|_q$ can be computed as follows:
\begin{align}
\| \nabla g(\bm{x}) \|_q &= \| \sum_{r \in A(\bm{x})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \|_q \label{eq:exact_grad_g} \\ \nonumber
&\leq \sum_{r \in A(\bm{x})} \| ( \bm{V}_{cr} - \bm{V}_{jr} ) \bm{w}_r^\top \|_q \\ \nonumber
&= \sum_{r \in A(\bm{x})} |\bm{V}_{cr} - \bm{V}_{jr}| \cdot \| \bm{w}_r^\top \|_q \\
&\leq \sum_{r=1}^{d} |\bm{V}_{cr} - \bm{V}_{jr}| \cdot \| \bm{w}_r^\top \|_q. \label{eq:global_bnd_grad_g}
\end{align}
Therefore, we can actually obtain a global upper bound of the Lipschitz constant $L_q$ by evaluating \eqref{eq:global_bnd_grad_g} and don't have to do sampling.
\subsubsection*{Tighten the bound of $L_{q,x_0}$}
In fact, if we can find out all possible $A(\bm{x})$ around $\bm{x}_0$, e.g. looking at the fixed ball $B_p(\bm{x}_0,R)$ and its corresponding $A(\bm{x})$, then we can obtain the exact value of $L_q$ by computing \eqref{eq:exact_grad_g}. This can be possibly realized in several ways:
\begin{enumerate}
\item[(a)] via sampling on the $B_p(\bm{x}_0,R)$
\item[(b)] construct the neighborhood geometry by computing the distance to the hyperplanes.
\item[(c)] via constructing bounds on the activation status (will be discussed in the next section)
\end{enumerate}
\section{Some Results on 3-layer NN \textcolor{red}{WIP}}
Given $u \in \mathbb{R}^t$, $v\in \mathbb{R}^{k \times t}$, $w\in \mathbb{R}^{d \times k}$, where $u_j \in \mathbb{R}$ is the $j$-th coordinate of $u$, $v_{j,i} \in \mathbb{R}$ is the $j,i$-th entry of $v$, and $w_i \in \mathbb{R}^d$ is the $i$-th column vector of $w$.
Suppose we are using a general activation function $\phi$.
\begin{align*}
f(x) = \sum_{j=1}^t u_j \cdot \phi \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right)
\end{align*}
We can compute the first derivative
\begin{align*}
\frac{\partial f(x)}{\partial x} = & \left[ \sum_{j=1}^t u_j \cdot \phi' \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right) \right. \\
& ~ \left. \sum_{i=1}^k v_{j,i} \phi'(w_i^\top \cdot x) w_i \right]
\end{align*}
For ReLU, the activation depends on the sign of both $w_i^\top \cdot x$ and $\sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x)$ (ignoring a bias). A global bound can be given, but will be even weaker than~\eqref{eq:global_bnd_grad_g} (roughly up to a factor of $\sum_{j=1}^t | u_j |$).
and compute the second derivative (For ReLU, the second derivative is 0)
\begin{align*}
\frac{\partial^2 f(x)}{ \partial x^2 } = & ~ \left[ \sum_{j=1}^t u_j \cdot \phi'' \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right) \right. \\
& ~ \cdot \sum_{i=1}^k v_{j,i} \phi'(w_i^\top \cdot x) \sum_{i'=1}^k v_{j,i'} \phi'(w_{i'}^\top x) w_i w_{i'}^\top \\
& ~ + \sum_{j=1}^t u_j \cdot \phi' \left( \sum_{i=1}^k v_{j,i} \cdot \phi(w_i^\top \cdot x) \right) \\
& ~ \left. \cdot \sum_{i=1}^k v_{j,i} \phi''(w_i^\top \cdot x) w_i w_i^\top \right]
\end{align*}
\section{Sufficient condition on activation not change}
\subsection{2-layer NN}
Without loss of generality, let's look at the 1st unit (node) in the hidden layer. If $x+\delta$ has the same activation as $x$, it means that either they are both activated
\begin{align}
& \sum_{i=1}^{d} W_{1i} x_i + b_1 > 0 \\
& \sum_{i=1}^{d} W_{1i} (x_i+\delta_i) + b_1 > 0
\end{align}
or they are both deactivated
\begin{align}
& \sum_{i=1}^{d} W_{1i} x_i + b_1 \leq 0 \\
& \sum_{i=1}^{d} W_{1i} (x_i+\delta_i) + b_1 \leq 0.
\end{align}
If $\delta$ satisfies the following condition, then the activation status will not change (i.e. the sufficient condition of remaining the same activation status):
\begin{equation}
| \sum_{i=1}^{d} W_{1i} \delta_i | < |\sum_{i=1}^{d} W_{1i} x_i + b_1 |.
\end{equation}
Meanwhile, with H\"older's inequality, we can bound $| \sum_{i=1}^{d} W_{1i} \delta_i |$ as follows
\begin{equation}
| \sum_{i=1}^{d} W_{1i} \delta_i | \leq \| W_{1:} \|_q \cdot \| \delta \|_{p} ,
\end{equation}
where $1/p + 1/q = 1$, $p,q \geq 1$. Therefore, if we have
\begin{equation}
\| W_{1:} \|_q \cdot \| \delta \|_{p} < |\sum_{i=1}^{d} W_{1i} x_i + b_1 | \Rightarrow \| \delta \|_{p} < \frac{|\sum_{i=1}^{d} W_{1i} x_i + b_1|}{\| W_{1:} \|_q},
\end{equation}
then we can make sure $x$ and $x+\delta$ have the same activation status on the node 1 in the hidden layer. Similarly, we can derive the sufficient condition for node $j$ and the sufficient condition for all the nodes remaining the same activation status is
\begin{equation}
\label{eq:sufficient_delta_same_act}
\| \delta \|_{p} \leq \inf_{j} \frac{|\sum_{i=1}^{d} W_{ji} x_i + b_j|}{\| W_{j:} \|_q}.
\end{equation}
\section{Combine LP formulation with activation status}
\begin{eqnarray}
\underset{\delta}{\text{minimize }} && \| \delta \|_p \nonumber\\
\text{subject to } && g(x_0 + \delta) < 0 \label{eq:general_opt}
\end{eqnarray}
where $g(x_0 + \delta) = f_c(x_0 + \delta) - f_j(x_0 + \delta)$ and $f(x_0 + \delta) = W^{(n)} \sigma(W^{(n-1)} \sigma(...W^{(1)}(x_0 + \delta)))$, where $W^{(n)}$ is the weight matrix of $n$th layer. The optimization formulation is exact -- if we can solve \eqref{eq:general_opt}, we get an effective attack with minimal norm. However, the constraint set in \eqref{eq:general_opt} is non-convex for general activation function. For ReLU, the constraint set can be non-convex as well. Below, we discuss the cases where the constraint set is linear in 2-layer NN.
\subsection{2-layer NN}
If we add the bound \eqref{eq:sufficient_delta_same_act} as an additional constraint in \eqref{eq:general_opt}, then we are guaranteed that the activation status won't change for $x_0 + \delta$ and we can directly solve the following without using the constraints in Bastani's paper:
\begin{eqnarray}
\underset{\delta}{\text{minimize }} && \| \delta \|_p \nonumber\\
\text{subject to } && \sum_{r \in A(\bm{x_0})} ( \bm{V}_{cr} - \bm{V}_{jr} ) \cdot (\bm{w}_r (\bm{x}_0+\bm{\delta}) + b_r) < 0 \label{eq:lily_bastani_opt} \\
&& \| \delta \|_{p} \leq \inf_{j} \frac{|\sum_{i=1}^{d} W_{ji} x_i + b_j|}{\| W_{j:} \|_q} \nonumber
\end{eqnarray}
If \eqref{eq:lily_bastani_opt} is feasible, then $\delta^*$ is the minimum distortion of $\bm{x}_0$ and we should get the same solution as Bastani's paper; if \eqref{eq:lily_bastani_opt} is infeasible, then $\inf_{j} \frac{|\sum_{i=1}^{d} W_{ji} x_i + b_j|}{\| W_{j:} \|_q}$ is a strict lower bound of minimum distortion (i.e. we won't over-approximate the lower bound as in the calculation in clever).
Alternatively, we can also combine the idea of Lipschitz constant with the activation status and obtain a strict analytic lower bound for $\bm{x}_0$ (to be illustrated later)
\section{Robustness guarantees for ReLU networks}\label{sec:result}
\paragraph{Overview of our results.}
We begin with a motivating theorem in Sec~\ref{sec3:hard} showing that there does NOT exist a polynomial time algorithm able to find the minimum adversarial distortion with a $(1 - o(1))\ln n$ approximation ratio. We then introduce notations in Sec~\ref{sec3:relu} and state our main results in Sec~\ref{sec3:convexbnd} and \ref{sec3:gradbnd}, where we develop two approaches that guarantee to obtain a lower bound of minimum adversarial distortion. In Sec~\ref{sec3:convexbnd}, we first demonstrate a general approach to \textit{directly} derive the output bounds of a ReLU network with linear approximations when inputs are perturbed by a general $\ell_p$ norm noise. The analytic output bounds allow us to develop a fast algorithm \textbf{Fast-Lin}\xspace to compute certified lower bound.
In Sec~\ref{sec3:gradbnd}, we present \textbf{Fast-Lip}\xspace to obtain a certified lower bound of minimum distortion by deriving upper bounds for the local Lipschitz constant. Both methods are highly efficient and allow fast computation of certified lower bounds on large ReLU networks.
\subsection{Finding the minimum distortion with a $0.99\ln n$ approximation ratio is hard}
\label{sec3:hard}
\cite{katz2017reluplex} shows that verifying robustness for ReLU networks is NP-complete; in other words, there is no efficient (polynomial time) algorithm to find the exact minimum adversarial distortion. Here, we further show that even \textit{approximately} finding the minimum adversarial distortion with a guaranteed approximation ratio can be hard. Suppose the $\ell_p$ norm of the true minimum adversarial distortion is $r_0$,
and a robustness verification program \textsf{A} gives a guarantee that no adversarial examples exist within an $\ell_p$ ball of radius $r$ ($r$ is a lower bound of $r_0$). The approximation ratio $\alpha \coloneqq \frac{r_0}{r} > 1$. We hope that $\alpha$ is close to 1 with a guarantee; for example, if $\alpha$ is a constant regardless of the scale of the network, we can always be sure that $r_0$ is at most $\alpha$ times as large as the lower bound $r$ found by $\textsf{A}$.
Here we relax this requirement and allow the approximation ratio to increase with the number of neurons $n$. In other words, when $n$ is larger, the approximation becomes more inaccurate, but this ``inaccuracy'' can be bounded. However, the following theorem shows that no efficient algorithms exist to give a $0.99\ln n$ approximation in the special case of $\ell_1$ robustness:
\begin{theorem}\label{thm:intro_hardness_np_p}
Unless $\mathsf{P}=\mathsf{NP}$, there is no polynomial time algorithm that gives $(1-o(1))\ln n$-approximation to the $\ell_1$ {\rm ReLU} robustness verification problem with $n$ neurons.
\end{theorem}
Our proof is based on a well-known in-approximability result of \textsf{SET-COVER} problem \cite{rs97,ams06,ds14} and a novel reduction from \textsf{SET-COVER} to our problem. We defer the proof into Appendix~\ref{app:hardness}. The formal definition of the $\ell_1$ ReLU robustness verification problem can be found in Definition~\ref{def:robust_net_real}. Theorem~\ref{thm:intro_hardness_np_p} implies that any efficient (polynomial time) algorithm cannot give better than $(1-o(1))\ln n$-approximation guarantee. Moreover, by making a stronger assumption of Exponential Time Hypothesis ($\mathsf{ETH}$), we can state an explicit result about running time using existing results from \textsf{SET-COVER}~\cite{m12,moshkovitz2012projection},
\begin{corollary}\label{cor:intro_hardness_eth_pgc}
Under $\mathsf{ETH}$, there is no $2^{o(n^c)}$ time algorithm that gives $(1-o(1))\ln n$-approximation to the $\ell_1$ {\rm ReLU} robustness verification problem with $n$ neurons, where $c\in (0,1)$ is some fixed constant.
\end{corollary}
\subsection{ReLU Networks and Their Activation Patterns}
\label{sec3:relu}
Let $\bm{x} \in \mathbb{R}^{n_0}$ be the input vector for an $m$-layer neural network with $m-1$ hidden layers and let the number of neurons in each layer be $n_k, \forall k \in [m]$. We use $[n]$ to denote set $\{1,2,\cdots,n\}$. The weight matrix $\W{(k)}$ and bias vector $\bias{(k)}$ for the $k$-th layer have dimension $n_k \times n_{k-1}$ and $n_k$, respectively. Let $\phi_k: \mathbb{R}^{n_0}\to\mathbb{R}^{n_k}$ be the operator mapping from input layer to layer $k$ and $\sigma(\bm{y})$ be the coordinate-wise activation function; for each $k\in [m-1]$, the relation between layer $k-1$ and layer $k$ can be written as $\phi_k(\bm{x}) = \sigma(\W{(k)}\phi_{k-1}(\bm{x})+\bias{(k)}),$
where $\W{(k)} \in \mathbb{R}^{n_k \times n_{k-1}}, \bias{(k)} \in \mathbb{R}^{n_k}$. For the input layer and the output layer, we have $\phi_0(\bm{x}) = \bm{x}$ and $\phi_m(\bm{x}) = \W{(m)}\phi_{m-1}(\bm{x})+\bias{(m)}$. The output of the neural network is $f(\bm{x}) = \phi_m(\bm{x})$, which is a vector of length $n_m$, and the $j$-th output is its $j$-th coordinate, denoted as $f_j(\bm{x}) = [\phi_m(\bm{x})]_j$. For ReLU activation, the activation function $\sigma(\bm{y}) = \max(\bm{y},\bm{0})$ is an element-wise operation on the input vector $\bm{y}$.
Given an input data point $\bm{x_0} \in \mathbb{R}^{n_0}$ and a bounded $\ell_p$-norm perturbation $\epsilon \in \mathbb{R}_{+}$, the input $\bm{x}$ is constrained in an $\ell_p$ ball $B_p(\bm{x_0},\epsilon) := \{ \bm{x} ~|~ \| \bm{x} - \bm{x_0} \|_{p} \leq \epsilon \}$. With all possible perturbations in $B_p(\bm{x_0},\epsilon)$, the pre-ReLU activation of each neuron has a lower and upper bound $l \in \mathbb{R}$ and $u \in \mathbb{R}$, where $l \leq u$. Let us use $\lwbnd{(k)}_r$ and $\upbnd{(k)}_r$ to denote the lower and upper bound for the $r$-th neuron in the $k$-th layer, and let $\bm{z}^{(k)}_r$ be its pre-ReLU activation, where $\bm{z}^{(k)}_r = \W{(k)}_{r,:}\phi_{k-1}(\bm{x})+\bias{(k)}_r$, $\lwbnd{(k)}_r \leq \bm{z}^{(k)}_r \leq \upbnd{(k)}_r$, and $\W{(k)}_{r,:}$ is the $r$-th row of $\W{(k)}$. There are three categories of possible activation patterns -- (i) the neuron is always activated: $\setIpos{k} \coloneqq \{ r \in [n_k] ~|~ \upbnd{(k)}_r \geq \lwbnd{(k)}_r \geq 0 \}$, (ii) the neuron is always inactivated: $\setIneg{k} \coloneqq \{ r \in [n_k] ~|~ \lwbnd{(k)}_r \leq \upbnd{(k)}_r \leq 0 \} $, and (iii) the neuron could be either activated or inactivated: $\setIuns{k} \coloneqq \{ r \in [n_k] ~|~ {\lwbnd{(k)}_r < 0 < \upbnd{(k)}_r} \}$. Obviously, $\{ \setIpos{k}, \setIneg{k},\setIuns{k} \}$ is a partition of set $[n_k]$.
\subsection{Approach 1 (Fast-Lin): Certified lower bounds via linear approximations}
\label{sec3:convexbnd}
\subsubsection{Derivation of the output bounds via linear upper and lower bounds for ReLU}
\label{sec:fastlin_derivation}
In this section, we propose a methodology to \textit{directly} derive upper bounds and lower bounds of the output of an $m$-layer feed-forward ReLU network. The central idea is to derive an \textit{explicit} upper/lower bound based on the linear approximations for the neurons in category (iii) and the signs of the weights associated with the activations.
\begin{figure*}[t]
\ifdef{}{
\includegraphics[width=0.7\textwidth]{figure_relu_network}
}{
\includegraphics[width=\textwidth]{figure_relu_network}
}
\centering
\caption{Illustration of deriving output bounds for ReLU networks in Section~\ref{sec3:convexbnd}. The final output upper bounds ($f_j^U$) and lower bounds ($f_j^L$) can be derived by considering the activation status of the neurons with input perturbation $\| \delta \|_p \leq \epsilon$. For neurons in $\setIpos{k}$, their outputs are identical to their inputs; for neurons in $\setIneg{k}$, they can be removed during computation as their outputs are always zero; for neurons in $\setIuns{k}$, their outputs can be bounded by corresponding linear upper bounds and lower bounds considering the signs of associated weights.}
\label{fig:activation}
\end{figure*}
We start with a 2-layers network and then extend it to $m$ layers. The $j$-th output of a 2-layer network is:
$$f_j(\bm{x}) = \sum_{r\in \setIpos{1}, \setIneg{1}, \setIuns{1}} \W{(2)}_{j,r} \sigma(\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r) + \bias{(2)}_j.$$For neurons $r \in \setIpos{1}$, we have $\sigma(\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r) = \W{(1)}_{r,:} \bm{x} + \bias{(1)}_r$; for neurons $r \in \setIneg{1}$, we have $\sigma(\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r) = 0.$ For the neurons in category (iii), we propose to use the following linear upper bound and a linear lower bound to replace the ReLU activation $\sigma(y)$:
\begin{equation}
\label{eq:our_cvx_approx}
\frac{u}{u-l} y \leq \sigma(y) \leq \frac{u}{u-l} (y-l).
\end{equation}
Let $\bm{d}^{(1)}_r := \frac{\upbnd{(1)}_r}{\upbnd{(1)}_r-\lwbnd{(1)}_r}$, we have
\begin{align}
\label{eq:2-layer-cvx-ours}
\bm{d}_r^{(1)}(\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r) & \leq \; \sigma(\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r)
\ifdef{}{\\ & }{}
\leq \bm{d}^{(1)}_r (\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r - \lwbnd{(1)}_r)
\ifdef{}{. \nonumber}{}
\end{align}
To obtain an upper bound and lower bound of $f_j(\bm{x})$ with~\eqref{eq:our_cvx_approx}, set $\bm{d}^{(1)}_r=1$ for $r \in \setIpos{1}$, and we have
\begin{align}
\label{eq:2-layer-newfju}
f_j^U(\bm{x}) & = \sum_{r\in \setIpos{1}, \setIuns{1}} \W{(2)}_{j,r} \bm{d}^{(1)}_r (\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r)
\ifdef{}{\\ & }{}
- \sum_{r\in \setIuns{1}, \W{(2)}_{j,r}>0} \W{(2)}_{j,r} \bm{d}^{(1)}_r \lwbnd{(1)}_r + \bias{(2)}_j
\ifdef{}{, \nonumber}{}
\end{align}
\vspace{-1em}
\begin{align}
\label{eq:2-layer-newfjl}
f_j^L(\bm{x}) & = \sum_{r\in \setIpos{1}, \setIuns{1}} \W{(2)}_{j,r} \bm{d}^{(1)}_r (\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r)
\ifdef{}{\\ & }{}
- \sum_{r\in \setIuns{1}, \W{(2)}_{j,r}<0} \W{(2)}_{j,r} \bm{d}^{(1)}_r \lwbnd{(1)}_r + \bias{(2)}_j
\ifdef{}{, \nonumber}{}
\end{align}
where $f_j^L(\bm{x}) \leq f_j(\bm{x}) \leq f_j^U(\bm{x})$.
To obtain $f_j^U(\bm{x})$, we take the upper bound of $\sigma(\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r)$ for $r \in \setIuns{1}, \W{(2)}_{j,r}>0$ and its lower bound for $r \in \setIuns{1}, \W{(2)}_{j,r} \leq 0$. Both cases share a common term of $\bm{d}^{(1)}_r (\W{(1)}_{r,:} \bm{x} + \bias{(1)}_r)$, which is combined into the first summation term in~\eqref{eq:2-layer-newfju} with $r \in \setIuns{1}$. Similarly we get the bound for $f_j^L(\bm{x})$.
For a general $m$-layer ReLU network with the linear approximation~\eqref{eq:our_cvx_approx}, we will show in Theorem~\ref{thm:cvx_bnd} that the network output can be bounded by two explicit functions when the input $\bm{x}$ is perturbed with a $\epsilon$-bounded $\ell_p$ noise. We start by defining the activation matrix $\DD{(k)}$ and the additional equivalent bias terms $\upbias{(k)}$ and $\lwbias{(k)}$ for the $k$-th layer in Definition \ref{def:A_eta_tau} and the two explicit functions in \ref{def:f_L_f_U}.
\begin{definition}[$\A{(k)}, \upbias{(k)}, \lwbias{(k)}$]\label{def:A_eta_tau}
Given matrices $\W{(k)} \in \mathbb{R}^{n_k \times n_{k-1}}$ and vectors $\bias{(k)} \in \mathbb{R}^{n_k}, \forall k \in [m]$.
We define ${\bf D}^{(0)} \in \mathbb{R}^{n_0 \times n_0}$ as an identity matrix. For each $k \in [m-1]$, we define matrix ${\bf D}^{(k)} \in \mathbb{R}^{n_k \times n_k}$ as follows
\begin{align}\label{eq:def_D}
\DD{(k)}_{r,r} & =
\begin{cases}
\frac{\upbnd{(k)}_r}{\upbnd{(k)}_r-\lwbnd{(k)}_r} & \mathrm{~if~} r \in {\cal I}_k; \\
1 & \mathrm{~if~} r \in {\cal I}_k^+; \\
0 & \mathrm{~if~} r \in {\cal I}_k^-.
\end{cases}
\end{align}
We define matrix $\A{(m-1)} \in \mathbb{R}^{n_m \times n_{m-1}}$ to be $\W{(m)} \DD{(m-1)}$, and for each $k \in \{m-1, m-2, \cdots, 1\}$, matrix $\A{(k-1)} \in \mathbb{R}^{n_m \times n_{k-1}}$ is defined recursively as $\A{(k-1)} = \A{(k)} \W{(k)} \DD{(k-1)}.$
For each $k \in [m-1]$, we define matrices $\upbias{(k)} , \lwbias{(k)} \in \mathbb{R}^{n_k \times n_m}$, where
\begin{align*}
\upbias{(k)}_{r,j} & =
\begin{cases}
\lwbnd{(k)}_r & \mathrm{~if~} r \in \setIuns{k}, \, \A{(k)}_{j,r} > 0 ; \\
0 & \mathrm{~otherwise~} .
\end{cases}
\\
\lwbias{(k)}_{r,j} & =
\begin{cases}
\lwbnd{(k)}_r & \mathrm{~if~} r \in \setIuns{k}, \, \A{(k)}_{j,r} < 0 ; \\
0 & \mathrm{~otherwise~} .
\end{cases}
\end{align*}
\end{definition}
\begin{definition}[Two explicit functions : $f^U(\cdot)$ and $f^L(\cdot)$]\label{def:f_L_f_U}
Let matrices $\A{(k)}$, $\upbias{(k)}$ and $\lwbias{(k)}$ be defined as in Definition~\ref{def:A_eta_tau}. We define two functions $f^U , f^L : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ as follows. For each input vector $\bm{x} \in \mathbb{R}^{n_0}$,
\begin{align*}
f^{U}_j(\bm{x}) = & ~ \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\upbias{(k)}_{:,j}), \\
f^{L}_j(\bm{x}) = & ~ \A{(0)}_{j,:}\bm{x}+ \bias{(m)}_j+\sum_{k=1}^{m-1}\A{(k)}_{j,:}(\bias{(k)}-\lwbias{(k)}_{:,j}).
\end{align*}
\end{definition}
Now, we are ready to state our main theorem,
\begin{theorem}[Explicit upper and lower bounds]\label{thm:approach_1}
\label{thm:cvx_bnd}
Given an $m$-layer {\rm ReLU} neural network function $f : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$, there exists two explicit functions $f^L : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ and $f^U :\mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$ (see Definition~\ref{def:f_L_f_U}) such that $\forall j \in [n_m], \; f_{j}^{L}(\bm{x}) \leq f_{j}(\bm{x}) \leq f_{j}^{U}(\bm{x}), \; \forall \bm{x} \in B_p(\bm{x_0},\epsilon)$.
\end{theorem}
The proof of Theorem \ref{thm:cvx_bnd} is in Appendix~\ref{app:approach1_explicit_function}. Since the input $\bm{x} \in B_p(\bm{x_0},\epsilon)$, we can maximize \eqref{eq:2-layer-newfju} and minimize \eqref{eq:2-layer-newfjl} within this set to obtain a global upper and lower bound of $f_j(\bm{x})$, which has analytical solutions for any $1 \leq p \leq \infty$ and the result is formally shown in Corollary \ref{cor:cvx_bnd} (proof in Appendix~\ref{app:approach1_fixed_value}). In other words, we have \textit{analytic} bounds that can be computed efficiently without resorting to any optimization solvers for general $\ell_p$ distortion, and this is the key to enable fast computation for layer-wise output bounds.
We first formally define the global upper bound $\gamma_j^U$ and lower bound $\gamma_j^L$ of $f_j(\bm{x})$, and then obtain Corollary~\ref{cor:cvx_bnd}.
\begin{definition}[$\gamma_j^L, \gamma_j^U$]\label{def:gamma_j_L_gamma_j_U}
Given a point $\bm{x_0} \in \mathbb{R}^{n_0}$, a neural network function $f : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$, parameters $p,\epsilon$. Let matrices $\A{(k)}$, $\upbias{(k)}$ and $\lwbias{(k)}$, $\forall k \in [m-1]$ be defined as in Definition~\ref{def:A_eta_tau}. We define $\gamma_j^L, \gamma_j^U, \, \forall j \in [n_m]$ as
\begin{align*}
\gamma^L_j = \mu_j^- + \nu_j - \epsilon \|\A{(0)}_{j,:}\|_q \mathrm{~and~}
\gamma^U_j = \mu_j^+ + \nu_j + \epsilon \|\A{(0)}_{j,:}\|_q ,
\end{align*}
where $1/p+1/q=1$ and $\nu_j, \mu_j^+, \mu_j^-$ are defined as
\begin{align}
\mu_j^+ = ~ - \sum_{k=1}^{m-1}&\A{(k)}_{j,:}\upbias{(k)}_{:,j} ,\quad
\mu_j^- = ~ - \sum_{k=1}^{m-1}\A{(k)}_{j,:}\lwbias{(k)}_{:,j} \label{eq:def_mu_j_plus_minus} \\
\nu_j = ~ &\A{(0)}_{j,:} \bm{x_0} + \bias{(m)}_j + \sum_{k=1}^{m-1}\A{(k)}_{j,:}\bias{(k)} \label{eq:def_nu_j}
\end{align}
\end{definition}
\begin{corollary}[Two side bounds in closed-form]
\label{cor:cvx_bnd}
Given a point $\bm{x_0} \in \mathbb{R}^{n_0}$, an $m$-layer neural network function $f : \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_m}$, parameters $p$ and $\epsilon$. For each $j\in [n_m]$, there exist two fixed values $\gamma^L_j$ and $\gamma^U_j$ (see Definition~\ref{def:gamma_j_L_gamma_j_U}) such that $\gamma^L_j \leq f_j (\bm{x}) \leq \gamma^U_j, \; \forall \bm{x} \in B_p(\bm{x}_0, \epsilon ).$
\end{corollary}
\ifdef{}{
\input{alg_fastlin}
}{}
\subsubsection{Computing pre-{\rm ReLU} activation bounds }
Theorem \ref{thm:cvx_bnd} and Corollary \ref{cor:cvx_bnd} give us a global lower bound $\gamma_j^L$ and upper bound $\gamma_j^U$ of the $j$-th neuron at the $m$-th layer if we know all the pre-ReLU activation bounds $\lwbnd{(k)}$ and $\upbnd{(k)}$, from layer $1$ to $m-1$, as the construction of $\DD{(k)}$, $\lwbias{(k)}$ and $\upbias{(k)}$ requires $\lwbnd{(k)}$ and $\upbnd{(k)}$ (see Definition~\ref{def:A_eta_tau}). Here, we show how this can be done easily and layer-by-layer. We start from $m = 1$ where $\A{(0)} = \W{(1)}, f^U(\bm{x}) = f^L(\bm{x}) = \A{(0)}\bm{x}+\bias{(1)}$. Then, we can apply Corollary \ref{cor:cvx_bnd} to get the output bounds of each neuron and set them as $\lwbnd{(1)}$ and $\upbnd{(1)}$. Then, we can proceed to $m = 2$ with $\lwbnd{(1)}$ and $\upbnd{(1)}$ and compute the output bounds of second layer by Corollary \ref{cor:cvx_bnd} and set them as $\lwbnd{(2)}$ and $\upbnd{(2)}$. Repeating this procedure for all $m-1$ layers, we will get all the $\lwbnd{(k)}$ and $\upbnd{(k)}$ needed to compute the output range of the $m$-th layer.
Note that when computing $\lwbnd{(k)}$ and $\upbnd{(k)}$, the constructed $\W{(k)} \DD{(k-1)}$ can be saved and reused for bounding the next layer, which facilitates efficient implementations. Moreover, the time complexity of computing the output bounds of an $m$-layer ReLU network with Theorem~\ref{thm:cvx_bnd} and Corollary~\ref{cor:cvx_bnd} is \textit{polynomial} time in contrast to the approaches in \cite{katz2017reluplex} and \cite{lomuscio2017approach} where SMT solvers and MIO solvers have \textit{exponential} time complexity. The major computation cost is to form $\A{(0)}$ for the $m$-th layer, which involves multiplications of layer weights in a \textit{similar cost of forward propagation}. See the ``ComputeTwoSideBounds'' procedure in Algorithm~\ref{alg:fast-lin} in Appendix~\ref{sec:app_algs}.
\subsubsection{Deriving maximum certified lower bounds of minimum adversarial distortion}
\label{sec:cal_last_layer}
Suppose $c$ is the predicted class of the input data point $\bm{x_0}$ and the class is $j$. With Theorem~\ref{thm:cvx_bnd}, the maximum possible lower bound for the targeted attacks $\tilde \epsilon_j$ and un-targeted attacks $\tilde \epsilon$ are
\begin{equation*}
\tilde \epsilon_j = \max_{\epsilon} \, \epsilon \; \text{s.t.} \; \gamma_c^L(\epsilon) - \gamma_j^U(\epsilon) > 0 \; \text{ and } \; \tilde \epsilon = \min_{j \neq c} \, \tilde \epsilon_j .
\end{equation*}
Though it is hard to get analytic forms of $\gamma_c^L(\epsilon)$ and $\gamma_j^U(\epsilon)$ in terms of $\epsilon$, fortunately, we can still obtain $\tilde \epsilon_j$ via a binary search. This is because Corollary~\ref{cor:cvx_bnd} allows us to efficiently compute the numerical values of $\gamma_c^L(\epsilon)$ and $\gamma_j^U(\epsilon)$ given $\epsilon$. It is worth noting that we can further improve the bound by considering $g(\bm{x}) := f_c(\bm{x}) - f_j(\bm{x})$ at the last layer and apply the same procedure to compute the lower bound of $g(\bm{x})$ (denoted as $\tilde \gamma^L$); this can be done easily by redefining the last layer's weights to be a row vector $\bm{\bar{w}} \coloneqq \W{(m)}_{c,:} - \W{(m)}_{j,:}$. The corresponding maximum possible lower bound for the targeted attacks is $\tilde \epsilon_j = \max \epsilon \; \text{s.t.} \; \tilde \gamma^L(\epsilon)> 0$.
\ifdef{}{Our proposed algorithm, \textbf{Fast-Lin}\xspace, is shown in Algorithm~\ref{alg:fast-lin}.
}
{We list our complete algorithm, \textbf{Fast-Lin}\xspace, in Appendix~\ref{sec:app_algs}.}
\subsubsection{Discussions}
\label{sec:discuss}
We have shown how to derive explicit output bounds of ReLU network (Theorem~\ref{thm:approach_1}) with the proposed linear approximations and obtain analytical certified lower bounds (Corollary~\ref{cor:cvx_bnd}), which is the key of our proposed algorithm \textbf{Fast-Lin}.
\cite{zico17convex} presents a similar algorithmic result on computing certified bounds, but our framework and theirs are entirely different -- we use direct computation of layer-wise linear upper/lower bounds in Sec \ref{sec3:convexbnd} with binary search on $\epsilon$, while their results is achieved via the lens of dual LP formulation with Newton's method.
Interestingly, when we choose a special set of lower and upper bounds as in~\eqref{eq:2-layer-cvx-ours} and they choose a special dual LP variable in their equation (8), the two different frameworks coincidentally produce the same procedure for computing layer-wise bounds (the ``ComputeTwoSideBounds'' procedure in \textbf{Fast-Lin}\xspace and Algorithm 1 in \cite{zico17convex}).
However, our choice of bounds~\eqref{eq:2-layer-cvx-ours} is due to computation efficiency, while~\cite{zico17convex} gives a quite different justification. We encourage the readers to read Appendix A.3 in their paper on the justifications for this specific selection of dual variables and understand this robustness verification problem from different perspectives.
\subsection{Approach 2 (Fast-Lip): Certified lower bounds via bounding the local Lipschitz constant}
\label{sec3:gradbnd}
\cite{weng2017evaluating} shows a non-trivial lower bound of minimum adversarial distortion for an input example $\bm{x_0}$ in targeted attacks is $\min \left( g(\bm{x_0})/L_{q,x_0}^j,\epsilon \right)$, where $g(\bm{x})=f_c(\bm{x}) - f_j(\bm{x}), \, L_{q,x_0}^j$ is the local Lipschitz constant of $g(\bm{x})$ in $B_p(\bm{x_0}, \epsilon)$, $\,j$ is the target class, $c$ is the original class, and $1/p + 1/q = 1$. For un-targeted attacks, the lower bound can be presented in a similar form. \cite{weng2017evaluating} uses sampling techniques to estimate the local Lipschitz constant and compute an estimated lower bound without certificates.
Here, we propose a new algorithm to compute a \textit{certified} lower bound of the minimum adversarial distortion by upper bounding the local Lipschitz constant. To start with, let us rewrite the relations of subsequent layers in the following form: $\phi_k(\bm{x}) = \Lam{(k)}(\W{(k)}\phi_{k-1}(\bm{x})+\bias{(k)})$, where $\sigma(\cdot)$ is replaced by the diagonal activation pattern matrix $\Lam{(k)}$ that encodes the status of neurons $r$ in $k$-th layer:
\begin{equation}
\Lam{(k)}_{r,r} =
\begin{cases}
1 \text{ or } 0 & \text{if $r \in \setIuns{k}$} \\
1 & \text{if $r \in \setIpos{k}$} \\
0 & \text{if $r \in \setIneg{k}$}
\end{cases}
\end{equation}
and $\Lam{(m)} = \bm{I}_{n_m}$. With a slight abuse of notation, let us define $\Lam{(k)}_a$ as a diagonal activation matrix for neurons in the $k$-th layer who are always \textbf{a}ctivated, i.e. the $r$-th diagonal is $1$ if $r \in \setIpos{k}$ and $0$ otherwise, and $\Lam{(k)}_u$ as the diagonal activation matrix for $k$-th layer neurons whose status are \textbf{u}ncertain, i.e. the $r$-th diagonal is $1$ or $0$ (to be determined) if $r \in \setIuns{k}$, and $0$ otherwise. Therefore, we have $\Lam{(k)} = \Lam{(k)}_a + \Lam{(k)}_u$.
We can obtain $\Lam{(k)}$ for $\bm{x} \in B_p(\bm{x_0},\epsilon)$ by applying Algorithm~\ref{alg:fast-lin} and check the lower and upper bounds for each neuron $r$ in layer $k$.
\subsubsection{A general upper bound of Lipschitz constant in $\ell_q$ norm}
The central idea is to compute upper bounds of $L_{q,x_0}^j$ by exploiting the three categories of activation patterns in ReLU networks when the allowable inputs are in $B_p(\bm{x_0},\epsilon)$. $L_{q, \bm{x}_0}^j$ can be defined as the maximum norm of directional derivative as shown in \cite{weng2017evaluating}.
For the ReLU network, the maximum directional derivative norm can be found by examining all the possible activation patterns and take the one (the worst-case) that results in the largest gradient norm. However, as all possible activation patterns grow exponentially with the number of the neurons, it is impossible to examine all of them in brute-force. Fortunately, computing the worst-case pattern on each element of $\nabla g(\bm{x})$ (i.e. $[\nabla g(x)]_k, \, k \in [n_0]$) is much easier and more efficient. In addition, we apply a simple fact that the maximum norm of a vector (which is $\nabla g(\bm{x}), \bm{x} \in B_p(\bm{x_0},\epsilon)$ in our case) is upper bounded by the norm of the maximum value for each components. By computing the worst-case pattern on $[\nabla g(\bm{x})]_k$ and its norm, we can obtain an upper bound of the local Lipschitz constant, which results in a certified lower bound of minimum distortion.
Below, we first show how to derive an upper bound of the Lipschitz constant by computing the worst-case activation pattern on $[\nabla g(\bm{x})]_k$ for $2$ layers. Next, we will show how to apply it repeatedly for a general $m$-layer network, and the algorithm is named \textbf{Fast-Lip}. Note that for simplicity, we will use $[\nabla f_j(\bm{x})]_k$ to illustrate our derivation; however, it is easy to extend to $[\nabla g(\bm{x})]_k$ as $g(\bm{x}) = f_c(\bm{x}) - f_j(\bm{x})$ by simply replacing last layer weight vector by $\W{(m)}_{c,:}-\W{(m)}_{j,:}$.
\ifdef{}{
\input{alg_fastlip}
}{}
\paragraph{Bounds for a 2-layer ReLU Network.} The gradient is:
$$[\nabla f_j(\bm{x})]_k = \W{(2)}_{j,:} \Lam{(1)}_a \W{(1)}_{:,k} + \W{(2)}_{j,:} \Lam{(1)}_u \W{(1)}_{:,k}.$$ The first term $\W{(2)}_{j,:} \Lam{(1)}_a \W{(1)}_{:,k}$ is a constant and all we need to bound is the second term $\W{(2)}_{j,:} \Lam{(1)}_u \W{(1)}_{:,k}$. Let $\gradC{(1)}_{j,k} = \W{(2)}_{j,:} \Lam{(1)}_a \W{(1)}_{:,k},\,$ $\gradL{(1)}_{j,k}$ and $\gradU{(1)}_{j,k}$ be the lower and upper bounds of the second term, we have
\begin{equation*}
\gradL{(1)}_{j,k} = \hspace{-4mm} \sum_{i \in \setIuns{1},\W{(2)}_{j,i}\W{(2)}_{i,k} < 0} \hspace{-8mm} \W{(2)}_{j,i}\W{(2)}_{i,k}, \; \gradU{(1)}_{j,k} = \hspace{-4mm} \sum_{i \in \setIuns{1},\W{(2)}_{j,i}\W{(2)}_{i,k} > 0} \hspace{-8mm} \W{(2)}_{j,i}\W{(2)}_{i,k}
\end{equation*}
\[\max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k| \leq \max (|\gradC{(1)}_{j,k} + \gradL{(1)}_{j,k}|, |\gradC{(1)}_{j,k} + \gradU{(1)}_{j,k}|).
\]
\paragraph{Bounds for 3 layers or more.} For 3 or more layers, we can apply the above 2-layer results recursively, layer-by-layer. For example, for a 3-layer ReLU network, $$[\nabla f_j(\bm{x})]_k = \W{(3)}_{j,:} \Lam{(2)} \W{(2)} \Lam{(1)} \W{(1)}_{:,k}, $$
if we let $\Y{(1)}_{:,k} = \W{(2)} \Lam{(1)} \W{(1)}_{:,k}$, then $[\nabla f_j(\bm{x})]_k$ is reduced to the following form that is similar to 2 layers:
\begin{align}
\label{eq:grad_3layer1}
[\nabla f_j(\bm{x})]_k & = \W{(3)}_{j,:} \Lam{(2)} \Y{(1)}_{:,k} \\
\label{eq:grad_3layer2}
& = \W{(3)}_{j,:} \Lam{(2)}_a \Y{(1)}_{:,k} + \W{(3)}_{j,:} \Lam{(2)}_u \Y{(1)}_{:,k}
\end{align}
To obtain the bound in \eqref{eq:grad_3layer1}, we first need to obtain a lower bound and upper bound of $\Y{(1)}_{:,k}$, where we can directly apply the 2-layer results to get an upper and an lower bound for each component $i$ as $\gradC{(1)}_{i,k} + \gradL{(1)}_{i,k} \leq \Y{(1)}_{i,k} \leq \gradC{(1)}_{i,k} + \gradU{(1)}_{i,k}$. Next, the first term $\W{(3)}_{j,:} \Lam{(2)}_a \Y{(1)}_{:,k}$ in \eqref{eq:grad_3layer2} can be lower bounded and upper bounded respectively by
\begin{align}
\label{eq:3-layer-LB1}
& \sum_{i \in \setIpos{2}} \W{(3)}_{j,i} \gradC{(1)}_{i,k} + \hspace{-3mm} \sum_{i \in \setIpos{2}, \W{(3)}_{j,i} > 0} \hspace{-4mm} \W{(3)}_{j,i} \gradL{(1)}_{i,k} + \hspace{-4mm} \sum_{i \in \setIpos{2}, \W{(3)}_{j,i} < 0} \hspace{-3.5mm} \W{(3)}_{j,i} \gradU{(1)}_{i,k} \\
\label{eq:3-layer-UB1}
& \sum_{i \in \setIpos{2}} \W{(3)}_{j,i} \gradC{(1)}_{i,k} + \hspace{-3mm} \sum_{i \in \setIpos{2}, \W{(3)}_{j,i} > 0} \hspace{-4mm} \W{(3)}_{j,i} \gradU{(1)}_{i,k} + \hspace{-4mm} \sum_{i \in \setIpos{2}, \W{(3)}_{j,i} < 0} \hspace{-3.5mm} \W{(3)}_{j,i} \gradL{(1)}_{i,k}
\end{align}
whereas the second term $\W{(3)}_{j,:} \Lam{(2)}_u \Y{(1)}_{:,k}$ in \eqref{eq:grad_3layer2} is bounded by $\sum_{i \in \mathcal{P}} \W{(3)}_{j,i} (\gradC{(1)}_{i,k} + \gradL{(1)}_{i,k}) + \sum_{i \in \mathcal{Q}} \W{(3)}_{j,i} (\gradC{(1)}_{i,k} + \gradU{(1)}_{i,k})$ with lower/upper bound index sets $\mathcal{P}_L,\mathcal{Q}_L$ and $\mathcal{P}_U,\mathcal{Q}_U$:
\begin{align}
\label{eq:3-layer-LB2}
& \mathcal{P}_L = \{i \mid i \in \setIuns{2}, \W{(3)}_{j,i} > 0, \gradC{(1)}_{i,k} + \gradL{(1)}_{i,k} < 0 \}, \nonumber \\
& \mathcal{Q}_L = \{i \mid i \in \setIuns{2}, \W{(3)}_{j,i} < 0, \gradC{(1)}_{i,k} + \gradU{(1)}_{i,k} > 0 \};\\
\label{eq:3-layer-UB2}
& \mathcal{P}_U = \{i \mid i \in \setIuns{2}, \W{(3)}_{j,i} < 0, \gradC{(1)}_{i,k} + \gradL{(1)}_{i,k} < 0 \}, \nonumber \\
& \mathcal{Q}_U = \{i \mid i \in \setIuns{2}, \W{(3)}_{j,i} > 0, \gradC{(1)}_{i,k} + \gradU{(1)}_{i,k} > 0 \}.
\end{align}
Let $\gradC{(2)}_{j,k} = \sum_{i \in \setIpos{2}} \W{(3)}_{j,i} \gradC{(1)}_{i,k}$, $\gradU{(2)}_{j,k}+\gradC{(2)}_{j,k}$ and $\gradL{(2)}_{j,k}+\gradC{(2)}_{j,k}$ be the upper and lower bound of $[\nabla f_j(\bm{x})]_k$, we have
\begin{equation*}
\gradU{(2)}_{j,k} + \gradC{(2)}_{j,k} = \eqref{eq:3-layer-UB1} + \eqref{eq:3-layer-UB2} \; \; \text{and} \; \; \gradL{(2)}_{j,k} + \gradC{(2)}_{j,k} = \eqref{eq:3-layer-LB1} + \eqref{eq:3-layer-LB2},
\end{equation*}
\[
\max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k| \hspace{-1mm} \leq \hspace{-1mm} \max (|\gradL{(2)}_{j,k}+\gradC{(2)}_{j,k}|, |\gradU{(2)}_{j,k}+\gradC{(2)}_{j,k}|).
\]
Thus, this technique can be used iteratively to solve $\max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k|$ for a general $m$-layer network, and we can easily bound any $q$ norm of $\nabla f_j(\bm{x})$ by the $q$ norm of the vector of maximum values. For example,
\ifdef{}{
\begin{align*}
& \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \| \nabla f_j(\bm{x}) \|_1 \leq \sum_j \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k| , \\
& \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \| \nabla f_j(\bm{x}) \|_2 \leq \sqrt{\sum_j ( \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k| )^2 }, \\
& \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \| \nabla f_j(\bm{x}) \|_\infty \leq \max_j \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k|.
\end{align*}
}{
\begin{equation*}
\max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} \| \nabla f_j(\bm{x}) \|_q \leq \left( {\sum_k ( \max_{\bm{x} \in B_p(\bm{x_0},\epsilon)} |[\nabla f_j(\bm{x})]_k| )^q } \right )^{\frac{1}{q}}
\end{equation*}
}
\ifdef{}{
The full procedure of \textbf{Fast-Lip}\xspace is in Algorithm~\ref{alg:fast-lip}.
}{We list our full procedure, \textbf{Fast-Lip}\xspace, in Appendix~\ref{sec:app_algs}.
}
\paragraph{Further speed-up.} Note that in the 3-layer example, we compute the bounds from right to left, i.e. we first get the bound of $\W{(2)} \Lam{(1)} \W{(1)}_{:,k}$, and then bound $\W{(3)}_{j,:} \Lam{(2)} \Y{(1)}_{:,k}$. Similarly, we can compute the bounds from left to right -- get the bound of $\W{(3)}_{j,:} \Lam{(2)} \W{(2)}$ first, and then bound $\Y{(2)}_{j,:} \Lam{(1)} \W{(1)}_{:,k}$, where $\Y{(2)}_{j,:} = \W{(3)}_{j,:} \Lam{(2)} \W{(2)}$. Since the dimension of the output layer ($n_m$) is typically far less than the dimension of the input vector ($n_0$), computing the bounds from left to right is more efficient as the matrix $\Y{}$ has a smaller dimension of $n_m \times n_k$ rather than $n_k \times n_0$.
\section*{Writing plans and Ideas}
\subsection*{Theory}
\begin{itemize}
\item Derived gradient norm bound for n-layer network [Lily]
\item Derived required activation change bound for n-layer network [Lily]
\item Try to deal with CNN (simple 2-layer is enough)
\end{itemize}
\subsection*{Experiments}
\begin{itemize}
\item Compare with our theoretical bound with
\begin{itemize}
\item adversarial distortion found by CW
\item lower bound by CLEVER
\item lower bound by Szegedy Lipschitz operator norm
\item Bastani's LP form [Hongge]
\item Reluplex [Hongge]
\end{itemize}
\item Target network: 2-4 layer ReLU MLP, possibly 2-3 layer CNN, for MNIST/Cifar
\end{itemize}
\fi
|
1,314,259,995,852 | arxiv | \section{Introduction}
Machine learning often requires an object to be represented as a point $x$ in a vector space $V$.
In many settings, there is a linear group $G$ of ambiguities for which the same object can be represented as $gx$ for any $g\in G$.
For example, when the object is a point cloud or a graph, the vector representation depends on how the points or vertices are labeled.
One is inclined to apply a $G$-invariant feature map $\Phi\colon V\to F$ to factor out these ambiguities before training a machine learning model.
Recently, \cite{CahillIMP:22}~introduced such a feature map that is well suited for groups of linear isometries:
\begin{definition}
Fix a real inner product space $V$ and a group $G\leq\operatorname{O}(V)$.
\begin{itemize}
\item[(a)]
The \textbf{max filtering map} $\langle\hspace{-2.5pt}\langle\cdot,\cdot\rangle\hspace{-2.5pt}\rangle\colon V/G\times V/G\to\mathbb{R}$ is defined by
\[
\langle\hspace{-2.5pt}\langle G\cdot x,G\cdot y\rangle\hspace{-2.5pt}\rangle
:=\sup_{\substack{p\in G\cdot x\\q\in G\cdot y}}\langle p,q\rangle.
\]
\item[(b)]
Given $\{t_i\}_{i=1}^n$ in $V$, the corresponding \textbf{max filter bank} $\Phi\colon V/G\to\mathbb{R}^n$ is defined by
\[
\Phi(G\cdot x):=\{\langle\hspace{-2.5pt}\langle G\cdot t_i,G\cdot x\rangle\hspace{-2.5pt}\rangle\}_{i=1}^n.
\]
\end{itemize}
\end{definition}
The vectors $\{t_i\}_{i=1}^n$ that determine a max filter bank are known as \textbf{templates}.
Max filter banks offer a noteworthy contrast to the polynomial invariants that are typically studied in the literature~\cite{BandeiraBKPWW:17,PerryWBRS:19,CahillCC:20,BendoryELS:22,BalanHS:22}.
In particular, every $G$-invariant polynomial map $V\to\mathbb{R}^n$ is either affine linear or not Lipschitz.
As a result, polynomial invariants are doomed to either fail to separate orbits or fail to exhibit numerical stability.
Meanwhile, \cite{CahillIMP:22}~establishes that for every finite $G\leq\operatorname{O}(d)$, there exists a \textit{bilipschitz} max filter bank $\mathbb{R}^d/G\to\mathbb{R}^n$.
We enunciate the quantitative details of this result in terms of the following definition:
\begin{definition}
Given a group $G\leq\operatorname{O}(d)$ whose action on $\mathbb{R}^d$ has closed orbits, the \textbf{optimal max filtering condition number} $\kappa(G)$ is the infimum of quotients $C/c$ for which there exist $n\in\mathbb{N}$ and templates $\{t_i\}_{i=1}^n$ in $\mathbb{R}^d$ such that the corresponding max filter bank (as a map from $\mathbb{R}^d/G$ with the quotient Euclidean metric to $\mathbb{R}^n$ with the Euclidean metric) has upper and lower Lipschitz bounds $C$ and $c$, respectively.
\end{definition}
\begin{proposition}[see Theorem~18 in~\cite{CahillIMP:22}]
\label{prop.general bound on kappa}
There exists a universal constant $C>0$ such that for every $d,m\in\mathbb{N}$ and every finite group $G\leq\operatorname{O}(d)$ of order $m$, it holds that
\[
\kappa(G)
\leq Cm^3d^{1/2}(d\log d+d\log m+\log^2m)^{1/2}.
\]
\end{proposition}
Proposition~\ref{prop.general bound on kappa} is the first of its kind, but it is not sharp and thus warrants further investigation.
In this paper, we derive the exact value of $\kappa(G)$ for every finite reflection group $G$.
To compare with Proposition~\ref{prop.general bound on kappa}, an easy-to-state consequence of our main results is that
\[
\kappa(G)\leq Cm
\]
whenever $G$ is a finite reflection group.
While this upper bound is saturated by the dihedral groups, $\kappa(G)$ is typically much smaller, frequently being on the order of $d$.
In addition to determining $\kappa(G)$ for each finite reflection group $G$, we also identify $n=d$ max filtering templates that attain this optimal condition number.
In the next section, we review some preliminaries on reflection groups.
In Section~\ref{sec.main results}, we formulate our main results and describe how our problem reduces to the study of Weyl chambers of essential and irreducible finite reflection groups.
Sections~\ref{sec.proof of thm.main} and~\ref{sec.proof of thm.asymptotic} contain the proofs of our main results.
\section{Background on reflection groups}
This section reviews basic information about reflection groups.
We encourage the interested reader to consult~\cite{Kane:01} for more information.
A \textbf{reflection} is an orthogonal transformation with exactly one negative eigenvalue.
Every nonzero vector $u\in\mathbb{R}^d$ determines a reflection with matrix representation
\[
R(u):=I-\frac{2}{\|u\|^2}uu^\top,
\]
where $I$ denotes the identity matrix.
A \textbf{reflection group} is any group that is generated by a set of reflections, implying that it is a subgroup of the orthogonal group $\operatorname{O}(d)$.
For example, the group of permutation matrices is generated by the transposition matrices $R(e_i-e_j)$, where $e_i$ and $e_j$ denote standard basis elements of $\mathbb{R}^d$.
Given nonzero vectors $\{u_i\}_{i=1}^k$, the fixed points of the reflection group generated by $\{R(u_i)\}_{i=1}^k$ form the orthogonal complement of $\operatorname{span}\{u_i\}_{i=1}^k$.
We say a reflection group is \textbf{essential} if the origin is its only fixed point.
The group of permutation matrices fixes the all-ones vector, and so this reflection group is not essential in $\mathbb{R}^d$ (however, it is essential when interpreted as a reflection group on the orthogonal complement of the all-ones vector).
Given a finite reflection group $G\leq\operatorname{O}(d)$, let $U$ denote the union of the hyperplanes fixed by each individual reflection in $G$.
The connected components of $\mathbb{R}^d\setminus U$ are known as the \textbf{Weyl chambers} of $G$.
Fix a Weyl chamber $C\subseteq\mathbb{R}^d$.
It turns out that $C$ is an open set whose boundary is contained in a union of $\ell\in\mathbb{N}$ hyperplanes corresponding to reflections in $G$ (and no fewer).
We denote these reflections by $\{R(\alpha_i)\}_{i=1}^\ell$ for unit vectors $\{\alpha_i\}_{i=1}^\ell$, which are signed so that they reside in the dual cone of $C$.
Then $\{\alpha_i\}_{i=1}^\ell$ is known as the \textbf{fundamental system} of $G$ corresponding to $C$.
Interestingly, the reflections $\{R(\alpha_i)\}_{i=1}^\ell$ generate the original reflection group $G$.
In the case where $G$ is essential, we have $\ell=d$, and furthermore, the dual basis $\{\beta_i\}_{i=1}^d$ of the fundamental system $\{\alpha_i\}_{i=1}^d$ generates the closed convex cone $\overline{C}$, i.e., $\overline{C}$ is an example of a simplicial cone.
As an example, consider the symmetries of the square, i.e., the dihedral group of order~$8$.
Figure~\ref{fig.weyl_chamber} illustrates the $1$-eigenspaces (in this case, lines) of the reflections in this group.
These lines carve the plane into eight Weyl chambers, one of which we call $C$.
The corresponding fundamental system consists of the unit vectors $\alpha_1:=(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$ and $\alpha_2:=(0,1)$, and the resulting dual basis, given by $\beta_1:=(\sqrt{2},0)$ and $\beta_2:=(1,1)$, generates $\overline{C}$.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=3]
\draw[green!30,fill=green!30] (0,0) -- (45:1) arc(45:0:1) -- cycle;
\draw[thick] (-1,0) -- (1,0);
\draw[thick] (-1/1.414,-1/1.414) -- (1/1.414,1/1.414);
\draw[thick] (0,-1) -- (0,1);
\draw[thick] (1/1.414,-1/1.414) -- (-1/1.414,1/1.414);
\draw (0.65,0.65*0.414) node {$C$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=2]
\draw[draw=none] (-1.5,0) -- (1.5,0);
\draw[draw=none] (0,-1.5) -- (0,1.5);
\draw[green!30,fill=green!30] (0,0) -- (45:1.35) arc(45:0:1.35) -- cycle;
\draw (0.65*1.414,0.65*0.414*1.414) node {$C$};
\draw [thick, -stealth](0,0) -- (0,1);
\draw (0,1.15) node {$\alpha_2$};
\draw [thick, -stealth](0,0) -- (1/1.414,-1/1.414);
\draw (1.15/1.414,-1.15/1.414) node {$\alpha_1$};
\draw [thick, -stealth](0,0) -- (1,1);
\draw (1.15,1.15) node {$\beta_2$};
\draw [thick, -stealth](0,0) -- (1.414,0);
\draw (1.414+0.15,0) node {$\beta_1$};
\end{tikzpicture}
\end{center}
\caption{\label{fig.weyl_chamber}
\textbf{(left)}
Weyl chamber arising from dihedral group.
\textbf{(right)}
Fundamental system and its dual basis.
}
\end{figure}
A reflection group in $\operatorname{O}(d)$ is said to be \textbf{reducible} if it can be decomposed as a direct product of nontrivial reflection groups in $\operatorname{O}(d)$, in which case $\mathbb{R}^d$ can be correspondingly decomposed into orthogonal invariant subspaces.
For example, the group of orthogonal diagonal matrices is reducible and can be decomposed along the lines spanned by the standard basis.
We note that this use of ``reducible'' differs slightly from its usage in representation theory.
While a reflection group may be viewed as the image of a representation, the representation is irreducible precisely when its image is both irreducible \textit{and} essential as a reflection group.
For example, the group of permutation matrices is reducible as a representation but irreducible as a reflection group (since it is not essential).
For each $d\in\mathbb{N}$, there are finitely many finite essential irreducible reflection groups in $\operatorname{O}(d)$ up to conjugation by an orthogonal matrix.
These were famously classified by Coxeter~\cite{Coxeter:35} in terms of abstract group presentations.
To motivate these presentations, note that for linearly independent unit vectors $u,v\in\mathbb{R}^d$, the composition $R(u)R(v)$ has a $1$-eigenspace of dimension $d-2$, and it acts as a rotation by $2\arccos\langle u,v\rangle$ radians in the $2$-dimensional orthogonal complement.
If $R(u)$ and $R(v)$ belong to a common finite reflection group, it must hold that $R(u)R(v)$ has finite order, and so $\arccos\langle u,v\rangle$ is a rational multiple of $\pi$.
If $u$ and $v$ belong to a fundamental system of the reflection group, it turns out that $\langle u,v\rangle=-\cos\frac{\pi}{m}$, where $m$ is the order of $R(u)R(v)$.
As an abstraction of this, a \textbf{Coxeter system} consists of generators $\{r_i\}_{i=1}^d$ and orders $\{m_{ij}\}_{i,j=1}^d$ such that $m_{ii}=1$ and $m_{ij}\geq 2$ for $i\neq j$.
The resulting \textbf{Coxeter group} is then given by
\[
\langle r_1,\ldots,r_d:(r_ir_j)^{m_{ij}}=1\rangle.
\]
Every Coxeter system has a \textbf{Coxeter diagram}, namely, a graph with vertex set $\{1,\ldots,d\}$ and adjacency $i\leftrightarrow j$ precisely when $m_{ij}\geq3$, and this edge is labeled by $m_{ij}$ whenever $m_{ij}\geq4$.
The Coxeter system is \textbf{irreducible} when the Coxeter diagram is connected.
For example, the connected Coxeter diagram on two vertices whose edge is labeled by $4$ corresponds to the Coxeter group
\[
\langle r_1,r_2:r_1^2=r_2^2=(r_1r_2)^{4}=1\rangle,
\]
which is isomorphic to the dihedral group of order $8$; see Figure~\ref{fig.weyl_chamber} for a relevant illustration.
Given a finite irreducible Coxeter system, one may use the geometry of fundamental systems to reconstruct the corresponding finite essential irreducible reflection group (which again is unique up to conjugation).
Indeed, the Gram matrix of the fundamental system $\{\alpha_i\}_{i=1}^d$ is given by
\[
\langle \alpha_i,\alpha_j\rangle
=\left\{\begin{array}{cl}1&\text{if }i=j\\
-\cos\frac{\pi}{m_{ij}}&\text{if }i\neq j,
\end{array}\right.
\]
which can then be Cholesky factored to determine $\{\alpha_i\}_{i=1}^d$ up to orthogonal transformation.
Then $\{R(\alpha_i)\}_{i=1}^d$ generates the desired reflection group.
In the above case where $d=2$ and $m_{12}=4$, we obtain the Gram matrix
\[
\left[\begin{array}{cc}
\phantom{-}1&-\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\phantom{-}1
\end{array}\right],
\]
which can be factored to recover the fundamental system illustrated in Figure~\ref{fig.weyl_chamber}.
In the case where $G$ is essential and irreducible, the closed fundamental chamber is an \textbf{acute cone}, i.e., a convex cone $K$ such that every nonzero $x,y\in K$ satisfies $\langle x,y\rangle>0$.
This geometric fact will prove useful in understanding the spectral properties associated with fundamental systems of $G$.
\begin{lemma}
\label{lem.acute cone}
For every essential and irreducible finite reflection group $G$, each closed chamber of $G$ is an acute cone.
\end{lemma}
\begin{proof}
Since the rows of $A$ form a fundamental system, the off-diagonal entries of the Gram matrix $AA^\top$ are nonpositive; see Lemma~B on page~39 in~\cite{Kane:01}.
Furthermore, since the reflection group is irreducible, the Gram matrix $AA^\top$ is irreducible as a matrix (i.e., it is not permutation equivalent to a block-diagonal matrix).
It follows from Corollary~3.24 in~\cite{Varga:62} that $B^\top B=(AA^\top)^{-1}$ is entrywise positive, which implies the result.
\end{proof}
All of our main results rely on the following spectral phenomena:
\begin{lemma}
\label{lem.spectral phenomenon}
Suppose $G\leq\operatorname{O}(d)$ is an essential and irreducible finite reflection group, and select any matrix $A\in\mathbb{R}^{d\times d}$ with unit-norm rows that form a fundamental system for $G$.
\begin{itemize}
\item[(a)]
The top and bottom eigenvectors of $AA^\top$ satisfy $(v_1)_i^2=(v_d)_i^2$ for all $i\in[d]$.
\end{itemize}
Next, consider the matrix $M:=AA^\top-I_d$.
For each $j\in[d]$, let $M^{(j)}$ denote the principal submatrix of $M$ obtained by discarding the $j$th row and column, and let $\lambda_k(\cdot)$ denote the $k$th largest eigenvalue of the argument, counted with multiplicity.
\begin{itemize}
\item[(b)]
For each $k\in[d]$, it holds that $\lambda_{d-k+1}(M)=-\lambda_k(M)$.
\item[(c)]
For each $j\in[d]$ and $k\in[d-1]$, it holds that $\lambda_{d-k}(M^{(j)})=-\lambda_k(M^{(j)})$.
\end{itemize}
\end{lemma}
We could not locate a proof of the above facts, and so we provide our own in Appendix~\ref{appendix.spectral phenomenon}.
Our proof argues by cases over Coxeter's classification.
\section{Main results and key reductions}
\label{sec.main results}
Our first main result describes minimal optimal max filter banks for every reflection group in terms of its decomposition into irreducible constituent representations:
\begin{theorem}
\label{thm.main}
Suppose $G\leq\operatorname{O}(d)$ is a finite reflection group.
\begin{itemize}
\item[(a)]
If $G$ is the trivial group, then $\kappa(G)=1$, and minimal optimal templates are given by any orthonormal basis.
\item[(b)]
If $G$ is essential and irreducible, then for any matrix $A\in\mathbb{R}^{d\times d}$ with unit-norm rows that form a fundamental system for $G$, it holds that $\kappa(G)=\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)}$, and minimal optimal templates are given by the columns of $A^{-1}$.
\item[(c)]
Otherwise, $G$ can be decomposed into irreducible constituents $G_i\leq\operatorname{O}(d_i)$, one of which is trivial precisely when $G$ is not essential, and the others are essential and irreducible as reflection groups.
Then $\kappa(G)=\max_i\kappa(G_i)$, and minimal optimal templates are given by an orthogonal direct sum of the minimal optimal templates for each irreducible constituent described in (a) and (b).
\end{itemize}
\end{theorem}
Theorem~\ref{thm.main}(a) is immediate.
The proof of part~(c) is straightforward after some preliminary analysis of Weyl chambers, and we provide it later in this section.
The proof of part~(b) occupies Section~\ref{sec.proof of thm.main}.
Our second main result (Theorem~\ref{thm.asymptotic}) leverages Theorem~\ref{thm.main}(b) to compute $\kappa(G)$ for the standard representation $G$ of any irreducible Coxeter group, which in turn determines $\kappa(G)$ for any finite reflection group $G$ by Theorem~\ref{thm.main}(c).
The proof of Theorem~\ref{thm.asymptotic} appears in Section~\ref{sec.proof of thm.asymptotic}.
\begin{theorem}
\label{thm.asymptotic}
The optimal max filtering condition numbers of the exceptional Coxeter groups are reported in Table~\ref{table.condition numbers}.
For the remaining Coxeter groups, we have
\begin{itemize}
\item[(a)]
$\kappa(A_\ell)=(\frac{2}{\pi}+o(1))\ell$,
\item[(b)]
$\kappa(B_\ell)=(\frac{4}{\pi}+o(1))\ell$,
\item[(c)]
$\kappa(D_\ell)=(\frac{4}{\pi}+o(1))\ell$, and
\item[(d)]
$\kappa(I_m)=\sqrt{\frac{1+\cos(\frac{\pi}{m})}{1-\cos(\frac{\pi}{m})}}=(\frac{2}{\pi}+o(1))m$.
\end{itemize}
\end{theorem}
\begin{table}
\caption{\label{table.condition numbers}
Optimal max filtering condition numbers of exceptional Coxeter groups
}
\begin{center}
\begingroup
\setlength{\tabcolsep}{12pt}
\begin{tabular}{|crl|}\hline
$G$ & $\kappa(G)$ & minimal polynomial \\ \hline\hline
$E_6$ & $7.5957$ & $x^4 - 8 x^3 + 2 x^2 + 8 x + 1$\\ \hline
$E_7$ & $11.4301$ & $x^6 - 12x^5 + 3x^4+40x^3+3x^2-12x+1$\\ \hline
$E_8$ & $19.0811$ & $x^8-16x^7-60x^6+16x^5+134x^4+16x^3-60x^2-16x+1$ \\ \hline
$F_4$ & $7.5957$ & $x^4 - 8 x^3 + 2 x^2 + 8 x + 1$\\ \hline
$H_3$ & $6.3137$ & $x^4-4x^3-14x^2-4x+1$\\ \hline
$H_4$ & $19.0811$ & $x^8-16x^7-60x^6+16x^5+134x^4+16x^3-60x^2-16x+1$ \\ \hline
\end{tabular}
\endgroup
\end{center}
\end{table}
Before proving Theorem~\ref{thm.main}(c), we first motivate our approach with an example.
When $G$ is the group of $d\times d$ permutation matrices, it is straightforward to verify that
\begin{equation}
\label{eq.sort}
\langle\hspace{-2.5pt}\langle G\cdot x,G\cdot y\rangle\hspace{-2.5pt}\rangle
=\langle \operatorname{sort}(x),\operatorname{sort}(y)\rangle.
\end{equation}
It turns out that every reflection group has a map analogous to sorting.
As motivation, we observe that the image of the sorting map is the polyhedral cone
\[
\{(x_1,\ldots,x_d)\in\mathbb{R}^d:x_1\geq\cdots\geq x_d\}.
\]
Every orbit $G\cdot x$ intersects this cone at a single point, namely, at $\operatorname{sort}(x)$.
One may show that the above cone is the closure of a Weyl chamber of $G$, which offers a hint for how to generalize the sorting map.
Given a finite reflection group $G\leq\operatorname{O}(d)$ and a corresponding Weyl chamber $C\subseteq\mathbb{R}^d$, we define $\operatorname{rep}\colon\mathbb{R}^d/G\to\overline{C}$ so that $\operatorname{rep}(G\cdot x)$ is the unique member of $(G\cdot x)\cap\overline{C}$; indeed, uniqueness is established on page~62 in~\cite{Kane:01}, which also demonstrates that $\operatorname{rep}$ is bijective.
In addition, $\operatorname{rep}$ preserves distances:
\begin{lemma}
\label{lem.isometry}
The bijection $\operatorname{rep}\colon \mathbb{R}^d/G\to\overline{C}$ defined above is an isometry.
\end{lemma}
This implies the following generalization of \eqref{eq.sort}:
\begin{align}
\langle\hspace{-2.5pt}\langle G\cdot x,G\cdot y\rangle\hspace{-2.5pt}\rangle
\nonumber
&=\frac{1}{2}\Big(\|x\|^2+\|y\|^2-d(G\cdot x,G\cdot y)^2\Big)\\
\nonumber
&=\frac{1}{2}\Big(\|\operatorname{rep}(G\cdot x)\|^2+\|\operatorname{rep}(G\cdot y)\|^2-\|\operatorname{rep}(G\cdot x)-\operatorname{rep}(G\cdot y)\|^2\Big)\\
\label{eq.representative representation}
&=\langle \operatorname{rep}(G\cdot x),\operatorname{rep}(G\cdot y)\rangle.
\end{align}
\begin{proof}[Proof of Lemma~\ref{lem.isometry}]
A change of variables gives
\[
d(G\cdot x,G\cdot y)
=\min_{g,h\in G}\|gx-hy\|
=\min_{y'\in G\cdot y}\|\operatorname{rep}(G\cdot x)-y'\|.
\]
We note for future use that $(x,y)\mapsto d(G\cdot x,G\cdot y)$ is continuous since the first equality above expresses it as a minimum of finitely many continuous functions.
Suppose $x,y\in C$.
For each $y''\in G\cdot y$ with $y''\neq y$, there is a hyperplane orthogonal to some $\alpha$ in the fundamental system of $C$ that strictly separates $y''$ from $y\in C$, and so
\[
\|x-y''\|
>\|x-R(\alpha)y''\|
\geq \min_{y'\in G\cdot y}\|x-y'\|.
\]
It follows that $\|x-y'\|$ is minimized over $y'\in G\cdot y$ at $y'=y$, i.e.,
\[
d(G\cdot x,G\cdot y)
=\|x-y\|
=\|\operatorname{rep}(G\cdot x)-\operatorname{rep}(G\cdot y)\|.
\]
Now consider any $x,y\in\mathbb{R}^d$, and take sequences $\{x_n\}_{n=1}^\infty$ and $\{y_n\}_{n=1}^\infty$ in $C$ such that $x_n\to \operatorname{rep}(G\cdot x)$ and $y_n\to \operatorname{rep}(G\cdot y)$.
Then continuity implies
\[
d(G\cdot x_n,G\cdot y_n)
\to d\Big(G\cdot \operatorname{rep}(G\cdot x),G\cdot \operatorname{rep}(G\cdot y)\Big)
= d(G\cdot x,G\cdot y).
\]
Meanwhile, the previous argument gives
\[
d(G\cdot x_n,G\cdot y_n)
=\|x_n-y_n\|
\to\|\operatorname{rep}(G\cdot x)-\operatorname{rep}(G\cdot y)\|.
\]
It follows that $d(G\cdot x,G\cdot y)=\|\operatorname{rep}(G\cdot x)-\operatorname{rep}(G\cdot y)\|$, as desired.
\end{proof}
By virtue of \eqref{eq.representative representation}, we may represent max filtering as a linear map precomposed with the $\operatorname{rep}$ map.
While the image of $\operatorname{rep}$ is $C$, the linear map is defined on the full space.
As the following lemma shows, extending the linear map does not alter its condition number.
\begin{lemma}
\label{lem.linearize}
Let $G\leq\operatorname{O}(d)$ be a finite reflection group.
Given $T:=\{t_i\}_{i=1}^n\in(\mathbb{R}^d)^n$, define $\Phi\colon\mathbb{R}^d/G\to\mathbb{R}^n$ and $L\colon\mathbb{R}^d\to\mathbb{R}^n$ by
\[
\Phi(G\cdot x):=\{\langle\hspace{-2.5pt}\langle G\cdot t_i,G\cdot x\rangle\hspace{-2.5pt}\rangle\}_{i=1}^n,
\qquad
L(x):=\{\langle \operatorname{rep}(G\cdot t_i),x\rangle\}_{i=1}^n.
\]
Then $\Phi$ and $L$ have identical upper and lower Lipschitz bounds.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem.isometry}, $\operatorname{rep}\colon \mathbb{R}^d/G\to\overline{C}$ is a bijective isometry.
Letting $L_{\overline{C}}\colon \overline{C}\to\mathbb{R}^n$ denote the restriction of $L$ to $\overline{C}\subseteq\mathbb{R}^d$, we have $\Phi=L_{\overline{C}}\circ \operatorname{rep}$, and so it suffices to show that $L_{\overline{C}}$ and $L$ have identical upper and lower Lipschitz bounds.
Since $L$ is linear, its Lipschitz bounds are precisely its top and bottom singular values, and by restriction, it holds that
\[
\sigma_{\min}(L)
\leq \frac{\|L_{\overline{C}}(x)-L_{\overline{C}}(y)\|}{\|x-y\|}
\leq \sigma_{\max}(L)
\]
for every $x,y\in\overline{C}$ with $x\neq y$.
To show that these bounds are optimal for $L_{\overline{C}}$, select any $y\in C$, and let $z$ be a top (respectively, bottom) right-singular unit vector of $L$.
Since $C$ is open, it holds that $x:=y+\epsilon z\in C$ for an appropriately small $\epsilon>0$, in which case
\[
\frac{\|L_{\overline{C}}(x)-L_{\overline{C}}(y)\|}{\|x-y\|}
=\frac{\|L(x)-L(y)\|}{\|x-y\|}
=\frac{\|L(x-y)\|}{\|x-y\|}
=\frac{\|L(\epsilon z)\|}{\|\epsilon z\|}
=\|L z\|,
\]
which achieves equality in the upper (respectively, lower) Lipschitz bound, as desired.
\end{proof}
To find the best conditioned max filtering map, Lemma~\ref{lem.linearize} suggests finding templates $\{t_i\}_{i=1}^{n}$ for which the vectors $\{\operatorname{rep}(G\cdot t_i)\}_{i=1}^n$ form the rows of an optimally conditioned matrix.
Since these vectors are constrained to reside in the closed Weyl chamber $\overline{C}$, this optimization problem motivates the following definition.
\begin{definition}
The \textbf{optimal condition number} $\kappa(A)$ of a set $A\subseteq\mathbb{R}^d$ is the infimum of quotients $C/c$ for which there exists $n\in\mathbb{N}$ and vectors $\{t_i\}_{i=1}^n$ in $A$ such that
\[
c\|x\|\leq\bigg(\sum_{i=1}^n|\langle t_i,x\rangle|^2\bigg)^{1/2}\leq C\|x\|
\qquad
\forall x\in\mathbb{R}^d.
\]
Notably, $C/c$ is the condition number of the $d\times n$ matrix with columns $\{t_i\}_{i=1}^n$.
\end{definition}
The optimal condition number of a (possibly nonconvex) cone interacts nicely with direct sums in the sense that the worst distortion is the distortion of the worst behaved summand.
This is made precise in the following lemma.
\begin{lemma}
\label{lem.direct sum condition number}
Given spanning sets $X\subseteq\mathbb{R}^{d_X}$ and $Y\subseteq\mathbb{R}^{d_Y}$ that are closed under positive scalar multiplication, the direct sum $X\oplus Y\subseteq\mathbb{R}^{d_X+d_Y}$ has optimal condition number
\[
\kappa(X\oplus Y)=\max\{\kappa(X),\kappa(Y)\}.
\]
\end{lemma}
\begin{proof}
To further overload notation, given $A\in\mathbb{R}^{d\times n}$, we let $\kappa(A)$ denote the condition number of $A$.
It suffices to show two things:
\begin{itemize}
\item[(a)]
For all matrices $U$ and $V$ with columns from $X$ and $Y$, respectively, there exists a matrix $W$ with columns from $X\oplus Y$ such that $\kappa(W)\leq\max\{\kappa(U),\kappa(V)\}$.
\item[(b)]
For every matrix $W$ with columns from $X\oplus Y$, there exist matrices $U$ and $V$ with columns from $X$ and $Y$, respectively, such that $\max\{\kappa(U),\kappa(V)\}\leq\kappa(W)$.
\end{itemize}
For (a), we may assume $\kappa(U),\kappa(V)<\infty$.
Scale $U$ and $V$ so that $\sigma_{\min}(U)=\sigma_{\min}(V)=1$.
Taking $W$ to be block diagonal with blocks $U$ and $V$, then $\sigma_{\min}(W)=1$ and so
\[
\kappa(W)
=\max\{\sigma_{\max}(U),\sigma_{\max}(V)\}
=\max\{\kappa(U),\kappa(V)\}.
\]
For (b), we may assume $\kappa(W)<\infty$.
We may decompose $W$ as
\[
W=\left[\begin{array}{c}
U\\
V
\end{array}\right],
\]
where $U$ has columns from $X$ and $V$ has columns from $Y$.
We claim that
\[
0
<\sigma_{\min}(W)
\leq\sigma_{\min}(U)
\leq\sigma_{\max}(U)
\leq\sigma_{\max}(W)
<\infty,
\]
and similarly for $V$; this in turn implies the desired inequality.
Indeed,
\[
\sigma_{\max}(W)
=\max_{\|x\|=1}\|W^\top x\|
\geq\max_{\|y\|=1} \left\|\left[\begin{array}{cc}
U^\top&V^\top
\end{array}\right]
\left[\begin{array}{c}y\\0\end{array}\right]\right\|
=\max_{\|y\|=1}\|U^\top y\|
=\sigma_{\max}(U),
\]
and a similar argument gives the desired bound for $\sigma_{\min}(U)$.
\end{proof}
We are now ready to prove Theorem~\ref{thm.main}(c).
\begin{proof}[Proof of Theorem~\ref{thm.main}(c)]
Combining Lemmas~\ref{lem.linearize} and~\ref{lem.direct sum condition number} gives $\kappa(G)=\max_i\kappa(G_i)$.
Next, we view $\mathbb{R}^d$ as a direct sum of $\mathbb{R}^{d_i}$ according to the factorization of $G$, and let $B_i$ denote the matrix whose columns are the minimal optimal templates for $\mathbb{R}^{d_i}$.
Then by Lemma~\ref{lem.spectral phenomenon}(b) and Theorem~\ref{thm.main}(a) and~(b),
\[
\frac{1}{\sigma_{\max}(B_i)^2}+\frac{1}{\sigma_{\min}(B_i)^2}=2,
\]
from which is follows that $\sigma_{\min}(B_i)$ and $\sigma_{\max}(B_i)$ are inversely related.
Take $B\in\mathbb{R}^{d\times d}$ to be block diagonal with blocks $B_i$.
Then for all $i$, it holds that
\[
\sigma_{\min}(B)
\leq\sigma_{\min}(B_i)
\leq\sigma_{\max}(B_i)
\leq\sigma_{\max}(B).
\]
Furthermore, since $\sigma_{\min}(B_i)$ and $\sigma_{\max}(B_i)$ are inversely related, there exists $j$ such that both $\sigma_{\min}(B_j)=\sigma_{\min}(B)$ and $\sigma_{\max}(B_j)=\sigma_{\max}(B)$.
It follows that
\[
\frac{\sigma_{\max}(B)}{\sigma_{\min}(B)}
=\frac{\sigma_{\max}(B_j)}{\sigma_{\min}(B_j)}
=\max_i\frac{\sigma_{\max}(B_i)}{\sigma_{\min}(B_i)}
=\max_i\kappa(G_i)
=\kappa(G),
\]
i.e., the columns of $B$ are optimal templates.
Finally, since a finite condition number requires $B$ to have at least $d$ columns, $B$ is minimal.
\end{proof}
\section{Proof of Theorem~\ref{thm.main}(b)}
\label{sec.proof of thm.main}
In this section, we leverage convex duality to establish Theorem~\ref{thm.main}(b).
Our reduction to convex programming is summarized by the following:
\begin{lemma}
\label{lem.convex programming reduction}
Let $G\leq\operatorname{O}(d)$ be a finite reflection group with Weyl chamber $C\subseteq\mathbb{R}^d$.
Denote
\[
\alpha:=\inf\Bigg\{\bigg\|\sum_{i=1}^n t_it_i^\top-I_d\bigg\|_{2\to2}:n\in\mathbb{N},t_1,\ldots,t_n\in\overline{C}\Bigg\}.
\]
Then $\kappa(G)=\sqrt{\frac{1+\alpha}{1-\alpha}}$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem.linearize}, every max filter bank shares upper and lower Lipschitz bounds with a linear map $L\colon\mathbb{R}^d\to\mathbb{R}^n$ of the form $L(x)=\{\langle t_i,x\rangle\}_{i=1}^n$ for some $t_1,\ldots,t_n\in\overline{C}$.
These bounds are the square roots of the top and bottom eigenvalues of $LL^*$, whose matrix representation is $\sum_{i=1}^nt_it_i^\top$.
The scaling of $L$ that makes the eigenvalues of $LL^*$ uniformly closest to $1$ makes the top and bottom eigenvalues exhibit a common distance $r$ from $1$.
Meanwhile, this scaling does not affect the quotient $\sqrt{\frac{1+r}{1-r}}$ of Lipschitz bounds.
The result follows.
\end{proof}
The definition of $\alpha$ in Lemma~\ref{lem.convex programming reduction} was directly inspired by Cahill and Chen's treatment of scalable frames in~\cite{CahillC:13}.
Scalable frames are concerned with best choices of vectors belonging to a particular union of $1$-dimensional cones (i.e., rays).
Our result replaces this union with a particular closed convex cone, namely, a Weyl chamber of a finite reflection group.
\textbf{For the remainder of this section, we assume that $G$ is an essential and irreducible finite reflection group, and we use the following notation:}
Take $A\in\mathbb{R}^{d\times d}$ to be the matrix whose unit-norm rows form the fundamental system corresponding to the Weyl chamber $C$, and put $B:=A^{-1}$.
Then
\[
\overline{C}
=\{x\in\mathbb{R}^d:Ax\geq0\}
=\{By:y\geq0\}.
\]
Observe that for each $t\in\overline{C}$, there exists $y\geq0$ such that $tt^\top = Byy^\top B^\top$.
By factoring out $B$ and $B^\top$, arbitrary sums of such $tt^\top$ correspond to arbitrary sums of $yy^\top$, which in turn make up the convex cone $\operatorname{CP}(d)$ of $d\times d$ \textbf{completely positive matrices}.
Thus,
\[
\bigg\{\sum_{i=1}^n t_it_i^\top:n\in\mathbb{N},t_1,\ldots,t_n\in\overline{C}\bigg\}
=\Big\{BMB^\top:M\in\operatorname{CP}(d)\Big\}.
\]
In particular, the program in Lemma~\ref{lem.convex programming reduction} can be reformulated as
\begin{equation}
\label{eq.main primal}
\text{minimize}
\quad
\|BMB^\top-I_d\|_{2\to2}
\quad
\text{subject to}
\quad
M\in\operatorname{CP}(d).
\end{equation}
Our approach is to prove a sequence of technical conditions on the optimizers of \eqref{eq.main primal}.
\begin{lemma}
\label{lem.opt identity}
There exists an optimizer of \eqref{eq.main primal} that is a positive multiple of $I_d$.
\end{lemma}
Indeed, Lemma~\ref{lem.opt identity} allows us to prove our result:
\begin{proof}[Proof of Theorem~\ref{thm.main}(b)]
By Lemmas~\ref{lem.opt identity}, \ref{lem.convex programming reduction}, and~\ref{lem.spectral phenomenon}(c), we have
\[
\kappa(G)=\sqrt{\frac{1+\|cBB^\top-I_d\|_{2\to2}}{1-\|cBB^\top-I_d\|_{2\to2}}}
=\sqrt{\frac{\lambda_{\max}(BB^\top)}{\lambda_{\min}(BB^\top)}}
=\frac{\sigma_{\max}(B)}{\sigma_{\min}(B)}
=\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)},
\]
with optimal templates given by the columns of $B$.
\end{proof}
To prove Lemma~\ref{lem.opt identity}, we first prove a weaker result:
\begin{lemma}
\label{lem.opt diag}
There exists an optimizer of \eqref{eq.main primal} that is diagonal.
\end{lemma}
We will prove Lemma~\ref{lem.opt diag} by observing an interplay between dual and restricted programs.
A restriction of the primal program (for instance to a diagonal primal variable) corresponds to a relaxation of the dual program.
To show the restriction does not change the value of the primal program, we can show that an optimizer of the relaxed dual program is still feasible in the original dual program.
The dual program of \eqref{eq.main primal} is naturally expressed in terms of the convex cone $\operatorname{COP}(d)$ of \textbf{copositive matrices}, that is, symmetric matrices $A\in\mathbb{R}^{d\times d}$ such that $x^\top Ax\geq0$ for all $x\geq0$:
\begin{equation}
\label{eq.main dual}
\text{maximize}
\quad
\operatorname{tr}W
\quad
\text{subject to}
\quad
\|W\|_*\leq 1,
\quad
-B^\top WB\in\operatorname{COP}(d).
\end{equation}
Consider the restriction of \eqref{eq.main primal} to diagonal $M\in\operatorname{CP}(d)$, i.e., $M=\operatorname{diag}(x)$ with $x\geq0$:
\begin{equation}
\label{eq.diagonal primal}
\text{minimize}
\quad
\|B\operatorname{diag}(x)B^\top-I_d\|_{2\to2}
\quad
\text{subject to}
\quad
x\geq0.
\end{equation}
The dual of this restriction is a relaxation of \eqref{eq.main dual}:
\begin{equation}
\label{eq.diagonal dual}
\text{maximize}
\quad
\operatorname{tr}Y
\quad
\text{subject to}
\quad
\|Y\|_*\leq 1,
\quad
Y^\top=Y,
\quad
\operatorname{diag}(B^\top YB)\leq0.
\end{equation}
Since the feasibility regions of \eqref{eq.main dual} and \eqref{eq.diagonal primal} each have nonempty interior, they satisfy strong duality by Slater's condition.
We will show that \eqref{eq.diagonal dual} has an optimizer that is feasible in \eqref{eq.main dual}.
By strong duality, this in turn implies Lemma~\ref{lem.opt diag}.
First, we use complementary slackness to derive useful statements about optimizers, for example:
\begin{lemma}\
\label{lem.opt in diag programs}
\begin{itemize}
\item[(a)]
Any optimal $x$ in \eqref{eq.diagonal primal} necessarily satisfies $\|B\operatorname{diag}(x)B^\top-I_d\|_{2\to2}<1$ and $x>0$.
\item[(b)]
Any optimal $Y$ in \eqref{eq.diagonal dual} necessarily satisfies $\operatorname{diag}(B^\top YB)=0$.
\end{itemize}
\end{lemma}
\begin{proof}
If $x$ has a zero entry, then $B\operatorname{diag}(x)B^\top$ is rank deficient, and so the objective in \eqref{eq.diagonal primal} at $x$ is at least $1$.
Meanwhile, taking $x:=\sigma_{\max}(B)^{-2}\cdot\mathbf{1}$ gives $0\prec B\operatorname{diag}(x)B^\top\preceq I_d$, and so the objective in \eqref{eq.diagonal primal} at $x$ is strictly less than $1$.
This implies (a), and then (b) follows from complementary slackness.
\end{proof}
We can say even more about the optimal $x$ and $Y$ by leveraging the following version of the von Neumann trace inequality:
\begin{proposition}[symmetric von Neumann trace inequality]
\label{prop.von neumann}
Suppose $A,B\in\mathbb{R}^{n\times n}$ are symmetric with eigenvalues $\lambda_1\leq\cdots\leq\lambda_n$ and $\mu_1\leq\cdots\leq\mu_n$, respectively.
Then
\[
\operatorname{tr}(AB)
\leq\sum_{i=1}^n\lambda_i\mu_i,
\]
with equality precisely when there exist orthonormal $\{u_i\}_{i=1}^n$ in $\mathbb{R}^n$ such that
\[
A=\sum_{i=1}^n\lambda_iu_iu_i^\top
\qquad
\text{and}
\qquad
B=\sum_{i=1}^n\mu_iu_iu_i^\top.
\]
\end{proposition}
\begin{proof}
First, $A$ and $B$ of the prescribed form achieve equality in the given bound.
It remains to prove the bound and then show that equality requires $A$ and $B$ to have this form.
Consider the case where $A$ and $B$ are both positive definite.
Then the eigenvalues coincide with singular values, and so Theorem~3.1 in~\cite{Carlsson:21} gives the bound, with equality precisely when there exist $V,W\in\operatorname{O}(n)$ such that $A=V\operatorname{diag}(\lambda)W^\top$ and $B=V\operatorname{diag}(\mu)W^\top$.
Since $A$ is positive definite, it has polar decomposition $A=(VW^\top)(W\operatorname{diag}(\lambda)W^\top)$, whose uniqueness implies $VW^\top=I_n$, i.e., $V=W$.
This resolves the positive definite case.
Now suppose $A$ and $B$ are symmetric, but not necessarily positive definite.
Then there exist $\alpha,\beta\in\mathbb{R}$ such that $A+\alpha I_n,B+\beta I_n\succ0$.
Our analysis above then gives
\begin{align*}
\operatorname{tr}(AB)
&=\operatorname{tr}((A+\alpha I_n)(B+\beta I_n))-\beta\operatorname{tr}A-\alpha\operatorname{tr}(B)-\alpha\beta n\\
&\leq\sum_{i=1}^n(\lambda_i+\alpha)(\mu_i+\beta)-\beta\sum_{i=1}^n\lambda_i-\alpha\sum_{i=1}^n\mu_i-\alpha\beta n
=\sum_{i=1}^n\lambda_i\mu_i,
\end{align*}
with equality precisely when there exist orthonormal $\{u_i\}_{i=1}^n$ in $\mathbb{R}^n$ such that
\[
A+\alpha I_n=\sum_{i=1}^n(\lambda_i+\alpha)u_iu_i^\top
\qquad
\text{and}
\qquad
B+\beta I_n=\sum_{i=1}^n(\mu_i+\beta)u_iu_i^\top.
\]
The result then follows since $\sum_iu_iu_i^\top=I_n$.
\end{proof}
Next, we show that for every optimal $x$ in \eqref{eq.diagonal primal} and $Y$ in \eqref{eq.diagonal dual}, the top eigenvector of $B\operatorname{diag}(x)B^\top$ is the bottom eigenvector of $Y$, and it resides in $C$.
\begin{lemma}\
\label{eq.eigenstructure of opt x and Y}
\begin{itemize}
\item[(a)]
For every optimal $x$ in \eqref{eq.diagonal primal}, the top eigenvalue of $B\operatorname{diag}(x)B^\top$ is simple, and the corresponding eigenspace is spanned by a positive combination of the columns of $B$.
\item[(b)]
Every optimal $Y$ in \eqref{eq.diagonal dual} has a unique negative eigenvalue, this eigenvalue is simple, and the corresponding eigenspace is identical to the top eigenspace in (a).
\end{itemize}
\end{lemma}
\begin{proof}
For (a), let $D$ denote the square root of $\operatorname{diag}(x)$, and put $S:=BD$.
Since $S$ is square, we have that $B\operatorname{diag}(x)B^\top=SS^\top$ has the same spectrum as $S^\top S=DB^\top BD$ with the same multiplicities.
By Lemma~\ref{lem.acute cone}, $B^\top B$ is entrywise positive.
Since $x>0$ by Lemma~\ref{lem.opt in diag programs}(a), it holds that $S^\top S$ is also entrywise positive, and so Perron--Frobenius gives that the top eigenvalue is simple with an entrywise positive eigenvector $v$.
Then the top eigenvector of $B\operatorname{diag}(x)B^\top=SS^\top$ is $Sv=BDv$, i.e., a positive combination of the columns of $B$.
For (b), we first give a proof of weak duality between \eqref{eq.diagonal primal} and \eqref{eq.diagonal dual}:
For every feasible $x$ in \eqref{eq.diagonal primal} and $Y$ in \eqref{eq.diagonal dual}, if we let $\lambda,\mu\in\mathbb{R}^d$ denote the vectors of sorted eigenvalues of $Y$ and $I_d-B\operatorname{diag}(x)B^\top$, respectively, then symmetric von Neumann and H\"{o}lder together give
\begin{align}
\operatorname{tr}Y
\nonumber
&\leq\operatorname{tr}Y - \langle B^\top YB,\operatorname{diag}(x)\rangle\\
\nonumber
&=\langle Y,I_d-B\operatorname{diag}(x)B^\top\rangle\\
\label{eq.weak duality between diag programs}
&\leq\langle \lambda,\mu\rangle
\leq\|\lambda\|_1\|\mu\|_\infty
=\|Y\|_*\|B\operatorname{diag}(x)B^\top-I_d\|_{2\to2}
\leq\|B\operatorname{diag}(x)B^\top-I_d\|_{2\to2}.
\end{align}
If $x$ is optimal, then we may combine a portion of \eqref{eq.weak duality between diag programs} with Lemma~\ref{lem.opt in diag programs}(a) to get
\[
\operatorname{tr}Y
\leq\|Y\|_*\|B\operatorname{diag}(x)B^\top-I_d\|_{2\to2}
<\|Y\|_*.
\]
Thus, every feasible $Y$ has a strictly negative eigenvalue, i.e., $\lambda_1<0$.
Next, any optimal $x$ and $Y$ together achieve equality in the following portion of \eqref{eq.weak duality between diag programs}:
\[
\langle \lambda,\mu\rangle
\leq\|\lambda\|_1\|\mu\|_\infty.
\]
Equality requires $\operatorname{supp}(\lambda)\subseteq\arg\max_i|\mu_i|$ and $\lambda_i\mu_i\geq0$ for all $i$.
Since $\lambda_1<0$, we have $1\in\arg\max_i|\mu_i|$.
As a consequence of the simplicity conclusion of part~(a), $\mu$ is nonzero, meaning $|\mu_1|=\max_i|\mu_i|>0$, which, combined with $\lambda_1<0$ and $\lambda_1\mu_1\geq0$, implies $\mu_1<0$.
Next, part~(a) gives $\mu_1<\mu_i$ for all $i>1$.
It follows that any $j\in\arg\max_i|\mu_i|$ with $j\neq 1$ must have $\mu_j>0$, and so $\lambda_j\geq0$.
As such, $\lambda_1$ is the only negative eigenvalue of $Y$.
Finally, any optimal $x$ and $Y$ achieve equality in the following portion of \eqref{eq.weak duality between diag programs}:
\[
\langle Y,I_d-B\operatorname{diag}(x)B^\top\rangle
\leq\langle \lambda,\mu\rangle.
\]
By part~(a), we have $\mu_1<\mu_2$, while the above gives $\lambda_1<\lambda_2$.
As such, Proposition~\ref{prop.von neumann} gives that the bottom eigenspace of $Y$ is identical to the top eigenspace of $B\operatorname{diag}(x)B^\top$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem.opt diag}]
Given an optimizer $x$ of \eqref{eq.diagonal primal}, we will show that $M:=\operatorname{diag}(x)$ is an optimizer of \eqref{eq.main primal}.
Such a choice of $M$ is feasible in \eqref{eq.main primal}.
By strong duality, it suffices to show that any optimal $Y$ in \eqref{eq.diagonal dual} is feasible in \eqref{eq.main dual}.
To this end, by Lemma~\ref{eq.eigenstructure of opt x and Y}, we may diagonalize $Y=UDU^\top$ so that the first column of $U$ is a positive combination of the columns of $B$ and $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_d)$ with $\lambda_1<0\leq\lambda_2\leq\cdots\leq\lambda_d$.
Consider the ellipsoid
\[
E
:=\{z\in\mathbb{R}^d:z^\top Dz\leq0,z_1=1\}
=\{z\in\mathbb{R}^d:\lambda_2z_2^2+\cdots+\lambda_dz_d^2\leq|\lambda_1|,z_1=1\}.
\]
Notably, $E$ is convex.
Fix $i\in[d]$ and let $b_i$ denote the $i$th column of $B$.
We claim that $v:=U^\top b_i\in\operatorname{cone}(E)$.
By Lemma~\ref{lem.opt in diag programs}(b), we have
\[
-|\lambda_1|v_1^2+\lambda_2v_2^2+\cdots+\lambda_dv_d^2
=v^\top Dv
=b_i^\top Yb_i
=(B^\top YB)_{ii}
=0.
\]
Rearranging then gives
\[
\lambda_2v_2^2+\cdots+\lambda_dv_d^2
=|\lambda_1|v_1^2.
\]
Lemmas~\ref{eq.eigenstructure of opt x and Y} and~\ref{lem.acute cone} together give $v_1>0$, and so $v\in\operatorname{cone}(E)$, as claimed.
Since our choice for $i\in[d]$ was arbitrary, it follows that the closed chamber $\overline{C}$ (which is generated by the columns of $B$) satisfies $U^\top\overline{C}\subseteq\operatorname{cone}(E)$.
To show that $Y$ is feasible in \eqref{eq.main dual}, i.e., $-B^\top YB\in\operatorname{COP}(d)$, take any vector $x\geq0$.
Then $Bx\in\overline{C}$, and so $v:=U^\top Bx\in\operatorname{cone}(E)$, meaning
\[
x^\top B^\top YBx
= x^\top B^\top UDU^\top Bx
= v^\top Dv
\leq 0,
\]
as desired.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem.opt identity}]
By Lemma~\ref{lem.opt diag}, there exists an optimizer of \eqref{eq.main primal} that is diagonal, i.e., any optimizer $x$ of the restriction \eqref{eq.diagonal primal} corresponds to an optimizer $\operatorname{diag}(x)$ of \eqref{eq.main primal}.
Consider a further restriction of \eqref{eq.diagonal primal}:
\begin{equation}
\label{eq.scalar primal}
\text{minimize}
\quad
\|cBB^\top-I_d\|_{2\to2}
\quad
\text{subject to}
\quad
c\geq0.
\end{equation}
In what follows, we show that any optimizer $c$ of \eqref{eq.scalar primal} is strictly positive and corresponds to an optimizer $c\mathbf{1}$ of \eqref{eq.diagonal primal}.
First, the optimal $c$ in \eqref{eq.scalar primal} is strictly positive, since $c=0$ has value $1$, while $c=\sigma_{\max}(B)^{-2}$ has value strictly less than $1$ since the columns of $B$ span.
Next, the dual program of \eqref{eq.scalar primal} is\begin{equation}
\label{eq.scalar dual}
\text{maximize}
\quad
\operatorname{tr}Z
\quad
\text{subject to}
\quad
\|Z\|_*\leq1,
\quad
Z^\top=Z,
\quad
\operatorname{tr}(B^\top ZB)\leq 0.
\end{equation}
Since \eqref{eq.scalar primal} satisfies Slater's condition, we have strong duality, and so it suffices to show that an optimizer of the dual program \eqref{eq.scalar dual} is feasible in \eqref{eq.diagonal dual}.
To this end, we first give a proof of weak duality between \eqref{eq.scalar primal} and \eqref{eq.scalar dual}:
For every feasible $c$ in \eqref{eq.scalar primal} and $Z$ in \eqref{eq.scalar dual}, if we let $\lambda,\mu\in\mathbb{R}^d$ denote the vectors of sorted eigenvalues of $Z$ and $I_d-cBB^\top$, respectively, then symmetric von Neumann and H\"{o}lder together give
\begin{align*}
\operatorname{tr}Z
&\leq\operatorname{tr}Z-c\operatorname{tr}(B^\top ZB)\\
&=\langle Z,I_d-cBB^\top\rangle\\
&\leq\langle\lambda,\mu\rangle
\leq\|\lambda\|_1\|\mu\|_\infty
=\|Z\|_*\|cBB^\top-I_d\|_{2\to2}
\leq\|cBB^\top-I_d\|_{2\to2}.
\end{align*}
Equality in $\operatorname{tr}Z\leq\operatorname{tr}Z-c\operatorname{tr}(B^\top ZB)$ requires $\operatorname{tr}(B^\top ZB)=0$.
Next, by Proposition~\ref{prop.von neumann}, equality in $\langle Z,I_d-cBB^\top\rangle\leq\langle\lambda,\mu\rangle$ implies that $Z$ and $I_d-cBB^\top$ are simultaneously diagonalizable.
Equality in $\langle\lambda,\mu\rangle\leq\|\lambda\|_1\|\mu\|_\infty$ then gives that $\lambda_i\neq0$ only if $i\in\{1,d\}$.
Select any optimal $c$ in \eqref{eq.scalar primal} and optimal $Z$ in \eqref{eq.scalar dual}.
Let $\{u_i\}_{i=1}^d$ denote the orthonormal basis of eigenvectors of $I_d-cBB^\top$ afforded by Proposition~\ref{prop.von neumann}.
These eigenvectors appear in the singular value decomposition
\[
B=\sum_{i=1}^d \sigma_iu_iv_i^\top
\]
as well as the eigenvalue decomposition
\[
Z=\lambda_1u_1u_1^\top+\lambda_du_du_d^\top.
\]
Then $B^\top ZB=\sigma_1^2\lambda_1 v_1v_1^\top+\sigma_d^2\lambda_d v_dv_d^\top$, the trace of which is $\sigma_1^2\lambda_1+\sigma_d^2\lambda_d=0$ by the above complementary slackness argument.
Thus, $\operatorname{diag}(B^\top ZB)=0$ precisely when $(v_1)_i^2=(v_d)_i^2$ for all $i\in[d]$, which in turn holds by Lemma~\ref{lem.spectral phenomenon}(a).
As such, an optimizer of the dual program~\eqref{eq.scalar dual} is feasible in \eqref{eq.diagonal dual}, as desired.
\end{proof}
\section{Proof of Theorem~\ref{thm.asymptotic}}
\label{sec.proof of thm.asymptotic}
Theorem~\ref{thm.main} reports the best choice of max filtering templates in terms of the matrix whose unit-norm rows form a fundamental system of $G$.
Furthermore, the singular values of this matrix determine the optimal condition number.
Recalling Coxeter's classification, we compute these optimal condition numbers in Mathematica\footnote{\url{https://github.com/Daniel-Packer/reflection-groups/blob/main/ExceptionalGroups.nb}} for finitely many reflection groups.
The optimal condition numbers for the dihedral groups are easy to compute by hand:
\[
AA^\top
=\left[\begin{array}{cc}1&-\cos(\frac{\pi}{m})\\-\cos(\frac{\pi}{m})&1\end{array}\right]
\]
has eigenvalues $1\pm\cos(\frac{\pi}{m})$, and so $\kappa(I_m)=\sqrt{\frac{1+\cos(\frac{\pi}{m})}{1-\cos(\frac{\pi}{m})}}$, which is $(\frac{2}{\pi}+o(1))m$ by the Taylor series expansion of cosine.
The remaining families $A_\ell$, $B_\ell$, and $D_\ell$ correspond to $\ell\times\ell$ matrices whose characteristic polynomials are cumbersome to interact with as $\ell\to\infty$.
Instead, we determine the asymptotic form of the optimal condition numbers in these cases.
To do so, we embed the relevant matrices as integral operators over a common Hilbert space, where we can characterize spectral convergence as the matrix size grows.
\begin{definition}
The \textbf{pixel embedding} of $A\in\mathbb{R}^{n\times n}$ is $A^\sharp\in L^2([0,1]^2)$ defined by
\[
A^\sharp(x,y)
=\sum_{i=1}^n\sum_{j=1}^n A_{ij} \cdot n\cdot 1_{(\frac{i-1}{n},\frac{i}{n})}(x)\cdot 1_{(\frac{j-1}{n},\frac{j}{n})}(y),
\]
i.e., we chop the unit square into squares of width $1/n$, and then we take $A^\sharp$ to be constant on each square according to appropriately scaled entries of $A$.
We interpret $A^\sharp$ as the kernel of an integral operator $L^2([0,1])\to L^2([0,1])$, which we also denote $A^\sharp$ by an abuse of notation:
\[
(A^\sharp f)(x):=\int_0^1 A^\sharp(x,y)f(y)dy.
\]
\end{definition}
\begin{lemma}
\label{lem.limit norm}
Given a sequence $\{A_n\}_{n=1}^\infty$ of matrices of possibly different sizes such that $A_n^\sharp\to K$ in $L^2([0,1]^2)$, it holds that $\|A_n\|_{2\to2}\to\|K\|_{L^2\to L^2}$.
\end{lemma}
\begin{proof}
First, we observe some relationships between $A\in\mathbb{R}^{n\times n}$ and its pixel embedding $A^\sharp$:
\begin{itemize}
\item
The operator $A^\sharp$ acts invariantly on the subspace $P_n\leq L^2([0,1])$ spanned by the characteristic functions of intervals $(\frac{i-1}{n},\frac{i}{n})$.
\item
$A$ is the matrix representation of the restriction $A^\sharp|_{P_n}$ with respect to this basis.
\item
The kernel of $A^\sharp$ contains the orthogonal complement of $P_n$ in $L^2([0,1])$.
\end{itemize}
Thus, $\|A\|_{2\to2}=\|A^\sharp\|_{L^2\to L^2}$.
The result then follows from Cauchy--Schwarz:
\[
\big|\|A_n^\sharp\|_{L^2\to L^2}-\|K\|_{L^2\to L^2}\big|
\leq\|A_n^\sharp-K\|_{L^2\to L^2}
=\sup_{\|f\|_{L^2}=1}\|(A_n^\sharp-K)f\|_{L^2}
\leq\|A_n^\sharp-K\|_{L^2([0,1]^2)}.
\qedhere
\]
\end{proof}
In the following subsections, we apply Lemma~\ref{lem.limit norm} to treat $A_\ell$, $B_\ell$, and $D_\ell$.
Unlike $B_\ell$ and $D_\ell$, the action of $A_\ell$ is more naturally represented in an $(\ell+1)$-dimensional space.
For this reason, our analysis of $A_\ell$ is more intricate, while our analysis of $B_\ell$ and $D_\ell$ is straightforward.
\subsection{$A_\ell$}
Fix $\ell\in\mathbb{N}$, put $d=\ell$, and select $A\in\mathbb{R}^{d\times d}$ with unit-norm rows that form a fundamental system for $A_\ell$.
In this subsection, we estimate
\[
\kappa(A_\ell)
=\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)}
=\sqrt{\frac{\lambda_{\max}(AA^\top)}{\lambda_{\min}(AA^\top)}}
=\sqrt{\frac{2-\lambda_{\min}(AA^\top)}{\lambda_{\min}(AA^\top)}},
\]
where the last equality applies Lemma~\ref{lem.spectral phenomenon}(b).
It will be convenient to analyze
\[
M:=\left[\begin{array}{cc}
AA^\top&0\\
0^\top &1
\end{array}\right],
\]
which satisfies $\lambda_{\min}(M)=\lambda_{\min}(AA^\top)$ by Lemma~\ref{lem.spectral phenomenon}(b).
From the Coxeter diagram of $A_\ell$, we have that $AA^\top$ is tridiagonal with $1$'s on the diagonal and $-\frac{1}{2}$'s above and below the diagonal.
Define $S\in\mathbb{R}^{(d+1)\times(d+1)}$ by
\[
S:=\left[\begin{array}{rrr|c}
\frac{1}{\sqrt{2}}&&&\frac{1}{\sqrt{d+1}}\\
-\frac{1}{\sqrt{2}}&\ddots&&\frac{1}{\sqrt{d+1}}\\
&\ddots&\frac{1}{\sqrt{2}}&\vdots\\
&&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{d+1}}\\
\end{array}\right],
\]
where the blank entries are zeros.
Then $S^\top S=M$, and furthermore, one may verify that
\[
(S^{-1})_{ij}
=\left\{\begin{array}{cl}
\sqrt{2}\cdot(1-\frac{i}{d+1})&\text{if }j\leq i\leq d\\
\frac{1}{\sqrt{d+1}}&\text{if }j\leq i=d+1\\
-\sqrt{2}\cdot\frac{i}{d+1}&\text{if }j>i.
\end{array}\right.
\]
We wish to estimate
\[
\sqrt{\lambda_{\min}(AA^\top)}
=\sqrt{\lambda_{\min}(M)}
=\sigma_{\min}(S)
=\sigma_{\max}(S^{-1})^{-1}
=\tfrac{1}{d+1}\cdot\|\tfrac{1}{d+1}S^{-1}\|_{2\to2}^{-1}.
\]
It is straightforward to verify that as $d\to\infty$, the pixel embedding $(\frac{1}{d+1}S^{-1})^\sharp$ converges in $L^2([0,1]^2)$ to the kernel $K$ defined by
\[
K(x,y):=\left\{\begin{array}{cl}
\sqrt{2}\cdot(1-x)&\text{if }y\leq x\\
-\sqrt{2}\cdot x&\text{if }y>x.
\end{array}\right.
\]
Lemma~\ref{lem.kernel norm} below gives that $\|K\|_{L^2\to L^2}=\sqrt{2}/\pi$, which combined with Lemma~\ref{lem.limit norm} implies
\[
(d+1)\cdot\sqrt{\lambda_{\min}(AA^\top)}
=\|\tfrac{1}{d+1}S^{-1}\|_{2\to2}^{-1}
\to\|K\|_{L^2\to L^2}^{-1}
=\tfrac{\pi}{\sqrt{2}},
\]
and so recalling $d=\ell$, we have
\[
\kappa(A_\ell)
=\sqrt{\frac{2-\lambda_{\min}(AA^\top)}{\lambda_{\min}(AA^\top)}}
=(\tfrac{2}{\pi}+o(1)) \ell,
\]
as desired.
\begin{lemma}
\label{lem.kernel norm}
For $K$ defined above, it holds that $\|K\|_{L^2\to L^2}=\sqrt{2}/\pi$.
\end{lemma}
\begin{proof}
We start by identifying all eigenvalues and eigenvectors of $KK^*$.
To this end, a straightforward calculation gives
\[
(KK^*f)(x)
=2\int_0^x\int_y^1 f(z)dzdy-2x\int_0^1\int_y^1 f(z)dzdy.
\]
Observe that $(KK^*f)(0)=0=(KK^*f)(1)$ for any $f\in L^2([0,1])$, and so any eigenvector $g$ of $KK^*$ with nonzero eigenvalue must satisfy the boundary conditions
\[
g(0)=0=g(1).
\]
Next, two applications of the fundamental theorem of calculus gives $(KK^*f)''=-2f$, meaning $g$ is an eigenvector with eigenvalue $\lambda$ only if $\lambda g''=-2g$.
Notably, this implies that $0$ is not an eigenvalue of $KK^*$.
Since $KK^*$ is positive definite, it follows that every eigenvalue is strictly positive.
As such, $g$ is an eigenvector with eigenvalue $\lambda>0$ only if $g''=-\omega^2 g$ with $\omega:=\sqrt{2/\lambda}$, i.e., there exist $a,b\in\mathbb{R}$ such that
\[
g(x)=a\cos(\omega x)+b\sin(\omega x).
\]
The boundary condition $g(0)=0$ then implies that $a=0$, and since $g$ is an eigenvector, it follows that $b\neq0$.
The boundary condition $g(1)=0$ then gives $b\sin(\omega)=0$, i.e., $\omega=k\pi$ for some $k\in\mathbb{N}$.
Thus, every eigenvalue of $KK^*$ takes the form $2/(k\pi)^2$ for some $k\in\mathbb{N}$.
Furthermore, an easy calculation confirms that $g(x):=\sin(\pi x)$ satisfies $KK^*g=(2/\pi^2)g$.
The $C^*$ identity and the spectral theorem for compact self-adjoint operators then gives $\|K\|_{L^2\to L^2}^2=\lambda_{\max}(KK^*)=2/\pi^2$.
\end{proof}
\subsection{$B_\ell$ and $D_\ell$}
First, we address $B_\ell$.
Fix $\ell\in\mathbb{N}$, put $d=\ell$, and consider $A,B\in\mathbb{R}^{d\times d}$ defined by
\[
A:=
\left[\begin{array}{cccc}
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&\phantom\ddots&\\
&\ddots&\ddots&\\
&\phantom\ddots&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\
&&\phantom\ddots&1
\end{array}\right],
\qquad
B:=
\left[\begin{array}{cccc}
\sqrt{2}&\cdots&\sqrt{2}&1\\
&\ddots&\vdots&\vdots\\
&&\sqrt{2}&1\\
&&&1
\end{array}\right].
\]
The rows of $A$ form a fundamental system for $B_\ell$, and $B=A^{-1}$.
As $d\to\infty$, the pixel embedding $(\frac{1}{d}B)^\sharp$ converges in $L^2([0,1]^2)$ to $K$ defined by
\[
K(x,y):=\left\{\begin{array}{cl}
\sqrt{2}&\text{if }y\geq x\\
0&\text{if }y<x.
\end{array}\right.
\]
The operator $\frac{1}{\sqrt{2}}K$ is the adjoint of the Volterra operator, whose operator norm is well known to be $2/\pi$ (by an argument similar to our analysis of $A_\ell$).
Then
\[
\lambda_{\min}(AA^\top)
=\lambda_{\max}(B^\top B)^{-1}
=\tfrac{1}{d^2}\cdot\|\tfrac{1}{d}B\|_{2\to2}^{-2}
=\tfrac{1}{d^2}\Big(\|K\|_{2\to2}^{-2}+o(1)\Big)
=\tfrac{1}{d^2}(\tfrac{\pi^2}{8}+o(1)),
\]
and so recalling $d=\ell$, we have
\[
\kappa(B_\ell)
=\sqrt{\frac{2-\lambda_{\min}(AA^\top)}{\lambda_{\min}(AA^\top)}}
=\sqrt{\frac{2-\tfrac{1}{d^2}(\tfrac{\pi^2}{8}+o(1))}{\tfrac{1}{d^2}(\tfrac{\pi^2}{8}+o(1))}}
=(\tfrac{4}{\pi}+o(1))\ell.
\]
To analyze $D_\ell$, we instead take
\[
A:=
\left[\begin{array}{cccr}
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&\phantom\ddots&\\
&\ddots&\ddots&\\
&\phantom\ddots&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\
&\phantom\ddots&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}
\end{array}\right],
\qquad
B:=
\left[\begin{array}{cccrc}
\sqrt{2}&\cdots&\sqrt{2}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
&\ddots&\vdots&\vdots\phantom{~}&\vdots\\
&&\sqrt{2}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
&&\phantom\vdots&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}
\end{array}\right].
\]
The rows of $A$ form a fundamental system for $D_\ell$, and $B=A^{-1}$.
As $d\to\infty$, the pixel embedding $(\frac{1}{d}B)^\sharp$ converges in $L^2([0,1]^2)$ to the same kernel $K$ defined above, and so the identical argument gives $\kappa(D_\ell)=(\tfrac{4}{\pi}+o(1))\ell$.
\section*{Acknowledgments}
The authors thank Jameson Cahill, Joey Iverson, John Jasper, and Yousef Qaddura for enlightening discussions.
|
1,314,259,995,853 | arxiv | \section{Introduction}
\ac{RA} \ac{MAC} protocols in the recent past have been intensively
investigated especially coupled with multi-packet per user sending
and \ac{SIC} procedure. It was first proposed in
\cite{Choudhury1983} to send the same packet twice in different
\ac{TDMA} slots. Benefits in terms of delay and throughput were
found under very moderate channel load conditions.
More recently, the idea to send multiple packets per user of
\cite{Choudhury1983} has been exploited for retrieving collided
packets with the more effective \ac{SIC} procedure in \cite{Casini2007},
\cite{Liva2011}, \cite{Kissling2011a} and \cite{Clazzer2012}. \ac{CRDSA}
presented in \cite{Casini2007} is the first which applies
\ac{SIC} for trying to resolve packet collisions within a frame in a
\ac{SA} like \ac{RA} scheme.
\ac{CRDSA} takes from \cite{Choudhury1983} the idea to send more
than one packet instance per user for each frame. But, on the other
hand \ac{CRDSA} is designed in a way to resolve most of packet
contentions by using \ac{SIC}. The key idea of \ac{CRDSA} is to provide
in each replica the signaling information of where the other replicas of
the corresponding user are placed in the frame. This signaling information
is stored in the so called \ac{RP} section of the packet header. Every time a
packet is recovered, this information can be exploited by the \ac{SIC}
procedure for removing the signal contribution of the other replicas
from the frame, thus possibly removing its interference contribution
to other packets. The main \ac{CRDSA} advantages lie in an improved
packet loss ratio and a much higher operational throughput.
The \ac{CRA} protocol \cite{Kissling2011a} exploits the same approach
of using \ac{SIC} as \ac{CRDSA}, but in an Aloha-like \ac{RA} scheme.
Here no slots are present in the frame and thus the replicas of the
users can be placed within the frame without constraints, except that
replicas of a user may not interfere each other. The avoidance of slots
results in significant advantages such
as relaxation in synchronization requirements among users and
possibility of varying packet length without padding overhead.
\ac{FEC} in \ac{CRA} is beneficial also when
no power unbalance among users is present, unlike in \ac{CRDSA}, because partial
interference is not only possible but also more probable than
complete interference.
\section{ECRA and the replicas' pointers}
In \cite{Clazzer2012} \ac{ECRA} was first presented. The protocol is an evolution of \ac{CRA}, where it is tempted to resolve \textit{loops} among packets. Both \ac{CRDSA} and \ac{CRA} are not able to resolve collisions where the packets involved belong to the same users and the inter-packet interference is too high. These situations are called \textit{loops}. Suppose we are sending two packets per user and we are using the \ac{CRDSA} protocol. Suppose further, a collision involving both the packets belonging to the two users happens. In this case there is no possibility to recover the two packets sent, and this results in a complete loss of the information stored in the users' packets. If the \ac{CRA} protocol is employed, the situation is more complex. In fact, the collision between the two users' packet can be partial, that is only a portion of the packets can be involved in the collision. For an example of loops see figure \ref{loops}. We can suppose in our case, although \ac{FEC} is used for counteracting the interference coming from the packets' collisions, the level of interference is too high for correct decoding of the packets. Also in this second case the loop cannot be resolved and the users' packets cannot be retrieved.
\begin{figure}
\centering
\includegraphics[width=8cm]{CRA-loop}
\caption{\ac{CRA} simplest loop, and \ac{ECRA} combined packet creation}
\label{loops}
\end{figure}
In \ac{ECRA} it can be observed that the partial collision of an Aloha like \ac{RA} scheme involves different parts of the packet, unlike \ac{CRDSA} where the collisions are always involving the packets entirely. Therefore, starting from the SIC algorithm of the \ac{CRA} protocol, a further step is introduced where a combined packet is created from the two (or more) user's replicas. The combined packet is composed in a way such that the resulting \ac{SNIR} is maximized, as presented in \cite{Clazzer2012}. In particular, for each symbol of one packet, the one belonging to the replica with less interference is selected to be used in the combined packet. In the case of figure \ref{loops} we can see how the combined packet of \ac{ECRA} will be created. Due to the higher \ac{SNIR} further packets can be retrived from the frame, increasing the throughput performance and decreasing the \ac{PER} w.r.t. \ac{CRA}. An increase up to 26\% in the peak throughput is achieved by \ac{ECRA} w.r.t. \ac{CRA} as shown in \cite{Clazzer2012}. The key assumption there is the perfect knowledge of all the packets positions within the frame after the first step of \ac{SIC}. Differently from \cite{ZigZag}, where an iterative \textit{chunk-by-chunk} decoding between the collided packets is employed, in \ac{ECRA} an entire combined packet is constructed and the decoding attempted on it in one step. Moreover, in \ac{ECRA} the replicas generation is made regardless of the decoding success, while in \cite{ZigZag} only the collided packets are replicated.
The aim of this work is two fold. First to investigate what is the best header and thus \ac{RP} positioning, and second to evaluate the behavior of \ac{ECRA} in terms of throughput when the aforementioned assumption is removed and the best \ac{RP} positioning is employed.
\section{Optimum replicas' pointers position}
In \ac{RA} protocols without slots and with \ac{SIC} such as \cite{Kissling2011a} and \cite{Clazzer2012}, the packets are sent randomly within a frame. Therefore, when a collision between two packets happens, the quantity of symbols involved in that collision is uniformly distributed between 1 and the length of the packet $P_{len}$. If we consider the whole set of collided packets, it can be shown that the probability of collision for every symbol within the generic packet, is equal for all the symbols, $p_i=p,$ for $i=1,...,P_{len}$. Since every symbol has the same probability to face interference, there is no way to optimize the header positioning in the packet in order to minimize the probability to have interference in these sensitive sections.
However an interesting question is, whether sections of the packets can be found which are less subject to interference, if interference cancellation is applied. To address this question let us first consider the simplest interference case of only two packets interfering each other.
\subsection{Case I: Two interfering packets}
First we assume that the colliding packets are symbol synchronized, hence, when a symbol is involved in a collision it is entirely collided. Second we assume perfect \ac{SIC}, therefore all the decodable packets are correctly received and ideally removed from the frame. Third, the received packet power is the same for all the packets. Fourth, the decoding threshold is assumed to be the Shannon capacity limit $SNIR_{SHA}$ and thus, we are considering Gaussian inputs. According to \cite{Kissling2011a} and \cite{Clazzer2012} we can write the \ac{SNIR} decoding threshold as $SNIR_{SHA} = 2^R-1$, given the rate $R$. We can also write the \ac{SNIR} of user $u$ and replica $r$ as $SNIR_{u,r}=\frac{P}{x_{u,r}\cdot P + N}$. The quantity $x_{u,r}$ is the interference ratio suffered by the packet of user $u$ and replica $r$, e.g. $x_{u,r}=1$ means that one packet is entirely colliding with the given packet. Every packet with $SNIR_{u,r}\geq SNIR_{SHA}$ is correctly received, thanks to the fourth hypothesis. We are interested in determining the quantity $x^*$ which is the maximum percentage of interference for which a packet is still decodable at the receiver. This value can be derived by the equation
\begin{equation}
\label{21}
SNIR_{SHA}=SNIR_{min} \Rightarrow 2^R-1=\frac{P/N}{x^*\cdot P/N + 1},
\end{equation}
where we can define \ac{SNR} as $SNR=P/N$. This leads to
\[
x^*=\frac{1}{2^R-1}-\frac{1}{SNR}.
\]
Thus, $x^*$ can be determined when the rate $R$ and the packet's $SNR$ are defined. For a better understanding of the scenario, let us divide the two interfering packets case in two subcases $x^*\geq0.5$ and $x^*<0.5$.
\subsubsection{\underline{$x^*\geq0.5$}}
We are interested in determining the number of possible cases $N_{int}$, where two packets are interfering. Since the packets are symbol synchronized, this number is finite and is $N_{int} = 2 \cdot P_{len} - 1$. This value can be derived observing that at least one symbol of the two packets needs to be interfered. In order to ensure this, we have $2 \cdot (P_{len} - 1) + 1$ possible interference cases among two packets, which lead to the value of $N_{int}$.
Let us now define $L=\lfloor P_{len}\cdot x^* \rfloor + 1$, where $L$ represents the minimum number of symbols that make a packet undecodable. Due to the second hypothesis, all the packets with $0,...,L-1$ symbols interfered were already correctly decoded. Therefore, after the \ac{SIC} process the number of possible cases $N_{int_{after-SIC}}$, where two packets are interfering reduces to
\[
N_{int_{after-SIC}}=2 \cdot (P_{len} - L) + 1.
\]
Intuitively, the formula derives directly from the previous reasoning, where instead of one, at least L symbols need to be interfered.
We now want to determine the probability of interference for each symbol in the packet $p_i,$ for $i=1,...,P_{len}$. Since $x^*\geq0.5$, it follows that $L>P_{len}/2$ and therefore some of the central symbols of the packet must face interference with probability equal to 1. The exact position of these symbols within the packet is related to $L$. We can observe that the least interference possible is when exactly $L$ symbols are involved in a collision. In this case the first $P_{len} - L$ symbols of the first packet collided and the last $P_{len} - L$ symbols of the second packet collided. Call $S_{c_1}$ the set of collided symbols of the first packet, and $S_{c_2}$ the set of collided symbols of the second packet. Explicitly
\[
S_{c_1} = \left\{ 1,...,L \right\} \ and \ S_{c_2} = \left\{ P_{len}-L+1,...,P_{len} \right\}.
\]
The intersection of symbols involved in the collision from both packets, are the ones with probability equal to 1 to be interfered. This can be derived by the observation that in all the other possible collisions, more than $L$ symbols are involved in the interference. Call $S_c$ the set given by $S_{c_1} \cap S_{c_2}$,
\[
S_c=S_{c_1} \cap S_{c_2}=\left\{ P_{len}-L+1,...,L \right\}.
\]
For every symbol outside $S_c$, the probability of interference is $<1$. The symbol immediately before symbol $P_{len}-L+1$, called $s_{P_{len}-L}$, is involved in collisions in all the cases except one. The case where this symbol is not involved in the collision is when exactly the last $L$ symbols of the packet are interfering. In a similar way, the symbol immediately after symbol $L$, called $s_{L+1}$, is also involved in all the cases except one. If we iterate this reasoning until symbol $s_1$ in the backward direction and symbol $s_{P_{len}}$ in the forward direction, we find $p_1 = p_{P_{len}} = \frac{P_{len}-L+1}{2 \cdot (P_{len} - L) + 1}$.
We are now able to formulate the interference probability $p_i$ for each symbol in the packet, $i=1,..,P_{len}$ as
\begin{equation}
\label{31}
p_i = \left\{
\begin{array}{rl}
\frac{P_{len}-L+i}{2 \cdot (P_{len} - L) + 1} & \mbox{for $i=1,...,P_{len}-L$}\\
1 & \mbox{for $i=P_{len}-L+1,...,L$}\\
\frac{2 \cdot (P_{len} - L) + 1 - (i-L)}{2 \cdot (P_{len} - L) + 1} & \mbox{for $i=L+1,...,P_{len}$}
\end{array}
\right .
\end{equation}
\subsubsection{\underline{$x^*<0.5$}}
If we now consider the case of $x^*<0.5$, the first observation is that the total number of cases where two packets are interfering $N_{int}$ is again, $N_{int_{after-SIC}}=2 \cdot (P_{len} - L) + 1$. It should be noted that the expression of $N_{int}$ is the same, but the actual value has changed because $L$ is different from the previous case. The second observation is that $S_c=\varnothing$, because
\begin{equation}
\label{32}
L<P_{len}-L+1, \ for \ each \ L \ with \ x^*<0.5
\end{equation}
The demonstration is the following, $L=\lfloor P_{len}\cdot x^* \rfloor + 1$, leads to $L\leq P_{len}/2$. Call the maximum of $L$, $L_{max}$
\begin{equation}
\label{33}
L_{max}=\frac{P_{len}}{2}.
\end{equation}
For $L=L_{max}$ we are maximizing the left hand and minimizing the right hand of inequality \eqref{32},
\[
L_{max}<P_{len}-L_{max}+1 \Rightarrow_{with \eqref{33}} \frac{P_{len}}{2}<\frac{P_{len}}{2}+1.
\]
Therefore, in this second case, no symbols have probability 1 to face interference. The maximum number of possible cases where a symbol is involved in collisions is $P_{len}$, and the symbols that are involved in all these cases are $s_i$ with $i=L,...,P_{len}-L+1$. This result can be found following the same reasoning presented in the previous paragraph with the appropriate differences. Moreover, the first and last symbol interference probability that can be computed applying the same procedure, is again $p_1 = p_{P_{len}} = \frac{P_{len}-L+1}{2 \cdot (P_{len} - L) + 1}$.
We are now able to formulate the interference probability $p_i$ for each symbol in the packet, $i=1,..,P_{len}$ as
\begin{equation}
\label{34}
p_i = \left\{
\begin{array}{rl}
\frac{P_{len}-L+i}{2 \cdot (P_{len} - L) + 1} & \mbox{for $i=1,...,L-1$}\\
\frac{P_{len}}{2 \cdot (P_{len} - L) + 1} & \mbox{for $i=L,...,P_{len}-L+1$}\\
\frac{2 \cdot (P_{len} - L) + 1 - (i-L)}{2 \cdot (P_{len} - L) + 1} & \mbox{for $i=P_{len}-L+2,...,P_{len}$}
\end{array}
\right .
\end{equation}
\begin{figure}
\centering
\includegraphics[width=8cm]{2Packets_0_9_0_3_5000_4times_1000symbols_2}
\caption{Collision of 2 packets after SIC, theoretical probability and empirical probability comparison}
\label{2Pack}
\end{figure}
In figure \ref{2Pack} the comparison between the average empirical probability of symbol interference among 5000 collisions and the theoretical probability is presented. A packet length $P_{len}=1000$ symbols and two representative thresholds, $x^*=0.9$ and $x^*=0.3$, have been selected. In order to have a quantitative evaluation of the difference between the empirical and the theoretical probabilities, the \ac{KL-d} has been evaluated. In particular
\[
KL_d = \sum_t V(t) \ln \frac{V(t)}{Q(t)},
\]
where $t=1,...,P_{len}$, $V$ is the theoretical probability and $Q$ is the empirical probability. The useful property of the \ac{KL-d} is that $KL_d = 0$ if and only if $V(t)=Q(t)$ for $\forall t$. Therefore, the closer $KL_d$ is to 0, the more the empirical and theoretical probabilities are matching. In our case the $KL_d$ distance gets $KL_d = -0.7019$ for the case of $x^*=0.3$ and $KL_d = -0.3053$ for $x^*=0.9$. In both cases \ac{KL-d} is very close to 0, which underlines the validity of the simulation results compared to the mathematical analysis.
In the case of $x^*=0.9$, the symbols at the very beginning and the symbols at the very end of the packet have quite half of the probability to face interference with respect to the central symbols. In the case of $x^*=0.3$ this ratio reduces to $0.30$, but it is still remarkable.
\subsection{Case II: Three or more interfering packets}
The evaluation of the cases of three or more packets interfering is carried out with the help of simulations. Moreover, some qualitative remarks are provided in the following. If we start considering the case of $x^*\geq 0.5$ and interference of three or more packets, we can make two considerations: first, there can be cases where the symbols in the packet center are free of interference, second, symbols in the beginning and in the end will have a higher probability of interference w.r.t. the two packets interference case as is shown in the simulations presented in figure \ref{2-3-4-5Pack_0_9}. These two remarks are the effects of the same cause, if the collision involves more than two packets, in some cases the central section of one or more packets can be free from interference, but at the same time higher interference must be found at the beginning or at the end of the packet. The case of $x^*< 0.5$ is slightly different, in fact not only the beginning and ending symbols will have an increased probability of interference w.r.t. the two interfering packets case, but also the central symbols will face interference more frequently. This is due to the increased number of combinations of interference that can be expected in the case of more than two packets interfering which lead to increased probability of interference among all the symbols in the packet.
\begin{figure}
\centering
\includegraphics[width=8cm]{2-3-4-5Packets_0_9_5000_4times_1000symbols_2}
\caption{Collision of 2, 3, 4 and 5 packets after SIC for $x^*=0.9$, $P_{len}=1000$ symbols and $L=901$ symbols - simulation results}
\label{2-3-4-5Pack_0_9}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{2-3-4-5Packets_0_3_5000_4times_1000symbols_2}
\caption{Collision of 2, 3, 4 and 5 packets after SIC for $x^*=0.3$, $P_{len}=1000$ symbols and $L=301$ symbols - simulation results}
\label{2-3-4-5Pack_0_3}
\end{figure}
The symbol probability of every packet symbol for 2, 3, 4 and 5 packets colliding with $x^*=0.9$ is presented in figure \ref{2-3-4-5Pack_0_9} while the same symbol probability but with $x^*=0.3$ is presented in figure \ref{2-3-4-5Pack_0_3}. All the aforementioned qualitative remarks are confirmed by the simulations. In figure \ref{2-3-4-5Pack_0_9} it can noted that already for three packets colliding, the probability of interference for every symbol is lower than 1. Furthermore, the beginning and ending symbols have more probability to be interfered w.r.t. the two packets colliding case. In case of $x^*=0.3$, the results presented in figure \ref{2-3-4-5Pack_0_3} show an increase of symbol interference probability for all the symbol positions. It can be also noted in the case of $x^*=0.3$, that for 3 packets colliding two peaks of interference probability can be found around $L$ and $P_{len}-L+1$. This effect is smothered in case of 5 packets colliding. This fact can be qualitatively explained in the following way: when 3 packets collide and the minimum level of interference is considered, in one packet only one of the two border parts, the beginning section or the ending section, can be involved in the collision. Otherwise, when 4 packets collide the constraint disappears and both sections can be interfered.
A fundamental conclusion can be drawn from the results presented here, regardless the number of packets involved in a collision and regardless the decoding threshold, the symbols in the beginning and in the end of a packet are less frequently involved in collisions. The property can be exploited for a smart header positioning within the packet.
\section{Numerical Results}
Thanks to the result of the previous section, we know that positioning the header and therefore the \ac{RP} in the beginning or in the end of a packet is the best choice for minimizing the probability of interference on it after the \ac{SIC} process.
In the following results are provided for the probability of interference in the packet headers after the \ac{SIC} procedure of \ac{CRA}. The aim is first to show how the symbol interference probability evaluated in the previous section translates to packet header interference probability after \ac{SIC} and second, to use this statistics for the \ac{ECRA} throughput and \ac{PER} simulations.
\subsection{Header interference probability}
The header length $H_l$ [symbols] is fixed to a certain value, in this example 10\% of the packet length. Bigger headers will have too much impact on the overhead of the protocol, hence this choice can be seen as a worst case assumption on the header length. The header has a \ac{CRC} code through which the decoder is able to recognize any impairment on this section of the packet. We assume that if one or more symbols of the packet header face interference, the header is not decodable and therefore we cannot retrieve the information about the replicas positioning of the user. The simulation assumptions are the following, the rate $R=2$ is selected and the \ac{SNR} is $SNR=10$ dB equal for each user generating traffic. The frame duration $T_f$ is $T_f=100$ ms and the symbol duration $T_s$ is $T_s=1$ $\mu$s. The packet length is $L_p=1000$ bits and is equal for each user. If we fix the number of users $N_u$, the offered traffic load $G$ is computed as $G=\frac{N_u\cdot L_p\cdot T_s}{T_f\cdot R}$.
Given $R$ and $L_p$ we can calculate the symbol length of the packet as $P_{len}=L_p/R=500$ symbols. The number of replicas sent by each user within a frame is $d=2$. The maximum number of \ac{SIC} iterations $I_{max}$ is chosen to be $I_{max}=10$. In this way, a first step of \ac{SIC} procedure is performed on the user packets at the decoder. The packets which have a sufficient \ac{SNIR} can be correctly decoded and removed from the frame. In our simulations the Shannon capacity limit is assumed as decoding threshold according to \eqref{21}. After the \ac{SIC} procedure the packets which are still not decoded are checked and the ratio between the number of remaining packets and the number of packets with interference in the header is evaluated. This ratio is the empirical probability of having interference in the packet header after the \ac{SIC} procedure.
In figure \ref{head_prob} the results for different header positions and different header lengths are provided. Two header lengths are assumed $H_{l-1}=10\%\cdot P_{len}$ (continuous curves) and $H_{l-2}=5\%\cdot P_{len}$ (dotted curves). The scenarios with $H_{l-1}$ can be seen as a worst case, in fact having a header which occupies more than 10\% of a packet is quite inefficient and would be an overhead hardly acceptable, while the choice $H_{l-2}$ appears more appealing. As expected the worst case is when the header is placed in the central part of the packet. In this case we have the highest probability that the header symbols face interference, (see figure \ref{2-3-4-5Pack_0_3}). If we call $p_{h-int}$ the probability of interference in the header, in this first case $p_{h-int} > 0.7$ for all the values of offered traffic load $G$ for $H_{l-1}$. For low values of $G$, if we decrease the header length to $H_{l-2}$ an observable difference in $p_{h-int}$ is found which decreases as $G$ grows.
When we move the header to the beginning of the packet, the interference probability decreases by more than 10\% for both header lengths cases and for $0.1\leq G \leq 0.6$ w.r.t. to the cases of a header in the center. Also for this second case, the decrease of the header length is beneficial from the probability of header interference point of view. Thanks to the previous section we are ensured that no other positioning of the header can achieve better probability of interference $p_{h-int}$. Moreover, thanks to the symmetry of the symbol interference probability w.r.t. the center of the packet, we know that positioning the header at the end of the packet will achieve the same probability of interference $p_{h-int}$.
Since the information carried by the header on the user replicas position is fundamental for \ac{ECRA}, it can be thought that replicating this information could improve the reliability. Therefore, two more cases are shown in the figure. The first supposes that the header is placed twice per packet in the beginning and in the end. Here the total overhead due to the header replication is 20\% of $P_{len}$ assuming $H_{l-1}$ and 10\% of $P_{len}$ assuming $H_{l-2}$. The second additional case assumes that the header is replicated three times inside each packet: at the beginning, in the center and at the end. In this last case the total header overhead is 30\% of $P_{len}$ assuming $H_{l-1}$ and 15\% of $P_{len}$ assuming $H_{l-2}$. The advantage is clear, the header cannot be correctly decoded only if at least one symbol for each of the two or three places where the header is replicated faces interference. Conversely to the other two cases, only one or more symbols interfering in one of the header locations will not prevent the retrieval of the header information and the \ac{RP}. On the other hand, given a fixed packet length, the replication of the header increases the overhead and decreases the throughput. The simulations of the next section will help to understand the benefits of the header replication. If we consider both cases for $H_{l-1}$, the header interference probability is below 0.2 for $G\leq 0.3$ and remains below 0.5 until $G=0.55$. The difference between the two cases is observable for $G\geq0.4$ but very limited in absolute terms ($<0.4\%$). Moreover, if we consider both the cases for $H_{l-2}$, the header interference probability is lower than 0.2 for $G\leq 0.4$ and remains under 0.5 until $G=0.65$. These results are very promising, especially for the case of header positioning in the beginning and in the end of the packet. The very limited decrease of header interference probability of the three headers case w.r.t the two headers case, cannot counteract the decrease of information sent per packet due to the increase of symbols required by the replication of the header, as will be shown in the next subsection.
\subsection{ECRA throughput simulations without perfect packets position knowledge}
After the first step where \ac{SIC} is performed as in \ac{CRA}, the \ac{ECRA} protocol tries to create a mix packet from the two user replicas which has the lowest level of interference possible and performs the decoding on it. If it is successful, a second round of \ac{SIC} is done. In \cite{Clazzer2012}, it was assumed that although the remaining packets were not decoded, it was always possible to retrieve the information of their position. Indeed this is not always the case, and here this assumption is removed. Instead, we assume that the information in the \ac{RP} can be recovered with probability $p_{h-int}$ (simulated in figure \ref{head_prob}) because $p_{h-int}$ represents the probability of no interference in the header(s) where the \ac{RP} is stored. Therefore with probability $p_{h-int}$ we are able to create the mix packet and to try the decoding. Since $p_{h-int}$ depends strongly on the simulation parameters, the throughput and \ac{PER} simulations of \ac{ECRA} will use the same parameters listed in the previous subsection.
The average packet error rate $\overline{PER}$, is evaluated as:
\[
\overline{PER} = \frac{P_{err}}{N_u \cdot N_f}
\]
\begin{figure}
\centering
\includegraphics[width=9.5cm]{Part_inter_pack_Prob_comparison_5_10}
\caption{Header interference probability for different header positions and different header lengths}
\label{head_prob}
\end{figure}
where $P_{err}$ is the number of lost packets at the receiver side, and $N_f$ is the number of simulated frames for the corresponding
$G$ which is $N_f=10^3$. The average throughput $\overline{T}$ is defined as the probability of successful reception of a packet, multiplied by the offered traffic load $G$. The average throughput here is related to the logical throughput, i.e. user packets, whereas the physical throughput would also consider the number of replicas generated per packet. Since the $\overline{PER}$ represents the average probability of a packet error, $\overline{T}$ is computed in the following way:
\begin{equation}
\label{41}
\overline{T} = \left[(1-\overline{PER}) \cdot G \right] \cdot \left[1-(n-1)\cdot \frac{H_l}{P_{len}} \right].
\end{equation}
Where $n$ is the number of headers per packet. With equation \eqref{41} we are taking into consideration the loss of information sent per packet due to the replication of the header. It can be noted that for $\lim_{H_l\rightarrow0} \overline{T} = (1-\overline{PER}) \cdot G$, e.g. the smaller the packet header w.r.t. to the packet length, the lower is the impact in terms of throughput of the header replication.
\begin{figure}
\centering
\includegraphics[width=9.5cm]{ECRA-2_T_5-10_comparison}
\caption{ECRA-2 Throughput simulations}
\label{Ecra_t}
\end{figure}
In figure \ref{Ecra_t} the throughput simulations for different header positions and header lengths are provided. In terms of maximum throughput $T_{max}$, the best choice appears positioning the header twice per packet at the beginning and at the end. In this case with the smaller header length $H_{l-1}$ the increase of $T_{max}$ w.r.t. \ac{CRA} is 13\%, while in the case of perfect knowledge of the packet positions the increase is 23\%. For all the other header positions and header lengths, the gain of \ac{ECRA} is reduced to 7\%. Please note that when the header is replicated within the packet, the throughput performance for $G<G_{max}$ are always slightly worse than the cases with no header replication. This is due to the decrease of information sent per packet.
\section{Conclusions}
In this work, the probability of symbol interference after \ac{SIC} procedure was mathematically derived for the case of two colliding packets and simulated up to 5 packets colliding. Interestingly, it is shown that regardless of the number of packets involved in the collision, the beginning and the end part of the packet has the lowest probability to face interference. Exploiting this property, the packet header and the \ac{RP} can be placed in the beginning of the packets. Moreover, it is shown that replicating twice the header in the beginning and in the end of each packet is beneficial also from the \ac{ECRA} throughput point of view although less information per packet is sent. \ac{ECRA} throughput simulations are provided when the hypothesis of perfect knowledge of the packets position within the frame is removed. \ac{ECRA} in the case of header positioning in the beginning and in the end of the packets, showed a throughput gain of 13\% w.r.t. \ac{CRA} if the header has a length of 5\% of the packets length. Moreover, it is shown that if the length of the header can be further decreased, a gain up to 23\% in the maximum throughput can be reached by \ac{ECRA} w.r.t. \ac{CRA}.
\section{Introduction}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,314,259,995,854 | arxiv | \section{Introduction}
The understanding of transition of a stochastic system between
disjoint regions in the phase space has applications in the study of
chemical reactions and thermally activated process (see \cite{MetzlerOshaninRedner:14, EVa:10} and references therein). For such physical
processes, the underlying dynamics can be often described by a
stochastic process such as the overdamped Langevin equation (also
known as the Brownian dynamics), which we will focus on in this paper
(while our methods can be generalized to other scenarios):
\begin{equation}\label{eq:overdamp}
\,\mathrm{d} X_t = - \nabla U(X_t) \,\mathrm{d} t + \sqrt{2 \beta^{-1}} \,\mathrm{d} W_t,
\end{equation}
where $X_t \in \Omega \subset \mathbb{R}^{d}$ denotes the current
configuration of the system at time $t$, $U: \Omega \to \mathbb{R}$ is a
potential function, $\beta = 1/(k_BT)$ is the inverse temperature
($k_B$ being the Boltzmann constant and $T$ the absolute temperature),
and $W_t$ is the standard $d$-dimensional Wiener process. Here
$\Omega$ is the configurational space of the system (with some
suitable boundary condition if $\Omega$ is not the full space). We are
interested in understanding the transition of the dynamics
\eqref{eq:overdamp} from a subset $A \subset \Omega$ (represents for
example the reactant state) to a disjoint subset $B \subset \Omega$
(represents for example the product state of the system). Direct
simulation of such transitions might be challenging since most of
these systems exhibit time scale separations: It takes much longer
time for the system to go from $A$ to $B$ compared to the intrinsic
time scale of the dynamics (which limits the time step size of
numerical simulation). As a result, the study of such rare transitions
requires novel methodology development, which has been a very active
area in physical chemistry and applied mathematics.
The transition path theory, proposed by E and Vanden-Eijnden in
\cite{EVa:06} and further developed in \cite{MeScVa:06, MeScVa:09,
LuNolen:15, CaVa:14}, is a framework to study the rare transitions
in the phase space. See also the review article \cite{EVa:10}. In the
transition path theory, the central role is played by the
\emph{committor function} $q$ \cite{Hummer:04, EVa:06}, which is the
probability that a trajectory starting from $x \in \Omega$ first hits
$B$ rather than $A$. Denote $\tau_{\Sigma}$ the first hitting time of a
subset $\Sigma \subset \Omega$, the committor function is defined by
\begin{equation}
q(x) = \mathbb{P}_x(\tau_B < \tau_A).
\end{equation}
Intuitively the committor function indicates the progression of a
transition: It takes the value $q = 0$ on $A$, $q = 1$ on $B$ and
increases going from $A$ to $B$. In chemical terms, the committor
function can be understood as a reaction coordinate of the transition (see \cite{Peters:16} for a recent review on reaction coordinates).
It follows that $q$ solves the following PDE on
$\Omega \backslash (A \cup B)$ with Dirichlet boundary conditions
given on $A$ and $B$ \cite{EVa:06, LuNolen:15}:
\begin{equation}\label{eq:committor}
\begin{cases}
L q = 0, & \text{in } \Omega \backslash (A \cup B); \\
q = 0, & \text{in } A; \\
q = 1, & \text{in } B,
\end{cases}
\end{equation}
where $L$ is the infinitesimal generator of the process \eqref{eq:overdamp}, given by
\begin{equation}
L = - \beta^{-1} \Delta + \nabla U \cdot \nabla.
\end{equation}
Note that some boundary
conditions are needed in the above equation if $\Omega$ is not a closed manifold, which we
will come back to later.
Once the committor function is provided, we can easily obtain
information on the reaction rate, density of transition paths, current
of transition paths \cite{EVa:06, EVa:10, LuNolen:15}, which help
understand the stochastic system. Moreover, we can also write down
the SDE for the transition path between $A$ and $B$ that only depends
on the committor function \cite{LuNolen:15}, which can be used for
transition path sampling \cite{DellagoBolhuisGeissler:02,
BolhuisChandlerDellagoGeissler:02}.
Solving the PDE \eqref{eq:committor} for the committor function is
however non-trivial due to the curse of dimensionality. As a result,
in the framework of transition path theory, further assumptions are
usually made to approximate the committor function: It is assumed that
the transition from $A$ to $B$ is concentrated in quasi-one
dimensional ``reaction tube'', which is the working assumption of the
finite temperature string method \cite{ERenVa:05, VaVe:09}. Due to the
usefulness of the committor function in understanding the transition,
other approaches have been also developed, mainly in the chemical
physics literature, for example methods based on statistical analysis
of an ensemble of trajectories \cite{MaDinner:05, PetersTrout:06,
LechnerRogal:10}, based on approximating the diffusion by a discrete
state space jump process (milestoning) \cite{MajekElber:10,
KirmizialtinElber:11} (see also \cite{CaVa:14} on direct computation
of committor function on a discrete Markov jump process). To the best
of our knowledge, none of the existing methods aims at approximating
the PDE~\eqref{eq:committor} directly.
The main contribution of this work is to propose a method to directly
solve the committor equation \eqref{eq:committor} based on point cloud
discretization, especially the technique of local mesh discretization
recently developed in \cite{LaiLiangZhao:13}. Assume a given point cloud
samples the equilibrium distribution of the dynamics
\eqref{eq:overdamp}, the idea is to discretize the PDE
\eqref{eq:committor} on the point cloud to approximate the committor
function. Here, the working assumption is that while the
configurational space of the stochastic system is high dimensional,
the transition between the interested regions $A$ and $B$ lies in an
intrinsically low-dimensional manifold (for simplicity, we assume that
the intrinsic dimension does not change). In particular, this
generalizes the ``reaction tube'' assumption of the finite temperature
string method to transition in higher than quasi-one dimension.
Our method is closely related in spirit to the method of diffusion map
\cite{CoifmanLafon:06}. In particular, in the work \cite{NaLaCoKe:06,
CoKeLaMaNa:08} and subsequently \cite{MaggioniClementi:11a,
MaggioniClementi:11b}, the diffusion map has been applied to obtain
an approximation of the infinitesimal generator to compute the first
few low-lying eigenfunctions of $L$, the Fokker-Planck operator. Those
eigenmodes are used to approximate the long time behavior and to
coarse-grain the dynamics \eqref{eq:overdamp}. As demonstrated in
\cite{MaggioniClementi:11a, MaggioniClementi:11b}, the assumption that
the stochastic dynamics can be approximated by low-dimensional
reaction modes is indeed valid for a variety of chemical systems. It
is in fact possible to use the idea of diffusion maps to approximate
the equation for the committor function \eqref{eq:committor} and we
will compare our method with the diffusion map based method. Let us
also remark that we do not focus on the low-lying eigenmodes of
Fokker-Planck equation as \cite{NaLaCoKe:06, MaggioniClementi:11a,
MaggioniClementi:11b}, but rather the committor function, which
provides the information on the transition from $A$ to $B$ more
directly. One potential advantage of focusing directly at the
transition region is that we do not require point cloud to well
represent the regions $A$ and $B$, which are typically of high
intrinsic dimension, as usually they correspond to regions of local
minima of the potential energy surfaces, and hence leads to challenges
for point cloud discretization. We refer the readers to \cite{EVa:10}
for further comparison between the committor function and low-lying
eigenmodes. Besides diffusion map and the local mesh method, other
numerical techniques have been also developed for solving PDEs on
point clouds without a mesh and this has been a very active area in
recent years; a brief review of those can be found in
\ref{sec:localmesh}.
The rest of the paper is organized as follows. The point cloud discretization method for the committor function is described in detail in \ref{sec:localmesh}. The algorithm is validated and compared with other approaches through numerical examples in \ref{sec:experiments}. We conclude the paper in \ref{sec:conclusions}.
\section{Local mesh method for Fokker-Planck operators}\label{sec:localmesh}
Given a potential function $U(x)$ defined on $\Omega$, recall that we
aim at solving the following equation on $\Omega$ represented as point
clouds:
\begin{equation}
\begin{cases}
-\beta^{-1} \Delta q(x) + \nabla U(x) \cdot \nabla q(x) = 0, & x\in \Omega \backslash (A\cup B); \\
q(x) = 0, & x\in A; \\
q(x) = 1, & x\in B; \\
\nabla q(x)\cdot \vec{n}(x) = 0, & x\in \partial\Omega.
\end{cases}
\label{eqn:FK}
\end{equation}
Note that compared to \eqref{eq:committor}, we have also specified the
Neumann boundary condition at the boundary of $\partial\Omega$, which
is the natural boundary condition from a variational point of view, as
discussed below.
Before we proceed to numerical algorithms, let us write down the weak formulation of \eqref{eqn:FK}. Given any test function $\eta(x)$, using the equation and integration by parts, we have:
\begin{equation}
\begin{aligned}
0 & = \int_{\Omega} \left(\Delta q(x) -\beta \nabla U(x) \cdot \nabla q(x) \right ) \eta(x) e^{-\beta U(x)} ~\mathrm{d} x \\
& = \int_{\Omega} \nabla \left(\nabla q(x) \eta(x) e^{-\beta U(x)} \right) - \nabla q(x) \nabla \eta(x) e^{-\beta U(x)} ~\mathrm{d} x \\
& = - \int_{\Omega}\nabla q(x) \nabla \eta(x) e^{-\beta U(x)} ~\mathrm{d} x,
\label{eqn:weakFK}
\end{aligned}
\end{equation}
where in the last equality, we have used the Neumann boundary
condition $\vec{n}(x) \cdot \nabla q(x) = 0$ on $\partial \Omega$ so
that the boundary contribution vanishes. Note that the Gibbs weight
$e^{-\beta U(x)}$ appeared in \eqref{eqn:weakFK} is the invariant measure $\rho(x) = Z^{-1}e^{-\beta U(x)}$ of the overdamped
equation \eqref{eq:overdamp} up to a normalization constant $Z = \int_{\Omega} e^{-\beta U(x)} ~\mathrm{d} x$.
In the transition path theory~\cite{EVa:10}, after we compute $q(x)$, we can immediately obtain the transition rate between $A$ and $B$ provided by
\begin{equation}\label{eq:nuR}
\nu_R = k_B T \int_{\Omega} \lvert \nabla q(x) \rvert^2 \rho(x) ~\mathrm{d} x,
\end{equation}
which we will later use to quantify the approximation quality of our numerical methods. The committor function also leads to other useful information to understand the dynamics, including the probability density of reactive trajectories
\begin{equation}
\rho_R(x) = q(x) (1 - q(x)) \rho(x)
\end{equation}
and also the reactive current
\begin{equation}
J_R(x) = k_B T \rho(x) \nabla q(x),
\end{equation}
which can be obtained through explicit formulas based on $q$.
Let us come back to numerical solution to \eqref{eqn:FK}. Since
$\Omega$ is potentially high dimensional while the transition between the interested regions A and B lies in an intrinsically low-dimensional manifold, our idea is to solve the
equation and get the approximation of $q$ instead on a point cloud well sampling the transition path.
More precisely, assume that we are given a point cloud
$\mathcal{P}=\{\bm{p}_i\in\mathbb{R}^N \mid i=1,\dots,n\}$, for instance from
snapshots of a long trajectory of the SDE
\eqref{eq:overdamp},\footnote{The advantage of using a long ergodic
trajectory of the SDE is to cover the important region of the
transition; on the other hand, let us emphasize that we do not need
to assume that the point cloud is distributed according to the
invariant measure of the SDE \eqref{eq:overdamp}, in particular, the
method works as long as the point cloud well cover the important
part of the configurational space.} the goal here is to approximate the
committor function, given by \eqref{eq:committor} based on $\mathcal{P}$. In particular, we
are aiming at the value of $q$ on the point clouds, instead of on the
whole configurational space $\Omega$. We will also interpret the sets
$A$ and $B$ as the collection of points in the point cloud $\mathcal{P}$ that
lies in the two chosen sets in the whole configurational space. For
simplicity of notation, when there is no danger of confusion, we will
not distinguish in the sequel the sets $A$, $B$, $\Omega$ etc. with
their point cloud interpretations.
In the rest of this section, we first give a brief review of the existing diffusion map based method~\cite{CoifmanLafon:06,MaggioniClementi:11a} for approximating
the Fokker-Planck equation on the given point clouds, which will be compared later with our method. After that, our method of solving the Fokker-Planck equation on point clouds will be presented based on the local mesh method discussed in~\cite{LaiLiangZhao:13}.
\subsection{Diffusion map discretization of Fokker-Planck operator}
In recent years, several numerical schemes have been proposed for
solving partial differential equations on point clouds without a
global mesh or grid in the ambient space. These methods are
particularly useful in high dimensions where a global triangulation or
grid is intractable. Among those methods, a popular class of methods
is kernel based, where the Laplace-Beltrami operator on point clouds
is approximated by heat diffusion in the ambient Euclidean
space~\cite{CoifmanLafon:06} or in the tangent
space~\cite{Belkin09clp} among nearby points. In other words, the
metric on the manifold is approximated by Euclidean metric
locally. The main advantage of such methods is their
simplicity and generality for approximating diffusion type operators.
Among those, the methods based on the ideas of diffusion map~\cite{CoifmanLafon:06} have been rather popular. For discretizing the Fokker-Planck operator $ -\beta^{-1}\Delta + \nabla U\cdot \nabla$ in \eqref{eq:committor} on the point cloud, one uses
$L = D^{-1} K - I_n$ where $K$ is the normalized kernel function according to the Gibbs weight $\exp(-\beta U)$:
\begin{equation}\label{eqn:diffKernel}
K(\bm{p}_i, \bm{p}_j) =\exp(-\beta U(\bm{p}_i)) \exp \left(-\frac{\|\bm{p}_i - \bm{p}_j\|^2}{2\epsilon_i\epsilon_j} \right)\exp(-\beta U(\bm{p}_j))
\end{equation}
and $D$ being a diagonal matrix with
\[
D_{ii} = \sum_{j} K(\bm{p}_i,\bm{p}_j),
\]
and $I_n$ being the identity matrix of size $n \times n$. Note that $\epsilon_i$ can be chosen either as a constant or adaptively depending on the local information of the data set using the multiscale SVD method proposed in \cite{little2009multiscale,MaggioniClementi:11a}.
Therefore, if we denote $C = \Omega - A\cup B$ (recall that these sets
are interpreted as the subsets of points in $\mathcal{P}$), then the committor
function $q = [q_A,q_B, q_C]$ can be solved by the linear equation
\[
L(C,C)q_C = - L(C, B) q_B,
\]
where $q_B \equiv 1$ and $q_A \equiv 0$ according to the boundary
conditions of the committor function. We remark that $L(C,C)$ and $L(C,B)$ are denoted as the restriction of the matrix $L$ on the index set of $(C,C)$ and $(C,B)$, respectively.
The above kernel based methods however usually render low order
approximation. More recently, two new
methods for solving PDEs on point clouds are introduced
in~\cite{Liang:CVPR2012,Liang2013solving,LaiLiangZhao:13}, where the
differential operators are approximated systematically and
intrinsically at each point from local approximation of the manifold
and its metric through nearby neighbors.
These methods can achieve
high order accuracy and can be used to approximate general differential
operators (\textit{i.e.}, other than diffusion type operators) on point clouds sampling manifolds with arbitrary dimensions
and co-dimensions. Moreover, the computational complexity depends
mainly on the intrinsic dimension of the manifold rather than the dimension
of the embedded space.
To solve \eqref{eqn:FK} on a point cloud, we will focus on
generalizing the local mesh method discussed in \cite{LaiLiangZhao:13}
to Fokker-Planck operators. The local mesh method is natural to
handle the Neumann boundary conditions, which is more advantageous in
the current scenario. The crucial part of solving the weak formula \eqref{eqn:weakFK} is to numerically approximate $\int_\Omega \nabla q \nabla \eta e^{-\beta U} \mathrm{d} x$ on the given point cloud. Note that the definition of differential operators only rely on local information of point clouds. Thus, our idea is composed of constructing local connectivity to approximate the gradient operator and estimating the stiffness matrix on point clouds. In the rest of this section, we will discuss technical details about local connectivity construction and the stiffness matrix approximation on point clouds. After that, a numerical method of solving the Fokker-Planck equation \eqref{eqn:FK} will be proposed.
\subsection{Local connectivity construction for point clouds}
\label{subsec:localmesh}
\begin{wrapfigure}{r}{6cm}
\vspace{-0.3cm}
\centering
\includegraphics[width=1\linewidth]{PC_localmesh_proj.png}\\
\label{fig:PC_localmesh_pro}
\vspace{-.5cm}
\caption{Red stars mark the KNN of $\bm{p}_i$, green stars mark the projection of red stars on the tangent plane at $\bm{p}_i$, and blue triangles color-code the connectivity of the first ring of $\bm{p}_i$.}
\end{wrapfigure}
Given a point cloud $\mathcal{P}=\{\bm{p}_i\in\mathbb{R}^N~|~ i=1,\dots,n\}$ sampled from a $d$-dimensional manifold $\mathcal{M}$ in $\mathbb{R}^N$, let us denote the indices set of K-nearest neighborhood (KNN) of each
point $\bm{p}_i \in \mathcal{P}$ by $\mathcal{I}(i)$ and write $\mathcal{P}(i) = \{\bm{p}_k \in \mathcal{P}~|~ k\in \mathcal{I}(i)\}$. Based on $\mathcal{P}(i)$, the tangent space and normal space of $\mathcal{M}$ at $\bm{p}_i$ can be approximated by the standard principle component analysis (PCA)~\cite{jolliffe2002principal}. More precisely, one can first construct the following covariance matrix $\mathrm{Cov}_i$ of $\mathcal{P}(i)$:
\begin{equation}
\mathrm{Cov}_i=\sum_{k\in \mathcal{I}(i)}(\bm{p}_k - \bm{c}_i)^T(\bm{p}_k-\bm{c}_i)
\end{equation}
where $\bm{c}_i$ is the local barycenter of $\mathcal{P}(i)$, given by
\begin{equation*}
\bm{c}_i=\frac{1}{|\mathcal{P}(i)|}\sum_{k\in \mathcal{I}(i)}\bm{p}_k.
\end{equation*}
As the point cloud is sampled from a $d$-dimensional manifold $\mathcal{M}$, if the local sampling is dense enough to resolve local features, then eigenvalues of $\mathrm{Cov}_i$ have a natural jump $\lambda_i^1 \geq \cdots \geq \lambda_i^d \gg \lambda_i^{d+1} \geq \cdots\geq \lambda_i^N \geq 0$
which guides the splitting $\mathbb{R}^N = \mathcal{T}_i \oplus \mathcal{N}_i$.
Here $\mathcal{T}_i$ represents the tangent space spanned by $\{\bm{e}_i^1,\cdots,\bm{e}_i^d\}$ corresponding to the $d$ largest eigenvectors of $\mathrm{Cov}_i$, and $\mathcal{N}_i$ represents the normal space spanned by $\{\bm{e}_i^{d+1},\cdots,\bm{e}_i^N\}$.
For noisy data set, a technique called multiscale SVD method~\cite{little2009multiscale} is an effective way to estimate the intrinsic dimension.
To construct the local connectivity and a mesh at the point $\bm{p}_i$, we project its K-nearest neighborhood $\mathcal{N}(i)$ on the tangent plane $\mathcal{T}_i$ of $\bm{p}_i$. Namely, we have the following construction:
\begin{equation}
\hat{\bm{p}}_k = \proj_{\mathcal{T}_i}(\bm{p}_k) = \bm{p}_k - \bm{p}_i - \sum_{\alpha=d+1}^N\langle \bm{p}_k-\bm{p}_i, \bm{e}_i^\alpha\rangle \bm{e}_i^\alpha,\quad k\in\mathcal{I}(i)
\end{equation}
For convenience, we have
translated the K-nearest neighborhood $\mathcal{P}(i)$ to
center $\bm{p}_i$ at the origin.
With this projection, all points in $\{\hat{\bm{p}}_k, k\in I(i)\}$ belong to the tangent plane $\mathcal{T}_i$. Then, the local mesh structure near $\bm{p}_i$ can be obtained by the standard Delaunay triangulation. Denote by $\mathcal{R}(i)=\{F_i^1,\cdots, F_i^{l_i}\}$ all simplexes adjacent to $\bm{p}_i$, referred as $\bm{p}_i$'s first ring and $\mathcal{V}(i)$ all vertices in the first ring of $\bm{p}_i$. The local connectivity of $\bm{p}_i$ is provided by $\mathcal{C}_i = \{\bm{p}_i; \mathcal{V}(i),\mathcal{R}(i)\}$. Figure~\ref{fig:PC_localmesh_pro} illustrates an example of a set of point sampled in a 2D manifold in $\mathbb{R}^3$.
After obtaining the local connectively, we are ready to discretize the weak formula \eqref{eqn:weakFK} to solve the Fokker-Planck equation \eqref{eqn:FK}. Our basic idea is to represent $q$ as a linear combination of a nodal basis on the point cloud. Derivatives of all nodal basis can be approximated by linear interpolation. After that, we approximate the stiffness matrix using the nodal basis and solve the weak equation.
\subsection{Stiffness matrix construction on point clouds}
\label{sec:stiffM}
Suppose that we have a point cloud $\mathcal{P}$ with local connectivity $\mathcal{C} = \{\mathcal{C}_i = \{\bm{p}_i; \mathcal{V}(i),\mathcal{R}(i)\} ~|~ i = 1,\cdots,n\}$ constructed in section~\ref{subsec:localmesh}, we define the set of nodal basis $\{\eta_j\}_{j=1}^{n}$ on $\mathcal{P}$ as:
\begin{eqnarray}
\eta_j: \mathcal{P} \rightarrow \mathbb{R},\quad \eta_j(\bm{p}_i) =
\begin{cases} 1, & \text{if }\quad i=j \\
0, & \text{otherwise }
\end{cases}
\quad i,j = 1, \cdots, n
\label{eqn:linearelements}
\end{eqnarray}
Inspired by the idea from the finite element method, we consider $\{\eta_i\}_{i=1}^n$ as the set of linear elements defined on $\mathcal{P}$ given in (\ref{eqn:linearelements}), and write $E= \text{Span}_{\mathbb{R}}\{\eta_i\}_{i=1}^{n}$. We approximate the desired committor function $q= (q(\bm{p}_1),\cdots,q(\bm{p}_n))^T$ defined on $\mathcal{P}$ as $q=\sum_i^n v_i\eta_i$.
Then the discrete version of the equation \eqref{eqn:weakFK} on $\mathcal{P}$
can be defined by the following weak formulation:
\begin{eqnarray}
\int_{\mathcal{P}}e^{-\beta U} \nabla_{\mathcal{P}}q \cdot \nabla_{\mathcal{P}}
\eta = 0,~\quad \forall~\eta\in E.
\label{formula:LBeigsdiscrete}
\end{eqnarray}
Therefore, the essential part of
the above problem is a numerical approximation of the stiffness matrix $S = (S_{ij})_{n\times n}$ for the given point cloud $\mathcal{P}$ with:
\begin{equation}\label{eq:stiffness}
S_{ij} = \int_{\mathcal{P}}e^{-\beta U} \nabla_{\mathcal{P}}\eta_i\cdot\nabla_{\mathcal{P}}\eta_j.
\end{equation}
To clearly indicate the proposed method, we would like to assume the point cloud is sampled from a two dimensional manifold in $\mathbb{R}^N$.\footnote{We emphasize that the proposed idea can be easily adapted to other cases that the intrinsic dimension $\mathcal{M}$ is not two.} In this case, the local connectivity is a set of local triangle mesh structure. Ideally, we would like $S$ to
be symmetric and non-negative definite, similar to the usual
properties of the stiffness matrix for a triangulated
surface~\cite{dziuk1988finite}.
Unfortunately, these global properties
might not be possible to achieve due to the use of only local mesh in our discretization. To be more specific, the first ring structure of $\bm{p}_i$ is not
necessary compatible with the first ring structure of $\bm{p}_j$ although
$\bm{p}_j$ belongs to the first ring of $\bm{p}_i$. To have a numerical
approximation of the stiffness matrix, we first define
\begin{eqnarray}
A_{ij}&=&\sum_{F\in\mathcal{R}(i)}\int_{F} e^{-\beta U}\nabla_{F}\eta_i\cdot\nabla_{F}\eta_j \nonumber \\
&=& \sum_{F\in\mathcal{R}(i),[\bm{p}_i,\bm{p}_j]\in F} -\frac{1}{2}w_{ij}^F \cot\alpha^{F}_{ij}(p_i) , \quad i\neq j
\end{eqnarray}
where $\int_{F} e^{-\beta U}\nabla_{F}\eta_i\cdot\nabla_{F}\eta_j$ is computed by linear interpolation of $U, \eta_j$ and $\eta_j$ on $F$,$w_{ij}^F$ is the average of $e^{-\beta U}$ on $F$ and $\alpha^F_{ij}(\bm{p}_i)$ are the angles opposite to the edge connecting points $\bm{p}_i$ and $\bm{p}_j$ in the face $F$.
Note that $A_{ij}$ may not be equal to $A_{ji}$ due to the possible incompatibility of the first ring structures of $\bm{p}_i$ and $\bm{p}_j$. One simple symmetrized definition of the stiffness matrix is the following:
\begin{equation}
S_{ij}=
\left\{
\begin{array}{ll}
\displaystyle\frac{1}{2}\left(A_{ij}+A_{ji}\right), &\quad \mbox{if} \quad i\ne j \vspace{0.2cm}\\
\displaystyle -\sum_{k\neq i} S_{ik}, &\quad \mbox{if} \quad i = j
\end{array}\right.
\label{eqn:Stiffness1}
\end{equation}
The above definition of the diagonal elements is to enforce the
consistency condition, \textit{i.e.}, constant function is an
eigenfunction of $S$ with zero eigenvalue. In particular, if
all triangles in the first ring structure are acute, off-diagonal
elements are non-positive and diagonal elements
$S_{ii}=|\sum_{k\neq i} S_{ik}|$ are positive. Hence all eigenvalues
are real and non-negative. When the density of points is
reasonably uniform on $\mathcal{M}$, this definition of stiffness matrix works
quite well. However, when the density of points are non-uniformly as the data produced from the SDE \eqref{eq:overdamp} used in our experiments, the
first ring structure of $\bm{p}_i$ is more likely incompatible with the
first ring structure of neighboring points $\bm{p}_j$. To overcome this issue, we use a similar
strategy used in \cite{LaiLiangZhao:13} to construct $S$ as follows:
\begin{equation}
S_{ij}=\left\{
\begin{array}{ll}
\max (A_{ij}, A_{ji}) & \mbox{if} \quad A_{ij}\le 0 \quad \mbox{and} \quad A_{ji}\le0 \\
\min(A_{ij}, A_{ji}) & \mbox{if} \quad A_{ij}\ge 0 \quad \mbox{and} \quad A_{ji}\ge0 \\
\min(A_{ij}, A_{ji}) & \mbox{if} \quad A_{ij}\cdot A_{ji} <0 \\
-\sum_{k\neq i} S_{ik} & \mbox{if} \quad i = j
\end{array}
\right.
\label{eqn:Stiffness2}
\end{equation}
As long as the stiffness matrix are constructed, we can approximate the committor function, the solution of the equation~\eqref{eqn:FK}, in the following way.
Remember that we write $C = \Omega - A\cup B$, then we obtain the following discretization of the equation~\eqref{eqn:FK}.
\begin{equation}
\left\{\begin{array}{c}
S(C, [A, B, C]) (q_A,~ q_B,~ q_C)^T = 0. \\
q_A = 0 \\
q_B = 1
\end{array} \right. \label{eqn:FK_LM}
\end{equation}
This provides a matrix equation $$S(C,C)q_{C} = -S_{C,B} q_B$$ which solves the committor function we desired.
\section{Numerical Experiments}
\label{sec:experiments}
In this section, we test the proposed method for committor functions
on several examples obtained from the stochastic differential equation
\eqref{eq:overdamp}. All experiments are impletmented by MATLAB in a
PC with a 32G RAM and a 2.7 GHz quad-core CPU.
\subsection{Comparison of methods for a 1D double well potential}
\label{subsec:diffusionmap}
We first conduct numerical experiments on 1D interval $[-1,~1]$ for
the standard double well potential $U = (x^2-1)^2$ with $\beta = 1$.
We choose the sets $A = [-1,~-1+0.1]$ and $B = [1-0.1, ~1]$. We will
compare our method with the diffusion map discretization of the
Fokker-Planck operator, following the
works~\cite{CoifmanLafon:06,CoKeLaMaNa:08,MaggioniClementi:11a,MaggioniClementi:11b}. The
numerical experiments illustrate that the local mesh method achieves
better accuracy and robustness.
We first compare results using our method and diffusion map method on
$1000$ equally sampled points distributed on $[-1,~1]$. In this case,
we also apply a standard finite element based method to solve the
weak equation \eqref{eqn:weakFK}, serving as a reference. Figure~\ref{fig:Comparison_FD_LM_DM}
illustrates comparisons among these three methods on the same point
cloud data set (for finite element, we interpret the points as grid
points). It is clear to see that solutions obtained from the finite
element method and the proposed method are nearly identical. The
diffusion map approach yields a similar result, however not quite as
accurate as the proposed method. To quantify this, we denote
$q_{\text{FE}}$, $q_{\text{LM}}$ and $q_{\text{DM}}$ as solutions
obtained from the finite element method, the proposed local mesh
method and the diffusion map method, respectively. The maximum
absolute error
\begin{equation}
\max \left\{\frac{|q_{\text{FE}}(x) - q_{\text{LM}}(x)|}{|q_{\text{FE}}(x)|} \right\}
\end{equation}
between solutions using finite element and the proposed method is
$3.3317e$-$7$, while the maximum absolute error between solutions using
finite element and the diffusion map method is $0.0112$.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKCurve_FD_LM_DM}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKCurve_FD_LM_DM_zoomin}
\end{minipage}
\caption{Left: Solutions obtained from a finite element method (FE), the proposed local mesh method (LM) and the diffusion map method (DM). Right: A zoom-in image. }
\label{fig:Comparison_FD_LM_DM}
\end{figure}
In the second experiment, we consider points simulated by the SDE
\eqref{eq:overdamp} using an Euler-Maruyama scheme. We take 1000 time snapshots from a long trajectory generated by the SDE~\eqref{eq:overdamp} and only keep those points inside $[-1,~1]$, which provides 596 points. We apply both
local mesh method and diffusion map method to this data set, and
compare to the reference solution obtained by the finite element
method based on $1000$ equally distributed points on $[-1,~1]$. As we
can see from the left panel in Figure \ref{fig:Comparison_randpt},
although the data does not well sample the invariant measure
associated with the double well as we only choose $1000$ points to
discretize the SDE (in particular, the empirical distribution is far
from symmetric), our method still provides almost identical
result as the solution obtained from the standard finite element
method. The diffusion map method on the other hand does not provide
satisfactory result in this case. Note that the information of the invariant measure has been also used in the diffusion map method as the way of constructing $K(\bm{p_i},\bm{p_j})$ in \eqref{eqn:diffKernel}.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{1D_SDEptDistribution}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKCurve_FD_LM_DM_SEDpt}
\end{minipage}
\caption{Left: histogram of distribution of 596 points sampled on $[-1,~1]$. Right: Numerical solutions using different methods.}
\label{fig:Comparison_randpt}
\end{figure}
It is natural to ask about the performance of the methods when more
data points are available; in particular, the accuracy of the
diffusion map discretization will improve. In the third experiment, we
compare our method with diffusion map method using more points
obtained from the same SDE \eqref{eq:overdamp} than the previous
experiment: For each test, we take $10000$ time snapshots from a long
trajectory generated by the SDE and only keep those points inside the
interval $[-1,~1]$, which provides around $6000$ points for each
test. The test is repeated for $10$ times with independent drawing of the point clouds. Figure
\ref{fig:FPCurve_reprod} reports the approximation of committor
functions as solution of equation \eqref{eq:committor}. The left
panel of Figure \ref{fig:FPCurve_reprod} shows results obtained from
the diffusion map method, where curves with different color-coding
indicate solutions for different realization of the test (note that
the point cloud changes from test to test). It is clear to see that
the diffusion map based method is quite sensitive to sampling quality
of the point clouds, thus the approximated solutions of the committor
function is not quite reproducible due to the different sampling
quality. As a comparison, the corresponding solutions using our method
are plotted in the right picture of Figure \ref{fig:FPCurve_reprod}
using exactly the same samples. We found that solutions
obtained from our method are stacked on top of each other as they are
all essentially identical to the result obtained from the finite
element method based regular grid. This shows
that our method is very robust to points sample quality, and thus
yields better reproducibility, besides provides more accurate
approximation of the committor function.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKCurve_DiffMap_10tests}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKCurve_LocalMesh_10tests}
\end{minipage}
\caption{Left:10 tests to the backward Fokker-Planck equation (sampled from 10000 points) using diffusion map. Right: 10 tests to the backward Fokker-Planck equation (sampled from 10000 points) using our method.}
\label{fig:FPCurve_reprod}
\end{figure}
\subsection{Rugged Mueller potential}
Next we test our methods on a $2D$ problem with the rugged Mueller
potential, which is a well-established test problem in chemical
physics. The point cloud is sampled in $2D$ domain
$\Omega = [-1.5,~1]\times [-0.5,~2]$ and the potential is given by
\begin{equation}
U_{rm}(x,y) = U(x,y) + \gamma\sin(2 k \pi x)\sin(2k\pi y)
\label{eqn:ruggedMueller}
\end{equation}
with $U$ being the original Mueller potential
\begin{equation*}
U(x,y)= \sum_{i=1}^4 D_i e^{a_i (x - X_i)^2 + b_i(x - X_i) (y - Y_i) + c_i (y - Y_i)^2}
\end{equation*}
with the parameters chosen as:
\begin{align*}
[a_1,a_2,a_3,a_4] & = [-1,-1,-6.5,0.7], \\
[b_1,b_2,b_3,b_4] & = [0,0,11,0.6], \\
[c_1,c_2,c_3,c_4] & = [-10,-10,-6.5,0.7], \\
[D_1,D_2,D_3,D_4] & = [-200,-100, -170,15], \\
[X_1,X_2,X_3,X_4] & = [1,0 ,-0.5, -1], \\
[Y_1,Y_2,Y_3,Y_4] & = [0,~ 0.5,~ 1.5,~ 1].
\end{align*}
In addition, we choose $\gamma = 9, k = 5$.
Thus the rugged Mueller potential increases the roughness of the potential.
The reactant and product sets are chosen to be
\begin{align*}
& A = \{ U(x,y) < -120 \} \cap \{ y > 0.75\} \\
& B = \{ U(x,y) < -82\} \cap \{ y < 0.35 \}.
\end{align*}
In the left
panel of Figure \ref{fig:ruggedMueller}, we plot the above rugged
Mueller potential and its level contours. As a reference, we also use the weak formula \eqref{eqn:weakFK} to
solve the Fokker-Planck equation using the finite element method in the domain
$[-1.5,~1]\times [-0.5,~2]$ and denote the resulting committor function as $q_{\text{FE}}$. The right panel of
Figure~\ref{fig:ruggedMueller} illustrates this committor function and
its level contours. Using the stiffness matrix and the mass matrix constructed in the finite element method, we also numerically evaluate the transition rate $\nu_R$ given by equation~\eqref{eq:nuR}, measures the total probability of reactive trajectories out of
the set $A$ to the set $B$. In this case, we obtain the transition
rate is $0.92960$, which we denote it as $\nu_R^{\text{FE}}$ for later
comparisons.
\begin{figure}[h]
\centering
\begin{minipage}{0.495\linewidth}
\includegraphics[width=1\linewidth]{RuggedMueller.png}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\includegraphics[width=1\linewidth]{FKSurf_FEM_square_solution}
\end{minipage}
\caption{Left: A rugged Mueller potential used in our
experiments. Right: The committor function with its level contours
obtained by the standard finite element method on the domain
$[-1.5,~1]\times [-0.5,~2]$ , $\nu_R^{\text{FE}} = 0.92960$. }
\label{fig:ruggedMueller}
\end{figure}
We now consider point clouds generated by numerically integrating the
overdamped Langevin equation \eqref{eq:overdamp} with the above rugged
Mueller potential using the Euler-Maruyama method. We only keep those
points inside the domain $ [-1.5,~1]\times [-0.5,~2]$. Based on an
input set of irregular data points, the proposed local mesh method
will be applied to solve for the committor function $q$. In
Figure~\ref{fig:2DFPsolution}, the left panel shows a realization of
the point cloud obtained
with $\beta = 1/22$. The numerical approximation of a committor
function $q$ is color-coded on the point cloud and illustrated in the
right panel of Figure \ref{fig:2DFPsolution}. Qualitatively, the
obtained function $q$ represents an increase trend of probability that
moving from the set $A$ to the set $B$, consistent with the intuition
behind the committor functions.
\begin{figure}[h]
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{FKSurfLocalMesh_SDEPC_data.png}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{FKSurfLocalMesh_SDEPC_solution.png}
\end{minipage}\hfill
\caption{Left: Data generated by the SDE \eqref{eq:overdamp} with $\beta = 1/22$. Right: A solution obtained by LM and its level contours.}
\label{fig:2DFPsolution}
\end{figure}
To test the accuracy and convergence of the proposed local mesh
method, we chose the number of snapshots in SDE \eqref{eq:overdamp}
from $20,000$ to $200,000$ with an incremental size
$10,000$. Thus, we obtain a sequence of point clouds which are sampled
in $ [-1.5,~1]\times [-0.5,~2]$ and accumulate near the minimal energy
path connecting two deep wells of the ragged Mueller potential.
The local mesh method thus produces a sequence approximation of the committor function. For
comparison, we use committor function $q_{\text{FE}}$ based on the regular grid
$[-1.5,~1]\times [-0.5,~2]$ as a reference. As can be seen from Figure~\ref{fig:Contours}, level contours, represented by black curve, of the
approximated committor function match very well to the blue curves
representing level contours of $q_{\text{FE}}$. Moreover, the transition rate $\nu_R$ defined in ~\eqref{eq:nuR} can be approximated by evaluating $\int_{\Omega} |\nabla q| e^{\beta U} \mathrm{d}x$ using the stiffness matrix constructed in~\eqref{eqn:Stiffness2} and approximating $Z = \int_{\Omega} e^{-\beta U} \mathrm{d}x$ using the method proposed in~\cite{LaiLiangZhao:13} based on a numerical approximation of the mass matrix. After that, we measure the
relative error
\[
E_{\nu_R} = \dfrac{|\nu_R -\nu_R^{\text{FE}}|}{\nu_R^{\text{FE}}}
\]
between the approximated transition rate $\nu_R$ and the transition
rate $\nu_R^{\text{FE}}$. We also compute the relative error
\[
E_{q} = \frac{\|q - \tilde{q}_{\text{FE}}\|_2}{\|\tilde{q}_{\text{FE}}\|_2}
\]
between $q$ and $q_{\text{FE}}$. We remark that the original $q_{\text{FE}}$ is defined on regular grid.
The above $\tilde{q}_{\text{FE}}$ is calculated using the interpolation
of $q_{\text{FE}}$ from regular grid to the input points and the standard $l_2$ norm is used here to measure difference between these two vectors. Our numerical results reported in Figure
\ref{fig:VrConvergence} indicate that the approximation error can be
controlled around $1\%$ for moderate size of points.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf__qComparison_LocalMesh_11372pt}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf__qComparison_LocalMesh_25711pt}
\end{minipage}\hfill\\
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf__qComparison_LocalMesh_45035pt}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf__qComparison_LocalMesh_69274pt}
\end{minipage}
\caption{Contour lines match, where black curves represent contours from our solutions with different number of points generated by SDE \eqref{eq:overdamp} and blue-dash curves represent contours from solutions based on regular grid. Left up: 11372 points. Right up: 25711 points. Left down: 45035 points. Right down: 69274 points.}
\label{fig:Contours}
\end{figure}
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurfLocalMesh_SDEPC_Vr_convergence}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurfLocalMesh_SDEPC_VrError_convergence}
\end{minipage}
\caption{Left: $\nu_R$ v.s number of points. Right: Relative error of $\nu_R$ and relative error of $q$ v.s. number of points. }
\label{fig:VrConvergence}
\end{figure}
As a direct application based on the committor function obtained from
the Fokker-Planck equation, we trace a deterministic reactive flow
$X(s)$ by solving the following ODE on the point clouds
\begin{equation}
\label{eqn:trajectorytracking}
\left\{\begin{array}{l}
\displaystyle \frac{dX(s)}{ds} = J_R(X(s)) = k_B T \rho(X(s)) \nabla q(X(s)),\vspace{0.2cm} \\
X(0)= p_0
\end{array}\right.
\end{equation}
The idea of solving the above ODE is to interpolate the point cloud locally using the moving least square method~\cite{Liang:CVPR2012,Liang2013solving}, then the solution curve can be extended based on the locally interpolated manifold.
\begin{wrapfigure}{r}{0.45\textwidth}
\vspace{-1cm}
\begin{center}
\includegraphics[width=.9\linewidth]{Reactiveflowtracing.png} \\
\end{center}
\vspace{-.7cm}
\caption{\small Reactive flow tracing. The current point $\bm{p}_c$, its KNN points and the new point $\bm{p}_{new}$ are marked as the red solid circle, the blue solid circles and the black solid star respectively, whose projections on the tangle plane at $\bm{p}_c$ are plotted as the corresponding hollow markers. The red line has direction $(-v_1,-v_2,0)$.}
\label{fig:ReactiveFlow}
\end{wrapfigure}
To clearly indicate our method of solving \eqref{eqn:trajectorytracking}, we assume that the given point cloud is sampled on a two dimensional manifold embedded in $\mathbb{R}^3$. We emphasize that the following idea can be straightforwardly extended to high dimension cases. Suppose a point $\bm{p}_c$ (current point on the reactive flow) has already been obtained, we intend to find the next point on the reactive flow. Without loss of generality, suppose $\bm{p}_1,\bm{p}_2, \cdots, \bm{p}_K \in\mathcal{P}$ are KNN of $\bm{p}_c$ in the point cloud $\mathcal{P}$. Using PCA, we can build a local coordinate system $\{\bm{e}_{\bm{p}_c}^1, \bm{e}_{\bm{p}_c}^2, \bm{e}_{\bm{p}_c}^3\}$ centered at $\bm{p}_c$ and the KNN of $\bm{p}_c$ has local coordinates $(x_i,y_i,z_i)$. We use moving least squares (MLS) to locally approximate the surface as $\Gamma=(x,y,z(x,y))$ and estimate $k_B T \rho(\bm{p}_c)\ \nabla q(\bm{p}_c)$ (more details can be found in \cite{Liang:CVPR2012,Liang2013solving}). We construct the Delaunay triangulation of the projections $\{\hat{\bm{p}}_c,\hat{\bm{p}}_1,\hat{\bm{p}}_2,\cdots,\hat{\bm{p}}_K\}$ and find the first ring $\mathcal{R}=\{T_c^1,\cdots, T_c^l\}$ of $p_c$, which is the same as we did in Section \ref{subsec:localmesh}. Suppose that $k_B T \rho(\bm{p}_c)\ \nabla q(\bm{p}_c)$ has a local coordinate $(v_1,v_2,v_3)$ in $\{\bm{e}_{\bm{p}_c}^1,\bm{e}_{\bm{p}_c}^2,\bm{e}_{\bm{p}_c}^3 \}$,
we find the intersection of line segment starting at $\bm{p}_c$ with the direction $(-v_1,-v_2,0)$ and the first ring. Notice that this computation is done within the tangent space of $\bm{p}_c$. Denote the intersection as $\hat{\bm{p}}_{new}=(x_0,y_0,0)$, we then project it back to the approximated surface to obtain the next point on the geodesic path $\bm{p}_{new}=(x_0,y_0,z(x_0,y_0))$. This process is illustrated in Figure \ref{fig:ReactiveFlow}. We refer~\cite{LaiLiangZhao:13} for more detailed discussion about solving the above equation on point clouds. Figure~\ref{fig:trajectory} plots the trajectory starting from the red star point $p_0$ in state $A$ to finally hit the region in state $B$, which clearly show that the reactive flow jump from state $A$ to state $B$.
\begin{figure}[h]
\centering
\includegraphics[width=.7\linewidth]{FKSurfLocalMesh_SDEPC_GradFlowPath.png}
\caption{Reactive trajectory tracing.}
\label{fig:trajectory}
\end{figure}
\subsection{Experiments in higher dimensions}
As an advantage of the proposed intrinsic method, the solver we designed can handle point clouds sampled from a low-dimensional manifold in a high dimension space. To illustrate the robustness of the proposed method, we next test the our solver for point clouds embedded in a high dimensional ambient space with an artificial Gaussian noise. In other words, we first simulate 2D point clouds as we conducted in the previous numerical experiments. After that, we embed the point cloud to $\mathbb{R}^{10}$ by setting the last eight coordinates to be zero. In addition, we also perturb this point cloud in $\mathbb{R}^{10}$ by adding Gaussian noise with variance $\sigma = \gamma d_{max}$. Here we choose $d$ to be the maximal number of the 50th smallest distance to each point. Namely, $d_{max} = \max_i\{d_i~|~ d_i = \mbox{the 50-th smallest distance to } \bm{p}_i\}$. In our experiments, we solve the Fokker-Planck equation based on the point cloud perturbed by Gaussian noise with different level $\gamma = 10\%, 20\%, 50\%, 100\%$, and also compute the reactive trajectory from the starting point. For better visualization, figure~\ref{fig:NoiseHD} shows 2D projection of the Gaussian noise perturbed point clouds color-coded with the resulting commitor function $q$. In addition, reactive trajectories are also plotted for different noise level with the same starting point. This figure clearly demonstrates the robustness of the proposed method.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf_LocalMesh_SDEPC_noise_10_GradFlowPath}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf_LocalMesh_SDEPC_noise_20_GradFlowPath}
\end{minipage}\hfill\\
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf_LocalMesh_SDEPC_noise_50_GradFlowPath}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\linewidth]{FKSurf_LocalMesh_SDEPC_noise_100_GradFlowPath}
\end{minipage}
\caption{The committor functions and reaction trajectories for point clouds embedded in $\mathbb{R}^{10}$ with noise levels $\gamma = 10\%, 20\%, 50\%, 100\%$ from top left to bottom right, respectively.}
\label{fig:NoiseHD}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this work, we develop a point cloud discretization for computing
committor functions of stochastic systems. Numerical examples on toy
model systems confirm that the method provides a promising tool to
analyze the stochastic system in the framework of the transition path
theory. In particular, the point cloud discretization extends the
applicability of the transition path theory beyond the ``tube
approximation''. As for future directions, an obvious next step is to
test the approach in thermally activated process in more complicated
and realistic examples arising from biophysics. In addition, our
method does not require that the point cloud samples exactly the
invariant measure. This provides advantages for considering point
clouds sampled locally rather than using a long trajectory and for
combining our method with advanced sampling strategies of the underlying
stochastic system. The numerical convergence analysis of the point
cloud discretization for Fokker-Planck operators is also an
interesting topic to pursue.
\section*{Acknowledgments}
This work was supported in part by the National Science Foundation
through the grants DMS-1522645 (R.L.) and DMS-1454939
(J.L.). Authors would like to thank Mauro Maggioni for helpful
discussions in various stages of this project.
|
1,314,259,995,855 | arxiv | \section{Introduction}
The unusual suppression of density fluctuations at large length scales is central
to the hyperuniformity concept, whose broad importance for condensed matter physics
and materials science was brought to the fore only about a decade ago
in a study that focused on fundamental theoretical aspects, including how it provides
a unified means to classify and categorize crystals, quasicrystals and special
disordered point configurations \cite{To03a}. Moreover, it was shown that the hyperuniform many-particle systems are poised at a unique type of critical point
in which (normalized) large-scale density fluctuations vanish
such that the direct correlation function of the Ornstein-Zernike relation is long-ranged \cite{To03a}.
This is to be contrasted
with a standard thermal critical point in which large-scale density fluctuations are
infinitely large and the total correlation function (not the direct correlation function) is long-ranged
\cite{Wi65,Ka66,Fi67,Wi74}.
Roughly speaking, a hyperuniform (or superhomoegeneous \cite{Ga02}) many-particle system in $d$-dimensional Euclidean space
$\mathbb{R}^d$ is one in which (normalized)
density fluctuations are completely suppressed at very large length scales,
implying that the structure factor $S({\bf k})$ tends to zero as the wavenumber $k\equiv |\bf k|$ tends to zero,
i.e.,
\begin{equation}
\lim_{|{\bf k}| \rightarrow 0} S({\bf k}) = 0.
\label{hyper}
\end{equation}
Equivalently, it is one in which the number variance of particles within a
spherical observation window of radius $R$, denoted by $\sigma^2_{_N}(R)$, grows more slowly than the window volume ($R^d$) in the large-$R$ limit. Typical disordered systems, such as liquids and structural glasses, have the standard volume scaling, that is, $\sigma^2_{_N}(R) \sim R^d$. By contrast, all perfect crystals and quasicrystals are hyperuniform with the surface-area
scaling $\sigma^2_{_N}(R)\sim R^{d-1}$. Surprisingly,
there are a special class of disordered particle configurations, such as the one shown in the right panel of
Fig. \ref{stealthy}, that have the same asymptotic behavior as crystals. There are scalings for the
number variance other than surface-area growth. When the structure factor
goes to zero in the limit $|{\bf k}| \rightarrow 0$ with the power-law form
\begin{equation}
S({\bf k}) \sim |{\bf k}|^\alpha,
\label{power}
\end{equation}
where $\alpha >0$, the number variance has the following large-$R$ asymptotic scaling \cite{To03a,Za09,Za11b}:
\begin{equation}
\sigma^2_{_N}(R) \sim\cases{
R^{d-1},
&$\alpha >1$, \cr
R^{d-1} \ln R,
& $\alpha = 1 \qquad (R \rightarrow \infty)$. \cr
R^{d-\alpha},
& $0 < \alpha < 1$
}
\label{sigma-N-asy}
\end{equation}
\begin{figure}
\begin{center}
{\includegraphics[ width=2.5in, keepaspectratio,clip=]{fig1a.eps}
\includegraphics[ width=2.5in, keepaspectratio,clip=]{fig1b.eps}}
\caption{A disordered non-hyperuniform many-particle configuration (left)
and a disordered hyperuniform many-particle configuration (right) \cite{To15}. The latter
is arrived at by very tiny collective displacements of the particles on the left.
These two examples show that it can be very difficult to detect hyperuniformity by eye,
and yet their large-scale structural properties are dramatically different.}
\label{stealthy}
\end{center}
\end{figure}
Disordered hyperuniform systems can be regarded to be exotic states
of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. In this sense, they can have a {\it hidden order} (see Fig. \ref{stealthy} for a vivid example) and
appear to be endowed with novel physical properties, as described below.
We knew of only a few examples of {\it disordered} hyperuniform systems
about a decade ago \cite{To03a,Leb83,Ga02,Do05d}. The importance of the hyperuniformity concept in the context of condensed matter started to become apparent
when it was shown that classical many-particle systems with certain long-ranged
pair potentials could counterintuitively freeze into disordered hyperuniform states
at absolute zero with singular scattering patterns, such as the one shown
in the right panel of Fig. \ref{pattern} \cite{Uc04b,Ba08}. This exotic situation runs counter to our everyday experience where
we expect liquids to freeze into crystal structures (like ice).
Mapping such configurations of particles to network structures, what was previously thought to
be impossible became possible, namely, the first disordered
dielectric networks to have large isotropic photonic band gaps comparable in
size to photonic crystals \cite{Fl09b}.
We now know that these exotic states of matter can exist as both {\it equilibrium} and {\it nonequilibrium} phases across
space dimensions,
including maximally random jammed particle packings \cite{Za11a,Ji11c,Ch14a}, jammed athermal granular media~\cite{Be11}, jammed thermal colloidal packings~\cite{Ku11,Dr15}, dynamical processes in ultracold atoms~\cite{Le14}, driven nonequilibrium systems \cite{Ja15,He15,We15,Tj15,Di15,Sc15}, avian photoreceptor patterns \cite{Ji14}, geometry of neuronal tracts \cite{Bur15},
certain quantum ground states (both fermionic and bosonic) \cite{To08c,Fe56}, classical disordered
(noncrystalline) ground states \cite{To15,Uc04b,Ba08,Zh15a,Zh15b}.
A variety of groups have recently fabricated disordered hyperuniform materials at the
micro- and nano-scales for various photonic applications \cite{Man13a,Ha13,Man13b},
surface-enhanced Raman spectroscopy \cite{De15}, the
realization of a terahertz quantum cascade laser \cite{Deg15} and self-assembly
of diblock copolymers \cite{Zi15b}. Moreover, a computational
study revealed that the electronic bandgap of amorphous silicon
widens as it tends toward a hyperuniform state \cite{He13}. Recent
X-ray scattering measurements indicate that amorphous-silicon samples
can be made to be nearly hyperuniform \cite{Xie13}.
Finally, we note that the hyperuniformity concept has suggested new correlation functions
from which one can extract relevant growing length scales as a function of temperature
as a liquid is supercooled below its glass transition temperature \cite{Ma13a},
a problem of intense interest in the glass physics community \cite{Lu07,Be07,Sc07,Ka09,Chand10,Hock12}.
\begin{figure}
\begin{center}
\includegraphics*[ width=2.in,clip=keepaspectratio]{fig2a.eps}\hspace{0.3in}
\includegraphics*[ width=2.in,clip=keepaspectratio]{fig2b.eps}
\caption{Left: Scattering pattern for a crystal. Right: Scattering pattern for a disordered ``stealthy" hyperuniform material
(defined in Sec. \ref{points}).
Notice that apart from forward scattering, there is a circle around the origin
in which there is no scattering, a highly exotic situation for an amorphous state of matter.}
\label{pattern}
\end{center}
\end{figure}
The hyperuniformity concept was generalized to the case of two-phase heterogeneous materials \cite{Za09},
which are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends,
granular media, cellular solids and colloids \cite{To02a,Sa03}. Here the phase volume fraction
fluctuates within a finite-sized spherical window of radius $R$ (see Fig. \ref{patterns}) and hence can be characterized by the
volume-fraction variance $\sigma_{_V}^2(R)$. For typical disordered two-phase media, the variance $\sigma_{_V}^2(R)$ for large $R$ goes to zero
like $R^{-d}$. However, for hyperuniform disordered two-phase media, $\sigma_{_V}^2(R)$ goes to zero asymptotically more
rapidly than the inverse of the window volume, i.e., faster than $R^{-d}$, which is equivalent to the following condition
on the spectral density (defined in Sec. \ref{back}):
\begin{eqnarray}
\lim_{|\mathbf{k}|\rightarrow 0}\tilde{\chi}_{_V}(\mathbf{k}) = 0.
\label{hyper-2}
\end{eqnarray}
As in the case of hyperuniform point configurations \cite{To03a,Za09,Za11b}, three different scaling regimes
when the spectral density goes to zero with the power-law form ${\tilde \chi}_{_V}({\bf k})\sim |{\bf k}|^\alpha$:
\begin{eqnarray}
\sigma^2_{_V}(R) \sim \left\{
\begin{array}{lr}
R^{-(d+1)}, \quad \alpha >1\\
R^{-(d+1)} \ln R, \quad \alpha = 1 \qquad (R \rightarrow \infty),\\
R^{-(d+\alpha)}, \quad 0 < \alpha < 1
\end{array}\right.
\label{sigma-V-asy}
\end{eqnarray}
where the exponent $\alpha$ is a positive constant.
\begin{figure}[bthp]
\centerline{\includegraphics[ width=3.in, keepaspectratio,clip=]{fig3.eps}}
\caption{A schematic indicating a circular observation window of radius $R$ that is centered at
position $\bf x_0$ in a disordered two-phase medium; one phase is depicted
as a green region and the other phase as a white region. The phase volume fractions within the window will fluctuate as the window position ${\bf x}_0$ is varied. }
\label{patterns}
\end{figure}
Much of our recent theoretical understanding of hyperuniform states of matter is based on many-particle
systems. The purpose of this paper is to delve more deeply into theoretical foundations
of disordered hyperuniform two-phase media by establishing new rigorous criteria that such systems must obey and exploring
their consequences.
In Sec. \ref{back}, we provide necessary mathematical definitions and background. In Sec. \ref{packing-1},
we derive some results concerning hyperuniformity of two-phase systems in $\mathbb{R}^d$ in which
one phase is a sphere packing and the spheres generally have different sizes.
We determine the necessary and sufficient conditions for a sphere packing
to be stealthy and hyperuniform, and prove that when each subpacking associated with each component is
hyperuniform, the entire packing is hyperuniform: a property called ``multihyperuniformity" \cite{Ji14}.
In Sec. \ref{two}, we consider hyperuniformity for general two-phase
media that lie outside the special class that are derived from sphere
packings in $d$-dimensional Euclidean space $\mathbb{R}^d$. Here we apply realizability conditions
for an autocovariance function and its associated spectral density of a two-phase
medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions.
We demonstrate that some functional forms
can immediately be eliminated from consideration, but also identify other forms that are allowable.
Specific examples and counterexamples are described, including remarks about well-known microstructural models (e.g., overlapping spheres and checkerboards) as
well as irregular phase-separation and Turing-type patterns. We also ascertain a family of autocovariance functions
that are realizable by disordered hyperuniform two-phase media in arbitrary space dimensions,
In Sec. \ref{con}, we close with some concluding remarks.
\section{Background}
\label{back}
\subsection{Point Configurations}
\label{points}
Consider statistically homogeneous point configurations in $d$-dimensional Euclidean
space $\mathbb{R}^d$. The standard pair correlation function $g_2({\bf r})$ is proportional
to the probability density associated with finding pairs of points separated
by the displacement vector $\bf r$, and is normalized in such a way
that it tends to unity in the limit $|{\bf r}| \rightarrow \infty$ in the absence
of long-range order. The {\it total correlation function} $h({\bf r})$ is defined as
\begin{equation}
h({\bf r})=g_2({\bf r})-1.
\label{total}
\end{equation}
The nonnegative structure factor $S(\bf k)$, which is proportional to the scattering intensity,
is trivially related to the Fourier transform of $h(\bf r)$:
\begin{equation}
S({\bf k})=1+\rho {\tilde h}({\bf k}).
\label{factor}
\end{equation}
\ref{FT} provides definitions of the $d$-dimensional Fourier transforms that we use in this paper.
The local number variance $\sigma_N^2(R)$ is determined entirely by pair correlations \cite{To03a}:\vspace{-0.1in}
\begin{eqnarray}
\sigma_N^2(R)&=&
\rho v_1(R)\Big[ 1+\rho\int_{\mathbb{R}^d} h({\bf r})
\alpha(r;R) d{\bf r}\Big] \nonumber \\
&=&\rho v_1(R)\Big[\frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} S({\bf k})
{\tilde \alpha}(k;R) d{\bf k}\Big] ,
\label{local}
\end{eqnarray}
where $v_1(R)= \pi^{d/2} R^d/\Gamma(1+d/2)$ is the $d$-dimensional volume of a spherical window,
$\alpha(r;R)$ is the intersection volume of two identical
hyperspheres of radius $R$ (scaled by the volume of a sphere) whose centers
are separated by a distance $r$, which is known analytically in any space dimension \cite{To03a,To06b},
and ${\tilde \alpha}(k;R)$ is its Fourier transform, which is nonnegative and explicitly given by
\begin{equation}
{\tilde \alpha}(k;R)= 2^d \pi^{d/2} \Gamma(1+d/2)\frac{[J_{d/2}(kR)]^2}{k^d}.
\label{alpha-k}
\end{equation}
Here $J_{\nu}(x)$ is the Bessel function of order $\nu$.
As mentioned earlier, the hyperuniformity property for point configurations is specified by the structure-factor
condition (\ref{hyper}).
{\it Stealthy} configurations are those in which
the structure factor is exactly zero for a subset of wave vectors, meaning that they completely suppress
single scattering of incident radiation for these wave vectors \cite{Ba08}.
Stealthy {\it hyperuniform} patterns \cite{To15,Uc04b,Ba08} are a subclass of hyperuniform
systems in which the structure factor is zero for a range
of wave vectors around the origin, i.e.,
\begin{equation}
S({\bf k})= 0 \qquad \mbox{for}\; 0 \le |{\bf k}| \le K,
\end{equation}
\label{stealth}
where $K$ is some positive number. An example of a stealthy disordered scattering pattern
is shown in the right panel of Fig. \ref{pattern}.
\subsection{Two-Phase Media}
A two-phase random medium is a domain of space $\mathcal{V} \subseteq \mathbb{R}^d$ of volume $V$
that is partitioned into two disjoint regions that make up $\mathcal{V}$:
a phase 1 region $\mathcal{V}_1$ of volume fraction $\phi_1$ and a phase 2 region $\mathcal{V}_2$ of volume fraction $\phi_2$ \cite{To02a}.
\subsubsection{Two-Point Statistics}
The phase indicator function ${\cal I}^{(i)}({\bf x})$ for a given realization is defined as
\begin{equation}
{\cal I}^{(i)}({\bf x}) = \left\{
{\begin{array}{*{20}c}
{1, \quad\quad {\bf x} \in {\cal V}_i,}\\
{0, \quad\quad {\bf x} \notin {\cal V}_i},
\end{array} }\right.
\label{phase-char}
\end{equation}
\noindent
The one-point correlation function $S_1^{(i)}({\bf x})= \langle {\cal I}^{(i)}({\bf x}) \rangle$
(where angular brackets indicate an ensemble average) is generally dependent on the position $\bf x$, but
is a constant for statistically homogeneous media, namely, the phase volume fraction, i.e.,
\begin{equation}
\phi_i = \langle {\cal I}^{(i)}({\bf x}) \rangle,
\end{equation}
such that $\phi_1+\phi_2=1$.
The two-point correlation function is defined as $S^{(i)}_2({\bf x}_1,{\bf x}_2) = \left\langle{{\cal I}^{(i)}({\bf x}_1){\cal I}^{(i)}({\bf x}_2)}\right\rangle$.
This function is the probability
of finding two points at positions ${\bf x}_1$ and ${\bf x}_2$ in phase $i$.
For statistically homogeneous media, the
two-point correlation function will
only depend on the relative displacement vector ${\bf r} \equiv {\bf x}_2-{\bf x}_1$
and hence $S_2^{(i)}({\bf x}_1,{\bf x}_2)=S_2^{(i)}({\bf r})$.
The autocovariance function $\chi_{_V}({\bf r})$ associated with the random variable ${\cal I}^{(i)}({\bf x})$
for phase 1 is equal to that for phase 2, i.e.,
\begin{equation}
\label{eq108}
\chi_{_V}({\bf r}) \equiv S^{(1)}_2({\bf r}) - {\phi_ 1}^2 = S^{(2)}_2({\bf r}) - {\phi_2}^2.
\end{equation}
At the extreme limits of its argument, $\chi_{_V}$ has the following asymptotic behavior
\begin{equation}
\chi_{_V}({\bf r}=0)=\phi_1\phi_2, \qquad \lim_{|{\bf r}| \rightarrow \infty} \chi_{_V}({\bf r})=0,
\label{limits}
\end{equation}
the latter limit applying when the medium possesses no long-range order. If the medium is statistically homogeneous and isotropic, then the autocovariance
function ${\chi}_{_V}({\bf r})$ depends only on the magnitude of its argument $r=|\bf r|$,
and hence is a radial function. In such instances, its slope at the origin is directly related
to the {\it specific surface} $s$ (interface area per unit volume); specifically, we have in any space
dimension $d$, the asymptotic form \cite{To02a},
\begin{equation}
\chi_{_V}({\bf r})= \phi_1\phi_2 - \beta(d) s \;|{\bf r}| + {\cal O}(|{\bf r}|^2),
\label{specific}
\end{equation}
where
\begin{equation}
\beta(d)= \frac{\Gamma(d/2)}{2\sqrt{\pi} \Gamma((d+1)/2)}.
\label{beta}
\end{equation}
The nonnegative spectral density ${\tilde \chi}_{_V}({\bf k})$, which can be obtained from scattering experiments \cite{De49,De57},
is the Fourier transform of $\chi_{_V}({\bf r})$, i.e.,
\begin{equation}
{\tilde \chi}_{_V}({\bf k}) = \int_{\mathbb{R}^d} \chi_{_V}({\bf r}) e^{-i{\bf k \cdot r}} {\rm d} {\bf r} \ge 0, \qquad \mbox{for all} \; {\bf k}.
\label{1}
\end{equation}
For isotropic media, the spectral density only depends
on $k=|{\bf k}|$ and, as a consequence of (\ref{specific}), its decay in the large-$k$ limit is controlled
by the exact following power-law form:
\begin{equation}
{\tilde \chi}_{_V}({\bf k}) \sim \frac{\gamma(d)\,s}{k^{d+1}}, \qquad k \rightarrow \infty,
\label{decay}
\end{equation}
where
\begin{equation}
\gamma(d)=2^d\,\pi^{(d-1)/2} \,\Gamma((d+1)/2)
\end{equation}
is a $d$-dimensional constant.
The higher-order correlation functions $S_3,S_4,\ldots$
\cite{To02a,To82b,To83a} will not be considered here, but we note that they arise in rigorous bounds
and exact expressions for effective transport \cite{To02a,To85f,Be85a,Be88b,Se89,Gi95a,Mi02,Ph03,To04a},
elastic \cite{To02a,Be88b,Gi95a,Mi02,To97b} and electromagnetic \cite{Re08a}
properties of two-phase media.
\subsubsection{Realizability Conditions on Autocovariance Functions of Two-Phase Media}
\label{realize}
A necessary and sufficient condition for the existence of a scalar autocovariance function of a stochastically
continuous homogeneous process is that its spectral function must be a nonnegative
bounded measure \cite{To02a,Pr81}. However, it is known that for
a two-phase system characterized by the phase indicator function (\ref{phase-char}), the nonnegativity
property of the spectral function [cf. (\ref{1})]
is a necessary but generally not sufficient
condition for the existence of an autocovariance function $\chi_{_V}({\bf r})$ corresponding
to a two-phase medium \cite{To02a,To06b,To99c,Ji07,Qu08,Lu15}.
The autocovariance function must also satisfy other conditions, which are most
conveniently stated in terms of the scaled autocovariance function $f({\bf r})$, which is defined by
\begin{equation}
f({\bf r})\equiv \frac{\chi_{_V}({\bf r})}{\phi_1\phi_2}.
\end{equation}
Comparing this to relation (\ref{limits}), we see that
\begin{equation}
f({\bf r}=0)=1, \qquad \lim_{|{\bf r}| \rightarrow \infty} f({\bf r})=0.
\label{limits-2}
\end{equation}
We let ${\tilde f}({\bf k})$ denote the Fourier transform of $f({\bf r})$, implying that
\begin{equation}
{\tilde f}({\bf k})=\frac{{\tilde \chi}_{_V}({\bf k})}{\phi_1\phi_2} \ge 0\qquad \mbox{for all} \; {\bf k}.
\end{equation}
Among other conditions, the scaled autocovariance function
must satisfy the following bounds for all $\bf r$:
\begin{equation}
\label{2}
-\min \left[\frac{\phi_1}{\phi_2},\frac{\phi_2}{\phi_1}\right] \le f({\bf r}) \le 1.
\end{equation}
Another necessary condition on $f({\bf r})$ in the case of statistically
homogeneous and isotropic media, i.e., when $f({\bf r})$ is dependent only
on the distance $r \equiv |{\bf r}|$, is that its derivative at $r = 0$ is strictly
negative for all $0<\phi_i<1$:
\begin{equation}
\label{4}
\frac{{\rm d}f}{{\rm d}r} \Bigg|_{r=0} < 0,
\end{equation}
which is consistent with the fact that slope at $r=0$ is proportional to
the negative of the specific surface $s$ [cf. (\ref{specific})]. Since
$f(|{\bf r}|)$ is an even function (i.e.,
$f({\bf r})=f(-{\bf r})$) that is linear in $|{\bf r}|$ at the origin, it is nonanalytic at the origin.
This is rather a strong restriction because it eliminates
any function that is analytic at the origin (which necessarily implies even
powers of $|{\bf r}|$); for example, it prohibits autocovariance functions
of a Gaussian form [e.g., $\exp(-(r/a)^2)$].
For statistically homogeneous media, another condition is the
so-called ``triangular inequality'':
\begin{equation}
f({\bf r}) \ge f({\bf s}) + f({\bf t}) - 1,
\label{5}
\end{equation}
where ${\bf r} = {\bf t}- {\bf s}$. If the autocovariance function
of a statistically homogeneous and isotropic medium is monotonically
decreasing, nonnegative and convex (i.e., ${\rm d}^2 f/ {\rm d}^2 r \ge 0$), then
it satisfies the triangular inequality (\ref{5}). The triangular inequality
implies several pointwise conditions on $f({\bf r})$. For
example, for statistically homogeneous and isotropic media, it
implies the condition (\ref{4})
and convexity at the origin:
\begin{equation}
\label{6}
\frac{{\rm d}^2f}{{\rm d}r^2} \Bigg|_{r=0} \ge 0.
\end{equation}
The triangular inequality is actually a
special case of the following more general condition:
\begin{equation}
\label{7}
\sum\limits_{i=1}^m\sum\limits_{j=1}^m \varepsilon_i\varepsilon_jf({\bf r}_i-{\bf r}_j)\ge 1,
\end{equation}
where $\varepsilon_i = \pm 1$ ($i = 1,...,m$ and $m$ is odd). Note that by
choosing $m = 3$; $\varepsilon_1\varepsilon_2 = 1$, $\varepsilon_1\varepsilon_3 =
\varepsilon_2\varepsilon_3 = -1$, Eq.~(\ref{5}) can be rediscovered. If $m = 3$;
$\varepsilon_1\varepsilon_2 = \varepsilon_1\varepsilon_3 = \varepsilon_2\varepsilon_3
= 1$ are chosen instead, another ``triangular inequality'' can be obtained, i.e.,
\begin{equation}
\label{8}
f({\bf r}) \ge -f({\bf s})- f({\bf t})-1,
\end{equation}
where ${\bf r} = {\bf t}- {\bf s}$. Equation~(\ref{8}) was first
derived by Quintanilla \cite{Qu08}.
Equation~(\ref{7}) is a much stronger necessary condition that implies that there
are other necessary conditions beyond those identified thus far. However,
Eq.~(\ref{7}) is difficult to check in practice, because it does not have a simple
spectral analog.
\subsubsection{Local Volume-Fraction Variance and Spectral Density}
It is known that the volume-fraction variance $\sigma_{_V}^2(R)$
within a $d$-dimensional spherical window of radius $R$ can be expressed in terms of the autocovariance function $\chi_{_V}({\bf r})$ \cite{Lu90b}:
\begin{eqnarray}
\sigma_{_V}^2(R) = \frac{1}{v_1(R)} \int_{\mathbb{R}^d} \chi_{_V}(\mathbf{r}) \alpha(r; R) d\mathbf{r},
\label{phi-var-1}
\end{eqnarray}
where
\begin{equation}
v_1(R) =\frac{\pi^{d/2} R^d}{\Gamma(1+d/2)}
\label{v1}
\end{equation}
is the volume of a $d$-dimensional sphere of radius $R$, and
$\alpha(r;R)$ is the scaled intersection volume, as defined in Eq. (\ref{local}). \footnote{Note that we have
changed the earlier notation for the volume-fraction variance used in Ref. \cite{Za09} from
$\sigma^2_{\tau}(R)$ to $\sigma^2_{_V}(R)$ to distinguish it from other variance functions
that have been introduced elsewhere \cite{To16a} to describe generalizations of the
hyperuniformity concept.}
The alternative Fourier representation of the volume-fraction variance
that is dual to the direct-space representation (\ref{phi-var-1}) is trivially obtained by applying Parseval's theorem to (\ref{phi-var-1})
under the assumption that the spectral density ${\tilde \chi}_{_V}({\bf k})$ [Fourier transform of $\chi_{_V}(\mathbf{r})$] exists:
\begin{eqnarray}
\sigma_{_V}^2(R) = \frac{1}{v_1(R)(2\pi)^d} \int_{\mathbb{R}^d} {\tilde \chi}_{_V}(\mathbf{k}) {\tilde \alpha}(k; R) d\mathbf{k}.
\label{phi-var-2}
\end{eqnarray}
Note that the hyperuniformity condition (\ref{hyper-2}) dictates that the direct-space autocovariance
function $\chi_{_V}({\bf r})$ exhibits both positive and negative correlations such that
its volume integral over all space is exactly zero, i.e.,
\begin{equation}
\int_{\mathbb{R}^d} \chi_{_V}({\bf r}) d{\bf r}=0,
\label{sum-1}
\end{equation}
which can be thought of as a sum rule. The generalization of the hyperuniformity concept to two-phase systems
has been fruitfully applied to characterize a variety of disordered sphere packings \cite{Za11a,Dr15,Za11c, Za11d,Ch15}.
\section{Hyperuniform Sphere Packings}
\label{packing-1}
Here we collect in one place various known results scattered throughout
the literature concerning the autocovariance function
$\chi_{_V}({\bf r})$ and spectral density ${\tilde \chi}_{_V}({\bf k})$ for two-phase
media in $\mathbb{R}^d$ in which one phase is a sphere packing in order to make some remarks about hyperuniformity and stealthiness.
A particle packing is a configuration of nonoverlapping (i.e., hard)
particles in $\mathbb{R}^d$.
For statistically homogeneous packings of congruent spheres of radius $a$ in $\mathbb{R}^d$ at number density $\rho$,
the two-point probability function $S_2({\bf r})$ of the particle (sphere) phase is known exactly in terms of the pair correlation function \cite{To02a,To85b},
yielding the autocovariance function as
\begin{eqnarray}
{\chi}_{_V}({\bf r}) &=& \rho\, m(r;a) \otimes m(r;a) +\rho^2 m(r;a) \otimes m(r;a) \otimes h({\bf r}) \nonumber \\
&=& \rho \,v_2^{int}(r;a) +\rho^2 v_2^{int}(r;a) \otimes h({\bf r}),
\label{S2-spheres}
\end{eqnarray}
where
\begin{equation}
m(r;a) =\Theta(a-r)=\Bigg\{{1, \quad r \le a,\atop{0, \quad r > a,}}
\label{indicator}
\end{equation}
is the sphere indicator function,
and $v_2^{int}(r;a)=v_1(a)\alpha(r;a)$ is the intersection volume of two spherical windows
of radius $a$ whose centers are separated by a distance $r$, where $v_1(a)$ and $\alpha(r;a)$
are defined as in (\ref{phi-var-1}), and $\otimes$ denotes the convolution of two
functions $F({\bf r})$ and $G({\bf r})$:
\begin{equation}
F({\bf r}) \otimes G({\bf r}) =\int_{\mathbb{R}^d} F({\bf x}) G({\bf r}-{\bf x}) d{\bf x}.
\end{equation}
Fourier transformation of (\ref{S2-spheres}) gives the corresponding spectral
density in terms of the structure factor \cite{Za09,To02a,To85b}:
\begin{eqnarray}
{\tilde \chi}_{_V}({\bf k})&=& \rho \,{\tilde m}^2(k;a)+ \rho^2 {\tilde m}^2(k;a) {\tilde h}({\bf k}) \nonumber \\
&=& \rho\, {\tilde m}^2(k;a) S({\bf k}) \nonumber \\
&=& \phi {\tilde \alpha}(k;a) S({\bf k})
\label{chi_V-S}
\end{eqnarray}
where
\begin{equation}
{\tilde \alpha}(k;a)= \frac{1}{v_1(a)} {\tilde m}^2(k;a)= \frac{1}{v_1(a)} \left(\frac{2\pi a}{k}\right)^{d} J_{d/2}^2(ka),
\end{equation}
\begin{equation}
\phi =\rho v_1(a),
\end{equation}
is the {\it packing fraction}, defined to be the fraction of space covered by the nonoverlapping spheres,
and $v_1(a)$ is the volume of a sphere of radius $a$ defined by (\ref{v1}).
We can bound the volume-fraction variance $\sigma^2_{_V}(R)$ from above in terms of
the number variance $\sigma^2_{_N}(R)$ for some fixed $R$. This is accomplished by
substituting the second line of (\ref{chi_V-S}) into the integral expression (\ref{phi-var-2}),
employing the number-variance relation (\ref{local}) and using the fact that ${\tilde m}(k;a)$ achieves its maximum value of $v_1(a)$
at $k=0$. This leads to the following upper bound:
\begin{equation}
\sigma^2_{_V}(R) \le \left(\frac{a}{R} \right)^{2d} \sigma^2_{_N}(R) \qquad \mbox{for all} \; R,
\label{bound}
\end{equation}
In Ref. \cite{Za09}, the same bound was given, but was derived for the large-$R$ asymptotic limit.
Bound (\ref{bound}) is in fact valid for any $R$.
We now show that the hyperuniformity of a sphere packing in terms
of volume-fraction fluctuations can only arise
if the underlying point configuration (determined by the sphere
centers) is itself hyperuniform. Since ${\tilde \alpha}(k;a)$ is analytic at $k=0$, we have
that in the limit $k \rightarrow 0$,
\begin{equation}
{\tilde \alpha}(k;a)= \frac{\pi^{d/2} R^d}{\Gamma(1+d/2)}
\left[ 1- \frac{(ka)^2}{d+2} + {\cal O}(k^4) \right],
\end{equation}
Because ${\tilde \alpha}(k;a)$ is a positive well-behaved function in the vicinity of the origin,
it immediately follows from expression (\ref{chi_V-S}) that if the underlying
point process is hyperuniform, as per the structure-factor condition (\ref{hyper}), then
the spectral density ${\tilde \chi}_{_V}({\bf k})$ inherits the hyperuniformity property (\ref{hyper-2})
only through the structure factor, not ${\tilde \alpha}(k;a)$.
The stealthiness property (no scattering at some finite subset of wave vectors) is a bit more subtle. We see from relation (\ref{chi_V-S})
that ${\tilde \chi}_{_V}({\bf k})$ is zero at those wave vectors where $S({\bf k})$ is zero as well as
at the zeros
of the function ${\tilde \alpha}(k;a)$, which is determined by the zeros of the Bessel function
$J_{d/2}(ka)$.
To illustrate the utility of these results, we now consider an example where the spectral density
as well as the volume-fraction variance can be calculated exactly for a sphere-packing
model as density increases up to a maximal value corresponding to hyperuniform state. Specifically, we compute these quantities
for sphere packings corresponding to a $g_2$-invariant process
introduced by Torquato and Stillinger \cite{To03a}. A $g_2$-invariant process
is one in which a chosen nonnegative form for
the pair correlation function $g_2$ remains
invariant over a nonvanishing density range while keeping
all other relevant macroscopic variables fixed \cite{To02b}. The upper
limiting ``terminal'' density is the point above which
the nonnegativity condition on the structure factor
[cf. (\ref{factor})] would be violated. Thus, whenever the structure
factor attains its minimum value of zero at ${\bf k}=0$ at the terminal
or critical density, the system, if realizable, is hyperuniform.
In Ref. \cite{To03a}, a variety of hyperuniform $g_2$-invariant processes
in which the number variance $\sigma^2_{_N}(R)$ grows like the window surface
area (i.e., $R^{d-1}$) were exactly studied in arbitrary space dimensions.
For our purposes, we employ the ``step-function" $g_2$-invariant process, namely, a
$g_2(r)$ that is defined by the unit step function $\Theta(r-D)$, where
$D=2a$ is the sphere diameter. The corresponding structure factor
in the density range $0 \le \rho \le \rho_c$ is given by
\begin{equation}
S(k)=1-\Gamma(1+d/2)
\left(\frac{2}{kD}\right)^{d/2}
\left(\frac{\rho}{\rho_c}\right) J_{d/2}(kD),
\label{invariant}
\end{equation}
where $\rho_c=[2^dv_1(D/2)]^{-1}$ is the terminal density
at which the packing is hyperuniform \cite{To03a}. For $\rho < \rho_c$, the packing
is not hyperuniform. Substitution of (\ref{invariant}) into relation (\ref{chi_V-S}) yields
the associated spectral density for this model in $d$ dimensions:
\begin{equation}
\hspace{-0.7in}{\tilde \chi}_{_V}({\bf k})=\rho \left(\frac{\pi D}{k}\right)^{d} J_{d/2}^2(kD/2)\Bigg[ 1-\Gamma(1+d/2)
\left(\frac{2}{kD}\right)^{d/2}
\left(\frac{\rho}{\rho_c}\right) J_{d/2}(kD)\Bigg].
\label{CHI}
\end{equation}
The top panel of Fig. \ref{specs} shows the spectral function ${\tilde \chi}_{_V}(k)$
for the aforementioned $g_2$-invariant packing process in three dimensions at two different
densities: one at a non-hyperuniform density $\rho=\rho_c/2$ and the other at the hyperuniform terminal density
$\rho_c$, where $\rho_c=\rho=3/(4\pi)$, as obtained from (\ref{CHI}). As noted above, the
degree of hyperuniformity reflected in ${\tilde \chi}_{_V}(k)$ is inherited from the properties
of the structure factor. Note that value of the spectral density at the origin for $\rho=\rho_c/2$
would monotonically decrease as the density increases up to the terminal density at which
point it is exactly zero.
The bottom panel of this figures depicts the associated local volume-fraction
variance $\sigma^2_{_V}(R)$ multiplied by $R^3$ for these two packings, as obtained
from relation (\ref{phi-var-2}). Observe that because $\sigma^2_{_V}(R)$ for the non-hyperuniform curve
decays like $R^{-3}$ for large $R$, the product $\sigma^2_{_V}(R) R^3$ asymptotes to a constant value.
By contrast, the product $\sigma^2_{_V}(R) R^3$ for $\rho=\rho_c$ decays like $R^{-1}$ for large $R$,
as it should for this three-dimensional hyperuniform
two-phase system.
\begin{figure}
\begin{center}
\includegraphics[ width=3.2in, keepaspectratio,clip=]{fig4a.eps}
\includegraphics[ width=3.2in, keepaspectratio,clip=]{fig4b.eps}
\caption{Top panel: A hyperuniform spectral density ${\tilde \chi}_{_V}(k)$
versus wavenumber $k$ for sphere packings corresponding to the step-function $g_2$-invariant process in three dimensions
at two different densities: one at a non-hyperuniform density $\rho=\rho_c/2$ and the other at the hyperuniform terminal density
$\rho_c$, where $\rho_c=\rho=3/(4\pi)$ \cite{To03a}. Bottom panel: The corresponding volume-fraction variance $\sigma^2_{_V}(R)$
versus window sphere radius $R$ for the non-hyperuniform and hyperuniform cases.
The diameter of a hard sphere is the unit distance that makes all relevant dimensional variables dimensionless.}
\label{specs}
\end{center}
\end{figure}
The aforementioned results for the pair statistics in both direct and Fourier spaces
for identical spheres have been generalized to the case of impenetrable spheres
with a continuous or discrete size distribution at overall number density $\rho$ \cite{To02a,Lu91}.
We collect these results in \ref{size} in order and prove there that when each subpacking associated with each component is
hyperuniform, the entire packing is hyperuniform,
what has been termed {\it multihyperuniformity} \cite{Ji14}.
It is important to note that examining the structure factor $S({\bf k})$ of the point
configurations derived from the centers of spheres with a polydispersity in size could lead one to incorrectly
conclude that the packings were not hyperuniform. It has been demonstrated \cite{Za11a,Za11c,Za11d}
that the proper means of investigating hyperuniformity in this case is through a packing's spectral
density ${\tilde \chi}_{_V}({\bf k})$. This has also been confirmed in experimental studies
of maximally random jammed packings of colloidal spherical particles with a size distribution \cite{Dr15}.
\section{Hyperuniformity Conditions for a General Class of Two-Phase Media}
\label{two}
Our interest here is to elucidate our understanding of hyperuniformity in general two-phase
media that lie outside the special class that are derived from sphere
packings, as per the previous section. This is accomplished by applying the
realizability conditions for an autocovariance function of a two-phase
medium that is also hyperuniform. We show that some functional forms
can immediately be eliminated from consideration and that other forms
are allowable. Specific examples and counterexamples are described.
We note that it trivially follows from (\ref{sum-1}) that the scaled autocovariance
$f({\bf r})$ obeys the sum rule
\begin{equation}
\int_{\mathbb{R}^d} f({\bf r}) d{\bf r}=0.
\label{sum-2}
\end{equation}
When $f({\bf r})$ is a function of the modulus $r=|\bf r|$, this sum rule
reduces to the following one-dimensional integral condition:
\begin{equation}
\int_{0}^{\infty} r^{d-1} f(r) dr=0.
\label{sum-3}
\end{equation}
\subsection{Monotonic Autocovariance Functions}
\begin{figure}
\begin{center}
{\includegraphics[ width=2.3in, keepaspectratio,clip=]{fig5a.eps}\hspace{0.2in}
\includegraphics[ width=2.2in, keepaspectratio,clip=]{fig5b.eps}}
\caption{Realizations of overlapping circular disks at $\phi_2=0.885$ (left) and
of a random checkerboard at $\phi_2=0.5$.}
\label{over}
\end{center}
\end{figure}
To begin, it is instructive to illustrate
the capacity of the sum rule (\ref{sum-2}) to eliminate an enormous set
of two-phase structures from the hyperuniform class. First, we make the
simple observation that any two-phase medium with a scaled autocovariance
function $f({\bf r})$ that {\it monotonically decreases} from its maximum
value of unity at the origin to its long-range value, such as the well-known overlapping-sphere
and symmetric-cell models \cite{To83b,To02a}, cannot be hyperuniform
at any positive volume fraction, since the sum rule (\ref{sum-2})
requires that the autocovariance function $\chi_{_V}({\bf r})$ possess both positive and negative values such that
its volume integral over all space be zero. The overlapping-sphere model in $\mathbb{R}^d$ consists
of the union of spheres that circumscribe the points generated
from a Poisson distribution. The symmetric-cell model is derived
from a tessellation of space into ``cells" with cells being randomly designated
as phase 1 and phase 2 with probability $\phi_1$ and $\phi_2$, respectively.
Figure \ref{over} shows two-dimensional realizations of each of these models. We note that while
these are idealized models, there are many real two-phase systems (e.g.,
sandstones and ceramic-metal composites) that have similar monotonic
autocovariance functions \cite{To02a,Co96} and hence can be immediately ruled out as hyperuniform structures.
Moreover, it is noteworthy that there is a huge class of two-phase systems that exhibit
strong positive and negative pair correlations at small pair distances
(e.g., equilibrium and nonequilibrium distributions of nonoverlapping
particles) that nonetheless are not hyperuniform by virtue of the fact that their autocovariance functions
violate the sum rule (\ref{sum-2}) \cite{To85a,To85b,To02a}.
\subsection{Remarks About Phase-Separation and Turing Patterns}
There are a variety of interesting spatial patterns that arise
in biological and chemical systems that result from a competition
between different pattern instabilities with selected wavelengths.
Such phenomena have been theoretically described by, for example,
Cahn-Hilliard equations \cite{Ca58} and Swift-Hohenberg equations \cite{Sw77}, whose solutions
can lead to irregular phase-separation and Turing patterns
with a well-defined characteristic wavelength. Thus, it is plausible that
binarized (two-phase) patterns
obtained by thresholding such scalar fields might be hyperuniform or even stealthy and hyperuniform. An example
of a Turing pattern with an irregular labyrinth-like structure \cite{Wiki} is shown
in Fig. \ref{turing1}. The distance between adjacent ``channels" of the labyrinth-type pattern is a physical
display of the wavelength (or wavenumber) that has been selected, which
is roughly equal to the mean chord length $\ell_C$ \cite{To02a}. Also, depicted in this
figure is the autocovariance function $\chi_{V}({\bf r})$ associated with the thresholded binarized (two-phase)
version of the Turing image. This function exhibits strong positive as well as negative correlations
at short distances.
The top panel of Figure \ref{turing2} shows the spectral density ${\tilde \chi}_{_V}({\bf k})$
obtained from the thresholded image. Note that it exhibits a well-defined
annulus in which scattering intensity is enhanced relative to that in the region outside
this annulus, which is radially centered at $k\, \ell_C \approx 7$. The bottom panel
of Fig. \ref{turing2} shows the angular-averaged spectral density from which we conclude
that the thresholded Turing pattern is neither stealthy nor hyperuniform.
\begin{figure}
\begin{center}
\includegraphics[ width=3.5in, keepaspectratio,clip=]{fig6a.eps}\vspace{0.2in}
\includegraphics[ width=3.5in, keepaspectratio,clip=]{fig6b.eps}
\caption{ Top left panel: Image of a Turing pattern with a labyrinth-like structure \cite{Wiki}.
Bottom panel: The autocovariance function $\chi_{_V}({\bf r})$ associated with the thresholded
version of the Turing image (with $\phi_1 \approx \phi_2 =1/2$), showing strong short-range order,
including anti-correlations (negative values).
The unit of distance is the mean chord length of the ``yellow" phase \cite{To02a}, which is roughly equal to the characteristic
width of the ``channels." }.
\label{turing1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[ width=3.5in, keepaspectratio,clip=]{fig7a.eps}\vspace{0.2in}
\includegraphics[ width=3.5in, keepaspectratio,clip=]{fig7b.eps}
\caption{ Top panel: Scattering pattern
as obtained from the spectral density ${\tilde \chi}_{_V}({\bf k})$ associated with the thresholded
version of the Turing image shown in Fig. \ref{turing1}. Bottom panel: Angular-averaged spectral
density, ${\tilde \chi}_{_V}(k)$, obtained from the 2D scattering pattern shown in the top panel. The unit of distance used in both spectral plots is the mean chord length
of the ``yellow" phase \cite{To02a}.}
\label{turing2}
\end{center}
\end{figure}
While this outcome does not mean that thresholded Turing-type patterns
can never be hyperuniform, it does lead to the following question: Are there disordered stealthy and hyperuniform two-phase systems
with spectral densities in which scattering is concentrated within some relatively thin annulus
defined by a small range of wavenumbers away from the origin?
To answer this question, we consider the following hypothetical, idealized scaled
spectral functions to see if they can fall within this possible stealthy and hyperuniform class:
\begin{equation}
{\tilde f}_A({\bf k})= c_A(d) \delta(k-K)
\label{A}
\end{equation}
and
\begin{equation}
{\tilde f}_B({\bf k}) =\Bigg\{{c_B(d), \quad K_1\le k \le K_2,\atop{\hspace{-0.1in} 0, \quad\mbox{otherwise},}}
\label{B}
\end{equation}
where $\delta(k)$ is a radial Dirac delta function is $d$-dimensional
Fourier space, $K_2 > K_1$,
\begin{equation}
c_A(d)= \frac{2^{d-1}\, \pi^{d/2}\, \Gamma(d/2)}{K^{d-1}}
\end{equation}
and
\begin{equation}
c_B(d)= (2 \pi)^{d/2} \,\Gamma(d/2+1) \frac{(K_1^d -K_2^d)}{K_1^d K_2^d}.
\end{equation}
Using the results of \ref{FT}, the corresponding
hypothetical scaled autocovariance function, which obeys the exact limiting conditions (\ref{limits-2}), are given by
\begin{equation}
f_A({\bf r})=
\left(\frac{2}{K r}\right)^{d/2-1} \, \Gamma(d/2) \,J_{d/2-1}(K r),
\label{AA}
\end{equation}
and
\begin{equation}
f_B({\bf r})=F(r;K_2)-F(r;K_1),
\label{BB}
\end{equation}
where
\begin{equation}
F(r;K)= c_B(d) \left(\frac{1}{2\pi K r}\right)^{d/2} K^d J_{d/2}(K r).
\end{equation}
Using the results of \ref{FT}, we can expand the aforementioned putative autocovariance functions
about $r=0$ to yield
\begin{equation}
f_A({\bf r})= 1- C_A(d) r^2 +{\cal O}(r^4)
\end{equation}
and
\begin{equation}
f_B({\bf r})= 1- C_B(d) r^2 +{\cal O}(r^4),
\end{equation}
where $C_A(d)$ and $C_B(d)$ are positive $d$-dimensional constants. It immediately follows
the autocovariance functions (\ref{AA}) and (\ref{BB}) cannot be realizable by two-phase media since
such systems would have a vanishing specific surface $s$, i.e., the small-$r$ expansion
of a valid autocovariance function must be nonanalytic at the origin such
that the slope is strictly negative [cf. (\ref{realize}]. This strongly suggests
that scattering patterns in which power is concentrated within some concentric
ring of the origin cannot be derived from a two-phase medium. Indeed, any function
that is analytic at the origin cannot be an autocovariance function that
corresponds to a two-phase medium.
\subsection{General Considerations}
A general formalism has been proposed that enables the functional form
of a realizable autocovariance function to be expressed by a set of chosen
realizable basis functions \cite{Ji07}. For our limited purposes
in this paper, we will make use of only some of these results.
It is known that convex combinations of a set of realizable scaled
autocovariance functions $f_1({\bf r}),f_2({\bf r}),\cdots,f_m({\bf r})$ is itself
a realizable autocovariance function $f({\bf r})$ \cite{Ji07}, i.e.,
\begin{equation}
f({\bf r})= \sum_{i=1}^m \alpha_i f_i({\bf r}),
\label{basis}
\end{equation}
where $0 \le \alpha_i \le 1$ ($i=1,2,\ldots,m$) such that $\sum_{i=1}^m \alpha_i=1$.
In what follows, we focus on basis functions that could correspond to statistically homogeneous and isotropic two-phase
media. A simple choice is the radial exponential function:
\begin{equation}
f_1({\bf r})=\exp(-r/a),
\label{a}
\end{equation}
which is itself a realizable autocovariance function for all positive and finite $a$ \cite{Ji07}.
For reasons discussed at the beginning of this section,
the monotonicity of $f_1$ precludes it from ever corresponding to a hyperuniform two-phase
system. It has been shown that a linear combination of $f_1$ and the basis function
\begin{equation}
f_2({\bf r}) = \exp(-r/b)\cos(q r + \theta)
\label{b}
\end{equation}
may be realizable for some parameters, but whether such a linear combination can ever correspond
to a disordered hyperuniform two-phase system has heretofore not been studied. Here $b$ can be thought of as a characteristic
correlation length and $q$ determines the characteristic wavelength associated with the oscillations.
Therefore, we explore here whether a disordered hyperuniform two-phase can have an autocovariance function of the form
\begin{equation}
f({\bf r})= \alpha_1 \exp(-r/a) +\alpha_2 \exp(-r/b)\cos(q r + \theta),
\label{case}
\end{equation}
where $\alpha_1+\alpha_2=1$. For simplicity, we examine two special cases. First, we consider the instance
in which $\alpha_1=0$, $\alpha_2=1$ and $\theta=0$, i.e.,
\begin{equation}
f({\bf r})= \exp(-r/b)\cos(q r).
\label{I}
\end{equation}
Notice that the specific surface corresponding to (\ref{I}) is given by $s=\beta(d)/(b\phi_1\phi_2)$,
where $\beta(d)$ is the $d$-dimensional constant specified in (\ref{beta}).
The hyperuniformity sum rule (\ref{sum-3}) provides conditions on the parameters $b$ and $q$,
which will depend on the dimension. For example, for $d=1$, we immediately conclude that
(\ref{I}) can never correspond to a hyperuniform medium because (\ref{sum-3}) cannot
be satisfied. On the other hand, for $d=2$ and
$d=3$, hyperuniformity requires that $(qb)^2=1$ and $3(qb)^2=1$, respectively, implying
that the autocovariance function for a hyperuniform system in a particular dimension generally does not correspond to a hyperuniform system in another dimension. Moreover,
these are only necessary conditions on the parameters $b$ and $q$ for the existence of a hyperuniform two-phase medium and one must still
check whether the known realizability conditions for two-phase media (described Sec. \ref{realize}) are satisfied. As it turns out,
all of these realizability conditions are satisfied, including the nonnegativity
of the spectral density [cf. \ref{1}]. Under these hyperuniform restrictions, the small-$k$ behavior
of the spectral density ${\tilde f}({\bf k})\equiv {\tilde \chi}_{_V}({\bf k})/(\phi_1\phi_2)$ associated
with (\ref{I}) for $d=2$ and $d=3$ are given
respectively by
\begin{equation}
\hspace{-1in}\frac{{\tilde f}({\bf k})}{b^2}= \frac{3}{4\pi} (kb)^2-\frac{35}{128 \pi} (kb)^6 +\frac{693}{8192 \pi} (kb)^{10} +{\cal O}(k^{14}), \quad [(qb)^2=1]
\label{exp-2}
\end{equation}
and
\begin{equation}
\hspace{-1in}\frac{{\tilde f}({\bf k})}{b^3}= \frac{27}{4\pi} (kb)^2-\frac{243}{32 \pi} (kb)^4 +\frac{3645}{512 \pi} (kb)^8- \frac{6561}{1024\pi} (kb)^{10} +{\cal O}(k^{14}), \quad [3(qb)^2=1],
\label{exp-3}
\end{equation}
where we have made use of the small-argument asymptotic expansion of the Bessel function $J_{\nu}(x)$ given in \ref{FT}.
Notice also that the hyperuniformity constraint prohibits multiple powers of four in the two-dimensional expansion (\ref{exp-2})
and multiple powers of six in the three-dimensional expansion (\ref{exp-3}). In the opposite asymptotic large-$k$ limit,
we respectively have for $d=2$ and $d=3$
\begin{equation}
\frac{{\tilde f}({\bf k})}{b^2} \sim \frac{2\pi}{(kb)^3}, \qquad k \rightarrow \infty
\end{equation}
and
\begin{equation}
\frac{{\tilde f}({\bf k})}{b^3} \sim \frac{8\pi}{(kb)^4}, \qquad k \rightarrow \infty.
\end{equation}
These results are consistent with the general asymptotic result (\ref{decay}).
\begin{figure}
\begin{center}
\includegraphics[ width=2.8in, keepaspectratio,clip=]{fig8a.eps}\vspace{0.2in}
\includegraphics[ width=2.8in, keepaspectratio,clip=]{fig8b.eps} \vspace{0.6in}
\includegraphics[ width=2.4in, keepaspectratio,clip=]{fig8c.eps} \vspace{-0.4in}
\caption{ Top panel: Hyperuniform autocovariance function $\chi_{_V}({\bf r})$ given by (\ref{I}) with $\phi_1=\phi_2=1/2$ in two dimensions where $b=1$, $q=1$ and
in three dimensions where $b=1$ and $q=1/\sqrt{3}$.
Middle panel: Corresponding hyperuniform spectral density ${\tilde \chi}_{_V}({\bf k})$ in two and three dimensions. Bottom panel: A realization of a disordered hyperuniform two-phase system that corresponds to (\ref{I}) in two dimensions with these parameters, as obtained using reconstruction
techniques \cite{To02a,Ye98a,Ji07}. The final ``energy" is smaller than $10^{-9}$, indicating that the targeted function is achieved
to very high accuracy.}
\label{hypo-1}
\end{center}
\end{figure}
Figure \ref{hypo-1} shows the autocovariance function and spectral density for a selected set of hyperuniform parameters
($b$, $q$) in both two and three dimensions.
Not surprisingly, the spectral densities associated with autocovariance functional form (\ref{I}) differ
across dimensions. To verify that there are indeed
disordered hyperuniform two-phase media that correspond to these autocovariance, well-established
``construction" (reconstruction) optimization techniques devised by Yeong and Torquato \cite{To02a,Ye98a,Ji07} are employed. Such procedures
utilize simulated-annealing methods that begin with a random initial guess for a digitized two-phase
system (hypercubic fundamental simulation box that is tessellated into finer hypercubic
cells) satisfying a prescribed volume fraction. The fictitious energy is a sum of squared
differences between a target correlation function (or corresponding spectral function)
and the correlation function (or corresponding spectral function) of the simulated structure
at any point along the evolution process to the global energy minimum (ground state)
as the fictitious temperature tends to zero. Here we target hyperuniform spectral densities
associated with (\ref{I}). The bottom panel of Fig. \ref{hypo-1}
shows a final construction in the case of two dimensions
that corresponds to (\ref{I}) with extremely high numerical
accuracy for a selected set of parameters. Apparently, the known realizability conditions on the function (\ref{I}) are sufficient to
ensure that it corresponds to a two-phase medium in two dimensions. It is noteworthy that it becomes easier
to ensure realizability of a hypothesized autocovariance function of specific functional form
as the space dimension increases for exactly the same reasons identified for point-configuration
realizability \cite{To06b}. Figure \ref{hypo-var} shows the volume-fraction variance $\sigma^2_{_V}(R)$
as a function of the window radius $R$, as obtained analytically from (\ref{phi-var-1}), in the case of three dimensions for a selected set
of parameters. We can analytically show that this specific three-dimensional volume-fraction variance has the following
asymptotic scaling:
\begin{equation}
\sigma^2_{_V}(R) \sim \frac{243}{256} \frac{1}{R^4} \qquad (R \rightarrow \infty).
\label{asy}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[ width=2.8in, keepaspectratio,clip=]{fig9.eps}
\caption{Volume-fraction variance $\sigma^2_V(R)$ (multiplied by $R^4$) as a function
of the window radius using (\ref{I}) with $\phi_1=\phi_2=1/2$ in three dimensions where $b=1$, $q=1/\sqrt{3}$.
The fact that this scaled variance asymptotes to a constant value ($243/256= 0.94921875\ldots$) for large $R$ implies
that this three-dimensional hyperuniform system has a variance that decays like $R^{-4}$, which is
consistent with the analytical formula (\ref{asy}).}
\label{hypo-var}
\end{center}
\end{figure}
As a second example, we consider the function (\ref{case}) in which $\alpha_1=1/2$, $\alpha_2=1/2$ and $\theta=0$, i.e.,
\begin{equation}
f({\bf r})= \frac{1}{2}\exp(-r/a) +\frac{1}{2}\exp(-r/b)\cos(q r),
\label{II}
\end{equation}
which provides greater degrees of freedom to achieve hyperuniformity relative to the form (\ref{I}).
Here $a$ and $b$ are taken to be positive and thus characteristic length scales.
The specific surface corresponding to (\ref{II}) is given by $s=(a+b)\beta(d)/(2ab\,\phi_1\phi_2)$,
where $\beta(d)$ is the $d$-dimensional constant specified in (\ref{beta}).
For $d=1$, we find that (\ref{II}) can never correspond to a hyperuniform medium because (\ref{sum-3}) cannot
be satisfied, which also was the case for the function (\ref{I}). This indicates that the hyperuniformity
condition is more difficult to achieve in one dimension than in higher dimensions.
For $d=2$ and $d=3$, the hyperuniformity sum rule requires that
\begin{equation}
a= \frac{b((qb)^2-1)^{1/2}}{(qb)^2+1}
\end{equation}
and
\begin{equation}
a= \frac{b(3(qb)^2-1)^{1/3}}{(qb)^2+1}\,
\end{equation}
respectively.
Even though these conditions ensure hyperuniformity in these dimensions, they are not sufficient
to guarantee the nonnegativity of the spectral density [cf. \ref{1}] for all $k$ because the leading term
in the series expansion of ${\tilde f}({\bf k})$ about $k=0$ is generally quadratic in $k$ but may have a negative coefficient.
For example, for $d=2$, to ensure positivity of the quadratic term, $qb$ must satisfy the following inequalities:
\begin{equation}
1 < qb \le \frac{1}{2}(\sqrt{6}+\sqrt{2}).
\label{67}
\end{equation}
If $qb$ is equal to the upper bound in (\ref{67}), the quadratic term vanishes identically such that the leading term
in the expansion of ${\tilde f}({\bf k})$ about $k=0$ is now quartic in $k$, which is to be contrasted
with the hyperuniform spectral density associated with (\ref{I}) that goes to zero quadratically in $k$
in the limit $k \rightarrow 0$.
Under the aforementioned restrictions on the parameters $a$, $b$ and $q$, all of the known realizability conditions described in Sec. \ref{realize} are satisfied.
Figure \ref{hypo-2} shows both the autocovariance function and spectral density for a set of hyperuniform parameters in two dimensions.
The bottom panel of Fig. \ref{hypo-2}
shows a realization obtained by the construction procedure \cite{To02a,Ye98a,Ji07} that corresponds to (\ref{II}) in two dimensions
with extremely high numerical accuracy for a selected set of parameters.
\begin{figure}[bthp]
\begin{center}
\includegraphics[ width=2.8in, keepaspectratio,clip=]{fig10a.eps}\vspace{0.2in}
\includegraphics[ width=2.8in, keepaspectratio,clip=]{fig10b.eps} \vspace{0.6in}
\includegraphics[ width=2.4in, keepaspectratio,clip=]{fig10c.eps} \vspace{-0.4in}
\caption{ Top panel: Hyperuniform autocovariance function $\chi_{_V}({\bf r})$ given by (\ref{II}) in two dimensions with $a=(1+\sqrt{3})^{3/2}/(\sqrt{2}(3+\sqrt{3}))=0.67479019\ldots$
$b=(\sqrt{6}+\sqrt{2})/2=1.9318516\ldots$, $q=1$ and $\phi_1=\phi_2=1/2$.
Middle panel: Corresponding hyperuniform spectral density ${\tilde \chi}_{_V}({\bf k})$ in two dimensions. Bottom panel: A realization of a disordered hyperuniform two-phase system that corresponds to (\ref{II}) in two dimensions with these parameters, as obtained using reconstruction
techniques \cite{To02a,Ye98a,Ji07}. The final ``energy" is smaller than $10^{-9}$, indicating that the targeted function is achieved
to very high accuracy.}
\label{hypo-2}
\end{center}
\end{figure}
\section{Conclusions and Discussion}
\label{con}
For two-phase media in $d$-dimensional Euclidean space $\mathbb{R}^d$ in which one
of the phases is a packing of spheres, we presented explicit exact expressions for the
autocovariance function and associated spectral density as well as upper bounds on the volume-fraction
variance in terms of the number variance for any window radius $R$. We used these results
to determine the necessary and sufficient conditions for a sphere packing
to be stealthy and hyperuniform as well as to establish rigorously the requirements
for a packing comprised of spheres of different sizes to be multihyperuniform.
We then considered hyperuniformity for general two-phase
media in $\mathbb{R}^d$ outside the class consisting of sphere packings. We applied realizability conditions for an autocovariance function and its
associated spectral density of a two-phase
medium and incorporated hyperuniformity as a constraint in order to derive new conditions. We showed that some functional forms
can immediately be eliminated from consideration and identified other forms that are allowable.
Contact was made with well-known two-phase microstructural models (e.g., overlapping spheres and checkerboards) as
well as irregular phase-separation and Turing-type patterns. We ascertained a family of autocovariance functions
that are realizable by disordered hyperuniform two-phase media in arbitrary space dimensions.
Realizations of disordered hyperuniform two-phase media with targeted spectral densities
were explicitly constructed. These studies
elucidate the nature of hyperuniformity in the context of heterogeneous materials.
In a subsequent work, we will explore more fully the explicit construction of disordered hyperuniform
two-phase media and characterize their higher-order statistics (beyond the two-point autocovariance function)
as well as host of other microstructural descriptors that are well-known in homogenization
theory \cite{To02a}. A particularly important goal of such studies will be to develop a deeper understanding
of the effect of space dimensionality on the microstructural descriptors, including the relevance
of the ``decorrelation principle" as the space dimension is increased \cite{To06b}.
A fruitful direction for future research would be the study and determination of the effective physical properties
of disordered hyperuniform two-phase systems. There is already evidence demonstrating that disordered hyperuniform cellular
network structures possess novel photonic properties \cite{Man13a,Ha13,Man13b}. However, the investigation
of the bulk properties of general disordered hyperuniform two-phase materials and their technological
relevance is essentially uncharted territory, and its exploration may offer great promise
for novel materials by design.
Very recently, the hyperuniformity concept was generalized to spin systems and shown
to exist as disordered spin ground states \cite{Ch16a}.
The implications and significance of the existence of such disordered spin systems warrants further study,
including whether their bulk physical properties, like their many-particle system counterparts,
are singularly remarkable, and can be experimentally realized. Finally, we note that the notion of
hyperuniformity has recently been generalized to include surface-area fluctuations
in two-phase media as well as fluctuations associated with random scalar and vector fields \cite{To16a}.
Now that we know what to look for, different varieties of disordered hyperuniform systems seem to
be arising in surprising places and contexts, and hence offer both intriguing fundamental and applied
research challenges and questions for the future.
|
1,314,259,995,856 | arxiv | \section{Introduction}
The use of non-Hermitian Hamiltonians in theoretical physics has a
long history. It extends from early attempts to construct
divergence-free relativistic quantum field theories
\cite{indefinite}, to more practical and successful applications in
nuclear and atomic physics \cite{nuclear-phys} and particularly
quantum optics \cite{optics,berry}. During the past ten years there
has been a renewed interest in the study of a special class of
non-Hermitian Hamiltonians that possess a real spectrum. The
best-known examples are the $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians
\cite{bender-prl} such as $p^2+ix^3$. These belong to the wider
class of pseudo-Hermitian Hamiltonians $H$ whose adjoint $H^\dagger$
is given by
\begin{equation}
H^\dagger=\eta\,H\eta^{-1},
\label{ph}
\end{equation}
for some Hermitian invertible operator $\eta$, \cite{p1,p2-p3}. What
makes pseudo-Hermitian Hamiltonians interesting is that they are
Hermitian with respect to a possibly indefinite inner
product\footnote{This means that $\langle \cdot, H\cdot\rangle_\eta=\langle
H\cdot,\cdot\rangle_\eta$.}, namely
\begin{equation}
\langle\cdot,\cdot\rangle_\eta:=\langle\cdot|\eta\cdot\rangle,
\label{inn-ph}
\end{equation}
where $\langle\cdot|\cdot\rangle$ is the inner product of the Hilbert space
in which all the relevant operators act. Most of the recent work on
the subject is concentrated on a particular class of
pseudo-Hermitian Hamiltonians, called quasi-Hermitian \cite{quasi},
that satisfy (\ref{ph}) for some positive-definite (metric) operator
$\eta$. In this case, (\ref{inn-ph}) is a positive-definite inner
product, and $H$ becomes Hermitian provided that we define the
physical Hilbert space of the system using the inner product
$\langle\cdot,\cdot\rangle_\eta$, \cite{jpa-2004b,review}. This allows for
formulating the pseudo-Hermitian representation of quantum mechanics
in which $\mathcal{P}\mathcal{T}$-symmetric as well as non-$\mathcal{P}\mathcal{T}$-symmetric
quasi-Hermitian Hamiltonians can be employed to describe unitary
quantum systems \cite{review}. The techniques developed in this
framework have so far found interesting applications in relativistic
quantum mechanics \cite{ann}, quantum cosmology \cite{qc}, quantum
field theory \cite{qft}, bound state scattering \cite{matzkin}, and
classical electrodynamics \cite{epl}. But these developments do not
undermine the importance of the requirement that the observables of
a unitary quantum system must be Hermitian with respect to the inner
product of the physical Hilbert space, \cite{review}.
Among the properties of Hermitian operators that make them
indispensable in quantum mechanics is their diagonalizability. For
Hermitian and more generally normal operators (those commuting with
their adjoint), diagonalizability is equivalent to the existence of
an orthonormal basis consisting of the eigenvectors of the operator.
This is more commonly referred to as \emph{completeness}. For a
non-normal (and hence non-Hermitian) operator $H$, diagonalizability
of $H$ means the existence of a basis $\mathcal{B}^\dagger$ consisting of
(scattering and bound state) eigenfunctions of the adjoint operator
$H^\dagger$ that is biorthonormal to some basis $\mathcal{B}$ consisting of
the eigenfunctions of $H$, i.e., $\mathcal{B}$ and $\mathcal{B}^\dagger$ form a
biorthonormal system of the Hilbert space, \cite{review}. For
brevity we shall call such a biorthonormal system a
\emph{biorthonormal eigensystem} for $H$.
Diagonalizability is a weaker condition than Hermiticity, but
diagonalizable operators with a real and discrete spectrum can be
related to Hermitian operators via similarity transformations
\cite{p2-p3}. This in turn implies that they are quasi-Hermitian
\cite{quasi}. The situation is more complicated when the spectrum is
continuous. A serious difficulty is the emergence of spectral
singularities that conflict with the diagonalizability of the
operator in question \cite{samsonov2}. The aim of this paper is to
elucidate the mechanism by which spectral singularities obstruct the
construction of a biorthonormal eigensystem for the operator. We
shall achieve this aim by obtaining a quantitative measure of lack
of a biorthonormal eigensystem and comparing the latter with the
mathematical condition for the presence of spectral singularities
that is based on the behavior of the Jost functions. In order to
clarify the meaning and consequences of spectral singularities we
shall offer a detailed investigation of the spectral properties of
the Hamiltonian operators of the form
\begin{equation}
H=-\frac{\hbar^2}{2m}\frac{d^2}{d{\rm x}^2}+
\zeta_+\delta({\rm x}-\alpha)+\zeta_-\delta({\rm x}+\alpha),
\label{H}
\end{equation}
where $\zeta_\pm$ are complex coupling constants, $\alpha$ is a real
parameter, and $\delta({\rm x})$ stands for the Dirac delta-function.
An alternative mechanism that can make a non-Hermitian operator
non-diagonalizable is the emergence of exceptional points. These
correspond to degeneracies where both eigenvalues and eigenvectors
coalesce \cite{ep}. Exceptional points have various physical
implications \cite{optics,berry,ep2}. But they must not be confused
with spectral singularities. Unlike exceptional points that can be
present for operators with a discrete spectrum (in particular matrix
Hamiltonians), spectral singularities are exclusive features of
certain operators having a continuous part in their spectrum. As we
will see in Sections 2 and 3, for an operator having a spectral
singularity we can still define two linearly independent
(scattering) eigenfunctions for each eigenvalue, nevertheless it is
impossible to construct a biorthonormal eigensystem for the
operator. To the best of our knowledge, physical meaning of spectral
singularities and their possible applications have not been
previously studied. This is the subject of Ref.~\cite{p89} where the
results of the present paper have been used to develop a physical
interpretation for spectral singularities.
Ref.~\cite{samsonov} uses the mathematical theory of spectral
singularities developed in \cite{naimark-review,ljance} to emphasize
their relevance to the recent attempts at using complex scattering
potentials to define unitary quantum systems. The results of
\cite{samsonov} are, however, confined to potentials defined on the
half-line ${\rm x}\geq 0$, where the Hamiltonian operator acts in the
Hilbert space of square-integrable functions $\psi: [0,\infty)\to\mathbb{C}$
satisfying the boundary condition $\psi(0)=0$. Furthermore, due to
the nature of the concrete potentials studied in \cite{samsonov}, it
has not been possible to construct bases of the corresponding
Hamiltonian and its adjoint and show by explicit calculation how the
presence of a spectral singularity obstructs the existence of a
biorthonormal eigensystem. This is quite essential, because for the
cases that the spectrum is real, the availability of a biorthonormal
eigensystem is a necessary condition for the existence of an
associated metric operator and the quasi-Hermiticity of the
Hamiltonian, \cite{review}.
The only thoroughly studied example of a complex scattering
potential that is defined on the whole real line and can lead to
spectral singularities is the single-delta-function potential with a
complex coupling \cite{jpa-2006b}. The Hamiltonian operator is given
by (\ref{H}) with $\alpha=\zeta_-=0$. It develops a spectral
singularity if and only if the coupling constant ($\zeta_+$) is
imaginary. In particular, for the cases that the real part of
$\zeta_+$ is positive, the bound states are also lacking and the
Hamiltonian is quasi-Hermitian. The complex single-delta-function
potentials provide a class of manifestly non-$\mathcal{P}\mathcal{T}$-symmetric
Hamiltonians with a continuous spectrum that happen to be
quasi-Hermitian. An advantage of considering complex
double-delta-function potentials is that their space of coupling
constants has a subspace, given by $\zeta_+=\zeta_-^*$, where the
Hamiltonian is $\mathcal{P}\mathcal{T}$-invariant. Therefore, these potentials
provide an opportunity to investigate the significance of
$\mathcal{P}\mathcal{T}$-symmetry \cite{bender-review}.
The spectral properties of the $\mathcal{P}\mathcal{T}$-symmetric double- and
multiple-delta-function potentials have been studied in
\cite{jones,ahmed,albaverio,demiralp}. The results are, however,
confined to the determination of the (scattering and bound-state)
spectrum of these potentials, and no attempt has been made to decide
if these potentials lead to spectral singularities.
In the present paper we will try to obtain a map of the space
$\mathbb{C}^2=\mathbb{R}^4$ of the coupling constants $\zeta_\pm$ that specifies the
regions corresponding to the existence of bound states and spectral
singularities. We will in particular investigate the intersection of
these regions with the two-dimensional $\mathcal{P}\mathcal{T}$-symmetric subspace:
$\zeta_-=\zeta_+^*$. The following is an outline of the results we
report in this paper. In Section~2, we obtain an explicit
quantitative measure of the existence of biorthonormal eigensystems
and compare the latter with the known condition of the presence of
spectral singularities. Here we also provide a useful
characterization of spectral singularities and bound states for
complex scattering potentials. Section~3 treats the spectral
properties of the double-delta-function potentials. It consists of
four subsections in which we obtain the regions in the space of
coupling constants where spectral singularities and bound states
exist, find their location in the spectrum of the operator, and
determine a lower bound on the size of certain regions in $\mathbb{C}^2$
where the operator (\ref{H}) is quasi-Hermitian. Section~4 presents
our concluding remarks.
\section{Spectral Singularities}
Consider a complex-valued potential $v:\mathbb{R}\to\mathbb{C}$ depending on a set
of complex coupling constants $z_1,z_2,\cdots,z_d$ such that $v^*$
is obtained by complex-conjugating the coupling constants in the
expression for $v$. Suppose that $v$ decays rapidly\footnote{For the
purpose of the present paper, we assume that as $|x|\to\infty$ we
have $|v(x)|\leq \exp[-\epsilon\sqrt{|x|}]$ for some
$\epsilon\in\mathbb{R}^+$. As far as the general properties related with
spectral singularities, all the results hold for the less rapidly
decaying potentials that satisfy $\int_{-\infty}^\infty dx\:
(1+|x|)|v(x)|<\infty$. See \cite{TB-1999,guseinov}.} as
$|x|\to\infty$ and that the spectrum of the corresponding
Hamiltonian operator,
\begin{equation}
H=-\frac{d^2}{dx^2}+v(x),~~~~~~~~~x\in\mathbb{R},
\label{second-order}
\end{equation}
is the set of nonnegative real numbers.\footnote{We shall consider
the more general case that the spectrum involves eigenvalues (with
square-integrable eigenfunctions) at the end of this section.} Let
$\psi^{\vec z}_{\mathfrak{a} k}(x)$ denote the (generalized) eigenfunctions
of $H$, i.e., linearly-independent bounded solutions of
\begin{equation}
H\psi^{\vec z}_{\mathfrak{a} k}(x)=k^2\psi^{\vec z}_{\mathfrak{a} k}(x),
\label{eg-va-zero}
\end{equation}
where $k\in\mathbb{R}^+$ and $\mathfrak{a}\in\{1,2\}$ are respectively the spectral
and degeneracy\footnote{The spectrum is necessarily
doubly-degenerate.} labels and $\vec z:=(z_1,z_2,\cdots,z_d)$.
By definition, $H$ is diagonalizable, if $\psi^{\vec z}_{\mathfrak{a} k}(x)$
together with a set of (generalized) eigenfunctions $\phi^{\vec
z}_{\mathfrak{a} k}(x)$ of $H^\dagger$ form a complete biorthonormal system
$\{\psi^{\vec z}_{\mathfrak{a} k},\phi^{\vec z}_{\mathfrak{a} k}\}$, i.e., they
satisfy
\begin{equation}
\langle\phi^{\vec z}_{\mathfrak{a} k}|\psi^{\vec z}_{\mathfrak{b}
q}\rangle=\delta_{\mathfrak{a}\mathfrak{b}}\:\delta(k-q),~~~~~~~
\sum_{\mathfrak{a}=1}^2\int_0^\infty dk\;|\psi^{\vec z}_{\mathfrak{a} k}\rangle\langle\phi^{\vec z}_{\mathfrak{a}
k}|=1,
\label{bi-ortho}
\end{equation}
where $\langle\cdot|\cdot\rangle$ is the usual $L^2$-inner product. The
biorthonormality relations (\ref{bi-ortho}) imply the spectral
representation of $H$,
\begin{equation}
H=\sum_{\mathfrak{a}=1}^2\int_0^\infty dk\;k^2|\psi^{\vec z}_{\mathfrak{a} k}\rangle\langle\phi^{\vec z}_{\mathfrak{a}
k}|,
\label{sp-rep}
\end{equation}
as well as the eigenfunction expansion:
\begin{equation}
f(x)=\sum_{\mathfrak{a}=1}^2\int_0^\infty dk\;f_{\mathfrak{a} k}\:\psi^{\vec z}_{\mathfrak{a}
k}(x),
\label{eg-fn-exp}
\end{equation}
where $f:\mathbb{R}\to\mathbb{C}$ is a test function and
\begin{equation}
f_{\mathfrak{a} k}:=\langle\phi^{\vec z}_{\mathfrak{a} k}|f\rangle.
\label{coeff}
\end{equation}
Because $H^\dagger=-\frac{d^2}{dx^2}+v(x)^*$, $\psi^{\vec z^*}_{\mathfrak{a}
k}$ are the eigenfunctions of $H^\dagger$. This in turn means that
$\phi^{\vec z}_{\mathfrak{a} k}$ must be a linear combination of $\psi^{\vec
z^*}_{\mathfrak{a} k}$, i.e., there are $J_{\mathfrak{a}\mathfrak{b}}\in\mathbb{C}$ satisfying
\begin{equation}
\phi^{\vec z}_{\mathfrak{a} k}=\sum_{\mathfrak{b}=1}^2 J_{\mathfrak{a}\mathfrak{b}}\,\psi^{\vec z^*}_{\mathfrak{b}
k}.
\label{phi-psi}
\end{equation}
In view of (\ref{bi-ortho}), there must exist $K_{\mathfrak{a}\mathfrak{b}}\in\mathbb{C}$ such
that
\begin{equation}
\langle \psi^{\vec z^*}_{\mathfrak{a} k}|\psi^{\vec z}_{\mathfrak{b}
q}\rangle=K_{\mathfrak{a}\mathfrak{b}}\,\delta(k-q).
\label{psi-psi-2}
\end{equation}
Furthermore, if we respectively denote by $I$, $J$ and $K$ the
$2\times 2$ identity matrix and the matrices with entries
$J_{\mathfrak{a}\mathfrak{b}}$ and $K_{\mathfrak{a}\mathfrak{b}}$, we find $J^*K=I$. In particular, $K$
must be an invertible matrix and $J_{\mathfrak{a}\mathfrak{b}}=K^{-1*}_{\mathfrak{a}\mathfrak{b}}$. We
can write this relation in the form
\begin{equation}
J_{\mathfrak{a}\mathfrak{b}}=\frac{\tilde K_{\mathfrak{a}\mathfrak{b}}^*}{\det(K)^*},
\label{JM}
\end{equation}
where $\tilde K$ is the transpose of the matrix of cofactors of $K$.
It satisfies
\[\left(\begin{array}{cc}
\langle \psi^{\vec z^*}_{2 k}|\psi^{\vec z}_{2 q}\rangle &
-\langle \psi^{\vec z^*}_{1 k}|\psi^{\vec z}_{2 q}\rangle\\
-\langle \psi^{\vec z^*}_{2 k}|\psi^{\vec z}_{1 q}\rangle &
\langle \psi^{\vec z^*}_{1 k}|\psi^{\vec z}_{1
q}\rangle\end{array}\right)=\tilde K\:\delta(k-q).\]
We can use (\ref{JM}) and (\ref{phi-psi}) to express (\ref{coeff})
as
\begin{equation}
f_{\mathfrak{a} k}=\frac{1}{\det(K)}\sum_{\mathfrak{b}=1}^2\tilde K_{\mathfrak{a}\mathfrak{b}}\langle
\psi^{\vec z^*}_{\mathfrak{b} k}|f\rangle.
\label{coeff-2}
\end{equation}
According to this equation if $\det(K)=0$, the eigenfunction
expansion (\ref{eg-fn-exp}) breaks down; the eigenfunctions
$\psi^{\vec z}_{\mathfrak{a} k}$ do not form a complete set; and $H$ is not
diagonalizable. We identify this situation with the presence of
spectral singularities:
\begin{equation}
\mbox{\begin{tabular}{|c|}
\hline
~\emph{Spectral singularities are points $k^2$ of the continuous
spectrum of $H$ where $\det(K)=0$}.\\
\hline
\end{tabular}}
\label{def-1}
\end{equation}
Because of (complex) analyticity property of the eigenfunctions
$\psi^{\vec z}_{\mathfrak{a} k}$, $\det(K)$ is an analytic function of $k$.
Therefore the (real) zeros of $\det(K)$ are isolated points forming
a countable subset of the real line. Moreover, because $v$ is a
bounded function decaying rapidly away from zero, the eigenfunctions
tend to plane waves as $k$ becomes large. This shows that $\det(K)$
does not have arbitrarily large zeros (for fixed $\vec z$). As a
result, the zeros of $\det(K)$ are not only isolated but actually
finite in number. In other words, depending on the values of the
coupling constants $z_1,z_2,\cdots,z_d$, $\det(K)$ may have no
(real) zeros in which case spectral singularities do not arise and
$H$ is diagonalizable, or a finite number of (non-vanishing real)
zeros $\kappa_1,\kappa_2,\cdots,\kappa_\mu$ in which case
$\kappa_1^2,\kappa_2^2,\cdots,\kappa_\mu^2$ are spectral
singularities and $H$ is not diagonalizable.
In general the space of coupling constants can be divided into two
regions, namely the \emph{singular region} where $H$ has spectral
singularities and the \emph{regular region} where it is
diagonalizable.
In mathematics literature a spectral singularity is defined as
follows:
\begin{itemize}
\item[] \textbf{Definition~1}: \emph{An element $E_\star$ of the
(continuous) spectrum of $H$ is called a spectral singularity
if the integral kernel of the resolvent operator:
$(H-E)^{-1}$, i.e., the Green's function $\langle x|(H-E)^{-1}|y\rangle$, is
an unbounded function in every small open neighborhood of $E_\star$,
but $E_\star$ is not an eigenvalue of $H$ with a square-integrable
eigenfunction }\cite{guseinov}.\footnote{Therefore spectral singularities are certain
poles of $\langle x|(H-E)^{-1}|y\rangle$.}
\end{itemize}
There is a rather general theory of spectral singularities for the
differential operators of the form (\ref{second-order}) where the
spectral singularities are characterized as the real zeros of
certain analytic functions
\cite{naimark-book2,naimark-review,kemp,schwartz,ljance,TB-1999}.
For the case that the operator acts in $L^2(\mathbb{R})$, this is the
Wronskian,
\begin{equation}
W[\psi_{k-},\psi_{k+}]:=
\psi_{k-}(x)\psi_{k+}'(x)-\psi_{k-}'(x)\psi_{k+}(x)=
\psi_{k-}(0)\psi_{k+}'(0)-\psi_{k-}'(0)\psi_{k+}(0),~~~~~~
\label{wronskian}
\end{equation}
of the Jost solutions $\psi_{k\pm}$ of the eigenvalue equation
$H\psi=k^2\psi$. These are defined in terms of their asymptotic
behavior
\begin{equation}
\psi_{k-}(x)\to e^{-ikx}~~{\rm
for}~~x\to-\infty,\quad\quad\quad\quad
\psi_{k+}(x)\to e^{ikx}~~{\rm for}~~x\to\infty.
\label{jost}
\end{equation}
More specifically, we have \cite{guseinov}
\begin{equation}
\mbox{\begin{tabular}{|c|}
\hline
~\emph{Spectral singularities are the real
(non-vanishing) zeros of $W[\psi_{k-},\psi_{k+}]$}.\\
\hline
\end{tabular}}
\label{def-2}
\end{equation}
This description of spectral singularities seems to differ from the
one given in (\ref{def-1}). In Section~3, we demonstrate the
equivalence of the two descriptions for the double-delta-function
potential by explicit calculations. The following calculation shows
how the description (\ref{def-1}) relates to Definition~1. Using
(\ref{bi-ortho}), (\ref{sp-rep}), (\ref{phi-psi}) and (\ref{JM}), we
have
\begin{eqnarray}
\langle x|(H-E)^{-1}|y\rangle&=&\sum_{\mathfrak{a}=1}^2\int_0^\infty dk\;
\frac{\psi^{\vec z}_{\mathfrak{a} k}(x)\phi^{\vec z}_{\mathfrak{a} k}(y)^*}{k^2-E}=
\sum_{\mathfrak{a},\mathfrak{b}=1}^2\int_0^\infty dk\;
\frac{J_{\mathfrak{a}\mathfrak{b}}^*\psi^{\vec z}_{\mathfrak{a} k}(x)
\psi^{\vec z^*}_{\mathfrak{b} k}(y)^*}{k^2-E}\nonumber\\
&=&
\sum_{\mathfrak{a},\mathfrak{b}=1}^2\int_0^\infty dk\;
\frac{\tilde K_{\mathfrak{a}\mathfrak{b}}\psi^{\vec z}_{\mathfrak{a} k}(x)
\psi^{\vec z^*}_{\mathfrak{b} k}(y)^*}{\det(K)(k^2-E)}.\nonumber
\end{eqnarray}
In the remainder of this section we provide a useful
characterization of the spectral singularities and bound states.
Because $|v(x)|$ decays rapidly as $|x|\to\infty$, solutions of
(\ref{eg-va-zero}) have the asymptotic behavior:
\begin{equation}
\psi_{k\mathfrak{a}}^{\vec z}(x)\to A_\pm e^{ikx}+B_\pm e^{-ikx}~~~~~
{\rm for}~~~~x\to\pm\infty,
\label{gen-asym}
\end{equation}
where $A_\pm$ and $B_\pm$ are possibly $k$-dependent complex
coefficients. If we denote the coefficients $A_\pm$ and $B_\pm$ for
the Jost solutions $\psi_{k\pm}$ by $A_\pm^{\pm}$ and $B_\pm^\pm$,
we can express (\ref{jost}) as
\begin{equation}
A_+^+=B_-^-=1,~~~~~A_-^-=B_+^+=0.
\label{jost2}
\end{equation}
Next, we let $M=({M}_{\mathfrak{a}\mathfrak{b}})$ be the possibly $k$-dependent
$2\times 2$ transfer matrix \cite{razavy} satisfying
\begin{equation}
\left(\begin{array}{c}
A_+\\
B_+\end{array}\right)=M\left(\begin{array}{c}
A_-\\
B_-\end{array}\right),
\label{S-matrix}
\end{equation}
and use this relation and Eqs.~(\ref{gen-asym}) and (\ref{jost2}) to
obtain
\begin{equation}
A_-^+=\frac{{M}_{22}}{\det M},~~~~B_-^+=-\frac{{M}_{21}}{\det M},
~~~~A_+^-={M}_{12},~~~~B_+^-={M}_{22}.
\label{S-mat-3}
\end{equation}
Inserting these equations in (\ref{gen-asym}), we find
\begin{eqnarray}
\psi_{k-}(x)&\to&
{M}_{12}\,e^{ikx}+{M}_{22}\,e^{-ikx}~~~~~
{\rm for}~~~~~x\to\infty,
\label{jost3-}\\
\psi_{k+}(x)&\to&
\frac{{M}_{22}\,e^{ikx}-{M}_{21}\,e^{-ikx}}{\det M}~~~~~
{\rm for}~~~~~x\to-\infty,
\label{jost3}
\end{eqnarray}
Because according to Abel's theorem \cite{boyce}, the Wronskian of
solutions of (\ref{eg-va-zero}) is independent of $x$, we can use
the asymptotic formulas for the Jost solutions to compute their
Wronskian. We use Eqs.~(\ref{jost}), (\ref{jost3-}) and
(\ref{jost3}), to perform this calculation first for $x\to\infty$
and then for $x\to-\infty$. This gives
\begin{eqnarray}
W[\psi_{k-},\psi_{k+}]&=& 2ik\,{M}_{22}(k),
\label{wronskian2}\\
W[\psi_{k-},\psi_{k+}]&=&\frac{2ik\,{M}_{22}(k)}{\det
M(k)},
\label{wronskian2-}
\end{eqnarray}
where we have made the $k$-dependence of ${M}_{22}$ and $M$
explicit. A direct consequence of (\ref{wronskian2}) and
(\ref{wronskian2-}) is
\begin{equation}
\det M(k)=1.
\label{det-S=}
\end{equation}
More importantly, we have the following characterization of spectral
singularities that follows from (\ref{def-2}) and
(\ref{wronskian2}).
\begin{equation}
\mbox{
\begin{tabular}{|c|}
\hline
~\emph{Spectral singularities are given by $k^2$ where $k$ is a
(non-vanishing) real zero of ${M}_{22}(k)$}.\\
\hline
\end{tabular}}
\label{desc-ss}
\end{equation}
Finally, consider the more general case that the Hamiltonian
operator (\ref{second-order}) has, in addition to a continuous
spectrum corresponding to $k\in\mathbb{R}^+$, a possibly complex discrete
spectrum. The latter corresponds to the square-integrable solutions
of (\ref{eg-va-zero}) that represent bound states. It is not
difficult to show that the spectral label corresponding to these
bound states are also zeros of ${M}_{22}(k)$, but unlike the zeros
associated with the spectral singularities these must have a
positive imaginary part. In other words, we have the following
characterization of the bound states.
\begin{equation}
\mbox{
\begin{tabular}{|c|}
\hline
~\emph{Bound state energies are
given by $k^2$ where $k$ is a zero of ${M}_{22}(k)$ with
$\Im(k)>0$}.\\
\hline
\end{tabular}}
\label{desc-bs}
\end{equation}
\section{The Double-Delta-Function Potential}
\subsection{Eigenfunctions for Scattering and Bound States}
Consider the time-independent Schr\"odinger equation
\begin{equation}
\left[-\frac{\hbar^2}{2m}\frac{d^2}{d{\rm x}^2}+
\zeta_+\delta({\rm x}-\alpha)+\zeta_-\delta({\rm x}+\alpha)\right]\psi=
{\rm E}\psi.
\label{eg-va}
\end{equation}
Let $\ell$ be an arbitrary length scale and introduce the
dimensionless quantities
\begin{equation}
z_\pm:=\frac{2m\ell\zeta_\pm}{\hbar^2},~~~x:=\frac{{\rm x}}{\ell},~~~
a:=\frac{\alpha}{\ell},
~~~E:=\frac{2m\ell^2{\rm E}}{\hbar^2}.
\end{equation}
Then (\ref{eg-va}) takes the form
\begin{equation}
-\psi''+[z_+\delta(x-a)+z_-\delta(x+a)]\psi=E\psi.
\label{eg-va-2}
\end{equation}
We can write solutions of (\ref{eg-va-2}) as
\begin{eqnarray}
\psi(x)&=&\left\{\begin{array}{ccc}
\psi^-(x)&{\rm for}& x<-a,\\
\psi^0(x)&{\rm for}& |x|\leq a,\\
\psi^+(x)&{\rm for}& x>a,\end{array}\right.
\label{wf1}\\
\psi^\nu(x)&=&A_\nu e^{ikx}+B_\nu
e^{-ikx},~~~~~~\nu\in\{-,0,+\},
\label{wf2}
\end{eqnarray}
where $k:=\sqrt{E}$ and without of loss of generality we require
that the principal argument of $k$ belongs to $[0,\pi)$.
To determine the matching conditions at $x=\pm a$, we demand that
$\psi$ be continuous, i.e.,
\begin{equation}
\psi^-(-a)=\psi^0(-a),~~~~~
\psi^0(a)=\psi^+(a).
\label{conti}
\end{equation}
Furthermore, we integrate both sides of (\ref{eg-va-2}) over the
intervals $[\mp a-\epsilon,\mp a+\epsilon]$ and take the limit
$\epsilon\to 0$ in the resulting formulas to find
\begin{equation}
{\psi^-}'(-a)-{\psi^0}'(-a)+z_-\psi^0(-a)=0,~~~~~
{\psi^0}'(a)-{\psi^+}'(a)+z_+\psi^0(a)=0.
\label{der}
\end{equation}
Introducing
\begin{equation}
w_\pm:=\frac{i z_\pm}{2k},
\label{w=}
\end{equation}
and inserting (\ref{wf1}) and (\ref{wf2}) in (\ref{conti}) and
(\ref{der}) yield the desired matching conditions that we can write
in the form
{\small\begin{equation}
\left(\begin{array}{c}
A_-\\
B_-\end{array}\right)=\left(\begin{array}{cc}
1+w_-& w_-e^{2iak}\\
-w_-e^{-2iak} & 1-w_-\end{array}\right)\left(\begin{array}{c}
A_0\\
B_0\end{array}\right),~~~
\left(\begin{array}{c}
A_+\\
B_+\end{array}\right)=\left(\begin{array}{cc}
1-w_+& -w_+e^{-2iak}\\
w_+e^{2iak} & 1+w_+\end{array}\right)\left(\begin{array}{c}
A_0\\
B_0\end{array}\right).
\label{match}
\end{equation}}
In light of these relations, the matrix $M$ satisfying
(\ref{S-matrix}) reads
\begin{equation}
M=
\left(\begin{array}{cc}
1-w_--w_++(1-e^{-4iak})w_-w_+ &
2iw_-w_+\sin(2ak)-w_-e^{2iak}-w_+e^{-2iak}\\
-2iw_-w_+\sin(2ak)+w_-e^{-2iak}+w_+e^{2iak} &
1+w_-+w_++(1-e^{4iak})w_-w_+\end{array}\right).
\label{S}
\end{equation}
It is easy to check that indeed $\det(M)=1$.
Next, we let $\vec z$ stand for $(z_-,z_+)$ and use $\psi^{\vec
z}_{1k}$ and $\psi_{2k}^{\vec z}$ to denote the eigenfunctions
obtained by setting $A_0=(2\pi)^{-1/2}, B_0=0$ and
$A_0=0,B_0=(2\pi)^{-1/2}$ respectively. Then,
\begin{eqnarray}
\psi_{1k}^{\vec z}(x)&=&(2\pi)^{-1/2}\times\left\{
\begin{array}{ccc}
(1+w_-)e^{ikx}-w_-e^{-ik(x+2a)}&{\rm for}& x<-a,\\
e^{ikx}&{\rm for}& |x|\leq a,\\
(1-w_+)e^{ikx}+w_+e^{-ik(x-2a)}&{\rm for}& x>a,\end{array}\right.
\label{psi1}\\ && \nonumber \\
\psi_{2k}^{\vec z}(x)&=&(2\pi)^{-1/2}\times\left\{
\begin{array}{ccc}
(1-w_-)e^{-ikx}+w_-e^{ik(x+2a)}&{\rm for}& x<-a,\\
e^{-ikx}&{\rm for}& |x|\leq a,\\
(1+w_+)e^{-ikx}-w_+e^{ik(x-2a)}&{\rm for}& x>a.\end{array}\right.
\label{psi2}
\end{eqnarray}}%
We can construct a set of eigenfunctions of $H^\dagger$ by taking
$z_\pm$ to $z_\pm^*$ or $w_\pm$ to $-w_\pm^*$ in these relations.
They are given by
\begin{eqnarray}
\psi_{1k}^{\vec z^*}(x)&=&(2\pi)^{-1/2}\times\left\{
\begin{array}{ccc}
(1-w_-^*)e^{ikx}+w_-^*e^{-ik(x+2a)}&{\rm for}& x<-a,\\
e^{ikx}&{\rm for}& |x|\leq a,\\
(1+w_+^*)e^{ikx}-w_+^*e^{-ik(x-2a)}&{\rm for}& x>a,\end{array}\right.
\label{phi1}\\ && \nonumber \\
\psi_{2k}^{\vec z^*}(x)&=&(2\pi)^{-1/2}\times\left\{
\begin{array}{ccc}
(1+w_-^*)e^{-ikx}-w_-^*e^{ik(x+2a)}&{\rm for}& x<-a,\\
e^{-ikx}&{\rm for}& |x|\leq a,\\
(1-w_+^*)e^{-ikx}+w_+^*e^{ik(x-2a)}&{\rm for}& x>a.\end{array}\right.
\label{phi2}
\end{eqnarray}}%
\subsection{Characterization of Spectral Singularities}
In this subsection we use (\ref{def-1}) to determine the spectral
singularities of the double-delta-function potential. This requires
computes $\langle\psi^{\vec z^*}_{\mathfrak{a},k}|\psi^{\vec z}_{\mathfrak{b},q}\rangle$ for
all $\mathfrak{a},\mathfrak{b}\in\{1,2\}$. Using (\ref{psi1}) and (\ref{psi2}) and the
identities
\[\int_\nu^\infty e^{i\mu x}dx=\pi\delta(\mu)+\frac{i
e^{i\mu\nu}}{\mu},~~~\int_{-\infty}^\nu e^{i\mu x}dx=\pi\delta(\mu)-\frac{i
e^{i\mu\nu}}{\mu},\]
we find
\begin{equation}
\left(\begin{array}{cc}
\langle\psi^{\vec z^*}_{1k}|\psi^{\vec z}_{1q}\rangle &
\langle\psi^{\vec z^*}_{1k}|\psi^{\vec z}_{2q}\rangle\\
\langle\psi^{\vec z^*}_{2k}|\psi^{\vec z}_{1q}\rangle &
\langle\psi^{\vec z^*}_{2k}|\psi^{\vec z}_{2q}\rangle\end{array}\right)=\delta(k-q)K,
\label{phi-psi-2}
\end{equation}
where $K=(K_{ij})$ is a $2\times 2$ matrix with entries
\begin{eqnarray}
K_{11}&=&K_{22}=1-w_-^2-w_+^2=1+\frac{z_-^2+z_+^2}{4k^2},
\label{e-K11}\\
K_{12}&=&w_-(1-w_-)e^{2iak}-w_+(1+w_+)e^{-2iak}\nonumber\\
&=&(4k^2)^{-1}\left[
iz_-(2k-iz_-)e^{2iak}-iz_+(2k+iz_+)e^{-2iak}\right],
\label{e-K12}\\
K_{21}&=&-w_-(1+w_-)e^{-2iak}+w_+(1-w_+)e^{2iak}\nonumber\\
&=&(4k^2)^{-1}\left[
-iz_-(2k+iz_-)e^{-2iak}+iz_+(2k-iz_+)e^{2iak}\right].
\label{e-K21}
\end{eqnarray}
In the $\mathcal{P}\mathcal{T}$-symmetric case, where $z_+=z_-^*=:z\neq 0$, $K$ is a
real matrix, and
\begin{eqnarray}
K_{11}&=&K_{22}=1+\frac{\Re(z^2)}{2k^2},
\label{k11-pt}\\
K_{12}&=&(2k^2)^{-1}\Im\left[z(2k+iz)e^{-2iak}\right],
\label{k12-pt}\\
K_{21}&=&
(2k^2)^{-1}\Im\left[z(-2k+iz)e^{2iak}\right].
\label{k21-pt}
\end{eqnarray}
The fact that $K$ is not generally diagonal shows that $\{\psi^{\vec
z}_{\mathfrak{a} k},\psi^{\vec z^*}_{\mathfrak{b} q}\}$ is not a biorthonormal system.
To construct the basis biorthonormal to $\{\psi^{\vec z}_{\mathfrak{a} k}\}$
we transform $\psi^{\vec z^*}_{\mathfrak{a} k}$ according to
$$\psi^{\vec z^*}_{\mathfrak{a} k}\to \phi^{\vec z}_{\mathfrak{a} k}:=\sum_{\mathfrak{b}=1}^2
J_{\mathfrak{a} \mathfrak{b}}\,\psi^{\vec z^*}_{\mathfrak{b} k},$$
and fix the coefficients $J_{\mathfrak{a} \mathfrak{b}}$ by demanding that
$\{\psi^{\vec z}_{\mathfrak{a} k},\phi_{\mathfrak{a} k}^{\vec z}\}$ be a biorthonormal
system. As we explained in Section~2, in terms of $K$ this condition
takes the form $\delta_{\mathfrak{a}\mathfrak{b}}=\sum_{\mathfrak{c}=1}^2
J_{\mathfrak{a}\mathfrak{c}}^*K_{\mathfrak{c}\mathfrak{b}}$. Therefore, a basis biorthonormal to
$\{\psi^{\vec z}_{\mathfrak{a} k}\}$ exists provided that the matrix $K$ is
invertible, and the matrix $J$ of coefficients $J_{\mathfrak{a}\mathfrak{b}}$ has the
form $J=K^{-1*}$.
The nonzero real values of $k$ for which $K$ is a singular matrix
give the spectral singularities of $H$. These are the non-vanishing
real zeros of $\det(K)$ that we can obtain using (\ref{e-K11}) --
(\ref{e-K21}):
\begin{equation}
\det(K)=1+\frac{z_-^2+z_+^2}{4k^2}+
\frac{z_-^2z_+^2}{8k^4}+\frac{z_-z_+}{2k^2}\left[
\left(1-\frac{z_-z_+}{4k^2}\right)\cos(4ak)+
\left(\frac{z_-+z_+}{2k}\right)\sin(4ak)\right]=0.
\label{det-K}
\end{equation}
If either $z_-=0$ and $z:=z_+$ or $z_+=0$ and $z:=z_-$, this
equation reduces to
\[ 1+\frac{z^2}{4k^2}=0.\]
Therefore, for pure imaginary $z$ there is a spectral singularity
located at $k=\pm iz/2=|z|/2$. This agrees with the results for the
single delta-function potential \cite{jpa-2006b}.
For the $\mathcal{P}\mathcal{T}$-symmetric case ($z_+=z_-^*=:z$), we have
\begin{equation}
\det(K)=1+\frac{\Re(z^2)}{2k^2}+
\frac{|z|^4}{8k^4}+\frac{|z|^2}{2k^2}\left[
\left(1-\frac{|z|^2}{4k^2}\right)\cos(4ak)+
\left(\frac{\Re(z)}{k}\right)\sin(4ak)\right]=0.
\label{det-K-PT}
\end{equation}
In particular if $z$ is purely imaginary, i.e., $z=i\sigma$ for some
$\sigma\in\mathbb{R}$,
\begin{equation}
\det(K)=\cos^2(2ak)+\Big(1-\frac{\sigma^2}{2k^2}\Big)^2\sin^2(2ak).
\label{det-K-PT-imaginary}
\end{equation}
Therefore, $\det(K)=0$ iff $\cos(2ak)=0$ and $k=|\sigma|/\sqrt 2$.
This implies that there is a spectral singularity for
$k=|\sigma|/\sqrt 2=|z|/\sqrt 2$ iff $\sigma$ takes one of the
following values
\begin{equation}
\sigma_n:=\frac{\pi(2n+1)}{2\sqrt 2~a},~~~~~n\in\mathbb{Z}.
\label{sigma-n}
\end{equation}
In summary, for the case that $z_+=-z_-=:z$ is purely imaginary, $H$
has a single spectral singularity given by
\begin{equation}
E_\star=\frac{\sigma_n^2}{2}=\left[\frac{(2n+1)\pi}{4a}\right]^2,
\label{ss-imaginary-pt}
\end{equation}
if $z=i\sigma_n$ for some $n\in\mathbb{Z}$. Otherwise it does not have any
spectral singularities.
Next, consider the general $\mathcal{P}\mathcal{T}$-symmetric case where
$z_+=z_-^*=:z$, and $z$ need not be purely imaginary. In this more
general case, we rewrite (\ref{det-K-PT}) in the form
\begin{equation}
\det(K)=|f(z,a,k)|^2,
\label{det-K-PT2}
\end{equation}
where
\begin{equation}
f(z,a,k):=\frac{|z|^2\sin(2ak)}{2k^2}+e^{-2iak}\left(\frac{\Re(z)}{k}-i\right).
\end{equation}
It is easy to compute
\begin{eqnarray}
\Re[f(z,a,k)]&=&\left(\frac{|z|^2}{2k^2}-1\right)\sin(2ak)+\frac{\Re(z)}{k}\,\cos(2ak),
\label{real}\\
\Im[f(z,a,k)]&=&-\left[\cos(2ak)+\frac{\Re(z)}{k}\,\sin(2ak)\right].
\label{imaginary}
\end{eqnarray}
In view of (\ref{det-K-PT2}), $\det(K)=0$ iff
$\Re[f(z,a,k)]=\Im[f(z,a,k)]=0$. Imposing $\Im[f(z,a,k)]=0$, we have
\begin{equation}
\cos(2ak)=-\frac{\Re(z)}{k}\,\sin(2ak),
\label{q1}
\end{equation}
which in particular implies $\sin(2ak)\neq 0$. Moreover,
$\cos(2ak)=0$ iff $\Re(z)=0$. In light of $\sin(2ak)\neq 0$ and
(\ref{q1}), $\Re[f(z,a,k)]=0$ gives
\begin{equation}
-\Re(z)^2+\Im(z)^2=2k^2.
\label{q2}
\end{equation}
This implies that \emph{if $|\Re(z)|\geq|\Im(z)|$, there is no
spectral singularity.}
Next, we solve (\ref{q2}) for $k$ to obtain
\begin{equation}
k=\sqrt{\frac{-\Re(z)^2+\Im(z)^2}{2}},
\label{q5}
\end{equation}
and express (\ref{q1}) as
\begin{equation}
\Re(z)=-k\cot(2ak).
\label{q3}
\end{equation}
Inserting (\ref{q5}) in (\ref{q3}) yields a necessary and sufficient
condition for the existence of a spectral singularity, namely
\begin{equation}
2\Re(z)\tan\left(a\sqrt{2[-\Re(z)^2+\Im(z)^2]}\right)+
\sqrt{2[-\Re(z)^2+\Im(z)^2]}=0.
\label{q6}
\end{equation}
Introducing the variables
\begin{equation}
r:=2a\Re(z),~~~~~~~~~s:=2a\Im(z),~~~~~~~~~
t:=a\sqrt{2[-\Re(z)^2+\Im(z)^2]},
\label{r-s}
\end{equation}
we can express (\ref{q6}) in the form
\begin{equation}
r=-t\cot t,
\label{r=}
\end{equation}
and establish
\begin{equation}
s=\pm t\sqrt{1+\csc^2t}.
\label{s=}
\end{equation}
Figure~\ref{fig1} shows a plot of the parametric curve defined by
(\ref{r=}) and (\ref{s=}). It consists of an infinite set of
disjoint open curves with asymptotes $s=\pm r$ in the $r$-$s$ plane.
The points on these curves correspond to the values of the coupling
constant $z$ for which a spectral singularity appears.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.75,clip]{fig1.eps}
\parbox{14cm}{\caption{Curves in the $r$-$s$ plane giving the location
of the spectral singularities for the general $\mathcal{P}\mathcal{T}$-symmetric
double-delta-function potential. The dashed lines are the asymptotes
$s=\pm r$. The intersection of the curves with the $s$-axis
corresponds to the spectral singularities given by
Eq.~(\ref{ss-imaginary-pt}). \label{fig1}}}\end{center}
\end{figure}
Next, consider the general not necessarily $\mathcal{P}\mathcal{T}$-symmetric case.
Generalizing our treatment of the $\mathcal{P}\mathcal{T}$-symmetric case, we use
(\ref{det-K}) to factorize $\det(K)$ as
\begin{equation}
\det(K)=f_-(z_-,z_+,a,k)f_+(z_-,z_+,a,k),
\label{det-K-factor}
\end{equation}
where
\begin{equation}
f_\pm(z_-,z_+,a,k):=\frac{u}{2k^2}\:\sin(2ak)+e^{\pm
2iak}\left(\frac{v}{k}\pm i\right),~~~~~
u:=z_-z_+,~~~~~v:=\frac{z_-+z_+}{2}.
\label{f-pm}
\end{equation}
Therefore $\det(K)=0$ if and only if at least one of
$f_\pm(z_-,z_+,a,k)$ vanishes.
Let us abbreviate $f_-(z_-,z_+,a,k)$ as $f(k)$, i.e., set
\begin{equation}
f(k):=\frac{u}{2k^2}\:\sin(2ak)+e^{-
2iak}\left(\frac{v}{k}- i\right).
\label{f=}
\end{equation}
Then it is easy to see that $f_+(z_-,z_+,a,k)=-f(-k)$. Therefore,
the positive zeros of $f_+(z_-,z_+,a,k)$ are identical with the
absolute-value of the negative zeros of $f_-(z_-,z_+,a,k)$. In other
words, the spectral singularities are given by positive and negative
real zeros of $f(k)$. Another interesting property of $f(k)$ is that
it satisfies
\begin{equation}
f(k)=-ie^{-2iak}\left[1+w_-+w_++w_-w_+(1-e^{4aik})\right]=
-ie^{-2iak}{M}_{22}(k),
\label{f=s22}
\end{equation}
where ${M}_{22}(k)$ is the the entry of the matrix $M$ of
(\ref{S}) with the row and column labels 2. According to
(\ref{f=s22}) the spectral singularities are the non-vanishing real
zeros of ${M}_{22}(k)$. This establishes the equivalence of
(\ref{def-1}) and (\ref{def-2}) for the double-delta-function
potentials.
In order to characterize the real zeros of $f(k)$, we set the real
and imaginary parts of the right-hand side of (\ref{f=}) equal to
zero. This gives
\begin{eqnarray}
\left(-1+\frac{\Re(u)}{2k^2}+\frac{\Im(v)}{k}\right)\sin(2ak)+
\left(\frac{\Re(v)}{k}\right)\cos(2ak)&=&0,
\label{re-f=zero}\\
\left(\frac{\Im(u)}{2k^2}-\frac{\Re(v)}{k}\right)\sin(2ak)+
\left(\frac{\Im(v)}{k}- 1\right)\cos(2ak)&=&0.
\label{im-f=zero}
\end{eqnarray}
Because $\sin(2ak)$ and $\cos(2ak)$ cannot vanish simultaneously,
these equations hold provided that the matrix of coefficients of
$\sin(2ak)$ and $\cos(2ak)$ is singular. Equating the determinant of
this matrix to zero and simplifying the resulting equation, we find
\begin{equation}
g(k):= k^3-2\Im(v)k^2+ \left(-\frac{\Re(u)}{2}+|v|^2\right)k+
\frac{1}{2}\left[\Re(u)\Im(v)-\Re(v)\Im(u)\right]=0.
\label{q21}
\end{equation}
Because $g$ is a real cubic polynomial, it always has at least one
real root $\kappa$. If $\kappa\neq 0$, $E_\star=\kappa^2$ is a
spectral singularity. Expressing $\kappa$ as a function of $u$ and
$v$ and inserting it in say (\ref{im-f=zero}) we find a sufficient
condition on the coupling constants $z_\pm$ for the existence of a
spectral singularity. Repeating this for all the roots of $g$ (for
the cases that (\ref{q21}) has other nonzero real solutions) we
obtain a complete characterization of the spectral singularities.
They lie on a three-dimensional surface ${\cal S}$ embedded in the
four-dimensional space $(\mathbb{C}^2)$ of the coupling constants
$(z_-,z_+)$. Figure~\ref{fig1} is a graphical demonstration of the
intersection of ${\cal S}$ with the plane $z_+=z_-^*$ that
represents the $\mathcal{P}\mathcal{T}$-symmetric region of $\mathbb{C}^2$. In the following
we examine some non-$\mathcal{P}\mathcal{T}$-symmetric regions of $\mathbb{C}^2$ and their
intersection with ${\cal S}$.
\begin{enumerate}
\item
Consider the plane $\Pi_1$ in $\mathbb{C}^2$ defined by
$z_+=-z_-^*=:z$ where $u=-|z|^2$ and $v=i\Im(z)$. Then
Eqs.~(\ref{re-f=zero}) and (\ref{im-f=zero}) take the form
\[\left[1+\left(\frac{\Re(z)}{k}\right)^2+\left(\frac{\Im(z)}
{k}- 1\right)^2\right]\sin(2ak)=0=
\left(\frac{\Im(z)}{k}- 1\right)\cos(2ak).\]
These are satisfied if and only if $\sin(2ak)=0$ and $k=|\Im(z)|$.
Therefore, we have a spectral singularity located at
$E_\star=\kappa^2=\Im(z)^2$ if and only if
\begin{equation}
\Im(z)=n\pi,~~~{\rm for~some}~~~n\in \mathbb{Z}-\{0\}.
\label{anti-PT}
\end{equation}
This shows that $\Pi_1$ intersects $\cal S$ along equidistant
lines parallel to the $\Re(z)$-axis in $\Pi_1$.
\item Consider the case that both $z_+$ and $z_-$ are
purely imaginary. This also defines a plane in $\mathbb{C}^2$ that we
denote by $\Pi_2$. In this case, we can express $z_\pm$ as
$$z_\pm=:\frac{iy_\pm}{a},$$
where $y_\pm$ are nonzero real numbers.
In terms of $y_\pm$, Eqs.~(\ref{re-f=zero}) and (\ref{im-f=zero})
take the form
\[\left(\frac{y_+ +y_-}{2ak}-1 \right) \cos(2ak)=0=
\left(\frac{y_+ +y_-}{2ak}-\frac{y_+ y_-}{2a^2k^2}-1 \right)
\sin(2ak).\]
There are two ways to satisfy these equations. Either
\begin{equation}
\frac{y_+ +y_-}{2ak}-1=\sin(2ak)=0,
\label{2-1}
\end{equation}
or
\begin{equation}
2a^2k^2-ak\left(y_+ +y_-\right)+y_+ y_-
=\cos(2ak)=0.
\label{2-2}
\end{equation}
We consider these two cases separately.
If (\ref{2-1}) holds, $E_\star=\kappa^2$ with
$\kappa:=(y_++y_-)/2a$ is a spectral singularity provided
that
\begin{equation}
y_++y_-=\frac{n\pi}{2},~~~~n\in\mathbb{Z}-\{0\}.
\label{pure-im-1}
\end{equation}
This defines a set of equidistance parallel lines
in $\Pi_2$ along which we have spectral singularities.
If (\ref{2-2}) holds, $k=\kappa_n$ where $\kappa_n:=(2n+1)\pi/(4a)$
for all $n\in \mathbb{Z}$, and
\begin{equation}
y_+=a\kappa_n\left( \frac{ y_- -2a\kappa_n}{y_- -a\kappa_n}\right)=
\frac{(2n+1)\pi}{2}\left[\frac{2y_--(2n+1)\pi}{
4y_--(2n+1)\pi}\right].
\label{pure-im-2}
\end{equation}
This equation gives the location of another set of spectral
singularities, namely $E_\star=\kappa_n^2$, in the plane $\Pi_2$.
Figure~\ref{fig2} shows the curves in $\Pi_2$ along which a
spectral singularity arises, i.e., $\Pi_2\cap\mathcal{S}$.
\begin{figure}[t]
\begin{center}
\vspace{-.5cm} \includegraphics[scale=.75,clip]{fig2.eps}
\parbox{14cm}{\caption{Curves in the $y_+$-$y_-$ plane
($\Pi_2$) along which one has a spectral singularity for
purely imaginary couplings. There are spectral singularities
along the $y_-$- and $y_+$-axes. The dashed line ($y_+=-y_-$)
represents the $\mathcal{P}\mathcal{T}$-symmetric double-delta function
potential with purely imaginary couplings. The intersection of
these lines with the full curves correspond to the spectral
singularities given by Eq.~(\ref{ss-imaginary-pt}).
\label{fig2}}}\end{center}
\end{figure}
\item Consider the plane $\Pi_3$ in $\mathbb{C}^2$ corresponding
to $z_+=-z_-=:z$. Then $v=0$ and $u=-z^2$. In particular,
$\Im(u)=-2\Re(z)\Im(z)$. We can confine our attention to the
subcase: $\Im(u)\neq 0$, because for $\Im(u)=0$ either $\Re(z)=0$,
in which case $z_\pm$ are purely imaginary and the results of
case~2 apply, or $\Im(z)=0$, in which case the potential is real
and there are no spectral singularities.
In view of $v=0$ and (\ref{q21}), $g(k)=k(k^2-\Re(u)/2)$.
Therefore, $k$ does not have a real zero and there is no
spectral singularities, if $\Re(u)\leq 0$. For $\Re(u)>0$, there
is a spectral singularity at $E_\star=\kappa_\pm^2=\Re(u)/2$,
where
\begin{equation}
\kappa_\pm:=\pm \sqrt{\frac{\Re(u)}{2}}.
\label{p3}
\end{equation}
Inserting this equation in (\ref{im-f=zero}) gives
\begin{equation}
\Im(u)=\pm\,\Re(u)\cot\left(a\sqrt{2\Re(u)}\right).
\label{5}
\end{equation}
Introducing the parameter $t:=a\sqrt{2\Re(u)}$,
we can use (\ref{5}) to obtain the following parametric
equations for the $r:=a\Re(z)$ and $s:=a\Im(z)$ values
that correspond to the spectral singularities.
\begin{equation}
|r(t)|=\frac{t}{2}\sqrt{|\csc t|-1},~~~~~
|s(t)|=\frac{t\cos t}{\sqrt{|\sin t|-\sin^2t}}.
\label{r-s=}
\end{equation}
Figure~\ref{fig3} shows the graph of the parametric curves defined by
(\ref{r-s=}). They form the intersection of the
plane $\Pi_3$ with the singular region $\mathcal{S}$ of $\mathbb{C}^2$.
\begin{figure}[t]
\vspace{-.5cm}
\begin{center}\includegraphics[scale=.75,clip]{fig3.eps}
\parbox{14cm}{\caption{Curves in the $r$-$s$ plane along which
spectral singularities occur for the coupling constants with
opposite sign. The origin $(s=r=0)$ does not actually lie on these
curves. The intersection of the curves with the $s$-axis corresponds
to the spectral singularities given by Eq.~(\ref{ss-imaginary-pt}).
The dashed lines are the lines $s=\pm r$.\label{fig3}}}\end{center}
\end{figure}
\item Consider the case that $z_\pm=(1+is_\pm)/a$ with
$s_\pm\in\mathbb{R}$ arbitrary. This corresponds to another plane in
$\mathbb{C}^2$ that we denote by $\Pi_4$. Introducing
\begin{equation}
s:=\frac{s_-+s_+}{2},~~~~~~t:=\frac{1+s_-s_+}{2},
\label{case4-1}
\end{equation}
we have
\begin{equation}
v=\frac{1+is}{a},~~~~~u=\frac{1-t+is}{a^2}.
\label{case4-2}
\end{equation}
Inserting these in (\ref{q21}) yields
\begin{equation}
a^3g(k)=(ak-s)(a^2k^2-ask+t)=0.
\label{case4-3}
\end{equation}
Therefore, we need to consider the following two possibilities.
\begin{itemize}
\item[]4.a) $k=s/a=(s_-+s_+)/(2a)$. In this case (\ref{im-f=zero}) is
satisfied automatically while (\ref{re-f=zero}) yields
\begin{equation}
t=s\cot(2s)+1.
\label{case4-4}
\end{equation}
We can use (\ref{case4-1}) and (\ref{case4-4})
to express $s_\pm$ in terms of $s$. This gives
\begin{eqnarray}
s_-&=&s\mp\sqrt{s^2+1-2(s\cot(2s)+1)},
\label{case4-4m}\\
s_+&=&2s-s_-=s\pm\sqrt{s^2+1-2(s\cot(2s)+1)}.
\label{case4-4p}
\end{eqnarray}
\item[]4.b) $k\neq s/a$. Then according to (\ref{case4-3}),
\begin{equation}
t=ask-a^2k^2.
\label{case4-5}
\end{equation}
Furthermore, both (\ref{re-f=zero}) and (\ref{im-f=zero}) become
\begin{equation}
\tan(2ak)+ak=0
\label{case4-6}
\end{equation}
This equation has a countably infinite set of real solutions
$\kappa_n$ that can be easily obtained numerically. Substituting
$\kappa_n$ for $k$ in (\ref{case4-5}) and using (\ref{case4-1}),
we find
\begin{eqnarray}
s_-&=&s\mp\sqrt{s^2+1-2(as\kappa_n-a^2\kappa_n^2)},
\label{case4-7b}\\
s_+&=&2s-s_-=s\pm\sqrt{s^2+1-2(as\kappa_n-a^2\kappa_n^2)}.
\label{case4-7a}
\end{eqnarray}
\end{itemize}
\begin{figure}[t]
\vspace{-.5cm}
\begin{center}\includegraphics[scale=.75,clip]{fig4.eps}
\parbox{14cm}{\caption{Curves in the $s_-$-$s_+$ plane along which
the spectral singularities occur for the coupling constants of the
form $z_\pm=1+is_\pm$. The solid (red) and dashed (blue) curves
correspond to the spectral singularities with $k=(s_++s_-)/2$ (Case
4.a) and $k\neq(s_++s_-)/2$ (Case 4.b), respectively. Also shown (in
green) is the unit disc: $s_-^2+s_+^2\leq 1$, where there are no
spectral singularities. \label{fig4-new}}}\end{center}
\end{figure}
Figure~\ref{fig4-new} shows the parametric plot of the curves in the
$s_-$-$s_+$ plane corresponding to the spectral singularities for
both cases 4.a and 4.b with $a=1$. As seen from this figure, there
are no spectral singularity in the unit disc defined by
$s_-^2+s_+^2\leq 1$.
\end{enumerate}
\subsection{Location of the Spectral Singularities and the Bound States}
As we noted in Section~2, the spectral singularities are given by
the real zeros of ${M}_{22}(k)$ while the bound states correspond
to zeroes of ${M}_{22}(k)$ with positive imaginary part. For the
double-delta-function potential, we can write ${M}_{22}(k)=0$ in
the following more compact form.
\begin{equation}
(\mathfrak{K}-\mathfrak{z}_-)(\mathfrak{K}-\mathfrak{z}_+)=\mathfrak{z}_-\mathfrak{z}_+\:e^{2\mathfrak{K}},
\label{e1}
\end{equation}
where we have used (\ref{S}) and introduced
\[\mathfrak{z}_\pm:=a z_\pm=\frac{2m\alpha\zeta_\pm}{\hbar^2},
~~~~~~\mathfrak{K}:=2iak.\]
In particular, the spectral singularities are give by
\begin{equation}
E_\star:=-\frac{\mathfrak{K}^2}{4a^2}
\label{E=ss-bs}
\end{equation}
where $\mathfrak{K}$ is a nonzero solution of (\ref{e1}) lying on the
imaginary axis in the complex $\mathfrak{K}$-plane, i.e.,
\[\mathfrak{K}\in\ell:=\{w\in\mathbb{C}~|~\Re(w)=0\neq w~\},\]
whereas the bound state ``energies'' are given by (\ref{E=ss-bs})
for solutions $\mathfrak{K}$ of (\ref{e1}) lying to the left of this axis,
i.e.,
\[\mathfrak{K}\in\Pi_-:=\{w\in\mathbb{C}~|~\Re(w)< 0~\}.\]
For both spectral singularities and bound states, we have
$\Re(\mathfrak{K})\leq 0$ which implies $|e^{2\mathfrak{K}}|\leq 1$. Taking the
modulus of both sides of (\ref{e1}), we find
\begin{equation}
|\mathfrak{K}-\mathfrak{z}_-||\mathfrak{K}-\mathfrak{z}_+|\leq |\mathfrak{z}_+||\mathfrak{z}_-|.
\label{e2}
\end{equation}
This is violated for any $\mathfrak{K}$ fulfilling
\begin{equation}
|\mathfrak{K}-\mathfrak{z}_-|>|\mathfrak{z}_-|~~~{\rm and}~~~|\mathfrak{K}-\mathfrak{z}_+|>|\mathfrak{z}_+|.
\label{e3}
\end{equation}
Therefore, the solutions of (\ref{e1}) with $\Re(\mathfrak{K})\leq 0$ must
belong to the union of the discs
\[D_{\pm}:=\big\{\mathfrak{K}\in\mathbb{C}~\big|~\big|\mathfrak{K}-\mathfrak{z}_\pm|
\leq |\mathfrak{z}_\pm|~\big\}.\]
This provides an upper bound on the size of the region in the
complex $\mathfrak{K}$-plane where bound state energies and spectral
singularities are located, namely
\[ \mathcal{R}_{\vec\mathfrak{z}}:=(\Pi_-\cup\ell)\cap (D_+\cup D_-).\]
Here we have used the index $\vec\mathfrak{z}:=(\mathfrak{z}_-,\mathfrak{z}_+)$ to emphasize
the $\mathfrak{z}_\pm$-dependence of $\mathcal{R}_{\vec\mathfrak{z}}$. Figure~\ref{fig4}
illustrates the discs $D_\pm$ and the region $\mathcal{R}_{\vec\mathfrak{z}}$ for a
generic choice of $\mathfrak{z}_\pm$ and also for the case that $\mathfrak{z}_\pm$ are
real and positive. It is easy to see that in the latter case
$\mathcal{R}_{\vec\mathfrak{z}}$ is empty and there are no spectral singularities or
bound states.
\begin{figure}[pt]
\begin{center}
\includegraphics[scale=.75,clip]{fig5.eps}
\parbox{15cm}{\caption{(a) Discs $D_\pm$ and $D_\sigma$
for generic values of $\mathfrak{z}_\pm$. The gray area with the origin
excluded is the region $\mathcal{R}_{\vec\mathfrak{z}}$ where the bound states and
spectral singularities are located (if any). (b) $D_\pm$
for $\mathfrak{z}_\pm\in\mathbb{R}^+$. In this case $\mathcal{R}_{\vec\mathfrak{z}}$ is empty.
\label{fig4}}}\end{center}
\end{figure}
Let $D_\sigma$ and $\mathfrak{D}_\sigma$ be the disc and half-disc defined by
\begin{equation}
D_\sigma:=\big\{\mathfrak{K}\in\mathbb{C}~\big|~|\mathfrak{K}|\leq\sigma~\big\},~~~~
\mathfrak{D}_{\sigma}:=\big\{\mathfrak{K}\in\mathbb{C}~\big|~|\mathfrak{K}|\leq\sigma,~\Re(\mathfrak{K})
\leq 0.~\big\},
\label{discs-z}
\end{equation}
where $\sigma$ is the largest of $2|\mathfrak{z}_\pm|$, i.e.,
\begin{equation}
\sigma:= 2\:{\rm max}(|\mathfrak{z}_-|,|\mathfrak{z}_+|).
\label{sigma=}
\end{equation}
Then, $\mathfrak{K}\in \mathfrak{D}_{\sigma}$ is a weaker necessary condition for the
existence of bound states and spectral singularities. This is simply
because $D_{\pm}\subseteq D_\sigma$. See Figure~\ref{fig4}(a).
Because the spectral singularities and bound states are given by the
zeros of
\begin{equation}
F_{\vec\mathfrak{z}}(\mathfrak{K}):=(\mathfrak{K}-\mathfrak{z}_-)(\mathfrak{K}-\mathfrak{z}_+)-\mathfrak{z}_-\mathfrak{z}_+\:e^{2\mathfrak{K}},
\label{F=}
\end{equation}
which is an entire (everywhere complex-analytic) function, and these
zeros are contained in $\mathfrak{D}_{\sigma}$ which is a compact subset of
the complex $\mathfrak{K}$-plane, we can determine the location of the
spectral singularities and bound states using the following
well-known result of complex analysis:
\begin{itemize}
\item[]\textbf{Theorem~1:} Let $C$ be a counterclockwise oriented
contour bounding a compact and simply-connected region $R$ in
complex plane and $h:\mathbb{C}\to\mathbb{C}$ be a function that is analytic on
an open subset containing $R$. Then $h$ has a finite number of
zeros in $R$. Moreover, if none of these zeros lie on $C$, the
contour integral
\begin{equation}
n_C:=\frac{1}{2\pi i}\oint_C\frac{h'(w)}{h(w)}\:dw
\label{n=}
\end{equation}
gives the sum of orders of zeros of $h$ contained in $R$.
In particular, if all of these zeros are simple (of order 1),
$n_C$ gives their number,
\cite[\S10]{howie}.
\end{itemize}
A proper use of this theorem requires a careful analysis of the
order of zeros of $F_{\vec\mathfrak{z}}$. It is not difficult to show that
the zeros of $F_{\vec\mathfrak{z}}$ can at most be of order three. Moreover,
$\mathfrak{K}$ is a third order zero of $F_{\vec\mathfrak{z}}$ if and only if $\mathfrak{K}=0$
and
\begin{equation}
\mathfrak{z}_-=\frac{-1\pm i}{2},~~~~\mathfrak{z}_+=\frac{1}{2\mathfrak{z}_-}=
\frac{-1\mp i}{2}.
\label{3rd-order}
\end{equation}
This does not correspond to a spectral singularity or a bound state.
$F_{\vec\mathfrak{z}}$ has a second order zero $\mathfrak{K}_2$ if and only if
\begin{eqnarray}
&&2\mathfrak{z}_-\mathfrak{z}_+\: e^{1+\mathfrak{z}_-+\mathfrak{z}_+\pm\sqrt{1+(\mathfrak{z}_--\mathfrak{z}_+)^2}}=
1\pm\sqrt{1+(\mathfrak{z}_--\mathfrak{z}_+)^2},
\label{2nd-1}\\
&&\mathfrak{K}_2=\frac{1}{2}\left[1+\mathfrak{z}_-+\mathfrak{z}_+
\pm\sqrt{1+(\mathfrak{z}_--\mathfrak{z}_+)^2}\right].
\label{2nd-2}
\end{eqnarray}
Requiring that $\Re(\mathfrak{K}_2)\leq 0$, we can use (\ref{2nd-1}) to show
that $|1\pm\sqrt{1+(\mathfrak{z}_--\mathfrak{z}_+)^2}|\leq 2|\mathfrak{z}_-\mathfrak{z}_+|$. Therefore,
there is no spectral singularity or bound state associated with a
second order zero of $F_{\vec\mathfrak{z}}$, if for both choices of the sign,
\begin{equation}
\left|1\pm\sqrt{1+(\mathfrak{z}_--\mathfrak{z}_+)^2}\right|> 2|\mathfrak{z}_-\mathfrak{z}_+|.
\label{2nd-3}
\end{equation}
This inequality in turn implies the following sufficient condition
for the lack of spectral singularities and bound states associated
with a second order zero of $F_{\vec\mathfrak{z}}$.
\begin{equation}
|\mathfrak{z}_-\mathfrak{z}_+|(|\mathfrak{z}_-\mathfrak{z}_+|-1)<\frac{|\mathfrak{z}_--\mathfrak{z}_+|^2}{4}.
\label{2nd-3n}
\end{equation}
In particular, such bound states or spectral singularities are
forbidden if $|\mathfrak{z}_-\mathfrak{z}_+|\leq 1$.
Next, we return to the idea of using Theorem~1 for locating the
spectral singularities and bound states. For this purpose we can use
the contours $C(\rho,\theta)$ and $c_\pm(\rho_\pm)$ depicted in
Figure~\ref{fig5} to compute
\begin{eqnarray}
n_\pm(\rho)&:=&\frac{1}{2\pi i}\oint_{c_\pm(\rho)}
\frac{F_{\vec\mathfrak{z}}'(\mathfrak{K})}{F_{\vec\mathfrak{z}}(\mathfrak{K})}\:d\mathfrak{K},
\label{n-ss-pm}\\
n(\rho_-,\rho_+)&:=&n_-(\rho_-)+n_+(\rho_+),
\label{n-ss}\\
N({\rho,\theta})&:=&\frac{1}{2\pi i}\oint_{C(\rho,\theta)}
\frac{F_{\vec\mathfrak{z}}'(\mathfrak{K})}{F_{\vec\mathfrak{z}}(\mathfrak{K})}\:d\mathfrak{K},
\label{n=bs}
\end{eqnarray}
where $\rho,\rho_\pm\in[\epsilon,\sigma]$, $\epsilon\ll 1$, and
$\theta\in[\epsilon,\pi-\epsilon]$. In the generic case where
$F_{\vec\mathfrak{z}}$ has no second order zeros, $n_\pm(\rho)$ and
$N({\rho,\theta})$ give the number of zeros of $F_{\vec\mathfrak{z}}$
enclosed by $c_\pm(\rho)$ and $C(\rho,\theta)$, respectively.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.90,clip]{fig6.eps}
\parbox{14cm}{\caption{(a) $c_\pm(\rho_\pm)$ are rectangular contours
of width $\epsilon\ll 1$ and hight $\rho_\pm\leq\sigma$; (b)
$C(\rho,\theta)$ is the boundary of the region lying between
circular arcs of side length $\epsilon\ll 1$ and
$\rho\leq\sigma$ and opening angle
$\theta\in[\epsilon,\pi-\epsilon]$.
\label{fig5}}}\end{center}
\end{figure}
Therefore, plotting $n({\rho_-,\rho_+})$ and $N({\rho,\theta})$ as
functions of $\rho_\pm$ and $({\rho,\theta})$, we can locate all the
spectral singularities and bound states of the double-delta-function
potential for given coupling constants $z_\pm$. In particular, for
$\epsilon\to 0$, $n_{\rm tot}:=n(\sigma,\sigma)$ and $N_{\rm
tot}:=N(\sigma,\pi-\epsilon)$ respectively give the total number of
spectral singularities and bound states, except for the cases that
for some imaginary $\mathfrak{K}$ both $\mathfrak{K}$ and $-\mathfrak{K}$ are zeros of
$F_{\vec\mathfrak{z}}$. In the latter case, $\mathfrak{K}$ and $-\mathfrak{K}$ give rise to the
same spectral singularity, and one must account for the
corresponding double counting in $n_{\rm tot}$.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.25,clip]{fig7a.eps}\hspace{1cm}
\includegraphics[scale=.25,clip]{fig7b.eps}
\parbox{15cm}{\caption{Density Plots of $N(\rho,\theta)$ for the
$\mathcal{P}\mathcal{T}$-symmetric case $\mathfrak{z}_\pm=-8\pm 3i$ (on the left)
and the non-${\mathcal{P}\mathcal{T}}$-symmetric case $\mathfrak{z}_-=-8 + 3 i$,
$\mathfrak{z}_+=-4 -2 i$ (on the right) in the complex $\mathfrak{K}$-plane.
$\mathfrak{K}_r$ and $\mathfrak{K}_i$ mark the real and imaginary axes.
As the color changes from the lightest to the darkest
$N(\rho,\theta)$ takes values 0,1, and 2, respectively.
The critical points marked by black spots are the $\mathfrak{K}$-values
corresponding to bound states. They are symmetric about the
$\mathfrak{K}_r$-axis for the ${\mathcal{P}\mathcal{T}}$-symmetric case.
\label{fig6}}}\end{center}
\end{figure}
Note that locating spectral singularities is most conveniently
carried out using (\ref{q21}). In the absence of an analogous
equation giving the $k$ values for the bound states, we use
$N({\rho,\theta})$ to locate the latter. Figure~\ref{fig6} shows the
density plots of $N({\rho,\theta})$ for the $\mathcal{P}\mathcal{T}$-symmetric case
$\mathfrak{z}_\pm=-8\pm 3i$ and the non-${\mathcal{P}\mathcal{T}}$-symmetric case $\mathfrak{z}_-=-8 +
3 i$, $\mathfrak{z}_+=-4 - 2 i$. These resemble the phase diagrams of
statistical mechanics where the critical points correspond to the
bound states. As we expect, for the ${\mathcal{P}\mathcal{T}}$-symmetric case the
location of these points is symmetric about the real axis in the
complex $\mathfrak{K}$-plane.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1.4,clip]{fig8a.eps}
\vspace{1cm}
\includegraphics[scale=1.4,clip]{fig8b.eps}
\parbox{15cm}{\caption{Plots of $N(\rho,\pi-.01)$ (top figure)
and $N(\sigma,\theta)$ (bottom figure) for the
$\mathcal{P}\mathcal{T}$-symmetric system defined by $\mathfrak{z}_\pm=-1\pm8i$ (the
solid curves) and the non-$\mathcal{P}\mathcal{T}$-symmetric system defined by
$\mathfrak{z}_-=-2 + 7 i$ and $\mathfrak{z}_+=-4 - 5 i$ (the dashed curves).
For the $\mathcal{P}\mathcal{T}$-symmetric model $N(\rho,\pi-.01)$ changes
in increments of 2 while $N(\sigma,\theta)$ is symmetric with
respect to the $\theta=\pi/2$ line.
\label{fig7}}}\end{center}
\end{figure}
Figure~\ref{fig7} shows the graphs of $N(\rho,\pi-\epsilon)$ for the
$\mathcal{P}\mathcal{T}$-symmetric case $\mathfrak{z}_\pm=-1\pm 8i$ and the
non-${\mathcal{P}\mathcal{T}}$-symmetric case $\mathfrak{z}_-=-2 + 7 i$, $\mathfrak{z}_+=-4 - 5 i$.
These show the distance between the bound states from the origin.
For the $\mathcal{P}\mathcal{T}$-symmetric case the bound states are created in
complex-conjugate pairs with the same distance from the origin. This
explains the fact that the number of bound states changes in
increments of 2. This is clearly not the case for the
non-$\mathcal{P}\mathcal{T}$-symmetric case. For both of the above choices of the
coupling constants, $\sigma<17$. Therefore, the maximum value of
each curve gives the total number of bound states for the
corresponding system.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.40,clip]{fig9.eps}
\parbox{15cm}{\caption{Contour plot of the number $N_{\rm tot}$
of bound states located in the region: $\pi/2+\epsilon\leq
{\rm arg}(\mathfrak{K})\leq 3\pi/2-\epsilon$ for $\mathfrak{z}_\pm=1+s_\pm i$
and $\epsilon=10^{-6}$. As the color changes from the lightest to
the darkest $N_{\rm tot}$ take values $0,1,2,3,4$, respectively.
The diagonal line $s_+=-s_-$ corresponds to the $\mathcal{P}\mathcal{T}$-symmetric
region along which the number of bound states change in increments of
2.
\label{fig8}}}\end{center}
\end{figure}
Figure~\ref{fig8} shows a contour plot of $N_{\rm
tot}:=N(\sigma,\pi-\epsilon)$ for $\mathfrak{z}_\pm=1+is_\pm$ and
$\epsilon=10^{-6}$ as functions of $s_\pm\in\mathbb{R}$. Although the real
part of the coupling constants are positive and equal, for large
enough values of their imaginary part the system develops bound
states. This is in contrast to the single delta-function potential
where there are no bound states for coupling constants with a
positive real part. We also see that in the $\mathcal{P}\mathcal{T}$-symmetric case
$s_+=-s_-$, which corresponds to the depicted diagonal line, the
number of bound states change in increments of 2. This is consistent
with the fact that these are produced in complex-conjugate pairs.
\subsection{Real Bound States and Quasi-Hermiticity}
An important feature of the graphical demonstration of the location
of spectral singularities and bound states in the complex
$\mathfrak{K}$-plane is that for the cases that $\Re(\mathfrak{z}_\pm)>0$ and
$|\Im(\mathfrak{z}_\pm)|$ are sufficiently small, the system does not have
any spectral singularities or bound states. Figures~\ref{fig4-new}
and~\ref{fig8} provide a clear demonstration of this phenomenon for
the case that $\Re(\mathfrak{z}_\pm)=1$.
The presence of spectral singularities is an obstruction to the
quasi-Hermiticity of the Hamiltonian operator. This is also true for
the bound states unless they happen to have real energies
(eigenvalues). We will refer to these bound states as ``\emph{real
bound states}.'' It is not difficult to see that generic bound
states are not real. In this subsection, we shall first derive
analytic expressions for the existence and location of real bound
states and then for fixed and positive values of $\Re(\mathfrak{z}_\pm)$ we
establish the existence of a positive lower bound on the size of a
region in the $\Im(\mathfrak{z}_-)$-$\Im(\mathfrak{z}_+)$ plane where the system is
free of both the spectral singularities and bound states. This is a
region where the Hamiltonian operator is quasi-Hermitian. It is in
this region that we can employ the machinery of pseudo-Hermitian
quantum mechanics \cite{jpa-2004b,review} to construct an associated
positive-definite metric operator and use the Hamiltonian operator
to define a unitary quantum system.
Excluding the case of a single-delta-function potential
\cite{jpa-2006b} where $\mathfrak{z}_+\mathfrak{z}_-=0$, we can express (\ref{e1}) as
\begin{equation}
\left(\frac{\mathfrak{K}}{\mathfrak{z}_+}-1\right)\left(\frac{\mathfrak{K}}{\mathfrak{z}_-}-1\right)=
e^{2\mathfrak{K}}.
\label{e1-n}
\end{equation}
According to (\ref{E=ss-bs}), the real bound states are given by
real and negative solutions of (\ref{e1-n}). For these solutions the
right-hand side of (\ref{e1-n}) is real, positive, and less than 1.
Equating the left-hand side with its complex-conjugate yields
\begin{equation}
\Im(\mathfrak{z}_-\mathfrak{z}_+)\mathfrak{K}=|\mathfrak{z}_-|^2\Im(\mathfrak{z}_+)+|\mathfrak{z}_+|^2\Im(\mathfrak{z}_-).
\label{r-bs1}
\end{equation}
In order to explore the consequences of this equation we introduce
the notation
\begin{equation}
r_\pm:=\Re(\mathfrak{z}_\pm),~~~~~~~~~s_\pm:=\Im(\mathfrak{z}_\pm),
\label{notation}
\end{equation}
and consider the following cases separately.
\begin{itemize}
\item[(i)] $\Im(\mathfrak{z}_-\mathfrak{z}_+)=0$: In this case,
\begin{equation}
r_-s_++r_+s_-=0,~~~~~~~
\frac{s_-}{|\mathfrak{z}_-|^2}+\frac{s_+}{|\mathfrak{z}_+|^2}=0.
\label{r-bs2}
\end{equation}
Therefore, either both $s_\pm$ vanish and the potential is real or
both $s_\pm$ are nonzero. In the latter case, (\ref{r-bs2}) implies
$\mathfrak{z}_-=\mathfrak{z}_+^*$. This is the $\mathcal{P}\mathcal{T}$-symmetric case for which
(\ref{e1}) reduces to
\begin{equation}
|\mathfrak{K}-\mathfrak{z}_+|=|\mathfrak{z}_+|\,e^{\mathfrak{K}}.
\label{e1-nn}
\end{equation}
Because $e^{\mathfrak{K}}<1$, this equation cannot be satisfied, if
$\Re(\mathfrak{z}_+)\geq 0$. This is consistent with the results of
\cite{demiralp}. Furthermore, for the non-${\mathcal{P}\mathcal{T}}$ cases with real
$\mathfrak{z}_-\mathfrak{z}_+$, such as $\mathfrak{z}_-=-\mathfrak{z}_+^*$ or imaginary $\mathfrak{z}_\pm$ with
$\mathfrak{z}_-\neq\mathfrak{z}_+^*$, there is no real bound states.
\item[(ii)] $\Im(\mathfrak{z}_-\mathfrak{z}_+)\neq 0$: In this case, we can write
(\ref{r-bs1}) as
\begin{equation}
\mathfrak{K}=\frac{|\mathfrak{z}_-|^2\Im(\mathfrak{z}_+)+|\mathfrak{z}_+|^2\Im(\mathfrak{z}_-)}{\Im(\mathfrak{z}_-\mathfrak{z}_+)}.
\label{r-bs3}
\end{equation}
Substituting this equation in (\ref{e1}) gives a rather
complicated relation between $\mathfrak{z}_-$ and $\mathfrak{z}_+$. This relation
together with the requirement that the right-hand side of
(\ref{r-bs3}) be negative provide the necessary and sufficient
condition for the existence of real bound states for
non-$\mathcal{P}\mathcal{T}$-symmetric cases. We have implemented this condition
to address the existence of real bound states for the special
cases where $\mathfrak{z}_+=\mathfrak{z}_-e^{i\nu}:=\mathfrak{z}\: e^{i\nu/2}$ with
$\nu\in[0,2\pi)$. Figure~\ref{fig9} shows the curves in the
complex $\mathfrak{z}$-plane along which real bound states exist for
various values of $\nu$. It is important to note that all
these curves are finite in length. Therefore, there are no real
bound states for sufficiently large values of $|\mathfrak{z}|$.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.9,clip]{fig10.eps}
\parbox{13cm}{\caption{Curves in the complex $\mathfrak{z}$-plane along
which real bound sates exist for $\mathfrak{z}_\pm=\mathfrak{z} e^{\pm i\nu/2}$,
$\nu=\pi(2n-1)/20$, and $n\in\{1,2,\cdots,10\}$. The numbers
attached to each curve segment is the corresponding value of $n$.
$r$ and $s$ respectively mark the $\Re(\mathfrak{z})$- and
$\Im(\mathfrak{z})$-axes. Note that all the curves have finite length.
\label{fig9}}}\end{center}
\end{figure}
\end{itemize}
Next, we wish to show the existence of regions in the space of the
coupling constants $\mathfrak{z}_\pm$ where there is no spectral
singularities or bound states. Our main tools are the following
basic theorems of real and complex analysis.
\begin{itemize}
\item[] \textbf{Theorem~2:} Let $n\in\mathbb{Z}^+$, $D$ be a compact
subset of $\mathbb{R}^n$ with its standard topology, and
$\varphi:\mathbb{R}^n\to\mathbb{R}$ be a function that is continuous on $D$.
Then $\{\varphi(\vec x)~\big|~\vec x\in D~\}$ has both
a minimum and a maximum, \cite[\S 8]{Edwards}.
\end{itemize}
\begin{itemize}
\item[] \textbf{Theorem~3 (Maximum Modulus Theorem):} Let $C$ be a
contour bounding a compact and simply-connected subset $R$ of the complex
plane and $h:\mathbb{C}\to\mathbb{C}$ be a function that is analytic on
an open subset containing $R$. Then $\{|h(w)|~\big|~w\in R~\}$
attains its maximum on $C$, \cite[\S III.1]{Lang}.
\end{itemize}
First, we use Theorem~3 to prove the following preliminary results.
\begin{itemize}
\item[] \textbf{Lemma~1:} Let $\mathfrak{D}_\rho$ denote the following
half disc of radius $\rho\in\mathbb{R}^+$:
\[\mathfrak{D}_\rho:=\left\{~\mathfrak{K}\in\mathbb{C}~\big|~|\mathfrak{K}|\leq \rho,~\Re(\mathfrak{K})\leq
0~\right\},\]
and $L:\mathbb{C}\to\mathbb{C}$ be the function defined by
\begin{equation}
L(\mathfrak{K}):=\left\{
\begin{array}{ccc}
\frac{\mbox{\large$\displaystyle 1-e^{2\mathfrak{K}}$}}{
\mbox{$\displaystyle\mathfrak{K}$}}
&{\rm for}&\mathfrak{K}\neq 0,\\
-2&{\rm for}&\mathfrak{K}= 0.\end{array}\right.
\label{L=}
\end{equation}
Then $|L|$ attains its maximum value on $\mathfrak{D}_\rho$ at
$\mathfrak{K}=0$, i.e., $2=|L(0)|$ is the maximum of
${\cal A}_\rho:=\{|L(\mathfrak{K})|~\big|~\mathfrak{K}\in \mathfrak{D}_\rho\}$ for all
$\rho\in\mathbb{R}^+$.
\item[] \textbf{Proof:} First, consider the case $\rho<1$. Then
$\mathfrak{D}_\rho\subsetneq D_1$, which implies ${\cal A}_\rho\subseteq {\cal A}_1$.
Therefore, the maximum of ${\cal A}_\rho$ is less than or equal
to that of ${\cal A}_1$. This shows that it is sufficient to
prove the lemma for the case $\rho\geq 1$. Because $L$ is an
entire function and $\mathfrak{D}_\rho$ is compact, according to Theorem~3,
${\cal A}_\rho$ has a maximum that is located on the boundary of
$\mathfrak{D}_\rho$. This is the union of the closed line segment
$\ell_\rho:=\{ iy~|~y\in[-\rho,\rho]~\}$ and the open
semicircle $C_\rho:=\{ i\rho~e^{i\varphi}~|~\varphi\in(0,\pi)~\}$.
The maximum of ${\cal A}_\rho$ is the largest of the values taken by
$|L|$ on $\ell_\rho$ and $C_\rho$. We will show that these values
are bounded from above by $2$. Because $0\in \mathfrak{D}_\rho$ and $|L(0)|=2$,
this is sufficient to prove the lemma. In the following we consider
the values of $|L|$ on $\ell_\rho$ and $C_\rho$ separately.
\begin{itemize}
\item For all $\mathfrak{K}\in\ell_\rho$, we can write $\mathfrak{K}=iy$ for
some $y\in[-\rho,\rho]$. Inserting $\mathfrak{K}=iy$ in (\ref{L=}) and
computing the modulus of both sides of the resulting expression
yields
\begin{equation}
|L(\mathfrak{K})|=\frac{2\sin y}{y}\leq 2.
\label{bound-L1}
\end{equation}
\item For all
$\mathfrak{K}\in C_\rho$, we can write $\mathfrak{K}=i\rho~e^{i\varphi}$ for some
$\varphi\in(0,\pi)$. Because $\rho\geq 1$ and
$\sin\varphi>0$, (\ref{L=}) implies
\begin{equation}
|L(\mathfrak{K})|=\frac{\left|1-\exp(2i\rho~e^{i\varphi})\right|}{\rho}
\leq 1+|\exp(2i\rho~e^{i\varphi})|=1+e^{-2\rho\sin\varphi}<2.
\label{bound-L2}
\end{equation}
This together with (\ref{bound-L1}) proves the lemma for
$\rho\geq 1$. As we explained above this establishes the statement
of the lemma also for the case $\rho<1$.~~~$\square$
\end{itemize}
\end{itemize}
\begin{itemize}
\item[] \textbf{Lemma~2:} Suppose that $r_\pm>0$. Then $\mathfrak{K}=0$
is a first order zero of the function $F_{\vec\mathfrak{z}}$ defined by
(\ref{F=}).
\item[] \textbf{Proof:} Recall that the zeros of $F_{\vec\mathfrak{z}}$ are at
most of order three and $F_{\vec\mathfrak{z}}(0)=0$. Therefore, it is
sufficient to show that $\mathfrak{K}=0$ is not a second or third order
zero of $F_{\vec\mathfrak{z}}$. Assume (by contradiction) that $\mathfrak{K}=0$ is
a second order zero of $F_{\vec\mathfrak{z}}$. Then $F'_{\vec\mathfrak{z}}(0)=0$, i.e.,
$\mathfrak{z}_-+\mathfrak{z}_-+2\mathfrak{z}_-\mathfrak{z}_+=0$. Equivalently, we have
\[r_++r_-+2(r_-r_+-s_-s_+)=0,~~~~~s_++s_-+2(r_-s_++r_+s_-)=0.\]
Solving the second of these for $s_-$ and inserting the result
in the first, we find
\[r_++r_-+2r_-r_++\frac{2(1+2r_-)s_+^2}{1+2r_+}=0.\]
But this equation cannot be satisfied for $r_\pm>0$. This shows
that the above assumption is false and $\mathfrak{K}=0$ is not a second
order zero of $F_{\vec\mathfrak{z}}$. Next, we recall that $\mathfrak{K}=0$ is a
third order zero of $F_{\vec\mathfrak{z}}$ if and only if (\ref{3rd-order})
hold. But these conflict with the condition $r_\pm>0$.
Hence $\mathfrak{K}=0$ is not a third order zero of $F_{\vec\mathfrak{z}}$.~~~$\square$
\end{itemize}
Next, we use Theorem~2 and Lemmas~1 and~2 to prove the following
desired result.
\begin{itemize}
\item[] \textbf{Theorem~4:} Suppose that $r_\pm>0$ and
$|s_\pm|<r_{\rm max}:={\rm max}(r_-,r_+)$. Then there is a positive
upper bound $B_{\vec r}$ on $|s_\pm|$ such that for all $s_\pm$
satisfying $|s_\pm|<B_{\vec r}$, the Hamiltonian (\ref{H}) does not
have any spectral singularities or bound
states.\footnote{Here and in what follows $\vec r:=(r_-,r_+)$.}
\item[] \textbf{Proof:} Recall that spectral
singularities and bound states are zeros $\mathfrak{K}$ of $F_{\vec\mathfrak{z}}$
with $\Re(\mathfrak{K})\leq 0$ and that they belong to $D_\sigma$, where
$\sigma:=2\,{\rm max}(|\mathfrak{z}_-|,|\mathfrak{z}_+|)$. The
latter is a subset of the half-disc
\[\mathfrak{D}:=\mathfrak{D}_{\sqrt 8\,r_{\rm max}}=\Big\{\mathfrak{K}\in\mathbb{C}~\Big|~|~
\mathfrak{K}|\leq \sqrt8\: r_{\rm max},~~\Re(\mathfrak{K})\leq 0\Big\},\]
because in view of $r_\pm\leq r_{\rm max}$ and $|s_\pm|<r_{\rm
max}$, we have $\sigma<\sqrt 8\:r_{\rm
max}$.
According to Lemma~2, $\mathfrak{K}=0$ is a first order zero of
$F_{\vec\mathfrak{z}}$. This implies that the function
$G_{\vec\mathfrak{z}}:\mathbb{C}\to\mathbb{C}$ defined by
\begin{equation}
G_{\vec\mathfrak{z}}(\mathfrak{K}):=\left\{\begin{array}{ccc}
\mathfrak{K}^{-1}F_{\vec\mathfrak{z}}(\mathfrak{K}) & {\rm for} & \mathfrak{K}\neq 0,\\
F'_{\vec\mathfrak{z}}(0)& {\rm for} & \mathfrak{K}= 0,\end{array}\right.
\label{G=}
\end{equation}
is an entire function and $G_{\vec\mathfrak{z}}(0)\neq 0$. Furthermore,
the spectral singularities and bound states of the
Hamiltonian~(\ref{H}) correspond to the zeros $\mathfrak{K}_0$ of
$G_{\vec\mathfrak{z}}$ lying in $\mathfrak{D}$. Another important
observation is that $G_{\vec r}$ has no zeros $\mathfrak{K}$ with
$\Re(\mathfrak{K})\leq 0$, because if existed these zeros would have
corresponded to the spectral singularities or bound states of the
Hamiltonian~(\ref{H}) with real and positive coupling constants
$(\mathfrak{z}_\pm\in \mathbb{R}^+)$. But as we argued above this Hamiltonian does not
have any spectral singularities or bound states. This observation
establishes the fact that
\begin{equation}
G_{\vec r}(\mathfrak{K})\neq 0,~~~~~\mbox{for all}~\mathfrak{K}\in \mathfrak{D}.
\label{G-r}
\end{equation}
Because $G_{\vec r}$ is an entire function, $|G_{\vec r}|$ is
continuous on $\mathfrak{D}$ which is a compact subset
of $\mathbb{C}=\mathbb{R}^2$. In view of Theorem~2, this implies that the set
$\{|G_{\vec r}(\mathfrak{K})|~\big|~\mathfrak{K}\in \mathfrak{D}~\}$ has a minimum
$m_{\vec r}$, i.e., there is $\mathfrak{K}_{\rm min}\in \mathfrak{D}$ such
$m_{\vec r}=|G_{\vec r}(\mathfrak{K}_{\rm min})|$. Because
$\mathfrak{K}_0,\mathfrak{K}_{\rm min}\in \mathfrak{D}$ and (\ref{G-r}) holds, we have
\begin{equation}
0<|G_{\vec r}(\mathfrak{K}_{\rm min})|=m_{\vec r}\leq |G_{\vec r}(\mathfrak{K}_0)|.
\label{bound-G}
\end{equation}
Next, we introduce $J_{\vec\mathfrak{z}}:\mathbb{C}\to\mathbb{C}$
as the function defined by
\begin{equation}
J_{\vec\mathfrak{z}}(\mathfrak{K}):=G_{\vec\mathfrak{z}}(\mathfrak{K})-G_{\vec r}(\mathfrak{K}).
\label{J=}
\end{equation}
Because $G_{\vec\mathfrak{z}}(\mathfrak{K}_0)=0$, we have
\begin{equation}
|J_{\vec\mathfrak{z}}(\mathfrak{K}_0)|=|G_{\vec r}(\mathfrak{K}_0)|.
\label{bound1}
\end{equation}
Furthermore, in view of (\ref{G=}), (\ref{J=}),
(\ref{L=}), and the fact that $\mathfrak{K}_0\neq 0$,
\begin{equation}
J_{\vec\mathfrak{z}}(\mathfrak{K}_0)=-i(s_-+s_+)+
\big[-s_-s_++i(r_-s_++r_+s_-)\big]L(\mathfrak{K}_0).
\label{J=2}
\end{equation}
This implies
\begin{eqnarray}
|J_{\vec\mathfrak{z}}(\mathfrak{K}_0)|&\leq &
|s_-|+|s_+|+
\Big(|s_-||s_+|+|r_-||s_+|+|r_+||s_-|\Big)|L(\mathfrak{K}_0)|\nonumber\\
&\leq &2\left(3r_{\rm max}+1\right)s_{\rm max},
\label{bound-J}
\end{eqnarray}
where $s_{\rm max}:={\rm max}(|s_-|,|s_+|)$ and we have used the
triangular inequality, the condition $|s_\pm|\leq r_{\rm max}$, and
$|L(\mathfrak{K}_0)|\leq 2$ that follows from Lemma~1.
If we combine (\ref{bound-J}) with (\ref{bound1}) and
(\ref{bound-G}), we obtain
\begin{equation}
0<\frac{m_{\vec r}}{2(3r_{\rm max}+1)}\leq s_{\rm max}.
\label{the-bound}
\end{equation}
This inequality is violated for the values of $s_\pm$ for which
\begin{equation}
|s_\pm|<\frac{m_{\vec r}}{2(3r_{\rm max}+1)}=:B_{\vec r}.
\label{violate-bound}
\end{equation}
Therefore, for the cases that $|s_\pm|<B_{\vec r}$ the existence of
$\mathfrak{K}_0$ leads to a contradiction; such a $\mathfrak{K}_0$ cannot exist; and
there are no spectral singularities or bound states.~~~$\square$
\end{itemize}
The upper bound $B_{\vec r}$ given in (\ref{violate-bound}) involves
the minimum $m_{\vec r}$ of $|G_{\vec r}|$ on the half-Disc $\mathfrak{D}$.
Because $G_{\vec r}$ is a nowhere-zero analytic function on $\mathfrak{D}$,
$1/G_{\vec r}$ is also analytic on $\mathfrak{D}$. Hence, according to
Theorem~3, $1/|G_{\vec r}|$ attains its maximum $M_{\vec r}$ on the
boundary of $\mathfrak{D}$. It is not difficult to see that $m_{\vec
r}=1/M_{\vec r}$. Therefore, in practice, for given values of
$r_\pm$, we can obtain $m_{\vec r}$ by exploring the values of
$|G_{\vec r}|$ on the boundary of $\mathfrak{D}$.
We can identify the boundary of $\mathfrak{D}$ with the graph $\Gamma_{\vec
r}$ of the parameterized curve:
\begin{equation}
\gamma_{\vec r}(t):=2ir_{\rm
max}\left[(2t+1)\Theta(-t)+e^{i\pi
t}\Theta(t)\right],~~~~t\in[-1,1],
\label{graph}
\end{equation}
where $\Theta$ is the unit step function: $\Theta(0):=1/2$ and
$\Theta(t):=(1+t/|t|)/2$ for $t\neq 0$. Figure~\ref{fig10} shows the
graphs of $|G_{\vec r}(\gamma_{\vec r}(t))|$ and $|L(\gamma_{\vec
r}(t))|$ for the case $r_\pm=1$ that is considered in
Figures~\ref{fig4-new} and~\ref{fig8}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.7,clip]{fig11.eps}
\parbox{15cm}{\caption{Plots of $|G_{\vec r}(\gamma_{\vec r}(t))|$
(the full curve) and $|L(\gamma_{\vec r}(t))|$ (the dashed curve)
as a function of $t\in[-1,1]$. $t_-\approx-0.949$ and $t_+\approx-.051$
are the minimum points of $|G_{\vec r}(\gamma_{\vec r}(t))|$
corresponding to $\mathfrak{K}_\pm\approx\pm 1.795 i$. These give the
minimum value $m_{\vec r}\approx 1.906$. The maximum value of
$|L(\gamma_{\vec r}(t))|$ is 2 that is attained at $t_0=-0.5$
corresponding to $\mathfrak{K}=0$.
\label{fig10}}}\end{center}
\end{figure}
In this case, $r_{\rm max}=1$ and $\mathfrak{D}$ is the half-disc of radius
$2$ lying in $\Pi_-$. As seen from the graph of values of $|L|$, it
attains its maximum at $t=-0.5$ (corresponding to $\mathfrak{K}=0$) and its
maximum value is 2. This is consistent with the statement of
Lemma~1. The minimum points of $|G_{\vec r}|$ are located at
$t=-0.949$ and $t=-.051$. These correspond to $\mathfrak{K}_{\rm min}\approx
\pm 1.795 i$ where $|G_{\vec r}|$ takes its minimum value: $m_{\vec
r}\approx 1.906$. According to (\ref{violate-bound}), this gives
$B_{\vec r}=m_{\vec r}/8\approx 0.238$. Therefore, for $\mathfrak{z}_\pm=1\pm
is_\pm$ with $|s_\pm|<0.238$ there should be no spectral
singularities or bound states. This is in complete agreement with
the graphical data depicted in Figures~\ref{fig4-new} and
\ref{fig8}; the disc with center $s_\pm=0$ and radius 0.238 lies in
the region with no spectral singularities or bound states.
\section{Concluding Remarks}
In this article we provided an explicit demonstration of how
spectral singularities obstruct the existence of a biorthonormal
eigensystem and render the Hamiltonian non-diagonalizable. We
achieved this by obtaining a characterization of spectral
singularities in terms of the ${M}_{22}$ entry of the matrix $M$
of Eq.~(\ref{S-matrix}). In particular we showed that while bound
states are zeros of ${M}_{22}(k)$ with $\Im(k)>0$, the spectral
singularities are the real zeros of ${M}_{22}(k)$. It is not
difficult to infer from this observation that, similarly to the
bound states, the spectral singularities are linked with
singularities of the scattering matrix \cite{p89}. However, unlike
the bound states, they lie on the real axis in the complex
$k$-plane. This in turn suggests interpreting spectral singularities
as resonances having a vanishing width. Ref.~\cite{p89} provides a
thorough description of this interpretation and its physical
implications.
We established the utility of our general results by providing a
thorough analysis of the spectral properties of a two-parameter
family of complex point interactions. We obtained various results on
the nature and location of the bound states and spectral
singularities for this family and proved the existence of regions in
the space of coupling constants where both bound states and spectral
singularities are lacking and the Hamiltonian is quasi-Hermitian.
Throughout our study we examined the consequences of imposing
$\mathcal{P}\mathcal{T}$-symmetry which corresponds to restricting the coupling
constants to a complex plane in the space $\mathbb{C}^2$ of coupling
constants. This revealed a previously unnoticed fact that
$\mathcal{P}\mathcal{T}$-symmetric double-delta-function potential can involve
spectral singularities.
The results of this paper may be extended to complex point
interactions corresponding to three or larger number of delta
function potentials. Another line of research is to try to compute a
metric operator $\eta_{_{+}}$ and the corresponding equivalent
Hermitian Hamiltonian $h$ and the pseudo-Hermitian position and
momentum operators $X$ and $P$ for the double-delta-function
potential whenever the Hamiltonian is quasi-Hermitian. Theorem~4
provides the mathematical basis for a perturbative calculation of
$\eta_{_{+}}$, $h$, $X$, and $P$. We plan to report the results of
this calculation in a forthcoming publication.
\section*{Acknowledgments}
This work has been supported by the Scientific and Technological
Research Council of Turkey (T\"UB\.{I}TAK) in the framework of the
project no: 108T009, and by the Turkish Academy of Sciences
(T\"UBA). We wish to express our gratitude to Prof.\ Gusein Guseinov
for preparing and sending us a detailed description of spectral
singularities \cite{guseinov}.
|
1,314,259,995,857 | arxiv | \section{Introduction}
Hydrogen is known to have a wide range of physical effects in semiconductors,
including the passivation of states associated with deep--level impurities,
enhancement of the diffusivity of oxygen, and the formation of large, planar
structures known as platelets.~\cite{Pat88} It is present in large quantities
during the processing stages of device manufacture and is one of the commonest
impurities in technologically important materials such as silicon and
germanium. Since such hydrogen impurities can have significant effects on
semiconductor electrical properties, a more complete understanding of their
behaviour at the microscopic level is desirable.
Paramagnetic hydrogen centres can in principle be studied using the electron
paramagnetic resonance (EPR) technique. Information on their local environment
is obtained by following the time evolution of the signal corresponding to the
coupling of the spin of the impurity with an external electromagnetic field.
However, few studies have been reported for hydrogen in semiconductors because
the hydrogen atoms are mobile and diffuse to defects where they form passivated
complexes. The transient centres of isolated hydrogen impurities are
nevertheless of significant interest because of their involvement in diffusion
processes, and in fact they may be studied using muon spin resonance ($\mu$SR)
techniques. Muons have the same charge as protons but only about one ninth of
the mass. They can capture an electron to form a hydrogen--like bound state
known as muonium (given the symbol Mu), and it is thus possible to
consider the muon as a proton analogue. Transient centres of implanted positive
muons in semiconductors may be studied as the muon has a lifetime of just 2.2
$\mu$s and diffuses to locally stable sites within a few nanoseconds. The
short lifetime also means that there is almost never more than one muon in the
sample at any one time, and that the distribution of muons does not reach true
thermal equilibrium. In a muon spin resonance experiment fully polarized
positive muons are injected into the sample, and by observing the positrons
produced by the decay of the muons one can obtain information about the
defect.~\cite{Pat88}
When $\mu$SR experiments are performed on silicon or germanium, two different
hyperfine signals are observed. One of these is entirely isotropic while the
other has an anisotropic (dipolar) component with uniaxial symmetry along the
[1 1 1] axis.~\cite{Pat88} The impurity responsible for the former signal is
usually referred to as normal muonium (Mu) and that for the latter, anomalous
muonium (Mu$^{*}$). Normal muonium has been identified as muonium in the
interstitial region, probably in the vicinity of the tetrahedral (T)
interstitial site. Symons and Cox~\cite{symons} first suggested that anomolous
muonium corresponds to a neutral muonium at the bond--centred (BC) site and
this has been borne out by a number of theoretical studies. The various
experimental data for muonium in silicon have been interpreted in terms of a
configuration--coordinate diagram.~\cite{config}
The majority of recent theoretical work in this area has been at the first
principles level within an adiabatic approximation, using the local spin
density (LSDA) or generalized gradient (GGA) approximations to density
functional theory (DFT).~\cite{parr&yang} Calculations using pseudopotentials
and plane--wave basis sets~\cite{VdW90,VdW_Blochl93,Luch97} have been
reasonably successful in reproducing the hyperfine and superhyperfine
parameters observed in experiments. The majority of such calculations appear
to demonstrate that hydrogen impurities at the T and BC sites have similar
energies.~\cite{Luch97,chang89}
The application of the Feynman path--integral~\cite{Feynman} method to the
study of these systems enables the effect of the quantum nature of the muon to
be studied directly at finite temperatures. However, the large computational
demands of such an approach have limited its use to date. Ram\'{\i}rez and
Herrero~\cite{Ramirez} used the path integral molecular dynamics method to
study hydrogen and muonium in silicon with the H/Mu-Si interaction described by
an empirical three--body potential. However, the results appear to be in
conflict with experiment. Recently, Miyake {\it et al.}~\cite{Miyake_etal98}
applied the path--integral Monte Carlo technique to the study of hydrogen and
muonium at the T site in silicon, with the electron-electron interactions
described within the LDA. Despite finding the T site to be a local maximum on
the potential energy surface, they found the muon distribution to be peaked at
that site because of the quantum motion.
In this work we employ all--electron DFT calculations within a double adiabatic
approximation to study muonium and hydrogen at the BC and T sites in silicon
and germanium. The use of all--electron calculations allows an assessment of
the accuracy of the correction procedures which are used to obtain the
hyperfine and superhyperfine parameters in pseudopotential
calculations.~\cite{VdW_Blochl93} The use of a double adiabatic approximation
allows us to obtain both the zero--point energy and wave function of the
impurity. Our inclusion of the zero--point motion is at a level beyond that in
previous first principles calculations since the positions of the host silicon
or germanium atoms are allowed to relax in the presence of the zero--point
motion of the impurity. At this level of approximation the relaxations of the
host lattice are different for a muon and a proton. Our calculations thus allow
an assessment of the differences in the potentials felt by the two impurities,
thereby testing one of the assumptions underlying the configuration--coordinate
diagram~\cite{config} used to interpret experimental data.
\section{Method}
\label{method}
\subsection{All--electron spin--polarized LSDA--DFT calculations}
All the first principles calculations reported here were performed with the
CRYSTAL95 software package~\cite{crystal}. The (zero temperature)
spin--polarized density functional method~\cite{HKtheorem,KohnSham} was used,
together with both local density and gradient corrected approximations to the
exchange--correlation functional (namely the Perdew--Zunger LSDA~\cite{PZ} and
the PW91 form of the GGA~\cite{GGA}). The calculations were performed within a
periodic supercell approach with a single hydrogen impurity in supercells
containing either sixteen or fifty--four silicon or germanium atoms.
Fig.~\ref{fig:geometries} shows the relaxed atomic environments of a single
muon at both bond--centred and tetrahedral impurity sites in silicon. The
measured lattice constants (5.429 \AA \ for silicon and 5.6579 \AA \ for
germanium) were used in all calculations.
Other approximations made were as follows. The use of local basis functions
requires the real space Coulomb and exchange series to be limited and
approximated as described in references~\onlinecite{crystal,pisani}; the
accuracy with which the various Gaussian integrals are computed is controlled
by classifying basis function pairs according to overlap or penetration
criteria defined by five parameters, which in this study were set to $10^{-7}$,
$10^{-6}$, $10^{-7}$, $10^{-7}$ and $10^{-14}$.~\cite{crystal} This is
normally sufficient to give a numerical error of less than 0.001~eV/atom in the
relative energies of different structures. The reciprocal space integrations
necessary to reconstruct the density matrix in real space at each
self-consistent cycle were approximated by summing over a set of k--points
belonging to a mesh of Monkhorst--Pack~\cite{Monk_Pack} type which was centred
on the origin in reciprocal space. The convergence of both the total energy and
the isotropic hyperfine parameter of the muon with respect to the reciprocal
space sampling density was investigated. A $4\!\times\!4\!\times\!4$ k--point
mesh was found to be sufficient for the 16--atom supercell. With this mesh, the
total energy and isotropic hyperfine parameter are within 0.0025~eV/atom and 3
MHz of their fully converged values, respectively. A $3\!\times\!3\!\times\!3$
k--point mesh was used for the 54--atom supercell, which also gives excellent
convergence. The convergence of various quantities with respect to the
supercell size and basis set is discussed in Section~\ref{results}.
A hydrogen impurity introduces a defect state into the band gap of the host
crystal. Finite supercell sizes give rise to interactions between the defects
in different cells and thus to a small but potentially significant dispersion
in the defect band. This dispersion could lead, for example, to overlap of the
majority spin defect band with the minority spin defect band and/or the silicon
valence/conduction bands. In either case an unphysical conducting state is
formed. This problem is not entirely eliminated even with the use of the
larger 54--atom supercell. However, judicious use of the level--shifting
convergence technique~\cite{crystal,levshift} allows a small decoupling of
unoccupied and occupied states which prevents the system entering a conducting
state. Population analysis of the final self--consistent wave function revealed
that each supercell contained a single extra majority spin electron as expected
on physical grounds. The calculations thus correctly model this aspect of the
behaviour of a single impurity in a large crystal, which is necessary in order
to obtain physical hyperfine and superhyperfine parameters.
\subsection{Gaussian Basis Sets}
The Bloch functions required to expand the Kohn-Sham orbitals in the
solid--state band structure problem are built from periodic arrays of
atom--centred Gaussian functions. One motivation for the use of such a basis
set is that all electrons in the system may be treated explicitly, allowing the
spin density at and around the nucleus (and hence the hyperfine parameters) to
be calculated directly without resorting to correction procedures of the sort
required in pseudopotential calculations.~\cite{VdW_Blochl93}
The basis set used for the majority of the silicon calculations was of the type
{\it s}(8){\it sp}(8){\it sp}(3){\it sp}(1) where the numbers in brackets refer
to the number of contracted primitive Gaussians making up each shell. For
convergence checking we also used a higher quality silicon set with an
additional {\it d} polarization function of the type {\it s}(8){\it sp}(8){\it
sp}(1){\it sp}(1){\it sp}(1){\it sp}(1){\it d}(1). The basis set used for the
germanium calculations was {\it s}(9){\it sp}(7){\it sp}(6){\it sp}(3){\it
sp}(1){\it d}(6){\it d}(1).
To describe the hydrogen atom, an uncontracted basis of eleven {\it s}
functions and a single {\it p} function was used. Such a large set (mainly
consisting of functions with relatively high exponents) was found to be
necessary to compute accurate hyperfine parameters. A spin--unrestricted
Hartree--Fock calculation of the total energy of the free atom with this basis
gave $-0.49988$~Ha which is close to the exact result of $-0.5$~Ha. The
isotropic hyperfine parameter was 1421.9~MHz compared with that obtained from
the exact wave function of 1422.8~MHz. The corresponding values obtained from
an LSDA--DFT calculation with this basis set were $-0.47833$ Ha (which is very
close to the value of $-0.47885$~Ha obtained from an atomic code using
integration on a very fine grid) and 1356.6~MHz.
Optimal Gaussian basis sets for use in close packed solids are significantly
different from those appropriate to the atomic and molecular cases. In
particular, careful optimization is required to avoid the problems of linear
dependence and basis set superposition error due to the overlap of diffuse
functions. In this study we used the following procedure. All basis set
parameters were first optimized in the free atom. The exponents and contraction
coefficients of the valence functions in silicon and germanium were then
reoptimized in the pure bulk material. Finally, a hydrogen atom was inserted at
a bond--centred site, the positions of the nearest--neighbour silicon/germanium
atoms relaxed, and the parameters of the valence functions of each atom again
reoptimized.
To test the transferability of the optimized basis sets the hydrogen was
displaced from the BC site along the bond by 0.27 \AA, and the basis function
parameters were reoptimized for the new geometry. The energy as a function of
displacement along the bond was calculated for each of these two basis sets.
The variation in energy was essentially the same.
The final exponents and contraction coefficients of all the basis sets employed
in this study are available elsewhere.~\cite{basis_set_library}
It is important to investigate the possibility of basis set superposition error
(BSSE) in defect energetics calculations for a system described by a localized
basis. The basis sets for the host lattice atoms are necessarily incomplete.
Insertion of an impurity atom allows additional variational freedom in the
description of the atoms adjacent to the defect site, particularly when the
impurity is described by a relatively diffuse basis set. This can distort the
relative stabilities of defects at impurity sites of differing local
coordination number and geometry. In the present case, the hydrogen impurity is
considerably closer to its neighbours at the BC site than at the T site, and
thus one might expect the BC site to be artificially stabilized with respect to
the T.
This expectation is confirmed by an estimate of the BSSE using a counterpoise
correction.~\cite{counterpoise} For the 16--atom supercell, addition of
``ghost'' hydrogen basis functions into the relaxed silicon lattice lowered the
energy per cell by 0.199~eV (BC site) and 0.068~eV (T site) with the smaller
silicon basis, and by 0.0533~eV (BC site) and 0.0243~eV (T site) with the
larger silicon basis. In germanium, the energy is lowered by 0.251 eV (BC site)
and 0.0534~eV (T site) by the same procedure. Inclusion of a lattice of
``ghost'' silicon/germanium functions around a hydrogen atom lowered the energy
by less than 0.0005~eV. These numbers may be taken to give a rough indication
of basis set incompleteness in each case. It can thus be concluded that in
silicon the BSSE lowers the energy of the BC site over that of the T site by
around 0.13~eV with the smaller basis, but by only 0.03~eV with the larger set.
The corresponding correction for the germanium case is 0.20~eV.
\subsection{Calculation of zero--point motion}
\label{zero}
For the calculation of the zero--point motion of the muon/proton a double
adiabatic approximation was used. This means that the motions of the electrons
and of the muon are considered to be decoupled from the motion of the atomic
nuclei, and furthermore that the electronic motion is decoupled from that of
the muon. The approximation is justified by the fact that a muon is roughly
207 times more massive than an electron and around 243 times less massive than
a silicon nucleus. For a proton the equivalent factors are respectively 1836
and 28; the decoupling of the proton and silicon motion is thus somewhat less
justified. The mass differences are of course more favourable for the heavier
germanium nucleus.
The positions of the silicon/germanium nuclei are denoted by ${\bf
r}_{n}$, the muon or proton positions by ${\bf
r}_{\mu}$, and the electron positions by ${\bf r}_{e}$. The double
adiabatic approximation is used to decouple the motions of the
particles by approximating the wave function as a product of nuclear,
muon/proton and electronic parts,
\begin{equation}
\Psi({\bf r}_{e},{\bf r}_{\mu},{\bf r}_{n})=\psi^{n}({\bf r}_{n})
X^{\mu}({\bf r}_{\mu};{\bf r}_{n}) \phi^{e}({\bf
r}_{e};{\bf r}_{\mu},{\bf r}_{n}) \ ,
\end{equation}
where the variables to the right of the semi--colons appear as
parameters and those to the left are dynamical variables. Within the
double adiabatic approximation the three wave functions each satisfy
separate Schr\"{o}dinger equations:
\begin{equation}
\hat{H}_{e}({\bf r}_{e};{\bf r}_{\mu},{\bf r}_{n})\, \phi^{e}({\bf r}_{e};{\bf r}_{\mu},{\bf r}_{n})= E^{e}({\bf r}_{\mu},{\bf r}_{n})\,\phi^{e}({\bf r}_{e};{\bf r}_{\mu},{\bf r}_{n}) \ ,
\label{eq:e_SE}
\end{equation}
\begin{equation}
\hat{H}_{\mu}({\bf r}_{\mu};{\bf r}_{n})
\,X^{\mu}_{\alpha}({\bf r}_{\mu};{\bf r}_{n})=
E^{\mu}_{\alpha}({\bf r}_{n})\,X^{\mu}_{\alpha}({\bf r}_{\mu};{\bf r}_{n}) \
,
\label{eq:mu_SE}
\end{equation}
\begin{equation}
\hat{H}_{n}\psi^{n}_{\alpha}({\bf r}_{n})= E^{n}_{\alpha} \psi^{n}_{\alpha}({\bf
r}_{n}) \ .
\label{eq:I_SE}
\end{equation}
The subscript $\alpha$ labels the different eigenstates of the muon. Although
only the ground state of the nuclear wave function is considered here, it is
also labelled by $\alpha$ since it depends on the chosen muon eigenstate. The
different electronic eigenstates are not labelled, since it is only the ground
state of the electronic wave function as a function of the muon and nuclear
positions that is of interest in the current work. The three Hamiltonians are:
\begin{eqnarray}
\hat{H}_{e}({\bf r}_{e};{\bf r}_{\mu},{\bf r}_{n})&= &\hat{T}_{e}({\bf r}_{e})+V_{ee}({\bf r}_{e}) \\ \nonumber
&+&V_{en}({\bf r}_{e},{\bf r}_{n})+V_{e\mu}({\bf
r}_{e},{\bf r}_{\mu}) \ ,
\label{eq:H_e}
\end{eqnarray}
\begin{eqnarray}
\hat{H}_{\mu}({\bf r}_{\mu};{\bf r}_{n})&=&\hat{T}_{\mu}({\bf r}_{\mu})+V_{\mu\mu}({\bf r}_{\mu})+ V_{\mu n}({\bf r}_{\mu},{\bf r}_{n}) \\ \nonumber
&+&E^{e}({\bf r}_{\mu},{\bf r}_{n}) \ ,
\label{eq:H_mu}
\end{eqnarray}
\begin{equation}
\hat{H}_{n}({\bf r}_{n})=
V_{nn}({\bf r}_{n}) +
E^{\mu}_{\alpha}({\bf r}_{n}) \ .
\label{eq:H_I}
\end{equation}
where $\hat{T}$ is the kinetic operator and $V_{ab}$ is the Ewald interaction
between particles of types {\it a} and {\it b}. The term $V_{\mu\mu}$ is a
constant describing the interactions between the impurity atoms in the
different supercells. The electronic energy, $E^{e}$, appears as an effective
potential in the muonic Hamiltonian, Eq.~\ref{eq:H_mu}. Hence, when
Eq.~\ref{eq:mu_SE} is solved, the resulting energy includes the electronic
contribution. This energy then appears in the nuclear Hamiltonian as an
effective potential. Thus the total energy of the system is given by the
eigenvalue in Eq.~\ref{eq:I_SE}.
In order to find a good starting point for the BC calculations we performed
LSDA calculations with the muon/proton fixed at the bond centre. The positions
of the nearest and next--nearest neighbour (NN and NNN) silicon/germanium atoms
were then relaxed. For the T site the muon/proton was held fixed at the T site
while the four NN silicon/germanium atoms were relaxed in the radial direction.
The next step is to calculate the potential experienced by the muon/proton in
the crystal by solving the electronic Schr\"{o}dinger equation (~\ref{eq:e_SE})
within the LSDA, as a function of the parameters ${\bf r}_{\mu}$ and ${\bf
r}_{n}$. For the BC site calculations, the parameters ${\bf r}_{n}$ were
varied by considering four different additional relaxations of the NN silicon
atoms along the [1 1 1] direction. For each of the positions of the NN silicon
atoms, the NNN atoms were relaxed. We then performed a further twelve LSDA
calculations as a function of the position of the muon/proton for each of these
nuclear configurations in order to map out the required potential energy
surfaces. A similar procedure was used for germanium. For the T site the
static relaxation of the NN atoms is very small and we assumed that the
zero--point motion of the muon/proton would not give any additional relaxation.
In order to solve Eq.~\ref{eq:mu_SE} for the muon/proton wave function we used
a fitted polynomial for the energy $E^{e}({\bf r}_{\mu},{\bf
r}_{n})$. For the BC site we use a cylindrical coordinate
system with the origin at the BC site and the $z$-- and $\rho$--coordinates
directed along the bond and in the plane perpendicular to the bond,
respectively. We have neglected the $\theta$ dependence of the potential.
This assumption was checked by displacing the muon by 0.53 \AA \ from the BC
site along the [-1 1 0] direction and then rotating it about the [1 1 1]
axis. The maximum variation seen in the energy of the 16--atom supercell
during the rotation was just 0.002~eV. The Taylor expansion of the
cylindrically symmetric potential, neglecting sixth--order terms and higher, is
\begin{equation} V_{\rm BC}(\rho,z)=V_{\rm BC}(0,0) + \beta \rho^{2} + \gamma
z^{2} + \delta \rho^2z^2 + \zeta \rho^{4} + \eta z^{4} \ . \label{eq:BCexpn}
\end{equation}
As a further simplification in order to avoid costly, low--symmetry
calculations, the $\delta \rho^2z^2$ term was neglected. The resulting
Schr\"odinger equation is separable. In order to check the assumption of
$\rho$--$z$ separability, a few calculations were performed with the muon at
points where both the $\rho$ and $z$ coordinates were non--zero. These
energies were then compared with those predicted by the fitted potential
neglecting the term in $\rho^2 z^2$. The errors due to neglecting the $\rho^2
z^2$ term increased only slowly away from the BC site, and the corresponding
error in the ground state energy is small because the wave function of the
muon/proton is localized around the BC site.
Our approximation formula was then obtained by a least--squares fit of the
twelve energies calculated at different values of ${\bf r}_{\mu}$ (for fixed
${\bf r}_{n}$) to the resulting polynomial. The fitted values of the
parameters in Eq.~\ref{eq:BCexpn} used in the calculations of the
zero--point energies for silicon and germanium are given in
Table~\ref{tble:BC_param}.
The polynomial expansion for the energy surface around the T site is the Taylor
expansion which is invariant under all of the 24 rotations forming the group
$T_d$. Cartesian coordinates centred at the T site were used, and in order to
obtain a good fit to the energy surface it was found necessary to include terms
up to sixth order:
\begin{eqnarray}
V_{\rm T}(x,y,z)& = & V_{\rm T}(0,0,0) + a_{2}(x^2 + y^2 + z^2) +
a_{3}\, xyz \nonumber \\
&+& a_{4}(x^4 + y^4 + z^4) + b_{4}(x^{2}y^{2} + x^{2}z^{2} + y^{2}z^{2}) \nonumber \\
&+& a_{5}\,xyz(x^2 + y^2 +
z^2) + a_{6}(x^{6}+y^{6}+z^{6}) \nonumber \\ \hbox{} & + &
b_{6}(x^{2}y^{4} + x^{2}z^{4} + x^{4}y^{2} + x^{4}z^{2} + y^{2}z^{4} \nonumber \\ &&+y^{4}z^{2}) + c_{6}\,x^{2}y^{2}z^{2} \ .
\label{eq:Texpn}
\end{eqnarray}
The fitted values of the parameters in this equation are given in
Table~\ref{tble:T_param}.
The solution of the muon/proton Schr\"{o}dinger equation (~\ref{eq:mu_SE}) was performed by diagonalizing within a basis of
harmonic oscillator eigenfunctions centred on either the BC or T site.
We constructed muon and proton basis sets consisting of the solutions
of the harmonic part of the calculated potentials:
\begin{equation}
V_{0_{\rm BC}}= V_{\rm BC}(0,0,0) + \beta(x^{2} + y^{2}) + \gamma z^{2} \ ,
\end{equation}
and
\begin{equation}
V_{0_{\rm T}} = V_{\rm T}(0,0,0) + a_{2}(x^2 + y^2 + z^2) \ .
\end{equation}
For a harmonic potential, the Schr\"{o}dinger equation may be solved
by separation of variables:
\begin{eqnarray}
X({{\mathbf r}_{\mu}}) & = & {\mathcal X}(x){\mathcal Y}(y){\mathcal Z}(z) \
. \nonumber
\end{eqnarray}
This gives three separate equations for $\mathcal{X}$, $\mathcal{Y}$,
and $\mathcal{Z}$, whose solutions are of the form:
\begin{equation}
{\mathcal X}_{l}(x^{\prime})=A_{l}H_{l}(x^{\prime})e^{-\frac{1}{2}
x^{\prime 2}},
\end{equation}
where $A_{l}=\frac{1}{\sqrt{2^{l}\pi^{\frac{1}{2}}l!}}$ is the
normalisation factor, $x^{\prime}=(2MC_{x})^{\frac{1}{4}}x$ is a
rescaled variable which allows the eigenfunctions to be written in
terms of the standard Hermite polynomials, $H_{l}(x^{\prime})$, and
$C_{x}$ is the appropriate harmonic coefficient. The associated
energy eigenvalues are
\begin{equation}
E_{\lambda_{l}}=\left(l+\frac{1}{2}\right) \left( \frac{2 C_{\lambda}}{M}
\right)^{\frac{1}{2}} \ ,
\end{equation}
where $\lambda$ runs over the three directions, $x$, $y$, and $z$, and
$C_{\lambda}$ is the harmonic coefficient corresponding to that
direction. The value of the mass, $M$, of the particle depends on
whether we are solving for the proton or muon wave function.
Having constructed our basis functions we now consider the full
potentials which are written as the sum of harmonic and anharmonic
terms:
\begin{eqnarray}
V_{\rm BC} & = & V_{0_{\rm BC}} + \Delta V_{\rm BC} \\
V_{\rm T} & = & V_{0_{\rm T}} + \Delta V_{\rm T} \ .
\end{eqnarray}
The Hamiltonian matrix elements were calculated in the basis of the harmonic
solutions and the resulting matrix equations were diagonalized. A basis set
constructed from all Hermite polynomials up to and including eighth order and
containing a total of 729 basis functions was found to be sufficient to obtain
converged values for at least the lowest six eigenvalues,
$E^{\mu}_{\alpha}({\bf r}_{n})$, of the system with the muon/proton at the BC
site. At the T site it was found necessary to increase the size of the basis
set to include all Hermite polynomials up to twelfth order, which gave 2197
functions.
The solution of Eq.~\ref{eq:H_I} is trivial as the operator is
multiplicative and the eigenfunctions are delta functions. The total
energy, $E^{n}_{\alpha}$, is therefore the sum of
$E^{\mu}_{\alpha}({\bf r}_{n})$ and the Ewald
energy of the lattice of host atoms, $V_{nn}({\bf
r}_{n})$.
\subsection{Hyperfine and superhyperfine parameters and motion averaging}
\label{hyperfine_mthd}
The components of the hyperfine tensor, ${\bf A}$, define the spin Hamiltonian
for the hyperfine interaction between the spins of an electron and a nucleus:
\begin{equation}
\hat{\mathcal H}_{s}={\bf S}_{e} \cdot {\bf A} \cdot {\bf
S}_{n} \ .
\end{equation}
The hyperfine tensor is normally split into isotropic and anisotropic
parts,
\begin{equation}
{\bf A} = A_s{\bf I} + {\bf A}_{p} \ ,
\end{equation}
\noindent where ${\bf I}$ is the ($3\times3$) unit matrix. The isotropic
hyperfine parameter (or {\it superhyperfine} parameter when it is
calculated at one of the nearest neighbours of the impurity) is given by
\begin{eqnarray}
A_s & = & \frac{2\mu_{0}}{3}g_{e}\mu^{e}g_{n}
\mu^{n}\rho_{\sigma} ({\mathbf
r}_{n}) \nonumber \\ & = &
104.982\gamma^{n} \rho_{\sigma}({\mathbf
r}_{n}) \ {\mathrm [MHz]} \ .
\end{eqnarray}
where $\mu_{0}$ is the permeability of free space, $\mu^{e}$ is the
Bohr magneton, $\mu^{n}$ is the nuclear magneton
and $g_{e}$ and $g_{n}$ are the electron and
nuclear {\it g} factors.~\cite{footnote} The position of the nucleus
is denoted by ${\mathbf r}_{n}$ and $\rho_{\sigma}
=\rho_{\uparrow} - \rho_{\downarrow}$ [bohr$^{-3}$].~\cite{footnote_2}
The anisotropic part of the hyperfine tensor is given by
\begin{eqnarray}
{\bf A}_{p} & = & \frac{\mu_{0}}{4\pi}g_{e}\mu^{e}g_{n}
\mu^{n} \int {\bf T}({\bf r})
\rho_{\sigma}({\bf r} + {\bf r}_{n})
d{\bf r} \\ \nonumber
& = & 12.531\gamma^{n} \int {\bf T}({\bf r})
\rho_{\sigma}({\bf r} + {\bf r}_{n})
d{\bf r} \ {\mathrm [MHz]} \nonumber \,
\end{eqnarray}
where ${\bf T}({\bf r})$ is a traceless tensor,
\begin{equation}
\label{eq:T}
{\bf T}({\bf r}) = \frac{1}{r^5} \left( \begin{array}{ccc}
3x^{2} - r^{2} & 3xy & 3xz \\
3xy & 3y^{2} - r^{2} & 3yz \\
3xz & 3yz & 3z^{2} - r^{2}
\end{array} \right) \ ,
\end{equation}
\noindent and the origin of coordinates is on the nucleus at ${\bf
r}_{n}$. For a particle located precisely at the
BC site the hyperfine interaction has axial symmetry with respect to
the [1 1 1] axis and thus ${\bf A}_{p}$ has the form
\begin{equation}
{\bf A}_{p} = A_{p} \left(
\begin{array}{c@{\hspace{5mm}}c@{\hspace{5mm}}c}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{array} \right) \ ,
\end{equation}
with $A_{p}$ being the anisotropic hyperfine parameter at this site.
For a particle located precisely at the T site, all elements of ${\bf
A}_{p}$ are zero and hence the hyperfine tensor is purely isotropic.
In reality the muon/proton will explore the environment around these sites by
virtue of its zero--point motion and thermal effects. In order to account for
the zero--point motion the hyperfine interaction tensor must be averaged over
the squared modulus of the muon/proton wave function:
\begin{equation}
\label{eqn:mtn_avrge}
\langle {\bf A} \rangle_{\mu} = \int |X({{\mathbf r}_{\mu}};{\mathbf
r}_{n})|^2 {\bf A}({{\mathbf r}_{\mu}}) d{{\mathbf
r}_{\mu}} \ .
\end{equation}
To evaluate the integral for each component of ${\bf A}$ we fit
$A_s(x_{\mu},y_{\mu},z_{\mu})$ and each of the six distinct elements of the
symmetric tensor ${\bf A}_p(x_{\mu},y_{\mu},z_{\mu})$ to polynomial expressions
of the correct symmetry. Since the muon/proton wave function is expanded in
terms of Hermite polynomials, analytic expressions for the elements of
$\langle{\bf A}\rangle_{\mu}$ may be obtained.
For the isotropic hyperfine parameters the polynomial expression for
$A_s(x_{\mu},y_{\mu},z_{\mu})$ has the same symmetry as the relevant potential
energy surface. These parameters were expanded in sixth--order polynomials.
The polynomial describing the isotropic superhyperfine parameter at the BC site
contains terms which are odd in $z_{\mu}$. (Superhyperfine parameters were not
calculated for the T site.) Each of the elements of ${\bf A}_{p}^{\rm T}$ and
${\bf A}_{p}^{\rm BC}$ were fitted to second--order polynomials of the correct
symmetry.
We now consider the symmetry of the hyperfine tensor including the effects of
zero--point motion. Within the double adiabatic approximation the muon motion
is described by the wave function $X({\bf r}_{\mu};{\bf r}_{n})$. The
motion--average of the $\alpha\beta$--component ($\alpha,\beta = x,y,z$) of the
anisotropic hyperfine tensor is given by:
\begin{equation}
\label{eq1}
\langle A_{\alpha\beta}\rangle_{\mu} = C \int \int |X({\bf r}_{\mu};
{\bf r}_{n})|^2 \rho_{\sigma}({\bf r}+ {\bf
r}_{\mu}) \, T_{\alpha\beta}({\bf r}) \, d{\bf r} \, d{\bf r}_{\mu} \;,
\end{equation}
\noindent where $C$ is a constant, $\rho_{\sigma}$ is the electron
spin density, and $T_{\alpha\beta} $ denotes the components of the
tensor ${\bf T}$ defined in Eq.~\ref{eq:T}.
The muon/proton may be said to be trapped in a potential well if its wave
function is negligibly small outside of an equi--potential--energy surface
enclosing the region. The muon wave function, $X$, is the non--degenerate
ground state of the potential well and therefore has the full point--group
symmetry of the well, {\it i.e.},
\begin{equation}
\label{eq2}
P(Q_i)X({\bf r}_{\mu};{\bf r}_{n}) = X({\bf
R}^{-1}_i{\bf r}_{\mu};{\bf r}_{n}) = X({\bf
r}_{\mu};{\bf r}_{n})\;,
\end{equation}
\noindent where $P$ is a scalar transformation operator, $Q_i$ is an
operation of the point group of the well, and ${\bf R}_i$ is the
corresponding transformation matrix. The electron spin density,
$\rho_{\sigma}({\bf r}+{\bf r}_{\mu})$, satisfies
\begin{equation}
\label{eq3}
P(Q_i) \rho_{\sigma}({\bf r}+{\bf r}_{\mu}) = \rho_{\sigma}({\bf
R}^{-1}_i({\bf r}+{\bf r}_{\mu})) = \rho_{\sigma}({\bf r}+{\bf
r}_{\mu}) \;.
\end{equation}
\noindent $\langle A_{\alpha\beta}\rangle_{\mu}$ is unchanged by a
scalar transformation of the integrand, {\it i.e.},
\begin{eqnarray}
\label{eq4}
\langle A_{\alpha\beta}\rangle_{\mu} = C \int \int P(Q_i) \left[|X({\bf
r}_{\mu}; {\bf r}_{n})|^2 \times \right. &&\nonumber \\
\left. \rho_{\sigma}({\bf
r}+{\bf r}_{\mu}) T_{\alpha\beta}({\bf r}) \right] \, d{\bf r} \,
d{\bf r}_{\mu} \;. &&
\end{eqnarray}
\noindent $\langle A_{\alpha\beta}\rangle_{\mu}$ is again unaltered if
we sum over the $i$ operations and divide by their number, $N$,
\begin{eqnarray}
\label{eq5}
\langle A_{\alpha\beta}\rangle_{\mu} = \frac{C}{N}\sum_i \int \int
P(Q_i)\left[|X({\bf r}_{\mu}; {\bf r}_{n})|^2
\times \right. &&\nonumber \\
\left. \rho_{\sigma}({\bf r}+{\bf r}_{\mu})T_{\alpha\beta}({\bf r})
\right] \, d{\bf r} \, d{\bf r}_{\mu} \;. &&
\end{eqnarray}
\noindent Using Eqs.~\ref{eq2} and~\ref{eq3} we have
\begin{eqnarray}
\label{eq6}
\langle A_{\alpha\beta}\rangle_{\mu} = \frac{C}{N} \int \int |X({\bf
r}_{\mu}; {\bf r}_{n})|^2 \rho_{\sigma}({\bf
r}+{\bf r}_{\mu}) \times && \nonumber \\
\left[ \sum_i P(Q_i) \, T_{\alpha\beta}({\bf r})
\right] \, d{\bf r} \, d{\bf r}_{\mu}\;. &&
\end{eqnarray}
The symmetry properties of $\langle A_{\alpha\beta}\rangle_{\mu}$ are
easily obtained from Eq.~\ref{eq6}. For example, $\langle
A_{xy}\rangle_{\mu}$ will be equal to $\langle A_{yz}\rangle_{\mu}$ if
$\sum_i P(Q_i)\,xy = \sum_i P(Q_i)\,yz$. Taking the specific
case of the T site, it is easily shown that $\sum_i P(Q_i)\,xy =
\sum_i P(Q_i)\,yz = 0$, where the sum is over the 24 operations of the
tetrahedral point group. Similar arguments show that all the elements
of $\langle A_{\alpha\beta}\rangle_{\mu}$ are zero for the T site.
Similarly, for the BC site we find that all off--diagonal elements of
$\langle A_{\alpha\beta}\rangle_{\mu}$ are equal, and the diagonal
elements are zero.
If the zero--point motion of the muon is neglected then $|X({\bf
r}_{\mu}; {\bf r}_{n})|^2 = \delta({\bf
r}_{\mu}-{\bf r}_0)$, where ${\bf r}_0$ is the position of the muon.
It follows that if the muon is placed at an invariant point of the
symmetry group of the well, then including the zero--point motion does
not change the symmetry of the anisotropic hyperfine tensor. This
result explains why the zero--point motion does not affect the
symmetry of the anisotropic hyperfine tensor for either the T or BC
sites considered here.
The presence of the muon could lead to a symmetry lowering distortion
of the host lattice, in which case the appropriate point group is the
lower symmetry one. We have not considered the possibility of
symmetry lowering distortions in our calculations because of the
computational cost of evaluating the energy $E^{e}({\bf r}_{\mu},{\bf
r}_{n})$ of Eq.~\ref{eq:e_SE} for the required
atomic configurations. However, we believe such distortions to be
unlikely for the cases considered here.
\section{Results}
\label{results}
\subsection{Static relaxations}
\label{static}
The static relaxations (neglecting zero--point motion) are, of course,
identical for the muon and proton. Calculations with the muon/proton fixed at
the BC site of the 16--atom silicon cell showed that the two nearest neighbours
of the muon/proton relax outwards from the muon/proton by 0.40 \AA \ along the
[1 1 1] axis with the NNNs relaxing by 0.01 \AA \ in the same direction. The
corresponding relaxations for the 54--atom supercell were 0.39 \AA\ and 0.02
\AA. These values are close to the plane--wave pseudopotential results of
Luchsinger {\it et al.}~\cite{Luch97} who obtained relaxations of 0.45 \AA \
for the NN silicon atoms and 0.07 \AA \ for the NNNs. For the 16--atom
germanium cell the corresponding relaxations were 0.44 \AA \ for the NNs and
0.02 \AA \ for the NNNs. Our NN relaxation is in good agreement with the value
of 0.42 \AA \ calculated by Vogel {\it et al.}~\cite{Vogel89}
At the T site, the NN atoms in the host lattice were allowed to relax in the
radial direction. The relaxations in silicon and germanium were approximately
equal and very small; just 0.02 \AA \ {\it towards} the muon/proton in the
16--atom supercell and 0.03 \AA \ (in the same direction) in the 54--atom
supercell. Again, this is in agreement with the ``negligible'' relaxation for
the T site in silicon found by Luchsinger {\it et al.}
\subsection{Relaxations including zero--point motion}
\label{dynamic}
The influence of the zero--point motion of the muon/proton on the relaxation of
the silicon/germanium host lattice was studied by calculating the total energy,
$E^{n}_{\alpha}$, of Eq.~\ref{eq:I_SE}, for different
relaxations of the NN host atoms, as described in Section~\ref{zero}.
For the BC site four different relaxations of the NNs were considered, and for
each of these the six NNNs were also relaxed. The NN relaxations are in
addition to the static relaxations given in Section~\ref{static}. The inclusion
of the zero--point energy of the muon was found to give only a small correction
to the static relaxations; the NN silicon atoms relaxed outwards by just an
additional 0.01 \AA \ in the [1 1 1] direction, so that the final separation of
the muon from a NN silicon atom is 1.58 \AA \ in the 16--atom supercell. The
much smaller zero--point energy of the heavier hydrogen impurity is swamped by
the increase in energy of the crystal as the separation of the NN atoms is
increased and thus there is no additional relaxation. As a check on the
finite--size errors, the energies of five geometries were recalculated using
the 54--atom supercell. These energies and the corresponding potential energy
curves calculated within the 16--atom supercell are shown in
Fig.~\ref{fig:finsize_pe}. The very small differences between the 16--atom and
54--atom results justify the use of the 16--atom supercell in calculations of
the shape of the potential well at the BC site.
The story is very similar for the BC site in germanium. When the zero--point
energy of the muon is included, the relaxation of the NN atoms again increases
by just 0.01 \AA, so that the final separation of the muon from a NN germanium
atom is 1.69 \AA \ in the 16--atom supercell. Once more, the smaller
zero--point energy of the proton means there is no additional relaxation due to
quantum effects. Fig.~\ref{fig:bc_wfns} shows the potential energy well and
calculated wave functions for the muon and proton at the BC site in germanium.
\subsection{Zero--point energies}
\label{zp_energies}
The zero--point energy of the muon at the BC site was calculated to be 0.63~eV
in silicon and 0.56~eV in germanium. It is perhaps surprising that such large
zero--point energies have so little effect on the relaxations. As shown in
Fig.~\ref{fig:bc_pot}, the potential well is narrow in the direction along the
bond and wider perpendicular to the bond. Within the harmonic approximation
one can decompose the zero--point energy into contributions from the well along
and perpendicular to the bond. For a muon at the BC site in silicon this gives
0.47~eV in the direction along the bond and 0.22~eV perpendicular to the bond.
(The sum of these differs from the full zero--point energy of 0.63~eV because
the latter does not assume the harmonic approximation.) The corresponding
energies for germanium are 0.37~eV and 0.22~eV in the directions along and
perpendicular to the bond, respectively. If we were to consider only the
zero--point energy in the direction along the bond then the outwards relaxation
of the silicon/germanium atoms would be larger; approximately 0.03 \AA \ in
silicon and more than 0.025 \AA \ in germanium. Although the component of the
zero--point energy along the bond is significantly reduced by further outward
relaxation of the silicon/germanium atoms, the potential well also gets
narrower in the plane perpendicular to the bond, which tends to increase the
zero--point energy. The narrowing of the potential well in the plane
perpendicular to the bond correlates with the narrowing of the bonding charge
cloud as the bond lengthens.
Our result of 0.63~eV for the zero--point energy of the muon at the BC site in
silicon is close to the value of 0.54~eV obtained by Claxton {\it et
al.}~\cite{Claxton92} from Hartree--Fock calculations on Si$_{26}$H$_{30}$
clusters. In that calculation the potential well at the BC site was assumed to
be cylindrically symmetric about the bond (as it is in this work) and the
resulting Schr\"{o}dinger equation was solved within the harmonic
approximation.
The larger mass of the proton significantly reduces the quantum effects. We
calculated the zero--point energy of a proton at the BC site to be 0.20~eV in
silicon and 0.18~eV in germanium. Our value for silicon is close to that of
0.18~eV obtained by Luchsinger {\it et al.}~\cite{Luch97}, which suggests that
the harmonic approximation to the potential well used in that work is quite
good for the proton ground state.
In contrast to the results of plane--wave pseudopotential
calculations~\cite{Luch97,VdW_Dent89}, we find the T site corresponds to a
local minimum in the potential energy surface. This is, however, in agreement
with a more recent plane--wave pseudopotential study.~\cite{Probert} The
calculated energy surface along the [1 1 1] direction is shown in
Fig.~\ref{fig:T_si_wfns}. It turns out that the muon/proton is not strongly
bound in our potential well, which turns over at the hexagonal site situated at
a distance of 1.18 \AA\ from the T site along the [1 1 1] direction. To
confine the muon/proton in the well we therefore constrained the fit to prevent
the potential turning over, as shown in Fig.~\ref{fig:T_si_wfns}. The
zero--point energy of the muon/proton calculated in such a well is then an
upper bound on the true value, but as the wave function decays quite rapidly
away from the T site this bound is accurate.
In contrast to the BC site, investigation of the finite size effects present in
the calculation of this energy surface showed that while the results in the
16-- and 54--atom supercells were qualitatively similar, they differed
significantly in the openness and depth of the potential well. As a result,
the calculation of the zero--point energy {\it etc.} of the muon/proton at this
site was carried out using the potential well obtained from the 54--atom
supercell. The expense of the LSDA calculations with this supercell made the
generation of data points off the [1 1 1] axis too costly. Therefore, the three
parameters left undetermined after the one--dimensional fit to the data on the
[1 1 1] axis were assigned the values obtained in the fit to the 16--atom
supercell data. This is not a critical choice since the one--dimensional fit
has already constrained the shape of the energy surface in eight directions
(due to the symmetry of the T site). This procedure was also used to generate
a three--dimensional potential--energy function from the fit to data along the
[1 1 1] axis (calculated using the 16--atom supercell) at the T site in
germanium. The parameters obtained from these fits are given in
Table~\ref{tble:T_param}.
For a muon at the T site the zero--point energy was calculated to be 0.28~eV in
silicon and 0.22~eV in germanium. For a proton the corresponding values are
0.09~eV and 0.06~eV. The ground state wave functions of the muon/proton along
the [1 1 1] axis in silicon are shown in Fig.~\ref{fig:T_si_wfns} and those in
germanium in Fig.~\ref{fig:T_ge_wfns}. The results for germanium must be
considered approximate since we have not calculated any data points off the [1
1 1] axis in this case. In addition, the 16--atom supercell was used for all
of the germanium calculations and therefore it follows from the behaviour found
in silicon that the true potential energy surface will be more open than the
one we have obtained.
\subsection{Excited states of the muon and proton}
\label{excited}
For a muon at the BC site in silicon our zero--point energy of 0.63~eV is
considerably smaller than the well depth of 1.37~eV, indicating the possibility
that excited states of the muon may be bound within the well. Numerical
calculations show a two--fold degenerate first excited state at an energy of
0.84~eV. The wave functions of these states are similar to those obtained from
a harmonic approximation, {\it i.e.}, they consist essentially of an excitation
within the plane perpendicular to the bond. The energy of the first excited
state is also reasonably well described within the harmonic approximation which
predicts the excited state to be 0.22~eV above the ground state. It is also
possible that some of the higher energy states are bound within the well. In
germanium, the potential well at the BC site is 1.51~eV deep. The first
excited state is two--fold degenerate with an energy of 0.78~eV and is of the
same character as in silicon.
For the more massive proton the excited states are of lower energy. At the BC
site in silicon, the two--fold degenerate first excited state of the proton has
an energy of 0.27~eV, which is 0.07~eV higher than the ground state, while in
germanium the excited state has an energy of 0.25~eV, which is also 0.07~eV
above the ground state.
Each excited state of the muon/proton defines a different adiabatic potential
for the nuclei ({\it i.e.}, a different $E^{\mu}_{\alpha}({\bf
r}_{n})$ in Eq.~\ref{eq:H_I}). It is therefore possible for
the lattice relaxations that occur when the muon/proton is in its first excited
state (say) to be different from those for the ground state. For instance, the
fact that the wave function of the first excited state of the muon/proton is
essentially an excitation in the plane perpendicular to the bond, combined with
the fact that the potential well in this plane becomes narrower as the
separation of the NN atoms increases, results in the NN atoms actually relaxing
towards the impurity. This relaxation is small for the muon, but effectively
zero for the proton due to the smaller zero--point energy. The effect on the
energies of the excited states is negligible.
At the T site in silicon, the potential well is considerably shallower with a
depth of only 0.20~eV. For the muon this means that even the ground state
energy (0.28~eV) of our constrained potential well (which is an upper bound on
the true ground state energy) is greater than the well depth. The proton,
however, has a (triply degenerate) first excited state with an energy of
0.14~eV which may therefore be bound at the T site. In germanium the potential
well at the T site has a depth of just 0.18~eV in our 16--atom supercell
calculations. It follows from the behaviour found in going from the 16--atom to
the 54--atom supercell in silicon that the true well depth in germanium will
probably be less than this. It is therefore unlikely that excited states of
either the muon or the proton will be bound at the T site in germanium.
\subsection{Energy barriers at the T and BC sites}
\label{barriers}
The heights of the energy barriers confining the muon and proton at the BC and
T sites are clearly of great importance in determining the dynamics of the
impurities within the lattice and hence are a significant part of the
configuration--coordinate diagram.
The static barriers ({\it i.e.}, excluding zero--point effects) experienced by
the muon and proton are identical. For the BC site in silicon we calculate the
static barrier to motion towards the hexagonal site (in the [$-$1 1 0]
direction) to be 1.37~eV while in germanium it is 1.51~eV. The effective
barrier height is reduced by the zero--point energy and therefore depends on
the nature of the impurity. Including this effect, the effective barrier
experienced by a muon at the BC site in silicon is 0.74~eV while in germanium
it is 0.95~eV. For the proton, the effective barriers (1.17~eV in silicon and
1.33~eV in germanium) are higher. The effective barrier height for the muon at
the BC site may be considered a measure of the barrier to the BC$\rightarrow$T
site transition. In reality this transition is believed to involve charged
states: muonium at the BC site is first ionized (with activation energy
0.22~eV~\cite{Kreitzman95}) and then moves to the T site while simultaneously
recapturing an electron to regain its neutral charge state. The sum of the
activation and barrier energies for these two processes as measured
experimentally is 0.60~eV.~\cite{Kreitzman95}
At the T site the energy barriers are very much lower. In silicon the static
barrier to motion of the muon towards the hexagonal site (in the [1 1 1]
direction) is calculated to be 0.20~eV while in germanium it is 0.18~eV. When
zero--point effects are taken into account, the effective barriers for the muon
at the T site in silicon and germanium are zero indicating that, even at
$T=0$K, the muon is free to diffuse through the interstitial region. However,
this barrier is not appropriate for the T$\rightarrow$BC site transition
because in our calculations for the muon in the interstitial region, the host
atoms around the BC site are unrelaxed. The process by which these atoms relax
the large distances required to allow the muon to move to the BC site is
unclear. Experimentally, the barrier for the T$\rightarrow$BC site transition
in silicon is 0.39~eV.~\cite{Kreitzman95}
Since the zero--point energy of the proton is around a third of that of the
muon, these calculations indicate that it will be bound at the T site in both
silicon and germanium with an effective barrier of around 0.12~eV in each case.
As previously discussed the true effective barrier in germanium will probably
be lower than this.
\subsection{Hyperfine and superhyperfine parameters}
\label{hyperfine}
The hyperfine parameters depend on the spin density in the region at and around
the atomic nuclei. More specific insight into the origin of the large measured
differences between hyperfine parameters for muons located at the two impurity
sites can be gained from a consideration of the spin density isosurfaces.
Fig.~\ref{fig:spin_density} shows spin density contour plots for silicon in
appropriate planes encompassing the BC and T sites. Evidently the majority spin
density around an impurity placed at the BC site is largely dispersed onto the
two nearest--neighbour silicon atoms; the spin density in a small region around
the hydrogen nucleus is comparatively small and of opposite sign. At the T
site, by contrast, almost all of the majority spin density is localized on the
defect. From these calculations one therefore expects the isotropic hyperfine
parameter at the two sites to be of opposite sign, with the magnitude of the
parameter at the BC site much smaller than at the T site.
It is of course necessary to check the dependence of calculated hyperfine and
superhyperfine parameters on the supercell size and basis set quality. A set of
computed numbers are shown in Table~\ref{tble:hyp_convrg}. The parameters
appear to be reasonably well converged with respect to the basis set, but the
convergence with increasing supercell size is less good, particularly for the
isotropic hyperfine and superhyperfine parameters at the BC site.
Table~\ref{tble:hyp_BC} gives the hyperfine and superhyperfine parameters
calculated at the BC site in both silicon and germanium together with the
results of other calculations for comparison. Without motion averaging, the
values obtained for silicon using the LSDA approximation are in reasonable
agreement with both experiment and other DFT calculations.
Both the hyperfine and superhyperfine motion--averaged tensors ($\langle {\bf
A}_{p_{\mu}}^{\rm BC} \rangle$ and $\langle {\bf A}_{p_{\rm Si}}^{\rm BC}
\rangle$) were found to be axially symmetric about the Si--Si bond ([1 1 1]
direction) in agreement with the experimental results. As Luchsinger {\it et
al.}~\cite{Luch97} found, motion averaging increases the values of all but one
(the anisotropic hyperfine parameter) of the hyperfine and superhyperfine
parameters, with the isotropic (contact) term on the muon being the most
sensitive. This is because of the very small contact charge density which
varies quite significantly with the muon position (Fig.~\ref{fig:hyper_bc}). In
agreement with Luchsinger {\it et al.}~\cite{Luch97}, use of the
Perdew--Wang~\cite{GGA} GGA functional did not consistently improve the values
of the parameters. The results obtained for the muon at the BC site in
germanium follow a similar pattern.
The calculated hyperfine parameters for the T site are given in
Table~\ref{tble:hyp_T}. For silicon our values are in good agreement with
both experiment and previous calculations. Again use of the
Perdew--Wang~\cite{GGA} GGA functional fails to improve this agreement. For
germanium, our value of the isotropic hyperfine parameter at the T site also
agrees quite well with the measured value.
The behaviour of the isotropic hyperfine parameter along the [1 1 1] axis in
the vicinity of the T site in silicon and germanium is shown in
Fig.~\ref{fig:hyper_t}. Motion averaging for the muon/proton at the T site
reduces the isotropic hyperfine parameter in both silicon and germanium. The
final motion--averaged results are in reasonable agreement with experiment.
Motion averaging of ${\bf A}_{p_{\mu}}^{\rm T}$ resulted in an isotropic
tensor, in agreement with the symmetry arguments presented in
Section~\ref{hyperfine_mthd} and experimental observations.
In a recent application of the path--integral Monte Carlo approach, Miyake {\it
et al.}~\cite{Miyake_etal98} studied hydrogen and muonium at the T site in
silicon, with the electron--electron interactions calculated within the LDA.
They found the T site to be a local maximum in the potential energy surface, in
agreement with Luchsinger {\it et al.}~\cite{Luch97} but in disagreement with
our results and a recent plane--wave pseudopotential
calculation.~\cite{Probert} Their path--integral Monte Carlo study showed that
quantum effects led to the muonium distribution being centred on the T site
while hydrogen behaved as a largely classical particle and was thus distributed
away from the local maximum on that site. Evaluating the motion--averaged
isotropic hyperfine parameter with our hyperfine data gives a value of 492~MHz
for the hydrogen distribution of Miyake {\it et al.}, but 685~MHz with our
hydrogen distribution. Therefore if one could measure the isotropic hyperfine
signal of {\it hydrogen} at the T site, one could deduce whether the T site is
a maximum or minimum in the potential energy surface.
\subsection{Energies of a muon/proton at the T and BC sites}
\label{energies}
The question of the relative stabilities of the muon and proton at the BC and T
sites is of considerable interest. For a particular impurity this energy
difference is the sum of contributions from the static--lattice energy and the
zero--point energy. The contribution from the static lattice
is sensitive to the size of the supercell and to the quality of the basis set.
We investigated this point using the 16-- and 54--atom supercells. We have
added a BSSE correction to each of the static--lattice energy differences
quoted here. With the 16--atom silicon cell and the standard basis set, the T
site was found to be 0.63~eV lower in energy than the BC site. Using the large
basis set reduced this to 0.32~eV. In the 54--atom supercell and using the
standard basis set the T site was 0.41~eV lower in energy than the BC site.
With the large basis set, this was reduced to just 0.07~eV. These results
indicate that a 16--atom supercell is too small to give reliable estimates of
the static--lattice energy difference between the two sites. A summary of the
computed energies that influence the relative stabilities is given in
Table~\ref{tble:energy_summary}.
There have been several previous calculations of the static--lattice energy
difference between the T and BC sites in silicon. Using a plane--wave
pseudopotential method and the LSDA, Chang and Chadi~\cite{chang89} found the T
site to be lower in energy, but only by an amount $\leq$0.25~eV. Luchsinger
{\it et al.}~\cite{Luch97}, also using a plane--wave pseudopotential method,
found the T site to be 0.15~eV higher in energy than the BC site within the
LSDA and 0.19~eV higher within the GGA. Note, however, that Luchsinger {\it et
al.}~\cite{Luch97} found the T site to be a local maximum in the energy and
that a nearby site has an energy about 0.05~eV lower. It is clear from the
various results that the static--lattice energy difference between the T and BC
sites in silicon is small within the LSDA/GGA, but its precise value has yet to
be settled.
The fact that the static--lattice energy difference is small means that the
zero--point energy of the impurity is crucial in determining the relative
stability of the T and BC sites. For a muon in silicon we have found the
zero--point energy at the BC site to be 0.35~eV larger than at the T site.
This difference is large enough to to make the BC site unfavourable for the
muon, irrespective of which of the above values for the static--lattice energy
difference is used. However, the zero--point energy of the proton at the BC
site in silicon is only 0.12~eV higher than at the T site. Therefore, for this
impurity the relative stability of the two sites depends on the precise value
of the static--lattice energy difference.
In germanium with a 16--atom supercell, the difference in static lattice
energies favours the T site by an energy of 0.57~eV. The convergence with
respect to supercell size found in silicon suggests that this difference in a
fully converged LSDA calculation would be smaller. We estimate that the
zero--point energy of a muon at the BC site is 0.34~eV larger than at the T
site (where the form of the potential energy surface was obtained from data
calculated along the [1 1 1] axis only). For a proton the corresponding value
is 0.12~eV. These results are similar to those obtained in silicon and thus it
is likely that for the muon the T site is lower in energy. Without a fully
converged value for the static lattice energy difference we are unable to draw
any conclusions on the lowest energy site of the proton.
\section{Conclusions}
\label{conclusions}
We have calculated the zero--point motions and energies as well as the
hyperfine parameters of both muonium and hydrogen when present as impurities in
silicon and germanium crystals at the BC and T sites. The electron,
muon/proton and ion motions were decoupled using a double adiabatic
approximation, and for the BC site we have included the effect of the
zero--point motion on the relaxation of the host lattice. The ground states of
both the muon and proton at the BC sites of silicon and germanium are strongly
confined within a potential well of depth 1.37~eV (silicon) and 1.51~eV
(germanium). The zero--point energy of a muon at the BC site is calculated to
be 0.63~eV for silicon and 0.56~eV for germanium. Despite the relatively large
zero--point energy of the muon at the BC site, it causes only a small
additional outwards relaxation of the nearest--neighbour silicon/germanium
atoms of about 0.01 \AA. For the proton the additional relaxations of the
nearest neighbours due to zero--point motion are negligible. At the T site the
static relaxations of the host atoms are very small and the zero--point energy
is considerably smaller than at the BC site, being 0.28~eV (0.22~eV) for a muon
in silicon (germanium). It is therefore reasonable to assume that the
additional relaxation due to the zero--point motion is negligible for either a
muon or proton at the T site.
The relaxation of the crystal around either the BC or T sites is practically
independent of whether the impurity is a muon or a proton. This result confirms
one of the underlying assumptions of the widely accepted configuration
coordinate model.~\cite{config} The potential well at the BC sites of both
silicon and germanium is reasonably well described by a harmonic approximation,
at least for the ground states of the muon and proton. The potential well at
the BC site in both materials is deep enough to bind several excited states of
the muon and proton, although we are not aware of any experimental evidence for
such states. The potential well at the T site in either silicon or germanium is
not deep enough to bind the muon which is free to diffuse through the
interstitial region, although our calculations suggest that the proton is bound
at this site at $T=0$~K.
Various LSDA and GGA calculations have indicated that the energies for a static
muon or proton at the BC and T sites in silicon are very similar. However, we
have calculated the difference in zero--point energies of a muon at the T and
BC sites in silicon (germanium) to be 0.35~eV (0.34~eV) which is sufficient to
make the T site more stable, whether we assume our value for the
static--lattice energy difference between the BC and T sites or those of
others.~\cite{chang89,Luch97} This result is in conflict with the
interpretation of experimental data.
The hyperfine parameters calculated for silicon in our all--electron
calculations are close to those obtained in plane--wave pseudopotential
calculations. This agreement confirms that the procedure used to correct for
the pseudopotential and for the incomplete plane--wave basis sets are
accurate. For silicon our static LSDA results are in reasonable agreement with
other LSDA results and experiment. Our hyperfine parameter for the muon at the
T site in germanium is in much better agreement with experiment than the only
previous calculation of which we are aware.~\cite{casarin91}
In our work the motion averages of the hyperfine and superhyperfine parameters
are evaluated by averaging over the squared modulus of the wave function
obtained from the full solution of the muon/proton Schr\"odinger equation in
the potential well. We note that the symmetry of the potential wells requires
that the symmetry of the motion--averaged hyperfine tensors at the T and BC
sites are the same as if the muon/proton was situated exactly at the sites. We
have obtained detailed information about the variation of the hyperfine and
superhyperfine parameters with the position of the muon/proton. Our results
show that motion averaging for the muon/proton at the BC site in silicon and
germanium increases the values of all of the hyperfine and superhyperfine
parameters apart from the anisotropic hyperfine term which decreases slightly,
in agreement with the conclusions of Luchsinger {\it et al}.~\cite{Luch97}
With the exception of the isotropic hyperfine term however, all of the changes
are small. At the T sites in silicon and germanium, motion averaging reduces
the isotropic hyperfine parameter.
\section{Acknowledgments}
We thank R.~Q.~Hood and M.~I.~J.~Probert for useful discussions.
Financial support was provided by the Engineering and Physical
Sciences Research Council (UK).
\bibliographystyle{plain}
|
1,314,259,995,858 | arxiv | \section{Introduction}\label{sec:intro}
A graph $G=(V,E)$ is \emph{$k$-thin} if there exist an ordering
$v_1, \dots , v_n$ of $V$ and a partition of $V$ into $k$ classes
$(V^1,\dots,V^k)$ such that, for each triple $(r,s,t)$ with
$r<s<t$, if $v_r$, $v_s$ belong to the same class and $v_t v_r \in
E$, then $v_t v_s \in E$. The minimum $k$ such that $G$ is
$k$-thin is called the \emph{thinness} of $G$ and denoted by
$\thin(G)$.
The thinness is unbounded on the class of all graphs, and graphs
with bounded thinness were introduced in~\cite{M-O-R-C-thinness}
as a generalization of interval graphs.
In~\cite{B-D-thinness}, the concept of \emph{proper thinness} is
defined in order to obtain an analogous generalization of proper
interval graphs, and it is proved that the proper thinness is
unbounded on the class of interval graphs. A graph $G=(V,E)$ is
\emph{proper $k$-thin} if there exist an ordering $v_1, \dots ,
v_n$ of $V$ and a partition of $V$ into $k$ classes
$(V^1,\dots,V^k)$ such that, for each triple $(r,s,t)$ with
$r<s<t$, if $v_r$, $v_s$ belong to the same class and $v_t v_r \in
E$, then $v_t v_s \in E$, and if $v_s$, $v_t$ belong to the same
class and $v_t v_r \in E$, then $v_s v_r \in E$. The minimum $k$
such that $G$ is proper $k$-thin is called the \emph{proper
thinness} of $G$ and denoted by $\pthin(G)$.
{The parameters of thinness and proper thinness represent how far
a graph is from being an interval and proper interval graph,
respectively. The class of (proper) $1$-thin graphs is that of
(proper) interval graphs. This is so because, considering a
$1$-partitioning, a (strongly) consistent ordering is sufficient
to characterize (proper) interval
graphs~\cite{Ola-interval,Rob-uig}.}
When a representation of the graph as a $k$-thin graph is given,
for a constant value $k$, a wide family of NP-complete problems
can be solved in polynomial time, and the family of problems can
be enlarged when a representation of the graph as a proper
$k$-thin graph is given, for a constant value
$k$~\cite{B-D-thinness,B-M-O-thin-tcs,M-O-R-C-thinness}. Hardness
results for relaxed versions of this family of problems are shown
in~\cite{Bentz-thin}, even for classes of graphs with thinness one
or two.
Many operations are defined over graphs, and some of them arise in
structural characterizations of particular graph families or graph
classes, for example union and join in the case of
cographs~\cite{CorneilLerchsStewart81}. The operations known as
\emph{graph products} are those for which the graph obtained by
the operation of graphs $G_1$ and $G_2$ has as vertex set the
Cartesian product $V(G_1) \times V(G_2)$. Different graph products
are determined by the rules that define the edge set of the
obtained graph. The main properties of these products are surveyed
in~\cite{I-K-prod}.
There is a wide literature about the behavior of graph parameters
under graph operations, and in particular graph products. For the
chromatic number, it includes the famous conjecture of Hedetniemi
(1966)~\cite{Hed-conj}, that remained open more than fifty years,
it was shown to hold for many particular classes, and was recently
disproved by Shitov~\cite{Shitov2019}. Other results on the
chromatic number and its variations in product graphs can be found
in~\cite{A-H-L-prod,Mat-product,B-K-T-V-lagos15-dam,C-G-H-L-M-prod,C-F-prod,G-S-lex,Har,H-M-prod,I-K-prod,I-K-R-Cart,J-P-prod,J-T-color,
Kla-col-prod,K-P-b-col,K-M-prod1,K-M-prod2,Sabidussi1964,V-V-col,Zhu-frac},
and on domination in product graphs
in~\cite{H-R-dom-prod,Hed-thesis,I-K-prod,I-K-R-Cart}. For width
parameters, there is a recent paper studying the boxicity and
cubicity of product graphs~\cite{C-I-M-R-box-prod}.
In~\cite{B-D-thinness}, the behavior of the thinness and proper
thinness under the graph operations union, join, and Cartesian
product is studied. These results allow, respectively, to fully
characterize $k$-thin graphs by forbidden induced subgraphs within
the class of cographs, and to show the polynomiality of the
$t$-rainbow domination problem for fixed $t$ on graphs with
bounded thinness.
In this paper, we give bounds for the thinness and proper thinness
of union and join of graphs, as well as the thinness and proper
thinness of the lexicographical, Cartesian, direct, strong,
disjunctive, modular, homomorphic and hom-products of graphs in
terms of invariants of the component graphs. We also show that in
some cases such bounds do not exist. Furthermore, we describe new
general lower and upper bounds for the thinness of graphs. Also,
we consider the concepts of independent and complete (proper)
thinness, corresponding to the situations in which the classes are
all independent or complete sets. Several of the results on the
bounds of products of graphs are given additionally for these
cases.
The organization of the paper is as follows. In
Section~\ref{sec:thin} we state the main definitions and present
some basic results on thinness and proper thinness. In
Section~\ref{sec:families}, we determine the (proper) thinness of
some graph families, and prove some lower and upper bounds for the
parameters. Section~\ref{sec:thin-and-oper} contains the main
results of the paper, namely bounds of (proper) thinness for
different binary operations, in terms of the (proper) thinness of
their factors. Some concluding remarks form the last section.
\section{Definitions and basic results}\label{sec:thin}
All graphs in this work are finite, undirected, and have no loops
or multiple edges. For all graph-theoretic notions and notation
not defined here, we refer to West~\cite{West}. Let $G$ be a
graph. Denote by $V(G)$ its vertex set, by $E(G)$ its edge set, by
$\overline G$ its complement, by $\Delta(G)$ (resp. $\delta(G)$)
the maximum (resp. minimum) degree of a vertex in $G$. A graph is
\emph{$k$-regular} if every vertex has degree $k$.
Denote by $N(v)$ the neighborhood of a vertex $v$ in $G$, and by
$N[v]$ the closed neighborhood $N(v)\cup\{v\}$. If $X \subseteq
V(G)$, denote by $N(X)$ the set of vertices of $G$ having at least
one neighbor in $X$. A vertex $v$ of $G$ is \emph{universal}
(resp. \emph{isolated}) if $N[v] = V(G)$ (resp. $N(v) =
\emptyset$).
{An \emph{homogeneous set}} is a proper subset $X \subset V(G)$ of
at least two vertices such that every vertex not in $X$ is
adjacent either to all the vertices in $X$ or to none of them.
Denote by $G[W]$ the subgraph of $G$ induced by $W\subseteq V(G)$,
and by $G - W$ or $G \setminus W$ the graph $G[V(G)\setminus W]$.
A subgraph $H$ (not necessarily induced) of $G$ is a
\emph{spanning subgraph} if $V(H)=V(G)$.
Denote the size of a set $S$ by $|S|$. A \emph{clique} or
\emph{complete set} (resp.\ \emph{stable set} or \emph{independent
set}) is a set of pairwise adjacent (resp.\ nonadjacent) vertices.
We use \emph{maximum} to mean maximum-sized, whereas
\emph{maximal} means inclusion-wise maximal. The use of
\emph{minimum} and \emph{minimal} is analogous. The size of a
maximum clique (resp.\ stable set) in a graph $G$ is denoted by
$\omega(G)$ (resp.\ $\alpha(G)$).
A \emph{vertex cover} is a set $S$ of vertices of a graph $G$ such
that each edge of $G$ has at least one endpoint in $S$. Denote by
$\tau(G)$ the size of a minimum vertex cover in a graph $G$.
A graph is called \emph{trivial} if it has only one vertex. A
graph is \emph{complete} if its vertices are pairwise adjacent.
Denote by $K_n$ the complete graph of size~$n$.
Let $H$ be a graph and $t$ a natural number. The disjoint union of
$t$ disjoint copies of the graph $H$ is denoted by $tH$. In
particular, $\overline{tK_2}$ is the complement of a matching of
size~$t$. Denote by $\mim(G)$ the size of a maximum induced
matching of a graph $G$.
{Denote by $P_n$ the path on $n$ vertices.} Given a connected
graph $G$, let $\lip(G)$ be the length of the longest induced path
of $G$, and $\diam(G)$ its diameter. A graph is a \emph{cograph}
if it contains no induced $P_4$.
For a positive integer $r$, the \emph{$(r \times r)$-grid} $GR_r$
is the graph whose vertex set is $\{(i,j) : 1 \leq i, j \leq r\}$
and whose edge set is $\{(i,j)(k,l) : |i - k| + |j - l| = 1,
\mbox{ where } 1 \leq i,j,k, l \leq r \}$.
The \emph{crown graph} $CR_n$ {(also known as Hiraguchi graph)} is
the graph on $2n$ vertices obtained from a complete bipartite
graph $K_{n,n}$ by removing a perfect matching.
A \emph{dominating set} in a graph is a set of vertices such that
each vertex outside the set has at least one neighbor in the set.
A \emph{coloring} of a graph is an assignment of colors to its
vertices such that any two adjacent vertices are assigned
different colors. The smallest number $t$ such that $G$ admits a
coloring with $t$ colors (a \emph{$t$-coloring}) is called the
\emph{chromatic number} of $G$ and is denoted by $\chi(G)$. A
coloring defines a partition of the vertices of the graph into
stable sets, called \emph{color classes}.
A graph $G(V,E)$ is a \emph{comparability graph} if there exists
an ordering $v_1, \dots , v_n$ of $V$ such that, for each triple
$(r,s,t)$ with $r<s<t$, if $v_r v_s$ and $v_s v_t$ are edges of
$G$, then so is $v_r v_t$.\ Such an ordering is a
\emph{comparability ordering}. A graph is a \emph{co-comparability
graph} if its complement is a comparability graph.
In the context of thinness, an ordering $v_1, \dots , v_n$ of
$V(G)$ and a partition of $V(G)$ satisfying that for each triple
$(r, s, t)$ with $r <s<t$, if $v_r$, $v_s$ belong to the same
class and $v_tv_r \in E(G)$, then $v_tv_s \in E(G)$, are said to
be \emph{consistent}. If both $v_1, \dots , v_n$ and $v_n, \dots ,
v_1$ are consistent with the partition, the partition and the
ordering $v_1,\dots,v_n$ are said to be \emph{strongly
consistent}. Notice that a graph is (proper) $k$-thin if and only
if it admits a vertex ordering and a vertex partition into $k$
classes that are (strongly) consistent.
We will often use the following definitions and results.
Let $G$ be a graph and ${<}$ an ordering of its vertices. The
graph $G_{<}$ has $V(G)$ as vertex set, and $E(G_{<})$ is such
that for $v < w$ in the ordering, $vw \in E(G_{<})$ if and only if
there is a vertex $z$ in $G$ such that $w < z$ in the ordering,
$zv \in E(G)$ and $zw \not \in E(G)$. Similarly, the graph
$\tilde{G}_{<}$ has $V(G)$ as vertex set, and $E(\tilde{G}_{<})$
is such that for $v < w$ in the ordering, $vw \in
E(\tilde{G}_{<})$ if and only if either $vw \in E(G_{<})$ or there
is a vertex $x$ in $G$ such that $x < v$ in the ordering, $xw \in
E(G)$ and $xv \not \in E(G)$. An edge of $G_{<}$ (respectively
$\tilde{G}_{<}$) represents that its endpoints cannot belong to
the same class in a vertex partition that is consistent
(respectively strongly consistent) with the ordering ${<}$.
\begin{theorem}\label{thm:thin-comp-order}\cite{B-D-thinness,B-M-O-thin-tcs} Given a graph $G$ and an ordering ${<}$ of its vertices,
the graphs $G_{<}$ and $\tilde{G}_{<}$ have the following
properties:
\begin{enumerate}[(1)]
\item\label{item:2} the chromatic number of $G_{<}$ (resp.
$\tilde{G}_{<}$) is equal to the minimum integer $k$ such that
there is a partition of $V(G)$ into $k$ sets that is consistent
(resp. strongly consistent) with the order ${<}$, and the color
classes of a valid coloring of $G_{<}$ (resp. $\tilde{G}_{<}$)
form a partition consistent (resp. strongly consistent) with
${<}$;
\item $G_{<}$ and $\tilde{G}_{<}$ are co-comparability graphs;
\item if $G$ is a co-comparability graph and ${<}$ a comparability
ordering of $\overline{G}$, then $G_{<}$ and $\tilde{G}_{<}$ are
spanning subgraphs of $G$.
\end{enumerate}
\end{theorem}
Since co-comparability graphs are perfect~\cite{Meyn-co-comp},
$\chi(G_{<})=\omega(G_{<})$ and
$\chi(\tilde{G}_{<})=\omega(\tilde{G}_{<})$. We thus have the
following.
\begin{corollary}\label{cor:thin-comp-order} Let $G$ be a graph, and $k$ a positive integer.
Then $\thin(G) \geq k$ (resp. $\pthin(G) \geq k$) if and only if,
for every ordering ${<}$ of $V(G)$, the graph $G_{<}$ (resp.
$\tilde{G}_{<}$) has a clique of size $k$.
\end{corollary}
We will define also two new concepts related to (proper) thinness:
independent (proper) thinness and complete (proper) thinness.
These concepts are involved in some of the bounds of
Section~\ref{sec:thin-and-oper}.
A graph $G=(V,E)$ is \emph{$k$-independent-thin} if there exist an
ordering of $V$ and a partition of $V$ into $k$ classes,
consistent with the ordering, and such that each class is an
independent set of the graph. The minimum $k$ such that $G$ is
$k$-independent-thin is called the \emph{independent thinness} of
$G$ and is denoted by $\indthin(G)$. Similarly, we can define the
concept of \emph{proper $k$-independent-thin} and
\emph{independent proper thinness} (denoted by $\indpthin(G)$),
where the partition has to be consistent with the ordering and its
reverse. Exchanging independent set by complete set, we define the
concepts of \emph{$k$-complete-thin}, \emph{complete thinness}
(denoted by $\compthin(G)$), \emph{proper $k$-complete-thin} and
\emph{complete proper thinness} (denoted by $\comppthin(G)$).
\begin{remark}\label{rem:co-comp-ind-thin} Notice that $\indpthin(G) \geq \indthin(G) \geq \chi(G)$
and, by Theorem~\ref{thm:thin-comp-order}, $\indpthin(G) =
\indthin(G) = \chi(G)$ when $G$ is a co-comparability graph.
Indeed, we can also see the independent (proper) thinness as a
coloring problem in a graph whose vertex set is $V(G)$ and whose
edge set is $E(G) \cup E(G_{<})$ (resp. $E(G) \cup
E(\tilde{G}_{<})$). Similarly, $\comppthin(G) \geq \compthin(G)
\geq \chi(\overline{G})$ and we can see the complete (proper)
thinness as a coloring problem in a graph whose vertex set is
$V(G)$ and whose edge set is $E(\overline{G}) \cup E(G_{<})$
(resp. $E(\overline{G}) \cup E(\tilde{G}_{<})$).
Theorem~\ref{thm:tK2} and Corollary~\ref{cor:crown} show that the
bounds $\compthin(G) \geq \chi(\overline{G})$ and $\indthin(G)
\geq \chi(G)$ can be arbitrarily bad. Notice also that, given a
(proper) $k$-thin representation of a graph, we can split each
class into independent sets and obtain a (proper)
$k$-independent-thin representation. Thus $\opindthin(G) \leq
\chi(G)\opthin(G)$. Analogously, $\opcompthin \leq
\chi(\overline{G})\opthin(G)$.
\end{remark}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\node (max) at (0,4) {$|V(G)|$};
\node (a) at (-2,2) {$\indpthin(G)$};
\node (c) at (2,2) {$\comppthin(G)$};
\node (d) at (-2,0) {$\indthin(G)$};
\node (e) at (0,0) {$\pthin(G)$};
\node (f) at (2,0) {$\compthin(G)$};
\node (min) at (0,-2) {$\thin(G)$};
\draw (min) -- (d) -- (a) -- (max)
(e) -- (min) -- (f) -- (c) -- (max);
\draw[preaction={draw=white, -,line width=6pt}] (a) -- (e) -- (c);
\end{tikzpicture}
\end{center}
\caption{Hasse diagram of the parameters involved (if $\alpha$
precedes $\beta$ in the diagram, then $\alpha\leq \beta$). We will
state the strongest results, and the consequences for other
parameters can be deduced from the diagram.}\label{fig:hasse}
\end{figure}
\section{Thinness of some graph families and general
bounds}\label{sec:families}
In this section, we determine or give lower bounds for the
thinness and proper thinness of families of graphs, as induced
matchings, crowns, and hypercubes. In addition, we determine both
general lower and upper bounds for the thinness and proper
thinness of graphs. Also, we relate the thinness to the
independence and clique numbers of graphs.
For the complement of an induced matching, the exact value of the
thinness is known.
\begin{theorem}\cite{C-M-O-thinness-man}\label{thm:tK2} For every $t \geq
1$, $\thin(\overline{tK_2})=t$. \end{theorem}
The vertex partition used when proving the part
``$\thin(\overline{tK_2})=t$'' of the equation of
Theorem~\ref{thm:tK2}, which is consistent with any vertex
ordering, is the one where each class consists of a pair of
nonadjacent vertices. It is easy to see that this partition is
also strongly consistent with any vertex ordering. So we have the
following corollary.
\begin{corollary}\label{thm:tK2prop} For every $t \geq
1$,
$\pthin(\overline{tK_2})=\indthin(\overline{tK_2})=\indpthin(\overline{tK_2})=t$.
\end{corollary}
\subsection{Lower bounds}
{ Let $G_1$ and $G_2$ be graphs on $n$ vertices, and $f: V(G_1)
\to V(G_2)$ a bijection. Let $G_1 \boxminus_f G_2$ with vertex set
$V(G_1) \cup V(G_2)$, such that $V(G_i)$ induces $G_i$ for $i =
1,2$, and the edges from $V(G_1)$ to $V(G_2)$ are exactly
$\{vf(v)\}_{v \in V(G_1)}$. When $G_1$ or $G_2$ is either the
complete graph $K_n$ or the empty graph $nK_1$, we can omit $f$ by
symmetry. Notice that $\overline{tK_2} = \overline{tK_1 \boxminus
tK_1}$, and the \emph{crown} $CR_n = \overline{K_n \boxminus
K_n}$.
\begin{theorem}\label{thm:gcrown}
For $G_1$ and $G_2$ graphs on $n$ vertices and $f: V(G_1) \to
V(G_2)$ a bijection, $\thin(\overline{G_1 \boxminus_f G_2}) \geq
n/2$.
\end{theorem}
\begin{proof}
Let $G = \overline{G_1 \boxminus_f G_2}$, and $<$ an arbitrary
ordering of its vertices. We will show that $\omega(G_{<}) \geq
n/2$. Let $A = V(G_1)$ and $A'=V(G_2)$, and for each vertex $v \in
A$, let $v' = f(v)$ (the only vertex in $A'$ that is not adjacent
to $v$ in $G$).
By definition of $G_{<}$, if $v < v'$ then $v$ is adjacent in
$G_{<}$ to every vertex $w$ in $A$ such that $w < v$, and to every
vertex $w$ in $A$ such that $v < w < w'$. Analogously, if $v' <
v$, then $v'$ is adjacent in $G_{<}$ to every vertex $w'$ in $A'$
such that $w' < v'$, and to every vertex $w'$ in $A'$ such that
$v' < w' < w$. Therefore, the vertices $v \in A$ such that $v <
v'$ form a clique in $G_{<}$, and the vertices $v' \in A'$ such
that $v' < v$ form a clique in $G_{<}$. Since for each of the $n$
pairs of vertices $v,v'$ one of the inequalities holds, by the
pigeonhole principle, $G_{<}$ has a clique of size at least $n/2$.
\end{proof}
\begin{corollary}\label{cor:crown}
For every $n \geq 1$, $\thin(CR_n) \geq n/2$.
\end{corollary}
Since a \emph{fat spider} is the graph $\overline{K_n \boxminus
nK_1}$, the theorem above implies that split graphs have unbounded
thinness.}
\medskip
The \emph{vertex isoperimetric peak} of a graph $G$, denoted as
$b_v(G)$, is defined as $b_v(G) = \max_s \min_{X\subset V, |X|=s}
|N(X) \cap (V(G) \setminus X)|$, i.e., the maximum over $s$ of the
lower bounds for the number of boundary vertices (vertices outside
the set with a neighbor in the set) in sets of size $s$.
\begin{theorem}\cite{C-M-O-thinness-man}\label{thm:peak}
For every graph $G$ with at least one edge, $\thin(G)\geq
b_v(G)/\Delta(G)$.
\end{theorem}
The thinness of the grid $GR_r$ was lower bounded by using
Theorem~\ref{thm:peak}.
\begin{corollary}\cite{C-M-O-thinness-man}\label{cor:grid} For every $r \geq
2$, $\thin(GR_r) \geq r/4$.
\end{corollary}
We will prove next some other lower bounds, which are very useful
for bounding the thinness of highly symmetric graphs, as is the
case of graph products of highly symmetric graphs.
\begin{theorem}\label{thm:degree}
Let $G$ be a graph. If $|N(u) \setminus N[v]| \geq k$ for all $u,v
\in V(G)$ then $\thin(G) \geq k+1$. Moreover, for every order $<$
of $V(G)$, the first $k+1$ vertices induce a complete graph in
$G_{<}$.
\end{theorem}
\begin{proof}
Let $v_1, \ldots, v_n$ be an ordering of the vertices of $G$. Let
$i,j$ be such that $1 \leq i < j \leq k+1$. We know that $|N(v_i)
\setminus N[v_j]| \geq k$. Hence, $|(N(v_i) \setminus N[v_j])
\setminus (\{v_1, \ldots, v_{j-1}\} \setminus \{v_i\})| \geq 1$.
Therefore, there exists a vertex $v_h$ with $h > j$ such that $v_h
\in N(v_i)$ and $v_h \notin N(v_j)$, implying that $v_i$ and $v_j$
are adjacent in $G_{<}$.
So, for every order $<$ of vertices of $G$, we have that the first
$k+1$ vertices induce a complete graph in $G_{<}$. By
Corollary~\ref{cor:thin-comp-order}, $\thin(G) \geq k+1$.
\end{proof}
\begin{corollary}\label{cor:k-reg}
Let $G$ be a graph with $\delta(G) \geq d$ and such that for all
$u,v \in V(G)$, $|N(u) \cap N(v)| \leq c < d$. Then $\thin(G) \geq
d-c$.
\end{corollary}
The class of hypercubes $Q_n$ consists of the graphs whose vertex
sets correspond to all binary strings with fixed size $n$ and two
vertices $u$ and $v$ are adjacent if $u$ and $v$ differ exactly in
one position. We say that $n$ is the \emph{dimension} of the
hypercube $Q_n$. Clearly, $Q_n$ is a $n$-regular graph.
\begin{lemma}\cite{Mulder-Qn}
For all $u,v\in V(Q_n)$, $|N(u) \cap N(v)| \leq 2$.
\end{lemma}
\begin{corollary}\label{cor:Qn}
For every $n \geq 1$, $\thin(Q_n) \geq n-2$.
\end{corollary}
\begin{theorem}
Let $G$ be a graph. Let $S \subseteq V(G)$ and $p = |S|$. If
$|N(u) \setminus N[v]| \geq k$ for all $u,v \in S$ and $|V(G)| - p
\leq k$ then $\thin(G) \geq 1 + k + p - |V(G)|$.
\end{theorem}
\begin{proof}
Let $v_1, \ldots, v_n$ be an ordering of the vertices of $G$. Let
{$v_i,v_j \in S$ be} such that $1 \leq i < j \leq k+1$.
We know that $|N(v_i) \setminus N[v_j]| \geq k$. Hence, $|(N(v_i)
\setminus N[v_j]) \setminus (\{v_1, \ldots, v_{j-1}\} \setminus
\{v_i\})| \geq 1$. Therefore, there exists a vertex $v_h$ with $h
> j$ such that $v_h \in N(v_i)$ and $v_h \notin N(v_j)$, implying
that $v_i$ and $v_j$ are adjacent in $G_{<}$.
So, for every order $<$ of vertices of $G$, we have that the
vertices in $S$ within the first $k+1$ vertices induce a complete
graph in $G_{<}$. Since they are at least {$k+1-|V(G) \setminus S|
= k+1-|V(G)|+p$, by Corollary~\ref{cor:thin-comp-order},
$\thin(G) \geq k+1-|V(G)|+p$.}
\end{proof}
\subsection{Upper Bounds}
{Two general upper bounds were known for the thinness of a graph.
\begin{theorem}\cite{C-M-O-thinness-man}\label{thm:n-log4}
Let $G$ be a graph. Then $\thin(G) \leq |V(G)| - \log(|V(G)|)/4$.
\end{theorem}
\begin{theorem}\cite{C-M-O-thinness-man}\label{thm:bound-delta}
Let $G$ be a graph. Then $\thin(G) \leq
|V(G)|(\Delta(G)+3)/(\Delta(G)+4)$. \end{theorem}
We will prove here other general upper bounds.}
\begin{lemma}\label{n-S+thin}
Let $S \subseteq V(G)$. Then $\thin(G) \leq |V(G)| - |S| +
\thin(G[S])$.
\end{lemma}
\begin{proof}
Consider an order $<$ of vertices of $G$ such that $v < s$ for all
$v \in V(G) - S$ and $s \in S$, and such that $\omega(G[S]_{<_S})
= \thin(G[S])$, where $<_S$ stands for the order restricted to
$S$. Such an order exists by Corollary~\ref{cor:thin-comp-order}.
Notice that, since $v < s$ for all $v \in V(G) - S$ and $s \in S$,
$G_{<}[S] = G[S]_{<_S}$. Then $\thin(G) \leq \omega(G_{<}) \leq
\omega(G_{<}[S]) + \omega(G_{<}[V(G) \setminus S]) \leq
\omega(G[S]_{<_S}) + |V(G)| - |S| = |V(G)| - |S| + \thin(G[S])$.
\end{proof}
\begin{corollary}\label{cor:subgint}
Let $S \subseteq V(G)$ be such that $G[S]$ is an interval graph.
Then $\thin(G) \leq |V(G)| - |S| + 1$.
\end{corollary}
An \emph{interval completion} of a graph $G$ is a spanning
supergraph of $G$ which is an interval graph.
\begin{lemma}
Let $G$ be a graph. Let $H$ be an interval completion of $G$. Let
$F$ be the subgraph of $H$ whose edges are $E(H) - E(G)$. Then
{the number of vertices of a maximum induced interval subgraph of
$G$} is at least $|V(G)| - \tau(F)$.
\end{lemma}
\begin{proof}
Let $H$ be an interval completion of $G$. Then $H$ has $|V(G)|$
vertices. Let $F$ be the subgraph of $H$ whose edges are $E(H) -
E(G)$. If we remove from $H$ the vertices of a vertex cover in
$F$, we get an interval graph that is an induced subgraph of $G$.
\end{proof}
\begin{corollary}
Let $G$ be a graph. Let $H$ be an interval completion of $G$. Let
$F$ be the subgraph of $H$ whose edges are $E(H) - E(G)$. Then
$\thin(G) \leq \tau(F) + 1$.
\end{corollary}
In particular, stable and complete sets induce interval graphs.
Moreover, if $\alpha(G) < |V(G)|$ (resp. $\omega(G) < |V(G)|$), we
can add one more vertex $v$ and reorder the vertices of the stable
or complete set $S$ such that $u < w$ for all $u \in S - N(v)$ and
$w \in N(s) \cap S$ and $s < v$ for all $s \in S$, so $G$ has an
induced interval graph of size at least $\alpha(G) + 1$ (resp.
$\omega(G) +1$). As corollaries of Corollary~\ref{cor:subgint}, we
have the following two results.
\begin{corollary}\label{cor:thin-alpha}
If $V(G)$ is not a stable set then $\thin(G) \leq |V(G)| -
\alpha(G)$.
\end{corollary}
\begin{corollary}\label{cor:thin-omega}
If $G$ is not a complete graph then $\thin(G) \leq |V(G)| -
\omega(G)$.
\end{corollary}
\begin{remark}\label{rem:bounds-indep}
In the same way, one can see that $\indthin(G) \leq |V(G)| -
\alpha(G)+1$ and $\compthin(G) \leq |V(G)| - \omega(G)+1$ (in
this case we cannot add another vertex to the class containing the
maximum stable set or maximum clique, respectively).
\end{remark}
We have also the following bound for co-comparability graphs. As
already noticed in~\cite{B-M-O-thin-tcs},
Theorem~\ref{thm:thin-comp-order} implies that if $G$ is a
co-comparability graph, then $\thin(G) \leq \chi(G)$. Recalling
that co-comparability graphs are perfect~\cite{Meyn-co-comp}, this
implies $\thin(G) \leq \omega(G)$.
We prove next a new upper bound for the thinness of
co-comparability graphs.
\begin{theorem}
If $G$ is a non trivial co-comparability graph, then $\thin(G)
\leq |V(G)|/2$.
\end{theorem}
\begin{proof}
If $G$ is complete and non trivial, then $\thin(G) = 1 \leq
|V(G)|/2$. If $G$ is not complete, by
Corollary~\ref{cor:thin-omega}, $\thin(G) \leq |V(G)| -
\omega(G)$. Adding this inequality to the inequality $\thin(G)
\leq \omega(G)$ that holds for co-comparability graphs, we have
$2\thin(G) \leq |V(G)|$, thus $\thin(G) \leq |V(G)|/2$.
\end{proof}
The bound is attained, for example, by the family
$\overline{tK_2}$ (Theorem~\ref{thm:tK2}).
\section{Thinness and binary graph operations}\label{sec:thin-and-oper}
In this section, we analyze the behavior of the thinness and
proper thinness under different binary graph operations. {Each one
of these operations will be defined over a pair of graphs $G_1 =
(V_1,E_1)$ and $G_2 = (V_2,E_2)$ such that $|V_1| = n_1$, $|V_2| =
n_2$ and $V_1 \cap V_2 = \emptyset$. Besides, for some of the
following proofs, we consider an implicit ordering and partition
for both $V_1$ and $V_2$, as defined next.
The ordering of $V_1$ will be denoted by $v_1,\dots, v_{n_1}$ and
that of $V_2$ by $w_1,\dots, w_{n_2}$. Moreover, if the value
$t_i$ of some variation of thinness of $G_i$ (for $i \in \{1,2\}$)
is involved in the bound to be proved, the implicit ordering is
one consistent, according to the specified variation of thinness,
with a partition $(V_i^1, \ldots, V_1^{t_i})$. If, otherwise, only
the cardinality $n_i$ of $V_i$ is involved in the bound, the
implicit ordering is an arbitrary one. For instance, if $G_1$ is a
proper $t_1$-independent-thin graph, and $t_1$ is involved in the
bound to be proved, it means that the implicit ordering and
partition of $V_1$ are strongly consistent and all the $t_1$ parts
of the partition are independent sets.}
{Although the proofs in this section are not exactly the same,
some of them indeed share a common structure in the reasoning. For
the sake of conciseness, some of them were omitted and can be
found in~\ref{apdx:sec4:proofs}. }
\subsection{Union and join}\label{sec:unionjoin}
The \emph{union} of $G_1$ and $G_2$ is the graph $G_1 \cup G_2 =
(V_1 \cup V_2, E_1 \cup E_2)$, and the \emph{join} of $G_1$ and
$G_2$ is the graph {$G_1 \vee G_2 = (V_1 \cup V_2, E_1 \cup E_2
\cup \{vv' : v \in V_1, v' \in V_2\})$} (i.e., $\overline{G_1\vee
G_2} = \overline{G_1} \cup \overline{G_2}$). (The join is
sometimes also noted by $G_1 \otimes G_2$, but we follow the
notation in~\cite{B-D-thinness}).
The class of \emph{cographs} can be defined as the graphs that can
be obtained from trivial graphs by the union and join
operations~\cite{CorneilLerchsStewart81}. Aiming to characterize
$k$-thin graphs by forbidden induced subgraphs within the class of
cographs, the following results were proved.
\begin{theorem}\label{thm:union}\cite{B-D-thinness}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \cup G_2) =
\max\{f(G_1),$ $f(G_2)\}$, for $f \in \{\thin,\pthin\}$.
\end{theorem}
\begin{theorem}\label{thm:join}\cite{B-D-thinness}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \vee G_2) \leq
f(G_1)+f(G_2)$, for $f \in \{\thin,\pthin\}$. Moreover, if $G_2$
is complete, then $\thin(G_1 \vee G_2) = \thin(G_1)$.
\end{theorem}
\begin{lemma}\label{lem:join}\cite{B-D-thinness}
If $G$ is not complete, then $\thin(G \vee 2K_1) = \thin(G)+1$.
\end{lemma}
Lemma~\ref{lem:join} implies that if there is some constant value
$k$ such that recognizing $k$-thin graphs is NP-complete, then for
every $k' > k$, recognizing $k'$-thin graphs is NP-complete. The
existence of such $k$ is still not known, and in general the
complexity of recognition of $k$-thin and proper $k$-thin graphs,
both with $k$ as a parameter and with constant $k$, is open.
\begin{remark}\label{rem:clique-top} By definition of $G_{<}$, every non-smallest vertex of
any non-trivial clique has a vertex in $V(G)$ greater than it and
non-adjacent to it in $G$.
\end{remark}
The following lemma is necessary to prove Theorem~\ref{thm:join2}.
\begin{lemma}\label{lem:clique-top}
Let $G=(V,E)$ be a graph with $\thin(G)=k$ and $v_1 < \dots < v_n$
be an ordering of $V$. If $G$ is not complete, then there exist a
clique of size $k$ of $G_{<}$, $v_{i_1} < \dots < v_{i_k}$, and
$v_j
> v_{i_1}$, such that $v_j v_{i_1} \not \in E$.
\end{lemma}
\begin{proof}
By Corollary~\ref{cor:thin-comp-order}, for every order $<$ of the
vertices of $G$, $\omega(G_{<}) \geq k$. If $\thin(G)=1$, the
statement follows because $G$ is not complete. Suppose $\thin(G) >
1$, and let $v_1 < \dots < v_n$ be an ordering of $V$. By
definition of $G_{<}$, for every clique of $G_{<}$, all the
vertices that are not the smallest one have a vertex in $V(G)$
which is greater than it and non-adjacent to it in $G$. So, if
$G_{<}$ contains a clique of size greater than $k$, the statement
follows. {Consider} now that $\omega(G_{<})=k$. {In order to reach
a contradiction, suppose that} no clique of size $k$ of $G_{<}$
satisfies the property, then each vertex in the {set
$S = \{v \in V(G_<):v\text{ is the first vertex in $<$ of a clique of size $k$ of $G_<$}\}$}
is adjacent to every vertex greater than it in $G$. So, modifying the
order by placing $S$ as the largest vertices produces a graph
$G_{<'}$ which is a subgraph of $G_{<}$ and {in which the vertices
of $S$ are isolated vertices}. In particular, $\omega(G_{<'}) <
\omega(G_{<}) = k$, a contradiction since, {by
Corollary~\ref{cor:thin-comp-order}, $\omega(G_{<'}) \geq k $}.
\end{proof}
We strength the result of Theorem~\ref{thm:join} for thinness.
\begin{theorem}\label{thm:join2}
Let $G_1$ and $G_2$ be graphs. If $G_1$ is complete, then
$\thin(G_1 \vee G_2) = \thin(G_2)$. If neither $G_1$ nor $G_2$ are
complete, then $\thin(G_1 \vee G_2) = \thin(G_1) + \thin(G_2)$.
\end{theorem}
\begin{proof}
Let $G = G_1 \vee G_2$. If one of them is complete (suppose
without loss of generality $G_1$), then, by Theorem
\ref{thm:join}, $\thin(G) = \thin(G_2)$. Otherwise, by Theorem
\ref{thm:join}, $\thin(G_1 \vee G_2) \leq \thin(G_1)+\thin(G_2)$.
Let us prove the equality. Let $k_1 = \thin(G_1)$, $k_2 = \thin
(G_2)$, and $k = \thin(G_1 \vee G_2)$. Let $<$ be an ordering
consistent with a $k$-partition of $V(G)$. Let ${G_1}_{<}$ and
${G_2}_{<}$ {be the incompatibility graphs} obtained from the
order $<$ restricted to $V_1$ and $V_2$, respectively.
By Lemma \ref{lem:clique-top}, there exist a $k_1$-clique of
${G_1}_{<}$, $v^1_{i_1} < \dots < v^1_{i_{k_1}}$, and $v^1_j
> v^1_{i_1}$, such that $v^1_j v^1_{i_1} \not \in E_1$. As
well, there exist a $k_2$-clique of ${G_2}_{<}$, $v^2_{i_1} <
\dots < v^2_{i_{k_2}}$, and $v^2_j
> v^2_{i_1}$, such that $v^2_j v^2_{i_1} \not \in E_2$.
Notice also that, by definition of ${G_i}_{<}$, $i=1,2$, every
non-smallest vertex of the clique of ${G_i}_{<}$ has a vertex in
$V_i$, greater than it and non-adjacent to it in $G_i$.
Considering this property and the fact that every vertex of $G_1$
is adjacent to every vertex of $G_2$ in $G$, it follows that every
vertex of $\{v^1_{i_1}, \dots, v^1_{i_{k_1}}\}$ is adjacent to
every vertex of $\{v^2_{i_1}, \dots, v^2_{i_{k_2}}\}$ in $G_{<}$,
hence $k \geq k_1+k_2$. This completes the proof of the theorem.
\end{proof}
\begin{remark}
The proper thinness of the join $G_1 \vee G_2$ cannot be expressed
as a function whose only parameters are the proper thinness of
$G_1$ and $G_2$ (even excluding simple particular cases, like
trivial or complete graphs). The graph $tK_1$ has proper
thinness~1 for every $t$. By Theorem~\ref{thm:join}, the proper
thinness of the join of two graphs of proper thinness~1 is either
1~or~2, and there are examples for both of the values. The graph
$P_3 = 2K_1 \vee K_1$ has proper thinness~1 but $3K_1 \vee K_1$,
known as \emph{claw}, or $C_4 = 2K_1 \vee 2K_1$ have proper
thinness~2 (the claw and $C_4$ are not proper interval graphs).
Similarly, by Theorem~\ref{thm:join}, the proper thinness of the
join of a graph of proper thinness~2 and a graph of proper
thinness~1 is either 2~or~3, and there are examples for both of
the values. The graph $(\mbox{claw} \cup tK_1) \vee K_1$ has
proper thinness~2, but the graph $3\mbox{claw} \vee K_1$ has
proper thinness~3~\cite{B-D-thinness}.
\end{remark}
Nevertheless, we have a lemma similar to Lemma~\ref{lem:join} for
proper thinness.
\begin{lemma}\label{lem:joinp}\cite{B-D-thinness}
For every graph $G$, $\pthin(3G \vee K_1) = \pthin(G)+1$.
\end{lemma}
\begin{proof}
Theorems~\ref{thm:union} and~\ref{thm:join} imply $\pthin(3G \vee
K_1) \leq \pthin(G)+1$. To show $\pthin(3G \vee K_1) \geq
\pthin(G)+1$, we will use Corollary~\ref{cor:thin-comp-order}. Let
$u$ be the corresponding vertex of the $K_1$ in $H=3G \vee K_1$,
and let $<$ be an ordering of the vertices of $H$. Let $w,w'$ be
the minimum and maximum vertices, respectively, according to $<$
restricted to $H \setminus \{u\}$. Let $S$ be the set of vertices
of the copy of $G$ in $H$ which contains neither $w$ nor $w'$. We
will show that in $\tilde{H}_{<}$, all the vertices of $S$ are
adjacent to $u$. Let $v$ be a vertex of $S$. If $u < v$, then $u <
v < w'$, $w'v \not \in V(H)$ and $w'u \in V(H)$. If $v < u$, then
$w < v < u$, $wv \not \in V(H)$ and $wu \in V(H)$. In either case,
by definition of $\tilde{H}_{<}$, $uv \in V(\tilde{H}_{<})$. By
Corollary~\ref{cor:thin-comp-order}, $\tilde{H}_{<}[S]$ contains a
clique of size $\pthin(G)$, and thus $\tilde{H}_{<}[S \cup \{u\}]$
contains a clique of size $\pthin(G)+1$. As the order $<$ was
arbitrary, again by Corollary~\ref{cor:thin-comp-order},
$\pthin(H) \geq \pthin(G)+1$.
\end{proof}
We will now study the behavior of independent and complete
(proper) thinness under the union and join operations.
\begin{theorem}\label{thm:union-ind}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \cup G_2) =
\max\{f(G_1),$ $f(G_2)\}$, for $f \in \{\indthin, \indpthin\}$.
\end{theorem}
\begin{proof}
Since both $G_1$ and $G_2$ are induced subgraphs of $G_1 \cup
G_2$, then $\indthin(G_1 \cup G_2) \geq
\max\{\indthin(G_1),\indthin(G_2)\}$ and the same holds for the
independent proper thinness.
Let $G_1$ and $G_2$ be two graphs with independent thinness (resp.
independent proper thinness) $t_1$ and $t_2$, respectively.
Suppose without loss of generality that $t_1 \leq t_2$. For $G =
G_1 \cup G_2$, define a partition $(V^1,\dots, V^{t_2})$ such that
$V^i = V_1^i \cup V_2^i$ for $i = 1, \dots, t_1$ and $V^i = V_2^i$
for $i = t_1+1, \dots, t_2$, and define $v_1,\dots,
v_{n_1},w_1,\dots, w_{n_2}$ as an ordering of the vertices. By
definition of union of graphs, the sets of the partition are
independent and, if three ordered vertices according to the order
defined in $V(G_1 \cup G_2)$ are such that the first and the third
are adjacent, either the three vertices belong to $V_1$ or the
three vertices belong to $V_2$. Since the order and the partition
restricted to each of $G_1$ and $G_2$ are the original ones, the
properties required for consistency (resp. strong consistency) are
satisfied.
\end{proof}
\begin{theorem}\label{thm:union-comp}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \cup G_2) =
f(G_1)+f(G_2)$, for $f \in \{\compthin,\comppthin\}$.
\end{theorem}
\begin{proof}
Let $G_1$ and $G_2$ be two graphs with complete thinness (resp.
complete proper thinness) $t_1$ and $t_2$, respectively. By
definition of $G_1 \cup G_2$, any vertex ordering and partition
into complete sets of $G_1 \cup G_2$ which are (strongly)
consistent are (strongly) consistent when restricted to $V_1$ and
$V_2$. Notice that no complete set of $G_1 \cup G_2$ contains both
a vertex of $V_1$ and a vertex of $V_2$. So, $\compthin(G_1 \cup
G_2)$ (resp. $\comppthin(G_1 \cup G_2)$) is at least $t_1 + t_2$.
On the other hand, consider orderings and partitions of $V_1$ and
$V_2$ into $t_1$ and $t_2$ complete sets, respectively, which are
consistent (resp. strongly consistent). For $G_1 \cup G_2$, define
a partition with $t_1+t_2$ complete sets as the union of the two
partitions and define as ordering of the vertices the
concatenation of the orderings of $V_1$ and $V_2$ ({i.e., $v < v'$
if either $v$ and $v'$ belong to $V_i$ and $v < v'$, for $i\in
\{1,2\}$, or $v \in V_1$ and $v'$ in $V_2$}). By definition of
union of graphs, if three ordered vertices according to the order
defined in $V(G_1 \cup G_2)$ are such that the first and the third
are adjacent, either the three vertices belong to $V_1$ or the
three vertices belong to $V_2$. Since the order and the partition
restricted to each of $G_1$ and $G_2$ are the original ones, the
properties required for consistency (resp. strong consistency) are
satisfied. Thus $\compthin(G_1 \cup G_2)$ (resp. $\comppthin(G_1
\cup G_2)$) is at most $t_1 + t_2$, which completes the proof.
\end{proof}
\begin{theorem}\label{thm:join-ind}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \vee G_2) =
f(G_1)+f(G_2)$, for $f \in \{\indthin, \indpthin\}$.
\end{theorem}
\begin{proof}
Let $G_1$ and $G_2$ be two graphs with independent thinness (resp.
independent proper thinness) $t_1$ and $t_2$, respectively. By
definition of $G_1 \vee G_2$, any vertex ordering and partition
into independent sets of $G_1 \vee G_2$ which are (strongly)
consistent are (strongly) consistent when restricted to $V_1$ and
$V_2$. Notice that no independent set of $G_1 \vee G_2$ contains
both a vertex of $V_1$ and a vertex of $V_2$. So, $\indthin(G_1
\vee G_2)$ (resp. $\indpthin(G_1 \vee G_2)$) is at least $t_1 +
t_2$.
On the other hand, consider orderings and partitions of $V_1$ and
$V_2$ into $t_1$ and $t_2$ independent sets, respectively, which
are consistent (resp. strongly consistent). For $G = G_1 \vee
G_2$, define a partition with $t_1+t_2$ {independent sets} as the
union of the two partitions and define as ordering of the vertices
the concatenation of the orderings of $V_1$ and $V_2$. Let $x,y,z$
be three vertices of $V(G)$ such that $x < y < z$, $xz \in E(G)$,
and $x$ and $y$ are in the same class of the partition of $V(G)$.
Then, in particular, $x$ and $y$ both belong either to $V_1$ or to
$V_2$. If $z$ belongs to the same graph, then $yz \in E(G)$
because the ordering and partition restricted to each of $G_1$ and
$G_2$ are consistent. Otherwise, $z$ is also adjacent to $y$ by
the definition of join. We have proved that the defined partition
and ordering are consistent. The respective proof of the strong
consistency, given the strong consistency of the partition and
ordering of each of $G_1$ and $G_2$, is symmetric. Then
$\indthin(G_1 \vee G_2)$ (resp. $\indpthin(G_1 \vee G_2))$ is at
most $t_1+t_2$, which completes the proof.
\end{proof}
Lemma~\ref{lem:joinp} and Theorems~\ref{thm:union-comp}
and~\ref{thm:join-ind} imply the following corollary.
\begin{corollary}
If there is some constant value $k$ such that recognizing proper
$k$-thin graphs (resp. (proper) $k$-independent-thin and (proper)
$k$-complete-thin graphs) is NP-complete, then for every $k' > k$,
recognizing proper $k'$-thin graphs (resp. (proper)
$k'$-independent-thin and (proper) $k'$-complete-thin graphs) is
NP-complete.
\end{corollary}
Recall that the existence of such $k$ is still not known for all
these classes.
\begin{theorem}\label{thm:join-comp}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \vee G_2) \leq
f(G_1)+f(G_2)$, for $f \in \{\compthin, \comppthin\}$. Moreover,
if $G_2$ is complete, then $\compthin(G_1 \vee G_2) =
\compthin(G_1)$.
\end{theorem}
\subsection{Graph composition or lexicographical product}\label{sec:lex}
{Let $v \in V_1$.} The \emph{lexicographical product} of $G_1$ and
$G_2$ \emph{over the vertex} $v$ is the graph $G_1 \bullet_v G_2$
obtained from $G_1$ by replacing vertex $v$ by graph $G_2$, i.e.,
$V(G_1 \bullet_v G_2) = V_2 \cup V_1 \setminus \{v\}$, and $x$,
$y$ are adjacent if either $x, y \in V_1 \setminus \{v\}$ and $xy
\in E_1$, or $x, y \in V_2$ and $xy \in E_2$, or $x \in V_1
\setminus \{v\}$, $y \in V_2$, and $xv \in E_1$.
\begin{theorem}\label{thm:lexv} Let $G_1$ and $G_2$ be two graphs.
Then $f(G_1 \bullet_v G_2) \leq f(G_1) + f(G_2)$, for $f \in
\{\thin,$ $\pthin,$ $\compthin,$ $\comppthin\}$, and $f(G_1
\bullet_v G_2) \leq f'(G_1) + f(G_2)-1$, for $(f,f') \in
\{(\thin,\indthin),$ $(\indthin,\indthin),$ $(\pthin,\indpthin),$
$(\indpthin,\indpthin)\}$. Moreover, if $G_2$ is complete, $f(G_1
\bullet_v G_2) = f(G_1)$, for $f \in \{\thin,$ $\pthin,$
$\compthin,$ $\comppthin\}$.
\end{theorem}
\begin{proof}
Let $G_1$ and $G_2$ be two graphs with thinness (resp. proper
thinness) $t_1$ and $t_2$, respectively.
For $G = G_1 \bullet_v G_2$, if $v$ is the $i$-th vertex in the
ordering of $V_1$ and belongs to the class $V_1^j$, define
$v_1,\dots,v_{i-1}, w_1,\dots, w_{n_2},v_{i+1}, \dots, v_{n_1}$ as
an ordering of the vertices of $G$, and a partition with at most
$t_1+t_2$ sets as the union of the two partitions, where $V_1^j$
is replaced by $V_1^j \setminus \{v\}$ (or eliminated if $v$ is
the only vertex in the class, justifying the partition to have
\emph{at most} $t_1+t_2$ classes).
Let $x,y,z$ be three vertices of $V(G)$ such that $x < y < z$, $xz
\in E(G)$, and $x$ and $y$ are in the same class of the partition
of $V(G)$. Then, in particular, $x$ and $y$ both belong either to
$V_1 \setminus \{v\}$ or to $V_2$. If $z$ belongs to the same
graph, then $yz \in E(G)$ because the ordering and partition
restricted to each of $G_1$ and $G_2$ are consistent.
Otherwise, if $x$ and $y$ belong to $V_2$ and $z$ belongs to $V_1
\setminus \{v\}$, then $z$ is adjacent to $y$ because $V_2$ is an
homogeneous set in $G$. If $x$ and $y$ belong to $V_1 \setminus
\{v\}$ and $z$ belongs to $V_2$, by the definition of the order in
$G$, $y < v$ in the order of $V_1$, so $v$ is adjacent to $y$ in
$G_1$. By the definition of $G$, $y$ is adjacent to $z$.
We have proved that the defined partition and ordering are
consistent, and thus that $\thin(G_1 \bullet_v G_2) \leq
\thin(G_1)+\thin(G_2)$. The proof of the strong consistency, given
the strong consistency of the partition and ordering of each of
$G_1$ and $G_2$, is symmetric and implies $\pthin(G_1 \bullet_v
G_2) \leq \pthin(G_1)+\pthin(G_2)$.
Notice that if the partitions of $G_1$ and $G_2$ are into complete
sets (resp. independent sets), so is the defined partition of $G_1
\bullet_v G_2$. Moreover, we will define a new partition with
$t_1+t_2-1$ sets as the union of the partitions of $V_1$ and
$V_2$, where $V_1^j$ and $V_2^1$ are replaced by $V_1^j \setminus
\{v\} \cup V_2^1$. Notice that if the partitions of $G_1$ and
$G_2$ are into complete sets (resp. independent sets), so is the
new class.
We will prove first that if the partition of $V_1$ consists of
independent sets (not necessarily the partition of $V_2$), then
the order and the new partition are consistent. For strongly
consistence the proof is symmetric.
The only cases that need to be re-considered are the ones in which
$x$ and $y$ belong to the new class and, moreover, one of them
belongs to $V_1$ and the other one belongs to $V_2$. If $x$
belongs to $V_2$ and $y$ belongs to $V_1$, since the vertices of
$G_2$ are consecutive in the order, this implies that $z$ belongs
to $V_1$. Since $xz \in E(G)$, $vz \in E_1$. By the order
definition, $v < y < z$ in $G_1$. Since the ordering and partition
of $G_1$ are consistent, $yz \in E_1$ and thus $yz \in E(G)$. If
$x$ belongs to $V_1$ and $y$ belongs to $V_2$, since the partition
of $G_1$ consists of independent sets, $xv \not \in V_1$. So, if
$xz \in E(G)$, $z$ belongs to $V_1$. By the order definition, $x <
v < z$ in $G_1$. Since the ordering and partition of $G_1$ are
consistent, $vz \in E_1$ and thus $yz \in E(G)$.
Now we will prove that if $G_2$ is complete, i.e., $V_2 = V_2^1$,
then the order and the new partition are consistent. For strongly
consistence, the proof is symmetric.
Again, the only cases that need to be re-considered are the ones
in which $x$ and $y$ belong to the new class and, moreover, one of
them belongs to $V_1$ and the other one belongs to $V_2$. We will
prove them for consistence, and for strongly consistence the proof
is symmetric.
If $x$ belongs to $V_2$ and $y$ belongs to $V_1$, since the
vertices of $G_2$ are consecutive in the order, this implies that
$z$ belongs to $V_1$. Since $xz \in E(G)$, $vz \in E_1$. By the
order definition, $v < y < z$ in $G_1$. Since the ordering and
partition of $G_1$ are consistent, $yz \in E_1$ and thus $yz \in
E(G)$. If $x$ belongs to $V_1$ and $y$ belongs to $V_2$, since
$G_2$ is complete, if $z \in V_2$, $zy \in V(G)$. So, assume $z$
belongs to $V_1$. By the order definition, $x < v < z$ in $G_1$.
Since the ordering and partition of $G_1$ are consistent, $xz \in
E_1$ implies $vz \in E_1$ and thus $yz \in E(G)$.
\end{proof}
\begin{remark} Notice that if $v$ is isolated in $G_1$, then $G_1 \bullet_v G_2 = G_1[V(G_1) \setminus \{v\}] \cup G_2$, and if $v$ is universal in $G_1$, then $G_1 \bullet_v G_2 = G_1[V(G_1) \setminus \{v\}] \vee G_2$, so we can obtain better bounds by using the results of Section~\ref{sec:unionjoin}. \end{remark}
An equivalent formulation of Theorem~\ref{thm:lexv} is the
following.
\begin{theorem}\label{thm:lexhom}
Let $H$ be an homogeneous set of $G$, and $G|_H$ be the graph
obtained by contracting $H$ into a vertex. Then $f(G) \leq f(G|_H)
+ f(H)$, for $f \in \{\thin,$ $\pthin,$ $\compthin,$
$\comppthin\}$, and $f(G) \leq f'(G|_H) + f(H)-1$, for $(f,f') \in
\{(\thin,\indthin),$ $(\indthin,\indthin),$ $(\pthin,\indpthin),$
$(\indpthin,\indpthin)\}$. Moreover, if $H$ is complete, $f(G) =
f(G|_H)$, for $f \in \{\thin,$ $\pthin,$ $\compthin,$
$\comppthin\}$.
\end{theorem}
The \emph{lexicographical product} of $G_1$ and $G_2$ (also known
as \emph{composition} of $G_1$ and $G_2$) is the graph $G_1
\bullet G_2$ (also noted as $G_1[G_2]$) whose vertex set is the
Cartesian product $V_1 \times V_2$, and such that two vertices
$(u_1,u_2)$ and $(v_1,v_2)$ are adjacent in $G_1 \bullet G_2$ if
and only if either $u_1 = v_1$ and $u_2$ is adjacent to $v_2$ in
$G_2$, or $u_1$ is adjacent to $v_1$ in $G_1$. It is not
necessarily commutative.
\begin{theorem}\label{thm:lex} Let $G_1$ and $G_2$ be two graphs.
Then, if $G_2$ is complete, $f(G_1 \bullet G_2) = f(G_1)$, for $f
\in \{\thin,$ $\pthin,$ $\compthin,$ $\comppthin\}$. Also, $f(G_1
\bullet G_2) \leq f'(G_1)f(G_2)$, for $(f,f') \in
\{(\thin,\indthin),$ $(\indthin,\indthin),$ $(\pthin,\indpthin),$
$(\indpthin,\indpthin)\}$, and $f(G_1 \bullet G_2) \leq
|V_1|f(G_2)$, for $f \in \{\compthin,$ $\comppthin\}$. If $G_2$ is
not complete, $\omega(G_1)f(G_2) \leq f(G_1 \bullet G_2)$, for $f
\in \{\thin,\indthin,\allowbreak \indpthin\}$.
\end{theorem}
\begin{proof}
If $G_2$ is complete, we can iteratively apply
Theorem~\ref{thm:lexv}, since $G_1 \bullet G_2 = ((\dots((G_1
\bullet_{v_1} G_2) \bullet_{v_2} G_2) \dots ) \bullet_{v_{n_1}}
G_2)$, with {$\{v_1,\dots, v_{n_1}\} = V_1$}. By induction in
$n_1$, $f(G_1 \bullet G_2) = f(G_1)$, for $f \in \{\thin,$
$\pthin,$ $\compthin,$ $\comppthin\}$.
So, let $G_1$ and $G_2$ be two graphs, such that $f'(G_1) = t_1$
and $f(G_2) = t_2$, and assume $G_2$ is not complete. In $G_1
\bullet G_2$, consider $V_1 \times V_2$ lexicographically ordered
with respect to the {defined} orderings of $V_1$ and $V_2$.
Consider first $(f,f') \in \{(\thin,\indthin),$
$(\indthin,\indthin),$ $(\pthin,\indpthin),$
$(\indpthin,\indpthin)\}$, and the partition $\{V^{i,j}\}_{1\leq i
\leq t_1,\ 1 \leq j \leq t_2}$ such that $V^{i,j} = \{(v,w) : v
\in V_1^i, w \in V_2^j\}$ for each $1\leq i \leq t_1$, $1 \leq j
\leq t_2$. Since the partition of $V_1$ consists of independent
sets, vertices $(v,w)$ and $(v',w)$ in the same partition are not
adjacent for $v \neq v'$, and if furthermore the partition of
$V_2$ consists of independent sets, the same property holds for
the defined partition of $V_1 \times V_2$ for $G_1 \bullet G_2$.
We will show now that this ordering and partition of $V_1 \times
V_2$ are consistent (resp. strongly consistent, when $f, f'$ are
proper). Let $(v_p,w_i), (v_q,w_j), (v_r,w_{\ell})$ be three
vertices appearing in that ordering in $V_1 \times V_2$.
\emph{Case 1: $p = q = r$.} In this case, $i < j < \ell$. Suppose
first that $(v_p,w_i), (v_q,w_j)$ belong to the same class, i.e.,
$w_i, w_j$ belong to the same class in $G_2$. Vertices $(v_p,w_i)$
and $(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$ if and only
if $w_i w_{\ell} \in E_2$. Since the order and partition of $G_2$
are consistent, $w_j w_{\ell} \in E_2$, so $(v_q,w_j)$ and
$(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$, as required.
If $f$ and $f'$ are proper, the proof for strongly consistence is
symmetric.
\emph{Case 2: $p = q < r$.} Suppose first that $(v_p,w_i),
(v_q,w_j)$ belong to the same class, i.e., $w_i, w_j$ belong to
the same class in $G_2$. Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$
are adjacent in $G_1 \bullet G_2$ if and only if $v_p v_r \in
E_1$. Since $p = q$, $v_q v_r \in E_1$, so $(v_q,w_j)$ and
$(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$, as required.
Suppose now $f$ and $f'$ are proper, and {$(v_q, w_j), (v_r,
w_{\ell})$} belong to the same class, and, in particular, $v_q,
v_r$ belong to the same class in $G_1$ thus they are not adjacent.
Since $p = q$, $(v_p,w_i)$ and $(v_r, w_{\ell})$ are not adjacent
in $G_1 \bullet G_2$.
\emph{Case 3: $p < q = r$.} Suppose first that $(v_p,w_i),
(v_q,w_j)$ belong to the same class, and, in particular, $v_p,
v_q$ belong to the same class in $G_1$. Thus, they are not
adjacent. Since $q = r$, $(v_p,w_i)$ and $(v_r, w_{\ell})$ are not
adjacent in $G_1 \bullet G_2$. Suppose now $f$ and $f'$ are
proper, and $(v_q, w_j), (v_r, w_{\ell})$ belong to the same
class, i.e., $w_i, w_j$ belong to the same class in $G_2$.
Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$ are adjacent in $G_1
\bullet G_2$ if and only if $v_p v_r \in E_1$. Since $q = r$, $v_p
v_q \in E_1$, so $(v_p,w_i)$ and $(v_q,w_j)$ are adjacent in $G_1
\bullet G_2$, as required.
\emph{Case 4: $p < q < r$.} Suppose first that $(v_p,w_i),
(v_q,w_j)$ belong to the same class, and, in particular, $v_p,
v_q$ belong to the same class in $G_1$. Since the ordering an the
partition of $G_1$ are consistent, if $(v_p,w_i), (v_r, w_{\ell})$
are adjacent in $G_1 \bullet G_2$, in particular $v_p, v_r$ are
adjacent in $G_1$, thus $v_q, v_r$ are adjacent in $G_1$ and
$(v_q,w_j), (v_r, w_{\ell})$, as required. If $f$ and $f'$ are
proper, the proof for strongly consistence is symmetric.
Consider $f \in \{\compthin,$ $\comppthin\}$, and the partition
$\{V^{i,j}\}_{1\leq i \leq n_1,\ 1 \leq j \leq t_2}$ such that
$V^{i,j} = \{(v_i,w) : w \in V_2^j\}$ for each $1\leq i \leq n_1$,
$1 \leq j \leq t_2$. Since the partition of $V_2$ consists of
complete sets, the same property holds for the defined partition
of $V_1 \times V_2$ for $G_1 \bullet G_2$.
We will show now that this ordering and partition of $V_1 \times
V_2$ are consistent (resp. strongly consistent). Let $(v_p,w_i),
(v_q,w_j), (v_r,w_{\ell})$ be three vertices appearing in that
ordering in $V_1 \times V_2$.
\emph{Case 1: $p = q = r$.} In this case, $i < j < \ell$. Suppose
first that $(v_p,w_i), (v_q,w_j)$ belong to the same class, i.e.,
$w_i, w_j$ belong to the same class in $G_2$. Vertices $(v_p,w_i)$
and $(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$ if and only
if $w_i w_{\ell} \in E_2$. Since the order and partition of $G_2$
are consistent, $w_j w_{\ell} \in E_2$, so $(v_q,w_j)$ and
$(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$, as required.
If $f = \comppthin$, the proof for strongly consistence is
symmetric.
\emph{Case 2: $p = q < r$.} Suppose first that $(v_p,w_i),
(v_q,w_j)$ belong to the same class, i.e., $w_i, w_j$ belong to
the same class in $G_2$. Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$
are adjacent in $G_1 \bullet G_2$ if and only if $v_p v_r \in
E_1$. Since $p = q$, $v_q v_r \in E_1$, so $(v_q,w_j)$ and
$(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$, as required.
No further restriction has to be satisfied if $f = \comppthin$,
since by definition of the classes $(v_q, w_j), (v_r, w_{\ell})$
belong to different classes.
\emph{Case 3: $p < q = r$.} No restriction has to be satisfied for
consistence, as $(v_p,w_i), (v_q,w_j)$ belong to different
classes. If $f = \comppthin$, suppose that $(v_q, w_j), (v_r,
w_{\ell})$ belong to the same class, i.e., $w_j, w_{\ell}$ belong
to the same class in $G_2$. Vertices $(v_p,w_i)$ and
$(v_r,w_{\ell})$ are adjacent in $G_1 \bullet G_2$ if and only if
$v_p v_r \in E_1$. Since $q = r$, $v_p v_q \in E_1$, so
$(v_p,w_i)$ and $(v_q,w_j)$ are adjacent in $G_1 \bullet G_2$, as
required.
\emph{Case 4: $p < q < r$.} In this case, the three vertices are
in different classes, so no restriction has to be satisfied.
To prove the lower bound when $G_2$ is not complete, notice that
$K_r \bullet G_2$ is isomorphic to $(((G_2 \vee G_2) \vee G_2)
\dots \vee G_2)$ ($r$ times). By Theorems~\ref{thm:join2}
and~\ref{thm:join-ind}, $\omega(G_1)f(G_2) \leq f(G_1 \bullet
G_2)$, for $f \in \{\thin,\indthin,\indpthin\}$.
\end{proof}
\begin{corollary}\label{thm:lexomega}
Let $G_1$ and $G_2$ be graphs. If $G_2$ is not complete, then
$\thin(G_1 \bullet G_2) \geq \omega(G_1)$.
\end{corollary}
Notice that $K_n \bullet 2K_1 = \overline{tK_2}$. So, we have the
following corollary of Theorem~\ref{thm:tK2}.
\begin{corollary}\label{thm:nblex}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \bullet G_2) \leq f(\comppthin(G_1),|V(G_2)|)$ for any
pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
\begin{corollary}
Let $G_1$ be a co-comparability graph. If $G_2$ is complete, then
$\thin(G_1 \bullet G_2) = \thin(G_1)$, and if not, then $\thin(G_1
\bullet G_2) = \omega(G_1)\thin(G_2)$.
\end{corollary}
\begin{theorem}\label{thm:pthin-indpthin}
Let $G$ be a graph and $t \geq 3$, $q \geq 1$. Then, $\pthin(G
\bullet tK_1) = \indpthin(G)$, and $\pthin((G \bullet tK_1) \vee
qK_1) = \pthin((G \bullet tK_1) \vee K_q) = \indpthin(G)+1$.
\end{theorem}
\begin{proof}
The upper bounds are a consequence of Theorems~\ref{thm:lex}
and~\ref{thm:join}.
For the lower bounds, we will prove the statement for $t = 3$ and
$q = 1$, since $(G \bullet 3K_1)$ (resp. $(G \bullet 3K_1) \vee
K_1$) is an induced subgraph of $(G \bullet tK_1)$ (resp. $(G
\bullet tK_1) \vee qK_1$ and $(G \bullet tK_1) \vee K_q$). Let $G'
= (G \bullet 3K_1)$ and $G'' = G' \vee K_1$. Let $V(G') = \{v_i^1
< v_i^2 < v_i^3 : v_i \in V(G)\}$, and $V(G'') = V(G') \cup
\{u\}$. Consider an ordering of the vertices of $G''$, and let $<$
be the vertex order of $V(G)$ induced by the order of
$\{v_i^2\}_{v_i \in V(G)}$. We will show the following three
statements, that are enough to prove the theorem: if $vw \in
E(\tilde{G}_{<})$ then $v^2w^2 \in E(\tilde{G'}_{<})$; if $vw \in
E(G)$ then $v^2w^2 \in E(\tilde{G'}_{<})$; for any $v \in V(G)$,
$v^2u \in E(\tilde{G''}_{<})$.
First, let $v < w$ be adjacent in $\tilde{G}_{<}$. Then either
there is a vertex $z$ such that $v < w < z$, $vz \in E(G)$ and $wz
\not \in E(G)$, or there is a vertex $z$ such that $z < v < w$,
$zw \in E(G)$ and $zv \not \in E(G)$. In either case, the same
holds for $v^2, w^2, z^2$, so $v^2w^2 \in E(\tilde{G'}_{<})$.
Next, let $v < w$ be adjacent in $G$. Then $v^2 < w^2 < w^3$,
$v^2w^3 \in E(G')$ and $w^2w^3 \not \in E(G')$, so $v^2w^2 \in
E(\tilde{G'}_{<})$.
Finally, for $v \in V(G)$, if $u < v^2$, then $u < v^2 < v^3$,
$uv^3 \in E(G')$ and $v^2v^3 \not \in E(G')$, so $uv^2 \in
E(\tilde{G'}_{<})$. If $v^2 < u$, then $v^1 < v^2 < u$, $v^1u \in
E(G')$ and $v^1v^2 \not \in E(G')$, so $uv^2 \in
E(\tilde{G'}_{<})$.
By Remark~\ref{rem:co-comp-ind-thin}, it holds that $\pthin(G')
\geq \indpthin(G)$, and that $\pthin(G'') \geq \indpthin(G)+1$.
\end{proof}
Theorem~\ref{thm:pthin-indpthin} implies that if recognizing
proper $k$-independent-thin graphs is NP-complete, then
recognizing proper $k$-thin graphs is NP-complete (both with $k$
as a parameter and with constant $k$).
\subsection{Cartesian product}
The \emph{Cartesian product} $G_1 \ \square \ G_2$ is a graph
whose vertex set is the Cartesian product $V_1 \times V_2$, and
such that two vertices $(u_1,u_2)$ and $(v_1,v_2)$ are adjacent in
$G_1 \ \square \ G_2$ if and only if either $u_1 = v_1$ and $u_2$
is adjacent to $v_2$ in $G_2$, or $u_2 = v_2$ and $u_1$ is
adjacent to $v_1$ in $G_1$.
The following result was proved in~\cite{B-D-thinness}. We include
the proof in order to make some remarks about it. Most of the
proofs for other graph products are structurally similar to this
one.
\begin{theorem}\label{thm:cart}\cite{B-D-thinness}
Let $G_1$ and $G_2$ be graphs. Then, for $f \in \{\thin,\pthin\}$, {$f(G_1 \ \square \ G_2) \leq f(G_1)|V(G_2)|$}.
\end{theorem}
\begin{proof}
Let $G_1$ be a $k$-thin (resp. proper $k$-thin) graph. Consider
$V_1 \times V_2$ lexicographically ordered with respect to the
{defined} orderings of $V_1$ and $V_2$. Consider now the partition
$\{V^{i,j}\}_{1\leq i \leq k,\ 1 \leq j \leq n_2}$ such that
$V^{i,j} = \{(v,w_j) : v \in V_1^i\}$ for each $1\leq i \leq k$,
$1 \leq j \leq n_2$. We will show that this ordering and partition
of $V_1 \times V_2$ are consistent (resp. strongly consistent).
Let $(v_p,w_i), (v_q,w_j), (v_r,w_{\ell})$ be three vertices
appearing in that ordering in $V_1 \times V_2$.
\emph{Case 1: $p = q = r$.} In this case, the three vertices are
in different classes, so no restriction has to be satisfied.
\emph{Case 2: $p = q < r$.} In this case, $(v_p,w_i)$ and
$(v_q,w_j)$ are in different classes. So suppose $G_1$ is proper
$k$-thin and $(v_q,w_j), (v_r,w_{\ell})$ belong to the same class,
i.e., $j=\ell$. Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$ are
adjacent in $G_1 \ \square \ G_2$ if and only if $i = \ell$ and
$v_pv_r \in E_1$. But then $(v_p,w_i)=(v_q,w_j)$, a contradiction.
\emph{Case 3: $p < q = r$.} In this case, $(v_q,w_j)$ and
$(v_r,w_{\ell})$ are in different classes. So suppose $G_1$ is
$k$-thin (resp. proper $k$-thin) and $(v_p,w_i), (v_q,w_j)$ belong
to the same class, i.e., $i=j$. Vertices $(v_p,w_i)$ and
$(v_r,w_{\ell})$ are adjacent in $G_1 \ \square \ G_2$ if and only
if $i = \ell$ and $v_pv_r \in E_1$. But then
$(v_r,w_{\ell})=(v_q,w_j)$, a contradiction.
\emph{Case 4: $p < q < r$.} Suppose first $G_1$ is $k$-thin (resp.
proper $k$-thin) and $(v_p,w_i), (v_q,w_j)$ belong to the same
class, i.e., $i=j$ and $v_p$, $v_q$ belong to the same class in
$G_1$. Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$ are adjacent in
$G_1 \ \square \ G_2$ if and only if $i = \ell$ and $v_pv_r \in
E_1$. But then $j=\ell$ and since the ordering and the partition
are consistent (resp. strongly consistent) in $G_1$, $v_rv_q \in
E_1$ and so $(v_r,w_{\ell})$ and $(v_q,w_j)$ are adjacent in $G_1
\ \square \ G_2$. Now suppose that $G_1$ is proper $k$-thin and
$(v_q,w_j), (v_r,w_{\ell})$ belong to the same class, i.e.,
$j=\ell$. Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$ are adjacent
in $G_1 \ \square \ G_2$ if and only if $i = \ell$ and $v_pv_r \in
E_1$. But then $i=j$ and since the ordering and the partition are
strongly consistent in $G_1$, $v_pv_q \in E_1$ and so $(v_p,w_i)$
and $(v_q,w_j)$ are adjacent in $G_1 \ \square \ G_2$.
\end{proof}
\begin{remark}\label{rem:cart}
Notice that if the partition of $G_1$ consists of independent sets
(respectively, complete sets), the partition defined for $G_1 \
\square \ G_2$ consists also of independent sets (respectively,
complete sets). So, $f(G_1 \ \square \ G_2) \leq f(G_1)|V(G_2)|$,
for $f \in \{\thin, \pthin, \indthin, \compthin, \indpthin,
\comppthin\}$.
\end{remark}
These results can be strengthened by replacing $|V(G_2)|$ by the
size of the largest connected component of $G_2$, by
Theorem~\ref{thm:union} and since $G \ \square \ (H \cup H') = (G
\ \square \ H) \cup (G \ \square \ H')$.
On the negative side, since $P_r$ has independent proper
thinness~2 (with the order given by the definition of path), but
$P_r \ \square \ P_r = GR_r$, we have the following corollary of
Corollary~\ref{cor:grid}.
\begin{corollary}\label{cor:nbcart}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \ \square \ G_2) \leq f(\indpthin(G_1),\indpthin(G_2))$
for any pair of graphs $G_1$, $G_2$.
\end{corollary}
\begin{lemma}\label{lem:Knsq}
For $n \geq 1$, $\thin(K_n \ \square \ K_n) = n$.
\end{lemma}
\begin{proof} For $n \geq 1$, $K_n \ \square \ K_n$ is $(2n-2)$-regular, and
for any pair of vertices $u$, $v$, $|N(u) \cap N(v)| \leq n-2$. By
Corollary~\ref{cor:k-reg}, $\thin(K_n \ \square \ K_n) \geq n$. By
Theorem~\ref{thm:cart}, {$\thin(K_n \ \square \ K_n) \leq n$}.
\end{proof}
\begin{corollary}\label{cor:nbcart2}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \ \square \ G_2) \leq
f(\comppthin(G_1),\comppthin(G_2))$ for any pair of graphs $G_1$,
$G_2$.
\end{corollary}
\begin{lemma}\label{lem:KnKnn}
For $n \geq 1$, $\thin(K_n \ \square \ K_{n,n}) \geq n-1$.
\end{lemma}
\begin{proof} For $n \geq 1$, $K_n \ \square \ K_{n,n}$ is $(2n-1)$-regular, and
for any pair of vertices $u$, $v$, $|N(u) \cap N(v)| \leq n$. By
Corollary~\ref{cor:k-reg}, $\thin(K_n \ \square \ K_{n,n}) \geq n-
1$.
\end{proof}
\begin{corollary}\label{cor:nbcart3}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \ \square \ G_2) \leq
f(\comppthin(G_1),\indpthin(G_2))$ for any pair of graphs $G_1$,
$G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
Further consequences of the examples above are the following.
\begin{corollary}\label{cor:lowercart}
Given two connected graphs $G_1$ and $G_2$, $(\min\{\diam(G_1),$
$\diam(G_2)\}+1)/4 \leq (\min\{\lip(G_1),\lip(G_2)\}+1)/4 \leq
\thin(G_1 \ \square \ G_2)$.
\end{corollary}
\begin{corollary}\label{cor:lowercart2}
Given two graphs $G_1$ and $G_2$, $\min\{\omega(G_1),\omega(G_2)\}
\leq $ $\thin(G_1 \ \square \ G_2)$.
\end{corollary}
\subsection{Tensor or direct or categorical product}
The \emph{tensor product} or \emph{direct product} or
\emph{categorical product} or \emph{Kronecker product} $G_1 \times
G_2$ is a graph whose vertex set is the Cartesian product $V_1
\times V_2$, and such that two vertices $(u_1,u_2)$ and
$(v_1,v_2)$ are adjacent in $G_1 \times G_2$ if and only if
$u_1$ is adjacent to $v_1$ in $G_1$ and $u_2$ is adjacent to
$v_2$ in $G_2$.
\begin{theorem}\label{thm:direct}
Let $G_1$ and $G_2$ be graphs. Then {$f(G_1 \times G_2) \leq
\indf(G_1 \times G_2) \leq \indf(G_1)|V(G_2)| \leq
f(G_1)\chi(G_1)|V(G_2)|$}, for $f \in \{\thin, \pthin\}$.
\end{theorem}
This result can be strengthened by replacing $|V(G_2)|$ by the
size of the largest connected component of $G_2$, by
Theorem~\ref{thm:union-ind} and since $G \times (H \cup H') = (G
\times H) \cup (G \times H')$.
Since $\comppthin(K_n) = 1$, $|K_2|=2$, and $K_n \times K_2 =
CR_n$, we have the following consequence of
Corollary~\ref{cor:crown}.
\begin{corollary}\label{cor:nbdirect}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \times G_2) \leq f(\comppthin(G_1),|V(G_2)|)$ for any
pair of graphs $G_1$, $G_2$.
\end{corollary}
The graph $P_{2r-1} \times P_{2r-1}$ contains $GR_r$ as an induced
subgraph. Since $\indpthin(P_{2r-1}) = 2$, we have also the
following.
\begin{corollary}\label{cor:nbdirect2}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \times G_2) \leq f(\indpthin(G_1),\indpthin(G_2))$ for
any pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
Further consequences of the examples above are the following.
\begin{corollary}\label{cor:lowerdirect}
Given two graphs $G_1$ and $G_2$, if $G_2$ has at least one edge,
then $\omega(G_1)/2 \leq \thin(G_1 \times G_2)$.
\end{corollary}
\begin{corollary}\label{cor:lowerdirect2}
Given two connected graphs $G_1$ and $G_2$, $(\min\{\diam(G_1),$
$\diam(G_2)\}+1)/8 \leq (\min\{\lip(G_1),\lip(G_2)\}+1)/8 \leq
\thin(G_1 \times G_2)$.
\end{corollary}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\vertex{0}{0}{v00};
\vertex{0}{1}{v01};
\vertex{0}{2}{v02};
\vertex{0}{3}{v03};
\vertex{0}{4}{v04};
\vertex{1}{0}{v10};
\vertex{1}{1}{v11};
\vertex{1}{2}{v12};
\vertex{1}{3}{v13};
\vertex{1}{4}{v14};
\vertex{2}{0}{v20};
\vertex{2}{1}{v21};
\vertex{2}{2}{v22};
\vertex{2}{3}{v23};
\vertex{2}{4}{v24};
\vertex{3}{0}{v30};
\vertex{3}{1}{v31};
\vertex{3}{2}{v32};
\vertex{3}{3}{v33};
\vertex{3}{4}{v34};
\vertex{4}{0}{v40};
\vertex{4}{1}{v41};
\vertex{4}{2}{v42};
\vertex{4}{3}{v43};
\vertex{4}{4}{v44};
\edge{v00}{v11};
\edge{v01}{v12};
\edge[ultra thick]{v02}{v13};
\edge{v03}{v14};
\edge{v01}{v10};
\edge[ultra thick]{v02}{v11};
\edge{v03}{v12};
\edge{v04}{v13};
\edge{v10}{v21};
\edge[ultra thick]{v11}{v22};
\edge{v12}{v23};
\edge[ultra thick]{v13}{v24};
\edge[ultra thick]{v11}{v20};
\edge{v12}{v21};
\edge[ultra thick]{v13}{v22};
\edge{v14}{v23};
\edge[ultra thick]{v20}{v31};
\edge{v21}{v32};
\edge[ultra thick]{v22}{v33};
\edge{v23}{v34};
\edge{v21}{v30};
\edge[ultra thick]{v22}{v31};
\edge{v23}{v32};
\edge[ultra thick]{v24}{v33};
\edge{v30}{v41};
\edge[ultra thick]{v31}{v42};
\edge{v32}{v43};
\edge{v33}{v44};
\edge{v31}{v40};
\edge{v32}{v41};
\edge[ultra thick]{v33}{v42};
\edge{v34}{v43};
\vertex{7}{0}{w00};
\vertex{7}{1}{w01};
\vertex{7}{2}{w02};
\vertex{7}{3}{w03};
\vertex{7}{4}{w04};
\vertex{8}{0}{w10};
\vertex{8}{1}{w11};
\vertex{8}{2}{w12};
\vertex{8}{3}{w13};
\vertex{8}{4}{w14};
\vertex{9}{0}{w20};
\vertex{9}{1}{w21};
\vertex{9}{2}{w22};
\vertex{9}{3}{w23};
\vertex{9}{4}{w24};
\vertex{10}{0}{w30};
\vertex{10}{1}{w31};
\vertex{10}{2}{w32};
\vertex{10}{3}{w33};
\vertex{10}{4}{w34};
\vertex{11}{0}{w40};
\vertex{11}{1}{w41};
\vertex{11}{2}{w42};
\vertex{11}{3}{w43};
\vertex{11}{4}{w44};
\edge{w00}{w11};
\edge{w01}{w12};
\edge[ultra thick]{w02}{w13};
\edge{w03}{w14};
\edge{w01}{w10};
\edge[ultra thick]{w02}{w11};
\edge{w03}{w12};
\edge{w04}{w13};
\edge{w10}{w21};
\edge[ultra thick]{w11}{w22};
\edge{w12}{w23};
\edge[ultra thick]{w13}{w24};
\edge[ultra thick]{w11}{w20};
\edge{w12}{w21};
\edge[ultra thick]{w13}{w22};
\edge{w14}{w23};
\edge[ultra thick]{w20}{w31};
\edge{w21}{w32};
\edge[ultra thick]{w22}{w33};
\edge{w23}{w34};
\edge{w21}{w30};
\edge[ultra thick]{w22}{w31};
\edge{w23}{w32};
\edge[ultra thick]{w24}{w33};
\edge{w30}{w41};
\edge[ultra thick]{w31}{w42};
\edge{w32}{w43};
\edge{w33}{w44};
\edge{w31}{w40};
\edge{w32}{w41};
\edge[ultra thick]{w33}{w42};
\edge{w34}{w43};
\edge{w00}{w01};
\edge{w01}{w02};
\edge{w02}{w03};
\edge{w03}{w04};
\edge{w10}{w11};
\edge{w11}{w12};
\edge{w12}{w13};
\edge{w13}{w14};
\edge{w20}{w21};
\edge{w21}{w22};
\edge{w22}{w23};
\edge{w23}{w24};
\edge{w30}{w31};
\edge{w31}{w32};
\edge{w32}{w33};
\edge{w33}{w34};
\edge{w40}{w41};
\edge{w41}{w42};
\edge{w42}{w43};
\edge{w43}{w44};
\edge{w00}{w10};
\edge{w10}{w20};
\edge{w20}{w30};
\edge{w30}{w40};
\edge{w01}{w11};
\edge{w11}{w21};
\edge{w21}{w31};
\edge{w31}{w41};
\edge{w02}{w12};
\edge{w12}{w22};
\edge{w22}{w32};
\edge{w32}{w42};
\edge{w03}{w13};
\edge{w13}{w23};
\edge{w23}{w33};
\edge{w33}{w43};
\edge{w04}{w14};
\edge{w14}{w24};
\edge{w24}{w34};
\edge{w34}{w44};
\end{tikzpicture}
\end{center}
\caption{The $(r \times r)$-grid $GR_r$ as an induced subgraph of $P_{2r-1} \times P_{2r-1}$ and of $P_{2r-1} \boxtimes P_{2r-1}$.}
\end{figure}
\subsection{Strong or normal product}
The \emph{strong product} (also known as \emph{normal product})
$G_1 \boxtimes G_2$ is a graph whose vertex set is the Cartesian
product $V_1 \times V_2$, and such that two vertices $(u_1,u_2)$
and $(v_1,v_2)$ are adjacent in $G_1 \boxtimes G_2$ if and only if
they are adjacent either in $G_1 \ \square \ G_2$ or in $G_1
\times G_2$.
\begin{theorem}\label{thm:strong}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \boxtimes G_2) \leq
f(G_1)|V(G_2)|$ for $f \in \{\thin, \pthin, \compthin, \comppthin,
\indthin, \indpthin\}$. Moreover, if $G_2$ is complete, then
$f(G_1 \boxtimes G_2) = f(G_1)$ for $f \in \{\thin, \pthin,
\compthin, \comppthin\}$.
\end{theorem}
This result can be strengthened for $f \in \{\thin, \pthin,
\indthin, \indpthin\}$, replacing $|V(G_2)|$ by the size of the
largest connected component of $G_2$, by Theorems~\ref{thm:union}
and~\ref{thm:union-ind}, and since $G \boxtimes (H \cup H') = (G
\boxtimes H) \cup (G \boxtimes H')$.
The graph $P_{2r-1} \boxtimes P_{2r-1}$ contains $GR_r$ as an
induced subgraph. Since $\indpthin(P_{2r-1}) = 2$, we have the
following corollary of Corollary~\ref{cor:grid}.
\begin{corollary}\label{cor:nbstrong}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \boxtimes G_2) \leq f(\indpthin(G_1),\indpthin(G_2))$
for any pair of graphs $G_1$, $G_2$.
\end{corollary}
\begin{lemma}\label{lem:Knsqbox}
For $n \geq 2$, $\thin((K_n \ \square \ K_2) \boxtimes (K_n \
\square \ K_2)) \geq n+2$.
\end{lemma}
\begin{proof} For $n \geq 2$, $(K_n \ \square \ K_2) \boxtimes (K_n \
\square \ K_2)$ is $(n^2+2n)$-regular, and for any pair of
vertices $u$, $v$, $|N(u) \cap N(v)| \leq n^2+n-2$. By
Corollary~\ref{cor:k-reg}, $\thin((K_n \ \square \ K_2) \boxtimes
(K_n \ \square \ K_2)) \geq n+2$.
\end{proof}
Remark~\ref{rem:cart} implies $\comppthin(K_n \ \square \ K_2) =
2$ for $n \geq 2$, since the graph contains an induced cycle of
length four, which is not an interval graph. So we have also the
following.
\begin{corollary}\label{cor:nbstrong2}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \boxtimes G_2) \leq f(\comppthin(G_1),\comppthin(G_2))$
for any pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
A further consequence of the example used for
Corollary~\ref{cor:nbstrong} is the following.
\begin{corollary}\label{cor:lowerstrong}
Given two connected graphs $G_1$ and $G_2$, $(\min\{\diam(G_1),$
$\diam(G_2)\}+1)/8 \leq (\min\{\lip(G_1),\lip(G_2)\}+1)/8 \leq
\thin(G_1 \boxtimes G_2)$.
\end{corollary}
\subsection{Co-normal or disjunctive product}
The \emph{co-normal product} or \emph{disjunctive product} $G_1
\ast G_2$ is a graph whose vertex set is the Cartesian product
$V_1 \times V_2$, and such that two vertices $(u_1,u_2)$ and
$(v_1,v_2)$ are adjacent in $G_1 \ast G_2$ if and only if either
$u_1$ is adjacent to $v_1$ in $G_1$ or $u_2$ is adjacent to $v_2$
in $G_2$.
Notice that $\overline{G \ast H} = \overline{G} \boxtimes
\overline{H}$.
\begin{theorem}\label{thm:conorm}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \ast G_2) \leq
\indf(G_1 \ast G_2) \leq \indf(G_1)|V(G_2)| \leq
f(G_1)\chi(G_1)|V(G_2)|$, for $f \in \{\thin, \pthin\}$.
\end{theorem}
Since $\comppthin(K_t) = 1$, $|2K_1|=2$, and $K_t \ast 2K_1 =
\overline{tK_2}$, we have the following corollary of
Theorem~\ref{thm:tK2}.
\begin{corollary}\label{cor:nbconorm}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \ast G_2) \leq f(\comppthin(G_1),|V(G_2)|)$ for any
pair of graphs $G_1$, $G_2$.
\end{corollary}
Consider the graph $tK_2 \ast tK_2$. It is $(4t-1)$-regular, and
for every pair of vertices $u,v$, it holds $|N(u) \cap N(v)| \leq
2t+1$. By Corollary~\ref{cor:k-reg}, $\thin(tK_2 \ast tK_2) \geq
2t-2$. Since $\indpthin(tK_2) = 2$, we have also the following
corollary.
\begin{corollary}\label{cor:nbconorm2}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \ast G_2) \leq f(\indpthin(G_1),\indpthin(G_2))$ for
any pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
Further consequences of the examples above are the following.
\begin{corollary}\label{cor:lowerconorm}
Given two graphs $G_1$ and $G_2$, if $G_2$ is not complete, then
$\omega(G_1) \leq \thin(G_1 \ast G_2)$.
\end{corollary}
\begin{corollary}\label{cor:lowerconorm2}
Given two graphs $G_1$ and $G_2$, $2\min\{\mim(G_1),\mim(G_2)\}-2
\leq \thin(G_1 \ast G_2)$.
\end{corollary}
\subsection{Modular product}
The \emph{modular product} $G_1 \diamond G_2$ is a graph whose
vertex set is the Cartesian product $V_1 \times V_2$, and such
that two vertices $(u_1,u_2)$ and $(v_1,v_2)$ are adjacent in $G_1
\diamond G_2$ if and only if either $u_1$ is adjacent to $v_1$ in
$G_1$ and $u_2$ is adjacent to $v_2$ in $G_2$, or $u_1$ is
nonadjacent to $v_1$ in $G_1$ and $u_2$ is nonadjacent to $v_2$ in
$G_2$.
Notice that $K_n \diamond K_2 = CR_n$ and $tK_2 \diamond K_1 =
\overline{tK_2}$, so we have the following.
\begin{corollary}\label{cor:nbmodular}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \diamond G_2) \leq f(h(G_1),|G_2|)$, for $h \in
\{\comppthin,\indpthin\}$ for any pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
Further consequences of the examples above are the following.
\begin{corollary}\label{cor:lowermodular}
Given two graphs $G_1$ and $G_2$, if $G_2$ has at least one edge,
then $\omega(G_1)/2 \leq \thin(G_1 \diamond G_2)$.
\end{corollary}
\begin{corollary}\label{cor:lowermodular2}
Given two graphs $G_1$ and $G_2$, $\mim(G_1) \leq \thin(G_1
\diamond G_2)$.
\end{corollary}
\subsection{Homomorphic product}
The \emph{homomorphic product} $G_1 \ltimes
G_2$~\cite{M-R-hom-prod} is a graph whose vertex set is the
Cartesian product $V_1 \times V_2$, and such that two vertices
$(u_1,u_2)$ and $(v_1,v_2)$ are adjacent in $G_1 \ltimes G_2$ if
and only if either $u_1 = v_1$ or $u_1$ is adjacent to $v_1$ in
$G_1$ and $u_2$ is nonadjacent to $v_2$ in $G_2$. It is not
necessarily commutative.
\begin{theorem}\label{thm:homo}
Let $G_1$ and $G_2$ be graphs. Then $f(G_1 \ltimes G_2) \leq
f(G_1)|V(G_2)|$ for $f \in \{\thin, \pthin, \compthin, \comppthin,
\indthin, \indpthin\}$.
\end{theorem}
Let $G_1$ be isomorphic to $K_2$, such that $V(G_1) =
\{v_1,v_2\}$, and $G_2$ be isomorphic to $tK_2$, for some $t \geq
1$, such that $V(G_2) = \{w_1,z_1,\dots,w_t,z_t\}$ and $E(G_2) =
\{w_iz_i : 1 \leq i \leq t\}$. In $G_1 \ltimes G_2$, the vertices
$\{(v_1,w_i) : 1 \leq i \leq t\} \cup \{(v_2,z_i) : 1 \leq i \leq
t\}$ induce the graph $\overline{tK_2}$. In other words, $K_2
\ltimes tK_2$ has $\overline{tK_2}$ as induced subgraph.
Now let $G_3$ be isomorphic to $K_t \ \square \ K_2$, for some $t
\geq 1$, such that $V(G_3) = \{w_1,z_1,\dots,w_t,z_t\}$, $\{w_1,
\dots, w_t\}$ is a clique, $\{z_1, \dots, z_t\}$ is a clique, and
$w_i$ is adjacent to $z_j$ if and only if $i = j$. In $G_1 \ltimes
G_3$, the vertices $\{(v_1,w_i) : 1 \leq i \leq t\} \cup
\{(v_2,z_i) : 1 \leq i \leq t\}$ induce the graph
$\overline{tK_2}$. In other words, $K_2 \ltimes (K_t \ \square \
K_2)$ has $\overline{tK_2}$ as induced subgraph.
Since $\indpthin(tK_2)=2$ and $\comppthin(K_n \ \square \ K_2)=2$
(Remark~\ref{rem:cart}), we have the following.
\begin{corollary}\label{cor:nbhomo}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \ltimes G_2) \leq f(|G_1|,h(G_2))$, for $h \in
\{\indpthin,\comppthin\}$ for any pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
Another consequence of the example above is the following.
\begin{corollary}\label{cor:lowerhomo}
Given two graphs $G_1$ and $G_2$, if $G_1$ has at least one edge,
then $\mim(G_2) \leq \thin(G_1 \ltimes G_2)$.
\end{corollary}
\subsection{Hom-product}
The \emph{hom-product} $G_1 \circ G_2$~\cite{Bacik-phd-hom-prod}
is a graph whose vertex set is the Cartesian product $V_1 \times
V_2$, and such that two vertices $(u_1,u_2)$ and $(v_1,v_2)$ are
adjacent in $G_1 \circ G_2$ if and only if $u_1 \neq v_1$ and
either $u_1$ is nonadjacent to $v_1$ in $G_1$ or $u_2$ is adjacent
to $v_2$ in $G_2$. It is not necessarily commutative, indeed, $G_1
\circ G_2 = \overline{G_1 \ltimes G_2}$.
\begin{theorem}\label{thm:hom}
Let $G_1$ and $G_2$ be graphs. Then {$f(G_1 \circ G_2) \leq
\indf(G_1 \circ G_2) \leq |V(G_1)|\indf(G_2)$}, for $f \in
\{\thin, \pthin\}$.
\end{theorem}
\begin{proof}
Let $G_2$ be a $k$-independent-thin (resp. proper
$k$-independent-thin) graph. Consider $V_1 \times V_2$
lexicographically ordered with respect to the {defined} orderings
of $V_2$ and $V_1$. Consider now the partition $\{V^{i,j}\}_{1\leq
i \leq n_1,\ 1 \leq j \leq k}$ such that $V^{i,j} = \{(v_i,w) : w
\in V_2^j\}$ for each $1\leq i \leq n_1$, $1 \leq j \leq k$. By
definition of the hom-product, each $V^{i,j}$ is an independent
set.
We will show now that this ordering and partition of $V_1 \times
V_2$ are consistent (resp. strongly consistent). Let $(v_p,w_i),
(v_q,w_j), (v_r,w_{\ell})$ be three vertices appearing in that
ordering in $V_1 \times V_2$.
\emph{Case 1: $i = j = \ell$.} In this case, the three vertices
are in different classes, so no restriction has to be satisfied.
\emph{Case 2: $i = j < \ell$.} In this case, $(v_p,w_i)$ and
$(v_q,w_j)$ are in different classes. So, suppose $G_2$ is proper
$k$-independent-thin and $(v_q,w_j), (v_r,w_{\ell})$ belong to the
same class, i.e., $q=r$ and $w_i = w_j$ and $w_{\ell}$ belong to
the same class in $G_2$. In particular, since the classes are
independent sets, $w_i w_{\ell} \not \in E_2$. Vertices
$(v_p,w_i)$ and $(v_r,w_{\ell})$ are adjacent in $G_1 \circ G_2$
if and only if $p\neq r$ and either $v_p v_r \not \in E_1$ or $w_i
w_{\ell} \in E_2$. So, assume $p \neq r$. Since $q=r$, $p \neq q$,
and since the graph is loopless and $i = j$, $w_i w_j \not \in
E_2$. So $(v_p,w_i)$ is adjacent to $(v_q,w_j)$ in $G_1 \circ
G_2$, as required.
\emph{Case 3: $i < j = \ell$.} In this case, $(v_q,w_j)$ and
$(v_r,w_{\ell})$ are in different classes. So suppose $G_2$ is
$k$-independent-thin (resp. proper $k$-independent-thin) and
$(v_p,w_i), (v_q,w_j)$ belong to the same class, i.e., $p=q$ and
$w_i$ and $w_j = w_{\ell}$ belong to the same class in $G_2$. In
particular, since the classes are independent sets, $w_i w_j \not
\in E_2$. Vertices $(v_p,w_i)$ and $(v_r,w_{\ell})$ are adjacent
in $G_1 \circ G_2$ if and only if $p \neq r$ and either $v_p v_r
\not \in E_1$ or $w_i w_{\ell} \in E_2$. So, assume $p \neq r$.
Since $p=q$, $q \neq r$, and since the graph is loopless and $j =
\ell$, $w_j w_{\ell} \not \in E_2$. So $(v_q,w_j)$ is adjacent to
$(v_r,w_{\ell})$ in $G_1 \circ G_2$, as required.
\emph{Case 4: $i < j < \ell$.} Suppose first that $G_2$ is
$k$-independent-thin (resp. proper $k$-independent-thin) and
$(v_p,w_i), (v_q,w_j)$ belong to the same class, i.e., $p=q$ and
$w_i$, $w_j$ belong to the same class in $G_2$. Since the classes
are independent, $w_i$ and $w_j$ are not adjacent. Vertices
$(v_p,w_i)$ and $(v_r,w_{\ell})$ are adjacent in $G_1 \circ G_2$
if and only if $p \neq r$ and either $v_p v_r \not \in E_1$ or
$w_i w_{\ell} \in E_2$. So, assume $p \neq r$, and since $p=q$, $q
\neq r$ too. If $v_p v_r \not \in E_1$, then since $p=q$, $v_q v_r
\not \in E_1$, and $(v_q,w_j)$ is adjacent to $(v_r,w_{\ell})$ in
$G_1 \circ G_2$, as required. If $w_i w_{\ell} \in E_2$, since
$w_i$, $w_j$ belong to the same class in $G_2$ and the partition
of $V_2$ is (strongly) consistent with the ordering, {$w_j
w_{\ell} \in E_2$}, and $(v_q,w_j)$ is adjacent to
$(v_r,w_{\ell})$ in $G_1 \circ G_2$, as required.
Now suppose that $G_1$ is proper $k$-thin and $(v_q,w_j),
(v_r,w_{\ell})$ belong to the same class, i.e., $q = r$ and $w_j$,
$w_{\ell}$ belong to the same class in $G_2$. Vertices $(v_p,w_i)$
and $(v_r,w_{\ell})$ are adjacent in $G_1 \circ G_2$ if and only
if $p \neq r$ and either $v_p v_r \not \in E_1$ or $w_i w_{\ell}
\in E_2$. So, assume $p \neq r$, and since $q = r$, $p \neq q$
too. If $v_p v_r \not \in E_1$, then since $q = r$, $v_p v_q \not
\in E_1$, and $(v_p,w_i)$ and $(v_q,w_j)$ are adjacent in $G_1
\circ G_2$, as required. If $w_i w_{\ell} \in E_2$, since $w_j$,
$w_{\ell}$ belong to the same class in $G_2$ and the partition of
$V_2$ is strongly consistent with the ordering, $w_i w_j \in E_2$,
and $(v_p,w_i)$ is adjacent to $(v_q,w_j)$ in $G_1 \circ G_2$, as
required.
\end{proof}
Notice that $K_2 \circ K_n = CR_n$. Also, $G \circ K_1 =
\overline{G}$, so, by taking $G = tK_2$ and $G = K_n \ \square \
K_2$, we have the following.
\begin{corollary}\label{cor:nbhom}
There is no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that
$\thin(G_1 \circ G_2) \leq f(|G_1|,\comppthin(G_2))$, and there is
no function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\thin(G_1
\circ G_2) \leq f(h(G_1),|G_2|)$ for $h \in
\{\indpthin,\comppthin\}$ for any pair of graphs $G_1$, $G_2$.
\end{corollary}
The non existence of bounds in terms of other parameters can be
deduced from diagram in Figure~\ref{fig:hasse}.
Further consequences of the examples above are the following.
\begin{corollary}\label{cor:lowerhom}
Given two graphs $G_1$ and $G_2$, if $G_1$ has at least one edge,
then $\omega(G_2)/2 \leq \thin(G_1 \circ G_2)$.
\end{corollary}
\begin{corollary}\label{cor:lowerhom2}
Given two graphs $G_1$ and $G_2$, $\mim(G_1) \leq \thin(G_1 \circ
G_2)$.
\end{corollary}
\section{Conclusion}\label{sec:conc}
In this paper, we give upper bounds for the thinness, complete
thinness, independent thinness and their proper versions for the
union and join of graphs, as well as the lexicographical,
Cartesian, direct, strong, disjunctive, modular, homomorphic and
hom-products of graphs. These bounds are given in terms of the
parameters (depicted in the Hasse diagram of
Figure~\ref{fig:hasse}) of the component graphs, and for each of
the cases, it is proved that no upper bound in terms of lower
parameters (from those in the diagram) of the component graphs
exists. The non existence proofs are based on the determination of
exact values or lower bounds for the thinness of some families of
graphs like complements of matchings, grids, crown graphs,
hypercubes, or products of simple graphs like complete graphs,
stable sets, induced matchings and induced paths. We summarize the
main bounds obtained and the graph families with high thinness in
Table~\ref{table1}.
\begin{table}
\footnotesize
\begin{tabular}{|l|l|}
\hline
\textbf{Upper bounds} & \textbf{High thinness} \\
\hline
$\opthin(G_1 \cup G_2) =\max\{\opthin(G_1),\opthin(G_2)\}$ & \\
$\opindthin(G_1 \cup G_2) =\max\{\opindthin(G_1),\opindthin(G_2)\}$ & \\
$\opcompthin(G_1 \cup G_2) =\opcompthin(G_1)+\opcompthin(G_2)$ & \\ \hline
$\thin(G_1^* \vee G_2^*)=\thin(G_1^*)+\thin(G_2^*)$ & \\
$\thin(G_1 \vee K_n)=\thin(G_1)$ & \\
$\pthin(G_1 \vee G_2) \leq \pthin(G_1)+\pthin(G_2)$ & \\
$\opindthin(G_1 \vee G_2)=\opindthin(G_1)+\opindthin(G_2)$ & \\
$\opcompthin(G_1 \vee G_2) \leq \opcompthin(G_1)+\opcompthin(G_2)$ & \\
$\compthin(G_1 \vee K_n) =\compthin(G_1)$ & \\ \hline
$\opthin(G_1 \bullet G_2) \leq \opindthin(G_1)\cdot \opthin(G_2)$ & $K_n \bullet 2K_1$ \\
$\opthin(G_1 \bullet K_n) =\opthin(G_1)$ & \\
$\opindthin(G_1 \bullet G_2) \leq \opindthin(G_1)\cdot \opindthin(G_2)$ & \\
$\opcompthin(G_1 \bullet G_2) \leq |V(G_1)|\cdot \opcompthin(G_2)$ & \\
$\opcompthin(G_1 \bullet K_n) =\opcompthin(G_1)$ & \\ \hline
$\opthin(G_1 \ \square \ G_2) \leq \opthin(G_1) \cdot |V(G_2)|$ & $P_n \ \square \ P_n$ \\
$\opindthin(G_1 \ \square \ G_2) \leq \opindthin(G_1) \cdot |V(G_2)|$ & $K_n \ \square \ K_n$ \\
$\opcompthin(G_1 \ \square \ G_2) \leq \opcompthin(G_1) \cdot |V(G_2)|$ & $K_n \ \square \ K_{n,n}$ \\ \hline
$\opthin(G_1 \times G_2) \leq \opindthin(G_1) \cdot |V(G_2)|$ & $K_n \times K_2$ \\
$\opindthin(G_1 \times G_2) \leq \opindthin(G_1) \cdot |V(G_2)|$ &
$P_n \times P_n$
\\ \hline
$\opthin(G_1 \boxtimes G_2) \leq \opthin(G_1)\cdot |V(G_2)|$ & $P_n \boxtimes P_n$ \\
$\opthin(G_1 \boxtimes K_n) = \opthin(G_1)$ & $(K_n \ \square \ K_2) \boxtimes (K_n \ \square \ K_2)$ \\
$\opindthin(G_1 \boxtimes G_2) \leq \opindthin(G_1)\cdot |V(G_2)|$ & \\
$\opcompthin(G_1 \boxtimes G_2) \leq \opcompthin(G_1)\cdot |V(G_2)|$ & \\
$\opcompthin(G_1 \boxtimes G_2) = \opcompthin(G_1)$ & \\ \hline
$\opthin(G_1 \ast G_2) \leq \opindthin(G_1) \cdot |V(G_2)|$ & $K_n \ast 2K_1$ \\
$\opindthin(G_1 \ast G_2) \leq \opindthin(G_1) \cdot |V(G_2)|$ &
$nK_2 \ast nK_2$
\\ \hline
& $K_n \diamond K_2$ \\
& $nK_2 \diamond K_1$ \\ \hline
$\opthin(G_1 \ltimes G_2) \leq \opthin(G_1)\cdot |V(G_2)|$ & $K_2
\ltimes nK_2$ \\
$\opindthin(G_1 \ltimes G_2) \leq \opindthin(G_1)\cdot |V(G_2)|$ &
$K_2
\ltimes (K_n \ \square \ K_2)$ \\
$\opcompthin(G_1 \ltimes G_2) \leq \opcompthin(G_1)\cdot |V(G_2)|$ & \\
\hline
$\opthin(G_1 \circ G_2) \leq |V(G_1)| \cdot \opindthin(G_2)$ & $K_2 \circ K_n$ \\
$\opindthin(G_1 \circ G_2) \leq |V(G_1)| \cdot \opindthin(G_2)$ & $nK_2 \circ K_1$ \\
& $(K_n \ \square \ K_2) \circ K_1$ \\ \hline
\end{tabular}
\caption{We summarize the upper bounds (when needed, graphs with
an asterisk are not complete). We also summarize the families of
graphs with bounded parameters whose product have high thinness,
used to show the nonexistence of bounds in terms of certain
parameters. Recall that the lexicographic, homomorphic, and
hom-product are not necessarily commutative.}\label{table1}
\end{table}
Furthermore, we describe new general lower and upper bounds for
the thinness of graphs, and some lower bounds for the graph
operations in terms of other well known graph invariants like
clique number, maximum induced matching, longest induced path, or
diameter.
Some open problems and possible research directions are:
\begin{itemize}
\item It would be interesting to find tighter bounds, in the case
in which it is possible.
\item It remains as an open problem the computational
complexity of computing the independent and complete (proper)
thinness for general graphs. For co-comparability graphs, both the
independent thinness and independent proper thinness are exactly
the chromatic number, which can be computed in polynomial
time~\cite{Go-comp2}.
\item Regarding lower and upper bounds in
Section~\ref{sec:families}, can we have some similar results for
proper thinness? Or for the independent and complete versions of
(proper) thinness?
\item Does there exist a graph $G$ such that $\thin(G) >
|V(G)|/2$?
\item Does there exist a co-comparability graph $G$ such that
$\pthin(G) > |V(G)|/2$?
\end{itemize}
\section*{Acknowledgements}
\label{sec:ack}
This work was done when Moys\'es S. Sampaio Jr. was visiting the
University of Buenos Aires, funded by a grant from FAPERJ. The
work was also partially supported by UBACyT Grants
20020170100495BA and 20020160100095BA (Argentina), FAPERJ, CAPES
and CNPq (Brazil), and Programa Regional MATHAMSUD MATH190013.
Carolina L. Gonzalez is partially supported by a CONICET doctoral
fellowship.
We want to thank the anonymous referees for their valuable
suggestions that helped us improving this work.
|
1,314,259,995,859 | arxiv | \section{Introduction}\label{Sec1}
SELEX Collaboration first reported evidence for the doubly charmed
baryon $\Xi^{+}_{cc}$(3520) in the decay mode
$\Xi^{+}_{cc}\rightarrow\Lambda_{c}^{+}K^{-}\pi^{+}$ with the mass
$M_{\Xi^{+}_{cc}}=3519\pm1\rm{MeV}$~\cite{Mattson:2002vu}, although
other experimental collaborations like FOCUS~\cite{Ratti:2003ez},
BABAR~\cite{Aubert:2006qw} and Belle~\cite{Chistov:2006zj} did not
find any evidence of the doubly charmed baryons. Recently, LHCb
collaboration observed $\Xi^{++}_{cc}$ in the
$\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ mass spectrum with the mass
$M_{\Xi^{++}_{cc}}=3621.40\pm0.72 (\rm stat)\pm0.27 (\rm
syst)\pm0.14 (\Lambda^{+}_{c}) \rm{MeV}$~\cite{LHCb}.
In the past decade, there have been many investigations of the
doubly charmed baryon masses
\cite{Bagan:1992za,Roncaglia:1995az,SilvestreBrac:1996bg,Ebert:1996ec,Tong:1999qs,Itoh:2000um,Gershtein:2000nx,
Kiselev:2001fw,Kiselev:2002iy,Narodetskii:2001bq,Lewis:2001iz,Faessler:2001mr,Ebert:2002ig,Mathur:2002ce,Flynn:2003vz,Vijande:2004at,Chiu:2005zc,
Migura:2006ep,Albertus:2006ya,Liu:2007fg,Roberts:2007ni,Valcarce:2008dr,Liu:2009jc,Namekawa:2012mp,Alexandrou:2012xk,
Aliev:2012ru,Aliev:2012iv,Namekawa:2013vu,Sun:2014aya,Chen:2015kpa,Sun:2016wzh,
Shah:2016vmd,Chen:2016spr,Kiselev:2017eic,Chen:2017sbg}. However,
the electromagnetic form factors, especially the magnetic moments
play a pivotal role in describing the inner structures of hadrons.
In the quark-model, the doubly charmed baryons are just like the
light baryons with two light quarks replaced by two charm quarks.
The magnetic moments of doubly charmed baryons were first
investigated by Lichtenberg in Ref.~\cite{Lichtenberg:1976fi} with
nonrelativistic qurak model. Since then, more elaborate quark models
have been developed to study the magnetic moments of doubly charmed
baryons. In Ref.~\cite{SilvestreBrac:1996bg}, various static
properties, including magnetic moments were studied within
non-relativistic quark model using the Faddeev formalism.
magnetic moments were also evaluated in the relativistic quark
model~\cite{JuliaDiaz:2004vh,Faessler:2006ft}. In Ref.~\cite{Branz:2010pq}, the radiative decays of double heavy baryons were studied in a relativistic constituent three-quark model including hyperfine mixing.
Besides the quark models, the magnetic moments of the doubly charmed
baryons have been studied with other approaches, such as the MIT bag
model \cite{Bose:1980vy,Bernotas:2012nz}, the Dirac equation
formalism \cite{Jena:1986xs}, the Skyrmion model \cite{Oh:1991ws},
the hyper central description of the three-body system
\cite{Patel:2008xs} and lattice QCD \cite{Can:2013zpa,Can:2013tna}.
In Refs.~\cite{Can:2013zpa,Can:2013tna}, the authors studied the
electromagnetic properties of baryons in 2+1 flavor lattice QCD.
They found that the magnetic moments of the singly charmed baryons
are dominantly determined by the light quarks, while the charm
quarks play a more important role in the doubly charmed baryons,
which is confirmed in this paper.
Unfortunately most of the above models miss the chiral corrections.
The Goldstone boson cloud effect can be taken into account through
Chiral perturbation theory (ChPT)~\cite{Weinberg:1978kz}, which
organizes the low-energy interactions order by order. Since the
baryon mass $M$ does not vanish in the chiral limit, the convergence
of the chiral expansion is destroyed by the large energy scale $M$.
To overcome the above difficulty, heavy baryon chiral perturbation
theory (HBChPT) was
proposed~\cite{Jenkins:1990jv,Jenkins:1992pi,Bernard:1992qa,Bernard:1995dp},
which has been successfully used in the investigation of baryons.
For the doubly charmed baryons, the two charmed quarks are so heavy
that they can be treated as spectators. Thus, the remaining light
quark dominates the chiral dynamics of the doubly charmed baryons.
In this work, we will investigate the magnetic moments of the
spin-$\frac{1}{2}$ doubly charmed or bottomed baryons with HBChPT.
Right now, there does not exist any experimental measurement of the
magnetic moments of the doubly charmed baryons. We use quark model
to estimate the corresponding low energy constants (LECs) and
calculate the chiral corrections to the magnetic moments order by
order. The numerical results are presented up to next-to-leading
order while the analytical results are calculated to
next-to-next-to-leading order.
Our work is organized as follows. In Section \ref{Sec3}, we discuss
the electromagnetic form factors of the spin-$\frac{1}{2}$ doubly
charmed baryons. In Section \ref{Sec2}, we introduce the effective
chiral Lagrangians. We calculate the chiral corrections to the
magnetic moments order by order in Section \ref{secFormalism} and
present our numerical results in Section \ref{Sec6}. A short summary
is given in Section \ref{Sec7}. We collect
the coefficients of the loop corrections in the
Appendix~\ref{appendix-B}.
\section{Electromagnetic form factors of spin-$\frac{1}{2}$ doubly charmed baryon baryon }\label{Sec3}
For the spin-$\frac{1}{2}$ doubly charmed baryons, the matrix
elements of the electromagnetic current is similar to that of the
nucleon,
\begin{equation}
<\Psi(p^{\prime})|J_{\mu}|\Psi(p)>=e\bar{u}(p^{\prime})\mathcal{O}_{\mu}(p^{\prime},p)u(p),
\end{equation}
with
\begin{equation}
\mathcal{O}_{\mu}(p^{\prime},p)=\frac{1}{M_H}[P_{\mu}G_{E}(q^{2})+\frac{i\sigma_{\mu\nu}q^{\nu}}{2}G_{M}(q^{2})].
\label{eq_new_current}
\end{equation}
where $P=\frac{1}{2}(p^{\prime}+p)$, $q=p^{\prime}-p$, $M_{H}$ is
the doubly charmed baryon mass.
As the doubly charmed baryons are very heavy compared to the chiral
symmetry breaking scale, we adopt the heavy-baryon formulation. In
the heavy baryon limit, the spin-$\frac{1}{2}$ doubly charmed baryon
field $B$ can be decomposed into the large component $H$ and the
small component $L$.
\begin{equation}
B=e^{-iM_{H}v\cdot x}(H+L),
\end{equation}
\begin{equation}
H=e^{iM_{H}v\cdot x}\frac{1+v\hspace{-0.5em}/}{2}B,~
L=e^{iM_{H}v\cdot x}\frac{1-v\hspace{-0.5em}/}{2}B,
\end{equation}
where $v_{\mu}=(1,\vec{0})$ is the velocity of the baryon. Now the
doubly charmed baryon matrix elements of the electromagnetic current
$J_{\mu}$ read
\begin{equation}
<H(p^{\prime})|J_{\mu}|H(p)>=e\bar{u}(p^{\prime})\mathcal{O}_{\mu}(p^{\prime},p)u(p)\label{eq:ocurrent}.
\end{equation}
The tensor $\mathcal{O}_{\mu}$ can be parameterized in terms of
electric and magnetic form factors.
\begin{eqnarray}
\mathcal{O}_{\mu}(p^{\prime},p)=v_{\mu}G_{E}(q^{2})+\frac{[S^{\mu},S^{\nu}]q^{\nu}}{M_H}G_{M}(q^{2}),
\label{eq_newnew_current}
\end{eqnarray}
where $G_{E}(q^{2})$ is the electric form factor and $G_{M}(q^{2})$
is the magnetic form factor. When $q^2=0$, we obtain the charge
($Q$) and magnetic moment ($\mu_{H}$),
\begin{eqnarray}
Q=G_{E}(0), \mu_{H}=\frac{e}{2M_H}G_{M}(0).
\label{eq_magneticcurrent}
\end{eqnarray}
\section{Chiral Lagrangians}\label{Sec2}
\subsection{The strong interaction chiral Lagrangians}
To calculate the chiral corrections to the magnetic moment, we
construct the relevant chiral Lagrangians. We follow
Refs.~\cite{Li:2016ezv,Scherer:2002tk,Bernard:1995dp} to define the
basic chiral effective Lagrangians of the pseudoscalar mesons.
The spin-$\frac{1}{2}$ doubly charmed baryon field reads
\begin{equation}
\Psi=\left(\begin{array}{c}
\Xi_{cc}^{++}\\
\Xi_{cc}^{+}\\
\Omega_{cc}^{+}
\end{array}\right)\Rightarrow\left(\begin{array}{c}
ccu\\
ccd\\
ccs
\end{array}\right).
\end{equation}
The leading order pseudoscalar meson and doubly charmed baryon
interaction Lagrangians read
\begin{eqnarray}
\mathcal{L}^{(1)}=\bar{\Psi}(iD\hspace{-0.6em}/-M_H)\Psi,
\label{Eq:baryon01}\\
\mathcal{L}_{{\rm
int}}^{(1)}=\frac{\tilde{g}_{A}}{2}\bar{\Psi}\gamma^{\mu}\gamma_{5}u_{\mu}\Psi
,\label{Eq:baryon02}
\end{eqnarray}
where $M_H$ is doublely charmed baryon mass,
\begin{eqnarray}
D_{\mu}\Psi&=&\partial_{\mu}\Psi+[\Gamma_{\mu},\Psi].
\end{eqnarray}
We also need the second order pseudoscalar meson and doubly charmed
baryon interaction Lagrangians. Recall that for SU(3) group
representations,
\begin{eqnarray}
3\otimes\bar{3} & = & 1\oplus8\label{Eq:flavor1},\\
8\otimes8 & = &
1\oplus8_{1}\oplus8_{2}\oplus10\oplus\bar{10}\oplus27.\label{Eq:flavor2}
\end{eqnarray}
Both $u_{\mu}$ and $u_{\nu}$ transform as the adjoint
representation. When the product of $u_{\mu}$ and $u_{\nu}$ belongs
to the $8_1$ and $8_2$ flavor representations, we can write down two
independent interaction terms of the second order pseudoscalar meson
and baryon Lagrangians:
\begin{eqnarray}
\hat{\mathcal{L}}_{\rm int}^{(2)}&=&\frac{ig_{h1}}{4M_{B}}
\bar{\Psi}\sigma^{\mu\nu}[u_{\mu},
u_{\nu}]\Psi+\frac{ig_{h2}}{4M_{B}}
\bar{\Psi}\sigma^{\mu\nu}\{u_{\mu}, u_{\nu}\}\Psi ,
\label{Eq:baryon03}
\end{eqnarray}
where the superscript denotes the chiral order, $M_B$ is the nucleon
mass and $g_{h1,h2}$ are the coupling constants. The $g_{h2}$ term
vanishes because of anti-symmetric lorentz structure. Thus, there is
only one linearly independent low energy constant (LEC) $g_{h1}$
which contributes to the present investigations of the doubly
charmed baryon magnetic moments up to $\mathcal{O}(p^4)$.
In the framework of HBChPT, the leading order nonrelativistic
pseudoscalar meson and doubly charmed baryon Lagrangians read
\begin{equation}
\mathcal{L}_{0}^{(1)}=\bar{H}(iv\cdot D)H, \label{Eq:baryon1}
\end{equation}
\begin{equation}
\mathcal{L}_{\rm int}^{(1)}=\tilde{g}_{A}{\rm Tr}\bar{H}S^{\mu}u_{\mu}H,
\label{Eq:baryon2}
\end{equation}
where $\mathcal{L}_{0}^{(1)}$ and $\mathcal{L}_{\rm int}^{(1)}$ are
the free and interaction parts respectively. $S_{\mu}$ is the
covariant spin-operator. We do not consider the mass differences
among different doubly charmed baryons. We estimated the $\phi H H$
coupling $\tilde{g}_{A}=0.5$ with the help of quark model in Section
\ref{Sec6}. For the pseudoscalar meson masses, we use
$m_{\pi}=0.140$ GeV, $m_{K}=0.494$ GeV, and $m_{\eta}=0.550$ GeV. We
use the nucleon masses $M_B=0.938\rm{GeV}$.
The second order pseudoscalar meson and baryon nonrelativistic
Lagrangians read
\begin{eqnarray}
\hat{\mathcal{L}}_{\rm int}^{(2)}&=&\frac{g_{h1}}{2M_{B}}
\bar{H}[S^\mu,S^\nu][u_{\mu}, u_{\nu}]H
.\label{Eq:HHUU}
\end{eqnarray}
The above Lagrangians contribute to the doubly charmed baryon
magnetic moments in diagram (e) of Fig.~\ref{fig:allloop}.
\subsection{The electromagnetic chiral Lagrangians at $\mathcal{O}(p^{2})$}
The lowest order $\mathcal{O}(p^{2})$ Lagrangian contributes to the
magnetic moments of the doubly charmed baryons at the tree level
\begin{equation}
\mathcal{L}_{\mu_{H}}^{(2)}=a_{1}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]\hat{F}_{\mu\nu}^{+}H+a_{2}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]H{\rm
Tr}(F_{\mu\nu}^{+}), \label{Eq:MM1}
\end{equation}
where the coefficients $a_{1,2}$ are the LECs. The chirally
covariant QED field strength tensor $F_{\mu\nu}^{\pm}$ is defined as
\begin{eqnarray} \nonumber
F_{\mu\nu}^{\pm} & = & u^{\dagger}F_{\mu\nu}^{R}u\pm
uF_{\mu\nu}^{L}u^{\dagger},\\
F_{\mu\nu}^{R} & = &
\partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}],\\
F_{\mu\nu}^{L} & = &
\partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}],
\end{eqnarray}
where $r_{\mu}=l_{\mu}=-eQ_HA_{\mu}$ and $Q_H=\rm{diag}(2,1,1)$. The
operator $\hat{F}_{\mu\nu}^{+}=F_{\mu\nu}^{+}-\frac{1}{3}\rm
Tr(F_{\mu\nu}^{+})$ is traceless and transforms as the adjoint
representation. Recall that the direct product $3\otimes\bar{3} =
1\oplus8$ . Therefore, there are two independent interaction terms
in the $\mathcal{O}(p^{2})$ Lagrangians for the magnetic moments of
the doubly charmed baryons.
\subsection{The higher order electromagnetic chiral Lagrangians }
To calculate the magnetic moments to $\mathcal{O}(p^{3})$, we also
need the $\mathcal{O}(p^{4})$ electromagnetic chiral Lagrangians at
the tree level. Recalling flavor representation in
Eqs.~(\ref{Eq:flavor1}), (\ref{Eq:flavor2}) and considering that we
only need the leading-order terms of the fields $F_{\mu\nu}^{+}$ and
$\chi^{+}$ which are diagonal matrices, only three independent terms
contribute to the magnetic moments of the doubly charmed baryons up
to $\mathcal{O}(p^{3})$,
\begin{eqnarray}
\mathcal{L}_{\mu_{H}}^{(4)}&=&d_{1}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]H{\rm
Tr}(\chi^{+}F_{\mu\nu}^{+})+d_{2}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]\{F_{\mu\nu}^{+},\chi^{+}\}H\nonumber
\\&&+d_{3}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]\chi^{+}H{\rm Tr}(F_{\mu\nu}^{+})\label{Eq:MM3}
\end{eqnarray}
where $\chi^{+}$=diag(0,0,1) at the leading order and the factor
$m_{s}$ has been absorbed in the LECs $d_{1,2,3}$.
There are two more terms which also contribute to the doubly charmed
baryon magnetic moments.
\begin{eqnarray}
\mathcal{L^{\prime}}_{\mu_{H}}^{(4)}&=&a_{1}^{\prime}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]F_{\mu\nu}^{+}H{\rm
Tr}(\chi^{+})+a_{2}^{\prime}\frac{-i}{4M_{B}}\bar{H}[S^{\mu},S^{\nu}]H{\rm
Tr}(F_{\mu\nu}^{+}){\rm Tr}(\chi^{+})
\end{eqnarray}
However, their contributions can be absorbed through the
renomalization of the LECs $a_{1,2}$, i.e.
\begin{eqnarray}
a_{1}&\rightarrow&a_{1}+{\rm Tr}(\chi^{+})a_{1}^{\prime},\\
a_{2}&\rightarrow&a_{2}+{\rm Tr}(\chi^{+})a_{2}^{\prime}.
\end{eqnarray}
\section{Formalism up to one-loop level}\label{secFormalism}
We follow the standard power counting scheme as in Ref.
\cite{Li:2017vmq}. The chiral order $D_{\chi}$ is given
by~\cite{Ecker:1994gg}
\begin{equation}
D_{\chi}=4N_{L}-2I_{M}-I_{B}+\sum_{n}nN_{n}, \label{Eq:Power
counting}
\end{equation}
where $N_{L}$ is the number of loops, $I_{M}$ is the number of the
internal pion lines, $I_{B}$ is the number of the internal baryon
lines and $N_{n}$ is the number of the vertices from the $n$th order
Lagrangians. The chiral order of the magnetic moments $\mu_{H}$ is
$(D_\chi-1)$ based on Eq. (\ref{eq_magneticcurrent}).
We assume the exact isospin symmetry with $m_{u}=m_{d}$ throughout
this work. The tree-level Lagrangians in Eqs.
~(\ref{Eq:MM1}),(\ref{Eq:MM3}) contribute to the doubly charmed
baryon magnetic moments at $\mathcal{O}(p^{1})$ and
$\mathcal{O}(p^{3})$ as shown in Fig.~\ref{fig:tree}. The
Clebsch-Gordan coefficients for the various doubly charmed baryons
are collected in Table~\ref{Magnetic moments}. All doubly charmed
baryon magnetic moments are given in terms of $a_{1}$, $a_{2}$,
$d_{1}$, $d_{2}$ and $d_{3}$.
\begin{figure}
\centering
\includegraphics[width=0.6\hsize]{treeccq}
\caption{The $\mathcal{O}(p^{2})$ and $\mathcal{O}(p^{4})$ tree
level diagrams where the doubly charmed baryon is denoted by the
solid line. The left dot and the right black square
represent second- and fourth-order couplings respectively.}
\label{fig:tree}
\end{figure}
\begin{figure}[tbh]
\centering
\includegraphics[width=0.9\hsize]{allloop_ccq}
\caption{The one-loop diagrams where the doubly charmed baryon is
denoted by the solid line. The dashed and wiggly lines represent the
pseudoscalar meson and photon respectively.}\label{fig:allloop}
\end{figure}
There are six Feynman diagrams contribute to the doubly charmed
baryon magnetic moments at one-loop level as shown in
Fig.~\ref{fig:allloop}. All the vertices in these diagrams come from
Eqs.~(\ref{Eq:baryon2}-\ref{Eq:MM1}). In diagrams (a), the meson
vertex is from the strong interaction terms while the photon vertex
is from the meson photon interaction term. In diagram (b), the
photon-meson-baryon vertex is from the $\mathcal{O}(p^{2})$ tree
level magnetic moment interaction in Eq.~(\ref{Eq:MM1}). In diagram
(c), the two vertices are from the strong interaction and seagull
terms respectively. In diagrams (d), the meson vertex is from the
strong interaction terms in Eq.~(\ref{Eq:baryon2}) while the photon
vertex from the $\mathcal{O}(p^{2})$ tree level magnetic moment
interaction in Eq.~(\ref{Eq:MM1}). In diagram (e), the meson-baryon
vertex is from the second order pseudoscalar meson and baryon
Lagrangian in Eq.~(\ref{Eq:HHUU}) while the photon vertex is also
from the meson photon interaction term. In diagram (f), the meson
vertex is from the strong interaction terms while the photon vertex
is from the $\mathcal{O}(p^{2})$ tree level magnetic moment
interaction in Eq.~(\ref{Eq:MM1}).
The diagram (a) contributes to the tensor $e \mathcal O_{\mu}$ in
Eq.~(\ref{eq:ocurrent}) at $\mathcal{O}(p^{3})$ while the diagrams
(b-f) contribute at $\mathcal{O}(p^{4})$. The diagram (c) vanishes
in the heavy baryon mass limit. In particular,
\begin{eqnarray}
J_{c}\propto\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-m^{2}+i\epsilon}(S\cdot
l)\frac{i}{v\cdot l+i\epsilon}S^{\mu}\propto&S\cdot v=0.
\end{eqnarray}
In other words, diagram (c) does not contribute to the magnetic
moments in the leading order of the heavy baryon expansion. The
diagram (f) indicates the corrections from the wave function
renormalization.
Summing all the contributions to the doubly charmed baryon magnetic
moments in Fig.~\ref{fig:allloop}, the leading and next-to-leading
order loop corrections can be expressed as
\begin{eqnarray}
\mu_{H}^{(2,\rm loop)}& = &-\sum_{\phi=\pi,K}\frac{\tilde{g}_{A}^2m_{\phi}M_{N}\beta_{a}^{\phi}}{64\pi f_{\phi}^2},\label{eq:mu2Loop}\\
\mu_{H}^{(3,\rm loop)}& =
&\sum_{\phi=\pi,K}[\frac{\beta^\phi_bm_{\phi}^2\ln
\frac{m_{\phi}^2}{\lambda ^2}}{128\pi^2 f_{\phi}^2}+
\frac{\beta^\phi_e m_{\phi}^2\ln \frac{m_{\phi}^2}{\lambda
^2}}{16\pi^2 f_{\phi}^2}]+\sum_{\phi=\pi,K,\eta}[\frac{-\beta^\phi_d
\tilde{g}_{A}^2m_{\phi}^2}{512\pi^2 f_{\phi}^2}(\ln
\frac{m_{\phi}^2}{\lambda ^2}-2)+ \frac{-3\beta^\phi_f
\tilde{g}_{A}^2m_{\phi}^2\ln \frac{m_{\phi}^2}{\lambda ^2}}{256\pi^2
f_{\phi}^2}] \label{eq:mu3Loop}
\end{eqnarray}
where $\mud=4\pi f_{\pi}$ is the renormalization scale. Here, we use
the number $n$ within the parenthesis in the superscript of $X^{(n,
...)}$ to indicate the chiral order of $X$. $\beta^\phi_{a-f}$ arise
from the corresponding diagrams in Fig.~\ref{fig:allloop}. We
collect their explicit expressions in Tables~\ref{table:abd} and
\ref{table:ef} in the Appendix \ref{appendix-B}.
With the low energy counter terms and loop contributions
(\ref{eq:mu2Loop}, \ref{eq:mu3Loop}), we obtain the magnetic
moments,
\begin{equation}
\mu_{H}=\left\{\mu_{H}^{(1)}\right\}+\left\{\mu_{H}^{(2,\rm
loop)}\right\}+\left\{\mu_{H}^{(3,\rm tree)}+\mu_{H}^{(3,\rm
loop)}\right\}
\end{equation}
where $\mu_{H}^{(1)}$ and $\mu_{H}^{(3,\rm tree)}$ are the
tree-level magnetic moments from Eqs.~(\ref{Eq:MM1}),(\ref{Eq:MM3}).
\section{NUMERICAL RESULTS AND DISCUSSIONS}\label{Sec6}
There are not any experimental data on the doubly charmed baryon
magnetic moments so far. We do not have any experimental inputs to
fit the LECs. In this paper, we use quark model to estimate the
leading-order low energy constants. At the leading order
$\mathcal{O}(p^{1})$, there are two unknown LECs $a_{1,2}$. The
charge matrix $Q_{H}$ is not traceless which is different from that
in the case of the light baryons. Notice that the $a_1$ parts are
proportional to the light quark charge within the doubly charmed
baryon. The $a_2$ parts are the same for the three doubly charmed
baryons and arise solely from the two charm quarks.
At the quark level, the flavor and spin wave function of the
$\Xi_{cc}^{++}$ reads:
\begin{eqnarray}
|\Xi_{cc}^{++};\uparrow\rangle&=&\frac{1}{3\sqrt{2}}[2c\uparrow
c\uparrow u\downarrow-c\uparrow c\downarrow u\uparrow-c\downarrow
c\uparrow u\uparrow +2c\uparrow u\downarrow c\uparrow-c\downarrow
u\uparrow c\uparrow\nonumber\\&&-c\downarrow u\downarrow
c\downarrow+2u\downarrow c\uparrow c\uparrow-u\downarrow c\downarrow
c\downarrow-u\uparrow c\downarrow c\uparrow], \label{xiwavefunc}
\end{eqnarray}
where the arrows denote the third-components of the spin. Replacing
the $u$ quark by the $d$ and $s$ quark, we get the wave functions of
the $\Xi_{cc}^{+}$ and $\Omega_{cc}^{+}$ respectively. The magnetic
moments of the doubly charmed baryons in the quark model are the
matrix elements of the following operator in Eq.~(\ref{xiwavefunc}),
\begin{equation}
\vec{\mu}=\sum_i\mu_i\vec{\sigma}^i, \label{magmomen}
\end{equation}
where $\mu_i$ is the magnetic moment of the quark.
\begin{equation}
\mu_i={e_i\over 2m_i},\quad i=u,d,s.
\end{equation}
We adopt the $m_u=m_d=336$ MeV, $m_s=540$ MeV, $m_c=1660$ MeV as the
constituent quark masses and give the results in the second column
in Table~\ref{various orders Magnetic moments}. The light quark
magnetic moments contributes to the LEC $a_{1}$, which is
proportional to the light quark charge. The heavy quark magnetic
moments contributes to the LEC $a_{2}$, which are the same for the
three doubly charmed baryons. The magnetic moments of the three
doubly charmed baryons are given in the second column in
Table~\ref{various orders Magnetic moments}.
Up to $\mathcal{O}(p^{2})$, we need include both the leading
tree-level magnetic moments and the $\mathcal{O}(p^{2})$ loop
corrections. At this order, there exists only one new LEC $\tilde{g}_A$. We
also use the quark model to estimate $\tilde{g}_A$. Considering the
$\pi^{0}$ coupling at the hadron level,
\begin{eqnarray}
\mathcal{L}_{\Xi_{cc}^{++}\Xi_{cc}^{++}\pi^{0}}=-\frac{1}{2F_{0}}\frac{\tilde{g}_{A}}{2}\bar{\Xi}_{cc}^{++}
\gamma^{\mu}\gamma_{5}\partial_{\mu}\pi^{0}\Xi_{cc}^{++}.
\end{eqnarray}
At the quark level, the $\pi^{0}$ quark interaction reads
\begin{eqnarray}
\mathcal{L}_{\rm quark}=\frac{1}{2}g_{0}\bar{\Psi}_q
\gamma^{\mu}\gamma_{5}\partial_{\mu}\pi^{0}\Psi_q.
\end{eqnarray}
With the help of the flavor wave functions of $\Xi_{cc}^{++}$, we
obtain the matrix elements at the hadron level
\begin{eqnarray}
\langle\Xi_{cc}^{++},s=\frac{1}{2}\mid
i\mathcal{L}_{\Xi_{cc}^{++}\Xi_{cc}^{++}\pi^{0}}\mid\Xi_{cc}^{++},s=\frac{1}{2};\pi^{0}\rangle
\sim-\frac{1}{2F_{0}}\frac{\tilde{g}_{A}}{2}q_3,
\end{eqnarray}
and at the quark level,
\begin{eqnarray}
\langle\Xi_{cc}^{++},s=\frac{1}{2}\mid i\mathcal{L}_{\rm
quark}\mid\Xi_{cc}^{++},s=\frac{1}{2};\pi^{0}\rangle
\sim-\frac{1}{6}g_0q_3.
\end{eqnarray}
After comparison with the axial charge of the nucleon,
\begin{eqnarray}
\frac{-\frac{1}{2}\frac{\tilde{g}_{A}}{2}}{\frac{-1}{6}g_0}=\frac{\frac{1}{2}g_{A}}{\frac{5}{6}g_0},\label{eq:ga}
\end{eqnarray}
one obtains $\tilde{g}_A=\frac{2}{5}g_A=0.5$. Thus, we obtain the numerical
results of $\mathcal{O}(p^{2})$ chiral loop corrections in the third
column in Table~\ref{various orders Magnetic moments}. We list the
numerical results of $\mathcal{O}(p^{2})$ magnetic moments of the
three doubly charmed baryons in the fourth column in
Table~\ref{various orders Magnetic moments}. We also compare the
numerical results of the magnetic moments when the chiral expansions
are truncated at $\mathcal{O}(p^{1})$ and $\mathcal{O}(p^{2})$
respectively in Table~\ref{various orders Magnetic moments}.
Up to $\mathcal{O}(p^{3})$, there are six unknown LECs: $a_{1,2}$,
$g_{h1}$, $d_{1,2,3}$. Unfortunately, we are not able to present
numerical results since it is impossible to to fix all these LECs
with the available experimental information. We present our
analytical results in Eqs. (\ref{eq:mu2Loop}),(\ref{eq:mu3Loop}) and
Table~\ref{Magnetic moments}. Our analytical results may be useful
to the possible chiral extrapolation of the lattice simulations of
the doubly charmed baryon electromagnetic properties.
\begin{table}
\centering
\begin{tabular}{c|ccccc}
\toprule[1pt]\toprule[1pt] Baryons & $\mathcal{O}(p^{1})$ tree &
$\mathcal{O}(p^{2})$ loop & $\mathcal{O}(p^{3})$ tree &
$\mathcal{O}(p^{3})$ loop \tabularnewline \midrule[1pt]
$\Xi_{cc}^{++}$ & $\frac{2}{3}a_{1}+4a_{2}$ & $-0.51\tilde{g}_{A}^2$ &
$-\frac{1}{3}d_{1}$ & $0.15a_{1}+0.21a_{2}-0.27g_{h1}$ &
\tabularnewline
$\Xi_{cc}^{+}$ & $-\frac{1}{3}$$a_{1}+4a_{2}$ &$0.15\tilde{g}_{A}^2$ &
$-\frac{1}{3}d_{1}$ & $-0.05a_{1}+0.21a_{2}+0.06g_{h1}$ &
\tabularnewline
$\Omega_{cc}^{+}$ & $-\frac{1}{3}a_{1}+4a_{2}$ & $0.36\tilde{g}_{A}^2$ &
$-\frac{1}{3}d_{1}-\frac{2}{3}d_{2}+4d_{3}$ &
$-0.12a_{1}+0.36a_{2}+0.21g_{h1}$ & \tabularnewline
\bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The doubly charmed baryon magnetic moments to the
next-to-next-to-leading order(in unit of $\mu_{N}$).}
\label{Magnetic moments}
\end{table}
\begin{table}
\centering
\begin{tabular}{c|cccc}
\hline\toprule[1pt]\toprule[1pt] Baryons& $\mathcal{O}(p^{1})$&
$\mathcal{O}(p^{2})$ loop & $\mathcal{O}(p^{2})$
total\tabularnewline \midrule[1pt] $\Xi_{cc}^{++}$ &
${4\over3}\mu_c-{1\over3}\mu_u=-0.12$ & $-0.13$ &
-0.25\tabularnewline \hline $\Xi_{cc}^{+}$ &
${4\over3}\mu_c-{1\over3}\mu_d=0.81$ & $0.04$ & 0.85\tabularnewline
\hline $\Omega_{cc}^{+}$ & ${4\over3}\mu_c-{1\over3}\mu_s=0.69$ &
$0.09$ & 0.78\tabularnewline \bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The doubly charmed baryon magnetic moments when the chiral
expansion is truncated at $\mathcal{O}(p^{1})$ and
$\mathcal{O}(p^{2})$, respectively (in unit of $\mu_{N}$).}
\label{various orders Magnetic moments}
\end{table}
We also calculate the magnetic moments of the other doubly heavy baryons. At the quark level, the flavor and spin wave functions of the doubly bottomed baryons are the same as those of the doubly charmed baryons after replacing the c quarks by the b quarks. After the similar calculations of Eqs. (\ref{xiwavefunc})-(\ref{eq:ga}), one obtains the axial charge of doubly bottomed baryons $\tilde{g}_A(bbq)=\frac{2}{5}g_A$ and the tree level magnetic moments of the three doubly bottomed baryons in the second column in Table \ref{table:bbq}. We collect the numerical results of doubly bottomed baryon magnetic moments to next-to-leading order in Table \ref{table:bbq}.
\begin{table}
\centering
\begin{tabular}{c|cccc}
\toprule[1pt]\toprule[1pt]
Baryons & $\mathcal{O}(p^{1})$ tree & $\mathcal{O}(p^{2})$ loop & Total \tabularnewline
\midrule[1pt]
$\Xi_{bb}^{0}$ & $\frac{4}{3}\mu_{b}-\frac{1}{3}\mu_{u}=-0.71$ & -0.13 & -0.84 \tabularnewline
$\Xi_{bb}^{-}$ & $\frac{4}{3}\mu_{b}-\frac{1}{3}\mu_{d}=0.22$ & 0.04 & 0.26 \tabularnewline
$\Omega_{bb}^{-}$ & $\frac{4}{3}\mu_{b}-\frac{1}{3}\mu_{s}=0.10$& 0.09 & 0.19 \tabularnewline
\bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The doubly bottomed baryon magnetic moments $(bbq)$ to the next-to-leading order(in
unit of $\mu_{N}$).} \label{table:bbq}
\end{table}
We also calculate the magnetic moments of the doubly heavy baryons containing a light quark, a charm quark and a bottom quark. We refer to the charm quark and the bottom quark as a diquark. There are two different multiplets of the doubly heavy baryons. The symmetric diquark $(\{bc\})$ has spin 1, while the antisymmetric diquark $([bc])$ has spin 0.
At the quark level, the flavor and spin wave function of the $(\{bc\}q)$ baryons reads,
\begin{eqnarray}
|\{bc\}q;\uparrow\rangle = \frac{1}{\sqrt{2}}(\mid cbq\rangle+\mid bcq\rangle)\otimes\frac{1}{\sqrt{6}}
(2\mid\uparrow\uparrow\downarrow\rangle-\mid\uparrow\downarrow\uparrow\rangle-\mid\downarrow\uparrow\uparrow\rangle),
\end{eqnarray}
while the flavor and spin wave function of the $([bc]q)$ baryons reads,
\begin{eqnarray}
|[bc]q;\uparrow\rangle = \frac{1}{\sqrt{2}}(\mid cbq\rangle-\mid bcq\rangle)\otimes\frac{1}{\sqrt{2}}
(\mid\uparrow\downarrow\uparrow\rangle-\mid\downarrow\uparrow\uparrow\rangle).
\end{eqnarray}
After the similar calculations, one obtains the axial charge of the $\{bc\}q$ baryons $\tilde{g}_A(\{bc\}q)=\frac{2}{5}g_A$ and the axial charge of the $[bc]q$ baryons $\tilde{g}_A([bc]q)=-\frac{6}{5}g_A$. We collect the tree level magnetic moments of the $\{bc\}q$ baryons in the second column in Table \ref{table:{bc}q} and the tree level magnetic moments of the three $[bc]q$ baryons in the second column in Table \ref{table:[bc]q}. We collect the numerical results of the $\{bc\}q$ and $[bc]q$ baryon magnetic moments to next-to-leading order in the fourth column in Table \ref{table:{bc}q} and Table \ref{table:[bc]q} respectively.
\begin{table}
\centering
\begin{tabular}{c|cccc}
\toprule[1pt]\toprule[1pt]
Baryons & $\mathcal{O}(p^{1})$ tree & $\mathcal{O}(p^{2})$ loop & Total \tabularnewline
\midrule[1pt]
$\Xi_{\{bc\}q}^{+}$ & $\frac{1}{3}(2\mu_{b}+2\mu_c-\mu_u)=-0.41$ & -0.13 & -0.54 \tabularnewline
$\Xi_{\{bc\}q}^{0}$ & $\frac{1}{3}(2\mu_{b}+2\mu_c-\mu_d)=0.52$ & 0.04 & 0.56 \tabularnewline
$\Omega_{\{bc\}u}^{0}$ & $\frac{1}{3}(2\mu_{b}+2\mu_c-\mu_s)=0.40$& 0.09 & 0.49 \tabularnewline
\bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The magnetic moments of doubly heavy baryons $(\{bc\}q)$ to the next-to-leading order(in
unit of $\mu_{N}$).} \label{table:{bc}q}
\end{table}
\begin{table}
\centering
\begin{tabular}{c|cccc}
\toprule[1pt]\toprule[1pt]
Baryons& $\mathcal{O}(p^{1})$ tree & $\mathcal{O}(p^{2})$ loop & Total \tabularnewline
\midrule[1pt]
$\Xi_{[bc]q}^{+}$ & $\mu_u=1.86$ & -1.17 & 0.69 \tabularnewline
$\Xi_{[bc]q}^{0}$ & $\mu_d=-0.93$ & 0.34 & -0.59 \tabularnewline
$\Omega_{[bc]u}^{0}$ & $\mu_s=-0.58$& 0.82 & 0.24 \tabularnewline
\bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The magnetic moments of doubly heavy baryons $([bc]q)$ to the next-to-leading order(in
unit of $\mu_{N}$).} \label{table:[bc]q}
\end{table}
\section{Conclusions}\label{Sec7}
The discovery of the $\Xi^{++}_{cc}$ inspired heated theoretical
investigation of the doubly charmed baryons. The doubly charmed
baryons are so special that the chiral dynamics is dominated by the
single light quark. The electromagnetic property of the doubly
charmed baryons encodes crucial information of their inner
structure. In this work, we have performed a systematical
calculations of the chiral corrections to the magnetic moments of
doubly charmed baryons up to the next-to-next-to-leading order in
the framework of heavy baryon chiral perturbation theory. We use
quark model to estimate the low energy constants and present the
numerical results up to next-to-leading order:
$\mu_{\Xi^{++}_{cc}}=-0.25\mu_{N}$,
$\mu_{\Xi^{+}_{cc}}=0.85\mu_{N}$,
$\mu_{\Omega^{+}_{cc}}=0.78\mu_{N}$.
From Table~\ref{various orders Magnetic moments}, the magnetic
moments of the $\Xi^{+}_{cc}$ and $\Omega^{+}_{cc}$ are dominated by
the leading order term while the chiral corrections are quite small.
To be specific, the numerical values of the $\mathcal{O}(p^{1})$
magnetic moments of the $\Xi^{+}_{cc}$ and $\Omega^{+}_{cc}$ are
enhanced since the charge of the down and strange quark is $-{1\over
3}$ while the charm quark charge is $+{2\over 3}$. Only the $\pi^+$
meson contributes to the chiral correction to $\mu_{\Xi^{+}_{cc}}$
at $\mathcal{O}(p^{2})$ while only $K^+$ contributes to
$\mu_{\Omega^{+}_{cc}}$ at this order.
For comparison, the up and charm quark contributions to the
$\mathcal{O}(p^{1})$ magnetic moment of the $\Xi^{++}_{cc}$ are
destructive. Such an accidental strong cancelation renders the
leading order magnetic moment of the $\Xi^{++}_{cc}$ is much smaller
than those of its partner states. In contrast, both the $\pi^+$ and
$K^+$ mesons contribute to the chiral corrections to
$\mu_{\Xi^{++}_{cc}}$ at $\mathcal{O}(p^{2})$. In other words, the
leading order magnetic moment of the $\Xi^{++}_{cc}$ is reduced
while the loop correction is enhanced. As a result, the loop
correction is numerically very important and even slightly larger
than the leading order term. Such a unique feature can be exposed by
future lattice QCD simulation.
In Table~\ref{Comparison of magnetic moments}, we compare our
results obtained in the HBChPT with those from other model
calculations such as quark model (QM)~\cite{Lichtenberg:1976fi},
relativistic three-quark model (RTQM)~\cite{Faessler:2006ft},
nonrelativistic quark model in Faddeev approach
(NQM)~\cite{SilvestreBrac:1996bg}, relativistic quark model
(RQM)~\cite{JuliaDiaz:2004vh}, skyrmion
description~\cite{Oh:1991ws}, confining logarithmic potential
(CLP)~\cite{Jena:1986xs}, MIT bag model~\cite{Bose:1980vy},
nonrelativistic quark model (NQM)~\cite{Patel:2008xs} and lattice
QCD(LQCD). All these approaches lead to roughly consistent results.
As the byproducts, we have also calculated the magnetic moments of
the other doubly heavy baryons, including the $bbq$ baryons, the
$\{bc\}q$ baryons and the $[bc]q$ baryons. Especially, the magnetic
moments of $[bc]q$ baryons are quite interesting as their magnetic
moments totally arise from the light quarks as shown in Table
\ref{table:[bc]q}.
We hope our calculation may be useful for future experimental
measurements. As there are several unknown LECs up to
next-to-next-to-leading order, we are looking forward to further
progresses in both theory and experiment so that we can check the
chiral expansion convergence of the three doubly charmed baryons.
Our results may be useful for future experimental measurement of the
magnetic moments. Our analytical results may also be useful to the
possible chiral extrapolation of the lattice simulations.
\begin{table}
\centering
\begin{tabular}{c|ccc}
\toprule[1pt]\toprule[1pt] Baryons & $\Xi_{cc}^{++}$ &
$\Xi_{cc}^{+}$ & $\Omega_{cc}^{+}$\tabularnewline \midrule[1pt]
QM~\cite{Lichtenberg:1976fi} & -0.124 & 0.806 & 0.688\tabularnewline
\hline RTQM \cite{Faessler:2006ft} & 0.13 & 0.72 &
0.67\tabularnewline \hline NRQM \cite{SilvestreBrac:1996bg} & -0.206
& 0.784 & 0.635\tabularnewline \hline RQM \cite{JuliaDiaz:2004vh} &
-0.10 & 0.86 & 0.72\tabularnewline \hline Skyrmion \cite{Oh:1991ws}
& -0.47 & 0.98 & 0.59\tabularnewline \hline CLP \cite{Jena:1986xs} &
-0.154 & 0.778 & 0.657\tabularnewline \hline MIT bag model
\cite{Bose:1980vy} & 0.17 & 0.86 & 0.84\tabularnewline \hline NQM
\cite{Patel:2008xs} & -0.208 & 0.785 & 0.635\tabularnewline \hline
LQCD \cite{Can:2013tna} & \textemdash{} & 0.425 &
0.413\tabularnewline \hline This work & -0.25 & 0.85 &
0.78\tabularnewline \bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{Comparison of the decuplet to octet baryon transition
magnetic moments in literature including quark model
(QM)~\cite{Lichtenberg:1976fi}, relativistic three-quark model
(RTQM)~\cite{Faessler:2006ft}, nonrelativistic quark model in
Faddeev approach (NQM)~\cite{SilvestreBrac:1996bg}, relativistic
quark model (RQM)~\cite{JuliaDiaz:2004vh}, skyrmion
description~\cite{Oh:1991ws}, confining logarithmic potential
(CLP)~\cite{Jena:1986xs}, MIT bag model~\cite{Bose:1980vy},
nonrelativistic quark model (NQM)~\cite{Patel:2008xs} and lattice
QCD(LQCD) \cite{Can:2013tna}(in unit of $\mu_{N}$).}
\label{Comparison of magnetic moments}
\end{table}
\section*{ACKNOWLEDGMENTS}
H. S. Li is very grateful to X. L. Chen and W. Z. Deng for very
helpful discussions. This project is supported by the National
Natural Science Foundation of China under Grants 11575008,
11621131001 and 973 program. This work is also supported by the
Fundamental Research Funds for the Central Universities of Lanzhou
University under Grants 223000--862637.
\begin{appendix}
\section{COEFFICIENTS OF THE LOOP CORRECTIONS} \label{appendix-B}
In this appendix, we collect the explicit formulae for the chiral
expansion of the doubly charmed baryon magnetic moments in Tables
\ref{table:abd} and \ref{table:ef}.
\begin{table}
\centering
\begin{tabular}{c|ccccccc}
\toprule[1pt]\toprule[1pt] Baryons & $\beta_{a}^{\pi}$ &
$\beta_{a}^{K}$ & $\beta_{b}^{\pi}$ & $\beta_{b}^{K}$ &
$\beta_{d}^{\pi}$ & $\beta_{d}^{K}$ &
$\beta_{d}^{\eta}$\tabularnewline \midrule[1pt] $\Xi_{cc}^{++}$ &
$2$ & $2$ & $-4a_{1}$ & $-4a_{1}$ & $24a_{2}$ &
$-\frac{4}{3}a_{1}+16a_{2}$ &
$\frac{4}{9}(a_{1}+6a_{2})$\tabularnewline \hline $\Xi_{cc}^{+}$ &
$-2$ & 0 & $4a_{1}$ & 0 & $2a_{1}+24a_{2}$ &
$-\frac{4}{3}a_{1}+16a_{2}$ &
$\frac{2}{9}(-a_{1}+12a_{2})$\tabularnewline \hline
$\Omega_{cc}^{+}$ & 0 & $-2$ & 0 & $4a_{1}$ & 0 &
$\frac{4}{3}(a_{1}+24a_{2})$ &
$-\frac{8}{9}(a_{1}-12a_{2})$\tabularnewline
\bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The coefficients of the loop corrections to the
doubly charmed baryon magnetic moments from Figs.
\ref{fig:allloop}(a), \ref{fig:allloop}(b) and
\ref{fig:allloop}(d).} \label{table:abd}
\end{table}
\begin{table}
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule[1pt]\toprule[1pt] Baryons & $\beta_{e}^{\pi}$ &
$\beta_{e}^{K}$ & $\beta_{f}^{\pi}$ & $\beta_{f}^{K}$ &
$\beta_{f}^{\eta}$\tabularnewline \midrule[1pt] $\Xi_{cc}^{++}$ &
$g_{h1}$ & $g_{h1}$ & $2(a_{1}+6a_{2})$ &
$\frac{4}{3}(a_{1}+6a_{2})$ &
$\frac{2}{9}(a_{1}+6a_{2})$\tabularnewline \hline $\Xi_{cc}^{+}$ &
$-g_{h1}$ & 0 & $-a_{1}+12a_{2}$ & $-\frac{2}{3}a_{1}+8a_{2}$ &
$-\frac{1}{9}a_{1}+\frac{4}{3}a_{2}$\tabularnewline \hline
$\Omega_{cc}^{+}$ & 0 & $-g_{h1}$ & 0 & $-\frac{4}{3}a_{1}+16a_{2}$
& $-\frac{4}{9}(a_{1}-12a_{2})$\tabularnewline
\bottomrule[1pt]\bottomrule[1pt]
\end{tabular}
\caption{The coefficients of the loop corrections to the
doubly charmed baryon magnetic moments from Figs.
\ref{fig:allloop}(e) and \ref{fig:allloop}(f).} \label{table:ef}
\end{table}
\end{appendix}
\vfil \thispagestyle{empty}
\newpage
|
1,314,259,995,860 | arxiv | \section{Introduction}
Characterizing the structure and evolution of the cold gas in dark
matter halos is a key element in current models of galaxy formation
(e.g.,\ Kere{\v s} \ensuremath{\mbox{et~al.}}\ 2005). The fraction of cold and hot gas within
dark matter halos and the rate at which gas is being accreted are
essential to our understanding of disc and star formation
(e.g.,\ \citealt{dekel2009a}). Extended gaseous envelopes around
galaxies were first predicted several decades ago
(\citealt{spitzer1956a}). Observations of H\,I maps around local
galaxies (e.g.,\ \citealt{thilker2004a,doyle2005a}) and comparisons of
galaxies and QSO absorption-line systems
(e.g.,\ \citealt{bergeron1986a,lanzetta1990a,steidel1994a,chen2001b,chen2008a})
have indeed shown the presence of extended cool gas ($T\sim 10^4$ K)
out to $50-100\ h^{-1}$ kpc radii. The physical mechanism that
explains the origin of the extended cold halo gas is, however,
unclear. Some of the most common scenarios are (i) outflows from starburst
systems (e.g.,\ \citealt{bond2001a}); (ii) stripping from the
accretion of gas-rich satellites \citep{wang1993a}; (iii) cold gas
bound to substructure within the host dark halo (e.g.,\ \citealt{sternberg2002a}),
and (iv) a two-phase medium composed of cold and hot
gas \citep{mo1996a,maller2004a}.
A potential probe of the cold halo gas is the Mg\,II $\lambda\lambda$
2796,2803 absorption features commonly seen in the spectra of
background QSOs. These absorbers are thought to originate in
photo-ionized gas of temperature $T\sim 10^4$ K and to trace
high-column density \mbox{${\rm H\,I}$}\ clouds of neutral hydrogen column density
$N(\mbox{${\rm H\,I}$}) \approx 10^{18}-10^{22}$ \mbox{${\rm cm^{-2}}$}\
\citep{bergeron1986a,rao2006a}. This large associated \mbox{${\rm H\,I}$}\ column
density indicates that Mg\,II absorbers arise in halo gas around
individual galaxies \citep{doyle2005a}. This is also supported by the
presence of luminous galaxies at projected distances $\rho = 50-100 \
h^{-1}$ kpc from known Mg\,II absorbers \citep{bergeron1986b,
lanzetta1990a,bergeron1991a,lanzetta1992a,steidel1992b,steidel1994a,zibetti2007a,nestor2007a,kacprzak2008a}.
In addition to the classical gas accretion scenario for the origin of
Mg\,II absorbers at larger galactic radii, there is a competing
scenario that has gained substantial attention recently. In this new
picture, strong Mg\,II absorbers of rest-frame absorption equivalent
width $W_r(2796)>1$ \AA\ originate in starburst driven outflows
(e.g.,\ \citealt{bond2001a,menard2008a,weiner2008a}). Under this scenario, the
Mg\,II absorbing gas orginates in the cold \emph{outflowing} material
surrounding starburst galaxies. An interesting recent finding is a
strong correlation between dust extinction $E(B-V)$ and $W_r(2796)$
by \citet{menard2008b}. While the observed $E(B-V)$ vs.\ $W_r(2796)$
is consistent with the expectation of the starburst scenario, this
observation is also expected if the Mg\,II absorbing galaxies exhibit
a metallicity gradient commonly seen in regular galaxies
(e.g.,\ \citealt{zaritsky1994a,vanzee1998a}). Dense clumps in starburst
driven outflows are expected to contribute to some fraction of the
observed Mg\,II absorbers, but the significance of this fraction and
how the fraction varies with $W_r(2796)$ are both uncertain.
As a first step toward a quantitative understanding of the physical
origin of the Mg\,II absorber population, we are carrying out a
cross-correlation analysis of Mg\,II absorbers with photometrically
identified luminous red galaxies (LRGs) in the Sloan Digital Sky
Survey (SDSS; \citealt{york2000a}). The primary goals are (1) to
determine the clustering amplitude of Mg\,II absorbers and (2) to
examine how the clustering amplitude depends on absorber strength
$W_r(2796)$. The clustering amplitude of Mg\,II absorbers is
determined based on their cross-correlation signals with LRGs on
projected co-moving distance scales of $r_p=1-30\ h^{-1}$ Mpc. Because the mean
halo\footnote{We define halo as a region of overdensity 200 with
respect to the mean mass density in the universe.} mass of the LRGs
can be calculated from their clustering signal
(e.g.,\ \citealt{zheng2008a,blake2008a,wake2008a,padmanabhan2008a}) the cross-correlation amplitude of LRGs
and Mg\,II absorbers provides a statistical estimate of the mean mass
of the host dark matter halos. A similar study was published by
Bouch\'e \ensuremath{\mbox{et~al.}}\ (2004, 2006), in which the authors attempted to
constrain the mean halo mass of Mg\,II absorbers using a flux-limited
($i'<21$ mag) sample of LRGs found in SDSS (see also Lundgren \ensuremath{\mbox{et~al.}}\
2009).
Our analysis differs from others in two important aspects. First, we
measure the clustering amplitude of Mg\,II absorbers using a {\it
volume-limited} (instead of flux-limited) LRG sample. A
flux-limited selection criterion forms an inhomogeneous sample of
LRGs, excluding progressively more intrinsically fainter LRGs (and
hence lower-mass halos) toward higher redshifts. Such inhomogeneous
samples of LRGs over a broad redshift range ($z=0.35-0.8$) include an
inherent uncertainty in the estimated mean halo mass that is difficult
to assess, but this systematic bias has not been addressed in previous
studies. We have directly compared the clustering amplitudes between
using a flux-limited and a volume-limited LRG sample. We will show in
the following sections that the relative halo bias of Mg\,II absorbers
may have been {\it overestimated} by as much as $\approx$ 20\% in
previous studies.
Second, the LRGs in SDSS have been identified using photometric
redshifts ($z_{\rm phot}$) that have associated uncertainties relative
to spectroscopic redshifts ($z_{\rm spec}$) of $\sigma_0=|z_{\rm
phot}-z_{\rm spec}|/(1+z_{\rm spec})\approx0.03$ at $i'\approx 19$
(\citealt{collister2007a}; \citealt{oyaizu2008a}). At fainter
magnitudes, $\sigma_0$ increases steeply to $\sigma_0\ge 0.1$ at
$i'>20.7$ (Collister \ensuremath{\mbox{et~al.}}\ 2007). It is clear from these studies
that the redshift uncertainties of the LRG sample vary from
$\sigma_z=\sigma_0(1+z)=0.04$ for galaxies of $i'<19$ at
$z=0.35$ to $\sigma_z=0.18$ for galaxies of $i'\approx 21$ at $z=0.8$.
On the other hand, Mg\,II absorbers are identified in QSO spectra with
a redshift precision better than $\sigma_z \approx 0.0004$, corresponding
to roughly half of the width of a resolution element. The
dependence of photometric redshift errors on galaxy brightness and
redshift are expected to introduce additional systematic uncertainties
in the estimate of the Mg\,II-LRG cross-correlation amplitude. To
reduce the systematic bias due to photometric redshift errors, we
first restrict our analysis to including only galaxies brighter than
$i'=20$\footnote{The volume-limited selection criterion is applied
after adopting this brightness limit.}. This allows us to maintain
an LRG sample of higher redshift precisions. In addition, we assess
the systematic uncertainty using a mock galaxy catalog with and
without an added $\sigma_0\times (1+z)$ redshift perturbation. We
will show in the following section that relative clustering amplitude
of Mg\,II absorbers measured without accounting for photometric
redshift uncertainties may have been {\it overestimated} by $\approx
10$\%.
This paper is organized as follows. We first describe our data
samples, Mg\,II and LRGs in Section 2. In Section 3.1, we describe the
method used to calculate the two-point correlation function and
associated errors.
Cross-correalation results are presented in section 3.2, including a
discussion of the effects of photometric redshift uncertainties in the
calculation. In Section 4, we summarize the routines adopted for
calculating the bias and mean halo mass of absorbers.
Finally, we discuss the results of our analysis in Section 5.
A more extensive halo occupation distribution analysis of the Mg\,II
absorbers will be the subject of a forthcoming paper. We adopt a
$\Lambda$CDM cosmology, $\Omega_{\rm M}=0.25$ and $\Omega_\Lambda =
0.75$, with a dimensionless Hubble parameter $h = H_0/(100 \ {\rm km} \
{\rm s}^{-1}\ {\rm Mpc}^{-1})$ throughout the paper. All masses are expressed in units
of $h^{-1}~M_{\odot}$ unless otherwise stated.
\section{Data}
\subsection{MgII Catalog}
The MgII absorbers catalog is based on an extension of the SDSS DR3
sample \citep{prochter2006a} to include DR5 QSO spectra (J.X.\
Prochaska, private communication). This sample of Mg\,II absorbers has
a 95\% completeness for absorbers of $W_r(2796) > 1$
\AA. The catalog
contains 11,254 Mg\,II absorbers detected at $z_{\rm Mg\,II}=0.37-2.3$ along 9,774 QSO
sightlines. From this sample, we selected absorbers with a separation
of at least 10,000 km/s from the QSO redshift to avoid contamination
by associated absorbers of the QSOs. We excluded absorbers that fall
outside of our survey mask (see Section 3.1.1) and we limited ourselves
to Mg\,II doublets in the redshift range of interest, $z=0.40-0.70$,
defined by the LRG sample. Note that none of the absorbers listed in
the Prochter et al.\ catalog are found in the Ly$\alpha$ forest of
their respective quasar. The procedure described above yielded 1,158
absorbers of $W_r(2796) > 1$ \AA. This contitutes our \emph{primary}
Mg\,II catalog. The short dashed histogram in Figure \ref{lrgs_z_catalog}
shows the redshift distribution of our primary Mg\,II sample.
\subsection{LRG Catalog}
LRGs are tracers of the large-scale structures in the universe.
Their clustering properties are well-known, and they reside in massive halos of
$M_h>10^{13} \mbox{$h^{-1}\,{\rm M}_\odot$}$ (e.g.,\ \citealt{blake2008a,zheng2008a,wake2008a,padmanabhan2008a}).
Because they are luminous and have strong 4,000-\AA\ break, their photometric redshift can be more
reliably estimated than blue star-forming galaxies.
We used the MegaZ-LRG catalog (hereafter MegaZ) \citep{collister2007a}
as our initial LRG sample. MegaZ is a photometric redshift catalog of
approximately $10^6$ LRGs found in the SDSS DR4 imaging footprint.
The catalog covers more than 5,000 $\rm{deg}^2$ in the redshift range
0.4 $<$ $z$ $<$ 0.7. The LRGs are selected following a series of cuts
in a multidimensional color diagram (e.g.,\ \citealt{scranton2003a,eisenstein2005a,collister2007a}).
The photometric redshifts of
LRGs are determined using artificial neural networks (ANNz) and the
2SLAQ training set of 13,000 available LRGs spectroscopic redshifts
\citep{cannon2006a}.
We applied a set of additional color selection criteria suggested by
\citet{blake2007a}. These modified criteria yielded less
contamination by blue galaxies and were used in clustering and halo
occupation analyses \citep{blake2008a}. We further limited ourselves
to galaxies with $i'<20$ for a higher photometric redshift precision.
We defined three LRG samples for our analysis. The first was a flux-limited sample of $i'<20$ LRGs at $z=0.4-0.7$. This primary sample covers the entire redshift range offered by the initial LRG sample. The second was a volume-limited sample of $M_{i'}-5\,\log\,h<-22$ at
$z=0.45-0.60$. The redshift range was selected to provide the largest number of LRGs available under a uniform minimum rest-frame absolute magnitude selection criterion. Extending to lower or higher redshifts would result in a significant reduction in the sample size. The third was
a flux-limited sample of $i'<20$ at $z=0.45-0.60$ to be directly compared with the volume-limited subsample from the same redshift range.
Our sample definition was motivated by the knowledge that a flux-limited sample contains galaxies that are progressively fainter at lower redshifts, whereas a volume-limited criterion identifies a uniform sample of galaxies that occupy the same luminosity interval at different redshifts.
In addition, the cross- and auto-correlation calculations are evaluated at different
redshift in a flux-limited sample. The redshift number density of LRGs peaks at $z\sim 0.45$, while the Mg\,II redshift distribution is flat.
This implies that the mean redshifts of the cross-correlation and auto-correlation calculations are different, introducing additional uncertainties in the estimated mean halo bias. This problem is alleviated in a
volume-limited sample because the cross- and auto-correlation terms have
similar redshift evolution (see Figure 1). Comparing the correlation functions determined using different subsamples allowed us to evaluate possible systematic uncertainties due to sample selections.
The limiting absolute magnitude of
the volume-limited sample was determined by calculating the absolute magnitude of
our faintest galaxies ($i'=20.0$) at the limiting redshift $z=0.60$
\begin{equation}
20.0-{M}_i = {\rm DM}(z) + k(z) \; ,
\label{absolute_magnitude}
\end{equation}
where ${\rm DM}(z)$ is the distance modulus and $k(z)$ represents the
$k$-correction. At $z=0.60$, the limiting magnitude is $M_i - 5\log h
= -22$ and we found roughly 197\,K galaxies more luminous than this
limit. A quantitative description of the three LRG samples including
the number of objects and redshift range can be found in Table
\ref{summary_calculations}. Figure \ref{lrgs_z_catalog} shows the
redshift distributions of the three LRG samples and the top panel of Figure
\ref{lrgs_photoz} shows the magnitude distributions of our
flux-limited LRG sample and of the initial LRG sample.
It is notoriously difficult to determine the photometric redshift of
faint galaxies. The bottom panel of Figure \ref{lrgs_photoz} shows a
subsample of the photometric redshift errors taken for 1,000
overlapping galaxies between MegaZ and the SDSS photometric redshift
table. The photoz's errors shown in this figure are taken in the
"photozcc2" table found on the SDSS skyserver archive
\citep{oyaizu2008a}. The photometric redshift errors obtained by the
MegaZ team are consistent with the ones in
\citet{oyaizu2008a} to within measurement uncertainties.
However, it is clear from Figure \ref{lrgs_photoz}
that for $i'>20$, the uncertainties in the photoz's increase rapidly. For this reason, we decided to restrict ourselves
to LRG candidates with $i' <20$.
This procedure yielded a total of 962,216 MegaZ objects satisfying our
additional selection criteria. This constitutes our \emph{primary}
LRG catalog.
\begin{figure}
\vspace{5pt} \centerline{\hbox{ \hspace{0.0in}
\includegraphics[angle=-90,scale=0.40]{figure2a_bw.eps}
}
} \vspace{5pt}
\caption{ Redshift distributions of the initial MegaZ LRG sample
and of the three LRG samples used in our correlation studies.
The initial MegaZ catalog is shown in the \emph{dotted} line. The
\emph{solid} line shows the flux-limited sample ($z=0.40-0.70$)
and the \emph{dashed} line shows the volume-limited sample
($z=0.45-0.6$). The redshift distribution of Mg\,II absorbers in
our primary sample is overplotted in \emph{short-dashed} line for
comparison. }
\label{lrgs_z_catalog}
\end{figure}
\begin{figure}
\vspace{0.5pt}
\centerline{\hbox{ \hspace{0.0in}
\includegraphics[angle=-90,scale=0.40]{figure3a_bw.eps}
}
}
\vspace{0.5pt}
\vspace{5pt}
\centerline{\hbox{ \hspace{0.0in}
\includegraphics[angle=-90,scale=0.40]{figure3b_bw.eps}
}
}
\vspace{5pt}
\caption{\emph{Top panel}:
Magnitude distribution of the LRGs satisfying our selection
criteria ($\approx 9.6\times10^5$ LRGs). The line
shows the $i'$ magnitude distribution. Only
galaxies with $i' < 20$ are included in our LRG sample.
\emph{Bottom panel}: Photometric redshift error versus $i'$
magnitude for a subsample of 1,000 galaxies in SDSS taken from
\citet{oyaizu2008a}. The photometric errors are listed in the SDSS
Table "photozerr2" under "CC2". Black diamond points with error
bars represent the mean and dispersion calculated in bins of $\Delta
i'=1$ mag. }
\label{lrgs_photoz}
\end{figure}
\section{Mg\,II absorbers-LRG cross-correlation}
\subsection{Method}
We used the \citet{landy1993a} (LS93) minimum variance estimator to calculate
the projected two-point correlation statistics between Mg\,II absorbers and LRGs.
Using the LS93 estimator, the real-space correlation function can be calculated
following
\begin{equation}
\xi(r_p,\pi) = \frac{D_{\rm a}D_{\rm g} - D_{\rm a}R_{\rm g}
- D_{\rm g}R_{\rm a} + R_{\rm a}R_{\rm g}} {R_{\rm a}R_{\rm g}}
\label{landy}
\end{equation}
where $R$ and $D$ are random points and data, the subscripts $a$ and
$g$ refer to absorbers and galaxies, $r_p$ is the projected comoving
separation between two objects on the sky and $\pi$ is their distance
parallel to the line of sight. Note that the above two-point
estimator has been successfully used in previous correlation study
based on QSO absorbers and galaxy data collected from a smaller area
survey \citep{adelberger2003a}. In practice, we calculated the
projected two-point correlation statistics by summing all pairs along
the sightline
\begin{equation}
w_p(r_p) = \int_l \xi(r_p,\pi) d\pi
\end{equation}
and inside of redshift limits $l$ of our three LRG samples.
Another commonly used estimator, $1+\xi(r_p,\pi) = \frac{D_{\rm a}D_{\rm
g}}{D_{\rm a}R_{\rm g}}$ has also been used in the initial stages
of this work for comparison and validation purposes. Even though this
estimator is easier (no random absorbers) and faster (two terms to
compute instead of four), the variance of $w_p$ is larger for this
estimator than for LS93. For this reason, we employed the LS93 estimator for
all $w_p$ calculations.
We divided the pairs into eight $r_p$ bins covering the range $0.2-35~h^{-1}$ Mpc.
The bins were equally separated in logarithm space. The bin size and the $r_p$ value
of the inner most bin were determined in such a way that at least 10 $DD$ pairs
were found in that bin. The upper limit of $35~h^{-1}$ Mpc was chosen to be a few times
smaller than the size of the jackknife cells (see section 3.1.3).
\subsubsection{Survey mask}
For the calculation of the two-point statistics (following equation \ref{landy})
both data and randoms were distributed over the exact same
survey mask.
Different masks for LRGs and MgII would alter the shape and amplitude of the
correlation signal in an undesirable fashion, especially at large separations.
To make sure the sky coverage was the same for Mg\,II, LRGs and their randoms,
we used the lowest common denominator for all : the DR4 spectroscopic sky survey mask.
Indeed, Mg\,II absorbers were taken
from the SDSS DR5 spectroscopic sample which includes the QSO sightlines inside the DR4 spectroscopic sky.
The LRGs were found in the DR4 imaging sky which encompasses the DR4 spectroscopic sky. The main disadvantage of using
this mask was that the number of rejected objects falling outside of the mask was large, $\approx 40\%$.
To be able to determine which Mg\,II absorbers and LRGs fell inside
this DR4 spectroscopic mask, we used the mask catalog provided by the
NYU Value-Added Galaxy Catalog team \citet{blanton2005a}. The mask
corresponds to the angular selection function describing the
completeness of the SDSS spectroscopic across the sky. It is defined
by spherical polygons. The completeness quantifies the fraction,
inside each polygon, of galaxies with spectrocopic redshift. We used
the angular selection function
of the spectroscopic SDSS DR4 sky ($\rm{sdssdr4safe0res6d}$).
\subsubsection{Generating random Mg\,II absorbers and LRGs}
The right ascension and declination of the LRGs were randomly selected
over the DR4 spectroscopic sky using the function \emph{ransack}
available in the Mangle software package
\citep{hamilton2004a,swanson2008a}. The redshifts of the random
galaxies were determined by sampling the redshift distribution of the
LRG dataset. Determining the sky positions and redshifts of the
random absorbers was not a straightforward process. The angular
selection function of quasars follows the DR4 spectroscopic mask, but
the mask defining the positions where absorbers can be found is
limited to the actual coordinates of the QSOs themselves. Thus, the
random absorbers must be distributed randomly among fixed QSO
sightlines for which SDSS spectra are available. Assigning random
Mg\,II this way eliminates any undesired bias due to the intrinsic
clustering of QSO sightlines. We identified these sightlines
from the \citet{schneider2007a} SDSS DR5 QSO catalog by selecting all
QSOs falling inside our DR4 spectroscopic mask with redshifts large
enough to allow for the detection of Mg\,II absorbers inside the
redshift range of interest for our calculations ($z=0.40-0.70$). We
found a total of $\approx 5.5~\times~10^4$ sightlines. The (ra,dec)
positions of the random absorbers were determined by randomly selecting
the coordinates of these sightlines. Redshifts were selected randomly
from a top hat probability distribution function over $z=0.4-0.7$.
The number of random LRGs, $R_g$, and the number of random Mg\,II absorbers, $R_a$, were
determined after running convergence tests with increasing number of
randoms. We varied the number of each random set (Mg\,II and LRGs)
independently until the measured $w_p$ approached asymptotic values. The numbers of randoms
used in our $w_p$ calculation were $R_a = 10^5$ and $R_g =
4\times10^6$ for Mg\,II and LRGs, respectively. It is important to
note that increasing the number of random Mg\,II absorbers to a very
large number in comparison with the number of available QSO sightlines
($\sim$ 55K) would not lead to a rapid convergence because we would
simply be overcounting the same $D_{\rm g}R_{\rm a}$ pairs and no
additional information would be gained.
\subsubsection{Relative contribution of cosmic variance and
photometric uncertainty to $w_p$ errors}
We looked at two independent sources contributing to the error bars on
$w_p$ : cosmic variance and photometric redshift errors. Photometric
redshift errors are expected to affect the measurements of $w_p$'s in
two different ways. First, redshift uncertainties are expected to
increase the noise in the $w_p$ measurements due to uncertainties in
the object positions. Second, redshift uncertainties are expected to
systematically alter the $w_p$ measurements to lower values due to an
inherent sample selection bias. A galaxy sample selected based on
imprecise photometric redshifts contain galaxies from a broader
redshift range than one selected based on precise spectroscopic
redshifts, and consequently reducing the observed correlation signal
by including additional uncorrelated pairs. In this section, we
address random errors in the $w_p$ measurements due to cosmic variance
and object distance uncertainties. We defer the discussion on the
systematic errors of $w_p$ due to the sample selection bias to \S\
3.2.2.
We estimated the cosmic variance using the jackknife resampling
technique. The sky was separated into $N=192$ (see section 3.2.1
for a justification of this number) cells of roughly equal
survey area.
The cosmic variance for each point $r_{p,i}$
corresponds to the $i$th-diagonal element of the covariance matrix was
calculated using a jackknife resampling technique :
\begin{equation}
{\rm COV}(w_i,w_j)=\frac{N-1}{N}\sum_{k=1}^{N}(w_i^k-\overline{w_i})(w_j^k-\overline{w_j})
\label{covariance}
\end{equation}
and $k$ represents the iteration in which box $k$ was removed.
The mean $\overline{w_{i}}$ was calculated for bin $i$ over all $w_{p}^k$'s.
The impact of the large uncertainties due to photometric redshifts on
the size of the $w_p$ errors is not taken into account by the
jackknife resampling technique alone. Previous works (e.g.,\
\citealt{bouche2006a,blake2008a}) have focused primarily on cosmic
variance in the calculation of the errors bars, but have not addressed
additional random noise due to photometric redshift errors which are
quite large ($\sigma_z \approx 0.05$ for $i'<20$) compared to the
redshift ($z=0.5$).
To account for the independent contribution of photoz's uncertainties
on the final $w_p$ error bars, we generated 100 independent
realizations of the MegaZ catalog. For each one of them, we resampled
the redshift of each individual galaxy according to a normal
distribution $N(z_{\rm phot},\sigma_{z})$, where $z_{\rm phot}$ is the
photometric redshift and $\sigma_z$ is the photometric redshift error
of each galaxy. In this case, we followed the error function
$\sigma_z=0.03(1+z)$ found in \citet{collister2007a} to assign photometric redshift
errors to galaxies. A new mock LRG sample was then established
according to the criteria discussed in \S\ 2.2. We calculated $w_p$
for each one of these realizations and assigned the error contribution
from redshift uncertainties to the final error budget of $w_p$ by
calculating the dispersion among these 100 independent realizations.
We found that the size of the $w_p$ error bars was dominated by cosmic
variance. The contribution of photometric redshift errors to the $w_p$
error budget was small, at most 20\% for the smallest separation
bin. It is negligible ($\sim 1\%$) at large separations where we
calculated the relative clustering strength and absolute bias. For
this reason, we decided to adopt the cosmic variance $\sigma_{i}$ as
the error on $w_p$ and neglect the contribution of the photometric
redshifts.
\subsection{Results}
This section addresses the cross- and auto-correlation results for the
three LRG samples and the effect of photometric redshifts on the
clustering amplitude. For each one of our three LRG samples, we
considered three subsamples of Mg\,II absorbers:
weak $W_r(2796)=1-1.5$ \AA, strong $W_r(2796)=1.5-5$ \AA, and all
absorbers. A description of each correlation calculation can be found
in Table \ref{summary_calculations}. We also include the number of
$DD$ pairs (equation \ref{landy}) in the first bin and the number of
data found in each subsample in columns (6), (7), and (8).
\begin{centering}
\begin{deluxetable*}{lccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Summary of the cross- and auto-correlation calculations}
\tablewidth{0pt}
\tablehead{
& & & \colhead{$W_{\rm{r,min}}(2796)^a$} &
\colhead{$W_{\rm{r,max}}(2796)$} & \colhead{Number} & \colhead{Number} & \colhead{Number} \\
\colhead{Sample} & \colhead{$z_{\rm min}$} & \colhead{$z_{\rm max}$} & \colhead{(\AA)} &
\colhead{\AA} & \colhead{DD pairs$_{\rm (1st bin)}$$^b$} & \colhead{LRGs} & \colhead{Mg\,II} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{(8)}
}
\startdata
\multicolumn{8}{c}{Volume-limited sample} \\
\cline{1-8}\\
V-weak(VW) & 0.45 & 0.60 & 1.0 & 1.5 & 21 & 197,968 & 279\\
V-strong(VS) & 0.45 & 0.60 & 1.5 & 5.0 & 13 & 197,968 & 257 \\
V-all(VA) & 0.45 & 0.60 & 1.0 & 5.0 & 34 & 197,968 & 536 \\
\cutinhead{Flux-limited samples}
F1-weak(F1W) & 0.45 & 0.60 & 1.0 & 1.5 & 47 & 517,549 & 279 \\
F1-strong(F1s) & 0.45 & 0.60 & 1.5 & 5.0 & 35 & 517,549 & 257 \\
F1-all(F1A) & 0.45 & 0.60 & 1.0 & 5.0 & 82 & 517,549 & 536 \\
F2-weak(F2W) & 0.40 & 0.70 & 1.0 & 1.5 & 88 & 618,086 & 541 \\
F2-strong(F2S) & 0.40 & 0.70 & 1.5 & 5.0 & 68 & 618,086 & 617 \\
F2-all(F2A) & 0.40 & 0.70 & 1.0 & 5.0 & 156 & 618,086 & 1158 \\
\cutinhead{LRGs auto-correlation}
V-LRGs(VG) & 0.45 & 0.60 & - & - & 6,977 & 197,968 & - \\
F1-LRGs(F1G) & 0.45 & 0.60 & - & - & 43,093 & 517,549 & - \\
F2-LRGs(F2G) & 0.40 & 0.70 & - & - & 55,062 & 618,086 & - \\
\enddata
\tablenotetext{a}{$W_r$ is the rest-frame equivalent width. }
\tablenotetext{b}{The first bin is centered on $0.312~h^{-1}$ Mpc}
\label{summary_calculations}
\end{deluxetable*}
\end{centering}
\renewcommand{\arraystretch}{1.5}
\subsubsection{Cross- and auto-correlation results}
Figure \ref{everything} shows the cross- and auto-clustering results
for the flux- and volume-limited samples of LRGs. For each one of
these samples, the results are shown for the three Mg\,II subsamples
in gray. We overplot the LRGs auto-correlation results in black. A
clear feature of Figure \ref{everything} is that, for the three LRG
samples considered, the clustering amplitude of weak absorbers is
systematically higher than for strong ones. It is also interesting to
note that after accounting for systematic bias due to redshift
uncertainties (see \S\ 3.2.2), weak Mg\,II absorbers appear to be
unbiased (sharing the same clustering amplitude) with respect to LRGs
in both flux-limited and volume-limited samples.
For each auto- and cross-correlation calculations, we estimated the
correlated uncertainties between different $r_p$'s using the
covariance matrix (see equation \ref{covariance}) and the
normalized correlation matrix
\begin{equation}
\rho(w_i,w_j)=\frac{{\rm COV}(w_i,w_j)}{\sqrt{{\rm COV}(w_i,w_i){\rm COV}(w_j,w_j)}} \; .
\label{correlation}
\end{equation}
We performed a convergence test on the number of jackknife boxes. We
made sure that the off-diagonal elements of the correlation matrix
varied by less than 10\% after doubling the number of boxes. We plot
in Figure \ref{cij} the normalized correlation matrix $\rho(w_i,w_j)$
for all calculations, including the LRGs auto-correlation shown in the
fourth column. Adjacent bins are strongly correlated for the
auto-correlation functions of LRGs and, in all cases of the
LRG--Mg\,II cross-correlation measurements, bins with large $r_p$ are
more correlated.
\begin{figure*}
\centerline{
\includegraphics[angle=-90,scale=0.90]{figure4a_bw.eps}
}
\caption{ Cross- and auto-correlation results for the flux and
volume-limited samples of LRGs. From \emph{top to bottom},
flux-limited sample $z=0.40-0.70$, flux-limited $z=0.40-0.60$, and
volume-limited $z=0.40-0.60$. From left to right, \emph{all}
absorbers, \emph{weak} absorbers and \emph{strong} absorbers. The
rest-frame equivalent-width coverage is listed at the top of each
column. In all panels, the gray points represent the
cross-correlation results (Mg\,II and LRGs) and the black points are
for the LRGs auto-correlation. In the bottom left panel, we show the
halo occupation distribution analysis done on the volume-limited sample
of LRGs. The solid line shows the full HOD model and the dotted line
the one-halo contribution (see section 4.2). Note that both cross- and
auto-correlation results were corrected for photometric redshifts (see section 3.2.2).
Error bars represent the 1-$\sigma$
jackknife errors calculated on $N=192$ sky boxes of equal area. Note
that the last cross-correlation points in two of the strong Mg\,II
absorber sample are negative and not displayed in these panels. The
weak absorbers are essentially unbiased with respect to the LRGs and
their clustering is strong, even down to 0.3 $ h^{-1}$ Mpc. }
\label{everything}
\end{figure*}
\begin{figure*}
\centerline{
\includegraphics[angle=-90,scale=0.70]{figure5a_bw.eps}
}
\caption{Gray-scale plot of the correlation matrix (see equation
\ref{correlation}) showing the degree of covariance between
different separation bins. The different matrices shown correspond
to the same calculations of Figure \ref{everything} for
$W_r(2796)=1-5$ \AA\ (left), $W_r(2795)=1-1.5$ \AA\ (middle-left),
and $W_r(2796)=1.5-5$ \AA\ (middle-right), respectively. The fourth
column shows the LRGs auto-correlation results.}
\label{cij}
\end{figure*}
\subsubsection{Effect of photometric redshifts on the amplitude of clustering}
An important effect of photometric redshift errors is the sample
selection uncertainty. Photometric redshift uncertainties not only
affect the precision of $w_p$, but also lower its accuracy that one
would normally get from using spectroscopic redshifts (e.g.,\
\citealt{brown2008a}). This needs to be accounted for in the bias
calculation. One can think of photometric redshift errors as adding
uncorrelated galaxies in the calculation, and the overall effect is an
unwanted widening of the galaxy redshift distribution.
This effect results in a systematic error in the clustering signal and the
amplitude of $w_p$ is comparatively smaller than expected in
calculations using only spectroscopic redshifts.
To estimate the attenuation of the clustering signal due to
photometric errors, we performed several tests on mock LRG galaxy
distributions. Our mock catalog is produced by populating the halos in
an N-body simulation with a halo occupation function determined from
the spectroscopic LRG sample in \citet{zheng2008a} (specifically, the
``faint'' sample). The simulation is large, 1 Gpc/h on a side, and
represents our fiducial cosmology at $z\sim 0.5$ (see
\citealt{tinker2008b} for more details on this simulation). Since the
box size is not big enough to cover the redshift range of our study,
we mirror-imaged three identical copies of the box along the redshift
direction. The mock LRG catalog has precise positions and therefore
mimic a spectroscopic sample.
Three tests were performed. First, we auto-correlated the mock LRG
sample without applying any modifications to their redshifts. This
first case mimics the results one would get by auto-correlating a
sample of LRGs with spectroscopic redshifts (the $w_{hh}$ case in Figure 6).
Second, we cross-correlated the mock LRG sample with one
that involved perturbed redshifts. The perturbed redshifts were
generated by sampling a normal distribution centered on the input
redshift in the mock catalog within $\sigma_z=0.03(1+z)$. That is
the $w_{hh'}$ term in Figure \ref{simbox}, which is in analogous to our
cross-correlation measurements betwen absorbers with spectroscopic
redshifts and galaxies with photometric redshifts. Third, we
auto-correlated the mock LRG sample with perturbed redshifts
($w_{h'h'}$ in Figure 6). This mimics our LRG auto-correlation function.
We then computed the $w_p$ ratios between the different calculations
and plotted the results in Figure \ref{simbox}. The three sets of
symbols correspond to ratios of $w_p$ calculated over the same series
of $r_p$ bins as in the correlation calculations shown in Figure
\ref{everything}. We show, by thick black lines, the best-fit
amplitude of the three ratios in two different regimes : small and
large separations. Best-fit values are listed in Table
\ref{summary_fit_mgii_perturbation} with their corresponding
uncertainties ($\Delta \chi^2 < 1$). Error bars on the ratios
themselves correspond to the dispersion obtained among 100
realizations of a mock catalog with perturbed redshifts. The best-fit
amplitude (of the light points) at large separation enters in the
calculation of the relative bias (see section 5.1).
The dark datapoints in Figure \ref{simbox} show that the
cross-correlation function between Mg\,II absorbers and photometric
redshift identified LRGs is underestimated by roughly 20\% due to
redshift errors. The gray points show that the auto-correlation
function of these LRGs is underestimated by roughly 30\%. Finally,
the light-gray points show that the mean bias calculated based on the
ratio of absorber-LRG cross-correlation and LRG auto-correlation
functions is {\it overestimated} by roughly 10\%.
Our estimated reduction in clustering strength of 10\% is smaller than
the value quoted in \citet{bouche2004a}. These authors found that, in
the case of a gaussian redshift distribution for galaxies, the
amplitude of the MgII-LRG cross-correlation is overestimated by 25
$\pm$ 10\%. They used numerical integration and mock galaxy catalog
with phometric redshift uncertainties $\Delta_z=0.1$ corresponding to
size of their redshift interval of interest. The discrepancy could be
partly attributed to the larger photometric redshift uncertainties
used by these authors.
\begin{centering}
\begin{deluxetable*}{lcccc}
\tabletypesize{\scriptsize}
\tablecaption{Estimated reduction factor of the cross-correlation amplitude due to photometric
redshift sampling errors}
\tablewidth{0pt}
\tablehead{ \colhead{Measurement} & \colhead{$r_p < 1~h^{-1}$ Mpc} & \colhead{$\chi^2$/D.O.F} & \colhead{$r_p>1~h^{-1}$ Mpc }
& \colhead{$\chi^2$/D.O.F}}
\startdata
$w_{\rm Mg\,II-LRG}$ & 0.77 $\pm 0.03$ & 1.35 & 0.79 $\pm 0.02$ & 0.26 \\
$w_{\rm LRG-LRG}$ & 0.66 $\pm 0.04$ & 0.97 & 0.71 $\pm 0.04$ & 0.12 \\
\enddata
\label{summary_fit_mgii_perturbation}
\end{deluxetable*}
\end{centering}
\begin{figure}
\vspace{5pt}
\centerline{\hbox{ \hspace{0.0in}
\includegraphics[angle=-90,scale=0.40]{figure6a_bw.eps}
}
}
\vspace{5pt}
\caption{Ratios of cross-correlation signals obtained from the
simulation boxes to address the impact of photometric redshift
uncertainties on the amplitude of $w_p$. \emph{Solid black} :
ratios of $w_p$'s where one of the two datasets has redshift
position perturbed (within its error bars) ($w_{hh'}$) over $w_p$'s
obtained where both datasets have spectroscopic redshift
($w_{hh}$); \emph{dashed gray} : ratios where both datasets have
perturbed redshifts ($w_{h'h'}$) over two datasets with unperturbed
redshifts ($w_{hh}$); \emph{Dotted light gray} : $w_p$ for
perturbed-perturbed datasets ($w_{h'h'}$) over
perturbed-unperturbed ($w_{hh'}$). The last set of points gives us
an estimate of the systematic bias in the observed clustering
amplitude of Mg\,II absorbers using photometric redshift identified
LRGs. Error bars show dispersion among 100 realizations of the
perturbed datasets ($h'$). The thick solid lines represent best-fit
amplitude obtained for the one ($r_p \leq 1.1 h^{-1}$ Mpc) and
two-halo ($r_p>1.1 h^{-1}$ Mpc) terms. Best-fit values are listed
in table \ref{summary_fit_mgii_perturbation}. The best-fit
amplitude obtained for the light gray points is used to correct for
the addtional systematic error introduced by using photoz's in the
calculation of the correlation functions. }
\label{simbox}
\end{figure}
\begin{figure*}
\vspace{5pt}
\centerline{\hbox{ \hspace{0.0in}
\includegraphics[angle=0,scale=0.9]{figure7a_bw.eps}
}
}
\vspace{5pt}
\caption{\emph{Panel (a)} shows the relative bias of Mg\,II for two of the three LRG samples.
\emph{Solid lines with triangles} are for the volume-limited sample, \emph{solid lines with squares} for the
flux-limited $z=0.40-0.70$, and \emph{open circles} are taken from \citet{bouche2006a} for direct comparison
with the flux-limited sample. For each LRG sample, the points are derived
from the direct ratio of the points for $r_p > 1~h^{-1}$ Mpc. The power-law technique yielded very similar results.
We shifted the results for the flux-limited samples
by $\langle \hat{b} \rangle+1.5$ for more clarity.
\emph{Panel (b)} : absolute bias for the volume-limited sample.
\emph{Panel (c)} bias-inverted halo mass derived from the absolute bias. The bias-weighted halo mass results of \citet{bouche2006a} are shown
in open circles. Error bars along the x-axis represent the binning used. }
\label{bias}
\end{figure*}
\begin{centering}
\begin{deluxetable*}{lcccc}
\tabletypesize{\scriptsize}
\tablecaption{Best-fit power-law $f(x)=ax^b$ results for the cross and auto-correlations.}
\tablewidth{0pt}
\tablehead{
\colhead{Sample} & \colhead{$a$} & \colhead{$b^{\rm i}$} & \colhead{$\chi^2$} & \colhead{$\chi^2/{\rm D.O.F}^{\rm ii}$}
}
\arraystretch
\startdata
VW & 0.337 $\pm 0.11 $ & -0.835 & 1.2 & 0.3 \\
VS & 0.206 $\pm 0.09 $ & -0.835 & 0.394 & 0.13 \\
VA & 0.275 $\pm 0.075 $ & -0.835 & 1.06 & 0.27 \\
F1W & 0.245 $\pm 0.080$ & -0.783 & 0.54 & 0.14 \\
F1S & 0.194 $\pm 0.070$ & -0.783 & 4.60 & 1.15 \\
F1A & 0.206 $\pm 0.056$ & -0.783 & 3.78 & 0.76 \\
F2W & 0.160 $\pm 0.046$ & -0.781 & 3.47 & 0.87 \\
F2S & 0.114 $\pm 0.047$ & -0.781 & 0.776 & 0.258\\
F2A & 0.136 $\pm 0.035$ & -0.781 & 3.68 & 0.92 \\
\cutinhead{LRGs auto-correlation}
VG & 0.356 $\pm 0.013$ & -0.835 $\pm 0.029$ & 11.25 & 3.75 \\
FG1 & 0.273 $\pm 0.008$ & -0.783 $\pm 0.026$ & 9.72 & 3.24 \\
FG2 & 0.241 $\pm 0.007$ & -0.781 $\pm 0.026$ & 6.79 & 2.26 \\
\enddata
\tablenotetext{i}{In the case of the LRGs auto-correlation calculation, both $a$ and $b$ are
free parameters. For cross-correlation Mg\,II-LRGs, $b$ is kept fixed and corresponds to the best-fit value
obtained for the auto-correlation.}
\tablenotetext{ii}{$\chi^2$ per degree of freedom. For auto-correlation calculations, D.O.F=4 and D.O.F=5 for cross-correlation.}
\label{power_law}
\end{deluxetable*}
\end{centering}
\begin{centering}
\begin{deluxetable*}{lcccccc}
\tabletypesize{\scriptsize}
\tablecaption{Relative and absolute bias. Bias-inferred masses.}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{direct} & \colhead{power-law} & \colhead{direct} & \colhead{power-law} & \colhead{direct} & \colhead{power-law} \\
\colhead{Sample} & \colhead{$\langle \widehat{b}_{\rm{DR}}^\dagger \rangle$} & \colhead{$\langle \widehat{b}_{\rm{RA}} \rangle $}
& \colhead{$\langle b_{\rm{DR}}^{\ddagger} \rangle $} & \colhead{$\langle b_{\rm{RA}} \rangle$}
& \colhead{$(\log M_{\rm{h}})_{\rm{DR}}$}
& \colhead{$(\log M_{\rm{h}})_{\rm{RA}}$}\\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)}
}
\arraystretch
\startdata
\colhead{} & \multicolumn{2}{c}{Relative bias} & \multicolumn{2}{c}{Absolute bias} & \multicolumn{2}{c}{Mean mass$^{i}$}\\
\cline{2-3} \cline{4-5} \cline{6-7} \\
VW & 0.77 $\pm 0.28 $ & 0.85 $\pm 0.25$ & 1.56 $\pm 0.57 $ & 1.72 $\pm 0.51 $ & $12.8^{+0.5}_{-1.1} $ & $13.0^{+0.4}_{-0.6} $ \\
VS & 0.54 $\pm 0.23 $ & 0.52 $\pm 0.21$ & 1.09 $\pm 0.47 $ & 1.05 $\pm 0.42 $ & $12.0^{+0.8} $ & $11.9^{+0.8} $ \\
VA & 0.67 $\pm 0.19 $ & 0.69 $\pm 0.16$ & 1.36 $\pm 0.38 $ & 1.41 $\pm 0.33 $ & $12.5^{+0.4}_{-0.8} $ & $12.6^{+0.4}_{-0.6} $ \\
F1W &0.80 $\pm 0.26 $ & 0.81 $\pm 0.24$ & - & - & - & - \\
F1S &0.75 $\pm 0.25 $ & 0.64 $\pm 0.21$ & - & - & - & - \\
F1A &0.74 $\pm 0.19 $ & 0.68 $\pm 0.16$ & - & - & - & - \\
F2W &0.69 $\pm 0.17 $ & 0.60 $\pm 0.15$ & - & - & - & - \\
F2S &0.44 $\pm 0.17 $ & 0.43 $\pm 0.16$ & - & - & - & - \\
F2A &0.56 $\pm 0.13 $ & 0.51 $\pm 0.12$ & - & - & - & - \\
\enddata
\tablenotetext{$\dagger$}{The subscript DR represents the results obtained with the mean ratio of the two-halo term data points and RA is calculated
from the ratio of the best-fit power-law amplitudes of the cross- and auto-correlations.}
\tablenotetext{$\ddagger$}{$b$ is the absolute bias.}
\tablenotetext{$i$}{Note that we did not quote a lower limit on the halo mass for strong absorbers. Indeed, the bias vs. halo
mass relationship flattens to $b_h(M) \sim 0.7$ for $\log M_h < 9$.
When the lower limit on the bias is $<$ 0.7, this gives us no lower limit on the halo mass.}
\label{bias_table}
\end{deluxetable*}
\end{centering}
\section{Determining the Absolute Bias and Mean Mass Scale of Absorber hosts}
\subsection{Theoretical Framework}
The bias of dark matter halos can be defined as the ratio between the
clustering of halos (at a fixed mass) and the underlying clustering of
dark matter,
\begin{equation}
b_h^2(M,r) = \frac{\xi_h(M,r)}{\xi_m(r)}
\end{equation}
\noindent where $\xi_m(r)$ is the correlation function of the dark
matter itself. At large scales, linear bias holds and $b_h$ is
independent of $r$. In the translinear regime, $r\lesssim 10$ $h^{-1}$ Mpc,
$b_h$ has a scale dependence (with respect to either $\xi_m(r)$
obtained from linear theory or the true non-linear
clustering). Although the scale dependence of halo bias varies with halo mass,
over the mass range
probed by LRGs and Mg\,II absorbers, $\sim 10^{12-13}$ \mbox{$h^{-1}\,{\rm M}_\odot$}, the scale
dependence is nearly independent of mass and will divide out in the
cross-correlation function \citep{tinker2009a}. The auto-correlation function
of LRGs can then be expressed as
\begin{equation}
\xi_g(r) = b^2_h(M_g,r)\xi_m(r) = b^2_g(M_g)f_g^2(r)\xi_m(r)
\end{equation}
\noindent where $M_g$ is the bias-weighted mean mass scale (see
equation \ref{bw_mass}) of galaxies, $b_h$ and $b_g$ are the large-scale linear
biases of dark matter halos and galaxies, and $f(r)$ is the
scale-dependent bias term (see, e.g.,
\citealt{tinker2005a,tinker2009a}). The cross-correlation is then
\begin{equation}
\xi_{ga}(r) = b_g(M_g)b_a(M_a)f_g(r)f_a(r)\xi_m(r).
\end{equation}
\noindent Thus the relative bias of absorbers to LRGs is the ratio of
the cross to auto-correlation functions, ie,
\begin{equation}
\hat{b} \equiv \frac{b_a}{b_g}=\frac{w_{ag}}{w_{gg}},
\label{relative_bias}
\end{equation}
\noindent which should be close to a constant at large separations.
We used the measured $w_{gg}(r_p)$ to obtain $b_g$, which we utilized
to obtain $b_a$. At $z=0.5$ and at masses above $M\sim 10^{11.5}$
\mbox{$h^{-1}\,{\rm M}_\odot$}, the bias of dark matter halos increases monotonically with
mass, thus the equivalent dark matter halo mass can be obtained from
inverting the $b_h(M)$ formula. We used the following halo bias
function \citep{tinker2009a} for halos defined at an overdensity of
200 times the background
\begin{equation}
b(\sigma)=1-A\frac{\sigma^{-a}}{\sigma^{-1}+1} + B\sigma^{-b}+C\sigma^{-c}
\label{halo_bias}
\end{equation}
\noindent where $\sigma$ is the linear matter variance on the
Lagrangian scale (radius of the halo in the initial
mass distribution when $\delta(\rho) = (\rho-\bar{\rho})/\bar{\rho}\sim 0$)
of the halo, $R=(3M/4\pi\bar{\rho})^{1/3}$,
and $a$, $b$, $c$, $A$, $B$, $C$ are constants ($a=0.132$, $b=1.5$,
$c=2.4$, $A=1.04$, $B=0.4$, $C=0.99$).
\subsection{The bias of LRGs}
We obtained the bias of LRGs through halo occupation modeling of the
$w_{gg}$ data and the number density of galaxies in the sample. Note
that $w_{gg}$ was corrected for the systematic error due to
photometric redshift uncertainties (see \S\ 3.2.2). Our modeling was
similar to the one performed in \citet{zheng2008a}, which is based on
the analytic halo occupation model developed in \citet{zheng2004a} and
\citet{tinker2005a}. The best-fit model is shown in the bottom left
panel of Figure \ref{everything}, which yielded a $\chi^2$ of 9.8
using the full covariance matrix. Halo occupation models separate
pairs from galaxies located inside the same dark matter halo (one-halo
term) and pairs from galaxies located in two different halos (two-halo
term). The one-halo contribution to the clustering amplitude of LRGs
is shown in Figure \ref{everything}. At small separations, ($r_p
\lesssim 1~h^{-1}$ Mpc) the one-halo term dominates but the two-halo
term shapes the clustering signal for $r_p \gtrsim 1~h^{-1}$ Mpc.
Because the analytic model fully incorporates the scale-dependent bias
of dark matter halos, we can obtain the linear bias directly from
these data. The bias of LRGs in our sample is $b_g=2.023\pm
0.006$. The high precision is due to the large volume of the sample,
yielding an excellent measurement of the amplitude of $w_{gg}$ in the
two-halo regime.
\subsection{The relative bias of absorbers}
We calculated the relative bias using equation (\ref{relative_bias})
on large scales only ($>1.0\, h^{-1}$ Mpc). Two methods were employed.
In the first case, we fitted the cross- and auto-correlation results
using a power-law model and estimated the relative bias using the
ratio of the best-fit amplitudes. This is a standard procedure that
has been commonly done in previous works (e.g.,\
\citealt{davis1983a}). However, the power-law model does not have a
physical justification (e.g.,\ \citealt{blake2008a,zehavi2004a}). It
simply provides an adequate fit to the data. In the second case, we
directly calculated the relative bias by taking a weighted mean ratio
of all points at $r_p>1~h^{-1}$ Mpc. Because the measurements and
measurement errors vary significantly between data points at different
$r_p$'s, we adopted the weights $\omega_i$'s that were designed to
maximize the significance of the mean relative bias
$\langle\hat{b}\rangle$,
\begin{equation}
\langle \hat{b} \rangle = \sum_{i=4}^8 \omega_i \frac{w_{ag,i}}{w_{gg,i}}
\label{dr}
\end{equation}
where
\begin{equation}
\omega_i=\frac{w_{ag,i}}{\sigma_i^2 w_{gg,i}} \;.
\end{equation}
the index $i$ denotes the $r_p$ bin and $\sigma_i$ is the associated error of
$w_{ag,i}/w_{gg,i}$ computed using the error propagation
technique.
The best-fit power-law parameters can be found in Table
\ref{power_law}. We first determined the best-fit parameters of the
LRG auto-correlation function by minimizing the $\chi^2$ function that
accounts for the correlated errors between adjacent bins:
\begin{equation}
\chi^2 = (w-\tilde{w})^T {\rm COV}^{-1}(w-\tilde{w}) \; .
\label{least-square}
\end{equation}
$\tilde{w}$ is the model and $w$ is the data vector. Next, we adopted
the best-fit slope of the LRG auto-correlation function for all
corresponding Mg\,II--LRGs cross-correlation calculations. For the
LRGs auto-correlation, the errors on the parameters were determined
from all values within $\Delta \chi^2 < 2.3$ (two parameters to fit)
from the minimum. The relative and absolute biases derived from the
ratio of the best-fit power-law amplitude are denoted by the subscript
"RA" in Table 4. In the cross-correlation cases, the error on the
best-fit amplitude $a$ corresponds to all models with $\Delta
\chi^2<1$ from the minimum $\chi^2$ value.
For the relative bias derived from the direct ratio "DR" of datapoints
in the two-halo regime (equation \ref{dr}), we excluded all negative
datapoints from the calculation. The final error was derived using the
standard error propagation technique and we kept the two most dominant
terms of the expansion. These two terms are at least two orders of
magnitude larger than any other term in the expansion,
\begin{eqnarray}
& &\sigma_{\rm DR}^2 = \sum_i \omega_i^2 \frac{1}{w_{\rm w_{ag,i}}^2}\sigma_{\rm w_{ag},i}^2 +{} \nonumber \\
& & {}+2\sum_i\sum_{j>i}
\omega_i \omega_j {\rm COV}(w_{\rm ag,i},w_{\rm ag,j})\frac{w_{\rm gg,i}w_{\rm gg,j}}{w_{\rm ag,i}w_{\rm ag,j}}
\label{sigma_dr}
\end{eqnarray}
where the sum runs over the bins $i,j$ with $r_p > 1.1~h^{-1}$
Mpc. $w_{\rm ag,i}$ represents the $i$th bin of the cross-correlation
function and $w_{\rm gg,j}$ is the $j$th bin of the LRGs
auto-correlation function. The relative bias values, calculated using
both methods, are listed in columns (2) and (3) of Table
\ref{bias_table}. We corrected the relative bias values for
photometric redshift errors by using the large scale correction factor
0.90$\pm 0.02$.
\subsection{The absolute bias and mass scale of absorber hosts}
Since the flux-limited sample of galaxies is not based on a homogeneous
sample of galaxies and would render the halo occupation analysis
much more challenging, we limited our absolute bias calculation
to the volume-limited data. The absolute biases obtained, for the
direct-ratio case, are $b=1.56\pm0.57$ for the weak Mg\,II sample
and $b=1.09\pm0.47$ for the strong one. Biases derived from the power-law
technique are found in column (5) of Table \ref{bias_table}.
We determined the corresponding halo mass in two different ways. In the
first case, we inverted equation (\ref{halo_bias}) to obtain the halo mass for these absorbers.
These masses are denoted by $\log M_h$, which we refer to as the \emph{bias-inverted} mass.
Using the direct-ratio evaluated according to equation (\ref{dr}), we derived
$\log M_h = 13.0^{+0.4}_{-0.6}$ for the weak absorbers and $\log M_h
= 12.0^{+0.8}_{-6.0}$ for the strong ones. The lower error bar we quote for
the weak absorbers is arbitrary. All halos with $\log M_h<12$ are consistent with our results.
This is because the lower error bar on the bias was less than 0.7
giving us no constraint on the halo mass. Indeed, $b(M)$ has a minimum value of
$\sim 0.7$ and becomes nearly independent of mass at $\log M_h \lesssim 9$.
The corresponding halo masses using
the power-law method are in column (7) of Table \ref{bias_table}.
In the second case, we first solved for the minimum halo mass ($M_{min}$) using
\begin{equation}
\langle b \rangle = n_h^{-1} \int_{M_{min}}^{\infty} d\log M_h \frac{dn(M_h)}{d\log M_h} b(M_h)
\end{equation}
where $\langle b \rangle$ is the absorbers bias, $dn(M_h)/d\log M_h$ is the
halo mass function and
\begin{equation}
n_h = \int_{M_{min}}^{\infty} d\log M_h \frac{dn(M_h)}{d\log M_h} \; .
\end{equation}
We then estimated the mass following
\begin{equation}
\langle \log M_h \rangle = n_h^{-1}\int_{M_{min}}^{\infty} d\log M_h \frac{dn(M_h)}{d\log M_h} \log M_h \; .
\label{bw_mass}
\end{equation}
We refer to the mass found using equation (\ref{bw_mass}) as the \emph{bias-weighted} halo mass.
$M_{min}$ corresponds to the minimum halo mass above which there is a single
absorber per halo. Of course, absorbers are distributed over a wide range of masses and
the covering fraction is likely to vary with halo mass. This method only
gives an approximate answer. As for the the bias-inverted mass, it is
a characteristic mass obtained by inverting the
$b(M)$ relation. Even though these two methods involve different
assumptions, they give similar (within 0.1 dex) results for both
LRG and absorbers masses. We used the bias-inverted masses as our LRG
and absorbers mass estimate.
The relative and absolute biases, calculated with the direct ratio
technique, and their corresponding halo masses are shown in Figure
\ref{bias}. Panel (a) shows the relative bias obtained for the volume-
and flux-limited ($z=0.40-0.70$) samples and allows for a direct
comparison with the \citet{bouche2006a} results; (b) shows the
absolute bias derived from the halo occupation analysis. The
bias-inverted halo masses are shown in panel (c) along with
\citet{bouche2006a} bias-weighted mass estimates. We did not quote a
lower limit for the halo mass when the lower limit on the absolute
bias is $\lesssim 0.7$.
We obtained similar relative biases for the flux-limited sample at
$z=0.40-0.70$ as Bouch\'e \ensuremath{\mbox{et~al.}}, who corrected their relative bias by
20\% to account for phototmetric redshifts. This large correction is
expected since these authors included fainter LRGs ($i<21$) in their
analysis that are expected to contain larger photometric redshift
uncertainties. It is, however, not clear how the authors accounted
for the varying redshift errors with galaxy brightness in their estimate.
In contrast, we applied a 10\% correction factor for the clustering
measurements according to our simulation studies described in \S\
3.2.2. The halo masses derived for weak/strong Mg\,II are larger than
the ones obtained by \citet{bouche2006a} over similar $W_r(2796)$
intervals.
The apparent discrepancy may be due to the absolute halo bias of LRGs
included in different analysis. We calculated $b_{\rm lrg}$ directly
from the LRG clustering signal observed in a volume-limited sample,
whereas \citet{bouche2006a} compared their LRG clustering strength
with previous studies that were carried out using a similar, but not
identical galaxy population. Additional uncertainties in the
estimated correction factor to account for photometric redshift errors
may also contribute to the differences in our findings.
\section{Discussion}
We have calculated the clustering amplitude of Mg\,II absorbers with
respect to three samples of LRGs. Using the volume-limited sample, we
have computed the absolute bias and typical halo mass of two
subsamples of Mg\,II absorbers: $W_r(2796)>1.5$ \AA\, and
$W_r(2796)=1-1.5$ \AA\. A $\sim 1\sigma$ anti-correlation between
$W_r(2796)$ and mean mass is seen in panel (c) of Figure \ref{bias}.
If a significant anti-correlation signal is confirmed by larger datasets,
this would imply that weaker Mg\,II absorbers are found to be
more strongly clustered than stronger ones. Our results show that a significant
fraction of the Mg\,II absorber population of $W_r(2796)=1-1.5$ \AA\
absorbers are found around massive galaxies
with $ \log M_h < 13.4$, whereas absorbers of $W_r(2796)>1.5$ \AA\
are primarily found in $\log M_h < 12.7$ galaxies. Larger datasets for both LRGs and
Mg\,II would improve the precision on the clustering measurements and the equivalent
width vs. mass relationship.
Corresponding galaxy luminosities can
be inferred from the bias-luminosity relation found for SDSS data at
$z\approx 0.1$ (\citealt{tegmark2004a}; see also
\citealt{zehavi2005a}). Assuming $b=1$ for $L_*$-galaxies (which
is reasonable since $L_*$-galaxies are found in halos of mass $\sim
10^{12.3}$ M$_{\odot}$; see \citealt{zheng2007a}), the bias-inferred
luminosity for weak and strong absorbers are $\approx 4.5 L_*$ and \
$\approx 1.5 L_*$ respectively. Note that these values are obtained
from the mean bias estimate of equation 15 where more massive galaxies
(higher luminosity) have higher weights (bias) than less massive (lower
luminosity) objects. These values should not
be interpreted as the luminosity of a typical galaxy producing absorbers
of a given strength. In addition, the relationship between bias and luminosity
is not a one-to-one relation since the bias of galaxies is affected by
the satellites within their dark matter halos. There is also
an expected redshift evolution of the bias-luminosity
relation. However, \citet{zheng2007a} found that there is little
evolution in the halo mass hosting the central galaxies between $z=0$
and $z=1$. These authors found that a typical $L_*$-galaxies reside in
halos only a few times more massive at $z=1$ than at $z=0$.
Therefore, we assume no redshift evolution for the bias-luminosity
relation and used the expression found at $z=0.1$ as a reasonable
guess for our $z=0.5$ sample.
Another important aspect of the results is that the
Mg\,II-LRG cross-correlation function continues to exhibit
a strong signal (even after correcting for photmetric redshifts) down
to $\sim 0.3\ h^{-1}$ Mpc, indicating that some of the Mg\,II
absorbers and the LRGs share a common dark matter halo. Here we
discuss the implications of these results.
\subsection{The bias vs $W_r(2796)$ relationship}
The $W_r$ vs.\ mean halo mass relationship found in our analysis
is qualitatively consistent with the previous report by Bouch\'e
\ensuremath{\mbox{et~al.}}\ (2006; see also \citealt{lundgren2009a}). It is a
1-$\sigma$ trend but it argues against the simple notion that more
massive halos might produce stronger absorbers because they contain a
larger volume of gas. Our results show that \emph{weaker} absorbers
are preferentially found in more massive halos.
Bouch\'e \ensuremath{\mbox{et~al.}}\ (2006) measured the Mg\,II-LRG cross-correlation
function using 1806 Mg\,II absorbers of $W_r(2796)>0.3$ \AA\ and
250,000 LRGs of $i'<21$ at $z=0.35-0.8$. They found that
Mg\,II absorbers of $W_r(2796)\lesssim 1$ \AA\ appeared to be more
strongly clustered than the $W_r(2796)\gtrsim 2$ \AA\ ones by
nearly a factor of two. The authors attributed the observed
anti-correlation to a starburst outflow origin for absorbers
with $W_r(2796)\gtrsim 2$ \AA, in order to explain the on-average lower
halo mass of these absorbers.
However, our analysis shows that absorbers with $W_r(2796)\gtrsim 1.5$
are essentially unbiased with respect to dark matter ($b=1$).
This indicates that the
halo population probed by these absorbers is consistent with a
random, unbiased sample of dark matter halos, and does not favor a
specific sub-population such as starbursting systems. For
absorbers of $W_r(2796)<1.5$ \AA, the mean halo bias was found to be
still higher. This large mass scale is at odds with previous
findings that Mg\,II absorbers of $W_r(2796)=0.3-1$ \AA\ are
associated with $L_*$-type galaxies (e.g.,\ Steidel \ensuremath{\mbox{et~al.}}\ 1994).
To understand the physical mechanisms that could explain a
$W_r(2796)$ vs.\ clustering amplitude anti-correlation,
\citealt{tinker2008a} (hereafter TC08) developed a halo occupation
model that constrains the cold gas content of dark matter halos based
on the observed number density and clustering amplitudes of the Mg\,II
absorbers. In the TC08 model, Mg\,II absorbers serve as a
representative tracer of cool gas ($T\sim 10^4$ K) in dark matter
halos, and the observed anti-correlation arises as a result of an
elevated clustering amplitude of $W_r(2796)=0.3-1.5$ \AA\ absorbers
due to the contributions of residual cold gas in high-mass halos
($M_h>10^{13}\,\mbox{$h^{-1}\,{\rm M}_\odot$}$). These massive halos are rare and are likely
missed in small samples (e.g., Steidel \ensuremath{\mbox{et~al.}}\ 1994). The halo
occupation model represents the first empirical constraint of the cold
gas content across the full spectrum of dark matter halos and provides
additional information for models of the growth of gaseous halos.
\subsection{Presence of cool gas in massive halos}
From Figure \ref{everything}, it is worth noting the strong clustering
of Mg\,II absorbers for all three LRG samples. Indeed, the
clustering strength is comparable to the LRGs auto-correlation signal
for the most inner bin ($r_p = 0.31~h^{-1}$ Mpc). In physical units,
this bin corresponds to 0.21$~h^{-1}$ Mpc. For the measured
clustering scale, the typical mass scale of LRGs is roughly
$\rm{log}(\rm{M}_h/\rm{M}_{\odot}) \approx 13.2$, implying a
typical virial radius of $R_{\rm vir} \sim 0.35~h^{-1}$
Mpc. The virial radius is larger than the physical separation
probed by the most inner bin. The strong cross-correlation
signal is thus indicative of the presence of cool gas well
inside the virial radius of massive galaxies.
To further investigate the presence of cool gas in massive halos, we
examined the effects that a varying cold gas covering fraction
($\kappa$) in LRG halos would have on the cross-correlation signal.
To do this, we constructed a mock LRG catalog based on our halo occupation
distribution fits to the volume-limited sample (see bottom left panel of Figure 4). Using
the best-fit halo occupation function, we populated the halos
identified in a z=0.5 output of an N-body simulation. This simulation
is smaller in volume (400 $h^{-1}$ Mpc on a side)
than the simulation used to test the photometric redshift errors, in order to probe lower-mass
halos. Details about this simulation can be found in \citet{tinker2007a}.
The cosmology of this simulation differs from our fiducial cosmology (WMAP1
vs WMAP5), so the large-scale bias of the LRG auto-correlation function
is lower in comparison to the data, but the one-halo clustering
is a good match to the data.
We then simulated a mock Mg\,II absorber catalog by selecting random
sightlines. Every halo of $M_h\ge 10^{12} \mbox{$h^{-1}\,{\rm M}_\odot$}$ was allowed to
produce a mock Mg\,II absorber if the impact parameter was less
than the virial radius, whether or not it contains a mock LRG. We then
measured the cross-correlation between mock absorbers and mock LRGs
for different values of $\kappa$ following different recipes. First,
all halos of containing an LRG yielded an absorber if intersected by
a sightline. Namely, all LRG halos have a gas covering fraction of $\kappa=1$.
Then, we varied the covering fraction of Mg\,II in the mock LRG halos.
The results for the two limiting cases of $\kappa=0$ and $\kappa=1$
are shown in the left panel of Figure \ref{cool_gas} along with the
mock LRG auto-correlation function calculated from the box. The right
panel shows four curves corresponding to four different $\kappa$
values on top of our volume-limited cross-correlation and
auto-correlation measurements for $W_r(2796)=1-5$ \AA\ absorbers.
The cross-correlation function is a probe of the relative covering
fraction of LRG halos with respect to lower mass halos. Lowering $\kappa$ by the
same amount for all halos, LRGs and $L_*$ alike, does not change the
resulting cross-correlation signal. The results from Figure 7 imply
that the covering fraction of LRG-hosting halos must be comparable
to that of halos that contain $L_*$-galaxies at their centers.
\begin{figure*}
\centerline{
\includegraphics[angle=-90,scale=0.80]{figure8a_bw.eps}
}
\caption{\emph{Left panel}: Cross- and auto-correlation results for the 400 $h^{-1}$ Mpc box.
The \emph{solid line} shows the auto-correlation of the mock LRG sample. The \emph{diamond} points
show the cross-correlation results for which $\kappa=1$ for all halos above $10^{12}~h^{-1}$
M$_{\odot}$. The \emph{square} points represent the case of no cold gas in the halos of the mock LRGs and
$\kappa=1$ for all remaining halos $>10^{12}~h^{-1}$ M$_{\odot}$. \emph{Right panel}: the cross- and
auto-correlation results for the volume-limited sample of LRGs ($W_r(2796)=1-5$ \AA) are shown in
gray and black points respectively. The four curves correspond to different values of $\kappa$ for the mock LRGs.
From bottom to top, $\kappa=0,0.33,0.5,1$.
}
\label{cool_gas}
\end{figure*}
The presence of cold gas in massive halos has been a debated subject
in recent numerical simulation studies.
In a series of high-resolution SPH simulations,
\citet{keres2008a} and \citet{brooks2008a} examined the temperature
history of gas accreted onto dark matter halos. They found that most
of the baryonic mass is acquired through filamentary cold mode
accretion that is never shock heated to the virial temperature for halos of
$\le 10^{12}$ \mbox{$h^{-1}\,{\rm M}_\odot$}. \citet{keres2008a} found that these cold flows
are not present in massive halos typically hosting LRGs.
At high redshift, cold flows may penetrate inside the virial shock
of $10^{13}~h^{-1}$ M$_{\odot}$, but this effect is highly redshift dependent,
and is not likely to yield to cross-correlation functions seen in Figure 4.
Other mechanisms such as thermal instability could generate pockets of
cold gas inside a hot medium (e.g.,\ \citealt{mo1996a}). \citet{maller2004a}
showed that the hot gas is thermally unstable and
prone to fragmentation. They also show that cooling proceeds via the
formation of cold $10^{4}$ K clouds in pressure equilibrium with the
hot halo gas. For a Milky-Way-size system, cool clouds of mass $\sim 5
\times 10^6$ $\rm{M}_{\odot}$ are expected to extend up to $\sim 150$
kpc from the galactic center and survive for several Gyrs. \citet{kaufmann2008a} showed
that cloud formation is viable in $M_*$ halos, but needs to be extended to higher mass. In a
similar argument developed by \citet{mo1996a}, a two-phase medium in
pressure equilibrium was used to explain observations of Lyman limit
systems. This model also makes predictions about the presence of C\,IV
around low-mass galaxies and at large impact parameters of massive
galaxies. These predictions were partially confirmed later by
\citet{chen2001b} who found C\,IV in galaxies of different
morphologies and luminosities. These authors also observed the sharp
boundary in the $W_r$-(projected separation) plane that was also
predicted. \citet{mo1996a} attributed the presence of Mg\,II absorbers
to cold pockets of photoionized gas in halos around massive galaxies.
The observed strong Mg\,II-LRGs cross-correlation signal on scales
smaller than the virial radii of halos hosting LRGs indicates the
presence of cold gas is more common around massive galaxies than
previously thought.
On the observational side, the detections of cool gas in
group size halos is uncertain. The challenges lie
in the limited sensitivities available to detect HI gas via 21-cm
observations (e.g.,\ \citealt{verdes-montenegro2001a,verdes-montenegro2006a}).
\citet{verdes-montenegro2006a} reported detections of HI column density
down to $\sim 10^{19}$ cm$^{-2}$ in some of the Hickson Compact Groups
\citep{hickson1982a}. Observations are not sensitive enough to
probe the low HI column density environment yet.
The difficulty in finding cool gas around groups using 21-cm observations
underscores the powerful application of QSO absorption-line studies.
We are currently conducting a follow-up imaging and spectroscopy
campaign to study the cool gas content of individual LRG halos
(Gauthier et al. 2009 in preparation).
\subsection{Future prospects : DR7 Mg\,II database \& HOD modeling}
Figure 7 demonstrates how a halo occupation approach to the galaxy-absorber
cross-correlation function at both large and small scales can put constraints
on the covering fraction of cold gas in LRG-hosting dark matter halos.
In a forthcoming paper, we will address the detailed halo occupation
distribution modeling of the Mg\,II absorber environment with a
particular focus on the one-halo term. This will be achieved by using
the SDSS DR7 Mg\,II absorber catalog. This catalog, currently in
preparation (Prochaska et al.\ 2009), will more than
double the number of absorbers from the current DR5 sample. It will
allow us to probe smaller projected separations
and improve the clustering measurements at $r_p < 1~h^{-1}$ Mpc.
\acknowledgments
We thank C.\ Blake, I.\ Zehavi, and A. Kravtsov for useful discussions and
the anonymous referee for useful comments
that improved the draft. We are grateful to the NYU-VAGC team
for generating the SDSS survey masks and
making them available through their website and to the MANGLE team for
making their software publicly available. We thank J.X. Prochaska for
providing an updated SDSS DR5 Mg\,II catalog, and H.\ Oyaizu for
providing the electronic table necessary for producing Figure 3 in
this paper. HWC acknowledges partial support from NASA Long Term
Space Astrophysics grant NNG06GC36G and an NSF grant AST-0607510.
\bibliographystyle{apj}
|
1,314,259,995,861 | arxiv | \section{Introduction}
The study of $p$-harmonic maps and in particular $p$-harmonic functions is
central to $p$-harmonic geometry and related problems.
A real-valued $C^{3}$ function on a Riemannian $m$-manifold $M$ with a
Riemannian metric $\langle \,\,,\,\rangle \,$ is said to be \textit{strongly
}$p$-\textit{harmonic} if $u$ is a (strong) solution of the $p$-Laplace
equation (\ref{1.0}), $p>1,$
\begin{equation}
\Delta _{p}u:=\text{\textrm{div}}\left( |\nabla u|^{p-2}\nabla u\right) =0.
\label{1.0}
\end{equation}%
where $\nabla u$ is the gradient vector field of $u$ on $M\,,$ and $|\nabla
u|=\langle \nabla u,\nabla u\rangle ^{\frac{1}{2}}\,.$
A function $u\in W_{loc}^{1,p}\left( M\right) $ is said to be \textit{weakly
}$p$-\textit{harmonic} if $u$ is a (Sobolev) weak solution of the $p$%
-Laplace equation (\ref{1.0}), i.e.
\begin{equation*}
\begin{array}{lll}
\int_{M}\left\vert \nabla u\right\vert ^{p-2}\left\langle \nabla u,\nabla
\phi \right\rangle dv & = & 0%
\end{array}%
\end{equation*}%
holds for every $\phi \in C_{0}^{\infty }\left( M\right) ,$ where $dv$ is
the volume element of $M\,.$
The $p$-Laplace equation (\ref{1.0}) arises as the Euler-Lagrange equation
of the $p$-energy $E_{p}$ functional given by $E_{p}(u) = \int_{M} |\nabla
u|^{p}\, dv\, . $ Ural'tseva \cite{U}, Evans \cite{E} and Uhlenbeck \cite{Ur}
proved that weak solutions of the equation (\ref{1.0}) have H\"{o}lder
continuous derivatives for $p\geq2$. Tolksdorff \cite{To}, Lewis \cite{Le}
and DiBenedetto \cite{D} extended the result to $p>1.$ In fact, weak
solutions of (\ref{1.0}), in general do not have any regularity better than $%
C_{loc}^{1,\alpha}.$
When $p=2,$ $p$-harmonic functions are simply harmonic functions. Liouville
type properties or topological end properties have been studied by a long
list of authors. We refer the reader to, for example \cite{L}\cite{LT}\cite%
{LT1}\cite{LT2}\cite{LT3}\cite{LW1}\cite{LW2}\cite{LW3}\cite{PRS}\cite{SY}
for further references. In particular, P. Li and J. Wang showed Liouville
type properties and splitting type properties on complete noncompact
manifolds with positive spectrum $\lambda $ when the Ricci curvature has a
lower bound depending on $\lambda .$ They also extended their work to a
complete noncompact manifold with weighted Poincar\'{e} inequality $(P_{\rho
}).$
For $p>1,$ We refer the works, for example \cite{CHS}\cite{H}\cite{H1}\cite%
{H2}\cite{H3}\cite{HK}\cite{HPV}\cite{HV}\cite{KN}\cite{P}\cite{PRS2}, to
the reader. In particular, I. Holopainen \cite{H2} proved a sharp $L^{q}$%
-Liouville properties for $p$-harmonic functions, i.e. if $u\in L^{q}\left(
M\right) $ is $p$-harmonic (or $\mathcal{A}$-harmonic) in $M$ with $q>p-1,$
then $u$ is constant. For $q=p-1$ and $m\geq 2,$ there exist a complete
Riemannian $m$-manifold $M$ and a nonconstant positive $p$-harmonic function
$f$ with $\left\Vert f\right\Vert _{L^{p-1}\left( M\right) }<\infty .$ In
\cite{HPV}, I. Holopainen and S. Pigola and G. Veronelli showed that if $%
u,v\in W_{loc}^{1,p}\left( M\right) \cap C^{0}\left( M\right) $ satisfy $%
\Delta _{p}u\geq \Delta _{p}v$ weakly and $\left\vert \nabla u\right\vert ,$
$\left\vert \nabla v\right\vert \in L^{p}\left( M\right) ,$ for $p>1,$ then $%
u-v$ is constant provided $M$ is connected, possibly incomplete, $p$%
-parabolic Riemannian manifold. They also discussed $L^{q}$ comparison
principles in the non-parabolic setting. In \cite{PRS2}, S. Pigola, M.
Rigoli and A.G. Setti showed the constancy of $p$-harmonic map homotopic to
a constant and with finite $p$-energy from $p$-parabolic manifolds to
non-positive sectional curvatures manifolds. Moreover, if manifold $M$ has
Poincar\'{e}-Sobolev inequality, and $Ric_{M}\geq -k\left( x\right) $ with $%
k\left( x\right) \geq 0$ and the integral type of $k$ has upper bound
depending on Poincar\'{e}-Sobolev constant and $p\geq 2$ and $q,$ then they
obtained constancy properties of $p$-harmonic map with some finite energy
types form $M$ to non-positive sectional curvatures manifolds.
In \cite{KN}, B. Kotschwar and L. Ni use a Bochner's formula on a
neighborhood of the maximum point (i.e. the $p$-Laplace operator is neither
degenerate nor singular elliptic on this neighborhood) to prove a gradient
estimate for positive $p$-harmonic functions. This also implies Liouville
type properties of positive $p$-harmonic functions on complete noncompact
manifolds with nonnegative Ricci curvature, and sectional curvature bounded
below.
However, the approach of Kotschwar-Ni 's gradient estimate for positive $p$%
-harmonic functions, does not seem to work in this paper, since we need a
Bochner's formula which is unambiguously defined at every point in the
manifold.
To overcome the difficulty, in this paper, we introduce and study an
approximate solution $u_{\epsilon }$ of the weakly $p$-harmonic function $u.$
This $u_{\epsilon }$ is the Euler-Lagrange equation of the $\left(
p,\epsilon \right) $-energy%
\begin{equation*}
\begin{array}{l}
E_{p,\epsilon }=\int_{\Omega }\left( \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon \right) ^{\frac{p}{2}}dv%
\end{array}%
\end{equation*}%
with $u-u_{\epsilon }\in W_{0}^{1,p}\left( \Omega \right) ,$ where $\Omega $
is a domain on $M.$ That is, $u_{\epsilon }$ is the weak solution of a
perturbed $p$-Laplace equation
\begin{equation}
\begin{array}{lllll}
\Delta _{p,\epsilon }u_{\epsilon } & = & \text{\textrm{div}}\left( \left(
\left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon \right) ^{\frac{p-2%
}{2}}\nabla u_{\epsilon }\right) & = & 0.%
\end{array}
\label{1.2}
\end{equation}%
Moreover, we consider a linearization $\mathcal{L}_{\epsilon }$ of the
perturbed operator $\Delta _{p,\epsilon }\,,$ given by
\begin{equation}
\begin{array}{lll}
\mathcal{L}_{\epsilon }\left( \Psi \right) & = & \text{\textrm{div}}\left(
f_{\epsilon }^{p-2}A_{\epsilon }\left( \nabla \Psi \right) \right) ,%
\end{array}
\label{1.3}
\end{equation}%
for $\Psi \in C^{2}\left( \Omega \right) ,$ where $p>1,$ $f_{\epsilon }=%
\sqrt{\left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon }$ and%
\begin{equation*}
\begin{array}{l}
A_{\epsilon }:=\mathrm{id}+\left( p-2\right) \frac{\nabla u_{\epsilon
}\otimes \nabla u_{\epsilon }}{f_{\epsilon }^{2}}.%
\end{array}%
\end{equation*}
We observe that since $\Delta _{p,\epsilon }$ is no longer degenerate, by
the existence and $\epsilon $-Regularization results (Proposition \ref{ex}
and Proposition \ref{3.1}), $u_{\epsilon }$ exists and is infinitely
differentiable. Then we can derive an $\mathcal{L}_{\epsilon }$-type
Bochner's formula and a Kato type inequality, and apply them to $u_{\epsilon
}.$ Hence, using the convergence of the approximate solutions $u_{\epsilon }$
in $W^{1,p}$ on every domain in $M$, as $\epsilon \rightarrow 0,$ we prove a
Liouville type property of weakly $p$-harmonic functions with finite $p$%
-energy. This nonexistence result, when combined with the result of
Proposition \ref{2 E}, yields in turn the topological information that such
manifold has at most one $p$-hyperbolic end.
We also note that, the perturbation method we employed in studying the $p$%
-Laplace equation is in contrast to the methods in \cite{SU} for harmonic
maps on surfaces, in \cite{ES} for the level-set formulation of the mean
curvature flow, in \cite{HI} for the inverse mean curvature flow, and in
\cite{KN} for certain parabolic equations associated to the $p$-Laplacian.
\begin{theorem}
\label{T1}Let $M$ be a complete noncompact Riemannian $m$-manifold, $m\geq 2$
supporting a weighted Poincar\'{e} inequality $\left( P_{\rho }\right) \,,$
with Ricci curvature
\begin{equation}
\begin{array}{lll}
Ric_{M}(x) & \geq & -\tau \rho \left( x\right)%
\end{array}
\label{1.4}
\end{equation}%
for all $x\in M,$ where $\tau $ is a constant such that%
\begin{equation*}
\begin{array}{l}
\tau <\frac{4\left( p-1+\kappa \right) }{p^{2}},\text{ }%
\end{array}%
\end{equation*}%
in which $p>1,$ and%
\begin{equation*}
\begin{array}{l}
\kappa =\left\{
\begin{array}{ll}
\max \left\{ \frac{1}{m-1},\min \left\{ \frac{\left( p-1\right) ^{2}}{m}%
,1\right\} \right\} & \text{if }p>2, \\
\frac{\left( p-1\right) ^{2}}{m-1} & \text{if }1<p\leq 2.%
\end{array}%
\right.%
\end{array}%
\end{equation*}
Then every weakly $p$-harmonic function $u$ with finite $p$-energy $E_{p}$
is constant. Moreover, $M$ has at most one $p$-hyperbolic end.
\end{theorem}
In Theorem \ref{T1}, we say that $M$ supports a weighted Poincar\'{e}
inequality $\left( P_{\rho }\right) $, if there exists a positive function $%
\rho (x)$ a.e. on $M$ such that, for every $\Psi \in W_{0}^{1.2}\left(
M\right) \,,$
\begin{equation}
\begin{array}{lll}
\int_{M}\rho\left( x\right) \Psi^{2}\left( x\right) dv & \leq & \int
_{M}\left\vert \nabla\Psi\left( x\right) \right\vert ^{2}dv.%
\end{array}
\label{WP}
\end{equation}
If $\rho (x)$ is no less than a positive constant $\lambda \,,$ then $M$ has
positive spectrum. For example, the hyperbolic space $H^{m}$ has positive
spectrum, and $\rho \left( x\right) =\frac{\left( m-1\right) ^{2}}{4}.$ In $%
\mathbb{R}^{m},$ if we select $\rho \left( x\right) =\frac{\left( m-2\right)
^{2}}{4r^{2}}\left( x\right) ,$ then (\ref{WP}) is Hardy's inequality. For
more examples, see \cite{CLW}\cite{LW3}\cite{WL}. \bigskip
\bigskip
If $u$ is a $C^{3}$ strongly $p$-harmonic function with finite $q$-energy,
then we have a Liouville type property as follows. \bigskip
\begin{theorem}
\label{T2}Let $M$ be a complete noncompact Riemannian $m$-manifold, $m\geq
2, $ satisfying $\left( P_{\rho }\right) \,,$ with Ricci curvature
\begin{equation}
\begin{array}{lll}
Ric_{M}(x) & \geq & -\tau \rho (x)%
\end{array}
\label{Rs}
\end{equation}%
for all $x\in M,$ where $\tau $ is a constant such that
\begin{equation*}
\begin{array}{l}
\tau <\frac{4\left( q-1+\kappa +b\right) }{q^{2}},\text{ }%
\end{array}%
\end{equation*}%
in which
\begin{equation*}
\begin{array}{l}
\kappa =\min \{\frac{\left( p-1\right) ^{2}}{m-1},1\}\text{ and}\text{ }%
b=\min \{0,(p-2)(q-p)\},\text{ where }p>1.%
\end{array}%
\end{equation*}%
Let $u\in C^{3}\left( M\right) $ be a strongly $p$-harmonic function with
finite $q$-energy $E_{q}\left( u\right) <\infty .$ \newline
(I). Then $u$ is constant under each one of the following conditions:
\newline
(1) $p=2$ and $q>\frac{m-2}{m-1},$ \newline
(2) $p=4,$ $q>\max \left\{ 1,1-\kappa -b\right\} ,$ \newline
(3) $p>2,$ $p\neq 4,$ and either%
\begin{equation*}
\begin{array}{l}
\max \left\{ 1,p-1-\frac{\kappa }{p-1}\right\} <q\leq \min \left\{ 2,p-\frac{%
\left( p-4\right) ^{2}m}{4\left( p-2\right) }\right\}%
\end{array}%
\end{equation*}%
or%
\begin{equation*}
\begin{array}{l}
\max \left\{ 2,1-\kappa -b\right\} <q,%
\end{array}%
\end{equation*}%
\newline
(II) $u$ does not exist for $1<p<2$ and $q>2.$
\end{theorem}
As an application, we also extend this theorem to $p$-harmonic morphisms and
conformal maps in Sections \ref{morphism} and \ref{Maps} respectively.%
\bigskip
The paper is organized as follows. In section $2$, we recall some facts
about $p$-hyperbolic and $p$-parabolic ends from \cite{LT} and \cite{H1},
and prove an existence theorem on manifolds with two $p$-hyperbolic ends. In
section $3$, we introduce the linearization $\mathcal{L}_{\epsilon }$ (\ref%
{1.3}) of the perturbed operator$\Delta _{p,\epsilon }\,,$ and derive the $%
\mathcal{L}_{\epsilon }$-type Bochner's formula (\ref{b}) and Kato type
inequality (\ref{k}) for the solution $u_{\epsilon }$ of the perturbed
equation (\ref{1.2}). In section $4$, by applying Bochner's formula and
Kato's inequality, we show a Liouville type theorem and one $p$-hyperbolic
end property for a weakly $p$-harmonic function with finite $p$-energy in a
complete noncompact manifold which supports a weighted Poincar\'{e}
inequality and satisfies a curvature assumption. In section $5$, we show
Liouville type theorems for strongly $p$-harmonic functions with finite $q$%
-energy, and we also extend our results to some $p$-harmonic maps such as $p$%
-harmonic morphisms and conformal maps between Riemannian manifolds. In
section $6$ of the Appendix, we prove the existence of the approximate
solution $u_{\epsilon },$ Proposition \ref{3.1}, and volume estimate of
complete noncompact manifolds with $p$-Poincar\'{e} inequality. We also
construct an example of non-trivial $p$-harmonic function with finite $q$%
-energy on manifolds with weighted Poincar\'{e} inequality.
\section{$p$-Hyperbolicity}
We recall some basic facts about capacities from \cite{H}, \cite{H1} and
\cite{Tr}.
Let $M$ be a Riemannian manifold, $G\subset M$ a connected open set in $M.$
If $D$ and $\Omega$ are nonempty, disjoint, and closed sets contained in the
closure of $G.$ A triple $\left( \Omega,D;G\right) $ is called a condenser.
The $p$-capacity of $\left( \Omega,D;G\right) $ is defined by%
\begin{equation*}
\begin{array}{lll}
\emph{Cap}_{p}\left( \Omega,D;G\right) & = & \displaystyle\inf\limits_{u}
\int_{G}\left\vert \nabla u\right\vert ^{p}dv,%
\end{array}%
\end{equation*}
for $1\leq p<\infty\, ,$ where the infimum is taken over all $u\in
W^{1,p}\left( G\right) \cap C^{0}(G) $ with $u=1$ in $\Omega$ and $u=0$ in $%
D.$
Above and in what follows, $W^{1,p}\left( M\right) $ is the Sobolev space of
all function \thinspace$u\in L^{p}\left( M\right) $ and whose distributional
gradient $\nabla u$ also belongs to $L^{p}\left( M\right) ,$ with respect to
the Sobolev norm
\begin{equation*}
\begin{array}{lll}
\left\Vert u\right\Vert _{1,p} & = & \left\Vert u\right\Vert
_{L_{p}}+\left\Vert \nabla u\right\Vert _{L_{p}}.%
\end{array}%
\end{equation*}
The space $W_{0}^{1,p}\left( M\right) $ is the closure of $C_{0}^{\infty
}\left( M\right) $ in $W^{1,p}\left( M\right) \, ,$ with respect to the $%
\left\Vert \quad\right\Vert _{1,p}$ norm.
The following properties of capacities are well known (see e.g. \cite{Tr}).
\begin{itemize}
\item $\Omega_{2}\subset\Omega_{1}\Longrightarrow\emph{Cap}_{p}\left(
\Omega_{2},D;G\right) \leq\emph{Cap}_{p}\left( \Omega_{1},D;G\right) ;$
\item $D_{2}\subset D_{1}\Longrightarrow\emph{Cap}_{p}\left( \Omega
,D_{2};G\right) \leq\emph{Cap}_{p}\left( \Omega,D_{1};G\right) ;$
\item If $\Omega_{1}\supset\Omega_{2}\cdots\supset\cap_{i}\Omega_{i}=\Omega$
and $D_{1}\supset D_{2}\cdots\supset\cap_{i}D_{i}=D,$ then $\emph{Cap}%
_{p}\left( \Omega,D;G\right) =\lim_{i\rightarrow\infty}\emph{Cap}_{p}\left(
\Omega_{i},D_{i};G\right) .$
\item If $\overline{G\backslash\left( \Omega\cup D\right) }$ is compact,
then there exists a unique weak solution $u:\overline{G\backslash\left(
\Omega\cup D\right) }\rightarrow\mathbb{R} $ to the Dirichlet problem%
\begin{equation*}
\begin{array}{l}
\left\{
\begin{array}{ll}
\Delta_{p}u=0\text{ \ \ \ \ \ } & \text{on }G\backslash\left( \Omega\cup
D\right) ,\text{ } \\
u=1 & \text{on }\Omega, \\
u=0 & \text{on }D,%
\end{array}
\right.%
\end{array}%
\end{equation*}
with $\emph{Cap}_{p}\left( \Omega,D;G\right) =\int_{G}\left\vert
du\right\vert ^{p}dv.$
\end{itemize}
Given a compact set $\Omega$ in $M$, an \textit{end} $E_{\Omega}$ with
respect to $\Omega$ is an unbounded connected component of $%
M\backslash\Omega\, .$ By a compactness argument, it is readily seen that
the number of ends with respect to $\Omega$ is finite, it is also clear that
if $\Omega\subset \Omega^{\prime}$ , then every end $E_{\Omega^{\prime}}$ is
contained in $E_{\Omega}$, so that the number of ends increases as the
compact $\Omega$ enlarges. Let $x_{0}\in\Omega.$ We denote $E_{\Omega}\left(
R\right) =B_{x_{0}}\left( R\right) \cap E_{\Omega},$ $\partial
E_{\Omega}\left( R\right) =\partial B_{x_{0}}\left( R\right) \cap E_{\Omega}$
and $\partial E_{\Omega}=\partial\Omega\cap E_{\Omega}.$
In \cite{LT} (or see e.g. \cite{L}, \cite{LT1}-\cite{LT3}, \cite{PRS}), $2$%
-parabolic and $2$-nonparabolic manifolds and ends are introduced. In \cite%
{H1}, I. Holopainen defined the $p$-parabolic end as follows:
\begin{definition}
Let $E$ be an end of $M$ with respect to $\Omega.$ $E$ is $p$-parabolic, or,
equivalently, has zero $p$-capacity at infinity if,%
\begin{equation*}
\begin{array}{lllll}
\emph{Cap}_{p}\left( \Omega,\infty;E\right) & := & \lim_{i\rightarrow\infty }%
\emph{Cap}_{p}\left( \Omega,\overline{E}\backslash\Omega_{i};E\right) & = &
0,%
\end{array}%
\end{equation*}
where $\left\{ \Omega_{i}\right\} _{i=1}^{\infty}$ is an exhaustion of $M$
by relatively compact open domains with smooth boundary and $\Omega_{i}
\subset\subset\Omega_{i+1}, $ for every integer $i\, .$
\end{definition}
This definition also implies: if $E$ is an end with respect to $\Omega,$
there are sequence of weakly $p$-harmonic functions $\left\{ u_{i}\right\} ,$
$u_{i}\in W^{1,p},$ defined on $E,$ satisfying%
\begin{equation}
\begin{array}{lllll}
\Delta_{p}u_{i} & = & 0 & \text{on} & E\left( r_{i}\right)%
\end{array}
\label{p-1}
\end{equation}
with boundary conditions%
\begin{equation}
\begin{array}{lll}
u_{i} & = & \left\{
\begin{array}{ll}
1\text{ \ \ \ \ \ \ \ \ } & \text{on }\Omega, \\
0 & \text{on }\overline{E\backslash\Omega_{i}},%
\end{array}
\right.%
\end{array}
\label{p-2}
\end{equation}
then $\left\{ u_{i}\right\} $ converges (converges uniformly on each compact
set of $E$) to the constant function $u=1$ on $E$ as $i\rightarrow\infty.$
\begin{definition}
An end $E$ is $p$-hyperbolic (or $p$-nonparabolic) if $E$ is not $p$%
-parabolic.
\end{definition}
If $h_{i}$ is a weakly $p$-harmonic function satisfying (\ref{p-1}) and \ref%
{p-2}, then $E$ is $p$-hyperbolic if and only if $\left\{ h_{i}\right\} $
converges to a weakly $p$-harmonic function $h$ with $h=1$ on $\partial E$, $%
\inf_{E}h=0$ and finite $p$-energy.
\begin{definition}
A manifold $M$ is $p$-parabolic, or, equivalently, has zero $p$-capacity at
infinity if, for each compact set $\Omega\subset M,$
\begin{equation*}
\begin{array}{lllll}
\emph{Cap}_{p}\left( \Omega,\infty;M\right) & := & \lim_{i\rightarrow\infty }%
\emph{Cap}_{p}\left( \Omega,M\backslash\Omega_{i};M\right) & = & 0,%
\end{array}%
\end{equation*}
where $\left\{ \Omega_{i}\right\} _{i=1}^{\infty}$ is an exhaustion of $M$
by domains with smooth boundary and $\Omega_{i} \subset\subset\Omega_{i+1}, $
for every integer $i\, .$
\end{definition}
\begin{definition}
A manifold $M$ is $p$-hyperbolic (or $p$-nonparabolic) if $M$ is not $p$%
-parabolic.
\end{definition}
This definition also implies that a manifold $M$ is $p$-parabolic if each
end of $M$ is $p$-parabolic, $M$ is $p$-hyperbolic if $M$ has at least one $%
p $-hyperbolic end.
Now we focus on manifold $M$ with two $p$-hyperbolic ends (cf. \cite{H}).
\begin{proposition}
\label{2 E}Let $M$ be a complete noncompact manifold, and assume $M$ has two
$p$-hyperbolic ends $E_{1}$ and $E_{2}.$ Then there exists a weakly $p$%
-harmonic function $h:M\rightarrow\mathbb{R}$ with finite $p$-energy such
that $0<h<1,$ $\sup_{E_{1}}h=1$ and $\inf_{E_{2}}h=0.$ Moreover, $h$ is $%
C^{1,\alpha}$.
\end{proposition}
\proof
Given $\Omega\subset M,$ we fix an exhaustion $\left\{ \Omega_{i}\right\} $
of $M$ by domains with smooth boundary and $\Omega_{i}\subset\subset
\Omega_{i+1}$ for every integer $i\, .$
Denote by $E_{A}$ the $p$-hyperbolic ends of $M$ with respect to $A\,.$ For
every $A$, let $u_{i}^{E_{A}}$ be the $p$-harmonic function satisfying%
\begin{equation*}
\left\{
\begin{array}{ll}
\Delta_{p}u_{i}^{E_{A}}=0 & \text{in }E_{A}\cap\Omega_{i}, \\
u_{i}^{E_{A}}=1 & \text{on }\partial E_{A}, \\
u_{i}^{E_{A}}=0 & \text{on }\partial E_{A,\Omega_{i}}=\partial\left(
E_{A}\cap\Omega_{i}\right) \backslash\partial E_{A}.%
\end{array}
\right.
\end{equation*}
By the monotone property, $u_{i}^{E_{A}}$ converges uniformly to $u^{E_{A}}$
on every compact subset of $E_{A}.$
For every $i$, let $h_{i}$ be the weak solution of the boundary value
problem
\begin{equation*}
\left\{
\begin{array}{ll}
\Delta _{p}h_{i}=0 & \text{in }\Omega _{i}, \\
h_{i}=1 & \text{on }\partial \Omega _{i}\cap E_{1}, \\
h_{i}=0 & \text{on }\partial \Omega _{i}\cap \left( M\backslash E_{1}\right)
.%
\end{array}%
\right.
\end{equation*}%
Then, $0\leq h_{i}\leq 1$, and by regularity, then there are subsequence,
say $\left\{ h_{i}\right\} ,$ converges, locally uniformly, to a weakly $p$%
-harmonic function $h$ on $M$, satisfying $0\leq h\leq 1.$
On $E_{1}$, the maximum principle implies $1-u_{i}^{E_{1}}\leq h_{i}<1.$
Hence $1-u^{E_{1}}\leq h<1$ on $E_{1}$, so that $\sup_{E_{1}}\left(
1-u^{E_{1}}\right) \leq\sup_{E_{A}}h=1$ gives $\sup_{E_{1}}h=1$ since $%
\inf_{E_{1}}u^{E_{1}}=0.$
On $E_{2},$ the maximum principle implies $0<h_{i}\leq u_{i}^{E_{2}}.$ Hence
we have $0<h\leq u^{E_{2}}$ on $E_{2}$, so that $0\leq\inf_{E_{2}}h\leq
\inf_{E_{2}}u^{E_{2}}=0$.
Now we have $\sup_{E_{1}}h=1$ and $\inf_{E_{2}}h=0,$ so $h$ is a nonconstant
$p$-harmonic function on $M.$
Finally, $h$ has finite $p$-energy by
\begin{equation*}
\begin{array}{lll}
\emph{Cap}_{p}\left( E_{1}\backslash\Omega_{i},M\backslash\left( \Omega
_{i}\cup E_{1}\right) ;M\right) & = & \int_{M}\left\vert \nabla
h_{i}\right\vert ^{p}dv\neq0,%
\end{array}%
\end{equation*}
and the monotonic properties of capacities. \endproof
\bigskip
\section{An $\mathcal{L}_{\protect\epsilon}$-type Bochner's formula and a
Kato type inequality}
As stated in the Introduction, for each solution $u_{\epsilon }$ of a
perturbed $p$-Laplace equation (\ref{1.2}), we study the linearized operator
$\mathcal{L}_{\epsilon }$ of the perturbed operator $\Delta _{p,\epsilon }\,$
(\ref{1.3}) form the argument of Kotschwar-Ni \cite{KN}. We note that $%
\mathcal{L}_{\epsilon }\left( f_{\epsilon }^{2}\right) \left( x\right) $ is
well define for all $x\in \Omega \,,$ since $f_{\epsilon }>0$ and $%
f_{\epsilon }^{2}\in C_{loc}^{\infty }\left( \Omega \right) .$
Now we use the operator $\mathcal{L}_{\epsilon}$ to derive a generalized
Bochner's formula (or an $\mathcal{L}_{\epsilon}$-type Bochner's formula)
for the solution $u_{\epsilon}$ of (\ref{1.2}).
\begin{lemma}
\label{Bo} Let $u_{\epsilon}$ be a solution of (\ref{1.2}) on $\Omega\subset
M\, ,$ $f_{\epsilon}=\sqrt{\left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon}\, ,$ and $\nabla du_{\epsilon}$ be the Hessian of $%
u_{\epsilon }\, .$ Then for every $p>1,$%
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{\epsilon}\left( f_{\epsilon}^{2}\right) & = & \frac{%
p-2}{4}f_{\epsilon}^{p-4}\left\vert \nabla f_{\epsilon}^{2}\right\vert ^{2}
+ f_{\epsilon}^{p-2}\left( \left\vert \nabla du_{\epsilon}\right\vert
^{2}+Ric\left( \nabla u_{\epsilon},\nabla u_{\epsilon}\right) \right) .%
\end{array}
\label{b}
\end{equation}
\end{lemma}
\proof
Since $f_{\epsilon} \ne0\, ,$ for every $p>1\, ,$ the perturbed $p$-Laplace
equation (\ref{1.2}) is equivalent to
\begin{equation}
\begin{array}{l}
\frac{p-2}{2}\left\langle \nabla f_{\epsilon}^{2},\nabla u_{\epsilon
}\right\rangle = - f_{\epsilon}^{2}\Delta u_{\epsilon}.%
\end{array}
\label{b0}
\end{equation}
On the one hand, (\ref{b0}) implies
\begin{equation}
\begin{array}{lll}
\frac{p-2}{2}f_{\epsilon}^{p-6}\left\langle \nabla u_{\epsilon},\nabla
f_{\epsilon}^{2}\right\rangle ^{2} & = & -f_{\epsilon}^{p-4}\left\langle
\nabla u_{\epsilon},\nabla f_{\epsilon}^{2}\right\rangle \Delta u_{\epsilon}.%
\end{array}
\label{b4}
\end{equation}
On the other hand, taking the gradient of both sides of (\ref{b0}), and then
taking the inner product with $\nabla u_{\epsilon}\,,$ we have
\begin{equation}
\begin{array}{lll}
\frac{p-2}{2}\left\langle \nabla\left\langle \nabla f_{\epsilon}^{2},\nabla
u_{\epsilon}\right\rangle ,\nabla u_{\epsilon}\right\rangle & = &
-\left\langle \nabla f_{\epsilon}^{2},\nabla u_{\epsilon}\right\rangle
\Delta u_{\epsilon}-f_{\epsilon}^{2}\left\langle \nabla\left( \Delta
u_{\epsilon }\right) ,\nabla u_{\epsilon}\right\rangle .%
\end{array}
\label{b1}
\end{equation}
Now we compute
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{\epsilon}\left( f_{\epsilon}^{2}\right) & = & \frac{1%
}{2}\text{\textrm{div}}\left( f_{\epsilon}^{p-2}\nabla f_{\epsilon
}^{2}+\left( p-2\right) f_{\epsilon}^{p-4}\left\langle \nabla u_{\epsilon
},\nabla f_{\epsilon}^{2}\right\rangle \nabla u_{\epsilon}\right) \\
& = & \frac{p-2}{4}f_{\epsilon}^{p-4}\left\vert \nabla
f_{\epsilon}^{2}\right\vert ^{2}+\frac{1}{2}f_{\epsilon}^{p-2}\Delta
f_{\epsilon}^{2}+\frac{\left( p-2\right) \left( p-4\right) }{4}%
f_{\epsilon}^{p-6}\left\langle \nabla u_{\epsilon},\nabla
f_{\epsilon}^{2}\right\rangle ^{2} \\
& & +\frac{p-2}{2}f_{\epsilon}^{p-4}\left\langle \nabla\left\langle \nabla
u_{\epsilon},\nabla f_{\epsilon}^{2}\right\rangle ,\nabla u_{\epsilon
}\right\rangle +\frac{p-2}{2}f_{\epsilon}^{p-4}\left\langle \nabla
u_{\epsilon},\nabla f_{\epsilon}^{2}\right\rangle \Delta u_{\epsilon}.%
\end{array}
\label{b2}
\end{equation}
Substituting (\ref{b1}) into (\ref{b2}), one has%
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{\epsilon}\left( f_{\epsilon}^{2}\right) & = & \frac{%
p-2}{4}f_{\epsilon}^{p-4}\left\vert \nabla f_{\epsilon}^{2}\right\vert ^{2}+%
\frac{1}{2}f_{\epsilon}^{p-2}\Delta f_{\epsilon}^{2}+\frac{p-4}{2}%
f_{\epsilon}^{p-4}\left\langle \nabla u_{\epsilon},\nabla
f_{\epsilon}^{2}\right\rangle \Delta u_{\epsilon} \\
& & -f_{\epsilon}^{p-4}f_{\epsilon}^{2}\left\langle \nabla\left( \Delta
u_{\epsilon}\right) ,\nabla u_{\epsilon}\right\rangle +\frac{\left(
p-2\right) \left( p-4\right) }{4}f_{\epsilon}^{p-6}\left\langle \nabla
u_{\epsilon},\nabla f_{\epsilon}^{2}\right\rangle ^{2}.%
\end{array}
\label{b3}
\end{equation}
Applying Bochner's formula%
\begin{equation*}
\begin{array}{l}
\frac{1}{2}\Delta f_{\epsilon}^{2}=\left\vert \nabla
du_{\epsilon}\right\vert ^{2}+\left\langle \nabla u_{\epsilon},\nabla\Delta
u_{\epsilon}\right\rangle +Ric\left( \nabla u_{\epsilon},\nabla
u_{\epsilon}\right)%
\end{array}%
\end{equation*}
and the equation (\ref{b4}) to the second term and the last term of right
hand side of (\ref{b3}) respectively, one obtains the desired formula (\ref%
{b}). \endproof
We derive the following Kato type inequality for the approximate solution $%
u_{\epsilon}:$
\begin{lemma}
\label{ka'}Let $u_{\epsilon }$ be a solution of (\ref{1.2}) on $\Omega
\subset M^{m},$ $p>1,$ and $\kappa $ be as Theorem \ref{T1}. Then the
Hessian of $u_{\epsilon }$ satisfies
\begin{equation}
\begin{array}{lll}
\left\vert \nabla du_{\epsilon }\right\vert ^{2} & \geq & \left( 1+\kappa
\right) \left\vert \nabla \left\vert du_{\epsilon }\right\vert \right\vert
^{2}%
\end{array}
\label{k'}
\end{equation}%
at $x\in \Omega $ with $du_{\epsilon }\left( x\right) \neq 0.$
\end{lemma}
\proof
Fix $x\in\Omega\subset M$ with $du_{\epsilon}\neq0,$ we select a local
orthonormal frame field $\left\{ e_{1},e_{2},\ldots e_{m}\right\} $ such
that, at $x,$ $\nabla_{e_{i}}e_{j}=0,$ $\nabla u_{\epsilon}=\left\vert
\nabla u_{\epsilon}\right\vert e_{1},$ and $u_{\epsilon,\alpha}=0$ for all $%
i,j=1,\ldots,m,$ $\alpha=2,\ldots\,,$ where $u_{\epsilon,\alpha}=\left%
\langle \nabla u_{\epsilon},e_{\alpha}\right\rangle .$
Let $f=\left\vert \nabla u_{\epsilon }\right\vert ,$ $f_{\epsilon }=\sqrt{%
\left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon }$ and the
directional derivative $f_{\epsilon ,i}=\left\langle \nabla f_{\epsilon
},e_{i}\right\rangle \,.$ Denote the directional derivative $\left\langle
\nabla u_{\epsilon ,i},e_{j}\right\rangle $ by $u_{\epsilon ,ij}\,.$ Then (%
\ref{b0}) implies
\begin{equation*}
\begin{array}{lllll}
\Delta u_{\epsilon } & = & -\frac{p-2}{2f_{\epsilon }^{2}}\left\langle
\nabla f_{\epsilon }^{2},\nabla u_{\epsilon }\right\rangle & = & -\frac{p-2}{%
2f_{\epsilon }^{2}}\sum_{i=1}^{m}\left( f_{\epsilon }^{2}\right)
_{,i}u_{\epsilon ,i} \\
& & & = & -\frac{p-2}{2f_{\epsilon }^{2}}\left( f_{\epsilon }^{2}\right)
_{,1}u_{\epsilon ,1} \\
& & & = & -\frac{p-2}{2f_{\epsilon }^{2}}\left( f_{\epsilon }^{2}\right)
_{,1}f.%
\end{array}%
\end{equation*}%
Moreover, by using the following property%
\begin{equation*}
\begin{array}{lllllll}
\left( f_{\epsilon }^{2}\right) _{,j} & = & \left( f^{2}\right) \,_{,j} & =
& \sum_{i=1}^{m}\left( u_{\epsilon ,i}^{2}\right) \,_{,j} & = &
2\sum_{i=1}^{m}u_{\epsilon ,i}u_{\epsilon ,ij} \\
& & & & & = & 2u_{\epsilon ,1}u_{\epsilon ,1j} \\
& & & & & = & 2fu_{\epsilon ,1j}.%
\end{array}%
\end{equation*}%
We have%
\begin{equation}
\begin{array}{l}
\Delta u_{\epsilon }=\frac{-\left( p-2\right) f^{2}}{f_{\epsilon }^{2}}%
u_{\epsilon ,11},%
\end{array}
\label{k3}
\end{equation}%
and
\begin{equation}
\begin{array}{l}
u_{\epsilon ,1j}=f_{,j}.%
\end{array}
\label{k4}
\end{equation}%
On the other hand,
\begin{equation}
\begin{array}{lll}
\sum_{i,j=1}^{m}\left( u_{\epsilon ,ij}\right) ^{2} & \geq & \left(
u_{\epsilon ,11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left( u_{\epsilon
,1\alpha }\right) ^{2}+\sum_{\alpha =2}^{m}\left( u_{\epsilon ,\alpha \alpha
}\right) ^{2} \\
& \geq & \left( u_{\epsilon ,11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left(
u_{\epsilon ,1\alpha }\right) ^{2}+\frac{\left( \sum_{\alpha
=2}^{m}u_{\epsilon ,\alpha \alpha }\right) ^{2}}{m-1} \\
& = & \left( u_{\epsilon ,11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left(
u_{\epsilon ,1\alpha }\right) ^{2}+\frac{\left( \Delta u_{\epsilon
}-u_{\epsilon ,11}\right) ^{2}}{m-1}.%
\end{array}
\label{k1}
\end{equation}%
Therefore, by using (\ref{k3}) and (\ref{k4}), the inequality (\ref{k1}) can
be written as%
\begin{equation*}
\begin{array}{lll}
\sum_{i,j=1}^{m}\left( u_{\epsilon ,ij}\right) ^{2} & \geq & \left(
u_{\epsilon ,11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left( u_{\epsilon
,1\alpha }\right) ^{2}+\frac{\left( \left( \frac{\left( p-2\right) f^{2}}{%
f_{\epsilon }^{2}}+1\right) u_{\epsilon ,11}\right) ^{2}}{m-1} \\
& = & \left( 1+\frac{\left( \left( p-1\right) f^{2}+\epsilon \right) ^{2}}{%
\left( m-1\right) f_{\epsilon }^{4}}\right) \left( u_{\epsilon ,11}\right)
^{2}+2\sum_{\alpha =2}^{m}\left( u_{\epsilon ,1\alpha }\right) ^{2} \\
& \geq & \left( 1+\kappa \right) \left\vert \nabla f\right\vert ^{2},%
\end{array}%
\end{equation*}%
where
\begin{equation*}
\begin{array}{lll}
\kappa & = & \left\{
\begin{array}{ll}
\frac{1}{m-1} & \text{if }p\geq 2, \\
\frac{\left( p-1\right) ^{2}}{m-1} & \text{if }1<p<2.%
\end{array}%
\right.%
\end{array}%
\end{equation*}%
This completes the proof. \endproof
\bigskip
In the following, we use the add-one-dimension method to refine kato's
inequality in the case $p>2.$ This argument is from referee.
We define $N=M\times
\mathbb{R}
$ with metric $g_{N}=g+dt^{2},$ and let
\begin{equation}
v_{\epsilon }\left( x,t\right) =u_{\epsilon }\left( x\right) +\sqrt{\epsilon
}t \label{1.2.1}
\end{equation}%
for $x\in M$ and $\epsilon >0,$ where $u_{\epsilon }$ is the solution of (%
\ref{1.2}). Then $v_{\epsilon }$ is a $p$-harmonic function on $\left(
N,g_{N}\right) ,$ i.e. if $\Delta _{p}^{N}$ is the $p$-Laplace operator on $%
\left( N,g_{N}\right) ,$ we have $\Delta _{p}^{N}v_{\epsilon }=0$ with $%
\left\vert \nabla ^{N}v_{\epsilon }\right\vert ^{2}>\epsilon >0$ and $%
Ric_{N}\left( \nabla ^{N}v_{\epsilon },\nabla ^{N}v_{\epsilon }\right)
=Ric\left( \nabla u_{\epsilon },\nabla u_{\epsilon }\right) .$ Moreover, if
we denote $f_{\epsilon }=\left\vert \nabla v_{\epsilon }\right\vert ,$ then $%
f_{\epsilon }$ is independing of $t,$ hence we have $\nabla ^{N}f_{\epsilon
}=\nabla f_{\epsilon }$ and $\Delta ^{N}f_{\epsilon }=\Delta f_{\epsilon }.$
\begin{lemma}
\label{ka}Let $u_{\epsilon }$ be a solution of (\ref{1.2}) on $\Omega
\subset M^{m},$ $p>1,$ and $\kappa $ be as Theorem \ref{T1}. Then the
Hessian of $u_{\epsilon }$ satisfies
\begin{equation}
\begin{array}{lll}
\left\vert \nabla du_{\epsilon }\right\vert ^{2} & \geq & \left( 1+\kappa
\right) \left\vert \nabla \left\vert du_{\epsilon }\right\vert \right\vert
^{2}%
\end{array}
\label{k}
\end{equation}%
at $x\in \Omega $ with $du_{\epsilon }\left( x\right) \neq 0.$
\end{lemma}
\proof For the case $1<p\leq 2,$ see Lemma \ref{ka'}. For $p>2,$ we use the
kato's inequality for $p$-harmonic function on $\left( N,g_{N}\right) $ (see
Lemma \ref{KS}) and Lemma \ref{ka'} to complete the proof. \ \ \ \endproof
\bigskip
\section{The Proof of Theorem \protect\ref{T1}}
Now we use Lemma \ref{ka}, Lemma \ref{Bo} and weighted Poincar\'{e}
inequality (\ref{WP}) to obtain the following inequality (\ref{v3}):
\begin{lemma}
\label{m1}Let $M$ be a manifold satisfying the hypothesis of Theorem \ref{T1}%
. Let $u_{\epsilon}$ be a solution of (\ref{1.2}) on $B\left( 2R\right)
\subset M.$ Then we have
\begin{equation}
\begin{array}{lll}
C\int_{B\left( R\right) }\rho\left\vert \nabla u_{\epsilon}\right\vert ^{p}dv
& \leq & \frac{100\cdot B}{R^{2}}\int_{B\left( 2R\right) \backslash B\left(
R\right) }\left( \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right) ^{\frac{p}{2}}dv,%
\end{array}
\label{v3}
\end{equation}
where $C\left( p,m,\kappa,\tau,\epsilon_{1},\epsilon_{2}\right) >0$ and $%
B\left( p,m,\kappa,\epsilon_{1},\epsilon_{2}\right) >0$ are positive
constants for sufficiently small constants $\epsilon_{1}, \epsilon_{2} > 0\,
.$
\end{lemma}
\proof
Let $\Omega=B\left( 2R\right) $ be a geodesic ball of radius $2R$ centered
at a fixed point.
Let $f=\left\vert \nabla u_{\epsilon }\right\vert $ and $f_{\epsilon }=\sqrt{%
f^{2}+\epsilon }.$ In view of Lemma \ref{ka},
\begin{equation*}
\begin{array}{ccccc}
f_{\epsilon }^{2}\left\vert \nabla du_{\epsilon }\right\vert ^{2} & \geq &
f^{2}\left\vert \nabla du_{\epsilon }\right\vert ^{2} & \geq & \frac{%
1+\kappa }{4}\left\vert \nabla f^{2}\right\vert ^{2}%
\end{array}%
\end{equation*}%
holds for all on $M.$ Then, by Lemma \ref{Bo}, we rewrite Bochner's formula
as%
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{\epsilon }\left( f_{\epsilon }^{2}\right) & \geq &
\left( p-1+\kappa \right) f_{\epsilon }^{p-2}\left\vert \nabla f_{\epsilon
}\right\vert ^{2}+f_{\epsilon }^{p-2}Ric\left( \nabla u_{\epsilon },\nabla
u_{\epsilon }\right) ,%
\end{array}
\label{v0}
\end{equation}%
here we use $\nabla f_{\epsilon }^{2}=\nabla f^{2}\,.$
We multiply both sides of (\ref{v0}) by $\eta ^{2}$ and integrate over $M,$
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\int_{M}\eta ^{2}\mathcal{L}_{\epsilon }\left( f_{\epsilon
}^{2}\right) dv & \geq & \left( p-1+\kappa \right) \int_{M}\eta
^{2}f_{\epsilon }^{p-2}\left\vert \nabla f_{\epsilon }\right\vert
^{2}dv+\int_{M}\eta ^{2}f_{\epsilon }^{p-2}Ric\left( \nabla u_{\epsilon
},\nabla u_{\epsilon }\right) dv%
\end{array}
\label{v1}
\end{equation}%
where $\eta \in C_{0}^{\infty }(M)$ is a cut-off function with $0\leq \eta
\left( x\right) \leq 1$ on $M$ satisfying%
\begin{equation*}
\left\{
\begin{array}{ll}
\eta \left( x\right) =1 & \text{if }x\in \overline{B\left( R\right) }, \\
\left\vert \nabla \eta \left( x\right) \right\vert \leq \frac{10}{R}\text{\
\ \ \ \ } & \text{if }x\in B\left( 2R\right) \backslash \overline{B\left(
R\right) }, \\
\eta \left( x\right) =0 & \text{if }x\in M\backslash B\left( 2R\right) .%
\end{array}%
\right.
\end{equation*}
On the other hand, applying integration by parts and Cauchy-Schwarz
inequality one has%
\begin{equation*}
\begin{array}{lll}
\frac{1}{2}\int_{M}\eta ^{2}\mathcal{L}_{\epsilon }\left( f_{\epsilon
}^{2}\right) dv & = & \frac{-1}{2}\int_{M}\left\langle \nabla \eta
^{2},f_{\epsilon }^{p-2}\nabla f_{\epsilon }^{2}+\left( p-2\right)
f_{\epsilon }^{p-4}\left\langle \nabla u_{\epsilon },\nabla f_{\epsilon
}^{2}\right\rangle \nabla u_{\epsilon }\right\rangle dv \\
& \leq & 2\int_{M}\eta \left\vert \nabla \eta \right\vert \left( f_{\epsilon
}^{p-1}\left\vert \nabla f_{\epsilon }\right\vert +\left\vert p-2\right\vert
f_{\epsilon }^{p-3}f^{2}\left\vert \nabla f_{\epsilon }\right\vert \right) dv
\\
& \leq & 2\left( 1+\left\vert p-2\right\vert \right) \int_{M}\eta \left\vert
\nabla \eta \right\vert f_{\epsilon }^{p-1}\left\vert \nabla f_{\epsilon
}\right\vert dv \\
& \leq & \epsilon _{1}\int_{M}\eta ^{2}f_{\epsilon }^{p-2}\left\vert \nabla
f_{\epsilon }\right\vert ^{2}dv+\frac{\left( 1+\left\vert p-2\right\vert
\right) ^{2}}{\epsilon _{1}}\int_{M}\left\vert \nabla \eta \right\vert
^{2}f_{\epsilon }^{p}dv,%
\end{array}%
\end{equation*}%
where $\epsilon _{1}$ is a positive constant satisfying%
\begin{equation*}
\begin{array}{l}
p-1-\epsilon _{1}>0.%
\end{array}%
\end{equation*}
Then (\ref{v1}) implies%
\begin{equation}
\begin{array}{lll}
\frac{\left( 1+\left\vert p-2\right\vert \right) ^{2}}{\epsilon _{1}}%
\int_{M}\left\vert \nabla \eta \right\vert ^{2}f_{\epsilon }^{p}dv & \geq &
\int_{M}\left( p-1+\kappa -\epsilon _{1}\right) \eta ^{2}f_{\epsilon
}^{p-2}\left\vert \nabla f_{\epsilon }\right\vert ^{2}dv \\
& & +\int_{M}\eta ^{2}f_{\epsilon }^{p-2}Ric\left( \nabla u_{\epsilon
},\nabla u_{\epsilon }\right) dv.%
\end{array}
\label{v1-2}
\end{equation}
Besides, we may rewrite the first term in the right hand side of (\ref{v1-2}%
) by
\begin{equation*}
\begin{array}{lll}
& & \left( p-1+\kappa -\epsilon _{1}\right) \int_{M}\eta ^{2}f_{\epsilon
}^{p-2}\left\vert \nabla f_{\epsilon }\right\vert ^{2}dv \\
& = & \frac{4\left( p-1+\kappa -\epsilon _{1}\right) }{p^{2}}\int_{M}\eta
^{2}\left\vert \nabla f_{\epsilon }^{\frac{p}{2}}\right\vert ^{2}dv \\
& = & \frac{4\left( p-1+\kappa -\epsilon _{1}\right) }{p^{2}}%
\int_{M}\left\vert \nabla \left( \eta f_{\epsilon }^{\frac{p}{2}}\right)
-\left( \nabla \eta \right) f_{\epsilon }^{\frac{p}{2}}\right\vert ^{2}dv \\
& = & \frac{4\left( p-1+\kappa -\epsilon _{1}\right) }{p^{2}}\int_{M}\left\{
\left\vert \nabla \left( \eta f_{\epsilon }^{\frac{p}{2}}\right) \right\vert
^{2}-2\left\langle \nabla \left( \eta f_{\epsilon }^{\frac{p}{2}}\right)
,f_{\epsilon }^{\frac{p}{2}}\nabla \eta \right\rangle +\left\vert \nabla
\eta \right\vert ^{2}f_{\epsilon }^{p}\right\} dv \\
& \geq & \frac{4\left( 1-\epsilon _{2}\right) \left( p-1+\kappa -\epsilon
_{1}\right) }{p^{2}}\int_{M}\left\vert \nabla \left( \eta f_{\epsilon }^{%
\frac{p}{2}}\right) \right\vert ^{2}+\frac{4\left( 1-\frac{1}{\epsilon _{2}}%
\right) \left( p-1+\kappa -\epsilon _{1}\right) }{p^{2}}\int_{M}\left\vert
\nabla \eta \right\vert ^{2}f_{\epsilon }^{p}dv.%
\end{array}%
\end{equation*}%
where $\epsilon _{2}$ is a positive constant satisfying $\epsilon _{2}<1.$
Thus, we have%
\begin{equation}
\begin{array}{lll}
& & \frac{4\left( 1-\epsilon _{2}\right) \left( p-1+\kappa -\epsilon
_{1}\right) }{p^{2}}\int_{M}\left\vert \nabla \left( \eta f_{\epsilon }^{%
\frac{p}{2}}\right) \right\vert ^{2}dv+\int_{M}\eta ^{2}f_{\epsilon
}^{p-2}Ric\left( \nabla u_{\epsilon },\nabla u_{\epsilon }\right) \,dv \\
& \leq & \left( \frac{\left( 1+\left\vert p-2\right\vert \right) ^{2}}{%
\epsilon _{1}}+\frac{4\left( \frac{1}{\epsilon _{2}}-1\right) \left(
p-1+\kappa -\epsilon _{1}\right) }{p^{2}}\right) \int_{M}\left\vert \nabla
\eta \right\vert ^{2}f_{\epsilon }^{p}dv.%
\end{array}
\label{v2}
\end{equation}
According to the weighted Poincar\'{e} inequality (\ref{WP})
\begin{equation*}
\begin{array}{lll}
\int_{M}\rho \Psi ^{2}dv & \leq & \int_{M}\left\vert \nabla \Psi \right\vert
^{2}dv%
\end{array}%
\end{equation*}%
with $\Psi =\eta f_{\epsilon }^{\frac{p}{2}},$ then (\ref{v2}) implies%
\begin{equation}
\begin{array}{lll}
\int_{B\left( R\right) }Af_{\epsilon }^{p-2}dv & \leq & \frac{100\cdot B}{%
R^{2}}\int_{B\left( 2R\right) \backslash B\left( R\right) }f_{\epsilon
}^{p}dv,%
\end{array}
\label{v3-1}
\end{equation}%
for all fixed $R>0,$ where
\begin{equation*}
\begin{array}{lll}
A & = & \frac{4\left( 1-\epsilon _{2}\right) \left( p-1+\kappa -\epsilon
_{1}\right) }{p^{2}}\rho f_{\epsilon }^{2}-Ric\left( \nabla u_{\epsilon
},\nabla u_{\epsilon }\right) \,,%
\end{array}%
\end{equation*}%
and%
\begin{equation*}
\begin{array}{lll}
B & = & \left( \frac{\left( 1+\left\vert p-2\right\vert \right) ^{2}}{%
\epsilon _{1}}+\frac{4\left( \frac{1}{\epsilon _{2}}-1\right) \left(
p-1+\kappa -\epsilon _{1}\right) }{p^{2}}\right) .%
\end{array}%
\end{equation*}
Since the curvature condition (\ref{1.4}) means that there exists a constant
$0<\tau<\frac{4\left( p-1+\kappa\right) }{p^{2}}$ such that
\begin{equation*}
\begin{array}{lll}
Ric_{M} & \geq & -\tau\rho,%
\end{array}%
\end{equation*}
Then%
\begin{equation*}
\begin{array}{lll}
A & \geq & C\left( p,m,\kappa,\tau,\epsilon_{1},\epsilon_{2}\right) \rho
f^{2}%
\end{array}%
\end{equation*}
with $C>0$ whenever we select $\epsilon_{1}$ and $\epsilon_{2}$ small enough.
Hence, (\ref{v3-1}) gives
\begin{equation*}
\begin{array}{lll}
C\int_{B\left( R\right) }\rho\left\vert \nabla u_{\epsilon}\right\vert ^{p}dv
& \leq & \frac{100\cdot B}{R^{2}}\int_{B\left( 2R\right) \backslash B\left(
R\right) }\left( \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right) ^{\frac p2}dv,%
\end{array}%
\end{equation*}
where $C\left( p,m,\kappa,\tau,\epsilon_{1},\epsilon_{2}\right) >0,$ and $%
B\left( p,m,\kappa,\epsilon_{1},\epsilon_{2}\right) >0.$
\endproof
\bigskip
\textbf{Proof of Theorem \ref{T1}.}
\proof
Given $B\left( R_{0}\right) \subset M,$ for every $a>0,$ we let $\Omega
_{a}=\left\{ x\in B\left( R_{0}\right) :\rho \left( x\right) >1/a\right\} .$
It is clear the measure of $\Omega _{\alpha }$ tends to zero as $%
a\rightarrow 0^{+}.$ If we are able to show $\int_{B\left( R_{0}\right)
\backslash \Omega _{\alpha }}\rho \left\vert \nabla u\right\vert
^{p}dv<\delta $ for any $\delta >0\,,$ then it implies $\nabla u=0$ on $%
B\left( R_{0}\right) $ almost everywhere. This also infers $\nabla u=0$ on $%
B\left( R_{0}\right) $ by the fact $u\in C_{loc}^{1,\alpha }\left( M\right)
. $ Moreover, since $B\left( R_{0}\right) $ is arbitrary, $u$ must be
constant on $M.$
Moreover, if we assume $M$ has at least two $p$-hyperbolic ends. By
Proposition \ref{2 E}, one may construct a nontrivial bounded $p$-harmonic
function with finite $p$-energy on $M$, this gives a contradiction to our
conclusion, hence $M$ has only one $p$-hyperbolic end.
Now we prove the claim. By using the finite $p$-energy of $u,$ we may select
$0<<R<\infty$ large enough such that $B\left( R\right) \supset B\left(
R_{0}\right) $ and
\begin{equation*}
\begin{array}{lll}
\frac{100B}{R^{2}C}\int_{B\left( 2R\right) \backslash B\left( R\right)
}\left\vert \nabla u\right\vert ^{p}dv & < & \delta%
\end{array}%
\end{equation*}
where $B$ and $C$ are defined as (\ref{v3}).
Now we construct $u_{\epsilon }\in C^{\infty }\left( B\left( 2R\right)
\right) $ such that $u_{\epsilon }=u$ on $\partial B\left( 2R\right) $ and $%
u_{\epsilon }$ satisfies (\ref{1.2}). Then (\ref{v3}) implies
\begin{equation*}
\begin{array}{lll}
C\int_{B\left( R_{0}\right) \backslash \Omega _{a}}\rho \left\vert \nabla
u\right\vert ^{p}dv & \leq & \frac{100\cdot B}{R^{2}}\int_{B\left( 2R\right)
\backslash B\left( R\right) }\left\vert \nabla u\right\vert ^{p}dv,%
\end{array}%
\end{equation*}%
as $\epsilon \rightarrow 0,$ we may therefore conclude that
\begin{equation*}
\begin{array}{lll}
\int_{B\left( R_{0}\right) \backslash \Omega _{a}}\rho \left\vert \nabla
u\right\vert ^{p}dv & < & \delta .%
\end{array}%
\end{equation*}
\endproof
\bigskip
\bigskip
If $M$ has positive spectrum $\lambda >0,$ then $M$ has $p$-Poincar\'{e}
inequality%
\begin{equation*}
\begin{array}{l}
\lambda _{p}\int_{M}\Psi ^{p}\leq \int_{M}\left\vert \nabla \Psi \right\vert
^{p},\text{ }\lambda _{p}>0%
\end{array}%
\end{equation*}%
for all $\Psi \in W_{0}^{1,p}\left( M\right) $ and $p\geq 2$ (cf. \cite{HKM}
Theorem 1.8). Since $p$-Poincar\'{e} inequality and Caccioppoli type
estimate imply decay estimate (see Lemma \ref{decay estimate on E} which is
similar to the work of \cite{LW1} Lemma 1.1 and Lemma 1.2), then $p$-Poincar%
\'{e} inequality infers that $M$ must be a $p$-hyperbolic manifold (see
Theorem \ref{hy}). So we have the following Corollary.
\begin{corollary}
Let $M^{m},m\geq2,$ be a complete noncompact Riemannian manifold with
positive spectrum $\lambda>0$ and
\begin{equation*}
\begin{array}{lll}
Ric_{M} & \geq & -\tau\lambda%
\end{array}%
\end{equation*}
where $p\geq2,$ and constant $\tau$ is the same as in Theorem \ref{T1}. Then
every weakly $p$-harmonic function $u$ with finite $p$-energy is constant.
Moreover, $M$ has only one $p$-hyperbolic end
\end{corollary}
\begin{remark}
Similarly, if $M$ has $p$-Poincar\'{e} inequality, $1<p<2,$ then $M$ has
positive spectrum $\lambda >0.$ Hence, if $M$ is a complete noncompact
Riemannian manifold with $p$-Poincar\'{e} inequality, $1<p<2,$ and $%
Ric_{M}\geq -\tau \lambda $ where $\tau <\frac{4\left( p-1\right) \left(
p+m-2\right) }{p^{2}\left( m-1\right) }.$ Then $M$ has only one $p$%
-hyperbolic end.
\end{remark}
\bigskip
\section{Strongly $p$-harmonic functions with applications}
\subsection{Bochner's formula}
Let $u$ be $C^{3}$ a strongly $p$-harmonic function for $p>1$, by using $%
\left\vert \nabla u\right\vert ^{p-2}\nabla u$ must be differentiable on $M,$
then $u$ is a solution of (\ref{pl}) as follows.
\begin{lemma}
\label{PL}If $u$ is a $C^{3}$ strongly $p$-harmonic function for $p>1$, then
$u$ is a solution of
\begin{equation}
\begin{array}{lll}
f^{2}\Delta u+\frac{p-2}{2}\left\langle \nabla f^{2},\nabla u\right\rangle &
= & 0,%
\end{array}
\label{pl}
\end{equation}
where $f=\left\vert \nabla u\right\vert .$
\end{lemma}
\proof
First, we multiply both side of (\ref{1.0}) by $f^{4},$ because of $f^{4}\in
C^{2}\left( M\right) ,$ then%
\begin{equation*}
\begin{array}{l}
f^{4}\text{\textrm{div}}(f^{p-2}\nabla u)=0%
\end{array}%
\end{equation*}%
implies%
\begin{equation*}
\begin{array}{lll}
0 & = & \text{\textrm{div}}(f^{p+2}\nabla u)-2f^{p}\left\langle \nabla
f^{2},\nabla u\right\rangle \\
& = & f^{p+2}\Delta u+\left\langle \nabla f^{p+2},\nabla u\right\rangle
-2f^{p}\left\langle \nabla f^{2},\nabla u\right\rangle .%
\end{array}%
\end{equation*}%
Since $p>1$ and%
\begin{equation*}
\begin{array}{l}
\nabla f^{p+2}=\nabla \left( \left( f^{2}\right) ^{\frac{p+2}{2}}\right) =%
\frac{p+2}{2}f^{p}\nabla f^{2},%
\end{array}%
\end{equation*}%
so we have%
\begin{equation*}
\begin{array}{l}
f^{p+2}\Delta u+\frac{p-2}{2}f^{p}\left\langle \nabla f^{2},\nabla
u\right\rangle =0.%
\end{array}%
\end{equation*}%
which implies%
\begin{equation*}
\begin{array}{l}
f^{2}\Delta u+\frac{p-2}{2}\left\langle \nabla f^{2},\nabla u\right\rangle =0%
\end{array}%
\end{equation*}%
on all of $M.$ \
\endproof
\begin{remark}
(1). If $u$ is a solution of (\ref{pl}), $u$ may be not a strongly $p$%
-harmonic function, e.g. constant function is a solution of (\ref{pl}), but
it is not a strongly $p$-harmonic function for $1<p<2.$ \newline
(2). For $p\geq 4, $ $u\in C^{2}\left( M\right) $ is a solution of (\ref{pl}%
) if and only if $u$ is a strongly $p$-harmonic function.
\end{remark}
Now we define an operator $\mathcal{L}_{s,\varepsilon}$ by%
\begin{equation*}
\begin{array}{l}
\mathcal{L}_{s,\varepsilon}\left( \Psi\right) =\text{\textrm{div}}\left(
f_{\varepsilon}^{s}A_{\varepsilon}\left( \nabla\Psi\right) \right) ,%
\end{array}%
\end{equation*}
for $\Psi\in C^{2}\left( M\right) ,$ where $s\in\mathbb{R},$ $p>1,$ $%
\varepsilon>0,$ $f_{\varepsilon}=\sqrt{f^{2}+\varepsilon}$ and%
\begin{equation*}
\begin{array}{l}
A_{\varepsilon}:=\mathrm{id}+\left( p-2\right) \frac{\nabla u\otimes\nabla u%
}{f_{\varepsilon}^{2}}.%
\end{array}%
\end{equation*}
Note that $\mathcal{L}_{s,\varepsilon}$ is a linearized operator of the
nonlinear equation (\ref{1.0}), and $\mathcal{L}_{s,\varepsilon}\left(
f_{\varepsilon}^{2}\right) \left( x\right) $ is well define for all $x\in M$
since $f_{\varepsilon}>0$ and $f_{\varepsilon}^{2}\in C^{2}\left( M\right) .$
\bigskip
Next we use the operator $\mathcal{L}_{s,\varepsilon }$ to derive the
Bochner's formula of strongly $p$-harmonic function, i.e. the Bochner's
formula for the solution of (\ref{pl}). We also note that this $\mathcal{L}%
_{s,\varepsilon }$-Bochner's formula is well defined on all of $x\in M.$
\begin{lemma}[Bochner's formula]
\label{BS}If $u\in C^{3}\left( M\right) $ is a strongly $p$-harmonic
function. Let $f=\left\vert \nabla u\right\vert $ and $f_{\varepsilon}=\sqrt{%
f^{2}+\varepsilon},$ then for all $p>1$ and $s\in\mathbb{R},$ the formula
\begin{equation*}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{s,\varepsilon}\left( f_{\varepsilon}^{2}\right) & =
& \frac{s}{4}f_{\varepsilon}^{s-2}\left\vert \nabla
f_{\varepsilon}^{2}\right\vert
^{2}+f_{\varepsilon}^{s}\sum_{i,j=1}^{m}\left(
u_{ij}^{2}+R_{ij}u_{i}u_{j}\right) \\
& & +\frac{\left( p-2\right) \left( s-p+2\right) }{4}f_{\varepsilon
}^{s-4}\left\langle \nabla u,\nabla f_{\varepsilon}^{2}\right\rangle ^{2} \\
& & +\varepsilon\left( f_{\varepsilon}^{s-2}\left\langle \nabla
u,\nabla\Delta u\right\rangle +\frac{p-4}{2}f_{\varepsilon}^{s-4}\left%
\langle \nabla u,\nabla f_{\varepsilon}^{2}\right\rangle \Delta u\right)%
\end{array}%
\end{equation*}
holds on all of $M.$ In particular, if $p=2,$ then%
\begin{equation*}
\begin{array}{l}
\frac{1}{2}\mathcal{L}_{s,\varepsilon}\left( f_{\varepsilon}^{2}\right) =%
\frac{s}{4}f_{\varepsilon}^{s-2}\left\vert \nabla
f_{\varepsilon}^{2}\right\vert
^{2}+f_{\varepsilon}^{s}\sum_{i,j=1}^{m}\left(
u_{ij}^{2}+R_{ij}u_{i}u_{j}\right)%
\end{array}%
\end{equation*}
holds on all of $M$ and for all $s\in\mathbb{R}.$
\end{lemma}
\proof
By Lemma \ref{PL}, $u$ must be a solution of (\ref{pl}). Taking the gradient
of both sides of (\ref{pl}), and then taking the inner product with $\nabla
u\,,$ we have
\begin{equation}
\begin{array}{lll}
0 & = & \frac{p-2}{2}\left\langle \nabla \left\langle \nabla f^{2},\nabla
u\right\rangle ,\nabla u\right\rangle +\left\langle \nabla f^{2},\nabla
u\right\rangle \Delta u \\
& & +f^{2}\left\langle \nabla \left( \Delta u\right) ,\nabla u\right\rangle
.%
\end{array}
\label{w1}
\end{equation}
Now we rewrite $\mathcal{L}_{s,\varepsilon}\left( f_{\varepsilon}^{2}\right)
$ as the following formula,
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{s,\varepsilon}\left( f_{\varepsilon}^{2}\right) & =
& \frac{1}{2}\text{\textrm{div}}\left( f_{\varepsilon}^{s}\nabla
f_{\varepsilon}^{2}+\left( p-2\right) f_{\varepsilon}^{s-2}\left\langle
\nabla u,\nabla f_{\varepsilon}^{2}\right\rangle \nabla u\right) \\
& = & \frac{s}{4}f_{\varepsilon}^{s-2}\left\vert \nabla
f_{\varepsilon}^{2}\right\vert ^{2}+\frac{1}{2}f_{\varepsilon}^{s}\Delta
f_{\varepsilon}^{2} \\
& & +\frac{\left( p-2\right) \left( s-2\right) }{4}f_{\varepsilon}^{s-4}%
\left\langle \nabla u,\nabla f_{\varepsilon}^{2}\right\rangle ^{2} \\
& & +\frac{p-2}{2}f_{\varepsilon}^{s-2}\left\langle \nabla\left\langle
\nabla u,\nabla f_{\varepsilon}^{2}\right\rangle ,\nabla u\right\rangle \\
& & +\frac{p-2}{2}f_{\varepsilon}^{s-2}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle \Delta u.%
\end{array}
\label{w2}
\end{equation}
Combining (\ref{w1}), one has%
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{s,\varepsilon}\left( f_{\varepsilon}^{2}\right) & =
& \frac{s}{4}f_{\varepsilon}^{s-2}\left\vert \nabla
f_{\varepsilon}^{2}\right\vert ^{2}+\frac{1}{2}f_{\varepsilon}^{s}\Delta
f_{\varepsilon}^{2} \\
& & +\frac{\left( p-2\right) \left( s-2\right) }{4}f_{\varepsilon}^{s-4}%
\left\langle \nabla u,\nabla f_{\varepsilon}^{2}\right\rangle ^{2} \\
& & -f_{\varepsilon}^{s-2}f^{2}\left\langle \nabla\left( \Delta u\right)
,\nabla u\right\rangle \\
& & +\frac{p-4}{2}f_{\varepsilon}^{s-2}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle \Delta u,%
\end{array}
\label{w3}
\end{equation}
here we use the fact $\nabla f_{\varepsilon}^{2}=\nabla f^{2}.$
According to (\ref{pl}), the last term of right hand side can be rewritten as%
\begin{equation*}
\begin{array}{lll}
f_{\varepsilon}^{s-2}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle \Delta u & = & f_{\varepsilon}^{s-4}\left(
f^{2}+\varepsilon\right) \left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle \Delta u \\
& = & f_{\varepsilon}^{s-4}f^{2}\left\langle \nabla u,\nabla f_{\varepsilon
}^{2}\right\rangle \Delta u+\varepsilon f_{\varepsilon}^{s-4}\left\langle
\nabla u,\nabla f_{\varepsilon}^{2}\right\rangle \Delta u \\
& = & -\frac{p-2}{2}f_{\varepsilon}^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle ^{2}+\varepsilon
f_{\varepsilon}^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle \Delta u.%
\end{array}%
\end{equation*}
Using Bochner's formula%
\begin{equation*}
\begin{array}{l}
\frac{1}{2}\Delta f^{2}=\sum_{i,j=1}^{m}u_{ij}^{2}+\left\langle \nabla
u,\nabla\Delta u\right\rangle +\sum_{i,j=1}^{m}R_{ij}u_{i}u_{j}%
\end{array}%
\end{equation*}
and the equality $\Delta f^{2}=\Delta f_{\varepsilon}^{2},$ then (\ref{w3})
gives the desired%
\begin{equation*}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{s,\varepsilon}\left( f_{\varepsilon}^{2}\right) & =
& \frac{s}{4}f_{\varepsilon}^{s-2}\left\vert \nabla
f_{\varepsilon}^{2}\right\vert
^{2}+f_{\varepsilon}^{s}\sum_{i,j=1}^{m}\left(
u_{ij}^{2}+R_{ij}u_{i}u_{j}\right) \\
& & +\frac{\left( p-2\right) \left( s-p+2\right) }{4}f_{\varepsilon
}^{s-4}\left\langle \nabla u,\nabla f_{\varepsilon}^{2}\right\rangle ^{2} \\
& & +\varepsilon\left( f_{\varepsilon}^{s-2}\left\langle \nabla
u,\nabla\Delta u\right\rangle +\frac{p-4}{2}f_{\varepsilon}^{s-4}\left%
\langle \nabla u,\nabla f_{\varepsilon}^{2}\right\rangle \Delta u\right) .%
\end{array}%
\end{equation*}
\endproof
We show a refined Kato's inequality for a$\ $strongly $p$-harmonic function,
generaling the work of \cite{LW1}(or \cite{LW2}) for the case $p=2,$ and
extending the work of \cite{KN} for a function $\log v,$ where $v$ is a
positive $p$-harmonic function.\
\begin{lemma}[Refined Kato's inequality]
\label{KS}Let $u\in C^{2}\left( M\right) $ be a $p$-harmonic function on a
complete manifold $M^{m},$ $p>1$ and $\kappa =\min \left\{ \frac{\left(
p-1\right) ^{2}}{m-1},1\right\} .$ Then at any $x\in M$ with $du\left(
x\right) \neq 0,$
\begin{equation}
\begin{array}{l}
\left\vert \nabla \left( du\right) \right\vert ^{2}\geq \left( 1+\kappa
\right) \left\vert \nabla \left\vert du\right\vert \right\vert ^{2},%
\end{array}
\label{ks}
\end{equation}%
and "$=$" holds if and only if
\begin{equation*}
\left\{
\begin{array}{ll}
u_{\alpha \beta }=0\text{ and }u_{11}=-\frac{m-1}{p-1}u_{\alpha \alpha }, &
\text{for }\left( p-1\right) ^{2}=m-1, \\
u_{\alpha \beta }=0,\text{ }u_{1\alpha }=0\text{ and }u_{11}=-\frac{m-1}{p-1}%
u_{\alpha \alpha }, & \text{for }\left( p-1\right) ^{2}<m-1, \\
u_{\alpha \beta }=0\text{ and }u_{ii}=0,\text{ } & \text{for }\left(
p-1\right) ^{2}>m-1,%
\end{array}%
\right.
\end{equation*}%
for all $\alpha ,\beta =2,\ldots ,m,$ $\alpha \neq \beta $ and $i=1,\ldots
,m.$
\end{lemma}
\proof
Fix a point $x\in M.$ If $du\neq0$ at $x,$ we are able to select a local
orthonormal frame field $\left\{ e_{1},e_{2},\ldots e_{m}\right\} $ such
that, at $x,$ $\nabla_{e_{i}}e_{j}=0,$ $\nabla u=\left\vert \nabla
u\right\vert e_{1},$ and $u_{\alpha}=0$ for all $i,j=1,\ldots,m,$ $%
\alpha=2,\ldots,m.$ Here we use the convenient notation $u_{i}=\left\langle
\nabla u,e_{i}\right\rangle .$
Observing that
\begin{equation}
\begin{array}{lll}
\sum_{i,j=1}^{m}\left( u_{ij}\right) ^{2} & \geq & \left( u_{11}\right)
^{2}+2\sum_{\alpha =2}^{m}\left( u_{1\alpha }\right) ^{2}+\sum_{\alpha
=2}^{m}\left( u_{\alpha \alpha }\right) ^{2} \\
& \geq & \left( u_{11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left( u_{1\alpha
}\right) ^{2}+\frac{\left( \sum_{\alpha =2}^{m}u_{\alpha \alpha }\right) ^{2}%
}{m-1} \\
& = & \left( u_{11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left( u_{1\alpha
}\right) ^{2}+\frac{\left( \Delta u-u_{11}\right) ^{2}}{m-1}.%
\end{array}
\label{ks1}
\end{equation}%
However, let $f=\left\vert \nabla u\right\vert ,$ using%
\begin{equation*}
\begin{array}{lll}
0 & = & div\left( f^{p-2}\nabla u\right) \\
& = & f^{p-2}\Delta u+\left( p-2\right) f^{p-3}\left\langle \nabla f,\nabla
u\right\rangle \\
& = & u_{1}^{p-2}\Delta u+\left( p-2\right) u_{1}^{p-2}f_{1},%
\end{array}%
\end{equation*}%
and%
\begin{equation}
\begin{array}{lll}
f_{j}=\frac{\left( f^{2}\right) _{,j}}{2f}=\frac{\left(
\sum_{i=1}^{m}u_{i}^{2}\right) _{,j}}{2f}=\frac{\sum_{i=1}^{m}u_{i}u_{ij}}{f}
& = & \frac{u_{1}u_{1j}}{f} \\
& = & u_{1j},%
\end{array}
\label{ks2}
\end{equation}%
then we obtain%
\begin{equation}
\begin{array}{l}
\Delta u=-\left( p-2\right) u_{11}.%
\end{array}
\label{ks3}
\end{equation}%
Therefore the inequality (\ref{ks1}) can be written as%
\begin{equation}
\begin{array}{lll}
\sum_{i,j=1}^{m}\left( u_{ij}\right) ^{2} & \geq & \left( u_{11}\right)
^{2}+2\sum_{\alpha =2}^{m}\left( u_{1\alpha }\right) ^{2}+\frac{\left(
p-1\right) ^{2}}{m-1}\left( u_{11}\right) ^{2} \\
& = & \left( 1+\frac{\left( p-1\right) ^{2}}{m-1}\right) \left(
u_{11}\right) ^{2}+2\sum_{\alpha =2}^{m}\left( u_{1\alpha }\right) ^{2} \\
& \geq & \left( 1+\kappa \right) \sum_{j=1}^{m}\left( u_{1j}\right) ^{2} \\
& = & \left( 1+\kappa \right) \left\vert \nabla f\right\vert ^{2}.%
\end{array}
\label{ks4}
\end{equation}%
Then (\ref{ks}) follows.
When "$=$" holds in the inequality (\ref{ks}), then by (\ref{ks1}), we have
\begin{equation*}
\begin{array}{ll}
u_{\alpha\beta}=0 & \text{for all }\alpha\neq\beta\text{ and }\alpha
,\beta=2,\ldots m,%
\end{array}%
\end{equation*}
and%
\begin{equation}
\begin{array}{ll}
u_{\alpha\alpha}=u_{\beta\beta}\text{ \ \ } & \text{for all }\alpha
,\beta=2,\ldots m.%
\end{array}
\label{ks5}
\end{equation}
Using (\ref{ks3}), then (\ref{ks5}) gives%
\begin{equation*}
\begin{array}{ll}
u_{11}=-\frac{m-1}{p-1}u_{\alpha\alpha}\text{ \ \ } & \text{for all }%
\alpha=2,\ldots m.%
\end{array}%
\end{equation*}
Moreover, by (\ref{ks4}),
\begin{itemize}
\item If $\left( p-1\right) ^{2}<m-1,$ then $u_{1\alpha}=0$ for all $%
\alpha=2,\ldots m.$
\item If $\left( p-1\right) ^{2}>m-1,$ then $u_{11}=0,$ i.e. $u_{ii}=0$ for
all $i=1,\ldots m.$
\end{itemize}
Hence we complete the proof. \
\endproof
\bigskip
Next, we show two examples to verify Lemma \ref{KS} is sharp in some case.
\begin{example}
If $u\left( x\right) =\log\left\vert x\right\vert $ in $\mathbb{R}
^{m}\backslash\left\{ 0\right\} ,$ then it is easy to check that $\Delta
_{m}u=0$ for all $m\geq2.$ Since%
\begin{equation*}
\begin{array}{lll}
\left\vert \nabla du\right\vert ^{2}=\sum_{i,j=1}^{m}\left( \frac{\delta
_{ij}}{\left\vert x\right\vert ^{2}}-\frac{2x_{i}x_{j}}{\left\vert
x\right\vert ^{4}}\right) ^{2} & \text{and} & \left\vert \nabla\left\vert
\nabla u\right\vert \right\vert ^{2}=\frac{1}{\left\vert x\right\vert ^{4}},%
\end{array}
\text{ }
\end{equation*}
then we obtain%
\begin{equation*}
\begin{array}{l}
\left\vert \nabla du\right\vert ^{2}=m\left\vert \nabla\left\vert \nabla
u\right\vert \right\vert ^{2}%
\end{array}%
\end{equation*}
for $m\geq2.$ This example implies Lemma \ref{KS} is sharp in the case of $%
p=m=2.$
\end{example}
\begin{example}
Let $u\left( x\right) =\left\vert x\right\vert ^{\frac{p-m}{p-1}}$ in $%
\mathbb{R} ^{m}\backslash\left\{ 0\right\} ,$ $p\neq m,$ then $u$ is a $p$%
-harmonic function. Since%
\begin{equation*}
\begin{array}{l}
\left\vert \nabla du\right\vert ^{2}=\left( \frac{p-m}{p-1}\right)
^{2}\left\vert x\right\vert ^{\frac{2\left( 1-m\right) }{p-1}-2}\sum
_{i,j=1}^{m}\left\{ \delta_{ij}+\left( \frac{1-m}{p-1}-1\right) \frac {%
x_{i}x_{j}}{\left\vert x\right\vert ^{2}}\right\} ^{2}%
\end{array}%
\end{equation*}
and%
\begin{equation*}
\begin{array}{l}
\left\vert \nabla\left\vert \nabla u\right\vert \right\vert ^{2}=\left(
\frac{\left( p-m\right) \left( 1-m\right) }{\left( p-1\right) ^{2}}\right)
^{2}\left\vert x\right\vert ^{\frac{2\left( 1-m\right) }{p-1}-2},%
\end{array}%
\end{equation*}
then we have%
\begin{equation*}
\begin{array}{l}
\left\vert \nabla du\right\vert ^{2}=\left( 1+\frac{\left( p-1\right) ^{2}}{%
m-1}\right) \left\vert \nabla\left\vert \nabla u\right\vert \right\vert ^{2}.%
\end{array}%
\end{equation*}
This example implies Lemma \ref{KS} is sharp in the case of $\left(
p-1\right) ^{2}\leq m-1.$
\end{example}
\bigskip
\subsection{The Proof of Theorem \protect\ref{T2}}
\smallskip
We need several Lemmas:
\begin{lemma}
\label{S-1}Suppose $M^{m}$ is a complete noncompact Riemannian manifold
satisfying $\left( P_{\rho }\right) $ and (\ref{Rs}). Let $u\in C^{3}\left(
M^{m}\right) $ be a strongly $p$-harmonic function, $p>1,$ $p\neq 2.$ Then,
for $0<\varepsilon ,\varepsilon _{1},\varepsilon _{2}<1,$
\begin{equation}
\begin{array}{l}
\int_{B\left( R\right) }A_{2}f_{\varepsilon }^{q-2}dv+\varepsilon B_{0}\leq
\frac{100\cdot B_{1}}{R^{2}}\int_{B\left( 2R\right) \backslash B\left(
R\right) }f_{\varepsilon }^{q}dv,%
\end{array}
\label{vs3-1}
\end{equation}%
for all fixed $R>0,$ where $f=\left\vert \nabla u\right\vert $ and $%
f_{\varepsilon }=\sqrt{f^{2}+\varepsilon },$ $q=s+2,$ $q-1+\kappa
+b>\varepsilon _{1},$ $b=\min \left\{ 0,\left( p-2\right) \left( q-p\right)
\right\} ,$ and
\begin{equation}
\begin{array}{lll}
B_{0} & = & \int_{M}\eta ^{2}\bigg(f_{\varepsilon
}^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}+f_{\varepsilon }^{s-2}\left\langle \nabla
u,\nabla \Delta u\right\rangle \\
& & +\frac{p-4}{2}f_{\varepsilon }^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon }^{2}\right\rangle \Delta u-bf_{\varepsilon }^{s-2}\left\vert
\nabla f_{\varepsilon }\right\vert ^{2}\bigg)dv%
\end{array}
\label{vs5}
\end{equation}%
and
\begin{equation*}
\begin{array}{l}
A_{2}=\frac{4\left( 1-\varepsilon _{2}\right) \left( q-1+\kappa
+b-\varepsilon _{1}\right) }{q^{2}}\rho f_{\varepsilon
}^{2}-\sum_{i,j=1}^{m}R_{ij}u_{i}u_{j}\,%
\end{array}%
\end{equation*}%
and%
\begin{equation*}
\begin{array}{l}
B_{1}=\frac{\left( 1+\left\vert p-2\right\vert \right) ^{2}}{\varepsilon _{1}%
}+\frac{4\left( \frac{1}{\varepsilon _{2}}-1\right) \left( q-1+\kappa
+b-\varepsilon _{1}\right) }{q^{2}}.%
\end{array}%
\end{equation*}
\end{lemma}
\proof
Combining Lemma \ref{KS} and Lemma \ref{BS}, and using the formula%
\begin{equation*}
\begin{array}{l}
f^{2}\left\vert \nabla \left( du\right) \right\vert ^{2}\geq \frac{1+\kappa
}{4}\left\vert \nabla \left\vert du\right\vert ^{2}\right\vert ^{2}%
\end{array}%
\end{equation*}%
holds on all of $M$, we have the following.%
\begin{equation}
\begin{array}{lll}
\frac{1}{2}\mathcal{L}_{s,\varepsilon }\left( f_{\varepsilon }^{2}\right) &
\geq & \left( s+1+\kappa \right) f_{\varepsilon }^{s}\left\vert \nabla
f_{\varepsilon }\right\vert ^{2}+f_{\varepsilon
}^{s}\sum_{i,j=1}^{m}R_{ij}u_{i}u_{j} \\
& & +\frac{\left( p-2\right) \left( s-p+2\right) }{4}f_{\varepsilon
}^{s-4}\left\langle \nabla u,\nabla f_{\varepsilon }^{2}\right\rangle ^{2}
\\
& & +\varepsilon \bigg(f_{\varepsilon }^{s-2}u_{ij}^{2}+f_{\varepsilon
}^{s-2}\left\langle \nabla u,\nabla \Delta u\right\rangle \\
& & +\frac{p-4}{2}f_{\varepsilon }^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon }^{2}\right\rangle \Delta u\bigg).%
\end{array}
\label{bs1}
\end{equation}
We multiply both sides of (\ref{bs1}) by \ a cut off function $\eta^{2}\in
C_{0}^{\infty}(M)$ and integrate over $M,$
\begin{equation}
\begin{array}{lll}
& & \frac{1}{2}\int_{M}\eta^{2}\mathcal{L}_{s,\varepsilon}\left(
f_{\varepsilon}^{2}\right) dv \\
& \geq & \left( s+1+\kappa\right) \int_{M}\eta^{2}f_{\varepsilon}^{s}\left(
\left\vert \nabla f_{\varepsilon}\right\vert
^{2}+\sum_{i,j=1}^{m}R_{ij}u_{i}u_{j}\right) dv \\
& & +\frac{\left( p-2\right) \left( s-p+2\right) }{4}\int_{M}\eta
^{2}f_{\varepsilon}^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle ^{2}dv \\
& & +\varepsilon\int_{M}\eta^{2}\bigg(f_{\varepsilon}^{s-2}%
\sum_{i,j=1}^{m}u_{ij}^{2}+f_{\varepsilon}^{s-2}\left\langle \nabla
u,\nabla\Delta u\right\rangle \\
& & +\frac{p-4}{2}f_{\varepsilon}^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle \Delta u\bigg)dv%
\end{array}
\label{vs1}
\end{equation}
where $\eta$ is a cut-off function on $M$ satisfying%
\begin{equation*}
\left\{
\begin{array}{ll}
\eta\left( x\right) =1 & \text{if }x\in\overline{B\left( R\right) }, \\
0<\eta\left( x\right) <1\text{\ \ \ \ \ } & \text{if }x\in B\left( 2R\right)
\backslash\overline{B\left( R\right) }, \\
\eta\left( x\right) =0 & \text{if }x\in M\backslash B\left( 2R\right) ,%
\end{array}
\right.
\end{equation*}
and%
\begin{equation*}
\left\{
\begin{array}{ll}
\left\vert \nabla\eta\left( x\right) \right\vert =0 & \text{if }x\in B\left(
R\right) \text{ or }x\in M\backslash B\left( 2R\right) , \\
\left\vert \nabla\eta\left( x\right) \right\vert \leq\frac{10}{R}\text{\ \ \
\ \ } & \text{if }x\in B\left( 2R\right) \backslash \overline{B\left(
R\right) },%
\end{array}
\right.
\end{equation*}
Since integration by parts and Cauchy-Schwarz inequality assert that%
\begin{equation*}
\begin{array}{lll}
\frac{1}{2}\int_{M}\eta ^{2}\mathcal{L}_{s,\varepsilon }\left(
f_{\varepsilon }^{2}\right) dv & = & \frac{-1}{2}\int_{M}\left\langle \nabla
\eta ^{2},f_{\varepsilon }^{s}\mathcal{\nabla }f_{\varepsilon }^{2}+\left(
p-2\right) f_{\varepsilon }^{s-2}\left\langle \nabla u,\mathcal{\nabla }%
f_{\varepsilon }^{2}\right\rangle \nabla u\right\rangle dv \\
& \leq & 2\int_{M}\eta \left\vert \nabla \eta \right\vert \left(
f_{\varepsilon }^{s+1}\left\vert \mathcal{\nabla }f_{\varepsilon
}\right\vert +\left( p-2\right) f_{\varepsilon }^{s-1}f^{2}\left\vert
\mathcal{\nabla }f_{\varepsilon }\right\vert \right) dv \\
& \leq & 2\left( 1+\left\vert p-2\right\vert \right) \int_{M}\eta \left\vert
\nabla \eta \right\vert f_{\varepsilon }^{s+1}\left\vert \mathcal{\nabla }%
f_{\varepsilon }\right\vert dv \\
& \leq & \varepsilon _{1}\int_{M}\eta ^{2}f_{\varepsilon }^{s}\left\vert
\mathcal{\nabla }f_{\varepsilon }\right\vert ^{2}dv+\frac{\left(
1+\left\vert p-2\right\vert \right) ^{2}}{\varepsilon _{1}}%
\int_{M}\left\vert \nabla \eta \right\vert ^{2}f_{\varepsilon }^{s+2}dv,%
\end{array}%
\end{equation*}%
where $0<\varepsilon _{1}<1$ is a positive constant such that $q-1+\kappa
+b>\varepsilon _{1}.$
Besides,%
\begin{equation*}
\begin{array}{lll}
\frac{\left( p-2\right) \left( s-p+2\right) }{4}\int_{M}\eta
^{2}f_{\varepsilon}^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon}^{2}\right\rangle ^{2}dv & \geq & \frac{b}{4}%
\int_{M}\eta^{2}f_{\varepsilon }^{s-4}\left\vert \nabla u\right\vert
^{2}\left\vert \nabla f_{\varepsilon }^{2}\right\vert ^{2}dv \\
& = & b\int_{M}\eta^{2}f_{\varepsilon}^{s-2}f^{2}\left\vert \nabla
f_{\varepsilon}\right\vert ^{2}dv \\
& = & b\int_{M}\eta^{2}f_{\varepsilon}^{s}\left\vert \nabla f_{\varepsilon
}\right\vert
^{2}dv-b\varepsilon\int_{M}\eta^{2}f_{\varepsilon}^{s-2}\left\vert \nabla
f_{\varepsilon}\right\vert ^{2}dv%
\end{array}%
\end{equation*}
where%
\begin{equation*}
\begin{array}{l}
b=\min\left\{ 0,\left( p-2\right) \left( s-p+2\right) \right\} .%
\end{array}%
\end{equation*}
Then (\ref{vs1}) implies
\begin{equation}
\begin{array}{lll}
& & A_{1}\int_{M}\eta ^{2}f_{\varepsilon }^{s}\left\vert \nabla
f_{\varepsilon }\right\vert ^{2}dv+\int_{M}\eta ^{2}f_{\varepsilon
}^{s}\sum_{i,j=1}^{m}R_{ij}u_{i}u_{j}dv+\varepsilon B_{0} \\
& \leq & \frac{\left( 1+\left\vert p-2\right\vert \right) ^{2}}{\varepsilon
_{1}}\int_{M}\left\vert \nabla \eta \right\vert ^{2}f_{\varepsilon }^{s+2}dv,%
\end{array}
\label{vs1-2}
\end{equation}%
where $A_{1}=s+1+\kappa +b-\varepsilon _{1}>0,$ since $q=s+2.$
Now we compute the first term in the left hand side of (\ref{vs1-2}),
\begin{equation*}
\begin{array}{lll}
\int_{M}\eta ^{2}f_{\varepsilon }^{s}\left\vert \nabla f_{\varepsilon
}\right\vert ^{2}dv & = & \frac{4}{q^{2}}\int_{M}\eta ^{2}\left\vert \nabla
f_{\varepsilon }^{q/2}\right\vert ^{2}dv \\
& = & \frac{4}{q^{2}}\int_{M}\left\vert \nabla \left( \eta f_{\varepsilon
}^{q/2}\right) -\left( \nabla \eta \right) f_{\varepsilon }^{q/2}\right\vert
^{2}dv \\
& = & \frac{4}{q^{2}}\int_{M}\left\vert \nabla \left( \eta f_{\varepsilon
}^{q/2}\right) \right\vert ^{2}-2\left\langle \nabla \left( \eta
f_{\varepsilon }^{q/2}\right) ,f_{\varepsilon }^{q/2}\nabla \eta
\right\rangle +\left\vert \nabla \eta \right\vert ^{2}f_{\varepsilon }^{q}dv
\\
& \geq & \frac{4\left( 1-\varepsilon _{2}\right) }{q^{2}}\int_{M}\left\vert
\nabla \left( \eta f_{\varepsilon }^{q/2}\right) \right\vert ^{2}+\frac{%
4\left( 1-\frac{1}{\varepsilon _{2}}\right) }{q^{2}}\int_{M^{m}}\left\vert
\nabla \eta \right\vert ^{2}f_{\varepsilon }^{q}dv.%
\end{array}%
\end{equation*}%
where $\varepsilon _{2}$ is a positive constant satisfying $0<\varepsilon
_{2}<1.$ Thus, we have%
\begin{equation}
\begin{array}{lll}
& & \frac{4\left( 1-\varepsilon _{2}\right) A_{1}}{q^{2}}\int_{M}\left\vert
\nabla \left( \eta f_{\varepsilon }^{q/2}\right) \right\vert
^{2}dv+\int_{M}\eta ^{2}f_{\varepsilon
}^{q-2}\sum_{i,j=1}^{m}R_{ij}u_{i}u_{j}\,dv+\varepsilon B_{0} \\
& \leq & \left( \frac{\left( 1+\left\vert p-2\right\vert \right) ^{2}}{%
\varepsilon _{1}}+\frac{4\left( \frac{1}{\varepsilon _{2}}-1\right) A_{1}}{%
q^{2}}\right) \int_{M}\left\vert \nabla \eta \right\vert ^{2}f_{\varepsilon
}^{q}dv.%
\end{array}
\label{vs2}
\end{equation}
According to weighted Poincar\'{e} inequality
\begin{equation*}
\begin{array}{l}
\int_{M}\rho \Psi ^{2}dv\leq \int_{M}\left\vert \nabla \Psi \right\vert
^{2}dv,%
\end{array}%
\end{equation*}%
if we select $\Psi =\eta f^{q/2},$ then (\ref{vs2}) implies%
\begin{equation*}
\begin{array}{l}
\int_{B\left( R\right) }A_{2}f_{\varepsilon }^{q-2}dv+\varepsilon B_{0}\leq
\frac{100\cdot B_{1}}{R^{2}}\int_{B\left( 2R\right) \backslash B\left(
R\right) }f_{\varepsilon }^{q}dv,%
\end{array}%
\end{equation*}%
for all fixed $R>0.$\ \endproof
\bigskip
\begin{lemma}
\label{S}Let $B_{0}$ be the formula (\ref{vs5}), $p>1,$ $p\neq 2,$ $q=s+2$
and
\begin{equation*}
\begin{array}{l}
b=\min \left\{ 0,\left( p-2\right) \left( q-p\right) \right\} .%
\end{array}%
\end{equation*}%
Then \newline
(i) if $q>2,$ then $\varepsilon B_{0}\rightarrow 0$ as $\varepsilon
\rightarrow 0,$ \newline
(ii) if $1<q\leq 2$ and $b\leq -\frac{\left( p-4\right) ^{2}m}{4},$ then $%
\varepsilon B_{0}\geq 0$ as $\varepsilon \rightarrow 0.$ \newline
\end{lemma}
\proof
First of all, we compute some properties.
For $s\geq 2,$ it is easy to check that
\begin{equation}
\begin{array}{l}
\varepsilon f_{\varepsilon }^{s-2}\rightarrow 0\text{ as }\varepsilon
\rightarrow 0.%
\end{array}
\label{s00}
\end{equation}
If $0<s<2,$ then we also have%
\begin{equation}
\begin{array}{l}
\varepsilon f_{\varepsilon }^{s-2}=\frac{\varepsilon }{f_{\varepsilon }^{2-s}%
}\leq \frac{\varepsilon }{\varepsilon ^{1-s/2}}=\varepsilon
^{s/2}\rightarrow 0\text{ as }\varepsilon \rightarrow 0,%
\end{array}
\label{s0}
\end{equation}
By using the estimates%
\begin{equation*}
\begin{array}{lll}
\varepsilon f_{\varepsilon }^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon }^{2}\right\rangle & = & 2\varepsilon f_{\varepsilon
}^{s-4}\sum_{i,j=1}^{m}u_{ij}u_{i}u_{j} \\
& \leq & 2\varepsilon f_{\varepsilon }^{s-4}\sup_{i,j=1,\cdots ,m}\left\vert
u_{ij}\right\vert \sum_{i,j=1}^{m}u_{i}u_{j} \\
& \leq & m\varepsilon f_{\varepsilon }^{s-4}f^{2}\sup_{i,j=1,\cdots
,m}\left\vert u_{ij}\right\vert \\
& \leq & m\varepsilon f_{\varepsilon }^{s-2}\sup_{i,j=1,\cdots ,m}\left\vert
u_{ij}\right\vert \text{ \ \ }%
\end{array}%
\end{equation*}%
and%
\begin{equation*}
\begin{array}{lll}
\varepsilon f_{\varepsilon }^{s-2}\left\vert \nabla f_{\varepsilon
}\right\vert ^{2} & = & \frac{\varepsilon }{4}f_{\varepsilon
}^{s-4}\left\vert \nabla f_{\varepsilon }^{2}\right\vert ^{2} \\
& = & \varepsilon f_{\varepsilon
}^{s-4}\sum_{i,j,k=1}^{m}u_{ik}u_{kj}u_{i}u_{j} \\
& \leq & m\varepsilon f_{\varepsilon }^{s-4}f^{2}\sup_{i,j,k=1,\cdots
,m}\left\vert u_{ik}\right\vert \left\vert u_{kj}\right\vert \\
& \leq & m\varepsilon f_{\varepsilon }^{s-2}\sup_{i,j,k=1,\cdots
,m}\left\vert u_{ik}\right\vert \left\vert u_{kj}\right\vert%
\end{array}%
\end{equation*}%
then (\ref{s00}) and (\ref{s0}) imply
\begin{equation}
\left\{
\begin{array}{l}
\varepsilon f_{\varepsilon }^{s-4}\left\vert \left\langle \nabla u,\nabla
f_{\varepsilon }^{2}\right\rangle \right\vert \rightarrow 0, \\
\varepsilon f_{\varepsilon }^{s-2}\left\vert \nabla f_{\varepsilon
}\right\vert ^{2}\rightarrow 0,%
\end{array}%
\right. \label{s1}
\end{equation}%
as $\varepsilon \rightarrow 0,$ for all $s>0.$
In the case $-1<s\leq 0,$%
\begin{equation}
\begin{array}{l}
\varepsilon ff_{\varepsilon }^{s-2}\leq \frac{\varepsilon }{\left(
f^{2}+\varepsilon \right) ^{1/2-s/2}}\leq \varepsilon ^{1/2+s/2}\rightarrow 0%
\text{ as }\varepsilon \rightarrow 0.%
\end{array}
\label{s3}
\end{equation}
Now we prove Lemma as follows.
For any fixed $s>0,$ by (\ref{s00}), (\ref{s0}) and (\ref{s1}), then we
obtain,
\begin{equation*}
\begin{array}{lll}
\varepsilon B_{0} & = & \varepsilon \int_{M}\eta ^{2}\bigg(f_{\varepsilon
}^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}+f_{\varepsilon }^{s-2}\left\langle \nabla
u,\nabla \Delta u\right\rangle \\
& & +\frac{p-4}{2}f_{\varepsilon }^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon }^{2}\right\rangle \Delta u-bf_{\varepsilon }^{s-2}\left\vert
\nabla f_{\varepsilon }\right\vert ^{2}\bigg)dv \\
& \geq & \int_{M}\eta ^{2}\bigg(\left( \varepsilon f_{\varepsilon
}^{s-2}\right) \sum_{i,j=1}^{m}u_{ij}^{2}+\left( \varepsilon f_{\varepsilon
}^{s-2}\right) \left\vert \left\langle \nabla u,\nabla \Delta u\right\rangle
\right\vert \\
& & +\frac{p-4}{2}\left( \varepsilon f_{\varepsilon }^{s-4}\left\vert
\left\langle \nabla u,\nabla f_{\varepsilon }^{2}\right\rangle \right\vert
\right) \left\vert \Delta u\right\vert -b\left( \varepsilon f_{\varepsilon
}^{s-2}\left\vert \nabla f_{\varepsilon }\right\vert ^{2}\right) \bigg)dv \\
& \rightarrow & 0,\text{ whenever }\varepsilon \rightarrow 0.%
\end{array}%
\end{equation*}
If $s>-1,$ since $b\leq -\frac{\left( p-4\right) ^{2}m}{4},$ then%
\begin{equation*}
\begin{array}{lll}
& & f_{\varepsilon }^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}-bf_{\varepsilon
}^{s-2}\left\vert \nabla f_{\varepsilon }\right\vert ^{2}+\frac{p-4}{2}%
f_{\varepsilon }^{s-4}\left\langle \nabla u,\nabla f_{\varepsilon
}^{2}\right\rangle \Delta u \\
& \geq & f_{\varepsilon }^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}-bf_{\varepsilon
}^{s-2}\left\vert \nabla f_{\varepsilon }\right\vert ^{2}-\left\vert
p-4\right\vert f_{\varepsilon }^{s-2}\left\vert \nabla f_{\varepsilon
}\right\vert \left\vert \Delta u\right\vert \\
& \geq & f_{\varepsilon }^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}-bf_{\varepsilon
}^{s-2}\left\vert \nabla f_{\varepsilon }\right\vert ^{2}-\frac{%
f_{\varepsilon }^{s-2}\left( \Delta u\right) ^{2}}{m}-\frac{\left(
p-4\right) ^{2}m}{4}f_{\varepsilon }^{s-2}\left\vert \nabla f_{\varepsilon
}\right\vert ^{2} \\
& \geq & 0,%
\end{array}%
\end{equation*}%
here we use $\sum_{i,j=1}^{m}u_{ij}^{2}\geq \frac{\left( \Delta u\right) ^{2}%
}{m}.$ Hence by (\ref{s00}), (\ref{s0}) and (\ref{s3}),%
\begin{equation*}
\begin{array}{lll}
\varepsilon B_{0} & = & \varepsilon \int_{M}\eta ^{2}\bigg(f_{\varepsilon
}^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}+f_{\varepsilon }^{s-2}\left\langle \nabla
u,\nabla \Delta u\right\rangle \\
& & +\frac{p-4}{2}f_{\varepsilon }^{s-4}\left\langle \nabla u,\nabla
f_{\varepsilon }^{2}\right\rangle \Delta u-bf_{\varepsilon }^{s-2}\left\vert
\nabla f_{\varepsilon }\right\vert ^{2}\bigg)dv \\
& \geq & \varepsilon \int_{M}\eta ^{2}f_{\varepsilon }^{s-2}\left\langle
\nabla u,\nabla \Delta u\right\rangle \\
& \geq & -\int_{M}\eta ^{2}\left( \varepsilon f_{\varepsilon }^{s-2}f\right)
\left\vert \nabla \Delta u\right\vert \\
& \rightarrow & 0\text{ whenever }s>-1\text{ and }\varepsilon \rightarrow 0.%
\end{array}%
\end{equation*}
In particular, if $s>-1$ and $p=4,$ by applying (\ref{s00}), (\ref{s0}), (%
\ref{s3}) and $b\leq 0$, then%
\begin{equation*}
\begin{array}{lll}
\varepsilon B_{0} & = & \varepsilon \int_{M}\eta ^{2}\bigg(f_{\varepsilon
}^{s-2}\sum_{i,j=1}^{m}u_{ij}^{2}+f_{\varepsilon }^{s-2}\left\langle \nabla
u,\nabla \Delta u\right\rangle -bf_{\varepsilon }^{s-2}\left\vert \nabla
f_{\varepsilon }\right\vert ^{2}\bigg)dv \\
& \geq & -\int_{M}\eta ^{2}\left( \varepsilon f_{\varepsilon }^{s-2}f\right)
\left\vert \nabla \Delta u\right\vert dv \\
& \rightarrow & 0\text{ whenever }s>-1\text{ and }\varepsilon \rightarrow 0.%
\end{array}%
\end{equation*}%
\endproof
\bigskip
\begin{remark}
In Lemma \ref{S}, if $p=4$ and $q>1,$ then $\varepsilon B_{0}\geq 0$ as $%
\varepsilon \rightarrow 0.$
\end{remark}
\textbf{Proof Theorem \ref{T2}:}
\proof
For the case of $p=2,$ the proof is similar to the case $p\neq2$ (or see
\cite{LW1}), so we assume $p\neq2.$
Since we assume $q-1+\kappa +b>0,$ the curvature condition (\ref{Rs}) means
that there exists a constant $0<\delta <1$ such that
\begin{equation*}
\begin{array}{lll}
Ric_{M} & \geq & -\frac{4\left( q-1+\kappa +b\right) }{q^{2}}\delta \rho .%
\end{array}%
\end{equation*}%
Combine Lemma \ref{S-1} and Lemma \ref{S}, under the following condition:
\begin{equation*}
\left\{
\begin{array}{l}
\text{if }1<q\leq 2\text{ and }q-1+\kappa +b>0\text{ and }b\leq -\frac{%
\left( p-4\right) ^{2}m}{4},\text{ or} \\
\text{if }q>2\text{ and }q-1+\kappa +b>0.%
\end{array}%
\right.
\end{equation*}%
For $p>2,$ it can be rewrite as
\begin{equation*}
\begin{array}{l}
\max \left\{ 1,p-1-\frac{\kappa }{p-1}\right\} <q\leq \min \left\{ 2,p-\frac{%
\left( p-4\right) ^{2}m}{4\left( p-2\right) }\right\} \text{ } \\
\text{or }\max \left\{ 2,1-\kappa -b\right\} <q%
\end{array}%
\end{equation*}%
For $1<p<2,$ since $q-1+\kappa +b>0$ for all $q>1,$ then we may rewrite the
condition as follows:%
\begin{equation*}
\left\{
\begin{array}{l}
p<q\leq 2\text{ and }b\leq -\frac{\left( p-4\right) ^{2}m}{4},\text{ or} \\
q>2.%
\end{array}%
\right.
\end{equation*}%
But $b\leq -\frac{\left( p-4\right) ^{2}m}{4}$ is not valid for each $%
1<p<q\leq 2.$ Hence in the case $1<p<2,$ we just need the condition%
\begin{equation*}
\begin{array}{l}
q>2.%
\end{array}%
\end{equation*}
Now (\ref{vs3-1}) can be rewritten as
\begin{equation}
\begin{array}{lll}
\int_{B\left( R\right) }A_{3}f^{q}dv & \leq & \frac{100\cdot B_{1}}{R^{2}}%
\int_{B\left( 2R\right) \backslash B\left( R\right) }f^{q}dv,%
\end{array}
\label{vs3}
\end{equation}%
where%
\begin{equation*}
\begin{array}{lll}
A_{3} & = & \left( \frac{4\left( 1-\varepsilon _{2}\right) \left( q-1+\kappa
+b-\varepsilon _{1}\right) }{q^{2}}-\frac{4\left( q-1+\kappa +b\right)
\delta }{q^{2}}\right) \rho .%
\end{array}%
\end{equation*}%
\newline
Hence one has $A_{3}>0$ whenever we select $\varepsilon _{1}$ and $%
\varepsilon _{2}$ small enough. Suppose $f\in L^{q}\left( M\right) $, then
the right hand side of (\ref{vs3}) tends to zero as $R\rightarrow \infty $,
and then we conclude that $f(x)$ equals to zero for all $x\in M$ and for all
$0<\delta <1,$ i.e. $u\left( x\right) $ is a constant on $M$ for all $%
0<\delta <1$.
In particular, if $1<p<2,$ since constant function is not a strongly $p$%
-harmonic function, then such $u$ does not exist. \endproof
\begin{remark}
If $p>2$ and $p\geq q,$ then
\begin{equation*}
\begin{array}{lll}
q-1+\kappa +b & = & q-1+\kappa +\left( p-2\right) \left( q-p\right) \\
& = & \left( p-1\right) q-\left( p-1\right) ^{2}+\kappa >0,%
\end{array}%
\end{equation*}%
whenever $q>p-1-\frac{\kappa }{p-1}.$
\end{remark}
\begin{remark}
If we replace the finite $q$-energy by $\int_{B\left( 2R\right) \backslash
B\left( R\right) }\left\vert \nabla u\right\vert ^{q}dv=o\left( R^{2}\right)
$ as $R\rightarrow \infty ,$ then Theorem \ref{T2} is still valid.
\end{remark}
\begin{remark}
Since $\left( P_{\lambda _{q}}\right) $ implies $\left( P_{\lambda
_{p}}\right) $ for all $p>q$ (cf. \cite{HKM}). If $M$ satisfies $\left(
P_{\lambda _{2}}\right) ,$ by using Lemma \ref{Volume estimate}, then $2$%
-hyperbolic end is equality to $p$-hyperbolic end since this end have
infinite volume. Hence we may use the method of Theorem 2.1 of \cite{LW1} to
refine the conditions of Theorem \ref{T2} whenever $M$ satisfies $\left(
P_{\lambda _{2}}\right) .$ But we omit it in this paper.
\end{remark}
\begin{corollary}
Let $M^{m}$ be a complete noncompact Riemannian manifold satisfying $\left(
P_{\rho }\right) $ and (\ref{Rs}), where%
\begin{equation*}
\begin{array}{lll}
\tau & < & \frac{4\left( \left( p-1\right) q-\left( p-1\right) ^{2}+\kappa
\right) }{q^{2}},%
\end{array}%
\end{equation*}%
$\kappa =\min \{\frac{\left( p-1\right) ^{2}}{m-1},1\},$ $p>2,$ $p\geq q.$
Let $u\in C^{3}\left( M^{m}\right) $ be a strongly $p$-harmonic function,
with finite $q$-energy $E_{q}\left( u\right) .$ Then $u$ is a constant if $p$
and $q$ satisfy one of the following: \newline
(1) $p=4,$ $q>\frac{9-\kappa }{3},$ \newline
(2) $p\neq 4,$ and either%
\begin{equation*}
\begin{array}{l}
\max \left\{ 1,p-1-\frac{\kappa }{p-1}\right\} <q\leq \min \left\{ 2,p-\frac{%
\left( p-4\right) ^{2}m}{4\left( p-2\right) }\right\}
\end{array}%
\end{equation*}%
or
\begin{equation*}
\begin{array}{l}
q>\max \{p-1-\frac{\kappa }{p-1},2\}.%
\end{array}%
\end{equation*}
\end{corollary}
\begin{corollary}
Let $M^{m}$ be a complete noncompact Riemannian manifold satisfying $\left(
P_{\rho}\right) $ and (\ref{Rs}), where%
\begin{equation*}
\begin{array}{lll}
\tau & < & \frac{4\left( p-1+\kappa\right) }{p^{2}},%
\end{array}%
\end{equation*}
$\kappa=\min\{\frac{\left( p-1\right) ^{2}}{m-1},1\}.$ If $u\in C^{3}\left(
M^{m}\right) $ is a strongly $p$-harmonic function for $p\geq2,$ with $%
E_{p}\left( u\right) <\infty,$ then $u$ is a constant.
\end{corollary}
\begin{remark}
According the following Lemma \ref{ws}, we can replace "Let $u\in
C^{3}\left( M^{m}\right) $ be a strongly $p$-harmonic function" in Theorem %
\ref{T2} by "Let $u\in C^{3}\left( M^{m}\right) $ be a weakly $p$-harmonic
function for $p\in\left\{ 2\right\} \cup\lbrack4,\infty),$ and is strongly $%
p $-harmonic for $p\in\left( 1,2\right) \cup\left( 2,4\right) $".
\end{remark}
\begin{lemma}
\label{ws}If $u\in C^{2}\left( M\right) \,($resp. $u\in C^{0}\left( M\right)
\,)$ is a weakly $p$-harmonic function for $p\in\lbrack4,\infty )\,($resp. $%
p=2\,)\,,$ then $u$ is a strongly $p$-harmonic function.
\end{lemma}
\proof
By assumption, $u$ satisfies
\begin{equation*}
\begin{array}{l}
\int_{M}\left\langle f^{p-2}\nabla u,\nabla\eta\right\rangle \,dv=0%
\end{array}%
\end{equation*}
for every $\eta\in C_{0}^{\infty}\left( M\right) ,$ where $f=\left\vert
\nabla u\right\vert .$ Since $u\in C^{2}\left( M\right) \,,$ and either $%
p=2\,,$ or $p\geq4,$ we have $f^{p-2}\in C^{1}\left( M\right) .$ Hence $%
f^{p-2}\nabla u\in C^{1}\left( M\right) ,$ and the divergence theorem implies%
\begin{equation*}
\begin{array}{l}
0=\int_{M}\left\langle f^{p-2}\nabla u,\nabla\eta\right\rangle \,dv=-\int
_{M}\text{div}\left( f^{p-2}\nabla u\right) \,\eta\,dv%
\end{array}%
\end{equation*}
for every $\eta\in C_{0}^{\infty}\left( M\right) .$ This completes the
proof. \
\endproof%
\subsection{Application to $p$-harmonic morphism\label{morphism}}
A $C^{2}$ map $u:M\rightarrow N$ is called a $p$-harmonic morphism if for
any $p$-harmonic function $f$ defined on an open set $V$ of $N$, the
composition $f\circ u$ is $p$-harmonic on $u^{-1}(V)$. Examples of $p$%
-harmonic morphisms include the Hopf fibrations. E. Loubeau and J. M. Burel (%
\cite{BL}) and E. Loubeau(\cite{Lo}) prove that a $C^{2}$ map $%
u:M\rightarrow N$ is a $p$-harmonic morphism with $p\in(1,\infty)$ if and
only if $u$ is a $p$-harmonic and horizontally weak conformal map.
\begin{theorem}
\label{T:13} Let $M^{m}$ be a complete noncompact Riemannian manifold,
satisfying $\left( P_{\rho }\right) $ and (\ref{Rs}), where $\tau <\frac{%
4\left( q-1+\kappa +b\right) }{q^{2}},$ $\kappa =\min \{\frac{\left(
p-1\right) ^{2}}{m-1},1\}$ and $b=\min \{0,(p-2)(q-p)\}.$ If $u\in
C^{3}\left( M^{m}\right) $ is a $p$-harmonic morphism $\,u:M^{m}\rightarrow
\mathbb{R}^{k},$ $k>0$ of finite $q$-energy $E_{q}\left( u\right) <\infty .$
\newline
(I). Then $u$ is constant under one of the following: \newline
(1) $p=2$ and $q>\frac{m-2}{m-1},$ \newline
(2) $p=4,$ $q>1$ and $q-1+\kappa +b>0,$ \newline
(3) $p>2,$ $p\neq 4,$ and either%
\begin{equation*}
\begin{array}{l}
\max \left\{ 1,p-1-\frac{\kappa }{p-1}\right\} <q\leq \min \left\{ 2,p-\frac{%
\left( p-4\right) ^{2}m}{4\left( p-2\right) }\right\}%
\end{array}%
\end{equation*}%
or%
\begin{equation*}
\begin{array}{l}
\max \left\{ 2,1-\kappa -b\right\} <q.%
\end{array}%
\end{equation*}%
\newline
(II). Then $u$ does not exit under \newline
(4) $1<p<2,$ $q>2.$
\end{theorem}
\begin{lemma}
\cite{WLW}Let $M,N$ and $K$ be manifolds of dimension $m,$ $n,$ and $k$
respectively, and $u:M\rightarrow N$, and $w:N\rightarrow K$ be $C^{2}$. If $%
u$ is horizontally weak conformal, then $|d(w\circ u)|^{p-2}=(\frac{1}{n})^{%
\frac{p-2}{2}}|dw|^{p-2}|du|^{p-2}.$\label{L:14}
\end{lemma}
\textbf{Proof of Theorem \ref{T:13}.} Let $u^{i}=\pi_{i}\circ u\,,$ where $%
\pi_{i}:\mathbb{R}^{k}\rightarrow\mathbb{R}$ is the $i$-th projection. Then
the linear function $\pi_{i}$ is a $p$-harmonic function (cf. 2.2 in \cite{W}
). Hence $u^{i}\,,$ a composition of a $p$-harmonic morphism and a $p$%
-harmonic function is $p$-harmonic. Since $u$ is horizontally weak
conformal, it follows from Lemma \ref{L:14} that $E_{p}(u)<\infty$ implies $%
E_{p}(u^{i})<\infty\,.$ Now apply $u^{i}$ to Theorem \ref{T2}, the assertion
follows.
\endproof%
These results are in contrast to the following:
\begin{theorem}
\noindent\cite{WLW} If $u:M^{m}\rightarrow\mathbb{R}^{k},$ $k>0,$ is a $p$%
-harmonic morphism, and if there exists $i$, such that $u^{i}=\pi_{i}\circ
u\,$ is $p$-finite, i.e.%
\begin{equation*}
\begin{array}{l}
\liminf_{r\rightarrow\infty}\frac{1}{r^{p}}\int_{B\left( r\right)
}\left\vert u^{i}\right\vert ^{q}dv<\infty%
\end{array}%
\end{equation*}
where $B\left( r\right) $ is a geodesic ball of radius $r$, for some $q>p-1.$
Then $u$ must be constant.
\end{theorem}
As further applications, one obtains
\begin{theorem}
\label{T:14} Let $M^{m}$ be a complete noncompact Riemannian manifold,
satisfying $\left( P_{\rho }\right) $ and (\ref{Rs}), where $\tau <\frac{%
4\left( q-1+\kappa +b\right) }{q^{2}},$ $\kappa =\min \{\frac{\left(
p-1\right) ^{2}}{m-1},1\}$ and $b=\min \{0,(p-2)(q-p)\}.$ Let $u\in
C^{3}\left( M^{m}\right) $ be a $p$-harmonic morphism $\,u:M^{m}\rightarrow
\mathbb{R}^{k},$ $k>0,$ and $f:u\left( M^{m}\right) \subset
\mathbb{R}
^{k}\rightarrow
\mathbb{R}
$ be a nonconstant $p$-harmonic function. Assume $f\circ u$ has finite $q$
energy $E_{q}\left( f\circ u\right) <0.$ \newline
(I). Then $u$ is constant under one of the following: \newline
(1) $p=2$ and $q>\frac{m-2}{m-1},$ \newline
(2) $p=4,$ $q>1$ and $q-1+\kappa +b>0,$ \newline
(3) $p>2,$ $p\neq 4,$ and either%
\begin{equation*}
\begin{array}{l}
\max \left\{ 1,p-1-\frac{\kappa }{p-1}\right\} <q\leq \min \left\{ 2,p-\frac{%
\left( p-4\right) ^{2}m}{4\left( p-2\right) }\right\}%
\end{array}%
\end{equation*}%
or%
\begin{equation*}
\begin{array}{l}
\max \left\{ 2,1-\kappa -b\right\} <q.%
\end{array}%
\end{equation*}%
\newline
(II). Then $u$ does not exit under \newline
(4) $1<p<2,$ $q>2.$
\end{theorem}
\begin{lemma}
\label{open}A nonconstant $p$-harmonic morphism $u:M^{m}\rightarrow \mathbb{R%
}^{k}$ is an open map.
\end{lemma}
\textbf{Proof of Theorem \ref{T:14}.} Since $u$ is a $p$-harmonic morphism,
then $f\circ u$ is a $p$-harmonic function on $M^{m}.$ According to Theorem %
\ref{T2}, then $f\circ u$ is a constant $c$. On the other hand, due to Lemma %
\ref{open}, $u$ and $f$ are open maps whenever they are not constant. Now we
assume that $u$ is not constant, then the image of $u$ is an open set $%
u\left( M\right) \subset%
\mathbb{R}
^{k}$. Hence $f\circ u\left( M^{m}\right) $ is an open set. This gives a
contradiction to $f\circ u\left( M^{m}\right) =c.$ Then we conclude that $u$
is a constant.
\endproof%
\begin{theorem}
\noindent(Picard Theorem for $p$-harmonic morphisms). Let $M^{m}$ be as in
Theorem \ref{T:14}. Suppose that $u\in C^{3}\left( M^{m}\right) $ is a $p$%
-harmonic morphism $\,u:M^{m}\rightarrow R^{k}\backslash\{y_{0}\},$ and the
function $x\mapsto|u(x)-y_{0}|^{\frac{p-m}{p-1}}$ has finite $q$-energy
where $p\neq m$, for $p$ and $q$ satisfying one of the following: $(1)$, $%
(2) $, and $(3)$ as in Theorem \ref{T:14}. Then $u$ is constant. For $p$ and
$q$ satisfying $(4)$ as in Theorem \ref{T:14}, then $u$ does not exist.%
\textit{\ }\smallskip
\end{theorem}
\proof
Since $|x|^{\frac{p-n}{n-1}}$ is a $p$-harmonic map. Then $|u(x)-y_{0}|^{%
\frac{p-n}{n-1}}:M\rightarrow\mathbb{R}$ is a $p$-harmonic function with
finite $q$-energy. By Theorem \ref{T:14}, when $p\neq n$, we obtain the
conclusion.
\endproof%
\subsection{Application to Conformal Maps\label{Maps}}
Our previous result can be applied to weakly conformal maps between equal
dimensional manifolds based on the following:\bigskip
\noindent\textbf{Theorem A} (\cite{OW}) $u:M\rightarrow N$ is an $m$%
-harmonic morphism, if and only if $u$ is weakly conformal, where $m=\dim
M=\dim N\,.$ \bigskip
\noindent For instance, stereographic projections $u:\mathbb{R}%
^{m}\rightarrow S^{m}$ are $m$-harmonic maps and $m$-harmonic morphisms, for
all $m \ge1\, .$
\begin{theorem}
Let $M^{m}$ be a complete noncompact $m$-manifold satisfying $\left( P_{\rho
}\right) $ and (\ref{Rs}), where $\tau <\frac{4\left( q+b\right) }{q^{2}}$
and $b=\min \{0,(m-2)(q-m)\}.$ If $u:M^{m}\rightarrow \mathbb{R}^{m}$ is a
weakly conformal map of finite $q$-energy $E_{q}\left( u\right) <\infty .$
Then $u$ is a constant if $m$ and $q$ satisfy one of the following: \newline
(1) $m=2$ and $q>0,$ \newline
(2) $m=4,$ $q>1$ and $q+b>0,$ \newline
(3) $m>2,$ $m\neq 4,$ and either $\frac{m\left( m-2\right) }{m-1}<q\leq \min
\left\{ 2,m-\frac{\left( m-4\right) ^{2}m}{4\left( m-2\right) }\right\} $ or%
\textrm{\ }$q>\max \{2,-b\}.$ \newline
\end{theorem}
\proof
By Theorem A (\cite{OW}), $u$ is an $m$-harmonic morphism. Now the result
follows immediately from Theorem \ref{T:13} in which $p=m.$ Since $\log|x|$
is an $n$-harmonic function. Then $\log|u(x)-y_{0}|:M\rightarrow\mathbb{R}$
is an $n$-harmonic function with finite $q$-energy. By Theorem \ref{T:14},
when $p=n$, we obtain the conclusion.
\endproof%
\bigskip
\section{Appendix}
\subsection{The existence of the approximate solution}
In this subsection, we study an approximate solution $u_{\epsilon}$ of the $%
p $-Laplace equation or a solution $u_{\epsilon}$ of a perturbed $p$-Laplace
equation
\begin{equation}
\begin{array}{lllll}
\Delta_{p,\epsilon}u_{\epsilon} & = & \text{\textrm{div}}\left( \left(
\left\vert \nabla u_{\epsilon}\right\vert ^{2}+\epsilon\right) ^{\frac {p-2}{%
2}}\nabla u_{\epsilon}\right) & = & 0%
\end{array}
\label{a1.2}
\end{equation}
on a domain $\Omega\subset M$ with boundary condition $u_{\epsilon}=u$ on $%
\partial\Omega.$ That is, $u_{\epsilon}$ is the Euler-Lagrange equation of
the $\left( p,\epsilon\right) $-energy $E_{p,\epsilon}$ functional given by
\begin{equation}
\begin{array}{l}
E_{p,\epsilon}(\Psi)=\int_{\Omega}\left( |\nabla\Psi|^{2}+\epsilon\right)
^{p/2}\,dv%
\end{array}
\, \label{a1.3}
\end{equation}
with $\Psi\in W^{1,p}\left( \Omega\right) ,$ and $\Psi=u$ on $\partial
\Omega.$
\begin{proposition}[ The existence of $u_{\protect\epsilon }$]
\label{ex}Let $u$ be a $W^{1,p}$ function on a domain $\bar{\Omega}\subset M$%
. Then there is a solution $u_{\epsilon }\in W^{1,p}\left( \Omega \right) $
of the Euler-Lagrange equation of the $\left( p,\epsilon \right) $-energy $%
E_{p,\epsilon }$ with $u_{\epsilon }=u$ on the boundary of $\Omega $ in the
trace sense.
\end{proposition}
\proof
Let $H$ be the set of functions $v\in W^{1,p}\left( \Omega\right) $ such
that $v=u$ on the boundary of $\Omega$ in the trace sense, and $I=\inf
\{E_{p,\epsilon}(v):v\in H\}.$ Then $u\in H$, $H$ is nonempty, and $I$
exists. Furthermore $I\leq E_{p,\epsilon}(u).$
Take a minimizing sequence $\{v_{i}\}_{i=1}^{\infty}$ such that $%
E_{p,\epsilon }(v_{i})$ tends to $I$ as $i$ tends to $\infty$.
Then $\{v_{i}\}_{i=1}^{\infty }$ is a bounded sequence in $W^{1,p}\left(
\Omega \right) $. Hence there exists a subsequence, say $\{u_{i}\}_{i=1}^{%
\infty }\,,$ converges weakly to $u_{\epsilon }$ in $W^{1,p}\left( \Omega
\right) $, strongly in $L^{p}\left( \Omega \right) $, and pointwise almost
everywhere. We infer $u_{\epsilon }$ is in $H$ since $H$ is closed. Thus $%
I\leq E_{p,\epsilon }\left( u_{\epsilon }\right) .$ \smallskip
To prove $I\geq E_{p,\epsilon }\left( u_{\epsilon }\right) \,,$ it suffices
to prove the lower semi-continuity of $E_{p,\epsilon }$ (two methods).
\textbf{Method 1}:
If $\dim M>2\,,$ we let $\nu_{i}(x)$ be a unit vector perpendicular to a
tangent plane containing $\nabla u_{i}$ and $\nabla u_{\epsilon}$ (in the
tangent space $T_{x}\left( M\right) $ to $M$ at $x$), for a.e. $x\in
\Omega\,.$ If $\dim M=2\,,$ we isometrically embed $M$ in a Euclidean space $%
\mathbb{R}^{k}$ for sufficiently large dimension $k$ to obtain such $\nu
_{i}(x)\,\in\mathbb{R}^{k}\,.$
We set $b=\nabla u_{i}(x)+\sqrt{\epsilon}{\nu}_{i}(x)$ and $a=\nabla
u_{\epsilon}(x)+\sqrt{\epsilon}\nu_{i}(x)$. Then on $\Omega\,,$ $|b|=\sqrt {%
|\nabla u_{i}|^{2}+\epsilon}$ and $|a|=\sqrt{|\nabla
u_{\epsilon}|^{2}+\epsilon}.$
If $m=2$, we use the dot product in $\mathbb{R}^{k}\,.$ \smallskip
Applying the inequality
\begin{equation*}
|b|^{p}\geq|a|^{p}+p\langle|a|^{p-2}a,b-a\rangle_{\mathbb{R}^{k}}
\end{equation*}
and integrating over $\Omega$, we have for $m=2$
\begin{equation*}
\begin{array}{lll}
E_{p,\epsilon }(u_{i}) & \geq & E_{p,\epsilon }(u_{\epsilon })+\int_{\Omega
}\langle (|\nabla u_{\epsilon }|^{2}+\epsilon )^{\frac{p-2}{2}}(\nabla
u_{\epsilon }+\sqrt{\epsilon }\nu _{i}),\nabla u_{i}-\nabla u_{\epsilon
}\rangle _{\mathbb{R}^{k}}dv \\
& = & E_{p,\epsilon }(u_{\epsilon })+\int_{\Omega }\langle (|\nabla
u_{\epsilon }|^{2}+\epsilon )^{\frac{p-2}{2}}\nabla u_{\epsilon },\nabla
u_{i}-\nabla u_{\epsilon }\rangle _{M}dv%
\end{array}%
\end{equation*}%
We note that in the last term, $(|\nabla u_{\epsilon }|^{2}+\epsilon )^{%
\frac{p-2}{2}}\nabla u_{\epsilon }$ is in $L^{\frac{p}{p-1}}(\Omega
)\,,\nabla u_{i}-\nabla u_{\epsilon }$ is in $L^{p}(\Omega )\,,$ and if $%
m>2\,,$ we do not need the intermediate step in the above inequality.
Since $\nabla u_{i}$ converges weakly to $\nabla u_{\epsilon}$ in $L^{p}$,
the last term tends to $0$ as $i$ tends to $\infty$.\smallskip
It follows that $E_{p,\epsilon}(u_{\epsilon})\leq\lim\inf_{i\rightarrow%
\infty }E_{p,\epsilon}(u_{i})=I$.
\textbf{Method 2}:
Since Banach-saks Theorem (see, e.g. \cite{Y} p. 120, \cite{RS} p. 80)
asserts there exists some subsequence, say it again $v_{i}$ for simplicity,
such that the average%
\begin{equation*}
\begin{array}{l}
w_{n}=\frac{v_{1}+v_{2}+\cdots+v_{n}}{n}%
\end{array}%
\end{equation*}
converges strongly to $u_{\epsilon}$ in $W^{1,p}\left( \Omega\right) $.
Combining this property and Lemma \ref{con}, we have $E_{p,\epsilon}\left(
w_{n}\right) \rightarrow E_{p,\epsilon}\left( u_{\epsilon}\right) $ as $%
n\rightarrow\infty.$
Moreover, according to the convexity of $E_{p,\epsilon},$ one has%
\begin{equation*}
\begin{array}{l}
E_{p,\epsilon}\left( w_{n}\right) \leq\frac{\sum_{i=1}^{n}E_{p,\epsilon
}\left( v_{i}\right) }{n}.%
\end{array}%
\end{equation*}
This implies $E_{p,\epsilon}\left( u_{\epsilon}\right) \leq I$ as $%
n\rightarrow\infty.$
So we obtain lower semi-continuity of $E_{p,\epsilon}.$
\endproof
\bigskip
\bigskip
\begin{lemma}
\label{con}If $v_{i}$ converges strongly to $v_{0}$ in $W^{1,p},$ then $%
E_{p,\epsilon}\left( v_{i}\right) $ converges to $E_{p,\epsilon}\left(
v_{0}\right) .$
\end{lemma}
\proof Step 1: Since $v_{i}$ converges strongly to $v_{0}$ in $W^{1,p}$ i.e.
$\int_{\Omega }\left\vert \nabla v_{i}-\nabla v_{0}\right\vert
^{p}dv\rightarrow 0$ as $i\rightarrow \infty .$ Then
\begin{equation*}
\begin{array}{l}
\int_{\left\vert \nabla v_{i}\right\vert \geq \left\vert \nabla
v_{0}\right\vert }\left\vert \nabla v_{i}-\nabla v_{0}\right\vert
^{p}dv\rightarrow 0\text{ and}\int_{\left\vert \nabla v_{i}\right\vert
<\left\vert \nabla v_{0}\right\vert }\left\vert \nabla v_{i}-\nabla
v_{0}\right\vert ^{p}dv\rightarrow 0%
\end{array}%
\end{equation*}%
as $i\rightarrow \infty .$ By using Minkowski's inequality, these also imply%
\begin{equation*}
\begin{array}{l}
\int_{\left\vert \nabla v_{i}\right\vert \geq \left\vert \nabla
v_{0}\right\vert }\left( \left\vert \nabla v_{i}\right\vert ^{p}-\left\vert
\nabla v_{0}\right\vert ^{p}\right) dv\rightarrow 0\text{ and }%
\int_{\left\vert \nabla v_{i}\right\vert <\left\vert \nabla v_{0}\right\vert
}\left( \left\vert \nabla v_{0}\right\vert ^{p}-\left\vert \nabla
v_{i}\right\vert ^{p}\right) dv\rightarrow 0%
\end{array}%
\end{equation*}%
as $i\rightarrow \infty .$ That is, $\int_{\Omega }\left\vert \left\vert
\nabla v_{0}\right\vert ^{p}-\left\vert \nabla v_{i}\right\vert
^{p}\right\vert dv\rightarrow 0$ as $i\rightarrow \infty .$
Step 2: If we show that, for any positive constant $\delta>0,$
\begin{equation}
\begin{array}{l}
\left\vert \left( \left\vert \nabla v_{i}\right\vert ^{2}+\epsilon\right) ^{%
\frac{p}{2}}-\left( \left\vert \nabla v_{0}\right\vert ^{2}+\epsilon \right)
^{\frac{p}{2}}\right\vert \leq a\left\vert \left\vert \nabla
v_{i}\right\vert ^{p}-\left\vert \nabla v_{0}\right\vert ^{p}\right\vert
+\delta%
\end{array}
\label{e0-1}
\end{equation}
where $a$ is a positive constant independing of $i,$ $v_{i}$ and $v_{0}.$
Then we have, by step 1,%
\begin{equation*}
\begin{array}{lll}
\left\vert E_{p,\epsilon}\left( v_{i}\right) -E_{p,\epsilon}\left(
v_{0}\right) \right\vert & \leq & \int_{\Omega}\left\vert \left( \left\vert
\nabla v_{i}\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}}-\left(
\left\vert \nabla v_{0}\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}%
}\right\vert dv \\
& \leq & a\int_{\Omega}\left\vert \left\vert \nabla v_{i}\right\vert
^{p}-\left\vert \nabla v_{0}\right\vert ^{p}\right\vert dv+\delta\left\vert
\Omega\right\vert \\
& \rightarrow & \delta\left\vert \Omega\right\vert \text{ as }i\rightarrow
\infty.%
\end{array}%
\end{equation*}
This implies $E_{p,\epsilon}\left( v_{i}\right) \rightarrow E_{p,\epsilon
}\left( v_{0}\right) .$
To show (\ref{e0-1}), we only claim that, $X,Y\in
\mathbb{R}
^{n}$ with $\left\vert X\right\vert \geq \left\vert Y\right\vert ,$
\begin{equation}
\begin{array}{l}
\left( \left\vert X\right\vert ^{2}+\epsilon \right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+\epsilon \right) ^{\frac{p}{2}}\leq a\left(
\left\vert X\right\vert ^{p}-\left\vert Y\right\vert ^{p}\right) +\delta .%
\end{array}
\label{e00}
\end{equation}
Let $f(t)=\left( \left\vert X\right\vert ^{2}+t\right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+t\right) ^{\frac{p}{2}},$ $t\geq0.$ Then we
have $f(0)=\left\vert X\right\vert ^{p}-\left\vert Y\right\vert ^{p}$ and $%
f(\epsilon)=\left( \left\vert X\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}%
}-\left( \left\vert Y\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}}.$
Since
\begin{equation*}
\begin{array}{l}
f^{\prime }(t)=\frac{p}{2}\left( \left( \left\vert X\right\vert
^{2}+t\right) ^{\frac{p-2}{2}}-\left( \left\vert Y\right\vert ^{2}+t\right)
^{\frac{p-2}{2}}\right) ,%
\end{array}%
\end{equation*}%
then $f(t)$ is a decreasing function for $1\leq p\leq 2.$ Hence we have $%
f(\epsilon )\leq f(0)$ whenever $1\leq p\leq 2.$
If $2<p\leq4,$ then, for $s>0,$%
\begin{equation}
\begin{array}{lll}
f(s)-f(0) & = & \int_{0}^{s}f^{\prime}(t)dt \\
& = & \frac{p}{2}\int_{0}^{s}\left( \left\vert X\right\vert ^{2}+t\right) ^{%
\frac{p-2}{2}}-\left( \left\vert Y\right\vert ^{2}+t\right) ^{\frac {p-2}{2}%
}dt, \\
& \leq & \frac{ps}{2}\left( \left\vert X\right\vert ^{p-2}-\left\vert
Y\right\vert ^{p-2}\right) ,%
\end{array}
\label{e1}
\end{equation}
since $1<p-2\leq2.$
For any $\delta_{1}>0,$
\begin{equation*}
\begin{array}{l}
\left\vert X\right\vert ^{p-2}-\left\vert Y\right\vert ^{p-2}\leq\left\{
\begin{array}{ll}
\left\vert X\right\vert ^{p-2}\leq\delta_{1}^{p-2} & \text{if }\left\vert
X\right\vert +\left\vert Y\right\vert <\delta_{1}, \\
\frac{\left( \left\vert X\right\vert +\left\vert Y\right\vert \right)
^{2}\left( \left\vert X\right\vert ^{p-2}-\left\vert Y\right\vert
^{p-2}\right) }{\delta_{1}^{2}}\leq\frac{2}{\delta_{1}^{2}}\left( \left\vert
X\right\vert ^{p}-\left\vert Y\right\vert ^{p}\right) & \text{if }\left\vert
X\right\vert +\left\vert Y\right\vert \geq\delta_{1}.%
\end{array}
\right.%
\end{array}%
\end{equation*}
So we have
\begin{equation}
\begin{array}{l}
\left\vert X\right\vert ^{p-2}-\left\vert Y\right\vert ^{p-2}\leq\frac {2}{%
\delta_{1}^{2}}\left( \left\vert X\right\vert ^{p}-\left\vert Y\right\vert
^{p}+\delta_{1}^{p}\right) ,%
\end{array}
\label{e3}
\end{equation}
and then (\ref{e1}) can be rewritten as%
\begin{equation}
\begin{array}{l}
\left( \left\vert X\right\vert ^{2}+s\right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+s\right) ^{\frac{p}{2}}\leq\left( 1+\frac {ps}{%
\delta_{1}^{2}}\right) \left( \left\vert X\right\vert ^{p}-\left\vert
Y\right\vert ^{p}\right) +\left( \frac{ps}{\delta_{1}^{2}}\right)
\delta_{1}^{p}.%
\end{array}
\label{e3-1}
\end{equation}
Hence we have%
\begin{equation*}
\begin{array}{l}
\left( \left\vert X\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}}\leq a\left(
\left\vert X\right\vert ^{p}-\left\vert Y\right\vert ^{p}\right) +\delta,%
\end{array}%
\end{equation*}
where $a=1+\frac{p\epsilon}{\delta_{1}^{2}}$ and $\delta=\left( \frac{%
p\epsilon}{\delta_{1}^{2}}\right) \delta_{1}^{p}.$
If $4<p\leq6,$ then one has $2<p-2\leq4,$ so (\ref{e3}) and (\ref{e3-1})
imply%
\begin{equation*}
\begin{array}{lll}
f(s)-f(0) & = & \frac{p}{2}\int_{0}^{s}\left( \left\vert X\right\vert
^{2}+t\right) ^{\frac{p-2}{2}}-\left( \left\vert Y\right\vert ^{2}+t\right)
^{\frac{p-2}{2}}dt \\
& \leq & \frac{p}{2}\int_{0}^{s}\left( 1+\frac{pt}{\delta_{1}^{2}}\right)
\left( \left\vert X\right\vert ^{p-2}-\left\vert Y\right\vert ^{p-2}\right)
+\left( \frac{pt}{\delta_{1}^{2}}\right) \delta_{1}^{p-2}dt \\
& \leq & \frac{p}{2}\left( s+\frac{ps^{2}}{2\delta_{1}^{2}}\right) \left(
\frac{2}{\delta_{1}^{2}}\left( \left\vert X\right\vert ^{p}-\left\vert
Y\right\vert ^{p}+\delta_{1}^{p}\right) \right) +\frac{p}{2}\left( \frac{%
ps^{2}}{2\delta_{1}^{2}}\right) \delta_{1}^{p-2} \\
& \leq & \left( \frac{ps}{\delta_{1}^{2}}+\frac{1}{2}\left( \frac{ps}{%
\delta_{1}^{2}}\right) ^{2}\right) \left( \left\vert X\right\vert
^{p}-\left\vert Y\right\vert ^{p}\right) +\left( \frac{ps}{\delta_{1}^{2}}%
+\left( \frac{ps}{\delta_{1}^{2}}\right) ^{2}\right) \delta_{1}^{p}.%
\end{array}%
\end{equation*}
Hence
\begin{equation*}
\begin{array}{lll}
\left( \left\vert X\right\vert ^{2}+s\right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+s\right) ^{\frac{p}{2}} & \leq & \left( 1+\frac{%
ps}{\delta_{1}^{2}}+\frac{1}{2}\left( \frac{ps}{\delta_{1}^{2}}\right)
^{2}\right) \left( \left\vert X\right\vert ^{p}-\left\vert Y\right\vert
^{p}\right) \\
& & +\left( \frac{ps}{\delta_{1}^{2}}+\left( \frac{ps}{\delta_{1}^{2}}%
\right) ^{2}\right) \delta_{1}^{p}.%
\end{array}%
\end{equation*}
In particular, we obtain%
\begin{equation*}
\begin{array}{lll}
\left( \left\vert X\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}} & \leq & a\left(
\left\vert X\right\vert ^{p}-\left\vert Y\right\vert ^{p}\right) +\delta,%
\end{array}%
\end{equation*}
where $a=1+\frac{p\epsilon}{\delta_{1}^{2}}+\frac{1}{2}\left( \frac {%
p\epsilon}{\delta_{1}^{2}}\right) ^{2}$ and $\delta=\left( \frac{p\epsilon }{%
\delta_{1}^{2}}+\left( \frac{p\epsilon}{\delta_{1}^{2}}\right) ^{2}\right)
\delta_{1}^{p}.$
By mathematical induction, we conclude that, for any $p>2$ satisfying $%
2q<p\leq2q+2,$ $q\in%
\mathbb{Z}
^{+},$%
\begin{equation*}
\begin{array}{lll}
\left( \left\vert X\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}}-\left(
\left\vert Y\right\vert ^{2}+\epsilon\right) ^{\frac{p}{2}} & \leq & \left(
1+\sum_{n=1}^{q}\frac{1}{n!}\left( \frac{p\epsilon}{\delta_{1}^{2}}\right)
^{n}\right) \left( \left\vert X\right\vert ^{p}-\left\vert Y\right\vert
^{p}\right) \\
& & +\left( \sum_{n=1}^{q}\left( \frac{p\epsilon}{\delta_{1}^{2}}\right)
^{n}\right) \delta_{1}^{p}.%
\end{array}%
\end{equation*}
If we select $\delta_{1}$ small enough such that $\left(
\sum_{n=1}^{q}\left( \frac{p\epsilon}{2\delta_{1}^{2}}\right) ^{n}\right)
\delta _{1}^{p}=\delta,$ then we have (\ref{e00}) with $a=\left(
1+\sum_{n=1}^{q}\left( \frac{p\epsilon}{2\delta_{1}^{2}}\right) ^{n}\right)
. $
\endproof
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\subsection{$\protect\epsilon $-regularization of $p$-Laplacian}
\begin{proposition}
\label{3.1}Let $u$ be a weak solution of the $p$-Laplace equation (\ref{1.0}%
). For every $\epsilon >0,$ let $u_{\epsilon }$ be a solution of the
Euler-Lagrange equation (\ref{a1.2}) with $u-u_{\epsilon }\in
W_{0}^{1,p}\left( \Omega \right) ,$ where $\Omega $ is a domain in $M\,.$
Then $u_{\epsilon }\in C_{loc}^{\infty }\left( \Omega \right) $ is a strong
solution of (\ref{a1.2}), and $u_{\epsilon }$ converges strongly to $u$ in $%
W^{1,p}\left( \Omega \right) $ as $\epsilon \rightarrow 0\,.$
\end{proposition}
\bigskip
\proof
Such solution $u_{\epsilon }$ exists (Proposition \ref{ex}), and $%
u_{\epsilon }\in C_{loc}^{\infty }\left( \Omega \right) $ by the usual
arguments of boot-strap (see, e.g. \cite{LU} Chapter 4, \cite{S} Theorem
3.3, \cite{HS} Theorem 14.2, \cite{Ho}). That is, $u_{\epsilon }$ is the
strong solution of the partial differential equation (\ref{1.2}).
Since $u_{\epsilon}$ and $u$ are the miniming of the Euler functions
\begin{equation*}
\begin{array}{c}
\int_{\Omega}\left\vert \left\vert \nabla\phi\right\vert ^{2}+\epsilon
\right\vert ^{p/2}dv\text{ and }\int_{\Omega}\left\vert
\nabla\phi\right\vert ^{p}dv,%
\end{array}%
\end{equation*}
respectively, over all functions $\phi\in W^{1,p}\left( \Omega\right) $ and $%
\phi=u$ on $\partial\Omega.$ Then one has
\begin{equation}
\begin{array}{c}
\int_{\Omega}\left\vert \nabla u\right\vert
^{p}dv\leq\int_{\Omega}\left\vert \nabla u_{\epsilon}\right\vert ^{p}dv%
\end{array}
\label{p1}
\end{equation}
and
\begin{equation}
\begin{array}{c}
\int_{\Omega}\left\vert \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right\vert ^{p/2}dv\text{ }\leq\int_{\Omega}\left\vert
\left\vert \nabla u\right\vert ^{2}+\epsilon\right\vert ^{p/2}dv.%
\end{array}
\label{p2}
\end{equation}
Combining (\ref{p1}) and (\ref{p2}),
\begin{equation*}
\begin{array}{c}
\int_{\Omega }\left\vert \nabla u\right\vert ^{p}dv\leq \int_{\Omega
}\left\vert \nabla u_{\epsilon }\right\vert ^{p}dv\leq \int_{\Omega
}\left\vert \left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon
\right\vert ^{p/2}dv\text{ }\leq \int_{\Omega }\left\vert \left\vert \nabla
u\right\vert ^{2}+\epsilon \right\vert ^{p/2}dv,%
\end{array}%
\end{equation*}%
one has $\left\Vert \nabla u_{\epsilon }\right\Vert _{p}\rightarrow
\left\Vert \nabla u\right\Vert _{p}$ as $\epsilon \rightarrow 0.$ Moreover,
by Lemma \ref{c1} $\nabla u_{\epsilon }\rightarrow \nabla u$ a.e. on $\Omega
$ for $p>1,$ we have $\nabla u_{\epsilon }\rightarrow \nabla u$ in $%
L^{p}\left( \Omega \right) ,$ and then $p$-Poincar\'{e} inequality implies $%
u_{\epsilon }\rightarrow u$ in $W^{1,p}\left( \Omega \right) .$
\endproof
\begin{lemma}
\label{c1}$\nabla u_{\epsilon}\rightarrow\nabla u$ a.e. on $\Omega$ for $%
p>1. $
\end{lemma}
\bigskip
First, we recall the following inequality (cf. \cite{Li} Chapter 10, or \cite%
{HPV} Lemma 4)
\begin{proposition}
\label{12}Consider the vector field $X$ and $Y.$Then%
\begin{equation}
\begin{array}{l}
\left\langle X-Y,\left\vert X\right\vert ^{p-2}X-\left\vert Y\right\vert
^{p-2}Y\right\rangle \geq C\Psi \left( X,Y\right) ,%
\end{array}
\label{c0}
\end{equation}%
where%
\begin{equation}
\begin{array}{l}
\Psi \left( X,Y\right) =\left\{
\begin{array}{ll}
\left\vert X-Y\right\vert ^{p} & \text{if }p\geq 2, \\
\frac{\left( p-1\right) \left\vert X-Y\right\vert ^{2}}{\left( 1+\left\vert
X\right\vert ^{2}+\left\vert Y\right\vert ^{2}\right) ^{\frac{2-p}{2}}} &
\text{if }1<p<2.%
\end{array}%
\right.%
\end{array}
\label{c0-1}
\end{equation}
\end{proposition}
\bigskip
\proof
Since $u-u_{\epsilon}\in W_{0}^{1,p}\left( \Omega\right) ,$ one has
\begin{equation*}
\begin{array}{l}
\int_{\Omega}\left\vert \nabla u\right\vert ^{p-2}\left\langle \nabla
u,\nabla\left( u-u_{\epsilon}\right) \right\rangle dv=0%
\end{array}%
\end{equation*}
and
\begin{equation*}
\begin{array}{l}
\int_{\Omega}\left\vert \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}\left\langle \nabla u_{\epsilon
},\nabla\left( u-u_{\epsilon}\right) \right\rangle dv=0.%
\end{array}%
\end{equation*}
Then%
\begin{equation*}
\begin{array}{lll}
0 & = & \int_{\Omega}\left\vert \nabla u\right\vert ^{p-2}\left\langle
\nabla u,\nabla\left( u-u_{\epsilon}\right) \right\rangle -\left\vert
\left\vert \nabla u_{\epsilon}\right\vert ^{2}+\epsilon\right\vert ^{\frac{%
p-2}{2}}\left\langle \nabla u_{\epsilon},\nabla\left( u-u_{\epsilon}\right)
\right\rangle dv \\
& = & \int_{\Omega}\left\vert \nabla u\right\vert ^{p}-\left\vert \nabla
u\right\vert ^{p-2}\left\langle \nabla u,\nabla u_{\epsilon}\right\rangle \\
& & -\left\vert \left\vert \nabla u_{\epsilon}\right\vert ^{2}+\epsilon
\right\vert ^{\frac{p-2}{2}}\left\langle \nabla u_{\epsilon},\nabla
u\right\rangle +\left\vert \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}\left\vert \nabla u_{\epsilon
}\right\vert ^{2}dv.%
\end{array}%
\end{equation*}
This equality can be rewritten as $LHS1=RHS,$ where%
\begin{equation*}
\begin{array}{l}
LHS1=\int_{\Omega}\left\vert \nabla u\right\vert ^{p}-\left\vert \nabla
u\right\vert ^{p-2}\left\langle \nabla u,\nabla u_{\epsilon}\right\rangle
-\left\vert \nabla u_{\epsilon}\right\vert ^{p-2}\left\langle \nabla
u_{\epsilon},\nabla u\right\rangle +\left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p}{2}}dv%
\end{array}%
\end{equation*}
and%
\begin{equation*}
\begin{array}{l}
RHS=\int_{\Omega}\left( \left\vert \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}-\left\vert \nabla u_{\epsilon
}\right\vert ^{p-2}\right) \left\langle \nabla u_{\epsilon},\nabla
u\right\rangle +\epsilon\left\vert \left\vert \nabla u_{\epsilon}\right\vert
^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}dv.%
\end{array}%
\end{equation*}
It is easy to see that%
\begin{equation*}
\begin{array}{l}
LHS1\geq\int_{\Omega}\left\vert \nabla u\right\vert ^{p}-\left\vert \nabla
u\right\vert ^{p-2}\left\langle \nabla u,\nabla u_{\epsilon}\right\rangle
-\left\vert \nabla u_{\epsilon}\right\vert ^{p-2}\left\langle \nabla
u_{\epsilon},\nabla u\right\rangle +\left\vert \nabla
u_{\epsilon}\right\vert ^{p}dv=LHS2.%
\end{array}%
\end{equation*}
So, we select $X=$ $\nabla u$ and $Y=\nabla u_{\epsilon }$, then Proposition %
\ref{12} implies%
\begin{equation*}
\begin{array}{l}
LHS2\geq C\int_{\Omega }\Psi \left( \nabla u,\nabla u_{\epsilon }\right)
dv\geq 0%
\end{array}%
\end{equation*}%
where%
\begin{equation*}
\begin{array}{l}
\Psi \left( \nabla u,\nabla u_{\epsilon }\right) =\left\{
\begin{array}{ll}
\left\vert \nabla u-\nabla u_{\epsilon }\right\vert ^{p} & \text{if }p\geq 2,
\\
\frac{\left( p-1\right) \left\vert \nabla u-\nabla u_{\epsilon }\right\vert
^{2}}{\left( 1+\left\vert \nabla u\right\vert ^{2}+\left\vert \nabla
u_{\epsilon }\right\vert ^{2}\right) ^{\frac{2-p}{2}}} & \text{if }1<p<2.%
\end{array}%
\right.%
\end{array}%
\end{equation*}
If we can show that%
\begin{equation*}
\begin{array}{l}
RHS\rightarrow0\text{ as }\epsilon\rightarrow0,%
\end{array}%
\end{equation*}
Then we have%
\begin{equation*}
\begin{array}{l}
\int_{\Omega }\Psi \left( \nabla u,\nabla u_{\epsilon }\right) dv\rightarrow
0\text{ as }\epsilon \rightarrow 0.%
\end{array}%
\end{equation*}%
Therefore $\nabla u_{\epsilon }\rightarrow \nabla u$ a.e. on $\Omega .$
Now we claim that
\begin{equation*}
\begin{array}{l}
RHS=RHS1+RHS2\rightarrow0%
\end{array}%
\end{equation*}
as $\epsilon\rightarrow0,$ where%
\begin{equation*}
\begin{array}{l}
RHS1=\int_{\Omega}\epsilon\left\vert \left\vert \nabla
u_{\epsilon}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}dv%
\end{array}%
\end{equation*}
and%
\begin{equation*}
\begin{array}{l}
RHS2=\int_{\Omega}\left( \left\vert \left\vert \nabla
u_{\epsilon}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}-\left\vert
\nabla u_{\epsilon }\right\vert ^{p-2}\right) \left\langle \nabla
u_{\epsilon},\nabla u\right\rangle dv.%
\end{array}%
\end{equation*}
It is easy to see that, if $\left\vert \nabla u_{\epsilon }\right\vert
^{2}\geq 1,$%
\begin{equation*}
\begin{array}{lll}
\int_{\Omega }\epsilon \left\vert \left\vert \nabla u_{\epsilon }\right\vert
^{2}+\epsilon \right\vert ^{\frac{p-2}{2}}dv & \leq & \int_{\Omega }\epsilon
\left\vert \nabla u_{\epsilon }\right\vert ^{2}\left\vert \left\vert \nabla
u_{\epsilon }\right\vert ^{2}+\epsilon \right\vert ^{\frac{p-2}{2}}dv \\
& \leq & \epsilon \int_{\Omega }\left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon \right\vert ^{\frac{p}{2}}dv,%
\end{array}%
\end{equation*}%
and if $\left\vert \nabla u_{\epsilon }\right\vert ^{2}<1,$%
\begin{equation*}
\begin{array}{l}
\int_{\Omega }\epsilon \left\vert \left\vert \nabla u_{\epsilon }\right\vert
^{2}+\epsilon \right\vert ^{\frac{p-2}{2}}dv\leq \left\{
\begin{array}{ll}
\epsilon \left( 1+\epsilon \right) ^{\frac{p-2}{2}}\cdot vol\left( \Omega
\right) & \text{if }p\geq 2 \\
\epsilon ^{\frac{p}{2}}\cdot vol\left( \Omega \right) & \text{if }p<2.%
\end{array}%
\right.%
\end{array}%
\end{equation*}
So we have $RHS1\rightarrow0$ as $\epsilon\rightarrow0.$
Now we focus on the term $RHS2$,%
\begin{equation*}
\begin{array}{lll}
RHS2 & = & \int_{\Omega}\left( \left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}-\left\vert \nabla
u_{\epsilon}\right\vert ^{p-2}\right) \left\langle \nabla
u_{\epsilon},\nabla u\right\rangle dv \\
& \leq & \int_{\Omega}\left\vert \left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}-\left\vert \nabla
u_{\epsilon}\right\vert ^{p-2}\right\vert \left\vert \nabla u_{\epsilon
}\right\vert \left\vert \nabla u\right\vert dv.%
\end{array}%
\end{equation*}
In the case $p\geq2,$ one may rewrite it as%
\begin{equation*}
\begin{array}{lll}
RHS2 & \leq & \int_{\Omega}\left( \left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-2}{2}}-\left\vert \nabla
u_{\epsilon}\right\vert ^{p-2}\right) \left\vert \nabla u_{\epsilon
}\right\vert \left\vert \nabla u\right\vert dv \\
& \leq & \int_{\Omega}\left( \left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-1}{2}}-\left\vert \nabla
u_{\epsilon}\right\vert ^{p-1}\right) \left\vert \nabla u\right\vert dv.%
\end{array}%
\end{equation*}
If $p\geq3,$ using mean value theorem, we have the inequality%
\begin{equation*}
\begin{array}{l}
\left( x+\epsilon\right) ^{q}-x^{q}=q\epsilon\left( x+\epsilon_{1}\right)
^{q-1}\leq q\epsilon\left( x+\epsilon\right) ^{q-1}%
\end{array}%
\end{equation*}
here $q=\frac{p-1}{2}\geq1,$ $x\geq0$ and $\epsilon_{1}\in\left(
0,\epsilon\right) .$ Hence%
\begin{equation*}
\begin{array}{lll}
RHS2 & \leq & \frac{\left( p-1\right) \epsilon}{2}\int_{\Omega}\left\vert
\left\vert \nabla u_{\epsilon}\right\vert ^{2}+\epsilon\right\vert ^{\frac{%
p-3}{2}}\left\vert \nabla u\right\vert dv \\
& \leq & \left\{
\begin{array}{ll}
\frac{\left( p-1\right) \epsilon}{2}\int_{\Omega}\left\vert \left\vert
\nabla u_{\epsilon}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-1}{2}%
}\left\vert \nabla u\right\vert dv\text{ \ \ } & \text{if }\left\vert \nabla
u_{\epsilon}\right\vert ^{2}>1 \\
\frac{\left( p-1\right) \epsilon}{2}\left( 1+\epsilon\right) ^{\frac {p-3}{2}%
}\int_{\Omega}\left\vert \nabla u\right\vert dv & \text{if }\left\vert
\nabla u_{\epsilon}\right\vert ^{2}\leq1%
\end{array}
\right. \\
& \leq & \left\{
\begin{array}{ll}
\frac{\left( p-1\right) \epsilon}{2}\left( \int_{\Omega}\left\vert
\left\vert \nabla u_{\epsilon}\right\vert ^{2}+\epsilon\right\vert ^{\frac {p%
}{2}}\right) ^{\frac{p-1}{p}}\left( \int_{\Omega}\left\vert \nabla
u\right\vert ^{p}\right) ^{\frac{1}{p}}dv & \text{if }\left\vert \nabla
u_{\epsilon}\right\vert ^{2}>1 \\
\frac{\left( p-1\right) \epsilon}{2}\left( 1+\epsilon\right) ^{\frac {p-3}{2}%
}\left( vol\left( \Omega\right) \right) ^{\frac{p-1}{p}}\left(
\int_{\Omega}\left\vert \nabla u\right\vert ^{p}\right) ^{\frac{1}{p}}dv%
\text{ \ } & \text{if }\left\vert \nabla u_{\epsilon}\right\vert ^{2}\leq1%
\end{array}
\right. \\
& \rightarrow & 0\text{ as }\epsilon\rightarrow0.%
\end{array}%
\end{equation*}
If $2\leq p\leq3,$ using the inequality%
\begin{equation*}
\begin{array}{l}
\left( x+\epsilon\right) ^{q}-x^{q}\leq\epsilon^{q}%
\end{array}%
\end{equation*}
here $\frac{1}{2}\leq q=\frac{p-1}{2}\leq1,$ $x\geq0,$ then%
\begin{equation*}
\begin{array}{lll}
RHS2 & \leq & \int_{\Omega}\left( \left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon\right\vert ^{\frac{p-1}{2}}-\left\vert \nabla
u_{\epsilon}\right\vert ^{p-1}\right) \left\vert \nabla u\right\vert \\
& \leq & \epsilon^{\frac{p-1}{2}}\int_{\Omega}\left\vert \nabla u\right\vert
\\
& \leq & \epsilon^{\frac{p-1}{2}}\left( vol\left( \Omega\right) \right) ^{%
\frac{p-1}{p}}\left( \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\right)
^{\frac{1}{p}} \\
& \rightarrow & 0\text{ as }\epsilon\rightarrow0.%
\end{array}%
\end{equation*}
In the case $1<p<2,$ one may rewrite $RHS2$ as%
\begin{equation*}
\begin{array}{l}
RHS2\leq \int_{\Omega }\left( \left\vert \nabla u_{\epsilon }\right\vert
^{p-2}-\left\vert \left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon
\right\vert ^{\frac{p-2}{2}}\right) \left\vert \nabla u_{\epsilon
}\right\vert \left\vert \nabla u\right\vert .%
\end{array}%
\end{equation*}%
Since $0<\frac{2-p}{2}<1,$ then we have%
\begin{equation*}
\begin{array}{lll}
RHS2 & = & \int_{\Omega }\frac{\left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon \right\vert ^{\frac{2-p}{2}}-\left\vert \nabla
u_{\epsilon }\right\vert ^{2-p}}{\left\vert \left\vert \nabla u_{\epsilon
}\right\vert ^{2}+\epsilon \right\vert ^{\frac{2-p}{2}}}\left\vert \nabla
u_{\epsilon }\right\vert ^{p-1}\left\vert \nabla u\right\vert dv \\
& \leq & \int_{\Omega }\frac{\epsilon ^{\frac{2-p}{2}}}{\left\vert
\left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon \right\vert ^{\frac{%
3-2p}{2}}}\cdot \frac{\left\vert \nabla u_{\epsilon }\right\vert ^{p-1}}{%
\left\vert \left\vert \nabla u_{\epsilon }\right\vert ^{2}+\epsilon
\right\vert ^{\frac{p-1}{2}}}\left\vert \nabla u\right\vert \\
& \leq & \epsilon ^{\frac{p-1}{2}}\int_{\Omega }\left\vert \nabla
u\right\vert dv \\
& \leq & \epsilon ^{\frac{p-1}{2}}\left( vol\left( \Omega \right) \right) ^{%
\frac{p-1}{p}}\left( \int_{\Omega }\left\vert \nabla u\right\vert
^{p}dv\right) ^{\frac{1}{p}} \\
& \rightarrow & 0\text{ as }\epsilon \rightarrow 0.%
\end{array}%
\end{equation*}
Hence we conclue that%
\begin{equation*}
\begin{array}{l}
RHS=RHS1+RHS2\rightarrow0\text{ as }\epsilon\rightarrow0.%
\end{array}%
\end{equation*}
\endproof
\bigskip
\subsection{Nontival $p$-harmonic function with finite $q$-energy on
manifold with weighted Poincar\'{e} inequality}
In this subsection, we construct an example of nontival $p$-harmonic
function $u,$ with finite $q$-energy, $q>p-1,$ on a complete noncompact
manifold with weighted Poincar\'{e} inequality $\left( P_{\rho }\right) .$
Let $M=%
\mathbb{R}
\times N^{m-1},$ $m\geq 3,$ with a metric $ds^{2}=dt^{2}+\eta ^{2}\left(
t\right) g_{N},$ where $\eta \left( t\right) :%
\mathbb{R}
\rightarrow \left( 0,\infty \right) $ is a smooth function with $\eta
^{\prime \prime }>0,$ $\left( m-2\right) \left( \log \eta \right) ^{\prime
\prime }+\eta ^{-2}Ric_{N}\geq 0,$ and $\left( N,g_{N}\right) $ is a compact
Riemannian manifold with $vol\left( N^{m-1}\right) =1.$
According to \cite{LW3} Proposition 6.1, then $M$ satisfies weighted Poincar%
\'{e} inequality $\left( P_{\rho }\right) $ and $Ric_{M}\geq -\frac{m-1}{m-2}%
\rho $ with $\rho =\left( m-2\right) \eta ^{\prime \prime }\eta ^{-1}.$
Let $A\left( t\right) $ be the volume of $\left\{ t\right\} \times N^{m-1},$
then $A\left( t\right) =$ $\eta ^{m-1}\left( t\right) .$
Now we select $\eta \left( t\right) $ such that each end of $M$ is $p$%
-hyperbolic, and%
\begin{equation*}
A\left( t\right) \geq d_{1}\left\vert t\right\vert ^{\frac{p-1}{q-p+1-\delta
}},\text{ if }\left\vert t\right\vert \geq 1,
\end{equation*}%
where $d_{1}>0$ and $0<\delta <q-p+1$ are positive constants.
By using \cite{Tr} Proposition 5.3,
\begin{equation*}
\begin{array}{l}
Cap_{p}\left( \left( -\infty ,a\right) \times N^{m-1},\left( b,\infty
\right) \times N^{m-1};M\right) =\left( \int_{a}^{b}\left( \frac{1}{A\left(
t\right) }\right) ^{1/\left( p-1\right) }dt\right) ^{1-p},%
\end{array}%
\end{equation*}%
for any $-\infty <a<b<\infty .$ \ If we define $u$ by%
\begin{equation*}
\begin{array}{l}
u\left( t\right) =\int_{-\infty }^{t}\left( \frac{1}{A\left( s\right) }%
\right) ^{1/\left( p-1\right) }ds%
\end{array}%
\end{equation*}%
then%
\begin{equation*}
\begin{array}{l}
Cap_{p}\left( \left( -\infty ,a\right) \times N^{n-1},\left( b,\infty
\right) \times N^{n-1};M\right) =\left( u\left( b\right) -u\left( a\right)
\right) ^{1-p},%
\end{array}%
\end{equation*}%
$u\left( t\right) \rightarrow 0$ as $t\rightarrow -\infty $, and $u$ is
uniformly bounded for all $t\in \left( -\infty ,\infty \right) .$
Moreover, define a function $v$ as follows,%
\begin{equation*}
v\left( t\right) =\left\{
\begin{array}{ll}
0, & \text{if }t\leq a, \\
\frac{u\left( t\right) -u\left( a\right) }{u\left( b\right) -u\left(
a\right) },\text{ \ \ } & \text{if }a<t<b, \\
1, & \text{if }t\geq b.%
\end{array}%
\right.
\end{equation*}%
then%
\begin{equation*}
\begin{array}{l}
\int_{M}\left\vert \nabla v\right\vert ^{p}dv=\int_{a}^{b}\frac{\left(
u^{\prime }\left( t\right) \right) ^{p}}{\left( u\left( b\right) -u\left(
a\right) \right) ^{p}}A\left( t\right) dt=\left( u\left( b\right) -u\left(
a\right) \right) ^{1-p}%
\end{array}%
\end{equation*}%
which implies $v$ is extremal of $p$-energy for every $-\infty <a<b<\infty .$
Hence $u\left( t\right) $ is $p$-harmonic in $M$ with finite $q$ energy
\begin{equation*}
\begin{array}{l}
\int_{M}\left\vert \nabla u\right\vert ^{q}dv=\int_{-\infty }^{\infty }A^{%
\frac{p-1-q}{p-1}}\left( t\right) dt<\infty%
\end{array}%
\end{equation*}%
for all $q>p-1.$ Moreover, by \cite{LW3} Proposition 6.1, we have
\begin{equation*}
\begin{array}{l}
Ric_{M}\left( \nabla u,\nabla u\right) =-\frac{m-1}{m-2}\rho \left\vert
\nabla u\right\vert ^{2}.%
\end{array}%
\end{equation*}
\bigskip
\bigskip
\bigskip
\subsection{Volume estimate and $p$-Poincar\'{e} inequality}
In this subsection, we study a complete noncompact manifold $M$ with the $p$%
-Poincar\'{e} inequality $\left( P_{\lambda_{p}}\right) ,$ $p>1,$ that is,
the inequality
\begin{equation}
\begin{array}{l}
\lambda_{p}\int_{M}\Psi^{p}\leq\int_{M}\left\vert \nabla\Psi\right\vert ^{p}%
\end{array}
\label{p poincare}
\end{equation}
holds for all $\Psi\in W_{0}^{1,p}\left( M\right) .$ In particular, if $p=2,$
this formula is the general Poincar\'{e} inequality, and $\lambda_{2}$ is
the spectrum of $M.$ In \cite{HKM}, they show that a complete manifold $M$
with positive spectrum $\lambda_{2}>0,$ then it must have $\lambda_{p}>0$
for all $p\geq2.$ In fact, the following inequality%
\begin{equation*}
\begin{array}{l}
p\left( \lambda_{p}\right) ^{1/p}\geq2\left( \lambda_{2}\right) ^{1/2}%
\end{array}%
\end{equation*}
holds on $M$ for all $p\geq2.$
\begin{lemma}
\label{decay estimate}Let $M$ be a complete noncompact manifold satisfying $%
\left( P_{\lambda_{p}}\right) ,$ $p>1$. Suppose $w$ is a positive,
p-subharmonic function with a finite p-energy on $M.$ If $w$ satisfies
\begin{equation}
\begin{array}{l}
\int_{B\left( 2R\right) \backslash B\left( R\right) }\exp(-\frac{\left(
\lambda_{p}\right) ^{1/p}r\left( x\right) }{p+1})\left\vert w\right\vert
^{p}dv=o\left( R\right) ,%
\end{array}
\label{growth control}
\end{equation}
where $R\geq R_{0}+1.$ Then,
\begin{equation*}
\begin{array}{l}
\left( 1-\delta\right) \left\Vert \exp(\frac{\delta\left( \lambda
_{p}\right) ^{1/p}r\left( x\right) }{p+1})w\right\Vert _{L_{p}\left(
M\backslash B\left( R_{0}+1\right) \right) }\leq C,%
\end{array}%
\end{equation*}
and
\begin{equation*}
\begin{array}{l}
\left( 1-\delta\right) \left\Vert \exp(\frac{\delta\left( \lambda
_{p}\right) ^{1/p}r\left( x\right) }{p+1})\nabla w\right\Vert _{L_{p}\left(
M\backslash B\left( R_{0}+1\right) \right) }\leq C,%
\end{array}%
\end{equation*}
for all $0<\delta<1,$ and for some constant $C$ depending on $p$ and $%
\lambda_{p}.$
\end{lemma}
\bigskip
\bigskip
\proof Let $\psi$ be a non-negative cut-off function, then we have%
\begin{equation}
\begin{array}{lll}
0 & \geq & \int_{M}\psi^{p}w\left( -\Delta_{p}w\right) \\
& = & \int_{M}\left\langle \nabla(\psi^{p}w),\left\vert \nabla w\right\vert
^{p-2}\nabla w\right\rangle \\
& = & \int_{M}\psi^{p}\left\vert \nabla w\right\vert ^{p}+pw\left\vert
\nabla w\right\vert ^{p-2}\psi^{p-1}\left\langle \nabla\psi,\nabla
w\right\rangle \\
& \geq & \int_{M}\psi^{p}\left\vert \nabla w\right\vert
^{p}-p\int_{M}w\psi^{p-1}\left\vert \nabla w\right\vert ^{p-1}\left\vert
\nabla \psi\right\vert .%
\end{array}
\label{ed1}
\end{equation}
By using H\"{o}lder inequality%
\begin{equation*}
\begin{array}{l}
\int_{M}w\psi^{p-1}\left\vert \nabla w\right\vert ^{p-1}\left\vert \nabla
\psi\right\vert \leq\left( \int_{M}\left\vert \nabla w\right\vert
^{p}\psi^{p}\right) ^{(p-1)/p}\left( \int_{M}w^{p}\left\vert \nabla
\psi\right\vert ^{p}\right) ^{1/p},%
\end{array}%
\end{equation*}
then (\ref{ed1}) can be rewritten as%
\begin{equation}
\begin{array}{l}
\left\Vert \psi\nabla w\right\Vert _{L_{p}}\leq p\left\Vert \nabla\psi\cdot
w\right\Vert _{L_{p}},%
\end{array}
\label{ed2}
\end{equation}
and this inequality is the Caccioppoli type estimate.
Since Minkowski inequality yields%
\begin{equation*}
\begin{array}{l}
\left\Vert \nabla(\psi w)\right\Vert _{L_{p}}\leq\left\Vert \nabla\psi\cdot
w\right\Vert _{L_{p}}+\left\Vert \psi\nabla w\right\Vert _{L_{p}},%
\end{array}%
\end{equation*}
then (\ref{ed2}) implies%
\begin{equation}
\begin{array}{l}
\left\Vert \nabla(\psi w)\right\Vert _{L_{p}}\leq\left( p+1\right)
\left\Vert \nabla\psi\cdot w\right\Vert _{L_{p}}.%
\end{array}
\label{de1}
\end{equation}
This inequality is not sharp enough whenever $p=2.$ In fact, if $p=2,$ one
can easy to show $\left\Vert \nabla\left( \psi w\right) \right\Vert
_{L_{2}}\leq\left\Vert \nabla\psi w\right\Vert _{L_{2}}$ by the similar
method (cf. \cite{LW1}\cite{LW3}).
By scaling the metric, we may assume $\lambda_{p}=1.$ Combining (\ref{de1})
and (\ref{p poincare}), then
\begin{equation}
\begin{array}{l}
\left\Vert \psi w\right\Vert _{L_{p}}\leq\left( p+1\right) \left\Vert
\nabla\psi\cdot w\right\Vert _{L_{p}},%
\end{array}
\label{de1-1}
\end{equation}
where $\psi$ is a cut off function on $M.$
Now we select $\psi=\phi(r(x))\exp(a\left( r(x)\right) )$, then%
\begin{equation}
\begin{array}{lll}
\frac{1}{p+1}\left\Vert \psi w\right\Vert _{L_{p}} & \leq & \left\Vert
\left( \nabla\phi+\phi\nabla a\right) \exp(a(x))w\right\Vert _{L_{p}} \\
& \leq & \left\Vert \left( \nabla\phi\right) \exp(a(x))w\right\Vert
_{L_{p}}+\left\Vert \left( \nabla a\right) \phi\exp(a(x))w\right\Vert
_{L_{p}}%
\end{array}
\label{de11}
\end{equation}
where $\phi$ is a non-negative cut-off function defined by $%
\phi=\phi_{+}+\phi_{-}$ where%
\begin{equation*}
\begin{array}{ll}
\phi_{+}(r)=\left\{
\begin{array}{ll}
r-R_{0} & \text{for }R_{0}\leq r\leq R_{0}+1,\vspace{3mm} \\
1 & \text{for }r>R_{0}+1,\vspace{3mm}%
\end{array}
\right. & \phi_{-}(r)=\left\{
\begin{array}{ll}
\frac{R-r}{R} & \text{for }R\leq r\leq2R,\vspace{3mm} \\
-1 & \text{for }r>2R,%
\end{array}
\right.%
\end{array}%
\end{equation*}
and we also choose $a=a_{+}(r(x))+a_{-}(r(x))$ as%
\begin{equation*}
\begin{array}{ll}
a_{+}(r)=\left\{
\begin{array}{ll}
\frac{\delta r\left( x\right) }{p+1} & \text{for }r\leq\frac{K}{1+\delta },%
\vspace{3mm} \\
\frac{\delta K}{\left( 1+\delta\right) \left( p+1\right) } & \text{for }r>%
\frac{K}{1+\delta},%
\end{array}
\right. &
\begin{array}{l}
a_{-}(r)=\left\{
\begin{array}{ll}
0 & \text{for }r\leq\frac{K}{1+\delta},\vspace{3mm} \\
\frac{1}{p+1}\left( \frac{2K}{1+\delta}-r\left( x\right) \right) & \text{for
}r>\frac{K}{1+\delta},%
\end{array}
\right.%
\end{array}%
\end{array}%
\end{equation*}
for some fixed $K>\left( R_{0}+1\right) \left( 1+\delta\right) ,$ $%
0<\delta<1,$ and $R\geq\frac{K}{1+\delta},$ it's easy to check that%
\begin{equation*}
\begin{array}{l}
\left\vert \nabla\phi\right\vert ^{2}\left( x\right) =\left\{
\begin{array}{ll}
1\text{ \ \ \ \ } & \text{on }B(R_{0}+1)\backslash B(R_{0}),\vspace{3mm} \\
0 & \text{on }B\left( R_{0}\right) ,\text{ }B\left( R\right) \backslash
B(R_{0}+1)\text{ and }M\backslash B\left( 2R\right) ,\vspace{3mm} \\
\frac{1}{R^{2}} & \text{on }B\left( 2R\right) \backslash B\left( R\right) ,%
\vspace{3mm}%
\end{array}
\right.%
\end{array}%
\end{equation*}
and%
\begin{equation*}
\begin{array}{l}
\left\vert \nabla a\right\vert ^{2}\left( x\right) =\left\{
\begin{array}{ll}
\frac{\delta^{2}}{\left( p+1\right) ^{2}} & \text{for }r<\frac{K}{1+\delta },%
\vspace{3mm} \\
\frac{1}{\left( p+1\right) ^{2}}\text{\ \ \ } & \text{for }r>\frac {K}{%
1+\delta}.%
\end{array}
\right.%
\end{array}%
\end{equation*}
Substituting into (\ref{de11}), we obtain%
\begin{equation*}
\begin{array}{lll}
& & \frac{1}{p+1}\left\Vert \phi\exp(a(x))w\right\Vert _{L_{p}\left(
M\right) } \\
& \leq & \left\Vert \left( \nabla\phi_{+}\right) \exp(a(x))w\right\Vert
_{L_{p}\left( M\right) }+\left\Vert \left( \nabla\phi_{-}\right)
\exp(a(x))w\right\Vert _{L_{p}\left( M\right) } \\
& & +\left\Vert \left( \nabla a_{+}\right) \phi\exp(a(x))w\right\Vert
_{L_{p}\left( M\right) }+\left\Vert \left( \nabla a_{-}\right) \phi
\exp(a(x))w\right\Vert _{L_{p}\left( M\right) } \\
& \leq & \left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
B(R_{0}+1)\backslash B(R_{0})\right) }+\frac{1}{R}\left\Vert
\exp(a(x))w\right\Vert _{L_{p}\left( B\left( 2R\right) \backslash B\left(
R\right) \right) } \\
& & +\frac{\delta}{p+1}\left\Vert \phi\exp(a(x))w\right\Vert _{L_{p}\left(
B\left( \frac{K}{1+\delta}\right) \right) }+\frac{1}{p+1}\left\Vert
\phi\exp(a(x))w\right\Vert _{L_{p}\left( M\backslash B\left( \frac {K}{%
1+\delta}\right) \right) },%
\end{array}%
\end{equation*}
hence%
\begin{equation*}
\begin{array}{lll}
& & \left( \frac{1-\delta}{p+1}\right) \left\Vert \phi\exp
(a(x))w\right\Vert _{L_{p}\left( B\left( \frac{K}{1+\delta}\right)
\backslash B\left( R_{0}+1\right) \right) } \\
& \leq & \left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
B(R_{0}+1)\backslash B(R_{0})\right) }+\frac{1}{R}\left\Vert
\exp(a(x))w\right\Vert _{L_{p}\left( B\left( 2R\right) \backslash B\left(
R\right) \right) }.%
\end{array}%
\end{equation*}
The definition of $a(x)$ and the growth condition (\ref{growth control})
imply that the last term on the right hand side tends to $0$ as $%
R\rightarrow\infty $. Thus one has the following inequality,%
\begin{equation}
\left( \frac{1-\delta}{p+1}\right) \left\Vert \exp(a(x))w\right\Vert
_{L_{p}\left( B\left( \frac{K}{1+\delta}\right) \backslash B\left(
R_{0}+1\right) \right) }\leq\left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
B(R_{0}+1)\backslash B(R_{0})\right) }. \label{ed3-0}
\end{equation}
Since the right hand side of (\ref{ed3-0}) is independent of $K$ and $%
0<\delta<1,$ by letting $K\rightarrow\infty$ we obtain that
\begin{equation}
\begin{array}{l}
\left( 1-\delta\right) \left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
M\backslash B\left( R_{0}+1\right) \right) }\leq C_{1},%
\end{array}
\label{ed3}
\end{equation}
for some constant $0<C_{1}=C_{1}\left( p\right) <\infty.$
Moreover, by (\ref{ed2}) and similar process as above, we have%
\begin{equation*}
\begin{array}{lll}
\frac{1}{p}\left\Vert \psi\nabla w\right\Vert _{L_{p}\left( M\right) } & \leq
& \left\Vert \nabla\psi\cdot w\right\Vert _{L_{p}\left( M\right) } \\
& \leq & \left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
B(R_{0}+1)\backslash B(R_{0})\right) }+\frac{1}{R}\left\Vert
\exp(a(x))w\right\Vert _{L_{p}\left( B\left( 2R\right) \backslash B\left(
R\right) \right) } \\
& & +\frac{\delta}{p+1}\left\Vert \phi\exp(a(x))w\right\Vert _{L_{p}\left(
B\left( \frac{K}{1+\delta}\right) \right) }+\frac{1}{p+1}\left\Vert
\phi\exp(a(x))w\right\Vert _{L_{p}\left( B\left( 2R\right) \backslash
B\left( \frac{K}{1+\delta}\right) \right) } \\
& \leq & 2\left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
B(R_{0}+1)\backslash B(R_{0})\right) }+3\left\Vert
\phi\exp(a(x))w\right\Vert _{L_{p}\left( B\left( 2R\right) \backslash
B\left( R_{0}+1\right) \right) } \\
& \leq & C_{2}+\frac{3C_{1}}{1-\delta}.%
\end{array}%
\end{equation*}
Hence, by letting $R\rightarrow\infty$ and then letting $K\rightarrow\infty,$
we conclude%
\begin{equation*}
\begin{array}{l}
\left( 1-\delta\right) \left\Vert \exp(\delta r\left( x\right) )\nabla
w\right\Vert _{L_{p}\left( M\backslash B\left( R_{0}+1\right) \right) }\leq
C_{3}%
\end{array}%
\end{equation*}
for some constant $0<C_{3}=C_{2}+\frac{3C_{1}}{1-\delta}<\infty.$
Then lemma now follows.
\endproof
\bigskip
\bigskip
\begin{lemma}
\label{decay estimate on E}Let $M$ be a complete noncompact manifold
satisfying $\left( P_{\lambda_{p}}\right) ,$ $p>1$. Suppose $E$ is an end of
$M$ respective to a compact set, $w_{i}$ is a positive, p-harmonic function
with a finite p-energy on $E\left( R_{i}\right) $ and $w_{i}=1$ on $\partial
E$ and $w_{i}=0$ on $S\left( R_{i}\right) =\partial E\left( R_{i}\right)
\backslash\partial E.$ If $R_{i}\rightarrow\infty$ and $w_{i}\rightarrow w$
as $i\rightarrow\infty.$ Then,
\begin{equation}
\begin{array}{l}
\int_{E\backslash E\left( R\right) }\left\vert \nabla w\right\vert
^{p}dv\leq C_{3}R^{p}\exp(\frac{-\left( \lambda_{p}\right) ^{1/p}\left(
R-1\right) }{\left( p+1\right) }),%
\end{array}
\label{DDw-2}
\end{equation}
and
\begin{equation}
\begin{array}{l}
\int_{E\left( kR\right) \backslash E\left( R\right) }\left\vert w\right\vert
^{p}dv\leq C_{1}R^{p}\exp(\frac{-\left( \lambda_{p}\right) ^{1/p}\left(
R-1\right) }{p+1}),%
\end{array}
\label{Dw-2}
\end{equation}
for some constant $C$ depending on $p.$
\end{lemma}
\bigskip
\proof As in the proof of Lemma \ref{decay estimate}. If $\phi$ is a
non-negative cut-off function defined by%
\begin{equation*}
\begin{array}{l}
\phi(r(x))=\left\{
\begin{array}{ll}
r(x)-R_{0}\text{ \ \ \ \ \ \ \ \ } & \text{on }E(R_{0}+1)\backslash E(R_{0}),%
\vspace{3mm} \\
1 & \text{on }E\backslash E(R_{0}+1),\vspace{3mm}%
\end{array}
\right.%
\end{array}%
\end{equation*}
$\newline
$and we choose $a=\frac{\delta r\left( x\right) }{p+1}$ for $0<\delta<1.$
It's easy to check that%
\begin{equation*}
\begin{array}{lll}
\left\vert \nabla\phi\right\vert ^{2}\left( x\right) =\left\{
\begin{array}{ll}
1\text{\ \ \ } & \text{on }E(R_{0}+1)\backslash E(R_{0}),\vspace{3mm} \\
0 & \text{on }E\backslash E\left( R_{0}+1\right) ,%
\end{array}
\right. & \text{and} & \left\vert \nabla a\right\vert ^{2}\left( x\right) =%
\frac{\delta^{2}}{\left( p+1\right) ^{2}}.%
\end{array}%
\end{equation*}
By the formula (\ref{de11}), we obtain%
\begin{equation*}
\begin{array}{lll}
\frac{1}{p+1}\left\Vert \phi\exp(a(x))w\right\Vert _{L_{p}} & \leq &
\left\Vert \left( \nabla\phi\right) \exp(a(x))w\right\Vert
_{L_{p}}+\left\Vert \left( \nabla a\right) \phi\exp(a(x))w\right\Vert
_{L_{p}} \\
& \leq & \left\Vert \exp(a(x))w\right\Vert _{L_{p}\left( E\left(
R_{0}+1\right) \backslash E\left( R_{0}\right) \right) }+\frac{\delta }{p+1}%
\left\Vert \phi\exp(a(x))w\right\Vert _{L_{p}\left( E\right) }%
\end{array}%
\end{equation*}
hence%
\begin{equation*}
\begin{array}{l}
\left( \frac{1-\delta}{p+1}\right) \left\Vert \phi\exp(a(x))w\right\Vert
_{L_{p}\left( E\backslash E\left( R_{0}+1\right) \right) }\leq\left\Vert
\exp(a(x))w\right\Vert _{L_{p}\left( E(R_{0}+1)\backslash E(R_{0})\right) }.%
\end{array}%
\end{equation*}
Then we obtain that
\begin{equation}
\begin{array}{l}
\left( 1-\delta\right) \left\Vert \exp(\delta r)w\right\Vert _{L_{p}\left(
E\backslash E\left( R_{0}+1\right) \right) }\leq C_{1},%
\end{array}
\label{DDw1}
\end{equation}
for some constant $0<C_{1}=C_{1}\left( p\right) <\infty.$
Moreover, since%
\begin{equation*}
\begin{array}{lll}
\frac{1}{p}\left\Vert \psi\nabla w\right\Vert _{L_{p}} & \leq & \left\Vert
\nabla\psi w\right\Vert _{L_{p}} \\
& \leq & \left\Vert \exp(a(x))w\right\Vert _{L_{p}\left(
E(R_{0}+1)\backslash E(R_{0})\right) }+\delta\left\Vert
\phi\exp(a(x))w\right\Vert _{L_{p}\left( E\right) } \\
& \leq & C_{2}+\frac{\delta C_{1}}{1-\delta}.%
\end{array}%
\end{equation*}
Hence, we conclude%
\begin{equation}
\begin{array}{l}
\left( 1-\delta\right) \left\Vert \exp(\delta r\left( x\right) )\nabla
w\right\Vert _{L_{p}\left( E\backslash E\left( R_{0}+1\right) \right) }\leq
C_{3}%
\end{array}
\label{DDw2}
\end{equation}
for some constant $0<C_{3}=C_{3}\left( p\right) <\infty.$
If we select $\delta=\left( 1-\frac{1}{R}\right) $ and $R_{0}>1,$ then \ref%
{DDw1}) gives%
\begin{equation*}
\begin{array}{lll}
C_{3} & \geq & \frac{1}{R^{p}}\int_{E\backslash E\left( R_{0}+1\right)
}\exp\left( (1-\frac{1}{R})\frac{\left( \lambda_{p}\right) ^{1/p}r}{p+1}%
\right) \left\vert \nabla w\right\vert ^{p}dv \\
& \geq & \frac{1}{R^{p}}\int_{E\left( kR\right) \backslash E\left(
R_{0}+1\right) }\exp\left( (1-\frac{1}{R})\frac{\left( \lambda_{p}\right)
^{1/p}r}{p+1}\right) \left\vert \nabla w\right\vert ^{p}dv.%
\end{array}%
\end{equation*}
Hence%
\begin{equation*}
\begin{array}{l}
\int_{E\left( kR\right) }\exp(\frac{\left( \lambda_{p}\right) ^{1/p}\left(
R-1\right) r}{\left( p+1\right) R})\left\vert \nabla w\right\vert ^{p}dv\leq
C_{3}R^{p},%
\end{array}%
\end{equation*}
and then we have%
\begin{equation*}
\begin{array}{l}
\int_{E\left( kR\right) \backslash E\left( R\right) }\left\vert \nabla
w\right\vert ^{p}dv\leq C_{3}R^{p}\exp(\frac{-\left( \lambda_{p}\right)
^{1/p}\left( R-1\right) }{\left( p+1\right) }),%
\end{array}%
\end{equation*}
for all constant $k>1.$
Similarly, \ref{DDw2} implies%
\begin{equation*}
\begin{array}{l}
\int_{E\left( kR\right) }\exp(\frac{\left( \lambda_{p}\right) ^{1/p}\left(
R-1\right) r}{\left( p+1\right) R})\left\vert w\right\vert ^{p}dv\leq
C_{1}R^{p}%
\end{array}%
\end{equation*}
and%
\begin{equation*}
\begin{array}{l}
\int_{E\left( kR\right) \backslash E\left( R\right) }\left\vert w\right\vert
^{p}dv\leq C_{1}R^{p}\exp(\frac{-\left( \lambda_{p}\right) ^{1/p}\left(
R-1\right) }{p+1}),%
\end{array}%
\end{equation*}
for any constant $k>1.$
\endproof
\bigskip
\begin{lemma}
\label{Volume estimate}Let $M$ be a complete noncompact manifold satisfying $%
\left( P_{\lambda_{p}}\right) ,$ $p>1$. If $E$ is a $p$-hyperbolic end of $%
M^{n}$, then
\begin{equation*}
\begin{array}{l}
V\left( E(R+1)\right) -V\left( E\left( R\right) \right) \geq CR^{-p\left(
p-1\right) }\exp(\frac{\left( p-1\right) \left( \lambda _{p}\right)
^{1/p}\left( R-1\right) }{p+1}).%
\end{array}%
\end{equation*}
for some constant $C>0,$ and for $R$ sufficiently large. If $E$ is $p$%
-parabolic, then%
\begin{equation*}
\begin{array}{l}
V\left( E\right) <\infty%
\end{array}%
\end{equation*}
and%
\begin{equation*}
\begin{array}{l}
V\left( E\right) -V\left( E\left( R\right) \right) \leq CR^{p}\exp (\frac{%
-\left( \lambda_{p}\right) ^{1/p}\left( R-1\right) }{p+1})%
\end{array}%
\end{equation*}
for some constant $C>0,$ for any $0<\delta<1,$ and for $R$ sufficiently
large.
\end{lemma}
\bigskip
\proof If $E$ is $p$-parabolic, we select the barrier function $w=1$ on $E,$
then (\ref{Dw-2}) implies%
\begin{equation*}
\begin{array}{l}
\int_{E\backslash E\left( R\right) }dv\leq CR^{p}\exp(\frac{-\left(
\lambda_{p}\right) ^{1/p}\left( R-1\right) }{p+1})%
\end{array}%
\end{equation*}
for all $R$ large enough and for any $\delta$ satisfying $0<\delta<1.$ This
implies $V\left( E\right) <\infty.$
If $E$ is $p$-hyperbolic. Let $w$ be the barrier function on $E$, and $%
S\left( R\right) =\partial E\left( R\right) \backslash\partial E,$ then%
\begin{equation}
\begin{array}{lll}
C & = & \int_{\partial E}\left\vert \nabla w\right\vert ^{p-2}\frac{\partial
w}{\partial\nu}dA \\
& \leq & \int_{S\left( r\right) }\left\vert \nabla w\right\vert ^{p-1}dA \\
& \leq & \left( \int_{S\left( r\right) }\left\vert \nabla w\right\vert
^{p}dA\right) ^{\left( p-1\right) /p}\left( \int_{S\left( r\right)
}dA\right) ^{1/p}.%
\end{array}
\label{13}
\end{equation}
Then (\ref{13}) imply%
\begin{equation*}
\begin{array}{lll}
\int_{R}^{R+1}\left( \int_{S\left( r\right) }dA\right) ^{-1/\left(
p-1\right) }dr & \leq & C\int_{R}^{R+1}\int_{S\left( r\right) }\left\vert
\nabla w\right\vert ^{p}dAdr \\
& = & C\int_{E\left( R+1\right) \backslash E\left( R\right) }\left\vert
\nabla w\right\vert ^{p}dv.%
\end{array}%
\end{equation*}
By using Schwarz inequality,%
\begin{equation*}
\begin{array}{lll}
1 & = & \int_{R}^{R+1}\left( \int_{S\left( r\right) }dA\right) ^{-\frac {1}{p%
}}\left( \int_{S\left( r\right) }dA\right) ^{\frac{1}{p}}dr \\
& \leq & \left( \int_{R}^{R+1}\left( \int_{S\left( r\right) }dA\right) ^{-%
\frac{1}{p-1}}dr\right) ^{\frac{p-1}{p}}\cdot\left(
\int_{R}^{R+1}\int_{S\left( r\right) }dAdr\right) ^{\frac{1}{p}} \\
& \leq & C\left( \int_{E\left( R+1\right) \backslash E\left( R\right)
}\left\vert \nabla w\right\vert ^{p}dv\right) ^{\frac{p-1}{p}}\cdot\left(
\int_{R}^{R+1}\int_{S\left( R\right) }dAdr\right) ^{\frac{1}{p}}.%
\end{array}%
\end{equation*}
Then co-area formula and (\ref{DDw-2}) give%
\begin{equation*}
\begin{array}{l}
\int_{E\left( R+1\right) \backslash E\left( R\right) }dv\geq CR^{-p\left(
p-1\right) }\exp(\frac{\left( p-1\right) \left( \lambda_{p}\right)
^{1/p}\left( R-1\right) }{p+1}).%
\end{array}%
\end{equation*}
\endproof
\bigskip
\bigskip
Since $\left( P_{\lambda_{p}}\right) $ implies the volume of $M$ is
infinity, then Lemma \ref{Volume estimate} implies the following property.
\begin{theorem}
\label{hy}If $M$ is a complete noncompact manifold satisfying $\left(
P_{\lambda_{p}}\right) ,$ then $M$ must be $p$-hyperbolic.
\end{theorem}
\textbf{Acknowledgments. }The authors would like to express their thanks to
the referee for valuable comments. S.C. Chang and J.T. Chen were partially
supported by NSC, and S.W. Wei was partially supported by NSF. Part of this
paper has been written while the second author was visiting University of
Oklahoma in summer 2011. He is grateful to Professor S.W. Wei and to the
Department of Mathematics for the kind hospitality.
\bigskip
|
1,314,259,995,862 | arxiv | \section{Introduction}
\label{sec:intro}
Magnetic fields have been observed at all scales in the Universe. In galaxies, the field strengths range between a few to tens of $\mu$Gauss, correlated on kpc scales~\cite{Bernet:2008qp}. Magnetic fields of similar amplitude have also been measured in galaxy clusters, ordered on scales up to the Mpc~\cite{Feretti:2012vk}, and they have also been detected in superclusters~\cite{Xu:2005rb} and filaments~\cite{2010A&A...511L...5G} at smaller strengths. Even the voids of the large-scale structure are expected to host magnetic fields at the level of $10^{-15}$ G. These are estimated from observations of TeV blazars, whose emitted gamma-rays are expected to produce electron-positron cascades when interacting with background light. Upon inverse Compton scattering with the CMB, the charged particles are converted into GeV gamma-rays, which should reach observers on Earth, unless a magnetic field deflects the cascade away from the line-of-sight~\cite{Durrer:2013pga}. The lack of observation of these GeV halos has been used by the FERMI satellite to put \emph{lower} bounds on the void magnetic field strength of $10^{-13}$ --- $10^{-15}$ G, depending on assumptions about the gamma-ray jet life-time~\cite{Biteau:2018tmv}.\footnote{Note, however, that these results have been put into question by Broderick et al. in Ref.~\cite{Broderick:2018nqf}. In this work, TeV sources are observed off-axis, for which the authors argue that GeV gamma-rays should be detected. Their non-detection instead implies an \emph{upper} bound on the strength of inter-galactic magnetic fields of $10^{-15}$~G.}
Galactic magnetic fields are explained by current theories via a dynamo mechanism, activated in the final stages of gravitational collapse. This mechanism amplifies a pre-existing magnetic field until it saturates at the equipartition level of $\mu$G, at which we observe it today. The origin of the magnetic seed required to activate this mechanism is so far unknown and its size is difficult to estimate from these measurements, given that our observations only probe the saturated magnetic field and modelling of the dynamo is non-trivial~\cite{Brandenburg:2004jv}. However, using simplified arguments and assuming the efficiency of the dynamo is well understood, lower bounds can be found, ranging from the optimistic $10^{-30}$ G, assuming equipartition was only reached today~\cite{Davis:1999bt}, to $10^{-15}$ G when this saturation occurs already at redshift $z\sim2$~\cite{Durrer:2013pga}.
Both magnetic fields in voids and those in galaxies and other structures could have a common origin. Astrophysical processes could explain these fields, in particular if generated during the complex stages of structure collapse, in which non-linear dynamics plays a role. Another alternative is that these seed magnetic fields could be generated primordially, before structure formation takes place. Should that be the case, this primordial magnetic field would affect the Cosmic Microwave Background (CMB) through many different effects. Since these effects have remained undetected, Planck~\cite{Ade:2015cva} has placed upper bounds of approximately $10^{-9}$ G on the amplitude of magnetic fields at Mpc scales. Upper limits have also been placed on the total, integrated, magnetic field, at the level of $10^{-12}$ G~\cite{Jedamzik:2018itu}. Faraday rotation measures have also been used to place constraints on extra-galactic magnetic fields at the $n$G level, which are independent on the origin and generation mechanism of magnetic fields~\cite{Blasi:1999hu,Pshirkov:2015tua}.
Many models for the generation of primordial magnetic fields exist~\cite{Grasso:2000wj, Widrow:2002ud, Durrer:2013pga}, which differ greatly in terms of the epoch of magnetogenesis as well as in the amount of exotic physics necessary. Inflationary models can generate appreciable magnetic fields on very large scales, but they require the introduction of new physics to break the conformal invariance of electromagnetism~\cite{Turner:1987bw,Dimopoulos:2001wx,Ashoorioon:2004rs,Ferreira:2013sqa,Caprini:2017vnn}. Phase transitions, such as the GUT, electroweak or QCD transitions can generate considerable amplitudes for the magnetic fields~\cite{Vachaspati:1991nm,Kamada:2018tcs}, but their correlation length is always very small, unless their magnetic helicity is substantial~\cite{Caprini:2009pr}.
A more conservative alternative is the generation of vortical currents in the early Universe, after electron-position annihilation~\cite{1970MNRAS.147..279H,Matarrese:2004kq, Gopal:2004ut,Takahashi:2005nd,Ichiki:2006cd,Siegel:2006px, Ichiki:2007hu,Kobayashi:2007wd, Maeda:2008dv, Fenu:2010kh,Maeda:2011uq, Nalson:2013jya, Saga:2015bna, Fidler:2015kkt}. This mechanism is always present, as it only requires standard electromagnetism and the dynamics of fluctuations in the baryon-photon plasma. However, magnetic fields generated through this mechanism have a rather small amplitude, since the effect appears only at second order in cosmological perturbations and is suppressed by the tight coupling of baryons and photons. The most recent calculation of the spectrum of magnetic fields generated through this mechanism are detailed in Ref.~\cite{Fidler:2015kkt} and found an amplitude of order $10^{-29}$ G on Mpc scales at $z=0$.
Most previous works have assumed that the initial conditions for cosmological fluctuations are adiabatic. However, it has been suggested by Maeda et al. in Ref.~\cite{Maeda:2011uq} and Nalson et al. in Ref.~\cite{Nalson:2013jya}, that non-adiabatic fluctuations could enhance the magnetic field produced via this mechanism. Those conclusions are based on analytical calculations, which do not include all effects due to recombination and therefore require further confirmation. In this paper, we use the Boltzmann solver SONG\footnote{https://github.com/coccoinomane/song} \cite{pettinari:2013a,pettinari:2015a} with isocurvature initial conditions to compute the enhancement of the spectrum of magnetic fields created by these non-adiabatic modes. We are informed by the most recent results from Planck regarding the possible amplitude of the spectrum of primordial isocurvature modes and their spectral index~\cite{Akrami:2018odb}, but also expand on those possibilities, to better understand the effect of generic isocurvatures. We are also particularly interested in exploring the existence of a compensated isocurvature mode~\cite{2010ApJ...716..907H,Grin:2011tf,Grin:2013uya,He:2015msa}. This mode is an anti-correlated mixture of the baryon and dark matter (DM) density isocurvatures, in which those modes compensate each other to avoid the existence of a matter isocurvature mode. This fact implies that its primordial amplitude is very difficult to constrain. However, this mode does generate an observable effect in the CMB by modulating the background baryon-to-DM ratio, which has allowed Planck to estimate its amplitude to be six orders of magnitude larger than that of the adiabatic mode~\cite{Akrami:2018odb}. In this work, we also investigate the effects of this mode on magnetic field generation, since its large amplitude should allow for a substantial enhancement of the magnetic power spectrum.
This paper is organized in the following way: in Section~\ref{sec:mag}, we review the mechanism responsible for generating magnetic fields and show its evolution equations; in Section~\ref{sec:res} we show our numerical results for the spectrum of magnetic fields generated with different types of isocurvature initial conditions, including the compensated isocurvature mode. Finally, in Section~\ref{sec:conc}, we discuss our findings and explain the relevance of our results for the study of the cosmological generation of magnetic fields, as well as for the study of the early Universe in general. In Appendix \ref{sec:ini}, we detail the initial conditions for the vector degrees of freedom required to initialize the numerical evolution, following the techniques laid out in Ref.~\cite{Carrilho:2018mqy}.
\section{Magnetogenesis from vortical currents}
\label{sec:mag}
The mechanism for magnetogenesis studied here acts in the early Universe, starting from around the end of electron-positron annihilation and being active almost until today. During this entire stage, the species of interest are photons ($\gamma$), electrons ($e$), and protons ($p$), which comprise the tightly coupled baryon-photon plasma, prior to recombination. Additionally, cold dark matter (c) and neutrinos ($\nu$) are also present, but do not affect the generation of magnetic fields directly. We assume here Einstein's general relativity to describe the geometry in which these species live. The metric tensor, $g_{\alpha\beta}$, evolves according to the Einstein field equations
\begin{equation}
R^{\alpha\beta}-\frac12 g^{\alpha\beta}R=8\pi G T^{\alpha\beta}\,,
\end{equation}
in which $R^{\alpha\beta}$ is the Ricci tensor, $R$ is the Ricci scalar, $G$ is Newton's constant and $T^{\alpha\beta}$ is the stress-energy tensor, which is given by the sum of the stress-energy tensors of all species (s),
\begin{equation}
T^{\alpha\beta}=\sum_s T_s^{\alpha\beta}\,,
\end{equation}
which are given by
\begin{align}
&T_{\text{c}}^{\alpha\beta}=\rho_{\text{c}} u_{\text{c}}^\alpha u_{\text{c}}^\beta\,,\\
&T_e^{\alpha\beta}=\rho_e u_e^\alpha u_e^\beta\,,\\
&T_p^{\alpha\beta}=\rho_p u_p^\alpha u_p^\beta\,,\\
&T_\gamma^{\alpha\beta}=\frac43\rho_\gamma u_\gamma^\alpha u_\gamma^\beta+\frac13 \rho_\gamma g^{\alpha\beta}+\pi_\gamma^{\alpha\beta}\,,\\
&T_\nu^{\alpha\beta}=\frac43\rho_\nu u_\nu^\alpha u_\nu^\beta+\frac13 \rho_\nu g^{\alpha\beta}+\pi_\nu^{\alpha\beta}\,.
\end{align}
We have here defined the energy densities of each species as $\rho_s$, the 4-velocity vectors as $u_s^\alpha$, and the anisotropic stress tensors for photons and neutrinos as $\pi_\gamma^{\alpha\beta}$ and $\pi_\nu^{\alpha\beta}$, respectively. We are treating electrons, protons and dark matter as pressureless perfect fluids, as is clear by the absence of pressure and anisotropic stress in the their stress-energy tensors. Photons and neutrinos are assumed to be relativistic species with $P_s=\rho_s/3$. We have written all species in their respective energy frames, as shown by the lack of an energy flux term, $q^\mu$ in the expressions above.
In addition to these species, an electromagnetic field is present, which is described by the Faraday tensor, $F_{\mu\lambda}$. Using a normalised time-like 4-vector field, $u^\alpha$, to represent a set of observers, one may then define an electric field, $E^\mu$, and a magnetic field, $B^\mu$, via
\begin{equation}
E^\mu=F^{\mu\lambda}u_\lambda\,, \ B^\mu=u_\alpha \epsilon^{\alpha\mu\lambda\beta}F_{\lambda\beta}\,,
\end{equation}
in which $\epsilon^{\alpha\mu\lambda\beta}$ is the totally antisymmetric tensor, with $\epsilon_{0123}=\sqrt{-g}$, with $g$ the determinant of the metric.
The particle species evolve according to their Boltzmann equations coupled to the Einstein and Maxwell equations. For the purposes of this paper, it suffices to display here only the equations for the first two multipoles of the distribution functions, which can be written in terms of the divergence of the stress-energy tensors of the different species:
\begin{align}
&\nabla_\alpha T_c^{\alpha\mu}=0\,,\\
&\nabla_\alpha T_e^{\alpha\mu}=F^\mu_{\ \lambda} j^\lambda_e+C^\mu_{e\gamma}+C^\mu_{ep}\\
&\nabla_\alpha T_p^{\alpha\mu}=F^\mu_{\ \lambda} j^\lambda_p+C^\mu_{p\gamma}-C^\mu_{ep}\\
&\nabla_\alpha T_\gamma^{\alpha\mu}=-C^\mu_{p\gamma}-C^\mu_{e\gamma}\,,\\
&\nabla_\alpha T_\nu^{\alpha\mu}=0\,.
\end{align}
The electric currents $j_s^\mu$ are given by
\begin{equation}
j_s^\mu=q_s n_s u_s^\mu\,,
\end{equation}
with $q_s$ the charge of the particles in question, being equal to the fundamental electronic charge, $e$, for protons and $-e$ for electrons. The symbol $n_s$ denotes the number density of the species in question as seen by an observer in the $u_s$ frame. $C_{sr}^\mu$ are the collision terms for the interactions between the species $s$ and $r$, which will be detailed below.
The electromagnetic field obeys Maxwell's equations
\begin{equation}
\nabla_\lambda F^{\mu\lambda}=j_e^\mu+j_p^\mu\,,\ \ \nabla_{[\alpha}F_{\mu\lambda]}=0\,,
\end{equation}
which can be written in terms of the electric and magnetic fields as
\begin{align}
\nabla_\alpha B^\alpha+u^\alpha u^\beta\nabla_\beta B_\alpha &=-\epsilon_{\alpha\beta\mu\lambda}E^\alpha u^\beta\nabla^\lambda u^\mu\,,\\
\nabla_\alpha E^\alpha+u^\alpha u^\beta\nabla_\beta E_\alpha &=\epsilon_{\alpha\beta\mu\lambda}B^\alpha u^\beta\nabla^\lambda u^\mu+\varrho\,,\\
u^\alpha\nabla_\alpha B^\mu+B^\mu\nabla_\alpha u^\alpha-B^\alpha\nabla_\alpha u^\mu+& u^\alpha u^\beta u^\mu\nabla_\beta B_\alpha \nonumber\\
&=-\epsilon^{\mu\alpha\beta\lambda} u_\alpha\nabla_\lambda E_\beta-h^{\lambda\mu}\epsilon_{\lambda\sigma\alpha\beta}\nabla^\sigma u^\alpha E^\beta\,,\label{FaradayFull}\\
u^\alpha\nabla_\alpha E^\mu+E^\mu\nabla_\alpha u^\alpha-E^\alpha\nabla_\alpha u^\mu+& u^\alpha u^\beta u^\mu\nabla_\beta E_\alpha \nonumber\\
&=\epsilon^{\mu\alpha\beta\lambda} u_\alpha\nabla_\lambda B_\beta+h^{\lambda\mu}\epsilon_{\lambda\sigma\alpha\beta}\nabla^\sigma u^\alpha B^\beta-J^\mu\,,
\end{align}
in which $h^{\mu\lambda}=g^{\mu\lambda}+u^\mu u^\lambda$ is the projection tensor to the space perpendicular to the frame $u^\mu$ and $\varrho$ and $J^\mu$ are projections of $j^\mu=j^\mu_e+j^\mu_p$ given by
\begin{equation}
\varrho=-u_\mu j^\mu\,,\ \ J^\mu=h^\mu_{\lambda}j^\lambda\,.
\end{equation}
We now expand these equations around the flat Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) spacetime, up to second order in cosmological fluctuations and we show them in Poisson gauge, for which the line element simplifies to
\begin{equation}
\text{d}s^2=a(\eta)^2\left[-(1+2\phi)\text{d}\eta^2-2S_i\text{d}x^i\text{d}\eta+(1-2\psi)\delta_{ij}\text{d}x^i\text{d}x^j\right]\,,
\end{equation}
which we have written in terms of conformal time and have performed a scalar-vector-tensor decomposition. We denote by $a(\eta)$ the scale factor, by $S_i$ the vector part of the shift --- the only non-zero vector potential present in the metric in this gauge --- by $\phi$ the perturbation to the lapse and by $\psi$ the curvature fluctuation in this gauge. These scalar potentials as defined in this gauge are equal to the two gauge invariant Bardeen potentials \cite{Bardeen:1980kt}. For simplicity, we have neglected the tensor mode as it will not be important for the magnetic field calculation and we will assume that the vector mode, $S_i$, is only non-zero at second order, as we are not considering primordial vector modes. Our normalization for second-order quantities includes a factor of $1/2$, so that the vector mode, being purely second order, is given by $S_i=\frac12S_i^{(2)}$.
Other variables are expanded and decomposed into scalars, vectors and tensors in the standard way, as given in Refs.~\cite{Malik:2008im,Carrilho:2015cma}. In particular, in the Poisson gauge, the 4-velocity is expanded as
\begin{align}
&u^{0}=a^{-1}\left(1-\phi+\frac32\phi^2+\frac12v_i v^i\right)\,,\\
&u^{i}=a^{-1}v^i=a^{-1}\left( v^{,i}+v^i_\text{v}\right)\,,
\end{align}
while the anisotropic stress is given by
\begin{align}
\pi_{00}=0,\ \ \ \pi_{i0}=-2\pi_{ij} v^j\,,\nonumber\\
\pi_{ij}=a^2\left[\Pi_{ij}+\Pi_{(i,j)}+\Pi_{,ij}-\frac{1}{3}\delta_{ij}\nabla^2\Pi\right]\,.
\end{align}
The electric and magnetic fields are also decomposed and expanded up to second order in perturbations. This results in
\begin{align}
E_0=- E_i v^i\,,\ E_i=a (E_{\text{v}\,i}+E_{,i})\,,\\
B_0=- B_i v^i\,,\ B_i=a (B_{\text{v}\,i}+B_{,i})\,.
\end{align}
The collision terms are given in Refs.~\cite{Fenu:2010kh,Maeda:2011uq,Saga:2015bna}. The interaction term for Coulomb scattering is proportional to the velocity difference between electrons and protons. This interaction between the charged species is very strong and completely dominates the dynamics at early times, giving rise to a tightly coupled fluid, in which the velocity fields of its constituents nearly match. This implies electrons and protons can be considered a single fluid of baryons with velocity, $v_{\text{b}}$, given by
\begin{equation}
v_{\text{b}}^i=\frac{m_p v_p^i+m_e v_e^i}{m_p+m_e}\,.
\end{equation}
As shown in Ref.~\cite{Fenu:2010kh}, for the reason above, the collision term between electrons and protons does not enter the calculation of the magnetic field and, for brevity, we do not show it here.
The momentum transfer rates for the Compton/Thomson interactions are obtained by projecting the collision terms for that interaction with $h_{\mu\lambda}$. For the charged species $r$, the momentum transfer rate is given by
\begin{equation}
C^{r\gamma}_{\, i}=\frac43 \rho_\gamma \left(\frac{m_e}{m_r}\right)^2\kappa' (1+\delta_r+\delta_\gamma-2\psi) (v_{r\,i}-v_{\gamma\,i})+ \left(\frac{m_e}{m_r}\right)^2\kappa' \frac{1}{a^2}v_r^j\pi_{\gamma\, ij}\,,\label{ThomsonC}
\end{equation}
where we have defined the interaction rate $\kappa'=-a n_e \sigma_T$, with $\sigma_T$ the Thomson cross section and $n_e$ the background number density of free electrons, which is equal to that of free protons and will often be denoted simply as $n$. We have also defined the density contrasts $\delta_s=\delta\rho_s/\rho_s$, in which $\rho_s$ is the background density of species $s$ and $\delta\rho_s$ is its density perturbation. The density contrast $\delta_r$ represents here only the free charged particles.
It is clear from Eq.~\eqref{ThomsonC} that the Compton interaction is far more effective for electrons than it is for protons, given their substantial mass difference. This gives rise to charge separation due to this imbalance. An electric field is thus generated, which is given by\footnote{This equation is derived in detail in Ref.~\cite{Fenu:2010kh}, in which an analysis of the different time-scales of the problem is included and the term shown here is concluded to be the dominant one.}
\begin{equation}
\label{Elefield}
E_i=-\frac{1-\beta^3}{1+\beta}\frac{a\sigma_T}{e}\left(\frac{4}{3}\rho_\gamma(1+\delta_\gamma-2\psi)\Delta v_{\text{b}\gamma\, i}+\frac{1}{a^2}v_{\text{b}}^j\pi_{\gamma\, ij}\right)\,,
\end{equation}
where we have introduced new notation for the mass ratio $\beta=m_e/m_p$ and the velocity difference $\Delta v_{\text{b}\gamma\, i}=v_{\text{b}\,i}-v_{\gamma\,i}$. As stated above, it is the mass difference between electrons and protons which gives rise to this electric field, as this would not be possible with $\beta=1$. It is for a similar reason that this electric field can only be generated after electron-positron annihilation, as, before that, the mass ratio of relevance is that of positrons and electrons, which is unity.
A magnetic field can thus be generated via Faraday's law, Eq.~\eqref{FaradayFull}. Expanding that equation up to second order in fluctuations, one finds
\begin{equation}
\label{Faradcoord}
\left( a B_i\right)'=-a\epsilon_{ij}^{\ \ k}\partial^j\left( (1+\phi+v')E_k\right)\,.
\end{equation}
We see the frame-dependence of this field very clearly in the term with $v'$. This frame is often chosen to be a local inertial frame with the observer's 4-velocity, $u^\mu$, being aligned with one of the axis of a tetrad basis, $e_{\underline{a}}^\mu$. Should this alignment be such that $e_{\underline{0}}^\mu=u^\mu$, and in the Poisson gauge, we have $v=0$, $E_k=a (1-\psi) E_{\underline{k}}$ and $B_i=a B_{\underline{i}}$, where the underlined indices label the components of tensors in the tetrad basis.\footnote{This is the tetrad used in Ref.~\cite{Fenu:2010kh}, and the components of the basis vectors are presented in its Appendix B. Other choices may be made, such as $e^{\underline{0}}_\mu=u_\mu$, as is done in Refs.~\cite{pettinari:2015a, Carrilho:2018mqy} and in the Boltzmann solver SONG, which we use here. It can be shown that both choices give the same evolution equation for the magnetic field.} In that case, Eq.~\eqref{Faradcoord} becomes
\begin{equation}
\label{Faradtetrad}
\left( a^2 B_{\underline{i}}\right)'=-a^2\epsilon_{ij}^{\ \ k}\partial^j\left( (1+\phi-\psi)E_{\underline{k}}\right)\,,
\end{equation}
which is the version used in Refs.~\cite{Fenu:2010kh,Fidler:2015kkt} and which we also use in our numerical studies below.
Different frame choices have been studied in Ref.~\cite{Fenu:2010kh} and the results show differences between the baryon frame and the fundamental frame, at early times and large scales, but no effect at $z=0$. One could also use an alternative frame, such as the energy frame, but the results are not expected to vary significantly, unless a frame is chosen with a very high velocity with respect to the frame comoving with the expansion.
Another well-known aspect of this mechanism is that it does not occur at first order in fluctuations, in the absence of primordial vector modes. This is transparent in Eqs.~\eqref{Faradcoord} and~\eqref{Faradtetrad}, since the magnetic field is sourced by the curl of the electric field, which would only be non-zero at linear level in the presence of linear vectors.
Finally, we are able to write the evolution equation for the magnetic field in terms of variables commonly computed in a Boltzmann solver:
\begin{equation}
\label{Faraddv}
\left( a^2 B_{\underline{i}}\right)'=a^2\frac{1-\beta^3}{1+\beta}\frac{\sigma_T}{e}\epsilon_{ijk}\partial^j\left( \frac43\rho_\gamma\left(\Delta v_{\text{b}\gamma\, \text{v}}^{\ \ k}+(\delta_\gamma+\phi-2\psi)\Delta v_{\text{b}\gamma}^{\ \ ,k}\right)+v_{\text{b}\,,l}D^{lk}\Pi_\gamma\right)\,,
\end{equation}
with $D^{kl}=\partial^k\partial^l-\frac13\delta^{kl}\nabla^2$ and where we have written the right-hand-side using the scalar-vector-tensor decomposition. It is clear from Eqs.\eqref{Elefield} and \eqref{Faraddv} that the electromagnetic field depends on $\Delta v_{\text{b}\gamma}$ and $\Pi_\gamma$, both of which are strongly suppressed by the strong Thomson interactions in the early Universe. However, this tight-coupling becomes less effective around the time of recombination and an appreciable magnetic field is generated at that time.
To solve Eq.~\eqref{Faraddv}, one needs to solve the full system of linear scalar fluctuations, as well as the system of second-order vector perturbations. As mentioned above, we will use the Boltzmann solver SONG~\cite{pettinari:2013a,pettinari:2015a} to solve the vector equations numerically, which makes use of the linear solver CLASS~\cite{Lesgourgues:2011re,Blas:2011rf} to solve for the linear evolution.
\subsection{Isocurvature modes}
The solution of the Einstein-Boltzmann system requires the specification of initial conditions. Different choices of initial conditions would result in distinct solutions to the evolution equations, which can be probed with experimental data. An analysis of the linear Einstein-Boltzmann system~\cite{Bucher:1999re} has led to the classification of its solution space into regular (growing) and singular (decaying) modes. While decaying modes have also been studied~\cite{Amendola:2004rt}, growing modes are certain to exist and we focus only on those. These growing modes can further be decomposed into an adiabatic and four isocurvature modes, at first order~\cite{Bucher:1999re}, but only three isocurvature modes source growing solutions at second order~\cite{Carrilho:2018mqy}. We define entropy fluctuations of species $r,s$ as
\begin{equation}
S_{rs}=\frac{\delta_r}{1+w_r}-\frac{\delta_s}{1+w_s}\,,
\end{equation}
with $w_s=P_s/\rho_s$ the equation of state parameter for species $s$. The adiabatic mode is that for which all entropy fluctuations vanish initially.
The three isocurvature modes are defined instead by the initial vanishing of the curvature fluctuation $\zeta$, defined as
\begin{equation}
\zeta=-\psi-\frac{\delta}{3(1+w)}\,,
\end{equation}
at first order. Each isocurvature mode is defined by having one of $S_{s\gamma}$ non-zero and are named the baryon, cold dark matter and neutrino density isocurvature modes, depending on which entropy fluctuation is non-zero. A more detailed definition of isocurvature modes is included in Ref.~\cite{Carrilho:2018mqy}, which is also consistently extended to second order.
Isocurvature modes can have an influence on the generation of magnetic fields. Maeda et al.~\cite{Maeda:2011uq} have shown that, in the presence of a baryon isocurvature mode ($S_{\text{b}\gamma}\neq0$), the evolution equation for the magnetic field, Eq.~\eqref{Faraddv}, can be written as
\begin{equation}
\label{FaraddvMaeda}
\left( a^2 B_{\underline{i}}\right)'=\frac{1-\beta^3}{1+\beta}\frac{4\sigma_T}{3e}a^2\rho_\gamma\left( C_\omega(\eta)\,\omega_i+C_S(\eta) \epsilon_{ijk}S_{\text{b}\gamma}^{\ \ ,j}\Delta v_{\text{b}\gamma}^{\ \ ,k}\right)\,,
\end{equation}
where the tight coupling expansion~\cite{Pitrou:2010ai} was used up to first order in $\mathcal{H}/\kappa'$ to approximate the right-hand-side analytically. For that reason, this expression is only valid at early times, long before recombination. The symbols $C_\omega$ and $C_S$ represent time-dependent functions, which we do not specify, and $\omega_i$ is the vorticity of the total fluid. It is clear that without the baryon isocurvature mode, the second term in Eq.~\eqref{FaraddvMaeda} would vanish. The same is also true for the first term, since vorticity can only be generated in the presence of non-adiabatic pressure, as is well known~\cite{Christopherson:2009bt,Christopherson:2010ek,Christopherson:2010dw}. Therefore, a magnetic field cannot be generated at first order in tight coupling in the absence of this isocurvature mode, as argued by Maeda et al.~\cite{Maeda:2011uq}. A similar conclusion appears in Appendix D of Ref.~\cite{Fenu:2010kh} by Fenu et al., in which it is noted that a large suppression exists in the tight coupling approximation when only the adiabatic mode in considered. Similar arguments are also made in Ref.~\cite{Nalson:2013jya}. These conclusions point to the fact that a non-adiabatic mode can provide a large contribution to the source of the magnetic field, at least at sufficiently early times, when the tight coupling expansion is valid. The magnetic field spectrum has been computed analytically in this approximation, up to matter-radiation equality, by Maeda et al. and found to be larger than in the adiabatic case. However, this calculation ignores all effects occurring at a higher order in tight coupling and, especially, it does not take recombination into account, which is the moment in which the sources of the magnetic field are more important.
Another relevant aspect regarding isocurvatures is mode mixing. Given that the adiabatic mode is certain to exist, the presence of an isocurvature mode implies that the two modes will mix due to the non-linear nature of the evolution equations at second order. Beyond generating source terms which do not exist when each of the modes is considered individually, this coupling between modes is particularly important for the sources of the magnetic field, as they include cross products between gradients of scalar variables, i.e. $\epsilon_{ijk}\partial^j A\, \partial^k B$. At sufficiently early times, scalar variables are proportional to the initial value of $\zeta$ or $S_{r\gamma}$, depending on the modes being considered. If a single mode is present, the cross products vanish, as $\epsilon_{ijk}\partial^j A \partial^k B\propto\epsilon_{ijk}\partial^j I \partial^k I=0$. Evidently, the mixing of multiple modes avoids this issue, as one would have, for example, $\epsilon_{ijk}\partial^j A \partial^k B\propto\epsilon_{ijk}\partial^j \zeta \partial^k S_{\text{b}\gamma}\neq0$, unless the two modes are fully correlated or anti-correlated i.e. $S_{\text{b}\gamma}\propto\zeta$.
The arguments made here are valid at early times and large scales, but require more general investigation. This is what motivates us to explore the evolution of magnetic fields in the presence of isocurvature modes numerically using the Boltzmann solver SONG. This has required a modification of the publicly available version of that code to include isocurvature initial conditions. This was studied in Ref.~\cite{Carrilho:2018mqy} for scalar modes and we use the same techniques here to compute approximate initial solutions for vector modes for all isocurvature modes under consideration. The results for the initial conditions are shown in Appendix~\ref{sec:ini}.
\section{Numerical results}
\label{sec:res}
\subsection{Set-up}
We are interested in computing the spectrum of the magnetic field defined as
\begin{equation}
\left\langle B_{\underline{i}}\left(\textit{\textbf{k}},\eta\right) B^{\underline{i}}\left(\textit{\textbf{k}'},\eta\right)\right\rangle=(2\pi)^3 \delta^{(3)}\left(\textit{\textbf{k}}+\textit{\textbf{k}'}\right) P_B(k,\eta)\,.
\end{equation}
It is convenient, for numerical reasons, to decompose the magnetic field with polarization vectors $e_{\pm}^{\underline{i}}$, perpendicular to the direction $\textit{\textbf{k}}$, such that $B^{\underline{i}}=B_+ e_{+}^{\underline{i}}+ B_- e_{-}^{\underline{i}}$. We write these magnetic field components in terms of transfer functions as
\begin{equation}
B_\pm(\textit{\textbf{k}},\eta)=\int \frac{\text{d}^3k_1\text{d}^3k_2}{(2\pi)^3}\delta^{(3)}(\textit{\textbf{k}}-\textit{\textbf{k}}_1-\textit{\textbf{k}}_2)\mathcal{T}_{B_\pm}^{ab}(\textit{\textbf{k}},\textit{\textbf{k}}_1,\textit{\textbf{k}}_2,\eta)I_a(\textit{\textbf{k}}_1) I_b(\textit{\textbf{k}}_2)\,,\label{Btransfer}
\end{equation}
in which $I_a$ denote the variables defining the mode under consideration, taking values in the set $\left\{\zeta,S_{\text{c}\gamma},S_{\text{b}\gamma},S_{\nu\gamma}\right\}$ for the adiabatic mode, cold dark matter, baryon and neutrino isocurvature modes, respectively. The indices $a,b,c,d$ represent those modes, taking four different values: $\zeta,\text{c},\text{b},\nu$ for their respective modes. The Einstein summation convention was employed for those indices. The spectra of the defining variables $I_a$ are defined as
\begin{equation}
\left\langle I_{a}\left(\textit{\textbf{k}},\eta\right) I_b\left(\textit{\textbf{k}'},\eta\right)\right\rangle=(2\pi)^3 \delta^{(3)}\left(\textit{\textbf{k}}+\textit{\textbf{k}'}\right) P_{ab}(k)\,.
\end{equation}
These spectra are parameterized by an amplitude $A_{ab}$ and a spectral index $n_{ab}$, so that they are given by
\begin{equation}
P_{ab}(k)=A_{ab}\frac{2\pi^2}{k^3}\left(\frac{k}{k_*}\right)^{n_{ab}}\,,
\end{equation}
in which we have denoted the pivot scale by $k_*$ and will choose it to be $k_*=0.05\ \text{Mpc}^{-1}$. For $a=b$, one has the standard power spectra, such as the power spectrum of the curvature perturbation $P_{\zeta\zeta}$ for the adiabatic mode, for which the amplitude is more commonly denoted $A_s$ and the spectral index is given by $n_{\zeta\zeta}=n_s-1$. The cases with $a\neq b$ represent the correlations between different modes, which is often parameterized by the correlation fraction
\begin{equation}
\cos\Delta_{ab}=\frac{P_{ab}}{\sqrt{P_{aa} P_{bb}}}\,,
\end{equation}
which, as the notation indicates, must obey $-1<\cos\Delta<1$. Given our power law parameterisation, this can be shown to imply that $n_{ab}=(n_{aa}+n_{bb})/2$.
The transfer functions obey the identities given by
\begin{align}
\mathcal{T}^{ab}(\textit{\textbf{k}},\textit{\textbf{k}}_1,\textit{\textbf{k}}_2,\eta)=\mathcal{T}^{ba}(\textit{\textbf{k}},\textit{\textbf{k}}_2,\textit{\textbf{k}}_1,\eta)\,,\\
\mathcal{T}^{ab}(-\textit{\textbf{k}},-\textit{\textbf{k}}_1,-\textit{\textbf{k}}_2,\eta)=\mathcal{T}^{ab}(\textit{\textbf{k}},\textit{\textbf{k}}_1,\textit{\textbf{k}}_2,\eta)\,.
\end{align}
Thus, one can show that, in this notation, the magnetic field spectrum is given by
\begin{equation}
\label{SpcTra}
P_B(k,\eta)=4\int\frac{\text{d}^3q}{(2\pi)^3}\mathcal{T}^{ab}_{B_+}(\textit{\textbf{k}},\textit{\textbf{q}},\textit{\textbf{k}}-\textit{\textbf{q}},\eta)\mathcal{T}^{cd}_{B_+}(\textit{\textbf{k}},\textit{\textbf{q}},\textit{\textbf{k}}-\textit{\textbf{q}},\eta)P_{ac}(\textit{\textbf{q}})P_{bd}(\textit{\textbf{k}}-\textit{\textbf{q}})\,,
\end{equation}
assuming Gaussian initial conditions and that helical magnetic fields are not generated~\cite{Saga:2015bna}.
We now present our results for the magnetic field generated via this mechanism in the presence of isocurvature modes. We begin by showing the results for the addition of single isocurvature modes and study how the amplitudes and spectral tilts of the different modes affect the magnetic field production via this mechanism. We then show the results obtained for a mixture of isocurvature modes which is particularly interesting --- the so-called compensated isocurvature mode. All cosmological parameters used that are not related to isocurvatures, are taken to be the best-fit values from Planck, as given in Ref.~\cite{Akrami:2018odb}.
\subsection{Magnetogenesis with single isocurvature mode}
We begin by showing results for the magnetic field power spectrum, $P_B$, generated when a single isocurvature mode is present in addition to the adiabatic mode. We assume, initially, that the isocurvature mode has a scale-invariant spectrum ($n_{aa}=0$) with the same amplitude as the adiabatic mode ($A_{aa}=A_s$), but uncorrelated with it ($\cos\Delta_{a\zeta}=0$). We will later change these assumptions to measure the importance of those parameters.
Figure~\ref{fig:ADBI} shows the results for the baryon isocurvature mode. We can see that the contribution from the pure baryon isocurvature mode (BI) is much smaller than the adiabatic mode (Ad). However, the contribution sourced by the mixture of the adiabatic and baryon isocurvature modes (Ad x BI) is similar in size to the adiabatic mode on Mpc scales and enhances the total result by approximately 60\%. On intermediate scales, the largest contribution is still the adiabatic one, but, on very large scales, the largest contribution is from the mixed mode once more. In particular, the spectral index of the mixed contribution on large scales differs by at least $\Delta n=1$ relative to that of the adiabatic mode. However, as we will see below, this is dependent on the spectral index of the isocurvature spectrum.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PlotPBkAdBI.pdf}
\includegraphics[width=0.49\textwidth]{PlotPBaAdBI.pdf}
\caption{Normalized magnetic field power spectrum sourced by the mixture of adiabatic and baryon isocurvature modes as a function of wave number, $k$, at $z=0$ (left) and as a function of scale factor, $a$, for $k=0.06\ \text{Mpc}^{-1}$ (right).}
\label{fig:ADBI}
\end{figure}
Regarding the time evolution shown on the right-hand plot of Fig.~\ref{fig:ADBI}, we see that both pure modes have a similar evolution, in spite of the amplitude difference. The mixed contribution shows substantial differences. In particular, at very early times, the mixed mode is several orders of magnitude larger than the pure modes. This is due to the effects described at the end of Section~\ref{sec:mag} and in Ref.~\cite{Maeda:2011uq}, which also explain the large contribution from the mixed mode on large scales. However, at late times, the mixed mode has a smaller contribution, even dropping below that of the adiabatic mode for this scale ($k=0.06\ \text{Mpc}^{-1}$). This clearly demonstrates that the results of Maeda et al.~\cite{Maeda:2011uq}, in which the evolution was not followed beyond matter-radiation equality, where too optimistic. It is clear from the case presented here, that, after recombination, the magnetic field generated by the adiabatic mode is comparable to that generated by the mixed mode, in spite of the latter being larger at all times prior to recombination.
The result for the cold dark matter isocurvature mode is shown in Fig.~\ref{fig:ADCDI}. We can see that the contribution from the mixed mode (Ad x CDI) is much smaller in this case than it was when the baryon mode was active. This is not surprising, as dark matter does not play a role in magnetogenesis. In spite of the adiabatic mode dominating on small scales, the mixed mode still dominates on the largest scales, for the same reasons as described above --- all mixed modes generate a redder spectrum for the magnetic field on large scales than the pure modes. The time evolution of this solution also shows the mixed mode to be dominant at very early times, but with a smaller amplitude in this case.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PlotPBkAdCDI.pdf}
\includegraphics[width=0.49\textwidth]{PlotPBaAdCDI.pdf}
\caption{Normalized magnetic field power spectrum sourced by the mixture of adiabatic and cold dark matter isocurvature modes as a function of wave number, $k$, at $z=0$ (left) and as a function of scale factor, $a$, for $k=0.06\ \text{Mpc}^{-1}$ (right).}
\label{fig:ADCDI}
\end{figure}
Finally, we show the plots for the neutrino isocurvature mode in Fig.~\ref{fig:ADNI}. Similarly to the dark matter mode, the contributions from the neutrino mode are smaller on small scales than those generated by the adiabatic mode. However, the difference is now $O(1)$, instead of an order of magnitude as before, and for that reason, the total result is slightly enhanced on Mpc scales. On very large scales, and early times, the results are similar to the other modes.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PlotPBkAdNI.pdf}
\includegraphics[width=0.49\textwidth]{PlotPBaAdNI.pdf}
\caption{Normalized magnetic field power spectrum sourced by the mixture of adiabatic and neutrino isocurvature modes as a function of wave number, $k$, at $z=0$ (left) and as a function of scale factor, $a$, for $k=0.06\ \text{Mpc}^{-1}$ (right).}
\label{fig:ADNI}
\end{figure}
We should note once more that all the results above depend crucially on the properties of the initial spectra. The dependence on the amplitudes $A_{aa}$ is simple to derive from Eq.~\eqref{SpcTra}. We note that, if the mixed mode dominates the isocurvature contribution, we have $P_B\propto A_{aa}$, while we have $P_B\propto A_{aa}^2$ if the pure mode is dominant. In all cases shown, the dominant contribution from isocurvatures is the mixed mode, and given that the amplitude of isocurvatures is not expected to be greater than $A_s$, we expect the dependence to be $P_B\propto A_{aa}$, in any realistic case. The upper bounds for the amplitudes of the isocurvature modes given by Planck are: $A_{\text{bb}}\approx 1.12 A_s$ for the baryon mode, $A_{\text{cc}}\approx 0.04 A_s$ for the dark matter mode and $A_{\nu\nu}\approx 0.07 A_s$ for the neutrino density isocurvature~\cite{Akrami:2018odb}. This implies that, when re-scaled to the appropriate amplitude, the contributions of both the dark matter and the neutrino mode to the magnetic field are almost negligible. However, the baryon isocurvature mode is allowed to increase slightly with respect to the plot, giving a result of $\sqrt{k^3 P_B/(2\pi^2)}=1.73\times10^{-29}\ \text{G}$ at $k=0.5\ \text{Mpc}^{-1}$, which should be compared to the result for the adiabatic mode of $1.05\times10^{-29}\ \text{G}$. This amounts to a relative enhancement of approximately $64\ \%$.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PlotPBkAdBIvsn.pdf}
\includegraphics[width=0.49\textwidth]{PlotPBkAdBIvscos.pdf}
\caption{Normalized magnetic field power spectrum sourced by the mixture of adiabatic and baryon isocurvature modes as a function of wave number, $k$, at $z=0$ for different spectral indices of the baryon mode, $n_{\text{bb}}$ (left), and for different correlation angles, $\cos\Delta_{\text{b}\zeta}$ (right).}
\label{fig:ADBIncos}
\end{figure}
The dependence on the spectral index can be seen on the left-hand side of Fig.~\ref{fig:ADBIncos}, for the baryon mode. We plot only the sum of all contributions, instead of the different mixtures of sourced modes. We see that on Mpc scales, the magnetic field is amplified further for bluer input spectra, with the opposite happening on very large scales. The case $n_{\text{bb}}=3$ is the one used by Maeda et al.~\cite{Maeda:2011uq} and results in a magnetic field which is approximately 30 times larger than the adiabatic case at $k=0.5\ \text{Mpc}^{-1}$ and with a bluer spectral index. The most recent Planck results prefer a spectral index for the baryon isocurvature mode that is closer to $n_{\text{bb}}=1$~\cite{Akrami:2018odb}, for which the enhancement factor of the magnetic field is only approximately 3.
The right-hand side of Fig.~\ref{fig:ADBIncos} shows how the resulting magnetic field varies when the adiabatic mode has a non-zero correlation with the baryon isocurvature mode. We note that partially and fully anti-correlated baryon isocurvatures give rise to a larger enhancement of the magnetic field at small scales, bringing the magnetic field produced to roughly double that generated with the adiabatic mode alone. A positive correlation always reduces the enhancement due to the presence of the isocurvature, even almost eliminating it for the fully correlated case. Additionally, we note that for both the fully correlated and anti-correlated cases, the behaviour approaches that of the adiabatic mode on large scales. This was already expected due to the arguments presented at the end of Section \ref{sec:mag} --- the curls of gradients of scalars are suppressed on large scales, if the scalars in question are correlated. One should note, however, that the level of correlation currently allowed by Planck is small, with the reported 95\% CL interval being $\cos\Delta_{\text{b}\zeta}\in[-0.12,0.15]$ and these levels would not affect the magnetic field substantially.
The combination of both a blue spectral index ($n_{\text{bb}}=3$) and a full anti-correlation between the isocurvature and adiabatic modes was also tested and was found not to yield a substantially larger improvement, resulting in an enhancement ratio of approximately 31 instead of the 30 found with only $n_{\text{bb}}=3$. For other modes, similar enhancements can be found when varying the spectral index and the correlation angle. However, the baryon isocurvature mode is the one that gives rise to the largest magnetic field in all comparable cases.
We can conclude here that the presence of single isocurvature modes does enhance the magnetic field produced during the pre-recombination stage. However, this improvement is at most of an order of magnitude in the most extreme case, bringing the magnetic field to $3.3\times 10^{-28}$ G in the most optimistic scenario. This is still far from the levels required to explain observations ($\sim10^{-15}$ G).
\subsection{Magnetogenesis with compensated isocurvature mode}
We now move on to the case of the compensated isocurvature mode. This is a mode in which the density fluctuations in the non-relativistic matter species compensate each other so that no total matter entropy fluctuation exists. This is achieved by the relation
\begin{equation}
\delta_{\text{b}}=-\frac{\Omega_{\text{c}}}{\Omega_{\text{b}}}\delta_{\text{c}}\,,
\end{equation}
which is equivalent to
\begin{equation}
A_{\text{bb}}=\left(\frac{\Omega_{\text{c}}}{\Omega_{\text{b}}}\right)^2A_{\text{cc}}\,,\ \cos\Delta_{\text{bc}}=-1\,.
\end{equation}
This mode is interesting because it has essentially no effect at the linear level. It has particularly little effect on the CMB, but can be constrained, as it can contribute to the smoothing of the peaks of the angular power spectrum of the CMB at small angular scales. This effect is similar to that generated by lensing and, for that reason, can help reduce the tension between different observables of the lensing potential. The most recent Planck results give a best-fit value for the amplitude of this mode of $2.2\substack{+1.0 \\ -1.5}\times 10^{-3}$, for the scale-invariant, uncorrelated case~\cite{Akrami:2018odb}, which is roughly $10^6$ times larger than the adiabatic mode, confirming similar results from other authors~\cite{Munoz:2015fdv,Valiviita:2017fbx}.
We have previously shown in Ref.~\cite{Carrilho:2018mqy} that this compensated mode can generate evolution of some second-order cosmological fluctuations at early times, when this mode is mixed with an adiabatic mode. It is thus expected that this mode will generate effects at second order, which may be measurable, given the large possible amplitude of this mode.
Figure~\ref{fig:ADCIP} shows our results for the magnetic field generated by this compensated mode, with the amplitude quoted above. In our notation we approximate this to
\begin{equation}
A_{\text{bb}}=10^6 A_s\,,\ A_{\text{cc}}=(186.417)^2 A_s\,.
\end{equation}
The most striking result is that the magnetic field is amplified by more than $10^3$ with respect to the purely adiabatic case on Mpc scales. Analysing Figs.~\ref{fig:ADBI} and \ref{fig:ADCDI}, we note that both the shapes and sizes of the contributions from the baryon and dark matter modes are different, so it is not so surprising to find that they do not cancel out when mixed together, even in this anti-correlated case. We see also that this result is dominated by the contribution from the mixed baryon-adiabatic mode, in spite of the pure baryon isocurvature mode being considerably larger. This is were the compensation between the two matter modes has an effect, with the two pure modes canceling each other out. Furthermore, this implies that the magnetic field power spectrum is proportional to the amplitude of the baryon isocurvature mode, thus explaining why the plotted magnetic field ($\propto \sqrt{P_B}$) is approximately $10^3$ times larger than that shown in Fig.~\ref{fig:ADBI}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PlotPBkAdCIP.pdf}
\includegraphics[width=0.49\textwidth]{PlotPBkAdCIPcos.pdf}
\caption{Normalized magnetic field power spectrum sourced by the mixture of adiabatic and compensated isocurvature modes as a function of wave number, $k$, at $z=0$, showing the contributions from different sources for the uncorrelated, scale-invariant case (left) and showing the effects of a non-zero spectral index of the compensated mode, $n_{\text{CI}}$ and of a non-zero correlation angle, $\cos\Delta_{\zeta\text{CI}}$ (right).}
\label{fig:ADCIP}
\end{figure}
The plot on the right-hand-side of Fig.~\ref{fig:ADCIP} shows the effects of changing the spectral index of the compensated mode and its correlation angle with the adiabatic mode. We note that, contrarily to what occurred for the baryon isocurvature contribution, the anti-correlation between the adiabatic and compensated modes does not give rise to an increased magnetic field at any scale in the range studied here. However, we see that the effect of a blue spectral index is similar to that shown before, enhancing the magnetic field by a factor of approximately 30 on small scales, bringing the magnetic field at $k=0.5\ \text{Mpc}^{-1}$ to $3\times10^{-25}$ G. It should me noted, though, that the compensated mode used by Planck~\cite{Akrami:2018odb} was the scale-invariant, uncorrelated mode, a fact that was important for the lensing degeneracy, and it is not clear how alternative scenarios would affect that. However, a fully correlated mode is better motivated from the theoretical point of view, as it can be generated in curvaton models of the early Universe and would, in principle, be easier to detect~\cite{He:2015msa}. Models with non-scale-invariant spectra have not been explored. They may be harder to constrain for blue spectra, as the current methods rely on a large-scale modulation of the baryon-dark matter fraction, which would require larger amplitudes to be detected with a blue spectral index.
We see from these results that a compensated isocurvature mode can generate considerably larger magnetic fields than the adiabatic mode, given its potential large amplitude. In our most optimistic scenario, we see that the magnetic field can be enhanced up to 4 orders of magnitude on Mpc scales, which certainly improves the prospects of this magnetic field being detected in the future.
\section{Conclusions}
\label{sec:conc}
We have computed the magnetic field generated in the pre-recombination epoch in the presence of isocurvature modes. We have shown how the different initial conditions can affect the magnetic field power spectrum and concluded that it is the non-linear mode mixing that provides the greatest contributions. In particular, the baryon isocurvature mode, when mixed with the adiabatic mode, has the greatest potential to enhance the magnetic field. We have also demonstrated that isocurvature modes with bluer spectral indices give rise to larger improvements than those of the scale-invariant case, resulting in a magnetic field approximately 30 times larger than with the adiabatic mode, for the most optimistic case. Furthermore, we have also seen that a variation of the correlation fraction between the adiabatic mode and the isocurvature mode in question can modify the results, slightly enhancing the Mpc-scale magnetic field in the fully anti-correlated case and suppressing it in the opposite limit.
The compensated isocurvature mode can have a much larger effect on the magnetic field, since its amplitude is not as constrained as the others. We find that the uncorrelated, scale-invariant compensated mode can source a magnetic field of order $10^{-26}$ G, 3 orders of magnitude larger than the adiabatic case. For very blue input spectra this can once again be improved by a factor of 30, reaching $3\times10^{-25}$ G at $k=0.5\ \text{Mpc}^{-1}$. While possibly not sufficient to explain the tentative observations of void magnetic fields of $10^{-15}$ G, this is certainly a step in the right direction. Furthermore, the mechanisms for generating isocurvature fluctuations are often the source of primordial non-Gaussianity, whose contribution to the magnetic field could also be appreciable, given its dependence on the four-point function of primordial fluctuations. Further enhancements may be possible in the presence of features in the primordial spectra, particularly in the compensated isocurvature case. Both of these conjectured contributions could play an important role in magnetic field generation and should be further investigated.
This result regarding the compensated isocurvature mode could have further consequences. Note that a mode with such a large amplitude has virtually no effect on the CMB, but does leave a huge imprint on the magnetic field. This indicates that other quantities that appear only at second order in cosmological perturbations could be similarly enhanced by this mode, which would allow us to better constrain a primordial compensated isocurvature fluctuation. We shall study some of those quantities in future publications, including gravitational waves generated at second order in fluctuations, as well as the bispectrum of the CMB.
\acknowledgments
PC is supported by STFC grant ST/P000592/1 and by the Funda\c{c}\~{a}o para a Ci\^{e}ncia e Tecnologia (FCT) grant SFRH/BD/118740/2016. KAM is supported, in part, by STFC grant ST/P000592/1.
The tensor algebra package xAct \cite{xAct}\footnote{\href{http://www.xact.es}{http://www.xact.es}}, as well as its
sub-package xPand \cite{xPand}\footnote{\href{http://www.xact.es/xPand/}{http://www.xact.es/xPand/}}, were used in the derivation of
many of the equations presented in this work.
The authors are grateful to David Mulryne for useful discussions.
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